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Theorem mdetunilem9 20187
Description: Lemma for mdetuni 20189. (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 4027 . . . 4 𝑤 ∈ ∅ (𝑎𝑤) = if(𝑤 ∈ ( I ↾ 𝑁), 1 , 0 )
2 simpr 475 . . . . 5 ((𝜑𝑎𝐵) → 𝑎𝐵)
3 f1oi 6071 . . . . . . . 8 ( I ↾ 𝑁):𝑁1-1-onto𝑁
4 f1of 6035 . . . . . . . 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 7735 . . . . . . 7 (𝜑 → (( I ↾ 𝑁) ∈ (𝑁𝑚 𝑁) ↔ ( I ↾ 𝑁):𝑁𝑁))
85, 7mpbird 245 . . . . . 6 (𝜑 → ( I ↾ 𝑁) ∈ (𝑁𝑚 𝑁))
98adantr 479 . . . . 5 ((𝜑𝑎𝐵) → ( I ↾ 𝑁) ∈ (𝑁𝑚 𝑁))
10 simplrl 795 . . . . . . . . . . . . 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 19989 . . . . . . . . . . . . . . . 16 (𝑦𝐵𝑦 ∈ (𝐾𝑚 (𝑁 × 𝑁)))
15 elmapi 7742 . . . . . . . . . . . . . . . 16 (𝑦 ∈ (𝐾𝑚 (𝑁 × 𝑁)) → 𝑦:(𝑁 × 𝑁)⟶𝐾)
1614, 15syl 17 . . . . . . . . . . . . . . 15 (𝑦𝐵𝑦:(𝑁 × 𝑁)⟶𝐾)
1716feqmptd 6144 . . . . . . . . . . . . . 14 (𝑦𝐵𝑦 = (𝑤 ∈ (𝑁 × 𝑁) ↦ (𝑦𝑤)))
1817fveq2d 6092 . . . . . . . . . . . . 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 2609 . . . . . . . . . . . . . 14 (𝑁 × 𝑁) = (𝑁 × 𝑁)
21 mpteq12 4658 . . . . . . . . . . . . . . 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 6092 . . . . . . . . . . . . . 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 701 . . . . . . . . . . . . 13 (∀𝑤 ∈ (𝑁 × 𝑁)(𝑦𝑤) = if(𝑤𝑧, 1 , 0 ) → (𝐷‘(𝑤 ∈ (𝑁 × 𝑁) ↦ (𝑦𝑤))) = (𝐷‘(𝑤 ∈ (𝑁 × 𝑁) ↦ if(𝑤𝑧, 1 , 0 ))))
2423adantl 480 . . . . . . . . . . . 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 2675 . . . . . . . . . . . . . . . . 17 (𝑎 = 𝑧 → (𝑎 ∈ (𝑁𝑚 𝑁) ↔ 𝑧 ∈ (𝑁𝑚 𝑁)))
2625anbi2d 735 . . . . . . . . . . . . . . . 16 (𝑎 = 𝑧 → ((𝜑𝑎 ∈ (𝑁𝑚 𝑁)) ↔ (𝜑𝑧 ∈ (𝑁𝑚 𝑁))))
27 elequ2 1990 . . . . . . . . . . . . . . . . . . . 20 (𝑎 = 𝑧 → (𝑤𝑎𝑤𝑧))
2827ifbid 4057 . . . . . . . . . . . . . . . . . . 19 (𝑎 = 𝑧 → if(𝑤𝑎, 1 , 0 ) = if(𝑤𝑧, 1 , 0 ))
2928mpteq2dv 4667 . . . . . . . . . . . . . . . . . 18 (𝑎 = 𝑧 → (𝑤 ∈ (𝑁 × 𝑁) ↦ if(𝑤𝑎, 1 , 0 )) = (𝑤 ∈ (𝑁 × 𝑁) ↦ if(𝑤𝑧, 1 , 0 )))
3029fveq2d 6092 . . . . . . . . . . . . . . . . 17 (𝑎 = 𝑧 → (𝐷‘(𝑤 ∈ (𝑁 × 𝑁) ↦ if(𝑤𝑎, 1 , 0 ))) = (𝐷‘(𝑤 ∈ (𝑁 × 𝑁) ↦ if(𝑤𝑧, 1 , 0 ))))
3130eqeq1d 2611 . . . . . . . . . . . . . . . 16 (𝑎 = 𝑧 → ((𝐷‘(𝑤 ∈ (𝑁 × 𝑁) ↦ if(𝑤𝑎, 1 , 0 ))) = 0 ↔ (𝐷‘(𝑤 ∈ (𝑁 × 𝑁) ↦ if(𝑤𝑧, 1 , 0 ))) = 0 ))
3226, 31imbi12d 332 . . . . . . . . . . . . . . 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 2675 . . . . . . . . . . . . . . . . . . . 20 (𝑤 = ⟨𝑏, 𝑐⟩ → (𝑤𝑎 ↔ ⟨𝑏, 𝑐⟩ ∈ 𝑎))
3433ifbid 4057 . . . . . . . . . . . . . . . . . . 19 (𝑤 = ⟨𝑏, 𝑐⟩ → if(𝑤𝑎, 1 , 0 ) = if(⟨𝑏, 𝑐⟩ ∈ 𝑎, 1 , 0 ))
3534mpt2mpt 6628 . . . . . . . . . . . . . . . . . 18 (𝑤 ∈ (𝑁 × 𝑁) ↦ if(𝑤𝑎, 1 , 0 )) = (𝑏𝑁, 𝑐𝑁 ↦ if(⟨𝑏, 𝑐⟩ ∈ 𝑎, 1 , 0 ))
36 elmapi 7742 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑎 ∈ (𝑁𝑚 𝑁) → 𝑎:𝑁𝑁)
3736adantl 480 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝜑𝑎 ∈ (𝑁𝑚 𝑁)) → 𝑎:𝑁𝑁)
38 ffn 5944 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑎:𝑁𝑁𝑎 Fn 𝑁)
3937, 38syl 17 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑𝑎 ∈ (𝑁𝑚 𝑁)) → 𝑎 Fn 𝑁)
40393ad2ant1 1074 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑𝑎 ∈ (𝑁𝑚 𝑁)) ∧ 𝑏𝑁𝑐𝑁) → 𝑎 Fn 𝑁)
41 simp2 1054 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑𝑎 ∈ (𝑁𝑚 𝑁)) ∧ 𝑏𝑁𝑐𝑁) → 𝑏𝑁)
42 fnopfvb 6132 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑎 Fn 𝑁𝑏𝑁) → ((𝑎𝑏) = 𝑐 ↔ ⟨𝑏, 𝑐⟩ ∈ 𝑎))
4340, 41, 42syl2anc 690 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑎 ∈ (𝑁𝑚 𝑁)) ∧ 𝑏𝑁𝑐𝑁) → ((𝑎𝑏) = 𝑐 ↔ ⟨𝑏, 𝑐⟩ ∈ 𝑎))
4443bicomd 211 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑎 ∈ (𝑁𝑚 𝑁)) ∧ 𝑏𝑁𝑐𝑁) → (⟨𝑏, 𝑐⟩ ∈ 𝑎 ↔ (𝑎𝑏) = 𝑐))
4544ifbid 4057 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑎 ∈ (𝑁𝑚 𝑁)) ∧ 𝑏𝑁𝑐𝑁) → if(⟨𝑏, 𝑐⟩ ∈ 𝑎, 1 , 0 ) = if((𝑎𝑏) = 𝑐, 1 , 0 ))
4645mpt2eq3dva 6595 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑎 ∈ (𝑁𝑚 𝑁)) → (𝑏𝑁, 𝑐𝑁 ↦ if(⟨𝑏, 𝑐⟩ ∈ 𝑎, 1 , 0 )) = (𝑏𝑁, 𝑐𝑁 ↦ if((𝑎𝑏) = 𝑐, 1 , 0 )))
4735, 46syl5eq 2655 . . . . . . . . . . . . . . . . 17 ((𝜑𝑎 ∈ (𝑁𝑚 𝑁)) → (𝑤 ∈ (𝑁 × 𝑁) ↦ if(𝑤𝑎, 1 , 0 )) = (𝑏𝑁, 𝑐𝑁 ↦ if((𝑎𝑏) = 𝑐, 1 , 0 )))
4847fveq2d 6092 . . . . . . . . . . . . . . . 16 ((𝜑𝑎 ∈ (𝑁𝑚 𝑁)) → (𝐷‘(𝑤 ∈ (𝑁 × 𝑁) ↦ if(𝑤𝑎, 1 , 0 ))) = (𝐷‘(𝑏𝑁, 𝑐𝑁 ↦ if((𝑎𝑏) = 𝑐, 1 , 0 ))))
49 mdetuni.0g . . . . . . . . . . . . . . . . . 18 0 = (0g𝑅)
50 mdetuni.1r . . . . . . . . . . . . . . . . . 18 1 = (1r𝑅)
51 mdetuni.pg . . . . . . . . . . . . . . . . . 18 + = (+g𝑅)
52 mdetuni.tg . . . . . . . . . . . . . . . . . 18 · = (.r𝑅)
53 mdetuni.r . . . . . . . . . . . . . . . . . 18 (𝜑𝑅 ∈ Ring)
54 mdetuni.ff . . . . . . . . . . . . . . . . . 18 (𝜑𝐷:𝐵𝐾)
55 mdetuni.al . . . . . . . . . . . . . . . . . 18 (𝜑 → ∀𝑥𝐵𝑦𝑁𝑧𝑁 ((𝑦𝑧 ∧ ∀𝑤𝑁 (𝑦𝑥𝑤) = (𝑧𝑥𝑤)) → (𝐷𝑥) = 0 ))
56 mdetuni.li . . . . . . . . . . . . . . . . . 18 (𝜑 → ∀𝑥𝐵𝑦𝐵𝑧𝐵𝑤𝑁 (((𝑥 ↾ ({𝑤} × 𝑁)) = ((𝑦 ↾ ({𝑤} × 𝑁)) ∘𝑓 + (𝑧 ↾ ({𝑤} × 𝑁))) ∧ (𝑥 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑦 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) ∧ (𝑥 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑧 ↾ ((𝑁 ∖ {𝑤}) × 𝑁))) → (𝐷𝑥) = ((𝐷𝑦) + (𝐷𝑧))))
57 mdetuni.sc . . . . . . . . . . . . . . . . . 18 (𝜑 → ∀𝑥𝐵𝑦𝐾𝑧𝐵𝑤𝑁 (((𝑥 ↾ ({𝑤} × 𝑁)) = ((({𝑤} × 𝑁) × {𝑦}) ∘𝑓 · (𝑧 ↾ ({𝑤} × 𝑁))) ∧ (𝑥 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑧 ↾ ((𝑁 ∖ {𝑤}) × 𝑁))) → (𝐷𝑥) = (𝑦 · (𝐷𝑧))))
58 mdetunilem9.id . . . . . . . . . . . . . . . . . 18 (𝜑 → (𝐷‘(1r𝐴)) = 0 )
5911, 13, 12, 49, 50, 51, 52, 6, 53, 54, 55, 56, 57, 58mdetunilem8 20186 . . . . . . . . . . . . . . . . 17 ((𝜑𝑎:𝑁𝑁) → (𝐷‘(𝑏𝑁, 𝑐𝑁 ↦ if((𝑎𝑏) = 𝑐, 1 , 0 ))) = 0 )
6036, 59sylan2 489 . . . . . . . . . . . . . . . 16 ((𝜑𝑎 ∈ (𝑁𝑚 𝑁)) → (𝐷‘(𝑏𝑁, 𝑐𝑁 ↦ if((𝑎𝑏) = 𝑐, 1 , 0 ))) = 0 )
6148, 60eqtrd 2643 . . . . . . . . . . . . . . 15 ((𝜑𝑎 ∈ (𝑁𝑚 𝑁)) → (𝐷‘(𝑤 ∈ (𝑁 × 𝑁) ↦ if(𝑤𝑎, 1 , 0 ))) = 0 )
6232, 61chvarv 2250 . . . . . . . . . . . . . 14 ((𝜑𝑧 ∈ (𝑁𝑚 𝑁)) → (𝐷‘(𝑤 ∈ (𝑁 × 𝑁) ↦ if(𝑤𝑧, 1 , 0 ))) = 0 )
6362adantrl 747 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑦𝐵𝑧 ∈ (𝑁𝑚 𝑁))) → (𝐷‘(𝑤 ∈ (𝑁 × 𝑁) ↦ if(𝑤𝑧, 1 , 0 ))) = 0 )
6463adantr 479 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑦𝐵𝑧 ∈ (𝑁𝑚 𝑁))) ∧ ∀𝑤 ∈ (𝑁 × 𝑁)(𝑦𝑤) = if(𝑤𝑧, 1 , 0 )) → (𝐷‘(𝑤 ∈ (𝑁 × 𝑁) ↦ if(𝑤𝑧, 1 , 0 ))) = 0 )
6519, 24, 643eqtrd 2647 . . . . . . . . . . 11 (((𝜑 ∧ (𝑦𝐵𝑧 ∈ (𝑁𝑚 𝑁))) ∧ ∀𝑤 ∈ (𝑁 × 𝑁)(𝑦𝑤) = if(𝑤𝑧, 1 , 0 )) → (𝐷𝑦) = 0 )
6665ex 448 . . . . . . . . . 10 ((𝜑 ∧ (𝑦𝐵𝑧 ∈ (𝑁𝑚 𝑁))) → (∀𝑤 ∈ (𝑁 × 𝑁)(𝑦𝑤) = if(𝑤𝑧, 1 , 0 ) → (𝐷𝑦) = 0 ))
6766ralrimivva 2953 . . . . . . . . 9 (𝜑 → ∀𝑦𝐵𝑧 ∈ (𝑁𝑚 𝑁)(∀𝑤 ∈ (𝑁 × 𝑁)(𝑦𝑤) = if(𝑤𝑧, 1 , 0 ) → (𝐷𝑦) = 0 ))
68 xpfi 8093 . . . . . . . . . . 11 ((𝑁 ∈ Fin ∧ 𝑁 ∈ Fin) → (𝑁 × 𝑁) ∈ Fin)
696, 6, 68syl2anc 690 . . . . . . . . . 10 (𝜑 → (𝑁 × 𝑁) ∈ Fin)
70 raleq 3114 . . . . . . . . . . . . 13 (𝑥 = (𝑁 × 𝑁) → (∀𝑤𝑥 (𝑦𝑤) = if(𝑤𝑧, 1 , 0 ) ↔ ∀𝑤 ∈ (𝑁 × 𝑁)(𝑦𝑤) = if(𝑤𝑧, 1 , 0 )))
7170imbi1d 329 . . . . . . . . . . . 12 (𝑥 = (𝑁 × 𝑁) → ((∀𝑤𝑥 (𝑦𝑤) = if(𝑤𝑧, 1 , 0 ) → (𝐷𝑦) = 0 ) ↔ (∀𝑤 ∈ (𝑁 × 𝑁)(𝑦𝑤) = if(𝑤𝑧, 1 , 0 ) → (𝐷𝑦) = 0 )))
72712ralbidv 2971 . . . . . . . . . . 11 (𝑥 = (𝑁 × 𝑁) → (∀𝑦𝐵𝑧 ∈ (𝑁𝑚 𝑁)(∀𝑤𝑥 (𝑦𝑤) = if(𝑤𝑧, 1 , 0 ) → (𝐷𝑦) = 0 ) ↔ ∀𝑦𝐵𝑧 ∈ (𝑁𝑚 𝑁)(∀𝑤 ∈ (𝑁 × 𝑁)(𝑦𝑤) = if(𝑤𝑧, 1 , 0 ) → (𝐷𝑦) = 0 )))
73 mdetunilem9.y . . . . . . . . . . 11 𝑌 = {𝑥 ∣ ∀𝑦𝐵𝑧 ∈ (𝑁𝑚 𝑁)(∀𝑤𝑥 (𝑦𝑤) = if(𝑤𝑧, 1 , 0 ) → (𝐷𝑦) = 0 )}
7472, 73elab2g 3321 . . . . . . . . . 10 ((𝑁 × 𝑁) ∈ Fin → ((𝑁 × 𝑁) ∈ 𝑌 ↔ ∀𝑦𝐵𝑧 ∈ (𝑁𝑚 𝑁)(∀𝑤 ∈ (𝑁 × 𝑁)(𝑦𝑤) = if(𝑤𝑧, 1 , 0 ) → (𝐷𝑦) = 0 )))
7569, 74syl 17 . . . . . . . . 9 (𝜑 → ((𝑁 × 𝑁) ∈ 𝑌 ↔ ∀𝑦𝐵𝑧 ∈ (𝑁𝑚 𝑁)(∀𝑤 ∈ (𝑁 × 𝑁)(𝑦𝑤) = if(𝑤𝑧, 1 , 0 ) → (𝐷𝑦) = 0 )))
7667, 75mpbird 245 . . . . . . . 8 (𝜑 → (𝑁 × 𝑁) ∈ 𝑌)
77 ssid 3586 . . . . . . . . 9 (𝑁 × 𝑁) ⊆ (𝑁 × 𝑁)
78693ad2ant1 1074 . . . . . . . . . . 11 ((𝜑 ∧ (𝑁 × 𝑁) ⊆ (𝑁 × 𝑁) ∧ ¬ ∅ ∈ 𝑌) → (𝑁 × 𝑁) ∈ Fin)
79 sseq1 3588 . . . . . . . . . . . . . 14 (𝑎 = ∅ → (𝑎 ⊆ (𝑁 × 𝑁) ↔ ∅ ⊆ (𝑁 × 𝑁)))
80793anbi2d 1395 . . . . . . . . . . . . 13 (𝑎 = ∅ → ((𝜑𝑎 ⊆ (𝑁 × 𝑁) ∧ ¬ ∅ ∈ 𝑌) ↔ (𝜑 ∧ ∅ ⊆ (𝑁 × 𝑁) ∧ ¬ ∅ ∈ 𝑌)))
81 eleq1 2675 . . . . . . . . . . . . . 14 (𝑎 = ∅ → (𝑎𝑌 ↔ ∅ ∈ 𝑌))
8281notbid 306 . . . . . . . . . . . . 13 (𝑎 = ∅ → (¬ 𝑎𝑌 ↔ ¬ ∅ ∈ 𝑌))
8380, 82imbi12d 332 . . . . . . . . . . . 12 (𝑎 = ∅ → (((𝜑𝑎 ⊆ (𝑁 × 𝑁) ∧ ¬ ∅ ∈ 𝑌) → ¬ 𝑎𝑌) ↔ ((𝜑 ∧ ∅ ⊆ (𝑁 × 𝑁) ∧ ¬ ∅ ∈ 𝑌) → ¬ ∅ ∈ 𝑌)))
84 sseq1 3588 . . . . . . . . . . . . . 14 (𝑎 = 𝑏 → (𝑎 ⊆ (𝑁 × 𝑁) ↔ 𝑏 ⊆ (𝑁 × 𝑁)))
85843anbi2d 1395 . . . . . . . . . . . . 13 (𝑎 = 𝑏 → ((𝜑𝑎 ⊆ (𝑁 × 𝑁) ∧ ¬ ∅ ∈ 𝑌) ↔ (𝜑𝑏 ⊆ (𝑁 × 𝑁) ∧ ¬ ∅ ∈ 𝑌)))
86 eleq1 2675 . . . . . . . . . . . . . 14 (𝑎 = 𝑏 → (𝑎𝑌𝑏𝑌))
8786notbid 306 . . . . . . . . . . . . 13 (𝑎 = 𝑏 → (¬ 𝑎𝑌 ↔ ¬ 𝑏𝑌))
8885, 87imbi12d 332 . . . . . . . . . . . 12 (𝑎 = 𝑏 → (((𝜑𝑎 ⊆ (𝑁 × 𝑁) ∧ ¬ ∅ ∈ 𝑌) → ¬ 𝑎𝑌) ↔ ((𝜑𝑏 ⊆ (𝑁 × 𝑁) ∧ ¬ ∅ ∈ 𝑌) → ¬ 𝑏𝑌)))
89 sseq1 3588 . . . . . . . . . . . . . 14 (𝑎 = (𝑏 ∪ {𝑐}) → (𝑎 ⊆ (𝑁 × 𝑁) ↔ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁)))
90893anbi2d 1395 . . . . . . . . . . . . 13 (𝑎 = (𝑏 ∪ {𝑐}) → ((𝜑𝑎 ⊆ (𝑁 × 𝑁) ∧ ¬ ∅ ∈ 𝑌) ↔ (𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ ¬ ∅ ∈ 𝑌)))
91 eleq1 2675 . . . . . . . . . . . . . 14 (𝑎 = (𝑏 ∪ {𝑐}) → (𝑎𝑌 ↔ (𝑏 ∪ {𝑐}) ∈ 𝑌))
9291notbid 306 . . . . . . . . . . . . 13 (𝑎 = (𝑏 ∪ {𝑐}) → (¬ 𝑎𝑌 ↔ ¬ (𝑏 ∪ {𝑐}) ∈ 𝑌))
9390, 92imbi12d 332 . . . . . . . . . . . 12 (𝑎 = (𝑏 ∪ {𝑐}) → (((𝜑𝑎 ⊆ (𝑁 × 𝑁) ∧ ¬ ∅ ∈ 𝑌) → ¬ 𝑎𝑌) ↔ ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ ¬ ∅ ∈ 𝑌) → ¬ (𝑏 ∪ {𝑐}) ∈ 𝑌)))
94 sseq1 3588 . . . . . . . . . . . . . 14 (𝑎 = (𝑁 × 𝑁) → (𝑎 ⊆ (𝑁 × 𝑁) ↔ (𝑁 × 𝑁) ⊆ (𝑁 × 𝑁)))
95943anbi2d 1395 . . . . . . . . . . . . 13 (𝑎 = (𝑁 × 𝑁) → ((𝜑𝑎 ⊆ (𝑁 × 𝑁) ∧ ¬ ∅ ∈ 𝑌) ↔ (𝜑 ∧ (𝑁 × 𝑁) ⊆ (𝑁 × 𝑁) ∧ ¬ ∅ ∈ 𝑌)))
96 eleq1 2675 . . . . . . . . . . . . . 14 (𝑎 = (𝑁 × 𝑁) → (𝑎𝑌 ↔ (𝑁 × 𝑁) ∈ 𝑌))
9796notbid 306 . . . . . . . . . . . . 13 (𝑎 = (𝑁 × 𝑁) → (¬ 𝑎𝑌 ↔ ¬ (𝑁 × 𝑁) ∈ 𝑌))
9895, 97imbi12d 332 . . . . . . . . . . . 12 (𝑎 = (𝑁 × 𝑁) → (((𝜑𝑎 ⊆ (𝑁 × 𝑁) ∧ ¬ ∅ ∈ 𝑌) → ¬ 𝑎𝑌) ↔ ((𝜑 ∧ (𝑁 × 𝑁) ⊆ (𝑁 × 𝑁) ∧ ¬ ∅ ∈ 𝑌) → ¬ (𝑁 × 𝑁) ∈ 𝑌)))
99 simp3 1055 . . . . . . . . . . . 12 ((𝜑 ∧ ∅ ⊆ (𝑁 × 𝑁) ∧ ¬ ∅ ∈ 𝑌) → ¬ ∅ ∈ 𝑌)
