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Theorem mdetunilem9 21517
Description: Lemma for mdetuni 21519. (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 4424 . . . 4 𝑤 ∈ ∅ (𝑎𝑤) = if(𝑤 ∈ ( I ↾ 𝑁), 1 , 0 )
2 simpr 488 . . . . 5 ((𝜑𝑎𝐵) → 𝑎𝐵)
3 f1oi 6698 . . . . . . . 8 ( I ↾ 𝑁):𝑁1-1-onto𝑁
4 f1of 6661 . . . . . . . 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 8522 . . . . . . 7 (𝜑 → (( I ↾ 𝑁) ∈ (𝑁m 𝑁) ↔ ( I ↾ 𝑁):𝑁𝑁))
85, 7mpbird 260 . . . . . 6 (𝜑 → ( I ↾ 𝑁) ∈ (𝑁m 𝑁))
98adantr 484 . . . . 5 ((𝜑𝑎𝐵) → ( I ↾ 𝑁) ∈ (𝑁m 𝑁))
10 simplrl 777 . . . . . . . . . . . . 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 21319 . . . . . . . . . . . . . . . 16 (𝑦𝐵𝑦 ∈ (𝐾m (𝑁 × 𝑁)))
15 elmapi 8530 . . . . . . . . . . . . . . . 16 (𝑦 ∈ (𝐾m (𝑁 × 𝑁)) → 𝑦:(𝑁 × 𝑁)⟶𝐾)
1614, 15syl 17 . . . . . . . . . . . . . . 15 (𝑦𝐵𝑦:(𝑁 × 𝑁)⟶𝐾)
1716feqmptd 6780 . . . . . . . . . . . . . 14 (𝑦𝐵𝑦 = (𝑤 ∈ (𝑁 × 𝑁) ↦ (𝑦𝑤)))
1817fveq2d 6721 . . . . . . . . . . . . 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 2737 . . . . . . . . . . . . . 14 (𝑁 × 𝑁) = (𝑁 × 𝑁)
21 mpteq12 5142 . . . . . . . . . . . . . . 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 6721 . . . . . . . . . . . . . 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 690 . . . . . . . . . . . . 13 (∀𝑤 ∈ (𝑁 × 𝑁)(𝑦𝑤) = if(𝑤𝑧, 1 , 0 ) → (𝐷‘(𝑤 ∈ (𝑁 × 𝑁) ↦ (𝑦𝑤))) = (𝐷‘(𝑤 ∈ (𝑁 × 𝑁) ↦ if(𝑤𝑧, 1 , 0 ))))
2423adantl 485 . . . . . . . . . . . 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 2825 . . . . . . . . . . . . . . . . 17 (𝑎 = 𝑧 → (𝑎 ∈ (𝑁m 𝑁) ↔ 𝑧 ∈ (𝑁m 𝑁)))
2625anbi2d 632 . . . . . . . . . . . . . . . 16 (𝑎 = 𝑧 → ((𝜑𝑎 ∈ (𝑁m 𝑁)) ↔ (𝜑𝑧 ∈ (𝑁m 𝑁))))
27 elequ2 2125 . . . . . . . . . . . . . . . . . . . 20 (𝑎 = 𝑧 → (𝑤𝑎𝑤𝑧))
2827ifbid 4462 . . . . . . . . . . . . . . . . . . 19 (𝑎 = 𝑧 → if(𝑤𝑎, 1 , 0 ) = if(𝑤𝑧, 1 , 0 ))
2928mpteq2dv 5151 . . . . . . . . . . . . . . . . . 18 (𝑎 = 𝑧 → (𝑤 ∈ (𝑁 × 𝑁) ↦ if(𝑤𝑎, 1 , 0 )) = (𝑤 ∈ (𝑁 × 𝑁) ↦ if(𝑤𝑧, 1 , 0 )))
3029fveq2d 6721 . . . . . . . . . . . . . . . . 17 (𝑎 = 𝑧 → (𝐷‘(𝑤 ∈ (𝑁 × 𝑁) ↦ if(𝑤𝑎, 1 , 0 ))) = (𝐷‘(𝑤 ∈ (𝑁 × 𝑁) ↦ if(𝑤𝑧, 1 , 0 ))))
3130eqeq1d 2739 . . . . . . . . . . . . . . . 16 (𝑎 = 𝑧 → ((𝐷‘(𝑤 ∈ (𝑁 × 𝑁) ↦ if(𝑤𝑎, 1 , 0 ))) = 0 ↔ (𝐷‘(𝑤 ∈ (𝑁 × 𝑁) ↦ if(𝑤𝑧, 1 , 0 ))) = 0 ))
3226, 31imbi12d 348 . . . . . . . . . . . . . . 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 2825 . . . . . . . . . . . . . . . . . . . 20 (𝑤 = ⟨𝑏, 𝑐⟩ → (𝑤𝑎 ↔ ⟨𝑏, 𝑐⟩ ∈ 𝑎))
3433ifbid 4462 . . . . . . . . . . . . . . . . . . 19 (𝑤 = ⟨𝑏, 𝑐⟩ → if(𝑤𝑎, 1 , 0 ) = if(⟨𝑏, 𝑐⟩ ∈ 𝑎, 1 , 0 ))
3534mpompt 7324 . . . . . . . . . . . . . . . . . 18 (𝑤 ∈ (𝑁 × 𝑁) ↦ if(𝑤𝑎, 1 , 0 )) = (𝑏𝑁, 𝑐𝑁 ↦ if(⟨𝑏, 𝑐⟩ ∈ 𝑎, 1 , 0 ))
36 elmapi 8530 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑎 ∈ (𝑁m 𝑁) → 𝑎:𝑁𝑁)
3736adantl 485 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝜑𝑎 ∈ (𝑁m 𝑁)) → 𝑎:𝑁𝑁)
3837ffnd 6546 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑𝑎 ∈ (𝑁m 𝑁)) → 𝑎 Fn 𝑁)
39383ad2ant1 1135 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑𝑎 ∈ (𝑁m 𝑁)) ∧ 𝑏𝑁𝑐𝑁) → 𝑎 Fn 𝑁)
40 simp2 1139 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑𝑎 ∈ (𝑁m 𝑁)) ∧ 𝑏𝑁𝑐𝑁) → 𝑏𝑁)
41 fnopfvb 6766 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑎 Fn 𝑁𝑏𝑁) → ((𝑎𝑏) = 𝑐 ↔ ⟨𝑏, 𝑐⟩ ∈ 𝑎))
4239, 40, 41syl2anc 587 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑎 ∈ (𝑁m 𝑁)) ∧ 𝑏𝑁𝑐𝑁) → ((𝑎𝑏) = 𝑐 ↔ ⟨𝑏, 𝑐⟩ ∈ 𝑎))
4342bicomd 226 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑎 ∈ (𝑁m 𝑁)) ∧ 𝑏𝑁𝑐𝑁) → (⟨𝑏, 𝑐⟩ ∈ 𝑎 ↔ (𝑎𝑏) = 𝑐))
4443ifbid 4462 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑎 ∈ (𝑁m 𝑁)) ∧ 𝑏𝑁𝑐𝑁) → if(⟨𝑏, 𝑐⟩ ∈ 𝑎, 1 , 0 ) = if((𝑎𝑏) = 𝑐, 1 , 0 ))
4544mpoeq3dva 7288 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑎 ∈ (𝑁m 𝑁)) → (𝑏𝑁, 𝑐𝑁 ↦ if(⟨𝑏, 𝑐⟩ ∈ 𝑎, 1 , 0 )) = (𝑏𝑁, 𝑐𝑁 ↦ if((𝑎𝑏) = 𝑐, 1 , 0 )))
4635, 45syl5eq 2790 . . . . . . . . . . . . . . . . 17 ((𝜑𝑎 ∈ (𝑁m 𝑁)) → (𝑤 ∈ (𝑁 × 𝑁) ↦ if(𝑤𝑎, 1 , 0 )) = (𝑏𝑁, 𝑐𝑁 ↦ if((𝑎𝑏) = 𝑐, 1 , 0 )))
4746fveq2d 6721 . . . . . . . . . . . . . . . 16 ((𝜑𝑎 ∈ (𝑁m 𝑁)) → (𝐷‘(𝑤 ∈ (𝑁 × 𝑁) ↦ if(𝑤𝑎, 1 , 0 ))) = (𝐷‘(𝑏𝑁, 𝑐𝑁 ↦ if((𝑎𝑏) = 𝑐, 1 , 0 ))))
48 mdetuni.0g . . . . . . . . . . . . . . . . . 18 0 = (0g𝑅)
49 mdetuni.1r . . . . . . . . . . . . . . . . . 18 1 = (1r𝑅)
50 mdetuni.pg . . . . . . . . . . . . . . . . . 18 + = (+g𝑅)
51 mdetuni.tg . . . . . . . . . . . . . . . . . 18 · = (.r𝑅)
52 mdetuni.r . . . . . . . . . . . . . . . . . 18 (𝜑𝑅 ∈ Ring)
53 mdetuni.ff . . . . . . . . . . . . . . . . . 18 (𝜑𝐷:𝐵𝐾)
54 mdetuni.al . . . . . . . . . . . . . . . . . 18 (𝜑 → ∀𝑥𝐵𝑦𝑁𝑧𝑁 ((𝑦𝑧 ∧ ∀𝑤𝑁 (𝑦𝑥𝑤) = (𝑧𝑥𝑤)) → (𝐷𝑥) = 0 ))
55 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))))
56 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))))
57 mdetunilem9.id . . . . . . . . . . . . . . . . . 18 (𝜑 → (𝐷‘(1r𝐴)) = 0 )
5811, 13, 12, 48, 49, 50, 51, 6, 52, 53, 54, 55, 56, 57mdetunilem8 21516 . . . . . . . . . . . . . . . . 17 ((𝜑𝑎:𝑁𝑁) → (𝐷‘(𝑏𝑁, 𝑐𝑁 ↦ if((𝑎𝑏) = 𝑐, 1 , 0 ))) = 0 )
5936, 58sylan2 596 . . . . . . . . . . . . . . . 16 ((𝜑𝑎 ∈ (𝑁m 𝑁)) → (𝐷‘(𝑏𝑁, 𝑐𝑁 ↦ if((𝑎𝑏) = 𝑐, 1 , 0 ))) = 0 )
6047, 59eqtrd 2777 . . . . . . . . . . . . . . 15 ((𝜑𝑎 ∈ (𝑁m 𝑁)) → (𝐷‘(𝑤 ∈ (𝑁 × 𝑁) ↦ if(𝑤𝑎, 1 , 0 ))) = 0 )
6132, 60chvarvv 2007 . . . . . . . . . . . . . 14 ((𝜑𝑧 ∈ (𝑁m 𝑁)) → (𝐷‘(𝑤 ∈ (𝑁 × 𝑁) ↦ if(𝑤𝑧, 1 , 0 ))) = 0 )
6261adantrl 716 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑦𝐵𝑧 ∈ (𝑁m 𝑁))) → (𝐷‘(𝑤 ∈ (𝑁 × 𝑁) ↦ if(𝑤𝑧, 1 , 0 ))) = 0 )
6362adantr 484 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑦𝐵𝑧 ∈ (𝑁m 𝑁))) ∧ ∀𝑤 ∈ (𝑁 × 𝑁)(𝑦𝑤) = if(𝑤𝑧, 1 , 0 )) → (𝐷‘(𝑤 ∈ (𝑁 × 𝑁) ↦ if(𝑤𝑧, 1 , 0 ))) = 0 )
6419, 24, 633eqtrd 2781 . . . . . . . . . . 11 (((𝜑 ∧ (𝑦𝐵𝑧 ∈ (𝑁m 𝑁))) ∧ ∀𝑤 ∈ (𝑁 × 𝑁)(𝑦𝑤) = if(𝑤𝑧, 1 , 0 )) → (𝐷𝑦) = 0 )
6564ex 416 . . . . . . . . . 10 ((𝜑 ∧ (𝑦𝐵𝑧 ∈ (𝑁m 𝑁))) → (∀𝑤 ∈ (𝑁 × 𝑁)(𝑦𝑤) = if(𝑤𝑧, 1 , 0 ) → (𝐷𝑦) = 0 ))
6665ralrimivva 3112 . . . . . . . . 9 (𝜑 → ∀𝑦𝐵𝑧 ∈ (𝑁m 𝑁)(∀𝑤 ∈ (𝑁 × 𝑁)(𝑦𝑤) = if(𝑤𝑧, 1 , 0 ) → (𝐷𝑦) = 0 ))
67 xpfi 8942 . . . . . . . . . . 11 ((𝑁 ∈ Fin ∧ 𝑁 ∈ Fin) → (𝑁 × 𝑁) ∈ Fin)
686, 6, 67syl2anc 587 . . . . . . . . . 10 (𝜑 → (𝑁 × 𝑁) ∈ Fin)
69 raleq 3319 . . . . . . . . . . . . 13 (𝑥 = (𝑁 × 𝑁) → (∀𝑤𝑥 (𝑦𝑤) = if(𝑤𝑧, 1 , 0 ) ↔ ∀𝑤 ∈ (𝑁 × 𝑁)(𝑦𝑤) = if(𝑤𝑧, 1 , 0 )))
7069imbi1d 345 . . . . . . . . . . . 12 (𝑥 = (𝑁 × 𝑁) → ((∀𝑤𝑥 (𝑦𝑤) = if(𝑤𝑧, 1 , 0 ) → (𝐷𝑦) = 0 ) ↔ (∀𝑤 ∈ (𝑁 × 𝑁)(𝑦𝑤) = if(𝑤𝑧, 1 , 0 ) → (𝐷𝑦) = 0 )))
71702ralbidv 3120 . . . . . . . . . . 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 )))
72 mdetunilem9.y . . . . . . . . . . 11 𝑌 = {𝑥 ∣ ∀𝑦𝐵𝑧 ∈ (𝑁m 𝑁)(∀𝑤𝑥 (𝑦𝑤) = if(𝑤𝑧, 1 , 0 ) → (𝐷𝑦) = 0 )}
7371, 72elab2g 3589 . . . . . . . . . 10 ((𝑁 × 𝑁) ∈ Fin → ((𝑁 × 𝑁) ∈ 𝑌 ↔ ∀𝑦𝐵𝑧 ∈ (𝑁m 𝑁)(∀𝑤 ∈ (𝑁 × 𝑁)(𝑦𝑤) = if(𝑤𝑧, 1 , 0 ) → (𝐷𝑦) = 0 )))
7468, 73syl 17 . . . . . . . . 9 (𝜑 → ((𝑁 × 𝑁) ∈ 𝑌 ↔ ∀𝑦𝐵𝑧 ∈ (𝑁m 𝑁)(∀𝑤 ∈ (𝑁 × 𝑁)(𝑦𝑤) = if(𝑤𝑧, 1 , 0 ) → (𝐷𝑦) = 0 )))
7566, 74mpbird 260 . . . . . . . 8 (𝜑 → (𝑁 × 𝑁) ∈ 𝑌)
76 ssid 3923 . . . . . . . . 9 (𝑁 × 𝑁) ⊆ (𝑁 × 𝑁)
77683ad2ant1 1135 . . . . . . . . . . 11 ((𝜑 ∧ (𝑁 × 𝑁) ⊆ (𝑁 × 𝑁) ∧ ¬ ∅ ∈ 𝑌) → (𝑁 × 𝑁) ∈ Fin)
78 sseq1 3926 . . . . . . . . . . . . . 14 (𝑎 = ∅ → (𝑎 ⊆ (𝑁 × 𝑁) ↔ ∅ ⊆ (𝑁 × 𝑁)))
79783anbi2d 1443 . . . . . . . . . . . . 13 (𝑎 = ∅ → ((𝜑𝑎 ⊆ (𝑁 × 𝑁) ∧ ¬ ∅ ∈ 𝑌) ↔ (𝜑 ∧ ∅ ⊆ (𝑁 × 𝑁) ∧ ¬ ∅ ∈ 𝑌)))
80 eleq1 2825 . . . . . . . . . . . . . 14 (𝑎 = ∅ → (𝑎𝑌 ↔ ∅ ∈ 𝑌))
8180notbid 321 . . . . . . . . . . . . 13 (𝑎 = ∅ → (¬ 𝑎𝑌 ↔ ¬ ∅ ∈ 𝑌))
8279, 81imbi12d 348 . . . . . . . . . . . 12 (𝑎 = ∅ → (((𝜑𝑎 ⊆ (𝑁 × 𝑁) ∧ ¬ ∅ ∈ 𝑌) → ¬ 𝑎𝑌) ↔ ((𝜑 ∧ ∅ ⊆ (𝑁 × 𝑁) ∧ ¬ ∅ ∈ 𝑌) → ¬ ∅ ∈ 𝑌)))
83 sseq1 3926 . . . . . . . . . . . . . 14 (𝑎 = 𝑏 → (𝑎 ⊆ (𝑁 × 𝑁) ↔ 𝑏 ⊆ (𝑁 × 𝑁)))
84833anbi2d 1443 . . . . . . . . . . . . 13 (𝑎 = 𝑏 → ((𝜑𝑎 ⊆ (𝑁 × 𝑁) ∧ ¬ ∅ ∈ 𝑌) ↔ (𝜑𝑏 ⊆ (𝑁 × 𝑁) ∧ ¬ ∅ ∈ 𝑌)))
85 eleq1 2825 . . . . . . . . . . . . . 14 (𝑎 = 𝑏 → (𝑎𝑌𝑏𝑌))
8685notbid 321 . . . . . . . . . . . . 13 (𝑎 = 𝑏 → (¬ 𝑎𝑌 ↔ ¬ 𝑏𝑌))
8784, 86imbi12d 348 . . . . . . . . . . . 12 (𝑎 = 𝑏 → (((𝜑𝑎 ⊆ (𝑁 × 𝑁) ∧ ¬ ∅ ∈ 𝑌) → ¬ 𝑎𝑌) ↔ ((𝜑𝑏 ⊆ (𝑁 × 𝑁) ∧ ¬ ∅ ∈ 𝑌) → ¬ 𝑏𝑌)))
88 sseq1 3926 . . . . . . . . . . . . . 14 (𝑎 = (𝑏 ∪ {𝑐}) → (𝑎 ⊆ (𝑁 × 𝑁) ↔ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁)))
89883anbi2d 1443 . . . . . . . . . . . . 13 (𝑎 = (𝑏 ∪ {𝑐}) → ((𝜑𝑎 ⊆ (𝑁 × 𝑁) ∧ ¬ ∅ ∈ 𝑌) ↔ (𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ ¬ ∅ ∈ 𝑌)))
90 eleq1 2825 . . . . . . . . . . . . . 14 (𝑎 = (𝑏 ∪ {𝑐}) → (𝑎𝑌 ↔ (𝑏 ∪ {𝑐}) ∈ 𝑌))
9190notbid 321 . . . . . . . . . . . . 13 (𝑎 = (𝑏 ∪ {𝑐}) → (¬ 𝑎𝑌 ↔ ¬ (𝑏 ∪ {𝑐}) ∈ 𝑌))
9289, 91imbi12d 348 . . . . . . . . . . . 12 (𝑎 = (𝑏 ∪ {𝑐}) → (((𝜑𝑎 ⊆ (𝑁 × 𝑁) ∧ ¬ ∅ ∈ 𝑌) → ¬ 𝑎𝑌) ↔ ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ ¬ ∅ ∈ 𝑌) → ¬ (𝑏 ∪ {𝑐}) ∈ 𝑌)))
93 sseq1 3926 . . . . . . . . . . . . . 14 (𝑎 = (𝑁 × 𝑁) → (𝑎 ⊆ (𝑁 × 𝑁) ↔ (𝑁 × 𝑁) ⊆ (𝑁 × 𝑁)))
94933anbi2d 1443 . . . . . . . . . . . . 13 (𝑎 = (𝑁 × 𝑁) → ((𝜑𝑎 ⊆ (𝑁 × 𝑁) ∧ ¬ ∅ ∈ 𝑌) ↔ (𝜑 ∧ (𝑁 × 𝑁) ⊆ (𝑁 × 𝑁) ∧ ¬ ∅ ∈ 𝑌)))
95 eleq1 2825 . . . . . . . . . . . . . 14 (𝑎 = (𝑁 × 𝑁) → (𝑎𝑌 ↔ (𝑁 × 𝑁) ∈ 𝑌))
9695notbid 321 . . . . . . . . . . . . 13 (𝑎 = (𝑁 × 𝑁) → (¬ 𝑎𝑌 ↔ ¬ (𝑁 × 𝑁) ∈ 𝑌))
9794, 96imbi12d 348 . . . . . . . . . . . 12 (𝑎 = (𝑁 × 𝑁) → (((𝜑𝑎 ⊆ (𝑁 × 𝑁) ∧ ¬ ∅ ∈ 𝑌) → ¬ 𝑎𝑌) ↔ ((𝜑 ∧ (𝑁 × 𝑁) ⊆ (𝑁 × 𝑁) ∧ ¬ ∅ ∈ 𝑌) → ¬ (𝑁 × 𝑁) ∈ 𝑌)))
98 simp3 1140 . . . . . . . . . . . 12 ((𝜑 ∧ ∅ ⊆ (𝑁 × 𝑁) ∧ ¬ ∅ ∈ 𝑌) → ¬ ∅ ∈ 𝑌)
99 ssun1 4086 . . . . . . . . . . . . . . . 16 𝑏 ⊆ (𝑏 ∪ {𝑐})
100 sstr2 3908 . . . . . . . . . . . . . . . 16 (𝑏 ⊆ (𝑏 ∪ {𝑐}) → ((𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) → 𝑏 ⊆ (𝑁 × 𝑁)))
10199, 100ax-mp 5 . . . . . . . . . . . . . . 15 ((𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) → 𝑏 ⊆ (𝑁 × 𝑁))
1021013anim2i 1155 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ ¬ ∅ ∈ 𝑌) → (𝜑𝑏 ⊆ (𝑁 × 𝑁) ∧ ¬ ∅ ∈ 𝑌))
103102imim1i 63 . . . . . . . . . . . . 13 (((𝜑𝑏 ⊆ (𝑁 × 𝑁) ∧ ¬ ∅ ∈ 𝑌) → ¬ 𝑏𝑌) → ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ ¬ ∅ ∈ 𝑌) → ¬ 𝑏𝑌))
104 simpl1 1193 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ (𝑏 ∪ {𝑐}) ∈ 𝑌) ∧ ((𝑎𝐵𝑑 ∈ (𝑁m 𝑁)) ∧ ∀𝑤𝑏 (𝑎𝑤) = if(𝑤𝑑, 1 , 0 ))) → 𝜑)
105 simpl2 1194 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ (𝑏 ∪ {𝑐}) ∈ 𝑌) ∧ ((𝑎𝐵𝑑 ∈ (𝑁m 𝑁)) ∧ ∀𝑤𝑏 (𝑎𝑤) = if(𝑤𝑑, 1 , 0 ))) → (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁))
106 simprll 779 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ (𝑏 ∪ {𝑐}) ∈ 𝑌) ∧ ((𝑎𝐵𝑑 ∈ (𝑁m 𝑁)) ∧ ∀𝑤𝑏 (𝑎𝑤) = if(𝑤𝑑, 1 , 0 ))) → 𝑎𝐵)
10711, 12, 13matbas2i 21319 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑎𝐵𝑎 ∈ (𝐾m (𝑁 × 𝑁)))
108 elmapi 8530 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑎 ∈ (𝐾m (𝑁 × 𝑁)) → 𝑎:(𝑁 × 𝑁)⟶𝐾)
109107, 108syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑎𝐵𝑎:(𝑁 × 𝑁)⟶𝐾)
1101093ad2ant3 1137 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) → 𝑎:(𝑁 × 𝑁)⟶𝐾)
111110feqmptd 6780 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) → 𝑎 = (𝑒 ∈ (𝑁 × 𝑁) ↦ (𝑎𝑒)))
112111reseq1d 5850 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) → (𝑎 ↾ ({(1st𝑐)} × 𝑁)) = ((𝑒 ∈ (𝑁 × 𝑁) ↦ (𝑎𝑒)) ↾ ({(1st𝑐)} × 𝑁)))
113523ad2ant1 1135 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) → 𝑅 ∈ Ring)
114 ringgrp 19567 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 (𝑅 ∈ Ring → 𝑅 ∈ Grp)
115113, 114syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) → 𝑅 ∈ Grp)
116115adantr 484 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) ∧ 𝑒 ∈ ({(1st𝑐)} × 𝑁)) → 𝑅 ∈ Grp)
117110adantr 484 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) ∧ 𝑒 ∈ ({(1st𝑐)} × 𝑁)) → 𝑎:(𝑁 × 𝑁)⟶𝐾)
