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Theorem mdetunilem9 21235
 Description: Lemma for mdetuni 21237. (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 4414 . . . 4 𝑤 ∈ ∅ (𝑎𝑤) = if(𝑤 ∈ ( I ↾ 𝑁), 1 , 0 )
2 simpr 488 . . . . 5 ((𝜑𝑎𝐵) → 𝑎𝐵)
3 f1oi 6628 . . . . . . . 8 ( I ↾ 𝑁):𝑁1-1-onto𝑁
4 f1of 6591 . . . . . . . 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 8406 . . . . . . 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 776 . . . . . . . . . . . . 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 21037 . . . . . . . . . . . . . . . 16 (𝑦𝐵𝑦 ∈ (𝐾m (𝑁 × 𝑁)))
15 elmapi 8414 . . . . . . . . . . . . . . . 16 (𝑦 ∈ (𝐾m (𝑁 × 𝑁)) → 𝑦:(𝑁 × 𝑁)⟶𝐾)
1614, 15syl 17 . . . . . . . . . . . . . . 15 (𝑦𝐵𝑦:(𝑁 × 𝑁)⟶𝐾)
1716feqmptd 6709 . . . . . . . . . . . . . 14 (𝑦𝐵𝑦 = (𝑤 ∈ (𝑁 × 𝑁) ↦ (𝑦𝑤)))
1817fveq2d 6650 . . . . . . . . . . . . 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 2798 . . . . . . . . . . . . . 14 (𝑁 × 𝑁) = (𝑁 × 𝑁)
21 mpteq12 5118 . . . . . . . . . . . . . . 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 6650 . . . . . . . . . . . . . 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 689 . . . . . . . . . . . . 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 2877 . . . . . . . . . . . . . . . . 17 (𝑎 = 𝑧 → (𝑎 ∈ (𝑁m 𝑁) ↔ 𝑧 ∈ (𝑁m 𝑁)))
2625anbi2d 631 . . . . . . . . . . . . . . . 16 (𝑎 = 𝑧 → ((𝜑𝑎 ∈ (𝑁m 𝑁)) ↔ (𝜑𝑧 ∈ (𝑁m 𝑁))))
27 elequ2 2126 . . . . . . . . . . . . . . . . . . . 20 (𝑎 = 𝑧 → (𝑤𝑎𝑤𝑧))
2827ifbid 4447 . . . . . . . . . . . . . . . . . . 19 (𝑎 = 𝑧 → if(𝑤𝑎, 1 , 0 ) = if(𝑤𝑧, 1 , 0 ))
2928mpteq2dv 5127 . . . . . . . . . . . . . . . . . 18 (𝑎 = 𝑧 → (𝑤 ∈ (𝑁 × 𝑁) ↦ if(𝑤𝑎, 1 , 0 )) = (𝑤 ∈ (𝑁 × 𝑁) ↦ if(𝑤𝑧, 1 , 0 )))
3029fveq2d 6650 . . . . . . . . . . . . . . . . 17 (𝑎 = 𝑧 → (𝐷‘(𝑤 ∈ (𝑁 × 𝑁) ↦ if(𝑤𝑎, 1 , 0 ))) = (𝐷‘(𝑤 ∈ (𝑁 × 𝑁) ↦ if(𝑤𝑧, 1 , 0 ))))
3130eqeq1d 2800 . . . . . . . . . . . . . . . 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 2877 . . . . . . . . . . . . . . . . . . . 20 (𝑤 = ⟨𝑏, 𝑐⟩ → (𝑤𝑎 ↔ ⟨𝑏, 𝑐⟩ ∈ 𝑎))
3433ifbid 4447 . . . . . . . . . . . . . . . . . . 19 (𝑤 = ⟨𝑏, 𝑐⟩ → if(𝑤𝑎, 1 , 0 ) = if(⟨𝑏, 𝑐⟩ ∈ 𝑎, 1 , 0 ))
3534mpompt 7246 . . . . . . . . . . . . . . . . . 18 (𝑤 ∈ (𝑁 × 𝑁) ↦ if(𝑤𝑎, 1 , 0 )) = (𝑏𝑁, 𝑐𝑁 ↦ if(⟨𝑏, 𝑐⟩ ∈ 𝑎, 1 , 0 ))
36 elmapi 8414 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑎 ∈ (𝑁m 𝑁) → 𝑎:𝑁𝑁)
3736adantl 485 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝜑𝑎 ∈ (𝑁m 𝑁)) → 𝑎:𝑁𝑁)
3837ffnd 6489 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑𝑎 ∈ (𝑁m 𝑁)) → 𝑎 Fn 𝑁)
39383ad2ant1 1130 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑𝑎 ∈ (𝑁m 𝑁)) ∧ 𝑏𝑁𝑐𝑁) → 𝑎 Fn 𝑁)
40 simp2 1134 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑𝑎 ∈ (𝑁m 𝑁)) ∧ 𝑏𝑁𝑐𝑁) → 𝑏𝑁)
41 fnopfvb 6695 . . . . . . . . . . . . . . . . . . . . . 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 4447 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑎 ∈ (𝑁m 𝑁)) ∧ 𝑏𝑁𝑐𝑁) → if(⟨𝑏, 𝑐⟩ ∈ 𝑎, 1 , 0 ) = if((𝑎𝑏) = 𝑐, 1 , 0 ))
4544mpoeq3dva 7211 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑎 ∈ (𝑁m 𝑁)) → (𝑏𝑁, 𝑐𝑁 ↦ if(⟨𝑏, 𝑐⟩ ∈ 𝑎, 1 , 0 )) = (𝑏𝑁, 𝑐𝑁 ↦ if((𝑎𝑏) = 𝑐, 1 , 0 )))
4635, 45syl5eq 2845 . . . . . . . . . . . . . . . . 17 ((𝜑𝑎 ∈ (𝑁m 𝑁)) → (𝑤 ∈ (𝑁 × 𝑁) ↦ if(𝑤𝑎, 1 , 0 )) = (𝑏𝑁, 𝑐𝑁 ↦ if((𝑎𝑏) = 𝑐, 1 , 0 )))
4746fveq2d 6650 . . . . . . . . . . . . . . . 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 21234 . . . . . . . . . . . . . . . . 17 ((𝜑𝑎:𝑁𝑁) → (𝐷‘(𝑏𝑁, 𝑐𝑁 ↦ if((𝑎𝑏) = 𝑐, 1 , 0 ))) = 0 )
5936, 58sylan2 595 . . . . . . . . . . . . . . . 16 ((𝜑𝑎 ∈ (𝑁m 𝑁)) → (𝐷‘(𝑏𝑁, 𝑐𝑁 ↦ if((𝑎𝑏) = 𝑐, 1 , 0 ))) = 0 )
6047, 59eqtrd 2833 . . . . . . . . . . . . . . 15 ((𝜑𝑎 ∈ (𝑁m 𝑁)) → (𝐷‘(𝑤 ∈ (𝑁 × 𝑁) ↦ if(𝑤𝑎, 1 , 0 ))) = 0 )
6132, 60chvarvv 2005 . . . . . . . . . . . . . 14 ((𝜑𝑧 ∈ (𝑁m 𝑁)) → (𝐷‘(𝑤 ∈ (𝑁 × 𝑁) ↦ if(𝑤𝑧, 1 , 0 ))) = 0 )
6261adantrl 715 . . . . . . . . . . . . 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 2837 . . . . . . . . . . 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 3156 . . . . . . . . 9 (𝜑 → ∀𝑦𝐵𝑧 ∈ (𝑁m 𝑁)(∀𝑤 ∈ (𝑁 × 𝑁)(𝑦𝑤) = if(𝑤𝑧, 1 , 0 ) → (𝐷𝑦) = 0 ))