100 ssun1 3737 . . . . . . . . . . . . . . . 16 𝑏 ⊆ (𝑏 ∪ {𝑐})
101 sstr2 3574 . . . . . . . . . . . . . . . 16 (𝑏 ⊆ (𝑏 ∪ {𝑐}) → ((𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) → 𝑏 ⊆ (𝑁 × 𝑁)))
102100, 101ax-mp 5 . . . . . . . . . . . . . . 15 ((𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) → 𝑏 ⊆ (𝑁 × 𝑁))
1031023anim2i 1241 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ ¬ ∅ ∈ 𝑌) → (𝜑𝑏 ⊆ (𝑁 × 𝑁) ∧ ¬ ∅ ∈ 𝑌))
104103imim1i 60 . . . . . . . . . . . . 13 (((𝜑𝑏 ⊆ (𝑁 × 𝑁) ∧ ¬ ∅ ∈ 𝑌) → ¬ 𝑏𝑌) → ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ ¬ ∅ ∈ 𝑌) → ¬ 𝑏𝑌))
105 simpl1 1056 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ (𝑏 ∪ {𝑐}) ∈ 𝑌) ∧ ((𝑎𝐵𝑑 ∈ (𝑁𝑚 𝑁)) ∧ ∀𝑤𝑏 (𝑎𝑤) = if(𝑤𝑑, 1 , 0 ))) → 𝜑)
106 simpl2 1057 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ (𝑏 ∪ {𝑐}) ∈ 𝑌) ∧ ((𝑎𝐵𝑑 ∈ (𝑁𝑚 𝑁)) ∧ ∀𝑤𝑏 (𝑎𝑤) = if(𝑤𝑑, 1 , 0 ))) → (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁))
107 simprll 797 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ (𝑏 ∪ {𝑐}) ∈ 𝑌) ∧ ((𝑎𝐵𝑑 ∈ (𝑁𝑚 𝑁)) ∧ ∀𝑤𝑏 (𝑎𝑤) = if(𝑤𝑑, 1 , 0 ))) → 𝑎𝐵)
10811, 12, 13matbas2i 19989 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑎𝐵𝑎 ∈ (𝐾𝑚 (𝑁 × 𝑁)))
109 elmapi 7742 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑎 ∈ (𝐾𝑚 (𝑁 × 𝑁)) → 𝑎:(𝑁 × 𝑁)⟶𝐾)
110108, 109syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑎𝐵𝑎:(𝑁 × 𝑁)⟶𝐾)
1111103ad2ant3 1076 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) → 𝑎:(𝑁 × 𝑁)⟶𝐾)
112111feqmptd 6144 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) → 𝑎 = (𝑒 ∈ (𝑁 × 𝑁) ↦ (𝑎𝑒)))
113112reseq1d 5303 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) → (𝑎 ↾ ({(1st𝑐)} × 𝑁)) = ((𝑒 ∈ (𝑁 × 𝑁) ↦ (𝑎𝑒)) ↾ ({(1st𝑐)} × 𝑁)))
114533ad2ant1 1074 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) → 𝑅 ∈ Ring)
115 ringgrp 18321 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 (𝑅 ∈ Ring → 𝑅 ∈ Grp)
116114, 115syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) → 𝑅 ∈ Grp)
117116adantr 479 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) ∧ 𝑒 ∈ ({(1st𝑐)} × 𝑁)) → 𝑅 ∈ Grp)
118111adantr 479 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) ∧ 𝑒 ∈ ({(1st𝑐)} × 𝑁)) → 𝑎:(𝑁 × 𝑁)⟶𝐾)
119 simp2 1054 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) → (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁))
120119unssbd 3752 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) → {𝑐} ⊆ (𝑁 × 𝑁))
121 vex 3175 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 𝑐 ∈ V
122121snss 4258 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 (𝑐 ∈ (𝑁 × 𝑁) ↔ {𝑐} ⊆ (𝑁 × 𝑁))
123120, 122sylibr 222 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) → 𝑐 ∈ (𝑁 × 𝑁))
124 xp1st 7066 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 (𝑐 ∈ (𝑁 × 𝑁) → (1st𝑐) ∈ 𝑁)
125123, 124syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) → (1st𝑐) ∈ 𝑁)
126125snssd 4280 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) → {(1st𝑐)} ⊆ 𝑁)
127 xpss1 5140 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ({(1st𝑐)} ⊆ 𝑁 → ({(1st𝑐)} × 𝑁) ⊆ (𝑁 × 𝑁))
128126, 127syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) → ({(1st𝑐)} × 𝑁) ⊆ (𝑁 × 𝑁))
129128sselda 3567 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) ∧ 𝑒 ∈ ({(1st𝑐)} × 𝑁)) → 𝑒 ∈ (𝑁 × 𝑁))
130118, 129ffvelrnd 6253 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) ∧ 𝑒 ∈ ({(1st𝑐)} × 𝑁)) → (𝑎𝑒) ∈ 𝐾)
13112, 50ringidcl 18337 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 (𝑅 ∈ Ring → 1𝐾)
132114, 131syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) → 1𝐾)
13312, 49ring0cl 18338 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 (𝑅 ∈ Ring → 0𝐾)
134114, 133syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) → 0𝐾)
135132, 134ifcld 4080 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) → if(𝑒𝑑, 1 , 0 ) ∈ 𝐾)
136135adantr 479 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) ∧ 𝑒 ∈ ({(1st𝑐)} × 𝑁)) → if(𝑒𝑑, 1 , 0 ) ∈ 𝐾)
137 eqid 2609 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 (-g𝑅) = (-g𝑅)
13812, 51, 137grpnpcan 17276 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ((𝑅 ∈ Grp ∧ (𝑎𝑒) ∈ 𝐾 ∧ if(𝑒𝑑, 1 , 0 ) ∈ 𝐾) → (((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )) + if(𝑒𝑑, 1 , 0 )) = (𝑎𝑒))
139117, 130, 136, 138syl3anc 1317 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) ∧ 𝑒 ∈ ({(1st𝑐)} × 𝑁)) → (((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )) + if(𝑒𝑑, 1 , 0 )) = (𝑎𝑒))
140139eqcomd 2615 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) ∧ 𝑒 ∈ ({(1st𝑐)} × 𝑁)) → (𝑎𝑒) = (((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )) + if(𝑒𝑑, 1 , 0 )))
141140adantr 479 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) ∧ 𝑒 ∈ ({(1st𝑐)} × 𝑁)) ∧ 𝑒 = 𝑐) → (𝑎𝑒) = (((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )) + if(𝑒𝑑, 1 , 0 )))
142 iftrue 4041 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (𝑒 = 𝑐 → if(𝑒 = 𝑐, ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )), 0 ) = ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )))
143 iftrue 4041 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (𝑒 = 𝑐 → if(𝑒 = 𝑐, if(𝑒𝑑, 1 , 0 ), (𝑎𝑒)) = if(𝑒𝑑, 1 , 0 ))
144142, 143oveq12d 6545 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (𝑒 = 𝑐 → (if(𝑒 = 𝑐, ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )), 0 ) + if(𝑒 = 𝑐, if(𝑒𝑑, 1 , 0 ), (𝑎𝑒))) = (((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )) + if(𝑒𝑑, 1 , 0 )))
145144adantl 480 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) ∧ 𝑒 ∈ ({(1st𝑐)} × 𝑁)) ∧ 𝑒 = 𝑐) → (if(𝑒 = 𝑐, ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )), 0 ) + if(𝑒 = 𝑐, if(𝑒𝑑, 1 , 0 ), (𝑎𝑒))) = (((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )) + if(𝑒𝑑, 1 , 0 )))
146141, 145eqtr4d 2646 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) ∧ 𝑒 ∈ ({(1st𝑐)} × 𝑁)) ∧ 𝑒 = 𝑐) → (𝑎𝑒) = (if(𝑒 = 𝑐, ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )), 0 ) + if(𝑒 = 𝑐, if(𝑒𝑑, 1 , 0 ), (𝑎𝑒))))
14712, 51, 49grplid 17221 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ((𝑅 ∈ Grp ∧ (𝑎𝑒) ∈ 𝐾) → ( 0 + (𝑎𝑒)) = (𝑎𝑒))
148117, 130, 147syl2anc 690 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) ∧ 𝑒 ∈ ({(1st𝑐)} × 𝑁)) → ( 0 + (𝑎𝑒)) = (𝑎𝑒))
149148eqcomd 2615 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) ∧ 𝑒 ∈ ({(1st𝑐)} × 𝑁)) → (𝑎𝑒) = ( 0 + (𝑎𝑒)))
150149adantr 479 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) ∧ 𝑒 ∈ ({(1st𝑐)} × 𝑁)) ∧ ¬ 𝑒 = 𝑐) → (𝑎𝑒) = ( 0 + (𝑎𝑒)))
151 iffalse 4044 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 𝑒 = 𝑐 → if(𝑒 = 𝑐, ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )), 0 ) = 0 )
152 iffalse 4044 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 𝑒 = 𝑐 → if(𝑒 = 𝑐, if(𝑒𝑑, 1 , 0 ), (𝑎𝑒)) = (𝑎𝑒))
153151, 152oveq12d 6545 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 𝑒 = 𝑐 → (if(𝑒 = 𝑐, ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )), 0 ) + if(𝑒 = 𝑐, if(𝑒𝑑, 1 , 0 ), (𝑎𝑒))) = ( 0 + (𝑎𝑒)))
154153adantl 480 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) ∧ 𝑒 ∈ ({(1st𝑐)} × 𝑁)) ∧ ¬ 𝑒 = 𝑐) → (if(𝑒 = 𝑐, ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )), 0 ) + if(𝑒 = 𝑐, if(𝑒𝑑, 1 , 0 ), (𝑎𝑒))) = ( 0 + (𝑎𝑒)))
155150, 154eqtr4d 2646 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) ∧ 𝑒 ∈ ({(1st𝑐)} × 𝑁)) ∧ ¬ 𝑒 = 𝑐) → (𝑎𝑒) = (if(𝑒 = 𝑐, ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )), 0 ) + if(𝑒 = 𝑐, if(𝑒𝑑, 1 , 0 ), (𝑎𝑒))))
156146, 155pm2.61dan 827 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) ∧ 𝑒 ∈ ({(1st𝑐)} × 𝑁)) → (𝑎𝑒) = (if(𝑒 = 𝑐, ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )), 0 ) + if(𝑒 = 𝑐, if(𝑒𝑑, 1 , 0 ), (𝑎𝑒))))
157156mpteq2dva 4666 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) → (𝑒 ∈ ({(1st𝑐)} × 𝑁) ↦ (𝑎𝑒)) = (𝑒 ∈ ({(1st𝑐)} × 𝑁) ↦ (if(𝑒 = 𝑐, ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )), 0 ) + if(𝑒 = 𝑐, if(𝑒𝑑, 1 , 0 ), (𝑎𝑒)))))
158 snfi 7900 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 {(1st𝑐)} ∈ Fin
15963ad2ant1 1074 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) → 𝑁 ∈ Fin)
160 xpfi 8093 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (({(1st𝑐)} ∈ Fin ∧ 𝑁 ∈ Fin) → ({(1st𝑐)} × 𝑁) ∈ Fin)
161158, 159, 160sylancr 693 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) → ({(1st𝑐)} × 𝑁) ∈ Fin)
162 ovex 6555 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )) ∈ V
163 fvex 6098 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (0g𝑅) ∈ V
16449, 163eqeltri 2683 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 0 ∈ V
165162, 164ifex 4105 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 if(𝑒 = 𝑐, ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )), 0 ) ∈ V
166165a1i 11 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) ∧ 𝑒 ∈ ({(1st𝑐)} × 𝑁)) → if(𝑒 = 𝑐, ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )), 0 ) ∈ V)
167 fvex 6098 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (1r𝑅) ∈ V
16850, 167eqeltri 2683 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 1 ∈ V
169168, 164ifex 4105 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 if(𝑒𝑑, 1 , 0 ) ∈ V
170 fvex 6098 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝑎𝑒) ∈ V
171169, 170ifex 4105 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 if(𝑒 = 𝑐, if(𝑒𝑑, 1 , 0 ), (𝑎𝑒)) ∈ V
172171a1i 11 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) ∧ 𝑒 ∈ ({(1st𝑐)} × 𝑁)) → if(𝑒 = 𝑐, if(𝑒𝑑, 1 , 0 ), (𝑎𝑒)) ∈ V)
173 xp1st 7066 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (𝑒 ∈ ({(1st𝑐)} × 𝑁) → (1st𝑒) ∈ {(1st𝑐)})
174 elsni 4141 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((1st𝑒) ∈ {(1st𝑐)} → (1st𝑒) = (1st𝑐))