118 simp2 1139 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) → (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁))
119118unssbd 4102 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) → {𝑐} ⊆ (𝑁 × 𝑁))
120 vex 3412 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 𝑐 ∈ V
121120snss 4699 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 (𝑐 ∈ (𝑁 × 𝑁) ↔ {𝑐} ⊆ (𝑁 × 𝑁))
122119, 121sylibr 237 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) → 𝑐 ∈ (𝑁 × 𝑁))
123 xp1st 7793 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 (𝑐 ∈ (𝑁 × 𝑁) → (1st𝑐) ∈ 𝑁)
124122, 123syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) → (1st𝑐) ∈ 𝑁)
125124snssd 4722 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) → {(1st𝑐)} ⊆ 𝑁)
126 xpss1 5570 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ({(1st𝑐)} ⊆ 𝑁 → ({(1st𝑐)} × 𝑁) ⊆ (𝑁 × 𝑁))
127125, 126syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) → ({(1st𝑐)} × 𝑁) ⊆ (𝑁 × 𝑁))
128127sselda 3901 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) ∧ 𝑒 ∈ ({(1st𝑐)} × 𝑁)) → 𝑒 ∈ (𝑁 × 𝑁))
129117, 128ffvelrnd 6905 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) ∧ 𝑒 ∈ ({(1st𝑐)} × 𝑁)) → (𝑎𝑒) ∈ 𝐾)
13012, 49ringidcl 19586 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 (𝑅 ∈ Ring → 1𝐾)
131113, 130syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) → 1𝐾)
13212, 48ring0cl 19587 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 (𝑅 ∈ Ring → 0𝐾)
133113, 132syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) → 0𝐾)
134131, 133ifcld 4485 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) → if(𝑒𝑑, 1 , 0 ) ∈ 𝐾)
135134adantr 484 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) ∧ 𝑒 ∈ ({(1st𝑐)} × 𝑁)) → if(𝑒𝑑, 1 , 0 ) ∈ 𝐾)
136 eqid 2737 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 (-g𝑅) = (-g𝑅)
13712, 50, 136grpnpcan 18455 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ((𝑅 ∈ Grp ∧ (𝑎𝑒) ∈ 𝐾 ∧ if(𝑒𝑑, 1 , 0 ) ∈ 𝐾) → (((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )) + if(𝑒𝑑, 1 , 0 )) = (𝑎𝑒))
138116, 129, 135, 137syl3anc 1373 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) ∧ 𝑒 ∈ ({(1st𝑐)} × 𝑁)) → (((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )) + if(𝑒𝑑, 1 , 0 )) = (𝑎𝑒))
139138eqcomd 2743 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) ∧ 𝑒 ∈ ({(1st𝑐)} × 𝑁)) → (𝑎𝑒) = (((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )) + if(𝑒𝑑, 1 , 0 )))
140139adantr 484 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) ∧ 𝑒 ∈ ({(1st𝑐)} × 𝑁)) ∧ 𝑒 = 𝑐) → (𝑎𝑒) = (((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )) + if(𝑒𝑑, 1 , 0 )))
141 iftrue 4445 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (𝑒 = 𝑐 → if(𝑒 = 𝑐, ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )), 0 ) = ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )))
142 iftrue 4445 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (𝑒 = 𝑐 → if(𝑒 = 𝑐, if(𝑒𝑑, 1 , 0 ), (𝑎𝑒)) = if(𝑒𝑑, 1 , 0 ))
143141, 142oveq12d 7231 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 )))
144143adantl 485 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 )))
145140, 144eqtr4d 2780 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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))))
14612, 50, 48grplid 18397 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ((𝑅 ∈ Grp ∧ (𝑎𝑒) ∈ 𝐾) → ( 0 + (𝑎𝑒)) = (𝑎𝑒))
147116, 129, 146syl2anc 587 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) ∧ 𝑒 ∈ ({(1st𝑐)} × 𝑁)) → ( 0 + (𝑎𝑒)) = (𝑎𝑒))
148147eqcomd 2743 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) ∧ 𝑒 ∈ ({(1st𝑐)} × 𝑁)) → (𝑎𝑒) = ( 0 + (𝑎𝑒)))
149148adantr 484 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) ∧ 𝑒 ∈ ({(1st𝑐)} × 𝑁)) ∧ ¬ 𝑒 = 𝑐) → (𝑎𝑒) = ( 0 + (𝑎𝑒)))
150 iffalse 4448 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 𝑒 = 𝑐 → if(𝑒 = 𝑐, ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )), 0 ) = 0 )
151 iffalse 4448 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 𝑒 = 𝑐 → if(𝑒 = 𝑐, if(𝑒𝑑, 1 , 0 ), (𝑎𝑒)) = (𝑎𝑒))
152150, 151oveq12d 7231 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 𝑒 = 𝑐 → (if(𝑒 = 𝑐, ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )), 0 ) + if(𝑒 = 𝑐, if(𝑒𝑑, 1 , 0 ), (𝑎𝑒))) = ( 0 + (𝑎𝑒)))
153152adantl 485 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)))
154149, 153eqtr4d 2780 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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))))
155145, 154pm2.61dan 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))))
156155mpteq2dva 5150 . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)))))
157 snfi 8721 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 {(1st𝑐)} ∈ Fin
15863ad2ant1 1135 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) → 𝑁 ∈ Fin)
159 xpfi 8942 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (({(1st𝑐)} ∈ Fin ∧ 𝑁 ∈ Fin) → ({(1st𝑐)} × 𝑁) ∈ Fin)
160157, 158, 159sylancr 590 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) → ({(1st𝑐)} × 𝑁) ∈ Fin)
161 ovex 7246 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )) ∈ V
16248fvexi 6731 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 0 ∈ V
163161, 162ifex 4489 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 if(𝑒 = 𝑐, ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )), 0 ) ∈ V
164163a1i 11 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) ∧ 𝑒 ∈ ({(1st𝑐)} × 𝑁)) → if(𝑒 = 𝑐, ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )), 0 ) ∈ V)
16549fvexi 6731 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 1 ∈ V
166165, 162ifex 4489 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 if(𝑒𝑑, 1 , 0 ) ∈ V
167 fvex 6730 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝑎𝑒) ∈ V
168166, 167ifex 4489 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 if(𝑒 = 𝑐, if(𝑒𝑑, 1 , 0 ), (𝑎𝑒)) ∈ V
169168a1i 11 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) ∧ 𝑒 ∈ ({(1st𝑐)} × 𝑁)) → if(𝑒 = 𝑐, if(𝑒𝑑, 1 , 0 ), (𝑎𝑒)) ∈ V)
170 xp1st 7793 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (𝑒 ∈ ({(1st𝑐)} × 𝑁) → (1st𝑒) ∈ {(1st𝑐)})
171 elsni 4558 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((1st𝑒) ∈ {(1st𝑐)} → (1st𝑒) = (1st𝑐))
172 iftrue 4445 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 ))
173170, 171, 1723syl 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 ))
174173mpteq2ia 5146 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 ))
175174a1i 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 )))
176 eqidd 2738 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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))))
177160, 164, 169, 175, 176offval2 7488 . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)))))
178156, 177eqtr4d 2780 . . . . . . . . . . . . . . . . . . . . . . . . . . 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)))))
179127resmptd 5908 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) → ((𝑒 ∈ (𝑁 × 𝑁) ↦ (𝑎𝑒)) ↾ ({(1st𝑐)} × 𝑁)) = (𝑒 ∈ ({(1st𝑐)} × 𝑁) ↦ (𝑎𝑒)))
180127resmptd 5908 . . . . . . . . . . . . . . . . . . . . . . . . . . . 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))))
181127resmptd 5908 . . . . . . . . . . . . . . . . . . . . . . . . . . . 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))))
182180, 181oveq12d 7231 . . . . . . . . . . . . . . . . . . . . . . . . . . 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)))))
183178, 179, 1823eqtr4d 2787 . . . . . . . . . . . . . . . . . . . . . . . . . 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))))
184112, 183eqtrd 2777 . . . . . . . . . . . . . . . . . . . . . . . . 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))))
185111reseq1d 5850 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) → (𝑎 ↾ ((𝑁 ∖ {(1st𝑐)}) × 𝑁)) = ((𝑒 ∈ (𝑁 × 𝑁) ↦ (𝑎𝑒)) ↾ ((𝑁 ∖ {(1st𝑐)}) × 𝑁)))
186 xp1st 7793 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (𝑒 ∈ ((𝑁 ∖ {(1st𝑐)}) × 𝑁) → (1st𝑒) ∈ (𝑁 ∖ {(1st𝑐)}))
187 eldifsni 4703 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((1st𝑒) ∈ (𝑁 ∖ {(1st𝑐)}) → (1st𝑒) ≠ (1st𝑐))
188186, 187syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝑒 ∈ ((𝑁 ∖ {(1st𝑐)}) × 𝑁) → (1st𝑒) ≠ (1st𝑐))
189188neneqd 2945 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑒 ∈ ((𝑁 ∖ {(1st𝑐)}) × 𝑁) → ¬ (1st𝑒) = (1st𝑐))
190189adantl 485 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) ∧ 𝑒 ∈ ((𝑁 ∖ {(1st𝑐)}) × 𝑁)) → ¬ (1st𝑒) = (1st𝑐))
191190iffalsed 4450 . . . . . . . . . . . . . . . . . . . . . . . . . . . 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))
192191mpteq2dva 5150 . . . . . . . . . . . . . . . . . . . . . . . . . . 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)))
193 difss 4046 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑁 ∖ {(1st𝑐)}) ⊆ 𝑁
194 xpss1 5570 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((𝑁 ∖ {(1st𝑐)}) ⊆ 𝑁 → ((𝑁 ∖ {(1st𝑐)}) × 𝑁) ⊆ (𝑁 × 𝑁))
195193, 194ax-mp 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝑁 ∖ {(1st𝑐)}) × 𝑁) ⊆ (𝑁 × 𝑁)
196 resmpt 5905 . . . . . . . . . . . . . . . . . . . . . . . . . . . 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))))
197195, 196mp1i 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))))
198 resmpt 5905 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (((𝑁 ∖ {(1st𝑐)}) × 𝑁) ⊆ (𝑁 × 𝑁) → ((𝑒 ∈ (𝑁 × 𝑁) ↦ (𝑎𝑒)) ↾ ((𝑁 ∖ {(1st𝑐)}) × 𝑁)) = (𝑒 ∈ ((𝑁 ∖ {(1st𝑐)}) × 𝑁) ↦ (𝑎𝑒)))
199195, 198mp1i 13 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) → ((𝑒 ∈ (𝑁 × 𝑁) ↦ (𝑎𝑒)) ↾ ((𝑁 ∖ {(1st𝑐)}) × 𝑁)) = (𝑒 ∈ ((𝑁 ∖ {(1st𝑐)}) × 𝑁) ↦ (𝑎𝑒)))
200192, 197, 1993eqtr4rd 2788 . . . . . . . . . . . . . . . . . . . . . . . . . 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)))
201185, 200eqtrd 2777 . . . . . . . . . . . . . . . . . . . . . . . . 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)))
202 fveq2 6717 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑒 = 𝑐 → (1st𝑒) = (1st𝑐))
203190, 202nsyl 142 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) ∧ 𝑒 ∈ ((𝑁 ∖ {(1st𝑐)}) × 𝑁)) → ¬ 𝑒 = 𝑐)
204203iffalsed 4450 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) ∧ 𝑒 ∈ ((𝑁 ∖ {(1st𝑐)}) × 𝑁)) → if(𝑒 = 𝑐, if(𝑒𝑑, 1 , 0 ), (𝑎𝑒)) = (𝑎𝑒))
205204mpteq2dva 5150 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) → (𝑒 ∈ ((𝑁 ∖ {(1st𝑐)}) × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒𝑑, 1 , 0 ), (𝑎𝑒))) = (𝑒 ∈ ((𝑁 ∖ {(1st𝑐)}) × 𝑁) ↦ (𝑎𝑒)))
206 resmpt 5905 . . . . . . . . . . . . . . . . . . . . . . . . . . . 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))))
207195, 206mp1i 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))))
208205, 207, 1993eqtr4rd 2788 . . . . . . . . . . . . . . . . . . . . . . . . . 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)))
209185, 208eqtrd 2777 . . . . . . . . . . . . . . . . . . . . . . . . 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)))
210134adantr 484 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) ∧ 𝑒 ∈ (𝑁 × 𝑁)) → if(𝑒𝑑, 1 , 0 ) ∈ 𝐾)
211110ffvelrnda 6904 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) ∧ 𝑒 ∈ (𝑁 × 𝑁)) → (𝑎𝑒) ∈ 𝐾)
212210, 211ifcld 4485 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) ∧ 𝑒 ∈ (𝑁 × 𝑁)) → if(𝑒 = 𝑐, if(𝑒𝑑, 1 , 0 ), (𝑎𝑒)) ∈ 𝐾)
