Step | Hyp | Ref
| Expression |
1 | | ral0 4217 |
. . . 4
⊢
∀𝑤 ∈
∅ (𝑎‘𝑤) = if(𝑤 ∈ ( I ↾ 𝑁), 1 , 0 ) |
2 | | simpr 471 |
. . . . 5
⊢ ((𝜑 ∧ 𝑎 ∈ 𝐵) → 𝑎 ∈ 𝐵) |
3 | | f1oi 6315 |
. . . . . . . 8
⊢ ( I
↾ 𝑁):𝑁–1-1-onto→𝑁 |
4 | | f1of 6278 |
. . . . . . . 8
⊢ (( I
↾ 𝑁):𝑁–1-1-onto→𝑁 → ( I ↾ 𝑁):𝑁⟶𝑁) |
5 | 3, 4 | mp1i 13 |
. . . . . . 7
⊢ (𝜑 → ( I ↾ 𝑁):𝑁⟶𝑁) |
6 | | mdetuni.n |
. . . . . . . 8
⊢ (𝜑 → 𝑁 ∈ Fin) |
7 | 6, 6 | elmapd 8023 |
. . . . . . 7
⊢ (𝜑 → (( I ↾ 𝑁) ∈ (𝑁 ↑𝑚 𝑁) ↔ ( I ↾ 𝑁):𝑁⟶𝑁)) |
8 | 5, 7 | mpbird 247 |
. . . . . 6
⊢ (𝜑 → ( I ↾ 𝑁) ∈ (𝑁 ↑𝑚 𝑁)) |
9 | 8 | adantr 466 |
. . . . 5
⊢ ((𝜑 ∧ 𝑎 ∈ 𝐵) → ( I ↾ 𝑁) ∈ (𝑁 ↑𝑚 𝑁)) |
10 | | simplrl 762 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ (𝑁 ↑𝑚 𝑁))) ∧ ∀𝑤 ∈ (𝑁 × 𝑁)(𝑦‘𝑤) = if(𝑤 ∈ 𝑧, 1 , 0 )) → 𝑦 ∈ 𝐵) |
11 | | mdetuni.a |
. . . . . . . . . . . . . . . . 17
⊢ 𝐴 = (𝑁 Mat 𝑅) |
12 | | mdetuni.k |
. . . . . . . . . . . . . . . . 17
⊢ 𝐾 = (Base‘𝑅) |
13 | | mdetuni.b |
. . . . . . . . . . . . . . . . 17
⊢ 𝐵 = (Base‘𝐴) |
14 | 11, 12, 13 | matbas2i 20445 |
. . . . . . . . . . . . . . . 16
⊢ (𝑦 ∈ 𝐵 → 𝑦 ∈ (𝐾 ↑𝑚 (𝑁 × 𝑁))) |
15 | | elmapi 8031 |
. . . . . . . . . . . . . . . 16
⊢ (𝑦 ∈ (𝐾 ↑𝑚 (𝑁 × 𝑁)) → 𝑦:(𝑁 × 𝑁)⟶𝐾) |
16 | 14, 15 | syl 17 |
. . . . . . . . . . . . . . 15
⊢ (𝑦 ∈ 𝐵 → 𝑦:(𝑁 × 𝑁)⟶𝐾) |
17 | 16 | feqmptd 6391 |
. . . . . . . . . . . . . 14
⊢ (𝑦 ∈ 𝐵 → 𝑦 = (𝑤 ∈ (𝑁 × 𝑁) ↦ (𝑦‘𝑤))) |
18 | 17 | fveq2d 6336 |
. . . . . . . . . . . . 13
⊢ (𝑦 ∈ 𝐵 → (𝐷‘𝑦) = (𝐷‘(𝑤 ∈ (𝑁 × 𝑁) ↦ (𝑦‘𝑤)))) |
19 | 10, 18 | syl 17 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ (𝑁 ↑𝑚 𝑁))) ∧ ∀𝑤 ∈ (𝑁 × 𝑁)(𝑦‘𝑤) = if(𝑤 ∈ 𝑧, 1 , 0 )) → (𝐷‘𝑦) = (𝐷‘(𝑤 ∈ (𝑁 × 𝑁) ↦ (𝑦‘𝑤)))) |
20 | | eqid 2771 |
. . . . . . . . . . . . . 14
⊢ (𝑁 × 𝑁) = (𝑁 × 𝑁) |
21 | | mpteq12 4870 |
. . . . . . . . . . . . . . 15
⊢ (((𝑁 × 𝑁) = (𝑁 × 𝑁) ∧ ∀𝑤 ∈ (𝑁 × 𝑁)(𝑦‘𝑤) = if(𝑤 ∈ 𝑧, 1 , 0 )) → (𝑤 ∈ (𝑁 × 𝑁) ↦ (𝑦‘𝑤)) = (𝑤 ∈ (𝑁 × 𝑁) ↦ if(𝑤 ∈ 𝑧, 1 , 0 ))) |
22 | 21 | fveq2d 6336 |
. . . . . . . . . . . . . 14
⊢ (((𝑁 × 𝑁) = (𝑁 × 𝑁) ∧ ∀𝑤 ∈ (𝑁 × 𝑁)(𝑦‘𝑤) = if(𝑤 ∈ 𝑧, 1 , 0 )) → (𝐷‘(𝑤 ∈ (𝑁 × 𝑁) ↦ (𝑦‘𝑤))) = (𝐷‘(𝑤 ∈ (𝑁 × 𝑁) ↦ if(𝑤 ∈ 𝑧, 1 , 0 )))) |
23 | 20, 22 | mpan 670 |
. . . . . . . . . . . . 13
⊢
(∀𝑤 ∈
(𝑁 × 𝑁)(𝑦‘𝑤) = if(𝑤 ∈ 𝑧, 1 , 0 ) → (𝐷‘(𝑤 ∈ (𝑁 × 𝑁) ↦ (𝑦‘𝑤))) = (𝐷‘(𝑤 ∈ (𝑁 × 𝑁) ↦ if(𝑤 ∈ 𝑧, 1 , 0 )))) |
24 | 23 | adantl 467 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ (𝑁 ↑𝑚 𝑁))) ∧ ∀𝑤 ∈ (𝑁 × 𝑁)(𝑦‘𝑤) = if(𝑤 ∈ 𝑧, 1 , 0 )) → (𝐷‘(𝑤 ∈ (𝑁 × 𝑁) ↦ (𝑦‘𝑤))) = (𝐷‘(𝑤 ∈ (𝑁 × 𝑁) ↦ if(𝑤 ∈ 𝑧, 1 , 0 )))) |
25 | | eleq1 2838 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑎 = 𝑧 → (𝑎 ∈ (𝑁 ↑𝑚 𝑁) ↔ 𝑧 ∈ (𝑁 ↑𝑚 𝑁))) |
26 | 25 | anbi2d 614 |
. . . . . . . . . . . . . . . 16
⊢ (𝑎 = 𝑧 → ((𝜑 ∧ 𝑎 ∈ (𝑁 ↑𝑚 𝑁)) ↔ (𝜑 ∧ 𝑧 ∈ (𝑁 ↑𝑚 𝑁)))) |
27 | | elequ2 2159 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑎 = 𝑧 → (𝑤 ∈ 𝑎 ↔ 𝑤 ∈ 𝑧)) |
28 | 27 | ifbid 4247 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑎 = 𝑧 → if(𝑤 ∈ 𝑎, 1 , 0 ) = if(𝑤 ∈ 𝑧, 1 , 0 )) |
29 | 28 | mpteq2dv 4879 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑎 = 𝑧 → (𝑤 ∈ (𝑁 × 𝑁) ↦ if(𝑤 ∈ 𝑎, 1 , 0 )) = (𝑤 ∈ (𝑁 × 𝑁) ↦ if(𝑤 ∈ 𝑧, 1 , 0 ))) |
30 | 29 | fveq2d 6336 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑎 = 𝑧 → (𝐷‘(𝑤 ∈ (𝑁 × 𝑁) ↦ if(𝑤 ∈ 𝑎, 1 , 0 ))) = (𝐷‘(𝑤 ∈ (𝑁 × 𝑁) ↦ if(𝑤 ∈ 𝑧, 1 , 0 )))) |
31 | 30 | eqeq1d 2773 |
. . . . . . . . . . . . . . . 16
⊢ (𝑎 = 𝑧 → ((𝐷‘(𝑤 ∈ (𝑁 × 𝑁) ↦ if(𝑤 ∈ 𝑎, 1 , 0 ))) = 0 ↔ (𝐷‘(𝑤 ∈ (𝑁 × 𝑁) ↦ if(𝑤 ∈ 𝑧, 1 , 0 ))) = 0 )) |
32 | 26, 31 | imbi12d 333 |
. . . . . . . . . . . . . . 15
⊢ (𝑎 = 𝑧 → (((𝜑 ∧ 𝑎 ∈ (𝑁 ↑𝑚 𝑁)) → (𝐷‘(𝑤 ∈ (𝑁 × 𝑁) ↦ if(𝑤 ∈ 𝑎, 1 , 0 ))) = 0 ) ↔ ((𝜑 ∧ 𝑧 ∈ (𝑁 ↑𝑚 𝑁)) → (𝐷‘(𝑤 ∈ (𝑁 × 𝑁) ↦ if(𝑤 ∈ 𝑧, 1 , 0 ))) = 0 ))) |
33 | | eleq1 2838 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑤 = 〈𝑏, 𝑐〉 → (𝑤 ∈ 𝑎 ↔ 〈𝑏, 𝑐〉 ∈ 𝑎)) |
34 | 33 | ifbid 4247 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑤 = 〈𝑏, 𝑐〉 → if(𝑤 ∈ 𝑎, 1 , 0 ) = if(〈𝑏, 𝑐〉 ∈ 𝑎, 1 , 0 )) |
35 | 34 | mpt2mpt 6899 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑤 ∈ (𝑁 × 𝑁) ↦ if(𝑤 ∈ 𝑎, 1 , 0 )) = (𝑏 ∈ 𝑁, 𝑐 ∈ 𝑁 ↦ if(〈𝑏, 𝑐〉 ∈ 𝑎, 1 , 0 )) |
36 | | elmapi 8031 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (𝑎 ∈ (𝑁 ↑𝑚 𝑁) → 𝑎:𝑁⟶𝑁) |
37 | 36 | adantl 467 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((𝜑 ∧ 𝑎 ∈ (𝑁 ↑𝑚 𝑁)) → 𝑎:𝑁⟶𝑁) |
38 | | ffn 6185 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝑎:𝑁⟶𝑁 → 𝑎 Fn 𝑁) |
39 | 37, 38 | syl 17 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((𝜑 ∧ 𝑎 ∈ (𝑁 ↑𝑚 𝑁)) → 𝑎 Fn 𝑁) |
40 | 39 | 3ad2ant1 1127 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (((𝜑 ∧ 𝑎 ∈ (𝑁 ↑𝑚 𝑁)) ∧ 𝑏 ∈ 𝑁 ∧ 𝑐 ∈ 𝑁) → 𝑎 Fn 𝑁) |
41 | | simp2 1131 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (((𝜑 ∧ 𝑎 ∈ (𝑁 ↑𝑚 𝑁)) ∧ 𝑏 ∈ 𝑁 ∧ 𝑐 ∈ 𝑁) → 𝑏 ∈ 𝑁) |
42 | | fnopfvb 6378 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝑎 Fn 𝑁 ∧ 𝑏 ∈ 𝑁) → ((𝑎‘𝑏) = 𝑐 ↔ 〈𝑏, 𝑐〉 ∈ 𝑎)) |
43 | 40, 41, 42 | syl2anc 573 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (((𝜑 ∧ 𝑎 ∈ (𝑁 ↑𝑚 𝑁)) ∧ 𝑏 ∈ 𝑁 ∧ 𝑐 ∈ 𝑁) → ((𝑎‘𝑏) = 𝑐 ↔ 〈𝑏, 𝑐〉 ∈ 𝑎)) |
44 | 43 | bicomd 213 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (((𝜑 ∧ 𝑎 ∈ (𝑁 ↑𝑚 𝑁)) ∧ 𝑏 ∈ 𝑁 ∧ 𝑐 ∈ 𝑁) → (〈𝑏, 𝑐〉 ∈ 𝑎 ↔ (𝑎‘𝑏) = 𝑐)) |
45 | 44 | ifbid 4247 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((𝜑 ∧ 𝑎 ∈ (𝑁 ↑𝑚 𝑁)) ∧ 𝑏 ∈ 𝑁 ∧ 𝑐 ∈ 𝑁) → if(〈𝑏, 𝑐〉 ∈ 𝑎, 1 , 0 ) = if((𝑎‘𝑏) = 𝑐, 1 , 0 )) |
46 | 45 | mpt2eq3dva 6866 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ 𝑎 ∈ (𝑁 ↑𝑚 𝑁)) → (𝑏 ∈ 𝑁, 𝑐 ∈ 𝑁 ↦ if(〈𝑏, 𝑐〉 ∈ 𝑎, 1 , 0 )) = (𝑏 ∈ 𝑁, 𝑐 ∈ 𝑁 ↦ if((𝑎‘𝑏) = 𝑐, 1 , 0 ))) |
47 | 35, 46 | syl5eq 2817 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ 𝑎 ∈ (𝑁 ↑𝑚 𝑁)) → (𝑤 ∈ (𝑁 × 𝑁) ↦ if(𝑤 ∈ 𝑎, 1 , 0 )) = (𝑏 ∈ 𝑁, 𝑐 ∈ 𝑁 ↦ if((𝑎‘𝑏) = 𝑐, 1 , 0 ))) |
48 | 47 | fveq2d 6336 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑎 ∈ (𝑁 ↑𝑚 𝑁)) → (𝐷‘(𝑤 ∈ (𝑁 × 𝑁) ↦ if(𝑤 ∈ 𝑎, 1 , 0 ))) = (𝐷‘(𝑏 ∈ 𝑁, 𝑐 ∈ 𝑁 ↦ if((𝑎‘𝑏) = 𝑐, 1 , 0 )))) |
49 | | mdetuni.0g |
. . . . . . . . . . . . . . . . . 18
⊢ 0 =
(0g‘𝑅) |
50 | | mdetuni.1r |
. . . . . . . . . . . . . . . . . 18
⊢ 1 =
(1r‘𝑅) |
51 | | mdetuni.pg |
. . . . . . . . . . . . . . . . . 18
⊢ + =
(+g‘𝑅) |
52 | | mdetuni.tg |
. . . . . . . . . . . . . . . . . 18
⊢ · =
(.r‘𝑅) |
53 | | mdetuni.r |
. . . . . . . . . . . . . . . . . 18
⊢ (𝜑 → 𝑅 ∈ Ring) |
54 | | mdetuni.ff |
. . . . . . . . . . . . . . . . . 18
⊢ (𝜑 → 𝐷:𝐵⟶𝐾) |
55 | | mdetuni.al |
. . . . . . . . . . . . . . . . . 18
⊢ (𝜑 → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝑁 ∀𝑧 ∈ 𝑁 ((𝑦 ≠ 𝑧 ∧ ∀𝑤 ∈ 𝑁 (𝑦𝑥𝑤) = (𝑧𝑥𝑤)) → (𝐷‘𝑥) = 0 )) |
56 | | mdetuni.li |
. . . . . . . . . . . . . . . . . 18
⊢ (𝜑 → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝑁 (((𝑥 ↾ ({𝑤} × 𝑁)) = ((𝑦 ↾ ({𝑤} × 𝑁)) ∘𝑓 + (𝑧 ↾ ({𝑤} × 𝑁))) ∧ (𝑥 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑦 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) ∧ (𝑥 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑧 ↾ ((𝑁 ∖ {𝑤}) × 𝑁))) → (𝐷‘𝑥) = ((𝐷‘𝑦) + (𝐷‘𝑧)))) |
57 | | mdetuni.sc |
. . . . . . . . . . . . . . . . . 18
⊢ (𝜑 → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐾 ∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝑁 (((𝑥 ↾ ({𝑤} × 𝑁)) = ((({𝑤} × 𝑁) × {𝑦}) ∘𝑓 · (𝑧 ↾ ({𝑤} × 𝑁))) ∧ (𝑥 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑧 ↾ ((𝑁 ∖ {𝑤}) × 𝑁))) → (𝐷‘𝑥) = (𝑦 · (𝐷‘𝑧)))) |
58 | | mdetunilem9.id |
. . . . . . . . . . . . . . . . . 18
⊢ (𝜑 → (𝐷‘(1r‘𝐴)) = 0 ) |
59 | 11, 13, 12, 49, 50, 51, 52, 6, 53, 54, 55, 56, 57, 58 | mdetunilem8 20643 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ 𝑎:𝑁⟶𝑁) → (𝐷‘(𝑏 ∈ 𝑁, 𝑐 ∈ 𝑁 ↦ if((𝑎‘𝑏) = 𝑐, 1 , 0 ))) = 0 ) |
60 | 36, 59 | sylan2 580 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑎 ∈ (𝑁 ↑𝑚 𝑁)) → (𝐷‘(𝑏 ∈ 𝑁, 𝑐 ∈ 𝑁 ↦ if((𝑎‘𝑏) = 𝑐, 1 , 0 ))) = 0 ) |
61 | 48, 60 | eqtrd 2805 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑎 ∈ (𝑁 ↑𝑚 𝑁)) → (𝐷‘(𝑤 ∈ (𝑁 × 𝑁) ↦ if(𝑤 ∈ 𝑎, 1 , 0 ))) = 0 ) |
62 | 32, 61 | chvarv 2425 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑧 ∈ (𝑁 ↑𝑚 𝑁)) → (𝐷‘(𝑤 ∈ (𝑁 × 𝑁) ↦ if(𝑤 ∈ 𝑧, 1 , 0 ))) = 0 ) |
63 | 62 | adantrl 695 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ (𝑁 ↑𝑚 𝑁))) → (𝐷‘(𝑤 ∈ (𝑁 × 𝑁) ↦ if(𝑤 ∈ 𝑧, 1 , 0 ))) = 0 ) |
64 | 63 | adantr 466 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ (𝑁 ↑𝑚 𝑁))) ∧ ∀𝑤 ∈ (𝑁 × 𝑁)(𝑦‘𝑤) = if(𝑤 ∈ 𝑧, 1 , 0 )) → (𝐷‘(𝑤 ∈ (𝑁 × 𝑁) ↦ if(𝑤 ∈ 𝑧, 1 , 0 ))) = 0 ) |
65 | 19, 24, 64 | 3eqtrd 2809 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ (𝑁 ↑𝑚 𝑁))) ∧ ∀𝑤 ∈ (𝑁 × 𝑁)(𝑦‘𝑤) = if(𝑤 ∈ 𝑧, 1 , 0 )) → (𝐷‘𝑦) = 0 ) |
66 | 65 | ex 397 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ (𝑁 ↑𝑚 𝑁))) → (∀𝑤 ∈ (𝑁 × 𝑁)(𝑦‘𝑤) = if(𝑤 ∈ 𝑧, 1 , 0 ) → (𝐷‘𝑦) = 0 )) |
67 | 66 | ralrimivva 3120 |
. . . . . . . . 9
⊢ (𝜑 → ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ (𝑁 ↑𝑚 𝑁)(∀𝑤 ∈ (𝑁 × 𝑁)(𝑦‘𝑤) = if(𝑤 ∈ 𝑧, 1 , 0 ) → (𝐷‘𝑦) = 0 )) |
68 | | xpfi 8387 |
. . . . . . . . . . 11
⊢ ((𝑁 ∈ Fin ∧ 𝑁 ∈ Fin) → (𝑁 × 𝑁) ∈ Fin) |
69 | 6, 6, 68 | syl2anc 573 |
. . . . . . . . . 10
⊢ (𝜑 → (𝑁 × 𝑁) ∈ Fin) |
70 | | raleq 3287 |
. . . . . . . . . . . . 13
⊢ (𝑥 = (𝑁 × 𝑁) → (∀𝑤 ∈ 𝑥 (𝑦‘𝑤) = if(𝑤 ∈ 𝑧, 1 , 0 ) ↔ ∀𝑤 ∈ (𝑁 × 𝑁)(𝑦‘𝑤) = if(𝑤 ∈ 𝑧, 1 , 0 ))) |
71 | 70 | imbi1d 330 |
. . . . . . . . . . . 12
⊢ (𝑥 = (𝑁 × 𝑁) → ((∀𝑤 ∈ 𝑥 (𝑦‘𝑤) = if(𝑤 ∈ 𝑧, 1 , 0 ) → (𝐷‘𝑦) = 0 ) ↔ (∀𝑤 ∈ (𝑁 × 𝑁)(𝑦‘𝑤) = if(𝑤 ∈ 𝑧, 1 , 0 ) → (𝐷‘𝑦) = 0 ))) |
72 | 71 | 2ralbidv 3138 |
. . . . . . . . . . 11
⊢ (𝑥 = (𝑁 × 𝑁) → (∀𝑦 ∈ 𝐵 ∀𝑧 ∈ (𝑁 ↑𝑚 𝑁)(∀𝑤 ∈ 𝑥 (𝑦‘𝑤) = if(𝑤 ∈ 𝑧, 1 , 0 ) → (𝐷‘𝑦) = 0 ) ↔ ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ (𝑁 ↑𝑚 𝑁)(∀𝑤 ∈ (𝑁 × 𝑁)(𝑦‘𝑤) = if(𝑤 ∈ 𝑧, 1 , 0 ) → (𝐷‘𝑦) = 0 ))) |
73 | | mdetunilem9.y |
. . . . . . . . . . 11
⊢ 𝑌 = {𝑥 ∣ ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ (𝑁 ↑𝑚 𝑁)(∀𝑤 ∈ 𝑥 (𝑦‘𝑤) = if(𝑤 ∈ 𝑧, 1 , 0 ) → (𝐷‘𝑦) = 0 )} |
74 | 72, 73 | elab2g 3504 |
. . . . . . . . . 10
⊢ ((𝑁 × 𝑁) ∈ Fin → ((𝑁 × 𝑁) ∈ 𝑌 ↔ ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ (𝑁 ↑𝑚 𝑁)(∀𝑤 ∈ (𝑁 × 𝑁)(𝑦‘𝑤) = if(𝑤 ∈ 𝑧, 1 , 0 ) → (𝐷‘𝑦) = 0 ))) |
75 | 69, 74 | syl 17 |
. . . . . . . . 9
⊢ (𝜑 → ((𝑁 × 𝑁) ∈ 𝑌 ↔ ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ (𝑁 ↑𝑚 𝑁)(∀𝑤 ∈ (𝑁 × 𝑁)(𝑦‘𝑤) = if(𝑤 ∈ 𝑧, 1 , 0 ) → (𝐷‘𝑦) = 0 ))) |
76 | 67, 75 | mpbird 247 |
. . . . . . . 8
⊢ (𝜑 → (𝑁 × 𝑁) ∈ 𝑌) |
77 | | ssid 3773 |
. . . . . . . . 9
⊢ (𝑁 × 𝑁) ⊆ (𝑁 × 𝑁) |
78 | 69 | 3ad2ant1 1127 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ (𝑁 × 𝑁) ⊆ (𝑁 × 𝑁) ∧ ¬ ∅ ∈ 𝑌) → (𝑁 × 𝑁) ∈ Fin) |
79 | | sseq1 3775 |
. . . . . . . . . . . . . 14
⊢ (𝑎 = ∅ → (𝑎 ⊆ (𝑁 × 𝑁) ↔ ∅ ⊆ (𝑁 × 𝑁))) |
80 | 79 | 3anbi2d 1552 |
. . . . . . . . . . . . 13
⊢ (𝑎 = ∅ → ((𝜑 ∧ 𝑎 ⊆ (𝑁 × 𝑁) ∧ ¬ ∅ ∈ 𝑌) ↔ (𝜑 ∧ ∅ ⊆ (𝑁 × 𝑁) ∧ ¬ ∅ ∈ 𝑌))) |
81 | | eleq1 2838 |
. . . . . . . . . . . . . 14
⊢ (𝑎 = ∅ → (𝑎 ∈ 𝑌 ↔ ∅ ∈ 𝑌)) |
82 | 81 | notbid 307 |
. . . . . . . . . . . . 13
⊢ (𝑎 = ∅ → (¬ 𝑎 ∈ 𝑌 ↔ ¬ ∅ ∈ 𝑌)) |
83 | 80, 82 | imbi12d 333 |
. . . . . . . . . . . 12
⊢ (𝑎 = ∅ → (((𝜑 ∧ 𝑎 ⊆ (𝑁 × 𝑁) ∧ ¬ ∅ ∈ 𝑌) → ¬ 𝑎 ∈ 𝑌) ↔ ((𝜑 ∧ ∅ ⊆ (𝑁 × 𝑁) ∧ ¬ ∅ ∈ 𝑌) → ¬ ∅ ∈
𝑌))) |
84 | | sseq1 3775 |
. . . . . . . . . . . . . 14
⊢ (𝑎 = 𝑏 → (𝑎 ⊆ (𝑁 × 𝑁) ↔ 𝑏 ⊆ (𝑁 × 𝑁))) |
85 | 84 | 3anbi2d 1552 |
. . . . . . . . . . . . 13
⊢ (𝑎 = 𝑏 → ((𝜑 ∧ 𝑎 ⊆ (𝑁 × 𝑁) ∧ ¬ ∅ ∈ 𝑌) ↔ (𝜑 ∧ 𝑏 ⊆ (𝑁 × 𝑁) ∧ ¬ ∅ ∈ 𝑌))) |
86 | | eleq1 2838 |
. . . . . . . . . . . . . 14
⊢ (𝑎 = 𝑏 → (𝑎 ∈ 𝑌 ↔ 𝑏 ∈ 𝑌)) |
87 | 86 | notbid 307 |
. . . . . . . . . . . . 13
⊢ (𝑎 = 𝑏 → (¬ 𝑎 ∈ 𝑌 ↔ ¬ 𝑏 ∈ 𝑌)) |
88 | 85, 87 | imbi12d 333 |
. . . . . . . . . . . 12
⊢ (𝑎 = 𝑏 → (((𝜑 ∧ 𝑎 ⊆ (𝑁 × 𝑁) ∧ ¬ ∅ ∈ 𝑌) → ¬ 𝑎 ∈ 𝑌) ↔ ((𝜑 ∧ 𝑏 ⊆ (𝑁 × 𝑁) ∧ ¬ ∅ ∈ 𝑌) → ¬ 𝑏 ∈ 𝑌))) |
89 | | sseq1 3775 |
. . . . . . . . . . . . . 14
⊢ (𝑎 = (𝑏 ∪ {𝑐}) → (𝑎 ⊆ (𝑁 × 𝑁) ↔ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁))) |
90 | 89 | 3anbi2d 1552 |
. . . . . . . . . . . . 13
⊢ (𝑎 = (𝑏 ∪ {𝑐}) → ((𝜑 ∧ 𝑎 ⊆ (𝑁 × 𝑁) ∧ ¬ ∅ ∈ 𝑌) ↔ (𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ ¬ ∅ ∈ 𝑌))) |
91 | | eleq1 2838 |
. . . . . . . . . . . . . 14
⊢ (𝑎 = (𝑏 ∪ {𝑐}) → (𝑎 ∈ 𝑌 ↔ (𝑏 ∪ {𝑐}) ∈ 𝑌)) |
92 | 91 | notbid 307 |
. . . . . . . . . . . . 13
⊢ (𝑎 = (𝑏 ∪ {𝑐}) → (¬ 𝑎 ∈ 𝑌 ↔ ¬ (𝑏 ∪ {𝑐}) ∈ 𝑌)) |
93 | 90, 92 | imbi12d 333 |
. . . . . . . . . . . 12
⊢ (𝑎 = (𝑏 ∪ {𝑐}) → (((𝜑 ∧ 𝑎 ⊆ (𝑁 × 𝑁) ∧ ¬ ∅ ∈ 𝑌) → ¬ 𝑎 ∈ 𝑌) ↔ ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ ¬ ∅ ∈ 𝑌) → ¬ (𝑏 ∪ {𝑐}) ∈ 𝑌))) |
94 | | sseq1 3775 |
. . . . . . . . . . . . . 14
⊢ (𝑎 = (𝑁 × 𝑁) → (𝑎 ⊆ (𝑁 × 𝑁) ↔ (𝑁 × 𝑁) ⊆ (𝑁 × 𝑁))) |
95 | 94 | 3anbi2d 1552 |
. . . . . . . . . . . . 13
⊢ (𝑎 = (𝑁 × 𝑁) → ((𝜑 ∧ 𝑎 ⊆ (𝑁 × 𝑁) ∧ ¬ ∅ ∈ 𝑌) ↔ (𝜑 ∧ (𝑁 × 𝑁) ⊆ (𝑁 × 𝑁) ∧ ¬ ∅ ∈ 𝑌))) |
96 | | eleq1 2838 |
. . . . . . . . . . . . . 14
⊢ (𝑎 = (𝑁 × 𝑁) → (𝑎 ∈ 𝑌 ↔ (𝑁 × 𝑁) ∈ 𝑌)) |
97 | 96 | notbid 307 |
. . . . . . . . . . . . 13
⊢ (𝑎 = (𝑁 × 𝑁) → (¬ 𝑎 ∈ 𝑌 ↔ ¬ (𝑁 × 𝑁) ∈ 𝑌)) |
98 | 95, 97 | imbi12d 333 |
. . . . . . . . . . . 12
⊢ (𝑎 = (𝑁 × 𝑁) → (((𝜑 ∧ 𝑎 ⊆ (𝑁 × 𝑁) ∧ ¬ ∅ ∈ 𝑌) → ¬ 𝑎 ∈ 𝑌) ↔ ((𝜑 ∧ (𝑁 × 𝑁) ⊆ (𝑁 × 𝑁) ∧ ¬ ∅ ∈ 𝑌) → ¬ (𝑁 × 𝑁) ∈ 𝑌))) |
99 | | simp3 1132 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ ∅ ⊆ (𝑁 × 𝑁) ∧ ¬ ∅ ∈ 𝑌) → ¬ ∅ ∈
𝑌) |
100 | | ssun1 3927 |
. . . . . . . . . . . . . . . 16
⊢ 𝑏 ⊆ (𝑏 ∪ {𝑐}) |
101 | | sstr2 3759 |
. . . . . . . . . . . . . . . 16
⊢ (𝑏 ⊆ (𝑏 ∪ {𝑐}) → ((𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) → 𝑏 ⊆ (𝑁 × 𝑁))) |
102 | 100, 101 | ax-mp 5 |
. . . . . . . . . . . . . . 15
⊢ ((𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) → 𝑏 ⊆ (𝑁 × 𝑁)) |
103 | 102 | 3anim2i 1156 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ ¬ ∅ ∈ 𝑌) → (𝜑 ∧ 𝑏 ⊆ (𝑁 × 𝑁) ∧ ¬ ∅ ∈ 𝑌)) |
104 | 103 | imim1i 63 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑏 ⊆ (𝑁 × 𝑁) ∧ ¬ ∅ ∈ 𝑌) → ¬ 𝑏 ∈ 𝑌) → ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ ¬ ∅ ∈ 𝑌) → ¬ 𝑏 ∈ 𝑌)) |
