Step | Hyp | Ref
| Expression |
1 | | poimirlem22.2 |
. . . . . 6
⊢ (𝜑 → 𝑇 ∈ 𝑆) |
2 | | elrabi 3391 |
. . . . . . 7
⊢ (𝑇 ∈ {𝑡 ∈ ((((0..^𝐾) ↑_{𝑚} (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝐹 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2^{nd}
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^{st}
‘(1^{st} ‘𝑡)) ∘_{𝑓} +
((((2^{nd} ‘(1^{st} ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))} → 𝑇 ∈ ((((0..^𝐾) ↑_{𝑚} (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁))) |
3 | | poimirlem22.s |
. . . . . . 7
⊢ 𝑆 = {𝑡 ∈ ((((0..^𝐾) ↑_{𝑚} (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝐹 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2^{nd}
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^{st}
‘(1^{st} ‘𝑡)) ∘_{𝑓} +
((((2^{nd} ‘(1^{st} ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))} |
4 | 2, 3 | eleq2s 2748 |
. . . . . 6
⊢ (𝑇 ∈ 𝑆 → 𝑇 ∈ ((((0..^𝐾) ↑_{𝑚} (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁))) |
5 | 1, 4 | syl 17 |
. . . . 5
⊢ (𝜑 → 𝑇 ∈ ((((0..^𝐾) ↑_{𝑚} (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁))) |
6 | | xp1st 7242 |
. . . . 5
⊢ (𝑇 ∈ ((((0..^𝐾) ↑_{𝑚}
(1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) → (1^{st} ‘𝑇) ∈ (((0..^𝐾) ↑_{𝑚}
(1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})) |
7 | | xp1st 7242 |
. . . . 5
⊢
((1^{st} ‘𝑇) ∈ (((0..^𝐾) ↑_{𝑚} (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) → (1^{st}
‘(1^{st} ‘𝑇)) ∈ ((0..^𝐾) ↑_{𝑚} (1...𝑁))) |
8 | 5, 6, 7 | 3syl 18 |
. . . 4
⊢ (𝜑 → (1^{st}
‘(1^{st} ‘𝑇)) ∈ ((0..^𝐾) ↑_{𝑚} (1...𝑁))) |
9 | | xp2nd 7243 |
. . . . . . . 8
⊢
((1^{st} ‘𝑇) ∈ (((0..^𝐾) ↑_{𝑚} (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) → (2^{nd}
‘(1^{st} ‘𝑇)) ∈ {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) |
10 | 5, 6, 9 | 3syl 18 |
. . . . . . 7
⊢ (𝜑 → (2^{nd}
‘(1^{st} ‘𝑇)) ∈ {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) |
11 | | fvex 6239 |
. . . . . . . 8
⊢
(2^{nd} ‘(1^{st} ‘𝑇)) ∈ V |
12 | | f1oeq1 6165 |
. . . . . . . 8
⊢ (𝑓 = (2^{nd}
‘(1^{st} ‘𝑇)) → (𝑓:(1...𝑁)–1-1-onto→(1...𝑁) ↔ (2^{nd}
‘(1^{st} ‘𝑇)):(1...𝑁)–1-1-onto→(1...𝑁))) |
13 | 11, 12 | elab 3382 |
. . . . . . 7
⊢
((2^{nd} ‘(1^{st} ‘𝑇)) ∈ {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)} ↔ (2^{nd}
‘(1^{st} ‘𝑇)):(1...𝑁)–1-1-onto→(1...𝑁)) |
14 | 10, 13 | sylib 208 |
. . . . . 6
⊢ (𝜑 → (2^{nd}
‘(1^{st} ‘𝑇)):(1...𝑁)–1-1-onto→(1...𝑁)) |
15 | | poimirlem15.3 |
. . . . . . . . . . . . 13
⊢ (𝜑 → (2^{nd}
‘𝑇) ∈
(1...(𝑁 −
1))) |
16 | | elfznn 12408 |
. . . . . . . . . . . . 13
⊢
((2^{nd} ‘𝑇) ∈ (1...(𝑁 − 1)) → (2^{nd}
‘𝑇) ∈
ℕ) |
17 | 15, 16 | syl 17 |
. . . . . . . . . . . 12
⊢ (𝜑 → (2^{nd}
‘𝑇) ∈
ℕ) |
18 | 17 | nnred 11073 |
. . . . . . . . . . 11
⊢ (𝜑 → (2^{nd}
‘𝑇) ∈
ℝ) |
19 | 18 | ltp1d 10992 |
. . . . . . . . . . 11
⊢ (𝜑 → (2^{nd}
‘𝑇) <
((2^{nd} ‘𝑇)
+ 1)) |
20 | 18, 19 | ltned 10211 |
. . . . . . . . . 10
⊢ (𝜑 → (2^{nd}
‘𝑇) ≠
((2^{nd} ‘𝑇)
+ 1)) |
21 | 20 | necomd 2878 |
. . . . . . . . . 10
⊢ (𝜑 → ((2^{nd}
‘𝑇) + 1) ≠
(2^{nd} ‘𝑇)) |
22 | | fvex 6239 |
. . . . . . . . . . 11
⊢
(2^{nd} ‘𝑇) ∈ V |
23 | | ovex 6718 |
. . . . . . . . . . 11
⊢
((2^{nd} ‘𝑇) + 1) ∈ V |
24 | | f1oprg 6219 |
. . . . . . . . . . 11
⊢
((((2^{nd} ‘𝑇) ∈ V ∧ ((2^{nd}
‘𝑇) + 1) ∈ V)
∧ (((2^{nd} ‘𝑇) + 1) ∈ V ∧ (2^{nd}
‘𝑇) ∈ V)) →
(((2^{nd} ‘𝑇)
≠ ((2^{nd} ‘𝑇) + 1) ∧ ((2^{nd} ‘𝑇) + 1) ≠ (2^{nd}
‘𝑇)) →
{⟨(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)⟩, ⟨((2^{nd}
‘𝑇) + 1),
(2^{nd} ‘𝑇)⟩}:{(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)}–1-1-onto→{((2^{nd} ‘𝑇) + 1), (2^{nd} ‘𝑇)})) |
25 | 22, 23, 23, 22, 24 | mp4an 709 |
. . . . . . . . . 10
⊢
(((2^{nd} ‘𝑇) ≠ ((2^{nd} ‘𝑇) + 1) ∧ ((2^{nd}
‘𝑇) + 1) ≠
(2^{nd} ‘𝑇))
→ {⟨(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)⟩, ⟨((2^{nd}
‘𝑇) + 1),
(2^{nd} ‘𝑇)⟩}:{(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)}–1-1-onto→{((2^{nd} ‘𝑇) + 1), (2^{nd} ‘𝑇)}) |
26 | 20, 21, 25 | syl2anc 694 |
. . . . . . . . 9
⊢ (𝜑 → {⟨(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)⟩,
⟨((2^{nd} ‘𝑇) + 1), (2^{nd} ‘𝑇)⟩}:{(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)}–1-1-onto→{((2^{nd} ‘𝑇) + 1), (2^{nd} ‘𝑇)}) |
27 | | prcom 4299 |
. . . . . . . . . 10
⊢
{((2^{nd} ‘𝑇) + 1), (2^{nd} ‘𝑇)} = {(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) +
1)} |
28 | | f1oeq3 6167 |
. . . . . . . . . 10
⊢
({((2^{nd} ‘𝑇) + 1), (2^{nd} ‘𝑇)} = {(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)} →
({⟨(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)⟩, ⟨((2^{nd}
‘𝑇) + 1),
(2^{nd} ‘𝑇)⟩}:{(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)}–1-1-onto→{((2^{nd} ‘𝑇) + 1), (2^{nd} ‘𝑇)} ↔ {⟨(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)⟩,
⟨((2^{nd} ‘𝑇) + 1), (2^{nd} ‘𝑇)⟩}:{(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)}–1-1-onto→{(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)})) |
29 | 27, 28 | ax-mp 5 |
. . . . . . . . 9
⊢
({⟨(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)⟩, ⟨((2^{nd}
‘𝑇) + 1),
(2^{nd} ‘𝑇)⟩}:{(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)}–1-1-onto→{((2^{nd} ‘𝑇) + 1), (2^{nd} ‘𝑇)} ↔ {⟨(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)⟩,
⟨((2^{nd} ‘𝑇) + 1), (2^{nd} ‘𝑇)⟩}:{(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)}–1-1-onto→{(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)}) |
30 | 26, 29 | sylib 208 |
. . . . . . . 8
⊢ (𝜑 → {⟨(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)⟩,
⟨((2^{nd} ‘𝑇) + 1), (2^{nd} ‘𝑇)⟩}:{(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)}–1-1-onto→{(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)}) |
31 | | f1oi 6212 |
. . . . . . . 8
⊢ ( I
↾ ((1...𝑁) ∖
{(2^{nd} ‘𝑇),
((2^{nd} ‘𝑇)
+ 1)})):((1...𝑁) ∖
{(2^{nd} ‘𝑇),
((2^{nd} ‘𝑇)
+ 1)})–1-1-onto→((1...𝑁) ∖ {(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)}) |
32 | | disjdif 4073 |
. . . . . . . . 9
⊢
({(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)} ∩ ((1...𝑁) ∖ {(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)})) =
∅ |
33 | | f1oun 6194 |
. . . . . . . . 9
⊢
((({⟨(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)⟩, ⟨((2^{nd}
‘𝑇) + 1),
(2^{nd} ‘𝑇)⟩}:{(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)}–1-1-onto→{(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)} ∧ ( I ↾ ((1...𝑁) ∖ {(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) +
1)})):((1...𝑁) ∖
{(2^{nd} ‘𝑇),
((2^{nd} ‘𝑇)
+ 1)})–1-1-onto→((1...𝑁) ∖ {(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)})) ∧
(({(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)} ∩ ((1...𝑁) ∖ {(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)})) = ∅ ∧
({(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)} ∩ ((1...𝑁) ∖ {(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)})) = ∅)) →
({⟨(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)⟩, ⟨((2^{nd}
‘𝑇) + 1),
(2^{nd} ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) +
1)}))):({(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)} ∪ ((1...𝑁) ∖ {(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)}))–1-1-onto→({(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)} ∪ ((1...𝑁) ∖ {(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)}))) |
34 | 32, 32, 33 | mpanr12 721 |
. . . . . . . 8
⊢
(({⟨(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)⟩, ⟨((2^{nd}
‘𝑇) + 1),
(2^{nd} ‘𝑇)⟩}:{(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)}–1-1-onto→{(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)} ∧ ( I ↾ ((1...𝑁) ∖ {(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) +
1)})):((1...𝑁) ∖
{(2^{nd} ‘𝑇),
((2^{nd} ‘𝑇)
+ 1)})–1-1-onto→((1...𝑁) ∖ {(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)})) →
({⟨(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)⟩, ⟨((2^{nd}
‘𝑇) + 1),
(2^{nd} ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) +
1)}))):({(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)} ∪ ((1...𝑁) ∖ {(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)}))–1-1-onto→({(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)} ∪ ((1...𝑁) ∖ {(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)}))) |
35 | 30, 31, 34 | sylancl 695 |
. . . . . . 7
⊢ (𝜑 → ({⟨(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)⟩,
⟨((2^{nd} ‘𝑇) + 1), (2^{nd} ‘𝑇)⟩} ∪ ( I ↾
((1...𝑁) ∖
{(2^{nd} ‘𝑇),
((2^{nd} ‘𝑇)
+ 1)}))):({(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)} ∪ ((1...𝑁) ∖ {(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)}))–1-1-onto→({(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)} ∪ ((1...𝑁) ∖ {(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)}))) |
36 | | poimir.0 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → 𝑁 ∈ ℕ) |
37 | 36 | nncnd 11074 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → 𝑁 ∈ ℂ) |
38 | | npcan1 10493 |
. . . . . . . . . . . . . 14
⊢ (𝑁 ∈ ℂ → ((𝑁 − 1) + 1) = 𝑁) |
39 | 37, 38 | syl 17 |
. . . . . . . . . . . . 13
⊢ (𝜑 → ((𝑁 − 1) + 1) = 𝑁) |
40 | 36 | nnzd 11519 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → 𝑁 ∈ ℤ) |
41 | | peano2zm 11458 |
. . . . . . . . . . . . . . 15
⊢ (𝑁 ∈ ℤ → (𝑁 − 1) ∈
ℤ) |
42 | 40, 41 | syl 17 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → (𝑁 − 1) ∈ ℤ) |
43 | | uzid 11740 |
. . . . . . . . . . . . . 14
⊢ ((𝑁 − 1) ∈ ℤ
→ (𝑁 − 1) ∈
(ℤ_{≥}‘(𝑁 − 1))) |
44 | | peano2uz 11779 |
. . . . . . . . . . . . . 14
⊢ ((𝑁 − 1) ∈
(ℤ_{≥}‘(𝑁 − 1)) → ((𝑁 − 1) + 1) ∈
(ℤ_{≥}‘(𝑁 − 1))) |
45 | 42, 43, 44 | 3syl 18 |
. . . . . . . . . . . . 13
⊢ (𝜑 → ((𝑁 − 1) + 1) ∈
(ℤ_{≥}‘(𝑁 − 1))) |
46 | 39, 45 | eqeltrrd 2731 |
. . . . . . . . . . . 12
⊢ (𝜑 → 𝑁 ∈ (ℤ_{≥}‘(𝑁 − 1))) |
47 | | fzss2 12419 |
. . . . . . . . . . . 12
⊢ (𝑁 ∈
(ℤ_{≥}‘(𝑁 − 1)) → (1...(𝑁 − 1)) ⊆ (1...𝑁)) |
48 | 46, 47 | syl 17 |
. . . . . . . . . . 11
⊢ (𝜑 → (1...(𝑁 − 1)) ⊆ (1...𝑁)) |
49 | 48, 15 | sseldd 3637 |
. . . . . . . . . 10
⊢ (𝜑 → (2^{nd}
‘𝑇) ∈ (1...𝑁)) |
50 | 17 | peano2nnd 11075 |
. . . . . . . . . . 11
⊢ (𝜑 → ((2^{nd}
‘𝑇) + 1) ∈
ℕ) |
51 | 42 | zred 11520 |
. . . . . . . . . . . . 13
⊢ (𝜑 → (𝑁 − 1) ∈ ℝ) |
52 | 36 | nnred 11073 |
. . . . . . . . . . . . 13
⊢ (𝜑 → 𝑁 ∈ ℝ) |
53 | | elfzle2 12383 |
. . . . . . . . . . . . . 14
⊢
((2^{nd} ‘𝑇) ∈ (1...(𝑁 − 1)) → (2^{nd}
‘𝑇) ≤ (𝑁 − 1)) |
54 | 15, 53 | syl 17 |
. . . . . . . . . . . . 13
⊢ (𝜑 → (2^{nd}
‘𝑇) ≤ (𝑁 − 1)) |
55 | 52 | ltm1d 10994 |
. . . . . . . . . . . . 13
⊢ (𝜑 → (𝑁 − 1) < 𝑁) |
56 | 18, 51, 52, 54, 55 | lelttrd 10233 |
. . . . . . . . . . . 12
⊢ (𝜑 → (2^{nd}
‘𝑇) < 𝑁) |
57 | 17 | nnzd 11519 |
. . . . . . . . . . . . 13
⊢ (𝜑 → (2^{nd}
‘𝑇) ∈
ℤ) |
58 | | zltp1le 11465 |
. . . . . . . . . . . . 13
⊢
(((2^{nd} ‘𝑇) ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((2^{nd}
‘𝑇) < 𝑁 ↔ ((2^{nd}
‘𝑇) + 1) ≤ 𝑁)) |
59 | 57, 40, 58 | syl2anc 694 |
. . . . . . . . . . . 12
⊢ (𝜑 → ((2^{nd}
‘𝑇) < 𝑁 ↔ ((2^{nd}
‘𝑇) + 1) ≤ 𝑁)) |
60 | 56, 59 | mpbid 222 |
. . . . . . . . . . 11
⊢ (𝜑 → ((2^{nd}
‘𝑇) + 1) ≤ 𝑁) |
61 | | fznn 12446 |
. . . . . . . . . . . 12
⊢ (𝑁 ∈ ℤ →
(((2^{nd} ‘𝑇)
+ 1) ∈ (1...𝑁) ↔
(((2^{nd} ‘𝑇)
+ 1) ∈ ℕ ∧ ((2^{nd} ‘𝑇) + 1) ≤ 𝑁))) |
62 | 40, 61 | syl 17 |
. . . . . . . . . . 11
⊢ (𝜑 → (((2^{nd}
‘𝑇) + 1) ∈
(1...𝑁) ↔
(((2^{nd} ‘𝑇)
+ 1) ∈ ℕ ∧ ((2^{nd} ‘𝑇) + 1) ≤ 𝑁))) |
63 | 50, 60, 62 | mpbir2and 977 |
. . . . . . . . . 10
⊢ (𝜑 → ((2^{nd}
‘𝑇) + 1) ∈
(1...𝑁)) |
64 | | prssi 4385 |
. . . . . . . . . 10
⊢
(((2^{nd} ‘𝑇) ∈ (1...𝑁) ∧ ((2^{nd} ‘𝑇) + 1) ∈ (1...𝑁)) → {(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)} ⊆
(1...𝑁)) |
65 | 49, 63, 64 | syl2anc 694 |
. . . . . . . . 9
⊢ (𝜑 → {(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)} ⊆
(1...𝑁)) |
66 | | undif 4082 |
. . . . . . . . 9
⊢
({(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)} ⊆ (1...𝑁) ↔ ({(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)} ∪ ((1...𝑁) ∖ {(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)})) =
(1...𝑁)) |
67 | 65, 66 | sylib 208 |
. . . . . . . 8
⊢ (𝜑 → ({(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)} ∪
((1...𝑁) ∖
{(2^{nd} ‘𝑇),
((2^{nd} ‘𝑇)
+ 1)})) = (1...𝑁)) |
68 | | f1oeq23 6168 |
. . . . . . . 8
⊢
((({(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)} ∪ ((1...𝑁) ∖ {(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)})) = (1...𝑁) ∧ ({(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)} ∪
((1...𝑁) ∖
{(2^{nd} ‘𝑇),
((2^{nd} ‘𝑇)
+ 1)})) = (1...𝑁)) →
(({⟨(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)⟩, ⟨((2^{nd}
‘𝑇) + 1),
(2^{nd} ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) +
1)}))):({(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)} ∪ ((1...𝑁) ∖ {(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)}))–1-1-onto→({(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)} ∪ ((1...𝑁) ∖ {(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)})) ↔
({⟨(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)⟩, ⟨((2^{nd}
‘𝑇) + 1),
(2^{nd} ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) +
1)}))):(1...𝑁)–1-1-onto→(1...𝑁))) |
69 | 67, 67, 68 | syl2anc 694 |
. . . . . . 7
⊢ (𝜑 → (({⟨(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)⟩,
⟨((2^{nd} ‘𝑇) + 1), (2^{nd} ‘𝑇)⟩} ∪ ( I ↾
((1...𝑁) ∖
{(2^{nd} ‘𝑇),
((2^{nd} ‘𝑇)
+ 1)}))):({(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)} ∪ ((1...𝑁) ∖ {(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)}))–1-1-onto→({(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)} ∪ ((1...𝑁) ∖ {(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)})) ↔
({⟨(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)⟩, ⟨((2^{nd}
‘𝑇) + 1),
(2^{nd} ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) +
1)}))):(1...𝑁)–1-1-onto→(1...𝑁))) |
70 | 35, 69 | mpbid 222 |
. . . . . 6
⊢ (𝜑 → ({⟨(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)⟩,
⟨((2^{nd} ‘𝑇) + 1), (2^{nd} ‘𝑇)⟩} ∪ ( I ↾
((1...𝑁) ∖
{(2^{nd} ‘𝑇),
((2^{nd} ‘𝑇)
+ 1)}))):(1...𝑁)–1-1-onto→(1...𝑁)) |
71 | | f1oco 6197 |
. . . . . 6
⊢
(((2^{nd} ‘(1^{st} ‘𝑇)):(1...𝑁)–1-1-onto→(1...𝑁) ∧ ({⟨(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)⟩,
⟨((2^{nd} ‘𝑇) + 1), (2^{nd} ‘𝑇)⟩} ∪ ( I ↾
((1...𝑁) ∖
{(2^{nd} ‘𝑇),
((2^{nd} ‘𝑇)
+ 1)}))):(1...𝑁)–1-1-onto→(1...𝑁)) → ((2^{nd}
‘(1^{st} ‘𝑇)) ∘ ({⟨(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)⟩,
⟨((2^{nd} ‘𝑇) + 1), (2^{nd} ‘𝑇)⟩} ∪ ( I ↾
((1...𝑁) ∖
{(2^{nd} ‘𝑇),
((2^{nd} ‘𝑇)
+ 1)})))):(1...𝑁)–1-1-onto→(1...𝑁)) |
72 | 14, 70, 71 | syl2anc 694 |
. . . . 5
⊢ (𝜑 → ((2^{nd}
‘(1^{st} ‘𝑇)) ∘ ({⟨(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)⟩,
⟨((2^{nd} ‘𝑇) + 1), (2^{nd} ‘𝑇)⟩} ∪ ( I ↾
((1...𝑁) ∖
{(2^{nd} ‘𝑇),
((2^{nd} ‘𝑇)
+ 1)})))):(1...𝑁)–1-1-onto→(1...𝑁)) |
73 | | prex 4939 |
. . . . . . . 8
⊢
{⟨(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)⟩, ⟨((2^{nd}
‘𝑇) + 1),
(2^{nd} ‘𝑇)⟩} ∈ V |
74 | | ovex 6718 |
. . . . . . . . 9
⊢
(1...𝑁) ∈
V |
75 | | difexg 4841 |
. . . . . . . . 9
⊢
((1...𝑁) ∈ V
→ ((1...𝑁) ∖
{(2^{nd} ‘𝑇),
((2^{nd} ‘𝑇)
+ 1)}) ∈ V) |
76 | | resiexg 7144 |
. . . . . . . . 9
⊢
(((1...𝑁) ∖
{(2^{nd} ‘𝑇),
((2^{nd} ‘𝑇)
+ 1)}) ∈ V → ( I ↾ ((1...𝑁) ∖ {(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)})) ∈
V) |
77 | 74, 75, 76 | mp2b 10 |
. . . . . . . 8
⊢ ( I
↾ ((1...𝑁) ∖
{(2^{nd} ‘𝑇),
((2^{nd} ‘𝑇)
+ 1)})) ∈ V |
78 | 73, 77 | unex 6998 |
. . . . . . 7
⊢
({⟨(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)⟩, ⟨((2^{nd}