213212fmpttd 6932 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) → (𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒𝑑, 1 , 0 ), (𝑎𝑒))):(𝑁 × 𝑁)⟶𝐾)
21412fvexi 6731 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 𝐾 ∈ V
21567anidms 570 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑁 ∈ Fin → (𝑁 × 𝑁) ∈ Fin)
216158, 215syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) → (𝑁 × 𝑁) ∈ Fin)
217 elmapg 8521 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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))
218214, 216, 217sylancr 590 . . . . . . . . . . . . . . . . . . . . . . . . . . . 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))
219213, 218mpbird 260 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) → (𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒𝑑, 1 , 0 ), (𝑎𝑒))) ∈ (𝐾m (𝑁 × 𝑁)))
22011, 12matbas2 21318 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (𝐾m (𝑁 × 𝑁)) = (Base‘𝐴))
221158, 113, 220syl2anc 587 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) → (𝐾m (𝑁 × 𝑁)) = (Base‘𝐴))
222221, 13eqtr4di 2796 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) → (𝐾m (𝑁 × 𝑁)) = 𝐵)
223219, 222eleqtrd 2840 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) → (𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒𝑑, 1 , 0 ), (𝑎𝑒))) ∈ 𝐵)
224 simp3 1140 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) → 𝑎𝐵)
225115adantr 484 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) ∧ 𝑒 ∈ (𝑁 × 𝑁)) → 𝑅 ∈ Grp)
22612, 136grpsubcl 18443 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ((𝑅 ∈ Grp ∧ (𝑎𝑒) ∈ 𝐾 ∧ if(𝑒𝑑, 1 , 0 ) ∈ 𝐾) → ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )) ∈ 𝐾)
227225, 211, 210, 226syl3anc 1373 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) ∧ 𝑒 ∈ (𝑁 × 𝑁)) → ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )) ∈ 𝐾)
228133adantr 484 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) ∧ 𝑒 ∈ (𝑁 × 𝑁)) → 0𝐾)
229227, 228ifcld 4485 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) ∧ 𝑒 ∈ (𝑁 × 𝑁)) → if(𝑒 = 𝑐, ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )), 0 ) ∈ 𝐾)
230229, 211ifcld 4485 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)
231230fmpttd 6932 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)
232 elmapg 8521 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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))
233214, 216, 232sylancr 590 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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))
234231, 233mpbird 260 . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)))
235234, 222eleqtrd 2840 . . . . . . . . . . . . . . . . . . . . . . . . . . 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)
236553ad2ant1 1135 . . . . . . . . . . . . . . . . . . . . . . . . . . 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))))
237 reseq1 5845 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (𝑥 = 𝑎 → (𝑥 ↾ ({𝑤} × 𝑁)) = (𝑎 ↾ ({𝑤} × 𝑁)))
238237eqeq1d 2739 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝑥 = 𝑎 → ((𝑥 ↾ ({𝑤} × 𝑁)) = ((𝑦 ↾ ({𝑤} × 𝑁)) ∘f + (𝑧 ↾ ({𝑤} × 𝑁))) ↔ (𝑎 ↾ ({𝑤} × 𝑁)) = ((𝑦 ↾ ({𝑤} × 𝑁)) ∘f + (𝑧 ↾ ({𝑤} × 𝑁)))))
239 reseq1 5845 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (𝑥 = 𝑎 → (𝑥 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑎 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)))
240239eqeq1d 2739 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝑥 = 𝑎 → ((𝑥 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑦 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) ↔ (𝑎 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑦 ↾ ((𝑁 ∖ {𝑤}) × 𝑁))))
241239eqeq1d 2739 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝑥 = 𝑎 → ((𝑥 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑧 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) ↔ (𝑎 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑧 ↾ ((𝑁 ∖ {𝑤}) × 𝑁))))
242238, 240, 2413anbi123d 1438 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)))))
243 fveqeq2 6726 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑥 = 𝑎 → ((𝐷𝑥) = ((𝐷𝑦) + (𝐷𝑧)) ↔ (𝐷𝑎) = ((𝐷𝑦) + (𝐷𝑧))))
244242, 243imbi12d 348 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)))))
2452442ralbidv 3120 . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)))))
246 reseq1 5845 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)))
247246oveq1d 7228 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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))))
248247eqeq2d 2748 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)))))
249 reseq1 5845 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)))
250249eqeq2d 2748 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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))))
251248, 2503anbi12d 1439 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)))))
252 fveq2 6717 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)))))
253252oveq1d 7228 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)))
254253eqeq2d 2748 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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))))
255251, 254imbi12d 348 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)))))
2562552ralbidv 3120 . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)))))
257245, 256rspc2va 3548 . . . . . . . . . . . . . . . . . . . . . . . . . . 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))))
258224, 235, 236, 257syl21anc 838 . . . . . . . . . . . . . . . . . . . . . . . . . 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))))
259 reseq1 5845 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)))
260259oveq2d 7229 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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))))
261260eqeq2d 2748 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)))))
262 reseq1 5845 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)))
263262eqeq2d 2748 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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))))
264261, 2633anbi13d 1440 . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)))))
265 fveq2 6717 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)))))
266265oveq2d 7229 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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))))))
267266eqeq2d 2748 . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)))))))
268264, 267imbi12d 348 . . . . . . . . . . . . . . . . . . . . . . . . . . 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))))))))
269 sneq 4551 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (𝑤 = (1st𝑐) → {𝑤} = {(1st𝑐)})
270269xpeq1d 5580 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝑤 = (1st𝑐) → ({𝑤} × 𝑁) = ({(1st𝑐)} × 𝑁))
271270reseq2d 5851 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑤 = (1st𝑐) → (𝑎 ↾ ({𝑤} × 𝑁)) = (𝑎 ↾ ({(1st𝑐)} × 𝑁)))
272270reseq2d 5851 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)))
273270reseq2d 5851 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)))
274272, 273oveq12d 7231 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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))))
275271, 274eqeq12d 2753 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)))))
276269difeq2d 4037 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (𝑤 = (1st𝑐) → (𝑁 ∖ {𝑤}) = (𝑁 ∖ {(1st𝑐)}))
277276xpeq1d 5580 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝑤 = (1st𝑐) → ((𝑁 ∖ {𝑤}) × 𝑁) = ((𝑁 ∖ {(1st𝑐)}) × 𝑁))
278277reseq2d 5851 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑤 = (1st𝑐) → (𝑎 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑎 ↾ ((𝑁 ∖ {(1st𝑐)}) × 𝑁)))
279277reseq2d 5851 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)))
280278, 279eqeq12d 2753 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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))))
281277reseq2d 5851 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)))
282278, 281eqeq12d 2753 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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))))
283275, 280, 2823anbi123d 1438 . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)))))
284283imbi1d 345 . . . . . . . . . . . . . . . . . . . . . . . . . . 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))))))))
285268, 284rspc2va 3548 . . . . . . . . . . . . . . . . . . . . . . . . . 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)))))))
286223, 124, 258, 285syl21anc 838 . . . . . . . . . . . . . . . . . . . . . . . . 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)))))))
287184, 201, 209, 286mp3and 1466 . . . . . . . . . . . . . . . . . . . . . . . 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))))))
288104, 105, 106, 287syl3anc 1373 . . . . . . . . . . . . . . . . . . . . . . 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))))))
289 fveq2 6717 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (𝑒 = 𝑐 → (𝑎𝑒) = (𝑎𝑐))
290 elequ1 2117 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 (𝑒 = 𝑐 → (𝑒𝑑𝑐𝑑))
291290ifbid 4462 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (𝑒 = 𝑐 → if(𝑒𝑑, 1 , 0 ) = if(𝑐𝑑, 1 , 0 ))
292289, 291oveq12d 7231 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (𝑒 = 𝑐 → ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )) = ((𝑎𝑐)(-g𝑅)if(𝑐𝑑, 1 , 0 )))
293292adantl 485 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) ∧ 𝑒 ∈ ({(1st𝑐)} × 𝑁)) ∧ 𝑒 = 𝑐) → ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )) = ((𝑎𝑐)(-g𝑅)if(𝑐𝑑, 1 , 0 )))
294110, 122ffvelrnd 6905 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) → (𝑎𝑐) ∈ 𝐾)
295131, 133ifcld 4485 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) → if(𝑐𝑑, 1 , 0 ) ∈ 𝐾)
29612, 136grpsubcl 18443 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ((𝑅 ∈ Grp ∧ (𝑎𝑐) ∈ 𝐾 ∧ if(𝑐𝑑, 1 , 0 ) ∈ 𝐾) → ((𝑎𝑐)(-g𝑅)if(𝑐𝑑, 1 , 0 )) ∈ 𝐾)
297115, 294, 295, 296syl3anc 1373 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) → ((𝑎𝑐)(-g𝑅)if(𝑐𝑑, 1 , 0 )) ∈ 𝐾)
29812, 51, 49ringridm 19590 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ((𝑅 ∈ Ring ∧ ((𝑎𝑐)(-g𝑅)if(𝑐𝑑, 1 , 0 )) ∈ 𝐾) → (((𝑎𝑐)(-g𝑅)if(𝑐𝑑, 1 , 0 )) · 1 ) = ((𝑎𝑐)(-g𝑅)if(𝑐𝑑, 1 , 0 )))
299113, 297, 298syl2anc 587 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) → (((𝑎𝑐)(-g𝑅)if(𝑐𝑑, 1 , 0 )) · 1 ) = ((𝑎𝑐)(-g𝑅)if(𝑐𝑑, 1 , 0 )))
300299ad2antrr 726 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 )))
301293, 300eqtr4d 2780 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 ))
302141adantl 485 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 )))
303 iftrue 4445 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (𝑒 = 𝑐 → if(𝑒 = 𝑐, 1 , 0 ) = 1 )
304303oveq2d 7229 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (𝑒 = 𝑐 → (((𝑎𝑐)(-g𝑅)if(𝑐𝑑, 1 , 0 )) · if(𝑒 = 𝑐, 1 , 0 )) = (((𝑎𝑐)(-g𝑅)if(𝑐𝑑, 1 , 0 )) · 1 ))
305304adantl 485 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 ))
306301, 302, 3053eqtr4d 2787 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 )))
30712, 51, 48ringrz 19606 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ((𝑅 ∈ Ring ∧ ((𝑎𝑐)(-g𝑅)if(𝑐𝑑, 1 , 0 )) ∈ 𝐾) → (((𝑎𝑐)(-g𝑅)if(𝑐𝑑, 1 , 0 )) · 0 ) = 0 )