67 xpfi 8776 . . . . . . . . . . 11 ((𝑁 ∈ Fin ∧ 𝑁 ∈ Fin) → (𝑁 × 𝑁) ∈ Fin)
686, 6, 67syl2anc 587 . . . . . . . . . 10 (𝜑 → (𝑁 × 𝑁) ∈ Fin)
69 raleq 3358 . . . . . . . . . . . . 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 3164 . . . . . . . . . . 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 3616 . . . . . . . . . 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 3937 . . . . . . . . 9 (𝑁 × 𝑁) ⊆ (𝑁 × 𝑁)
77683ad2ant1 1130 . . . . . . . . . . 11 ((𝜑 ∧ (𝑁 × 𝑁) ⊆ (𝑁 × 𝑁) ∧ ¬ ∅ ∈ 𝑌) → (𝑁 × 𝑁) ∈ Fin)
78 sseq1 3940 . . . . . . . . . . . . . 14 (𝑎 = ∅ → (𝑎 ⊆ (𝑁 × 𝑁) ↔ ∅ ⊆ (𝑁 × 𝑁)))
79783anbi2d 1438 . . . . . . . . . . . . 13 (𝑎 = ∅ → ((𝜑𝑎 ⊆ (𝑁 × 𝑁) ∧ ¬ ∅ ∈ 𝑌) ↔ (𝜑 ∧ ∅ ⊆ (𝑁 × 𝑁) ∧ ¬ ∅ ∈ 𝑌)))
80 eleq1 2877 . . . . . . . . . . . . . 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 3940 . . . . . . . . . . . . . 14 (𝑎 = 𝑏 → (𝑎 ⊆ (𝑁 × 𝑁) ↔ 𝑏 ⊆ (𝑁 × 𝑁)))
84833anbi2d 1438 . . . . . . . . . . . . 13 (𝑎 = 𝑏 → ((𝜑𝑎 ⊆ (𝑁 × 𝑁) ∧ ¬ ∅ ∈ 𝑌) ↔ (𝜑𝑏 ⊆ (𝑁 × 𝑁) ∧ ¬ ∅ ∈ 𝑌)))
85 eleq1 2877 . . . . . . . . . . . . . 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 3940 . . . . . . . . . . . . . 14 (𝑎 = (𝑏 ∪ {𝑐}) → (𝑎 ⊆ (𝑁 × 𝑁) ↔ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁)))
89883anbi2d 1438 . . . . . . . . . . . . 13 (𝑎 = (𝑏 ∪ {𝑐}) → ((𝜑𝑎 ⊆ (𝑁 × 𝑁) ∧ ¬ ∅ ∈ 𝑌) ↔ (𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ ¬ ∅ ∈ 𝑌)))
90 eleq1 2877 . . . . . . . . . . . . . 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 3940 . . . . . . . . . . . . . 14 (𝑎 = (𝑁 × 𝑁) → (𝑎 ⊆ (𝑁 × 𝑁) ↔ (𝑁 × 𝑁) ⊆ (𝑁 × 𝑁)))
94933anbi2d 1438 . . . . . . . . . . . . 13 (𝑎 = (𝑁 × 𝑁) → ((𝜑𝑎 ⊆ (𝑁 × 𝑁) ∧ ¬ ∅ ∈ 𝑌) ↔ (𝜑 ∧ (𝑁 × 𝑁) ⊆ (𝑁 × 𝑁) ∧ ¬ ∅ ∈ 𝑌)))
95 eleq1 2877 . . . . . . . . . . . . . 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 1135 . . . . . . . . . . . 12 ((𝜑 ∧ ∅ ⊆ (𝑁 × 𝑁) ∧ ¬ ∅ ∈ 𝑌) → ¬ ∅ ∈ 𝑌)
99 ssun1 4099 . . . . . . . . . . . . . . . 16 𝑏 ⊆ (𝑏 ∪ {𝑐})
100 sstr2 3922 . . . . . . . . . . . . . . . 16 (𝑏 ⊆ (𝑏 ∪ {𝑐}) → ((𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) → 𝑏 ⊆ (𝑁 × 𝑁)))
10199, 100ax-mp 5 . . . . . . . . . . . . . . 15 ((𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) → 𝑏 ⊆ (𝑁 × 𝑁))
1021013anim2i 1150 . . . . . . . . . . . . . 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 1188 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ (𝑏 ∪ {𝑐}) ∈ 𝑌) ∧ ((𝑎𝐵𝑑 ∈ (𝑁m 𝑁)) ∧ ∀𝑤𝑏 (𝑎𝑤) = if(𝑤𝑑, 1 , 0 ))) → 𝜑)
105 simpl2 1189 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ (𝑏 ∪ {𝑐}) ∈ 𝑌) ∧ ((𝑎𝐵𝑑 ∈ (𝑁m 𝑁)) ∧ ∀𝑤𝑏 (𝑎𝑤) = if(𝑤𝑑, 1 , 0 ))) → (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁))
106 simprll 778 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ (𝑏 ∪ {𝑐}) ∈ 𝑌) ∧ ((𝑎𝐵𝑑 ∈ (𝑁m 𝑁)) ∧ ∀𝑤𝑏 (𝑎𝑤) = if(𝑤𝑑, 1 , 0 ))) → 𝑎𝐵)
10711, 12, 13matbas2i 21037 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑎𝐵𝑎 ∈ (𝐾m (𝑁 × 𝑁)))
108 elmapi 8414 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑎 ∈ (𝐾m (𝑁 × 𝑁)) → 𝑎:(𝑁 × 𝑁)⟶𝐾)
109107, 108syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑎𝐵𝑎:(𝑁 × 𝑁)⟶𝐾)
1101093ad2ant3 1132 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) → 𝑎:(𝑁 × 𝑁)⟶𝐾)
111110feqmptd 6709 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) → 𝑎 = (𝑒 ∈ (𝑁 × 𝑁) ↦ (𝑎𝑒)))
112111reseq1d 5818 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) → (𝑎 ↾ ({(1st𝑐)} × 𝑁)) = ((𝑒 ∈ (𝑁 × 𝑁) ↦ (𝑎𝑒)) ↾ ({(1st𝑐)} × 𝑁)))
113523ad2ant1 1130 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) → 𝑅 ∈ Ring)
114 ringgrp 19299 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 1134 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) → (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁))
119118unssbd 4115 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) → {𝑐} ⊆ (𝑁 × 𝑁))
120 vex 3444 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 𝑐 ∈ V
121120snss 4679 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 (𝑐 ∈ (𝑁 × 𝑁) ↔ {𝑐} ⊆ (𝑁 × 𝑁))
122119, 121sylibr 237 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) → 𝑐 ∈ (𝑁 × 𝑁))
123 xp1st 7706 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 (𝑐 ∈ (𝑁 × 𝑁) → (1st𝑐) ∈ 𝑁)
