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Theorem mdetunilem9 20644
 Description: Lemma for mdetuni 20646. (Contributed by SO, 15-Jul-2018.)
Hypotheses
Ref Expression
mdetuni.a 𝐴 = (𝑁 Mat 𝑅)
mdetuni.b 𝐵 = (Base‘𝐴)
mdetuni.k 𝐾 = (Base‘𝑅)
mdetuni.0g 0 = (0g𝑅)
mdetuni.1r 1 = (1r𝑅)
mdetuni.pg + = (+g𝑅)
mdetuni.tg · = (.r𝑅)
mdetuni.n (𝜑𝑁 ∈ Fin)
mdetuni.r (𝜑𝑅 ∈ Ring)
mdetuni.ff (𝜑𝐷:𝐵𝐾)
mdetuni.al (𝜑 → ∀𝑥𝐵𝑦𝑁𝑧𝑁 ((𝑦𝑧 ∧ ∀𝑤𝑁 (𝑦𝑥𝑤) = (𝑧𝑥𝑤)) → (𝐷𝑥) = 0 ))
mdetuni.li (𝜑 → ∀𝑥𝐵𝑦𝐵𝑧𝐵𝑤𝑁 (((𝑥 ↾ ({𝑤} × 𝑁)) = ((𝑦 ↾ ({𝑤} × 𝑁)) ∘𝑓 + (𝑧 ↾ ({𝑤} × 𝑁))) ∧ (𝑥 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑦 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) ∧ (𝑥 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑧 ↾ ((𝑁 ∖ {𝑤}) × 𝑁))) → (𝐷𝑥) = ((𝐷𝑦) + (𝐷𝑧))))
mdetuni.sc (𝜑 → ∀𝑥𝐵𝑦𝐾𝑧𝐵𝑤𝑁 (((𝑥 ↾ ({𝑤} × 𝑁)) = ((({𝑤} × 𝑁) × {𝑦}) ∘𝑓 · (𝑧 ↾ ({𝑤} × 𝑁))) ∧ (𝑥 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑧 ↾ ((𝑁 ∖ {𝑤}) × 𝑁))) → (𝐷𝑥) = (𝑦 · (𝐷𝑧))))
mdetunilem9.id (𝜑 → (𝐷‘(1r𝐴)) = 0 )
mdetunilem9.y 𝑌 = {𝑥 ∣ ∀𝑦𝐵𝑧 ∈ (𝑁𝑚 𝑁)(∀𝑤𝑥 (𝑦𝑤) = if(𝑤𝑧, 1 , 0 ) → (𝐷𝑦) = 0 )}
Assertion
Ref Expression
mdetunilem9 (𝜑𝐷 = (𝐵 × { 0 }))
Distinct variable groups:   𝜑,𝑥,𝑦,𝑧,𝑤   𝑥,𝐵,𝑦,𝑧,𝑤   𝑥,𝐾,𝑦,𝑧,𝑤   𝑥,𝑁,𝑦,𝑧,𝑤   𝑥,𝐷,𝑦,𝑧,𝑤   𝑥, · ,𝑦,𝑧,𝑤   𝑥, + ,𝑦,𝑧,𝑤   𝑥, 0 ,𝑦,𝑧,𝑤   𝑥, 1 ,𝑦,𝑧,𝑤   𝑥,𝑅,𝑦,𝑧,𝑤   𝑥,𝐴,𝑦,𝑧,𝑤
Allowed substitution hints:   𝑌(𝑥,𝑦,𝑧,𝑤)

Proof of Theorem mdetunilem9
Dummy variables 𝑎 𝑏 𝑐 𝑑 𝑒 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ral0 4217 . . . 4 𝑤 ∈ ∅ (𝑎𝑤) = if(𝑤 ∈ ( I ↾ 𝑁), 1 , 0 )
2 simpr 471 . . . . 5 ((𝜑𝑎𝐵) → 𝑎𝐵)
3 f1oi 6315 . . . . . . . 8 ( I ↾ 𝑁):𝑁1-1-onto𝑁
4 f1of 6278 . . . . . . . 8 (( I ↾ 𝑁):𝑁1-1-onto𝑁 → ( I ↾ 𝑁):𝑁𝑁)
53, 4mp1i 13 . . . . . . 7 (𝜑 → ( I ↾ 𝑁):𝑁𝑁)
6 mdetuni.n . . . . . . . 8 (𝜑𝑁 ∈ Fin)
76, 6elmapd 8023 . . . . . . 7 (𝜑 → (( I ↾ 𝑁) ∈ (𝑁𝑚 𝑁) ↔ ( I ↾ 𝑁):𝑁𝑁))
85, 7mpbird 247 . . . . . 6 (𝜑 → ( I ↾ 𝑁) ∈ (𝑁𝑚 𝑁))
98adantr 466 . . . . 5 ((𝜑𝑎𝐵) → ( I ↾ 𝑁) ∈ (𝑁𝑚 𝑁))
10 simplrl 762 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑦𝐵𝑧 ∈ (𝑁𝑚 𝑁))) ∧ ∀𝑤 ∈ (𝑁 × 𝑁)(𝑦𝑤) = if(𝑤𝑧, 1 , 0 )) → 𝑦𝐵)
11 mdetuni.a . . . . . . . . . . . . . . . . 17 𝐴 = (𝑁 Mat 𝑅)
12 mdetuni.k . . . . . . . . . . . . . . . . 17 𝐾 = (Base‘𝑅)
13 mdetuni.b . . . . . . . . . . . . . . . . 17 𝐵 = (Base‘𝐴)
1411, 12, 13matbas2i 20445 . . . . . . . . . . . . . . . 16 (𝑦𝐵𝑦 ∈ (𝐾𝑚 (𝑁 × 𝑁)))
15 elmapi 8031 . . . . . . . . . . . . . . . 16 (𝑦 ∈ (𝐾𝑚 (𝑁 × 𝑁)) → 𝑦:(𝑁 × 𝑁)⟶𝐾)
1614, 15syl 17 . . . . . . . . . . . . . . 15 (𝑦𝐵𝑦:(𝑁 × 𝑁)⟶𝐾)
1716feqmptd 6391 . . . . . . . . . . . . . 14 (𝑦𝐵𝑦 = (𝑤 ∈ (𝑁 × 𝑁) ↦ (𝑦𝑤)))
1817fveq2d 6336 . . . . . . . . . . . . 13 (𝑦𝐵 → (𝐷𝑦) = (𝐷‘(𝑤 ∈ (𝑁 × 𝑁) ↦ (𝑦𝑤))))
1910, 18syl 17 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑦𝐵𝑧 ∈ (𝑁𝑚 𝑁))) ∧ ∀𝑤 ∈ (𝑁 × 𝑁)(𝑦𝑤) = if(𝑤𝑧, 1 , 0 )) → (𝐷𝑦) = (𝐷‘(𝑤 ∈ (𝑁 × 𝑁) ↦ (𝑦𝑤))))
20 eqid 2771 . . . . . . . . . . . . . 14 (𝑁 × 𝑁) = (𝑁 × 𝑁)
21 mpteq12 4870 . . . . . . . . . . . . . . 15 (((𝑁 × 𝑁) = (𝑁 × 𝑁) ∧ ∀𝑤 ∈ (𝑁 × 𝑁)(𝑦𝑤) = if(𝑤𝑧, 1 , 0 )) → (𝑤 ∈ (𝑁 × 𝑁) ↦ (𝑦𝑤)) = (𝑤 ∈ (𝑁 × 𝑁) ↦ if(𝑤𝑧, 1 , 0 )))
2221fveq2d 6336 . . . . . . . . . . . . . 14 (((𝑁 × 𝑁) = (𝑁 × 𝑁) ∧ ∀𝑤 ∈ (𝑁 × 𝑁)(𝑦𝑤) = if(𝑤𝑧, 1 , 0 )) → (𝐷‘(𝑤 ∈ (𝑁 × 𝑁) ↦ (𝑦𝑤))) = (𝐷‘(𝑤 ∈ (𝑁 × 𝑁) ↦ if(𝑤𝑧, 1 , 0 ))))
2320, 22mpan 670 . . . . . . . . . . . . 13 (∀𝑤 ∈ (𝑁 × 𝑁)(𝑦𝑤) = if(𝑤𝑧, 1 , 0 ) → (𝐷‘(𝑤 ∈ (𝑁 × 𝑁) ↦ (𝑦𝑤))) = (𝐷‘(𝑤 ∈ (𝑁 × 𝑁) ↦ if(𝑤𝑧, 1 , 0 ))))
2423adantl 467 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑦𝐵𝑧 ∈ (𝑁𝑚 𝑁))) ∧ ∀𝑤 ∈ (𝑁 × 𝑁)(𝑦𝑤) = if(𝑤𝑧, 1 , 0 )) → (𝐷‘(𝑤 ∈ (𝑁 × 𝑁) ↦ (𝑦𝑤))) = (𝐷‘(𝑤 ∈ (𝑁 × 𝑁) ↦ if(𝑤𝑧, 1 , 0 ))))
25 eleq1 2838 . . . . . . . . . . . . . . . . 17 (𝑎 = 𝑧 → (𝑎 ∈ (𝑁𝑚 𝑁) ↔ 𝑧 ∈ (𝑁𝑚 𝑁)))
2625anbi2d 614 . . . . . . . . . . . . . . . 16 (𝑎 = 𝑧 → ((𝜑𝑎 ∈ (𝑁𝑚 𝑁)) ↔ (𝜑𝑧 ∈ (𝑁𝑚 𝑁))))
27 elequ2 2159 . . . . . . . . . . . . . . . . . . . 20 (𝑎 = 𝑧 → (𝑤𝑎𝑤𝑧))
2827ifbid 4247 . . . . . . . . . . . . . . . . . . 19 (𝑎 = 𝑧 → if(𝑤𝑎, 1 , 0 ) = if(𝑤𝑧, 1 , 0 ))
2928mpteq2dv 4879 . . . . . . . . . . . . . . . . . 18 (𝑎 = 𝑧 → (𝑤 ∈ (𝑁 × 𝑁) ↦ if(𝑤𝑎, 1 , 0 )) = (𝑤 ∈ (𝑁 × 𝑁) ↦ if(𝑤𝑧, 1 , 0 )))
3029fveq2d 6336 . . . . . . . . . . . . . . . . 17 (𝑎 = 𝑧 → (𝐷‘(𝑤 ∈ (𝑁 × 𝑁) ↦ if(𝑤𝑎, 1 , 0 ))) = (𝐷‘(𝑤 ∈ (𝑁 × 𝑁) ↦ if(𝑤𝑧, 1 , 0 ))))
3130eqeq1d 2773 . . . . . . . . . . . . . . . 16 (𝑎 = 𝑧 → ((𝐷‘(𝑤 ∈ (𝑁 × 𝑁) ↦ if(𝑤𝑎, 1 , 0 ))) = 0 ↔ (𝐷‘(𝑤 ∈ (𝑁 × 𝑁) ↦ if(𝑤𝑧, 1 , 0 ))) = 0 ))
3226, 31imbi12d 333 . . . . . . . . . . . . . . 15 (𝑎 = 𝑧 → (((𝜑𝑎 ∈ (𝑁𝑚 𝑁)) → (𝐷‘(𝑤 ∈ (𝑁 × 𝑁) ↦ if(𝑤𝑎, 1 , 0 ))) = 0 ) ↔ ((𝜑𝑧 ∈ (𝑁𝑚 𝑁)) → (𝐷‘(𝑤 ∈ (𝑁 × 𝑁) ↦ if(𝑤𝑧, 1 , 0 ))) = 0 )))
33 eleq1 2838 . . . . . . . . . . . . . . . . . . . 20 (𝑤 = ⟨𝑏, 𝑐⟩ → (𝑤𝑎 ↔ ⟨𝑏, 𝑐⟩ ∈ 𝑎))
3433ifbid 4247 . . . . . . . . . . . . . . . . . . 19 (𝑤 = ⟨𝑏, 𝑐⟩ → if(𝑤𝑎, 1 , 0 ) = if(⟨𝑏, 𝑐⟩ ∈ 𝑎, 1 , 0 ))
3534mpt2mpt 6899 . . . . . . . . . . . . . . . . . 18 (𝑤 ∈ (𝑁 × 𝑁) ↦ if(𝑤𝑎, 1 , 0 )) = (𝑏𝑁, 𝑐𝑁 ↦ if(⟨𝑏, 𝑐⟩ ∈ 𝑎, 1 , 0 ))
36 elmapi 8031 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑎 ∈ (𝑁𝑚 𝑁) → 𝑎:𝑁𝑁)
3736adantl 467 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝜑𝑎 ∈ (𝑁𝑚 𝑁)) → 𝑎:𝑁𝑁)
38 ffn 6185 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑎:𝑁𝑁𝑎 Fn 𝑁)
3937, 38syl 17 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑𝑎 ∈ (𝑁𝑚 𝑁)) → 𝑎 Fn 𝑁)
40393ad2ant1 1127 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑𝑎 ∈ (𝑁𝑚 𝑁)) ∧ 𝑏𝑁𝑐𝑁) → 𝑎 Fn 𝑁)
41 simp2 1131 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑𝑎 ∈ (𝑁𝑚 𝑁)) ∧ 𝑏𝑁𝑐𝑁) → 𝑏𝑁)
42 fnopfvb 6378 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑎 Fn 𝑁𝑏𝑁) → ((𝑎𝑏) = 𝑐 ↔ ⟨𝑏, 𝑐⟩ ∈ 𝑎))
4340, 41, 42syl2anc 573 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑎 ∈ (𝑁𝑚 𝑁)) ∧ 𝑏𝑁𝑐𝑁) → ((𝑎𝑏) = 𝑐 ↔ ⟨𝑏, 𝑐⟩ ∈ 𝑎))
4443bicomd 213 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑎 ∈ (𝑁𝑚 𝑁)) ∧ 𝑏𝑁𝑐𝑁) → (⟨𝑏, 𝑐⟩ ∈ 𝑎 ↔ (𝑎𝑏) = 𝑐))
4544ifbid 4247 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑎 ∈ (𝑁𝑚 𝑁)) ∧ 𝑏𝑁𝑐𝑁) → if(⟨𝑏, 𝑐⟩ ∈ 𝑎, 1 , 0 ) = if((𝑎𝑏) = 𝑐, 1 , 0 ))
4645mpt2eq3dva 6866 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑎 ∈ (𝑁𝑚 𝑁)) → (𝑏𝑁, 𝑐𝑁 ↦ if(⟨𝑏, 𝑐⟩ ∈ 𝑎, 1 , 0 )) = (𝑏𝑁, 𝑐𝑁 ↦ if((𝑎𝑏) = 𝑐, 1 , 0 )))
4735, 46syl5eq 2817 . . . . . . . . . . . . . . . . 17 ((𝜑𝑎 ∈ (𝑁𝑚 𝑁)) → (𝑤 ∈ (𝑁 × 𝑁) ↦ if(𝑤𝑎, 1 , 0 )) = (𝑏𝑁, 𝑐𝑁 ↦ if((𝑎𝑏) = 𝑐, 1 , 0 )))
4847fveq2d 6336 . . . . . . . . . . . . . . . 16 ((𝜑𝑎 ∈ (𝑁𝑚 𝑁)) → (𝐷‘(𝑤 ∈ (𝑁 × 𝑁) ↦ if(𝑤𝑎, 1 , 0 ))) = (𝐷‘(𝑏𝑁, 𝑐𝑁 ↦ if((𝑎𝑏) = 𝑐, 1 , 0 ))))
49 mdetuni.0g . . . . . . . . . . . . . . . . . 18 0 = (0g𝑅)
50 mdetuni.1r . . . . . . . . . . . . . . . . . 18 1 = (1r𝑅)
51 mdetuni.pg . . . . . . . . . . . . . . . . . 18 + = (+g𝑅)
52 mdetuni.tg . . . . . . . . . . . . . . . . . 18 · = (.r𝑅)
53 mdetuni.r . . . . . . . . . . . . . . . . . 18 (𝜑𝑅 ∈ Ring)
54 mdetuni.ff . . . . . . . . . . . . . . . . . 18 (𝜑𝐷:𝐵𝐾)
55 mdetuni.al . . . . . . . . . . . . . . . . . 18 (𝜑 → ∀𝑥𝐵𝑦𝑁𝑧𝑁 ((𝑦𝑧 ∧ ∀𝑤𝑁 (𝑦𝑥𝑤) = (𝑧𝑥𝑤)) → (𝐷𝑥) = 0 ))
56 mdetuni.li . . . . . . . . . . . . . . . . . 18 (𝜑 → ∀𝑥𝐵𝑦𝐵𝑧𝐵𝑤𝑁 (((𝑥 ↾ ({𝑤} × 𝑁)) = ((𝑦 ↾ ({𝑤} × 𝑁)) ∘𝑓 + (𝑧 ↾ ({𝑤} × 𝑁))) ∧ (𝑥 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑦 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) ∧ (𝑥 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑧 ↾ ((𝑁 ∖ {𝑤}) × 𝑁))) → (𝐷𝑥) = ((𝐷𝑦) + (𝐷𝑧))))
57 mdetuni.sc . . . . . . . . . . . . . . . . . 18 (𝜑 → ∀𝑥𝐵𝑦𝐾𝑧𝐵𝑤𝑁 (((𝑥 ↾ ({𝑤} × 𝑁)) = ((({𝑤} × 𝑁) × {𝑦}) ∘𝑓 · (𝑧 ↾ ({𝑤} × 𝑁))) ∧ (𝑥 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑧 ↾ ((𝑁 ∖ {𝑤}) × 𝑁))) → (𝐷𝑥) = (𝑦 · (𝐷𝑧))))
58 mdetunilem9.id . . . . . . . . . . . . . . . . . 18 (𝜑 → (𝐷‘(1r𝐴)) = 0 )
5911, 13, 12, 49, 50, 51, 52, 6, 53, 54, 55, 56, 57, 58mdetunilem8 20643 . . . . . . . . . . . . . . . . 17 ((𝜑𝑎:𝑁𝑁) → (𝐷‘(𝑏𝑁, 𝑐𝑁 ↦ if((𝑎𝑏) = 𝑐, 1 , 0 ))) = 0 )
6036, 59sylan2 580 . . . . . . . . . . . . . . . 16 ((𝜑𝑎 ∈ (𝑁𝑚 𝑁)) → (𝐷‘(𝑏𝑁, 𝑐𝑁 ↦ if((𝑎𝑏) = 𝑐, 1 , 0 ))) = 0 )
6148, 60eqtrd 2805 . . . . . . . . . . . . . . 15 ((𝜑𝑎 ∈ (𝑁𝑚 𝑁)) → (𝐷‘(𝑤 ∈ (𝑁 × 𝑁) ↦ if(𝑤𝑎, 1 , 0 ))) = 0 )
6232, 61chvarv 2425 . . . . . . . . . . . . . 14 ((𝜑𝑧 ∈ (𝑁𝑚 𝑁)) → (𝐷‘(𝑤 ∈ (𝑁 × 𝑁) ↦ if(𝑤𝑧, 1 , 0 ))) = 0 )
6362adantrl 695 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑦𝐵𝑧 ∈ (𝑁𝑚 𝑁))) → (𝐷‘(𝑤 ∈ (𝑁 × 𝑁) ↦ if(𝑤𝑧, 1 , 0 ))) = 0 )
6463adantr 466 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑦𝐵𝑧 ∈ (𝑁𝑚 𝑁))) ∧ ∀𝑤 ∈ (𝑁 × 𝑁)(𝑦𝑤) = if(𝑤𝑧, 1 , 0 )) → (𝐷‘(𝑤 ∈ (𝑁 × 𝑁) ↦ if(𝑤𝑧, 1 , 0 ))) = 0 )
6519, 24, 643eqtrd 2809 . . . . . . . . . . 11 (((𝜑 ∧ (𝑦𝐵𝑧 ∈ (𝑁𝑚 𝑁))) ∧ ∀𝑤 ∈ (𝑁 × 𝑁)(𝑦𝑤) = if(𝑤𝑧, 1 , 0 )) → (𝐷𝑦) = 0 )
6665ex 397 . . . . . . . . . 10 ((𝜑 ∧ (𝑦𝐵𝑧 ∈ (𝑁𝑚 𝑁))) → (∀𝑤 ∈ (𝑁 × 𝑁)(𝑦𝑤) = if(𝑤𝑧, 1 , 0 ) → (𝐷𝑦) = 0 ))
6766ralrimivva 3120 . . . . . . . . 9 (𝜑 → ∀𝑦𝐵𝑧 ∈ (𝑁𝑚 𝑁)(∀𝑤 ∈ (𝑁 × 𝑁)(𝑦𝑤) = if(𝑤𝑧, 1 , 0 ) → (𝐷𝑦) = 0 ))
68 xpfi 8387 . . . . . . . . . . 11 ((𝑁 ∈ Fin ∧ 𝑁 ∈ Fin) → (𝑁 × 𝑁) ∈ Fin)
696, 6, 68syl2anc 573 . . . . . . . . . 10 (𝜑 → (𝑁 × 𝑁) ∈ Fin)
70 raleq 3287 . . . . . . . . . . . . 13 (𝑥 = (𝑁 × 𝑁) → (∀𝑤𝑥 (𝑦𝑤) = if(𝑤𝑧, 1 , 0 ) ↔ ∀𝑤 ∈ (𝑁 × 𝑁)(𝑦𝑤) = if(𝑤𝑧, 1 , 0 )))
7170imbi1d 330 . . . . . . . . . . . 12 (𝑥 = (𝑁 × 𝑁) → ((∀𝑤𝑥 (𝑦𝑤) = if(𝑤𝑧, 1 , 0 ) → (𝐷𝑦) = 0 ) ↔ (∀𝑤 ∈ (𝑁 × 𝑁)(𝑦𝑤) = if(𝑤𝑧, 1 , 0 ) → (𝐷𝑦) = 0 )))
72712ralbidv 3138 . . . . . . . . . . 11 (𝑥 = (𝑁 × 𝑁) → (∀𝑦𝐵𝑧 ∈ (𝑁𝑚 𝑁)(∀𝑤𝑥 (𝑦𝑤) = if(𝑤𝑧, 1 , 0 ) → (𝐷𝑦) = 0 ) ↔ ∀𝑦𝐵𝑧 ∈ (𝑁𝑚 𝑁)(∀𝑤 ∈ (𝑁 × 𝑁)(𝑦𝑤) = if(𝑤𝑧, 1 , 0 ) → (𝐷𝑦) = 0 )))
73 mdetunilem9.y . . . . . . . . . . 11 𝑌 = {𝑥 ∣ ∀𝑦𝐵𝑧 ∈ (𝑁𝑚 𝑁)(∀𝑤𝑥 (𝑦𝑤) = if(𝑤𝑧, 1 , 0 ) → (𝐷𝑦) = 0 )}
7472, 73elab2g 3504 . . . . . . . . . 10 ((𝑁 × 𝑁) ∈ Fin → ((𝑁 × 𝑁) ∈ 𝑌 ↔ ∀𝑦𝐵𝑧 ∈ (𝑁𝑚 𝑁)(∀𝑤 ∈ (𝑁 × 𝑁)(𝑦𝑤) = if(𝑤𝑧, 1 , 0 ) → (𝐷𝑦) = 0 )))
7569, 74syl 17 . . . . . . . . 9 (𝜑 → ((𝑁 × 𝑁) ∈ 𝑌 ↔ ∀𝑦𝐵𝑧 ∈ (𝑁𝑚 𝑁)(∀𝑤 ∈ (𝑁 × 𝑁)(𝑦𝑤) = if(𝑤𝑧, 1 , 0 ) → (𝐷𝑦) = 0 )))
7667, 75mpbird 247 . . . . . . . 8 (𝜑 → (𝑁 × 𝑁) ∈ 𝑌)
77 ssid 3773 . . . . . . . . 9 (𝑁 × 𝑁) ⊆ (𝑁 × 𝑁)
78693ad2ant1 1127 . . . . . . . . . . 11 ((𝜑 ∧ (𝑁 × 𝑁) ⊆ (𝑁 × 𝑁) ∧ ¬ ∅ ∈ 𝑌) → (𝑁 × 𝑁) ∈ Fin)
79 sseq1 3775 . . . . . . . . . . . . . 14 (𝑎 = ∅ → (𝑎 ⊆ (𝑁 × 𝑁) ↔ ∅ ⊆ (𝑁 × 𝑁)))
80793anbi2d 1552 . . . . . . . . . . . . 13 (𝑎 = ∅ → ((𝜑𝑎 ⊆ (𝑁 × 𝑁) ∧ ¬ ∅ ∈ 𝑌) ↔ (𝜑 ∧ ∅ ⊆ (𝑁 × 𝑁) ∧ ¬ ∅ ∈ 𝑌)))
81 eleq1 2838 . . . . . . . . . . . . . 14 (𝑎 = ∅ → (𝑎𝑌 ↔ ∅ ∈ 𝑌))
8281notbid 307 . . . . . . . . . . . . 13 (𝑎 = ∅ → (¬ 𝑎𝑌 ↔ ¬ ∅ ∈ 𝑌))
8380, 82imbi12d 333 . . . . . . . . . . . 12 (𝑎 = ∅ → (((𝜑𝑎 ⊆ (𝑁 × 𝑁) ∧ ¬ ∅ ∈ 𝑌) → ¬ 𝑎𝑌) ↔ ((𝜑 ∧ ∅ ⊆ (𝑁 × 𝑁) ∧ ¬ ∅ ∈ 𝑌) → ¬ ∅ ∈ 𝑌)))
84 sseq1 3775 . . . . . . . . . . . . . 14 (𝑎 = 𝑏 → (𝑎 ⊆ (𝑁 × 𝑁) ↔ 𝑏 ⊆ (𝑁 × 𝑁)))
85843anbi2d 1552 . . . . . . . . . . . . 13 (𝑎 = 𝑏 → ((𝜑𝑎 ⊆ (𝑁 × 𝑁) ∧ ¬ ∅ ∈ 𝑌) ↔ (𝜑𝑏 ⊆ (𝑁 × 𝑁) ∧ ¬ ∅ ∈ 𝑌)))
86 eleq1 2838 . . . . . . . . . . . . . 14 (𝑎 = 𝑏 → (𝑎𝑌𝑏𝑌))
8786notbid 307 . . . . . . . . . . . . 13 (𝑎 = 𝑏 → (¬ 𝑎𝑌 ↔ ¬ 𝑏𝑌))
8885, 87imbi12d 333 . . . . . . . . . . . 12 (𝑎 = 𝑏 → (((𝜑𝑎 ⊆ (𝑁 × 𝑁) ∧ ¬ ∅ ∈ 𝑌) → ¬ 𝑎𝑌) ↔ ((𝜑𝑏 ⊆ (𝑁 × 𝑁) ∧ ¬ ∅ ∈ 𝑌) → ¬ 𝑏𝑌)))
89 sseq1 3775 . . . . . . . . . . . . . 14 (𝑎 = (𝑏 ∪ {𝑐}) → (𝑎 ⊆ (𝑁 × 𝑁) ↔ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁)))
90893anbi2d 1552 . . . . . . . . . . . . 13 (𝑎 = (𝑏 ∪ {𝑐}) → ((𝜑𝑎 ⊆ (𝑁 × 𝑁) ∧ ¬ ∅ ∈ 𝑌) ↔ (𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ ¬ ∅ ∈ 𝑌)))
91 eleq1 2838 . . . . . . . . . . . . . 14 (𝑎 = (𝑏 ∪ {𝑐}) → (𝑎𝑌 ↔ (𝑏 ∪ {𝑐}) ∈ 𝑌))
9291notbid 307 . . . . . . . . . . . . 13 (𝑎 = (𝑏 ∪ {𝑐}) → (¬ 𝑎𝑌 ↔ ¬ (𝑏 ∪ {𝑐}) ∈ 𝑌))
9390, 92imbi12d 333 . . . . . . . . . . . 12 (𝑎 = (𝑏 ∪ {𝑐}) → (((𝜑𝑎 ⊆ (𝑁 × 𝑁) ∧ ¬ ∅ ∈ 𝑌) → ¬ 𝑎𝑌) ↔ ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ ¬ ∅ ∈ 𝑌) → ¬ (𝑏 ∪ {𝑐}) ∈ 𝑌)))
94 sseq1 3775 . . . . . . . . . . . . . 14 (𝑎 = (𝑁 × 𝑁) → (𝑎 ⊆ (𝑁 × 𝑁) ↔ (𝑁 × 𝑁) ⊆ (𝑁 × 𝑁)))
95943anbi2d 1552 . . . . . . . . . . . . 13 (𝑎 = (𝑁 × 𝑁) → ((𝜑𝑎 ⊆ (𝑁 × 𝑁) ∧ ¬ ∅ ∈ 𝑌) ↔ (𝜑 ∧ (𝑁 × 𝑁) ⊆ (𝑁 × 𝑁) ∧ ¬ ∅ ∈ 𝑌)))
96 eleq1 2838 . . . . . . . . . . . . . 14 (𝑎 = (𝑁 × 𝑁) → (𝑎𝑌 ↔ (𝑁 × 𝑁) ∈ 𝑌))
9796notbid 307 . . . . . . . . . . . . 13 (𝑎 = (𝑁 × 𝑁) → (¬ 𝑎𝑌 ↔ ¬ (𝑁 × 𝑁) ∈ 𝑌))
9895, 97imbi12d 333 . . . . . . . . . . . 12 (𝑎 = (𝑁 × 𝑁) → (((𝜑𝑎 ⊆ (𝑁 × 𝑁) ∧ ¬ ∅ ∈ 𝑌) → ¬ 𝑎𝑌) ↔ ((𝜑 ∧ (𝑁 × 𝑁) ⊆ (𝑁 × 𝑁) ∧ ¬ ∅ ∈ 𝑌) → ¬ (𝑁 × 𝑁) ∈ 𝑌)))
99 simp3 1132 . . . . . . . . . . . 12 ((𝜑 ∧ ∅ ⊆ (𝑁 × 𝑁) ∧ ¬ ∅ ∈ 𝑌) → ¬ ∅ ∈ 𝑌)
100 ssun1 3927 . . . . . . . . . . . . . . . 16 𝑏 ⊆ (𝑏 ∪ {𝑐})
101 sstr2 3759 . . . . . . . . . . . . . . . 16 (𝑏 ⊆ (𝑏 ∪ {𝑐}) → ((𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) → 𝑏 ⊆ (𝑁 × 𝑁)))
102100, 101ax-mp 5 . . . . . . . . . . . . . . 15 ((𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) → 𝑏 ⊆ (𝑁 × 𝑁))
1031023anim2i 1156 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ ¬ ∅ ∈ 𝑌) → (𝜑𝑏 ⊆ (𝑁 × 𝑁) ∧ ¬ ∅ ∈ 𝑌))