‘𝑇) + 1),
(2^{nd} ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)}))) ∈
V |
79 | 11, 78 | coex 7160 |
. . . . . 6
⊢
((2^{nd} ‘(1^{st} ‘𝑇)) ∘ ({⟨(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)⟩,
⟨((2^{nd} ‘𝑇) + 1), (2^{nd} ‘𝑇)⟩} ∪ ( I ↾
((1...𝑁) ∖
{(2^{nd} ‘𝑇),
((2^{nd} ‘𝑇)
+ 1)})))) ∈ V |
80 | | f1oeq1 6165 |
. . . . . 6
⊢ (𝑓 = ((2^{nd}
‘(1^{st} ‘𝑇)) ∘ ({⟨(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)⟩,
⟨((2^{nd} ‘𝑇) + 1), (2^{nd} ‘𝑇)⟩} ∪ ( I ↾
((1...𝑁) ∖
{(2^{nd} ‘𝑇),
((2^{nd} ‘𝑇)
+ 1)})))) → (𝑓:(1...𝑁)–1-1-onto→(1...𝑁) ↔ ((2^{nd}
‘(1^{st} ‘𝑇)) ∘ ({⟨(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)⟩,
⟨((2^{nd} ‘𝑇) + 1), (2^{nd} ‘𝑇)⟩} ∪ ( I ↾
((1...𝑁) ∖
{(2^{nd} ‘𝑇),
((2^{nd} ‘𝑇)
+ 1)})))):(1...𝑁)–1-1-onto→(1...𝑁))) |
81 | 79, 80 | elab 3382 |
. . . . 5
⊢
(((2^{nd} ‘(1^{st} ‘𝑇)) ∘ ({⟨(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)⟩,
⟨((2^{nd} ‘𝑇) + 1), (2^{nd} ‘𝑇)⟩} ∪ ( I ↾
((1...𝑁) ∖
{(2^{nd} ‘𝑇),
((2^{nd} ‘𝑇)
+ 1)})))) ∈ {𝑓 ∣
𝑓:(1...𝑁)–1-1-onto→(1...𝑁)} ↔ ((2^{nd}
‘(1^{st} ‘𝑇)) ∘ ({⟨(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)⟩,
⟨((2^{nd} ‘𝑇) + 1), (2^{nd} ‘𝑇)⟩} ∪ ( I ↾
((1...𝑁) ∖
{(2^{nd} ‘𝑇),
((2^{nd} ‘𝑇)
+ 1)})))):(1...𝑁)–1-1-onto→(1...𝑁)) |
82 | 72, 81 | sylibr 224 |
. . . 4
⊢ (𝜑 → ((2^{nd}
‘(1^{st} ‘𝑇)) ∘ ({⟨(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)⟩,
⟨((2^{nd} ‘𝑇) + 1), (2^{nd} ‘𝑇)⟩} ∪ ( I ↾
((1...𝑁) ∖
{(2^{nd} ‘𝑇),
((2^{nd} ‘𝑇)
+ 1)})))) ∈ {𝑓 ∣
𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) |
83 | | opelxpi 5182 |
. . . 4
⊢
(((1^{st} ‘(1^{st} ‘𝑇)) ∈ ((0..^𝐾) ↑_{𝑚} (1...𝑁)) ∧ ((2^{nd}
‘(1^{st} ‘𝑇)) ∘ ({⟨(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)⟩,
⟨((2^{nd} ‘𝑇) + 1), (2^{nd} ‘𝑇)⟩} ∪ ( I ↾
((1...𝑁) ∖
{(2^{nd} ‘𝑇),
((2^{nd} ‘𝑇)
+ 1)})))) ∈ {𝑓 ∣
𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) → ⟨(1^{st}
‘(1^{st} ‘𝑇)), ((2^{nd} ‘(1^{st}
‘𝑇)) ∘
({⟨(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)⟩, ⟨((2^{nd}
‘𝑇) + 1),
(2^{nd} ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)}))))⟩
∈ (((0..^𝐾)
↑_{𝑚} (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})) |
84 | 8, 82, 83 | syl2anc 694 |
. . 3
⊢ (𝜑 → ⟨(1^{st}
‘(1^{st} ‘𝑇)), ((2^{nd} ‘(1^{st}
‘𝑇)) ∘
({⟨(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)⟩, ⟨((2^{nd}
‘𝑇) + 1),
(2^{nd} ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)}))))⟩
∈ (((0..^𝐾)
↑_{𝑚} (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})) |
85 | | fz1ssfz0 12474 |
. . . . 5
⊢
(1...𝑁) ⊆
(0...𝑁) |
86 | 48, 85 | syl6ss 3648 |
. . . 4
⊢ (𝜑 → (1...(𝑁 − 1)) ⊆ (0...𝑁)) |
87 | 86, 15 | sseldd 3637 |
. . 3
⊢ (𝜑 → (2^{nd}
‘𝑇) ∈ (0...𝑁)) |
88 | | opelxpi 5182 |
. . 3
⊢
((⟨(1^{st} ‘(1^{st} ‘𝑇)), ((2^{nd} ‘(1^{st}
‘𝑇)) ∘
({⟨(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)⟩, ⟨((2^{nd}
‘𝑇) + 1),
(2^{nd} ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)}))))⟩
∈ (((0..^𝐾)
↑_{𝑚} (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∧ (2^{nd} ‘𝑇) ∈ (0...𝑁)) → ⟨⟨(1^{st}
‘(1^{st} ‘𝑇)), ((2^{nd} ‘(1^{st}
‘𝑇)) ∘
({⟨(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)⟩, ⟨((2^{nd}
‘𝑇) + 1),
(2^{nd} ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)}))))⟩,
(2^{nd} ‘𝑇)⟩ ∈ ((((0..^𝐾) ↑_{𝑚} (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁))) |
89 | 84, 87, 88 | syl2anc 694 |
. 2
⊢ (𝜑 →
⟨⟨(1^{st} ‘(1^{st} ‘𝑇)), ((2^{nd} ‘(1^{st}
‘𝑇)) ∘
({⟨(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)⟩, ⟨((2^{nd}
‘𝑇) + 1),
(2^{nd} ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)}))))⟩,
(2^{nd} ‘𝑇)⟩ ∈ ((((0..^𝐾) ↑_{𝑚} (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁))) |
90 | | fveq2 6229 |
. . . . . . . . . . . 12
⊢ (𝑡 = 𝑇 → (2^{nd} ‘𝑡) = (2^{nd} ‘𝑇)) |
91 | 90 | breq2d 4697 |
. . . . . . . . . . 11
⊢ (𝑡 = 𝑇 → (𝑦 < (2^{nd} ‘𝑡) ↔ 𝑦 < (2^{nd} ‘𝑇))) |
92 | 91 | ifbid 4141 |
. . . . . . . . . 10
⊢ (𝑡 = 𝑇 → if(𝑦 < (2^{nd} ‘𝑡), 𝑦, (𝑦 + 1)) = if(𝑦 < (2^{nd} ‘𝑇), 𝑦, (𝑦 + 1))) |
93 | 92 | csbeq1d 3573 |
. . . . . . . . 9
⊢ (𝑡 = 𝑇 → ⦋if(𝑦 < (2^{nd} ‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^{st}
‘(1^{st} ‘𝑡)) ∘_{𝑓} +
((((2^{nd} ‘(1^{st} ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))) = ⦋if(𝑦 < (2^{nd}
‘𝑇), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^{st}
‘(1^{st} ‘𝑡)) ∘_{𝑓} +
((((2^{nd} ‘(1^{st} ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) |
94 | | fveq2 6229 |
. . . . . . . . . . . 12
⊢ (𝑡 = 𝑇 → (1^{st} ‘𝑡) = (1^{st} ‘𝑇)) |
95 | 94 | fveq2d 6233 |
. . . . . . . . . . 11
⊢ (𝑡 = 𝑇 → (1^{st}
‘(1^{st} ‘𝑡)) = (1^{st} ‘(1^{st}
‘𝑇))) |
96 | 94 | fveq2d 6233 |
. . . . . . . . . . . . . 14
⊢ (𝑡 = 𝑇 → (2^{nd}
‘(1^{st} ‘𝑡)) = (2^{nd} ‘(1^{st}
‘𝑇))) |
97 | 96 | imaeq1d 5500 |
. . . . . . . . . . . . 13
⊢ (𝑡 = 𝑇 → ((2^{nd}
‘(1^{st} ‘𝑡)) “ (1...𝑗)) = ((2^{nd} ‘(1^{st}
‘𝑇)) “
(1...𝑗))) |
98 | 97 | xpeq1d 5172 |
. . . . . . . . . . . 12
⊢ (𝑡 = 𝑇 → (((2^{nd}
‘(1^{st} ‘𝑡)) “ (1...𝑗)) × {1}) = (((2^{nd}
‘(1^{st} ‘𝑇)) “ (1...𝑗)) × {1})) |
99 | 96 | imaeq1d 5500 |
. . . . . . . . . . . . 13
⊢ (𝑡 = 𝑇 → ((2^{nd}
‘(1^{st} ‘𝑡)) “ ((𝑗 + 1)...𝑁)) = ((2^{nd} ‘(1^{st}
‘𝑇)) “ ((𝑗 + 1)...𝑁))) |
100 | 99 | xpeq1d 5172 |
. . . . . . . . . . . 12
⊢ (𝑡 = 𝑇 → (((2^{nd}
‘(1^{st} ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}) = (((2^{nd}
‘(1^{st} ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0})) |
101 | 98, 100 | uneq12d 3801 |
. . . . . . . . . . 11
⊢ (𝑡 = 𝑇 → ((((2^{nd}
‘(1^{st} ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})) = ((((2^{nd}
‘(1^{st} ‘𝑇)) “ (1...𝑗)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}))) |
102 | 95, 101 | oveq12d 6708 |
. . . . . . . . . 10
⊢ (𝑡 = 𝑇 → ((1^{st}
‘(1^{st} ‘𝑡)) ∘_{𝑓} +
((((2^{nd} ‘(1^{st} ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))) = ((1^{st}
‘(1^{st} ‘𝑇)) ∘_{𝑓} +
((((2^{nd} ‘(1^{st} ‘𝑇)) “ (1...𝑗)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0})))) |
103 | 102 | csbeq2dv 4025 |
. . . . . . . . 9
⊢ (𝑡 = 𝑇 → ⦋if(𝑦 < (2^{nd} ‘𝑇), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^{st}
‘(1^{st} ‘𝑡)) ∘_{𝑓} +
((((2^{nd} ‘(1^{st} ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))) = ⦋if(𝑦 < (2^{nd}
‘𝑇), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^{st}
‘(1^{st} ‘𝑇)) ∘_{𝑓} +
((((2^{nd} ‘(1^{st} ‘𝑇)) “ (1...𝑗)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0})))) |
104 | 93, 103 | eqtrd 2685 |
. . . . . . . 8
⊢ (𝑡 = 𝑇 → ⦋if(𝑦 < (2^{nd} ‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^{st}
‘(1^{st} ‘𝑡)) ∘_{𝑓} +
((((2^{nd} ‘(1^{st} ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))) = ⦋if(𝑦 < (2^{nd}
‘𝑇), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^{st}
‘(1^{st} ‘𝑇)) ∘_{𝑓} +
((((2^{nd} ‘(1^{st} ‘𝑇)) “ (1...𝑗)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0})))) |
105 | 104 | mpteq2dv 4778 |
. . . . . . 7
⊢ (𝑡 = 𝑇 → (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2^{nd}
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^{st}
‘(1^{st} ‘𝑡)) ∘_{𝑓} +
((((2^{nd} ‘(1^{st} ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2^{nd}
‘𝑇), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^{st}
‘(1^{st} ‘𝑇)) ∘_{𝑓} +
((((2^{nd} ‘(1^{st} ‘𝑇)) “ (1...𝑗)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}))))) |
106 | 105 | eqeq2d 2661 |
. . . . . 6
⊢ (𝑡 = 𝑇 → (𝐹 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2^{nd}
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^{st}
‘(1^{st} ‘𝑡)) ∘_{𝑓} +
((((2^{nd} ‘(1^{st} ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) ↔ 𝐹 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2^{nd}
‘𝑇), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^{st}
‘(1^{st} ‘𝑇)) ∘_{𝑓} +
((((2^{nd} ‘(1^{st} ‘𝑇)) “ (1...𝑗)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0})))))) |
107 | 106, 3 | elrab2 3399 |
. . . . 5
⊢ (𝑇 ∈ 𝑆 ↔ (𝑇 ∈ ((((0..^𝐾) ↑_{𝑚} (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∧ 𝐹 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2^{nd}
‘𝑇), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^{st}
‘(1^{st} ‘𝑇)) ∘_{𝑓} +
((((2^{nd} ‘(1^{st} ‘𝑇)) “ (1...𝑗)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0})))))) |
108 | 107 | simprbi 479 |
. . . 4
⊢ (𝑇 ∈ 𝑆 → 𝐹 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2^{nd}
‘𝑇), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^{st}
‘(1^{st} ‘𝑇)) ∘_{𝑓} +
((((2^{nd} ‘(1^{st} ‘𝑇)) “ (1...𝑗)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}))))) |
109 | 1, 108 | syl 17 |
. . 3
⊢ (𝜑 → 𝐹 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2^{nd}
‘𝑇), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^{st}
‘(1^{st} ‘𝑇)) ∘_{𝑓} +
((((2^{nd} ‘(1^{st} ‘𝑇)) “ (1...𝑗)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}))))) |
110 | | imaco 5678 |
. . . . . . . . . 10
⊢
(((2^{nd} ‘(1^{st} ‘𝑇)) ∘ ({⟨(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)⟩,
⟨((2^{nd} ‘𝑇) + 1), (2^{nd} ‘𝑇)⟩} ∪ ( I ↾
((1...𝑁) ∖
{(2^{nd} ‘𝑇),
((2^{nd} ‘𝑇)
+ 1)})))) “ (1...𝑦))
= ((2^{nd} ‘(1^{st} ‘𝑇)) “ (({⟨(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)⟩,
⟨((2^{nd} ‘𝑇) + 1), (2^{nd} ‘𝑇)⟩} ∪ ( I ↾
((1...𝑁) ∖
{(2^{nd} ‘𝑇),
((2^{nd} ‘𝑇)
+ 1)}))) “ (1...𝑦))) |
111 | | f1ofn 6176 |
. . . . . . . . . . . . . . . 16
⊢
({⟨(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)⟩, ⟨((2^{nd}
‘𝑇) + 1),
(2^{nd} ‘𝑇)⟩}:{(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)}–1-1-onto→{((2^{nd} ‘𝑇) + 1), (2^{nd} ‘𝑇)} → {⟨(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)⟩,
⟨((2^{nd} ‘𝑇) + 1), (2^{nd} ‘𝑇)⟩} Fn {(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) +
1)}) |
112 | 26, 111 | syl 17 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → {⟨(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)⟩,
⟨((2^{nd} ‘𝑇) + 1), (2^{nd} ‘𝑇)⟩} Fn {(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) +
1)}) |
113 | 112 | ad2antrr 762 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑦 < (2^{nd} ‘𝑇)) → {⟨(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)⟩,
⟨((2^{nd} ‘𝑇) + 1), (2^{nd} ‘𝑇)⟩} Fn {(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) +
1)}) |
114 | | incom 3838 |
. . . . . . . . . . . . . . 15
⊢
({(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)} ∩ (1...𝑦)) = ((1...𝑦) ∩ {(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)}) |
115 | | elfznn0 12471 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑦 ∈ (0...(𝑁 − 1)) → 𝑦 ∈ ℕ_{0}) |
116 | 115 | nn0red 11390 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑦 ∈ (0...(𝑁 − 1)) → 𝑦 ∈ ℝ) |
117 | | ltnle 10155 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝑦 ∈ ℝ ∧
(2^{nd} ‘𝑇)
∈ ℝ) → (𝑦
< (2^{nd} ‘𝑇) ↔ ¬ (2^{nd} ‘𝑇) ≤ 𝑦)) |
118 | 116, 18, 117 | syl2anr 494 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (𝑦 < (2^{nd} ‘𝑇) ↔ ¬ (2^{nd}
‘𝑇) ≤ 𝑦)) |
119 | 118 | biimpa 500 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑦 < (2^{nd} ‘𝑇)) → ¬ (2^{nd}
‘𝑇) ≤ 𝑦) |
120 | | elfzle2 12383 |
. . . . . . . . . . . . . . . . . . 19
⊢
((2^{nd} ‘𝑇) ∈ (1...𝑦) → (2^{nd} ‘𝑇) ≤ 𝑦) |
121 | 119, 120 | nsyl 135 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑦 < (2^{nd} ‘𝑇)) → ¬ (2^{nd}
‘𝑇) ∈ (1...𝑦)) |
122 | | disjsn 4278 |
. . . . . . . . . . . . . . . . . 18
⊢
(((1...𝑦) ∩
{(2^{nd} ‘𝑇)}) = ∅ ↔ ¬ (2^{nd}
‘𝑇) ∈ (1...𝑦)) |
123 | 121, 122 | sylibr 224 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑦 < (2^{nd} ‘𝑇)) → ((1...𝑦) ∩ {(2^{nd}
‘𝑇)}) =
∅) |
124 | 116 | ad2antlr 763 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑦 < (2^{nd} ‘𝑇)) → 𝑦 ∈ ℝ) |
125 | 18 | ad2antrr 762 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑦 < (2^{nd} ‘𝑇)) → (2^{nd}
‘𝑇) ∈
ℝ) |
126 | 50 | nnred 11073 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝜑 → ((2^{nd}
‘𝑇) + 1) ∈
ℝ) |
127 | 126 | ad2antrr 762 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑦 < (2^{nd} ‘𝑇)) → ((2^{nd}
‘𝑇) + 1) ∈
ℝ) |
128 | | simpr 476 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑦 < (2^{nd} ‘𝑇)) → 𝑦 < (2^{nd} ‘𝑇)) |
129 | 19 | ad2antrr 762 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑦 < (2^{nd} ‘𝑇)) → (2^{nd}
‘𝑇) <
((2^{nd} ‘𝑇)
+ 1)) |
130 | 124, 125,
127, 128, 129 | lttrd 10236 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑦 < (2^{nd} ‘𝑇)) → 𝑦 < ((2^{nd} ‘𝑇) + 1)) |
131 | | ltnle 10155 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝑦 ∈ ℝ ∧
((2^{nd} ‘𝑇)
+ 1) ∈ ℝ) → (𝑦 < ((2^{nd} ‘𝑇) + 1) ↔ ¬
((2^{nd} ‘𝑇)
+ 1) ≤ 𝑦)) |
132 | 116, 126,
131 | syl2anr 494 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (𝑦 < ((2^{nd} ‘𝑇) + 1) ↔ ¬
((2^{nd} ‘𝑇)
+ 1) ≤ 𝑦)) |
133 | 132 | adantr 480 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑦 < (2^{nd} ‘𝑇)) → (𝑦 < ((2^{nd} ‘𝑇) + 1) ↔ ¬
((2^{nd} ‘𝑇)
+ 1) ≤ 𝑦)) |
134 | 130, 133 | mpbid 222 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑦 < (2^{nd} ‘𝑇)) → ¬ ((2^{nd}
‘𝑇) + 1) ≤ 𝑦) |
135 | | elfzle2 12383 |
. . . . . . . . . . . . . . . . . . 19
⊢
(((2^{nd} ‘𝑇) + 1) ∈ (1...𝑦) → ((2^{nd} ‘𝑇) + 1) ≤ 𝑦) |
136 | 134, 135 | nsyl 135 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑦 < (2^{nd} ‘𝑇)) → ¬ ((2^{nd}
‘𝑇) + 1) ∈
(1...𝑦)) |
137 | | disjsn 4278 |
. . . . . . . . . . . . . . . . . 18
⊢
(((1...𝑦) ∩
{((2^{nd} ‘𝑇)
+ 1)}) = ∅ ↔ ¬ ((2^{nd} ‘𝑇) + 1) ∈ (1...𝑦)) |
138 | 136, 137 | sylibr 224 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑦 < (2^{nd} ‘𝑇)) → ((1...𝑦) ∩ {((2^{nd}
‘𝑇) + 1)}) =
∅) |
139 | 123, 138 | uneq12d 3801 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑦 < (2^{nd} ‘𝑇)) → (((1...𝑦) ∩ {(2^{nd}
‘𝑇)}) ∪
((1...𝑦) ∩
{((2^{nd} ‘𝑇)
+ 1)})) = (∅ ∪ ∅)) |
140 | | df-pr 4213 |
. . . . . . . . . . . . . . . . . 18
⊢
{(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)} = ({(2^{nd} ‘𝑇)} ∪ {((2^{nd}
‘𝑇) +
1)}) |
141 | 140 | ineq2i 3844 |
. . . . . . . . . . . . . . . . 17
⊢
((1...𝑦) ∩
{(2^{nd} ‘𝑇),
((2^{nd} ‘𝑇)
+ 1)}) = ((1...𝑦) ∩
({(2^{nd} ‘𝑇)} ∪ {((2^{nd} ‘𝑇) + 1)})) |
142 | | indi 3906 |
. . . . . . . . . . . . . . . . 17
⊢
((1...𝑦) ∩
({(2^{nd} ‘𝑇)} ∪ {((2^{nd} ‘𝑇) + 1)})) = (((1...𝑦) ∩ {(2^{nd}
‘𝑇)}) ∪
((1...𝑦) ∩
{((2^{nd} ‘𝑇)
+ 1)})) |
143 | 141, 142 | eqtr2i 2674 |
. . . . . . . . . . . . . . . 16
⊢
(((1...𝑦) ∩
{(2^{nd} ‘𝑇)}) ∪ ((1...𝑦) ∩ {((2^{nd} ‘𝑇) + 1)})) = ((1...𝑦) ∩ {(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) +
1)}) |
144 | | un0 4000 |
. . . . . . . . . . . . . . . 16
⊢ (∅
∪ ∅) = ∅ |
145 | 139, 143,
144 | 3eqtr3g 2708 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑦 < (2^{nd} ‘𝑇)) → ((1...𝑦) ∩ {(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)}) =
∅) |
146 | 114, 145 | syl5eq 2697 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑦 < (2^{nd} ‘𝑇)) → ({(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)} ∩
(1...𝑦)) =
∅) |
147 | | fnimadisj 6050 |
. . . . . . . . . . . . . 14
⊢
(({⟨(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)⟩, ⟨((2^{nd}
‘𝑇) + 1),
(2^{nd} ‘𝑇)⟩} Fn {(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)} ∧ ({(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)} ∩
(1...𝑦)) = ∅) →
({⟨(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)⟩, ⟨((2^{nd}
‘𝑇) + 1),
(2^{nd} ‘𝑇)⟩} “ (1...𝑦)) = ∅) |
148 | 113, 146,
147 | syl2anc 694 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑦 < (2^{nd} ‘𝑇)) →
({⟨(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)⟩, ⟨((2^{nd}
‘𝑇) + 1),
(2^{nd} ‘𝑇)⟩} “ (1...𝑦)) = ∅) |
