Theorem mdetunilem9 22444

Description: Lemma for mdetuni 22446. (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 = (0g‘𝑅)
mdetuni.1r	⊢ 1 = (1r‘𝑅)
mdetuni.pg	⊢ + = (+g‘𝑅)
mdetuni.tg	⊢ · = (.r‘𝑅)
mdetuni.n	⊢ (𝜑 → 𝑁 ∈ Fin)
mdetuni.r	⊢ (𝜑 → 𝑅 ∈ Ring)
mdetuni.ff	⊢ (𝜑 → 𝐷:𝐵⟶𝐾)
mdetuni.al	⊢ (𝜑 → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝑁 ∀𝑧 ∈ 𝑁 ((𝑦 ≠ 𝑧 ∧ ∀𝑤 ∈ 𝑁 (𝑦𝑥𝑤) = (𝑧𝑥𝑤)) → (𝐷‘𝑥) = 0 ))
mdetuni.li	⊢ (𝜑 → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝑁 (((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))) → (𝐷‘𝑥) = ((𝐷‘𝑦) + (𝐷‘𝑧))))
mdetuni.sc	⊢ (𝜑 → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐾 ∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝑁 (((𝑥 ↾ ({𝑤} × 𝑁)) = ((({𝑤} × 𝑁) × {𝑦}) ∘f · (𝑧 ↾ ({𝑤} × 𝑁))) ∧ (𝑥 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑧 ↾ ((𝑁 ∖ {𝑤}) × 𝑁))) → (𝐷‘𝑥) = (𝑦 · (𝐷‘𝑧))))
mdetunilem9.id	⊢ (𝜑 → (𝐷‘(1r‘𝐴)) = 0 )
mdetunilem9.y	⊢ 𝑌 = {𝑥 ∣ ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ (𝑁 ↑m 𝑁)(∀𝑤 ∈ 𝑥 (𝑦‘𝑤) = if(𝑤 ∈ 𝑧, 1 , 0 ) → (𝐷‘𝑦) = 0 )}

Assertion

Ref	Expression
mdetunilem9	⊢ (𝜑 → 𝐷 = (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.

Step	Hyp	Ref	Expression
1		ral0 4504	. . . 4 ⊢ ∀𝑤 ∈ ∅ (𝑎‘𝑤) = if(𝑤 ∈ ( I ↾ 𝑁), 1 , 0 )
2		simpr 484	. . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐵) → 𝑎 ∈ 𝐵)
3		f1oi 6861	. . . . . . . 8 ⊢ ( I ↾ 𝑁):𝑁–1-1-onto→𝑁
4		f1of 6823	. . . . . . . 8 ⊢ (( I ↾ 𝑁):𝑁–1-1-onto→𝑁 → ( I ↾ 𝑁):𝑁⟶𝑁)
5	3, 4	mp1i 13	. . . . . . 7 ⊢ (𝜑 → ( I ↾ 𝑁):𝑁⟶𝑁)
6		mdetuni.n	. . . . . . . 8 ⊢ (𝜑 → 𝑁 ∈ Fin)
7	6, 6	elmapd 8830	. . . . . . 7 ⊢ (𝜑 → (( I ↾ 𝑁) ∈ (𝑁 ↑m 𝑁) ↔ ( I ↾ 𝑁):𝑁⟶𝑁))
8	5, 7	mpbird 257	. . . . . 6 ⊢ (𝜑 → ( I ↾ 𝑁) ∈ (𝑁 ↑m 𝑁))
9	8	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐵) → ( I ↾ 𝑁) ∈ (𝑁 ↑m 𝑁))
10		simplrl 774	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ (𝑁 ↑m 𝑁))) ∧ ∀𝑤 ∈ (𝑁 × 𝑁)(𝑦‘𝑤) = if(𝑤 ∈ 𝑧, 1 , 0 )) → 𝑦 ∈ 𝐵)
11		mdetuni.a	. . . . . . . . . . . . . . . . 17 ⊢ 𝐴 = (𝑁 Mat 𝑅)
12		mdetuni.k	. . . . . . . . . . . . . . . . 17 ⊢ 𝐾 = (Base‘𝑅)
13		mdetuni.b	. . . . . . . . . . . . . . . . 17 ⊢ 𝐵 = (Base‘𝐴)
14	11, 12, 13	matbas2i 22246	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 ∈ 𝐵 → 𝑦 ∈ (𝐾 ↑m (𝑁 × 𝑁)))
15		elmapi 8839	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 ∈ (𝐾 ↑m (𝑁 × 𝑁)) → 𝑦:(𝑁 × 𝑁)⟶𝐾)
16	14, 15	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝑦 ∈ 𝐵 → 𝑦:(𝑁 × 𝑁)⟶𝐾)
17	16	feqmptd 6950	. . . . . . . . . . . . . 14 ⊢ (𝑦 ∈ 𝐵 → 𝑦 = (𝑤 ∈ (𝑁 × 𝑁) ↦ (𝑦‘𝑤)))
18	17	fveq2d 6885	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ 𝐵 → (𝐷‘𝑦) = (𝐷‘(𝑤 ∈ (𝑁 × 𝑁) ↦ (𝑦‘𝑤))))
19	10, 18	syl 17	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ (𝑁 ↑m 𝑁))) ∧ ∀𝑤 ∈ (𝑁 × 𝑁)(𝑦‘𝑤) = if(𝑤 ∈ 𝑧, 1 , 0 )) → (𝐷‘𝑦) = (𝐷‘(𝑤 ∈ (𝑁 × 𝑁) ↦ (𝑦‘𝑤))))
20		eqid 2724	. . . . . . . . . . . . . 14 ⊢ (𝑁 × 𝑁) = (𝑁 × 𝑁)
21		mpteq12 5230	. . . . . . . . . . . . . . 15 ⊢ (((𝑁 × 𝑁) = (𝑁 × 𝑁) ∧ ∀𝑤 ∈ (𝑁 × 𝑁)(𝑦‘𝑤) = if(𝑤 ∈ 𝑧, 1 , 0 )) → (𝑤 ∈ (𝑁 × 𝑁) ↦ (𝑦‘𝑤)) = (𝑤 ∈ (𝑁 × 𝑁) ↦ if(𝑤 ∈ 𝑧, 1 , 0 )))
22	21	fveq2d 6885	. . . . . . . . . . . . . 14 ⊢ (((𝑁 × 𝑁) = (𝑁 × 𝑁) ∧ ∀𝑤 ∈ (𝑁 × 𝑁)(𝑦‘𝑤) = if(𝑤 ∈ 𝑧, 1 , 0 )) → (𝐷‘(𝑤 ∈ (𝑁 × 𝑁) ↦ (𝑦‘𝑤))) = (𝐷‘(𝑤 ∈ (𝑁 × 𝑁) ↦ if(𝑤 ∈ 𝑧, 1 , 0 ))))
23	20, 22	mpan 687	. . . . . . . . . . . . 13 ⊢ (∀𝑤 ∈ (𝑁 × 𝑁)(𝑦‘𝑤) = if(𝑤 ∈ 𝑧, 1 , 0 ) → (𝐷‘(𝑤 ∈ (𝑁 × 𝑁) ↦ (𝑦‘𝑤))) = (𝐷‘(𝑤 ∈ (𝑁 × 𝑁) ↦ if(𝑤 ∈ 𝑧, 1 , 0 ))))
24	23	adantl 481	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ (𝑁 ↑m 𝑁))) ∧ ∀𝑤 ∈ (𝑁 × 𝑁)(𝑦‘𝑤) = if(𝑤 ∈ 𝑧, 1 , 0 )) → (𝐷‘(𝑤 ∈ (𝑁 × 𝑁) ↦ (𝑦‘𝑤))) = (𝐷‘(𝑤 ∈ (𝑁 × 𝑁) ↦ if(𝑤 ∈ 𝑧, 1 , 0 ))))
25		eleq1 2813	. . . . . . . . . . . . . . . . 17 ⊢ (𝑎 = 𝑧 → (𝑎 ∈ (𝑁 ↑m 𝑁) ↔ 𝑧 ∈ (𝑁 ↑m 𝑁)))
26	25	anbi2d 628	. . . . . . . . . . . . . . . 16 ⊢ (𝑎 = 𝑧 → ((𝜑 ∧ 𝑎 ∈ (𝑁 ↑m 𝑁)) ↔ (𝜑 ∧ 𝑧 ∈ (𝑁 ↑m 𝑁))))
27		elequ2 2113	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑎 = 𝑧 → (𝑤 ∈ 𝑎 ↔ 𝑤 ∈ 𝑧))
28	27	ifbid 4543	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑎 = 𝑧 → if(𝑤 ∈ 𝑎, 1 , 0 ) = if(𝑤 ∈ 𝑧, 1 , 0 ))
29	28	mpteq2dv 5240	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑎 = 𝑧 → (𝑤 ∈ (𝑁 × 𝑁) ↦ if(𝑤 ∈ 𝑎, 1 , 0 )) = (𝑤 ∈ (𝑁 × 𝑁) ↦ if(𝑤 ∈ 𝑧, 1 , 0 )))
30	29	fveq2d 6885	. . . . . . . . . . . . . . . . 17 ⊢ (𝑎 = 𝑧 → (𝐷‘(𝑤 ∈ (𝑁 × 𝑁) ↦ if(𝑤 ∈ 𝑎, 1 , 0 ))) = (𝐷‘(𝑤 ∈ (𝑁 × 𝑁) ↦ if(𝑤 ∈ 𝑧, 1 , 0 ))))
31	30	eqeq1d 2726	. . . . . . . . . . . . . . . 16 ⊢ (𝑎 = 𝑧 → ((𝐷‘(𝑤 ∈ (𝑁 × 𝑁) ↦ if(𝑤 ∈ 𝑎, 1 , 0 ))) = 0 ↔ (𝐷‘(𝑤 ∈ (𝑁 × 𝑁) ↦ if(𝑤 ∈ 𝑧, 1 , 0 ))) = 0 ))
32	26, 31	imbi12d 344	. . . . . . . . . . . . . . 15 ⊢ (𝑎 = 𝑧 → (((𝜑 ∧ 𝑎 ∈ (𝑁 ↑m 𝑁)) → (𝐷‘(𝑤 ∈ (𝑁 × 𝑁) ↦ if(𝑤 ∈ 𝑎, 1 , 0 ))) = 0 ) ↔ ((𝜑 ∧ 𝑧 ∈ (𝑁 ↑m 𝑁)) → (𝐷‘(𝑤 ∈ (𝑁 × 𝑁) ↦ if(𝑤 ∈ 𝑧, 1 , 0 ))) = 0 )))
33		eleq1 2813	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑤 = ⟨𝑏, 𝑐⟩ → (𝑤 ∈ 𝑎 ↔ ⟨𝑏, 𝑐⟩ ∈ 𝑎))
34	33	ifbid 4543	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑤 = ⟨𝑏, 𝑐⟩ → if(𝑤 ∈ 𝑎, 1 , 0 ) = if(⟨𝑏, 𝑐⟩ ∈ 𝑎, 1 , 0 ))
35	34	mpompt 7514	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑤 ∈ (𝑁 × 𝑁) ↦ if(𝑤 ∈ 𝑎, 1 , 0 )) = (𝑏 ∈ 𝑁, 𝑐 ∈ 𝑁 ↦ if(⟨𝑏, 𝑐⟩ ∈ 𝑎, 1 , 0 ))
36		elmapi 8839	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑎 ∈ (𝑁 ↑m 𝑁) → 𝑎:𝑁⟶𝑁)
37	36	adantl 481	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ 𝑎 ∈ (𝑁 ↑m 𝑁)) → 𝑎:𝑁⟶𝑁)
38	37	ffnd 6708	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ 𝑎 ∈ (𝑁 ↑m 𝑁)) → 𝑎 Fn 𝑁)
39	38	3ad2ant1 1130	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ 𝑎 ∈ (𝑁 ↑m 𝑁)) ∧ 𝑏 ∈ 𝑁 ∧ 𝑐 ∈ 𝑁) → 𝑎 Fn 𝑁)
40		simp2 1134	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ 𝑎 ∈ (𝑁 ↑m 𝑁)) ∧ 𝑏 ∈ 𝑁 ∧ 𝑐 ∈ 𝑁) → 𝑏 ∈ 𝑁)
41		fnopfvb 6935	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑎 Fn 𝑁 ∧ 𝑏 ∈ 𝑁) → ((𝑎‘𝑏) = 𝑐 ↔ ⟨𝑏, 𝑐⟩ ∈ 𝑎))
42	39, 40, 41	syl2anc 583	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑎 ∈ (𝑁 ↑m 𝑁)) ∧ 𝑏 ∈ 𝑁 ∧ 𝑐 ∈ 𝑁) → ((𝑎‘𝑏) = 𝑐 ↔ ⟨𝑏, 𝑐⟩ ∈ 𝑎))
43	42	bicomd 222	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑎 ∈ (𝑁 ↑m 𝑁)) ∧ 𝑏 ∈ 𝑁 ∧ 𝑐 ∈ 𝑁) → (⟨𝑏, 𝑐⟩ ∈ 𝑎 ↔ (𝑎‘𝑏) = 𝑐))
44	43	ifbid 4543	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑎 ∈ (𝑁 ↑m 𝑁)) ∧ 𝑏 ∈ 𝑁 ∧ 𝑐 ∈ 𝑁) → if(⟨𝑏, 𝑐⟩ ∈ 𝑎, 1 , 0 ) = if((𝑎‘𝑏) = 𝑐, 1 , 0 ))
45	44	mpoeq3dva 7478	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑎 ∈ (𝑁 ↑m 𝑁)) → (𝑏 ∈ 𝑁, 𝑐 ∈ 𝑁 ↦ if(⟨𝑏, 𝑐⟩ ∈ 𝑎, 1 , 0 )) = (𝑏 ∈ 𝑁, 𝑐 ∈ 𝑁 ↦ if((𝑎‘𝑏) = 𝑐, 1 , 0 )))
46	35, 45	eqtrid 2776	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑎 ∈ (𝑁 ↑m 𝑁)) → (𝑤 ∈ (𝑁 × 𝑁) ↦ if(𝑤 ∈ 𝑎, 1 , 0 )) = (𝑏 ∈ 𝑁, 𝑐 ∈ 𝑁 ↦ if((𝑎‘𝑏) = 𝑐, 1 , 0 )))
47	46	fveq2d 6885	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑎 ∈ (𝑁 ↑m 𝑁)) → (𝐷‘(𝑤 ∈ (𝑁 × 𝑁) ↦ if(𝑤 ∈ 𝑎, 1 , 0 ))) = (𝐷‘(𝑏 ∈ 𝑁, 𝑐 ∈ 𝑁 ↦ if((𝑎‘𝑏) = 𝑐, 1 , 0 ))))
48		mdetuni.0g	. . . . . . . . . . . . . . . . . 18 ⊢ 0 = (0g‘𝑅)
49		mdetuni.1r	. . . . . . . . . . . . . . . . . 18 ⊢ 1 = (1r‘𝑅)
50		mdetuni.pg	. . . . . . . . . . . . . . . . . 18 ⊢ + = (+g‘𝑅)
51		mdetuni.tg	. . . . . . . . . . . . . . . . . 18 ⊢ · = (.r‘𝑅)
52		mdetuni.r	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝑅 ∈ Ring)
53		mdetuni.ff	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝐷:𝐵⟶𝐾)
54		mdetuni.al	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝑁 ∀𝑧 ∈ 𝑁 ((𝑦 ≠ 𝑧 ∧ ∀𝑤 ∈ 𝑁 (𝑦𝑥𝑤) = (𝑧𝑥𝑤)) → (𝐷‘𝑥) = 0 ))
55		mdetuni.li	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝑁 (((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))) → (𝐷‘𝑥) = ((𝐷‘𝑦) + (𝐷‘𝑧))))
56		mdetuni.sc	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐾 ∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝑁 (((𝑥 ↾ ({𝑤} × 𝑁)) = ((({𝑤} × 𝑁) × {𝑦}) ∘f · (𝑧 ↾ ({𝑤} × 𝑁))) ∧ (𝑥 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑧 ↾ ((𝑁 ∖ {𝑤}) × 𝑁))) → (𝐷‘𝑥) = (𝑦 · (𝐷‘𝑧))))
57		mdetunilem9.id	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (𝐷‘(1r‘𝐴)) = 0 )
58	11, 13, 12, 48, 49, 50, 51, 6, 52, 53, 54, 55, 56, 57	mdetunilem8 22443	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑎:𝑁⟶𝑁) → (𝐷‘(𝑏 ∈ 𝑁, 𝑐 ∈ 𝑁 ↦ if((𝑎‘𝑏) = 𝑐, 1 , 0 ))) = 0 )
59	36, 58	sylan2 592	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑎 ∈ (𝑁 ↑m 𝑁)) → (𝐷‘(𝑏 ∈ 𝑁, 𝑐 ∈ 𝑁 ↦ if((𝑎‘𝑏) = 𝑐, 1 , 0 ))) = 0 )
60	47, 59	eqtrd 2764	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑎 ∈ (𝑁 ↑m 𝑁)) → (𝐷‘(𝑤 ∈ (𝑁 × 𝑁) ↦ if(𝑤 ∈ 𝑎, 1 , 0 ))) = 0 )
61	32, 60	chvarvv 1994	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝑁 ↑m 𝑁)) → (𝐷‘(𝑤 ∈ (𝑁 × 𝑁) ↦ if(𝑤 ∈ 𝑧, 1 , 0 ))) = 0 )
62	61	adantrl 713	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ (𝑁 ↑m 𝑁))) → (𝐷‘(𝑤 ∈ (𝑁 × 𝑁) ↦ if(𝑤 ∈ 𝑧, 1 , 0 ))) = 0 )
63	62	adantr 480	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ (𝑁 ↑m 𝑁))) ∧ ∀𝑤 ∈ (𝑁 × 𝑁)(𝑦‘𝑤) = if(𝑤 ∈ 𝑧, 1 , 0 )) → (𝐷‘(𝑤 ∈ (𝑁 × 𝑁) ↦ if(𝑤 ∈ 𝑧, 1 , 0 ))) = 0 )
64	19, 24, 63	3eqtrd 2768	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ (𝑁 ↑m 𝑁))) ∧ ∀𝑤 ∈ (𝑁 × 𝑁)(𝑦‘𝑤) = if(𝑤 ∈ 𝑧, 1 , 0 )) → (𝐷‘𝑦) = 0 )
65	64	ex 412	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ (𝑁 ↑m 𝑁))) → (∀𝑤 ∈ (𝑁 × 𝑁)(𝑦‘𝑤) = if(𝑤 ∈ 𝑧, 1 , 0 ) → (𝐷‘𝑦) = 0 ))
66	65	ralrimivva 3192	. . . . . . . . 9 ⊢ (𝜑 → ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ (𝑁 ↑m 𝑁)(∀𝑤 ∈ (𝑁 × 𝑁)(𝑦‘𝑤) = if(𝑤 ∈ 𝑧, 1 , 0 ) → (𝐷‘𝑦) = 0 ))
67		xpfi 9313	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ Fin ∧ 𝑁 ∈ Fin) → (𝑁 × 𝑁) ∈ Fin)
68	6, 6, 67	syl2anc 583	. . . . . . . . . 10 ⊢ (𝜑 → (𝑁 × 𝑁) ∈ Fin)
69		raleq 3314	. . . . . . . . . . . . 13 ⊢ (𝑥 = (𝑁 × 𝑁) → (∀𝑤 ∈ 𝑥 (𝑦‘𝑤) = if(𝑤 ∈ 𝑧, 1 , 0 ) ↔ ∀𝑤 ∈ (𝑁 × 𝑁)(𝑦‘𝑤) = if(𝑤 ∈ 𝑧, 1 , 0 )))
70	69	imbi1d 341	. . . . . . . . . . . 12 ⊢ (𝑥 = (𝑁 × 𝑁) → ((∀𝑤 ∈ 𝑥 (𝑦‘𝑤) = if(𝑤 ∈ 𝑧, 1 , 0 ) → (𝐷‘𝑦) = 0 ) ↔ (∀𝑤 ∈ (𝑁 × 𝑁)(𝑦‘𝑤) = if(𝑤 ∈ 𝑧, 1 , 0 ) → (𝐷‘𝑦) = 0 )))
71	70	2ralbidv 3210	. . . . . . . . . . 11 ⊢ (𝑥 = (𝑁 × 𝑁) → (∀𝑦 ∈ 𝐵 ∀𝑧 ∈ (𝑁 ↑m 𝑁)(∀𝑤 ∈ 𝑥 (𝑦‘𝑤) = if(𝑤 ∈ 𝑧, 1 , 0 ) → (𝐷‘𝑦) = 0 ) ↔ ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ (𝑁 ↑m 𝑁)(∀𝑤 ∈ (𝑁 × 𝑁)(𝑦‘𝑤) = if(𝑤 ∈ 𝑧, 1 , 0 ) → (𝐷‘𝑦) = 0 )))
72		mdetunilem9.y	. . . . . . . . . . 11 ⊢ 𝑌 = {𝑥 ∣ ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ (𝑁 ↑m 𝑁)(∀𝑤 ∈ 𝑥 (𝑦‘𝑤) = if(𝑤 ∈ 𝑧, 1 , 0 ) → (𝐷‘𝑦) = 0 )}
73	71, 72	elab2g 3662	. . . . . . . . . 10 ⊢ ((𝑁 × 𝑁) ∈ Fin → ((𝑁 × 𝑁) ∈ 𝑌 ↔ ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ (𝑁 ↑m 𝑁)(∀𝑤 ∈ (𝑁 × 𝑁)(𝑦‘𝑤) = if(𝑤 ∈ 𝑧, 1 , 0 ) → (𝐷‘𝑦) = 0 )))
74	68, 73	syl 17	. . . . . . . . 9 ⊢ (𝜑 → ((𝑁 × 𝑁) ∈ 𝑌 ↔ ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ (𝑁 ↑m 𝑁)(∀𝑤 ∈ (𝑁 × 𝑁)(𝑦‘𝑤) = if(𝑤 ∈ 𝑧, 1 , 0 ) → (𝐷‘𝑦) = 0 )))
75	66, 74	mpbird 257	. . . . . . . 8 ⊢ (𝜑 → (𝑁 × 𝑁) ∈ 𝑌)
76		ssid 3996	. . . . . . . . 9 ⊢ (𝑁 × 𝑁) ⊆ (𝑁 × 𝑁)
77	68	3ad2ant1 1130	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑁 × 𝑁) ⊆ (𝑁 × 𝑁) ∧ ¬ ∅ ∈ 𝑌) → (𝑁 × 𝑁) ∈ Fin)
78		sseq1 3999	. . . . . . . . . . . . . 14 ⊢ (𝑎 = ∅ → (𝑎 ⊆ (𝑁 × 𝑁) ↔ ∅ ⊆ (𝑁 × 𝑁)))
79	78	3anbi2d 1437	. . . . . . . . . . . . 13 ⊢ (𝑎 = ∅ → ((𝜑 ∧ 𝑎 ⊆ (𝑁 × 𝑁) ∧ ¬ ∅ ∈ 𝑌) ↔ (𝜑 ∧ ∅ ⊆ (𝑁 × 𝑁) ∧ ¬ ∅ ∈ 𝑌)))
80		eleq1 2813	. . . . . . . . . . . . . 14 ⊢ (𝑎 = ∅ → (𝑎 ∈ 𝑌 ↔ ∅ ∈ 𝑌))
81	80	notbid 318	. . . . . . . . . . . . 13 ⊢ (𝑎 = ∅ → (¬ 𝑎 ∈ 𝑌 ↔ ¬ ∅ ∈ 𝑌))
82	79, 81	imbi12d 344	. . . . . . . . . . . 12 ⊢ (𝑎 = ∅ → (((𝜑 ∧ 𝑎 ⊆ (𝑁 × 𝑁) ∧ ¬ ∅ ∈ 𝑌) → ¬ 𝑎 ∈ 𝑌) ↔ ((𝜑 ∧ ∅ ⊆ (𝑁 × 𝑁) ∧ ¬ ∅ ∈ 𝑌) → ¬ ∅ ∈ 𝑌)))
83		sseq1 3999	. . . . . . . . . . . . . 14 ⊢ (𝑎 = 𝑏 → (𝑎 ⊆ (𝑁 × 𝑁) ↔ 𝑏 ⊆ (𝑁 × 𝑁)))
84	83	3anbi2d 1437	. . . . . . . . . . . . 13 ⊢ (𝑎 = 𝑏 → ((𝜑 ∧ 𝑎 ⊆ (𝑁 × 𝑁) ∧ ¬ ∅ ∈ 𝑌) ↔ (𝜑 ∧ 𝑏 ⊆ (𝑁 × 𝑁) ∧ ¬ ∅ ∈ 𝑌)))
85		eleq1 2813	. . . . . . . . . . . . . 14 ⊢ (𝑎 = 𝑏 → (𝑎 ∈ 𝑌 ↔ 𝑏 ∈ 𝑌))
86	85	notbid 318	. . . . . . . . . . . . 13 ⊢ (𝑎 = 𝑏 → (¬ 𝑎 ∈ 𝑌 ↔ ¬ 𝑏 ∈ 𝑌))
87	84, 86	imbi12d 344	. . . . . . . . . . . 12 ⊢ (𝑎 = 𝑏 → (((𝜑 ∧ 𝑎 ⊆ (𝑁 × 𝑁) ∧ ¬ ∅ ∈ 𝑌) → ¬ 𝑎 ∈ 𝑌) ↔ ((𝜑 ∧ 𝑏 ⊆ (𝑁 × 𝑁) ∧ ¬ ∅ ∈ 𝑌) → ¬ 𝑏 ∈ 𝑌)))