104103imim1i 63 . . . . . . . . . . . . 13 (((𝜑𝑏 ⊆ (𝑁 × 𝑁) ∧ ¬ ∅ ∈ 𝑌) → ¬ 𝑏𝑌) → ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ ¬ ∅ ∈ 𝑌) → ¬ 𝑏𝑌))
105 simpl1 1227 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ (𝑏 ∪ {𝑐}) ∈ 𝑌) ∧ ((𝑎𝐵𝑑 ∈ (𝑁𝑚 𝑁)) ∧ ∀𝑤𝑏 (𝑎𝑤) = if(𝑤𝑑, 1 , 0 ))) → 𝜑)
106 simpl2 1229 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ (𝑏 ∪ {𝑐}) ∈ 𝑌) ∧ ((𝑎𝐵𝑑 ∈ (𝑁𝑚 𝑁)) ∧ ∀𝑤𝑏 (𝑎𝑤) = if(𝑤𝑑, 1 , 0 ))) → (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁))
107 simprll 764 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ (𝑏 ∪ {𝑐}) ∈ 𝑌) ∧ ((𝑎𝐵𝑑 ∈ (𝑁𝑚 𝑁)) ∧ ∀𝑤𝑏 (𝑎𝑤) = if(𝑤𝑑, 1 , 0 ))) → 𝑎𝐵)
10811, 12, 13matbas2i 20445 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑎𝐵𝑎 ∈ (𝐾𝑚 (𝑁 × 𝑁)))
109 elmapi 8031 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑎 ∈ (𝐾𝑚 (𝑁 × 𝑁)) → 𝑎:(𝑁 × 𝑁)⟶𝐾)
110108, 109syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑎𝐵𝑎:(𝑁 × 𝑁)⟶𝐾)
1111103ad2ant3 1129 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) → 𝑎:(𝑁 × 𝑁)⟶𝐾)
112111feqmptd 6391 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) → 𝑎 = (𝑒 ∈ (𝑁 × 𝑁) ↦ (𝑎𝑒)))
113112reseq1d 5533 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) → (𝑎 ↾ ({(1st𝑐)} × 𝑁)) = ((𝑒 ∈ (𝑁 × 𝑁) ↦ (𝑎𝑒)) ↾ ({(1st𝑐)} × 𝑁)))
114533ad2ant1 1127 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) → 𝑅 ∈ Ring)
115 ringgrp 18760 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 (𝑅 ∈ Ring → 𝑅 ∈ Grp)
116114, 115syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) → 𝑅 ∈ Grp)
117116adantr 466 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) ∧ 𝑒 ∈ ({(1st𝑐)} × 𝑁)) → 𝑅 ∈ Grp)
118111adantr 466 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) ∧ 𝑒 ∈ ({(1st𝑐)} × 𝑁)) → 𝑎:(𝑁 × 𝑁)⟶𝐾)
119 simp2 1131 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) → (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁))
120119unssbd 3942 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) → {𝑐} ⊆ (𝑁 × 𝑁))
121 vex 3354 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 𝑐 ∈ V
122121snss 4451 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 (𝑐 ∈ (𝑁 × 𝑁) ↔ {𝑐} ⊆ (𝑁 × 𝑁))
123120, 122sylibr 224 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) → 𝑐 ∈ (𝑁 × 𝑁))
124 xp1st 7347 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 (𝑐 ∈ (𝑁 × 𝑁) → (1st𝑐) ∈ 𝑁)
125123, 124syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) → (1st𝑐) ∈ 𝑁)
126125snssd 4475 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) → {(1st𝑐)} ⊆ 𝑁)
127 xpss1 5267 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ({(1st𝑐)} ⊆ 𝑁 → ({(1st𝑐)} × 𝑁) ⊆ (𝑁 × 𝑁))
128126, 127syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) → ({(1st𝑐)} × 𝑁) ⊆ (𝑁 × 𝑁))
129128sselda 3752 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) ∧ 𝑒 ∈ ({(1st𝑐)} × 𝑁)) → 𝑒 ∈ (𝑁 × 𝑁))
130118, 129ffvelrnd 6503 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) ∧ 𝑒 ∈ ({(1st𝑐)} × 𝑁)) → (𝑎𝑒) ∈ 𝐾)
13112, 50ringidcl 18776 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 (𝑅 ∈ Ring → 1𝐾)
132114, 131syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) → 1𝐾)
13312, 49ring0cl 18777 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 (𝑅 ∈ Ring → 0𝐾)
134114, 133syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) → 0𝐾)
135132, 134ifcld 4270 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) → if(𝑒𝑑, 1 , 0 ) ∈ 𝐾)
136135adantr 466 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) ∧ 𝑒 ∈ ({(1st𝑐)} × 𝑁)) → if(𝑒𝑑, 1 , 0 ) ∈ 𝐾)
137 eqid 2771 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 (-g𝑅) = (-g𝑅)
13812, 51, 137grpnpcan 17715 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ((𝑅 ∈ Grp ∧ (𝑎𝑒) ∈ 𝐾 ∧ if(𝑒𝑑, 1 , 0 ) ∈ 𝐾) → (((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )) + if(𝑒𝑑, 1 , 0 )) = (𝑎𝑒))
139117, 130, 136, 138syl3anc 1476 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) ∧ 𝑒 ∈ ({(1st𝑐)} × 𝑁)) → (((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )) + if(𝑒𝑑, 1 , 0 )) = (𝑎𝑒))
140139eqcomd 2777 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) ∧ 𝑒 ∈ ({(1st𝑐)} × 𝑁)) → (𝑎𝑒) = (((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )) + if(𝑒𝑑, 1 , 0 )))
141140adantr 466 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) ∧ 𝑒 ∈ ({(1st𝑐)} × 𝑁)) ∧ 𝑒 = 𝑐) → (𝑎𝑒) = (((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )) + if(𝑒𝑑, 1 , 0 )))
142 iftrue 4231 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (𝑒 = 𝑐 → if(𝑒 = 𝑐, ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )), 0 ) = ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )))
143 iftrue 4231 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (𝑒 = 𝑐 → if(𝑒 = 𝑐, if(𝑒𝑑, 1 , 0 ), (𝑎𝑒)) = if(𝑒𝑑, 1 , 0 ))
144142, 143oveq12d 6811 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (𝑒 = 𝑐 → (if(𝑒 = 𝑐, ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )), 0 ) + if(𝑒 = 𝑐, if(𝑒𝑑, 1 , 0 ), (𝑎𝑒))) = (((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )) + if(𝑒𝑑, 1 , 0 )))
145144adantl 467 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) ∧ 𝑒 ∈ ({(1st𝑐)} × 𝑁)) ∧ 𝑒 = 𝑐) → (if(𝑒 = 𝑐, ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )), 0 ) + if(𝑒 = 𝑐, if(𝑒𝑑, 1 , 0 ), (𝑎𝑒))) = (((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )) + if(𝑒𝑑, 1 , 0 )))
146141, 145eqtr4d 2808 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) ∧ 𝑒 ∈ ({(1st𝑐)} × 𝑁)) ∧ 𝑒 = 𝑐) → (𝑎𝑒) = (if(𝑒 = 𝑐, ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )), 0 ) + if(𝑒 = 𝑐, if(𝑒𝑑, 1 , 0 ), (𝑎𝑒))))
14712, 51, 49grplid 17660 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ((𝑅 ∈ Grp ∧ (𝑎𝑒) ∈ 𝐾) → ( 0 + (𝑎𝑒)) = (𝑎𝑒))
148117, 130, 147syl2anc 573 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) ∧ 𝑒 ∈ ({(1st𝑐)} × 𝑁)) → ( 0 + (𝑎𝑒)) = (𝑎𝑒))
149148eqcomd 2777 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) ∧ 𝑒 ∈ ({(1st𝑐)} × 𝑁)) → (𝑎𝑒) = ( 0 + (𝑎𝑒)))
150149adantr 466 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) ∧ 𝑒 ∈ ({(1st𝑐)} × 𝑁)) ∧ ¬ 𝑒 = 𝑐) → (𝑎𝑒) = ( 0 + (𝑎𝑒)))
151 iffalse 4234 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 𝑒 = 𝑐 → if(𝑒 = 𝑐, ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )), 0 ) = 0 )
152 iffalse 4234 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 𝑒 = 𝑐 → if(𝑒 = 𝑐, if(𝑒𝑑, 1 , 0 ), (𝑎𝑒)) = (𝑎𝑒))
153151, 152oveq12d 6811 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 𝑒 = 𝑐 → (if(𝑒 = 𝑐, ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )), 0 ) + if(𝑒 = 𝑐, if(𝑒𝑑, 1 , 0 ), (𝑎𝑒))) = ( 0 + (𝑎𝑒)))
154153adantl 467 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) ∧ 𝑒 ∈ ({(1st𝑐)} × 𝑁)) ∧ ¬ 𝑒 = 𝑐) → (if(𝑒 = 𝑐, ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )), 0 ) + if(𝑒 = 𝑐, if(𝑒𝑑, 1 , 0 ), (𝑎𝑒))) = ( 0 + (𝑎𝑒)))
155150, 154eqtr4d 2808 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) ∧ 𝑒 ∈ ({(1st𝑐)} × 𝑁)) ∧ ¬ 𝑒 = 𝑐) → (𝑎𝑒) = (if(𝑒 = 𝑐, ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )), 0 ) + if(𝑒 = 𝑐, if(𝑒𝑑, 1 , 0 ), (𝑎𝑒))))
156146, 155pm2.61dan 813 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) ∧ 𝑒 ∈ ({(1st𝑐)} × 𝑁)) → (𝑎𝑒) = (if(𝑒 = 𝑐, ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )), 0 ) + if(𝑒 = 𝑐, if(𝑒𝑑, 1 , 0 ), (𝑎𝑒))))
157156mpteq2dva 4878 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) → (𝑒 ∈ ({(1st𝑐)} × 𝑁) ↦ (𝑎𝑒)) = (𝑒 ∈ ({(1st𝑐)} × 𝑁) ↦ (if(𝑒 = 𝑐, ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )), 0 ) + if(𝑒 = 𝑐, if(𝑒𝑑, 1 , 0 ), (𝑎𝑒)))))
158 snfi 8194 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 {(1st𝑐)} ∈ Fin
15963ad2ant1 1127 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) → 𝑁 ∈ Fin)
160 xpfi 8387 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (({(1st𝑐)} ∈ Fin ∧ 𝑁 ∈ Fin) → ({(1st𝑐)} × 𝑁) ∈ Fin)
161158, 159, 160sylancr 575 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) → ({(1st𝑐)} × 𝑁) ∈ Fin)
162 ovex 6823 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )) ∈ V
163 fvex 6342 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (0g𝑅) ∈ V
16449, 163eqeltri 2846 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 0 ∈ V
165162, 164ifex 4295 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 if(𝑒 = 𝑐, ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )), 0 ) ∈ V
166165a1i 11 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) ∧ 𝑒 ∈ ({(1st𝑐)} × 𝑁)) → if(𝑒 = 𝑐, ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )), 0 ) ∈ V)
167 fvex 6342 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (1r𝑅) ∈ V
16850, 167eqeltri 2846 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 1 ∈ V
169168, 164ifex 4295 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 if(𝑒𝑑, 1 , 0 ) ∈ V
170 fvex 6342 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝑎𝑒) ∈ V
171169, 170ifex 4295 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 if(𝑒 = 𝑐, if(𝑒𝑑, 1 , 0 ), (𝑎𝑒)) ∈ V
172171a1i 11 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) ∧ 𝑒 ∈ ({(1st𝑐)} × 𝑁)) → if(𝑒 = 𝑐, if(𝑒𝑑, 1 , 0 ), (𝑎𝑒)) ∈ V)
173 xp1st 7347 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (𝑒 ∈ ({(1st𝑐)} × 𝑁) → (1st𝑒) ∈ {(1st𝑐)})
174 elsni 4333 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((1st𝑒) ∈ {(1st𝑐)} → (1st𝑒) = (1st𝑐))
175 iftrue 4231 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((1st𝑒) = (1st𝑐) → if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )), 0 ), (𝑎𝑒)) = if(𝑒 = 𝑐, ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )), 0 ))
176173, 174, 1753syl 18 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝑒 ∈ ({(1st𝑐)} × 𝑁) → if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )), 0 ), (𝑎𝑒)) = if(𝑒 = 𝑐, ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )), 0 ))
177176mpteq2ia 4874 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑒 ∈ ({(1st𝑐)} × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )), 0 ), (𝑎𝑒))) = (𝑒 ∈ ({(1st𝑐)} × 𝑁) ↦ if(𝑒 = 𝑐, ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )), 0 ))
178177a1i 11 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) → (𝑒 ∈ ({(1st𝑐)} × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )), 0 ), (𝑎𝑒))) = (𝑒 ∈ ({(1st𝑐)} × 𝑁) ↦ if(𝑒 = 𝑐, ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )), 0 )))
179 eqidd 2772 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) → (𝑒 ∈ ({(1st𝑐)} × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒𝑑, 1 , 0 ), (𝑎𝑒))) = (𝑒 ∈ ({(1st𝑐)} × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒𝑑, 1 , 0 ), (𝑎𝑒))))
180161, 166, 172, 178, 179offval2 7061 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) → ((𝑒 ∈ ({(1st𝑐)} × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )), 0 ), (𝑎𝑒))) ∘𝑓 + (𝑒 ∈ ({(1st𝑐)} × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒𝑑, 1 , 0 ), (𝑎𝑒)))) = (𝑒 ∈ ({(1st𝑐)} × 𝑁) ↦ (if(𝑒 = 𝑐, ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )), 0 ) + if(𝑒 = 𝑐, if(𝑒𝑑, 1 , 0 ), (𝑎𝑒)))))
181157, 180eqtr4d 2808 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) → (𝑒 ∈ ({(1st𝑐)} × 𝑁) ↦ (𝑎𝑒)) = ((𝑒 ∈ ({(1st𝑐)} × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )), 0 ), (𝑎𝑒))) ∘𝑓 + (𝑒 ∈ ({(1st𝑐)} × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒𝑑, 1 , 0 ), (𝑎𝑒)))))
182128resmptd 5593 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) → ((𝑒 ∈ (𝑁 × 𝑁) ↦ (𝑎𝑒)) ↾ ({(1st𝑐)} × 𝑁)) = (𝑒 ∈ ({(1st𝑐)} × 𝑁) ↦ (𝑎𝑒)))
183128resmptd 5593 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) → ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )), 0 ), (𝑎𝑒))) ↾ ({(1st𝑐)} × 𝑁)) = (𝑒 ∈ ({(1st𝑐)} × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )), 0 ), (𝑎𝑒))))
184128resmptd 5593 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) → ((𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒𝑑, 1 , 0 ), (𝑎𝑒))) ↾ ({(1st𝑐)} × 𝑁)) = (𝑒 ∈ ({(1st𝑐)} × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒𝑑, 1 , 0 ), (𝑎𝑒))))
185183, 184oveq12d 6811 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) → (((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )), 0 ), (𝑎𝑒))) ↾ ({(1st𝑐)} × 𝑁)) ∘𝑓 + ((𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒𝑑, 1 , 0 ), (𝑎𝑒))) ↾ ({(1st𝑐)} × 𝑁))) = ((𝑒 ∈ ({(1st𝑐)} × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )), 0 ), (𝑎𝑒))) ∘𝑓 + (𝑒 ∈ ({(1st𝑐)} × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒𝑑, 1 , 0 ), (𝑎𝑒)))))