175 iftrue 4041 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((1st𝑒) = (1st𝑐) → if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )), 0 ), (𝑎𝑒)) = if(𝑒 = 𝑐, ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )), 0 ))
176173, 174, 1753syl 18 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝑒 ∈ ({(1st𝑐)} × 𝑁) → if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )), 0 ), (𝑎𝑒)) = if(𝑒 = 𝑐, ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )), 0 ))
177176mpteq2ia 4662 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑒 ∈ ({(1st𝑐)} × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )), 0 ), (𝑎𝑒))) = (𝑒 ∈ ({(1st𝑐)} × 𝑁) ↦ if(𝑒 = 𝑐, ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )), 0 ))
178177a1i 11 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) → (𝑒 ∈ ({(1st𝑐)} × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )), 0 ), (𝑎𝑒))) = (𝑒 ∈ ({(1st𝑐)} × 𝑁) ↦ if(𝑒 = 𝑐, ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )), 0 )))
179 eqidd 2610 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) → (𝑒 ∈ ({(1st𝑐)} × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒𝑑, 1 , 0 ), (𝑎𝑒))) = (𝑒 ∈ ({(1st𝑐)} × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒𝑑, 1 , 0 ), (𝑎𝑒))))
180161, 166, 172, 178, 179offval2 6789 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) → ((𝑒 ∈ ({(1st𝑐)} × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )), 0 ), (𝑎𝑒))) ∘𝑓 + (𝑒 ∈ ({(1st𝑐)} × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒𝑑, 1 , 0 ), (𝑎𝑒)))) = (𝑒 ∈ ({(1st𝑐)} × 𝑁) ↦ (if(𝑒 = 𝑐, ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )), 0 ) + if(𝑒 = 𝑐, if(𝑒𝑑, 1 , 0 ), (𝑎𝑒)))))
181157, 180eqtr4d 2646 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) → (𝑒 ∈ ({(1st𝑐)} × 𝑁) ↦ (𝑎𝑒)) = ((𝑒 ∈ ({(1st𝑐)} × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )), 0 ), (𝑎𝑒))) ∘𝑓 + (𝑒 ∈ ({(1st𝑐)} × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒𝑑, 1 , 0 ), (𝑎𝑒)))))
182128resmptd 5358 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) → ((𝑒 ∈ (𝑁 × 𝑁) ↦ (𝑎𝑒)) ↾ ({(1st𝑐)} × 𝑁)) = (𝑒 ∈ ({(1st𝑐)} × 𝑁) ↦ (𝑎𝑒)))
183128resmptd 5358 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) → ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )), 0 ), (𝑎𝑒))) ↾ ({(1st𝑐)} × 𝑁)) = (𝑒 ∈ ({(1st𝑐)} × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )), 0 ), (𝑎𝑒))))
184128resmptd 5358 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) → ((𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒𝑑, 1 , 0 ), (𝑎𝑒))) ↾ ({(1st𝑐)} × 𝑁)) = (𝑒 ∈ ({(1st𝑐)} × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒𝑑, 1 , 0 ), (𝑎𝑒))))
185183, 184oveq12d 6545 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) → (((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )), 0 ), (𝑎𝑒))) ↾ ({(1st𝑐)} × 𝑁)) ∘𝑓 + ((𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒𝑑, 1 , 0 ), (𝑎𝑒))) ↾ ({(1st𝑐)} × 𝑁))) = ((𝑒 ∈ ({(1st𝑐)} × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )), 0 ), (𝑎𝑒))) ∘𝑓 + (𝑒 ∈ ({(1st𝑐)} × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒𝑑, 1 , 0 ), (𝑎𝑒)))))
186181, 182, 1853eqtr4d 2653 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) → ((𝑒 ∈ (𝑁 × 𝑁) ↦ (𝑎𝑒)) ↾ ({(1st𝑐)} × 𝑁)) = (((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )), 0 ), (𝑎𝑒))) ↾ ({(1st𝑐)} × 𝑁)) ∘𝑓 + ((𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒𝑑, 1 , 0 ), (𝑎𝑒))) ↾ ({(1st𝑐)} × 𝑁))))
187113, 186eqtrd 2643 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) → (𝑎 ↾ ({(1st𝑐)} × 𝑁)) = (((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )), 0 ), (𝑎𝑒))) ↾ ({(1st𝑐)} × 𝑁)) ∘𝑓 + ((𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒𝑑, 1 , 0 ), (𝑎𝑒))) ↾ ({(1st𝑐)} × 𝑁))))
188112reseq1d 5303 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) → (𝑎 ↾ ((𝑁 ∖ {(1st𝑐)}) × 𝑁)) = ((𝑒 ∈ (𝑁 × 𝑁) ↦ (𝑎𝑒)) ↾ ((𝑁 ∖ {(1st𝑐)}) × 𝑁)))
189 xp1st 7066 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (𝑒 ∈ ((𝑁 ∖ {(1st𝑐)}) × 𝑁) → (1st𝑒) ∈ (𝑁 ∖ {(1st𝑐)}))
190 eldifsni 4260 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((1st𝑒) ∈ (𝑁 ∖ {(1st𝑐)}) → (1st𝑒) ≠ (1st𝑐))
191189, 190syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝑒 ∈ ((𝑁 ∖ {(1st𝑐)}) × 𝑁) → (1st𝑒) ≠ (1st𝑐))
192191neneqd 2786 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑒 ∈ ((𝑁 ∖ {(1st𝑐)}) × 𝑁) → ¬ (1st𝑒) = (1st𝑐))
193192adantl 480 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) ∧ 𝑒 ∈ ((𝑁 ∖ {(1st𝑐)}) × 𝑁)) → ¬ (1st𝑒) = (1st𝑐))
194193iffalsed 4046 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) ∧ 𝑒 ∈ ((𝑁 ∖ {(1st𝑐)}) × 𝑁)) → if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )), 0 ), (𝑎𝑒)) = (𝑎𝑒))
195194mpteq2dva 4666 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) → (𝑒 ∈ ((𝑁 ∖ {(1st𝑐)}) × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )), 0 ), (𝑎𝑒))) = (𝑒 ∈ ((𝑁 ∖ {(1st𝑐)}) × 𝑁) ↦ (𝑎𝑒)))
196 difss 3698 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑁 ∖ {(1st𝑐)}) ⊆ 𝑁
197 xpss1 5140 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((𝑁 ∖ {(1st𝑐)}) ⊆ 𝑁 → ((𝑁 ∖ {(1st𝑐)}) × 𝑁) ⊆ (𝑁 × 𝑁))
198196, 197ax-mp 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝑁 ∖ {(1st𝑐)}) × 𝑁) ⊆ (𝑁 × 𝑁)
199 resmpt 5356 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (((𝑁 ∖ {(1st𝑐)}) × 𝑁) ⊆ (𝑁 × 𝑁) → ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )), 0 ), (𝑎𝑒))) ↾ ((𝑁 ∖ {(1st𝑐)}) × 𝑁)) = (𝑒 ∈ ((𝑁 ∖ {(1st𝑐)}) × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )), 0 ), (𝑎𝑒))))
200198, 199mp1i 13 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) → ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )), 0 ), (𝑎𝑒))) ↾ ((𝑁 ∖ {(1st𝑐)}) × 𝑁)) = (𝑒 ∈ ((𝑁 ∖ {(1st𝑐)}) × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )), 0 ), (𝑎𝑒))))
201 resmpt 5356 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (((𝑁 ∖ {(1st𝑐)}) × 𝑁) ⊆ (𝑁 × 𝑁) → ((𝑒 ∈ (𝑁 × 𝑁) ↦ (𝑎𝑒)) ↾ ((𝑁 ∖ {(1st𝑐)}) × 𝑁)) = (𝑒 ∈ ((𝑁 ∖ {(1st𝑐)}) × 𝑁) ↦ (𝑎𝑒)))
202198, 201mp1i 13 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) → ((𝑒 ∈ (𝑁 × 𝑁) ↦ (𝑎𝑒)) ↾ ((𝑁 ∖ {(1st𝑐)}) × 𝑁)) = (𝑒 ∈ ((𝑁 ∖ {(1st𝑐)}) × 𝑁) ↦ (𝑎𝑒)))
203195, 200, 2023eqtr4rd 2654 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) → ((𝑒 ∈ (𝑁 × 𝑁) ↦ (𝑎𝑒)) ↾ ((𝑁 ∖ {(1st𝑐)}) × 𝑁)) = ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )), 0 ), (𝑎𝑒))) ↾ ((𝑁 ∖ {(1st𝑐)}) × 𝑁)))
204188, 203eqtrd 2643 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) → (𝑎 ↾ ((𝑁 ∖ {(1st𝑐)}) × 𝑁)) = ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )), 0 ), (𝑎𝑒))) ↾ ((𝑁 ∖ {(1st𝑐)}) × 𝑁)))
205 fveq2 6088 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑒 = 𝑐 → (1st𝑒) = (1st𝑐))
206193, 205nsyl 133 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) ∧ 𝑒 ∈ ((𝑁 ∖ {(1st𝑐)}) × 𝑁)) → ¬ 𝑒 = 𝑐)
207206iffalsed 4046 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) ∧ 𝑒 ∈ ((𝑁 ∖ {(1st𝑐)}) × 𝑁)) → if(𝑒 = 𝑐, if(𝑒𝑑, 1 , 0 ), (𝑎𝑒)) = (𝑎𝑒))
208207mpteq2dva 4666 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) → (𝑒 ∈ ((𝑁 ∖ {(1st𝑐)}) × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒𝑑, 1 , 0 ), (𝑎𝑒))) = (𝑒 ∈ ((𝑁 ∖ {(1st𝑐)}) × 𝑁) ↦ (𝑎𝑒)))
209 resmpt 5356 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (((𝑁 ∖ {(1st𝑐)}) × 𝑁) ⊆ (𝑁 × 𝑁) → ((𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒𝑑, 1 , 0 ), (𝑎𝑒))) ↾ ((𝑁 ∖ {(1st𝑐)}) × 𝑁)) = (𝑒 ∈ ((𝑁 ∖ {(1st𝑐)}) × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒𝑑, 1 , 0 ), (𝑎𝑒))))
210198, 209mp1i 13 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) → ((𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒𝑑, 1 , 0 ), (𝑎𝑒))) ↾ ((𝑁 ∖ {(1st𝑐)}) × 𝑁)) = (𝑒 ∈ ((𝑁 ∖ {(1st𝑐)}) × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒𝑑, 1 , 0 ), (𝑎𝑒))))
211208, 210, 2023eqtr4rd 2654 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) → ((𝑒 ∈ (𝑁 × 𝑁) ↦ (𝑎𝑒)) ↾ ((𝑁 ∖ {(1st𝑐)}) × 𝑁)) = ((𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒𝑑, 1 , 0 ), (𝑎𝑒))) ↾ ((𝑁 ∖ {(1st𝑐)}) × 𝑁)))
212188, 211eqtrd 2643 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) → (𝑎 ↾ ((𝑁 ∖ {(1st𝑐)}) × 𝑁)) = ((𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒𝑑, 1 , 0 ), (𝑎𝑒))) ↾ ((𝑁 ∖ {(1st𝑐)}) × 𝑁)))
213135adantr 479 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) ∧ 𝑒 ∈ (𝑁 × 𝑁)) → if(𝑒𝑑, 1 , 0 ) ∈ 𝐾)
214111ffvelrnda 6252 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) ∧ 𝑒 ∈ (𝑁 × 𝑁)) → (𝑎𝑒) ∈ 𝐾)
215213, 214ifcld 4080 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) ∧ 𝑒 ∈ (𝑁 × 𝑁)) → if(𝑒 = 𝑐, if(𝑒𝑑, 1 , 0 ), (𝑎𝑒)) ∈ 𝐾)
216 eqid 2609 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒𝑑, 1 , 0 ), (𝑎𝑒))) = (𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒𝑑, 1 , 0 ), (𝑎𝑒)))
217215, 216fmptd 6277 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) → (𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒𝑑, 1 , 0 ), (𝑎𝑒))):(𝑁 × 𝑁)⟶𝐾)
218 fvex 6098 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (Base‘𝑅) ∈ V
21912, 218eqeltri 2683 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 𝐾 ∈ V
22068anidms 674 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑁 ∈ Fin → (𝑁 × 𝑁) ∈ Fin)
221159, 220syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) → (𝑁 × 𝑁) ∈ Fin)
222 elmapg 7734 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((𝐾 ∈ V ∧ (𝑁 × 𝑁) ∈ Fin) → ((𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒𝑑, 1 , 0 ), (𝑎𝑒))) ∈ (𝐾𝑚 (𝑁 × 𝑁)) ↔ (𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒𝑑, 1 , 0 ), (𝑎𝑒))):(𝑁 × 𝑁)⟶𝐾))
223219, 221, 222sylancr 693 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) → ((𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒𝑑, 1 , 0 ), (𝑎𝑒))) ∈ (𝐾𝑚 (𝑁 × 𝑁)) ↔ (𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒𝑑, 1 , 0 ), (𝑎𝑒))):(𝑁 × 𝑁)⟶𝐾))
224217, 223mpbird 245 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) → (𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒𝑑, 1 , 0 ), (𝑎𝑒))) ∈ (𝐾𝑚 (𝑁 × 𝑁)))