88		sseq1 3999	. . . . . . . . . . . . . 14 ⊢ (𝑎 = (𝑏 ∪ {𝑐}) → (𝑎 ⊆ (𝑁 × 𝑁) ↔ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁)))
89	88	3anbi2d 1437	. . . . . . . . . . . . 13 ⊢ (𝑎 = (𝑏 ∪ {𝑐}) → ((𝜑 ∧ 𝑎 ⊆ (𝑁 × 𝑁) ∧ ¬ ∅ ∈ 𝑌) ↔ (𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ ¬ ∅ ∈ 𝑌)))
90		eleq1 2813	. . . . . . . . . . . . . 14 ⊢ (𝑎 = (𝑏 ∪ {𝑐}) → (𝑎 ∈ 𝑌 ↔ (𝑏 ∪ {𝑐}) ∈ 𝑌))
91	90	notbid 318	. . . . . . . . . . . . 13 ⊢ (𝑎 = (𝑏 ∪ {𝑐}) → (¬ 𝑎 ∈ 𝑌 ↔ ¬ (𝑏 ∪ {𝑐}) ∈ 𝑌))
92	89, 91	imbi12d 344	. . . . . . . . . . . 12 ⊢ (𝑎 = (𝑏 ∪ {𝑐}) → (((𝜑 ∧ 𝑎 ⊆ (𝑁 × 𝑁) ∧ ¬ ∅ ∈ 𝑌) → ¬ 𝑎 ∈ 𝑌) ↔ ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ ¬ ∅ ∈ 𝑌) → ¬ (𝑏 ∪ {𝑐}) ∈ 𝑌)))
93		sseq1 3999	. . . . . . . . . . . . . 14 ⊢ (𝑎 = (𝑁 × 𝑁) → (𝑎 ⊆ (𝑁 × 𝑁) ↔ (𝑁 × 𝑁) ⊆ (𝑁 × 𝑁)))
94	93	3anbi2d 1437	. . . . . . . . . . . . 13 ⊢ (𝑎 = (𝑁 × 𝑁) → ((𝜑 ∧ 𝑎 ⊆ (𝑁 × 𝑁) ∧ ¬ ∅ ∈ 𝑌) ↔ (𝜑 ∧ (𝑁 × 𝑁) ⊆ (𝑁 × 𝑁) ∧ ¬ ∅ ∈ 𝑌)))
95		eleq1 2813	. . . . . . . . . . . . . 14 ⊢ (𝑎 = (𝑁 × 𝑁) → (𝑎 ∈ 𝑌 ↔ (𝑁 × 𝑁) ∈ 𝑌))
96	95	notbid 318	. . . . . . . . . . . . 13 ⊢ (𝑎 = (𝑁 × 𝑁) → (¬ 𝑎 ∈ 𝑌 ↔ ¬ (𝑁 × 𝑁) ∈ 𝑌))
97	94, 96	imbi12d 344	. . . . . . . . . . . 12 ⊢ (𝑎 = (𝑁 × 𝑁) → (((𝜑 ∧ 𝑎 ⊆ (𝑁 × 𝑁) ∧ ¬ ∅ ∈ 𝑌) → ¬ 𝑎 ∈ 𝑌) ↔ ((𝜑 ∧ (𝑁 × 𝑁) ⊆ (𝑁 × 𝑁) ∧ ¬ ∅ ∈ 𝑌) → ¬ (𝑁 × 𝑁) ∈ 𝑌)))
98		simp3 1135	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ ∅ ⊆ (𝑁 × 𝑁) ∧ ¬ ∅ ∈ 𝑌) → ¬ ∅ ∈ 𝑌)
99		ssun1 4164	. . . . . . . . . . . . . . . 16 ⊢ 𝑏 ⊆ (𝑏 ∪ {𝑐})
100		sstr2 3981	. . . . . . . . . . . . . . . 16 ⊢ (𝑏 ⊆ (𝑏 ∪ {𝑐}) → ((𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) → 𝑏 ⊆ (𝑁 × 𝑁)))
101	99, 100	ax-mp 5	. . . . . . . . . . . . . . 15 ⊢ ((𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) → 𝑏 ⊆ (𝑁 × 𝑁))
102	101	3anim2i 1150	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ ¬ ∅ ∈ 𝑌) → (𝜑 ∧ 𝑏 ⊆ (𝑁 × 𝑁) ∧ ¬ ∅ ∈ 𝑌))
103	102	imim1i 63	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑏 ⊆ (𝑁 × 𝑁) ∧ ¬ ∅ ∈ 𝑌) → ¬ 𝑏 ∈ 𝑌) → ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ ¬ ∅ ∈ 𝑌) → ¬ 𝑏 ∈ 𝑌))
104		simpl1 1188	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ (𝑏 ∪ {𝑐}) ∈ 𝑌) ∧ ((𝑎 ∈ 𝐵 ∧ 𝑑 ∈ (𝑁 ↑m 𝑁)) ∧ ∀𝑤 ∈ 𝑏 (𝑎‘𝑤) = if(𝑤 ∈ 𝑑, 1 , 0 ))) → 𝜑)
105		simpl2 1189	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ (𝑏 ∪ {𝑐}) ∈ 𝑌) ∧ ((𝑎 ∈ 𝐵 ∧ 𝑑 ∈ (𝑁 ↑m 𝑁)) ∧ ∀𝑤 ∈ 𝑏 (𝑎‘𝑤) = if(𝑤 ∈ 𝑑, 1 , 0 ))) → (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁))
106		simprll 776	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ (𝑏 ∪ {𝑐}) ∈ 𝑌) ∧ ((𝑎 ∈ 𝐵 ∧ 𝑑 ∈ (𝑁 ↑m 𝑁)) ∧ ∀𝑤 ∈ 𝑏 (𝑎‘𝑤) = if(𝑤 ∈ 𝑑, 1 , 0 ))) → 𝑎 ∈ 𝐵)
107	11, 12, 13	matbas2i 22246	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑎 ∈ 𝐵 → 𝑎 ∈ (𝐾 ↑m (𝑁 × 𝑁)))
108		elmapi 8839	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑎 ∈ (𝐾 ↑m (𝑁 × 𝑁)) → 𝑎:(𝑁 × 𝑁)⟶𝐾)
109	107, 108	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑎 ∈ 𝐵 → 𝑎:(𝑁 × 𝑁)⟶𝐾)
110	109	3ad2ant3 1132	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) → 𝑎:(𝑁 × 𝑁)⟶𝐾)
111	110	feqmptd 6950	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) → 𝑎 = (𝑒 ∈ (𝑁 × 𝑁) ↦ (𝑎‘𝑒)))
112	111	reseq1d 5970	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) → (𝑎 ↾ ({(1st ‘𝑐)} × 𝑁)) = ((𝑒 ∈ (𝑁 × 𝑁) ↦ (𝑎‘𝑒)) ↾ ({(1st ‘𝑐)} × 𝑁)))
113	52	3ad2ant1 1130	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) → 𝑅 ∈ Ring)
114		ringgrp 20133	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ (𝑅 ∈ Ring → 𝑅 ∈ Grp)
115	113, 114	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) → 𝑅 ∈ Grp)
116	115	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) ∧ 𝑒 ∈ ({(1st ‘𝑐)} × 𝑁)) → 𝑅 ∈ Grp)
117	110	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) ∧ 𝑒 ∈ ({(1st ‘𝑐)} × 𝑁)) → 𝑎:(𝑁 × 𝑁)⟶𝐾)
118		simp2 1134	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 ⊢ ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) → (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁))
119	118	unssbd 4180	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 ⊢ ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) → {𝑐} ⊆ (𝑁 × 𝑁))
120		vex 3470	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 ⊢ 𝑐 ∈ V
121	120	snss 4781	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 ⊢ (𝑐 ∈ (𝑁 × 𝑁) ↔ {𝑐} ⊆ (𝑁 × 𝑁))
122	119, 121	sylibr 233	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ⊢ ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) → 𝑐 ∈ (𝑁 × 𝑁))
123		xp1st 8000	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ⊢ (𝑐 ∈ (𝑁 × 𝑁) → (1st ‘𝑐) ∈ 𝑁)
124	122, 123	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ⊢ ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) → (1st ‘𝑐) ∈ 𝑁)
125	124	snssd 4804	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) → {(1st ‘𝑐)} ⊆ 𝑁)
126		xpss1 5685	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ ({(1st ‘𝑐)} ⊆ 𝑁 → ({(1st ‘𝑐)} × 𝑁) ⊆ (𝑁 × 𝑁))
127	125, 126	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) → ({(1st ‘𝑐)} × 𝑁) ⊆ (𝑁 × 𝑁))
128	127	sselda 3974	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) ∧ 𝑒 ∈ ({(1st ‘𝑐)} × 𝑁)) → 𝑒 ∈ (𝑁 × 𝑁))
129	117, 128	ffvelcdmd 7077	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) ∧ 𝑒 ∈ ({(1st ‘𝑐)} × 𝑁)) → (𝑎‘𝑒) ∈ 𝐾)
130	12, 49	ringidcl 20155	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ (𝑅 ∈ Ring → 1 ∈ 𝐾)
131	113, 130	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) → 1 ∈ 𝐾)
132	12, 48	ring0cl 20156	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ (𝑅 ∈ Ring → 0 ∈ 𝐾)
133	113, 132	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) → 0 ∈ 𝐾)
134	131, 133	ifcld 4566	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) → if(𝑒 ∈ 𝑑, 1 , 0 ) ∈ 𝐾)
135	134	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) ∧ 𝑒 ∈ ({(1st ‘𝑐)} × 𝑁)) → if(𝑒 ∈ 𝑑, 1 , 0 ) ∈ 𝐾)
136		eqid 2724	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ (-g‘𝑅) = (-g‘𝑅)
137	12, 50, 136	grpnpcan 18950	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ ((𝑅 ∈ Grp ∧ (𝑎‘𝑒) ∈ 𝐾 ∧ if(𝑒 ∈ 𝑑, 1 , 0 ) ∈ 𝐾) → (((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )) + if(𝑒 ∈ 𝑑, 1 , 0 )) = (𝑎‘𝑒))
138	116, 129, 135, 137	syl3anc 1368	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) ∧ 𝑒 ∈ ({(1st ‘𝑐)} × 𝑁)) → (((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )) + if(𝑒 ∈ 𝑑, 1 , 0 )) = (𝑎‘𝑒))
139	138	eqcomd 2730	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) ∧ 𝑒 ∈ ({(1st ‘𝑐)} × 𝑁)) → (𝑎‘𝑒) = (((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )) + if(𝑒 ∈ 𝑑, 1 , 0 )))
140	139	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) ∧ 𝑒 ∈ ({(1st ‘𝑐)} × 𝑁)) ∧ 𝑒 = 𝑐) → (𝑎‘𝑒) = (((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )) + if(𝑒 ∈ 𝑑, 1 , 0 )))
141		iftrue 4526	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (𝑒 = 𝑐 → if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ) = ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )))
142		iftrue 4526	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (𝑒 = 𝑐 → if(𝑒 = 𝑐, if(𝑒 ∈ 𝑑, 1 , 0 ), (𝑎‘𝑒)) = if(𝑒 ∈ 𝑑, 1 , 0 ))
143	141, 142	oveq12d 7419	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝑒 = 𝑐 → (if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ) + if(𝑒 = 𝑐, if(𝑒 ∈ 𝑑, 1 , 0 ), (𝑎‘𝑒))) = (((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )) + if(𝑒 ∈ 𝑑, 1 , 0 )))
144	143	adantl 481	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) ∧ 𝑒 ∈ ({(1st ‘𝑐)} × 𝑁)) ∧ 𝑒 = 𝑐) → (if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ) + if(𝑒 = 𝑐, if(𝑒 ∈ 𝑑, 1 , 0 ), (𝑎‘𝑒))) = (((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )) + if(𝑒 ∈ 𝑑, 1 , 0 )))
145	140, 144	eqtr4d 2767	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) ∧ 𝑒 ∈ ({(1st ‘𝑐)} × 𝑁)) ∧ 𝑒 = 𝑐) → (𝑎‘𝑒) = (if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ) + if(𝑒 = 𝑐, if(𝑒 ∈ 𝑑, 1 , 0 ), (𝑎‘𝑒))))
146	12, 50, 48	grplid 18887	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ ((𝑅 ∈ Grp ∧ (𝑎‘𝑒) ∈ 𝐾) → ( 0 + (𝑎‘𝑒)) = (𝑎‘𝑒))
147	116, 129, 146	syl2anc 583	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) ∧ 𝑒 ∈ ({(1st ‘𝑐)} × 𝑁)) → ( 0 + (𝑎‘𝑒)) = (𝑎‘𝑒))
148	147	eqcomd 2730	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) ∧ 𝑒 ∈ ({(1st ‘𝑐)} × 𝑁)) → (𝑎‘𝑒) = ( 0 + (𝑎‘𝑒)))
149	148	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) ∧ 𝑒 ∈ ({(1st ‘𝑐)} × 𝑁)) ∧ ¬ 𝑒 = 𝑐) → (𝑎‘𝑒) = ( 0 + (𝑎‘𝑒)))
150		iffalse 4529	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (¬ 𝑒 = 𝑐 → if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ) = 0 )
151		iffalse 4529	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (¬ 𝑒 = 𝑐 → if(𝑒 = 𝑐, if(𝑒 ∈ 𝑑, 1 , 0 ), (𝑎‘𝑒)) = (𝑎‘𝑒))
152	150, 151	oveq12d 7419	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (¬ 𝑒 = 𝑐 → (if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ) + if(𝑒 = 𝑐, if(𝑒 ∈ 𝑑, 1 , 0 ), (𝑎‘𝑒))) = ( 0 + (𝑎‘𝑒)))
153	152	adantl 481	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) ∧ 𝑒 ∈ ({(1st ‘𝑐)} × 𝑁)) ∧ ¬ 𝑒 = 𝑐) → (if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ) + if(𝑒 = 𝑐, if(𝑒 ∈ 𝑑, 1 , 0 ), (𝑎‘𝑒))) = ( 0 + (𝑎‘𝑒)))
154	149, 153	eqtr4d 2767	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) ∧ 𝑒 ∈ ({(1st ‘𝑐)} × 𝑁)) ∧ ¬ 𝑒 = 𝑐) → (𝑎‘𝑒) = (if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ) + if(𝑒 = 𝑐, if(𝑒 ∈ 𝑑, 1 , 0 ), (𝑎‘𝑒))))
155	145, 154	pm2.61dan 810	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) ∧ 𝑒 ∈ ({(1st ‘𝑐)} × 𝑁)) → (𝑎‘𝑒) = (if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ) + if(𝑒 = 𝑐, if(𝑒 ∈ 𝑑, 1 , 0 ), (𝑎‘𝑒))))
156	155	mpteq2dva 5238	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) → (𝑒 ∈ ({(1st ‘𝑐)} × 𝑁) ↦ (𝑎‘𝑒)) = (𝑒 ∈ ({(1st ‘𝑐)} × 𝑁) ↦ (if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ) + if(𝑒 = 𝑐, if(𝑒 ∈ 𝑑, 1 , 0 ), (𝑎‘𝑒)))))
157		snfi 9040	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ {(1st ‘𝑐)} ∈ Fin
158	6	3ad2ant1 1130	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) → 𝑁 ∈ Fin)
159		xpfi 9313	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (({(1st ‘𝑐)} ∈ Fin ∧ 𝑁 ∈ Fin) → ({(1st ‘𝑐)} × 𝑁) ∈ Fin)
160	157, 158, 159	sylancr 586	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) → ({(1st ‘𝑐)} × 𝑁) ∈ Fin)
161		ovex 7434	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )) ∈ V
162	48	fvexi 6895	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ 0 ∈ V
163	161, 162	ifex 4570	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ) ∈ V
164	163	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) ∧ 𝑒 ∈ ({(1st ‘𝑐)} × 𝑁)) → if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ) ∈ V)
165	49	fvexi 6895	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ 1 ∈ V
166	165, 162	ifex 4570	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ if(𝑒 ∈ 𝑑, 1 , 0 ) ∈ V
167		fvex 6894	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝑎‘𝑒) ∈ V
168	166, 167	ifex 4570	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ if(𝑒 = 𝑐, if(𝑒 ∈ 𝑑, 1 , 0 ), (𝑎‘𝑒)) ∈ V
169	168	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) ∧ 𝑒 ∈ ({(1st ‘𝑐)} × 𝑁)) → if(𝑒 = 𝑐, if(𝑒 ∈ 𝑑, 1 , 0 ), (𝑎‘𝑒)) ∈ V)
170		xp1st 8000	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝑒 ∈ ({(1st ‘𝑐)} × 𝑁) → (1st ‘𝑒) ∈ {(1st ‘𝑐)})
171		elsni 4637	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((1st ‘𝑒) ∈ {(1st ‘𝑐)} → (1st ‘𝑒) = (1st ‘𝑐))
172		iftrue 4526	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((1st ‘𝑒) = (1st ‘𝑐) → if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒)) = if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ))