124122, 123syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) → (1st𝑐) ∈ 𝑁)
125124snssd 4702 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) → {(1st𝑐)} ⊆ 𝑁)
126 xpss1 5539 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ({(1st𝑐)} ⊆ 𝑁 → ({(1st𝑐)} × 𝑁) ⊆ (𝑁 × 𝑁))
127125, 126syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) → ({(1st𝑐)} × 𝑁) ⊆ (𝑁 × 𝑁))
128127sselda 3915 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) ∧ 𝑒 ∈ ({(1st𝑐)} × 𝑁)) → 𝑒 ∈ (𝑁 × 𝑁))
129117, 128ffvelrnd 6830 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) ∧ 𝑒 ∈ ({(1st𝑐)} × 𝑁)) → (𝑎𝑒) ∈ 𝐾)
13012, 49ringidcl 19318 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 (𝑅 ∈ Ring → 1𝐾)
131113, 130syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) → 1𝐾)
13212, 48ring0cl 19319 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 (𝑅 ∈ Ring → 0𝐾)
133113, 132syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) → 0𝐾)
134131, 133ifcld 4470 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 2798 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 (-g𝑅) = (-g𝑅)
13712, 50, 136grpnpcan 18187 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 1368 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) ∧ 𝑒 ∈ ({(1st𝑐)} × 𝑁)) → (((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )) + if(𝑒𝑑, 1 , 0 )) = (𝑎𝑒))
139138eqcomd 2804 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 4431 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (𝑒 = 𝑐 → if(𝑒 = 𝑐, ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )), 0 ) = ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )))
142 iftrue 4431 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (𝑒 = 𝑐 → if(𝑒 = 𝑐, if(𝑒𝑑, 1 , 0 ), (𝑎𝑒)) = if(𝑒𝑑, 1 , 0 ))
143141, 142oveq12d 7154 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 2836 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 18129 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 2804 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 4434 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 𝑒 = 𝑐 → if(𝑒 = 𝑐, ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )), 0 ) = 0 )
151 iffalse 4434 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 𝑒 = 𝑐 → if(𝑒 = 𝑐, if(𝑒𝑑, 1 , 0 ), (𝑎𝑒)) = (𝑎𝑒))
152150, 151oveq12d 7154 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 2836 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 812 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 5126 . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 8580 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 {(1st𝑐)} ∈ Fin
15863ad2ant1 1130 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) → 𝑁 ∈ Fin)
159 xpfi 8776 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (({(1st𝑐)} ∈ Fin ∧ 𝑁 ∈ Fin) → ({(1st𝑐)} × 𝑁) ∈ Fin)
160157, 158, 159sylancr 590 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) → ({(1st𝑐)} × 𝑁) ∈ Fin)
161 ovex 7169 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )) ∈ V
16248fvexi 6660 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 0 ∈ V
163161, 162ifex 4473 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 6660 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 1 ∈ V
166165, 162ifex 4473 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 if(𝑒𝑑, 1 , 0 ) ∈ V
167 fvex 6659 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝑎𝑒) ∈ V
168166, 167ifex 4473 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 7706 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (𝑒 ∈ ({(1st𝑐)} × 𝑁) → (1st𝑒) ∈ {(1st𝑐)})
171 elsni 4542 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((1st𝑒) ∈ {(1st𝑐)} → (1st𝑒) = (1st𝑐))
172 iftrue 4431 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 5122 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 2799 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 7409 . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 2836 . . . . . . . . . . . . . . . . . . . . . . . . . . 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 5876 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) → ((𝑒 ∈ (𝑁 × 𝑁) ↦ (𝑎𝑒)) ↾ ({(1st𝑐)} × 𝑁)) = (𝑒 ∈ ({(1st𝑐)} × 𝑁) ↦ (𝑎𝑒)))