22511, 12matbas2 19988 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (𝐾𝑚 (𝑁 × 𝑁)) = (Base‘𝐴))
226159, 114, 225syl2anc 690 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) → (𝐾𝑚 (𝑁 × 𝑁)) = (Base‘𝐴))
227226, 13syl6eqr 2661 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) → (𝐾𝑚 (𝑁 × 𝑁)) = 𝐵)
228224, 227eleqtrd 2689 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) → (𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒𝑑, 1 , 0 ), (𝑎𝑒))) ∈ 𝐵)
229 simp3 1055 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) → 𝑎𝐵)
230116adantr 479 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) ∧ 𝑒 ∈ (𝑁 × 𝑁)) → 𝑅 ∈ Grp)
23112, 137grpsubcl 17264 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ((𝑅 ∈ Grp ∧ (𝑎𝑒) ∈ 𝐾 ∧ if(𝑒𝑑, 1 , 0 ) ∈ 𝐾) → ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )) ∈ 𝐾)
232230, 214, 213, 231syl3anc 1317 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) ∧ 𝑒 ∈ (𝑁 × 𝑁)) → ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )) ∈ 𝐾)
233134adantr 479 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) ∧ 𝑒 ∈ (𝑁 × 𝑁)) → 0𝐾)
234232, 233ifcld 4080 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) ∧ 𝑒 ∈ (𝑁 × 𝑁)) → if(𝑒 = 𝑐, ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )), 0 ) ∈ 𝐾)
235234, 214ifcld 4080 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) ∧ 𝑒 ∈ (𝑁 × 𝑁)) → if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )), 0 ), (𝑎𝑒)) ∈ 𝐾)
236 eqid 2609 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )), 0 ), (𝑎𝑒))) = (𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )), 0 ), (𝑎𝑒)))
237235, 236fmptd 6277 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) → (𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )), 0 ), (𝑎𝑒))):(𝑁 × 𝑁)⟶𝐾)
238 elmapg 7734 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((𝐾 ∈ V ∧ (𝑁 × 𝑁) ∈ Fin) → ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )), 0 ), (𝑎𝑒))) ∈ (𝐾𝑚 (𝑁 × 𝑁)) ↔ (𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )), 0 ), (𝑎𝑒))):(𝑁 × 𝑁)⟶𝐾))
239219, 221, 238sylancr 693 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) → ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )), 0 ), (𝑎𝑒))) ∈ (𝐾𝑚 (𝑁 × 𝑁)) ↔ (𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )), 0 ), (𝑎𝑒))):(𝑁 × 𝑁)⟶𝐾))
240237, 239mpbird 245 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) → (𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )), 0 ), (𝑎𝑒))) ∈ (𝐾𝑚 (𝑁 × 𝑁)))
241240, 227eleqtrd 2689 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) → (𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )), 0 ), (𝑎𝑒))) ∈ 𝐵)
242563ad2ant1 1074 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) → ∀𝑥𝐵𝑦𝐵𝑧𝐵𝑤𝑁 (((𝑥 ↾ ({𝑤} × 𝑁)) = ((𝑦 ↾ ({𝑤} × 𝑁)) ∘𝑓 + (𝑧 ↾ ({𝑤} × 𝑁))) ∧ (𝑥 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑦 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) ∧ (𝑥 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑧 ↾ ((𝑁 ∖ {𝑤}) × 𝑁))) → (𝐷𝑥) = ((𝐷𝑦) + (𝐷𝑧))))
243 reseq1 5298 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (𝑥 = 𝑎 → (𝑥 ↾ ({𝑤} × 𝑁)) = (𝑎 ↾ ({𝑤} × 𝑁)))
244243eqeq1d 2611 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝑥 = 𝑎 → ((𝑥 ↾ ({𝑤} × 𝑁)) = ((𝑦 ↾ ({𝑤} × 𝑁)) ∘𝑓 + (𝑧 ↾ ({𝑤} × 𝑁))) ↔ (𝑎 ↾ ({𝑤} × 𝑁)) = ((𝑦 ↾ ({𝑤} × 𝑁)) ∘𝑓 + (𝑧 ↾ ({𝑤} × 𝑁)))))
245 reseq1 5298 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (𝑥 = 𝑎 → (𝑥 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑎 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)))
246245eqeq1d 2611 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝑥 = 𝑎 → ((𝑥 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑦 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) ↔ (𝑎 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑦 ↾ ((𝑁 ∖ {𝑤}) × 𝑁))))
247245eqeq1d 2611 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝑥 = 𝑎 → ((𝑥 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑧 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) ↔ (𝑎 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑧 ↾ ((𝑁 ∖ {𝑤}) × 𝑁))))
248244, 246, 2473anbi123d 1390 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑥 = 𝑎 → (((𝑥 ↾ ({𝑤} × 𝑁)) = ((𝑦 ↾ ({𝑤} × 𝑁)) ∘𝑓 + (𝑧 ↾ ({𝑤} × 𝑁))) ∧ (𝑥 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑦 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) ∧ (𝑥 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑧 ↾ ((𝑁 ∖ {𝑤}) × 𝑁))) ↔ ((𝑎 ↾ ({𝑤} × 𝑁)) = ((𝑦 ↾ ({𝑤} × 𝑁)) ∘𝑓 + (𝑧 ↾ ({𝑤} × 𝑁))) ∧ (𝑎 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑦 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) ∧ (𝑎 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑧 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)))))
249 fveq2 6088 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝑥 = 𝑎 → (𝐷𝑥) = (𝐷𝑎))
250249eqeq1d 2611 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑥 = 𝑎 → ((𝐷𝑥) = ((𝐷𝑦) + (𝐷𝑧)) ↔ (𝐷𝑎) = ((𝐷𝑦) + (𝐷𝑧))))
251248, 250imbi12d 332 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑥 = 𝑎 → ((((𝑥 ↾ ({𝑤} × 𝑁)) = ((𝑦 ↾ ({𝑤} × 𝑁)) ∘𝑓 + (𝑧 ↾ ({𝑤} × 𝑁))) ∧ (𝑥 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑦 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) ∧ (𝑥 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑧 ↾ ((𝑁 ∖ {𝑤}) × 𝑁))) → (𝐷𝑥) = ((𝐷𝑦) + (𝐷𝑧))) ↔ (((𝑎 ↾ ({𝑤} × 𝑁)) = ((𝑦 ↾ ({𝑤} × 𝑁)) ∘𝑓 + (𝑧 ↾ ({𝑤} × 𝑁))) ∧ (𝑎 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑦 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) ∧ (𝑎 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑧 ↾ ((𝑁 ∖ {𝑤}) × 𝑁))) → (𝐷𝑎) = ((𝐷𝑦) + (𝐷𝑧)))))
2522512ralbidv 2971 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑥 = 𝑎 → (∀𝑧𝐵𝑤𝑁 (((𝑥 ↾ ({𝑤} × 𝑁)) = ((𝑦 ↾ ({𝑤} × 𝑁)) ∘𝑓 + (𝑧 ↾ ({𝑤} × 𝑁))) ∧ (𝑥 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑦 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) ∧ (𝑥 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑧 ↾ ((𝑁 ∖ {𝑤}) × 𝑁))) → (𝐷𝑥) = ((𝐷𝑦) + (𝐷𝑧))) ↔ ∀𝑧𝐵𝑤𝑁 (((𝑎 ↾ ({𝑤} × 𝑁)) = ((𝑦 ↾ ({𝑤} × 𝑁)) ∘𝑓 + (𝑧 ↾ ({𝑤} × 𝑁))) ∧ (𝑎 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑦 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) ∧ (𝑎 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑧 ↾ ((𝑁 ∖ {𝑤}) × 𝑁))) → (𝐷𝑎) = ((𝐷𝑦) + (𝐷𝑧)))))
253 reseq1 5298 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (𝑦 = (𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )), 0 ), (𝑎𝑒))) → (𝑦 ↾ ({𝑤} × 𝑁)) = ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )), 0 ), (𝑎𝑒))) ↾ ({𝑤} × 𝑁)))
254253oveq1d 6542 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (𝑦 = (𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )), 0 ), (𝑎𝑒))) → ((𝑦 ↾ ({𝑤} × 𝑁)) ∘𝑓 + (𝑧 ↾ ({𝑤} × 𝑁))) = (((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )), 0 ), (𝑎𝑒))) ↾ ({𝑤} × 𝑁)) ∘𝑓 + (𝑧 ↾ ({𝑤} × 𝑁))))
255254eqeq2d 2619 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝑦 = (𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )), 0 ), (𝑎𝑒))) → ((𝑎 ↾ ({𝑤} × 𝑁)) = ((𝑦 ↾ ({𝑤} × 𝑁)) ∘𝑓 + (𝑧 ↾ ({𝑤} × 𝑁))) ↔ (𝑎 ↾ ({𝑤} × 𝑁)) = (((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )), 0 ), (𝑎𝑒))) ↾ ({𝑤} × 𝑁)) ∘𝑓 + (𝑧 ↾ ({𝑤} × 𝑁)))))
256 reseq1 5298 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (𝑦 = (𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )), 0 ), (𝑎𝑒))) → (𝑦 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )), 0 ), (𝑎𝑒))) ↾ ((𝑁 ∖ {𝑤}) × 𝑁)))
257256eqeq2d 2619 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝑦 = (𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )), 0 ), (𝑎𝑒))) → ((𝑎 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑦 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) ↔ (𝑎 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )), 0 ), (𝑎𝑒))) ↾ ((𝑁 ∖ {𝑤}) × 𝑁))))
258255, 2573anbi12d 1391 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑦 = (𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )), 0 ), (𝑎𝑒))) → (((𝑎 ↾ ({𝑤} × 𝑁)) = ((𝑦 ↾ ({𝑤} × 𝑁)) ∘𝑓 + (𝑧 ↾ ({𝑤} × 𝑁))) ∧ (𝑎 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑦 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) ∧ (𝑎 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑧 ↾ ((𝑁 ∖ {𝑤}) × 𝑁))) ↔ ((𝑎 ↾ ({𝑤} × 𝑁)) = (((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )), 0 ), (𝑎𝑒))) ↾ ({𝑤} × 𝑁)) ∘𝑓 + (𝑧 ↾ ({𝑤} × 𝑁))) ∧ (𝑎 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )), 0 ), (𝑎𝑒))) ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) ∧ (𝑎 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑧 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)))))
259 fveq2 6088 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (𝑦 = (𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )), 0 ), (𝑎𝑒))) → (𝐷𝑦) = (𝐷‘(𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )), 0 ), (𝑎𝑒)))))
260259oveq1d 6542 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝑦 = (𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )), 0 ), (𝑎𝑒))) → ((𝐷𝑦) + (𝐷𝑧)) = ((𝐷‘(𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )), 0 ), (𝑎𝑒)))) + (𝐷𝑧)))
261260eqeq2d 2619 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑦 = (𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )), 0 ), (𝑎𝑒))) → ((𝐷𝑎) = ((𝐷𝑦) + (𝐷𝑧)) ↔ (𝐷𝑎) = ((𝐷‘(𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )), 0 ), (𝑎𝑒)))) + (𝐷𝑧))))