173	170, 171, 172	3syl 18	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝑒 ∈ ({(1st ‘𝑐)} × 𝑁) → if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒)) = if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ))
174	173	mpteq2ia 5241	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑒 ∈ ({(1st ‘𝑐)} × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) = (𝑒 ∈ ({(1st ‘𝑐)} × 𝑁) ↦ if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ))
175	174	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) → (𝑒 ∈ ({(1st ‘𝑐)} × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) = (𝑒 ∈ ({(1st ‘𝑐)} × 𝑁) ↦ if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 )))
176		eqidd 2725	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) → (𝑒 ∈ ({(1st ‘𝑐)} × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒 ∈ 𝑑, 1 , 0 ), (𝑎‘𝑒))) = (𝑒 ∈ ({(1st ‘𝑐)} × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒 ∈ 𝑑, 1 , 0 ), (𝑎‘𝑒))))
177	160, 164, 169, 175, 176	offval2 7683	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) → ((𝑒 ∈ ({(1st ‘𝑐)} × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ∘f + (𝑒 ∈ ({(1st ‘𝑐)} × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒 ∈ 𝑑, 1 , 0 ), (𝑎‘𝑒)))) = (𝑒 ∈ ({(1st ‘𝑐)} × 𝑁) ↦ (if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ) + if(𝑒 = 𝑐, if(𝑒 ∈ 𝑑, 1 , 0 ), (𝑎‘𝑒)))))
178	156, 177	eqtr4d 2767	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) → (𝑒 ∈ ({(1st ‘𝑐)} × 𝑁) ↦ (𝑎‘𝑒)) = ((𝑒 ∈ ({(1st ‘𝑐)} × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ∘f + (𝑒 ∈ ({(1st ‘𝑐)} × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒 ∈ 𝑑, 1 , 0 ), (𝑎‘𝑒)))))
179	127	resmptd 6030	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) → ((𝑒 ∈ (𝑁 × 𝑁) ↦ (𝑎‘𝑒)) ↾ ({(1st ‘𝑐)} × 𝑁)) = (𝑒 ∈ ({(1st ‘𝑐)} × 𝑁) ↦ (𝑎‘𝑒)))
180	127	resmptd 6030	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) → ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ↾ ({(1st ‘𝑐)} × 𝑁)) = (𝑒 ∈ ({(1st ‘𝑐)} × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))))
181	127	resmptd 6030	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) → ((𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒 ∈ 𝑑, 1 , 0 ), (𝑎‘𝑒))) ↾ ({(1st ‘𝑐)} × 𝑁)) = (𝑒 ∈ ({(1st ‘𝑐)} × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒 ∈ 𝑑, 1 , 0 ), (𝑎‘𝑒))))
182	180, 181	oveq12d 7419	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) → (((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ↾ ({(1st ‘𝑐)} × 𝑁)) ∘f + ((𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒 ∈ 𝑑, 1 , 0 ), (𝑎‘𝑒))) ↾ ({(1st ‘𝑐)} × 𝑁))) = ((𝑒 ∈ ({(1st ‘𝑐)} × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ∘f + (𝑒 ∈ ({(1st ‘𝑐)} × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒 ∈ 𝑑, 1 , 0 ), (𝑎‘𝑒)))))
183	178, 179, 182	3eqtr4d 2774	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) → ((𝑒 ∈ (𝑁 × 𝑁) ↦ (𝑎‘𝑒)) ↾ ({(1st ‘𝑐)} × 𝑁)) = (((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ↾ ({(1st ‘𝑐)} × 𝑁)) ∘f + ((𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒 ∈ 𝑑, 1 , 0 ), (𝑎‘𝑒))) ↾ ({(1st ‘𝑐)} × 𝑁))))
184	112, 183	eqtrd 2764	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) → (𝑎 ↾ ({(1st ‘𝑐)} × 𝑁)) = (((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ↾ ({(1st ‘𝑐)} × 𝑁)) ∘f + ((𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒 ∈ 𝑑, 1 , 0 ), (𝑎‘𝑒))) ↾ ({(1st ‘𝑐)} × 𝑁))))
185	111	reseq1d 5970	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) → (𝑎 ↾ ((𝑁 ∖ {(1st ‘𝑐)}) × 𝑁)) = ((𝑒 ∈ (𝑁 × 𝑁) ↦ (𝑎‘𝑒)) ↾ ((𝑁 ∖ {(1st ‘𝑐)}) × 𝑁)))
186		xp1st 8000	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝑒 ∈ ((𝑁 ∖ {(1st ‘𝑐)}) × 𝑁) → (1st ‘𝑒) ∈ (𝑁 ∖ {(1st ‘𝑐)}))
187		eldifsni 4785	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((1st ‘𝑒) ∈ (𝑁 ∖ {(1st ‘𝑐)}) → (1st ‘𝑒) ≠ (1st ‘𝑐))
188	186, 187	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝑒 ∈ ((𝑁 ∖ {(1st ‘𝑐)}) × 𝑁) → (1st ‘𝑒) ≠ (1st ‘𝑐))
189	188	neneqd 2937	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑒 ∈ ((𝑁 ∖ {(1st ‘𝑐)}) × 𝑁) → ¬ (1st ‘𝑒) = (1st ‘𝑐))
190	189	adantl 481	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) ∧ 𝑒 ∈ ((𝑁 ∖ {(1st ‘𝑐)}) × 𝑁)) → ¬ (1st ‘𝑒) = (1st ‘𝑐))
191	190	iffalsed 4531	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) ∧ 𝑒 ∈ ((𝑁 ∖ {(1st ‘𝑐)}) × 𝑁)) → if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒)) = (𝑎‘𝑒))
192	191	mpteq2dva 5238	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) → (𝑒 ∈ ((𝑁 ∖ {(1st ‘𝑐)}) × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) = (𝑒 ∈ ((𝑁 ∖ {(1st ‘𝑐)}) × 𝑁) ↦ (𝑎‘𝑒)))
193		difss 4123	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑁 ∖ {(1st ‘𝑐)}) ⊆ 𝑁
194		xpss1 5685	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((𝑁 ∖ {(1st ‘𝑐)}) ⊆ 𝑁 → ((𝑁 ∖ {(1st ‘𝑐)}) × 𝑁) ⊆ (𝑁 × 𝑁))
195	193, 194	ax-mp 5	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝑁 ∖ {(1st ‘𝑐)}) × 𝑁) ⊆ (𝑁 × 𝑁)
196		resmpt 6027	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (((𝑁 ∖ {(1st ‘𝑐)}) × 𝑁) ⊆ (𝑁 × 𝑁) → ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ↾ ((𝑁 ∖ {(1st ‘𝑐)}) × 𝑁)) = (𝑒 ∈ ((𝑁 ∖ {(1st ‘𝑐)}) × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))))
197	195, 196	mp1i 13	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) → ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ↾ ((𝑁 ∖ {(1st ‘𝑐)}) × 𝑁)) = (𝑒 ∈ ((𝑁 ∖ {(1st ‘𝑐)}) × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))))
198		resmpt 6027	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (((𝑁 ∖ {(1st ‘𝑐)}) × 𝑁) ⊆ (𝑁 × 𝑁) → ((𝑒 ∈ (𝑁 × 𝑁) ↦ (𝑎‘𝑒)) ↾ ((𝑁 ∖ {(1st ‘𝑐)}) × 𝑁)) = (𝑒 ∈ ((𝑁 ∖ {(1st ‘𝑐)}) × 𝑁) ↦ (𝑎‘𝑒)))
199	195, 198	mp1i 13	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) → ((𝑒 ∈ (𝑁 × 𝑁) ↦ (𝑎‘𝑒)) ↾ ((𝑁 ∖ {(1st ‘𝑐)}) × 𝑁)) = (𝑒 ∈ ((𝑁 ∖ {(1st ‘𝑐)}) × 𝑁) ↦ (𝑎‘𝑒)))
200	192, 197, 199	3eqtr4rd 2775	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) → ((𝑒 ∈ (𝑁 × 𝑁) ↦ (𝑎‘𝑒)) ↾ ((𝑁 ∖ {(1st ‘𝑐)}) × 𝑁)) = ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ↾ ((𝑁 ∖ {(1st ‘𝑐)}) × 𝑁)))
201	185, 200	eqtrd 2764	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) → (𝑎 ↾ ((𝑁 ∖ {(1st ‘𝑐)}) × 𝑁)) = ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ↾ ((𝑁 ∖ {(1st ‘𝑐)}) × 𝑁)))
202		fveq2 6881	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑒 = 𝑐 → (1st ‘𝑒) = (1st ‘𝑐))
203	190, 202	nsyl 140	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) ∧ 𝑒 ∈ ((𝑁 ∖ {(1st ‘𝑐)}) × 𝑁)) → ¬ 𝑒 = 𝑐)
204	203	iffalsed 4531	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) ∧ 𝑒 ∈ ((𝑁 ∖ {(1st ‘𝑐)}) × 𝑁)) → if(𝑒 = 𝑐, if(𝑒 ∈ 𝑑, 1 , 0 ), (𝑎‘𝑒)) = (𝑎‘𝑒))
205	204	mpteq2dva 5238	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) → (𝑒 ∈ ((𝑁 ∖ {(1st ‘𝑐)}) × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒 ∈ 𝑑, 1 , 0 ), (𝑎‘𝑒))) = (𝑒 ∈ ((𝑁 ∖ {(1st ‘𝑐)}) × 𝑁) ↦ (𝑎‘𝑒)))
206		resmpt 6027	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (((𝑁 ∖ {(1st ‘𝑐)}) × 𝑁) ⊆ (𝑁 × 𝑁) → ((𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒 ∈ 𝑑, 1 , 0 ), (𝑎‘𝑒))) ↾ ((𝑁 ∖ {(1st ‘𝑐)}) × 𝑁)) = (𝑒 ∈ ((𝑁 ∖ {(1st ‘𝑐)}) × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒 ∈ 𝑑, 1 , 0 ), (𝑎‘𝑒))))
207	195, 206	mp1i 13	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) → ((𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒 ∈ 𝑑, 1 , 0 ), (𝑎‘𝑒))) ↾ ((𝑁 ∖ {(1st ‘𝑐)}) × 𝑁)) = (𝑒 ∈ ((𝑁 ∖ {(1st ‘𝑐)}) × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒 ∈ 𝑑, 1 , 0 ), (𝑎‘𝑒))))
208	205, 207, 199	3eqtr4rd 2775	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) → ((𝑒 ∈ (𝑁 × 𝑁) ↦ (𝑎‘𝑒)) ↾ ((𝑁 ∖ {(1st ‘𝑐)}) × 𝑁)) = ((𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒 ∈ 𝑑, 1 , 0 ), (𝑎‘𝑒))) ↾ ((𝑁 ∖ {(1st ‘𝑐)}) × 𝑁)))
209	185, 208	eqtrd 2764	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) → (𝑎 ↾ ((𝑁 ∖ {(1st ‘𝑐)}) × 𝑁)) = ((𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒 ∈ 𝑑, 1 , 0 ), (𝑎‘𝑒))) ↾ ((𝑁 ∖ {(1st ‘𝑐)}) × 𝑁)))
210	134	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) ∧ 𝑒 ∈ (𝑁 × 𝑁)) → if(𝑒 ∈ 𝑑, 1 , 0 ) ∈ 𝐾)
211	110	ffvelcdmda 7076	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) ∧ 𝑒 ∈ (𝑁 × 𝑁)) → (𝑎‘𝑒) ∈ 𝐾)
212	210, 211	ifcld 4566	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) ∧ 𝑒 ∈ (𝑁 × 𝑁)) → if(𝑒 = 𝑐, if(𝑒 ∈ 𝑑, 1 , 0 ), (𝑎‘𝑒)) ∈ 𝐾)
213	212	fmpttd 7106	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) → (𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒 ∈ 𝑑, 1 , 0 ), (𝑎‘𝑒))):(𝑁 × 𝑁)⟶𝐾)
214	12	fvexi 6895	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ 𝐾 ∈ V
215	67	anidms 566	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑁 ∈ Fin → (𝑁 × 𝑁) ∈ Fin)
216	158, 215	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) → (𝑁 × 𝑁) ∈ Fin)
217		elmapg 8829	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((𝐾 ∈ V ∧ (𝑁 × 𝑁) ∈ Fin) → ((𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒 ∈ 𝑑, 1 , 0 ), (𝑎‘𝑒))) ∈ (𝐾 ↑m (𝑁 × 𝑁)) ↔ (𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒 ∈ 𝑑, 1 , 0 ), (𝑎‘𝑒))):(𝑁 × 𝑁)⟶𝐾))
218	214, 216, 217	sylancr 586	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) → ((𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒 ∈ 𝑑, 1 , 0 ), (𝑎‘𝑒))) ∈ (𝐾 ↑m (𝑁 × 𝑁)) ↔ (𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒 ∈ 𝑑, 1 , 0 ), (𝑎‘𝑒))):(𝑁 × 𝑁)⟶𝐾))
219	213, 218	mpbird 257	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) → (𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒 ∈ 𝑑, 1 , 0 ), (𝑎‘𝑒))) ∈ (𝐾 ↑m (𝑁 × 𝑁)))
220	11, 12	matbas2 22245	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (𝐾 ↑m (𝑁 × 𝑁)) = (Base‘𝐴))
221	158, 113, 220	syl2anc 583	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) → (𝐾 ↑m (𝑁 × 𝑁)) = (Base‘𝐴))
222	221, 13	eqtr4di 2782	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) → (𝐾 ↑m (𝑁 × 𝑁)) = 𝐵)
223	219, 222	eleqtrd 2827	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) → (𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒 ∈ 𝑑, 1 , 0 ), (𝑎‘𝑒))) ∈ 𝐵)
224		simp3 1135	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) → 𝑎 ∈ 𝐵)
225	115	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) ∧ 𝑒 ∈ (𝑁 × 𝑁)) → 𝑅 ∈ Grp)
226	12, 136	grpsubcl 18938	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ ((𝑅 ∈ Grp ∧ (𝑎‘𝑒) ∈ 𝐾 ∧ if(𝑒 ∈ 𝑑, 1 , 0 ) ∈ 𝐾) → ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )) ∈ 𝐾)
227	225, 211, 210, 226	syl3anc 1368	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) ∧ 𝑒 ∈ (𝑁 × 𝑁)) → ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )) ∈ 𝐾)
228	133	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) ∧ 𝑒 ∈ (𝑁 × 𝑁)) → 0 ∈ 𝐾)
229	227, 228	ifcld 4566	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) ∧ 𝑒 ∈ (𝑁 × 𝑁)) → if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ) ∈ 𝐾)
230	229, 211	ifcld 4566	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) ∧ 𝑒 ∈ (𝑁 × 𝑁)) → if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒)) ∈ 𝐾)
231	230	fmpttd 7106	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) → (𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))):(𝑁 × 𝑁)⟶𝐾)
232		elmapg 8829	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((𝐾 ∈ V ∧ (𝑁 × 𝑁) ∈ Fin) → ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ∈ (𝐾 ↑m (𝑁 × 𝑁)) ↔ (𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))):(𝑁 × 𝑁)⟶𝐾))
233	214, 216, 232	sylancr 586	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) → ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ∈ (𝐾 ↑m (𝑁 × 𝑁)) ↔ (𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))):(𝑁 × 𝑁)⟶𝐾))
234	231, 233	mpbird 257	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) → (𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ∈ (𝐾 ↑m (𝑁 × 𝑁)))