186181, 182, 1853eqtr4d 2815 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) → ((𝑒 ∈ (𝑁 × 𝑁) ↦ (𝑎𝑒)) ↾ ({(1st𝑐)} × 𝑁)) = (((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )), 0 ), (𝑎𝑒))) ↾ ({(1st𝑐)} × 𝑁)) ∘𝑓 + ((𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒𝑑, 1 , 0 ), (𝑎𝑒))) ↾ ({(1st𝑐)} × 𝑁))))
187113, 186eqtrd 2805 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) → (𝑎 ↾ ({(1st𝑐)} × 𝑁)) = (((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )), 0 ), (𝑎𝑒))) ↾ ({(1st𝑐)} × 𝑁)) ∘𝑓 + ((𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒𝑑, 1 , 0 ), (𝑎𝑒))) ↾ ({(1st𝑐)} × 𝑁))))
188112reseq1d 5533 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) → (𝑎 ↾ ((𝑁 ∖ {(1st𝑐)}) × 𝑁)) = ((𝑒 ∈ (𝑁 × 𝑁) ↦ (𝑎𝑒)) ↾ ((𝑁 ∖ {(1st𝑐)}) × 𝑁)))
189 xp1st 7347 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (𝑒 ∈ ((𝑁 ∖ {(1st𝑐)}) × 𝑁) → (1st𝑒) ∈ (𝑁 ∖ {(1st𝑐)}))
190 eldifsni 4457 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((1st𝑒) ∈ (𝑁 ∖ {(1st𝑐)}) → (1st𝑒) ≠ (1st𝑐))
191189, 190syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝑒 ∈ ((𝑁 ∖ {(1st𝑐)}) × 𝑁) → (1st𝑒) ≠ (1st𝑐))
192191neneqd 2948 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑒 ∈ ((𝑁 ∖ {(1st𝑐)}) × 𝑁) → ¬ (1st𝑒) = (1st𝑐))
193192adantl 467 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) ∧ 𝑒 ∈ ((𝑁 ∖ {(1st𝑐)}) × 𝑁)) → ¬ (1st𝑒) = (1st𝑐))
194193iffalsed 4236 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) ∧ 𝑒 ∈ ((𝑁 ∖ {(1st𝑐)}) × 𝑁)) → if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )), 0 ), (𝑎𝑒)) = (𝑎𝑒))
195194mpteq2dva 4878 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) → (𝑒 ∈ ((𝑁 ∖ {(1st𝑐)}) × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )), 0 ), (𝑎𝑒))) = (𝑒 ∈ ((𝑁 ∖ {(1st𝑐)}) × 𝑁) ↦ (𝑎𝑒)))
196 difss 3888 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑁 ∖ {(1st𝑐)}) ⊆ 𝑁
197 xpss1 5267 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((𝑁 ∖ {(1st𝑐)}) ⊆ 𝑁 → ((𝑁 ∖ {(1st𝑐)}) × 𝑁) ⊆ (𝑁 × 𝑁))
198196, 197ax-mp 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝑁 ∖ {(1st𝑐)}) × 𝑁) ⊆ (𝑁 × 𝑁)
199 resmpt 5590 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (((𝑁 ∖ {(1st𝑐)}) × 𝑁) ⊆ (𝑁 × 𝑁) → ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )), 0 ), (𝑎𝑒))) ↾ ((𝑁 ∖ {(1st𝑐)}) × 𝑁)) = (𝑒 ∈ ((𝑁 ∖ {(1st𝑐)}) × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )), 0 ), (𝑎𝑒))))
200198, 199mp1i 13 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) → ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )), 0 ), (𝑎𝑒))) ↾ ((𝑁 ∖ {(1st𝑐)}) × 𝑁)) = (𝑒 ∈ ((𝑁 ∖ {(1st𝑐)}) × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )), 0 ), (𝑎𝑒))))
201 resmpt 5590 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (((𝑁 ∖ {(1st𝑐)}) × 𝑁) ⊆ (𝑁 × 𝑁) → ((𝑒 ∈ (𝑁 × 𝑁) ↦ (𝑎𝑒)) ↾ ((𝑁 ∖ {(1st𝑐)}) × 𝑁)) = (𝑒 ∈ ((𝑁 ∖ {(1st𝑐)}) × 𝑁) ↦ (𝑎𝑒)))
202198, 201mp1i 13 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) → ((𝑒 ∈ (𝑁 × 𝑁) ↦ (𝑎𝑒)) ↾ ((𝑁 ∖ {(1st𝑐)}) × 𝑁)) = (𝑒 ∈ ((𝑁 ∖ {(1st𝑐)}) × 𝑁) ↦ (𝑎𝑒)))
203195, 200, 2023eqtr4rd 2816 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) → ((𝑒 ∈ (𝑁 × 𝑁) ↦ (𝑎𝑒)) ↾ ((𝑁 ∖ {(1st𝑐)}) × 𝑁)) = ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )), 0 ), (𝑎𝑒))) ↾ ((𝑁 ∖ {(1st𝑐)}) × 𝑁)))
204188, 203eqtrd 2805 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) → (𝑎 ↾ ((𝑁 ∖ {(1st𝑐)}) × 𝑁)) = ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )), 0 ), (𝑎𝑒))) ↾ ((𝑁 ∖ {(1st𝑐)}) × 𝑁)))
205 fveq2 6332 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑒 = 𝑐 → (1st𝑒) = (1st𝑐))
206193, 205nsyl 137 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) ∧ 𝑒 ∈ ((𝑁 ∖ {(1st𝑐)}) × 𝑁)) → ¬ 𝑒 = 𝑐)
207206iffalsed 4236 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) ∧ 𝑒 ∈ ((𝑁 ∖ {(1st𝑐)}) × 𝑁)) → if(𝑒 = 𝑐, if(𝑒𝑑, 1 , 0 ), (𝑎𝑒)) = (𝑎𝑒))
208207mpteq2dva 4878 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) → (𝑒 ∈ ((𝑁 ∖ {(1st𝑐)}) × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒𝑑, 1 , 0 ), (𝑎𝑒))) = (𝑒 ∈ ((𝑁 ∖ {(1st𝑐)}) × 𝑁) ↦ (𝑎𝑒)))
209 resmpt 5590 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (((𝑁 ∖ {(1st𝑐)}) × 𝑁) ⊆ (𝑁 × 𝑁) → ((𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒𝑑, 1 , 0 ), (𝑎𝑒))) ↾ ((𝑁 ∖ {(1st𝑐)}) × 𝑁)) = (𝑒 ∈ ((𝑁 ∖ {(1st𝑐)}) × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒𝑑, 1 , 0 ), (𝑎𝑒))))
210198, 209mp1i 13 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) → ((𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒𝑑, 1 , 0 ), (𝑎𝑒))) ↾ ((𝑁 ∖ {(1st𝑐)}) × 𝑁)) = (𝑒 ∈ ((𝑁 ∖ {(1st𝑐)}) × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒𝑑, 1 , 0 ), (𝑎𝑒))))
211208, 210, 2023eqtr4rd 2816 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) → ((𝑒 ∈ (𝑁 × 𝑁) ↦ (𝑎𝑒)) ↾ ((𝑁 ∖ {(1st𝑐)}) × 𝑁)) = ((𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒𝑑, 1 , 0 ), (𝑎𝑒))) ↾ ((𝑁 ∖ {(1st𝑐)}) × 𝑁)))
212188, 211eqtrd 2805 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) → (𝑎 ↾ ((𝑁 ∖ {(1st𝑐)}) × 𝑁)) = ((𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒𝑑, 1 , 0 ), (𝑎𝑒))) ↾ ((𝑁 ∖ {(1st𝑐)}) × 𝑁)))
213135adantr 466 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) ∧ 𝑒 ∈ (𝑁 × 𝑁)) → if(𝑒𝑑, 1 , 0 ) ∈ 𝐾)
214111ffvelrnda 6502 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) ∧ 𝑒 ∈ (𝑁 × 𝑁)) → (𝑎𝑒) ∈ 𝐾)
215213, 214ifcld 4270 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) ∧ 𝑒 ∈ (𝑁 × 𝑁)) → if(𝑒 = 𝑐, if(𝑒𝑑, 1 , 0 ), (𝑎𝑒)) ∈ 𝐾)
216 eqid 2771 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒𝑑, 1 , 0 ), (𝑎𝑒))) = (𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒𝑑, 1 , 0 ), (𝑎𝑒)))
217215, 216fmptd 6527 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) → (𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒𝑑, 1 , 0 ), (𝑎𝑒))):(𝑁 × 𝑁)⟶𝐾)
218 fvex 6342 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (Base‘𝑅) ∈ V
21912, 218eqeltri 2846 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 𝐾 ∈ V
22068anidms 556 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑁 ∈ Fin → (𝑁 × 𝑁) ∈ Fin)
221159, 220syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) → (𝑁 × 𝑁) ∈ Fin)
222 elmapg 8022 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((𝐾 ∈ V ∧ (𝑁 × 𝑁) ∈ Fin) → ((𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒𝑑, 1 , 0 ), (𝑎𝑒))) ∈ (𝐾𝑚 (𝑁 × 𝑁)) ↔ (𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒𝑑, 1 , 0 ), (𝑎𝑒))):(𝑁 × 𝑁)⟶𝐾))
223219, 221, 222sylancr 575 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) → ((𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒𝑑, 1 , 0 ), (𝑎𝑒))) ∈ (𝐾𝑚 (𝑁 × 𝑁)) ↔ (𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒𝑑, 1 , 0 ), (𝑎𝑒))):(𝑁 × 𝑁)⟶𝐾))
224217, 223mpbird 247 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) → (𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒𝑑, 1 , 0 ), (𝑎𝑒))) ∈ (𝐾𝑚 (𝑁 × 𝑁)))
22511, 12matbas2 20444 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (𝐾𝑚 (𝑁 × 𝑁)) = (Base‘𝐴))
226159, 114, 225syl2anc 573 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) → (𝐾𝑚 (𝑁 × 𝑁)) = (Base‘𝐴))
227226, 13syl6eqr 2823 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) → (𝐾𝑚 (𝑁 × 𝑁)) = 𝐵)
228224, 227eleqtrd 2852 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) → (𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒𝑑, 1 , 0 ), (𝑎𝑒))) ∈ 𝐵)
229 simp3 1132 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) → 𝑎𝐵)
230116adantr 466 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) ∧ 𝑒 ∈ (𝑁 × 𝑁)) → 𝑅 ∈ Grp)
23112, 137grpsubcl 17703 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ((𝑅 ∈ Grp ∧ (𝑎𝑒) ∈ 𝐾 ∧ if(𝑒𝑑, 1 , 0 ) ∈ 𝐾) → ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )) ∈ 𝐾)
232230, 214, 213, 231syl3anc 1476 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) ∧ 𝑒 ∈ (𝑁 × 𝑁)) → ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )) ∈ 𝐾)
233134adantr 466 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) ∧ 𝑒 ∈ (𝑁 × 𝑁)) → 0𝐾)
234232, 233ifcld 4270 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) ∧ 𝑒 ∈ (𝑁 × 𝑁)) → if(𝑒 = 𝑐, ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )), 0 ) ∈ 𝐾)
235234, 214ifcld 4270 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) ∧ 𝑒 ∈ (𝑁 × 𝑁)) → if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )), 0 ), (𝑎𝑒)) ∈ 𝐾)
236 eqid 2771 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )), 0 ), (𝑎𝑒))) = (𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )), 0 ), (𝑎𝑒)))
237235, 236fmptd 6527 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) → (𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )), 0 ), (𝑎𝑒))):(𝑁 × 𝑁)⟶𝐾)
238 elmapg 8022 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((𝐾 ∈ V ∧ (𝑁 × 𝑁) ∈ Fin) → ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )), 0 ), (𝑎𝑒))) ∈ (𝐾𝑚 (𝑁 × 𝑁)) ↔ (𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )), 0 ), (𝑎𝑒))):(𝑁 × 𝑁)⟶𝐾))
239219, 221, 238sylancr 575 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) → ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )), 0 ), (𝑎𝑒))) ∈ (𝐾𝑚 (𝑁 × 𝑁)) ↔ (𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )), 0 ), (𝑎𝑒))):(𝑁 × 𝑁)⟶𝐾))
240237, 239mpbird 247 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) → (𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )), 0 ), (𝑎𝑒))) ∈ (𝐾𝑚 (𝑁 × 𝑁)))
241240, 227eleqtrd 2852 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) → (𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )), 0 ), (𝑎𝑒))) ∈ 𝐵)
242563ad2ant1 1127 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) → ∀𝑥𝐵𝑦𝐵𝑧𝐵𝑤𝑁 (((𝑥 ↾ ({𝑤} × 𝑁)) = ((𝑦 ↾ ({𝑤} × 𝑁)) ∘𝑓 + (𝑧 ↾ ({𝑤} × 𝑁))) ∧ (𝑥 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑦 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) ∧ (𝑥 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑧 ↾ ((𝑁 ∖ {𝑤}) × 𝑁))) → (𝐷𝑥) = ((𝐷𝑦) + (𝐷𝑧))))
243 reseq1 5528 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (𝑥 = 𝑎 → (𝑥 ↾ ({𝑤} × 𝑁)) = (𝑎 ↾ ({𝑤} × 𝑁)))
244243eqeq1d 2773 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝑥 = 𝑎 → ((𝑥 ↾ ({𝑤} × 𝑁)) = ((𝑦 ↾ ({𝑤} × 𝑁)) ∘𝑓 + (𝑧 ↾ ({𝑤} × 𝑁))) ↔ (𝑎 ↾ ({𝑤} × 𝑁)) = ((𝑦 ↾ ({𝑤} × 𝑁)) ∘𝑓 + (𝑧 ↾ ({𝑤} × 𝑁)))))
245 reseq1 5528 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (𝑥 = 𝑎 → (𝑥 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑎 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)))
246245eqeq1d 2773 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝑥 = 𝑎 → ((𝑥 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑦 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) ↔ (𝑎 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑦 ↾ ((𝑁 ∖ {𝑤}) × 𝑁))))
247245eqeq1d 2773 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝑥 = 𝑎 → ((𝑥 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑧 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) ↔ (𝑎 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑧 ↾ ((𝑁 ∖ {𝑤}) × 𝑁))))
248244, 246, 2473anbi123d 1547 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑥 = 𝑎 → (((𝑥 ↾ ({𝑤} × 𝑁)) = ((𝑦 ↾ ({𝑤} × 𝑁)) ∘𝑓 + (𝑧 ↾ ({𝑤} × 𝑁))) ∧ (𝑥 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑦 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) ∧ (𝑥 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑧 ↾ ((𝑁 ∖ {𝑤}) × 𝑁))) ↔ ((𝑎 ↾ ({𝑤} × 𝑁)) = ((𝑦 ↾ ({𝑤} × 𝑁)) ∘𝑓 + (𝑧 ↾ ({𝑤} × 𝑁))) ∧ (𝑎 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑦 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) ∧ (𝑎 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑧 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)))))
249 fveq2 6332 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝑥 = 𝑎 → (𝐷𝑥) = (𝐷𝑎))
250249eqeq1d 2773 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑥 = 𝑎 → ((𝐷𝑥) = ((𝐷𝑦) + (𝐷𝑧)) ↔ (𝐷𝑎) = ((𝐷𝑦) + (𝐷𝑧))))
251248, 250imbi12d 333 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑥 = 𝑎 → ((((𝑥 ↾ ({𝑤} × 𝑁)) = ((𝑦 ↾ ({𝑤} × 𝑁)) ∘𝑓 + (𝑧 ↾ ({𝑤} × 𝑁))) ∧ (𝑥 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑦 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) ∧ (𝑥 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑧 ↾ ((𝑁 ∖ {𝑤}) × 𝑁))) → (𝐷𝑥) = ((𝐷𝑦) + (𝐷𝑧))) ↔ (((𝑎 ↾ ({𝑤} × 𝑁)) = ((𝑦 ↾ ({𝑤} × 𝑁)) ∘𝑓 + (𝑧 ↾ ({𝑤} × 𝑁))) ∧ (𝑎 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑦 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) ∧ (𝑎 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑧 ↾ ((𝑁 ∖ {𝑤}) × 𝑁))) → (𝐷𝑎) = ((𝐷𝑦) + (𝐷𝑧)))))