105 | | simpl1 1227 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ (𝑏 ∪ {𝑐}) ∈ 𝑌) ∧ ((𝑎 ∈ 𝐵 ∧ 𝑑 ∈ (𝑁 ↑𝑚 𝑁)) ∧ ∀𝑤 ∈ 𝑏 (𝑎‘𝑤) = if(𝑤 ∈ 𝑑, 1 , 0 ))) → 𝜑) |
106 | | simpl2 1229 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ (𝑏 ∪ {𝑐}) ∈ 𝑌) ∧ ((𝑎 ∈ 𝐵 ∧ 𝑑 ∈ (𝑁 ↑𝑚 𝑁)) ∧ ∀𝑤 ∈ 𝑏 (𝑎‘𝑤) = if(𝑤 ∈ 𝑑, 1 , 0 ))) → (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁)) |
107 | | simprll 764 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ (𝑏 ∪ {𝑐}) ∈ 𝑌) ∧ ((𝑎 ∈ 𝐵 ∧ 𝑑 ∈ (𝑁 ↑𝑚 𝑁)) ∧ ∀𝑤 ∈ 𝑏 (𝑎‘𝑤) = if(𝑤 ∈ 𝑑, 1 , 0 ))) → 𝑎 ∈ 𝐵) |
108 | 11, 12, 13 | matbas2i 20445 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
⊢ (𝑎 ∈ 𝐵 → 𝑎 ∈ (𝐾 ↑𝑚 (𝑁 × 𝑁))) |
109 | | elmapi 8031 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
⊢ (𝑎 ∈ (𝐾 ↑𝑚 (𝑁 × 𝑁)) → 𝑎:(𝑁 × 𝑁)⟶𝐾) |
110 | 108, 109 | syl 17 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
⊢ (𝑎 ∈ 𝐵 → 𝑎:(𝑁 × 𝑁)⟶𝐾) |
111 | 110 | 3ad2ant3 1129 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) → 𝑎:(𝑁 × 𝑁)⟶𝐾) |
112 | 111 | feqmptd 6391 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) → 𝑎 = (𝑒 ∈ (𝑁 × 𝑁) ↦ (𝑎‘𝑒))) |
113 | 112 | reseq1d 5533 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) → (𝑎 ↾ ({(1st ‘𝑐)} × 𝑁)) = ((𝑒 ∈ (𝑁 × 𝑁) ↦ (𝑎‘𝑒)) ↾ ({(1st ‘𝑐)} × 𝑁))) |
114 | 53 | 3ad2ant1 1127 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . 36
⊢ ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) → 𝑅 ∈ Ring) |
115 | | ringgrp 18760 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . 36
⊢ (𝑅 ∈ Ring → 𝑅 ∈ Grp) |
116 | 114, 115 | syl 17 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. 35
⊢ ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) → 𝑅 ∈ Grp) |
117 | 116 | adantr 466 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
34
⊢ (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) ∧ 𝑒 ∈ ({(1st ‘𝑐)} × 𝑁)) → 𝑅 ∈ Grp) |
118 | 111 | adantr 466 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. 35
⊢ (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) ∧ 𝑒 ∈ ({(1st ‘𝑐)} × 𝑁)) → 𝑎:(𝑁 × 𝑁)⟶𝐾) |
119 | | simp2 1131 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . 41
⊢ ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) → (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁)) |
120 | 119 | unssbd 3942 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . 40
⊢ ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) → {𝑐} ⊆ (𝑁 × 𝑁)) |
121 | | vex 3354 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . 41
⊢ 𝑐 ∈ V |
122 | 121 | snss 4451 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . 40
⊢ (𝑐 ∈ (𝑁 × 𝑁) ↔ {𝑐} ⊆ (𝑁 × 𝑁)) |
123 | 120, 122 | sylibr 224 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . 39
⊢ ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) → 𝑐 ∈ (𝑁 × 𝑁)) |
124 | | xp1st 7347 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . 39
⊢ (𝑐 ∈ (𝑁 × 𝑁) → (1st ‘𝑐) ∈ 𝑁) |
125 | 123, 124 | syl 17 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . 38
⊢ ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) → (1st ‘𝑐) ∈ 𝑁) |
126 | 125 | snssd 4475 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . 37
⊢ ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) → {(1st ‘𝑐)} ⊆ 𝑁) |
127 | | xpss1 5267 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . 37
⊢
({(1st ‘𝑐)} ⊆ 𝑁 → ({(1st ‘𝑐)} × 𝑁) ⊆ (𝑁 × 𝑁)) |
128 | 126, 127 | syl 17 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . 36
⊢ ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) → ({(1st ‘𝑐)} × 𝑁) ⊆ (𝑁 × 𝑁)) |
129 | 128 | sselda 3752 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. 35
⊢ (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) ∧ 𝑒 ∈ ({(1st ‘𝑐)} × 𝑁)) → 𝑒 ∈ (𝑁 × 𝑁)) |
130 | 118, 129 | ffvelrnd 6503 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
34
⊢ (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) ∧ 𝑒 ∈ ({(1st ‘𝑐)} × 𝑁)) → (𝑎‘𝑒) ∈ 𝐾) |
131 | 12, 50 | ringidcl 18776 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . 37
⊢ (𝑅 ∈ Ring → 1 ∈ 𝐾) |
132 | 114, 131 | syl 17 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . 36
⊢ ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) → 1 ∈ 𝐾) |
133 | 12, 49 | ring0cl 18777 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . 37
⊢ (𝑅 ∈ Ring → 0 ∈ 𝐾) |
134 | 114, 133 | syl 17 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . 36
⊢ ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) → 0 ∈ 𝐾) |
135 | 132, 134 | ifcld 4270 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. 35
⊢ ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) → if(𝑒 ∈ 𝑑, 1 , 0 ) ∈ 𝐾) |
136 | 135 | adantr 466 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
34
⊢ (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) ∧ 𝑒 ∈ ({(1st ‘𝑐)} × 𝑁)) → if(𝑒 ∈ 𝑑, 1 , 0 ) ∈ 𝐾) |
137 | | eqid 2771 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. 35
⊢
(-g‘𝑅) = (-g‘𝑅) |
138 | 12, 51, 137 | grpnpcan 17715 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
34
⊢ ((𝑅 ∈ Grp ∧ (𝑎‘𝑒) ∈ 𝐾 ∧ if(𝑒 ∈ 𝑑, 1 , 0 ) ∈ 𝐾) → (((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )) + if(𝑒 ∈ 𝑑, 1 , 0 )) = (𝑎‘𝑒)) |
139 | 117, 130,
136, 138 | syl3anc 1476 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
33
⊢ (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) ∧ 𝑒 ∈ ({(1st ‘𝑐)} × 𝑁)) → (((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )) + if(𝑒 ∈ 𝑑, 1 , 0 )) = (𝑎‘𝑒)) |
140 | 139 | eqcomd 2777 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
32
⊢ (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) ∧ 𝑒 ∈ ({(1st ‘𝑐)} × 𝑁)) → (𝑎‘𝑒) = (((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )) + if(𝑒 ∈ 𝑑, 1 , 0 ))) |
141 | 140 | adantr 466 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
31
⊢ ((((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) ∧ 𝑒 ∈ ({(1st ‘𝑐)} × 𝑁)) ∧ 𝑒 = 𝑐) → (𝑎‘𝑒) = (((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )) + if(𝑒 ∈ 𝑑, 1 , 0 ))) |
142 | | iftrue 4231 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
33
⊢ (𝑒 = 𝑐 → if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ) = ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 ))) |
143 | | iftrue 4231 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
33
⊢ (𝑒 = 𝑐 → if(𝑒 = 𝑐, if(𝑒 ∈ 𝑑, 1 , 0 ), (𝑎‘𝑒)) = if(𝑒 ∈ 𝑑, 1 , 0 )) |
144 | 142, 143 | oveq12d 6811 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
32
⊢ (𝑒 = 𝑐 → (if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ) + if(𝑒 = 𝑐, if(𝑒 ∈ 𝑑, 1 , 0 ), (𝑎‘𝑒))) = (((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )) + if(𝑒 ∈ 𝑑, 1 , 0 ))) |
145 | 144 | adantl 467 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
31
⊢ ((((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) ∧ 𝑒 ∈ ({(1st ‘𝑐)} × 𝑁)) ∧ 𝑒 = 𝑐) → (if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ) + if(𝑒 = 𝑐, if(𝑒 ∈ 𝑑, 1 , 0 ), (𝑎‘𝑒))) = (((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )) + if(𝑒 ∈ 𝑑, 1 , 0 ))) |
146 | 141, 145 | eqtr4d 2808 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
⊢ ((((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) ∧ 𝑒 ∈ ({(1st ‘𝑐)} × 𝑁)) ∧ 𝑒 = 𝑐) → (𝑎‘𝑒) = (if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ) + if(𝑒 = 𝑐, if(𝑒 ∈ 𝑑, 1 , 0 ), (𝑎‘𝑒)))) |
147 | 12, 51, 49 | grplid 17660 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
34
⊢ ((𝑅 ∈ Grp ∧ (𝑎‘𝑒) ∈ 𝐾) → ( 0 + (𝑎‘𝑒)) = (𝑎‘𝑒)) |
148 | 117, 130,
147 | syl2anc 573 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
33
⊢ (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) ∧ 𝑒 ∈ ({(1st ‘𝑐)} × 𝑁)) → ( 0 + (𝑎‘𝑒)) = (𝑎‘𝑒)) |
149 | 148 | eqcomd 2777 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
32
⊢ (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) ∧ 𝑒 ∈ ({(1st ‘𝑐)} × 𝑁)) → (𝑎‘𝑒) = ( 0 + (𝑎‘𝑒))) |
150 | 149 | adantr 466 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
31
⊢ ((((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) ∧ 𝑒 ∈ ({(1st ‘𝑐)} × 𝑁)) ∧ ¬ 𝑒 = 𝑐) → (𝑎‘𝑒) = ( 0 + (𝑎‘𝑒))) |
151 | | iffalse 4234 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
33
⊢ (¬
𝑒 = 𝑐 → if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ) = 0 ) |
152 | | iffalse 4234 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
33
⊢ (¬
𝑒 = 𝑐 → if(𝑒 = 𝑐, if(𝑒 ∈ 𝑑, 1 , 0 ), (𝑎‘𝑒)) = (𝑎‘𝑒)) |
153 | 151, 152 | oveq12d 6811 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
32
⊢ (¬
𝑒 = 𝑐 → (if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ) + if(𝑒 = 𝑐, if(𝑒 ∈ 𝑑, 1 , 0 ), (𝑎‘𝑒))) = ( 0 + (𝑎‘𝑒))) |
154 | 153 | adantl 467 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
31
⊢ ((((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) ∧ 𝑒 ∈ ({(1st ‘𝑐)} × 𝑁)) ∧ ¬ 𝑒 = 𝑐) → (if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ) + if(𝑒 = 𝑐, if(𝑒 ∈ 𝑑, 1 , 0 ), (𝑎‘𝑒))) = ( 0 + (𝑎‘𝑒))) |
155 | 150, 154 | eqtr4d 2808 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
⊢ ((((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) ∧ 𝑒 ∈ ({(1st ‘𝑐)} × 𝑁)) ∧ ¬ 𝑒 = 𝑐) → (𝑎‘𝑒) = (if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ) + if(𝑒 = 𝑐, if(𝑒 ∈ 𝑑, 1 , 0 ), (𝑎‘𝑒)))) |
156 | 146, 155 | pm2.61dan 813 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
⊢ (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) ∧ 𝑒 ∈ ({(1st ‘𝑐)} × 𝑁)) → (𝑎‘𝑒) = (if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ) + if(𝑒 = 𝑐, if(𝑒 ∈ 𝑑, 1 , 0 ), (𝑎‘𝑒)))) |
157 | 156 | mpteq2dva 4878 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) → (𝑒 ∈ ({(1st ‘𝑐)} × 𝑁) ↦ (𝑎‘𝑒)) = (𝑒 ∈ ({(1st ‘𝑐)} × 𝑁) ↦ (if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ) + if(𝑒 = 𝑐, if(𝑒 ∈ 𝑑, 1 , 0 ), (𝑎‘𝑒))))) |
158 | | snfi 8194 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
⊢
{(1st ‘𝑐)} ∈ Fin |
159 | 6 | 3ad2ant1 1127 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
⊢ ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) → 𝑁 ∈ Fin) |
160 | | xpfi 8387 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
⊢
(({(1st ‘𝑐)} ∈ Fin ∧ 𝑁 ∈ Fin) → ({(1st
‘𝑐)} × 𝑁) ∈ Fin) |
161 | 158, 159,
160 | sylancr 575 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
⊢ ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) → ({(1st ‘𝑐)} × 𝑁) ∈ Fin) |
162 | | ovex 6823 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
31
⊢ ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )) ∈
V |
163 | | fvex 6342 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
32
⊢
(0g‘𝑅) ∈ V |
164 | 49, 163 | eqeltri 2846 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
31
⊢ 0 ∈
V |
165 | 162, 164 | ifex 4295 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
⊢ if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ) ∈
V |
166 | 165 | a1i 11 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
⊢ (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) ∧ 𝑒 ∈ ({(1st ‘𝑐)} × 𝑁)) → if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ) ∈
V) |
167 | | fvex 6342 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
33
⊢
(1r‘𝑅) ∈ V |
168 | 50, 167 | eqeltri 2846 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
32
⊢ 1 ∈
V |
169 | 168, 164 | ifex 4295 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
31
⊢ if(𝑒 ∈ 𝑑, 1 , 0 ) ∈
V |
170 | | fvex 6342 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
31
⊢ (𝑎‘𝑒) ∈ V |
171 | 169, 170 | ifex 4295 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
⊢ if(𝑒 = 𝑐, if(𝑒 ∈ 𝑑, 1 , 0 ), (𝑎‘𝑒)) ∈ V |
172 | 171 | a1i 11 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
⊢ (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) ∧ 𝑒 ∈ ({(1st ‘𝑐)} × 𝑁)) → if(𝑒 = 𝑐, if(𝑒 ∈ 𝑑, 1 , 0 ), (𝑎‘𝑒)) ∈ V) |
173 | | xp1st 7347 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
32
⊢ (𝑒 ∈ ({(1st
‘𝑐)} × 𝑁) → (1st
‘𝑒) ∈
{(1st ‘𝑐)}) |
174 | | elsni 4333 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
32
⊢
((1st ‘𝑒) ∈ {(1st ‘𝑐)} → (1st
‘𝑒) = (1st
‘𝑐)) |
175 | | iftrue 4231 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
32
⊢
((1st ‘𝑒) = (1st ‘𝑐) → if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒)) = if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 )) |
176 | 173, 174,
175 | 3syl 18 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
31
⊢ (𝑒 ∈ ({(1st
‘𝑐)} × 𝑁) → if((1st
‘𝑒) = (1st
‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒)) = if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 )) |
177 | 176 | mpteq2ia 4874 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
⊢ (𝑒 ∈ ({(1st
‘𝑐)} × 𝑁) ↦ if((1st
‘𝑒) = (1st
‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) = (𝑒 ∈ ({(1st ‘𝑐)} × 𝑁) ↦ if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 )) |
178 | 177 | a1i 11 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
⊢ ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) → (𝑒 ∈ ({(1st ‘𝑐)} × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) = (𝑒 ∈ ({(1st ‘𝑐)} × 𝑁) ↦ if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ))) |
179 | | eqidd 2772 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
⊢ ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) → (𝑒 ∈ ({(1st ‘𝑐)} × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒 ∈ 𝑑, 1 , 0 ), (𝑎‘𝑒))) = (𝑒 ∈ ({(1st ‘𝑐)} × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒 ∈ 𝑑, 1 , 0 ), (𝑎‘𝑒)))) |
180 | 161, 166,