149 | 39 | adantr 480 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ((𝑁 − 1) + 1) = 𝑁) |
150 | | elfzuz3 12377 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑦 ∈ (0...(𝑁 − 1)) → (𝑁 − 1) ∈
(ℤ_{≥}‘𝑦)) |
151 | | peano2uz 11779 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝑁 − 1) ∈
(ℤ_{≥}‘𝑦) → ((𝑁 − 1) + 1) ∈
(ℤ_{≥}‘𝑦)) |
152 | 150, 151 | syl 17 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑦 ∈ (0...(𝑁 − 1)) → ((𝑁 − 1) + 1) ∈
(ℤ_{≥}‘𝑦)) |
153 | 152 | adantl 481 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ((𝑁 − 1) + 1) ∈
(ℤ_{≥}‘𝑦)) |
154 | 149, 153 | eqeltrrd 2731 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → 𝑁 ∈ (ℤ_{≥}‘𝑦)) |
155 | | fzss2 12419 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑁 ∈
(ℤ_{≥}‘𝑦) → (1...𝑦) ⊆ (1...𝑁)) |
156 | | reldisj 4053 |
. . . . . . . . . . . . . . . . 17
⊢
((1...𝑦) ⊆
(1...𝑁) → (((1...𝑦) ∩ {(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)}) = ∅
↔ (1...𝑦) ⊆
((1...𝑁) ∖
{(2^{nd} ‘𝑇),
((2^{nd} ‘𝑇)
+ 1)}))) |
157 | 154, 155,
156 | 3syl 18 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (((1...𝑦) ∩ {(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)}) = ∅
↔ (1...𝑦) ⊆
((1...𝑁) ∖
{(2^{nd} ‘𝑇),
((2^{nd} ‘𝑇)
+ 1)}))) |
158 | 157 | adantr 480 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑦 < (2^{nd} ‘𝑇)) → (((1...𝑦) ∩ {(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)}) = ∅
↔ (1...𝑦) ⊆
((1...𝑁) ∖
{(2^{nd} ‘𝑇),
((2^{nd} ‘𝑇)
+ 1)}))) |
159 | 145, 158 | mpbid 222 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑦 < (2^{nd} ‘𝑇)) → (1...𝑦) ⊆ ((1...𝑁) ∖ {(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) +
1)})) |
160 | | resiima 5515 |
. . . . . . . . . . . . . 14
⊢
((1...𝑦) ⊆
((1...𝑁) ∖
{(2^{nd} ‘𝑇),
((2^{nd} ‘𝑇)
+ 1)}) → (( I ↾ ((1...𝑁) ∖ {(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)})) “ (1...𝑦)) = (1...𝑦)) |
161 | 159, 160 | syl 17 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑦 < (2^{nd} ‘𝑇)) → (( I ↾
((1...𝑁) ∖
{(2^{nd} ‘𝑇),
((2^{nd} ‘𝑇)
+ 1)})) “ (1...𝑦)) =
(1...𝑦)) |
162 | 148, 161 | uneq12d 3801 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑦 < (2^{nd} ‘𝑇)) →
(({⟨(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)⟩, ⟨((2^{nd}
‘𝑇) + 1),
(2^{nd} ‘𝑇)⟩} “ (1...𝑦)) ∪ (( I ↾ ((1...𝑁) ∖ {(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)})) “ (1...𝑦))) = (∅ ∪ (1...𝑦))) |
163 | | imaundir 5581 |
. . . . . . . . . . . 12
⊢
(({⟨(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)⟩, ⟨((2^{nd}
‘𝑇) + 1),
(2^{nd} ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)}))) “
(1...𝑦)) =
(({⟨(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)⟩, ⟨((2^{nd}
‘𝑇) + 1),
(2^{nd} ‘𝑇)⟩} “ (1...𝑦)) ∪ (( I ↾ ((1...𝑁) ∖ {(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)})) “ (1...𝑦))) |
164 | | uncom 3790 |
. . . . . . . . . . . . 13
⊢ (∅
∪ (1...𝑦)) =
((1...𝑦) ∪
∅) |
165 | | un0 4000 |
. . . . . . . . . . . . 13
⊢
((1...𝑦) ∪
∅) = (1...𝑦) |
166 | 164, 165 | eqtr2i 2674 |
. . . . . . . . . . . 12
⊢
(1...𝑦) = (∅
∪ (1...𝑦)) |
167 | 162, 163,
166 | 3eqtr4g 2710 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑦 < (2^{nd} ‘𝑇)) →
(({⟨(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)⟩, ⟨((2^{nd}
‘𝑇) + 1),
(2^{nd} ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)}))) “
(1...𝑦)) = (1...𝑦)) |
168 | 167 | imaeq2d 5501 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑦 < (2^{nd} ‘𝑇)) → ((2^{nd}
‘(1^{st} ‘𝑇)) “ (({⟨(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)⟩,
⟨((2^{nd} ‘𝑇) + 1), (2^{nd} ‘𝑇)⟩} ∪ ( I ↾
((1...𝑁) ∖
{(2^{nd} ‘𝑇),
((2^{nd} ‘𝑇)
+ 1)}))) “ (1...𝑦)))
= ((2^{nd} ‘(1^{st} ‘𝑇)) “ (1...𝑦))) |
169 | 110, 168 | syl5eq 2697 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑦 < (2^{nd} ‘𝑇)) → (((2^{nd}
‘(1^{st} ‘𝑇)) ∘ ({⟨(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)⟩,
⟨((2^{nd} ‘𝑇) + 1), (2^{nd} ‘𝑇)⟩} ∪ ( I ↾
((1...𝑁) ∖
{(2^{nd} ‘𝑇),
((2^{nd} ‘𝑇)
+ 1)})))) “ (1...𝑦))
= ((2^{nd} ‘(1^{st} ‘𝑇)) “ (1...𝑦))) |
170 | 169 | xpeq1d 5172 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑦 < (2^{nd} ‘𝑇)) → ((((2^{nd}
‘(1^{st} ‘𝑇)) ∘ ({⟨(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)⟩,
⟨((2^{nd} ‘𝑇) + 1), (2^{nd} ‘𝑇)⟩} ∪ ( I ↾
((1...𝑁) ∖
{(2^{nd} ‘𝑇),
((2^{nd} ‘𝑇)
+ 1)})))) “ (1...𝑦))
× {1}) = (((2^{nd} ‘(1^{st} ‘𝑇)) “ (1...𝑦)) ×
{1})) |
171 | | imaco 5678 |
. . . . . . . . . 10
⊢
(((2^{nd} ‘(1^{st} ‘𝑇)) ∘ ({⟨(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)⟩,
⟨((2^{nd} ‘𝑇) + 1), (2^{nd} ‘𝑇)⟩} ∪ ( I ↾
((1...𝑁) ∖
{(2^{nd} ‘𝑇),
((2^{nd} ‘𝑇)
+ 1)})))) “ ((𝑦 +
1)...𝑁)) = ((2^{nd}
‘(1^{st} ‘𝑇)) “ (({⟨(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)⟩,
⟨((2^{nd} ‘𝑇) + 1), (2^{nd} ‘𝑇)⟩} ∪ ( I ↾
((1...𝑁) ∖
{(2^{nd} ‘𝑇),
((2^{nd} ‘𝑇)
+ 1)}))) “ ((𝑦 +
1)...𝑁))) |
172 | | imaundir 5581 |
. . . . . . . . . . . . 13
⊢
(({⟨(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)⟩, ⟨((2^{nd}
‘𝑇) + 1),
(2^{nd} ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)}))) “
((𝑦 + 1)...𝑁)) = (({⟨(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)⟩,
⟨((2^{nd} ‘𝑇) + 1), (2^{nd} ‘𝑇)⟩} “ ((𝑦 + 1)...𝑁)) ∪ (( I ↾ ((1...𝑁) ∖ {(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)})) “
((𝑦 + 1)...𝑁))) |
173 | | imassrn 5512 |
. . . . . . . . . . . . . . . . 17
⊢
({⟨(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)⟩, ⟨((2^{nd}
‘𝑇) + 1),
(2^{nd} ‘𝑇)⟩} “ ((𝑦 + 1)...𝑁)) ⊆ ran {⟨(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)⟩,
⟨((2^{nd} ‘𝑇) + 1), (2^{nd} ‘𝑇)⟩} |
174 | 173 | a1i 11 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑦 < (2^{nd} ‘𝑇)) →
({⟨(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)⟩, ⟨((2^{nd}
‘𝑇) + 1),
(2^{nd} ‘𝑇)⟩} “ ((𝑦 + 1)...𝑁)) ⊆ ran {⟨(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)⟩,
⟨((2^{nd} ‘𝑇) + 1), (2^{nd} ‘𝑇)⟩}) |
175 | | fnima 6048 |
. . . . . . . . . . . . . . . . . . 19
⊢
({⟨(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)⟩, ⟨((2^{nd}
‘𝑇) + 1),
(2^{nd} ‘𝑇)⟩} Fn {(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)} →
({⟨(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)⟩, ⟨((2^{nd}
‘𝑇) + 1),
(2^{nd} ‘𝑇)⟩} “ {(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)}) = ran
{⟨(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)⟩, ⟨((2^{nd}
‘𝑇) + 1),
(2^{nd} ‘𝑇)⟩}) |
176 | 26, 111, 175 | 3syl 18 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝜑 → ({⟨(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)⟩,
⟨((2^{nd} ‘𝑇) + 1), (2^{nd} ‘𝑇)⟩} “
{(2^{nd} ‘𝑇),
((2^{nd} ‘𝑇)
+ 1)}) = ran {⟨(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)⟩, ⟨((2^{nd}
‘𝑇) + 1),
(2^{nd} ‘𝑇)⟩}) |
177 | 176 | ad2antrr 762 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑦 < (2^{nd} ‘𝑇)) →
({⟨(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)⟩, ⟨((2^{nd}
‘𝑇) + 1),
(2^{nd} ‘𝑇)⟩} “ {(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)}) = ran
{⟨(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)⟩, ⟨((2^{nd}
‘𝑇) + 1),
(2^{nd} ‘𝑇)⟩}) |
178 | | elfzelz 12380 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑦 ∈ (0...(𝑁 − 1)) → 𝑦 ∈ ℤ) |
179 | | zltp1le 11465 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝑦 ∈ ℤ ∧
(2^{nd} ‘𝑇)
∈ ℤ) → (𝑦
< (2^{nd} ‘𝑇) ↔ (𝑦 + 1) ≤ (2^{nd} ‘𝑇))) |
180 | 178, 57, 179 | syl2anr 494 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (𝑦 < (2^{nd} ‘𝑇) ↔ (𝑦 + 1) ≤ (2^{nd} ‘𝑇))) |
181 | 180 | biimpa 500 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑦 < (2^{nd} ‘𝑇)) → (𝑦 + 1) ≤ (2^{nd} ‘𝑇)) |
182 | 18, 52, 56 | ltled 10223 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝜑 → (2^{nd}
‘𝑇) ≤ 𝑁) |
183 | 182 | ad2antrr 762 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑦 < (2^{nd} ‘𝑇)) → (2^{nd}
‘𝑇) ≤ 𝑁) |
184 | 57 | adantr 480 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (2^{nd}
‘𝑇) ∈
ℤ) |
185 | | nn0p1nn 11370 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (𝑦 ∈ ℕ_{0}
→ (𝑦 + 1) ∈
ℕ) |
186 | 115, 185 | syl 17 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝑦 ∈ (0...(𝑁 − 1)) → (𝑦 + 1) ∈ ℕ) |
187 | 186 | nnzd 11519 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑦 ∈ (0...(𝑁 − 1)) → (𝑦 + 1) ∈ ℤ) |
188 | 187 | adantl 481 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (𝑦 + 1) ∈ ℤ) |
189 | 40 | adantr 480 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → 𝑁 ∈ ℤ) |
190 | | elfz 12370 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢
(((2^{nd} ‘𝑇) ∈ ℤ ∧ (𝑦 + 1) ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((2^{nd}
‘𝑇) ∈ ((𝑦 + 1)...𝑁) ↔ ((𝑦 + 1) ≤ (2^{nd} ‘𝑇) ∧ (2^{nd}
‘𝑇) ≤ 𝑁))) |
191 | 184, 188,
189, 190 | syl3anc 1366 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ((2^{nd}
‘𝑇) ∈ ((𝑦 + 1)...𝑁) ↔ ((𝑦 + 1) ≤ (2^{nd} ‘𝑇) ∧ (2^{nd}
‘𝑇) ≤ 𝑁))) |
192 | 191 | adantr 480 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑦 < (2^{nd} ‘𝑇)) → ((2^{nd}
‘𝑇) ∈ ((𝑦 + 1)...𝑁) ↔ ((𝑦 + 1) ≤ (2^{nd} ‘𝑇) ∧ (2^{nd}
‘𝑇) ≤ 𝑁))) |
193 | 181, 183,
192 | mpbir2and 977 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑦 < (2^{nd} ‘𝑇)) → (2^{nd}
‘𝑇) ∈ ((𝑦 + 1)...𝑁)) |
194 | | 1red 10093 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑦 < (2^{nd} ‘𝑇)) → 1 ∈
ℝ) |
195 | | ltle 10164 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((𝑦 ∈ ℝ ∧
(2^{nd} ‘𝑇)
∈ ℝ) → (𝑦
< (2^{nd} ‘𝑇) → 𝑦 ≤ (2^{nd} ‘𝑇))) |
196 | 116, 18, 195 | syl2anr 494 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (𝑦 < (2^{nd} ‘𝑇) → 𝑦 ≤ (2^{nd} ‘𝑇))) |
197 | 196 | imp 444 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑦 < (2^{nd} ‘𝑇)) → 𝑦 ≤ (2^{nd} ‘𝑇)) |
198 | 124, 125,
194, 197 | leadd1dd 10679 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑦 < (2^{nd} ‘𝑇)) → (𝑦 + 1) ≤ ((2^{nd} ‘𝑇) + 1)) |
199 | 60 | ad2antrr 762 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑦 < (2^{nd} ‘𝑇)) → ((2^{nd}
‘𝑇) + 1) ≤ 𝑁) |
200 | 57 | peano2zd 11523 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝜑 → ((2^{nd}
‘𝑇) + 1) ∈
ℤ) |
201 | 200 | adantr 480 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ((2^{nd}
‘𝑇) + 1) ∈
ℤ) |
202 | | elfz 12370 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢
((((2^{nd} ‘𝑇) + 1) ∈ ℤ ∧ (𝑦 + 1) ∈ ℤ ∧ 𝑁 ∈ ℤ) →
(((2^{nd} ‘𝑇)
+ 1) ∈ ((𝑦 +
1)...𝑁) ↔ ((𝑦 + 1) ≤ ((2^{nd}
‘𝑇) + 1) ∧
((2^{nd} ‘𝑇)
+ 1) ≤ 𝑁))) |
203 | 201, 188,
189, 202 | syl3anc 1366 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (((2^{nd}
‘𝑇) + 1) ∈
((𝑦 + 1)...𝑁) ↔ ((𝑦 + 1) ≤ ((2^{nd} ‘𝑇) + 1) ∧ ((2^{nd}
‘𝑇) + 1) ≤ 𝑁))) |
204 | 203 | adantr 480 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑦 < (2^{nd} ‘𝑇)) → (((2^{nd}
‘𝑇) + 1) ∈
((𝑦 + 1)...𝑁) ↔ ((𝑦 + 1) ≤ ((2^{nd} ‘𝑇) + 1) ∧ ((2^{nd}
‘𝑇) + 1) ≤ 𝑁))) |
205 | 198, 199,
204 | mpbir2and 977 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑦 < (2^{nd} ‘𝑇)) → ((2^{nd}
‘𝑇) + 1) ∈
((𝑦 + 1)...𝑁)) |
206 | | prssi 4385 |
. . . . . . . . . . . . . . . . . . 19
⊢
(((2^{nd} ‘𝑇) ∈ ((𝑦 + 1)...𝑁) ∧ ((2^{nd} ‘𝑇) + 1) ∈ ((𝑦 + 1)...𝑁)) → {(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)} ⊆ ((𝑦 + 1)...𝑁)) |
207 | 193, 205,
206 | syl2anc 694 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑦 < (2^{nd} ‘𝑇)) → {(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)} ⊆
((𝑦 + 1)...𝑁)) |
208 | | imass2 5536 |
. . . . . . . . . . . . . . . . . 18
⊢
({(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)} ⊆ ((𝑦 + 1)...𝑁) → ({⟨(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)⟩,
⟨((2^{nd} ‘𝑇) + 1), (2^{nd} ‘𝑇)⟩} “
{(2^{nd} ‘𝑇),
((2^{nd} ‘𝑇)
+ 1)}) ⊆ ({⟨(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)⟩, ⟨((2^{nd}
‘𝑇) + 1),
(2^{nd} ‘𝑇)⟩} “ ((𝑦 + 1)...𝑁))) |
209 | 207, 208 | syl 17 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑦 < (2^{nd} ‘𝑇)) →
({⟨(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)⟩, ⟨((2^{nd}
‘𝑇) + 1),
(2^{nd} ‘𝑇)⟩} “ {(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)}) ⊆
({⟨(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)⟩, ⟨((2^{nd}
‘𝑇) + 1),
(2^{nd} ‘𝑇)⟩} “ ((𝑦 + 1)...𝑁))) |
210 | 177, 209 | eqsstr3d 3673 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑦 < (2^{nd} ‘𝑇)) → ran
{⟨(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)⟩, ⟨((2^{nd}
‘𝑇) + 1),
(2^{nd} ‘𝑇)⟩} ⊆ ({⟨(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)⟩,
⟨((2^{nd} ‘𝑇) + 1), (2^{nd} ‘𝑇)⟩} “ ((𝑦 + 1)...𝑁))) |
211 | 174, 210 | eqssd 3653 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑦 < (2^{nd} ‘𝑇)) →
({⟨(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)⟩, ⟨((2^{nd}
‘𝑇) + 1),
(2^{nd} ‘𝑇)⟩} “ ((𝑦 + 1)...𝑁)) = ran {⟨(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)⟩,
⟨((2^{nd} ‘𝑇) + 1), (2^{nd} ‘𝑇)⟩}) |
212 | | f1ofo 6182 |
. . . . . . . . . . . . . . . . . 18
⊢
({⟨(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)⟩, ⟨((2^{nd}
‘𝑇) + 1),
(2^{nd} ‘𝑇)⟩}:{(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)}–1-1-onto→{((2^{nd} ‘𝑇) + 1), (2^{nd} ‘𝑇)} → {⟨(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)⟩,
⟨((2^{nd} ‘𝑇) + 1), (2^{nd} ‘𝑇)⟩}:{(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)}–onto→{((2^{nd} ‘𝑇) + 1), (2^{nd}
‘𝑇)}) |
213 | | forn 6156 |
. . . . . . . . . . . . . . . . . 18
⊢
({⟨(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)⟩, ⟨((2^{nd}
‘𝑇) + 1),
(2^{nd} ‘𝑇)⟩}:{(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)}–onto→{((2^{nd} ‘𝑇) + 1), (2^{nd} ‘𝑇)} → ran
{⟨(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)⟩, ⟨((2^{nd}
‘𝑇) + 1),
(2^{nd} ‘𝑇)⟩} = {((2^{nd} ‘𝑇) + 1), (2^{nd}
‘𝑇)}) |
214 | 26, 212, 213 | 3syl 18 |
. . . . . . . . . . . . . . . . 17
⊢ (𝜑 → ran {⟨(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)⟩,
⟨((2^{nd} ‘𝑇) + 1), (2^{nd} ‘𝑇)⟩} = {((2^{nd}
‘𝑇) + 1),
(2^{nd} ‘𝑇)}) |
215 | 214, 27 | syl6eq 2701 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → ran {⟨(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)⟩,
⟨((2^{nd} ‘𝑇) + 1), (2^{nd} ‘𝑇)⟩} = {(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) +
1)}) |
216 | 215 | ad2antrr 762 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑦 < (2^{nd} ‘𝑇)) → ran
{⟨(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)⟩, ⟨((2^{nd}
‘𝑇) + 1),
(2^{nd} ‘𝑇)⟩} = {(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)}) |
217 | 211, 216 | eqtrd 2685 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑦 < (2^{nd} ‘𝑇)) →
({⟨(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)⟩, ⟨((2^{nd}
‘𝑇) + 1),
(2^{nd} ‘𝑇)⟩} “ ((𝑦 + 1)...𝑁)) = {(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)}) |
218 | | undif 4082 |
. . . . . . . . . . . . . . . . 17
⊢
({(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)} ⊆ ((𝑦 + 1)...𝑁) ↔ ({(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)} ∪ (((𝑦 + 1)...𝑁) ∖ {(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)})) = ((𝑦 + 1)...𝑁)) |
219 | 207, 218 | sylib 208 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑦 < (2^{nd} ‘𝑇)) → ({(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)} ∪
(((𝑦 + 1)...𝑁) ∖ {(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)})) = ((𝑦 + 1)...𝑁)) |
220 | 219 | imaeq2d 5501 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑦 < (2^{nd} ‘𝑇)) → (( I ↾
((1...𝑁) ∖
{(2^{nd} ‘𝑇),
((2^{nd} ‘𝑇)
+ 1)})) “ ({(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)} ∪ (((𝑦 + 1)...𝑁) ∖ {(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)}))) = (( I ↾