2522512ralbidv 3138 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑥 = 𝑎 → (∀𝑧𝐵𝑤𝑁 (((𝑥 ↾ ({𝑤} × 𝑁)) = ((𝑦 ↾ ({𝑤} × 𝑁)) ∘𝑓 + (𝑧 ↾ ({𝑤} × 𝑁))) ∧ (𝑥 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑦 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) ∧ (𝑥 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑧 ↾ ((𝑁 ∖ {𝑤}) × 𝑁))) → (𝐷𝑥) = ((𝐷𝑦) + (𝐷𝑧))) ↔ ∀𝑧𝐵𝑤𝑁 (((𝑎 ↾ ({𝑤} × 𝑁)) = ((𝑦 ↾ ({𝑤} × 𝑁)) ∘𝑓 + (𝑧 ↾ ({𝑤} × 𝑁))) ∧ (𝑎 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑦 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) ∧ (𝑎 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑧 ↾ ((𝑁 ∖ {𝑤}) × 𝑁))) → (𝐷𝑎) = ((𝐷𝑦) + (𝐷𝑧)))))
253 reseq1 5528 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (𝑦 = (𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )), 0 ), (𝑎𝑒))) → (𝑦 ↾ ({𝑤} × 𝑁)) = ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )), 0 ), (𝑎𝑒))) ↾ ({𝑤} × 𝑁)))
254253oveq1d 6808 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (𝑦 = (𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )), 0 ), (𝑎𝑒))) → ((𝑦 ↾ ({𝑤} × 𝑁)) ∘𝑓 + (𝑧 ↾ ({𝑤} × 𝑁))) = (((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )), 0 ), (𝑎𝑒))) ↾ ({𝑤} × 𝑁)) ∘𝑓 + (𝑧 ↾ ({𝑤} × 𝑁))))
255254eqeq2d 2781 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝑦 = (𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )), 0 ), (𝑎𝑒))) → ((𝑎 ↾ ({𝑤} × 𝑁)) = ((𝑦 ↾ ({𝑤} × 𝑁)) ∘𝑓 + (𝑧 ↾ ({𝑤} × 𝑁))) ↔ (𝑎 ↾ ({𝑤} × 𝑁)) = (((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )), 0 ), (𝑎𝑒))) ↾ ({𝑤} × 𝑁)) ∘𝑓 + (𝑧 ↾ ({𝑤} × 𝑁)))))
256 reseq1 5528 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (𝑦 = (𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )), 0 ), (𝑎𝑒))) → (𝑦 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )), 0 ), (𝑎𝑒))) ↾ ((𝑁 ∖ {𝑤}) × 𝑁)))
257256eqeq2d 2781 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝑦 = (𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )), 0 ), (𝑎𝑒))) → ((𝑎 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑦 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) ↔ (𝑎 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )), 0 ), (𝑎𝑒))) ↾ ((𝑁 ∖ {𝑤}) × 𝑁))))
258255, 2573anbi12d 1548 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑦 = (𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )), 0 ), (𝑎𝑒))) → (((𝑎 ↾ ({𝑤} × 𝑁)) = ((𝑦 ↾ ({𝑤} × 𝑁)) ∘𝑓 + (𝑧 ↾ ({𝑤} × 𝑁))) ∧ (𝑎 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑦 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) ∧ (𝑎 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑧 ↾ ((𝑁 ∖ {𝑤}) × 𝑁))) ↔ ((𝑎 ↾ ({𝑤} × 𝑁)) = (((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )), 0 ), (𝑎𝑒))) ↾ ({𝑤} × 𝑁)) ∘𝑓 + (𝑧 ↾ ({𝑤} × 𝑁))) ∧ (𝑎 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )), 0 ), (𝑎𝑒))) ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) ∧ (𝑎 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑧 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)))))
259 fveq2 6332 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (𝑦 = (𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )), 0 ), (𝑎𝑒))) → (𝐷𝑦) = (𝐷‘(𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )), 0 ), (𝑎𝑒)))))
260259oveq1d 6808 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝑦 = (𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )), 0 ), (𝑎𝑒))) → ((𝐷𝑦) + (𝐷𝑧)) = ((𝐷‘(𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )), 0 ), (𝑎𝑒)))) + (𝐷𝑧)))
261260eqeq2d 2781 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑦 = (𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )), 0 ), (𝑎𝑒))) → ((𝐷𝑎) = ((𝐷𝑦) + (𝐷𝑧)) ↔ (𝐷𝑎) = ((𝐷‘(𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )), 0 ), (𝑎𝑒)))) + (𝐷𝑧))))
262258, 261imbi12d 333 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑦 = (𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )), 0 ), (𝑎𝑒))) → ((((𝑎 ↾ ({𝑤} × 𝑁)) = ((𝑦 ↾ ({𝑤} × 𝑁)) ∘𝑓 + (𝑧 ↾ ({𝑤} × 𝑁))) ∧ (𝑎 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑦 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) ∧ (𝑎 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑧 ↾ ((𝑁 ∖ {𝑤}) × 𝑁))) → (𝐷𝑎) = ((𝐷𝑦) + (𝐷𝑧))) ↔ (((𝑎 ↾ ({𝑤} × 𝑁)) = (((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )), 0 ), (𝑎𝑒))) ↾ ({𝑤} × 𝑁)) ∘𝑓 + (𝑧 ↾ ({𝑤} × 𝑁))) ∧ (𝑎 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )), 0 ), (𝑎𝑒))) ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) ∧ (𝑎 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑧 ↾ ((𝑁 ∖ {𝑤}) × 𝑁))) → (𝐷𝑎) = ((𝐷‘(𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )), 0 ), (𝑎𝑒)))) + (𝐷𝑧)))))
2632622ralbidv 3138 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑦 = (𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )), 0 ), (𝑎𝑒))) → (∀𝑧𝐵𝑤𝑁 (((𝑎 ↾ ({𝑤} × 𝑁)) = ((𝑦 ↾ ({𝑤} × 𝑁)) ∘𝑓 + (𝑧 ↾ ({𝑤} × 𝑁))) ∧ (𝑎 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑦 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) ∧ (𝑎 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑧 ↾ ((𝑁 ∖ {𝑤}) × 𝑁))) → (𝐷𝑎) = ((𝐷𝑦) + (𝐷𝑧))) ↔ ∀𝑧𝐵𝑤𝑁 (((𝑎 ↾ ({𝑤} × 𝑁)) = (((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )), 0 ), (𝑎𝑒))) ↾ ({𝑤} × 𝑁)) ∘𝑓 + (𝑧 ↾ ({𝑤} × 𝑁))) ∧ (𝑎 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )), 0 ), (𝑎𝑒))) ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) ∧ (𝑎 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑧 ↾ ((𝑁 ∖ {𝑤}) × 𝑁))) → (𝐷𝑎) = ((𝐷‘(𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )), 0 ), (𝑎𝑒)))) + (𝐷𝑧)))))
264252, 263rspc2va 3473 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((𝑎𝐵 ∧ (𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )), 0 ), (𝑎𝑒))) ∈ 𝐵) ∧ ∀𝑥𝐵𝑦𝐵𝑧𝐵𝑤𝑁 (((𝑥 ↾ ({𝑤} × 𝑁)) = ((𝑦 ↾ ({𝑤} × 𝑁)) ∘𝑓 + (𝑧 ↾ ({𝑤} × 𝑁))) ∧ (𝑥 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑦 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) ∧ (𝑥 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑧 ↾ ((𝑁 ∖ {𝑤}) × 𝑁))) → (𝐷𝑥) = ((𝐷𝑦) + (𝐷𝑧)))) → ∀𝑧𝐵𝑤𝑁 (((𝑎 ↾ ({𝑤} × 𝑁)) = (((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )), 0 ), (𝑎𝑒))) ↾ ({𝑤} × 𝑁)) ∘𝑓 + (𝑧 ↾ ({𝑤} × 𝑁))) ∧ (𝑎 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )), 0 ), (𝑎𝑒))) ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) ∧ (𝑎 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑧 ↾ ((𝑁 ∖ {𝑤}) × 𝑁))) → (𝐷𝑎) = ((𝐷‘(𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )), 0 ), (𝑎𝑒)))) + (𝐷𝑧))))
265229, 241, 242, 264syl21anc 1475 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) → ∀𝑧𝐵𝑤𝑁 (((𝑎 ↾ ({𝑤} × 𝑁)) = (((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )), 0 ), (𝑎𝑒))) ↾ ({𝑤} × 𝑁)) ∘𝑓 + (𝑧 ↾ ({𝑤} × 𝑁))) ∧ (𝑎 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )), 0 ), (𝑎𝑒))) ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) ∧ (𝑎 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑧 ↾ ((𝑁 ∖ {𝑤}) × 𝑁))) → (𝐷𝑎) = ((𝐷‘(𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )), 0 ), (𝑎𝑒)))) + (𝐷𝑧))))
266 reseq1 5528 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝑧 = (𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒𝑑, 1 , 0 ), (𝑎𝑒))) → (𝑧 ↾ ({𝑤} × 𝑁)) = ((𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒𝑑, 1 , 0 ), (𝑎𝑒))) ↾ ({𝑤} × 𝑁)))
267266oveq2d 6809 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑧 = (𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒𝑑, 1 , 0 ), (𝑎𝑒))) → (((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )), 0 ), (𝑎𝑒))) ↾ ({𝑤} × 𝑁)) ∘𝑓 + (𝑧 ↾ ({𝑤} × 𝑁))) = (((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )), 0 ), (𝑎𝑒))) ↾ ({𝑤} × 𝑁)) ∘𝑓 + ((𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒𝑑, 1 , 0 ), (𝑎𝑒))) ↾ ({𝑤} × 𝑁))))
268267eqeq2d 2781 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑧 = (𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒𝑑, 1 , 0 ), (𝑎𝑒))) → ((𝑎 ↾ ({𝑤} × 𝑁)) = (((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )), 0 ), (𝑎𝑒))) ↾ ({𝑤} × 𝑁)) ∘𝑓 + (𝑧 ↾ ({𝑤} × 𝑁))) ↔ (𝑎 ↾ ({𝑤} × 𝑁)) = (((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )), 0 ), (𝑎𝑒))) ↾ ({𝑤} × 𝑁)) ∘𝑓 + ((𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒𝑑, 1 , 0 ), (𝑎𝑒))) ↾ ({𝑤} × 𝑁)))))
269 reseq1 5528 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑧 = (𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒𝑑, 1 , 0 ), (𝑎𝑒))) → (𝑧 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = ((𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒𝑑, 1 , 0 ), (𝑎𝑒))) ↾ ((𝑁 ∖ {𝑤}) × 𝑁)))
270269eqeq2d 2781 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑧 = (𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒𝑑, 1 , 0 ), (𝑎𝑒))) → ((𝑎 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑧 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) ↔ (𝑎 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = ((𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒𝑑, 1 , 0 ), (𝑎𝑒))) ↾ ((𝑁 ∖ {𝑤}) × 𝑁))))
271268, 2703anbi13d 1549 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑧 = (𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒𝑑, 1 , 0 ), (𝑎𝑒))) → (((𝑎 ↾ ({𝑤} × 𝑁)) = (((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )), 0 ), (𝑎𝑒))) ↾ ({𝑤} × 𝑁)) ∘𝑓 + (𝑧 ↾ ({𝑤} × 𝑁))) ∧ (𝑎 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )), 0 ), (𝑎𝑒))) ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) ∧ (𝑎 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑧 ↾ ((𝑁 ∖ {𝑤}) × 𝑁))) ↔ ((𝑎 ↾ ({𝑤} × 𝑁)) = (((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )), 0 ), (𝑎𝑒))) ↾ ({𝑤} × 𝑁)) ∘𝑓 + ((𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒𝑑, 1 , 0 ), (𝑎𝑒))) ↾ ({𝑤} × 𝑁))) ∧ (𝑎 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )), 0 ), (𝑎𝑒))) ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) ∧ (𝑎 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = ((𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒𝑑, 1 , 0 ), (𝑎𝑒))) ↾ ((𝑁 ∖ {𝑤}) × 𝑁)))))
272 fveq2 6332 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑧 = (𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒𝑑, 1 , 0 ), (𝑎𝑒))) → (𝐷𝑧) = (𝐷‘(𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒𝑑, 1 , 0 ), (𝑎𝑒)))))
273272oveq2d 6809 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑧 = (𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒𝑑, 1 , 0 ), (𝑎𝑒))) → ((𝐷‘(𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )), 0 ), (𝑎𝑒)))) + (𝐷𝑧)) = ((𝐷‘(𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )), 0 ), (𝑎𝑒)))) + (𝐷‘(𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒𝑑, 1 , 0 ), (𝑎𝑒))))))
274273eqeq2d 2781 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑧 = (𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒𝑑, 1 , 0 ), (𝑎𝑒))) → ((𝐷𝑎) = ((𝐷‘(𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )), 0 ), (𝑎𝑒)))) + (𝐷𝑧)) ↔ (𝐷𝑎) = ((𝐷‘(𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )), 0 ), (𝑎𝑒)))) + (𝐷‘(𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒𝑑, 1 , 0 ), (𝑎𝑒)))))))
275271, 274imbi12d 333 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑧 = (𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒𝑑, 1 , 0 ), (𝑎𝑒))) → ((((𝑎 ↾ ({𝑤} × 𝑁)) = (((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )), 0 ), (𝑎𝑒))) ↾ ({𝑤} × 𝑁)) ∘𝑓 + (𝑧 ↾ ({𝑤} × 𝑁))) ∧ (𝑎 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )), 0 ), (𝑎𝑒))) ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) ∧ (𝑎 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑧 ↾ ((𝑁 ∖ {𝑤}) × 𝑁))) → (𝐷𝑎) = ((𝐷‘(𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )), 0 ), (𝑎𝑒)))) + (𝐷𝑧))) ↔ (((𝑎 ↾ ({𝑤} × 𝑁)) = (((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )), 0 ), (𝑎𝑒))) ↾ ({𝑤} × 𝑁)) ∘𝑓 + ((𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒𝑑, 1 , 0 ), (𝑎𝑒))) ↾ ({𝑤} × 𝑁))) ∧ (𝑎 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )), 0 ), (𝑎𝑒))) ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) ∧ (𝑎 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = ((𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒𝑑, 1 , 0 ), (𝑎𝑒))) ↾ ((𝑁 ∖ {𝑤}) × 𝑁))) → (𝐷𝑎) = ((𝐷‘(𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )), 0 ), (𝑎𝑒)))) + (𝐷‘(𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒𝑑, 1 , 0 ), (𝑎𝑒))))))))
276 sneq 4326 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (𝑤 = (1st𝑐) → {𝑤} = {(1st𝑐)})
277276xpeq1d 5278 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝑤 = (1st𝑐) → ({𝑤} × 𝑁) = ({(1st𝑐)} × 𝑁))