262258, 261imbi12d 332 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑦 = (𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )), 0 ), (𝑎𝑒))) → ((((𝑎 ↾ ({𝑤} × 𝑁)) = ((𝑦 ↾ ({𝑤} × 𝑁)) ∘𝑓 + (𝑧 ↾ ({𝑤} × 𝑁))) ∧ (𝑎 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑦 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) ∧ (𝑎 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑧 ↾ ((𝑁 ∖ {𝑤}) × 𝑁))) → (𝐷𝑎) = ((𝐷𝑦) + (𝐷𝑧))) ↔ (((𝑎 ↾ ({𝑤} × 𝑁)) = (((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )), 0 ), (𝑎𝑒))) ↾ ({𝑤} × 𝑁)) ∘𝑓 + (𝑧 ↾ ({𝑤} × 𝑁))) ∧ (𝑎 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )), 0 ), (𝑎𝑒))) ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) ∧ (𝑎 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑧 ↾ ((𝑁 ∖ {𝑤}) × 𝑁))) → (𝐷𝑎) = ((𝐷‘(𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )), 0 ), (𝑎𝑒)))) + (𝐷𝑧)))))
2632622ralbidv 2971 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑦 = (𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )), 0 ), (𝑎𝑒))) → (∀𝑧𝐵𝑤𝑁 (((𝑎 ↾ ({𝑤} × 𝑁)) = ((𝑦 ↾ ({𝑤} × 𝑁)) ∘𝑓 + (𝑧 ↾ ({𝑤} × 𝑁))) ∧ (𝑎 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑦 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) ∧ (𝑎 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑧 ↾ ((𝑁 ∖ {𝑤}) × 𝑁))) → (𝐷𝑎) = ((𝐷𝑦) + (𝐷𝑧))) ↔ ∀𝑧𝐵𝑤𝑁 (((𝑎 ↾ ({𝑤} × 𝑁)) = (((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )), 0 ), (𝑎𝑒))) ↾ ({𝑤} × 𝑁)) ∘𝑓 + (𝑧 ↾ ({𝑤} × 𝑁))) ∧ (𝑎 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )), 0 ), (𝑎𝑒))) ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) ∧ (𝑎 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑧 ↾ ((𝑁 ∖ {𝑤}) × 𝑁))) → (𝐷𝑎) = ((𝐷‘(𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )), 0 ), (𝑎𝑒)))) + (𝐷𝑧)))))
264252, 263rspc2va 3293 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((𝑎𝐵 ∧ (𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )), 0 ), (𝑎𝑒))) ∈ 𝐵) ∧ ∀𝑥𝐵𝑦𝐵𝑧𝐵𝑤𝑁 (((𝑥 ↾ ({𝑤} × 𝑁)) = ((𝑦 ↾ ({𝑤} × 𝑁)) ∘𝑓 + (𝑧 ↾ ({𝑤} × 𝑁))) ∧ (𝑥 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑦 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) ∧ (𝑥 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑧 ↾ ((𝑁 ∖ {𝑤}) × 𝑁))) → (𝐷𝑥) = ((𝐷𝑦) + (𝐷𝑧)))) → ∀𝑧𝐵𝑤𝑁 (((𝑎 ↾ ({𝑤} × 𝑁)) = (((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )), 0 ), (𝑎𝑒))) ↾ ({𝑤} × 𝑁)) ∘𝑓 + (𝑧 ↾ ({𝑤} × 𝑁))) ∧ (𝑎 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )), 0 ), (𝑎𝑒))) ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) ∧ (𝑎 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑧 ↾ ((𝑁 ∖ {𝑤}) × 𝑁))) → (𝐷𝑎) = ((𝐷‘(𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )), 0 ), (𝑎𝑒)))) + (𝐷𝑧))))
265229, 241, 242, 264syl21anc 1316 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) → ∀𝑧𝐵𝑤𝑁 (((𝑎 ↾ ({𝑤} × 𝑁)) = (((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )), 0 ), (𝑎𝑒))) ↾ ({𝑤} × 𝑁)) ∘𝑓 + (𝑧 ↾ ({𝑤} × 𝑁))) ∧ (𝑎 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )), 0 ), (𝑎𝑒))) ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) ∧ (𝑎 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑧 ↾ ((𝑁 ∖ {𝑤}) × 𝑁))) → (𝐷𝑎) = ((𝐷‘(𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )), 0 ), (𝑎𝑒)))) + (𝐷𝑧))))
266 reseq1 5298 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝑧 = (𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒𝑑, 1 , 0 ), (𝑎𝑒))) → (𝑧 ↾ ({𝑤} × 𝑁)) = ((𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒𝑑, 1 , 0 ), (𝑎𝑒))) ↾ ({𝑤} × 𝑁)))
267266oveq2d 6543 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑧 = (𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒𝑑, 1 , 0 ), (𝑎𝑒))) → (((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )), 0 ), (𝑎𝑒))) ↾ ({𝑤} × 𝑁)) ∘𝑓 + (𝑧 ↾ ({𝑤} × 𝑁))) = (((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )), 0 ), (𝑎𝑒))) ↾ ({𝑤} × 𝑁)) ∘𝑓 + ((𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒𝑑, 1 , 0 ), (𝑎𝑒))) ↾ ({𝑤} × 𝑁))))
268267eqeq2d 2619 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑧 = (𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒𝑑, 1 , 0 ), (𝑎𝑒))) → ((𝑎 ↾ ({𝑤} × 𝑁)) = (((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )), 0 ), (𝑎𝑒))) ↾ ({𝑤} × 𝑁)) ∘𝑓 + (𝑧 ↾ ({𝑤} × 𝑁))) ↔ (𝑎 ↾ ({𝑤} × 𝑁)) = (((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )), 0 ), (𝑎𝑒))) ↾ ({𝑤} × 𝑁)) ∘𝑓 + ((𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒𝑑, 1 , 0 ), (𝑎𝑒))) ↾ ({𝑤} × 𝑁)))))
269 reseq1 5298 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑧 = (𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒𝑑, 1 , 0 ), (𝑎𝑒))) → (𝑧 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = ((𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒𝑑, 1 , 0 ), (𝑎𝑒))) ↾ ((𝑁 ∖ {𝑤}) × 𝑁)))
270269eqeq2d 2619 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑧 = (𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒𝑑, 1 , 0 ), (𝑎𝑒))) → ((𝑎 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑧 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) ↔ (𝑎 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = ((𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒𝑑, 1 , 0 ), (𝑎𝑒))) ↾ ((𝑁 ∖ {𝑤}) × 𝑁))))
271268, 2703anbi13d 1392 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑧 = (𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒𝑑, 1 , 0 ), (𝑎𝑒))) → (((𝑎 ↾ ({𝑤} × 𝑁)) = (((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )), 0 ), (𝑎𝑒))) ↾ ({𝑤} × 𝑁)) ∘𝑓 + (𝑧 ↾ ({𝑤} × 𝑁))) ∧ (𝑎 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )), 0 ), (𝑎𝑒))) ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) ∧ (𝑎 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑧 ↾ ((𝑁 ∖ {𝑤}) × 𝑁))) ↔ ((𝑎 ↾ ({𝑤} × 𝑁)) = (((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )), 0 ), (𝑎𝑒))) ↾ ({𝑤} × 𝑁)) ∘𝑓 + ((𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒𝑑, 1 , 0 ), (𝑎𝑒))) ↾ ({𝑤} × 𝑁))) ∧ (𝑎 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )), 0 ), (𝑎𝑒))) ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) ∧ (𝑎 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = ((𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒𝑑, 1 , 0 ), (𝑎𝑒))) ↾ ((𝑁 ∖ {𝑤}) × 𝑁)))))
272 fveq2 6088 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑧 = (𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒𝑑, 1 , 0 ), (𝑎𝑒))) → (𝐷𝑧) = (𝐷‘(𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒𝑑, 1 , 0 ), (𝑎𝑒)))))
273272oveq2d 6543 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑧 = (𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒𝑑, 1 , 0 ), (𝑎𝑒))) → ((𝐷‘(𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )), 0 ), (𝑎𝑒)))) + (𝐷𝑧)) = ((𝐷‘(𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )), 0 ), (𝑎𝑒)))) + (𝐷‘(𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒𝑑, 1 , 0 ), (𝑎𝑒))))))
274273eqeq2d 2619 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑧 = (𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒𝑑, 1 , 0 ), (𝑎𝑒))) → ((𝐷𝑎) = ((𝐷‘(𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )), 0 ), (𝑎𝑒)))) + (𝐷𝑧)) ↔ (𝐷𝑎) = ((𝐷‘(𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )), 0 ), (𝑎𝑒)))) + (𝐷‘(𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒𝑑, 1 , 0 ), (𝑎𝑒)))))))
275271, 274imbi12d 332 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑧 = (𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒𝑑, 1 , 0 ), (𝑎𝑒))) → ((((𝑎 ↾ ({𝑤} × 𝑁)) = (((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )), 0 ), (𝑎𝑒))) ↾ ({𝑤} × 𝑁)) ∘𝑓 + (𝑧 ↾ ({𝑤} × 𝑁))) ∧ (𝑎 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )), 0 ), (𝑎𝑒))) ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) ∧ (𝑎 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑧 ↾ ((𝑁 ∖ {𝑤}) × 𝑁))) → (𝐷𝑎) = ((𝐷‘(𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )), 0 ), (𝑎𝑒)))) + (𝐷𝑧))) ↔ (((𝑎 ↾ ({𝑤} × 𝑁)) = (((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )), 0 ), (𝑎𝑒))) ↾ ({𝑤} × 𝑁)) ∘𝑓 + ((𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒𝑑, 1 , 0 ), (𝑎𝑒))) ↾ ({𝑤} × 𝑁))) ∧ (𝑎 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )), 0 ), (𝑎𝑒))) ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) ∧ (𝑎 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = ((𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒𝑑, 1 , 0 ), (𝑎𝑒))) ↾ ((𝑁 ∖ {𝑤}) × 𝑁))) → (𝐷𝑎) = ((𝐷‘(𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )), 0 ), (𝑎𝑒)))) + (𝐷‘(𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒𝑑, 1 , 0 ), (𝑎𝑒))))))))
276 sneq 4134 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (𝑤 = (1st𝑐) → {𝑤} = {(1st𝑐)})
277276xpeq1d 5052 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝑤 = (1st𝑐) → ({𝑤} × 𝑁) = ({(1st𝑐)} × 𝑁))
278277reseq2d 5304 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑤 = (1st𝑐) → (𝑎 ↾ ({𝑤} × 𝑁)) = (𝑎 ↾ ({(1st𝑐)} × 𝑁)))
279277reseq2d 5304 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝑤 = (1st𝑐) → ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )), 0 ), (𝑎𝑒))) ↾ ({𝑤} × 𝑁)) = ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )), 0 ), (𝑎𝑒))) ↾ ({(1st𝑐)} × 𝑁)))
280277reseq2d 5304 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝑤 = (1st𝑐) → ((𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒𝑑, 1 , 0 ), (𝑎𝑒))) ↾ ({𝑤} × 𝑁)) = ((𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒𝑑, 1 , 0 ), (𝑎𝑒))) ↾ ({(1st𝑐)} × 𝑁)))
281279, 280oveq12d 6545 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑤 = (1st𝑐) → (((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )), 0 ), (𝑎𝑒))) ↾ ({𝑤} × 𝑁)) ∘𝑓 + ((𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒𝑑, 1 , 0 ), (𝑎𝑒))) ↾ ({𝑤} × 𝑁))) = (((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )), 0 ), (𝑎𝑒))) ↾ ({(1st𝑐)} × 𝑁)) ∘𝑓 + ((𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒𝑑, 1 , 0 ), (𝑎𝑒))) ↾ ({(1st𝑐)} × 𝑁))))