308113, 297, 307syl2anc 587 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) → (((𝑎𝑐)(-g𝑅)if(𝑐𝑑, 1 , 0 )) · 0 ) = 0 )
309308eqcomd 2743 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) → 0 = (((𝑎𝑐)(-g𝑅)if(𝑐𝑑, 1 , 0 )) · 0 ))
310309ad2antrr 726 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) ∧ 𝑒 ∈ ({(1st𝑐)} × 𝑁)) ∧ ¬ 𝑒 = 𝑐) → 0 = (((𝑎𝑐)(-g𝑅)if(𝑐𝑑, 1 , 0 )) · 0 ))
311150adantl 485 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) ∧ 𝑒 ∈ ({(1st𝑐)} × 𝑁)) ∧ ¬ 𝑒 = 𝑐) → if(𝑒 = 𝑐, ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )), 0 ) = 0 )
312 iffalse 4448 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 𝑒 = 𝑐 → if(𝑒 = 𝑐, 1 , 0 ) = 0 )
313312oveq2d 7229 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 𝑒 = 𝑐 → (((𝑎𝑐)(-g𝑅)if(𝑐𝑑, 1 , 0 )) · if(𝑒 = 𝑐, 1 , 0 )) = (((𝑎𝑐)(-g𝑅)if(𝑐𝑑, 1 , 0 )) · 0 ))
314313adantl 485 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 ))
315310, 311, 3143eqtr4d 2787 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 )))
316306, 315pm2.61dan 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 )))
317170adantl 485 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) ∧ 𝑒 ∈ ({(1st𝑐)} × 𝑁)) → (1st𝑒) ∈ {(1st𝑐)})
318317, 171syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) ∧ 𝑒 ∈ ({(1st𝑐)} × 𝑁)) → (1st𝑒) = (1st𝑐))
319318iftrued 4447 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 ))
320318iftrued 4447 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) ∧ 𝑒 ∈ ({(1st𝑐)} × 𝑁)) → if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎𝑒)) = if(𝑒 = 𝑐, 1 , 0 ))
321320oveq2d 7229 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 )))
322316, 319, 3213eqtr4d 2787 . . . . . . . . . . . . . . . . . . . . . . . . . . . 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))))
323322mpteq2dva 5150 . . . . . . . . . . . . . . . . . . . . . . . . . . 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)))))
324 ovexd 7248 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) ∧ 𝑒 ∈ ({(1st𝑐)} × 𝑁)) → ((𝑎𝑐)(-g𝑅)if(𝑐𝑑, 1 , 0 )) ∈ V)
325165, 162ifex 4489 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 if(𝑒 = 𝑐, 1 , 0 ) ∈ V
326325, 167ifex 4489 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎𝑒)) ∈ V
327326a1i 11 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) ∧ 𝑒 ∈ ({(1st𝑐)} × 𝑁)) → if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎𝑒)) ∈ V)
328 fconstmpt 5611 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (({(1st𝑐)} × 𝑁) × {((𝑎𝑐)(-g𝑅)if(𝑐𝑑, 1 , 0 ))}) = (𝑒 ∈ ({(1st𝑐)} × 𝑁) ↦ ((𝑎𝑐)(-g𝑅)if(𝑐𝑑, 1 , 0 )))
329328a1i 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 ))))
330127resmptd 5908 . . . . . . . . . . . . . . . . . . . . . . . . . . . 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))))
331160, 324, 327, 329, 330offval2 7488 . . . . . . . . . . . . . . . . . . . . . . . . . . 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)))))
332323, 180, 3313eqtr4d 2787 . . . . . . . . . . . . . . . . . . . . . . . . . 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))))
333 iffalse 4448 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (¬ (1st𝑒) = (1st𝑐) → if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )), 0 ), (𝑎𝑒)) = (𝑎𝑒))
334 iffalse 4448 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (¬ (1st𝑒) = (1st𝑐) → if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎𝑒)) = (𝑎𝑒))
335333, 334eqtr4d 2780 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)))
336190, 335syl 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)))
337336mpteq2dva 5150 . . . . . . . . . . . . . . . . . . . . . . . . . . 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))))
338 resmpt 5905 . . . . . . . . . . . . . . . . . . . . . . . . . . . 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))))
339195, 338mp1i 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))))
340337, 197, 3393eqtr4d 2787 . . . . . . . . . . . . . . . . . . . . . . . . . 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)))
341131, 133ifcld 4485 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) → if(𝑒 = 𝑐, 1 , 0 ) ∈ 𝐾)
342341adantr 484 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) ∧ 𝑒 ∈ (𝑁 × 𝑁)) → if(𝑒 = 𝑐, 1 , 0 ) ∈ 𝐾)
343342, 211ifcld 4485 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) ∧ 𝑒 ∈ (𝑁 × 𝑁)) → if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎𝑒)) ∈ 𝐾)
344343fmpttd 6932 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) → (𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎𝑒))):(𝑁 × 𝑁)⟶𝐾)
345 elmapg 8521 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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))
346214, 216, 345sylancr 590 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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))
347344, 346mpbird 260 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) → (𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎𝑒))) ∈ (𝐾m (𝑁 × 𝑁)))
348347, 222eleqtrd 2840 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) → (𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎𝑒))) ∈ 𝐵)
349563ad2ant1 1135 . . . . . . . . . . . . . . . . . . . . . . . . . . . 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))))
350 reseq1 5845 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)))
351350eqeq1d 2739 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)))))
352 reseq1 5845 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)))
353352eqeq1d 2739 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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))))
354351, 353anbi12d 634 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)))))
355 fveqeq2 6726 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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))))
356354, 355imbi12d 348 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)))))
3573562ralbidv 3120 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)))))
358 sneq 4551 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 (𝑦 = ((𝑎𝑐)(-g𝑅)if(𝑐𝑑, 1 , 0 )) → {𝑦} = {((𝑎𝑐)(-g𝑅)if(𝑐𝑑, 1 , 0 ))})
359358xpeq2d 5581 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (𝑦 = ((𝑎𝑐)(-g𝑅)if(𝑐𝑑, 1 , 0 )) → (({𝑤} × 𝑁) × {𝑦}) = (({𝑤} × 𝑁) × {((𝑎𝑐)(-g𝑅)if(𝑐𝑑, 1 , 0 ))}))
360359oveq1d 7228 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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))))
361360eqeq2d 2748 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)))))
362361anbi1d 633 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)))))
363 oveq1 7220 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (𝑦 = ((𝑎𝑐)(-g𝑅)if(𝑐𝑑, 1 , 0 )) → (𝑦 · (𝐷𝑧)) = (((𝑎𝑐)(-g𝑅)if(𝑐𝑑, 1 , 0 )) · (𝐷𝑧)))
364363eqeq2d 2748 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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))))
365362, 364imbi12d 348 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)))))
3663652ralbidv 3120 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)))))
367357, 366rspc2va 3548 . . . . . . . . . . . . . . . . . . . . . . . . . . . 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))))
368235, 297, 349, 367syl21anc 838 . . . . . . . . . . . . . . . . . . . . . . . . . . 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))))
369 reseq1 5845 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)))
370369oveq2d 7229 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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))))
371370eqeq2d 2748 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)))))
372 reseq1 5845 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)))
373372eqeq2d 2748 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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))))
374371, 373anbi12d 634 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)))))
375 fveq2 6717 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)))))
376375oveq2d 7229 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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))))))
377376eqeq2d 2748 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)))))))
378374, 377imbi12d 348 . . . . . . . . . . . . . . . . . . . . . . . . . . . 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))))))))
379270xpeq1d 5580 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (𝑤 = (1st𝑐) → (({𝑤} × 𝑁) × {((𝑎𝑐)(-g𝑅)if(𝑐𝑑, 1 , 0 ))}) = (({(1st𝑐)} × 𝑁) × {((𝑎𝑐)(-g𝑅)if(𝑐𝑑, 1 , 0 ))}))
380270reseq2d 5851 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)))
381379, 380oveq12d 7231 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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))))
382272, 381eqeq12d 2753 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)))))
383277reseq2d 5851 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)))
384279, 383eqeq12d 2753 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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))))
385382, 384anbi12d 634 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)))))
386385imbi1d 345 . . . . . . . . . . . . . . . . . . . . . . . . . . . 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))))))))
387378, 386rspc2va 3548 . . . . . . . . . . . . . . . . . . . . . . . . . . 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)))))))
388348, 124, 368, 387syl21anc 838 . . . . . . . . . . . . . . . . . . . . . . . . . 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)))))))
389332, 340, 388mp2and 699 . . . . . . . . . . . . . . . . . . . . . . . . 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))))))
390389oveq1d 7228 . . . . . . . . . . . . . . . . . . . . . . . 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))))))
391104, 105, 106, 390syl3anc 1373 . . . . . . . . . . . . . . . . . . . . . . 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))))))
392 simpl3 1195 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ (𝑏 ∪ {𝑐}) ∈ 𝑌) ∧ ((𝑎𝐵𝑑 ∈ (𝑁m 𝑁)) ∧ ∀𝑤𝑏 (𝑎𝑤) = if(𝑤𝑑, 1 , 0 ))) → (𝑏 ∪ {𝑐}) ∈ 𝑌)
393 simprlr 780 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ (𝑏 ∪ {𝑐}) ∈ 𝑌) ∧ ((𝑎𝐵𝑑 ∈ (𝑁m 𝑁)) ∧ ∀𝑤𝑏 (𝑎𝑤) = if(𝑤𝑑, 1 , 0 ))) → 𝑑 ∈ (𝑁m 𝑁))
394 simprr 773 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ (𝑏 ∪ {𝑐}) ∈ 𝑌) ∧ ((𝑎𝐵𝑑 ∈ (𝑁m 𝑁)) ∧ ∀𝑤𝑏 (𝑎𝑤) = if(𝑤𝑑, 1 , 0 ))) → ∀𝑤𝑏 (𝑎𝑤) = if(𝑤𝑑, 1 , 0 ))
395 ralss 3971 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (𝑏 ⊆ (𝑏 ∪ {𝑐}) → (∀𝑤𝑏 (𝑎𝑤) = if(𝑤𝑑, 1 , 0 ) ↔ ∀𝑤 ∈ (𝑏 ∪ {𝑐})(𝑤𝑏 → (𝑎𝑤) = if(𝑤𝑑, 1 , 0 ))))
39699, 395ax-mp 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (∀𝑤𝑏 (𝑎𝑤) = if(𝑤𝑑, 1 , 0 ) ↔ ∀𝑤 ∈ (𝑏 ∪ {𝑐})(𝑤𝑏 → (𝑎𝑤) = if(𝑤𝑑, 1 , 0 )))
397 iftrue 4445 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ((1st𝑤) = (1st𝑐) → if((1st𝑤) = (1st𝑐), if(𝑤 = 𝑐, 1 , 0 ), (𝑎𝑤)) = if(𝑤 = 𝑐, 1 , 0 ))
398397adantl 485 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 ))
399 ibar 532 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ((1st𝑤) = (1st𝑐) → ((2nd𝑤) = (2nd𝑐) ↔ ((1st𝑤) = (1st𝑐) ∧ (2nd𝑤) = (2nd𝑐))))
400399adantl 485 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 (((((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) ∧ 𝑑 ∈ (𝑁m 𝑁)) ∧ 𝑤 ∈ (𝑏 ∪ {𝑐})) ∧ (1st𝑤) = (1st𝑐)) → ((2nd𝑤) = (2nd𝑐) ↔ ((1st𝑤) = (1st𝑐) ∧ (2nd𝑤) = (2nd𝑐))))