278277reseq2d 5534 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑤 = (1st𝑐) → (𝑎 ↾ ({𝑤} × 𝑁)) = (𝑎 ↾ ({(1st𝑐)} × 𝑁)))
279277reseq2d 5534 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝑤 = (1st𝑐) → ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )), 0 ), (𝑎𝑒))) ↾ ({𝑤} × 𝑁)) = ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )), 0 ), (𝑎𝑒))) ↾ ({(1st𝑐)} × 𝑁)))
280277reseq2d 5534 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝑤 = (1st𝑐) → ((𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒𝑑, 1 , 0 ), (𝑎𝑒))) ↾ ({𝑤} × 𝑁)) = ((𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒𝑑, 1 , 0 ), (𝑎𝑒))) ↾ ({(1st𝑐)} × 𝑁)))
281279, 280oveq12d 6811 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑤 = (1st𝑐) → (((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )), 0 ), (𝑎𝑒))) ↾ ({𝑤} × 𝑁)) ∘𝑓 + ((𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒𝑑, 1 , 0 ), (𝑎𝑒))) ↾ ({𝑤} × 𝑁))) = (((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )), 0 ), (𝑎𝑒))) ↾ ({(1st𝑐)} × 𝑁)) ∘𝑓 + ((𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒𝑑, 1 , 0 ), (𝑎𝑒))) ↾ ({(1st𝑐)} × 𝑁))))
282278, 281eqeq12d 2786 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑤 = (1st𝑐) → ((𝑎 ↾ ({𝑤} × 𝑁)) = (((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )), 0 ), (𝑎𝑒))) ↾ ({𝑤} × 𝑁)) ∘𝑓 + ((𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒𝑑, 1 , 0 ), (𝑎𝑒))) ↾ ({𝑤} × 𝑁))) ↔ (𝑎 ↾ ({(1st𝑐)} × 𝑁)) = (((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )), 0 ), (𝑎𝑒))) ↾ ({(1st𝑐)} × 𝑁)) ∘𝑓 + ((𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒𝑑, 1 , 0 ), (𝑎𝑒))) ↾ ({(1st𝑐)} × 𝑁)))))
283276difeq2d 3879 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (𝑤 = (1st𝑐) → (𝑁 ∖ {𝑤}) = (𝑁 ∖ {(1st𝑐)}))
284283xpeq1d 5278 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝑤 = (1st𝑐) → ((𝑁 ∖ {𝑤}) × 𝑁) = ((𝑁 ∖ {(1st𝑐)}) × 𝑁))
285284reseq2d 5534 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑤 = (1st𝑐) → (𝑎 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑎 ↾ ((𝑁 ∖ {(1st𝑐)}) × 𝑁)))
286284reseq2d 5534 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑤 = (1st𝑐) → ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )), 0 ), (𝑎𝑒))) ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )), 0 ), (𝑎𝑒))) ↾ ((𝑁 ∖ {(1st𝑐)}) × 𝑁)))
287285, 286eqeq12d 2786 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑤 = (1st𝑐) → ((𝑎 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )), 0 ), (𝑎𝑒))) ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) ↔ (𝑎 ↾ ((𝑁 ∖ {(1st𝑐)}) × 𝑁)) = ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )), 0 ), (𝑎𝑒))) ↾ ((𝑁 ∖ {(1st𝑐)}) × 𝑁))))
288284reseq2d 5534 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑤 = (1st𝑐) → ((𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒𝑑, 1 , 0 ), (𝑎𝑒))) ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = ((𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒𝑑, 1 , 0 ), (𝑎𝑒))) ↾ ((𝑁 ∖ {(1st𝑐)}) × 𝑁)))
289285, 288eqeq12d 2786 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑤 = (1st𝑐) → ((𝑎 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = ((𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒𝑑, 1 , 0 ), (𝑎𝑒))) ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) ↔ (𝑎 ↾ ((𝑁 ∖ {(1st𝑐)}) × 𝑁)) = ((𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒𝑑, 1 , 0 ), (𝑎𝑒))) ↾ ((𝑁 ∖ {(1st𝑐)}) × 𝑁))))
290282, 287, 2893anbi123d 1547 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑤 = (1st𝑐) → (((𝑎 ↾ ({𝑤} × 𝑁)) = (((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )), 0 ), (𝑎𝑒))) ↾ ({𝑤} × 𝑁)) ∘𝑓 + ((𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒𝑑, 1 , 0 ), (𝑎𝑒))) ↾ ({𝑤} × 𝑁))) ∧ (𝑎 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )), 0 ), (𝑎𝑒))) ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) ∧ (𝑎 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = ((𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒𝑑, 1 , 0 ), (𝑎𝑒))) ↾ ((𝑁 ∖ {𝑤}) × 𝑁))) ↔ ((𝑎 ↾ ({(1st𝑐)} × 𝑁)) = (((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )), 0 ), (𝑎𝑒))) ↾ ({(1st𝑐)} × 𝑁)) ∘𝑓 + ((𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒𝑑, 1 , 0 ), (𝑎𝑒))) ↾ ({(1st𝑐)} × 𝑁))) ∧ (𝑎 ↾ ((𝑁 ∖ {(1st𝑐)}) × 𝑁)) = ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )), 0 ), (𝑎𝑒))) ↾ ((𝑁 ∖ {(1st𝑐)}) × 𝑁)) ∧ (𝑎 ↾ ((𝑁 ∖ {(1st𝑐)}) × 𝑁)) = ((𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒𝑑, 1 , 0 ), (𝑎𝑒))) ↾ ((𝑁 ∖ {(1st𝑐)}) × 𝑁)))))
291290imbi1d 330 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑤 = (1st𝑐) → ((((𝑎 ↾ ({𝑤} × 𝑁)) = (((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )), 0 ), (𝑎𝑒))) ↾ ({𝑤} × 𝑁)) ∘𝑓 + ((𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒𝑑, 1 , 0 ), (𝑎𝑒))) ↾ ({𝑤} × 𝑁))) ∧ (𝑎 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )), 0 ), (𝑎𝑒))) ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) ∧ (𝑎 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = ((𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒𝑑, 1 , 0 ), (𝑎𝑒))) ↾ ((𝑁 ∖ {𝑤}) × 𝑁))) → (𝐷𝑎) = ((𝐷‘(𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )), 0 ), (𝑎𝑒)))) + (𝐷‘(𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒𝑑, 1 , 0 ), (𝑎𝑒)))))) ↔ (((𝑎 ↾ ({(1st𝑐)} × 𝑁)) = (((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )), 0 ), (𝑎𝑒))) ↾ ({(1st𝑐)} × 𝑁)) ∘𝑓 + ((𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒𝑑, 1 , 0 ), (𝑎𝑒))) ↾ ({(1st𝑐)} × 𝑁))) ∧ (𝑎 ↾ ((𝑁 ∖ {(1st𝑐)}) × 𝑁)) = ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )), 0 ), (𝑎𝑒))) ↾ ((𝑁 ∖ {(1st𝑐)}) × 𝑁)) ∧ (𝑎 ↾ ((𝑁 ∖ {(1st𝑐)}) × 𝑁)) = ((𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒𝑑, 1 , 0 ), (𝑎𝑒))) ↾ ((𝑁 ∖ {(1st𝑐)}) × 𝑁))) → (𝐷𝑎) = ((𝐷‘(𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )), 0 ), (𝑎𝑒)))) + (𝐷‘(𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒𝑑, 1 , 0 ), (𝑎𝑒))))))))
292275, 291rspc2va 3473 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((((𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒𝑑, 1 , 0 ), (𝑎𝑒))) ∈ 𝐵 ∧ (1st𝑐) ∈ 𝑁) ∧ ∀𝑧𝐵𝑤𝑁 (((𝑎 ↾ ({𝑤} × 𝑁)) = (((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )), 0 ), (𝑎𝑒))) ↾ ({𝑤} × 𝑁)) ∘𝑓 + (𝑧 ↾ ({𝑤} × 𝑁))) ∧ (𝑎 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )), 0 ), (𝑎𝑒))) ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) ∧ (𝑎 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑧 ↾ ((𝑁 ∖ {𝑤}) × 𝑁))) → (𝐷𝑎) = ((𝐷‘(𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )), 0 ), (𝑎𝑒)))) + (𝐷𝑧)))) → (((𝑎 ↾ ({(1st𝑐)} × 𝑁)) = (((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )), 0 ), (𝑎𝑒))) ↾ ({(1st𝑐)} × 𝑁)) ∘𝑓 + ((𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒𝑑, 1 , 0 ), (𝑎𝑒))) ↾ ({(1st𝑐)} × 𝑁))) ∧ (𝑎 ↾ ((𝑁 ∖ {(1st𝑐)}) × 𝑁)) = ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )), 0 ), (𝑎𝑒))) ↾ ((𝑁 ∖ {(1st𝑐)}) × 𝑁)) ∧ (𝑎 ↾ ((𝑁 ∖ {(1st𝑐)}) × 𝑁)) = ((𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒𝑑, 1 , 0 ), (𝑎𝑒))) ↾ ((𝑁 ∖ {(1st𝑐)}) × 𝑁))) → (𝐷𝑎) = ((𝐷‘(𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )), 0 ), (𝑎𝑒)))) + (𝐷‘(𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒𝑑, 1 , 0 ), (𝑎𝑒)))))))
293228, 125, 265, 292syl21anc 1475 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) → (((𝑎 ↾ ({(1st𝑐)} × 𝑁)) = (((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )), 0 ), (𝑎𝑒))) ↾ ({(1st𝑐)} × 𝑁)) ∘𝑓 + ((𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒𝑑, 1 , 0 ), (𝑎𝑒))) ↾ ({(1st𝑐)} × 𝑁))) ∧ (𝑎 ↾ ((𝑁 ∖ {(1st𝑐)}) × 𝑁)) = ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )), 0 ), (𝑎𝑒))) ↾ ((𝑁 ∖ {(1st𝑐)}) × 𝑁)) ∧ (𝑎 ↾ ((𝑁 ∖ {(1st𝑐)}) × 𝑁)) = ((𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒𝑑, 1 , 0 ), (𝑎𝑒))) ↾ ((𝑁 ∖ {(1st𝑐)}) × 𝑁))) → (𝐷𝑎) = ((𝐷‘(𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )), 0 ), (𝑎𝑒)))) + (𝐷‘(𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒𝑑, 1 , 0 ), (𝑎𝑒)))))))
294187, 204, 212, 293mp3and 1575 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) → (𝐷𝑎) = ((𝐷‘(𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )), 0 ), (𝑎𝑒)))) + (𝐷‘(𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒𝑑, 1 , 0 ), (𝑎𝑒))))))
295105, 106, 107, 294syl3anc 1476 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ (𝑏 ∪ {𝑐}) ∈ 𝑌) ∧ ((𝑎𝐵𝑑 ∈ (𝑁𝑚 𝑁)) ∧ ∀𝑤𝑏 (𝑎𝑤) = if(𝑤𝑑, 1 , 0 ))) → (𝐷𝑎) = ((𝐷‘(𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )), 0 ), (𝑎𝑒)))) + (𝐷‘(𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒𝑑, 1 , 0 ), (𝑎𝑒))))))
296 fveq2 6332 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (𝑒 = 𝑐 → (𝑎𝑒) = (𝑎𝑐))
297 elequ1 2152 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 (𝑒 = 𝑐 → (𝑒𝑑𝑐𝑑))
298297ifbid 4247 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (𝑒 = 𝑐 → if(𝑒𝑑, 1 , 0 ) = if(𝑐𝑑, 1 , 0 ))
299296, 298oveq12d 6811 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (𝑒 = 𝑐 → ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )) = ((𝑎𝑐)(-g𝑅)if(𝑐𝑑, 1 , 0 )))
300299adantl 467 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) ∧ 𝑒 ∈ ({(1st𝑐)} × 𝑁)) ∧ 𝑒 = 𝑐) → ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )) = ((𝑎𝑐)(-g𝑅)if(𝑐𝑑, 1 , 0 )))
301111, 123ffvelrnd 6503 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) → (𝑎𝑐) ∈ 𝐾)
302132, 134ifcld 4270 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) → if(𝑐𝑑, 1 , 0 ) ∈ 𝐾)
30312, 137grpsubcl 17703 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ((𝑅 ∈ Grp ∧ (𝑎𝑐) ∈ 𝐾 ∧ if(𝑐𝑑, 1 , 0 ) ∈ 𝐾) → ((𝑎𝑐)(-g𝑅)if(𝑐𝑑, 1 , 0 )) ∈ 𝐾)
304116, 301, 302, 303syl3anc 1476 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) → ((𝑎𝑐)(-g𝑅)if(𝑐𝑑, 1 , 0 )) ∈ 𝐾)
30512, 52, 50ringridm 18780 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ((𝑅 ∈ Ring ∧ ((𝑎𝑐)(-g𝑅)if(𝑐𝑑, 1 , 0 )) ∈ 𝐾) → (((𝑎𝑐)(-g𝑅)if(𝑐𝑑, 1 , 0 )) · 1 ) = ((𝑎𝑐)(-g𝑅)if(𝑐𝑑, 1 , 0 )))
306114, 304, 305syl2anc 573 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) → (((𝑎𝑐)(-g𝑅)if(𝑐𝑑, 1 , 0 )) · 1 ) = ((𝑎𝑐)(-g𝑅)if(𝑐𝑑, 1 , 0 )))
307306ad2antrr 705 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) ∧ 𝑒 ∈ ({(1st𝑐)} × 𝑁)) ∧ 𝑒 = 𝑐) → (((𝑎𝑐)(-g𝑅)if(𝑐𝑑, 1 , 0 )) · 1 ) = ((𝑎𝑐)(-g𝑅)if(𝑐𝑑, 1 , 0 )))
308300, 307eqtr4d 2808 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) ∧ 𝑒 ∈ ({(1st𝑐)} × 𝑁)) ∧ 𝑒 = 𝑐) → ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )) = (((𝑎𝑐)(-g𝑅)if(𝑐𝑑, 1 , 0 )) · 1 ))
309142adantl 467 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) ∧ 𝑒 ∈ ({(1st𝑐)} × 𝑁)) ∧ 𝑒 = 𝑐) → if(𝑒 = 𝑐, ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )), 0 ) = ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )))
310 iftrue 4231 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (𝑒 = 𝑐 → if(𝑒 = 𝑐, 1 , 0 ) = 1 )
311310oveq2d 6809 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (𝑒 = 𝑐 → (((𝑎𝑐)(-g𝑅)if(𝑐𝑑, 1 , 0 )) · if(𝑒 = 𝑐, 1 , 0 )) = (((𝑎𝑐)(-g𝑅)if(𝑐𝑑, 1 , 0 )) · 1 ))
312311adantl 467 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) ∧ 𝑒 ∈ ({(1st𝑐)} × 𝑁)) ∧ 𝑒 = 𝑐) → (((𝑎𝑐)(-g𝑅)if(𝑐𝑑, 1 , 0 )) · if(𝑒 = 𝑐, 1 , 0 )) = (((𝑎𝑐)(-g𝑅)if(𝑐𝑑, 1 , 0 )) · 1 ))
313308, 309, 3123eqtr4d 2815 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) ∧ 𝑒 ∈ ({(1st𝑐)} × 𝑁)) ∧ 𝑒 = 𝑐) → if(𝑒 = 𝑐, ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )), 0 ) = (((𝑎𝑐)(-g𝑅)if(𝑐𝑑, 1 , 0 )) · if(𝑒 = 𝑐, 1 , 0 )))
31412, 52, 49ringrz 18796 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ((𝑅 ∈ Ring ∧ ((𝑎𝑐)(-g𝑅)if(𝑐𝑑, 1 , 0 )) ∈ 𝐾) → (((𝑎𝑐)(-g𝑅)if(𝑐𝑑, 1 , 0 )) · 0 ) = 0 )
315114, 304, 314syl2anc 573 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) → (((𝑎𝑐)(-g𝑅)if(𝑐𝑑, 1 , 0 )) · 0 ) = 0 )
316315eqcomd 2777 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) → 0 = (((𝑎𝑐)(-g𝑅)if(𝑐𝑑, 1 , 0 )) · 0 ))
317316ad2antrr 705 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) ∧ 𝑒 ∈ ({(1st𝑐)} × 𝑁)) ∧ ¬ 𝑒 = 𝑐) → 0 = (((𝑎𝑐)(-g𝑅)if(𝑐𝑑, 1 , 0 )) · 0 ))
318151adantl 467 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) ∧ 𝑒 ∈ ({(1st𝑐)} × 𝑁)) ∧ ¬ 𝑒 = 𝑐) → if(𝑒 = 𝑐, ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )), 0 ) = 0 )
319 iffalse 4234 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 𝑒 = 𝑐 → if(𝑒 = 𝑐, 1 , 0 ) = 0 )
320319oveq2d 6809 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 𝑒 = 𝑐 → (((𝑎𝑐)(-g𝑅)if(𝑐𝑑, 1 , 0 )) · if(𝑒 = 𝑐, 1 , 0 )) = (((𝑎𝑐)(-g𝑅)if(𝑐𝑑, 1 , 0 )) · 0 ))
321320adantl 467 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) ∧ 𝑒 ∈ ({(1st𝑐)} × 𝑁)) ∧ ¬ 𝑒 = 𝑐) → (((𝑎𝑐)(-g𝑅)if(𝑐𝑑, 1 , 0 )) · if(𝑒 = 𝑐, 1 , 0 )) = (((𝑎𝑐)(-g𝑅)if(𝑐𝑑, 1 , 0 )) · 0 ))