235	234, 222	eleqtrd 2827	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) → (𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ∈ 𝐵)
236	55	3ad2ant1 1130	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝑁 (((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))) → (𝐷‘𝑥) = ((𝐷‘𝑦) + (𝐷‘𝑧))))
237		reseq1 5965	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝑥 = 𝑎 → (𝑥 ↾ ({𝑤} × 𝑁)) = (𝑎 ↾ ({𝑤} × 𝑁)))
238	237	eqeq1d 2726	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝑥 = 𝑎 → ((𝑥 ↾ ({𝑤} × 𝑁)) = ((𝑦 ↾ ({𝑤} × 𝑁)) ∘f + (𝑧 ↾ ({𝑤} × 𝑁))) ↔ (𝑎 ↾ ({𝑤} × 𝑁)) = ((𝑦 ↾ ({𝑤} × 𝑁)) ∘f + (𝑧 ↾ ({𝑤} × 𝑁)))))
239		reseq1 5965	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝑥 = 𝑎 → (𝑥 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑎 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)))
240	239	eqeq1d 2726	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝑥 = 𝑎 → ((𝑥 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑦 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) ↔ (𝑎 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑦 ↾ ((𝑁 ∖ {𝑤}) × 𝑁))))
241	239	eqeq1d 2726	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝑥 = 𝑎 → ((𝑥 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑧 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) ↔ (𝑎 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑧 ↾ ((𝑁 ∖ {𝑤}) × 𝑁))))
242	238, 240, 241	3anbi123d 1432	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑥 = 𝑎 → (((𝑥 ↾ ({𝑤} × 𝑁)) = ((𝑦 ↾ ({𝑤} × 𝑁)) ∘f + (𝑧 ↾ ({𝑤} × 𝑁))) ∧ (𝑥 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑦 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) ∧ (𝑥 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑧 ↾ ((𝑁 ∖ {𝑤}) × 𝑁))) ↔ ((𝑎 ↾ ({𝑤} × 𝑁)) = ((𝑦 ↾ ({𝑤} × 𝑁)) ∘f + (𝑧 ↾ ({𝑤} × 𝑁))) ∧ (𝑎 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑦 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) ∧ (𝑎 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑧 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)))))
243		fveqeq2 6890	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑥 = 𝑎 → ((𝐷‘𝑥) = ((𝐷‘𝑦) + (𝐷‘𝑧)) ↔ (𝐷‘𝑎) = ((𝐷‘𝑦) + (𝐷‘𝑧))))
244	242, 243	imbi12d 344	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 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))) → (𝐷‘𝑥) = ((𝐷‘𝑦) + (𝐷‘𝑧))) ↔ (((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))) → (𝐷‘𝑎) = ((𝐷‘𝑦) + (𝐷‘𝑧)))))
245	244	2ralbidv 3210	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑥 = 𝑎 → (∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝑁 (((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))) → (𝐷‘𝑥) = ((𝐷‘𝑦) + (𝐷‘𝑧))) ↔ ∀𝑧 ∈ 𝐵 ∀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))) → (𝐷‘𝑎) = ((𝐷‘𝑦) + (𝐷‘𝑧)))))
246		reseq1 5965	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (𝑦 = (𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) → (𝑦 ↾ ({𝑤} × 𝑁)) = ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ↾ ({𝑤} × 𝑁)))
247	246	oveq1d 7416	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝑦 = (𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) → ((𝑦 ↾ ({𝑤} × 𝑁)) ∘f + (𝑧 ↾ ({𝑤} × 𝑁))) = (((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ↾ ({𝑤} × 𝑁)) ∘f + (𝑧 ↾ ({𝑤} × 𝑁))))
248	247	eqeq2d 2735	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝑦 = (𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) → ((𝑎 ↾ ({𝑤} × 𝑁)) = ((𝑦 ↾ ({𝑤} × 𝑁)) ∘f + (𝑧 ↾ ({𝑤} × 𝑁))) ↔ (𝑎 ↾ ({𝑤} × 𝑁)) = (((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ↾ ({𝑤} × 𝑁)) ∘f + (𝑧 ↾ ({𝑤} × 𝑁)))))
249		reseq1 5965	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝑦 = (𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) → (𝑦 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ↾ ((𝑁 ∖ {𝑤}) × 𝑁)))
250	249	eqeq2d 2735	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝑦 = (𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) → ((𝑎 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑦 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) ↔ (𝑎 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ↾ ((𝑁 ∖ {𝑤}) × 𝑁))))
251	248, 250	3anbi12d 1433	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑦 = (𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) → (((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((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ↾ ({𝑤} × 𝑁)) ∘f + (𝑧 ↾ ({𝑤} × 𝑁))) ∧ (𝑎 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) ∧ (𝑎 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑧 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)))))
252		fveq2 6881	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝑦 = (𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) → (𝐷‘𝑦) = (𝐷‘(𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒)))))
253	252	oveq1d 7416	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝑦 = (𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) → ((𝐷‘𝑦) + (𝐷‘𝑧)) = ((𝐷‘(𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒)))) + (𝐷‘𝑧)))
254	253	eqeq2d 2735	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑦 = (𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) → ((𝐷‘𝑎) = ((𝐷‘𝑦) + (𝐷‘𝑧)) ↔ (𝐷‘𝑎) = ((𝐷‘(𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒)))) + (𝐷‘𝑧))))
255	251, 254	imbi12d 344	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑦 = (𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) → ((((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))) → (𝐷‘𝑎) = ((𝐷‘𝑦) + (𝐷‘𝑧))) ↔ (((𝑎 ↾ ({𝑤} × 𝑁)) = (((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ↾ ({𝑤} × 𝑁)) ∘f + (𝑧 ↾ ({𝑤} × 𝑁))) ∧ (𝑎 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) ∧ (𝑎 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑧 ↾ ((𝑁 ∖ {𝑤}) × 𝑁))) → (𝐷‘𝑎) = ((𝐷‘(𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒)))) + (𝐷‘𝑧)))))
256	255	2ralbidv 3210	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑦 = (𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) → (∀𝑧 ∈ 𝐵 ∀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))) → (𝐷‘𝑎) = ((𝐷‘𝑦) + (𝐷‘𝑧))) ↔ ∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝑁 (((𝑎 ↾ ({𝑤} × 𝑁)) = (((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ↾ ({𝑤} × 𝑁)) ∘f + (𝑧 ↾ ({𝑤} × 𝑁))) ∧ (𝑎 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) ∧ (𝑎 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑧 ↾ ((𝑁 ∖ {𝑤}) × 𝑁))) → (𝐷‘𝑎) = ((𝐷‘(𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒)))) + (𝐷‘𝑧)))))
257	245, 256	rspc2va 3615	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((𝑎 ∈ 𝐵 ∧ (𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ∈ 𝐵) ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝑁 (((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))) → (𝐷‘𝑥) = ((𝐷‘𝑦) + (𝐷‘𝑧)))) → ∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝑁 (((𝑎 ↾ ({𝑤} × 𝑁)) = (((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ↾ ({𝑤} × 𝑁)) ∘f + (𝑧 ↾ ({𝑤} × 𝑁))) ∧ (𝑎 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) ∧ (𝑎 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑧 ↾ ((𝑁 ∖ {𝑤}) × 𝑁))) → (𝐷‘𝑎) = ((𝐷‘(𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒)))) + (𝐷‘𝑧))))
258	224, 235, 236, 257	syl21anc 835	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) → ∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝑁 (((𝑎 ↾ ({𝑤} × 𝑁)) = (((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ↾ ({𝑤} × 𝑁)) ∘f + (𝑧 ↾ ({𝑤} × 𝑁))) ∧ (𝑎 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) ∧ (𝑎 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑧 ↾ ((𝑁 ∖ {𝑤}) × 𝑁))) → (𝐷‘𝑎) = ((𝐷‘(𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒)))) + (𝐷‘𝑧))))
259		reseq1 5965	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝑧 = (𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒 ∈ 𝑑, 1 , 0 ), (𝑎‘𝑒))) → (𝑧 ↾ ({𝑤} × 𝑁)) = ((𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒 ∈ 𝑑, 1 , 0 ), (𝑎‘𝑒))) ↾ ({𝑤} × 𝑁)))
260	259	oveq2d 7417	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑧 = (𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒 ∈ 𝑑, 1 , 0 ), (𝑎‘𝑒))) → (((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ↾ ({𝑤} × 𝑁)) ∘f + (𝑧 ↾ ({𝑤} × 𝑁))) = (((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ↾ ({𝑤} × 𝑁)) ∘f + ((𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒 ∈ 𝑑, 1 , 0 ), (𝑎‘𝑒))) ↾ ({𝑤} × 𝑁))))
261	260	eqeq2d 2735	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑧 = (𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒 ∈ 𝑑, 1 , 0 ), (𝑎‘𝑒))) → ((𝑎 ↾ ({𝑤} × 𝑁)) = (((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ↾ ({𝑤} × 𝑁)) ∘f + (𝑧 ↾ ({𝑤} × 𝑁))) ↔ (𝑎 ↾ ({𝑤} × 𝑁)) = (((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ↾ ({𝑤} × 𝑁)) ∘f + ((𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒 ∈ 𝑑, 1 , 0 ), (𝑎‘𝑒))) ↾ ({𝑤} × 𝑁)))))
262		reseq1 5965	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑧 = (𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒 ∈ 𝑑, 1 , 0 ), (𝑎‘𝑒))) → (𝑧 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = ((𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒 ∈ 𝑑, 1 , 0 ), (𝑎‘𝑒))) ↾ ((𝑁 ∖ {𝑤}) × 𝑁)))
263	262	eqeq2d 2735	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑧 = (𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒 ∈ 𝑑, 1 , 0 ), (𝑎‘𝑒))) → ((𝑎 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑧 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) ↔ (𝑎 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = ((𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒 ∈ 𝑑, 1 , 0 ), (𝑎‘𝑒))) ↾ ((𝑁 ∖ {𝑤}) × 𝑁))))
264	261, 263	3anbi13d 1434	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑧 = (𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒 ∈ 𝑑, 1 , 0 ), (𝑎‘𝑒))) → (((𝑎 ↾ ({𝑤} × 𝑁)) = (((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ↾ ({𝑤} × 𝑁)) ∘f + (𝑧 ↾ ({𝑤} × 𝑁))) ∧ (𝑎 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) ∧ (𝑎 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑧 ↾ ((𝑁 ∖ {𝑤}) × 𝑁))) ↔ ((𝑎 ↾ ({𝑤} × 𝑁)) = (((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ↾ ({𝑤} × 𝑁)) ∘f + ((𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒 ∈ 𝑑, 1 , 0 ), (𝑎‘𝑒))) ↾ ({𝑤} × 𝑁))) ∧ (𝑎 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) ∧ (𝑎 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = ((𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒 ∈ 𝑑, 1 , 0 ), (𝑎‘𝑒))) ↾ ((𝑁 ∖ {𝑤}) × 𝑁)))))
265		fveq2 6881	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑧 = (𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒 ∈ 𝑑, 1 , 0 ), (𝑎‘𝑒))) → (𝐷‘𝑧) = (𝐷‘(𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒 ∈ 𝑑, 1 , 0 ), (𝑎‘𝑒)))))
266	265	oveq2d 7417	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑧 = (𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒 ∈ 𝑑, 1 , 0 ), (𝑎‘𝑒))) → ((𝐷‘(𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒)))) + (𝐷‘𝑧)) = ((𝐷‘(𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒)))) + (𝐷‘(𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒 ∈ 𝑑, 1 , 0 ), (𝑎‘𝑒))))))
267	266	eqeq2d 2735	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑧 = (𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒 ∈ 𝑑, 1 , 0 ), (𝑎‘𝑒))) → ((𝐷‘𝑎) = ((𝐷‘(𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒)))) + (𝐷‘𝑧)) ↔ (𝐷‘𝑎) = ((𝐷‘(𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒)))) + (𝐷‘(𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒 ∈ 𝑑, 1 , 0 ), (𝑎‘𝑒)))))))