((1...𝑁) ∖
{(2^{nd} ‘𝑇),
((2^{nd} ‘𝑇)
+ 1)})) “ ((𝑦 +
1)...𝑁))) |
221 | | fnresi 6046 |
. . . . . . . . . . . . . . . . . . 19
⊢ ( I
↾ ((1...𝑁) ∖
{(2^{nd} ‘𝑇),
((2^{nd} ‘𝑇)
+ 1)})) Fn ((1...𝑁) ∖
{(2^{nd} ‘𝑇),
((2^{nd} ‘𝑇)
+ 1)}) |
222 | | incom 3838 |
. . . . . . . . . . . . . . . . . . . 20
⊢
(((1...𝑁) ∖
{(2^{nd} ‘𝑇),
((2^{nd} ‘𝑇)
+ 1)}) ∩ {(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)}) = ({(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)} ∩ ((1...𝑁) ∖ {(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) +
1)})) |
223 | 222, 32 | eqtri 2673 |
. . . . . . . . . . . . . . . . . . 19
⊢
(((1...𝑁) ∖
{(2^{nd} ‘𝑇),
((2^{nd} ‘𝑇)
+ 1)}) ∩ {(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)}) = ∅ |
224 | | fnimadisj 6050 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((( I
↾ ((1...𝑁) ∖
{(2^{nd} ‘𝑇),
((2^{nd} ‘𝑇)
+ 1)})) Fn ((1...𝑁) ∖
{(2^{nd} ‘𝑇),
((2^{nd} ‘𝑇)
+ 1)}) ∧ (((1...𝑁)
∖ {(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)}) ∩ {(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)}) = ∅) → (( I
↾ ((1...𝑁) ∖
{(2^{nd} ‘𝑇),
((2^{nd} ‘𝑇)
+ 1)})) “ {(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)}) = ∅) |
225 | 221, 223,
224 | mp2an 708 |
. . . . . . . . . . . . . . . . . 18
⊢ (( I
↾ ((1...𝑁) ∖
{(2^{nd} ‘𝑇),
((2^{nd} ‘𝑇)
+ 1)})) “ {(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)}) = ∅ |
226 | 225 | a1i 11 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑦 < (2^{nd} ‘𝑇)) → (( I ↾
((1...𝑁) ∖
{(2^{nd} ‘𝑇),
((2^{nd} ‘𝑇)
+ 1)})) “ {(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)}) = ∅) |
227 | | nnuz 11761 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ℕ =
(ℤ_{≥}‘1) |
228 | 186, 227 | syl6eleq 2740 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑦 ∈ (0...(𝑁 − 1)) → (𝑦 + 1) ∈
(ℤ_{≥}‘1)) |
229 | | fzss1 12418 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝑦 + 1) ∈
(ℤ_{≥}‘1) → ((𝑦 + 1)...𝑁) ⊆ (1...𝑁)) |
230 | 228, 229 | syl 17 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑦 ∈ (0...(𝑁 − 1)) → ((𝑦 + 1)...𝑁) ⊆ (1...𝑁)) |
231 | 230 | ssdifd 3779 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑦 ∈ (0...(𝑁 − 1)) → (((𝑦 + 1)...𝑁) ∖ {(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)}) ⊆ ((1...𝑁) ∖ {(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) +
1)})) |
232 | | resiima 5515 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((((𝑦 + 1)...𝑁) ∖ {(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)}) ⊆ ((1...𝑁) ∖ {(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)}) → ((
I ↾ ((1...𝑁) ∖
{(2^{nd} ‘𝑇),
((2^{nd} ‘𝑇)
+ 1)})) “ (((𝑦 +
1)...𝑁) ∖
{(2^{nd} ‘𝑇),
((2^{nd} ‘𝑇)
+ 1)})) = (((𝑦 + 1)...𝑁) ∖ {(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) +
1)})) |
233 | 231, 232 | syl 17 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑦 ∈ (0...(𝑁 − 1)) → (( I ↾ ((1...𝑁) ∖ {(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)})) “
(((𝑦 + 1)...𝑁) ∖ {(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)})) =
(((𝑦 + 1)...𝑁) ∖ {(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) +
1)})) |
234 | 233 | ad2antlr 763 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑦 < (2^{nd} ‘𝑇)) → (( I ↾
((1...𝑁) ∖
{(2^{nd} ‘𝑇),
((2^{nd} ‘𝑇)
+ 1)})) “ (((𝑦 +
1)...𝑁) ∖
{(2^{nd} ‘𝑇),
((2^{nd} ‘𝑇)
+ 1)})) = (((𝑦 + 1)...𝑁) ∖ {(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) +
1)})) |
235 | 226, 234 | uneq12d 3801 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑦 < (2^{nd} ‘𝑇)) → ((( I ↾
((1...𝑁) ∖
{(2^{nd} ‘𝑇),
((2^{nd} ‘𝑇)
+ 1)})) “ {(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)}) ∪ (( I ↾ ((1...𝑁) ∖ {(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)})) “
(((𝑦 + 1)...𝑁) ∖ {(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)}))) =
(∅ ∪ (((𝑦 +
1)...𝑁) ∖
{(2^{nd} ‘𝑇),
((2^{nd} ‘𝑇)
+ 1)}))) |
236 | | imaundi 5580 |
. . . . . . . . . . . . . . . 16
⊢ (( I
↾ ((1...𝑁) ∖
{(2^{nd} ‘𝑇),
((2^{nd} ‘𝑇)
+ 1)})) “ ({(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)} ∪ (((𝑦 + 1)...𝑁) ∖ {(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)}))) = ((( I ↾
((1...𝑁) ∖
{(2^{nd} ‘𝑇),
((2^{nd} ‘𝑇)
+ 1)})) “ {(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)}) ∪ (( I ↾ ((1...𝑁) ∖ {(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)})) “
(((𝑦 + 1)...𝑁) ∖ {(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) +
1)}))) |
237 | | uncom 3790 |
. . . . . . . . . . . . . . . . 17
⊢ (∅
∪ (((𝑦 + 1)...𝑁) ∖ {(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)})) =
((((𝑦 + 1)...𝑁) ∖ {(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)}) ∪
∅) |
238 | | un0 4000 |
. . . . . . . . . . . . . . . . 17
⊢ ((((𝑦 + 1)...𝑁) ∖ {(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)}) ∪ ∅) =
(((𝑦 + 1)...𝑁) ∖ {(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) +
1)}) |
239 | 237, 238 | eqtr2i 2674 |
. . . . . . . . . . . . . . . 16
⊢ (((𝑦 + 1)...𝑁) ∖ {(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)}) = (∅ ∪
(((𝑦 + 1)...𝑁) ∖ {(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) +
1)})) |
240 | 235, 236,
239 | 3eqtr4g 2710 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑦 < (2^{nd} ‘𝑇)) → (( I ↾
((1...𝑁) ∖
{(2^{nd} ‘𝑇),
((2^{nd} ‘𝑇)
+ 1)})) “ ({(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)} ∪ (((𝑦 + 1)...𝑁) ∖ {(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)}))) = (((𝑦 + 1)...𝑁) ∖ {(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)})) |
241 | 220, 240 | eqtr3d 2687 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑦 < (2^{nd} ‘𝑇)) → (( I ↾
((1...𝑁) ∖
{(2^{nd} ‘𝑇),
((2^{nd} ‘𝑇)
+ 1)})) “ ((𝑦 +
1)...𝑁)) = (((𝑦 + 1)...𝑁) ∖ {(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)})) |
242 | 217, 241 | uneq12d 3801 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑦 < (2^{nd} ‘𝑇)) →
(({⟨(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)⟩, ⟨((2^{nd}
‘𝑇) + 1),
(2^{nd} ‘𝑇)⟩} “ ((𝑦 + 1)...𝑁)) ∪ (( I ↾ ((1...𝑁) ∖ {(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)})) “
((𝑦 + 1)...𝑁))) = ({(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)} ∪
(((𝑦 + 1)...𝑁) ∖ {(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) +
1)}))) |
243 | 172, 242 | syl5eq 2697 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑦 < (2^{nd} ‘𝑇)) →
(({⟨(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)⟩, ⟨((2^{nd}
‘𝑇) + 1),
(2^{nd} ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)}))) “
((𝑦 + 1)...𝑁)) = ({(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)} ∪
(((𝑦 + 1)...𝑁) ∖ {(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) +
1)}))) |
244 | 243, 219 | eqtrd 2685 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑦 < (2^{nd} ‘𝑇)) →
(({⟨(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)⟩, ⟨((2^{nd}
‘𝑇) + 1),
(2^{nd} ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)}))) “
((𝑦 + 1)...𝑁)) = ((𝑦 + 1)...𝑁)) |
245 | 244 | imaeq2d 5501 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑦 < (2^{nd} ‘𝑇)) → ((2^{nd}
‘(1^{st} ‘𝑇)) “ (({⟨(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)⟩,
⟨((2^{nd} ‘𝑇) + 1), (2^{nd} ‘𝑇)⟩} ∪ ( I ↾
((1...𝑁) ∖
{(2^{nd} ‘𝑇),
((2^{nd} ‘𝑇)
+ 1)}))) “ ((𝑦 +
1)...𝑁))) =
((2^{nd} ‘(1^{st} ‘𝑇)) “ ((𝑦 + 1)...𝑁))) |
246 | 171, 245 | syl5eq 2697 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑦 < (2^{nd} ‘𝑇)) → (((2^{nd}
‘(1^{st} ‘𝑇)) ∘ ({⟨(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)⟩,
⟨((2^{nd} ‘𝑇) + 1), (2^{nd} ‘𝑇)⟩} ∪ ( I ↾
((1...𝑁) ∖
{(2^{nd} ‘𝑇),
((2^{nd} ‘𝑇)
+ 1)})))) “ ((𝑦 +
1)...𝑁)) = ((2^{nd}
‘(1^{st} ‘𝑇)) “ ((𝑦 + 1)...𝑁))) |
247 | 246 | xpeq1d 5172 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑦 < (2^{nd} ‘𝑇)) → ((((2^{nd}
‘(1^{st} ‘𝑇)) ∘ ({⟨(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)⟩,
⟨((2^{nd} ‘𝑇) + 1), (2^{nd} ‘𝑇)⟩} ∪ ( I ↾
((1...𝑁) ∖
{(2^{nd} ‘𝑇),
((2^{nd} ‘𝑇)
+ 1)})))) “ ((𝑦 +
1)...𝑁)) × {0}) =
(((2^{nd} ‘(1^{st} ‘𝑇)) “ ((𝑦 + 1)...𝑁)) × {0})) |
248 | 170, 247 | uneq12d 3801 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑦 < (2^{nd} ‘𝑇)) → (((((2^{nd}
‘(1^{st} ‘𝑇)) ∘ ({⟨(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)⟩,
⟨((2^{nd} ‘𝑇) + 1), (2^{nd} ‘𝑇)⟩} ∪ ( I ↾
((1...𝑁) ∖
{(2^{nd} ‘𝑇),
((2^{nd} ‘𝑇)
+ 1)})))) “ (1...𝑦))
× {1}) ∪ ((((2^{nd} ‘(1^{st} ‘𝑇)) ∘
({⟨(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)⟩, ⟨((2^{nd}
‘𝑇) + 1),
(2^{nd} ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)})))) “
((𝑦 + 1)...𝑁)) × {0})) =
((((2^{nd} ‘(1^{st} ‘𝑇)) “ (1...𝑦)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑇)) “ ((𝑦 + 1)...𝑁)) × {0}))) |
249 | 248 | oveq2d 6706 |
. . . . . 6
⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑦 < (2^{nd} ‘𝑇)) → ((1^{st}
‘(1^{st} ‘𝑇)) ∘_{𝑓} +
(((((2^{nd} ‘(1^{st} ‘𝑇)) ∘ ({⟨(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)⟩,
⟨((2^{nd} ‘𝑇) + 1), (2^{nd} ‘𝑇)⟩} ∪ ( I ↾
((1...𝑁) ∖
{(2^{nd} ‘𝑇),
((2^{nd} ‘𝑇)
+ 1)})))) “ (1...𝑦))
× {1}) ∪ ((((2^{nd} ‘(1^{st} ‘𝑇)) ∘
({⟨(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)⟩, ⟨((2^{nd}
‘𝑇) + 1),
(2^{nd} ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)})))) “
((𝑦 + 1)...𝑁)) × {0}))) =
((1^{st} ‘(1^{st} ‘𝑇)) ∘_{𝑓} +
((((2^{nd} ‘(1^{st} ‘𝑇)) “ (1...𝑦)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑇)) “ ((𝑦 + 1)...𝑁)) × {0})))) |
250 | | iftrue 4125 |
. . . . . . . . 9
⊢ (𝑦 < (2^{nd}
‘𝑇) → if(𝑦 < (2^{nd}
‘𝑇), 𝑦, (𝑦 + 1)) = 𝑦) |
251 | 250 | csbeq1d 3573 |
. . . . . . . 8
⊢ (𝑦 < (2^{nd}
‘𝑇) →
⦋if(𝑦 <
(2^{nd} ‘𝑇),
𝑦, (𝑦 + 1)) / 𝑗⦌((1^{st}
‘(1^{st} ‘𝑇)) ∘_{𝑓} +
(((((2^{nd} ‘(1^{st} ‘𝑇)) ∘ ({⟨(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)⟩,
⟨((2^{nd} ‘𝑇) + 1), (2^{nd} ‘𝑇)⟩} ∪ ( I ↾
((1...𝑁) ∖
{(2^{nd} ‘𝑇),
((2^{nd} ‘𝑇)
+ 1)})))) “ (1...𝑗))
× {1}) ∪ ((((2^{nd} ‘(1^{st} ‘𝑇)) ∘
({⟨(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)⟩, ⟨((2^{nd}
‘𝑇) + 1),
(2^{nd} ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)})))) “
((𝑗 + 1)...𝑁)) × {0}))) =
⦋𝑦 / 𝑗⦌((1^{st}
‘(1^{st} ‘𝑇)) ∘_{𝑓} +
(((((2^{nd} ‘(1^{st} ‘𝑇)) ∘ ({⟨(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)⟩,
⟨((2^{nd} ‘𝑇) + 1), (2^{nd} ‘𝑇)⟩} ∪ ( I ↾
((1...𝑁) ∖
{(2^{nd} ‘𝑇),
((2^{nd} ‘𝑇)
+ 1)})))) “ (1...𝑗))
× {1}) ∪ ((((2^{nd} ‘(1^{st} ‘𝑇)) ∘
({⟨(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)⟩, ⟨((2^{nd}
‘𝑇) + 1),
(2^{nd} ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)})))) “
((𝑗 + 1)...𝑁)) ×
{0})))) |
252 | | vex 3234 |
. . . . . . . . 9
⊢ 𝑦 ∈ V |
253 | | oveq2 6698 |
. . . . . . . . . . . . 13
⊢ (𝑗 = 𝑦 → (1...𝑗) = (1...𝑦)) |
254 | 253 | imaeq2d 5501 |
. . . . . . . . . . . 12
⊢ (𝑗 = 𝑦 → (((2^{nd}
‘(1^{st} ‘𝑇)) ∘ ({⟨(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)⟩,
⟨((2^{nd} ‘𝑇) + 1), (2^{nd} ‘𝑇)⟩} ∪ ( I ↾
((1...𝑁) ∖
{(2^{nd} ‘𝑇),
((2^{nd} ‘𝑇)
+ 1)})))) “ (1...𝑗))
= (((2^{nd} ‘(1^{st} ‘𝑇)) ∘ ({⟨(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)⟩,
⟨((2^{nd} ‘𝑇) + 1), (2^{nd} ‘𝑇)⟩} ∪ ( I ↾
((1...𝑁) ∖
{(2^{nd} ‘𝑇),
((2^{nd} ‘𝑇)
+ 1)})))) “ (1...𝑦))) |
255 | 254 | xpeq1d 5172 |
. . . . . . . . . . 11
⊢ (𝑗 = 𝑦 → ((((2^{nd}
‘(1^{st} ‘𝑇)) ∘ ({⟨(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)⟩,
⟨((2^{nd} ‘𝑇) + 1), (2^{nd} ‘𝑇)⟩} ∪ ( I ↾
((1...𝑁) ∖
{(2^{nd} ‘𝑇),
((2^{nd} ‘𝑇)
+ 1)})))) “ (1...𝑗))
× {1}) = ((((2^{nd} ‘(1^{st} ‘𝑇)) ∘
({⟨(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)⟩, ⟨((2^{nd}
‘𝑇) + 1),
(2^{nd} ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)})))) “
(1...𝑦)) ×
{1})) |
256 | | oveq1 6697 |
. . . . . . . . . . . . . 14
⊢ (𝑗 = 𝑦 → (𝑗 + 1) = (𝑦 + 1)) |
257 | 256 | oveq1d 6705 |
. . . . . . . . . . . . 13
⊢ (𝑗 = 𝑦 → ((𝑗 + 1)...𝑁) = ((𝑦 + 1)...𝑁)) |
258 | 257 | imaeq2d 5501 |
. . . . . . . . . . . 12
⊢ (𝑗 = 𝑦 → (((2^{nd}
‘(1^{st} ‘𝑇)) ∘ ({⟨(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)⟩,
⟨((2^{nd} ‘𝑇) + 1), (2^{nd} ‘𝑇)⟩} ∪ ( I ↾
((1...𝑁) ∖
{(2^{nd} ‘𝑇),
((2^{nd} ‘𝑇)
+ 1)})))) “ ((𝑗 +
1)...𝑁)) =
(((2^{nd} ‘(1^{st} ‘𝑇)) ∘ ({⟨(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)⟩,
⟨((2^{nd} ‘𝑇) + 1), (2^{nd} ‘𝑇)⟩} ∪ ( I ↾
((1...𝑁) ∖
{(2^{nd} ‘𝑇),
((2^{nd} ‘𝑇)
+ 1)})))) “ ((𝑦 +
1)...𝑁))) |
259 | 258 | xpeq1d 5172 |
. . . . . . . . . . 11
⊢ (𝑗 = 𝑦 → ((((2^{nd}
‘(1^{st} ‘𝑇)) ∘ ({⟨(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)⟩,
⟨((2^{nd} ‘𝑇) + 1), (2^{nd} ‘𝑇)⟩} ∪ ( I ↾
((1...𝑁) ∖
{(2^{nd} ‘𝑇),
((2^{nd} ‘𝑇)
+ 1)})))) “ ((𝑗 +
1)...𝑁)) × {0}) =
((((2^{nd} ‘(1^{st} ‘𝑇)) ∘ ({⟨(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)⟩,
⟨((2^{nd} ‘𝑇) + 1), (2^{nd} ‘𝑇)⟩} ∪ ( I ↾
((1...𝑁) ∖
{(2^{nd} ‘𝑇),
((2^{nd} ‘𝑇)
+ 1)})))) “ ((𝑦 +
1)...𝑁)) ×
{0})) |
260 | 255, 259 | uneq12d 3801 |
. . . . . . . . . 10
⊢ (𝑗 = 𝑦 → (((((2^{nd}
‘(1^{st} ‘𝑇)) ∘ ({⟨(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)⟩,
⟨((2^{nd} ‘𝑇) + 1), (2^{nd} ‘𝑇)⟩} ∪ ( I ↾
((1...𝑁) ∖
{(2^{nd} ‘𝑇),
((2^{nd} ‘𝑇)
+ 1)})))) “ (1...𝑗))
× {1}) ∪ ((((2^{nd} ‘(1^{st} ‘𝑇)) ∘
({⟨(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)⟩, ⟨((2^{nd}
‘𝑇) + 1),
(2^{nd} ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)})))) “
((𝑗 + 1)...𝑁)) × {0})) =
(((((2^{nd} ‘(1^{st} ‘𝑇)) ∘ ({⟨(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)⟩,
⟨((2^{nd} ‘𝑇) + 1), (2^{nd} ‘𝑇)⟩} ∪ ( I ↾
((1...𝑁) ∖
{(2^{nd} ‘𝑇),
((2^{nd} ‘𝑇)
+ 1)})))) “ (1...𝑦))
× {1}) ∪ ((((2^{nd} ‘(1^{st} ‘𝑇)) ∘
({⟨(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)⟩, ⟨((2^{nd}
‘𝑇) + 1),
(2^{nd} ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)})))) “
((𝑦 + 1)...𝑁)) ×
{0}))) |
261 | 260 | oveq2d 6706 |
. . . . . . . . 9
⊢ (𝑗 = 𝑦 → ((1^{st}
‘(1^{st} ‘𝑇)) ∘_{𝑓} +
(((((2^{nd} ‘(1^{st} ‘𝑇)) ∘ ({⟨(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)⟩,
⟨((2^{nd} ‘𝑇) + 1), (2^{nd} ‘𝑇)⟩} ∪ ( I ↾
((1...𝑁) ∖
{(2^{nd} ‘𝑇),
((2^{nd} ‘𝑇)
+ 1)})))) “ (1...𝑗))
× {1}) ∪ ((((2^{nd} ‘(1^{st} ‘𝑇)) ∘
({⟨(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)⟩, ⟨((2^{nd}
‘𝑇) + 1),
(2^{nd} ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)})))) “
((𝑗 + 1)...𝑁)) × {0}))) =
((1^{st} ‘(1^{st} ‘𝑇)) ∘_{𝑓} +
(((((2^{nd} ‘(1^{st} ‘𝑇)) ∘ ({⟨(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)⟩,
⟨((2^{nd} ‘𝑇) + 1), (2^{nd} ‘𝑇)⟩} ∪ ( I ↾
((1...𝑁) ∖
{(2^{nd} ‘𝑇),
((2^{nd} ‘𝑇)
+ 1)})))) “ (1...𝑦))
× {1}) ∪ ((((2^{nd} ‘(1^{st} ‘𝑇)) ∘
({⟨(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)⟩, ⟨((2^{nd}
‘𝑇) + 1),
(2^{nd} ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)})))) “
((𝑦 + 1)...𝑁)) ×
{0})))) |
262 | 252, 261 | csbie 3592 |
. . . . . . . 8
⊢
⦋𝑦 /
𝑗⦌((1^{st}
‘(1^{st} ‘𝑇)) ∘_{𝑓} +
(((((2^{nd} ‘(1^{st} ‘𝑇)) ∘ ({⟨(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)⟩,
⟨((2^{nd} ‘𝑇) + 1), (2^{nd} ‘𝑇)⟩} ∪ ( I ↾
((1...𝑁) ∖
{(2^{nd} ‘𝑇),
((2^{nd} ‘𝑇)
+ 1)})))) “ (1...𝑗))
× {1}) ∪ ((((2^{nd} ‘(1^{st} ‘𝑇)) ∘
({⟨(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)⟩, ⟨((2^{nd}
‘𝑇) + 1),
(2^{nd} ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)})))) “
((𝑗 + 1)...𝑁)) × {0}))) =
((1^{st} ‘(1^{st} ‘𝑇)) ∘_{𝑓} +
(((((2^{nd} ‘(1^{st} ‘𝑇)) ∘ ({⟨(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)⟩,
⟨((2^{nd} ‘𝑇) + 1), (2^{nd} ‘𝑇)⟩} ∪ ( I ↾
((1...𝑁) ∖
{(2^{nd} ‘𝑇),
((2^{nd} ‘𝑇)
+ 1)})))) “ (1...𝑦))
× {1}) ∪ ((((2^{nd} ‘(1^{st} ‘𝑇)) ∘
({⟨(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)⟩, ⟨((2^{nd}
‘𝑇) + 1),
(2^{nd} ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)})))) “