322317, 318, 3213eqtr4d 2815 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) ∧ 𝑒 ∈ ({(1st𝑐)} × 𝑁)) ∧ ¬ 𝑒 = 𝑐) → if(𝑒 = 𝑐, ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )), 0 ) = (((𝑎𝑐)(-g𝑅)if(𝑐𝑑, 1 , 0 )) · if(𝑒 = 𝑐, 1 , 0 )))
323313, 322pm2.61dan 813 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) ∧ 𝑒 ∈ ({(1st𝑐)} × 𝑁)) → if(𝑒 = 𝑐, ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )), 0 ) = (((𝑎𝑐)(-g𝑅)if(𝑐𝑑, 1 , 0 )) · if(𝑒 = 𝑐, 1 , 0 )))
324173adantl 467 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) ∧ 𝑒 ∈ ({(1st𝑐)} × 𝑁)) → (1st𝑒) ∈ {(1st𝑐)})
325324, 174syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) ∧ 𝑒 ∈ ({(1st𝑐)} × 𝑁)) → (1st𝑒) = (1st𝑐))
326325iftrued 4233 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) ∧ 𝑒 ∈ ({(1st𝑐)} × 𝑁)) → if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )), 0 ), (𝑎𝑒)) = if(𝑒 = 𝑐, ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )), 0 ))
327325iftrued 4233 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) ∧ 𝑒 ∈ ({(1st𝑐)} × 𝑁)) → if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎𝑒)) = if(𝑒 = 𝑐, 1 , 0 ))
328327oveq2d 6809 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) ∧ 𝑒 ∈ ({(1st𝑐)} × 𝑁)) → (((𝑎𝑐)(-g𝑅)if(𝑐𝑑, 1 , 0 )) · if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎𝑒))) = (((𝑎𝑐)(-g𝑅)if(𝑐𝑑, 1 , 0 )) · if(𝑒 = 𝑐, 1 , 0 )))
329323, 326, 3283eqtr4d 2815 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) ∧ 𝑒 ∈ ({(1st𝑐)} × 𝑁)) → if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )), 0 ), (𝑎𝑒)) = (((𝑎𝑐)(-g𝑅)if(𝑐𝑑, 1 , 0 )) · if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎𝑒))))
330329mpteq2dva 4878 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) → (𝑒 ∈ ({(1st𝑐)} × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )), 0 ), (𝑎𝑒))) = (𝑒 ∈ ({(1st𝑐)} × 𝑁) ↦ (((𝑎𝑐)(-g𝑅)if(𝑐𝑑, 1 , 0 )) · if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎𝑒)))))
331 ovexd 6825 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) ∧ 𝑒 ∈ ({(1st𝑐)} × 𝑁)) → ((𝑎𝑐)(-g𝑅)if(𝑐𝑑, 1 , 0 )) ∈ V)
332168, 164ifex 4295 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 if(𝑒 = 𝑐, 1 , 0 ) ∈ V
333332, 170ifex 4295 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎𝑒)) ∈ V
334333a1i 11 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) ∧ 𝑒 ∈ ({(1st𝑐)} × 𝑁)) → if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎𝑒)) ∈ V)
335 fconstmpt 5303 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (({(1st𝑐)} × 𝑁) × {((𝑎𝑐)(-g𝑅)if(𝑐𝑑, 1 , 0 ))}) = (𝑒 ∈ ({(1st𝑐)} × 𝑁) ↦ ((𝑎𝑐)(-g𝑅)if(𝑐𝑑, 1 , 0 )))
336335a1i 11 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) → (({(1st𝑐)} × 𝑁) × {((𝑎𝑐)(-g𝑅)if(𝑐𝑑, 1 , 0 ))}) = (𝑒 ∈ ({(1st𝑐)} × 𝑁) ↦ ((𝑎𝑐)(-g𝑅)if(𝑐𝑑, 1 , 0 ))))
337128resmptd 5593 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) → ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎𝑒))) ↾ ({(1st𝑐)} × 𝑁)) = (𝑒 ∈ ({(1st𝑐)} × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎𝑒))))
338161, 331, 334, 336, 337offval2 7061 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) → ((({(1st𝑐)} × 𝑁) × {((𝑎𝑐)(-g𝑅)if(𝑐𝑑, 1 , 0 ))}) ∘𝑓 · ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎𝑒))) ↾ ({(1st𝑐)} × 𝑁))) = (𝑒 ∈ ({(1st𝑐)} × 𝑁) ↦ (((𝑎𝑐)(-g𝑅)if(𝑐𝑑, 1 , 0 )) · if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎𝑒)))))
339330, 183, 3383eqtr4d 2815 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) → ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )), 0 ), (𝑎𝑒))) ↾ ({(1st𝑐)} × 𝑁)) = ((({(1st𝑐)} × 𝑁) × {((𝑎𝑐)(-g𝑅)if(𝑐𝑑, 1 , 0 ))}) ∘𝑓 · ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎𝑒))) ↾ ({(1st𝑐)} × 𝑁))))
340 iffalse 4234 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (¬ (1st𝑒) = (1st𝑐) → if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )), 0 ), (𝑎𝑒)) = (𝑎𝑒))
341 iffalse 4234 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (¬ (1st𝑒) = (1st𝑐) → if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎𝑒)) = (𝑎𝑒))
342340, 341eqtr4d 2808 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (¬ (1st𝑒) = (1st𝑐) → if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )), 0 ), (𝑎𝑒)) = if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎𝑒)))
343193, 342syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) ∧ 𝑒 ∈ ((𝑁 ∖ {(1st𝑐)}) × 𝑁)) → if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )), 0 ), (𝑎𝑒)) = if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎𝑒)))
344343mpteq2dva 4878 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) → (𝑒 ∈ ((𝑁 ∖ {(1st𝑐)}) × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )), 0 ), (𝑎𝑒))) = (𝑒 ∈ ((𝑁 ∖ {(1st𝑐)}) × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎𝑒))))
345 resmpt 5590 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (((𝑁 ∖ {(1st𝑐)}) × 𝑁) ⊆ (𝑁 × 𝑁) → ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎𝑒))) ↾ ((𝑁 ∖ {(1st𝑐)}) × 𝑁)) = (𝑒 ∈ ((𝑁 ∖ {(1st𝑐)}) × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎𝑒))))
346198, 345mp1i 13 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) → ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎𝑒))) ↾ ((𝑁 ∖ {(1st𝑐)}) × 𝑁)) = (𝑒 ∈ ((𝑁 ∖ {(1st𝑐)}) × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎𝑒))))
347344, 200, 3463eqtr4d 2815 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) → ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )), 0 ), (𝑎𝑒))) ↾ ((𝑁 ∖ {(1st𝑐)}) × 𝑁)) = ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎𝑒))) ↾ ((𝑁 ∖ {(1st𝑐)}) × 𝑁)))
348132, 134ifcld 4270 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) → if(𝑒 = 𝑐, 1 , 0 ) ∈ 𝐾)
349348adantr 466 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) ∧ 𝑒 ∈ (𝑁 × 𝑁)) → if(𝑒 = 𝑐, 1 , 0 ) ∈ 𝐾)
350349, 214ifcld 4270 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) ∧ 𝑒 ∈ (𝑁 × 𝑁)) → if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎𝑒)) ∈ 𝐾)
351 eqid 2771 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎𝑒))) = (𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎𝑒)))
352350, 351fmptd 6527 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) → (𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎𝑒))):(𝑁 × 𝑁)⟶𝐾)
353 elmapg 8022 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((𝐾 ∈ V ∧ (𝑁 × 𝑁) ∈ Fin) → ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎𝑒))) ∈ (𝐾𝑚 (𝑁 × 𝑁)) ↔ (𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎𝑒))):(𝑁 × 𝑁)⟶𝐾))
354219, 221, 353sylancr 575 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) → ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎𝑒))) ∈ (𝐾𝑚 (𝑁 × 𝑁)) ↔ (𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎𝑒))):(𝑁 × 𝑁)⟶𝐾))
355352, 354mpbird 247 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) → (𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎𝑒))) ∈ (𝐾𝑚 (𝑁 × 𝑁)))
356355, 227eleqtrd 2852 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) → (𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎𝑒))) ∈ 𝐵)
357573ad2ant1 1127 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) → ∀𝑥𝐵𝑦𝐾𝑧𝐵𝑤𝑁 (((𝑥 ↾ ({𝑤} × 𝑁)) = ((({𝑤} × 𝑁) × {𝑦}) ∘𝑓 · (𝑧 ↾ ({𝑤} × 𝑁))) ∧ (𝑥 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑧 ↾ ((𝑁 ∖ {𝑤}) × 𝑁))) → (𝐷𝑥) = (𝑦 · (𝐷𝑧))))
358 reseq1 5528 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (𝑥 = (𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )), 0 ), (𝑎𝑒))) → (𝑥 ↾ ({𝑤} × 𝑁)) = ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )), 0 ), (𝑎𝑒))) ↾ ({𝑤} × 𝑁)))
359358eqeq1d 2773 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (𝑥 = (𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )), 0 ), (𝑎𝑒))) → ((𝑥 ↾ ({𝑤} × 𝑁)) = ((({𝑤} × 𝑁) × {𝑦}) ∘𝑓 · (𝑧 ↾ ({𝑤} × 𝑁))) ↔ ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )), 0 ), (𝑎𝑒))) ↾ ({𝑤} × 𝑁)) = ((({𝑤} × 𝑁) × {𝑦}) ∘𝑓 · (𝑧 ↾ ({𝑤} × 𝑁)))))
360 reseq1 5528 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (𝑥 = (𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )), 0 ), (𝑎𝑒))) → (𝑥 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )), 0 ), (𝑎𝑒))) ↾ ((𝑁 ∖ {𝑤}) × 𝑁)))
361360eqeq1d 2773 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (𝑥 = (𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )), 0 ), (𝑎𝑒))) → ((𝑥 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑧 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) ↔ ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )), 0 ), (𝑎𝑒))) ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑧 ↾ ((𝑁 ∖ {𝑤}) × 𝑁))))
362359, 361anbi12d 616 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝑥 = (𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )), 0 ), (𝑎𝑒))) → (((𝑥 ↾ ({𝑤} × 𝑁)) = ((({𝑤} × 𝑁) × {𝑦}) ∘𝑓 · (𝑧 ↾ ({𝑤} × 𝑁))) ∧ (𝑥 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑧 ↾ ((𝑁 ∖ {𝑤}) × 𝑁))) ↔ (((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )), 0 ), (𝑎𝑒))) ↾ ({𝑤} × 𝑁)) = ((({𝑤} × 𝑁) × {𝑦}) ∘𝑓 · (𝑧 ↾ ({𝑤} × 𝑁))) ∧ ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )), 0 ), (𝑎𝑒))) ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑧 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)))))
363 fveq2 6332 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (𝑥 = (𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )), 0 ), (𝑎𝑒))) → (𝐷𝑥) = (𝐷‘(𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )), 0 ), (𝑎𝑒)))))
364363eqeq1d 2773 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝑥 = (𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )), 0 ), (𝑎𝑒))) → ((𝐷𝑥) = (𝑦 · (𝐷𝑧)) ↔ (𝐷‘(𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )), 0 ), (𝑎𝑒)))) = (𝑦 · (𝐷𝑧))))
365362, 364imbi12d 333 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑥 = (𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )), 0 ), (𝑎𝑒))) → ((((𝑥 ↾ ({𝑤} × 𝑁)) = ((({𝑤} × 𝑁) × {𝑦}) ∘𝑓 · (𝑧 ↾ ({𝑤} × 𝑁))) ∧ (𝑥 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑧 ↾ ((𝑁 ∖ {𝑤}) × 𝑁))) → (𝐷𝑥) = (𝑦 · (𝐷𝑧))) ↔ ((((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )), 0 ), (𝑎𝑒))) ↾ ({𝑤} × 𝑁)) = ((({𝑤} × 𝑁) × {𝑦}) ∘𝑓 · (𝑧 ↾ ({𝑤} × 𝑁))) ∧ ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )), 0 ), (𝑎𝑒))) ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑧 ↾ ((𝑁 ∖ {𝑤}) × 𝑁))) → (𝐷‘(𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )), 0 ), (𝑎𝑒)))) = (𝑦 · (𝐷𝑧)))))
3663652ralbidv 3138 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑥 = (𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )), 0 ), (𝑎𝑒))) → (∀𝑧𝐵𝑤𝑁 (((𝑥 ↾ ({𝑤} × 𝑁)) = ((({𝑤} × 𝑁) × {𝑦}) ∘𝑓 · (𝑧 ↾ ({𝑤} × 𝑁))) ∧ (𝑥 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑧 ↾ ((𝑁 ∖ {𝑤}) × 𝑁))) → (𝐷𝑥) = (𝑦 · (𝐷𝑧))) ↔ ∀𝑧𝐵𝑤𝑁 ((((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )), 0 ), (𝑎𝑒))) ↾ ({𝑤} × 𝑁)) = ((({𝑤} × 𝑁) × {𝑦}) ∘𝑓 · (𝑧 ↾ ({𝑤} × 𝑁))) ∧ ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )), 0 ), (𝑎𝑒))) ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑧 ↾ ((𝑁 ∖ {𝑤}) × 𝑁))) → (𝐷‘(𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )), 0 ), (𝑎𝑒)))) = (𝑦 · (𝐷𝑧)))))
367 sneq 4326 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 (𝑦 = ((𝑎𝑐)(-g𝑅)if(𝑐𝑑, 1 , 0 )) → {𝑦} = {((𝑎𝑐)(-g𝑅)if(𝑐𝑑, 1 , 0 ))})
368367xpeq2d 5279 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (𝑦 = ((𝑎𝑐)(-g𝑅)if(𝑐𝑑, 1 , 0 )) → (({𝑤} × 𝑁) × {𝑦}) = (({𝑤} × 𝑁) × {((𝑎𝑐)(-g𝑅)if(𝑐𝑑, 1 , 0 ))}))
369368oveq1d 6808 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (𝑦 = ((𝑎𝑐)(-g𝑅)if(𝑐𝑑, 1 , 0 )) → ((({𝑤} × 𝑁) × {𝑦}) ∘𝑓 · (𝑧 ↾ ({𝑤} × 𝑁))) = ((({𝑤} × 𝑁) × {((𝑎𝑐)(-g𝑅)if(𝑐𝑑, 1 , 0 ))}) ∘𝑓 · (𝑧 ↾ ({𝑤} × 𝑁))))
370369eqeq2d 2781 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (𝑦 = ((𝑎𝑐)(-g𝑅)if(𝑐𝑑, 1 , 0 )) → (((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )), 0 ), (𝑎𝑒))) ↾ ({𝑤} × 𝑁)) = ((({𝑤} × 𝑁) × {𝑦}) ∘𝑓 · (𝑧 ↾ ({𝑤} × 𝑁))) ↔ ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )), 0 ), (𝑎𝑒))) ↾ ({𝑤} × 𝑁)) = ((({𝑤} × 𝑁) × {((𝑎𝑐)(-g𝑅)if(𝑐𝑑, 1 , 0 ))}) ∘𝑓 · (𝑧 ↾ ({𝑤} × 𝑁)))))
371370anbi1d 615 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝑦 = ((𝑎𝑐)(-g𝑅)if(𝑐𝑑, 1 , 0 )) → ((((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )), 0 ), (𝑎𝑒))) ↾ ({𝑤} × 𝑁)) = ((({𝑤} × 𝑁) × {𝑦}) ∘𝑓 · (𝑧 ↾ ({𝑤} × 𝑁))) ∧ ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )), 0 ), (𝑎𝑒))) ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑧 ↾ ((𝑁 ∖ {𝑤}) × 𝑁))) ↔ (((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )), 0 ), (𝑎𝑒))) ↾ ({𝑤} × 𝑁)) = ((({𝑤} × 𝑁) × {((𝑎𝑐)(-g𝑅)if(𝑐𝑑, 1 , 0 ))}) ∘𝑓 · (𝑧 ↾ ({𝑤} × 𝑁))) ∧ ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )), 0 ), (𝑎𝑒))) ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑧 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)))))