180127resmptd 5876 . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 5876 . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 7154 . . . . . . . . . . . . . . . . . . . . . . . . . . 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 2843 . . . . . . . . . . . . . . . . . . . . . . . . . 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 2833 . . . . . . . . . . . . . . . . . . . . . . . . 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 5818 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) → (𝑎 ↾ ((𝑁 ∖ {(1st𝑐)}) × 𝑁)) = ((𝑒 ∈ (𝑁 × 𝑁) ↦ (𝑎𝑒)) ↾ ((𝑁 ∖ {(1st𝑐)}) × 𝑁)))
186 xp1st 7706 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (𝑒 ∈ ((𝑁 ∖ {(1st𝑐)}) × 𝑁) → (1st𝑒) ∈ (𝑁 ∖ {(1st𝑐)}))
187 eldifsni 4683 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((1st𝑒) ∈ (𝑁 ∖ {(1st𝑐)}) → (1st𝑒) ≠ (1st𝑐))
188186, 187syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝑒 ∈ ((𝑁 ∖ {(1st𝑐)}) × 𝑁) → (1st𝑒) ≠ (1st𝑐))
189188neneqd 2992 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑒 ∈ ((𝑁 ∖ {(1st𝑐)}) × 𝑁) → ¬ (1st𝑒) = (1st𝑐))
190189adantl 485 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) ∧ 𝑒 ∈ ((𝑁 ∖ {(1st𝑐)}) × 𝑁)) → ¬ (1st𝑒) = (1st𝑐))
191190iffalsed 4436 . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 5126 . . . . . . . . . . . . . . . . . . . . . . . . . . 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 4059 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑁 ∖ {(1st𝑐)}) ⊆ 𝑁
194 xpss1 5539 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((𝑁 ∖ {(1st𝑐)}) ⊆ 𝑁 → ((𝑁 ∖ {(1st𝑐)}) × 𝑁) ⊆ (𝑁 × 𝑁))
195193, 194ax-mp 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝑁 ∖ {(1st𝑐)}) × 𝑁) ⊆ (𝑁 × 𝑁)
196 resmpt 5873 . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 5873 . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 2844 . . . . . . . . . . . . . . . . . . . . . . . . . 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 2833 . . . . . . . . . . . . . . . . . . . . . . . . 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 6646 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑒 = 𝑐 → (1st𝑒) = (1st𝑐))
203190, 202nsyl 142 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) ∧ 𝑒 ∈ ((𝑁 ∖ {(1st𝑐)}) × 𝑁)) → ¬ 𝑒 = 𝑐)
204203iffalsed 4436 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) ∧ 𝑒 ∈ ((𝑁 ∖ {(1st𝑐)}) × 𝑁)) → if(𝑒 = 𝑐, if(𝑒𝑑, 1 , 0 ), (𝑎𝑒)) = (𝑎𝑒))
205204mpteq2dva 5126 . . . . . . . . . . . . . . . . . . . . . . . . . . 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 5873 . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 2844 . . . . . . . . . . . . . . . . . . . . . . . . . 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 2833 . . . . . . . . . . . . . . . . . . . . . . . . 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 6829 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) ∧ 𝑒 ∈ (𝑁 × 𝑁)) → (𝑎𝑒) ∈ 𝐾)
212210, 211ifcld 4470 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) ∧ 𝑒 ∈ (𝑁 × 𝑁)) → if(𝑒 = 𝑐, if(𝑒𝑑, 1 , 0 ), (𝑎𝑒)) ∈ 𝐾)
213212fmpttd 6857 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) → (𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒𝑑, 1 , 0 ), (𝑎𝑒))):(𝑁 × 𝑁)⟶𝐾)
21412fvexi 6660 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 8405 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 21036 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 2851 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) → (𝐾m (𝑁 × 𝑁)) = 𝐵)
223219, 222eleqtrd 2892 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) → (𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒𝑑, 1 , 0 ), (𝑎𝑒))) ∈ 𝐵)
224 simp3 1135 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) → 𝑎𝐵)
225115adantr 484 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) ∧ 𝑒 ∈ (𝑁 × 𝑁)) → 𝑅 ∈ Grp)
22612, 136grpsubcl 18175 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ((𝑅 ∈ Grp ∧ (𝑎𝑒) ∈ 𝐾 ∧ if(𝑒𝑑, 1 , 0 ) ∈ 𝐾) → ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )) ∈ 𝐾)