401 relxp 5569 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 Rel (𝑁 × 𝑁)
402 simpl2 1194 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) ∧ 𝑑 ∈ (𝑁m 𝑁)) → (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁))
403402sselda 3901 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 ((((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) ∧ 𝑑 ∈ (𝑁m 𝑁)) ∧ 𝑤 ∈ (𝑏 ∪ {𝑐})) → 𝑤 ∈ (𝑁 × 𝑁))
404403adantr 484 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 (((((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) ∧ 𝑑 ∈ (𝑁m 𝑁)) ∧ 𝑤 ∈ (𝑏 ∪ {𝑐})) ∧ (1st𝑤) = (1st𝑐)) → 𝑤 ∈ (𝑁 × 𝑁))
405 1st2nd 7810 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 ((Rel (𝑁 × 𝑁) ∧ 𝑤 ∈ (𝑁 × 𝑁)) → 𝑤 = ⟨(1st𝑤), (2nd𝑤)⟩)
406401, 404, 405sylancr 590 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 (((((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) ∧ 𝑑 ∈ (𝑁m 𝑁)) ∧ 𝑤 ∈ (𝑏 ∪ {𝑐})) ∧ (1st𝑤) = (1st𝑐)) → 𝑤 = ⟨(1st𝑤), (2nd𝑤)⟩)
407406eleq1d 2822 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)⟩)))
408 simpr 488 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) ∧ 𝑑 ∈ (𝑁m 𝑁)) → 𝑑 ∈ (𝑁m 𝑁))
409 elmapi 8530 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 (𝑑 ∈ (𝑁m 𝑁) → 𝑑:𝑁𝑁)
410409adantl 485 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) ∧ 𝑑 ∈ (𝑁m 𝑁)) → 𝑑:𝑁𝑁)
411124adantr 484 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) ∧ 𝑑 ∈ (𝑁m 𝑁)) → (1st𝑐) ∈ 𝑁)
412 xp2nd 7794 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 (𝑐 ∈ (𝑁 × 𝑁) → (2nd𝑐) ∈ 𝑁)
413122, 412syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) → (2nd𝑐) ∈ 𝑁)
414413adantr 484 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) ∧ 𝑑 ∈ (𝑁m 𝑁)) → (2nd𝑐) ∈ 𝑁)
415 fsets 16722 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 (((𝑑 ∈ (𝑁m 𝑁) ∧ 𝑑:𝑁𝑁) ∧ (1st𝑐) ∈ 𝑁 ∧ (2nd𝑐) ∈ 𝑁) → (𝑑 sSet ⟨(1st𝑐), (2nd𝑐)⟩):𝑁𝑁)
416408, 410, 411, 414, 415syl211anc 1378 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) ∧ 𝑑 ∈ (𝑁m 𝑁)) → (𝑑 sSet ⟨(1st𝑐), (2nd𝑐)⟩):𝑁𝑁)
417416ffnd 6546 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) ∧ 𝑑 ∈ (𝑁m 𝑁)) → (𝑑 sSet ⟨(1st𝑐), (2nd𝑐)⟩) Fn 𝑁)
418417ad2antrr 726 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 (((((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) ∧ 𝑑 ∈ (𝑁m 𝑁)) ∧ 𝑤 ∈ (𝑏 ∪ {𝑐})) ∧ (1st𝑤) = (1st𝑐)) → (𝑑 sSet ⟨(1st𝑐), (2nd𝑐)⟩) Fn 𝑁)
419 xp1st 7793 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 (𝑤 ∈ (𝑁 × 𝑁) → (1st𝑤) ∈ 𝑁)
420403, 419syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 ((((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) ∧ 𝑑 ∈ (𝑁m 𝑁)) ∧ 𝑤 ∈ (𝑏 ∪ {𝑐})) → (1st𝑤) ∈ 𝑁)
421420adantr 484 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 (((((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) ∧ 𝑑 ∈ (𝑁m 𝑁)) ∧ 𝑤 ∈ (𝑏 ∪ {𝑐})) ∧ (1st𝑤) = (1st𝑐)) → (1st𝑤) ∈ 𝑁)
422 fnopfvb 6766 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 (((𝑑 sSet ⟨(1st𝑐), (2nd𝑐)⟩) Fn 𝑁 ∧ (1st𝑤) ∈ 𝑁) → (((𝑑 sSet ⟨(1st𝑐), (2nd𝑐)⟩)‘(1st𝑤)) = (2nd𝑤) ↔ ⟨(1st𝑤), (2nd𝑤)⟩ ∈ (𝑑 sSet ⟨(1st𝑐), (2nd𝑐)⟩)))
423418, 421, 422syl2anc 587 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)⟩)))
424 fveq2 6717 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 ((1st𝑤) = (1st𝑐) → ((𝑑 sSet ⟨(1st𝑐), (2nd𝑐)⟩)‘(1st𝑤)) = ((𝑑 sSet ⟨(1st𝑐), (2nd𝑐)⟩)‘(1st𝑐)))
425424adantl 485 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)))
426 vex 3412 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 𝑑 ∈ V
427 fvex 6730 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 (1st𝑐) ∈ V
428 fvex 6730 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 (2nd𝑐) ∈ V
429 fvsetsid 16721 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 ((𝑑 ∈ V ∧ (1st𝑐) ∈ V ∧ (2nd𝑐) ∈ V) → ((𝑑 sSet ⟨(1st𝑐), (2nd𝑐)⟩)‘(1st𝑐)) = (2nd𝑐))
430426, 427, 428, 429mp3an 1463 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 ((𝑑 sSet ⟨(1st𝑐), (2nd𝑐)⟩)‘(1st𝑐)) = (2nd𝑐)
431425, 430eqtrdi 2794 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 (((((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) ∧ 𝑑 ∈ (𝑁m 𝑁)) ∧ 𝑤 ∈ (𝑏 ∪ {𝑐})) ∧ (1st𝑤) = (1st𝑐)) → ((𝑑 sSet ⟨(1st𝑐), (2nd𝑐)⟩)‘(1st𝑤)) = (2nd𝑐))
432431eqeq1d 2739 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 (((((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) ∧ 𝑑 ∈ (𝑁m 𝑁)) ∧ 𝑤 ∈ (𝑏 ∪ {𝑐})) ∧ (1st𝑤) = (1st𝑐)) → (((𝑑 sSet ⟨(1st𝑐), (2nd𝑐)⟩)‘(1st𝑤)) = (2nd𝑤) ↔ (2nd𝑐) = (2nd𝑤)))
433 eqcom 2744 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 ((2nd𝑐) = (2nd𝑤) ↔ (2nd𝑤) = (2nd𝑐))
434432, 433bitrdi 290 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 (((((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) ∧ 𝑑 ∈ (𝑁m 𝑁)) ∧ 𝑤 ∈ (𝑏 ∪ {𝑐})) ∧ (1st𝑤) = (1st𝑐)) → (((𝑑 sSet ⟨(1st𝑐), (2nd𝑐)⟩)‘(1st𝑤)) = (2nd𝑤) ↔ (2nd𝑤) = (2nd𝑐)))
435407, 423, 4343bitr2rd 311 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 (((((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) ∧ 𝑑 ∈ (𝑁m 𝑁)) ∧ 𝑤 ∈ (𝑏 ∪ {𝑐})) ∧ (1st𝑤) = (1st𝑐)) → ((2nd𝑤) = (2nd𝑐) ↔ 𝑤 ∈ (𝑑 sSet ⟨(1st𝑐), (2nd𝑐)⟩)))
436122ad3antrrr 730 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 (((((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) ∧ 𝑑 ∈ (𝑁m 𝑁)) ∧ 𝑤 ∈ (𝑏 ∪ {𝑐})) ∧ (1st𝑤) = (1st𝑐)) → 𝑐 ∈ (𝑁 × 𝑁))
437 xpopth 7802 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ((𝑤 ∈ (𝑁 × 𝑁) ∧ 𝑐 ∈ (𝑁 × 𝑁)) → (((1st𝑤) = (1st𝑐) ∧ (2nd𝑤) = (2nd𝑐)) ↔ 𝑤 = 𝑐))
438404, 436, 437syl2anc 587 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 (((((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) ∧ 𝑑 ∈ (𝑁m 𝑁)) ∧ 𝑤 ∈ (𝑏 ∪ {𝑐})) ∧ (1st𝑤) = (1st𝑐)) → (((1st𝑤) = (1st𝑐) ∧ (2nd𝑤) = (2nd𝑐)) ↔ 𝑤 = 𝑐))
439400, 435, 4383bitr3rd 313 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 (((((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) ∧ 𝑑 ∈ (𝑁m 𝑁)) ∧ 𝑤 ∈ (𝑏 ∪ {𝑐})) ∧ (1st𝑤) = (1st𝑐)) → (𝑤 = 𝑐𝑤 ∈ (𝑑 sSet ⟨(1st𝑐), (2nd𝑐)⟩)))
440439ifbid 4462 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 ))
441398, 440eqtrd 2777 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 ))
442441a1d 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 )))
443 elsni 4558 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 (𝑤 ∈ {𝑐} → 𝑤 = 𝑐)
444443fveq2d 6721 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 (𝑤 ∈ {𝑐} → (1st𝑤) = (1st𝑐))
445444con3i 157 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 (¬ (1st𝑤) = (1st𝑐) → ¬ 𝑤 ∈ {𝑐})
446445adantl 485 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ((𝑤 ∈ (𝑏 ∪ {𝑐}) ∧ ¬ (1st𝑤) = (1st𝑐)) → ¬ 𝑤 ∈ {𝑐})
447 elun 4063 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 (𝑤 ∈ (𝑏 ∪ {𝑐}) ↔ (𝑤𝑏𝑤 ∈ {𝑐}))
448447biimpi 219 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 (𝑤 ∈ (𝑏 ∪ {𝑐}) → (𝑤𝑏𝑤 ∈ {𝑐}))
449448adantr 484 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ((𝑤 ∈ (𝑏 ∪ {𝑐}) ∧ ¬ (1st𝑤) = (1st𝑐)) → (𝑤𝑏𝑤 ∈ {𝑐}))
450 orel2 891 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 𝑤 ∈ {𝑐} → ((𝑤𝑏𝑤 ∈ {𝑐}) → 𝑤𝑏))
451446, 449, 450sylc 65 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ((𝑤 ∈ (𝑏 ∪ {𝑐}) ∧ ¬ (1st𝑤) = (1st𝑐)) → 𝑤𝑏)
452451adantll 714 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 (((((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) ∧ 𝑑 ∈ (𝑁m 𝑁)) ∧ 𝑤 ∈ (𝑏 ∪ {𝑐})) ∧ ¬ (1st𝑤) = (1st𝑐)) → 𝑤𝑏)
453 iffalse 4448 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 (¬ (1st𝑤) = (1st𝑐) → if((1st𝑤) = (1st𝑐), if(𝑤 = 𝑐, 1 , 0 ), if(𝑤𝑑, 1 , 0 )) = if(𝑤𝑑, 1 , 0 ))
454453adantl 485 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 (((((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) ∧ 𝑑 ∈ (𝑁m 𝑁)) ∧ 𝑤 ∈ (𝑏 ∪ {𝑐})) ∧ ¬ (1st𝑤) = (1st𝑐)) → if((1st𝑤) = (1st𝑐), if(𝑤 = 𝑐, 1 , 0 ), if(𝑤𝑑, 1 , 0 )) = if(𝑤𝑑, 1 , 0 ))
455 setsres 16731 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 (𝑑 ∈ V → ((𝑑 sSet ⟨(1st𝑐), (2nd𝑐)⟩) ↾ (V ∖ {(1st𝑐)})) = (𝑑 ↾ (V ∖ {(1st𝑐)})))
456455eleq2d 2823 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 (𝑑 ∈ V → (⟨(1st𝑤), (2nd𝑤)⟩ ∈ ((𝑑 sSet ⟨(1st𝑐), (2nd𝑐)⟩) ↾ (V ∖ {(1st𝑐)})) ↔ ⟨(1st𝑤), (2nd𝑤)⟩ ∈ (𝑑 ↾ (V ∖ {(1st𝑐)}))))
457426, 456mp1i 13 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 (((((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) ∧ 𝑑 ∈ (𝑁m 𝑁)) ∧ 𝑤 ∈ (𝑏 ∪ {𝑐})) ∧ ¬ (1st𝑤) = (1st𝑐)) → (⟨(1st𝑤), (2nd𝑤)⟩ ∈ ((𝑑 sSet ⟨(1st𝑐), (2nd𝑐)⟩) ↾ (V ∖ {(1st𝑐)})) ↔ ⟨(1st𝑤), (2nd𝑤)⟩ ∈ (𝑑 ↾ (V ∖ {(1st𝑐)}))))
458 fvex 6730 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 (1st𝑤) ∈ V
459458a1i 11 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 (¬ (1st𝑤) = (1st𝑐) → (1st𝑤) ∈ V)
460 neqne 2948 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 (¬ (1st𝑤) = (1st𝑐) → (1st𝑤) ≠ (1st𝑐))
461 eldifsn 4700 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 ((1st𝑤) ∈ (V ∖ {(1st𝑐)}) ↔ ((1st𝑤) ∈ V ∧ (1st𝑤) ≠ (1st𝑐)))
462459, 460, 461sylanbrc 586 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 (¬ (1st𝑤) = (1st𝑐) → (1st𝑤) ∈ (V ∖ {(1st𝑐)}))
463 fvex 6730 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 (2nd𝑤) ∈ V
464463opres 5861 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 ((1st𝑤) ∈ (V ∖ {(1st𝑐)}) → (⟨(1st𝑤), (2nd𝑤)⟩ ∈ ((𝑑 sSet ⟨(1st𝑐), (2nd𝑐)⟩) ↾ (V ∖ {(1st𝑐)})) ↔ ⟨(1st𝑤), (2nd𝑤)⟩ ∈ (𝑑 sSet ⟨(1st𝑐), (2nd𝑐)⟩)))
465464adantl 485 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 ((𝑤 ∈ (𝑁 × 𝑁) ∧ (1st𝑤) ∈ (V ∖ {(1st𝑐)})) → (⟨(1st𝑤), (2nd𝑤)⟩ ∈ ((𝑑 sSet ⟨(1st𝑐), (2nd𝑐)⟩) ↾ (V ∖ {(1st𝑐)})) ↔ ⟨(1st𝑤), (2nd𝑤)⟩ ∈ (𝑑 sSet ⟨(1st𝑐), (2nd𝑐)⟩)))
466 1st2nd2 7800 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 (𝑤 ∈ (𝑁 × 𝑁) → 𝑤 = ⟨(1st𝑤), (2nd𝑤)⟩)
467466eleq1d 2822 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 (𝑤 ∈ (𝑁 × 𝑁) → (𝑤 ∈ (𝑑 sSet ⟨(1st𝑐), (2nd𝑐)⟩) ↔ ⟨(1st𝑤), (2nd𝑤)⟩ ∈ (𝑑 sSet ⟨(1st𝑐), (2nd𝑐)⟩)))
468467adantr 484 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 ((𝑤 ∈ (𝑁 × 𝑁) ∧ (1st𝑤) ∈ (V ∖ {(1st𝑐)})) → (𝑤 ∈ (𝑑 sSet ⟨(1st𝑐), (2nd𝑐)⟩) ↔ ⟨(1st𝑤), (2nd𝑤)⟩ ∈ (𝑑 sSet ⟨(1st𝑐), (2nd𝑐)⟩)))
469465, 468bitr4d 285 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 ((𝑤 ∈ (𝑁 × 𝑁) ∧ (1st𝑤) ∈ (V ∖ {(1st𝑐)})) → (⟨(1st𝑤), (2nd𝑤)⟩ ∈ ((𝑑 sSet ⟨(1st𝑐), (2nd𝑐)⟩) ↾ (V ∖ {(1st𝑐)})) ↔ 𝑤 ∈ (𝑑 sSet ⟨(1st𝑐), (2nd𝑐)⟩)))
470403, 462, 469syl2an 599 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 (((((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) ∧ 𝑑 ∈ (𝑁m 𝑁)) ∧ 𝑤 ∈ (𝑏 ∪ {𝑐})) ∧ ¬ (1st𝑤) = (1st𝑐)) → (⟨(1st𝑤), (2nd𝑤)⟩ ∈ ((𝑑 sSet ⟨(1st𝑐), (2nd𝑐)⟩) ↾ (V ∖ {(1st𝑐)})) ↔ 𝑤 ∈ (𝑑 sSet ⟨(1st𝑐), (2nd𝑐)⟩)))
471463opres 5861 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 ((1st𝑤) ∈ (V ∖ {(1st𝑐)}) → (⟨(1st𝑤), (2nd𝑤)⟩ ∈ (𝑑 ↾ (V ∖ {(1st𝑐)})) ↔ ⟨(1st𝑤), (2nd𝑤)⟩ ∈ 𝑑))
472471adantl 485 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 ((𝑤 ∈ (𝑁 × 𝑁) ∧ (1st𝑤) ∈ (V ∖ {(1st𝑐)})) → (⟨(1st𝑤), (2nd𝑤)⟩ ∈ (𝑑 ↾ (V ∖ {(1st𝑐)})) ↔ ⟨(1st𝑤), (2nd𝑤)⟩ ∈ 𝑑))
473466eleq1d 2822 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 (𝑤 ∈ (𝑁 × 𝑁) → (𝑤𝑑 ↔ ⟨(1st𝑤), (2nd𝑤)⟩ ∈ 𝑑))
474473adantr 484 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 ((𝑤 ∈ (𝑁 × 𝑁) ∧ (1st𝑤) ∈ (V ∖ {(1st𝑐)})) → (𝑤𝑑 ↔ ⟨(1st𝑤), (2nd𝑤)⟩ ∈ 𝑑))
475472, 474bitr4d 285 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 ((𝑤 ∈ (𝑁 × 𝑁) ∧ (1st𝑤) ∈ (V ∖ {(1st𝑐)})) → (⟨(1st𝑤), (2nd𝑤)⟩ ∈ (𝑑 ↾ (V ∖ {(1st𝑐)})) ↔ 𝑤𝑑))
476403, 462, 475syl2an 599 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 (((((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) ∧ 𝑑 ∈ (𝑁m 𝑁)) ∧ 𝑤 ∈ (𝑏 ∪ {𝑐})) ∧ ¬ (1st𝑤) = (1st𝑐)) → (⟨(1st𝑤), (2nd𝑤)⟩ ∈ (𝑑 ↾ (V ∖ {(1st𝑐)})) ↔ 𝑤𝑑))