268	264, 267	imbi12d 344	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑧 = (𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒 ∈ 𝑑, 1 , 0 ), (𝑎‘𝑒))) → ((((𝑎 ↾ ({𝑤} × 𝑁)) = (((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ↾ ({𝑤} × 𝑁)) ∘f + (𝑧 ↾ ({𝑤} × 𝑁))) ∧ (𝑎 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) ∧ (𝑎 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑧 ↾ ((𝑁 ∖ {𝑤}) × 𝑁))) → (𝐷‘𝑎) = ((𝐷‘(𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒)))) + (𝐷‘𝑧))) ↔ (((𝑎 ↾ ({𝑤} × 𝑁)) = (((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ↾ ({𝑤} × 𝑁)) ∘f + ((𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒 ∈ 𝑑, 1 , 0 ), (𝑎‘𝑒))) ↾ ({𝑤} × 𝑁))) ∧ (𝑎 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) ∧ (𝑎 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = ((𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒 ∈ 𝑑, 1 , 0 ), (𝑎‘𝑒))) ↾ ((𝑁 ∖ {𝑤}) × 𝑁))) → (𝐷‘𝑎) = ((𝐷‘(𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒)))) + (𝐷‘(𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒 ∈ 𝑑, 1 , 0 ), (𝑎‘𝑒))))))))
269		sneq 4630	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝑤 = (1st ‘𝑐) → {𝑤} = {(1st ‘𝑐)})
270	269	xpeq1d 5695	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝑤 = (1st ‘𝑐) → ({𝑤} × 𝑁) = ({(1st ‘𝑐)} × 𝑁))
271	270	reseq2d 5971	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑤 = (1st ‘𝑐) → (𝑎 ↾ ({𝑤} × 𝑁)) = (𝑎 ↾ ({(1st ‘𝑐)} × 𝑁)))
272	270	reseq2d 5971	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝑤 = (1st ‘𝑐) → ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ↾ ({𝑤} × 𝑁)) = ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ↾ ({(1st ‘𝑐)} × 𝑁)))
273	270	reseq2d 5971	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝑤 = (1st ‘𝑐) → ((𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒 ∈ 𝑑, 1 , 0 ), (𝑎‘𝑒))) ↾ ({𝑤} × 𝑁)) = ((𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒 ∈ 𝑑, 1 , 0 ), (𝑎‘𝑒))) ↾ ({(1st ‘𝑐)} × 𝑁)))
274	272, 273	oveq12d 7419	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑤 = (1st ‘𝑐) → (((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ↾ ({𝑤} × 𝑁)) ∘f + ((𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒 ∈ 𝑑, 1 , 0 ), (𝑎‘𝑒))) ↾ ({𝑤} × 𝑁))) = (((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ↾ ({(1st ‘𝑐)} × 𝑁)) ∘f + ((𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒 ∈ 𝑑, 1 , 0 ), (𝑎‘𝑒))) ↾ ({(1st ‘𝑐)} × 𝑁))))
275	271, 274	eqeq12d 2740	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑤 = (1st ‘𝑐) → ((𝑎 ↾ ({𝑤} × 𝑁)) = (((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ↾ ({𝑤} × 𝑁)) ∘f + ((𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒 ∈ 𝑑, 1 , 0 ), (𝑎‘𝑒))) ↾ ({𝑤} × 𝑁))) ↔ (𝑎 ↾ ({(1st ‘𝑐)} × 𝑁)) = (((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ↾ ({(1st ‘𝑐)} × 𝑁)) ∘f + ((𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒 ∈ 𝑑, 1 , 0 ), (𝑎‘𝑒))) ↾ ({(1st ‘𝑐)} × 𝑁)))))
276	269	difeq2d 4114	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝑤 = (1st ‘𝑐) → (𝑁 ∖ {𝑤}) = (𝑁 ∖ {(1st ‘𝑐)}))
277	276	xpeq1d 5695	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝑤 = (1st ‘𝑐) → ((𝑁 ∖ {𝑤}) × 𝑁) = ((𝑁 ∖ {(1st ‘𝑐)}) × 𝑁))
278	277	reseq2d 5971	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑤 = (1st ‘𝑐) → (𝑎 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑎 ↾ ((𝑁 ∖ {(1st ‘𝑐)}) × 𝑁)))
279	277	reseq2d 5971	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑤 = (1st ‘𝑐) → ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ↾ ((𝑁 ∖ {(1st ‘𝑐)}) × 𝑁)))
280	278, 279	eqeq12d 2740	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑤 = (1st ‘𝑐) → ((𝑎 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) ↔ (𝑎 ↾ ((𝑁 ∖ {(1st ‘𝑐)}) × 𝑁)) = ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ↾ ((𝑁 ∖ {(1st ‘𝑐)}) × 𝑁))))
281	277	reseq2d 5971	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑤 = (1st ‘𝑐) → ((𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒 ∈ 𝑑, 1 , 0 ), (𝑎‘𝑒))) ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = ((𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒 ∈ 𝑑, 1 , 0 ), (𝑎‘𝑒))) ↾ ((𝑁 ∖ {(1st ‘𝑐)}) × 𝑁)))
282	278, 281	eqeq12d 2740	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑤 = (1st ‘𝑐) → ((𝑎 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = ((𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒 ∈ 𝑑, 1 , 0 ), (𝑎‘𝑒))) ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) ↔ (𝑎 ↾ ((𝑁 ∖ {(1st ‘𝑐)}) × 𝑁)) = ((𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒 ∈ 𝑑, 1 , 0 ), (𝑎‘𝑒))) ↾ ((𝑁 ∖ {(1st ‘𝑐)}) × 𝑁))))
283	275, 280, 282	3anbi123d 1432	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑤 = (1st ‘𝑐) → (((𝑎 ↾ ({𝑤} × 𝑁)) = (((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ↾ ({𝑤} × 𝑁)) ∘f + ((𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒 ∈ 𝑑, 1 , 0 ), (𝑎‘𝑒))) ↾ ({𝑤} × 𝑁))) ∧ (𝑎 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) ∧ (𝑎 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = ((𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒 ∈ 𝑑, 1 , 0 ), (𝑎‘𝑒))) ↾ ((𝑁 ∖ {𝑤}) × 𝑁))) ↔ ((𝑎 ↾ ({(1st ‘𝑐)} × 𝑁)) = (((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ↾ ({(1st ‘𝑐)} × 𝑁)) ∘f + ((𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒 ∈ 𝑑, 1 , 0 ), (𝑎‘𝑒))) ↾ ({(1st ‘𝑐)} × 𝑁))) ∧ (𝑎 ↾ ((𝑁 ∖ {(1st ‘𝑐)}) × 𝑁)) = ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ↾ ((𝑁 ∖ {(1st ‘𝑐)}) × 𝑁)) ∧ (𝑎 ↾ ((𝑁 ∖ {(1st ‘𝑐)}) × 𝑁)) = ((𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒 ∈ 𝑑, 1 , 0 ), (𝑎‘𝑒))) ↾ ((𝑁 ∖ {(1st ‘𝑐)}) × 𝑁)))))
284	283	imbi1d 341	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑤 = (1st ‘𝑐) → ((((𝑎 ↾ ({𝑤} × 𝑁)) = (((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ↾ ({𝑤} × 𝑁)) ∘f + ((𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒 ∈ 𝑑, 1 , 0 ), (𝑎‘𝑒))) ↾ ({𝑤} × 𝑁))) ∧ (𝑎 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) ∧ (𝑎 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = ((𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒 ∈ 𝑑, 1 , 0 ), (𝑎‘𝑒))) ↾ ((𝑁 ∖ {𝑤}) × 𝑁))) → (𝐷‘𝑎) = ((𝐷‘(𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒)))) + (𝐷‘(𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒 ∈ 𝑑, 1 , 0 ), (𝑎‘𝑒)))))) ↔ (((𝑎 ↾ ({(1st ‘𝑐)} × 𝑁)) = (((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ↾ ({(1st ‘𝑐)} × 𝑁)) ∘f + ((𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒 ∈ 𝑑, 1 , 0 ), (𝑎‘𝑒))) ↾ ({(1st ‘𝑐)} × 𝑁))) ∧ (𝑎 ↾ ((𝑁 ∖ {(1st ‘𝑐)}) × 𝑁)) = ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ↾ ((𝑁 ∖ {(1st ‘𝑐)}) × 𝑁)) ∧ (𝑎 ↾ ((𝑁 ∖ {(1st ‘𝑐)}) × 𝑁)) = ((𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒 ∈ 𝑑, 1 , 0 ), (𝑎‘𝑒))) ↾ ((𝑁 ∖ {(1st ‘𝑐)}) × 𝑁))) → (𝐷‘𝑎) = ((𝐷‘(𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒)))) + (𝐷‘(𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒 ∈ 𝑑, 1 , 0 ), (𝑎‘𝑒))))))))
285	268, 284	rspc2va 3615	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((((𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒 ∈ 𝑑, 1 , 0 ), (𝑎‘𝑒))) ∈ 𝐵 ∧ (1st ‘𝑐) ∈ 𝑁) ∧ ∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝑁 (((𝑎 ↾ ({𝑤} × 𝑁)) = (((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ↾ ({𝑤} × 𝑁)) ∘f + (𝑧 ↾ ({𝑤} × 𝑁))) ∧ (𝑎 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) ∧ (𝑎 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑧 ↾ ((𝑁 ∖ {𝑤}) × 𝑁))) → (𝐷‘𝑎) = ((𝐷‘(𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒)))) + (𝐷‘𝑧)))) → (((𝑎 ↾ ({(1st ‘𝑐)} × 𝑁)) = (((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ↾ ({(1st ‘𝑐)} × 𝑁)) ∘f + ((𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒 ∈ 𝑑, 1 , 0 ), (𝑎‘𝑒))) ↾ ({(1st ‘𝑐)} × 𝑁))) ∧ (𝑎 ↾ ((𝑁 ∖ {(1st ‘𝑐)}) × 𝑁)) = ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ↾ ((𝑁 ∖ {(1st ‘𝑐)}) × 𝑁)) ∧ (𝑎 ↾ ((𝑁 ∖ {(1st ‘𝑐)}) × 𝑁)) = ((𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒 ∈ 𝑑, 1 , 0 ), (𝑎‘𝑒))) ↾ ((𝑁 ∖ {(1st ‘𝑐)}) × 𝑁))) → (𝐷‘𝑎) = ((𝐷‘(𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒)))) + (𝐷‘(𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒 ∈ 𝑑, 1 , 0 ), (𝑎‘𝑒)))))))
286	223, 124, 258, 285	syl21anc 835	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) → (((𝑎 ↾ ({(1st ‘𝑐)} × 𝑁)) = (((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ↾ ({(1st ‘𝑐)} × 𝑁)) ∘f + ((𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒 ∈ 𝑑, 1 , 0 ), (𝑎‘𝑒))) ↾ ({(1st ‘𝑐)} × 𝑁))) ∧ (𝑎 ↾ ((𝑁 ∖ {(1st ‘𝑐)}) × 𝑁)) = ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ↾ ((𝑁 ∖ {(1st ‘𝑐)}) × 𝑁)) ∧ (𝑎 ↾ ((𝑁 ∖ {(1st ‘𝑐)}) × 𝑁)) = ((𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒 ∈ 𝑑, 1 , 0 ), (𝑎‘𝑒))) ↾ ((𝑁 ∖ {(1st ‘𝑐)}) × 𝑁))) → (𝐷‘𝑎) = ((𝐷‘(𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒)))) + (𝐷‘(𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒 ∈ 𝑑, 1 , 0 ), (𝑎‘𝑒)))))))
287	184, 201, 209, 286	mp3and 1460	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) → (𝐷‘𝑎) = ((𝐷‘(𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒)))) + (𝐷‘(𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒 ∈ 𝑑, 1 , 0 ), (𝑎‘𝑒))))))
288	104, 105, 106, 287	syl3anc 1368	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ (𝑏 ∪ {𝑐}) ∈ 𝑌) ∧ ((𝑎 ∈ 𝐵 ∧ 𝑑 ∈ (𝑁 ↑m 𝑁)) ∧ ∀𝑤 ∈ 𝑏 (𝑎‘𝑤) = if(𝑤 ∈ 𝑑, 1 , 0 ))) → (𝐷‘𝑎) = ((𝐷‘(𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒)))) + (𝐷‘(𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒 ∈ 𝑑, 1 , 0 ), (𝑎‘𝑒))))))
289		fveq2 6881	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (𝑒 = 𝑐 → (𝑎‘𝑒) = (𝑎‘𝑐))
290		elequ1 2105	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ (𝑒 = 𝑐 → (𝑒 ∈ 𝑑 ↔ 𝑐 ∈ 𝑑))
291	290	ifbid 4543	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (𝑒 = 𝑐 → if(𝑒 ∈ 𝑑, 1 , 0 ) = if(𝑐 ∈ 𝑑, 1 , 0 ))
292	289, 291	oveq12d 7419	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (𝑒 = 𝑐 → ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )) = ((𝑎‘𝑐)(-g‘𝑅)if(𝑐 ∈ 𝑑, 1 , 0 )))
293	292	adantl 481	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) ∧ 𝑒 ∈ ({(1st ‘𝑐)} × 𝑁)) ∧ 𝑒 = 𝑐) → ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )) = ((𝑎‘𝑐)(-g‘𝑅)if(𝑐 ∈ 𝑑, 1 , 0 )))
294	110, 122	ffvelcdmd 7077	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) → (𝑎‘𝑐) ∈ 𝐾)
295	131, 133	ifcld 4566	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) → if(𝑐 ∈ 𝑑, 1 , 0 ) ∈ 𝐾)
296	12, 136	grpsubcl 18938	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ ((𝑅 ∈ Grp ∧ (𝑎‘𝑐) ∈ 𝐾 ∧ if(𝑐 ∈ 𝑑, 1 , 0 ) ∈ 𝐾) → ((𝑎‘𝑐)(-g‘𝑅)if(𝑐 ∈ 𝑑, 1 , 0 )) ∈ 𝐾)
297	115, 294, 295, 296	syl3anc 1368	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) → ((𝑎‘𝑐)(-g‘𝑅)if(𝑐 ∈ 𝑑, 1 , 0 )) ∈ 𝐾)
298	12, 51, 49	ringridm 20159	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ ((𝑅 ∈ Ring ∧ ((𝑎‘𝑐)(-g‘𝑅)if(𝑐 ∈ 𝑑, 1 , 0 )) ∈ 𝐾) → (((𝑎‘𝑐)(-g‘𝑅)if(𝑐 ∈ 𝑑, 1 , 0 )) · 1 ) = ((𝑎‘𝑐)(-g‘𝑅)if(𝑐 ∈ 𝑑, 1 , 0 )))
299	113, 297, 298	syl2anc 583	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) → (((𝑎‘𝑐)(-g‘𝑅)if(𝑐 ∈ 𝑑, 1 , 0 )) · 1 ) = ((𝑎‘𝑐)(-g‘𝑅)if(𝑐 ∈ 𝑑, 1 , 0 )))
300	299	ad2antrr 723	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) ∧ 𝑒 ∈ ({(1st ‘𝑐)} × 𝑁)) ∧ 𝑒 = 𝑐) → (((𝑎‘𝑐)(-g‘𝑅)if(𝑐 ∈ 𝑑, 1 , 0 )) · 1 ) = ((𝑎‘𝑐)(-g‘𝑅)if(𝑐 ∈ 𝑑, 1 , 0 )))
301	293, 300	eqtr4d 2767	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) ∧ 𝑒 ∈ ({(1st ‘𝑐)} × 𝑁)) ∧ 𝑒 = 𝑐) → ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )) = (((𝑎‘𝑐)(-g‘𝑅)if(𝑐 ∈ 𝑑, 1 , 0 )) · 1 ))