172, 178, 179 | offval2 7061 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) → ((𝑒 ∈ ({(1st ‘𝑐)} × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ∘𝑓 + (𝑒 ∈ ({(1st
‘𝑐)} × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒 ∈ 𝑑, 1 , 0 ), (𝑎‘𝑒)))) = (𝑒 ∈ ({(1st ‘𝑐)} × 𝑁) ↦ (if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ) + if(𝑒 = 𝑐, if(𝑒 ∈ 𝑑, 1 , 0 ), (𝑎‘𝑒))))) |
181 | 157, 180 | eqtr4d 2808 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) → (𝑒 ∈ ({(1st ‘𝑐)} × 𝑁) ↦ (𝑎‘𝑒)) = ((𝑒 ∈ ({(1st ‘𝑐)} × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ∘𝑓 + (𝑒 ∈ ({(1st
‘𝑐)} × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒 ∈ 𝑑, 1 , 0 ), (𝑎‘𝑒))))) |
182 | 128 | resmptd 5593 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) → ((𝑒 ∈ (𝑁 × 𝑁) ↦ (𝑎‘𝑒)) ↾ ({(1st ‘𝑐)} × 𝑁)) = (𝑒 ∈ ({(1st ‘𝑐)} × 𝑁) ↦ (𝑎‘𝑒))) |
183 | 128 | resmptd 5593 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) → ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ↾ ({(1st ‘𝑐)} × 𝑁)) = (𝑒 ∈ ({(1st ‘𝑐)} × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒)))) |
184 | 128 | resmptd 5593 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) → ((𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒 ∈ 𝑑, 1 , 0 ), (𝑎‘𝑒))) ↾ ({(1st ‘𝑐)} × 𝑁)) = (𝑒 ∈ ({(1st ‘𝑐)} × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒 ∈ 𝑑, 1 , 0 ), (𝑎‘𝑒)))) |
185 | 183, 184 | oveq12d 6811 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) → (((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ↾ ({(1st ‘𝑐)} × 𝑁)) ∘𝑓 + ((𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒 ∈ 𝑑, 1 , 0 ), (𝑎‘𝑒))) ↾ ({(1st ‘𝑐)} × 𝑁))) = ((𝑒 ∈ ({(1st ‘𝑐)} × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ∘𝑓 + (𝑒 ∈ ({(1st
‘𝑐)} × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒 ∈ 𝑑, 1 , 0 ), (𝑎‘𝑒))))) |
186 | 181, 182,
185 | 3eqtr4d 2815 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) → ((𝑒 ∈ (𝑁 × 𝑁) ↦ (𝑎‘𝑒)) ↾ ({(1st ‘𝑐)} × 𝑁)) = (((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ↾ ({(1st ‘𝑐)} × 𝑁)) ∘𝑓 + ((𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒 ∈ 𝑑, 1 , 0 ), (𝑎‘𝑒))) ↾ ({(1st ‘𝑐)} × 𝑁)))) |
187 | 113, 186 | eqtrd 2805 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) → (𝑎 ↾ ({(1st ‘𝑐)} × 𝑁)) = (((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ↾ ({(1st ‘𝑐)} × 𝑁)) ∘𝑓 + ((𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒 ∈ 𝑑, 1 , 0 ), (𝑎‘𝑒))) ↾ ({(1st ‘𝑐)} × 𝑁)))) |
188 | 112 | reseq1d 5533 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) → (𝑎 ↾ ((𝑁 ∖ {(1st ‘𝑐)}) × 𝑁)) = ((𝑒 ∈ (𝑁 × 𝑁) ↦ (𝑎‘𝑒)) ↾ ((𝑁 ∖ {(1st ‘𝑐)}) × 𝑁))) |
189 | | xp1st 7347 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
32
⊢ (𝑒 ∈ ((𝑁 ∖ {(1st ‘𝑐)}) × 𝑁) → (1st ‘𝑒) ∈ (𝑁 ∖ {(1st ‘𝑐)})) |
190 | | eldifsni 4457 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
32
⊢
((1st ‘𝑒) ∈ (𝑁 ∖ {(1st ‘𝑐)}) → (1st
‘𝑒) ≠
(1st ‘𝑐)) |
191 | 189, 190 | syl 17 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
31
⊢ (𝑒 ∈ ((𝑁 ∖ {(1st ‘𝑐)}) × 𝑁) → (1st ‘𝑒) ≠ (1st
‘𝑐)) |
192 | 191 | neneqd 2948 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
⊢ (𝑒 ∈ ((𝑁 ∖ {(1st ‘𝑐)}) × 𝑁) → ¬ (1st ‘𝑒) = (1st ‘𝑐)) |
193 | 192 | adantl 467 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
⊢ (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) ∧ 𝑒 ∈ ((𝑁 ∖ {(1st ‘𝑐)}) × 𝑁)) → ¬ (1st ‘𝑒) = (1st ‘𝑐)) |
194 | 193 | iffalsed 4236 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) ∧ 𝑒 ∈ ((𝑁 ∖ {(1st ‘𝑐)}) × 𝑁)) → if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒)) = (𝑎‘𝑒)) |
195 | 194 | mpteq2dva 4878 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) → (𝑒 ∈ ((𝑁 ∖ {(1st ‘𝑐)}) × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) = (𝑒 ∈ ((𝑁 ∖ {(1st ‘𝑐)}) × 𝑁) ↦ (𝑎‘𝑒))) |
196 | | difss 3888 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
⊢ (𝑁 ∖ {(1st
‘𝑐)}) ⊆ 𝑁 |
197 | | xpss1 5267 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
⊢ ((𝑁 ∖ {(1st
‘𝑐)}) ⊆ 𝑁 → ((𝑁 ∖ {(1st ‘𝑐)}) × 𝑁) ⊆ (𝑁 × 𝑁)) |
198 | 196, 197 | ax-mp 5 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ ((𝑁 ∖ {(1st
‘𝑐)}) × 𝑁) ⊆ (𝑁 × 𝑁) |
199 | | resmpt 5590 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ (((𝑁 ∖ {(1st
‘𝑐)}) × 𝑁) ⊆ (𝑁 × 𝑁) → ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ↾ ((𝑁 ∖ {(1st ‘𝑐)}) × 𝑁)) = (𝑒 ∈ ((𝑁 ∖ {(1st ‘𝑐)}) × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒)))) |
200 | 198, 199 | mp1i 13 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) → ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ↾ ((𝑁 ∖ {(1st ‘𝑐)}) × 𝑁)) = (𝑒 ∈ ((𝑁 ∖ {(1st ‘𝑐)}) × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒)))) |
201 | | resmpt 5590 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ (((𝑁 ∖ {(1st
‘𝑐)}) × 𝑁) ⊆ (𝑁 × 𝑁) → ((𝑒 ∈ (𝑁 × 𝑁) ↦ (𝑎‘𝑒)) ↾ ((𝑁 ∖ {(1st ‘𝑐)}) × 𝑁)) = (𝑒 ∈ ((𝑁 ∖ {(1st ‘𝑐)}) × 𝑁) ↦ (𝑎‘𝑒))) |
202 | 198, 201 | mp1i 13 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) → ((𝑒 ∈ (𝑁 × 𝑁) ↦ (𝑎‘𝑒)) ↾ ((𝑁 ∖ {(1st ‘𝑐)}) × 𝑁)) = (𝑒 ∈ ((𝑁 ∖ {(1st ‘𝑐)}) × 𝑁) ↦ (𝑎‘𝑒))) |
203 | 195, 200,
202 | 3eqtr4rd 2816 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) → ((𝑒 ∈ (𝑁 × 𝑁) ↦ (𝑎‘𝑒)) ↾ ((𝑁 ∖ {(1st ‘𝑐)}) × 𝑁)) = ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ↾ ((𝑁 ∖ {(1st ‘𝑐)}) × 𝑁))) |
204 | 188, 203 | eqtrd 2805 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) → (𝑎 ↾ ((𝑁 ∖ {(1st ‘𝑐)}) × 𝑁)) = ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ↾ ((𝑁 ∖ {(1st ‘𝑐)}) × 𝑁))) |
205 | | fveq2 6332 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
⊢ (𝑒 = 𝑐 → (1st ‘𝑒) = (1st ‘𝑐)) |
206 | 193, 205 | nsyl 137 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
⊢ (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) ∧ 𝑒 ∈ ((𝑁 ∖ {(1st ‘𝑐)}) × 𝑁)) → ¬ 𝑒 = 𝑐) |
207 | 206 | iffalsed 4236 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) ∧ 𝑒 ∈ ((𝑁 ∖ {(1st ‘𝑐)}) × 𝑁)) → if(𝑒 = 𝑐, if(𝑒 ∈ 𝑑, 1 , 0 ), (𝑎‘𝑒)) = (𝑎‘𝑒)) |
208 | 207 | mpteq2dva 4878 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) → (𝑒 ∈ ((𝑁 ∖ {(1st ‘𝑐)}) × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒 ∈ 𝑑, 1 , 0 ), (𝑎‘𝑒))) = (𝑒 ∈ ((𝑁 ∖ {(1st ‘𝑐)}) × 𝑁) ↦ (𝑎‘𝑒))) |
209 | | resmpt 5590 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ (((𝑁 ∖ {(1st
‘𝑐)}) × 𝑁) ⊆ (𝑁 × 𝑁) → ((𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒 ∈ 𝑑, 1 , 0 ), (𝑎‘𝑒))) ↾ ((𝑁 ∖ {(1st ‘𝑐)}) × 𝑁)) = (𝑒 ∈ ((𝑁 ∖ {(1st ‘𝑐)}) × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒 ∈ 𝑑, 1 , 0 ), (𝑎‘𝑒)))) |
210 | 198, 209 | mp1i 13 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) → ((𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒 ∈ 𝑑, 1 , 0 ), (𝑎‘𝑒))) ↾ ((𝑁 ∖ {(1st ‘𝑐)}) × 𝑁)) = (𝑒 ∈ ((𝑁 ∖ {(1st ‘𝑐)}) × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒 ∈ 𝑑, 1 , 0 ), (𝑎‘𝑒)))) |
211 | 208, 210,
202 | 3eqtr4rd 2816 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) → ((𝑒 ∈ (𝑁 × 𝑁) ↦ (𝑎‘𝑒)) ↾ ((𝑁 ∖ {(1st ‘𝑐)}) × 𝑁)) = ((𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒 ∈ 𝑑, 1 , 0 ), (𝑎‘𝑒))) ↾ ((𝑁 ∖ {(1st ‘𝑐)}) × 𝑁))) |
212 | 188, 211 | eqtrd 2805 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) → (𝑎 ↾ ((𝑁 ∖ {(1st ‘𝑐)}) × 𝑁)) = ((𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒 ∈ 𝑑, 1 , 0 ), (𝑎‘𝑒))) ↾ ((𝑁 ∖ {(1st ‘𝑐)}) × 𝑁))) |
213 | 135 | adantr 466 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
⊢ (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) ∧ 𝑒 ∈ (𝑁 × 𝑁)) → if(𝑒 ∈ 𝑑, 1 , 0 ) ∈ 𝐾) |
214 | 111 | ffvelrnda 6502 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
⊢ (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) ∧ 𝑒 ∈ (𝑁 × 𝑁)) → (𝑎‘𝑒) ∈ 𝐾) |
215 | 213, 214 | ifcld 4270 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
⊢ (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) ∧ 𝑒 ∈ (𝑁 × 𝑁)) → if(𝑒 = 𝑐, if(𝑒 ∈ 𝑑, 1 , 0 ), (𝑎‘𝑒)) ∈ 𝐾) |
216 | | eqid 2771 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
⊢ (𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒 ∈ 𝑑, 1 , 0 ), (𝑎‘𝑒))) = (𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒 ∈ 𝑑, 1 , 0 ), (𝑎‘𝑒))) |
217 | 215, 216 | fmptd 6527 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) → (𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒 ∈ 𝑑, 1 , 0 ), (𝑎‘𝑒))):(𝑁 × 𝑁)⟶𝐾) |
218 | | fvex 6342 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
⊢
(Base‘𝑅)
∈ V |
219 | 12, 218 | eqeltri 2846 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
⊢ 𝐾 ∈ V |
220 | 68 | anidms 556 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
⊢ (𝑁 ∈ Fin → (𝑁 × 𝑁) ∈ Fin) |
221 | 159, 220 | syl 17 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
⊢ ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) → (𝑁 × 𝑁) ∈ Fin) |
222 | | elmapg 8022 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
⊢ ((𝐾 ∈ V ∧ (𝑁 × 𝑁) ∈ Fin) → ((𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒 ∈ 𝑑, 1 , 0 ), (𝑎‘𝑒))) ∈ (𝐾 ↑𝑚 (𝑁 × 𝑁)) ↔ (𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒 ∈ 𝑑, 1 , 0 ), (𝑎‘𝑒))):(𝑁 × 𝑁)⟶𝐾)) |
223 | 219, 221,
222 | sylancr 575 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) → ((𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒 ∈ 𝑑, 1 , 0 ), (𝑎‘𝑒))) ∈ (𝐾 ↑𝑚 (𝑁 × 𝑁)) ↔ (𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒 ∈ 𝑑, 1 , 0 ), (𝑎‘𝑒))):(𝑁 × 𝑁)⟶𝐾)) |
224 | 217, 223 | mpbird 247 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) → (𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒 ∈ 𝑑, 1 , 0 ), (𝑎‘𝑒))) ∈ (𝐾 ↑𝑚 (𝑁 × 𝑁))) |
225 | 11, 12 | matbas2 20444 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (𝐾 ↑𝑚
(𝑁 × 𝑁)) = (Base‘𝐴)) |
226 | 159, 114,
225 | syl2anc 573 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) → (𝐾 ↑𝑚 (𝑁 × 𝑁)) = (Base‘𝐴)) |
227 | 226, 13 | syl6eqr 2823 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) → (𝐾 ↑𝑚 (𝑁 × 𝑁)) = 𝐵) |
228 | 224, 227 | eleqtrd 2852 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) → (𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒 ∈ 𝑑, 1 , 0 ), (𝑎‘𝑒))) ∈ 𝐵) |
229 | | simp3 1132 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) → 𝑎 ∈ 𝐵) |
230 | 116 | adantr 466 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
33
⊢ (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) ∧ 𝑒 ∈ (𝑁 × 𝑁)) → 𝑅 ∈ Grp) |
231 | 12, 137 | grpsubcl 17703 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
33
⊢ ((𝑅 ∈ Grp ∧ (𝑎‘𝑒) ∈ 𝐾 ∧ if(𝑒 ∈ 𝑑, 1 , 0 ) ∈ 𝐾) → ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )) ∈ 𝐾) |
232 | 230, 214,
213, 231 | syl3anc 1476 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
32
⊢ (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) ∧ 𝑒 ∈ (𝑁 × 𝑁)) → ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )) ∈ 𝐾) |
233 | 134 | adantr 466 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
32
⊢ (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) ∧ 𝑒 ∈ (𝑁 × 𝑁)) → 0 ∈ 𝐾) |
234 | 232, 233 | ifcld 4270 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
31
⊢ (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) ∧ 𝑒 ∈ (𝑁 × 𝑁)) → if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ) ∈ 𝐾) |
235 | 234, 214 | ifcld 4270 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
⊢ (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) ∧ 𝑒 ∈ (𝑁 × 𝑁)) → if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒)) ∈ 𝐾) |
236 | | eqid 2771 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
⊢ (𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) = (𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) |
237 | 235, 236 | fmptd 6527 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
⊢ ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) → (𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))):(𝑁 × 𝑁)⟶𝐾) |
238 | | elmapg 8022 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
⊢ ((𝐾 ∈ V ∧ (𝑁 × 𝑁) ∈ Fin) → ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ∈ (𝐾 ↑𝑚 (𝑁 × 𝑁)) ↔ (𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))):(𝑁 × 𝑁)⟶𝐾)) |
239 | 219, 221,
238 | sylancr 575 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
⊢ ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) → ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ∈ (𝐾 ↑𝑚 (𝑁 × 𝑁)) ↔ (𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))):(𝑁 × 𝑁)⟶𝐾)) |
240 | 237, 239 | mpbird 247 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) → (𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ∈ (𝐾 ↑𝑚 (𝑁 × 𝑁))) |
241 | 240, 227 | eleqtrd 2852 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) → (𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ∈ 𝐵) |
242 | 56 | 3ad2ant1 1127 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝑁 (((𝑥 ↾ ({𝑤} × 𝑁)) = ((𝑦 ↾ ({𝑤} × 𝑁)) ∘𝑓 + (𝑧 ↾ ({𝑤} × 𝑁))) ∧ (𝑥 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑦 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) ∧ (𝑥 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑧 ↾ ((𝑁 ∖ {𝑤}) × 𝑁))) → (𝐷‘𝑥) = ((𝐷‘𝑦) + (𝐷‘𝑧)))) |
243 | | reseq1 5528 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
32
⊢ (𝑥 = 𝑎 → (𝑥 ↾ ({𝑤} × 𝑁)) = (𝑎 ↾ ({𝑤} × 𝑁))) |
244 | 243 | eqeq1d 2773 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
31
⊢ (𝑥 = 𝑎 → ((𝑥 ↾ ({𝑤} × 𝑁)) = ((𝑦 ↾ ({𝑤} × 𝑁)) ∘𝑓 + (𝑧 ↾ ({𝑤} × 𝑁))) ↔ (𝑎 ↾ ({𝑤} × 𝑁)) = ((𝑦 ↾ ({𝑤} × 𝑁)) ∘𝑓 + (𝑧 ↾ ({𝑤} × 𝑁))))) |
245 | | reseq1 5528 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
32