282278, 281eqeq12d 2624 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑤 = (1st𝑐) → ((𝑎 ↾ ({𝑤} × 𝑁)) = (((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )), 0 ), (𝑎𝑒))) ↾ ({𝑤} × 𝑁)) ∘𝑓 + ((𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒𝑑, 1 , 0 ), (𝑎𝑒))) ↾ ({𝑤} × 𝑁))) ↔ (𝑎 ↾ ({(1st𝑐)} × 𝑁)) = (((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )), 0 ), (𝑎𝑒))) ↾ ({(1st𝑐)} × 𝑁)) ∘𝑓 + ((𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒𝑑, 1 , 0 ), (𝑎𝑒))) ↾ ({(1st𝑐)} × 𝑁)))))
283276difeq2d 3689 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (𝑤 = (1st𝑐) → (𝑁 ∖ {𝑤}) = (𝑁 ∖ {(1st𝑐)}))
284283xpeq1d 5052 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝑤 = (1st𝑐) → ((𝑁 ∖ {𝑤}) × 𝑁) = ((𝑁 ∖ {(1st𝑐)}) × 𝑁))
285284reseq2d 5304 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑤 = (1st𝑐) → (𝑎 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑎 ↾ ((𝑁 ∖ {(1st𝑐)}) × 𝑁)))
286284reseq2d 5304 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑤 = (1st𝑐) → ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )), 0 ), (𝑎𝑒))) ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )), 0 ), (𝑎𝑒))) ↾ ((𝑁 ∖ {(1st𝑐)}) × 𝑁)))
287285, 286eqeq12d 2624 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑤 = (1st𝑐) → ((𝑎 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )), 0 ), (𝑎𝑒))) ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) ↔ (𝑎 ↾ ((𝑁 ∖ {(1st𝑐)}) × 𝑁)) = ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )), 0 ), (𝑎𝑒))) ↾ ((𝑁 ∖ {(1st𝑐)}) × 𝑁))))
288284reseq2d 5304 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑤 = (1st𝑐) → ((𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒𝑑, 1 , 0 ), (𝑎𝑒))) ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = ((𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒𝑑, 1 , 0 ), (𝑎𝑒))) ↾ ((𝑁 ∖ {(1st𝑐)}) × 𝑁)))
289285, 288eqeq12d 2624 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑤 = (1st𝑐) → ((𝑎 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = ((𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒𝑑, 1 , 0 ), (𝑎𝑒))) ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) ↔ (𝑎 ↾ ((𝑁 ∖ {(1st𝑐)}) × 𝑁)) = ((𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒𝑑, 1 , 0 ), (𝑎𝑒))) ↾ ((𝑁 ∖ {(1st𝑐)}) × 𝑁))))
290282, 287, 2893anbi123d 1390 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑤 = (1st𝑐) → (((𝑎 ↾ ({𝑤} × 𝑁)) = (((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )), 0 ), (𝑎𝑒))) ↾ ({𝑤} × 𝑁)) ∘𝑓 + ((𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒𝑑, 1 , 0 ), (𝑎𝑒))) ↾ ({𝑤} × 𝑁))) ∧ (𝑎 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )), 0 ), (𝑎𝑒))) ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) ∧ (𝑎 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = ((𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒𝑑, 1 , 0 ), (𝑎𝑒))) ↾ ((𝑁 ∖ {𝑤}) × 𝑁))) ↔ ((𝑎 ↾ ({(1st𝑐)} × 𝑁)) = (((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )), 0 ), (𝑎𝑒))) ↾ ({(1st𝑐)} × 𝑁)) ∘𝑓 + ((𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒𝑑, 1 , 0 ), (𝑎𝑒))) ↾ ({(1st𝑐)} × 𝑁))) ∧ (𝑎 ↾ ((𝑁 ∖ {(1st𝑐)}) × 𝑁)) = ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )), 0 ), (𝑎𝑒))) ↾ ((𝑁 ∖ {(1st𝑐)}) × 𝑁)) ∧ (𝑎 ↾ ((𝑁 ∖ {(1st𝑐)}) × 𝑁)) = ((𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒𝑑, 1 , 0 ), (𝑎𝑒))) ↾ ((𝑁 ∖ {(1st𝑐)}) × 𝑁)))))
291290imbi1d 329 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑤 = (1st𝑐) → ((((𝑎 ↾ ({𝑤} × 𝑁)) = (((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )), 0 ), (𝑎𝑒))) ↾ ({𝑤} × 𝑁)) ∘𝑓 + ((𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒𝑑, 1 , 0 ), (𝑎𝑒))) ↾ ({𝑤} × 𝑁))) ∧ (𝑎 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )), 0 ), (𝑎𝑒))) ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) ∧ (𝑎 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = ((𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒𝑑, 1 , 0 ), (𝑎𝑒))) ↾ ((𝑁 ∖ {𝑤}) × 𝑁))) → (𝐷𝑎) = ((𝐷‘(𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )), 0 ), (𝑎𝑒)))) + (𝐷‘(𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒𝑑, 1 , 0 ), (𝑎𝑒)))))) ↔ (((𝑎 ↾ ({(1st𝑐)} × 𝑁)) = (((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )), 0 ), (𝑎𝑒))) ↾ ({(1st𝑐)} × 𝑁)) ∘𝑓 + ((𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒𝑑, 1 , 0 ), (𝑎𝑒))) ↾ ({(1st𝑐)} × 𝑁))) ∧ (𝑎 ↾ ((𝑁 ∖ {(1st𝑐)}) × 𝑁)) = ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )), 0 ), (𝑎𝑒))) ↾ ((𝑁 ∖ {(1st𝑐)}) × 𝑁)) ∧ (𝑎 ↾ ((𝑁 ∖ {(1st𝑐)}) × 𝑁)) = ((𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒𝑑, 1 , 0 ), (𝑎𝑒))) ↾ ((𝑁 ∖ {(1st𝑐)}) × 𝑁))) → (𝐷𝑎) = ((𝐷‘(𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )), 0 ), (𝑎𝑒)))) + (𝐷‘(𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒𝑑, 1 , 0 ), (𝑎𝑒))))))))
292275, 291rspc2va 3293 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((((𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒𝑑, 1 , 0 ), (𝑎𝑒))) ∈ 𝐵 ∧ (1st𝑐) ∈ 𝑁) ∧ ∀𝑧𝐵𝑤𝑁 (((𝑎 ↾ ({𝑤} × 𝑁)) = (((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )), 0 ), (𝑎𝑒))) ↾ ({𝑤} × 𝑁)) ∘𝑓 + (𝑧 ↾ ({𝑤} × 𝑁))) ∧ (𝑎 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )), 0 ), (𝑎𝑒))) ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) ∧ (𝑎 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑧 ↾ ((𝑁 ∖ {𝑤}) × 𝑁))) → (𝐷𝑎) = ((𝐷‘(𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )), 0 ), (𝑎𝑒)))) + (𝐷𝑧)))) → (((𝑎 ↾ ({(1st𝑐)} × 𝑁)) = (((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )), 0 ), (𝑎𝑒))) ↾ ({(1st𝑐)} × 𝑁)) ∘𝑓 + ((𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒𝑑, 1 , 0 ), (𝑎𝑒))) ↾ ({(1st𝑐)} × 𝑁))) ∧ (𝑎 ↾ ((𝑁 ∖ {(1st𝑐)}) × 𝑁)) = ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )), 0 ), (𝑎𝑒))) ↾ ((𝑁 ∖ {(1st𝑐)}) × 𝑁)) ∧ (𝑎 ↾ ((𝑁 ∖ {(1st𝑐)}) × 𝑁)) = ((𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒𝑑, 1 , 0 ), (𝑎𝑒))) ↾ ((𝑁 ∖ {(1st𝑐)}) × 𝑁))) → (𝐷𝑎) = ((𝐷‘(𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )), 0 ), (𝑎𝑒)))) + (𝐷‘(𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒𝑑, 1 , 0 ), (𝑎𝑒)))))))
293228, 125, 265, 292syl21anc 1316 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) → (((𝑎 ↾ ({(1st𝑐)} × 𝑁)) = (((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )), 0 ), (𝑎𝑒))) ↾ ({(1st𝑐)} × 𝑁)) ∘𝑓 + ((𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒𝑑, 1 , 0 ), (𝑎𝑒))) ↾ ({(1st𝑐)} × 𝑁))) ∧ (𝑎 ↾ ((𝑁 ∖ {(1st𝑐)}) × 𝑁)) = ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )), 0 ), (𝑎𝑒))) ↾ ((𝑁 ∖ {(1st𝑐)}) × 𝑁)) ∧ (𝑎 ↾ ((𝑁 ∖ {(1st𝑐)}) × 𝑁)) = ((𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒𝑑, 1 , 0 ), (𝑎𝑒))) ↾ ((𝑁 ∖ {(1st𝑐)}) × 𝑁))) → (𝐷𝑎) = ((𝐷‘(𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )), 0 ), (𝑎𝑒)))) + (𝐷‘(𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒𝑑, 1 , 0 ), (𝑎𝑒)))))))
294187, 204, 212, 293mp3and 1418 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) → (𝐷𝑎) = ((𝐷‘(𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )), 0 ), (𝑎𝑒)))) + (𝐷‘(𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒𝑑, 1 , 0 ), (𝑎𝑒))))))
295105, 106, 107, 294syl3anc 1317 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ (𝑏 ∪ {𝑐}) ∈ 𝑌) ∧ ((𝑎𝐵𝑑 ∈ (𝑁𝑚 𝑁)) ∧ ∀𝑤𝑏 (𝑎𝑤) = if(𝑤𝑑, 1 , 0 ))) → (𝐷𝑎) = ((𝐷‘(𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )), 0 ), (𝑎𝑒)))) + (𝐷‘(𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒𝑑, 1 , 0 ), (𝑎𝑒))))))
296 fveq2 6088 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (𝑒 = 𝑐 → (𝑎𝑒) = (𝑎𝑐))
297 elequ1 1983 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 (𝑒 = 𝑐 → (𝑒𝑑𝑐𝑑))
298297ifbid 4057 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (𝑒 = 𝑐 → if(𝑒𝑑, 1 , 0 ) = if(𝑐𝑑, 1 , 0 ))
299296, 298oveq12d 6545 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (𝑒 = 𝑐 → ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )) = ((𝑎𝑐)(-g𝑅)if(𝑐𝑑, 1 , 0 )))
300299adantl 480 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) ∧ 𝑒 ∈ ({(1st𝑐)} × 𝑁)) ∧ 𝑒 = 𝑐) → ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )) = ((𝑎𝑐)(-g𝑅)if(𝑐𝑑, 1 , 0 )))
301111, 123ffvelrnd 6253 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) → (𝑎𝑐) ∈ 𝐾)
302132, 134ifcld 4080 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) → if(𝑐𝑑, 1 , 0 ) ∈ 𝐾)
30312, 137grpsubcl 17264 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ((𝑅 ∈ Grp ∧ (𝑎𝑐) ∈ 𝐾 ∧ if(𝑐𝑑, 1 , 0 ) ∈ 𝐾) → ((𝑎𝑐)(-g𝑅)if(𝑐𝑑, 1 , 0 )) ∈ 𝐾)
304116, 301, 302, 303syl3anc 1317 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) → ((𝑎𝑐)(-g𝑅)if(𝑐𝑑, 1 , 0 )) ∈ 𝐾)
30512, 52, 50ringridm 18341 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ((𝑅 ∈ Ring ∧ ((𝑎𝑐)(-g𝑅)if(𝑐𝑑, 1 , 0 )) ∈ 𝐾) → (((𝑎𝑐)(-g𝑅)if(𝑐𝑑, 1 , 0 )) · 1 ) = ((𝑎𝑐)(-g𝑅)if(𝑐𝑑, 1 , 0 )))
306114, 304, 305syl2anc 690 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) → (((𝑎𝑐)(-g𝑅)if(𝑐𝑑, 1 , 0 )) · 1 ) = ((𝑎𝑐)(-g𝑅)if(𝑐𝑑, 1 , 0 )))
307306ad2antrr 757 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) ∧ 𝑒 ∈ ({(1st𝑐)} × 𝑁)) ∧ 𝑒 = 𝑐) → (((𝑎𝑐)(-g𝑅)if(𝑐𝑑, 1 , 0 )) · 1 ) = ((𝑎𝑐)(-g𝑅)if(𝑐𝑑, 1 , 0 )))
308300, 307eqtr4d 2646 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) ∧ 𝑒 ∈ ({(1st𝑐)} × 𝑁)) ∧ 𝑒 = 𝑐) → ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )) = (((𝑎𝑐)(-g𝑅)if(𝑐𝑑, 1 , 0 )) · 1 ))
309142adantl 480 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) ∧ 𝑒 ∈ ({(1st𝑐)} × 𝑁)) ∧ 𝑒 = 𝑐) → if(𝑒 = 𝑐, ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )), 0 ) = ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )))
310 iftrue 4041 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (𝑒 = 𝑐 → if(𝑒 = 𝑐, 1 , 0 ) = 1 )
311310oveq2d 6543 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (𝑒 = 𝑐 → (((𝑎𝑐)(-g𝑅)if(𝑐𝑑, 1 , 0 )) · if(𝑒 = 𝑐, 1 , 0 )) = (((𝑎𝑐)(-g𝑅)if(𝑐𝑑, 1 , 0 )) · 1 ))
312311adantl 480 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) ∧ 𝑒 ∈ ({(1st𝑐)} × 𝑁)) ∧ 𝑒 = 𝑐) → (((𝑎𝑐)(-g𝑅)if(𝑐𝑑, 1 , 0 )) · if(𝑒 = 𝑐, 1 , 0 )) = (((𝑎𝑐)(-g𝑅)if(𝑐𝑑, 1 , 0 )) · 1 ))