477457, 470, 4763bitr3rd 313 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 (((((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) ∧ 𝑑 ∈ (𝑁m 𝑁)) ∧ 𝑤 ∈ (𝑏 ∪ {𝑐})) ∧ ¬ (1st𝑤) = (1st𝑐)) → (𝑤𝑑𝑤 ∈ (𝑑 sSet ⟨(1st𝑐), (2nd𝑐)⟩)))
478477ifbid 4462 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 (((((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) ∧ 𝑑 ∈ (𝑁m 𝑁)) ∧ 𝑤 ∈ (𝑏 ∪ {𝑐})) ∧ ¬ (1st𝑤) = (1st𝑐)) → if(𝑤𝑑, 1 , 0 ) = if(𝑤 ∈ (𝑑 sSet ⟨(1st𝑐), (2nd𝑐)⟩), 1 , 0 ))
479454, 478eqtrd 2777 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 (((((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) ∧ 𝑑 ∈ (𝑁m 𝑁)) ∧ 𝑤 ∈ (𝑏 ∪ {𝑐})) ∧ ¬ (1st𝑤) = (1st𝑐)) → if((1st𝑤) = (1st𝑐), if(𝑤 = 𝑐, 1 , 0 ), if(𝑤𝑑, 1 , 0 )) = if(𝑤 ∈ (𝑑 sSet ⟨(1st𝑐), (2nd𝑐)⟩), 1 , 0 ))
480 ifeq2 4444 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ((𝑎𝑤) = if(𝑤𝑑, 1 , 0 ) → if((1st𝑤) = (1st𝑐), if(𝑤 = 𝑐, 1 , 0 ), (𝑎𝑤)) = if((1st𝑤) = (1st𝑐), if(𝑤 = 𝑐, 1 , 0 ), if(𝑤𝑑, 1 , 0 )))
481480eqeq1d 2739 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ((𝑎𝑤) = if(𝑤𝑑, 1 , 0 ) → (if((1st𝑤) = (1st𝑐), if(𝑤 = 𝑐, 1 , 0 ), (𝑎𝑤)) = if(𝑤 ∈ (𝑑 sSet ⟨(1st𝑐), (2nd𝑐)⟩), 1 , 0 ) ↔ if((1st𝑤) = (1st𝑐), if(𝑤 = 𝑐, 1 , 0 ), if(𝑤𝑑, 1 , 0 )) = if(𝑤 ∈ (𝑑 sSet ⟨(1st𝑐), (2nd𝑐)⟩), 1 , 0 )))
482479, 481syl5ibrcom 250 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 (((((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) ∧ 𝑑 ∈ (𝑁m 𝑁)) ∧ 𝑤 ∈ (𝑏 ∪ {𝑐})) ∧ ¬ (1st𝑤) = (1st𝑐)) → ((𝑎𝑤) = if(𝑤𝑑, 1 , 0 ) → if((1st𝑤) = (1st𝑐), if(𝑤 = 𝑐, 1 , 0 ), (𝑎𝑤)) = if(𝑤 ∈ (𝑑 sSet ⟨(1st𝑐), (2nd𝑐)⟩), 1 , 0 )))
483452, 482embantd 59 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 )))
484442, 483pm2.61dan 813 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ((((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) ∧ 𝑑 ∈ (𝑁m 𝑁)) ∧ 𝑤 ∈ (𝑏 ∪ {𝑐})) → ((𝑤𝑏 → (𝑎𝑤) = if(𝑤𝑑, 1 , 0 )) → if((1st𝑤) = (1st𝑐), if(𝑤 = 𝑐, 1 , 0 ), (𝑎𝑤)) = if(𝑤 ∈ (𝑑 sSet ⟨(1st𝑐), (2nd𝑐)⟩), 1 , 0 )))
485 fveqeq2 6726 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 (𝑒 = 𝑤 → ((1st𝑒) = (1st𝑐) ↔ (1st𝑤) = (1st𝑐)))
486 equequ1 2033 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 (𝑒 = 𝑤 → (𝑒 = 𝑐𝑤 = 𝑐))
487486ifbid 4462 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 (𝑒 = 𝑤 → if(𝑒 = 𝑐, 1 , 0 ) = if(𝑤 = 𝑐, 1 , 0 ))
488 fveq2 6717 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 (𝑒 = 𝑤 → (𝑎𝑒) = (𝑎𝑤))
489485, 487, 488ifbieq12d 4467 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 (𝑒 = 𝑤 → if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎𝑒)) = if((1st𝑤) = (1st𝑐), if(𝑤 = 𝑐, 1 , 0 ), (𝑎𝑤)))
490 eqid 2737 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 (𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎𝑒))) = (𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎𝑒)))
491165, 162ifex 4489 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 if(𝑤 = 𝑐, 1 , 0 ) ∈ V
492 fvex 6730 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 (𝑎𝑤) ∈ V
493491, 492ifex 4489 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 if((1st𝑤) = (1st𝑐), if(𝑤 = 𝑐, 1 , 0 ), (𝑎𝑤)) ∈ V
494489, 490, 493fvmpt 6818 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 (𝑤 ∈ (𝑁 × 𝑁) → ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎𝑒)))‘𝑤) = if((1st𝑤) = (1st𝑐), if(𝑤 = 𝑐, 1 , 0 ), (𝑎𝑤)))
495494eqeq1d 2739 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (𝑤 ∈ (𝑁 × 𝑁) → (((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎𝑒)))‘𝑤) = if(𝑤 ∈ (𝑑 sSet ⟨(1st𝑐), (2nd𝑐)⟩), 1 , 0 ) ↔ if((1st𝑤) = (1st𝑐), if(𝑤 = 𝑐, 1 , 0 ), (𝑎𝑤)) = if(𝑤 ∈ (𝑑 sSet ⟨(1st𝑐), (2nd𝑐)⟩), 1 , 0 )))
496403, 495syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ((((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) ∧ 𝑑 ∈ (𝑁m 𝑁)) ∧ 𝑤 ∈ (𝑏 ∪ {𝑐})) → (((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎𝑒)))‘𝑤) = if(𝑤 ∈ (𝑑 sSet ⟨(1st𝑐), (2nd𝑐)⟩), 1 , 0 ) ↔ if((1st𝑤) = (1st𝑐), if(𝑤 = 𝑐, 1 , 0 ), (𝑎𝑤)) = if(𝑤 ∈ (𝑑 sSet ⟨(1st𝑐), (2nd𝑐)⟩), 1 , 0 )))
497484, 496sylibrd 262 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) ∧ 𝑑 ∈ (𝑁m 𝑁)) ∧ 𝑤 ∈ (𝑏 ∪ {𝑐})) → ((𝑤𝑏 → (𝑎𝑤) = if(𝑤𝑑, 1 , 0 )) → ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎𝑒)))‘𝑤) = if(𝑤 ∈ (𝑑 sSet ⟨(1st𝑐), (2nd𝑐)⟩), 1 , 0 )))
498497ralimdva 3100 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) ∧ 𝑑 ∈ (𝑁m 𝑁)) → (∀𝑤 ∈ (𝑏 ∪ {𝑐})(𝑤𝑏 → (𝑎𝑤) = if(𝑤𝑑, 1 , 0 )) → ∀𝑤 ∈ (𝑏 ∪ {𝑐})((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎𝑒)))‘𝑤) = if(𝑤 ∈ (𝑑 sSet ⟨(1st𝑐), (2nd𝑐)⟩), 1 , 0 )))
499396, 498syl5bi 245 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) ∧ 𝑑 ∈ (𝑁m 𝑁)) → (∀𝑤𝑏 (𝑎𝑤) = if(𝑤𝑑, 1 , 0 ) → ∀𝑤 ∈ (𝑏 ∪ {𝑐})((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎𝑒)))‘𝑤) = if(𝑤 ∈ (𝑑 sSet ⟨(1st𝑐), (2nd𝑐)⟩), 1 , 0 )))
500499impr 458 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) ∧ (𝑑 ∈ (𝑁m 𝑁) ∧ ∀𝑤𝑏 (𝑎𝑤) = if(𝑤𝑑, 1 , 0 ))) → ∀𝑤 ∈ (𝑏 ∪ {𝑐})((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎𝑒)))‘𝑤) = if(𝑤 ∈ (𝑑 sSet ⟨(1st𝑐), (2nd𝑐)⟩), 1 , 0 ))
5015003adantr1 1171 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) ∧ ((𝑏 ∪ {𝑐}) ∈ 𝑌𝑑 ∈ (𝑁m 𝑁) ∧ ∀𝑤𝑏 (𝑎𝑤) = if(𝑤𝑑, 1 , 0 ))) → ∀𝑤 ∈ (𝑏 ∪ {𝑐})((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎𝑒)))‘𝑤) = if(𝑤 ∈ (𝑑 sSet ⟨(1st𝑐), (2nd𝑐)⟩), 1 , 0 ))
502348adantr 484 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) ∧ ((𝑏 ∪ {𝑐}) ∈ 𝑌𝑑 ∈ (𝑁m 𝑁) ∧ ∀𝑤𝑏 (𝑎𝑤) = if(𝑤𝑑, 1 , 0 ))) → (𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎𝑒))) ∈ 𝐵)
503 simpr2 1197 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) ∧ ((𝑏 ∪ {𝑐}) ∈ 𝑌𝑑 ∈ (𝑁m 𝑁) ∧ ∀𝑤𝑏 (𝑎𝑤) = if(𝑤𝑑, 1 , 0 ))) → 𝑑 ∈ (𝑁m 𝑁))
504503, 409syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) ∧ ((𝑏 ∪ {𝑐}) ∈ 𝑌𝑑 ∈ (𝑁m 𝑁) ∧ ∀𝑤𝑏 (𝑎𝑤) = if(𝑤𝑑, 1 , 0 ))) → 𝑑:𝑁𝑁)
505124adantr 484 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) ∧ ((𝑏 ∪ {𝑐}) ∈ 𝑌𝑑 ∈ (𝑁m 𝑁) ∧ ∀𝑤𝑏 (𝑎𝑤) = if(𝑤𝑑, 1 , 0 ))) → (1st𝑐) ∈ 𝑁)
506413adantr 484 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) ∧ ((𝑏 ∪ {𝑐}) ∈ 𝑌𝑑 ∈ (𝑁m 𝑁) ∧ ∀𝑤𝑏 (𝑎𝑤) = if(𝑤𝑑, 1 , 0 ))) → (2nd𝑐) ∈ 𝑁)
507503, 504, 505, 506, 415syl211anc 1378 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) ∧ ((𝑏 ∪ {𝑐}) ∈ 𝑌𝑑 ∈ (𝑁m 𝑁) ∧ ∀𝑤𝑏 (𝑎𝑤) = if(𝑤𝑑, 1 , 0 ))) → (𝑑 sSet ⟨(1st𝑐), (2nd𝑐)⟩):𝑁𝑁)
508158, 158elmapd 8522 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) → ((𝑑 sSet ⟨(1st𝑐), (2nd𝑐)⟩) ∈ (𝑁m 𝑁) ↔ (𝑑 sSet ⟨(1st𝑐), (2nd𝑐)⟩):𝑁𝑁))
509508adantr 484 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) ∧ ((𝑏 ∪ {𝑐}) ∈ 𝑌𝑑 ∈ (𝑁m 𝑁) ∧ ∀𝑤𝑏 (𝑎𝑤) = if(𝑤𝑑, 1 , 0 ))) → ((𝑑 sSet ⟨(1st𝑐), (2nd𝑐)⟩) ∈ (𝑁m 𝑁) ↔ (𝑑 sSet ⟨(1st𝑐), (2nd𝑐)⟩):𝑁𝑁))
510507, 509mpbird 260 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) ∧ ((𝑏 ∪ {𝑐}) ∈ 𝑌𝑑 ∈ (𝑁m 𝑁) ∧ ∀𝑤𝑏 (𝑎𝑤) = if(𝑤𝑑, 1 , 0 ))) → (𝑑 sSet ⟨(1st𝑐), (2nd𝑐)⟩) ∈ (𝑁m 𝑁))
511 simpr1 1196 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) ∧ ((𝑏 ∪ {𝑐}) ∈ 𝑌𝑑 ∈ (𝑁m 𝑁) ∧ ∀𝑤𝑏 (𝑎𝑤) = if(𝑤𝑑, 1 , 0 ))) → (𝑏 ∪ {𝑐}) ∈ 𝑌)
512 raleq 3319 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (𝑥 = (𝑏 ∪ {𝑐}) → (∀𝑤𝑥 (𝑦𝑤) = if(𝑤𝑧, 1 , 0 ) ↔ ∀𝑤 ∈ (𝑏 ∪ {𝑐})(𝑦𝑤) = if(𝑤𝑧, 1 , 0 )))
513512imbi1d 345 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (𝑥 = (𝑏 ∪ {𝑐}) → ((∀𝑤𝑥 (𝑦𝑤) = if(𝑤𝑧, 1 , 0 ) → (𝐷𝑦) = 0 ) ↔ (∀𝑤 ∈ (𝑏 ∪ {𝑐})(𝑦𝑤) = if(𝑤𝑧, 1 , 0 ) → (𝐷𝑦) = 0 )))
5145132ralbidv 3120 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (𝑥 = (𝑏 ∪ {𝑐}) → (∀𝑦𝐵𝑧 ∈ (𝑁m 𝑁)(∀𝑤𝑥 (𝑦𝑤) = if(𝑤𝑧, 1 , 0 ) → (𝐷𝑦) = 0 ) ↔ ∀𝑦𝐵𝑧 ∈ (𝑁m 𝑁)(∀𝑤 ∈ (𝑏 ∪ {𝑐})(𝑦𝑤) = if(𝑤𝑧, 1 , 0 ) → (𝐷𝑦) = 0 )))
515514, 72elab2g 3589 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((𝑏 ∪ {𝑐}) ∈ 𝑌 → ((𝑏 ∪ {𝑐}) ∈ 𝑌 ↔ ∀𝑦𝐵𝑧 ∈ (𝑁m 𝑁)(∀𝑤 ∈ (𝑏 ∪ {𝑐})(𝑦𝑤) = if(𝑤𝑧, 1 , 0 ) → (𝐷𝑦) = 0 )))
516515ibi 270 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((𝑏 ∪ {𝑐}) ∈ 𝑌 → ∀𝑦𝐵𝑧 ∈ (𝑁m 𝑁)(∀𝑤 ∈ (𝑏 ∪ {𝑐})(𝑦𝑤) = if(𝑤𝑧, 1 , 0 ) → (𝐷𝑦) = 0 ))
517511, 516syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) ∧ ((𝑏 ∪ {𝑐}) ∈ 𝑌𝑑 ∈ (𝑁m 𝑁) ∧ ∀𝑤𝑏 (𝑎𝑤) = if(𝑤𝑑, 1 , 0 ))) → ∀𝑦𝐵𝑧 ∈ (𝑁m 𝑁)(∀𝑤 ∈ (𝑏 ∪ {𝑐})(𝑦𝑤) = if(𝑤𝑧, 1 , 0 ) → (𝐷𝑦) = 0 ))
518 fveq1 6716 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (𝑦 = (𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎𝑒))) → (𝑦𝑤) = ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎𝑒)))‘𝑤))
519518eqeq1d 2739 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (𝑦 = (𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎𝑒))) → ((𝑦𝑤) = if(𝑤𝑧, 1 , 0 ) ↔ ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎𝑒)))‘𝑤) = if(𝑤𝑧, 1 , 0 )))
520519ralbidv 3118 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝑦 = (𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎𝑒))) → (∀𝑤 ∈ (𝑏 ∪ {𝑐})(𝑦𝑤) = if(𝑤𝑧, 1 , 0 ) ↔ ∀𝑤 ∈ (𝑏 ∪ {𝑐})((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎𝑒)))‘𝑤) = if(𝑤𝑧, 1 , 0 )))
521 fveqeq2 6726 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝑦 = (𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎𝑒))) → ((𝐷𝑦) = 0 ↔ (𝐷‘(𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎𝑒)))) = 0 ))
522520, 521imbi12d 348 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑦 = (𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎𝑒))) → ((∀𝑤 ∈ (𝑏 ∪ {𝑐})(𝑦𝑤) = if(𝑤𝑧, 1 , 0 ) → (𝐷𝑦) = 0 ) ↔ (∀𝑤 ∈ (𝑏 ∪ {𝑐})((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎𝑒)))‘𝑤) = if(𝑤𝑧, 1 , 0 ) → (𝐷‘(𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎𝑒)))) = 0 )))
523 eleq2 2826 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (𝑧 = (𝑑 sSet ⟨(1st𝑐), (2nd𝑐)⟩) → (𝑤𝑧𝑤 ∈ (𝑑 sSet ⟨(1st𝑐), (2nd𝑐)⟩)))
524523ifbid 4462 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (𝑧 = (𝑑 sSet ⟨(1st𝑐), (2nd𝑐)⟩) → if(𝑤𝑧, 1 , 0 ) = if(𝑤 ∈ (𝑑 sSet ⟨(1st𝑐), (2nd𝑐)⟩), 1 , 0 ))
525524eqeq2d 2748 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (𝑧 = (𝑑 sSet ⟨(1st𝑐), (2nd𝑐)⟩) → (((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎𝑒)))‘𝑤) = if(𝑤𝑧, 1 , 0 ) ↔ ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎𝑒)))‘𝑤) = if(𝑤 ∈ (𝑑 sSet ⟨(1st𝑐), (2nd𝑐)⟩), 1 , 0 )))
526525ralbidv 3118 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝑧 = (𝑑 sSet ⟨(1st𝑐), (2nd𝑐)⟩) → (∀𝑤 ∈ (𝑏 ∪ {𝑐})((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎𝑒)))‘𝑤) = if(𝑤𝑧, 1 , 0 ) ↔ ∀𝑤 ∈ (𝑏 ∪ {𝑐})((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎𝑒)))‘𝑤) = if(𝑤 ∈ (𝑑 sSet ⟨(1st𝑐), (2nd𝑐)⟩), 1 , 0 )))
527526imbi1d 345 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑧 = (𝑑 sSet ⟨(1st𝑐), (2nd𝑐)⟩) → ((∀𝑤 ∈ (𝑏 ∪ {𝑐})((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎𝑒)))‘𝑤) = if(𝑤𝑧, 1 , 0 ) → (𝐷‘(𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎𝑒)))) = 0 ) ↔ (∀𝑤 ∈ (𝑏 ∪ {𝑐})((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎𝑒)))‘𝑤) = if(𝑤 ∈ (𝑑 sSet ⟨(1st𝑐), (2nd𝑐)⟩), 1 , 0 ) → (𝐷‘(𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎𝑒)))) = 0 )))
528522, 527rspc2va 3548 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎𝑒))) ∈ 𝐵 ∧ (𝑑 sSet ⟨(1st𝑐), (2nd𝑐)⟩) ∈ (𝑁m 𝑁)) ∧ ∀𝑦𝐵𝑧 ∈ (𝑁m 𝑁)(∀𝑤 ∈ (𝑏 ∪ {𝑐})(𝑦𝑤) = if(𝑤𝑧, 1 , 0 ) → (𝐷𝑦) = 0 )) → (∀𝑤 ∈ (𝑏 ∪ {𝑐})((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎𝑒)))‘𝑤) = if(𝑤 ∈ (𝑑 sSet ⟨(1st𝑐), (2nd𝑐)⟩), 1 , 0 ) → (𝐷‘(𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎𝑒)))) = 0 ))