⊢ (𝑥 = 𝑎 → (𝑥 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑎 ↾ ((𝑁 ∖ {𝑤}) × 𝑁))) |
246 | 245 | eqeq1d 2773 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
31
⊢ (𝑥 = 𝑎 → ((𝑥 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑦 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) ↔ (𝑎 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑦 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)))) |
247 | 245 | eqeq1d 2773 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
31
⊢ (𝑥 = 𝑎 → ((𝑥 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑧 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) ↔ (𝑎 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑧 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)))) |
248 | 244, 246,
247 | 3anbi123d 1547 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
⊢ (𝑥 = 𝑎 → (((𝑥 ↾ ({𝑤} × 𝑁)) = ((𝑦 ↾ ({𝑤} × 𝑁)) ∘𝑓 + (𝑧 ↾ ({𝑤} × 𝑁))) ∧ (𝑥 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑦 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) ∧ (𝑥 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑧 ↾ ((𝑁 ∖ {𝑤}) × 𝑁))) ↔ ((𝑎 ↾ ({𝑤} × 𝑁)) = ((𝑦 ↾ ({𝑤} × 𝑁)) ∘𝑓 + (𝑧 ↾ ({𝑤} × 𝑁))) ∧ (𝑎 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑦 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) ∧ (𝑎 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑧 ↾ ((𝑁 ∖ {𝑤}) × 𝑁))))) |
249 | | fveq2 6332 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
31
⊢ (𝑥 = 𝑎 → (𝐷‘𝑥) = (𝐷‘𝑎)) |
250 | 249 | eqeq1d 2773 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
⊢ (𝑥 = 𝑎 → ((𝐷‘𝑥) = ((𝐷‘𝑦) + (𝐷‘𝑧)) ↔ (𝐷‘𝑎) = ((𝐷‘𝑦) + (𝐷‘𝑧)))) |
251 | 248, 250 | imbi12d 333 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
⊢ (𝑥 = 𝑎 → ((((𝑥 ↾ ({𝑤} × 𝑁)) = ((𝑦 ↾ ({𝑤} × 𝑁)) ∘𝑓 + (𝑧 ↾ ({𝑤} × 𝑁))) ∧ (𝑥 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑦 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) ∧ (𝑥 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑧 ↾ ((𝑁 ∖ {𝑤}) × 𝑁))) → (𝐷‘𝑥) = ((𝐷‘𝑦) + (𝐷‘𝑧))) ↔ (((𝑎 ↾ ({𝑤} × 𝑁)) = ((𝑦 ↾ ({𝑤} × 𝑁)) ∘𝑓 + (𝑧 ↾ ({𝑤} × 𝑁))) ∧ (𝑎 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑦 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) ∧ (𝑎 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑧 ↾ ((𝑁 ∖ {𝑤}) × 𝑁))) → (𝐷‘𝑎) = ((𝐷‘𝑦) + (𝐷‘𝑧))))) |
252 | 251 | 2ralbidv 3138 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ (𝑥 = 𝑎 → (∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝑁 (((𝑥 ↾ ({𝑤} × 𝑁)) = ((𝑦 ↾ ({𝑤} × 𝑁)) ∘𝑓 + (𝑧 ↾ ({𝑤} × 𝑁))) ∧ (𝑥 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑦 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) ∧ (𝑥 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑧 ↾ ((𝑁 ∖ {𝑤}) × 𝑁))) → (𝐷‘𝑥) = ((𝐷‘𝑦) + (𝐷‘𝑧))) ↔ ∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝑁 (((𝑎 ↾ ({𝑤} × 𝑁)) = ((𝑦 ↾ ({𝑤} × 𝑁)) ∘𝑓 + (𝑧 ↾ ({𝑤} × 𝑁))) ∧ (𝑎 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑦 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) ∧ (𝑎 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑧 ↾ ((𝑁 ∖ {𝑤}) × 𝑁))) → (𝐷‘𝑎) = ((𝐷‘𝑦) + (𝐷‘𝑧))))) |
253 | | reseq1 5528 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
33
⊢ (𝑦 = (𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) → (𝑦 ↾ ({𝑤} × 𝑁)) = ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ↾ ({𝑤} × 𝑁))) |
254 | 253 | oveq1d 6808 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
32
⊢ (𝑦 = (𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) → ((𝑦 ↾ ({𝑤} × 𝑁)) ∘𝑓 + (𝑧 ↾ ({𝑤} × 𝑁))) = (((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ↾ ({𝑤} × 𝑁)) ∘𝑓 + (𝑧 ↾ ({𝑤} × 𝑁)))) |
255 | 254 | eqeq2d 2781 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
31
⊢ (𝑦 = (𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) → ((𝑎 ↾ ({𝑤} × 𝑁)) = ((𝑦 ↾ ({𝑤} × 𝑁)) ∘𝑓 + (𝑧 ↾ ({𝑤} × 𝑁))) ↔ (𝑎 ↾ ({𝑤} × 𝑁)) = (((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ↾ ({𝑤} × 𝑁)) ∘𝑓 + (𝑧 ↾ ({𝑤} × 𝑁))))) |
256 | | reseq1 5528 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
32
⊢ (𝑦 = (𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) → (𝑦 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ↾ ((𝑁 ∖ {𝑤}) × 𝑁))) |
257 | 256 | eqeq2d 2781 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
31
⊢ (𝑦 = (𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) → ((𝑎 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑦 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) ↔ (𝑎 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ↾ ((𝑁 ∖ {𝑤}) × 𝑁)))) |
258 | 255, 257 | 3anbi12d 1548 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
⊢ (𝑦 = (𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) → (((𝑎 ↾ ({𝑤} × 𝑁)) = ((𝑦 ↾ ({𝑤} × 𝑁)) ∘𝑓 + (𝑧 ↾ ({𝑤} × 𝑁))) ∧ (𝑎 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑦 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) ∧ (𝑎 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑧 ↾ ((𝑁 ∖ {𝑤}) × 𝑁))) ↔ ((𝑎 ↾ ({𝑤} × 𝑁)) = (((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ↾ ({𝑤} × 𝑁)) ∘𝑓 + (𝑧 ↾ ({𝑤} × 𝑁))) ∧ (𝑎 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) ∧ (𝑎 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑧 ↾ ((𝑁 ∖ {𝑤}) × 𝑁))))) |
259 | | fveq2 6332 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
32
⊢ (𝑦 = (𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) → (𝐷‘𝑦) = (𝐷‘(𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))))) |
260 | 259 | oveq1d 6808 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
31
⊢ (𝑦 = (𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) → ((𝐷‘𝑦) + (𝐷‘𝑧)) = ((𝐷‘(𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒)))) + (𝐷‘𝑧))) |
261 | 260 | eqeq2d 2781 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
⊢ (𝑦 = (𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) → ((𝐷‘𝑎) = ((𝐷‘𝑦) + (𝐷‘𝑧)) ↔ (𝐷‘𝑎) = ((𝐷‘(𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒)))) + (𝐷‘𝑧)))) |
262 | 258, 261 | imbi12d 333 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
⊢ (𝑦 = (𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) → ((((𝑎 ↾ ({𝑤} × 𝑁)) = ((𝑦 ↾ ({𝑤} × 𝑁)) ∘𝑓 + (𝑧 ↾ ({𝑤} × 𝑁))) ∧ (𝑎 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑦 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) ∧ (𝑎 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑧 ↾ ((𝑁 ∖ {𝑤}) × 𝑁))) → (𝐷‘𝑎) = ((𝐷‘𝑦) + (𝐷‘𝑧))) ↔ (((𝑎 ↾ ({𝑤} × 𝑁)) = (((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ↾ ({𝑤} × 𝑁)) ∘𝑓 + (𝑧 ↾ ({𝑤} × 𝑁))) ∧ (𝑎 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) ∧ (𝑎 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑧 ↾ ((𝑁 ∖ {𝑤}) × 𝑁))) → (𝐷‘𝑎) = ((𝐷‘(𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒)))) + (𝐷‘𝑧))))) |
263 | 262 | 2ralbidv 3138 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ (𝑦 = (𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) → (∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝑁 (((𝑎 ↾ ({𝑤} × 𝑁)) = ((𝑦 ↾ ({𝑤} × 𝑁)) ∘𝑓 + (𝑧 ↾ ({𝑤} × 𝑁))) ∧ (𝑎 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑦 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) ∧ (𝑎 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑧 ↾ ((𝑁 ∖ {𝑤}) × 𝑁))) → (𝐷‘𝑎) = ((𝐷‘𝑦) + (𝐷‘𝑧))) ↔ ∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝑁 (((𝑎 ↾ ({𝑤} × 𝑁)) = (((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ↾ ({𝑤} × 𝑁)) ∘𝑓 + (𝑧 ↾ ({𝑤} × 𝑁))) ∧ (𝑎 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) ∧ (𝑎 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑧 ↾ ((𝑁 ∖ {𝑤}) × 𝑁))) → (𝐷‘𝑎) = ((𝐷‘(𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒)))) + (𝐷‘𝑧))))) |
264 | 252, 263 | rspc2va 3473 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ (((𝑎 ∈ 𝐵 ∧ (𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ∈ 𝐵) ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝑁 (((𝑥 ↾ ({𝑤} × 𝑁)) = ((𝑦 ↾ ({𝑤} × 𝑁)) ∘𝑓 + (𝑧 ↾ ({𝑤} × 𝑁))) ∧ (𝑥 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑦 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) ∧ (𝑥 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑧 ↾ ((𝑁 ∖ {𝑤}) × 𝑁))) → (𝐷‘𝑥) = ((𝐷‘𝑦) + (𝐷‘𝑧)))) → ∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝑁 (((𝑎 ↾ ({𝑤} × 𝑁)) = (((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ↾ ({𝑤} × 𝑁)) ∘𝑓 + (𝑧 ↾ ({𝑤} × 𝑁))) ∧ (𝑎 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) ∧ (𝑎 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑧 ↾ ((𝑁 ∖ {𝑤}) × 𝑁))) → (𝐷‘𝑎) = ((𝐷‘(𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒)))) + (𝐷‘𝑧)))) |
265 | 229, 241,
242, 264 | syl21anc 1475 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) → ∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝑁 (((𝑎 ↾ ({𝑤} × 𝑁)) = (((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ↾ ({𝑤} × 𝑁)) ∘𝑓 + (𝑧 ↾ ({𝑤} × 𝑁))) ∧ (𝑎 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) ∧ (𝑎 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑧 ↾ ((𝑁 ∖ {𝑤}) × 𝑁))) → (𝐷‘𝑎) = ((𝐷‘(𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒)))) + (𝐷‘𝑧)))) |
266 | | reseq1 5528 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
31
⊢ (𝑧 = (𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒 ∈ 𝑑, 1 , 0 ), (𝑎‘𝑒))) → (𝑧 ↾ ({𝑤} × 𝑁)) = ((𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒 ∈ 𝑑, 1 , 0 ), (𝑎‘𝑒))) ↾ ({𝑤} × 𝑁))) |
267 | 266 | oveq2d 6809 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
⊢ (𝑧 = (𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒 ∈ 𝑑, 1 , 0 ), (𝑎‘𝑒))) → (((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ↾ ({𝑤} × 𝑁)) ∘𝑓 + (𝑧 ↾ ({𝑤} × 𝑁))) = (((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ↾ ({𝑤} × 𝑁)) ∘𝑓 + ((𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒 ∈ 𝑑, 1 , 0 ), (𝑎‘𝑒))) ↾ ({𝑤} × 𝑁)))) |
268 | 267 | eqeq2d 2781 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
⊢ (𝑧 = (𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒 ∈ 𝑑, 1 , 0 ), (𝑎‘𝑒))) → ((𝑎 ↾ ({𝑤} × 𝑁)) = (((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ↾ ({𝑤} × 𝑁)) ∘𝑓 + (𝑧 ↾ ({𝑤} × 𝑁))) ↔ (𝑎 ↾ ({𝑤} × 𝑁)) = (((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ↾ ({𝑤} × 𝑁)) ∘𝑓 + ((𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒 ∈ 𝑑, 1 , 0 ), (𝑎‘𝑒))) ↾ ({𝑤} × 𝑁))))) |
269 | | reseq1 5528 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
⊢ (𝑧 = (𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒 ∈ 𝑑, 1 , 0 ), (𝑎‘𝑒))) → (𝑧 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = ((𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒 ∈ 𝑑, 1 , 0 ), (𝑎‘𝑒))) ↾ ((𝑁 ∖ {𝑤}) × 𝑁))) |
270 | 269 | eqeq2d 2781 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
⊢ (𝑧 = (𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒 ∈ 𝑑, 1 , 0 ), (𝑎‘𝑒))) → ((𝑎 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑧 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) ↔ (𝑎 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = ((𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒 ∈ 𝑑, 1 , 0 ), (𝑎‘𝑒))) ↾ ((𝑁 ∖ {𝑤}) × 𝑁)))) |
271 | 268, 270 | 3anbi13d 1549 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ (𝑧 = (𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒 ∈ 𝑑, 1 , 0 ), (𝑎‘𝑒))) → (((𝑎 ↾ ({𝑤} × 𝑁)) = (((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ↾ ({𝑤} × 𝑁)) ∘𝑓 + (𝑧 ↾ ({𝑤} × 𝑁))) ∧ (𝑎 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) ∧ (𝑎 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑧 ↾ ((𝑁 ∖ {𝑤}) × 𝑁))) ↔ ((𝑎 ↾ ({𝑤} × 𝑁)) = (((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ↾ ({𝑤} × 𝑁)) ∘𝑓 + ((𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒 ∈ 𝑑, 1 , 0 ), (𝑎‘𝑒))) ↾ ({𝑤} × 𝑁))) ∧ (𝑎 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) ∧ (𝑎 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = ((𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒 ∈ 𝑑, 1 , 0 ), (𝑎‘𝑒))) ↾ ((𝑁 ∖ {𝑤}) × 𝑁))))) |
272 | | fveq2 6332 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
⊢ (𝑧 = (𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒 ∈ 𝑑, 1 , 0 ), (𝑎‘𝑒))) → (𝐷‘𝑧) = (𝐷‘(𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒 ∈ 𝑑, 1 , 0 ), (𝑎‘𝑒))))) |
273 | 272 | oveq2d 6809 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
⊢ (𝑧 = (𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒 ∈ 𝑑, 1 , 0 ), (𝑎‘𝑒))) → ((𝐷‘(𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒)))) + (𝐷‘𝑧)) = ((𝐷‘(𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒)))) + (𝐷‘(𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒 ∈ 𝑑, 1 , 0 ), (𝑎‘𝑒)))))) |
274 | 273 | eqeq2d 2781 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ (𝑧 = (𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒 ∈ 𝑑, 1 , 0 ), (𝑎‘𝑒))) → ((𝐷‘𝑎) = ((𝐷‘(𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒)))) + (𝐷‘𝑧)) ↔ (𝐷‘𝑎) = ((𝐷‘(𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒)))) + (𝐷‘(𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒 ∈ 𝑑, 1 , 0 ), (𝑎‘𝑒))))))) |