313308, 309, 3123eqtr4d 2653 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) ∧ 𝑒 ∈ ({(1st𝑐)} × 𝑁)) ∧ 𝑒 = 𝑐) → if(𝑒 = 𝑐, ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )), 0 ) = (((𝑎𝑐)(-g𝑅)if(𝑐𝑑, 1 , 0 )) · if(𝑒 = 𝑐, 1 , 0 )))
31412, 52, 49ringrz 18357 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ((𝑅 ∈ Ring ∧ ((𝑎𝑐)(-g𝑅)if(𝑐𝑑, 1 , 0 )) ∈ 𝐾) → (((𝑎𝑐)(-g𝑅)if(𝑐𝑑, 1 , 0 )) · 0 ) = 0 )
315114, 304, 314syl2anc 690 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) → (((𝑎𝑐)(-g𝑅)if(𝑐𝑑, 1 , 0 )) · 0 ) = 0 )
316315eqcomd 2615 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) → 0 = (((𝑎𝑐)(-g𝑅)if(𝑐𝑑, 1 , 0 )) · 0 ))
317316ad2antrr 757 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) ∧ 𝑒 ∈ ({(1st𝑐)} × 𝑁)) ∧ ¬ 𝑒 = 𝑐) → 0 = (((𝑎𝑐)(-g𝑅)if(𝑐𝑑, 1 , 0 )) · 0 ))
318151adantl 480 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) ∧ 𝑒 ∈ ({(1st𝑐)} × 𝑁)) ∧ ¬ 𝑒 = 𝑐) → if(𝑒 = 𝑐, ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )), 0 ) = 0 )
319 iffalse 4044 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 𝑒 = 𝑐 → if(𝑒 = 𝑐, 1 , 0 ) = 0 )
320319oveq2d 6543 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 𝑒 = 𝑐 → (((𝑎𝑐)(-g𝑅)if(𝑐𝑑, 1 , 0 )) · if(𝑒 = 𝑐, 1 , 0 )) = (((𝑎𝑐)(-g𝑅)if(𝑐𝑑, 1 , 0 )) · 0 ))
321320adantl 480 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) ∧ 𝑒 ∈ ({(1st𝑐)} × 𝑁)) ∧ ¬ 𝑒 = 𝑐) → (((𝑎𝑐)(-g𝑅)if(𝑐𝑑, 1 , 0 )) · if(𝑒 = 𝑐, 1 , 0 )) = (((𝑎𝑐)(-g𝑅)if(𝑐𝑑, 1 , 0 )) · 0 ))
322317, 318, 3213eqtr4d 2653 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) ∧ 𝑒 ∈ ({(1st𝑐)} × 𝑁)) ∧ ¬ 𝑒 = 𝑐) → if(𝑒 = 𝑐, ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )), 0 ) = (((𝑎𝑐)(-g𝑅)if(𝑐𝑑, 1 , 0 )) · if(𝑒 = 𝑐, 1 , 0 )))
323313, 322pm2.61dan 827 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) ∧ 𝑒 ∈ ({(1st𝑐)} × 𝑁)) → if(𝑒 = 𝑐, ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )), 0 ) = (((𝑎𝑐)(-g𝑅)if(𝑐𝑑, 1 , 0 )) · if(𝑒 = 𝑐, 1 , 0 )))
324173adantl 480 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) ∧ 𝑒 ∈ ({(1st𝑐)} × 𝑁)) → (1st𝑒) ∈ {(1st𝑐)})
325324, 174syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) ∧ 𝑒 ∈ ({(1st𝑐)} × 𝑁)) → (1st𝑒) = (1st𝑐))
326325iftrued 4043 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) ∧ 𝑒 ∈ ({(1st𝑐)} × 𝑁)) → if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )), 0 ), (𝑎𝑒)) = if(𝑒 = 𝑐, ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )), 0 ))
327325iftrued 4043 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) ∧ 𝑒 ∈ ({(1st𝑐)} × 𝑁)) → if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎𝑒)) = if(𝑒 = 𝑐, 1 , 0 ))
328327oveq2d 6543 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) ∧ 𝑒 ∈ ({(1st𝑐)} × 𝑁)) → (((𝑎𝑐)(-g𝑅)if(𝑐𝑑, 1 , 0 )) · if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎𝑒))) = (((𝑎𝑐)(-g𝑅)if(𝑐𝑑, 1 , 0 )) · if(𝑒 = 𝑐, 1 , 0 )))
329323, 326, 3283eqtr4d 2653 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) ∧ 𝑒 ∈ ({(1st𝑐)} × 𝑁)) → if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )), 0 ), (𝑎𝑒)) = (((𝑎𝑐)(-g𝑅)if(𝑐𝑑, 1 , 0 )) · if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎𝑒))))
330329mpteq2dva 4666 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) → (𝑒 ∈ ({(1st𝑐)} × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )), 0 ), (𝑎𝑒))) = (𝑒 ∈ ({(1st𝑐)} × 𝑁) ↦ (((𝑎𝑐)(-g𝑅)if(𝑐𝑑, 1 , 0 )) · if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎𝑒)))))
331 ovex 6555 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((𝑎𝑐)(-g𝑅)if(𝑐𝑑, 1 , 0 )) ∈ V
332331a1i 11 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) ∧ 𝑒 ∈ ({(1st𝑐)} × 𝑁)) → ((𝑎𝑐)(-g𝑅)if(𝑐𝑑, 1 , 0 )) ∈ V)
333168, 164ifex 4105 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 if(𝑒 = 𝑐, 1 , 0 ) ∈ V
334333, 170ifex 4105 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎𝑒)) ∈ V
335334a1i 11 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) ∧ 𝑒 ∈ ({(1st𝑐)} × 𝑁)) → if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎𝑒)) ∈ V)
336 fconstmpt 5075 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (({(1st𝑐)} × 𝑁) × {((𝑎𝑐)(-g𝑅)if(𝑐𝑑, 1 , 0 ))}) = (𝑒 ∈ ({(1st𝑐)} × 𝑁) ↦ ((𝑎𝑐)(-g𝑅)if(𝑐𝑑, 1 , 0 )))
337336a1i 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 ))))
338128resmptd 5358 . . . . . . . . . . . . . . . . . . . . . . . . . . . 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))))
339161, 332, 335, 337, 338offval2 6789 . . . . . . . . . . . . . . . . . . . . . . . . . . 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)))))
340330, 183, 3393eqtr4d 2653 . . . . . . . . . . . . . . . . . . . . . . . . . 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))))
341 iffalse 4044 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (¬ (1st𝑒) = (1st𝑐) → if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )), 0 ), (𝑎𝑒)) = (𝑎𝑒))
342 iffalse 4044 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (¬ (1st𝑒) = (1st𝑐) → if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎𝑒)) = (𝑎𝑒))
343341, 342eqtr4d 2646 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)))
344193, 343syl 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)))
345344mpteq2dva 4666 . . . . . . . . . . . . . . . . . . . . . . . . . . 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))))
346 resmpt 5356 . . . . . . . . . . . . . . . . . . . . . . . . . . . 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))))
347198, 346mp1i 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))))
348345, 200, 3473eqtr4d 2653 . . . . . . . . . . . . . . . . . . . . . . . . . 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)))
349132, 134ifcld 4080 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) → if(𝑒 = 𝑐, 1 , 0 ) ∈ 𝐾)
350349adantr 479 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) ∧ 𝑒 ∈ (𝑁 × 𝑁)) → if(𝑒 = 𝑐, 1 , 0 ) ∈ 𝐾)
351350, 214ifcld 4080 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) ∧ 𝑒 ∈ (𝑁 × 𝑁)) → if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎𝑒)) ∈ 𝐾)
352 eqid 2609 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎𝑒))) = (𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎𝑒)))
353351, 352fmptd 6277 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) → (𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎𝑒))):(𝑁 × 𝑁)⟶𝐾)
354 elmapg 7734 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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))
355219, 221, 354sylancr 693 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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))
356353, 355mpbird 245 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) → (𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎𝑒))) ∈ (𝐾𝑚 (𝑁 × 𝑁)))
357356, 227eleqtrd 2689 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) → (𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎𝑒))) ∈ 𝐵)
358573ad2ant1 1074 . . . . . . . . . . . . . . . . . . . . . . . . . . . 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))))
359 reseq1 5298 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)))
360359eqeq1d 2611 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)))))
361 reseq1 5298 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)))
362361eqeq1d 2611 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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))))
363360, 362anbi12d 742 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)))))
364 fveq2 6088 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (𝑥 = (𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )), 0 ), (𝑎𝑒))) → (𝐷𝑥) = (𝐷‘(𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st𝑒) = (1st𝑐), if(𝑒 = 𝑐, ((𝑎𝑒)(-g𝑅)if(𝑒𝑑, 1 , 0 )), 0 ), (𝑎𝑒)))))
365364eqeq1d 2611 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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))))
366363, 365imbi12d 332 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)))))
3673662ralbidv 2971 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)))))
368 sneq 4134 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 (𝑦 = ((𝑎𝑐)(-g𝑅)if(𝑐𝑑, 1 , 0 )) → {𝑦} = {((𝑎𝑐)(-g𝑅)if(𝑐𝑑, 1 , 0 ))})
369368xpeq2d 5053 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (𝑦 = ((𝑎𝑐)(-g𝑅)if(𝑐𝑑, 1 , 0 )) → (({𝑤} × 𝑁) × {𝑦}) = (({𝑤} × 𝑁) × {((𝑎𝑐)(-g𝑅)if(𝑐𝑑, 1 , 0 ))}))
370369oveq1d 6542 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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))))
371370eqeq2d 2619 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)))))
372371anbi1d 736 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)))))
373 oveq1 6534 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (𝑦 = ((𝑎𝑐)(-g𝑅)if(𝑐𝑑, 1 , 0 )) → (𝑦 · (𝐷𝑧)) = (((𝑎𝑐)(-g𝑅)if(𝑐𝑑, 1 , 0 )) · (𝐷𝑧)))
374373eqeq2d 2619 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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))))
375372, 374imbi12d 332 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)))))
3763752ralbidv 2971 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)))))
377367, 376rspc2va 3293 . . . . . . . . . . . . . . . . . . . . . . . . . . . 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))))
378241, 304, 358, 377syl21anc 1316 . . . . . . . . . . . . . . . . . . . . . . . . . . 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))))
379 reseq1 5298 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)))
380379oveq2d 6543 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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))))
381380eqeq2d 2619 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)))))
382 reseq1 5298 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)))
383382eqeq2d 2619 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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))))