((𝑦 + 1)...𝑁)) ×
{0}))) |
263 | 251, 262 | syl6eq 2701 |
. . . . . . 7
⊢ (𝑦 < (2^{nd}
‘𝑇) →
⦋if(𝑦 <
(2^{nd} ‘𝑇),
𝑦, (𝑦 + 1)) / 𝑗⦌((1^{st}
‘(1^{st} ‘𝑇)) ∘_{𝑓} +
(((((2^{nd} ‘(1^{st} ‘𝑇)) ∘ ({⟨(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)⟩,
⟨((2^{nd} ‘𝑇) + 1), (2^{nd} ‘𝑇)⟩} ∪ ( I ↾
((1...𝑁) ∖
{(2^{nd} ‘𝑇),
((2^{nd} ‘𝑇)
+ 1)})))) “ (1...𝑗))
× {1}) ∪ ((((2^{nd} ‘(1^{st} ‘𝑇)) ∘
({⟨(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)⟩, ⟨((2^{nd}
‘𝑇) + 1),
(2^{nd} ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)})))) “
((𝑗 + 1)...𝑁)) × {0}))) =
((1^{st} ‘(1^{st} ‘𝑇)) ∘_{𝑓} +
(((((2^{nd} ‘(1^{st} ‘𝑇)) ∘ ({⟨(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)⟩,
⟨((2^{nd} ‘𝑇) + 1), (2^{nd} ‘𝑇)⟩} ∪ ( I ↾
((1...𝑁) ∖
{(2^{nd} ‘𝑇),
((2^{nd} ‘𝑇)
+ 1)})))) “ (1...𝑦))
× {1}) ∪ ((((2^{nd} ‘(1^{st} ‘𝑇)) ∘
({⟨(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)⟩, ⟨((2^{nd}
‘𝑇) + 1),
(2^{nd} ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)})))) “
((𝑦 + 1)...𝑁)) ×
{0})))) |
264 | 263 | adantl 481 |
. . . . . 6
⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑦 < (2^{nd} ‘𝑇)) →
⦋if(𝑦 <
(2^{nd} ‘𝑇),
𝑦, (𝑦 + 1)) / 𝑗⦌((1^{st}
‘(1^{st} ‘𝑇)) ∘_{𝑓} +
(((((2^{nd} ‘(1^{st} ‘𝑇)) ∘ ({⟨(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)⟩,
⟨((2^{nd} ‘𝑇) + 1), (2^{nd} ‘𝑇)⟩} ∪ ( I ↾
((1...𝑁) ∖
{(2^{nd} ‘𝑇),
((2^{nd} ‘𝑇)
+ 1)})))) “ (1...𝑗))
× {1}) ∪ ((((2^{nd} ‘(1^{st} ‘𝑇)) ∘
({⟨(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)⟩, ⟨((2^{nd}
‘𝑇) + 1),
(2^{nd} ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)})))) “
((𝑗 + 1)...𝑁)) × {0}))) =
((1^{st} ‘(1^{st} ‘𝑇)) ∘_{𝑓} +
(((((2^{nd} ‘(1^{st} ‘𝑇)) ∘ ({⟨(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)⟩,
⟨((2^{nd} ‘𝑇) + 1), (2^{nd} ‘𝑇)⟩} ∪ ( I ↾
((1...𝑁) ∖
{(2^{nd} ‘𝑇),
((2^{nd} ‘𝑇)
+ 1)})))) “ (1...𝑦))
× {1}) ∪ ((((2^{nd} ‘(1^{st} ‘𝑇)) ∘
({⟨(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)⟩, ⟨((2^{nd}
‘𝑇) + 1),
(2^{nd} ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)})))) “
((𝑦 + 1)...𝑁)) ×
{0})))) |
265 | 250 | csbeq1d 3573 |
. . . . . . . 8
⊢ (𝑦 < (2^{nd}
‘𝑇) →
⦋if(𝑦 <
(2^{nd} ‘𝑇),
𝑦, (𝑦 + 1)) / 𝑗⦌((1^{st}
‘(1^{st} ‘𝑇)) ∘_{𝑓} +
((((2^{nd} ‘(1^{st} ‘𝑇)) “ (1...𝑗)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}))) = ⦋𝑦 / 𝑗⦌((1^{st}
‘(1^{st} ‘𝑇)) ∘_{𝑓} +
((((2^{nd} ‘(1^{st} ‘𝑇)) “ (1...𝑗)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0})))) |
266 | 253 | imaeq2d 5501 |
. . . . . . . . . . . 12
⊢ (𝑗 = 𝑦 → ((2^{nd}
‘(1^{st} ‘𝑇)) “ (1...𝑗)) = ((2^{nd} ‘(1^{st}
‘𝑇)) “
(1...𝑦))) |
267 | 266 | xpeq1d 5172 |
. . . . . . . . . . 11
⊢ (𝑗 = 𝑦 → (((2^{nd}
‘(1^{st} ‘𝑇)) “ (1...𝑗)) × {1}) = (((2^{nd}
‘(1^{st} ‘𝑇)) “ (1...𝑦)) × {1})) |
268 | 257 | imaeq2d 5501 |
. . . . . . . . . . . 12
⊢ (𝑗 = 𝑦 → ((2^{nd}
‘(1^{st} ‘𝑇)) “ ((𝑗 + 1)...𝑁)) = ((2^{nd} ‘(1^{st}
‘𝑇)) “ ((𝑦 + 1)...𝑁))) |
269 | 268 | xpeq1d 5172 |
. . . . . . . . . . 11
⊢ (𝑗 = 𝑦 → (((2^{nd}
‘(1^{st} ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}) = (((2^{nd}
‘(1^{st} ‘𝑇)) “ ((𝑦 + 1)...𝑁)) × {0})) |
270 | 267, 269 | uneq12d 3801 |
. . . . . . . . . 10
⊢ (𝑗 = 𝑦 → ((((2^{nd}
‘(1^{st} ‘𝑇)) “ (1...𝑗)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0})) = ((((2^{nd}
‘(1^{st} ‘𝑇)) “ (1...𝑦)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑇)) “ ((𝑦 + 1)...𝑁)) × {0}))) |
271 | 270 | oveq2d 6706 |
. . . . . . . . 9
⊢ (𝑗 = 𝑦 → ((1^{st}
‘(1^{st} ‘𝑇)) ∘_{𝑓} +
((((2^{nd} ‘(1^{st} ‘𝑇)) “ (1...𝑗)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}))) = ((1^{st}
‘(1^{st} ‘𝑇)) ∘_{𝑓} +
((((2^{nd} ‘(1^{st} ‘𝑇)) “ (1...𝑦)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑇)) “ ((𝑦 + 1)...𝑁)) × {0})))) |
272 | 252, 271 | csbie 3592 |
. . . . . . . 8
⊢
⦋𝑦 /
𝑗⦌((1^{st}
‘(1^{st} ‘𝑇)) ∘_{𝑓} +
((((2^{nd} ‘(1^{st} ‘𝑇)) “ (1...𝑗)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}))) = ((1^{st}
‘(1^{st} ‘𝑇)) ∘_{𝑓} +
((((2^{nd} ‘(1^{st} ‘𝑇)) “ (1...𝑦)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑇)) “ ((𝑦 + 1)...𝑁)) × {0}))) |
273 | 265, 272 | syl6eq 2701 |
. . . . . . 7
⊢ (𝑦 < (2^{nd}
‘𝑇) →
⦋if(𝑦 <
(2^{nd} ‘𝑇),
𝑦, (𝑦 + 1)) / 𝑗⦌((1^{st}
‘(1^{st} ‘𝑇)) ∘_{𝑓} +
((((2^{nd} ‘(1^{st} ‘𝑇)) “ (1...𝑗)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}))) = ((1^{st}
‘(1^{st} ‘𝑇)) ∘_{𝑓} +
((((2^{nd} ‘(1^{st} ‘𝑇)) “ (1...𝑦)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑇)) “ ((𝑦 + 1)...𝑁)) × {0})))) |
274 | 273 | adantl 481 |
. . . . . 6
⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑦 < (2^{nd} ‘𝑇)) →
⦋if(𝑦 <
(2^{nd} ‘𝑇),
𝑦, (𝑦 + 1)) / 𝑗⦌((1^{st}
‘(1^{st} ‘𝑇)) ∘_{𝑓} +
((((2^{nd} ‘(1^{st} ‘𝑇)) “ (1...𝑗)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}))) = ((1^{st}
‘(1^{st} ‘𝑇)) ∘_{𝑓} +
((((2^{nd} ‘(1^{st} ‘𝑇)) “ (1...𝑦)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑇)) “ ((𝑦 + 1)...𝑁)) × {0})))) |
275 | 249, 264,
274 | 3eqtr4d 2695 |
. . . . 5
⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑦 < (2^{nd} ‘𝑇)) →
⦋if(𝑦 <
(2^{nd} ‘𝑇),
𝑦, (𝑦 + 1)) / 𝑗⦌((1^{st}
‘(1^{st} ‘𝑇)) ∘_{𝑓} +
(((((2^{nd} ‘(1^{st} ‘𝑇)) ∘ ({⟨(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)⟩,
⟨((2^{nd} ‘𝑇) + 1), (2^{nd} ‘𝑇)⟩} ∪ ( I ↾
((1...𝑁) ∖
{(2^{nd} ‘𝑇),
((2^{nd} ‘𝑇)
+ 1)})))) “ (1...𝑗))
× {1}) ∪ ((((2^{nd} ‘(1^{st} ‘𝑇)) ∘
({⟨(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)⟩, ⟨((2^{nd}
‘𝑇) + 1),
(2^{nd} ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)})))) “
((𝑗 + 1)...𝑁)) × {0}))) =
⦋if(𝑦 <
(2^{nd} ‘𝑇),
𝑦, (𝑦 + 1)) / 𝑗⦌((1^{st}
‘(1^{st} ‘𝑇)) ∘_{𝑓} +
((((2^{nd} ‘(1^{st} ‘𝑇)) “ (1...𝑗)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0})))) |
276 | | lenlt 10154 |
. . . . . . . . . 10
⊢
(((2^{nd} ‘𝑇) ∈ ℝ ∧ 𝑦 ∈ ℝ) → ((2^{nd}
‘𝑇) ≤ 𝑦 ↔ ¬ 𝑦 < (2^{nd} ‘𝑇))) |
277 | 18, 116, 276 | syl2an 493 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ((2^{nd}
‘𝑇) ≤ 𝑦 ↔ ¬ 𝑦 < (2^{nd} ‘𝑇))) |
278 | 277 | biimpar 501 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ ¬ 𝑦 < (2^{nd} ‘𝑇)) → (2^{nd}
‘𝑇) ≤ 𝑦) |
279 | | imaco 5678 |
. . . . . . . . . . 11
⊢
(((2^{nd} ‘(1^{st} ‘𝑇)) ∘ ({⟨(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)⟩,
⟨((2^{nd} ‘𝑇) + 1), (2^{nd} ‘𝑇)⟩} ∪ ( I ↾
((1...𝑁) ∖
{(2^{nd} ‘𝑇),
((2^{nd} ‘𝑇)
+ 1)})))) “ (1...(𝑦 +
1))) = ((2^{nd} ‘(1^{st} ‘𝑇)) “ (({⟨(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)⟩,
⟨((2^{nd} ‘𝑇) + 1), (2^{nd} ‘𝑇)⟩} ∪ ( I ↾
((1...𝑁) ∖
{(2^{nd} ‘𝑇),
((2^{nd} ‘𝑇)
+ 1)}))) “ (1...(𝑦 +
1)))) |
280 | | imaundir 5581 |
. . . . . . . . . . . . . 14
⊢
(({⟨(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)⟩, ⟨((2^{nd}
‘𝑇) + 1),
(2^{nd} ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)}))) “
(1...(𝑦 + 1))) =
(({⟨(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)⟩, ⟨((2^{nd}
‘𝑇) + 1),
(2^{nd} ‘𝑇)⟩} “ (1...(𝑦 + 1))) ∪ (( I ↾ ((1...𝑁) ∖ {(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)})) “
(1...(𝑦 +
1)))) |
281 | | imassrn 5512 |
. . . . . . . . . . . . . . . . . 18
⊢
({⟨(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)⟩, ⟨((2^{nd}
‘𝑇) + 1),
(2^{nd} ‘𝑇)⟩} “ (1...(𝑦 + 1))) ⊆ ran {⟨(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)⟩,
⟨((2^{nd} ‘𝑇) + 1), (2^{nd} ‘𝑇)⟩} |
282 | 281 | a1i 11 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ (2^{nd}
‘𝑇) ≤ 𝑦) → ({⟨(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)⟩,
⟨((2^{nd} ‘𝑇) + 1), (2^{nd} ‘𝑇)⟩} “ (1...(𝑦 + 1))) ⊆ ran
{⟨(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)⟩, ⟨((2^{nd}
‘𝑇) + 1),
(2^{nd} ‘𝑇)⟩}) |
283 | 176 | ad2antrr 762 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ (2^{nd}
‘𝑇) ≤ 𝑦) → ({⟨(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)⟩,
⟨((2^{nd} ‘𝑇) + 1), (2^{nd} ‘𝑇)⟩} “
{(2^{nd} ‘𝑇),
((2^{nd} ‘𝑇)
+ 1)}) = ran {⟨(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)⟩, ⟨((2^{nd}
‘𝑇) + 1),
(2^{nd} ‘𝑇)⟩}) |
284 | 17 | ad2antrr 762 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ (2^{nd}
‘𝑇) ≤ 𝑦) → (2^{nd}
‘𝑇) ∈
ℕ) |
285 | 18 | ad2antrr 762 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ (2^{nd}
‘𝑇) ≤ 𝑦) → (2^{nd}
‘𝑇) ∈
ℝ) |
286 | 116 | ad2antlr 763 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ (2^{nd}
‘𝑇) ≤ 𝑦) → 𝑦 ∈ ℝ) |
287 | 186 | nnred 11073 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑦 ∈ (0...(𝑁 − 1)) → (𝑦 + 1) ∈ ℝ) |
288 | 287 | ad2antlr 763 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ (2^{nd}
‘𝑇) ≤ 𝑦) → (𝑦 + 1) ∈ ℝ) |
289 | | simpr 476 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ (2^{nd}
‘𝑇) ≤ 𝑦) → (2^{nd}
‘𝑇) ≤ 𝑦) |
290 | 116 | lep1d 10993 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑦 ∈ (0...(𝑁 − 1)) → 𝑦 ≤ (𝑦 + 1)) |
291 | 290 | ad2antlr 763 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ (2^{nd}
‘𝑇) ≤ 𝑦) → 𝑦 ≤ (𝑦 + 1)) |
292 | 285, 286,
288, 289, 291 | letrd 10232 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ (2^{nd}
‘𝑇) ≤ 𝑦) → (2^{nd}
‘𝑇) ≤ (𝑦 + 1)) |
293 | | fznn 12446 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((𝑦 + 1) ∈ ℤ →
((2^{nd} ‘𝑇)
∈ (1...(𝑦 + 1)) ↔
((2^{nd} ‘𝑇)
∈ ℕ ∧ (2^{nd} ‘𝑇) ≤ (𝑦 + 1)))) |
294 | 187, 293 | syl 17 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑦 ∈ (0...(𝑁 − 1)) → ((2^{nd}
‘𝑇) ∈
(1...(𝑦 + 1)) ↔
((2^{nd} ‘𝑇)
∈ ℕ ∧ (2^{nd} ‘𝑇) ≤ (𝑦 + 1)))) |
295 | 294 | ad2antlr 763 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ (2^{nd}
‘𝑇) ≤ 𝑦) → ((2^{nd}
‘𝑇) ∈
(1...(𝑦 + 1)) ↔
((2^{nd} ‘𝑇)
∈ ℕ ∧ (2^{nd} ‘𝑇) ≤ (𝑦 + 1)))) |
296 | 284, 292,
295 | mpbir2and 977 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ (2^{nd}
‘𝑇) ≤ 𝑦) → (2^{nd}
‘𝑇) ∈
(1...(𝑦 +
1))) |
297 | 50 | ad2antrr 762 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ (2^{nd}
‘𝑇) ≤ 𝑦) → ((2^{nd}
‘𝑇) + 1) ∈
ℕ) |
298 | | 1red 10093 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ (2^{nd}
‘𝑇) ≤ 𝑦) → 1 ∈
ℝ) |
299 | 285, 286,
298, 289 | leadd1dd 10679 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ (2^{nd}
‘𝑇) ≤ 𝑦) → ((2^{nd}
‘𝑇) + 1) ≤ (𝑦 + 1)) |
300 | | fznn 12446 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((𝑦 + 1) ∈ ℤ →
(((2^{nd} ‘𝑇)
+ 1) ∈ (1...(𝑦 + 1))
↔ (((2^{nd} ‘𝑇) + 1) ∈ ℕ ∧ ((2^{nd}
‘𝑇) + 1) ≤ (𝑦 + 1)))) |
301 | 187, 300 | syl 17 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑦 ∈ (0...(𝑁 − 1)) → (((2^{nd}
‘𝑇) + 1) ∈
(1...(𝑦 + 1)) ↔
(((2^{nd} ‘𝑇)
+ 1) ∈ ℕ ∧ ((2^{nd} ‘𝑇) + 1) ≤ (𝑦 + 1)))) |
302 | 301 | ad2antlr 763 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ (2^{nd}
‘𝑇) ≤ 𝑦) → (((2^{nd}
‘𝑇) + 1) ∈
(1...(𝑦 + 1)) ↔
(((2^{nd} ‘𝑇)
+ 1) ∈ ℕ ∧ ((2^{nd} ‘𝑇) + 1) ≤ (𝑦 + 1)))) |
303 | 297, 299,
302 | mpbir2and 977 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ (2^{nd}
‘𝑇) ≤ 𝑦) → ((2^{nd}
‘𝑇) + 1) ∈
(1...(𝑦 +
1))) |
304 | | prssi 4385 |
. . . . . . . . . . . . . . . . . . . 20
⊢
(((2^{nd} ‘𝑇) ∈ (1...(𝑦 + 1)) ∧ ((2^{nd} ‘𝑇) + 1) ∈ (1...(𝑦 + 1))) → {(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)} ⊆
(1...(𝑦 +
1))) |
305 | 296, 303,
304 | syl2anc 694 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ (2^{nd}
‘𝑇) ≤ 𝑦) → {(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)} ⊆
(1...(𝑦 +
1))) |
306 | | imass2 5536 |
. . . . . . . . . . . . . . . . . . 19
⊢
({(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)} ⊆ (1...(𝑦 + 1)) → ({⟨(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)⟩,
⟨((2^{nd} ‘𝑇) + 1), (2^{nd} ‘𝑇)⟩} “
{(2^{nd} ‘𝑇),
((2^{nd} ‘𝑇)
+ 1)}) ⊆ ({⟨(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)⟩, ⟨((2^{nd}
‘𝑇) + 1),
(2^{nd} ‘𝑇)⟩} “ (1...(𝑦 + 1)))) |
307 | 305, 306 | syl 17 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ (2^{nd}
‘𝑇) ≤ 𝑦) → ({⟨(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)⟩,
⟨((2^{nd} ‘𝑇) + 1), (2^{nd} ‘𝑇)⟩} “
{(2^{nd} ‘𝑇),
((2^{nd} ‘𝑇)
+ 1)}) ⊆ ({⟨(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)⟩, ⟨((2^{nd}
‘𝑇) + 1),
(2^{nd} ‘𝑇)⟩} “ (1...(𝑦 + 1)))) |
308 | 283, 307 | eqsstr3d 3673 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ (2^{nd}
‘𝑇) ≤ 𝑦) → ran
{⟨(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)⟩, ⟨((2^{nd}
‘𝑇) + 1),
(2^{nd} ‘𝑇)⟩} ⊆ ({⟨(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)⟩,
⟨((2^{nd} ‘𝑇) + 1), (2^{nd} ‘𝑇)⟩} “ (1...(𝑦 + 1)))) |
309 | 282, 308 | eqssd 3653 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ (2^{nd}
‘𝑇) ≤ 𝑦) → ({⟨(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)⟩,
⟨((2^{nd} ‘𝑇) + 1), (2^{nd} ‘𝑇)⟩} “ (1...(𝑦 + 1))) = ran
{⟨(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)⟩, ⟨((2^{nd}
‘𝑇) + 1),
(2^{nd} ‘𝑇)⟩}) |
310 | 215 | ad2antrr 762 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ (2^{nd}
‘𝑇) ≤ 𝑦) → ran
{⟨(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)⟩, ⟨((2^{nd}
‘𝑇) + 1),
(2^{nd} ‘𝑇)⟩} = {(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)}) |
311 | 309, 310 | eqtrd 2685 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ (2^{nd}
‘𝑇) ≤ 𝑦) → ({⟨(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)⟩,
⟨((2^{nd} ‘𝑇) + 1), (2^{nd} ‘𝑇)⟩} “ (1...(𝑦 + 1))) = {(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) +
1)}) |
312 | | undif 4082 |
. . . . . . . . . . . . . . . . . 18
⊢
({(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)} ⊆ (1...(𝑦 + 1)) ↔ ({(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)} ∪ ((1...(𝑦 + 1)) ∖ {(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)})) =
(1...(𝑦 +
1))) |
313 | 305, 312 | sylib 208 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ (2^{nd}
‘𝑇) ≤ 𝑦) → ({(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)} ∪
((1...(𝑦 + 1)) ∖
{(2^{nd} ‘𝑇),
((2^{nd} ‘𝑇)
+ 1)})) = (1...(𝑦 +
1))) |
314 | 313 | imaeq2d 5501 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ (2^{nd}
‘𝑇) ≤ 𝑦) → (( I ↾
((1...𝑁) ∖
{(2^{nd} ‘𝑇),
((2^{nd} ‘𝑇)
+ 1)})) “ ({(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)} ∪ ((1...(𝑦 + 1)) ∖ {(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)}))) = (( I ↾
((1...𝑁) ∖
{(2^{nd} ‘𝑇),
((2^{nd} ‘𝑇)
+ 1)})) “ (1...(𝑦 +
1)))) |
315 | 225 | a1i 11 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ (2^{nd}
‘𝑇) ≤ 𝑦) → (( I ↾
((1...𝑁) ∖
{(2^{nd} ‘𝑇),
((2^{nd} ‘𝑇)
+ 1)})) “ {(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)}) = ∅) |
316 | | eluzp1p1 11751 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ ((𝑁 − 1) ∈
(ℤ_{≥}‘𝑦) → ((𝑁 − 1) + 1) ∈
(ℤ_{≥}‘(𝑦 + 1))) |
317 | 150, 316 | syl 17 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝑦 ∈ (0...(𝑁 − 1)) → ((𝑁 − 1) + 1) ∈
(ℤ_{≥}‘(𝑦 + 1))) |
318 | 317 | adantl 481 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ((𝑁 − 1) + 1) ∈
(ℤ_{≥}‘(𝑦 + 1))) |
319 | 149, 318 | eqeltrrd 2731 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → 𝑁 ∈ (ℤ_{≥}‘(𝑦 + 1))) |
320 | | fzss2 12419 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑁 ∈
(ℤ_{≥}‘(𝑦 + 1)) → (1...(𝑦 + 1)) ⊆ (1...𝑁)) |
321 | 319, 320 | syl 17 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (1...(𝑦 + 1)) ⊆ (1...𝑁)) |
322 | 321 | ssdifd 3779 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ((1...(𝑦 + 1)) ∖ {(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)}) ⊆
((1...𝑁) ∖
{(2^{nd} ‘𝑇),
((2^{nd} ‘𝑇)
+ 1)})) |
323 | 322 | adantr 480 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ (2^{nd}
‘𝑇) ≤ 𝑦) → ((1...(𝑦 + 1)) ∖ {(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)}) ⊆
((1...𝑁) ∖
{(2^{nd} ‘𝑇),
((2^{nd} ‘𝑇)
+ 1)})) |
324 | | resiima 5515 |
. . . . . . . . . . . . . . . . . . 19