275 | 271, 274 | imbi12d 333 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ (𝑧 = (𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒 ∈ 𝑑, 1 , 0 ), (𝑎‘𝑒))) → ((((𝑎 ↾ ({𝑤} × 𝑁)) = (((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ↾ ({𝑤} × 𝑁)) ∘𝑓 + (𝑧 ↾ ({𝑤} × 𝑁))) ∧ (𝑎 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) ∧ (𝑎 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑧 ↾ ((𝑁 ∖ {𝑤}) × 𝑁))) → (𝐷‘𝑎) = ((𝐷‘(𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒)))) + (𝐷‘𝑧))) ↔ (((𝑎 ↾ ({𝑤} × 𝑁)) = (((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ↾ ({𝑤} × 𝑁)) ∘𝑓 + ((𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒 ∈ 𝑑, 1 , 0 ), (𝑎‘𝑒))) ↾ ({𝑤} × 𝑁))) ∧ (𝑎 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) ∧ (𝑎 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = ((𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒 ∈ 𝑑, 1 , 0 ), (𝑎‘𝑒))) ↾ ((𝑁 ∖ {𝑤}) × 𝑁))) → (𝐷‘𝑎) = ((𝐷‘(𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒)))) + (𝐷‘(𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒 ∈ 𝑑, 1 , 0 ), (𝑎‘𝑒)))))))) |
276 | | sneq 4326 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
32
⊢ (𝑤 = (1st ‘𝑐) → {𝑤} = {(1st ‘𝑐)}) |
277 | 276 | xpeq1d 5278 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
31
⊢ (𝑤 = (1st ‘𝑐) → ({𝑤} × 𝑁) = ({(1st ‘𝑐)} × 𝑁)) |
278 | 277 | reseq2d 5534 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
⊢ (𝑤 = (1st ‘𝑐) → (𝑎 ↾ ({𝑤} × 𝑁)) = (𝑎 ↾ ({(1st ‘𝑐)} × 𝑁))) |
279 | 277 | reseq2d 5534 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
31
⊢ (𝑤 = (1st ‘𝑐) → ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ↾ ({𝑤} × 𝑁)) = ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ↾ ({(1st ‘𝑐)} × 𝑁))) |
280 | 277 | reseq2d 5534 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
31
⊢ (𝑤 = (1st ‘𝑐) → ((𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒 ∈ 𝑑, 1 , 0 ), (𝑎‘𝑒))) ↾ ({𝑤} × 𝑁)) = ((𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒 ∈ 𝑑, 1 , 0 ), (𝑎‘𝑒))) ↾ ({(1st ‘𝑐)} × 𝑁))) |
281 | 279, 280 | oveq12d 6811 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
⊢ (𝑤 = (1st ‘𝑐) → (((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ↾ ({𝑤} × 𝑁)) ∘𝑓 + ((𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒 ∈ 𝑑, 1 , 0 ), (𝑎‘𝑒))) ↾ ({𝑤} × 𝑁))) = (((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ↾ ({(1st ‘𝑐)} × 𝑁)) ∘𝑓 + ((𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒 ∈ 𝑑, 1 , 0 ), (𝑎‘𝑒))) ↾ ({(1st ‘𝑐)} × 𝑁)))) |
282 | 278, 281 | eqeq12d 2786 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
⊢ (𝑤 = (1st ‘𝑐) → ((𝑎 ↾ ({𝑤} × 𝑁)) = (((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ↾ ({𝑤} × 𝑁)) ∘𝑓 + ((𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒 ∈ 𝑑, 1 , 0 ), (𝑎‘𝑒))) ↾ ({𝑤} × 𝑁))) ↔ (𝑎 ↾ ({(1st ‘𝑐)} × 𝑁)) = (((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ↾ ({(1st ‘𝑐)} × 𝑁)) ∘𝑓 + ((𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒 ∈ 𝑑, 1 , 0 ), (𝑎‘𝑒))) ↾ ({(1st ‘𝑐)} × 𝑁))))) |
283 | 276 | difeq2d 3879 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
32
⊢ (𝑤 = (1st ‘𝑐) → (𝑁 ∖ {𝑤}) = (𝑁 ∖ {(1st ‘𝑐)})) |
284 | 283 | xpeq1d 5278 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
31
⊢ (𝑤 = (1st ‘𝑐) → ((𝑁 ∖ {𝑤}) × 𝑁) = ((𝑁 ∖ {(1st ‘𝑐)}) × 𝑁)) |
285 | 284 | reseq2d 5534 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
⊢ (𝑤 = (1st ‘𝑐) → (𝑎 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑎 ↾ ((𝑁 ∖ {(1st ‘𝑐)}) × 𝑁))) |
286 | 284 | reseq2d 5534 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
⊢ (𝑤 = (1st ‘𝑐) → ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ↾ ((𝑁 ∖ {(1st ‘𝑐)}) × 𝑁))) |
287 | 285, 286 | eqeq12d 2786 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
⊢ (𝑤 = (1st ‘𝑐) → ((𝑎 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) ↔ (𝑎 ↾ ((𝑁 ∖ {(1st ‘𝑐)}) × 𝑁)) = ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ↾ ((𝑁 ∖ {(1st ‘𝑐)}) × 𝑁)))) |
288 | 284 | reseq2d 5534 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
⊢ (𝑤 = (1st ‘𝑐) → ((𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒 ∈ 𝑑, 1 , 0 ), (𝑎‘𝑒))) ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = ((𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒 ∈ 𝑑, 1 , 0 ), (𝑎‘𝑒))) ↾ ((𝑁 ∖ {(1st ‘𝑐)}) × 𝑁))) |
289 | 285, 288 | eqeq12d 2786 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
⊢ (𝑤 = (1st ‘𝑐) → ((𝑎 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = ((𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒 ∈ 𝑑, 1 , 0 ), (𝑎‘𝑒))) ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) ↔ (𝑎 ↾ ((𝑁 ∖ {(1st ‘𝑐)}) × 𝑁)) = ((𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒 ∈ 𝑑, 1 , 0 ), (𝑎‘𝑒))) ↾ ((𝑁 ∖ {(1st ‘𝑐)}) × 𝑁)))) |
290 | 282, 287,
289 | 3anbi123d 1547 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ (𝑤 = (1st ‘𝑐) → (((𝑎 ↾ ({𝑤} × 𝑁)) = (((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ↾ ({𝑤} × 𝑁)) ∘𝑓 + ((𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒 ∈ 𝑑, 1 , 0 ), (𝑎‘𝑒))) ↾ ({𝑤} × 𝑁))) ∧ (𝑎 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) ∧ (𝑎 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = ((𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒 ∈ 𝑑, 1 , 0 ), (𝑎‘𝑒))) ↾ ((𝑁 ∖ {𝑤}) × 𝑁))) ↔ ((𝑎 ↾ ({(1st ‘𝑐)} × 𝑁)) = (((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ↾ ({(1st ‘𝑐)} × 𝑁)) ∘𝑓 + ((𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒 ∈ 𝑑, 1 , 0 ), (𝑎‘𝑒))) ↾ ({(1st ‘𝑐)} × 𝑁))) ∧ (𝑎 ↾ ((𝑁 ∖ {(1st ‘𝑐)}) × 𝑁)) = ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ↾ ((𝑁 ∖ {(1st ‘𝑐)}) × 𝑁)) ∧ (𝑎 ↾ ((𝑁 ∖ {(1st ‘𝑐)}) × 𝑁)) = ((𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒 ∈ 𝑑, 1 , 0 ), (𝑎‘𝑒))) ↾ ((𝑁 ∖ {(1st ‘𝑐)}) × 𝑁))))) |
291 | 290 | imbi1d 330 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ (𝑤 = (1st ‘𝑐) → ((((𝑎 ↾ ({𝑤} × 𝑁)) = (((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ↾ ({𝑤} × 𝑁)) ∘𝑓 + ((𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒 ∈ 𝑑, 1 , 0 ), (𝑎‘𝑒))) ↾ ({𝑤} × 𝑁))) ∧ (𝑎 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) ∧ (𝑎 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = ((𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒 ∈ 𝑑, 1 , 0 ), (𝑎‘𝑒))) ↾ ((𝑁 ∖ {𝑤}) × 𝑁))) → (𝐷‘𝑎) = ((𝐷‘(𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒)))) + (𝐷‘(𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒 ∈ 𝑑, 1 , 0 ), (𝑎‘𝑒)))))) ↔ (((𝑎 ↾ ({(1st ‘𝑐)} × 𝑁)) = (((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ↾ ({(1st ‘𝑐)} × 𝑁)) ∘𝑓 + ((𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒 ∈ 𝑑, 1 , 0 ), (𝑎‘𝑒))) ↾ ({(1st ‘𝑐)} × 𝑁))) ∧ (𝑎 ↾ ((𝑁 ∖ {(1st ‘𝑐)}) × 𝑁)) = ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ↾ ((𝑁 ∖ {(1st ‘𝑐)}) × 𝑁)) ∧ (𝑎 ↾ ((𝑁 ∖ {(1st ‘𝑐)}) × 𝑁)) = ((𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒 ∈ 𝑑, 1 , 0 ), (𝑎‘𝑒))) ↾ ((𝑁 ∖ {(1st ‘𝑐)}) × 𝑁))) → (𝐷‘𝑎) = ((𝐷‘(𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒)))) + (𝐷‘(𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒 ∈ 𝑑, 1 , 0 ), (𝑎‘𝑒)))))))) |
292 | 275, 291 | rspc2va 3473 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ ((((𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒 ∈ 𝑑, 1 , 0 ), (𝑎‘𝑒))) ∈ 𝐵 ∧ (1st ‘𝑐) ∈ 𝑁) ∧ ∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝑁 (((𝑎 ↾ ({𝑤} × 𝑁)) = (((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ↾ ({𝑤} × 𝑁)) ∘𝑓 + (𝑧 ↾ ({𝑤} × 𝑁))) ∧ (𝑎 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) ∧ (𝑎 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑧 ↾ ((𝑁 ∖ {𝑤}) × 𝑁))) → (𝐷‘𝑎) = ((𝐷‘(𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒)))) + (𝐷‘𝑧)))) → (((𝑎 ↾ ({(1st ‘𝑐)} × 𝑁)) = (((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ↾ ({(1st ‘𝑐)} × 𝑁)) ∘𝑓 + ((𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒 ∈ 𝑑, 1 , 0 ), (𝑎‘𝑒))) ↾ ({(1st ‘𝑐)} × 𝑁))) ∧ (𝑎 ↾ ((𝑁 ∖ {(1st ‘𝑐)}) × 𝑁)) = ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ↾ ((𝑁 ∖ {(1st ‘𝑐)}) × 𝑁)) ∧ (𝑎 ↾ ((𝑁 ∖ {(1st ‘𝑐)}) × 𝑁)) = ((𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒 ∈ 𝑑, 1 , 0 ), (𝑎‘𝑒))) ↾ ((𝑁 ∖ {(1st ‘𝑐)}) × 𝑁))) → (𝐷‘𝑎) = ((𝐷‘(𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒)))) + (𝐷‘(𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒 ∈ 𝑑, 1 , 0 ), (𝑎‘𝑒))))))) |
293 | 228, 125,
265, 292 | syl21anc 1475 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) → (((𝑎 ↾ ({(1st ‘𝑐)} × 𝑁)) = (((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ↾ ({(1st ‘𝑐)} × 𝑁)) ∘𝑓 + ((𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒 ∈ 𝑑, 1 , 0 ), (𝑎‘𝑒))) ↾ ({(1st ‘𝑐)} × 𝑁))) ∧ (𝑎 ↾ ((𝑁 ∖ {(1st ‘𝑐)}) × 𝑁)) = ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ↾ ((𝑁 ∖ {(1st ‘𝑐)}) × 𝑁)) ∧ (𝑎 ↾ ((𝑁 ∖ {(1st ‘𝑐)}) × 𝑁)) = ((𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒 ∈ 𝑑, 1 , 0 ), (𝑎‘𝑒))) ↾ ((𝑁 ∖ {(1st ‘𝑐)}) × 𝑁))) → (𝐷‘𝑎) = ((𝐷‘(𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒)))) + (𝐷‘(𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒 ∈ 𝑑, 1 , 0 ), (𝑎‘𝑒))))))) |
294 | 187, 204,
212, 293 | mp3and 1575 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) → (𝐷‘𝑎) = ((𝐷‘(𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒)))) + (𝐷‘(𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒 ∈ 𝑑, 1 , 0 ), (𝑎‘𝑒)))))) |
295 | 105, 106,
107, 294 | syl3anc 1476 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ (𝑏 ∪ {𝑐}) ∈ 𝑌) ∧ ((𝑎 ∈ 𝐵 ∧ 𝑑 ∈ (𝑁 ↑𝑚 𝑁)) ∧ ∀𝑤 ∈ 𝑏 (𝑎‘𝑤) = if(𝑤 ∈ 𝑑, 1 , 0 ))) → (𝐷‘𝑎) = ((𝐷‘(𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒)))) + (𝐷‘(𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒 ∈ 𝑑, 1 , 0 ), (𝑎‘𝑒)))))) |
296 | | fveq2 6332 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
34
⊢ (𝑒 = 𝑐 → (𝑎‘𝑒) = (𝑎‘𝑐)) |
297 | | elequ1 2152 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. 35
⊢ (𝑒 = 𝑐 → (𝑒 ∈ 𝑑 ↔ 𝑐 ∈ 𝑑)) |
298 | 297 | ifbid 4247 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
34
⊢ (𝑒 = 𝑐 → if(𝑒 ∈ 𝑑, 1 , 0 ) = if(𝑐 ∈ 𝑑, 1 , 0 )) |
299 | 296, 298 | oveq12d 6811 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
33
⊢ (𝑒 = 𝑐 → ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )) = ((𝑎‘𝑐)(-g‘𝑅)if(𝑐 ∈ 𝑑, 1 , 0 ))) |
300 | 299 | adantl 467 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
32
⊢ ((((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) ∧ 𝑒 ∈ ({(1st ‘𝑐)} × 𝑁)) ∧ 𝑒 = 𝑐) → ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )) = ((𝑎‘𝑐)(-g‘𝑅)if(𝑐 ∈ 𝑑, 1 , 0 ))) |
301 | 111, 123 | ffvelrnd 6503 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. 35
⊢ ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) → (𝑎‘𝑐) ∈ 𝐾) |
302 | 132, 134 | ifcld 4270 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. 35
⊢ ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) → if(𝑐 ∈ 𝑑, 1 , 0 ) ∈ 𝐾) |
303 | 12, 137 | grpsubcl 17703 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. 35
⊢ ((𝑅 ∈ Grp ∧ (𝑎‘𝑐) ∈ 𝐾 ∧ if(𝑐 ∈ 𝑑, 1 , 0 ) ∈ 𝐾) → ((𝑎‘𝑐)(-g‘𝑅)if(𝑐 ∈ 𝑑, 1 , 0 )) ∈ 𝐾) |
304 | 116, 301,
302, 303 | syl3anc 1476 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
34
⊢ ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) → ((𝑎‘𝑐)(-g‘𝑅)if(𝑐 ∈ 𝑑, 1 , 0 )) ∈ 𝐾) |
305 | 12, 52, 50 | ringridm 18780 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
34
⊢ ((𝑅 ∈ Ring ∧ ((𝑎‘𝑐)(-g‘𝑅)if(𝑐 ∈ 𝑑, 1 , 0 )) ∈ 𝐾) → (((𝑎‘𝑐)(-g‘𝑅)if(𝑐 ∈ 𝑑, 1 , 0 )) · 1 ) = ((𝑎‘𝑐)(-g‘𝑅)if(𝑐 ∈ 𝑑, 1 , 0 ))) |
306 | 114, 304,
305 | syl2anc 573 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
33
⊢ ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) → (((𝑎‘𝑐)(-g‘𝑅)if(𝑐 ∈ 𝑑, 1 , 0 )) · 1 ) = ((𝑎‘𝑐)(-g‘𝑅)if(𝑐 ∈ 𝑑, 1 , 0 ))) |
307 | 306 | ad2antrr 705 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
32
⊢ ((((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) ∧ 𝑒 ∈ ({(1st ‘𝑐)} × 𝑁)) ∧ 𝑒 = 𝑐) → (((𝑎‘𝑐)(-g‘𝑅)if(𝑐 ∈ 𝑑, 1 , 0 )) · 1 ) = ((𝑎‘𝑐)(-g‘𝑅)if(𝑐 ∈ 𝑑, 1 , 0 ))) |
308 | 300, 307 | eqtr4d 2808 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
31
⊢ ((((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) ∧ 𝑒 ∈ ({(1st ‘𝑐)} × 𝑁)) ∧ 𝑒 = 𝑐) → ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )) = (((𝑎‘𝑐)(-g‘𝑅)if(𝑐 ∈ 𝑑, 1 , 0 )) · 1 )) |
309 | 142 | adantl 467 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
31
⊢ ((((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) ∧ 𝑒 ∈ ({(1st ‘𝑐)} × 𝑁)) ∧ 𝑒 = 𝑐) → if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ) = ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 ))) |
310 | | iftrue 4231 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
33
⊢ (𝑒 = 𝑐 → if(𝑒 = 𝑐, 1 , 0 ) = 1 ) |
311 | 310 | oveq2d 6809 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
32
⊢ (𝑒 = 𝑐 → (((𝑎‘𝑐)(-g‘𝑅)if(𝑐 ∈ 𝑑, 1 , 0 )) · if(𝑒 = 𝑐, 1 , 0 )) = (((𝑎‘𝑐)(-g‘𝑅)if(𝑐 ∈ 𝑑, 1 , 0 )) · 1 )) |
312 | 311 | adantl 467 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
31
⊢ ((((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) ∧ 𝑒 ∈ ({(1st ‘𝑐)} × 𝑁)) ∧ 𝑒 = 𝑐) → (((𝑎‘𝑐)(-g‘𝑅)if(𝑐 ∈ 𝑑, 1 , 0 )) · if(𝑒 = 𝑐, 1 , 0 )) = (((𝑎‘𝑐)(-g‘𝑅)if(𝑐 ∈ 𝑑, 1 , 0 )) · 1 )) |
313 | 308, 309,
312 | 3eqtr4d 2815 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
⊢ ((((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) ∧ 𝑒 ∈ ({(1st ‘𝑐)} × 𝑁)) ∧ 𝑒 = 𝑐) → if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ) = (((𝑎‘𝑐)(-g‘𝑅)if(𝑐 ∈ 𝑑, 1 , 0 )) · if(𝑒 = 𝑐, 1 , 0 ))) |
314 | 12, 52, 49 | ringrz 18796 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
34
⊢ ((𝑅 ∈ Ring ∧ ((𝑎‘𝑐)(-g‘𝑅)if(𝑐 ∈ 𝑑, 1 , 0 )) ∈ 𝐾) → (((𝑎‘𝑐)(-g‘𝑅)if(𝑐 ∈ 𝑑, 1 , 0 )) · 0 ) = 0 ) |
315 | 114, 304,
314 | syl2anc 573 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
33
⊢ ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) → (((𝑎‘𝑐)(-g‘𝑅)if(𝑐 ∈ 𝑑, 1 , 0 )) · 0 ) = 0 ) |
316 | 315 | eqcomd 2777 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
32