529502, 510, 517, 528syl21anc 838 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) ∧ ((𝑏 ∪ {𝑐}) ∈ 𝑌𝑑 ∈ (𝑁m 𝑁) ∧ ∀𝑤𝑏 (𝑎𝑤) = if(𝑤𝑑, 1 , 0 ))) → (∀𝑤 ∈ (𝑏 ∪ {𝑐})((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎𝑒)))‘𝑤) = if(𝑤 ∈ (𝑑 sSet ⟨(1st𝑐), (2nd𝑐)⟩), 1 , 0 ) → (𝐷‘(𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎𝑒)))) = 0 ))
530501, 529mpd 15 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) ∧ ((𝑏 ∪ {𝑐}) ∈ 𝑌𝑑 ∈ (𝑁m 𝑁) ∧ ∀𝑤𝑏 (𝑎𝑤) = if(𝑤𝑑, 1 , 0 ))) → (𝐷‘(𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎𝑒)))) = 0 )
531530oveq2d 7229 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) ∧ ((𝑏 ∪ {𝑐}) ∈ 𝑌𝑑 ∈ (𝑁m 𝑁) ∧ ∀𝑤𝑏 (𝑎𝑤) = if(𝑤𝑑, 1 , 0 ))) → (((𝑎𝑐)(-g𝑅)if(𝑐𝑑, 1 , 0 )) · (𝐷‘(𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎𝑒))))) = (((𝑎𝑐)(-g𝑅)if(𝑐𝑑, 1 , 0 )) · 0 ))
532118unssad 4101 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) → 𝑏 ⊆ (𝑁 × 𝑁))
533532adantr 484 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) ∧ ((𝑏 ∪ {𝑐}) ∈ 𝑌𝑑 ∈ (𝑁m 𝑁) ∧ ∀𝑤𝑏 (𝑎𝑤) = if(𝑤𝑑, 1 , 0 ))) → 𝑏 ⊆ (𝑁 × 𝑁))
534 simpr3 1198 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) ∧ ((𝑏 ∪ {𝑐}) ∈ 𝑌𝑑 ∈ (𝑁m 𝑁) ∧ ∀𝑤𝑏 (𝑎𝑤) = if(𝑤𝑑, 1 , 0 ))) → ∀𝑤𝑏 (𝑎𝑤) = if(𝑤𝑑, 1 , 0 ))
535 ssel2 3895 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ((𝑏 ⊆ (𝑁 × 𝑁) ∧ 𝑤𝑏) → 𝑤 ∈ (𝑁 × 𝑁))
536535adantr 484 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (((𝑏 ⊆ (𝑁 × 𝑁) ∧ 𝑤𝑏) ∧ (𝑎𝑤) = if(𝑤𝑑, 1 , 0 )) → 𝑤 ∈ (𝑁 × 𝑁))
537 elequ1 2117 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 (𝑒 = 𝑤 → (𝑒𝑑𝑤𝑑))
538537ifbid 4462 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 (𝑒 = 𝑤 → if(𝑒𝑑, 1 , 0 ) = if(𝑤𝑑, 1 , 0 ))
539486, 538, 488ifbieq12d 4467 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (𝑒 = 𝑤 → if(𝑒 = 𝑐, if(𝑒𝑑, 1 , 0 ), (𝑎𝑒)) = if(𝑤 = 𝑐, if(𝑤𝑑, 1 , 0 ), (𝑎𝑤)))
540 eqid 2737 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒𝑑, 1 , 0 ), (𝑎𝑒))) = (𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒𝑑, 1 , 0 ), (𝑎𝑒)))
541165, 162ifex 4489 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 if(𝑤𝑑, 1 , 0 ) ∈ V
542541, 492ifex 4489 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 if(𝑤 = 𝑐, if(𝑤𝑑, 1 , 0 ), (𝑎𝑤)) ∈ V
543539, 540, 542fvmpt 6818 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (𝑤 ∈ (𝑁 × 𝑁) → ((𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒𝑑, 1 , 0 ), (𝑎𝑒)))‘𝑤) = if(𝑤 = 𝑐, if(𝑤𝑑, 1 , 0 ), (𝑎𝑤)))
544536, 543syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (((𝑏 ⊆ (𝑁 × 𝑁) ∧ 𝑤𝑏) ∧ (𝑎𝑤) = if(𝑤𝑑, 1 , 0 )) → ((𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒𝑑, 1 , 0 ), (𝑎𝑒)))‘𝑤) = if(𝑤 = 𝑐, if(𝑤𝑑, 1 , 0 ), (𝑎𝑤)))
545 ifeq2 4444 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ((𝑎𝑤) = if(𝑤𝑑, 1 , 0 ) → if(𝑤 = 𝑐, if(𝑤𝑑, 1 , 0 ), (𝑎𝑤)) = if(𝑤 = 𝑐, if(𝑤𝑑, 1 , 0 ), if(𝑤𝑑, 1 , 0 )))
546545adantl 485 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (((𝑏 ⊆ (𝑁 × 𝑁) ∧ 𝑤𝑏) ∧ (𝑎𝑤) = if(𝑤𝑑, 1 , 0 )) → if(𝑤 = 𝑐, if(𝑤𝑑, 1 , 0 ), (𝑎𝑤)) = if(𝑤 = 𝑐, if(𝑤𝑑, 1 , 0 ), if(𝑤𝑑, 1 , 0 )))
547 ifid 4479 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 if(𝑤 = 𝑐, if(𝑤𝑑, 1 , 0 ), if(𝑤𝑑, 1 , 0 )) = if(𝑤𝑑, 1 , 0 )
548546, 547eqtrdi 2794 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (((𝑏 ⊆ (𝑁 × 𝑁) ∧ 𝑤𝑏) ∧ (𝑎𝑤) = if(𝑤𝑑, 1 , 0 )) → if(𝑤 = 𝑐, if(𝑤𝑑, 1 , 0 ), (𝑎𝑤)) = if(𝑤𝑑, 1 , 0 ))
549544, 548eqtrd 2777 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (((𝑏 ⊆ (𝑁 × 𝑁) ∧ 𝑤𝑏) ∧ (𝑎𝑤) = if(𝑤𝑑, 1 , 0 )) → ((𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒𝑑, 1 , 0 ), (𝑎𝑒)))‘𝑤) = if(𝑤𝑑, 1 , 0 ))
550549ex 416 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((𝑏 ⊆ (𝑁 × 𝑁) ∧ 𝑤𝑏) → ((𝑎𝑤) = if(𝑤𝑑, 1 , 0 ) → ((𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒𝑑, 1 , 0 ), (𝑎𝑒)))‘𝑤) = if(𝑤𝑑, 1 , 0 )))
551550ralimdva 3100 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑏 ⊆ (𝑁 × 𝑁) → (∀𝑤𝑏 (𝑎𝑤) = if(𝑤𝑑, 1 , 0 ) → ∀𝑤𝑏 ((𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒𝑑, 1 , 0 ), (𝑎𝑒)))‘𝑤) = if(𝑤𝑑, 1 , 0 )))
552533, 534, 551sylc 65 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) ∧ ((𝑏 ∪ {𝑐}) ∈ 𝑌𝑑 ∈ (𝑁m 𝑁) ∧ ∀𝑤𝑏 (𝑎𝑤) = if(𝑤𝑑, 1 , 0 ))) → ∀𝑤𝑏 ((𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒𝑑, 1 , 0 ), (𝑎𝑒)))‘𝑤) = if(𝑤𝑑, 1 , 0 ))
553142, 291eqtrd 2777 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝑒 = 𝑐 → if(𝑒 = 𝑐, if(𝑒𝑑, 1 , 0 ), (𝑎𝑒)) = if(𝑐𝑑, 1 , 0 ))
554165, 162ifex 4489 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 if(𝑐𝑑, 1 , 0 ) ∈ V
555553, 540, 554fvmpt 6818 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑐 ∈ (𝑁 × 𝑁) → ((𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒𝑑, 1 , 0 ), (𝑎𝑒)))‘𝑐) = if(𝑐𝑑, 1 , 0 ))
556122, 555syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) → ((𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒𝑑, 1 , 0 ), (𝑎𝑒)))‘𝑐) = if(𝑐𝑑, 1 , 0 ))
557556adantr 484 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) ∧ ((𝑏 ∪ {𝑐}) ∈ 𝑌𝑑 ∈ (𝑁m 𝑁) ∧ ∀𝑤𝑏 (𝑎𝑤) = if(𝑤𝑑, 1 , 0 ))) → ((𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒𝑑, 1 , 0 ), (𝑎𝑒)))‘𝑐) = if(𝑐𝑑, 1 , 0 ))
558 fveq2 6717 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝑤 = 𝑐 → ((𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒𝑑, 1 , 0 ), (𝑎𝑒)))‘𝑤) = ((𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒𝑑, 1 , 0 ), (𝑎𝑒)))‘𝑐))
559 elequ1 2117 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (𝑤 = 𝑐 → (𝑤𝑑𝑐𝑑))
560559ifbid 4462 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝑤 = 𝑐 → if(𝑤𝑑, 1 , 0 ) = if(𝑐𝑑, 1 , 0 ))
561558, 560eqeq12d 2753 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑤 = 𝑐 → (((𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒𝑑, 1 , 0 ), (𝑎𝑒)))‘𝑤) = if(𝑤𝑑, 1 , 0 ) ↔ ((𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒𝑑, 1 , 0 ), (𝑎𝑒)))‘𝑐) = if(𝑐𝑑, 1 , 0 )))
562561ralunsn 4805 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑐 ∈ V → (∀𝑤 ∈ (𝑏 ∪ {𝑐})((𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒𝑑, 1 , 0 ), (𝑎𝑒)))‘𝑤) = if(𝑤𝑑, 1 , 0 ) ↔ (∀𝑤𝑏 ((𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒𝑑, 1 , 0 ), (𝑎𝑒)))‘𝑤) = if(𝑤𝑑, 1 , 0 ) ∧ ((𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒𝑑, 1 , 0 ), (𝑎𝑒)))‘𝑐) = if(𝑐𝑑, 1 , 0 ))))
563562elv 3414 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (∀𝑤 ∈ (𝑏 ∪ {𝑐})((𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒𝑑, 1 , 0 ), (𝑎𝑒)))‘𝑤) = if(𝑤𝑑, 1 , 0 ) ↔ (∀𝑤𝑏 ((𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒𝑑, 1 , 0 ), (𝑎𝑒)))‘𝑤) = if(𝑤𝑑, 1 , 0 ) ∧ ((𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒𝑑, 1 , 0 ), (𝑎𝑒)))‘𝑐) = if(𝑐𝑑, 1 , 0 )))
564552, 557, 563sylanbrc 586 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) ∧ ((𝑏 ∪ {𝑐}) ∈ 𝑌𝑑 ∈ (𝑁m 𝑁) ∧ ∀𝑤𝑏 (𝑎𝑤) = if(𝑤𝑑, 1 , 0 ))) → ∀𝑤 ∈ (𝑏 ∪ {𝑐})((𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒𝑑, 1 , 0 ), (𝑎𝑒)))‘𝑤) = if(𝑤𝑑, 1 , 0 ))
565223adantr 484 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) ∧ ((𝑏 ∪ {𝑐}) ∈ 𝑌𝑑 ∈ (𝑁m 𝑁) ∧ ∀𝑤𝑏 (𝑎𝑤) = if(𝑤𝑑, 1 , 0 ))) → (𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒𝑑, 1 , 0 ), (𝑎𝑒))) ∈ 𝐵)
566 fveq1 6716 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (𝑦 = (𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒𝑑, 1 , 0 ), (𝑎𝑒))) → (𝑦𝑤) = ((𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒𝑑, 1 , 0 ), (𝑎𝑒)))‘𝑤))
567566eqeq1d 2739 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝑦 = (𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒𝑑, 1 , 0 ), (𝑎𝑒))) → ((𝑦𝑤) = if(𝑤𝑧, 1 , 0 ) ↔ ((𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒𝑑, 1 , 0 ), (𝑎𝑒)))‘𝑤) = if(𝑤𝑧, 1 , 0 )))
568567ralbidv 3118 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑦 = (𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒𝑑, 1 , 0 ), (𝑎𝑒))) → (∀𝑤 ∈ (𝑏 ∪ {𝑐})(𝑦𝑤) = if(𝑤𝑧, 1 , 0 ) ↔ ∀𝑤 ∈ (𝑏 ∪ {𝑐})((𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒𝑑, 1 , 0 ), (𝑎𝑒)))‘𝑤) = if(𝑤𝑧, 1 , 0 )))
569 fveqeq2 6726 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑦 = (𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒𝑑, 1 , 0 ), (𝑎𝑒))) → ((𝐷𝑦) = 0 ↔ (𝐷‘(𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒𝑑, 1 , 0 ), (𝑎𝑒)))) = 0 ))
570568, 569imbi12d 348 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑦 = (𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒𝑑, 1 , 0 ), (𝑎𝑒))) → ((∀𝑤 ∈ (𝑏 ∪ {𝑐})(𝑦𝑤) = if(𝑤𝑧, 1 , 0 ) → (𝐷𝑦) = 0 ) ↔ (∀𝑤 ∈ (𝑏 ∪ {𝑐})((𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒𝑑, 1 , 0 ), (𝑎𝑒)))‘𝑤) = if(𝑤𝑧, 1 , 0 ) → (𝐷‘(𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒𝑑, 1 , 0 ), (𝑎𝑒)))) = 0 )))
571 elequ2 2125 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (𝑧 = 𝑑 → (𝑤𝑧𝑤𝑑))
572571ifbid 4462 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (𝑧 = 𝑑 → if(𝑤𝑧, 1 , 0 ) = if(𝑤𝑑, 1 , 0 ))
573572eqeq2d 2748 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝑧 = 𝑑 → (((𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒𝑑, 1 , 0 ), (𝑎𝑒)))‘𝑤) = if(𝑤𝑧, 1 , 0 ) ↔ ((𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒𝑑, 1 , 0 ), (𝑎𝑒)))‘𝑤) = if(𝑤𝑑, 1 , 0 )))
574573ralbidv 3118 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑧 = 𝑑 → (∀𝑤 ∈ (𝑏 ∪ {𝑐})((𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒𝑑, 1 , 0 ), (𝑎𝑒)))‘𝑤) = if(𝑤𝑧, 1 , 0 ) ↔ ∀𝑤 ∈ (𝑏 ∪ {𝑐})((𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒𝑑, 1 , 0 ), (𝑎𝑒)))‘𝑤) = if(𝑤𝑑, 1 , 0 )))
575574imbi1d 345 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑧 = 𝑑 → ((∀𝑤 ∈ (𝑏 ∪ {𝑐})((𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒𝑑, 1 , 0 ), (𝑎𝑒)))‘𝑤) = if(𝑤𝑧, 1 , 0 ) → (𝐷‘(𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒𝑑, 1 , 0 ), (𝑎𝑒)))) = 0 ) ↔ (∀𝑤 ∈ (𝑏 ∪ {𝑐})((𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒𝑑, 1 , 0 ), (𝑎𝑒)))‘𝑤) = if(𝑤𝑑, 1 , 0 ) → (𝐷‘(𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒𝑑, 1 , 0 ), (𝑎𝑒)))) = 0 )))
576570, 575rspc2va 3548 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((((𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒𝑑, 1 , 0 ), (𝑎𝑒))) ∈ 𝐵𝑑 ∈ (𝑁m 𝑁)) ∧ ∀𝑦𝐵𝑧 ∈ (𝑁m 𝑁)(∀𝑤 ∈ (𝑏 ∪ {𝑐})(𝑦𝑤) = if(𝑤𝑧, 1 , 0 ) → (𝐷𝑦) = 0 )) → (∀𝑤 ∈ (𝑏 ∪ {𝑐})((𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒𝑑, 1 , 0 ), (𝑎𝑒)))‘𝑤) = if(𝑤𝑑, 1 , 0 ) → (𝐷‘(𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒𝑑, 1 , 0 ), (𝑎𝑒)))) = 0 ))
577565, 503, 517, 576syl21anc 838 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) ∧ ((𝑏 ∪ {𝑐}) ∈ 𝑌𝑑 ∈ (𝑁m 𝑁) ∧ ∀𝑤𝑏 (𝑎𝑤) = if(𝑤𝑑, 1 , 0 ))) → (∀𝑤 ∈ (𝑏 ∪ {𝑐})((𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒𝑑, 1 , 0 ), (𝑎𝑒)))‘𝑤) = if(𝑤𝑑, 1 , 0 ) → (𝐷‘(𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒𝑑, 1 , 0 ), (𝑎𝑒)))) = 0 ))
578564, 577mpd 15 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) ∧ ((𝑏 ∪ {𝑐}) ∈ 𝑌𝑑 ∈ (𝑁m 𝑁) ∧ ∀𝑤𝑏 (𝑎𝑤) = if(𝑤𝑑, 1 , 0 ))) → (𝐷‘(𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒𝑑, 1 , 0 ), (𝑎𝑒)))) = 0 )
579531, 578oveq12d 7231 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) ∧ ((𝑏 ∪ {𝑐}) ∈ 𝑌𝑑 ∈ (𝑁m 𝑁) ∧ ∀𝑤𝑏 (𝑎𝑤) = if(𝑤𝑑, 1 , 0 ))) → ((((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))))) = ((((𝑎𝑐)(-g𝑅)if(𝑐𝑑, 1 , 0 )) · 0 ) + 0 ))
580308oveq1d 7228 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) → ((((𝑎𝑐)(-g𝑅)if(𝑐𝑑, 1 , 0 )) · 0 ) + 0 ) = ( 0 + 0 ))
58112, 50, 48grplid 18397 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝑅 ∈ Grp ∧ 0𝐾) → ( 0 + 0 ) = 0 )
582115, 133, 581syl2anc 587 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) → ( 0 + 0 ) = 0 )