384381, 383anbi12d 742 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)))))
385 fveq2 6088 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)))))
386385oveq2d 6543 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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))))))
387386eqeq2d 2619 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)))))))
388384, 387imbi12d 332 . . . . . . . . . . . . . . . . . . . . . . . . . . . 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))))))))
389277xpeq1d 5052 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (𝑤 = (1st𝑐) → (({𝑤} × 𝑁) × {((𝑎𝑐)(-g𝑅)if(𝑐𝑑, 1 , 0 ))}) = (({(1st𝑐)} × 𝑁) × {((𝑎𝑐)(-g𝑅)if(𝑐𝑑, 1 , 0 ))}))
390277reseq2d 5304 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)))
391389, 390oveq12d 6545 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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))))
392279, 391eqeq12d 2624 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)))))
393284reseq2d 5304 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)))
394286, 393eqeq12d 2624 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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))))
395392, 394anbi12d 742 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)))))
396395imbi1d 329 . . . . . . . . . . . . . . . . . . . . . . . . . . . 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))))))))
397388, 396rspc2va 3293 . . . . . . . . . . . . . . . . . . . . . . . . . . 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)))))))
398357, 125, 378, 397syl21anc 1316 . . . . . . . . . . . . . . . . . . . . . . . . . 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)))))))
399340, 348, 398mp2and 710 . . . . . . . . . . . . . . . . . . . . . . . . 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))))))
400399oveq1d 6542 . . . . . . . . . . . . . . . . . . . . . . . 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))))))
401105, 106, 107, 400syl3anc 1317 . . . . . . . . . . . . . . . . . . . . . . 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))))))
402 simpl3 1058 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ (𝑏 ∪ {𝑐}) ∈ 𝑌) ∧ ((𝑎𝐵𝑑 ∈ (𝑁𝑚 𝑁)) ∧ ∀𝑤𝑏 (𝑎𝑤) = if(𝑤𝑑, 1 , 0 ))) → (𝑏 ∪ {𝑐}) ∈ 𝑌)
403 simprlr 798 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ (𝑏 ∪ {𝑐}) ∈ 𝑌) ∧ ((𝑎𝐵𝑑 ∈ (𝑁𝑚 𝑁)) ∧ ∀𝑤𝑏 (𝑎𝑤) = if(𝑤𝑑, 1 , 0 ))) → 𝑑 ∈ (𝑁𝑚 𝑁))
404 simprr 791 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ (𝑏 ∪ {𝑐}) ∈ 𝑌) ∧ ((𝑎𝐵𝑑 ∈ (𝑁𝑚 𝑁)) ∧ ∀𝑤𝑏 (𝑎𝑤) = if(𝑤𝑑, 1 , 0 ))) → ∀𝑤𝑏 (𝑎𝑤) = if(𝑤𝑑, 1 , 0 ))
405 ralss 3630 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (𝑏 ⊆ (𝑏 ∪ {𝑐}) → (∀𝑤𝑏 (𝑎𝑤) = if(𝑤𝑑, 1 , 0 ) ↔ ∀𝑤 ∈ (𝑏 ∪ {𝑐})(𝑤𝑏 → (𝑎𝑤) = if(𝑤𝑑, 1 , 0 ))))
406100, 405ax-mp 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (∀𝑤𝑏 (𝑎𝑤) = if(𝑤𝑑, 1 , 0 ) ↔ ∀𝑤 ∈ (𝑏 ∪ {𝑐})(𝑤𝑏 → (𝑎𝑤) = if(𝑤𝑑, 1 , 0 )))
407 iftrue 4041 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ((1st𝑤) = (1st𝑐) → if((1st𝑤) = (1st𝑐), if(𝑤 = 𝑐, 1 , 0 ), (𝑎𝑤)) = if(𝑤 = 𝑐, 1 , 0 ))
408407adantl 480 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 ))
409 ibar 523 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ((1st𝑤) = (1st𝑐) → ((2nd𝑤) = (2nd𝑐) ↔ ((1st𝑤) = (1st𝑐) ∧ (2nd𝑤) = (2nd𝑐))))
410409adantl 480 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 (((((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) ∧ 𝑑 ∈ (𝑁𝑚 𝑁)) ∧ 𝑤 ∈ (𝑏 ∪ {𝑐})) ∧ (1st𝑤) = (1st𝑐)) → ((2nd𝑤) = (2nd𝑐) ↔ ((1st𝑤) = (1st𝑐) ∧ (2nd𝑤) = (2nd𝑐))))
411 relxp 5139 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 Rel (𝑁 × 𝑁)
412 simpl2 1057 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) ∧ 𝑑 ∈ (𝑁𝑚 𝑁)) → (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁))
413412sselda 3567 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 ((((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) ∧ 𝑑 ∈ (𝑁𝑚 𝑁)) ∧ 𝑤 ∈ (𝑏 ∪ {𝑐})) → 𝑤 ∈ (𝑁 × 𝑁))
414413adantr 479 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 (((((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) ∧ 𝑑 ∈ (𝑁𝑚 𝑁)) ∧ 𝑤 ∈ (𝑏 ∪ {𝑐})) ∧ (1st𝑤) = (1st𝑐)) → 𝑤 ∈ (𝑁 × 𝑁))
415 1st2nd 7082 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 ((Rel (𝑁 × 𝑁) ∧ 𝑤 ∈ (𝑁 × 𝑁)) → 𝑤 = ⟨(1st𝑤), (2nd𝑤)⟩)
416411, 414, 415sylancr 693 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 (((((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) ∧ 𝑑 ∈ (𝑁𝑚 𝑁)) ∧ 𝑤 ∈ (𝑏 ∪ {𝑐})) ∧ (1st𝑤) = (1st𝑐)) → 𝑤 = ⟨(1st𝑤), (2nd𝑤)⟩)
417416eleq1d 2671 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)⟩)))
418 simpr 475 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) ∧ 𝑑 ∈ (𝑁𝑚 𝑁)) → 𝑑 ∈ (𝑁𝑚 𝑁))
419 elmapi 7742 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 (𝑑 ∈ (𝑁𝑚 𝑁) → 𝑑:𝑁𝑁)
420419adantl 480 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) ∧ 𝑑 ∈ (𝑁𝑚 𝑁)) → 𝑑:𝑁𝑁)
421125adantr 479 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) ∧ 𝑑 ∈ (𝑁𝑚 𝑁)) → (1st𝑐) ∈ 𝑁)
422 xp2nd 7067 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 (𝑐 ∈ (𝑁 × 𝑁) → (2nd𝑐) ∈ 𝑁)
423123, 422syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) → (2nd𝑐) ∈ 𝑁)
424423adantr 479 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) ∧ 𝑑 ∈ (𝑁𝑚 𝑁)) → (2nd𝑐) ∈ 𝑁)
425 fsets 15669 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 (((𝑑 ∈ (𝑁𝑚 𝑁) ∧ 𝑑:𝑁𝑁) ∧ (1st𝑐) ∈ 𝑁 ∧ (2nd𝑐) ∈ 𝑁) → (𝑑 sSet ⟨(1st𝑐), (2nd𝑐)⟩):𝑁𝑁)
426418, 420, 421, 424, 425syl211anc 1323 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) ∧ 𝑑 ∈ (𝑁𝑚 𝑁)) → (𝑑 sSet ⟨(1st𝑐), (2nd𝑐)⟩):𝑁𝑁)
427 ffn 5944 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 ((𝑑 sSet ⟨(1st𝑐), (2nd𝑐)⟩):𝑁𝑁 → (𝑑 sSet ⟨(1st𝑐), (2nd𝑐)⟩) Fn 𝑁)
428426, 427syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) ∧ 𝑑 ∈ (𝑁𝑚 𝑁)) → (𝑑 sSet ⟨(1st𝑐), (2nd𝑐)⟩) Fn 𝑁)
429428ad2antrr 757 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 (((((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) ∧ 𝑑 ∈ (𝑁𝑚 𝑁)) ∧ 𝑤 ∈ (𝑏 ∪ {𝑐})) ∧ (1st𝑤) = (1st𝑐)) → (𝑑 sSet ⟨(1st𝑐), (2nd𝑐)⟩) Fn 𝑁)
430 xp1st 7066 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 (𝑤 ∈ (𝑁 × 𝑁) → (1st𝑤) ∈ 𝑁)
431413, 430syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 ((((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) ∧ 𝑑 ∈ (𝑁𝑚 𝑁)) ∧ 𝑤 ∈ (𝑏 ∪ {𝑐})) → (1st𝑤) ∈ 𝑁)
432431adantr 479 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 (((((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) ∧ 𝑑 ∈ (𝑁𝑚 𝑁)) ∧ 𝑤 ∈ (𝑏 ∪ {𝑐})) ∧ (1st𝑤) = (1st𝑐)) → (1st𝑤) ∈ 𝑁)
433 fnopfvb 6132 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 (((𝑑 sSet ⟨(1st𝑐), (2nd𝑐)⟩) Fn 𝑁 ∧ (1st𝑤) ∈ 𝑁) → (((𝑑 sSet ⟨(1st𝑐), (2nd𝑐)⟩)‘(1st𝑤)) = (2nd𝑤) ↔ ⟨(1st𝑤), (2nd𝑤)⟩ ∈ (𝑑 sSet ⟨(1st𝑐), (2nd𝑐)⟩)))
434429, 432, 433syl2anc 690 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)⟩)))
435 fveq2 6088 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 ((1st𝑤) = (1st𝑐) → ((𝑑 sSet ⟨(1st𝑐), (2nd𝑐)⟩)‘(1st𝑤)) = ((𝑑 sSet ⟨(1st𝑐), (2nd𝑐)⟩)‘(1st𝑐)))
436435adantl 480 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)))
437 vex 3175 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 𝑑 ∈ V
438 fvex 6098 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 (1st𝑐) ∈ V
439 fvex 6098 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 (2nd𝑐) ∈ V
440 fvsetsid 15668 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 ((𝑑 ∈ V ∧ (1st𝑐) ∈ V ∧ (2nd𝑐) ∈ V) → ((𝑑 sSet ⟨(1st𝑐), (2nd𝑐)⟩)‘(1st𝑐)) = (2nd𝑐))
441437, 438, 439, 440mp3an 1415 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 ((𝑑 sSet ⟨(1st𝑐), (2nd𝑐)⟩)‘(1st𝑐)) = (2nd𝑐)
442436, 441syl6eq 2659 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 (((((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) ∧ 𝑑 ∈ (𝑁𝑚 𝑁)) ∧ 𝑤 ∈ (𝑏 ∪ {𝑐})) ∧ (1st𝑤) = (1st𝑐)) → ((𝑑 sSet ⟨(1st𝑐), (2nd𝑐)⟩)‘(1st𝑤)) = (2nd𝑐))
443442eqeq1d 2611 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 (((((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) ∧ 𝑑 ∈ (𝑁𝑚 𝑁)) ∧ 𝑤 ∈ (𝑏 ∪ {𝑐})) ∧ (1st𝑤) = (1st𝑐)) → (((𝑑 sSet ⟨(1st𝑐), (2nd𝑐)⟩)‘(1st𝑤)) = (2nd𝑤) ↔ (2nd𝑐) = (2nd𝑤)))
444 eqcom 2616 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 ((2nd𝑐) = (2nd𝑤) ↔ (2nd𝑤) = (2nd𝑐))
445443, 444syl6bb 274 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 (((((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) ∧ 𝑑 ∈ (𝑁𝑚 𝑁)) ∧ 𝑤 ∈ (𝑏 ∪ {𝑐})) ∧ (1st𝑤) = (1st𝑐)) → (((𝑑 sSet ⟨(1st𝑐), (2nd𝑐)⟩)‘(1st𝑤)) = (2nd𝑤) ↔ (2nd𝑤) = (2nd𝑐)))
446417, 434, 4453bitr2rd 295 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 (((((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) ∧ 𝑑 ∈ (𝑁𝑚 𝑁)) ∧ 𝑤 ∈ (𝑏 ∪ {𝑐})) ∧ (1st𝑤) = (1st𝑐)) → ((2nd𝑤) = (2nd𝑐) ↔ 𝑤 ∈ (𝑑 sSet ⟨(1st𝑐), (2nd𝑐)⟩)))
447123ad3antrrr 761 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 (((((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) ∧ 𝑑 ∈ (𝑁𝑚 𝑁)) ∧ 𝑤 ∈ (𝑏 ∪ {𝑐})) ∧ (1st𝑤) = (1st𝑐)) → 𝑐 ∈ (𝑁 × 𝑁))
448 xpopth 7075 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ((𝑤 ∈ (𝑁 × 𝑁) ∧ 𝑐 ∈ (𝑁 × 𝑁)) → (((1st𝑤) = (1st𝑐) ∧ (2nd𝑤) = (2nd𝑐)) ↔ 𝑤 = 𝑐))
449414, 447, 448syl2anc 690 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 (((((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) ∧ 𝑑 ∈ (𝑁𝑚 𝑁)) ∧ 𝑤 ∈ (𝑏 ∪ {𝑐})) ∧ (1st𝑤) = (1st𝑐)) → (((1st𝑤) = (1st𝑐) ∧ (2nd𝑤) = (2nd𝑐)) ↔ 𝑤 = 𝑐))
450410, 446, 4493bitr3rd 297 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 (((((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎𝐵) ∧ 𝑑 ∈ (𝑁𝑚 𝑁)) ∧ 𝑤 ∈ (𝑏 ∪ {𝑐})) ∧ (1st𝑤) = (1st𝑐)) → (𝑤 = 𝑐𝑤 ∈ (𝑑 sSet ⟨(1st𝑐), (2nd𝑐)⟩)))
451450ifbid 4057 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 ))
452408, 451eqtrd 2643 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 ))
453452a1d 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 )))
454 elsni 4141 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 (𝑤 ∈ {𝑐} → 𝑤 = 𝑐)
455454