⊢ ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) → 0 = (((𝑎‘𝑐)(-g‘𝑅)if(𝑐 ∈ 𝑑, 1 , 0 )) · 0 )) |
317 | 316 | ad2antrr 705 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
31
⊢ ((((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) ∧ 𝑒 ∈ ({(1st ‘𝑐)} × 𝑁)) ∧ ¬ 𝑒 = 𝑐) → 0 = (((𝑎‘𝑐)(-g‘𝑅)if(𝑐 ∈ 𝑑, 1 , 0 )) · 0 )) |
318 | 151 | adantl 467 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
31
⊢ ((((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) ∧ 𝑒 ∈ ({(1st ‘𝑐)} × 𝑁)) ∧ ¬ 𝑒 = 𝑐) → if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ) = 0 ) |
319 | | iffalse 4234 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
33
⊢ (¬
𝑒 = 𝑐 → if(𝑒 = 𝑐, 1 , 0 ) = 0 ) |
320 | 319 | oveq2d 6809 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
32
⊢ (¬
𝑒 = 𝑐 → (((𝑎‘𝑐)(-g‘𝑅)if(𝑐 ∈ 𝑑, 1 , 0 )) · if(𝑒 = 𝑐, 1 , 0 )) = (((𝑎‘𝑐)(-g‘𝑅)if(𝑐 ∈ 𝑑, 1 , 0 )) · 0 )) |
321 | 320 | adantl 467 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
31
⊢ ((((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) ∧ 𝑒 ∈ ({(1st ‘𝑐)} × 𝑁)) ∧ ¬ 𝑒 = 𝑐) → (((𝑎‘𝑐)(-g‘𝑅)if(𝑐 ∈ 𝑑, 1 , 0 )) · if(𝑒 = 𝑐, 1 , 0 )) = (((𝑎‘𝑐)(-g‘𝑅)if(𝑐 ∈ 𝑑, 1 , 0 )) · 0 )) |
322 | 317, 318,
321 | 3eqtr4d 2815 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
⊢ ((((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) ∧ 𝑒 ∈ ({(1st ‘𝑐)} × 𝑁)) ∧ ¬ 𝑒 = 𝑐) → if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ) = (((𝑎‘𝑐)(-g‘𝑅)if(𝑐 ∈ 𝑑, 1 , 0 )) · if(𝑒 = 𝑐, 1 , 0 ))) |
323 | 313, 322 | pm2.61dan 813 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
⊢ (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) ∧ 𝑒 ∈ ({(1st ‘𝑐)} × 𝑁)) → if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ) = (((𝑎‘𝑐)(-g‘𝑅)if(𝑐 ∈ 𝑑, 1 , 0 )) · if(𝑒 = 𝑐, 1 , 0 ))) |
324 | 173 | adantl 467 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
31
⊢ (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) ∧ 𝑒 ∈ ({(1st ‘𝑐)} × 𝑁)) → (1st ‘𝑒) ∈ {(1st
‘𝑐)}) |
325 | 324, 174 | syl 17 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
⊢ (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) ∧ 𝑒 ∈ ({(1st ‘𝑐)} × 𝑁)) → (1st ‘𝑒) = (1st ‘𝑐)) |
326 | 325 | iftrued 4233 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
⊢ (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) ∧ 𝑒 ∈ ({(1st ‘𝑐)} × 𝑁)) → if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒)) = if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 )) |
327 | 325 | iftrued 4233 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
⊢ (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) ∧ 𝑒 ∈ ({(1st ‘𝑐)} × 𝑁)) → if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒)) = if(𝑒 = 𝑐, 1 , 0 )) |
328 | 327 | oveq2d 6809 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
⊢ (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) ∧ 𝑒 ∈ ({(1st ‘𝑐)} × 𝑁)) → (((𝑎‘𝑐)(-g‘𝑅)if(𝑐 ∈ 𝑑, 1 , 0 )) · if((1st
‘𝑒) = (1st
‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒))) = (((𝑎‘𝑐)(-g‘𝑅)if(𝑐 ∈ 𝑑, 1 , 0 )) · if(𝑒 = 𝑐, 1 , 0 ))) |
329 | 323, 326,
328 | 3eqtr4d 2815 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) ∧ 𝑒 ∈ ({(1st ‘𝑐)} × 𝑁)) → if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒)) = (((𝑎‘𝑐)(-g‘𝑅)if(𝑐 ∈ 𝑑, 1 , 0 )) · if((1st
‘𝑒) = (1st
‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒)))) |
330 | 329 | mpteq2dva 4878 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) → (𝑒 ∈ ({(1st ‘𝑐)} × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) = (𝑒 ∈ ({(1st ‘𝑐)} × 𝑁) ↦ (((𝑎‘𝑐)(-g‘𝑅)if(𝑐 ∈ 𝑑, 1 , 0 )) · if((1st
‘𝑒) = (1st
‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒))))) |
331 | | ovexd 6825 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) ∧ 𝑒 ∈ ({(1st ‘𝑐)} × 𝑁)) → ((𝑎‘𝑐)(-g‘𝑅)if(𝑐 ∈ 𝑑, 1 , 0 )) ∈
V) |
332 | 168, 164 | ifex 4295 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
⊢ if(𝑒 = 𝑐, 1 , 0 ) ∈
V |
333 | 332, 170 | ifex 4295 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
⊢
if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒)) ∈ V |
334 | 333 | a1i 11 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) ∧ 𝑒 ∈ ({(1st ‘𝑐)} × 𝑁)) → if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒)) ∈ V) |
335 | | fconstmpt 5303 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
⊢
(({(1st ‘𝑐)} × 𝑁) × {((𝑎‘𝑐)(-g‘𝑅)if(𝑐 ∈ 𝑑, 1 , 0 ))}) = (𝑒 ∈ ({(1st ‘𝑐)} × 𝑁) ↦ ((𝑎‘𝑐)(-g‘𝑅)if(𝑐 ∈ 𝑑, 1 , 0 ))) |
336 | 335 | a1i 11 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) → (({(1st ‘𝑐)} × 𝑁) × {((𝑎‘𝑐)(-g‘𝑅)if(𝑐 ∈ 𝑑, 1 , 0 ))}) = (𝑒 ∈ ({(1st ‘𝑐)} × 𝑁) ↦ ((𝑎‘𝑐)(-g‘𝑅)if(𝑐 ∈ 𝑑, 1 , 0 )))) |
337 | 128 | resmptd 5593 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) → ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒))) ↾ ({(1st ‘𝑐)} × 𝑁)) = (𝑒 ∈ ({(1st ‘𝑐)} × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒)))) |
338 | 161, 331,
334, 336, 337 | offval2 7061 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) → ((({(1st ‘𝑐)} × 𝑁) × {((𝑎‘𝑐)(-g‘𝑅)if(𝑐 ∈ 𝑑, 1 , 0 ))})
∘𝑓 · ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒))) ↾ ({(1st ‘𝑐)} × 𝑁))) = (𝑒 ∈ ({(1st ‘𝑐)} × 𝑁) ↦ (((𝑎‘𝑐)(-g‘𝑅)if(𝑐 ∈ 𝑑, 1 , 0 )) · if((1st
‘𝑒) = (1st
‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒))))) |
339 | 330, 183,
338 | 3eqtr4d 2815 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) → ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ↾ ({(1st ‘𝑐)} × 𝑁)) = ((({(1st ‘𝑐)} × 𝑁) × {((𝑎‘𝑐)(-g‘𝑅)if(𝑐 ∈ 𝑑, 1 , 0 ))})
∘𝑓 · ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒))) ↾ ({(1st ‘𝑐)} × 𝑁)))) |
340 | | iffalse 4234 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
⊢ (¬
(1st ‘𝑒) =
(1st ‘𝑐)
→ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒)) = (𝑎‘𝑒)) |
341 | | iffalse 4234 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
⊢ (¬
(1st ‘𝑒) =
(1st ‘𝑐)
→ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒)) = (𝑎‘𝑒)) |
342 | 340, 341 | eqtr4d 2808 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
⊢ (¬
(1st ‘𝑒) =
(1st ‘𝑐)
→ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒)) = if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒))) |
343 | 193, 342 | syl 17 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) ∧ 𝑒 ∈ ((𝑁 ∖ {(1st ‘𝑐)}) × 𝑁)) → if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒)) = if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒))) |
344 | 343 | mpteq2dva 4878 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) → (𝑒 ∈ ((𝑁 ∖ {(1st ‘𝑐)}) × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) = (𝑒 ∈ ((𝑁 ∖ {(1st ‘𝑐)}) × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒)))) |
345 | | resmpt 5590 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ (((𝑁 ∖ {(1st
‘𝑐)}) × 𝑁) ⊆ (𝑁 × 𝑁) → ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒))) ↾ ((𝑁 ∖ {(1st ‘𝑐)}) × 𝑁)) = (𝑒 ∈ ((𝑁 ∖ {(1st ‘𝑐)}) × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒)))) |
346 | 198, 345 | mp1i 13 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) → ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒))) ↾ ((𝑁 ∖ {(1st ‘𝑐)}) × 𝑁)) = (𝑒 ∈ ((𝑁 ∖ {(1st ‘𝑐)}) × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒)))) |
347 | 344, 200,
346 | 3eqtr4d 2815 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) → ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ↾ ((𝑁 ∖ {(1st ‘𝑐)}) × 𝑁)) = ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒))) ↾ ((𝑁 ∖ {(1st ‘𝑐)}) × 𝑁))) |
348 | 132, 134 | ifcld 4270 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
32
⊢ ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) → if(𝑒 = 𝑐, 1 , 0 ) ∈ 𝐾) |
349 | 348 | adantr 466 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
31
⊢ (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) ∧ 𝑒 ∈ (𝑁 × 𝑁)) → if(𝑒 = 𝑐, 1 , 0 ) ∈ 𝐾) |
350 | 349, 214 | ifcld 4270 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
⊢ (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) ∧ 𝑒 ∈ (𝑁 × 𝑁)) → if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒)) ∈ 𝐾) |
351 | | eqid 2771 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
⊢ (𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒))) = (𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒))) |
352 | 350, 351 | fmptd 6527 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
⊢ ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) → (𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒))):(𝑁 × 𝑁)⟶𝐾) |
353 | | elmapg 8022 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
⊢ ((𝐾 ∈ V ∧ (𝑁 × 𝑁) ∈ Fin) → ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒))) ∈ (𝐾 ↑𝑚 (𝑁 × 𝑁)) ↔ (𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒))):(𝑁 × 𝑁)⟶𝐾)) |
354 | 219, 221,
353 | sylancr 575 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
⊢ ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) → ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒))) ∈ (𝐾 ↑𝑚 (𝑁 × 𝑁)) ↔ (𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒))):(𝑁 × 𝑁)⟶𝐾)) |
355 | 352, 354 | mpbird 247 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) → (𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒))) ∈ (𝐾 ↑𝑚 (𝑁 × 𝑁))) |
356 | 355, 227 | eleqtrd 2852 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) → (𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒))) ∈ 𝐵) |
357 | 57 | 3ad2ant1 1127 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐾 ∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝑁 (((𝑥 ↾ ({𝑤} × 𝑁)) = ((({𝑤} × 𝑁) × {𝑦}) ∘𝑓 · (𝑧 ↾ ({𝑤} × 𝑁))) ∧ (𝑥 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑧 ↾ ((𝑁 ∖ {𝑤}) × 𝑁))) → (𝐷‘𝑥) = (𝑦 · (𝐷‘𝑧)))) |
358 | | reseq1 5528 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
33
⊢ (𝑥 = (𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) → (𝑥 ↾ ({𝑤} × 𝑁)) = ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ↾ ({𝑤} × 𝑁))) |
359 | 358 | eqeq1d 2773 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
32
⊢ (𝑥 = (𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) → ((𝑥 ↾ ({𝑤} × 𝑁)) = ((({𝑤} × 𝑁) × {𝑦}) ∘𝑓 · (𝑧 ↾ ({𝑤} × 𝑁))) ↔ ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ↾ ({𝑤} × 𝑁)) = ((({𝑤} × 𝑁) × {𝑦}) ∘𝑓 · (𝑧 ↾ ({𝑤} × 𝑁))))) |
360 | | reseq1 5528 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
33
⊢ (𝑥 = (𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) → (𝑥 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ↾ ((𝑁 ∖ {𝑤}) × 𝑁))) |
361 | 360 | eqeq1d 2773 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
32
⊢ (𝑥 = (𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) → ((𝑥 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑧 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) ↔ ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑧 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)))) |
362 | 359, 361 | anbi12d 616 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
31
⊢ (𝑥 = (𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) → (((𝑥 ↾ ({𝑤} × 𝑁)) = ((({𝑤} × 𝑁) × {𝑦}) ∘𝑓 · (𝑧 ↾ ({𝑤} × 𝑁))) ∧ (𝑥 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑧 ↾ ((𝑁 ∖ {𝑤}) × 𝑁))) ↔ (((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ↾ ({𝑤} × 𝑁)) = ((({𝑤} × 𝑁) × {𝑦}) ∘𝑓 · (𝑧 ↾ ({𝑤} × 𝑁))) ∧ ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑧 ↾ ((𝑁 ∖ {𝑤}) × 𝑁))))) |
363 | | fveq2 6332 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
32
⊢ (𝑥 = (𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) → (𝐷‘𝑥) = (𝐷‘(𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))))) |
364 | 363 | eqeq1d 2773 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
31
⊢ (𝑥 = (𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) → ((𝐷‘𝑥) = (𝑦 · (𝐷‘𝑧)) ↔ (𝐷‘(𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒)))) = (𝑦 · (𝐷‘𝑧)))) |
365 | 362, 364 | imbi12d 333 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
⊢ (𝑥 = (𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) → ((((𝑥 ↾ ({𝑤} × 𝑁)) = ((({𝑤} × 𝑁) × {𝑦}) ∘𝑓 · (𝑧 ↾ ({𝑤} × 𝑁))) ∧ (𝑥 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑧 ↾ ((𝑁 ∖ {𝑤}) × 𝑁))) → (𝐷‘𝑥) = (𝑦 · (𝐷‘𝑧))) ↔ ((((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ↾ ({𝑤} × 𝑁)) = ((({𝑤} × 𝑁) × {𝑦}) ∘𝑓 · (𝑧 ↾ ({𝑤} × 𝑁))) ∧ ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑧 ↾ ((𝑁 ∖ {𝑤}) × 𝑁))) → (𝐷‘(𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒)))) = (𝑦 · (𝐷‘𝑧))))) |
366 | 365 | 2ralbidv 3138 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
⊢ (𝑥 = (𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) → (∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝑁 (((𝑥 ↾ ({𝑤} × 𝑁)) = ((({𝑤} × 𝑁) × {𝑦}) ∘𝑓 · (𝑧 ↾ ({𝑤} × 𝑁))) ∧ (𝑥 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑧 ↾ ((𝑁 ∖ {𝑤}) × 𝑁))) → (𝐷‘𝑥) = (𝑦 · (𝐷‘𝑧))) ↔ ∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝑁 ((((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ↾ ({𝑤} × 𝑁)) = ((({𝑤} × 𝑁) × {𝑦}) ∘𝑓 · (𝑧 ↾ ({𝑤} × 𝑁))) ∧ ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑧 ↾ ((𝑁 ∖ {𝑤}) × 𝑁))) → (𝐷‘(𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒)))) = (𝑦 · (𝐷‘𝑧))))) |
367 | | sneq 4326 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. 35
⊢ (𝑦 = ((𝑎‘𝑐)(-g‘𝑅)if(𝑐 ∈ 𝑑, 1 , 0 )) → {𝑦} = {((𝑎‘𝑐)(-g‘𝑅)if(𝑐 ∈ 𝑑, 1 , 0 ))}) |
368 | 367 | xpeq2d 5279 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
34