227225, 211, 210, 226syl3anc 1368 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 4470 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) ∧ 𝑒 ∈ (𝑁 × 𝑁)) → if(𝑒 = 𝑐, ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )), 0 ) ∈ 𝐾)
230229, 211ifcld 4470 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 6857 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 8405 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 2892 . . . . . . . . . . . . . . . . . . . . . . . . . . 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 1130 . . . . . . . . . . . . . . . . . . . . . . . . . . 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 5813 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (𝑥 = 𝑎 → (𝑥 ↾ ({𝑤} × 𝑁)) = (𝑎 ↾ ({𝑤} × 𝑁)))
238237eqeq1d 2800 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 5813 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (𝑥 = 𝑎 → (𝑥 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑎 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)))
240239eqeq1d 2800 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝑥 = 𝑎 → ((𝑥 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑦 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) ↔ (𝑎 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑦 ↾ ((𝑁 ∖ {𝑤}) × 𝑁))))
241239eqeq1d 2800 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝑥 = 𝑎 → ((𝑥 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑧 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) ↔ (𝑎 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑧 ↾ ((𝑁 ∖ {𝑤}) × 𝑁))))
242238, 240, 2413anbi123d 1433 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 6655 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 3164 . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 5813 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 7151 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 2809 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 5813 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 2809 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 1434 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 6646 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 7151 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 2809 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 3164 . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 3582 . . . . . . . . . . . . . . . . . . . . . . . . . . 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 836 . . . . . . . . . . . . . . . . . . . . . . . . . 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 5813 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 7152 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 2809 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 5813 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 2809 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 1435 . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 6646 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 7152 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 2809 . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 4535 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (𝑤 = (1st𝑐) → {𝑤} = {(1st𝑐)})
270269xpeq1d 5549 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝑤 = (1st𝑐) → ({𝑤} × 𝑁) = ({(1st𝑐)} × 𝑁))
271270reseq2d 5819 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑤 = (1st𝑐) → (𝑎 ↾ ({𝑤} × 𝑁)) = (𝑎 ↾ ({(1st𝑐)} × 𝑁)))
272270reseq2d 5819 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 5819 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 7154 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 2814 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 4050 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (𝑤 = (1st𝑐) → (𝑁 ∖ {𝑤}) = (𝑁 ∖ {(1st𝑐)}))
277276xpeq1d 5549 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝑤 = (1st𝑐) → ((𝑁 ∖ {𝑤}) × 𝑁) = ((𝑁 ∖ {(1st𝑐)}) × 𝑁))
278277reseq2d 5819 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑤 = (1st𝑐) → (𝑎 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑎 ↾ ((𝑁 ∖ {(1st𝑐)}) × 𝑁)))