372 oveq1 6800 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (𝑦 = ((𝑎𝑐)(-g𝑅)if(𝑐𝑑, 1 , 0 )) → (𝑦 · (𝐷𝑧)) = (((𝑎𝑐)(-g𝑅)if(𝑐𝑑, 1 , 0 )) · (𝐷𝑧)))
373372eqeq2d 2781 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝑦 = ((𝑎𝑐)(-g𝑅)if(𝑐𝑑, 1 , 0 )) → ((𝐷‘(𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )), 0 ), (𝑎𝑒)))) = (𝑦 · (𝐷𝑧)) ↔ (𝐷‘(𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )), 0 ), (𝑎𝑒)))) = (((𝑎𝑐)(-g𝑅)if(𝑐𝑑, 1 , 0 )) · (𝐷𝑧))))
374371, 373imbi12d 333 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑦 = ((𝑎𝑐)(-g𝑅)if(𝑐𝑑, 1 , 0 )) → (((((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )), 0 ), (𝑎𝑒))) ↾ ({𝑤} × 𝑁)) = ((({𝑤} × 𝑁) × {𝑦}) ∘𝑓 · (𝑧 ↾ ({𝑤} × 𝑁))) ∧ ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )), 0 ), (𝑎𝑒))) ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑧 ↾ ((𝑁 ∖ {𝑤}) × 𝑁))) → (𝐷‘(𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )), 0 ), (𝑎𝑒)))) = (𝑦 · (𝐷𝑧))) ↔ ((((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )), 0 ), (𝑎𝑒))) ↾ ({𝑤} × 𝑁)) = ((({𝑤} × 𝑁) × {((𝑎𝑐)(-g𝑅)if(𝑐𝑑, 1 , 0 ))}) ∘𝑓 · (𝑧 ↾ ({𝑤} × 𝑁))) ∧ ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )), 0 ), (𝑎𝑒))) ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑧 ↾ ((𝑁 ∖ {𝑤}) × 𝑁))) → (𝐷‘(𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )), 0 ), (𝑎𝑒)))) = (((𝑎𝑐)(-g𝑅)if(𝑐𝑑, 1 , 0 )) · (𝐷𝑧)))))
3753742ralbidv 3138 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑦 = ((𝑎𝑐)(-g𝑅)if(𝑐𝑑, 1 , 0 )) → (∀𝑧𝐵𝑤𝑁 ((((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )), 0 ), (𝑎𝑒))) ↾ ({𝑤} × 𝑁)) = ((({𝑤} × 𝑁) × {𝑦}) ∘𝑓 · (𝑧 ↾ ({𝑤} × 𝑁))) ∧ ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )), 0 ), (𝑎𝑒))) ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑧 ↾ ((𝑁 ∖ {𝑤}) × 𝑁))) → (𝐷‘(𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )), 0 ), (𝑎𝑒)))) = (𝑦 · (𝐷𝑧))) ↔ ∀𝑧𝐵𝑤𝑁 ((((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )), 0 ), (𝑎𝑒))) ↾ ({𝑤} × 𝑁)) = ((({𝑤} × 𝑁) × {((𝑎𝑐)(-g𝑅)if(𝑐𝑑, 1 , 0 ))}) ∘𝑓 · (𝑧 ↾ ({𝑤} × 𝑁))) ∧ ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )), 0 ), (𝑎𝑒))) ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑧 ↾ ((𝑁 ∖ {𝑤}) × 𝑁))) → (𝐷‘(𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )), 0 ), (𝑎𝑒)))) = (((𝑎𝑐)(-g𝑅)if(𝑐𝑑, 1 , 0 )) · (𝐷𝑧)))))
376366, 375rspc2va 3473 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )), 0 ), (𝑎𝑒))) ∈ 𝐵 ∧ ((𝑎𝑐)(-g𝑅)if(𝑐𝑑, 1 , 0 )) ∈ 𝐾) ∧ ∀𝑥𝐵𝑦𝐾𝑧𝐵𝑤𝑁 (((𝑥 ↾ ({𝑤} × 𝑁)) = ((({𝑤} × 𝑁) × {𝑦}) ∘𝑓 · (𝑧 ↾ ({𝑤} × 𝑁))) ∧ (𝑥 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑧 ↾ ((𝑁 ∖ {𝑤}) × 𝑁))) → (𝐷𝑥) = (𝑦 · (𝐷𝑧)))) → ∀𝑧𝐵𝑤𝑁 ((((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )), 0 ), (𝑎𝑒))) ↾ ({𝑤} × 𝑁)) = ((({𝑤} × 𝑁) × {((𝑎𝑐)(-g𝑅)if(𝑐𝑑, 1 , 0 ))}) ∘𝑓 · (𝑧 ↾ ({𝑤} × 𝑁))) ∧ ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )), 0 ), (𝑎𝑒))) ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑧 ↾ ((𝑁 ∖ {𝑤}) × 𝑁))) → (𝐷‘(𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )), 0 ), (𝑎𝑒)))) = (((𝑎𝑐)(-g𝑅)if(𝑐𝑑, 1 , 0 )) · (𝐷𝑧))))
377241, 304, 357, 376syl21anc 1475 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) → ∀𝑧𝐵𝑤𝑁 ((((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )), 0 ), (𝑎𝑒))) ↾ ({𝑤} × 𝑁)) = ((({𝑤} × 𝑁) × {((𝑎𝑐)(-g𝑅)if(𝑐𝑑, 1 , 0 ))}) ∘𝑓 · (𝑧 ↾ ({𝑤} × 𝑁))) ∧ ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )), 0 ), (𝑎𝑒))) ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑧 ↾ ((𝑁 ∖ {𝑤}) × 𝑁))) → (𝐷‘(𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )), 0 ), (𝑎𝑒)))) = (((𝑎𝑐)(-g𝑅)if(𝑐𝑑, 1 , 0 )) · (𝐷𝑧))))
378 reseq1 5528 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (𝑧 = (𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎𝑒))) → (𝑧 ↾ ({𝑤} × 𝑁)) = ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎𝑒))) ↾ ({𝑤} × 𝑁)))
379378oveq2d 6809 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝑧 = (𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎𝑒))) → ((({𝑤} × 𝑁) × {((𝑎𝑐)(-g𝑅)if(𝑐𝑑, 1 , 0 ))}) ∘𝑓 · (𝑧 ↾ ({𝑤} × 𝑁))) = ((({𝑤} × 𝑁) × {((𝑎𝑐)(-g𝑅)if(𝑐𝑑, 1 , 0 ))}) ∘𝑓 · ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎𝑒))) ↾ ({𝑤} × 𝑁))))
380379eqeq2d 2781 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑧 = (𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎𝑒))) → (((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )), 0 ), (𝑎𝑒))) ↾ ({𝑤} × 𝑁)) = ((({𝑤} × 𝑁) × {((𝑎𝑐)(-g𝑅)if(𝑐𝑑, 1 , 0 ))}) ∘𝑓 · (𝑧 ↾ ({𝑤} × 𝑁))) ↔ ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )), 0 ), (𝑎𝑒))) ↾ ({𝑤} × 𝑁)) = ((({𝑤} × 𝑁) × {((𝑎𝑐)(-g𝑅)if(𝑐𝑑, 1 , 0 ))}) ∘𝑓 · ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎𝑒))) ↾ ({𝑤} × 𝑁)))))
381 reseq1 5528 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝑧 = (𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎𝑒))) → (𝑧 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎𝑒))) ↾ ((𝑁 ∖ {𝑤}) × 𝑁)))
382381eqeq2d 2781 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑧 = (𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎𝑒))) → (((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )), 0 ), (𝑎𝑒))) ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑧 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) ↔ ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )), 0 ), (𝑎𝑒))) ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎𝑒))) ↾ ((𝑁 ∖ {𝑤}) × 𝑁))))
383380, 382anbi12d 616 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑧 = (𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎𝑒))) → ((((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )), 0 ), (𝑎𝑒))) ↾ ({𝑤} × 𝑁)) = ((({𝑤} × 𝑁) × {((𝑎𝑐)(-g𝑅)if(𝑐𝑑, 1 , 0 ))}) ∘𝑓 · (𝑧 ↾ ({𝑤} × 𝑁))) ∧ ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )), 0 ), (𝑎𝑒))) ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑧 ↾ ((𝑁 ∖ {𝑤}) × 𝑁))) ↔ (((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )), 0 ), (𝑎𝑒))) ↾ ({𝑤} × 𝑁)) = ((({𝑤} × 𝑁) × {((𝑎𝑐)(-g𝑅)if(𝑐𝑑, 1 , 0 ))}) ∘𝑓 · ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎𝑒))) ↾ ({𝑤} × 𝑁))) ∧ ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )), 0 ), (𝑎𝑒))) ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎𝑒))) ↾ ((𝑁 ∖ {𝑤}) × 𝑁)))))
384 fveq2 6332 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝑧 = (𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎𝑒))) → (𝐷𝑧) = (𝐷‘(𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎𝑒)))))
385384oveq2d 6809 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑧 = (𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎𝑒))) → (((𝑎𝑐)(-g𝑅)if(𝑐𝑑, 1 , 0 )) · (𝐷𝑧)) = (((𝑎𝑐)(-g𝑅)if(𝑐𝑑, 1 , 0 )) · (𝐷‘(𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎𝑒))))))
386385eqeq2d 2781 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑧 = (𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎𝑒))) → ((𝐷‘(𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )), 0 ), (𝑎𝑒)))) = (((𝑎𝑐)(-g𝑅)if(𝑐𝑑, 1 , 0 )) · (𝐷𝑧)) ↔ (𝐷‘(𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )), 0 ), (𝑎𝑒)))) = (((𝑎𝑐)(-g𝑅)if(𝑐𝑑, 1 , 0 )) · (𝐷‘(𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎𝑒)))))))
387383, 386imbi12d 333 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑧 = (𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎𝑒))) → (((((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )), 0 ), (𝑎𝑒))) ↾ ({𝑤} × 𝑁)) = ((({𝑤} × 𝑁) × {((𝑎𝑐)(-g𝑅)if(𝑐𝑑, 1 , 0 ))}) ∘𝑓 · (𝑧 ↾ ({𝑤} × 𝑁))) ∧ ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )), 0 ), (𝑎𝑒))) ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑧 ↾ ((𝑁 ∖ {𝑤}) × 𝑁))) → (𝐷‘(𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )), 0 ), (𝑎𝑒)))) = (((𝑎𝑐)(-g𝑅)if(𝑐𝑑, 1 , 0 )) · (𝐷𝑧))) ↔ ((((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )), 0 ), (𝑎𝑒))) ↾ ({𝑤} × 𝑁)) = ((({𝑤} × 𝑁) × {((𝑎𝑐)(-g𝑅)if(𝑐𝑑, 1 , 0 ))}) ∘𝑓 · ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎𝑒))) ↾ ({𝑤} × 𝑁))) ∧ ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )), 0 ), (𝑎𝑒))) ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎𝑒))) ↾ ((𝑁 ∖ {𝑤}) × 𝑁))) → (𝐷‘(𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )), 0 ), (𝑎𝑒)))) = (((𝑎𝑐)(-g𝑅)if(𝑐𝑑, 1 , 0 )) · (𝐷‘(𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎𝑒))))))))
388277xpeq1d 5278 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (𝑤 = (1st𝑐) → (({𝑤} × 𝑁) × {((𝑎𝑐)(-g𝑅)if(𝑐𝑑, 1 , 0 ))}) = (({(1st𝑐)} × 𝑁) × {((𝑎𝑐)(-g𝑅)if(𝑐𝑑, 1 , 0 ))}))
389277reseq2d 5534 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (𝑤 = (1st𝑐) → ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎𝑒))) ↾ ({𝑤} × 𝑁)) = ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎𝑒))) ↾ ({(1st𝑐)} × 𝑁)))
390388, 389oveq12d 6811 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝑤 = (1st𝑐) → ((({𝑤} × 𝑁) × {((𝑎𝑐)(-g𝑅)if(𝑐𝑑, 1 , 0 ))}) ∘𝑓 · ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎𝑒))) ↾ ({𝑤} × 𝑁))) = ((({(1st𝑐)} × 𝑁) × {((𝑎𝑐)(-g𝑅)if(𝑐𝑑, 1 , 0 ))}) ∘𝑓 · ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎𝑒))) ↾ ({(1st𝑐)} × 𝑁))))
391279, 390eqeq12d 2786 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑤 = (1st𝑐) → (((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )), 0 ), (𝑎𝑒))) ↾ ({𝑤} × 𝑁)) = ((({𝑤} × 𝑁) × {((𝑎𝑐)(-g𝑅)if(𝑐𝑑, 1 , 0 ))}) ∘𝑓 · ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎𝑒))) ↾ ({𝑤} × 𝑁))) ↔ ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )), 0 ), (𝑎𝑒))) ↾ ({(1st𝑐)} × 𝑁)) = ((({(1st𝑐)} × 𝑁) × {((𝑎𝑐)(-g𝑅)if(𝑐𝑑, 1 , 0 ))}) ∘𝑓 · ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎𝑒))) ↾ ({(1st𝑐)} × 𝑁)))))
392284reseq2d 5534 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝑤 = (1st𝑐) → ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎𝑒))) ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎𝑒))) ↾ ((𝑁 ∖ {(1st𝑐)}) × 𝑁)))
393286, 392eqeq12d 2786 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑤 = (1st𝑐) → (((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )), 0 ), (𝑎𝑒))) ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎𝑒))) ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) ↔ ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )), 0 ), (𝑎𝑒))) ↾ ((𝑁 ∖ {(1st𝑐)}) × 𝑁)) = ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎𝑒))) ↾ ((𝑁 ∖ {(1st𝑐)}) × 𝑁))))
394391, 393anbi12d 616 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑤 = (1st𝑐) → ((((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )), 0 ), (𝑎𝑒))) ↾ ({𝑤} × 𝑁)) = ((({𝑤} × 𝑁) × {((𝑎𝑐)(-g𝑅)if(𝑐𝑑, 1 , 0 ))}) ∘𝑓 · ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎𝑒))) ↾ ({𝑤} × 𝑁))) ∧ ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )), 0 ), (𝑎𝑒))) ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎𝑒))) ↾ ((𝑁 ∖ {𝑤}) × 𝑁))) ↔ (((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )), 0 ), (𝑎𝑒))) ↾ ({(1st𝑐)} × 𝑁)) = ((({(1st𝑐)} × 𝑁) × {((𝑎𝑐)(-g𝑅)if(𝑐𝑑, 1 , 0 ))}) ∘𝑓 · ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎𝑒))) ↾ ({(1st𝑐)} × 𝑁))) ∧ ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )), 0 ), (𝑎𝑒))) ↾ ((𝑁 ∖ {(1st𝑐)}) × 𝑁)) = ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎𝑒))) ↾ ((𝑁 ∖ {(1st𝑐)}) × 𝑁)))))
395394imbi1d 330 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑤 = (1st𝑐) → (((((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )), 0 ), (𝑎𝑒))) ↾ ({𝑤} × 𝑁)) = ((({𝑤} × 𝑁) × {((𝑎𝑐)(-g𝑅)if(𝑐𝑑, 1 , 0 ))}) ∘𝑓 · ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎𝑒))) ↾ ({𝑤} × 𝑁))) ∧ ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )), 0 ), (𝑎𝑒))) ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎𝑒))) ↾ ((𝑁 ∖ {𝑤}) × 𝑁))) → (𝐷‘(𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )), 0 ), (𝑎𝑒)))) = (((𝑎𝑐)(-g𝑅)if(𝑐𝑑, 1 , 0 )) · (𝐷‘(𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎𝑒)))))) ↔ ((((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )), 0 ), (𝑎𝑒))) ↾ ({(1st𝑐)} × 𝑁)) = ((({(1st𝑐)} × 𝑁) × {((𝑎𝑐)(-g𝑅)if(𝑐𝑑, 1 , 0 ))}) ∘𝑓 · ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎𝑒))) ↾ ({(1st𝑐)} × 𝑁))) ∧ ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )), 0 ), (𝑎𝑒))) ↾ ((𝑁 ∖ {(1st𝑐)}) × 𝑁)) = ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎𝑒))) ↾ ((𝑁 ∖ {(1st𝑐)}) × 𝑁))) → (𝐷‘(𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )), 0 ), (𝑎𝑒)))) = (((𝑎𝑐)(-g𝑅)if(𝑐𝑑, 1 , 0 )) · (𝐷‘(𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎𝑒))))))))