302	141	adantl 481	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) ∧ 𝑒 ∈ ({(1st ‘𝑐)} × 𝑁)) ∧ 𝑒 = 𝑐) → if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ) = ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )))
303		iftrue 4526	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (𝑒 = 𝑐 → if(𝑒 = 𝑐, 1 , 0 ) = 1 )
304	303	oveq2d 7417	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝑒 = 𝑐 → (((𝑎‘𝑐)(-g‘𝑅)if(𝑐 ∈ 𝑑, 1 , 0 )) · if(𝑒 = 𝑐, 1 , 0 )) = (((𝑎‘𝑐)(-g‘𝑅)if(𝑐 ∈ 𝑑, 1 , 0 )) · 1 ))
305	304	adantl 481	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) ∧ 𝑒 ∈ ({(1st ‘𝑐)} × 𝑁)) ∧ 𝑒 = 𝑐) → (((𝑎‘𝑐)(-g‘𝑅)if(𝑐 ∈ 𝑑, 1 , 0 )) · if(𝑒 = 𝑐, 1 , 0 )) = (((𝑎‘𝑐)(-g‘𝑅)if(𝑐 ∈ 𝑑, 1 , 0 )) · 1 ))
306	301, 302, 305	3eqtr4d 2774	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) ∧ 𝑒 ∈ ({(1st ‘𝑐)} × 𝑁)) ∧ 𝑒 = 𝑐) → if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ) = (((𝑎‘𝑐)(-g‘𝑅)if(𝑐 ∈ 𝑑, 1 , 0 )) · if(𝑒 = 𝑐, 1 , 0 )))
307	12, 51, 48	ringrz 20183	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ ((𝑅 ∈ Ring ∧ ((𝑎‘𝑐)(-g‘𝑅)if(𝑐 ∈ 𝑑, 1 , 0 )) ∈ 𝐾) → (((𝑎‘𝑐)(-g‘𝑅)if(𝑐 ∈ 𝑑, 1 , 0 )) · 0 ) = 0 )
308	113, 297, 307	syl2anc 583	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) → (((𝑎‘𝑐)(-g‘𝑅)if(𝑐 ∈ 𝑑, 1 , 0 )) · 0 ) = 0 )
309	308	eqcomd 2730	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) → 0 = (((𝑎‘𝑐)(-g‘𝑅)if(𝑐 ∈ 𝑑, 1 , 0 )) · 0 ))
310	309	ad2antrr 723	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) ∧ 𝑒 ∈ ({(1st ‘𝑐)} × 𝑁)) ∧ ¬ 𝑒 = 𝑐) → 0 = (((𝑎‘𝑐)(-g‘𝑅)if(𝑐 ∈ 𝑑, 1 , 0 )) · 0 ))
311	150	adantl 481	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) ∧ 𝑒 ∈ ({(1st ‘𝑐)} × 𝑁)) ∧ ¬ 𝑒 = 𝑐) → if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ) = 0 )
312		iffalse 4529	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (¬ 𝑒 = 𝑐 → if(𝑒 = 𝑐, 1 , 0 ) = 0 )
313	312	oveq2d 7417	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (¬ 𝑒 = 𝑐 → (((𝑎‘𝑐)(-g‘𝑅)if(𝑐 ∈ 𝑑, 1 , 0 )) · if(𝑒 = 𝑐, 1 , 0 )) = (((𝑎‘𝑐)(-g‘𝑅)if(𝑐 ∈ 𝑑, 1 , 0 )) · 0 ))
314	313	adantl 481	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) ∧ 𝑒 ∈ ({(1st ‘𝑐)} × 𝑁)) ∧ ¬ 𝑒 = 𝑐) → (((𝑎‘𝑐)(-g‘𝑅)if(𝑐 ∈ 𝑑, 1 , 0 )) · if(𝑒 = 𝑐, 1 , 0 )) = (((𝑎‘𝑐)(-g‘𝑅)if(𝑐 ∈ 𝑑, 1 , 0 )) · 0 ))
315	310, 311, 314	3eqtr4d 2774	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) ∧ 𝑒 ∈ ({(1st ‘𝑐)} × 𝑁)) ∧ ¬ 𝑒 = 𝑐) → if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ) = (((𝑎‘𝑐)(-g‘𝑅)if(𝑐 ∈ 𝑑, 1 , 0 )) · if(𝑒 = 𝑐, 1 , 0 )))
316	306, 315	pm2.61dan 810	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) ∧ 𝑒 ∈ ({(1st ‘𝑐)} × 𝑁)) → if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ) = (((𝑎‘𝑐)(-g‘𝑅)if(𝑐 ∈ 𝑑, 1 , 0 )) · if(𝑒 = 𝑐, 1 , 0 )))
317	170	adantl 481	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) ∧ 𝑒 ∈ ({(1st ‘𝑐)} × 𝑁)) → (1st ‘𝑒) ∈ {(1st ‘𝑐)})
318	317, 171	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) ∧ 𝑒 ∈ ({(1st ‘𝑐)} × 𝑁)) → (1st ‘𝑒) = (1st ‘𝑐))
319	318	iftrued 4528	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) ∧ 𝑒 ∈ ({(1st ‘𝑐)} × 𝑁)) → if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒)) = if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ))
320	318	iftrued 4528	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) ∧ 𝑒 ∈ ({(1st ‘𝑐)} × 𝑁)) → if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒)) = if(𝑒 = 𝑐, 1 , 0 ))
321	320	oveq2d 7417	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) ∧ 𝑒 ∈ ({(1st ‘𝑐)} × 𝑁)) → (((𝑎‘𝑐)(-g‘𝑅)if(𝑐 ∈ 𝑑, 1 , 0 )) · if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒))) = (((𝑎‘𝑐)(-g‘𝑅)if(𝑐 ∈ 𝑑, 1 , 0 )) · if(𝑒 = 𝑐, 1 , 0 )))
322	316, 319, 321	3eqtr4d 2774	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) ∧ 𝑒 ∈ ({(1st ‘𝑐)} × 𝑁)) → if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒)) = (((𝑎‘𝑐)(-g‘𝑅)if(𝑐 ∈ 𝑑, 1 , 0 )) · if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒))))
323	322	mpteq2dva 5238	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) → (𝑒 ∈ ({(1st ‘𝑐)} × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) = (𝑒 ∈ ({(1st ‘𝑐)} × 𝑁) ↦ (((𝑎‘𝑐)(-g‘𝑅)if(𝑐 ∈ 𝑑, 1 , 0 )) · if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒)))))
324		ovexd 7436	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) ∧ 𝑒 ∈ ({(1st ‘𝑐)} × 𝑁)) → ((𝑎‘𝑐)(-g‘𝑅)if(𝑐 ∈ 𝑑, 1 , 0 )) ∈ V)
325	165, 162	ifex 4570	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ if(𝑒 = 𝑐, 1 , 0 ) ∈ V
326	325, 167	ifex 4570	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒)) ∈ V
327	326	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) ∧ 𝑒 ∈ ({(1st ‘𝑐)} × 𝑁)) → if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒)) ∈ V)
328		fconstmpt 5728	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (({(1st ‘𝑐)} × 𝑁) × {((𝑎‘𝑐)(-g‘𝑅)if(𝑐 ∈ 𝑑, 1 , 0 ))}) = (𝑒 ∈ ({(1st ‘𝑐)} × 𝑁) ↦ ((𝑎‘𝑐)(-g‘𝑅)if(𝑐 ∈ 𝑑, 1 , 0 )))
329	328	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) → (({(1st ‘𝑐)} × 𝑁) × {((𝑎‘𝑐)(-g‘𝑅)if(𝑐 ∈ 𝑑, 1 , 0 ))}) = (𝑒 ∈ ({(1st ‘𝑐)} × 𝑁) ↦ ((𝑎‘𝑐)(-g‘𝑅)if(𝑐 ∈ 𝑑, 1 , 0 ))))
330	127	resmptd 6030	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) → ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒))) ↾ ({(1st ‘𝑐)} × 𝑁)) = (𝑒 ∈ ({(1st ‘𝑐)} × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒))))
331	160, 324, 327, 329, 330	offval2 7683	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) → ((({(1st ‘𝑐)} × 𝑁) × {((𝑎‘𝑐)(-g‘𝑅)if(𝑐 ∈ 𝑑, 1 , 0 ))}) ∘f · ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒))) ↾ ({(1st ‘𝑐)} × 𝑁))) = (𝑒 ∈ ({(1st ‘𝑐)} × 𝑁) ↦ (((𝑎‘𝑐)(-g‘𝑅)if(𝑐 ∈ 𝑑, 1 , 0 )) · if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒)))))
332	323, 180, 331	3eqtr4d 2774	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) → ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ↾ ({(1st ‘𝑐)} × 𝑁)) = ((({(1st ‘𝑐)} × 𝑁) × {((𝑎‘𝑐)(-g‘𝑅)if(𝑐 ∈ 𝑑, 1 , 0 ))}) ∘f · ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒))) ↾ ({(1st ‘𝑐)} × 𝑁))))
333		iffalse 4529	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (¬ (1st ‘𝑒) = (1st ‘𝑐) → if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒)) = (𝑎‘𝑒))
334		iffalse 4529	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (¬ (1st ‘𝑒) = (1st ‘𝑐) → if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒)) = (𝑎‘𝑒))
335	333, 334	eqtr4d 2767	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (¬ (1st ‘𝑒) = (1st ‘𝑐) → if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒)) = if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒)))
336	190, 335	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) ∧ 𝑒 ∈ ((𝑁 ∖ {(1st ‘𝑐)}) × 𝑁)) → if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒)) = if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒)))
337	336	mpteq2dva 5238	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) → (𝑒 ∈ ((𝑁 ∖ {(1st ‘𝑐)}) × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) = (𝑒 ∈ ((𝑁 ∖ {(1st ‘𝑐)}) × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒))))
338		resmpt 6027	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (((𝑁 ∖ {(1st ‘𝑐)}) × 𝑁) ⊆ (𝑁 × 𝑁) → ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒))) ↾ ((𝑁 ∖ {(1st ‘𝑐)}) × 𝑁)) = (𝑒 ∈ ((𝑁 ∖ {(1st ‘𝑐)}) × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒))))
339	195, 338	mp1i 13	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) → ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒))) ↾ ((𝑁 ∖ {(1st ‘𝑐)}) × 𝑁)) = (𝑒 ∈ ((𝑁 ∖ {(1st ‘𝑐)}) × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒))))
340	337, 197, 339	3eqtr4d 2774	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) → ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ↾ ((𝑁 ∖ {(1st ‘𝑐)}) × 𝑁)) = ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒))) ↾ ((𝑁 ∖ {(1st ‘𝑐)}) × 𝑁)))
341	131, 133	ifcld 4566	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) → if(𝑒 = 𝑐, 1 , 0 ) ∈ 𝐾)
342	341	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) ∧ 𝑒 ∈ (𝑁 × 𝑁)) → if(𝑒 = 𝑐, 1 , 0 ) ∈ 𝐾)
343	342, 211	ifcld 4566	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) ∧ 𝑒 ∈ (𝑁 × 𝑁)) → if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒)) ∈ 𝐾)
344	343	fmpttd 7106	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) → (𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒))):(𝑁 × 𝑁)⟶𝐾)
345		elmapg 8829	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((𝐾 ∈ V ∧ (𝑁 × 𝑁) ∈ Fin) → ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒))) ∈ (𝐾 ↑m (𝑁 × 𝑁)) ↔ (𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒))):(𝑁 × 𝑁)⟶𝐾))
346	214, 216, 345	sylancr 586	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) → ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒))) ∈ (𝐾 ↑m (𝑁 × 𝑁)) ↔ (𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒))):(𝑁 × 𝑁)⟶𝐾))
347	344, 346	mpbird 257	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) → (𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒))) ∈ (𝐾 ↑m (𝑁 × 𝑁)))
348	347, 222	eleqtrd 2827	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) → (𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒))) ∈ 𝐵)
349	56	3ad2ant1 1130	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐾 ∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝑁 (((𝑥 ↾ ({𝑤} × 𝑁)) = ((({𝑤} × 𝑁) × {𝑦}) ∘f · (𝑧 ↾ ({𝑤} × 𝑁))) ∧ (𝑥 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑧 ↾ ((𝑁 ∖ {𝑤}) × 𝑁))) → (𝐷‘𝑥) = (𝑦 · (𝐷‘𝑧))))
350		reseq1 5965	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (𝑥 = (𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) → (𝑥 ↾ ({𝑤} × 𝑁)) = ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ↾ ({𝑤} × 𝑁)))
351	350	eqeq1d 2726	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝑥 = (𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) → ((𝑥 ↾ ({𝑤} × 𝑁)) = ((({𝑤} × 𝑁) × {𝑦}) ∘f · (𝑧 ↾ ({𝑤} × 𝑁))) ↔ ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ↾ ({𝑤} × 𝑁)) = ((({𝑤} × 𝑁) × {𝑦}) ∘f · (𝑧 ↾ ({𝑤} × 𝑁)))))
352		reseq1 5965	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (𝑥 = (𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) → (𝑥 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ↾ ((𝑁 ∖ {𝑤}) × 𝑁)))
353	352	eqeq1d 2726	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝑥 = (𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) → ((𝑥 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑧 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) ↔ ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑧 ↾ ((𝑁 ∖ {𝑤}) × 𝑁))))
354	351, 353	anbi12d 630	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝑥 = (𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) → (((𝑥 ↾ ({𝑤} × 𝑁)) = ((({𝑤} × 𝑁) × {𝑦}) ∘f · (𝑧 ↾ ({𝑤} × 𝑁))) ∧ (𝑥 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑧 ↾ ((𝑁 ∖ {𝑤}) × 𝑁))) ↔ (((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ↾ ({𝑤} × 𝑁)) = ((({𝑤} × 𝑁) × {𝑦}) ∘f · (𝑧 ↾ ({𝑤} × 𝑁))) ∧ ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑧 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)))))
355		fveqeq2 6890	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝑥 = (𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) → ((𝐷‘𝑥) = (𝑦 · (𝐷‘𝑧)) ↔ (𝐷‘(𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒)))) = (𝑦 · (𝐷‘𝑧))))
356	354, 355	imbi12d 344	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑥 = (𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) → ((((𝑥 ↾ ({𝑤} × 𝑁)) = ((({𝑤} × 𝑁) × {𝑦}) ∘f · (𝑧 ↾ ({𝑤} × 𝑁))) ∧ (𝑥 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑧 ↾ ((𝑁 ∖ {𝑤}) × 𝑁))) → (𝐷‘𝑥) = (𝑦 · (𝐷‘𝑧))) ↔ ((((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ↾ ({𝑤} × 𝑁)) = ((({𝑤} × 𝑁) × {𝑦}) ∘f · (𝑧 ↾ ({𝑤} × 𝑁))) ∧ ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑧 ↾ ((𝑁 ∖ {𝑤}) × 𝑁))) → (𝐷‘(𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒)))) = (𝑦 · (𝐷‘𝑧)))))