⊢
(((1...(𝑦 + 1))
∖ {(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)}) ⊆ ((1...𝑁) ∖ {(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)}) → (( I ↾
((1...𝑁) ∖
{(2^{nd} ‘𝑇),
((2^{nd} ‘𝑇)
+ 1)})) “ ((1...(𝑦 +
1)) ∖ {(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)})) = ((1...(𝑦 + 1)) ∖ {(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)})) |
325 | 323, 324 | syl 17 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ (2^{nd}
‘𝑇) ≤ 𝑦) → (( I ↾
((1...𝑁) ∖
{(2^{nd} ‘𝑇),
((2^{nd} ‘𝑇)
+ 1)})) “ ((1...(𝑦 +
1)) ∖ {(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)})) = ((1...(𝑦 + 1)) ∖ {(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)})) |
326 | 315, 325 | uneq12d 3801 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ (2^{nd}
‘𝑇) ≤ 𝑦) → ((( I ↾
((1...𝑁) ∖
{(2^{nd} ‘𝑇),
((2^{nd} ‘𝑇)
+ 1)})) “ {(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)}) ∪ (( I ↾ ((1...𝑁) ∖ {(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)})) “
((1...(𝑦 + 1)) ∖
{(2^{nd} ‘𝑇),
((2^{nd} ‘𝑇)
+ 1)}))) = (∅ ∪ ((1...(𝑦 + 1)) ∖ {(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)}))) |
327 | | imaundi 5580 |
. . . . . . . . . . . . . . . . 17
⊢ (( I
↾ ((1...𝑁) ∖
{(2^{nd} ‘𝑇),
((2^{nd} ‘𝑇)
+ 1)})) “ ({(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)} ∪ ((1...(𝑦 + 1)) ∖ {(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)}))) = ((( I ↾
((1...𝑁) ∖
{(2^{nd} ‘𝑇),
((2^{nd} ‘𝑇)
+ 1)})) “ {(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)}) ∪ (( I ↾ ((1...𝑁) ∖ {(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)})) “
((1...(𝑦 + 1)) ∖
{(2^{nd} ‘𝑇),
((2^{nd} ‘𝑇)
+ 1)}))) |
328 | | uncom 3790 |
. . . . . . . . . . . . . . . . . 18
⊢ (∅
∪ ((1...(𝑦 + 1))
∖ {(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)})) = (((1...(𝑦 + 1)) ∖ {(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)}) ∪
∅) |
329 | | un0 4000 |
. . . . . . . . . . . . . . . . . 18
⊢
(((1...(𝑦 + 1))
∖ {(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)}) ∪ ∅) = ((1...(𝑦 + 1)) ∖ {(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) +
1)}) |
330 | 328, 329 | eqtr2i 2674 |
. . . . . . . . . . . . . . . . 17
⊢
((1...(𝑦 + 1))
∖ {(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)}) = (∅ ∪ ((1...(𝑦 + 1)) ∖ {(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) +
1)})) |
331 | 326, 327,
330 | 3eqtr4g 2710 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ (2^{nd}
‘𝑇) ≤ 𝑦) → (( I ↾
((1...𝑁) ∖
{(2^{nd} ‘𝑇),
((2^{nd} ‘𝑇)
+ 1)})) “ ({(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)} ∪ ((1...(𝑦 + 1)) ∖ {(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)}))) = ((1...(𝑦 + 1)) ∖ {(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) +
1)})) |
332 | 314, 331 | eqtr3d 2687 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ (2^{nd}
‘𝑇) ≤ 𝑦) → (( I ↾
((1...𝑁) ∖
{(2^{nd} ‘𝑇),
((2^{nd} ‘𝑇)
+ 1)})) “ (1...(𝑦 +
1))) = ((1...(𝑦 + 1))
∖ {(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)})) |
333 | 311, 332 | uneq12d 3801 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ (2^{nd}
‘𝑇) ≤ 𝑦) →
(({⟨(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)⟩, ⟨((2^{nd}
‘𝑇) + 1),
(2^{nd} ‘𝑇)⟩} “ (1...(𝑦 + 1))) ∪ (( I ↾ ((1...𝑁) ∖ {(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)})) “
(1...(𝑦 + 1)))) =
({(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)} ∪ ((1...(𝑦 + 1)) ∖ {(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)}))) |
334 | 280, 333 | syl5eq 2697 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ (2^{nd}
‘𝑇) ≤ 𝑦) →
(({⟨(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)⟩, ⟨((2^{nd}
‘𝑇) + 1),
(2^{nd} ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)}))) “
(1...(𝑦 + 1))) =
({(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)} ∪ ((1...(𝑦 + 1)) ∖ {(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)}))) |
335 | 334, 313 | eqtrd 2685 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ (2^{nd}
‘𝑇) ≤ 𝑦) →
(({⟨(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)⟩, ⟨((2^{nd}
‘𝑇) + 1),
(2^{nd} ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)}))) “
(1...(𝑦 + 1))) =
(1...(𝑦 +
1))) |
336 | 335 | imaeq2d 5501 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ (2^{nd}
‘𝑇) ≤ 𝑦) → ((2^{nd}
‘(1^{st} ‘𝑇)) “ (({⟨(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)⟩,
⟨((2^{nd} ‘𝑇) + 1), (2^{nd} ‘𝑇)⟩} ∪ ( I ↾
((1...𝑁) ∖
{(2^{nd} ‘𝑇),
((2^{nd} ‘𝑇)
+ 1)}))) “ (1...(𝑦 +
1)))) = ((2^{nd} ‘(1^{st} ‘𝑇)) “ (1...(𝑦 + 1)))) |
337 | 279, 336 | syl5eq 2697 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ (2^{nd}
‘𝑇) ≤ 𝑦) → (((2^{nd}
‘(1^{st} ‘𝑇)) ∘ ({⟨(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)⟩,
⟨((2^{nd} ‘𝑇) + 1), (2^{nd} ‘𝑇)⟩} ∪ ( I ↾
((1...𝑁) ∖
{(2^{nd} ‘𝑇),
((2^{nd} ‘𝑇)
+ 1)})))) “ (1...(𝑦 +
1))) = ((2^{nd} ‘(1^{st} ‘𝑇)) “ (1...(𝑦 + 1)))) |
338 | 337 | xpeq1d 5172 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ (2^{nd}
‘𝑇) ≤ 𝑦) → ((((2^{nd}
‘(1^{st} ‘𝑇)) ∘ ({⟨(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)⟩,
⟨((2^{nd} ‘𝑇) + 1), (2^{nd} ‘𝑇)⟩} ∪ ( I ↾
((1...𝑁) ∖
{(2^{nd} ‘𝑇),
((2^{nd} ‘𝑇)
+ 1)})))) “ (1...(𝑦 +
1))) × {1}) = (((2^{nd} ‘(1^{st} ‘𝑇)) “ (1...(𝑦 + 1))) ×
{1})) |
339 | | imaco 5678 |
. . . . . . . . . . 11
⊢
(((2^{nd} ‘(1^{st} ‘𝑇)) ∘ ({⟨(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)⟩,
⟨((2^{nd} ‘𝑇) + 1), (2^{nd} ‘𝑇)⟩} ∪ ( I ↾
((1...𝑁) ∖
{(2^{nd} ‘𝑇),
((2^{nd} ‘𝑇)
+ 1)})))) “ (((𝑦 + 1)
+ 1)...𝑁)) =
((2^{nd} ‘(1^{st} ‘𝑇)) “ (({⟨(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)⟩,
⟨((2^{nd} ‘𝑇) + 1), (2^{nd} ‘𝑇)⟩} ∪ ( I ↾
((1...𝑁) ∖
{(2^{nd} ‘𝑇),
((2^{nd} ‘𝑇)
+ 1)}))) “ (((𝑦 + 1)
+ 1)...𝑁))) |
340 | 112 | ad2antrr 762 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ (2^{nd}
‘𝑇) ≤ 𝑦) → {⟨(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)⟩,
⟨((2^{nd} ‘𝑇) + 1), (2^{nd} ‘𝑇)⟩} Fn {(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) +
1)}) |
341 | | incom 3838 |
. . . . . . . . . . . . . . . 16
⊢
({(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)} ∩ (((𝑦 + 1) + 1)...𝑁)) = ((((𝑦 + 1) + 1)...𝑁) ∩ {(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)}) |
342 | 126 | ad2antrr 762 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ (2^{nd}
‘𝑇) ≤ 𝑦) → ((2^{nd}
‘𝑇) + 1) ∈
ℝ) |
343 | 186 | peano2nnd 11075 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝑦 ∈ (0...(𝑁 − 1)) → ((𝑦 + 1) + 1) ∈ ℕ) |
344 | 343 | nnred 11073 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑦 ∈ (0...(𝑁 − 1)) → ((𝑦 + 1) + 1) ∈ ℝ) |
345 | 344 | ad2antlr 763 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ (2^{nd}
‘𝑇) ≤ 𝑦) → ((𝑦 + 1) + 1) ∈ ℝ) |
346 | 19 | ad2antrr 762 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ (2^{nd}
‘𝑇) ≤ 𝑦) → (2^{nd}
‘𝑇) <
((2^{nd} ‘𝑇)
+ 1)) |
347 | 116 | ltp1d 10992 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (𝑦 ∈ (0...(𝑁 − 1)) → 𝑦 < (𝑦 + 1)) |
348 | 347 | ad2antlr 763 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ (2^{nd}
‘𝑇) ≤ 𝑦) → 𝑦 < (𝑦 + 1)) |
349 | 285, 286,
288, 289, 348 | lelttrd 10233 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ (2^{nd}
‘𝑇) ≤ 𝑦) → (2^{nd}
‘𝑇) < (𝑦 + 1)) |
350 | 285, 288,
298, 349 | ltadd1dd 10676 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ (2^{nd}
‘𝑇) ≤ 𝑦) → ((2^{nd}
‘𝑇) + 1) < ((𝑦 + 1) + 1)) |
351 | 285, 342,
345, 346, 350 | lttrd 10236 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ (2^{nd}
‘𝑇) ≤ 𝑦) → (2^{nd}
‘𝑇) < ((𝑦 + 1) + 1)) |
352 | | ltnle 10155 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢
(((2^{nd} ‘𝑇) ∈ ℝ ∧ ((𝑦 + 1) + 1) ∈ ℝ) →
((2^{nd} ‘𝑇)
< ((𝑦 + 1) + 1) ↔
¬ ((𝑦 + 1) + 1) ≤
(2^{nd} ‘𝑇))) |
353 | 18, 344, 352 | syl2an 493 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ((2^{nd}
‘𝑇) < ((𝑦 + 1) + 1) ↔ ¬ ((𝑦 + 1) + 1) ≤ (2^{nd}
‘𝑇))) |
354 | 353 | adantr 480 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ (2^{nd}
‘𝑇) ≤ 𝑦) → ((2^{nd}
‘𝑇) < ((𝑦 + 1) + 1) ↔ ¬ ((𝑦 + 1) + 1) ≤ (2^{nd}
‘𝑇))) |
355 | 351, 354 | mpbid 222 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ (2^{nd}
‘𝑇) ≤ 𝑦) → ¬ ((𝑦 + 1) + 1) ≤ (2^{nd}
‘𝑇)) |
356 | | elfzle1 12382 |
. . . . . . . . . . . . . . . . . . . 20
⊢
((2^{nd} ‘𝑇) ∈ (((𝑦 + 1) + 1)...𝑁) → ((𝑦 + 1) + 1) ≤ (2^{nd} ‘𝑇)) |
357 | 355, 356 | nsyl 135 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ (2^{nd}
‘𝑇) ≤ 𝑦) → ¬ (2^{nd}
‘𝑇) ∈ (((𝑦 + 1) + 1)...𝑁)) |
358 | | disjsn 4278 |
. . . . . . . . . . . . . . . . . . 19
⊢
(((((𝑦 + 1) +
1)...𝑁) ∩
{(2^{nd} ‘𝑇)}) = ∅ ↔ ¬ (2^{nd}
‘𝑇) ∈ (((𝑦 + 1) + 1)...𝑁)) |
359 | 357, 358 | sylibr 224 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ (2^{nd}
‘𝑇) ≤ 𝑦) → ((((𝑦 + 1) + 1)...𝑁) ∩ {(2^{nd} ‘𝑇)}) = ∅) |
360 | | ltnle 10155 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢
((((2^{nd} ‘𝑇) + 1) ∈ ℝ ∧ ((𝑦 + 1) + 1) ∈ ℝ)
→ (((2^{nd} ‘𝑇) + 1) < ((𝑦 + 1) + 1) ↔ ¬ ((𝑦 + 1) + 1) ≤ ((2^{nd} ‘𝑇) + 1))) |
361 | 126, 344,
360 | syl2an 493 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (((2^{nd}
‘𝑇) + 1) < ((𝑦 + 1) + 1) ↔ ¬ ((𝑦 + 1) + 1) ≤ ((2^{nd}
‘𝑇) +
1))) |
362 | 361 | adantr 480 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ (2^{nd}
‘𝑇) ≤ 𝑦) → (((2^{nd}
‘𝑇) + 1) < ((𝑦 + 1) + 1) ↔ ¬ ((𝑦 + 1) + 1) ≤ ((2^{nd}
‘𝑇) +
1))) |
363 | 350, 362 | mpbid 222 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ (2^{nd}
‘𝑇) ≤ 𝑦) → ¬ ((𝑦 + 1) + 1) ≤ ((2^{nd}
‘𝑇) +
1)) |
364 | | elfzle1 12382 |
. . . . . . . . . . . . . . . . . . . 20
⊢
(((2^{nd} ‘𝑇) + 1) ∈ (((𝑦 + 1) + 1)...𝑁) → ((𝑦 + 1) + 1) ≤ ((2^{nd} ‘𝑇) + 1)) |
365 | 363, 364 | nsyl 135 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ (2^{nd}
‘𝑇) ≤ 𝑦) → ¬ ((2^{nd}
‘𝑇) + 1) ∈
(((𝑦 + 1) + 1)...𝑁)) |
366 | | disjsn 4278 |
. . . . . . . . . . . . . . . . . . 19
⊢
(((((𝑦 + 1) +
1)...𝑁) ∩
{((2^{nd} ‘𝑇)
+ 1)}) = ∅ ↔ ¬ ((2^{nd} ‘𝑇) + 1) ∈ (((𝑦 + 1) + 1)...𝑁)) |
367 | 365, 366 | sylibr 224 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ (2^{nd}
‘𝑇) ≤ 𝑦) → ((((𝑦 + 1) + 1)...𝑁) ∩ {((2^{nd} ‘𝑇) + 1)}) =
∅) |
368 | 359, 367 | uneq12d 3801 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ (2^{nd}
‘𝑇) ≤ 𝑦) → (((((𝑦 + 1) + 1)...𝑁) ∩ {(2^{nd} ‘𝑇)}) ∪ ((((𝑦 + 1) + 1)...𝑁) ∩ {((2^{nd} ‘𝑇) + 1)})) = (∅ ∪
∅)) |
369 | 140 | ineq2i 3844 |
. . . . . . . . . . . . . . . . . 18
⊢ ((((𝑦 + 1) + 1)...𝑁) ∩ {(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)}) = ((((𝑦 + 1) + 1)...𝑁) ∩ ({(2^{nd} ‘𝑇)} ∪ {((2^{nd}
‘𝑇) +
1)})) |
370 | | indi 3906 |
. . . . . . . . . . . . . . . . . 18
⊢ ((((𝑦 + 1) + 1)...𝑁) ∩ ({(2^{nd} ‘𝑇)} ∪ {((2^{nd}
‘𝑇) + 1)})) =
(((((𝑦 + 1) + 1)...𝑁) ∩ {(2^{nd}
‘𝑇)}) ∪ ((((𝑦 + 1) + 1)...𝑁) ∩ {((2^{nd} ‘𝑇) + 1)})) |
371 | 369, 370 | eqtr2i 2674 |
. . . . . . . . . . . . . . . . 17
⊢
(((((𝑦 + 1) +
1)...𝑁) ∩
{(2^{nd} ‘𝑇)}) ∪ ((((𝑦 + 1) + 1)...𝑁) ∩ {((2^{nd} ‘𝑇) + 1)})) = ((((𝑦 + 1) + 1)...𝑁) ∩ {(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)}) |
372 | 368, 371,
144 | 3eqtr3g 2708 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ (2^{nd}
‘𝑇) ≤ 𝑦) → ((((𝑦 + 1) + 1)...𝑁) ∩ {(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)}) =
∅) |
373 | 341, 372 | syl5eq 2697 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ (2^{nd}
‘𝑇) ≤ 𝑦) → ({(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)} ∩
(((𝑦 + 1) + 1)...𝑁)) = ∅) |
374 | | fnimadisj 6050 |
. . . . . . . . . . . . . . 15
⊢
(({⟨(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)⟩, ⟨((2^{nd}
‘𝑇) + 1),
(2^{nd} ‘𝑇)⟩} Fn {(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)} ∧ ({(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)} ∩
(((𝑦 + 1) + 1)...𝑁)) = ∅) →
({⟨(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)⟩, ⟨((2^{nd}
‘𝑇) + 1),
(2^{nd} ‘𝑇)⟩} “ (((𝑦 + 1) + 1)...𝑁)) = ∅) |
375 | 340, 373,
374 | syl2anc 694 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ (2^{nd}
‘𝑇) ≤ 𝑦) → ({⟨(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)⟩,
⟨((2^{nd} ‘𝑇) + 1), (2^{nd} ‘𝑇)⟩} “ (((𝑦 + 1) + 1)...𝑁)) = ∅) |
376 | 343, 227 | syl6eleq 2740 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑦 ∈ (0...(𝑁 − 1)) → ((𝑦 + 1) + 1) ∈
(ℤ_{≥}‘1)) |
377 | | fzss1 12418 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝑦 + 1) + 1) ∈
(ℤ_{≥}‘1) → (((𝑦 + 1) + 1)...𝑁) ⊆ (1...𝑁)) |
378 | | reldisj 4053 |
. . . . . . . . . . . . . . . . . 18
⊢ ((((𝑦 + 1) + 1)...𝑁) ⊆ (1...𝑁) → (((((𝑦 + 1) + 1)...𝑁) ∩ {(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)}) = ∅ ↔
(((𝑦 + 1) + 1)...𝑁) ⊆ ((1...𝑁) ∖ {(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) +
1)}))) |
379 | 376, 377,
378 | 3syl 18 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑦 ∈ (0...(𝑁 − 1)) → (((((𝑦 + 1) + 1)...𝑁) ∩ {(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)}) = ∅ ↔
(((𝑦 + 1) + 1)...𝑁) ⊆ ((1...𝑁) ∖ {(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) +
1)}))) |
380 | 379 | ad2antlr 763 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ (2^{nd}
‘𝑇) ≤ 𝑦) → (((((𝑦 + 1) + 1)...𝑁) ∩ {(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)}) = ∅ ↔
(((𝑦 + 1) + 1)...𝑁) ⊆ ((1...𝑁) ∖ {(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) +
1)}))) |
381 | 372, 380 | mpbid 222 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ (2^{nd}
‘𝑇) ≤ 𝑦) → (((𝑦 + 1) + 1)...𝑁) ⊆ ((1...𝑁) ∖ {(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)})) |
382 | | resiima 5515 |
. . . . . . . . . . . . . . 15
⊢ ((((𝑦 + 1) + 1)...𝑁) ⊆ ((1...𝑁) ∖ {(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)}) → (( I ↾
((1...𝑁) ∖
{(2^{nd} ‘𝑇),
((2^{nd} ‘𝑇)
+ 1)})) “ (((𝑦 + 1) +
1)...𝑁)) = (((𝑦 + 1) + 1)...𝑁)) |
383 | 381, 382 | syl 17 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ (2^{nd}
‘𝑇) ≤ 𝑦) → (( I ↾
((1...𝑁) ∖
{(2^{nd} ‘𝑇),
((2^{nd} ‘𝑇)
+ 1)})) “ (((𝑦 + 1) +
1)...𝑁)) = (((𝑦 + 1) + 1)...𝑁)) |
384 | 375, 383 | uneq12d 3801 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ (2^{nd}
‘𝑇) ≤ 𝑦) →
(({⟨(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)⟩, ⟨((2^{nd}
‘𝑇) + 1),
(2^{nd} ‘𝑇)⟩} “ (((𝑦 + 1) + 1)...𝑁)) ∪ (( I ↾ ((1...𝑁) ∖ {(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)})) “
(((𝑦 + 1) + 1)...𝑁))) = (∅ ∪ (((𝑦 + 1) + 1)...𝑁))) |
385 | | imaundir 5581 |
. . . . . . . . . . . . 13
⊢
(({⟨(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)⟩, ⟨((2^{nd}
‘𝑇) + 1),
(2^{nd} ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)}))) “
(((𝑦 + 1) + 1)...𝑁)) = (({⟨(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)⟩,
⟨((2^{nd} ‘𝑇) + 1), (2^{nd} ‘𝑇)⟩} “ (((𝑦 + 1) + 1)...𝑁)) ∪ (( I ↾ ((1...𝑁) ∖ {(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)})) “
(((𝑦 + 1) + 1)...𝑁))) |
386 | | uncom 3790 |
. . . . . . . . . . . . . 14
⊢ (∅
∪ (((𝑦 + 1) +
1)...𝑁)) = ((((𝑦 + 1) + 1)...𝑁) ∪ ∅) |
387 | | un0 4000 |
. . . . . . . . . . . . . 14
⊢ ((((𝑦 + 1) + 1)...𝑁) ∪ ∅) = (((𝑦 + 1) + 1)...𝑁) |
388 | 386, 387 | eqtr2i 2674 |
. . . . . . . . . . . . 13
⊢ (((𝑦 + 1) + 1)...𝑁) = (∅ ∪ (((𝑦 + 1) + 1)...𝑁)) |
389 | 384, 385,
388 | 3eqtr4g 2710 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ (2^{nd}
‘𝑇) ≤ 𝑦) →
(({⟨(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)⟩, ⟨((2^{nd}
‘𝑇) + 1),
(2^{nd} ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)}))) “
(((𝑦 + 1) + 1)...𝑁)) = (((𝑦 + 1) + 1)...𝑁)) |
390 | 389 | imaeq2d 5501 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ (2^{nd}