583580, 582eqtrd 2777 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) → ((((𝑎𝑐)(-g𝑅)if(𝑐𝑑, 1 , 0 )) · 0 ) + 0 ) = 0 )
584583adantr 484 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) ∧ ((𝑏 ∪ {𝑐}) ∈ 𝑌𝑑 ∈ (𝑁m 𝑁) ∧ ∀𝑤𝑏 (𝑎𝑤) = if(𝑤𝑑, 1 , 0 ))) → ((((𝑎𝑐)(-g𝑅)if(𝑐𝑑, 1 , 0 )) · 0 ) + 0 ) = 0 )
585579, 584eqtrd 2777 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) ∧ ((𝑏 ∪ {𝑐}) ∈ 𝑌𝑑 ∈ (𝑁m 𝑁) ∧ ∀𝑤𝑏 (𝑎𝑤) = if(𝑤𝑑, 1 , 0 ))) → ((((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))))) = 0 )
586104, 105, 106, 392, 393, 394, 585syl33anc 1387 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ (𝑏 ∪ {𝑐}) ∈ 𝑌) ∧ ((𝑎𝐵𝑑 ∈ (𝑁m 𝑁)) ∧ ∀𝑤𝑏 (𝑎𝑤) = if(𝑤𝑑, 1 , 0 ))) → ((((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))))) = 0 )
587288, 391, 5863eqtrd 2781 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ (𝑏 ∪ {𝑐}) ∈ 𝑌) ∧ ((𝑎𝐵𝑑 ∈ (𝑁m 𝑁)) ∧ ∀𝑤𝑏 (𝑎𝑤) = if(𝑤𝑑, 1 , 0 ))) → (𝐷𝑎) = 0 )
588587expr 460 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ (𝑏 ∪ {𝑐}) ∈ 𝑌) ∧ (𝑎𝐵𝑑 ∈ (𝑁m 𝑁))) → (∀𝑤𝑏 (𝑎𝑤) = if(𝑤𝑑, 1 , 0 ) → (𝐷𝑎) = 0 ))
589588ralrimivva 3112 . . . . . . . . . . . . . . . . . . . 20 ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ (𝑏 ∪ {𝑐}) ∈ 𝑌) → ∀𝑎𝐵𝑑 ∈ (𝑁m 𝑁)(∀𝑤𝑏 (𝑎𝑤) = if(𝑤𝑑, 1 , 0 ) → (𝐷𝑎) = 0 ))
590 fveq1 6716 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑎 = 𝑦 → (𝑎𝑤) = (𝑦𝑤))
591590eqeq1d 2739 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑎 = 𝑦 → ((𝑎𝑤) = if(𝑤𝑑, 1 , 0 ) ↔ (𝑦𝑤) = if(𝑤𝑑, 1 , 0 )))
592591ralbidv 3118 . . . . . . . . . . . . . . . . . . . . . 22 (𝑎 = 𝑦 → (∀𝑤𝑏 (𝑎𝑤) = if(𝑤𝑑, 1 , 0 ) ↔ ∀𝑤𝑏 (𝑦𝑤) = if(𝑤𝑑, 1 , 0 )))
593 fveqeq2 6726 . . . . . . . . . . . . . . . . . . . . . 22 (𝑎 = 𝑦 → ((𝐷𝑎) = 0 ↔ (𝐷𝑦) = 0 ))
594592, 593imbi12d 348 . . . . . . . . . . . . . . . . . . . . 21 (𝑎 = 𝑦 → ((∀𝑤𝑏 (𝑎𝑤) = if(𝑤𝑑, 1 , 0 ) → (𝐷𝑎) = 0 ) ↔ (∀𝑤𝑏 (𝑦𝑤) = if(𝑤𝑑, 1 , 0 ) → (𝐷𝑦) = 0 )))
595 elequ2 2125 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑑 = 𝑧 → (𝑤𝑑𝑤𝑧))
596595ifbid 4462 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑑 = 𝑧 → if(𝑤𝑑, 1 , 0 ) = if(𝑤𝑧, 1 , 0 ))
597596eqeq2d 2748 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑑 = 𝑧 → ((𝑦𝑤) = if(𝑤𝑑, 1 , 0 ) ↔ (𝑦𝑤) = if(𝑤𝑧, 1 , 0 )))
598597ralbidv 3118 . . . . . . . . . . . . . . . . . . . . . 22 (𝑑 = 𝑧 → (∀𝑤𝑏 (𝑦𝑤) = if(𝑤𝑑, 1 , 0 ) ↔ ∀𝑤𝑏 (𝑦𝑤) = if(𝑤𝑧, 1 , 0 )))
599598imbi1d 345 . . . . . . . . . . . . . . . . . . . . 21 (𝑑 = 𝑧 → ((∀𝑤𝑏 (𝑦𝑤) = if(𝑤𝑑, 1 , 0 ) → (𝐷𝑦) = 0 ) ↔ (∀𝑤𝑏 (𝑦𝑤) = if(𝑤𝑧, 1 , 0 ) → (𝐷𝑦) = 0 )))
600594, 599cbvral2vw 3371 . . . . . . . . . . . . . . . . . . . 20 (∀𝑎𝐵𝑑 ∈ (𝑁m 𝑁)(∀𝑤𝑏 (𝑎𝑤) = if(𝑤𝑑, 1 , 0 ) → (𝐷𝑎) = 0 ) ↔ ∀𝑦𝐵𝑧 ∈ (𝑁m 𝑁)(∀𝑤𝑏 (𝑦𝑤) = if(𝑤𝑧, 1 , 0 ) → (𝐷𝑦) = 0 ))
601589, 600sylib 221 . . . . . . . . . . . . . . . . . . 19 ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ (𝑏 ∪ {𝑐}) ∈ 𝑌) → ∀𝑦𝐵𝑧 ∈ (𝑁m 𝑁)(∀𝑤𝑏 (𝑦𝑤) = if(𝑤𝑧, 1 , 0 ) → (𝐷𝑦) = 0 ))
602 vex 3412 . . . . . . . . . . . . . . . . . . . 20 𝑏 ∈ V
603 raleq 3319 . . . . . . . . . . . . . . . . . . . . . 22 (𝑥 = 𝑏 → (∀𝑤𝑥 (𝑦𝑤) = if(𝑤𝑧, 1 , 0 ) ↔ ∀𝑤𝑏 (𝑦𝑤) = if(𝑤𝑧, 1 , 0 )))
604603imbi1d 345 . . . . . . . . . . . . . . . . . . . . 21 (𝑥 = 𝑏 → ((∀𝑤𝑥 (𝑦𝑤) = if(𝑤𝑧, 1 , 0 ) → (𝐷𝑦) = 0 ) ↔ (∀𝑤𝑏 (𝑦𝑤) = if(𝑤𝑧, 1 , 0 ) → (𝐷𝑦) = 0 )))
6056042ralbidv 3120 . . . . . . . . . . . . . . . . . . . 20 (𝑥 = 𝑏 → (∀𝑦𝐵𝑧 ∈ (𝑁m 𝑁)(∀𝑤𝑥 (𝑦𝑤) = if(𝑤𝑧, 1 , 0 ) → (𝐷𝑦) = 0 ) ↔ ∀𝑦𝐵𝑧 ∈ (𝑁m 𝑁)(∀𝑤𝑏 (𝑦𝑤) = if(𝑤𝑧, 1 , 0 ) → (𝐷𝑦) = 0 )))
606602, 605, 72elab2 3591 . . . . . . . . . . . . . . . . . . 19 (𝑏𝑌 ↔ ∀𝑦𝐵𝑧 ∈ (𝑁m 𝑁)(∀𝑤𝑏 (𝑦𝑤) = if(𝑤𝑧, 1 , 0 ) → (𝐷𝑦) = 0 ))
607601, 606sylibr 237 . . . . . . . . . . . . . . . . . 18 ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ (𝑏 ∪ {𝑐}) ∈ 𝑌) → 𝑏𝑌)
6086073expia 1123 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁)) → ((𝑏 ∪ {𝑐}) ∈ 𝑌𝑏𝑌))
609608con3d 155 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁)) → (¬ 𝑏𝑌 → ¬ (𝑏 ∪ {𝑐}) ∈ 𝑌))
6106093adant3 1134 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ ¬ ∅ ∈ 𝑌) → (¬ 𝑏𝑌 → ¬ (𝑏 ∪ {𝑐}) ∈ 𝑌))
611610a1i 11 . . . . . . . . . . . . . 14 ((𝑏 ∈ Fin ∧ ¬ 𝑐𝑏) → ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ ¬ ∅ ∈ 𝑌) → (¬ 𝑏𝑌 → ¬ (𝑏 ∪ {𝑐}) ∈ 𝑌)))
612611a2d 29 . . . . . . . . . . . . 13 ((𝑏 ∈ Fin ∧ ¬ 𝑐𝑏) → (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ ¬ ∅ ∈ 𝑌) → ¬ 𝑏𝑌) → ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ ¬ ∅ ∈ 𝑌) → ¬ (𝑏 ∪ {𝑐}) ∈ 𝑌)))
613103, 612syl5 34 . . . . . . . . . . . 12 ((𝑏 ∈ Fin ∧ ¬ 𝑐𝑏) → (((𝜑𝑏 ⊆ (𝑁 × 𝑁) ∧ ¬ ∅ ∈ 𝑌) → ¬ 𝑏𝑌) → ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ ¬ ∅ ∈ 𝑌) → ¬ (𝑏 ∪ {𝑐}) ∈ 𝑌)))
61482, 87, 92, 97, 98, 613findcard2s 8843 . . . . . . . . . . 11 ((𝑁 × 𝑁) ∈ Fin → ((𝜑 ∧ (𝑁 × 𝑁) ⊆ (𝑁 × 𝑁) ∧ ¬ ∅ ∈ 𝑌) → ¬ (𝑁 × 𝑁) ∈ 𝑌))
61577, 614mpcom 38 . . . . . . . . . 10 ((𝜑 ∧ (𝑁 × 𝑁) ⊆ (𝑁 × 𝑁) ∧ ¬ ∅ ∈ 𝑌) → ¬ (𝑁 × 𝑁) ∈ 𝑌)
6166153exp 1121 . . . . . . . . 9 (𝜑 → ((𝑁 × 𝑁) ⊆ (𝑁 × 𝑁) → (¬ ∅ ∈ 𝑌 → ¬ (𝑁 × 𝑁) ∈ 𝑌)))
61776, 616mpi 20 . . . . . . . 8 (𝜑 → (¬ ∅ ∈ 𝑌 → ¬ (𝑁 × 𝑁) ∈ 𝑌))
61875, 617mt4d 117 . . . . . . 7 (𝜑 → ∅ ∈ 𝑌)
619618adantr 484 . . . . . 6 ((𝜑𝑎𝐵) → ∅ ∈ 𝑌)
620 0ex 5200 . . . . . . 7 ∅ ∈ V
621 raleq 3319 . . . . . . . . 9 (𝑥 = ∅ → (∀𝑤𝑥 (𝑦𝑤) = if(𝑤𝑧, 1 , 0 ) ↔ ∀𝑤 ∈ ∅ (𝑦𝑤) = if(𝑤𝑧, 1 , 0 )))
622621imbi1d 345 . . . . . . . 8 (𝑥 = ∅ → ((∀𝑤𝑥 (𝑦𝑤) = if(𝑤𝑧, 1 , 0 ) → (𝐷𝑦) = 0 ) ↔ (∀𝑤 ∈ ∅ (𝑦𝑤) = if(𝑤𝑧, 1 , 0 ) → (𝐷𝑦) = 0 )))
6236222ralbidv 3120 . . . . . . 7 (𝑥 = ∅ → (∀𝑦𝐵𝑧 ∈ (𝑁m 𝑁)(∀𝑤𝑥 (𝑦𝑤) = if(𝑤𝑧, 1 , 0 ) → (𝐷𝑦) = 0 ) ↔ ∀𝑦𝐵𝑧 ∈ (𝑁m 𝑁)(∀𝑤 ∈ ∅ (𝑦𝑤) = if(𝑤𝑧, 1 , 0 ) → (𝐷𝑦) = 0 )))
624620, 623, 72elab2 3591 . . . . . 6 (∅ ∈ 𝑌 ↔ ∀𝑦𝐵𝑧 ∈ (𝑁m 𝑁)(∀𝑤 ∈ ∅ (𝑦𝑤) = if(𝑤𝑧, 1 , 0 ) → (𝐷𝑦) = 0 ))
625619, 624sylib 221 . . . . 5 ((𝜑𝑎𝐵) → ∀𝑦𝐵𝑧 ∈ (𝑁m 𝑁)(∀𝑤 ∈ ∅ (𝑦𝑤) = if(𝑤𝑧, 1 , 0 ) → (𝐷𝑦) = 0 ))
626 fveq1 6716 . . . . . . . . 9 (𝑦 = 𝑎 → (𝑦𝑤) = (𝑎𝑤))
627626eqeq1d 2739 . . . . . . . 8 (𝑦 = 𝑎 → ((𝑦𝑤) = if(𝑤𝑧, 1 , 0 ) ↔ (𝑎𝑤) = if(𝑤𝑧, 1 , 0 )))
628627ralbidv 3118 . . . . . . 7 (𝑦 = 𝑎 → (∀𝑤 ∈ ∅ (𝑦𝑤) = if(𝑤𝑧, 1 , 0 ) ↔ ∀𝑤 ∈ ∅ (𝑎𝑤) = if(𝑤𝑧, 1 , 0 )))
629 fveqeq2 6726 . . . . . . 7 (𝑦 = 𝑎 → ((𝐷𝑦) = 0 ↔ (𝐷𝑎) = 0 ))
630628, 629imbi12d 348 . . . . . 6 (𝑦 = 𝑎 → ((∀𝑤 ∈ ∅ (𝑦𝑤) = if(𝑤𝑧, 1 , 0 ) → (𝐷𝑦) = 0 ) ↔ (∀𝑤 ∈ ∅ (𝑎𝑤) = if(𝑤𝑧, 1 , 0 ) → (𝐷𝑎) = 0 )))
631 eleq2 2826 . . . . . . . . . 10 (𝑧 = ( I ↾ 𝑁) → (𝑤𝑧𝑤 ∈ ( I ↾ 𝑁)))
632631ifbid 4462 . . . . . . . . 9 (𝑧 = ( I ↾ 𝑁) → if(𝑤𝑧, 1 , 0 ) = if(𝑤 ∈ ( I ↾ 𝑁), 1 , 0 ))
633632eqeq2d 2748 . . . . . . . 8 (𝑧 = ( I ↾ 𝑁) → ((𝑎𝑤) = if(𝑤𝑧, 1 , 0 ) ↔ (𝑎𝑤) = if(𝑤 ∈ ( I ↾ 𝑁), 1 , 0 )))
634633ralbidv 3118 . . . . . . 7 (𝑧 = ( I ↾ 𝑁) → (∀𝑤 ∈ ∅ (𝑎𝑤) = if(𝑤𝑧, 1 , 0 ) ↔ ∀𝑤 ∈ ∅ (𝑎𝑤) = if(𝑤 ∈ ( I ↾ 𝑁), 1 , 0 )))
635634imbi1d 345 . . . . . 6 (𝑧 = ( I ↾ 𝑁) → ((∀𝑤 ∈ ∅ (𝑎𝑤) = if(𝑤𝑧, 1 , 0 ) → (𝐷𝑎) = 0 ) ↔ (∀𝑤 ∈ ∅ (𝑎𝑤) = if(𝑤 ∈ ( I ↾ 𝑁), 1 , 0 ) → (𝐷𝑎) = 0 )))
636630, 635rspc2va 3548 . . . . 5 (((𝑎𝐵 ∧ ( I ↾ 𝑁) ∈ (𝑁m 𝑁)) ∧ ∀𝑦𝐵𝑧 ∈ (𝑁m 𝑁)(∀𝑤 ∈ ∅ (𝑦𝑤) = if(𝑤𝑧, 1 , 0 ) → (𝐷𝑦) = 0 )) → (∀𝑤 ∈ ∅ (𝑎𝑤) = if(𝑤 ∈ ( I ↾ 𝑁), 1 , 0 ) → (𝐷𝑎) = 0 ))
6372, 9, 625, 636syl21anc 838 . . . 4 ((𝜑𝑎𝐵) → (∀𝑤 ∈ ∅ (𝑎𝑤) = if(𝑤 ∈ ( I ↾ 𝑁), 1 , 0 ) → (𝐷𝑎) = 0 ))
6381, 637mpi 20 . . 3 ((𝜑𝑎𝐵) → (𝐷𝑎) = 0 )
639638mpteq2dva 5150 . 2 (𝜑 → (𝑎𝐵 ↦ (𝐷𝑎)) = (𝑎𝐵0 ))
64053feqmptd 6780 . 2 (𝜑𝐷 = (𝑎𝐵 ↦ (𝐷𝑎)))
641 fconstmpt 5611 . . 3 (𝐵 × { 0 }) = (𝑎𝐵0 )
642641a1i 11 . 2 (𝜑 → (𝐵 × { 0 }) = (𝑎𝐵0 ))
643639, 640, 6423eqtr4d 2787 1 (𝜑𝐷 = (𝐵 × { 0 }))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 209  wa 399  wo 847  w3a 1089   = wceq 1543  wcel 2110  {cab 2714  wne 2940  wral 3061  Vcvv 3408  cdif 3863  cun 3864  wss 3866  c0 4237  ifcif 4439  {csn 4541  cop 4547  cmpt 5135   I cid 5454   × cxp 5549  cres 5553  Rel wrel 5556   Fn wfn 6375  wf 6376  1-1-ontowf1o 6379  cfv 6380  (class class class)co 7213  cmpo 7215  f cof 7467  1st c1st 7759  2nd c2nd 7760  m cmap 8508  Fincfn 8626   sSet csts 16716  Basecbs 16760  +gcplusg 16802  .rcmulr 16803  0gc0g 16944  Grpcgrp 18365  -gcsg 18367  1rcur 19516  Ringcrg 19562   Mat cmat 21304
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708  ax-rep 5179  ax-sep 5192  ax-nul 5199  ax-pow 5258  ax-pr 5322  ax-un 7523  ax-cnex 10785  ax-resscn 10786  ax-1cn 10787  ax-icn 10788  ax-addcl 10789  ax-addrcl 10790  ax-mulcl 10791  ax-mulrcl 10792  ax-mulcom 10793  ax-addass 10794  ax-mulass 10795  ax-distr 10796  ax-i2m1 10797  ax-1ne0 10798  ax-1rid 10799  ax-rnegex 10800  ax-rrecex 10801  ax-cnre 10802  ax-pre-lttri 10803  ax-pre-lttrn 10804  ax-pre-ltadd 10805  ax-pre-mulgt0 10806  ax-addf 10808  ax-mulf 10809
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3or 1090  df-3an 1091  df-xor 1508  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ne 2941  df-nel 3047  df-ral 3066  df-rex 3067  df-reu 3068  df-rmo 3069  df-rab 3070  df-v 3410  df-sbc 3695  df-csb 3812  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-pss 3885  df-nul 4238  df-if 4440  df-pw 4515  df-sn 4542  df-pr 4544  df-tp 4546  df-op 4548  df-ot 4550  df-uni 4820  df-int 4860  df-iun 4906  df-iin 4907  df-br 5054  df-opab 5116  df-mpt 5136  df-tr 5162  df-id 5455  df-eprel 5460  df-po 5468  df-so 5469  df-fr 5509  df-se 5510  df-we 5511  df-xp 5557  df-rel 5558  df-cnv 5559  df-co 5560  df-dm 5561  df-rn 5562  df-res 5563  df-ima 5564  df-pred 6160  df-ord 6216  df-on 6217  df-lim 6218  df-suc 6219  df-iota 6338  df-fun 6382  df-fn 6383  df-f 6384  df-f1 6385  df-fo 6386  df-f1o 6387  df-fv 6388  df-isom 6389  df-riota 7170  df-ov 7216  df-oprab 7217  df-mpo 7218  df-of 7469  df-om 7645  df-1st 7761  df-2nd 7762  df-supp 7904  df-tpos 7968  df-wrecs 8047  df-recs 8108  df-rdg 8146  df-1o 8202  df-2o 8203  df-er 8391  df-map 8510  df-ixp 8579  df-en 8627  df-dom 8628  df-sdom 8629  df-fin 8630  df-fsupp 8986  df-sup 9058  df-oi 9126  df-card 9555  df-pnf 10869  df-mnf 10870  df-xr 10871  df-ltxr 10872  df-le 10873  df-sub 11064  df-neg 11065  df-div 11490  df-nn 11831  df-2 11893  df-3 11894  df-4 11895  df-5 11896  df-6 11897  df-7 11898  df-8 11899  df-9 11900  df-n0 12091  df-xnn0 12163  df-z 12177  df-dec 12294  df-uz 12439  df-rp 12587  df-fz 13096  df-fzo 13239  df-seq 13575  df-exp 13636  df-hash 13897  df-word 14070  df-lsw 14118  df-concat 14126  df-s1 14153  df-substr 14206  df-pfx 14236  df-splice 14315  df-reverse 14324  df-s2 14413  df-struct 16700  df-sets 16717  df-slot 16735  df-ndx 16745  df-base 16761  df-ress 16785  df-plusg 16815  df-mulr 16816  df-starv 16817  df-sca 16818  df-vsca 16819  df-ip 16820  df-tset 16821  df-ple 16822  df-ds 16824  df-unif 16825  df-hom 16826  df-cco 16827  df-0g 16946  df-gsum 16947  df-prds 16952  df-pws 16954  df-mre 17089  df-mrc 17090  df-acs 17092  df-mgm 18114  df-sgrp 18163  df-mnd 18174  df-mhm 18218  df-submnd 18219  df-efmnd 18296  df-grp 18368  df-minusg 18369  df-sbg 18370  df-mulg 18489  df-subg 18540  df-ghm 18620  df-gim 18663  df-cntz 18711  df-oppg 18738  df-symg 18760  df-pmtr 18834  df-psgn 18883  df-evpm 18884  df-cmn 19172  df-abl 19173  df-mgp 19505  df-ur 19517  df-ring 19564  df-cring 19565  df-oppr 19641  df-dvdsr 19659  df-unit 19660  df-invr 19690  df-dvr 19701  df-rnghom 19735  df-drng 19769  df-subrg 19798  df-lmod 19901  df-lss 19969  df-sra 20209  df-rgmod 20210  df-cnfld 20364  df-zring 20436  df-zrh 20470  df-dsmm 20694  df-frlm 20709  df-mamu 21283  df-mat 21305
This theorem is referenced by:  mdetuni0  21518
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