⊢ (𝑦 = ((𝑎‘𝑐)(-g‘𝑅)if(𝑐 ∈ 𝑑, 1 , 0 )) → (({𝑤} × 𝑁) × {𝑦}) = (({𝑤} × 𝑁) × {((𝑎‘𝑐)(-g‘𝑅)if(𝑐 ∈ 𝑑, 1 , 0 ))})) |
369 | 368 | oveq1d 6808 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
33
⊢ (𝑦 = ((𝑎‘𝑐)(-g‘𝑅)if(𝑐 ∈ 𝑑, 1 , 0 )) → ((({𝑤} × 𝑁) × {𝑦}) ∘𝑓 · (𝑧 ↾ ({𝑤} × 𝑁))) = ((({𝑤} × 𝑁) × {((𝑎‘𝑐)(-g‘𝑅)if(𝑐 ∈ 𝑑, 1 , 0 ))})
∘𝑓 · (𝑧 ↾ ({𝑤} × 𝑁)))) |
370 | 369 | eqeq2d 2781 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
32
⊢ (𝑦 = ((𝑎‘𝑐)(-g‘𝑅)if(𝑐 ∈ 𝑑, 1 , 0 )) → (((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ↾ ({𝑤} × 𝑁)) = ((({𝑤} × 𝑁) × {𝑦}) ∘𝑓 · (𝑧 ↾ ({𝑤} × 𝑁))) ↔ ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ↾ ({𝑤} × 𝑁)) = ((({𝑤} × 𝑁) × {((𝑎‘𝑐)(-g‘𝑅)if(𝑐 ∈ 𝑑, 1 , 0 ))})
∘𝑓 · (𝑧 ↾ ({𝑤} × 𝑁))))) |
371 | 370 | anbi1d 615 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
31
⊢ (𝑦 = ((𝑎‘𝑐)(-g‘𝑅)if(𝑐 ∈ 𝑑, 1 , 0 )) → ((((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ↾ ({𝑤} × 𝑁)) = ((({𝑤} × 𝑁) × {𝑦}) ∘𝑓 · (𝑧 ↾ ({𝑤} × 𝑁))) ∧ ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑧 ↾ ((𝑁 ∖ {𝑤}) × 𝑁))) ↔ (((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ↾ ({𝑤} × 𝑁)) = ((({𝑤} × 𝑁) × {((𝑎‘𝑐)(-g‘𝑅)if(𝑐 ∈ 𝑑, 1 , 0 ))})
∘𝑓 · (𝑧 ↾ ({𝑤} × 𝑁))) ∧ ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑧 ↾ ((𝑁 ∖ {𝑤}) × 𝑁))))) |
372 | | oveq1 6800 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
32
⊢ (𝑦 = ((𝑎‘𝑐)(-g‘𝑅)if(𝑐 ∈ 𝑑, 1 , 0 )) → (𝑦 · (𝐷‘𝑧)) = (((𝑎‘𝑐)(-g‘𝑅)if(𝑐 ∈ 𝑑, 1 , 0 )) · (𝐷‘𝑧))) |
373 | 372 | eqeq2d 2781 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
31
⊢ (𝑦 = ((𝑎‘𝑐)(-g‘𝑅)if(𝑐 ∈ 𝑑, 1 , 0 )) → ((𝐷‘(𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒)))) = (𝑦 · (𝐷‘𝑧)) ↔ (𝐷‘(𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒)))) = (((𝑎‘𝑐)(-g‘𝑅)if(𝑐 ∈ 𝑑, 1 , 0 )) · (𝐷‘𝑧)))) |
374 | 371, 373 | imbi12d 333 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
⊢ (𝑦 = ((𝑎‘𝑐)(-g‘𝑅)if(𝑐 ∈ 𝑑, 1 , 0 )) → (((((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ↾ ({𝑤} × 𝑁)) = ((({𝑤} × 𝑁) × {𝑦}) ∘𝑓 · (𝑧 ↾ ({𝑤} × 𝑁))) ∧ ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑧 ↾ ((𝑁 ∖ {𝑤}) × 𝑁))) → (𝐷‘(𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒)))) = (𝑦 · (𝐷‘𝑧))) ↔ ((((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ↾ ({𝑤} × 𝑁)) = ((({𝑤} × 𝑁) × {((𝑎‘𝑐)(-g‘𝑅)if(𝑐 ∈ 𝑑, 1 , 0 ))})
∘𝑓 · (𝑧 ↾ ({𝑤} × 𝑁))) ∧ ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑧 ↾ ((𝑁 ∖ {𝑤}) × 𝑁))) → (𝐷‘(𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒)))) = (((𝑎‘𝑐)(-g‘𝑅)if(𝑐 ∈ 𝑑, 1 , 0 )) · (𝐷‘𝑧))))) |
375 | 374 | 2ralbidv 3138 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
⊢ (𝑦 = ((𝑎‘𝑐)(-g‘𝑅)if(𝑐 ∈ 𝑑, 1 , 0 )) → (∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝑁 ((((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ↾ ({𝑤} × 𝑁)) = ((({𝑤} × 𝑁) × {𝑦}) ∘𝑓 · (𝑧 ↾ ({𝑤} × 𝑁))) ∧ ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑧 ↾ ((𝑁 ∖ {𝑤}) × 𝑁))) → (𝐷‘(𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒)))) = (𝑦 · (𝐷‘𝑧))) ↔ ∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝑁 ((((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ↾ ({𝑤} × 𝑁)) = ((({𝑤} × 𝑁) × {((𝑎‘𝑐)(-g‘𝑅)if(𝑐 ∈ 𝑑, 1 , 0 ))})
∘𝑓 · (𝑧 ↾ ({𝑤} × 𝑁))) ∧ ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑧 ↾ ((𝑁 ∖ {𝑤}) × 𝑁))) → (𝐷‘(𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒)))) = (((𝑎‘𝑐)(-g‘𝑅)if(𝑐 ∈ 𝑑, 1 , 0 )) · (𝐷‘𝑧))))) |
376 | 366, 375 | rspc2va 3473 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ ((((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ∈ 𝐵 ∧ ((𝑎‘𝑐)(-g‘𝑅)if(𝑐 ∈ 𝑑, 1 , 0 )) ∈ 𝐾) ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐾 ∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝑁 (((𝑥 ↾ ({𝑤} × 𝑁)) = ((({𝑤} × 𝑁) × {𝑦}) ∘𝑓 · (𝑧 ↾ ({𝑤} × 𝑁))) ∧ (𝑥 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑧 ↾ ((𝑁 ∖ {𝑤}) × 𝑁))) → (𝐷‘𝑥) = (𝑦 · (𝐷‘𝑧)))) → ∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝑁 ((((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ↾ ({𝑤} × 𝑁)) = ((({𝑤} × 𝑁) × {((𝑎‘𝑐)(-g‘𝑅)if(𝑐 ∈ 𝑑, 1 , 0 ))})
∘𝑓 · (𝑧 ↾ ({𝑤} × 𝑁))) ∧ ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑧 ↾ ((𝑁 ∖ {𝑤}) × 𝑁))) → (𝐷‘(𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒)))) = (((𝑎‘𝑐)(-g‘𝑅)if(𝑐 ∈ 𝑑, 1 , 0 )) · (𝐷‘𝑧)))) |
377 | 241, 304,
357, 376 | syl21anc 1475 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) → ∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝑁 ((((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ↾ ({𝑤} × 𝑁)) = ((({𝑤} × 𝑁) × {((𝑎‘𝑐)(-g‘𝑅)if(𝑐 ∈ 𝑑, 1 , 0 ))})
∘𝑓 · (𝑧 ↾ ({𝑤} × 𝑁))) ∧ ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑧 ↾ ((𝑁 ∖ {𝑤}) × 𝑁))) → (𝐷‘(𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒)))) = (((𝑎‘𝑐)(-g‘𝑅)if(𝑐 ∈ 𝑑, 1 , 0 )) · (𝐷‘𝑧)))) |
378 | | reseq1 5528 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
32
⊢ (𝑧 = (𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒))) → (𝑧 ↾ ({𝑤} × 𝑁)) = ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒))) ↾ ({𝑤} × 𝑁))) |
379 | 378 | oveq2d 6809 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
31
⊢ (𝑧 = (𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒))) → ((({𝑤} × 𝑁) × {((𝑎‘𝑐)(-g‘𝑅)if(𝑐 ∈ 𝑑, 1 , 0 ))})
∘𝑓 · (𝑧 ↾ ({𝑤} × 𝑁))) = ((({𝑤} × 𝑁) × {((𝑎‘𝑐)(-g‘𝑅)if(𝑐 ∈ 𝑑, 1 , 0 ))})
∘𝑓 · ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒))) ↾ ({𝑤} × 𝑁)))) |
380 | 379 | eqeq2d 2781 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
⊢ (𝑧 = (𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒))) → (((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ↾ ({𝑤} × 𝑁)) = ((({𝑤} × 𝑁) × {((𝑎‘𝑐)(-g‘𝑅)if(𝑐 ∈ 𝑑, 1 , 0 ))})
∘𝑓 · (𝑧 ↾ ({𝑤} × 𝑁))) ↔ ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ↾ ({𝑤} × 𝑁)) = ((({𝑤} × 𝑁) × {((𝑎‘𝑐)(-g‘𝑅)if(𝑐 ∈ 𝑑, 1 , 0 ))})
∘𝑓 · ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒))) ↾ ({𝑤} × 𝑁))))) |
381 | | reseq1 5528 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
31
⊢ (𝑧 = (𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒))) → (𝑧 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒))) ↾ ((𝑁 ∖ {𝑤}) × 𝑁))) |
382 | 381 | eqeq2d 2781 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
⊢ (𝑧 = (𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒))) → (((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑧 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) ↔ ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒))) ↾ ((𝑁 ∖ {𝑤}) × 𝑁)))) |
383 | 380, 382 | anbi12d 616 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
⊢ (𝑧 = (𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒))) → ((((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ↾ ({𝑤} × 𝑁)) = ((({𝑤} × 𝑁) × {((𝑎‘𝑐)(-g‘𝑅)if(𝑐 ∈ 𝑑, 1 , 0 ))})
∘𝑓 · (𝑧 ↾ ({𝑤} × 𝑁))) ∧ ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑧 ↾ ((𝑁 ∖ {𝑤}) × 𝑁))) ↔ (((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ↾ ({𝑤} × 𝑁)) = ((({𝑤} × 𝑁) × {((𝑎‘𝑐)(-g‘𝑅)if(𝑐 ∈ 𝑑, 1 , 0 ))})
∘𝑓 · ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒))) ↾ ({𝑤} × 𝑁))) ∧ ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒))) ↾ ((𝑁 ∖ {𝑤}) × 𝑁))))) |
384 | | fveq2 6332 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
31
⊢ (𝑧 = (𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒))) → (𝐷‘𝑧) = (𝐷‘(𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒))))) |
385 | 384 | oveq2d 6809 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
⊢ (𝑧 = (𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒))) → (((𝑎‘𝑐)(-g‘𝑅)if(𝑐 ∈ 𝑑, 1 , 0 )) · (𝐷‘𝑧)) = (((𝑎‘𝑐)(-g‘𝑅)if(𝑐 ∈ 𝑑, 1 , 0 )) · (𝐷‘(𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒)))))) |
386 | 385 | eqeq2d 2781 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
⊢ (𝑧 = (𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒))) → ((𝐷‘(𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒)))) = (((𝑎‘𝑐)(-g‘𝑅)if(𝑐 ∈ 𝑑, 1 , 0 )) · (𝐷‘𝑧)) ↔ (𝐷‘(𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒)))) = (((𝑎‘𝑐)(-g‘𝑅)if(𝑐 ∈ 𝑑, 1 , 0 )) · (𝐷‘(𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒))))))) |
387 | 383, 386 | imbi12d 333 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ (𝑧 = (𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒))) → (((((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ↾ ({𝑤} × 𝑁)) = ((({𝑤} × 𝑁) × {((𝑎‘𝑐)(-g‘𝑅)if(𝑐 ∈ 𝑑, 1 , 0 ))})
∘𝑓 · (𝑧 ↾ ({𝑤} × 𝑁))) ∧ ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑧 ↾ ((𝑁 ∖ {𝑤}) × 𝑁))) → (𝐷‘(𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒)))) = (((𝑎‘𝑐)(-g‘𝑅)if(𝑐 ∈ 𝑑, 1 , 0 )) · (𝐷‘𝑧))) ↔ ((((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ↾ ({𝑤} × 𝑁)) = ((({𝑤} × 𝑁) × {((𝑎‘𝑐)(-g‘𝑅)if(𝑐 ∈ 𝑑, 1 , 0 ))})
∘𝑓 · ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒))) ↾ ({𝑤} × 𝑁))) ∧ ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒))) ↾ ((𝑁 ∖ {𝑤}) × 𝑁))) → (𝐷‘(𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒)))) = (((𝑎‘𝑐)(-g‘𝑅)if(𝑐 ∈ 𝑑, 1 , 0 )) · (𝐷‘(𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒)))))))) |
388 | 277 | xpeq1d 5278 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
32
⊢ (𝑤 = (1st ‘𝑐) → (({𝑤} × 𝑁) × {((𝑎‘𝑐)(-g‘𝑅)if(𝑐 ∈ 𝑑, 1 , 0 ))}) = (({(1st
‘𝑐)} × 𝑁) × {((𝑎‘𝑐)(-g‘𝑅)if(𝑐 ∈ 𝑑, 1 , 0 ))})) |
389 | 277 | reseq2d 5534 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
32
⊢ (𝑤 = (1st ‘𝑐) → ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒))) ↾ ({𝑤} × 𝑁)) = ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒))) ↾ ({(1st ‘𝑐)} × 𝑁))) |
390 | 388, 389 | oveq12d 6811 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
31
⊢ (𝑤 = (1st ‘𝑐) → ((({𝑤} × 𝑁) × {((𝑎‘𝑐)(-g‘𝑅)if(𝑐 ∈ 𝑑, 1 , 0 ))})
∘𝑓 · ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒))) ↾ ({𝑤} × 𝑁))) = ((({(1st ‘𝑐)} × 𝑁) × {((𝑎‘𝑐)(-g‘𝑅)if(𝑐 ∈ 𝑑, 1 , 0 ))})
∘𝑓 · ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒))) ↾ ({(1st ‘𝑐)} × 𝑁)))) |
391 | 279, 390 | eqeq12d 2786 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
⊢ (𝑤 = (1st ‘𝑐) → (((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ↾ ({𝑤} × 𝑁)) = ((({𝑤} × 𝑁) × {((𝑎‘𝑐)(-g‘𝑅)if(𝑐 ∈ 𝑑, 1 , 0 ))})
∘𝑓 · ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒))) ↾ ({𝑤} × 𝑁))) ↔ ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ↾ ({(1st ‘𝑐)} × 𝑁)) = ((({(1st ‘𝑐)} × 𝑁) × {((𝑎‘𝑐)(-g‘𝑅)if(𝑐 ∈ 𝑑, 1 , 0 ))})
∘𝑓 · ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒))) ↾ ({(1st ‘𝑐)} × 𝑁))))) |
392 | 284 | reseq2d 5534 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
31
⊢ (𝑤 = (1st ‘𝑐) → ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒))) ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒))) ↾ ((𝑁 ∖ {(1st ‘𝑐)}) × 𝑁))) |
393 | 286, 392 | eqeq12d 2786 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
⊢ (𝑤 = (1st ‘𝑐) → (((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒))) ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) ↔ ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ↾ ((𝑁 ∖ {(1st ‘𝑐)}) × 𝑁)) = ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒))) ↾ ((𝑁 ∖ {(1st ‘𝑐)}) × 𝑁)))) |
394 | 391, 393 | anbi12d 616 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
⊢ (𝑤 = (1st ‘𝑐) → ((((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ↾ ({𝑤} × 𝑁)) = ((({𝑤} × 𝑁) × {((𝑎‘𝑐)(-g‘𝑅)if(𝑐 ∈ 𝑑, 1 , 0 ))})
∘𝑓 · ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒))) ↾ ({𝑤} × 𝑁))) ∧ ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒))) ↾ ((𝑁 ∖ {𝑤}) × 𝑁))) ↔ (((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ↾ ({(1st ‘𝑐)} × 𝑁)) = ((({(1st ‘𝑐)} × 𝑁) × {((𝑎‘𝑐)(-g‘𝑅)if(𝑐 ∈ 𝑑, 1 , 0 ))})
∘𝑓 · ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒))) ↾ ({(1st ‘𝑐)} × 𝑁))) ∧ ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ↾ ((𝑁 ∖ {(1st ‘𝑐)}) × 𝑁)) = ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒))) ↾ ((𝑁 ∖ {(1st ‘𝑐)}) × 𝑁))))) |
395 | 394 | imbi1d 330 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ (𝑤 = (1st ‘𝑐) → (((((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ↾ ({𝑤} × 𝑁)) = ((({𝑤} × 𝑁) × {((𝑎‘𝑐)(-g‘𝑅)if(𝑐 ∈ 𝑑, 1 , 0 ))})
∘𝑓 · ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒))) ↾ ({𝑤} × 𝑁))) ∧ ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒))) ↾ ((𝑁 ∖ {𝑤}) × 𝑁))) → (𝐷‘(𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒)))) = (((𝑎‘𝑐)(-g‘𝑅)if(𝑐 ∈ 𝑑, 1 , 0 )) · (𝐷‘(𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒)))))) ↔ ((((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ↾ ({(1st ‘𝑐)} × 𝑁)) = ((({(1st ‘𝑐)} × 𝑁) × {((𝑎‘𝑐)(-g‘𝑅)if(𝑐 ∈ 𝑑, 1 , 0 ))})
∘𝑓 · ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒))) ↾ ({(1st ‘𝑐)} × 𝑁))) ∧ ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ↾ ((𝑁 ∖ {(1st ‘𝑐)}) × 𝑁)) = ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒))) ↾ ((𝑁 ∖ {(1st ‘𝑐)}) × 𝑁))) → (𝐷‘(𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒)))) = (((𝑎‘𝑐)(-g‘𝑅)if(𝑐 ∈ 𝑑, 1 , 0 )) · (𝐷‘(𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒)))))))) |
396 | 387, 395 | rspc2va 3473 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ ((((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒))) ∈ 𝐵 ∧ (1st ‘𝑐) ∈ 𝑁) ∧ ∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝑁 ((((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ↾ ({𝑤} × 𝑁)) = ((({𝑤} × 𝑁) × {((𝑎‘𝑐)(-g‘𝑅)if(𝑐 ∈ 𝑑, 1 , 0 ))})