‘𝑇) ≤ 𝑦) → ((2^{nd}
‘(1^{st} ‘𝑇)) “ (({⟨(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)⟩,
⟨((2^{nd} ‘𝑇) + 1), (2^{nd} ‘𝑇)⟩} ∪ ( I ↾
((1...𝑁) ∖
{(2^{nd} ‘𝑇),
((2^{nd} ‘𝑇)
+ 1)}))) “ (((𝑦 + 1)
+ 1)...𝑁))) =
((2^{nd} ‘(1^{st} ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁))) |
391 | 339, 390 | syl5eq 2697 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ (2^{nd}
‘𝑇) ≤ 𝑦) → (((2^{nd}
‘(1^{st} ‘𝑇)) ∘ ({⟨(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)⟩,
⟨((2^{nd} ‘𝑇) + 1), (2^{nd} ‘𝑇)⟩} ∪ ( I ↾
((1...𝑁) ∖
{(2^{nd} ‘𝑇),
((2^{nd} ‘𝑇)
+ 1)})))) “ (((𝑦 + 1)
+ 1)...𝑁)) =
((2^{nd} ‘(1^{st} ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁))) |
392 | 391 | xpeq1d 5172 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ (2^{nd}
‘𝑇) ≤ 𝑦) → ((((2^{nd}
‘(1^{st} ‘𝑇)) ∘ ({⟨(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)⟩,
⟨((2^{nd} ‘𝑇) + 1), (2^{nd} ‘𝑇)⟩} ∪ ( I ↾
((1...𝑁) ∖
{(2^{nd} ‘𝑇),
((2^{nd} ‘𝑇)
+ 1)})))) “ (((𝑦 + 1)
+ 1)...𝑁)) × {0}) =
(((2^{nd} ‘(1^{st} ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0})) |
393 | 338, 392 | uneq12d 3801 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ (2^{nd}
‘𝑇) ≤ 𝑦) → (((((2^{nd}
‘(1^{st} ‘𝑇)) ∘ ({⟨(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)⟩,
⟨((2^{nd} ‘𝑇) + 1), (2^{nd} ‘𝑇)⟩} ∪ ( I ↾
((1...𝑁) ∖
{(2^{nd} ‘𝑇),
((2^{nd} ‘𝑇)
+ 1)})))) “ (1...(𝑦 +
1))) × {1}) ∪ ((((2^{nd} ‘(1^{st} ‘𝑇)) ∘
({⟨(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)⟩, ⟨((2^{nd}
‘𝑇) + 1),
(2^{nd} ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)})))) “
(((𝑦 + 1) + 1)...𝑁)) × {0})) =
((((2^{nd} ‘(1^{st} ‘𝑇)) “ (1...(𝑦 + 1))) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0}))) |
394 | 278, 393 | syldan 486 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ ¬ 𝑦 < (2^{nd} ‘𝑇)) → (((((2^{nd}
‘(1^{st} ‘𝑇)) ∘ ({⟨(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)⟩,
⟨((2^{nd} ‘𝑇) + 1), (2^{nd} ‘𝑇)⟩} ∪ ( I ↾
((1...𝑁) ∖
{(2^{nd} ‘𝑇),
((2^{nd} ‘𝑇)
+ 1)})))) “ (1...(𝑦 +
1))) × {1}) ∪ ((((2^{nd} ‘(1^{st} ‘𝑇)) ∘
({⟨(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)⟩, ⟨((2^{nd}
‘𝑇) + 1),
(2^{nd} ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)})))) “
(((𝑦 + 1) + 1)...𝑁)) × {0})) =
((((2^{nd} ‘(1^{st} ‘𝑇)) “ (1...(𝑦 + 1))) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0}))) |
395 | 394 | oveq2d 6706 |
. . . . . 6
⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ ¬ 𝑦 < (2^{nd} ‘𝑇)) → ((1^{st}
‘(1^{st} ‘𝑇)) ∘_{𝑓} +
(((((2^{nd} ‘(1^{st} ‘𝑇)) ∘ ({⟨(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)⟩,
⟨((2^{nd} ‘𝑇) + 1), (2^{nd} ‘𝑇)⟩} ∪ ( I ↾
((1...𝑁) ∖
{(2^{nd} ‘𝑇),
((2^{nd} ‘𝑇)
+ 1)})))) “ (1...(𝑦 +
1))) × {1}) ∪ ((((2^{nd} ‘(1^{st} ‘𝑇)) ∘
({⟨(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)⟩, ⟨((2^{nd}
‘𝑇) + 1),
(2^{nd} ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)})))) “
(((𝑦 + 1) + 1)...𝑁)) × {0}))) =
((1^{st} ‘(1^{st} ‘𝑇)) ∘_{𝑓} +
((((2^{nd} ‘(1^{st} ‘𝑇)) “ (1...(𝑦 + 1))) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0})))) |
396 | | iffalse 4128 |
. . . . . . . . 9
⊢ (¬
𝑦 < (2^{nd}
‘𝑇) → if(𝑦 < (2^{nd}
‘𝑇), 𝑦, (𝑦 + 1)) = (𝑦 + 1)) |
397 | 396 | csbeq1d 3573 |
. . . . . . . 8
⊢ (¬
𝑦 < (2^{nd}
‘𝑇) →
⦋if(𝑦 <
(2^{nd} ‘𝑇),
𝑦, (𝑦 + 1)) / 𝑗⦌((1^{st}
‘(1^{st} ‘𝑇)) ∘_{𝑓} +
(((((2^{nd} ‘(1^{st} ‘𝑇)) ∘ ({⟨(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)⟩,
⟨((2^{nd} ‘𝑇) + 1), (2^{nd} ‘𝑇)⟩} ∪ ( I ↾
((1...𝑁) ∖
{(2^{nd} ‘𝑇),
((2^{nd} ‘𝑇)
+ 1)})))) “ (1...𝑗))
× {1}) ∪ ((((2^{nd} ‘(1^{st} ‘𝑇)) ∘
({⟨(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)⟩, ⟨((2^{nd}
‘𝑇) + 1),
(2^{nd} ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)})))) “
((𝑗 + 1)...𝑁)) × {0}))) =
⦋(𝑦 + 1) /
𝑗⦌((1^{st}
‘(1^{st} ‘𝑇)) ∘_{𝑓} +
(((((2^{nd} ‘(1^{st} ‘𝑇)) ∘ ({⟨(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)⟩,
⟨((2^{nd} ‘𝑇) + 1), (2^{nd} ‘𝑇)⟩} ∪ ( I ↾
((1...𝑁) ∖
{(2^{nd} ‘𝑇),
((2^{nd} ‘𝑇)
+ 1)})))) “ (1...𝑗))
× {1}) ∪ ((((2^{nd} ‘(1^{st} ‘𝑇)) ∘
({⟨(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)⟩, ⟨((2^{nd}
‘𝑇) + 1),
(2^{nd} ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)})))) “
((𝑗 + 1)...𝑁)) ×
{0})))) |
398 | | ovex 6718 |
. . . . . . . . 9
⊢ (𝑦 + 1) ∈ V |
399 | | oveq2 6698 |
. . . . . . . . . . . . 13
⊢ (𝑗 = (𝑦 + 1) → (1...𝑗) = (1...(𝑦 + 1))) |
400 | 399 | imaeq2d 5501 |
. . . . . . . . . . . 12
⊢ (𝑗 = (𝑦 + 1) → (((2^{nd}
‘(1^{st} ‘𝑇)) ∘ ({⟨(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)⟩,
⟨((2^{nd} ‘𝑇) + 1), (2^{nd} ‘𝑇)⟩} ∪ ( I ↾
((1...𝑁) ∖
{(2^{nd} ‘𝑇),
((2^{nd} ‘𝑇)
+ 1)})))) “ (1...𝑗))
= (((2^{nd} ‘(1^{st} ‘𝑇)) ∘ ({⟨(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)⟩,
⟨((2^{nd} ‘𝑇) + 1), (2^{nd} ‘𝑇)⟩} ∪ ( I ↾
((1...𝑁) ∖
{(2^{nd} ‘𝑇),
((2^{nd} ‘𝑇)
+ 1)})))) “ (1...(𝑦 +
1)))) |
401 | 400 | xpeq1d 5172 |
. . . . . . . . . . 11
⊢ (𝑗 = (𝑦 + 1) → ((((2^{nd}
‘(1^{st} ‘𝑇)) ∘ ({⟨(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)⟩,
⟨((2^{nd} ‘𝑇) + 1), (2^{nd} ‘𝑇)⟩} ∪ ( I ↾
((1...𝑁) ∖
{(2^{nd} ‘𝑇),
((2^{nd} ‘𝑇)
+ 1)})))) “ (1...𝑗))
× {1}) = ((((2^{nd} ‘(1^{st} ‘𝑇)) ∘
({⟨(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)⟩, ⟨((2^{nd}
‘𝑇) + 1),
(2^{nd} ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)})))) “
(1...(𝑦 + 1))) ×
{1})) |
402 | | oveq1 6697 |
. . . . . . . . . . . . . 14
⊢ (𝑗 = (𝑦 + 1) → (𝑗 + 1) = ((𝑦 + 1) + 1)) |
403 | 402 | oveq1d 6705 |
. . . . . . . . . . . . 13
⊢ (𝑗 = (𝑦 + 1) → ((𝑗 + 1)...𝑁) = (((𝑦 + 1) + 1)...𝑁)) |
404 | 403 | imaeq2d 5501 |
. . . . . . . . . . . 12
⊢ (𝑗 = (𝑦 + 1) → (((2^{nd}
‘(1^{st} ‘𝑇)) ∘ ({⟨(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)⟩,
⟨((2^{nd} ‘𝑇) + 1), (2^{nd} ‘𝑇)⟩} ∪ ( I ↾
((1...𝑁) ∖
{(2^{nd} ‘𝑇),
((2^{nd} ‘𝑇)
+ 1)})))) “ ((𝑗 +
1)...𝑁)) =
(((2^{nd} ‘(1^{st} ‘𝑇)) ∘ ({⟨(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)⟩,
⟨((2^{nd} ‘𝑇) + 1), (2^{nd} ‘𝑇)⟩} ∪ ( I ↾
((1...𝑁) ∖
{(2^{nd} ‘𝑇),
((2^{nd} ‘𝑇)
+ 1)})))) “ (((𝑦 + 1)
+ 1)...𝑁))) |
405 | 404 | xpeq1d 5172 |
. . . . . . . . . . 11
⊢ (𝑗 = (𝑦 + 1) → ((((2^{nd}
‘(1^{st} ‘𝑇)) ∘ ({⟨(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)⟩,
⟨((2^{nd} ‘𝑇) + 1), (2^{nd} ‘𝑇)⟩} ∪ ( I ↾
((1...𝑁) ∖
{(2^{nd} ‘𝑇),
((2^{nd} ‘𝑇)
+ 1)})))) “ ((𝑗 +
1)...𝑁)) × {0}) =
((((2^{nd} ‘(1^{st} ‘𝑇)) ∘ ({⟨(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)⟩,
⟨((2^{nd} ‘𝑇) + 1), (2^{nd} ‘𝑇)⟩} ∪ ( I ↾
((1...𝑁) ∖
{(2^{nd} ‘𝑇),
((2^{nd} ‘𝑇)
+ 1)})))) “ (((𝑦 + 1)
+ 1)...𝑁)) ×
{0})) |
406 | 401, 405 | uneq12d 3801 |
. . . . . . . . . 10
⊢ (𝑗 = (𝑦 + 1) → (((((2^{nd}
‘(1^{st} ‘𝑇)) ∘ ({⟨(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)⟩,
⟨((2^{nd} ‘𝑇) + 1), (2^{nd} ‘𝑇)⟩} ∪ ( I ↾
((1...𝑁) ∖
{(2^{nd} ‘𝑇),
((2^{nd} ‘𝑇)
+ 1)})))) “ (1...𝑗))
× {1}) ∪ ((((2^{nd} ‘(1^{st} ‘𝑇)) ∘
({⟨(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)⟩, ⟨((2^{nd}
‘𝑇) + 1),
(2^{nd} ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)})))) “
((𝑗 + 1)...𝑁)) × {0})) =
(((((2^{nd} ‘(1^{st} ‘𝑇)) ∘ ({⟨(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)⟩,
⟨((2^{nd} ‘𝑇) + 1), (2^{nd} ‘𝑇)⟩} ∪ ( I ↾
((1...𝑁) ∖
{(2^{nd} ‘𝑇),
((2^{nd} ‘𝑇)
+ 1)})))) “ (1...(𝑦 +
1))) × {1}) ∪ ((((2^{nd} ‘(1^{st} ‘𝑇)) ∘
({⟨(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)⟩, ⟨((2^{nd}
‘𝑇) + 1),
(2^{nd} ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)})))) “
(((𝑦 + 1) + 1)...𝑁)) ×
{0}))) |
407 | 406 | oveq2d 6706 |
. . . . . . . . 9
⊢ (𝑗 = (𝑦 + 1) → ((1^{st}
‘(1^{st} ‘𝑇)) ∘_{𝑓} +
(((((2^{nd} ‘(1^{st} ‘𝑇)) ∘ ({⟨(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)⟩,
⟨((2^{nd} ‘𝑇) + 1), (2^{nd} ‘𝑇)⟩} ∪ ( I ↾
((1...𝑁) ∖
{(2^{nd} ‘𝑇),
((2^{nd} ‘𝑇)
+ 1)})))) “ (1...𝑗))
× {1}) ∪ ((((2^{nd} ‘(1^{st} ‘𝑇)) ∘
({⟨(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)⟩, ⟨((2^{nd}
‘𝑇) + 1),
(2^{nd} ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)})))) “
((𝑗 + 1)...𝑁)) × {0}))) =
((1^{st} ‘(1^{st} ‘𝑇)) ∘_{𝑓} +
(((((2^{nd} ‘(1^{st} ‘𝑇)) ∘ ({⟨(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)⟩,
⟨((2^{nd} ‘𝑇) + 1), (2^{nd} ‘𝑇)⟩} ∪ ( I ↾
((1...𝑁) ∖
{(2^{nd} ‘𝑇),
((2^{nd} ‘𝑇)
+ 1)})))) “ (1...(𝑦 +
1))) × {1}) ∪ ((((2^{nd} ‘(1^{st} ‘𝑇)) ∘
({⟨(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)⟩, ⟨((2^{nd}
‘𝑇) + 1),
(2^{nd} ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)})))) “
(((𝑦 + 1) + 1)...𝑁)) ×
{0})))) |
408 | 398, 407 | csbie 3592 |
. . . . . . . 8
⊢
⦋(𝑦 +
1) / 𝑗⦌((1^{st}
‘(1^{st} ‘𝑇)) ∘_{𝑓} +
(((((2^{nd} ‘(1^{st} ‘𝑇)) ∘ ({⟨(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)⟩,
⟨((2^{nd} ‘𝑇) + 1), (2^{nd} ‘𝑇)⟩} ∪ ( I ↾
((1...𝑁) ∖
{(2^{nd} ‘𝑇),
((2^{nd} ‘𝑇)
+ 1)})))) “ (1...𝑗))
× {1}) ∪ ((((2^{nd} ‘(1^{st} ‘𝑇)) ∘
({⟨(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)⟩, ⟨((2^{nd}
‘𝑇) + 1),
(2^{nd} ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)})))) “
((𝑗 + 1)...𝑁)) × {0}))) =
((1^{st} ‘(1^{st} ‘𝑇)) ∘_{𝑓} +
(((((2^{nd} ‘(1^{st} ‘𝑇)) ∘ ({⟨(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)⟩,
⟨((2^{nd} ‘𝑇) + 1), (2^{nd} ‘𝑇)⟩} ∪ ( I ↾
((1...𝑁) ∖
{(2^{nd} ‘𝑇),
((2^{nd} ‘𝑇)
+ 1)})))) “ (1...(𝑦 +
1))) × {1}) ∪ ((((2^{nd} ‘(1^{st} ‘𝑇)) ∘
({⟨(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)⟩, ⟨((2^{nd}
‘𝑇) + 1),
(2^{nd} ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)})))) “
(((𝑦 + 1) + 1)...𝑁)) ×
{0}))) |
409 | 397, 408 | syl6eq 2701 |
. . . . . . 7
⊢ (¬
𝑦 < (2^{nd}
‘𝑇) →
⦋if(𝑦 <
(2^{nd} ‘𝑇),
𝑦, (𝑦 + 1)) / 𝑗⦌((1^{st}
‘(1^{st} ‘𝑇)) ∘_{𝑓} +
(((((2^{nd} ‘(1^{st} ‘𝑇)) ∘ ({⟨(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)⟩,
⟨((2^{nd} ‘𝑇) + 1), (2^{nd} ‘𝑇)⟩} ∪ ( I ↾
((1...𝑁) ∖
{(2^{nd} ‘𝑇),
((2^{nd} ‘𝑇)
+ 1)})))) “ (1...𝑗))
× {1}) ∪ ((((2^{nd} ‘(1^{st} ‘𝑇)) ∘
({⟨(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)⟩, ⟨((2^{nd}
‘𝑇) + 1),
(2^{nd} ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)})))) “
((𝑗 + 1)...𝑁)) × {0}))) =
((1^{st} ‘(1^{st} ‘𝑇)) ∘_{𝑓} +
(((((2^{nd} ‘(1^{st} ‘𝑇)) ∘ ({⟨(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)⟩,
⟨((2^{nd} ‘𝑇) + 1), (2^{nd} ‘𝑇)⟩} ∪ ( I ↾
((1...𝑁) ∖
{(2^{nd} ‘𝑇),
((2^{nd} ‘𝑇)
+ 1)})))) “ (1...(𝑦 +
1))) × {1}) ∪ ((((2^{nd} ‘(1^{st} ‘𝑇)) ∘
({⟨(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)⟩, ⟨((2^{nd}
‘𝑇) + 1),
(2^{nd} ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)})))) “
(((𝑦 + 1) + 1)...𝑁)) ×
{0})))) |
410 | 409 | adantl 481 |
. . . . . 6
⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ ¬ 𝑦 < (2^{nd} ‘𝑇)) →
⦋if(𝑦 <
(2^{nd} ‘𝑇),
𝑦, (𝑦 + 1)) / 𝑗⦌((1^{st}
‘(1^{st} ‘𝑇)) ∘_{𝑓} +
(((((2^{nd} ‘(1^{st} ‘𝑇)) ∘ ({⟨(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)⟩,
⟨((2^{nd} ‘𝑇) + 1), (2^{nd} ‘𝑇)⟩} ∪ ( I ↾
((1...𝑁) ∖
{(2^{nd} ‘𝑇),
((2^{nd} ‘𝑇)
+ 1)})))) “ (1...𝑗))
× {1}) ∪ ((((2^{nd} ‘(1^{st} ‘𝑇)) ∘
({⟨(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)⟩, ⟨((2^{nd}
‘𝑇) + 1),
(2^{nd} ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)})))) “
((𝑗 + 1)...𝑁)) × {0}))) =
((1^{st} ‘(1^{st} ‘𝑇)) ∘_{𝑓} +
(((((2^{nd} ‘(1^{st} ‘𝑇)) ∘ ({⟨(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)⟩,
⟨((2^{nd} ‘𝑇) + 1), (2^{nd} ‘𝑇)⟩} ∪ ( I ↾
((1...𝑁) ∖
{(2^{nd} ‘𝑇),
((2^{nd} ‘𝑇)
+ 1)})))) “ (1...(𝑦 +
1))) × {1}) ∪ ((((2^{nd} ‘(1^{st} ‘𝑇)) ∘
({⟨(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)⟩, ⟨((2^{nd}
‘𝑇) + 1),
(2^{nd} ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)})))) “
(((𝑦 + 1) + 1)...𝑁)) ×
{0})))) |
411 | 396 | csbeq1d 3573 |
. . . . . . . 8
⊢ (¬
𝑦 < (2^{nd}
‘𝑇) →
⦋if(𝑦 <
(2^{nd} ‘𝑇),
𝑦, (𝑦 + 1)) / 𝑗⦌((1^{st}
‘(1^{st} ‘𝑇)) ∘_{𝑓} +
((((2^{nd} ‘(1^{st} ‘𝑇)) “ (1...𝑗)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}))) = ⦋(𝑦 + 1) / 𝑗⦌((1^{st}
‘(1^{st} ‘𝑇)) ∘_{𝑓} +
((((2^{nd} ‘(1^{st} ‘𝑇)) “ (1...𝑗)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0})))) |
412 | 399 | imaeq2d 5501 |
. . . . . . . . . . . 12
⊢ (𝑗 = (𝑦 + 1) → ((2^{nd}
‘(1^{st} ‘𝑇)) “ (1...𝑗)) = ((2^{nd} ‘(1^{st}
‘𝑇)) “
(1...(𝑦 +
1)))) |
413 | 412 | xpeq1d 5172 |
. . . . . . . . . . 11
⊢ (𝑗 = (𝑦 + 1) → (((2^{nd}
‘(1^{st} ‘𝑇)) “ (1...𝑗)) × {1}) = (((2^{nd}
‘(1^{st} ‘𝑇)) “ (1...(𝑦 + 1))) × {1})) |
414 | 403 | imaeq2d 5501 |
. . . . . . . . . . . 12
⊢ (𝑗 = (𝑦 + 1) → ((2^{nd}
‘(1^{st} ‘𝑇)) “ ((𝑗 + 1)...𝑁)) = ((2^{nd} ‘(1^{st}
‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁))) |
415 | 414 | xpeq1d 5172 |
. . . . . . . . . . 11
⊢ (𝑗 = (𝑦 + 1) → (((2^{nd}
‘(1^{st} ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}) = (((2^{nd}
‘(1^{st} ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0})) |
416 | 413, 415 | uneq12d 3801 |
. . . . . . . . . 10
⊢ (𝑗 = (𝑦 + 1) → ((((2^{nd}
‘(1^{st} ‘𝑇)) “ (1...𝑗)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0})) = ((((2^{nd}
‘(1^{st} ‘𝑇)) “ (1...(𝑦 + 1))) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0}))) |
417 | 416 | oveq2d 6706 |
. . . . . . . . 9
⊢ (𝑗 = (𝑦 + 1) → ((1^{st}
‘(1^{st} ‘𝑇)) ∘_{𝑓} +
((((2^{nd} ‘(1^{st} ‘𝑇)) “ (1...𝑗)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}))) = ((1^{st}
‘(1^{st} ‘𝑇)) ∘_{𝑓} +
((((2^{nd} ‘(1^{st} ‘𝑇)) “ (1...(𝑦 + 1))) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0})))) |
418 | 398, 417 | csbie 3592 |
. . . . . . . 8
⊢
⦋(𝑦 +
1) / 𝑗⦌((1^{st}
‘(1^{st} ‘𝑇)) ∘_{𝑓} +
((((2^{nd} ‘(1^{st} ‘𝑇)) “ (1...𝑗)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}))) = ((1^{st}
‘(1^{st} ‘𝑇)) ∘_{𝑓} +
((((2^{nd} ‘(1^{st} ‘𝑇)) “ (1...(𝑦 + 1))) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0}))) |
419 | 411, 418 | syl6eq 2701 |
. . . . . . 7
⊢ (¬
𝑦 < (2^{nd}
‘𝑇) →
⦋if(𝑦 <
(2^{nd} ‘𝑇),
𝑦, (𝑦 + 1)) / 𝑗⦌((1^{st}
‘(1^{st} ‘𝑇)) ∘_{𝑓} +
((((2^{nd} ‘(1^{st} ‘𝑇)) “ (1...𝑗)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}))) = ((1^{st}
‘(1^{st} ‘𝑇)) ∘_{𝑓} +
((((2^{nd} ‘(1^{st} ‘𝑇)) “ (1...(𝑦 + 1))) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0})))) |
420 | 419 | adantl 481 |
. . . . . 6
⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ ¬ 𝑦 < (2^{nd} ‘𝑇)) →
⦋if(𝑦 <
(2^{nd} ‘𝑇),
𝑦, (𝑦 + 1)) / 𝑗⦌((1^{st}
‘(1^{st} ‘𝑇)) ∘_{𝑓} +
((((2^{nd} ‘(1^{st} ‘𝑇)) “ (1...𝑗)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}))) = ((1^{st}
‘(1^{st} ‘𝑇)) ∘_{𝑓} +
((((2^{nd} ‘(1^{st} ‘𝑇)) “ (1...(𝑦 + 1))) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0})))) |
421 | 395, 410,
420 | 3eqtr4d 2695 |
. . . . 5
⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ ¬ 𝑦 < (2^{nd} ‘𝑇)) →
⦋if(𝑦 <
(2^{nd} ‘𝑇),
𝑦, (𝑦 + 1)) / 𝑗⦌((1^{st}
‘(1^{st} ‘𝑇)) ∘_{𝑓} +
(((((2^{nd} ‘(1^{st} ‘𝑇)) ∘ ({⟨(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)⟩,
⟨((2^{nd} ‘𝑇) + 1), (2^{nd} ‘𝑇)⟩} ∪ ( I ↾
((1...𝑁) ∖
{(2^{nd} ‘𝑇),
((2^{nd} ‘𝑇)
+ 1)})))) “ (1...𝑗))
× {1}) ∪ ((((2^{nd} ‘(1^{st} ‘𝑇)) ∘
({⟨(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)⟩, ⟨((2^{nd}
‘𝑇) + 1),
(2^{nd} ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)})))) “
((𝑗 + 1)...𝑁)) × {0}))) =
⦋if(𝑦 <
(2^{nd} ‘𝑇),
𝑦, (𝑦 + 1)) / 𝑗⦌((1^{st}
‘(1^{st} ‘𝑇)) ∘_{𝑓} +
((((2^{nd} ‘(1^{st} ‘𝑇)) “ (1...𝑗)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0})))) |
422 | 275, 421 | pm2.61dan 849 |
. . . 4
⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ⦋if(𝑦 < (2^{nd}
‘𝑇), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^{st}
‘(1^{st} ‘𝑇)) ∘_{𝑓} +
(((((2^{nd} ‘(1^{st} ‘𝑇)) ∘ ({⟨(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)⟩,
⟨((2^{nd} ‘𝑇) + 1), (2^{nd} ‘𝑇)⟩} ∪ ( I ↾
((1...𝑁) ∖
{(2^{nd} ‘𝑇),
((2^{nd} ‘𝑇)
+ 1)})))) “ (1...𝑗))
× {1}) ∪ ((((2^{nd} ‘(1^{st} ‘𝑇)) ∘