279277reseq2d 5819 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 2814 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 5819 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 2814 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 1433 . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 3582 . . . . . . . . . . . . . . . . . . . . . . . . . 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 836 . . . . . . . . . . . . . . . . . . . . . . . . 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 1461 . . . . . . . . . . . . . . . . . . . . . . . 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 1368 . . . . . . . . . . . . . . . . . . . . . . 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 6646 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (𝑒 = 𝑐 → (𝑎𝑒) = (𝑎𝑐))
290 elequ1 2118 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 (𝑒 = 𝑐 → (𝑒𝑑𝑐𝑑))
291290ifbid 4447 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (𝑒 = 𝑐 → if(𝑒𝑑, 1 , 0 ) = if(𝑐𝑑, 1 , 0 ))
292289, 291oveq12d 7154 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 6830 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) → (𝑎𝑐) ∈ 𝐾)
295131, 133ifcld 4470 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) → if(𝑐𝑑, 1 , 0 ) ∈ 𝐾)
29612, 136grpsubcl 18175 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ((𝑅 ∈ Grp ∧ (𝑎𝑐) ∈ 𝐾 ∧ if(𝑐𝑑, 1 , 0 ) ∈ 𝐾) → ((𝑎𝑐)(-g𝑅)if(𝑐𝑑, 1 , 0 )) ∈ 𝐾)
297115, 294, 295, 296syl3anc 1368 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) → ((𝑎𝑐)(-g𝑅)if(𝑐𝑑, 1 , 0 )) ∈ 𝐾)
29812, 51, 49ringridm 19322 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 725 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 2836 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 4431 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (𝑒 = 𝑐 → if(𝑒 = 𝑐, 1 , 0 ) = 1 )
304303oveq2d 7152 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 2843 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 19338 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 2804 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) → 0 = (((𝑎𝑐)(-g𝑅)if(𝑐𝑑, 1 , 0 )) · 0 ))
310309ad2antrr 725 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 4434 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 𝑒 = 𝑐 → if(𝑒 = 𝑐, 1 , 0 ) = 0 )
313312oveq2d 7152 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 2843 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 812 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 4433 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 4433 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 7152 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 2843 . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 5126 . . . . . . . . . . . . . . . . . . . . . . . . . . 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 7171 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) ∧ 𝑒 ∈ ({(1st𝑐)} × 𝑁)) → ((𝑎𝑐)(-g𝑅)if(𝑐𝑑, 1 , 0 )) ∈ V)
325165, 162ifex 4473 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 if(𝑒 = 𝑐, 1 , 0 ) ∈ V
326325, 167ifex 4473 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 5579 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 5876 . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 7409 . . . . . . . . . . . . . . . . . . . . . . . . . . 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 2843 . . . . . . . . . . . . . . . . . . . . . . . . . 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 4434 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (¬ (1st𝑒) = (1st𝑐) → if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )), 0 ), (𝑎𝑒)) = (𝑎𝑒))
334 iffalse 4434 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (¬ (1st𝑒) = (1st𝑐) → if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎𝑒)) = (𝑎𝑒))