∘𝑓 · (𝑧 ↾ ({𝑤} × 𝑁))) ∧ ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑧 ↾ ((𝑁 ∖ {𝑤}) × 𝑁))) → (𝐷‘(𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒)))) = (((𝑎‘𝑐)(-g‘𝑅)if(𝑐 ∈ 𝑑, 1 , 0 )) · (𝐷‘𝑧)))) → ((((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ↾ ({(1st ‘𝑐)} × 𝑁)) = ((({(1st ‘𝑐)} × 𝑁) × {((𝑎‘𝑐)(-g‘𝑅)if(𝑐 ∈ 𝑑, 1 , 0 ))})
∘𝑓 · ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒))) ↾ ({(1st ‘𝑐)} × 𝑁))) ∧ ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ↾ ((𝑁 ∖ {(1st ‘𝑐)}) × 𝑁)) = ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒))) ↾ ((𝑁 ∖ {(1st ‘𝑐)}) × 𝑁))) → (𝐷‘(𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒)))) = (((𝑎‘𝑐)(-g‘𝑅)if(𝑐 ∈ 𝑑, 1 , 0 )) · (𝐷‘(𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒))))))) |
397 | 356, 125,
377, 396 | syl21anc 1475 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) → ((((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ↾ ({(1st ‘𝑐)} × 𝑁)) = ((({(1st ‘𝑐)} × 𝑁) × {((𝑎‘𝑐)(-g‘𝑅)if(𝑐 ∈ 𝑑, 1 , 0 ))})
∘𝑓 · ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒))) ↾ ({(1st ‘𝑐)} × 𝑁))) ∧ ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ↾ ((𝑁 ∖ {(1st ‘𝑐)}) × 𝑁)) = ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒))) ↾ ((𝑁 ∖ {(1st ‘𝑐)}) × 𝑁))) → (𝐷‘(𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒)))) = (((𝑎‘𝑐)(-g‘𝑅)if(𝑐 ∈ 𝑑, 1 , 0 )) · (𝐷‘(𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒))))))) |
398 | 339, 347,
397 | mp2and 679 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) → (𝐷‘(𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒)))) = (((𝑎‘𝑐)(-g‘𝑅)if(𝑐 ∈ 𝑑, 1 , 0 )) · (𝐷‘(𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒)))))) |
399 | 398 | oveq1d 6808 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) → ((𝐷‘(𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒)))) + (𝐷‘(𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒 ∈ 𝑑, 1 , 0 ), (𝑎‘𝑒))))) = ((((𝑎‘𝑐)(-g‘𝑅)if(𝑐 ∈ 𝑑, 1 , 0 )) · (𝐷‘(𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒))))) + (𝐷‘(𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒 ∈ 𝑑, 1 , 0 ), (𝑎‘𝑒)))))) |
400 | 105, 106,
107, 399 | syl3anc 1476 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ (𝑏 ∪ {𝑐}) ∈ 𝑌) ∧ ((𝑎 ∈ 𝐵 ∧ 𝑑 ∈ (𝑁 ↑𝑚 𝑁)) ∧ ∀𝑤 ∈ 𝑏 (𝑎‘𝑤) = if(𝑤 ∈ 𝑑, 1 , 0 ))) → ((𝐷‘(𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒)))) + (𝐷‘(𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒 ∈ 𝑑, 1 , 0 ), (𝑎‘𝑒))))) = ((((𝑎‘𝑐)(-g‘𝑅)if(𝑐 ∈ 𝑑, 1 , 0 )) · (𝐷‘(𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒))))) + (𝐷‘(𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒 ∈ 𝑑, 1 , 0 ), (𝑎‘𝑒)))))) |
401 | | simpl3 1231 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ (𝑏 ∪ {𝑐}) ∈ 𝑌) ∧ ((𝑎 ∈ 𝐵 ∧ 𝑑 ∈ (𝑁 ↑𝑚 𝑁)) ∧ ∀𝑤 ∈ 𝑏 (𝑎‘𝑤) = if(𝑤 ∈ 𝑑, 1 , 0 ))) → (𝑏 ∪ {𝑐}) ∈ 𝑌) |
402 | | simprlr 765 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ (𝑏 ∪ {𝑐}) ∈ 𝑌) ∧ ((𝑎 ∈ 𝐵 ∧ 𝑑 ∈ (𝑁 ↑𝑚 𝑁)) ∧ ∀𝑤 ∈ 𝑏 (𝑎‘𝑤) = if(𝑤 ∈ 𝑑, 1 , 0 ))) → 𝑑 ∈ (𝑁 ↑𝑚 𝑁)) |
403 | | simprr 756 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ (𝑏 ∪ {𝑐}) ∈ 𝑌) ∧ ((𝑎 ∈ 𝐵 ∧ 𝑑 ∈ (𝑁 ↑𝑚 𝑁)) ∧ ∀𝑤 ∈ 𝑏 (𝑎‘𝑤) = if(𝑤 ∈ 𝑑, 1 , 0 ))) → ∀𝑤 ∈ 𝑏 (𝑎‘𝑤) = if(𝑤 ∈ 𝑑, 1 , 0 )) |
404 | | ralss 3817 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
32
⊢ (𝑏 ⊆ (𝑏 ∪ {𝑐}) → (∀𝑤 ∈ 𝑏 (𝑎‘𝑤) = if(𝑤 ∈ 𝑑, 1 , 0 ) ↔ ∀𝑤 ∈ (𝑏 ∪ {𝑐})(𝑤 ∈ 𝑏 → (𝑎‘𝑤) = if(𝑤 ∈ 𝑑, 1 , 0 )))) |
405 | 100, 404 | ax-mp 5 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
31
⊢
(∀𝑤 ∈
𝑏 (𝑎‘𝑤) = if(𝑤 ∈ 𝑑, 1 , 0 ) ↔ ∀𝑤 ∈ (𝑏 ∪ {𝑐})(𝑤 ∈ 𝑏 → (𝑎‘𝑤) = if(𝑤 ∈ 𝑑, 1 , 0 ))) |
406 | | iftrue 4231 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . 37
⊢
((1st ‘𝑤) = (1st ‘𝑐) → if((1st ‘𝑤) = (1st ‘𝑐), if(𝑤 = 𝑐, 1 , 0 ), (𝑎‘𝑤)) = if(𝑤 = 𝑐, 1 , 0 )) |
407 | 406 | adantl 467 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . 36
⊢
(((((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) ∧ 𝑑 ∈ (𝑁 ↑𝑚 𝑁)) ∧ 𝑤 ∈ (𝑏 ∪ {𝑐})) ∧ (1st ‘𝑤) = (1st ‘𝑐)) → if((1st
‘𝑤) = (1st
‘𝑐), if(𝑤 = 𝑐, 1 , 0 ), (𝑎‘𝑤)) = if(𝑤 = 𝑐, 1 , 0 )) |
408 | | ibar 518 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . 39
⊢
((1st ‘𝑤) = (1st ‘𝑐) → ((2nd ‘𝑤) = (2nd ‘𝑐) ↔ ((1st
‘𝑤) = (1st
‘𝑐) ∧
(2nd ‘𝑤) =
(2nd ‘𝑐)))) |
409 | 408 | adantl 467 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . 38
⊢
(((((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) ∧ 𝑑 ∈ (𝑁 ↑𝑚 𝑁)) ∧ 𝑤 ∈ (𝑏 ∪ {𝑐})) ∧ (1st ‘𝑤) = (1st ‘𝑐)) → ((2nd
‘𝑤) = (2nd
‘𝑐) ↔
((1st ‘𝑤)
= (1st ‘𝑐)
∧ (2nd ‘𝑤) = (2nd ‘𝑐)))) |
410 | | relxp 5266 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . 41
⊢ Rel
(𝑁 × 𝑁) |
411 | | simpl2 1229 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . 43
⊢ (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) ∧ 𝑑 ∈ (𝑁 ↑𝑚 𝑁)) → (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁)) |
412 | 411 | sselda 3752 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . 42
⊢ ((((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) ∧ 𝑑 ∈ (𝑁 ↑𝑚 𝑁)) ∧ 𝑤 ∈ (𝑏 ∪ {𝑐})) → 𝑤 ∈ (𝑁 × 𝑁)) |
413 | 412 | adantr 466 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . 41
⊢
(((((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) ∧ 𝑑 ∈ (𝑁 ↑𝑚 𝑁)) ∧ 𝑤 ∈ (𝑏 ∪ {𝑐})) ∧ (1st ‘𝑤) = (1st ‘𝑐)) → 𝑤 ∈ (𝑁 × 𝑁)) |
414 | | 1st2nd 7363 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . 41
⊢ ((Rel
(𝑁 × 𝑁) ∧ 𝑤 ∈ (𝑁 × 𝑁)) → 𝑤 = 〈(1st ‘𝑤), (2nd ‘𝑤)〉) |
415 | 410, 413,
414 | sylancr 575 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . 40
⊢
(((((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) ∧ 𝑑 ∈ (𝑁 ↑𝑚 𝑁)) ∧ 𝑤 ∈ (𝑏 ∪ {𝑐})) ∧ (1st ‘𝑤) = (1st ‘𝑐)) → 𝑤 = 〈(1st ‘𝑤), (2nd ‘𝑤)〉) |
416 | 415 | eleq1d 2835 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . 39
⊢
(((((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) ∧ 𝑑 ∈ (𝑁 ↑𝑚 𝑁)) ∧ 𝑤 ∈ (𝑏 ∪ {𝑐})) ∧ (1st ‘𝑤) = (1st ‘𝑐)) → (𝑤 ∈ (𝑑 sSet 〈(1st ‘𝑐), (2nd ‘𝑐)〉) ↔
〈(1st ‘𝑤), (2nd ‘𝑤)〉 ∈ (𝑑 sSet 〈(1st ‘𝑐), (2nd ‘𝑐)〉))) |
417 | | simpr 471 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . 43
⊢ (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) ∧ 𝑑 ∈ (𝑁 ↑𝑚 𝑁)) → 𝑑 ∈ (𝑁 ↑𝑚 𝑁)) |
418 | | elmapi 8031 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . 44
⊢ (𝑑 ∈ (𝑁 ↑𝑚 𝑁) → 𝑑:𝑁⟶𝑁) |
419 | 418 | adantl 467 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . 43
⊢ (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) ∧ 𝑑 ∈ (𝑁 ↑𝑚 𝑁)) → 𝑑:𝑁⟶𝑁) |
420 | 125 | adantr 466 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . 43
⊢ (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) ∧ 𝑑 ∈ (𝑁 ↑𝑚 𝑁)) → (1st
‘𝑐) ∈ 𝑁) |
421 | | xp2nd 7348 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . 45
⊢ (𝑐 ∈ (𝑁 × 𝑁) → (2nd ‘𝑐) ∈ 𝑁) |
422 | 123, 421 | syl 17 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . 44
⊢ ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) → (2nd ‘𝑐) ∈ 𝑁) |
423 | 422 | adantr 466 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . 43
⊢ (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) ∧ 𝑑 ∈ (𝑁 ↑𝑚 𝑁)) → (2nd
‘𝑐) ∈ 𝑁) |
424 | | fsets 16098 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . 43
⊢ (((𝑑 ∈ (𝑁 ↑𝑚 𝑁) ∧ 𝑑:𝑁⟶𝑁) ∧ (1st ‘𝑐) ∈ 𝑁 ∧ (2nd ‘𝑐) ∈ 𝑁) → (𝑑 sSet 〈(1st ‘𝑐), (2nd ‘𝑐)〉):𝑁⟶𝑁) |
425 | 417, 419,
420, 423, 424 | syl211anc 1482 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . 42
⊢ (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) ∧ 𝑑 ∈ (𝑁 ↑𝑚 𝑁)) → (𝑑 sSet 〈(1st ‘𝑐), (2nd ‘𝑐)〉):𝑁⟶𝑁) |
426 | | ffn 6185 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . 42
⊢ ((𝑑 sSet 〈(1st
‘𝑐), (2nd
‘𝑐)〉):𝑁⟶𝑁 → (𝑑 sSet 〈(1st ‘𝑐), (2nd ‘𝑐)〉) Fn 𝑁) |
427 | 425, 426 | syl 17 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . 41
⊢ (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) ∧ 𝑑 ∈ (𝑁 ↑𝑚 𝑁)) → (𝑑 sSet 〈(1st ‘𝑐), (2nd ‘𝑐)〉) Fn 𝑁) |
428 | 427 | ad2antrr 705 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . 40
⊢
(((((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) ∧ 𝑑 ∈ (𝑁 ↑𝑚 𝑁)) ∧ 𝑤 ∈ (𝑏 ∪ {𝑐})) ∧ (1st ‘𝑤) = (1st ‘𝑐)) → (𝑑 sSet 〈(1st ‘𝑐), (2nd ‘𝑐)〉) Fn 𝑁) |
429 | | xp1st 7347 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . 42
⊢ (𝑤 ∈ (𝑁 × 𝑁) → (1st ‘𝑤) ∈ 𝑁) |
430 | 412, 429 | syl 17 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . 41
⊢ ((((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) ∧ 𝑑 ∈ (𝑁 ↑𝑚 𝑁)) ∧ 𝑤 ∈ (𝑏 ∪ {𝑐})) → (1st ‘𝑤) ∈ 𝑁) |
431 | 430 | adantr 466 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . 40
⊢
(((((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) ∧ 𝑑 ∈ (𝑁 ↑𝑚 𝑁)) ∧ 𝑤 ∈ (𝑏 ∪ {𝑐})) ∧ (1st ‘𝑤) = (1st ‘𝑐)) → (1st
‘𝑤) ∈ 𝑁) |
432 | | fnopfvb 6378 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . 40
⊢ (((𝑑 sSet 〈(1st
‘𝑐), (2nd
‘𝑐)〉) Fn 𝑁 ∧ (1st
‘𝑤) ∈ 𝑁) → (((𝑑 sSet 〈(1st ‘𝑐), (2nd ‘𝑐)〉)‘(1st
‘𝑤)) =
(2nd ‘𝑤)
↔ 〈(1st ‘𝑤), (2nd ‘𝑤)〉 ∈ (𝑑 sSet 〈(1st ‘𝑐), (2nd ‘𝑐)〉))) |
433 | 428, 431,
432 | syl2anc 573 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . 39
⊢
(((((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) ∧ 𝑑 ∈ (𝑁 ↑𝑚 𝑁)) ∧ 𝑤 ∈ (𝑏 ∪ {𝑐})) ∧ (1st ‘𝑤) = (1st ‘𝑐)) → (((𝑑 sSet 〈(1st ‘𝑐), (2nd ‘𝑐)〉)‘(1st
‘𝑤)) =
(2nd ‘𝑤)
↔ 〈(1st ‘𝑤), (2nd ‘𝑤)〉 ∈ (𝑑 sSet 〈(1st ‘𝑐), (2nd ‘𝑐)〉))) |
434 | | fveq2 6332 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . 43
⊢
((1st ‘𝑤) = (1st ‘𝑐) → ((𝑑 sSet 〈(1st ‘𝑐), (2nd ‘𝑐)〉)‘(1st
‘𝑤)) = ((𝑑 sSet 〈(1st
‘𝑐), (2nd
‘𝑐)〉)‘(1st ‘𝑐))) |
435 | 434 | adantl 467 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . 42
⊢
(((((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) ∧ 𝑑 ∈ (𝑁 ↑𝑚 𝑁)) ∧ 𝑤 ∈ (𝑏 ∪ {𝑐})) ∧ (1st ‘𝑤) = (1st ‘𝑐)) → ((𝑑 sSet 〈(1st ‘𝑐), (2nd ‘𝑐)〉)‘(1st
‘𝑤)) = ((𝑑 sSet 〈(1st
‘𝑐), (2nd
‘𝑐)〉)‘(1st ‘𝑐))) |
436 | | vex 3354 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . 43
⊢ 𝑑 ∈ V |
437 | | fvex 6342 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . 43
⊢
(1st ‘𝑐) ∈ V |
438 | | fvex 6342 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . 43
⊢
(2nd ‘𝑐) ∈ V |
439 | | fvsetsid 16097 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . 43
⊢ ((𝑑 ∈ V ∧ (1st
‘𝑐) ∈ V ∧
(2nd ‘𝑐)
∈ V) → ((𝑑 sSet
〈(1st ‘𝑐), (2nd ‘𝑐)〉)‘(1st ‘𝑐)) = (2nd
‘𝑐)) |
440 | 436, 437,
438, 439 | mp3an 1572 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . 42
⊢ ((𝑑 sSet 〈(1st
‘𝑐), (2nd
‘𝑐)〉)‘(1st ‘𝑐)) = (2nd
‘𝑐) |
441 | 435, 440 | syl6eq 2821 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . 41
⊢
(((((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) ∧ 𝑑 ∈ (𝑁 ↑𝑚 𝑁)) ∧ 𝑤 ∈ (𝑏 ∪ {𝑐})) ∧ (1st ‘𝑤) = (1st ‘𝑐)) → ((𝑑 sSet 〈(1st ‘𝑐), (2nd ‘𝑐)〉)‘(1st
‘𝑤)) =
(2nd ‘𝑐)) |
442 | 441 | eqeq1d 2773 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . 40
⊢
(((((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) ∧ 𝑑 ∈ (𝑁 ↑𝑚 𝑁)) ∧ 𝑤 ∈ (𝑏 ∪ {𝑐})) ∧ (1st ‘𝑤) = (1st ‘𝑐)) → (((𝑑 sSet 〈(1st ‘𝑐), (2nd ‘𝑐)〉)‘(1st
‘𝑤)) =
(2nd ‘𝑤)
↔ (2nd ‘𝑐) = (2nd ‘𝑤))) |
443 | | eqcom 2778 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . 40
⊢
((2nd ‘𝑐) = (2nd ‘𝑤) ↔ (2nd ‘𝑤) = (2nd ‘𝑐)) |
444 | 442, 443 | syl6bb 276 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . 39
⊢
(((((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) ∧ 𝑑 ∈ (𝑁 ↑𝑚 𝑁)) ∧ 𝑤 ∈ (𝑏 ∪ {𝑐})) ∧ (1st ‘𝑤) = (1st ‘𝑐)) → (((𝑑 sSet 〈(1st ‘𝑐), (2nd ‘𝑐)〉)‘(1st
‘𝑤)) =
(2nd ‘𝑤)
↔ (2nd ‘𝑤) = (2nd ‘𝑐))) |
445 | 416, 433,
444 | 3bitr2rd 297 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . 38
⊢
(((((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) ∧ 𝑑 ∈ (𝑁 ↑𝑚 𝑁)) ∧ 𝑤 ∈ (𝑏 ∪ {𝑐})) ∧ (1st ‘𝑤) = (1st ‘𝑐)) → ((2nd
‘𝑤) = (2nd
‘𝑐) ↔ 𝑤 ∈ (𝑑 sSet 〈(1st ‘𝑐), (2nd ‘𝑐)〉))) |
446 | 123 | ad3antrrr 709 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . 39
⊢
(((((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) ∧ 𝑑 ∈ (𝑁 ↑𝑚 𝑁)) ∧ 𝑤 ∈ (𝑏 ∪ {𝑐})) ∧ (1st ‘𝑤) = (1st ‘𝑐)) → 𝑐 ∈ (𝑁 × 𝑁)) |
447 | | xpopth 7356 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . 39
⊢ ((𝑤 ∈ (𝑁 × 𝑁) ∧ 𝑐 ∈ (𝑁 × 𝑁)) → (((1st ‘𝑤) = (1st ‘𝑐) ∧ (2nd
‘𝑤) = (2nd
‘𝑐)) ↔ 𝑤 = 𝑐)) |
448 | 413, 446,
447 | syl2anc 573 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . 38
⊢
(((((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) ∧ 𝑑 ∈ (𝑁 ↑𝑚 𝑁)) ∧ 𝑤 ∈ (𝑏 ∪ {𝑐})) ∧ (1st ‘𝑤) = (1st ‘𝑐)) → (((1st
‘𝑤) = (1st
‘𝑐) ∧
(2nd ‘𝑤) =
(2nd ‘𝑐))
↔ 𝑤 = 𝑐)) |
449 | 409, 445,
448 | 3bitr3rd 299 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . 37
⊢
(((((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) ∧ 𝑑 ∈ (𝑁 ↑𝑚 𝑁)) ∧ 𝑤 ∈ (𝑏 ∪ {𝑐})) ∧ (1st ‘𝑤) = (1st ‘𝑐)) → (𝑤 = 𝑐 ↔ 𝑤 ∈ (𝑑 sSet 〈(1st ‘𝑐), (2nd ‘𝑐)〉))) |
450 | 449 | ifbid 4247 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . 36
⊢
(((((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) ∧ 𝑑 ∈ (𝑁 ↑𝑚 𝑁)) ∧ 𝑤 ∈ (𝑏 ∪ {𝑐})) ∧ (1st ‘𝑤) = (1st ‘𝑐)) → if(𝑤 = 𝑐, 1 , 0 ) = if(𝑤 ∈ (𝑑 sSet 〈(1st ‘𝑐), (2nd ‘𝑐)〉), 1 , 0 )) |
451 | 407, 450 | eqtrd 2805 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. 35
⊢
(((((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) ∧ 𝑑 ∈ (𝑁 ↑𝑚 𝑁)) ∧ 𝑤 ∈ (𝑏 ∪ {𝑐})) ∧ (1st ‘𝑤) = (1st ‘𝑐)) → if((1st
‘𝑤) = (1st
‘𝑐), if(𝑤 = 𝑐, 1 , 0 ), (𝑎‘𝑤)) = if(𝑤 ∈ (𝑑 sSet 〈(1st ‘𝑐), (2nd ‘𝑐)〉), 1 , 0 )) |
452 | 451 | a1d 25 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
34
⊢
(((((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) ∧ 𝑑 ∈ (𝑁 ↑𝑚 𝑁)) ∧ 𝑤 ∈ (𝑏 ∪ {𝑐})) ∧ (1st ‘𝑤) = (1st ‘𝑐)) → ((𝑤 ∈ 𝑏 → (𝑎‘𝑤) = if(𝑤 ∈ 𝑑, 1 , 0 )) →
if((1st ‘𝑤) = (1st ‘𝑐), if(𝑤 = 𝑐, 1 , 0 ), (𝑎‘𝑤)) = if(𝑤 ∈ (𝑑 sSet 〈(1st ‘𝑐), (2nd ‘𝑐)〉), 1 , 0 ))) |
453 | | elsni 4333 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . 40
⊢ (𝑤 ∈ {𝑐} → 𝑤 = 𝑐) |
454 | 453 | fveq2d 6336 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . 39
⊢ (𝑤 ∈ {𝑐} → (1st ‘𝑤) = (1st ‘𝑐)) |
455 | 454 | |