396387, 395rspc2va 3473 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎𝑒))) ∈ 𝐵 ∧ (1st𝑐) ∈ 𝑁) ∧ ∀𝑧𝐵𝑤𝑁 ((((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )), 0 ), (𝑎𝑒))) ↾ ({𝑤} × 𝑁)) = ((({𝑤} × 𝑁) × {((𝑎𝑐)(-g𝑅)if(𝑐𝑑, 1 , 0 ))}) ∘𝑓 · (𝑧 ↾ ({𝑤} × 𝑁))) ∧ ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )), 0 ), (𝑎𝑒))) ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑧 ↾ ((𝑁 ∖ {𝑤}) × 𝑁))) → (𝐷‘(𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )), 0 ), (𝑎𝑒)))) = (((𝑎𝑐)(-g𝑅)if(𝑐𝑑, 1 , 0 )) · (𝐷𝑧)))) → ((((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )), 0 ), (𝑎𝑒))) ↾ ({(1st𝑐)} × 𝑁)) = ((({(1st𝑐)} × 𝑁) × {((𝑎𝑐)(-g𝑅)if(𝑐𝑑, 1 , 0 ))}) ∘𝑓 · ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎𝑒))) ↾ ({(1st𝑐)} × 𝑁))) ∧ ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )), 0 ), (𝑎𝑒))) ↾ ((𝑁 ∖ {(1st𝑐)}) × 𝑁)) = ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎𝑒))) ↾ ((𝑁 ∖ {(1st𝑐)}) × 𝑁))) → (𝐷‘(𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )), 0 ), (𝑎𝑒)))) = (((𝑎𝑐)(-g𝑅)if(𝑐𝑑, 1 , 0 )) · (𝐷‘(𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎𝑒)))))))
397356, 125, 377, 396syl21anc 1475 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) → ((((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )), 0 ), (𝑎𝑒))) ↾ ({(1st𝑐)} × 𝑁)) = ((({(1st𝑐)} × 𝑁) × {((𝑎𝑐)(-g𝑅)if(𝑐𝑑, 1 , 0 ))}) ∘𝑓 · ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎𝑒))) ↾ ({(1st𝑐)} × 𝑁))) ∧ ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )), 0 ), (𝑎𝑒))) ↾ ((𝑁 ∖ {(1st𝑐)}) × 𝑁)) = ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎𝑒))) ↾ ((𝑁 ∖ {(1st𝑐)}) × 𝑁))) → (𝐷‘(𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )), 0 ), (𝑎𝑒)))) = (((𝑎𝑐)(-g𝑅)if(𝑐𝑑, 1 , 0 )) · (𝐷‘(𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎𝑒)))))))
398339, 347, 397mp2and 679 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) → (𝐷‘(𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )), 0 ), (𝑎𝑒)))) = (((𝑎𝑐)(-g𝑅)if(𝑐𝑑, 1 , 0 )) · (𝐷‘(𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎𝑒))))))
399398oveq1d 6808 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) → ((𝐷‘(𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )), 0 ), (𝑎𝑒)))) + (𝐷‘(𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒𝑑, 1 , 0 ), (𝑎𝑒))))) = ((((𝑎𝑐)(-g𝑅)if(𝑐𝑑, 1 , 0 )) · (𝐷‘(𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎𝑒))))) + (𝐷‘(𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒𝑑, 1 , 0 ), (𝑎𝑒))))))
400105, 106, 107, 399syl3anc 1476 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ (𝑏 ∪ {𝑐}) ∈ 𝑌) ∧ ((𝑎𝐵𝑑 ∈ (𝑁𝑚 𝑁)) ∧ ∀𝑤𝑏 (𝑎𝑤) = if(𝑤𝑑, 1 , 0 ))) → ((𝐷‘(𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )), 0 ), (𝑎𝑒)))) + (𝐷‘(𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒𝑑, 1 , 0 ), (𝑎𝑒))))) = ((((𝑎𝑐)(-g𝑅)if(𝑐𝑑, 1 , 0 )) · (𝐷‘(𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎𝑒))))) + (𝐷‘(𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒𝑑, 1 , 0 ), (𝑎𝑒))))))
401 simpl3 1231 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ (𝑏 ∪ {𝑐}) ∈ 𝑌) ∧ ((𝑎𝐵𝑑 ∈ (𝑁𝑚 𝑁)) ∧ ∀𝑤𝑏 (𝑎𝑤) = if(𝑤𝑑, 1 , 0 ))) → (𝑏 ∪ {𝑐}) ∈ 𝑌)
402 simprlr 765 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ (𝑏 ∪ {𝑐}) ∈ 𝑌) ∧ ((𝑎𝐵𝑑 ∈ (𝑁𝑚 𝑁)) ∧ ∀𝑤𝑏 (𝑎𝑤) = if(𝑤𝑑, 1 , 0 ))) → 𝑑 ∈ (𝑁𝑚 𝑁))
403 simprr 756 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ (𝑏 ∪ {𝑐}) ∈ 𝑌) ∧ ((𝑎𝐵𝑑 ∈ (𝑁𝑚 𝑁)) ∧ ∀𝑤𝑏 (𝑎𝑤) = if(𝑤𝑑, 1 , 0 ))) → ∀𝑤𝑏 (𝑎𝑤) = if(𝑤𝑑, 1 , 0 ))
404 ralss 3817 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (𝑏 ⊆ (𝑏 ∪ {𝑐}) → (∀𝑤𝑏 (𝑎𝑤) = if(𝑤𝑑, 1 , 0 ) ↔ ∀𝑤 ∈ (𝑏 ∪ {𝑐})(𝑤𝑏 → (𝑎𝑤) = if(𝑤𝑑, 1 , 0 ))))
405100, 404ax-mp 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (∀𝑤𝑏 (𝑎𝑤) = if(𝑤𝑑, 1 , 0 ) ↔ ∀𝑤 ∈ (𝑏 ∪ {𝑐})(𝑤𝑏 → (𝑎𝑤) = if(𝑤𝑑, 1 , 0 )))
406 iftrue 4231 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ((1st𝑤) = (1st𝑐) → if((1st𝑤) = (1st𝑐), if(𝑤 = 𝑐, 1 , 0 ), (𝑎𝑤)) = if(𝑤 = 𝑐, 1 , 0 ))
407406adantl 467 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 (((((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) ∧ 𝑑 ∈ (𝑁𝑚 𝑁)) ∧ 𝑤 ∈ (𝑏 ∪ {𝑐})) ∧ (1st𝑤) = (1st𝑐)) → if((1st𝑤) = (1st𝑐), if(𝑤 = 𝑐, 1 , 0 ), (𝑎𝑤)) = if(𝑤 = 𝑐, 1 , 0 ))
408 ibar 518 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ((1st𝑤) = (1st𝑐) → ((2nd𝑤) = (2nd𝑐) ↔ ((1st𝑤) = (1st𝑐) ∧ (2nd𝑤) = (2nd𝑐))))
409408adantl 467 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 (((((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) ∧ 𝑑 ∈ (𝑁𝑚 𝑁)) ∧ 𝑤 ∈ (𝑏 ∪ {𝑐})) ∧ (1st𝑤) = (1st𝑐)) → ((2nd𝑤) = (2nd𝑐) ↔ ((1st𝑤) = (1st𝑐) ∧ (2nd𝑤) = (2nd𝑐))))
410 relxp 5266 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 Rel (𝑁 × 𝑁)
411 simpl2 1229 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) ∧ 𝑑 ∈ (𝑁𝑚 𝑁)) → (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁))
412411sselda 3752 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 ((((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) ∧ 𝑑 ∈ (𝑁𝑚 𝑁)) ∧ 𝑤 ∈ (𝑏 ∪ {𝑐})) → 𝑤 ∈ (𝑁 × 𝑁))
413412adantr 466 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 (((((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) ∧ 𝑑 ∈ (𝑁𝑚 𝑁)) ∧ 𝑤 ∈ (𝑏 ∪ {𝑐})) ∧ (1st𝑤) = (1st𝑐)) → 𝑤 ∈ (𝑁 × 𝑁))
414 1st2nd 7363 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 ((Rel (𝑁 × 𝑁) ∧ 𝑤 ∈ (𝑁 × 𝑁)) → 𝑤 = ⟨(1st𝑤), (2nd𝑤)⟩)
415410, 413, 414sylancr 575 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 (((((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) ∧ 𝑑 ∈ (𝑁𝑚 𝑁)) ∧ 𝑤 ∈ (𝑏 ∪ {𝑐})) ∧ (1st𝑤) = (1st𝑐)) → 𝑤 = ⟨(1st𝑤), (2nd𝑤)⟩)
416415eleq1d 2835 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 (((((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) ∧ 𝑑 ∈ (𝑁𝑚 𝑁)) ∧ 𝑤 ∈ (𝑏 ∪ {𝑐})) ∧ (1st𝑤) = (1st𝑐)) → (𝑤 ∈ (𝑑 sSet ⟨(1st𝑐), (2nd𝑐)⟩) ↔ ⟨(1st𝑤), (2nd𝑤)⟩ ∈ (𝑑 sSet ⟨(1st𝑐), (2nd𝑐)⟩)))
417 simpr 471 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) ∧ 𝑑 ∈ (𝑁𝑚 𝑁)) → 𝑑 ∈ (𝑁𝑚 𝑁))
418 elmapi 8031 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 (𝑑 ∈ (𝑁𝑚 𝑁) → 𝑑:𝑁𝑁)
419418adantl 467 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) ∧ 𝑑 ∈ (𝑁𝑚 𝑁)) → 𝑑:𝑁𝑁)
420125adantr 466 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) ∧ 𝑑 ∈ (𝑁𝑚 𝑁)) → (1st𝑐) ∈ 𝑁)
421 xp2nd 7348 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 (𝑐 ∈ (𝑁 × 𝑁) → (2nd𝑐) ∈ 𝑁)
422123, 421syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) → (2nd𝑐) ∈ 𝑁)
423422adantr 466 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) ∧ 𝑑 ∈ (𝑁𝑚 𝑁)) → (2nd𝑐) ∈ 𝑁)
424 fsets 16098 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 (((𝑑 ∈ (𝑁𝑚 𝑁) ∧ 𝑑:𝑁𝑁) ∧ (1st𝑐) ∈ 𝑁 ∧ (2nd𝑐) ∈ 𝑁) → (𝑑 sSet ⟨(1st𝑐), (2nd𝑐)⟩):𝑁𝑁)
425417, 419, 420, 423, 424syl211anc 1482 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) ∧ 𝑑 ∈ (𝑁𝑚 𝑁)) → (𝑑 sSet ⟨(1st𝑐), (2nd𝑐)⟩):𝑁𝑁)
426 ffn 6185 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 ((𝑑 sSet ⟨(1st𝑐), (2nd𝑐)⟩):𝑁𝑁 → (𝑑 sSet ⟨(1st𝑐), (2nd𝑐)⟩) Fn 𝑁)
427425, 426syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) ∧ 𝑑 ∈ (𝑁𝑚 𝑁)) → (𝑑 sSet ⟨(1st𝑐), (2nd𝑐)⟩) Fn 𝑁)
428427ad2antrr 705 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 (((((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) ∧ 𝑑 ∈ (𝑁𝑚 𝑁)) ∧ 𝑤 ∈ (𝑏 ∪ {𝑐})) ∧ (1st𝑤) = (1st𝑐)) → (𝑑 sSet ⟨(1st𝑐), (2nd𝑐)⟩) Fn 𝑁)
429 xp1st 7347 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 (𝑤 ∈ (𝑁 × 𝑁) → (1st𝑤) ∈ 𝑁)
430412, 429syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 ((((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) ∧ 𝑑 ∈ (𝑁𝑚 𝑁)) ∧ 𝑤 ∈ (𝑏 ∪ {𝑐})) → (1st𝑤) ∈ 𝑁)
431430adantr 466 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 (((((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) ∧ 𝑑 ∈ (𝑁𝑚 𝑁)) ∧ 𝑤 ∈ (𝑏 ∪ {𝑐})) ∧ (1st𝑤) = (1st𝑐)) → (1st𝑤) ∈ 𝑁)
432 fnopfvb 6378 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 (((𝑑 sSet ⟨(1st𝑐), (2nd𝑐)⟩) Fn 𝑁 ∧ (1st𝑤) ∈ 𝑁) → (((𝑑 sSet ⟨(1st𝑐), (2nd𝑐)⟩)‘(1st𝑤)) = (2nd𝑤) ↔ ⟨(1st𝑤), (2nd𝑤)⟩ ∈ (𝑑 sSet ⟨(1st𝑐), (2nd𝑐)⟩)))
433428, 431, 432syl2anc 573 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 (((((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) ∧ 𝑑 ∈ (𝑁𝑚 𝑁)) ∧ 𝑤 ∈ (𝑏 ∪ {𝑐})) ∧ (1st𝑤) = (1st𝑐)) → (((𝑑 sSet ⟨(1st𝑐), (2nd𝑐)⟩)‘(1st𝑤)) = (2nd𝑤) ↔ ⟨(1st𝑤), (2nd𝑤)⟩ ∈ (𝑑 sSet ⟨(1st𝑐), (2nd𝑐)⟩)))
434 fveq2 6332 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 ((1st𝑤) = (1st𝑐) → ((𝑑 sSet ⟨(1st𝑐), (2nd𝑐)⟩)‘(1st𝑤)) = ((𝑑 sSet ⟨(1st𝑐), (2nd𝑐)⟩)‘(1st𝑐)))
435434adantl 467 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 (((((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) ∧ 𝑑 ∈ (𝑁𝑚 𝑁)) ∧ 𝑤 ∈ (𝑏 ∪ {𝑐})) ∧ (1st𝑤) = (1st𝑐)) → ((𝑑 sSet ⟨(1st𝑐), (2nd𝑐)⟩)‘(1st𝑤)) = ((𝑑 sSet ⟨(1st𝑐), (2nd𝑐)⟩)‘(1st𝑐)))
436 vex 3354 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 𝑑 ∈ V
437 fvex 6342 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 (1st𝑐) ∈ V
438 fvex 6342 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 (2nd𝑐) ∈ V
439 fvsetsid 16097 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 ((𝑑 ∈ V ∧ (1st𝑐) ∈ V ∧ (2nd𝑐) ∈ V) → ((𝑑 sSet ⟨(1st𝑐), (2nd𝑐)⟩)‘(1st𝑐)) = (2nd𝑐))
440436, 437, 438, 439mp3an 1572 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 ((𝑑 sSet ⟨(1st𝑐), (2nd𝑐)⟩)‘(1st𝑐)) = (2nd𝑐)
441435, 440syl6eq 2821 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 (((((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) ∧ 𝑑 ∈ (𝑁𝑚 𝑁)) ∧ 𝑤 ∈ (𝑏 ∪ {𝑐})) ∧ (1st𝑤) = (1st𝑐)) → ((𝑑 sSet ⟨(1st𝑐), (2nd𝑐)⟩)‘(1st𝑤)) = (2nd𝑐))
442441eqeq1d 2773 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 (((((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) ∧ 𝑑 ∈ (𝑁𝑚 𝑁)) ∧ 𝑤 ∈ (𝑏 ∪ {𝑐})) ∧ (1st𝑤) = (1st𝑐)) → (((𝑑 sSet ⟨(1st𝑐), (2nd𝑐)⟩)‘(1st𝑤)) = (2nd𝑤) ↔ (2nd𝑐) = (2nd𝑤)))
443 eqcom 2778 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 ((2nd𝑐) = (2nd𝑤) ↔ (2nd𝑤) = (2nd𝑐))
444442, 443syl6bb 276 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 (((((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) ∧ 𝑑 ∈ (𝑁𝑚 𝑁)) ∧ 𝑤 ∈ (𝑏 ∪ {𝑐})) ∧ (1st𝑤) = (1st𝑐)) → (((𝑑 sSet ⟨(1st𝑐), (2nd𝑐)⟩)‘(1st𝑤)) = (2nd𝑤) ↔ (2nd𝑤) = (2nd𝑐)))
445416, 433, 4443bitr2rd 297 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 (((((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) ∧ 𝑑 ∈ (𝑁𝑚 𝑁)) ∧ 𝑤 ∈ (𝑏 ∪ {𝑐})) ∧ (1st𝑤) = (1st𝑐)) → ((2nd𝑤) = (2nd𝑐) ↔ 𝑤 ∈ (𝑑 sSet ⟨(1st𝑐), (2nd𝑐)⟩)))
446123ad3antrrr 709 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 (((((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) ∧ 𝑑 ∈ (𝑁𝑚 𝑁)) ∧ 𝑤 ∈ (𝑏 ∪ {𝑐})) ∧ (1st𝑤) = (1st𝑐)) → 𝑐 ∈ (𝑁 × 𝑁))
447 xpopth 7356 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ((𝑤 ∈ (𝑁 × 𝑁) ∧ 𝑐 ∈ (𝑁 × 𝑁)) → (((1st𝑤) = (1st𝑐) ∧ (2nd𝑤) = (2nd𝑐)) ↔ 𝑤 = 𝑐))
448413, 446, 447syl2anc 573 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 (((((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) ∧ 𝑑 ∈ (𝑁𝑚 𝑁)) ∧ 𝑤 ∈ (𝑏 ∪ {𝑐})) ∧ (1st𝑤) = (1st𝑐)) → (((1st𝑤) = (1st𝑐) ∧ (2nd𝑤) = (2nd𝑐)) ↔ 𝑤 = 𝑐))
449409, 445, 4483bitr3rd 299 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 (((((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) ∧ 𝑑 ∈ (𝑁𝑚 𝑁)) ∧ 𝑤 ∈ (𝑏 ∪ {𝑐})) ∧ (1st𝑤) = (1st𝑐)) → (𝑤 = 𝑐𝑤 ∈ (𝑑 sSet ⟨(1st𝑐), (2nd𝑐)⟩)))
450449ifbid 4247 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 (((((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) ∧ 𝑑 ∈ (𝑁𝑚 𝑁)) ∧ 𝑤 ∈ (𝑏 ∪ {𝑐})) ∧ (1st𝑤) = (1st𝑐)) → if(𝑤 = 𝑐, 1 , 0 ) = if(𝑤 ∈ (𝑑 sSet ⟨(1st𝑐), (2nd𝑐)⟩), 1 , 0 ))
451407, 450eqtrd 2805 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 (((((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) ∧ 𝑑 ∈ (𝑁𝑚 𝑁)) ∧ 𝑤 ∈ (𝑏 ∪ {𝑐})) ∧ (1st𝑤) = (1st𝑐)) → if((1st𝑤) = (1st𝑐), if(𝑤 = 𝑐, 1 , 0 ), (𝑎𝑤)) = if(𝑤 ∈ (𝑑 sSet ⟨(1st𝑐), (2nd𝑐)⟩), 1 , 0 ))
452451a1d 25 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (((((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) ∧ 𝑑 ∈ (𝑁𝑚 𝑁)) ∧ 𝑤 ∈ (𝑏 ∪ {𝑐})) ∧ (1st𝑤) = (1st𝑐)) → ((𝑤𝑏 → (𝑎𝑤) = if(𝑤𝑑, 1 , 0 )) → if((1st𝑤) = (1st𝑐), if(𝑤 = 𝑐, 1 , 0 ), (𝑎𝑤)) = if(𝑤 ∈ (𝑑 sSet ⟨(1st𝑐), (2nd𝑐)⟩), 1 , 0 )))
453 elsni 4333 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 (𝑤 ∈ {𝑐} → 𝑤 = 𝑐)
454453fveq2d 6336 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 (𝑤 ∈ {𝑐} → (1st𝑤) = (1st𝑐))
455454