335333, 334eqtr4d 2836 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 5126 . . . . . . . . . . . . . . . . . . . . . . . . . . 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 5873 . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 2843 . . . . . . . . . . . . . . . . . . . . . . . . . 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 4470 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 4470 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) ∧ 𝑒 ∈ (𝑁 × 𝑁)) → if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎𝑒)) ∈ 𝐾)
344343fmpttd 6857 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) → (𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎𝑒))):(𝑁 × 𝑁)⟶𝐾)
345 elmapg 8405 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 2892 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) → (𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎𝑒))) ∈ 𝐵)
349563ad2ant1 1130 . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 5813 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 2800 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 5813 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 2800 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 633 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 6655 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 3164 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 4535 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 (𝑦 = ((𝑎𝑐)(-g𝑅)if(𝑐𝑑, 1 , 0 )) → {𝑦} = {((𝑎𝑐)(-g𝑅)if(𝑐𝑑, 1 , 0 ))})
359358xpeq2d 5550 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (𝑦 = ((𝑎𝑐)(-g𝑅)if(𝑐𝑑, 1 , 0 )) → (({𝑤} × 𝑁) × {𝑦}) = (({𝑤} × 𝑁) × {((𝑎𝑐)(-g𝑅)if(𝑐𝑑, 1 , 0 ))}))
360359oveq1d 7151 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 2809 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 632 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 7143 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (𝑦 = ((𝑎𝑐)(-g𝑅)if(𝑐𝑑, 1 , 0 )) → (𝑦 · (𝐷𝑧)) = (((𝑎𝑐)(-g𝑅)if(𝑐𝑑, 1 , 0 )) · (𝐷𝑧)))
364363eqeq2d 2809 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 3164 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 3582 . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 836 . . . . . . . . . . . . . . . . . . . . . . . . . . 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 5813 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 7152 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 2809 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 5813 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 2809 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 633 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 6646 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 7152 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 2809 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 5549 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (𝑤 = (1st𝑐) → (({𝑤} × 𝑁) × {((𝑎𝑐)(-g𝑅)if(𝑐𝑑, 1 , 0 ))}) = (({(1st𝑐)} × 𝑁) × {((𝑎𝑐)(-g𝑅)if(𝑐𝑑, 1 , 0 ))}))
380270reseq2d 5819 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 7154 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 2814 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 5819 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 2814 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 633 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 3582 . . . . . . . . . . . . . . . . . . . . . . . . . . 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 836 . . . . . . . . . . . . . . . . . . . . . . . . . 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 698 . . . . . . . . . . . . . . . . . . . . . . . . 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 7151 . . . . . . . . . . . . . . . . . . . . . . . 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 1368 . . . . . . . . . . . . . . . . . . . . . . 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 1190 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ (𝑏 ∪ {𝑐}) ∈ 𝑌) ∧ ((𝑎𝐵𝑑 ∈ (𝑁m 𝑁)) ∧ ∀𝑤𝑏 (𝑎𝑤) = if(𝑤𝑑, 1 , 0 ))) → (𝑏 ∪ {𝑐}) ∈ 𝑌)
393 simprlr 779 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ (𝑏 ∪ {𝑐}) ∈ 𝑌) ∧ ((𝑎𝐵𝑑 ∈ (𝑁m 𝑁)) ∧ ∀𝑤𝑏 (𝑎𝑤) = if(𝑤𝑑, 1 , 0 ))) → 𝑑 ∈ (𝑁m 𝑁))