357	356	2ralbidv 3210	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑥 = (𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) → (∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝑁 (((𝑥 ↾ ({𝑤} × 𝑁)) = ((({𝑤} × 𝑁) × {𝑦}) ∘f · (𝑧 ↾ ({𝑤} × 𝑁))) ∧ (𝑥 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑧 ↾ ((𝑁 ∖ {𝑤}) × 𝑁))) → (𝐷‘𝑥) = (𝑦 · (𝐷‘𝑧))) ↔ ∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝑁 ((((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ↾ ({𝑤} × 𝑁)) = ((({𝑤} × 𝑁) × {𝑦}) ∘f · (𝑧 ↾ ({𝑤} × 𝑁))) ∧ ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑧 ↾ ((𝑁 ∖ {𝑤}) × 𝑁))) → (𝐷‘(𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒)))) = (𝑦 · (𝐷‘𝑧)))))
358		sneq 4630	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ (𝑦 = ((𝑎‘𝑐)(-g‘𝑅)if(𝑐 ∈ 𝑑, 1 , 0 )) → {𝑦} = {((𝑎‘𝑐)(-g‘𝑅)if(𝑐 ∈ 𝑑, 1 , 0 ))})
359	358	xpeq2d 5696	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (𝑦 = ((𝑎‘𝑐)(-g‘𝑅)if(𝑐 ∈ 𝑑, 1 , 0 )) → (({𝑤} × 𝑁) × {𝑦}) = (({𝑤} × 𝑁) × {((𝑎‘𝑐)(-g‘𝑅)if(𝑐 ∈ 𝑑, 1 , 0 ))}))
360	359	oveq1d 7416	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (𝑦 = ((𝑎‘𝑐)(-g‘𝑅)if(𝑐 ∈ 𝑑, 1 , 0 )) → ((({𝑤} × 𝑁) × {𝑦}) ∘f · (𝑧 ↾ ({𝑤} × 𝑁))) = ((({𝑤} × 𝑁) × {((𝑎‘𝑐)(-g‘𝑅)if(𝑐 ∈ 𝑑, 1 , 0 ))}) ∘f · (𝑧 ↾ ({𝑤} × 𝑁))))
361	360	eqeq2d 2735	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝑦 = ((𝑎‘𝑐)(-g‘𝑅)if(𝑐 ∈ 𝑑, 1 , 0 )) → (((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ↾ ({𝑤} × 𝑁)) = ((({𝑤} × 𝑁) × {𝑦}) ∘f · (𝑧 ↾ ({𝑤} × 𝑁))) ↔ ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ↾ ({𝑤} × 𝑁)) = ((({𝑤} × 𝑁) × {((𝑎‘𝑐)(-g‘𝑅)if(𝑐 ∈ 𝑑, 1 , 0 ))}) ∘f · (𝑧 ↾ ({𝑤} × 𝑁)))))
362	361	anbi1d 629	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝑦 = ((𝑎‘𝑐)(-g‘𝑅)if(𝑐 ∈ 𝑑, 1 , 0 )) → ((((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ↾ ({𝑤} × 𝑁)) = ((({𝑤} × 𝑁) × {𝑦}) ∘f · (𝑧 ↾ ({𝑤} × 𝑁))) ∧ ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑧 ↾ ((𝑁 ∖ {𝑤}) × 𝑁))) ↔ (((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ↾ ({𝑤} × 𝑁)) = ((({𝑤} × 𝑁) × {((𝑎‘𝑐)(-g‘𝑅)if(𝑐 ∈ 𝑑, 1 , 0 ))}) ∘f · (𝑧 ↾ ({𝑤} × 𝑁))) ∧ ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑧 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)))))
363		oveq1 7408	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝑦 = ((𝑎‘𝑐)(-g‘𝑅)if(𝑐 ∈ 𝑑, 1 , 0 )) → (𝑦 · (𝐷‘𝑧)) = (((𝑎‘𝑐)(-g‘𝑅)if(𝑐 ∈ 𝑑, 1 , 0 )) · (𝐷‘𝑧)))
364	363	eqeq2d 2735	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝑦 = ((𝑎‘𝑐)(-g‘𝑅)if(𝑐 ∈ 𝑑, 1 , 0 )) → ((𝐷‘(𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒)))) = (𝑦 · (𝐷‘𝑧)) ↔ (𝐷‘(𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒)))) = (((𝑎‘𝑐)(-g‘𝑅)if(𝑐 ∈ 𝑑, 1 , 0 )) · (𝐷‘𝑧))))
365	362, 364	imbi12d 344	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑦 = ((𝑎‘𝑐)(-g‘𝑅)if(𝑐 ∈ 𝑑, 1 , 0 )) → (((((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ↾ ({𝑤} × 𝑁)) = ((({𝑤} × 𝑁) × {𝑦}) ∘f · (𝑧 ↾ ({𝑤} × 𝑁))) ∧ ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑧 ↾ ((𝑁 ∖ {𝑤}) × 𝑁))) → (𝐷‘(𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒)))) = (𝑦 · (𝐷‘𝑧))) ↔ ((((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ↾ ({𝑤} × 𝑁)) = ((({𝑤} × 𝑁) × {((𝑎‘𝑐)(-g‘𝑅)if(𝑐 ∈ 𝑑, 1 , 0 ))}) ∘f · (𝑧 ↾ ({𝑤} × 𝑁))) ∧ ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑧 ↾ ((𝑁 ∖ {𝑤}) × 𝑁))) → (𝐷‘(𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒)))) = (((𝑎‘𝑐)(-g‘𝑅)if(𝑐 ∈ 𝑑, 1 , 0 )) · (𝐷‘𝑧)))))
366	365	2ralbidv 3210	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑦 = ((𝑎‘𝑐)(-g‘𝑅)if(𝑐 ∈ 𝑑, 1 , 0 )) → (∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝑁 ((((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ↾ ({𝑤} × 𝑁)) = ((({𝑤} × 𝑁) × {𝑦}) ∘f · (𝑧 ↾ ({𝑤} × 𝑁))) ∧ ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑧 ↾ ((𝑁 ∖ {𝑤}) × 𝑁))) → (𝐷‘(𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒)))) = (𝑦 · (𝐷‘𝑧))) ↔ ∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝑁 ((((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ↾ ({𝑤} × 𝑁)) = ((({𝑤} × 𝑁) × {((𝑎‘𝑐)(-g‘𝑅)if(𝑐 ∈ 𝑑, 1 , 0 ))}) ∘f · (𝑧 ↾ ({𝑤} × 𝑁))) ∧ ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑧 ↾ ((𝑁 ∖ {𝑤}) × 𝑁))) → (𝐷‘(𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒)))) = (((𝑎‘𝑐)(-g‘𝑅)if(𝑐 ∈ 𝑑, 1 , 0 )) · (𝐷‘𝑧)))))
367	357, 366	rspc2va 3615	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ∈ 𝐵 ∧ ((𝑎‘𝑐)(-g‘𝑅)if(𝑐 ∈ 𝑑, 1 , 0 )) ∈ 𝐾) ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐾 ∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝑁 (((𝑥 ↾ ({𝑤} × 𝑁)) = ((({𝑤} × 𝑁) × {𝑦}) ∘f · (𝑧 ↾ ({𝑤} × 𝑁))) ∧ (𝑥 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑧 ↾ ((𝑁 ∖ {𝑤}) × 𝑁))) → (𝐷‘𝑥) = (𝑦 · (𝐷‘𝑧)))) → ∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝑁 ((((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ↾ ({𝑤} × 𝑁)) = ((({𝑤} × 𝑁) × {((𝑎‘𝑐)(-g‘𝑅)if(𝑐 ∈ 𝑑, 1 , 0 ))}) ∘f · (𝑧 ↾ ({𝑤} × 𝑁))) ∧ ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑧 ↾ ((𝑁 ∖ {𝑤}) × 𝑁))) → (𝐷‘(𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒)))) = (((𝑎‘𝑐)(-g‘𝑅)if(𝑐 ∈ 𝑑, 1 , 0 )) · (𝐷‘𝑧))))
368	235, 297, 349, 367	syl21anc 835	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) → ∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝑁 ((((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ↾ ({𝑤} × 𝑁)) = ((({𝑤} × 𝑁) × {((𝑎‘𝑐)(-g‘𝑅)if(𝑐 ∈ 𝑑, 1 , 0 ))}) ∘f · (𝑧 ↾ ({𝑤} × 𝑁))) ∧ ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑧 ↾ ((𝑁 ∖ {𝑤}) × 𝑁))) → (𝐷‘(𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒)))) = (((𝑎‘𝑐)(-g‘𝑅)if(𝑐 ∈ 𝑑, 1 , 0 )) · (𝐷‘𝑧))))
369		reseq1 5965	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝑧 = (𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒))) → (𝑧 ↾ ({𝑤} × 𝑁)) = ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒))) ↾ ({𝑤} × 𝑁)))
370	369	oveq2d 7417	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝑧 = (𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒))) → ((({𝑤} × 𝑁) × {((𝑎‘𝑐)(-g‘𝑅)if(𝑐 ∈ 𝑑, 1 , 0 ))}) ∘f · (𝑧 ↾ ({𝑤} × 𝑁))) = ((({𝑤} × 𝑁) × {((𝑎‘𝑐)(-g‘𝑅)if(𝑐 ∈ 𝑑, 1 , 0 ))}) ∘f · ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒))) ↾ ({𝑤} × 𝑁))))
371	370	eqeq2d 2735	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑧 = (𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒))) → (((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ↾ ({𝑤} × 𝑁)) = ((({𝑤} × 𝑁) × {((𝑎‘𝑐)(-g‘𝑅)if(𝑐 ∈ 𝑑, 1 , 0 ))}) ∘f · (𝑧 ↾ ({𝑤} × 𝑁))) ↔ ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ↾ ({𝑤} × 𝑁)) = ((({𝑤} × 𝑁) × {((𝑎‘𝑐)(-g‘𝑅)if(𝑐 ∈ 𝑑, 1 , 0 ))}) ∘f · ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒))) ↾ ({𝑤} × 𝑁)))))
372		reseq1 5965	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝑧 = (𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒))) → (𝑧 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒))) ↾ ((𝑁 ∖ {𝑤}) × 𝑁)))
373	372	eqeq2d 2735	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑧 = (𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒))) → (((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑧 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) ↔ ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒))) ↾ ((𝑁 ∖ {𝑤}) × 𝑁))))
374	371, 373	anbi12d 630	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑧 = (𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒))) → ((((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ↾ ({𝑤} × 𝑁)) = ((({𝑤} × 𝑁) × {((𝑎‘𝑐)(-g‘𝑅)if(𝑐 ∈ 𝑑, 1 , 0 ))}) ∘f · (𝑧 ↾ ({𝑤} × 𝑁))) ∧ ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑧 ↾ ((𝑁 ∖ {𝑤}) × 𝑁))) ↔ (((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ↾ ({𝑤} × 𝑁)) = ((({𝑤} × 𝑁) × {((𝑎‘𝑐)(-g‘𝑅)if(𝑐 ∈ 𝑑, 1 , 0 ))}) ∘f · ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒))) ↾ ({𝑤} × 𝑁))) ∧ ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒))) ↾ ((𝑁 ∖ {𝑤}) × 𝑁)))))
375		fveq2 6881	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝑧 = (𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒))) → (𝐷‘𝑧) = (𝐷‘(𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒)))))
376	375	oveq2d 7417	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑧 = (𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒))) → (((𝑎‘𝑐)(-g‘𝑅)if(𝑐 ∈ 𝑑, 1 , 0 )) · (𝐷‘𝑧)) = (((𝑎‘𝑐)(-g‘𝑅)if(𝑐 ∈ 𝑑, 1 , 0 )) · (𝐷‘(𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒))))))
377	376	eqeq2d 2735	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑧 = (𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒))) → ((𝐷‘(𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒)))) = (((𝑎‘𝑐)(-g‘𝑅)if(𝑐 ∈ 𝑑, 1 , 0 )) · (𝐷‘𝑧)) ↔ (𝐷‘(𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒)))) = (((𝑎‘𝑐)(-g‘𝑅)if(𝑐 ∈ 𝑑, 1 , 0 )) · (𝐷‘(𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒)))))))
378	374, 377	imbi12d 344	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑧 = (𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒))) → (((((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ↾ ({𝑤} × 𝑁)) = ((({𝑤} × 𝑁) × {((𝑎‘𝑐)(-g‘𝑅)if(𝑐 ∈ 𝑑, 1 , 0 ))}) ∘f · (𝑧 ↾ ({𝑤} × 𝑁))) ∧ ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑧 ↾ ((𝑁 ∖ {𝑤}) × 𝑁))) → (𝐷‘(𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒)))) = (((𝑎‘𝑐)(-g‘𝑅)if(𝑐 ∈ 𝑑, 1 , 0 )) · (𝐷‘𝑧))) ↔ ((((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ↾ ({𝑤} × 𝑁)) = ((({𝑤} × 𝑁) × {((𝑎‘𝑐)(-g‘𝑅)if(𝑐 ∈ 𝑑, 1 , 0 ))}) ∘f · ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒))) ↾ ({𝑤} × 𝑁))) ∧ ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒))) ↾ ((𝑁 ∖ {𝑤}) × 𝑁))) → (𝐷‘(𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒)))) = (((𝑎‘𝑐)(-g‘𝑅)if(𝑐 ∈ 𝑑, 1 , 0 )) · (𝐷‘(𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒))))))))
379	270	xpeq1d 5695	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝑤 = (1st ‘𝑐) → (({𝑤} × 𝑁) × {((𝑎‘𝑐)(-g‘𝑅)if(𝑐 ∈ 𝑑, 1 , 0 ))}) = (({(1st ‘𝑐)} × 𝑁) × {((𝑎‘𝑐)(-g‘𝑅)if(𝑐 ∈ 𝑑, 1 , 0 ))}))
380	270	reseq2d 5971	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝑤 = (1st ‘𝑐) → ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒))) ↾ ({𝑤} × 𝑁)) = ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒))) ↾ ({(1st ‘𝑐)} × 𝑁)))
381	379, 380	oveq12d 7419	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝑤 = (1st ‘𝑐) → ((({𝑤} × 𝑁) × {((𝑎‘𝑐)(-g‘𝑅)if(𝑐 ∈ 𝑑, 1 , 0 ))}) ∘f · ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒))) ↾ ({𝑤} × 𝑁))) = ((({(1st ‘𝑐)} × 𝑁) × {((𝑎‘𝑐)(-g‘𝑅)if(𝑐 ∈ 𝑑, 1 , 0 ))}) ∘f · ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒))) ↾ ({(1st ‘𝑐)} × 𝑁))))
382	272, 381	eqeq12d 2740	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑤 = (1st ‘𝑐) → (((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ↾ ({𝑤} × 𝑁)) = ((({𝑤} × 𝑁) × {((𝑎‘𝑐)(-g‘𝑅)if(𝑐 ∈ 𝑑, 1 , 0 ))}) ∘f · ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒))) ↾ ({𝑤} × 𝑁))) ↔ ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ↾ ({(1st ‘𝑐)} × 𝑁)) = ((({(1st ‘𝑐)} × 𝑁) × {((𝑎‘𝑐)(-g‘𝑅)if(𝑐 ∈ 𝑑, 1 , 0 ))}) ∘f · ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒))) ↾ ({(1st ‘𝑐)} × 𝑁)))))
383	277	reseq2d 5971	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝑤 = (1st ‘𝑐) → ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒))) ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒))) ↾ ((𝑁 ∖ {(1st ‘𝑐)}) × 𝑁)))