({⟨(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)⟩, ⟨((2^{nd}
‘𝑇) + 1),
(2^{nd} ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)})))) “
((𝑗 + 1)...𝑁)) × {0}))) =
⦋if(𝑦 <
(2^{nd} ‘𝑇),
𝑦, (𝑦 + 1)) / 𝑗⦌((1^{st}
‘(1^{st} ‘𝑇)) ∘_{𝑓} +
((((2^{nd} ‘(1^{st} ‘𝑇)) “ (1...𝑗)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0})))) |
423 | 422 | mpteq2dva 4777 |
. . 3
⊢ (𝜑 → (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2^{nd}
‘𝑇), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^{st}
‘(1^{st} ‘𝑇)) ∘_{𝑓} +
(((((2^{nd} ‘(1^{st} ‘𝑇)) ∘ ({⟨(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)⟩,
⟨((2^{nd} ‘𝑇) + 1), (2^{nd} ‘𝑇)⟩} ∪ ( I ↾
((1...𝑁) ∖
{(2^{nd} ‘𝑇),
((2^{nd} ‘𝑇)
+ 1)})))) “ (1...𝑗))
× {1}) ∪ ((((2^{nd} ‘(1^{st} ‘𝑇)) ∘
({⟨(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)⟩, ⟨((2^{nd}
‘𝑇) + 1),
(2^{nd} ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)})))) “
((𝑗 + 1)...𝑁)) × {0})))) = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2^{nd}
‘𝑇), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^{st}
‘(1^{st} ‘𝑇)) ∘_{𝑓} +
((((2^{nd} ‘(1^{st} ‘𝑇)) “ (1...𝑗)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}))))) |
424 | 109, 423 | eqtr4d 2688 |
. 2
⊢ (𝜑 → 𝐹 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2^{nd}
‘𝑇), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^{st}
‘(1^{st} ‘𝑇)) ∘_{𝑓} +
(((((2^{nd} ‘(1^{st} ‘𝑇)) ∘ ({⟨(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)⟩,
⟨((2^{nd} ‘𝑇) + 1), (2^{nd} ‘𝑇)⟩} ∪ ( I ↾
((1...𝑁) ∖
{(2^{nd} ‘𝑇),
((2^{nd} ‘𝑇)
+ 1)})))) “ (1...𝑗))
× {1}) ∪ ((((2^{nd} ‘(1^{st} ‘𝑇)) ∘
({⟨(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)⟩, ⟨((2^{nd}
‘𝑇) + 1),
(2^{nd} ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)})))) “
((𝑗 + 1)...𝑁)) ×
{0}))))) |
425 | | opex 4962 |
. . . . . . 7
⊢
⟨(1^{st} ‘(1^{st} ‘𝑇)), ((2^{nd} ‘(1^{st}
‘𝑇)) ∘
({⟨(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)⟩, ⟨((2^{nd}
‘𝑇) + 1),
(2^{nd} ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)}))))⟩
∈ V |
426 | 425, 22 | op1std 7220 |
. . . . . 6
⊢ (𝑡 = ⟨⟨(1^{st}
‘(1^{st} ‘𝑇)), ((2^{nd} ‘(1^{st}
‘𝑇)) ∘
({⟨(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)⟩, ⟨((2^{nd}
‘𝑇) + 1),
(2^{nd} ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)}))))⟩,
(2^{nd} ‘𝑇)⟩ → (1^{st} ‘𝑡) = ⟨(1^{st}
‘(1^{st} ‘𝑇)), ((2^{nd} ‘(1^{st}
‘𝑇)) ∘
({⟨(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)⟩, ⟨((2^{nd}
‘𝑇) + 1),
(2^{nd} ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) +
1)}))))⟩) |
427 | 425, 22 | op2ndd 7221 |
. . . . . 6
⊢ (𝑡 = ⟨⟨(1^{st}
‘(1^{st} ‘𝑇)), ((2^{nd} ‘(1^{st}
‘𝑇)) ∘
({⟨(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)⟩, ⟨((2^{nd}
‘𝑇) + 1),
(2^{nd} ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)}))))⟩,
(2^{nd} ‘𝑇)⟩ → (2^{nd} ‘𝑡) = (2^{nd} ‘𝑇)) |
428 | | breq2 4689 |
. . . . . . . . 9
⊢
((2^{nd} ‘𝑡) = (2^{nd} ‘𝑇) → (𝑦 < (2^{nd} ‘𝑡) ↔ 𝑦 < (2^{nd} ‘𝑇))) |
429 | 428 | ifbid 4141 |
. . . . . . . 8
⊢
((2^{nd} ‘𝑡) = (2^{nd} ‘𝑇) → if(𝑦 < (2^{nd} ‘𝑡), 𝑦, (𝑦 + 1)) = if(𝑦 < (2^{nd} ‘𝑇), 𝑦, (𝑦 + 1))) |
430 | 429 | csbeq1d 3573 |
. . . . . . 7
⊢
((2^{nd} ‘𝑡) = (2^{nd} ‘𝑇) → ⦋if(𝑦 < (2^{nd}
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^{st}
‘(1^{st} ‘𝑡)) ∘_{𝑓} +
((((2^{nd} ‘(1^{st} ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))) = ⦋if(𝑦 < (2^{nd}
‘𝑇), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^{st}
‘(1^{st} ‘𝑡)) ∘_{𝑓} +
((((2^{nd} ‘(1^{st} ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) |
431 | | fvex 6239 |
. . . . . . . . . 10
⊢
(1^{st} ‘(1^{st} ‘𝑇)) ∈ V |
432 | 431, 79 | op1std 7220 |
. . . . . . . . 9
⊢
((1^{st} ‘𝑡) = ⟨(1^{st}
‘(1^{st} ‘𝑇)), ((2^{nd} ‘(1^{st}
‘𝑇)) ∘
({⟨(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)⟩, ⟨((2^{nd}
‘𝑇) + 1),
(2^{nd} ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)}))))⟩
→ (1^{st} ‘(1^{st} ‘𝑡)) = (1^{st} ‘(1^{st}
‘𝑇))) |
433 | 431, 79 | op2ndd 7221 |
. . . . . . . . 9
⊢
((1^{st} ‘𝑡) = ⟨(1^{st}
‘(1^{st} ‘𝑇)), ((2^{nd} ‘(1^{st}
‘𝑇)) ∘
({⟨(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)⟩, ⟨((2^{nd}
‘𝑇) + 1),
(2^{nd} ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)}))))⟩
→ (2^{nd} ‘(1^{st} ‘𝑡)) = ((2^{nd} ‘(1^{st}
‘𝑇)) ∘
({⟨(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)⟩, ⟨((2^{nd}
‘𝑇) + 1),
(2^{nd} ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) +
1)}))))) |
434 | | id 22 |
. . . . . . . . . 10
⊢
((1^{st} ‘(1^{st} ‘𝑡)) = (1^{st} ‘(1^{st}
‘𝑇)) →
(1^{st} ‘(1^{st} ‘𝑡)) = (1^{st} ‘(1^{st}
‘𝑇))) |
435 | | imaeq1 5496 |
. . . . . . . . . . . 12
⊢
((2^{nd} ‘(1^{st} ‘𝑡)) = ((2^{nd} ‘(1^{st}
‘𝑇)) ∘
({⟨(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)⟩, ⟨((2^{nd}
‘𝑇) + 1),
(2^{nd} ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)})))) →
((2^{nd} ‘(1^{st} ‘𝑡)) “ (1...𝑗)) = (((2^{nd} ‘(1^{st}
‘𝑇)) ∘
({⟨(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)⟩, ⟨((2^{nd}
‘𝑇) + 1),
(2^{nd} ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)})))) “
(1...𝑗))) |
436 | 435 | xpeq1d 5172 |
. . . . . . . . . . 11
⊢
((2^{nd} ‘(1^{st} ‘𝑡)) = ((2^{nd} ‘(1^{st}
‘𝑇)) ∘
({⟨(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)⟩, ⟨((2^{nd}
‘𝑇) + 1),
(2^{nd} ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)})))) →
(((2^{nd} ‘(1^{st} ‘𝑡)) “ (1...𝑗)) × {1}) = ((((2^{nd}
‘(1^{st} ‘𝑇)) ∘ ({⟨(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)⟩,
⟨((2^{nd} ‘𝑇) + 1), (2^{nd} ‘𝑇)⟩} ∪ ( I ↾
((1...𝑁) ∖
{(2^{nd} ‘𝑇),
((2^{nd} ‘𝑇)
+ 1)})))) “ (1...𝑗))
× {1})) |
437 | | imaeq1 5496 |
. . . . . . . . . . . 12
⊢
((2^{nd} ‘(1^{st} ‘𝑡)) = ((2^{nd} ‘(1^{st}
‘𝑇)) ∘
({⟨(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)⟩, ⟨((2^{nd}
‘𝑇) + 1),
(2^{nd} ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)})))) →
((2^{nd} ‘(1^{st} ‘𝑡)) “ ((𝑗 + 1)...𝑁)) = (((2^{nd}
‘(1^{st} ‘𝑇)) ∘ ({⟨(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)⟩,
⟨((2^{nd} ‘𝑇) + 1), (2^{nd} ‘𝑇)⟩} ∪ ( I ↾
((1...𝑁) ∖
{(2^{nd} ‘𝑇),
((2^{nd} ‘𝑇)
+ 1)})))) “ ((𝑗 +
1)...𝑁))) |
438 | 437 | xpeq1d 5172 |
. . . . . . . . . . 11
⊢
((2^{nd} ‘(1^{st} ‘𝑡)) = ((2^{nd} ‘(1^{st}
‘𝑇)) ∘
({⟨(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)⟩, ⟨((2^{nd}
‘𝑇) + 1),
(2^{nd} ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)})))) →
(((2^{nd} ‘(1^{st} ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}) = ((((2^{nd}
‘(1^{st} ‘𝑇)) ∘ ({⟨(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)⟩,
⟨((2^{nd} ‘𝑇) + 1), (2^{nd} ‘𝑇)⟩} ∪ ( I ↾
((1...𝑁) ∖
{(2^{nd} ‘𝑇),
((2^{nd} ‘𝑇)
+ 1)})))) “ ((𝑗 +
1)...𝑁)) ×
{0})) |
439 | 436, 438 | uneq12d 3801 |
. . . . . . . . . 10
⊢
((2^{nd} ‘(1^{st} ‘𝑡)) = ((2^{nd} ‘(1^{st}
‘𝑇)) ∘
({⟨(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)⟩, ⟨((2^{nd}
‘𝑇) + 1),
(2^{nd} ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)})))) →
((((2^{nd} ‘(1^{st} ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})) = (((((2^{nd}
‘(1^{st} ‘𝑇)) ∘ ({⟨(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)⟩,
⟨((2^{nd} ‘𝑇) + 1), (2^{nd} ‘𝑇)⟩} ∪ ( I ↾
((1...𝑁) ∖
{(2^{nd} ‘𝑇),
((2^{nd} ‘𝑇)
+ 1)})))) “ (1...𝑗))
× {1}) ∪ ((((2^{nd} ‘(1^{st} ‘𝑇)) ∘
({⟨(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)⟩, ⟨((2^{nd}
‘𝑇) + 1),
(2^{nd} ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)})))) “
((𝑗 + 1)...𝑁)) ×
{0}))) |
440 | 434, 439 | oveqan12d 6709 |
. . . . . . . . 9
⊢
(((1^{st} ‘(1^{st} ‘𝑡)) = (1^{st} ‘(1^{st}
‘𝑇)) ∧
(2^{nd} ‘(1^{st} ‘𝑡)) = ((2^{nd} ‘(1^{st}
‘𝑇)) ∘
({⟨(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)⟩, ⟨((2^{nd}
‘𝑇) + 1),
(2^{nd} ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)}))))) →
((1^{st} ‘(1^{st} ‘𝑡)) ∘_{𝑓} +
((((2^{nd} ‘(1^{st} ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))) = ((1^{st}
‘(1^{st} ‘𝑇)) ∘_{𝑓} +
(((((2^{nd} ‘(1^{st} ‘𝑇)) ∘ ({⟨(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)⟩,
⟨((2^{nd} ‘𝑇) + 1), (2^{nd} ‘𝑇)⟩} ∪ ( I ↾
((1...𝑁) ∖
{(2^{nd} ‘𝑇),
((2^{nd} ‘𝑇)
+ 1)})))) “ (1...𝑗))
× {1}) ∪ ((((2^{nd} ‘(1^{st} ‘𝑇)) ∘
({⟨(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)⟩, ⟨((2^{nd}
‘𝑇) + 1),
(2^{nd} ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)})))) “
((𝑗 + 1)...𝑁)) ×
{0})))) |
441 | 432, 433,
440 | syl2anc 694 |
. . . . . . . 8
⊢
((1^{st} ‘𝑡) = ⟨(1^{st}
‘(1^{st} ‘𝑇)), ((2^{nd} ‘(1^{st}
‘𝑇)) ∘
({⟨(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)⟩, ⟨((2^{nd}
‘𝑇) + 1),
(2^{nd} ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)}))))⟩
→ ((1^{st} ‘(1^{st} ‘𝑡)) ∘_{𝑓} +
((((2^{nd} ‘(1^{st} ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))) = ((1^{st}
‘(1^{st} ‘𝑇)) ∘_{𝑓} +
(((((2^{nd} ‘(1^{st} ‘𝑇)) ∘ ({⟨(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)⟩,
⟨((2^{nd} ‘𝑇) + 1), (2^{nd} ‘𝑇)⟩} ∪ ( I ↾
((1...𝑁) ∖
{(2^{nd} ‘𝑇),
((2^{nd} ‘𝑇)
+ 1)})))) “ (1...𝑗))
× {1}) ∪ ((((2^{nd} ‘(1^{st} ‘𝑇)) ∘
({⟨(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)⟩, ⟨((2^{nd}
‘𝑇) + 1),
(2^{nd} ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)})))) “
((𝑗 + 1)...𝑁)) ×
{0})))) |
442 | 441 | csbeq2dv 4025 |
. . . . . . 7
⊢
((1^{st} ‘𝑡) = ⟨(1^{st}
‘(1^{st} ‘𝑇)), ((2^{nd} ‘(1^{st}
‘𝑇)) ∘
({⟨(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)⟩, ⟨((2^{nd}
‘𝑇) + 1),
(2^{nd} ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)}))))⟩
→ ⦋if(𝑦
< (2^{nd} ‘𝑇), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^{st}
‘(1^{st} ‘𝑡)) ∘_{𝑓} +
((((2^{nd} ‘(1^{st} ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))) = ⦋if(𝑦 < (2^{nd}
‘𝑇), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^{st}
‘(1^{st} ‘𝑇)) ∘_{𝑓} +
(((((2^{nd} ‘(1^{st} ‘𝑇)) ∘ ({⟨(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)⟩,
⟨((2^{nd} ‘𝑇) + 1), (2^{nd} ‘𝑇)⟩} ∪ ( I ↾
((1...𝑁) ∖
{(2^{nd} ‘𝑇),
((2^{nd} ‘𝑇)
+ 1)})))) “ (1...𝑗))
× {1}) ∪ ((((2^{nd} ‘(1^{st} ‘𝑇)) ∘
({⟨(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)⟩, ⟨((2^{nd}
‘𝑇) + 1),
(2^{nd} ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)})))) “
((𝑗 + 1)...𝑁)) ×
{0})))) |
443 | 430, 442 | sylan9eqr 2707 |
. . . . . 6
⊢
(((1^{st} ‘𝑡) = ⟨(1^{st}
‘(1^{st} ‘𝑇)), ((2^{nd} ‘(1^{st}
‘𝑇)) ∘
({⟨(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)⟩, ⟨((2^{nd}
‘𝑇) + 1),
(2^{nd} ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)}))))⟩
∧ (2^{nd} ‘𝑡) = (2^{nd} ‘𝑇)) → ⦋if(𝑦 < (2^{nd}
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^{st}
‘(1^{st} ‘𝑡)) ∘_{𝑓} +
((((2^{nd} ‘(1^{st} ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))) = ⦋if(𝑦 < (2^{nd}
‘𝑇), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^{st}
‘(1^{st} ‘𝑇)) ∘_{𝑓} +
(((((2^{nd} ‘(1^{st} ‘𝑇)) ∘ ({⟨(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)⟩,
⟨((2^{nd} ‘𝑇) + 1), (2^{nd} ‘𝑇)⟩} ∪ ( I ↾
((1...𝑁) ∖
{(2^{nd} ‘𝑇),
((2^{nd} ‘𝑇)
+ 1)})))) “ (1...𝑗))
× {1}) ∪ ((((2^{nd} ‘(1^{st} ‘𝑇)) ∘
({⟨(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)⟩, ⟨((2^{nd}
‘𝑇) + 1),
(2^{nd} ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)})))) “
((𝑗 + 1)...𝑁)) ×
{0})))) |
444 | 426, 427,
443 | syl2anc 694 |
. . . . 5
⊢ (𝑡 = ⟨⟨(1^{st}
‘(1^{st} ‘𝑇)), ((2^{nd} ‘(1^{st}
‘𝑇)) ∘
({⟨(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)⟩, ⟨((2^{nd}
‘𝑇) + 1),
(2^{nd} ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)}))))⟩,
(2^{nd} ‘𝑇)⟩ → ⦋if(𝑦 < (2^{nd}
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^{st}
‘(1^{st} ‘𝑡)) ∘_{𝑓} +
((((2^{nd} ‘(1^{st} ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))) = ⦋if(𝑦 < (2^{nd}
‘𝑇), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^{st}
‘(1^{st} ‘𝑇)) ∘_{𝑓} +
(((((2^{nd} ‘(1^{st} ‘𝑇)) ∘ ({⟨(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)⟩,
⟨((2^{nd} ‘𝑇) + 1), (2^{nd} ‘𝑇)⟩} ∪ ( I ↾
((1...𝑁) ∖
{(2^{nd} ‘𝑇),
((2^{nd} ‘𝑇)
+ 1)})))) “ (1...𝑗))
× {1}) ∪ ((((2^{nd} ‘(1^{st} ‘𝑇)) ∘
({⟨(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)⟩, ⟨((2^{nd}
‘𝑇) + 1),
(2^{nd} ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)})))) “
((𝑗 + 1)...𝑁)) ×
{0})))) |
445 | 444 | mpteq2dv 4778 |
. . . 4
⊢ (𝑡 = ⟨⟨(1^{st}
‘(1^{st} ‘𝑇)), ((2^{nd} ‘(1^{st}
‘𝑇)) ∘
({⟨(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)⟩, ⟨((2^{nd}
‘𝑇) + 1),
(2^{nd} ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)}))))⟩,
(2^{nd} ‘𝑇)⟩ → (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2^{nd}
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^{st}
‘(1^{st} ‘𝑡)) ∘_{𝑓} +
((((2^{nd} ‘(1^{st} ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2^{nd}
‘𝑇), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^{st}
‘(1^{st} ‘𝑇)) ∘_{𝑓} +
(((((2^{nd} ‘(1^{st} ‘𝑇)) ∘ ({⟨(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)⟩,
⟨((2^{nd} ‘𝑇) + 1), (2^{nd} ‘𝑇)⟩} ∪ ( I ↾
((1...𝑁) ∖
{(2^{nd} ‘𝑇),
((2^{nd} ‘𝑇)
+ 1)})))) “ (1...𝑗))
× {1}) ∪ ((((2^{nd} ‘(1^{st} ‘𝑇)) ∘
({⟨(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)⟩, ⟨((2^{nd}
‘𝑇) + 1),
(2^{nd} ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)})))) “
((𝑗 + 1)...𝑁)) ×
{0}))))) |
446 | 445 | eqeq2d 2661 |
. . 3
⊢ (𝑡 = ⟨⟨(1^{st}
‘(1^{st} ‘𝑇)), ((2^{nd} ‘(1^{st}
‘𝑇)) ∘
({⟨(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)⟩, ⟨((2^{nd}
‘𝑇) + 1),
(2^{nd} ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)}))))⟩,
(2^{nd} ‘𝑇)⟩ → (𝐹 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2^{nd}
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^{st}
‘(1^{st} ‘𝑡)) ∘_{𝑓} +
((((2^{nd} ‘(1^{st} ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) ↔ 𝐹 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2^{nd}
‘𝑇), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^{st}
‘(1^{st} ‘𝑇)) ∘_{𝑓} +
(((((2^{nd} ‘(1^{st} ‘𝑇)) ∘ ({⟨(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)⟩,
⟨((2^{nd} ‘𝑇) + 1), (2^{nd} ‘𝑇)⟩} ∪ ( I ↾
((1...𝑁) ∖
{(2^{nd} ‘𝑇),
((2^{nd} ‘𝑇)
+ 1)})))) “ (1...𝑗))
× {1}) ∪ ((((2^{nd} ‘(1^{st} ‘𝑇)) ∘
({⟨(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)⟩, ⟨((2^{nd}
‘𝑇) + 1),
(2^{nd} ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)})))) “
((𝑗 + 1)...𝑁)) ×
{0})))))) |
447 | 446, 3 | elrab2 3399 |
. 2
⊢
(⟨⟨(1^{st} ‘(1^{st} ‘𝑇)), ((2^{nd}
‘(1^{st} ‘𝑇)) ∘ ({⟨(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)⟩,
⟨((2^{nd} ‘𝑇) + 1), (2^{nd} ‘𝑇)⟩} ∪ ( I ↾
((1...𝑁) ∖
{(2^{nd} ‘𝑇),
((2^{nd} ‘𝑇)
+ 1)}))))⟩, (2^{nd} ‘𝑇)⟩ ∈ 𝑆 ↔ (⟨⟨(1^{st}
‘(1^{st} ‘𝑇)), ((2^{nd} ‘(1^{st}
‘𝑇)) ∘
({⟨(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)⟩, ⟨((2^{nd}
‘𝑇) + 1),
(2^{nd} ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)}))))⟩,
(2^{nd} ‘𝑇)⟩ ∈ ((((0..^𝐾) ↑_{𝑚} (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∧ 𝐹 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2^{nd}
‘𝑇), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^{st}
‘(1^{st} ‘𝑇)) ∘_{𝑓} +
(((((2^{nd} ‘(1^{st} ‘𝑇)) ∘ ({⟨(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)⟩,
⟨((2^{nd} ‘𝑇) + 1), (2^{nd} ‘𝑇)⟩} ∪ ( I ↾
((1...𝑁) ∖
{(2^{nd} ‘𝑇),
((2^{nd} ‘𝑇)
+ 1)})))) “ (1...𝑗))
× {1}) ∪ ((((2^{nd} ‘(1^{st} ‘𝑇)) ∘
({⟨(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)⟩, ⟨((2^{nd}
‘𝑇) + 1),
(2^{nd} ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)})))) “
((𝑗 + 1)...𝑁)) ×
{0})))))) |
448 | 89, 424, 447 | sylanbrc 699 |
1
⊢ (𝜑 →
⟨⟨(1^{st} ‘(1^{st} ‘𝑇)), ((2^{nd} ‘(1^{st}
‘𝑇)) ∘
({⟨(2^{nd} ‘𝑇), ((2^{nd} ‘𝑇) + 1)⟩, ⟨((2^{nd}
‘𝑇) + 1),
(2^{nd} ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^{nd}
‘𝑇), ((2^{nd}
‘𝑇) + 1)}))))⟩,
(2^{nd} ‘𝑇)⟩ ∈ 𝑆) |