394 simprr 772 . . . . . . . . . . . . . . . . . . . . . . . 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 3985 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 4431 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 5538 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 Rel (𝑁 × 𝑁)
402 simpl2 1189 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) ∧ 𝑑 ∈ (𝑁m 𝑁)) → (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁))
403402sselda 3915 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 7723 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 2874 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 8414 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 7707 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 16511 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 (((𝑑 ∈ (𝑁m 𝑁) ∧ 𝑑:𝑁𝑁) ∧ (1st𝑐) ∈ 𝑁 ∧ (2nd𝑐) ∈ 𝑁) → (𝑑 sSet ⟨(1st𝑐), (2nd𝑐)⟩):𝑁𝑁)
416408, 410, 411, 414, 415syl211anc 1373 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) ∧ 𝑑 ∈ (𝑁m 𝑁)) → (𝑑 sSet ⟨(1st𝑐), (2nd𝑐)⟩):𝑁𝑁)
417416ffnd 6489 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) ∧ 𝑑 ∈ (𝑁m 𝑁)) → (𝑑 sSet ⟨(1st𝑐), (2nd𝑐)⟩) Fn 𝑁)
418417ad2antrr 725 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 (((((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) ∧ 𝑑 ∈ (𝑁m 𝑁)) ∧ 𝑤 ∈ (𝑏 ∪ {𝑐})) ∧ (1st𝑤) = (1st𝑐)) → (𝑑 sSet ⟨(1st𝑐), (2nd𝑐)⟩) Fn 𝑁)
419 xp1st 7706 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 6695 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 6646 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 3444 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 𝑑 ∈ V
427 fvex 6659 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 (1st𝑐) ∈ V
428 fvex 6659 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 (2nd𝑐) ∈ V
429 fvsetsid 16510 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 ((𝑑 ∈ V ∧ (1st𝑐) ∈ V ∧ (2nd𝑐) ∈ V) → ((𝑑 sSet ⟨(1st𝑐), (2nd𝑐)⟩)‘(1st𝑐)) = (2nd𝑐))
430426, 427, 428, 429mp3an 1458 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 ((𝑑 sSet ⟨(1st𝑐), (2nd𝑐)⟩)‘(1st𝑐)) = (2nd𝑐)
431425, 430eqtrdi 2849 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 (((((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) ∧ 𝑑 ∈ (𝑁m 𝑁)) ∧ 𝑤 ∈ (𝑏 ∪ {𝑐})) ∧ (1st𝑤) = (1st𝑐)) → ((𝑑 sSet ⟨(1st𝑐), (2nd𝑐)⟩)‘(1st𝑤)) = (2nd𝑐))
432431eqeq1d 2800 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 2805 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 ((2nd𝑐) = (2nd𝑤) ↔ (2nd𝑤) = (2nd𝑐))
434432, 433syl6bb 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 729 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 (((((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) ∧ 𝑑 ∈ (𝑁m 𝑁)) ∧ 𝑤 ∈ (𝑏 ∪ {𝑐})) ∧ (1st𝑤) = (1st𝑐)) → 𝑐 ∈ (𝑁 × 𝑁))
437 xpopth 7715 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 4447 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 2833 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 4542 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 (𝑤 ∈ {𝑐} → 𝑤 = 𝑐)
444443fveq2d 6650 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 (𝑤 ∈ {𝑐} → (1st𝑤) = (1st𝑐))
445444con3i 157 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 (¬ (1st𝑤) = (1st𝑐) → ¬ 𝑤 ∈ {𝑐})
446445adantl 485 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ((𝑤 ∈ (𝑏 ∪ {𝑐}) ∧ ¬ (1st𝑤) = (1st𝑐)) → ¬ 𝑤 ∈ {𝑐})
447 elun 4076 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 888 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 𝑤 ∈ {𝑐} → ((𝑤𝑏𝑤 ∈ {𝑐}) → 𝑤𝑏))
451446, 449, 450sylc 65 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ((𝑤 ∈ (𝑏 ∪ {𝑐}) ∧ ¬ (1st𝑤) = (1st𝑐)) → 𝑤𝑏)
452451adantll 713 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 (((((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) ∧ 𝑑 ∈ (𝑁m 𝑁)) ∧ 𝑤 ∈ (𝑏 ∪ {𝑐})) ∧ ¬ (1st𝑤) = (1st𝑐)) → 𝑤𝑏)
453 iffalse 4434 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 16520 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 (𝑑 ∈ V → ((𝑑 sSet ⟨(1st𝑐), (2nd𝑐)⟩) ↾ (V ∖ {(1st𝑐)})) = (𝑑 ↾ (V ∖ {(1st𝑐)})))
456455eleq2d 2875 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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) ∧ 𝑎