384	279, 383	eqeq12d 2740	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑤 = (1st ‘𝑐) → (((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒))) ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) ↔ ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ↾ ((𝑁 ∖ {(1st ‘𝑐)}) × 𝑁)) = ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒))) ↾ ((𝑁 ∖ {(1st ‘𝑐)}) × 𝑁))))
385	382, 384	anbi12d 630	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑤 = (1st ‘𝑐) → ((((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ↾ ({𝑤} × 𝑁)) = ((({𝑤} × 𝑁) × {((𝑎‘𝑐)(-g‘𝑅)if(𝑐 ∈ 𝑑, 1 , 0 ))}) ∘f · ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒))) ↾ ({𝑤} × 𝑁))) ∧ ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒))) ↾ ((𝑁 ∖ {𝑤}) × 𝑁))) ↔ (((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ↾ ({(1st ‘𝑐)} × 𝑁)) = ((({(1st ‘𝑐)} × 𝑁) × {((𝑎‘𝑐)(-g‘𝑅)if(𝑐 ∈ 𝑑, 1 , 0 ))}) ∘f · ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒))) ↾ ({(1st ‘𝑐)} × 𝑁))) ∧ ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ↾ ((𝑁 ∖ {(1st ‘𝑐)}) × 𝑁)) = ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒))) ↾ ((𝑁 ∖ {(1st ‘𝑐)}) × 𝑁)))))
386	385	imbi1d 341	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑤 = (1st ‘𝑐) → (((((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ↾ ({𝑤} × 𝑁)) = ((({𝑤} × 𝑁) × {((𝑎‘𝑐)(-g‘𝑅)if(𝑐 ∈ 𝑑, 1 , 0 ))}) ∘f · ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒))) ↾ ({𝑤} × 𝑁))) ∧ ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒))) ↾ ((𝑁 ∖ {𝑤}) × 𝑁))) → (𝐷‘(𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒)))) = (((𝑎‘𝑐)(-g‘𝑅)if(𝑐 ∈ 𝑑, 1 , 0 )) · (𝐷‘(𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒)))))) ↔ ((((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ↾ ({(1st ‘𝑐)} × 𝑁)) = ((({(1st ‘𝑐)} × 𝑁) × {((𝑎‘𝑐)(-g‘𝑅)if(𝑐 ∈ 𝑑, 1 , 0 ))}) ∘f · ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒))) ↾ ({(1st ‘𝑐)} × 𝑁))) ∧ ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ↾ ((𝑁 ∖ {(1st ‘𝑐)}) × 𝑁)) = ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒))) ↾ ((𝑁 ∖ {(1st ‘𝑐)}) × 𝑁))) → (𝐷‘(𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒)))) = (((𝑎‘𝑐)(-g‘𝑅)if(𝑐 ∈ 𝑑, 1 , 0 )) · (𝐷‘(𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒))))))))
387	378, 386	rspc2va 3615	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒))) ∈ 𝐵 ∧ (1st ‘𝑐) ∈ 𝑁) ∧ ∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝑁 ((((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ↾ ({𝑤} × 𝑁)) = ((({𝑤} × 𝑁) × {((𝑎‘𝑐)(-g‘𝑅)if(𝑐 ∈ 𝑑, 1 , 0 ))}) ∘f · (𝑧 ↾ ({𝑤} × 𝑁))) ∧ ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑧 ↾ ((𝑁 ∖ {𝑤}) × 𝑁))) → (𝐷‘(𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒)))) = (((𝑎‘𝑐)(-g‘𝑅)if(𝑐 ∈ 𝑑, 1 , 0 )) · (𝐷‘𝑧)))) → ((((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ↾ ({(1st ‘𝑐)} × 𝑁)) = ((({(1st ‘𝑐)} × 𝑁) × {((𝑎‘𝑐)(-g‘𝑅)if(𝑐 ∈ 𝑑, 1 , 0 ))}) ∘f · ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒))) ↾ ({(1st ‘𝑐)} × 𝑁))) ∧ ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ↾ ((𝑁 ∖ {(1st ‘𝑐)}) × 𝑁)) = ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒))) ↾ ((𝑁 ∖ {(1st ‘𝑐)}) × 𝑁))) → (𝐷‘(𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒)))) = (((𝑎‘𝑐)(-g‘𝑅)if(𝑐 ∈ 𝑑, 1 , 0 )) · (𝐷‘(𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒)))))))
388	348, 124, 368, 387	syl21anc 835	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) → ((((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ↾ ({(1st ‘𝑐)} × 𝑁)) = ((({(1st ‘𝑐)} × 𝑁) × {((𝑎‘𝑐)(-g‘𝑅)if(𝑐 ∈ 𝑑, 1 , 0 ))}) ∘f · ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒))) ↾ ({(1st ‘𝑐)} × 𝑁))) ∧ ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ↾ ((𝑁 ∖ {(1st ‘𝑐)}) × 𝑁)) = ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒))) ↾ ((𝑁 ∖ {(1st ‘𝑐)}) × 𝑁))) → (𝐷‘(𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒)))) = (((𝑎‘𝑐)(-g‘𝑅)if(𝑐 ∈ 𝑑, 1 , 0 )) · (𝐷‘(𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒)))))))
389	332, 340, 388	mp2and 696	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) → (𝐷‘(𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒)))) = (((𝑎‘𝑐)(-g‘𝑅)if(𝑐 ∈ 𝑑, 1 , 0 )) · (𝐷‘(𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒))))))
390	389	oveq1d 7416	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) → ((𝐷‘(𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒)))) + (𝐷‘(𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒 ∈ 𝑑, 1 , 0 ), (𝑎‘𝑒))))) = ((((𝑎‘𝑐)(-g‘𝑅)if(𝑐 ∈ 𝑑, 1 , 0 )) · (𝐷‘(𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒))))) + (𝐷‘(𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒 ∈ 𝑑, 1 , 0 ), (𝑎‘𝑒))))))
391	104, 105, 106, 390	syl3anc 1368	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ (𝑏 ∪ {𝑐}) ∈ 𝑌) ∧ ((𝑎 ∈ 𝐵 ∧ 𝑑 ∈ (𝑁 ↑m 𝑁)) ∧ ∀𝑤 ∈ 𝑏 (𝑎‘𝑤) = if(𝑤 ∈ 𝑑, 1 , 0 ))) → ((𝐷‘(𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒)))) + (𝐷‘(𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒 ∈ 𝑑, 1 , 0 ), (𝑎‘𝑒))))) = ((((𝑎‘𝑐)(-g‘𝑅)if(𝑐 ∈ 𝑑, 1 , 0 )) · (𝐷‘(𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒))))) + (𝐷‘(𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒 ∈ 𝑑, 1 , 0 ), (𝑎‘𝑒))))))
392		simpl3 1190	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ (𝑏 ∪ {𝑐}) ∈ 𝑌) ∧ ((𝑎 ∈ 𝐵 ∧ 𝑑 ∈ (𝑁 ↑m 𝑁)) ∧ ∀𝑤 ∈ 𝑏 (𝑎‘𝑤) = if(𝑤 ∈ 𝑑, 1 , 0 ))) → (𝑏 ∪ {𝑐}) ∈ 𝑌)
393		simprlr 777	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ (𝑏 ∪ {𝑐}) ∈ 𝑌) ∧ ((𝑎 ∈ 𝐵 ∧ 𝑑 ∈ (𝑁 ↑m 𝑁)) ∧ ∀𝑤 ∈ 𝑏 (𝑎‘𝑤) = if(𝑤 ∈ 𝑑, 1 , 0 ))) → 𝑑 ∈ (𝑁 ↑m 𝑁))
394		simprr 770	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ (𝑏 ∪ {𝑐}) ∈ 𝑌) ∧ ((𝑎 ∈ 𝐵 ∧ 𝑑 ∈ (𝑁 ↑m 𝑁)) ∧ ∀𝑤 ∈ 𝑏 (𝑎‘𝑤) = if(𝑤 ∈ 𝑑, 1 , 0 ))) → ∀𝑤 ∈ 𝑏 (𝑎‘𝑤) = if(𝑤 ∈ 𝑑, 1 , 0 ))
395		ralss 4046	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝑏 ⊆ (𝑏 ∪ {𝑐}) → (∀𝑤 ∈ 𝑏 (𝑎‘𝑤) = if(𝑤 ∈ 𝑑, 1 , 0 ) ↔ ∀𝑤 ∈ (𝑏 ∪ {𝑐})(𝑤 ∈ 𝑏 → (𝑎‘𝑤) = if(𝑤 ∈ 𝑑, 1 , 0 ))))
396	99, 395	ax-mp 5	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (∀𝑤 ∈ 𝑏 (𝑎‘𝑤) = if(𝑤 ∈ 𝑑, 1 , 0 ) ↔ ∀𝑤 ∈ (𝑏 ∪ {𝑐})(𝑤 ∈ 𝑏 → (𝑎‘𝑤) = if(𝑤 ∈ 𝑑, 1 , 0 )))
397		iftrue 4526	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ ((1st ‘𝑤) = (1st ‘𝑐) → if((1st ‘𝑤) = (1st ‘𝑐), if(𝑤 = 𝑐, 1 , 0 ), (𝑎‘𝑤)) = if(𝑤 = 𝑐, 1 , 0 ))
398	397	adantl 481	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ (((((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) ∧ 𝑑 ∈ (𝑁 ↑m 𝑁)) ∧ 𝑤 ∈ (𝑏 ∪ {𝑐})) ∧ (1st ‘𝑤) = (1st ‘𝑐)) → if((1st ‘𝑤) = (1st ‘𝑐), if(𝑤 = 𝑐, 1 , 0 ), (𝑎‘𝑤)) = if(𝑤 = 𝑐, 1 , 0 ))
399		ibar 528	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ⊢ ((1st ‘𝑤) = (1st ‘𝑐) → ((2nd ‘𝑤) = (2nd ‘𝑐) ↔ ((1st ‘𝑤) = (1st ‘𝑐) ∧ (2nd ‘𝑤) = (2nd ‘𝑐))))
400	399	adantl 481	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ⊢ (((((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) ∧ 𝑑 ∈ (𝑁 ↑m 𝑁)) ∧ 𝑤 ∈ (𝑏 ∪ {𝑐})) ∧ (1st ‘𝑤) = (1st ‘𝑐)) → ((2nd ‘𝑤) = (2nd ‘𝑐) ↔ ((1st ‘𝑤) = (1st ‘𝑐) ∧ (2nd ‘𝑤) = (2nd ‘𝑐))))
401		relxp 5684	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 ⊢ Rel (𝑁 × 𝑁)
402		simpl2 1189	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 ⊢ (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) ∧ 𝑑 ∈ (𝑁 ↑m 𝑁)) → (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁))
403	402	sselda 3974	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 ⊢ ((((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) ∧ 𝑑 ∈ (𝑁 ↑m 𝑁)) ∧ 𝑤 ∈ (𝑏 ∪ {𝑐})) → 𝑤 ∈ (𝑁 × 𝑁))
404	403	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 ⊢ (((((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) ∧ 𝑑 ∈ (𝑁 ↑m 𝑁)) ∧ 𝑤 ∈ (𝑏 ∪ {𝑐})) ∧ (1st ‘𝑤) = (1st ‘𝑐)) → 𝑤 ∈ (𝑁 × 𝑁))
405		1st2nd 8018	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 ⊢ ((Rel (𝑁 × 𝑁) ∧ 𝑤 ∈ (𝑁 × 𝑁)) → 𝑤 = ⟨(1st ‘𝑤), (2nd ‘𝑤)⟩)
406	401, 404, 405	sylancr 586	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 ⊢ (((((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) ∧ 𝑑 ∈ (𝑁 ↑m 𝑁)) ∧ 𝑤 ∈ (𝑏 ∪ {𝑐})) ∧ (1st ‘𝑤) = (1st ‘𝑐)) → 𝑤 = ⟨(1st ‘𝑤), (2nd ‘𝑤)⟩)
407	406	eleq1d 2810	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ⊢ (((((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) ∧ 𝑑 ∈ (𝑁 ↑m 𝑁)) ∧ 𝑤 ∈ (𝑏 ∪ {𝑐})) ∧ (1st ‘𝑤) = (1st ‘𝑐)) → (𝑤 ∈ (𝑑 sSet ⟨(1st ‘𝑐), (2nd ‘𝑐)⟩) ↔ ⟨(1st ‘𝑤), (2nd ‘𝑤)⟩ ∈ (𝑑 sSet ⟨(1st ‘𝑐), (2nd ‘𝑐)⟩)))
408		simpr 484	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 ⊢ (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) ∧ 𝑑 ∈ (𝑁 ↑m 𝑁)) → 𝑑 ∈ (𝑁 ↑m 𝑁))
409		elmapi 8839	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 ⊢ (𝑑 ∈ (𝑁 ↑m 𝑁) → 𝑑:𝑁⟶𝑁)
410	409	adantl 481	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 ⊢ (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) ∧ 𝑑 ∈ (𝑁 ↑m 𝑁)) → 𝑑:𝑁⟶𝑁)
411	124	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 ⊢ (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) ∧ 𝑑 ∈ (𝑁 ↑m 𝑁)) → (1st ‘𝑐) ∈ 𝑁)
412		xp2nd 8001	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 ⊢ (𝑐 ∈ (𝑁 × 𝑁) → (2nd ‘𝑐) ∈ 𝑁)
413	122, 412	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 ⊢ ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) → (2nd ‘𝑐) ∈ 𝑁)
414	413	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 ⊢ (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) ∧ 𝑑 ∈ (𝑁 ↑m 𝑁)) → (2nd ‘𝑐) ∈ 𝑁)
415		fsets 17101	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 ⊢ (((𝑑 ∈ (𝑁 ↑m 𝑁) ∧ 𝑑:𝑁⟶𝑁) ∧ (1st ‘𝑐) ∈ 𝑁 ∧ (2nd ‘𝑐) ∈ 𝑁) → (𝑑 sSet ⟨(1st ‘𝑐), (2nd ‘𝑐)⟩):𝑁⟶𝑁)
416	408, 410, 411, 414, 415	syl211anc 1373	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 ⊢ (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) ∧ 𝑑 ∈ (𝑁 ↑m 𝑁)) → (𝑑 sSet ⟨(1st ‘𝑐), (2nd ‘𝑐)⟩):𝑁⟶𝑁)
417	416	ffnd 6708	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 ⊢ (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) ∧ 𝑑 ∈ (𝑁 ↑m 𝑁)) → (𝑑 sSet ⟨(1st ‘𝑐), (2nd ‘𝑐)⟩) Fn 𝑁)
418	417	ad2antrr 723	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 ⊢ (((((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) ∧ 𝑑 ∈ (𝑁 ↑m 𝑁)) ∧ 𝑤 ∈ (𝑏 ∪ {𝑐})) ∧ (1st ‘𝑤) = (1st ‘𝑐)) → (𝑑 sSet ⟨(1st ‘𝑐), (2nd ‘𝑐)⟩) Fn 𝑁)
419		xp1st 8000	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 ⊢ (𝑤 ∈ (𝑁 × 𝑁) → (1st ‘𝑤) ∈ 𝑁)
420	403, 419	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 ⊢ ((((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) ∧ 𝑑 ∈ (𝑁 ↑m 𝑁)) ∧ 𝑤 ∈ (𝑏 ∪ {𝑐})) → (1st ‘𝑤) ∈ 𝑁)
421	420	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 ⊢ (((((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) ∧ 𝑑 ∈ (𝑁 ↑m 𝑁)) ∧ 𝑤 ∈ (𝑏 ∪ {𝑐})) ∧ (1st ‘𝑤) = (1st ‘𝑐)) → (1st ‘𝑤) ∈ 𝑁)
422		fnopfvb 6935	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 ⊢ (((𝑑 sSet ⟨(1st ‘𝑐), (2nd ‘𝑐)⟩) Fn 𝑁 ∧ (1st ‘𝑤) ∈ 𝑁) → (((𝑑 sSet ⟨(1st ‘𝑐), (2nd ‘𝑐)⟩)‘(1st ‘𝑤)) = (2nd ‘𝑤) ↔ ⟨(1st ‘𝑤), (2nd ‘𝑤)⟩ ∈ (𝑑 sSet ⟨(1st ‘𝑐), (2nd ‘𝑐)⟩)))
423	418, 421, 422	syl2anc 583	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ⊢ (((((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) ∧ 𝑑 ∈ (𝑁 ↑m 𝑁)) ∧ 𝑤 ∈ (𝑏 ∪ {𝑐})) ∧ (1st ‘𝑤) = (1st ‘𝑐)) → (((𝑑 sSet ⟨(1st ‘𝑐), (2nd ‘𝑐)⟩)‘(1st ‘𝑤)) = (2nd ‘𝑤) ↔ ⟨(1st ‘𝑤), (2nd ‘𝑤)⟩ ∈ (𝑑 sSet ⟨(1st ‘𝑐), (2nd ‘𝑐)⟩)))
424		fveq2 6881	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 ⊢ ((1st ‘𝑤) = (1st ‘𝑐) → ((𝑑 sSet ⟨(1st ‘𝑐), (2nd ‘𝑐)⟩)‘(1st ‘𝑤)) = ((𝑑 sSet ⟨(1st ‘𝑐), (2nd ‘𝑐)⟩)‘(1st ‘𝑐)))
425	424	adantl 481	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 ⊢ (((((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) ∧ 𝑑 ∈ (𝑁 ↑m 𝑁)) ∧ 𝑤 ∈ (𝑏 ∪ {𝑐})) ∧ (1st ‘𝑤) = (1st ‘𝑐)) → ((𝑑 sSet ⟨(1st ‘𝑐), (2nd ‘𝑐)⟩)‘(1st ‘𝑤)) = ((𝑑 sSet ⟨(1st ‘𝑐), (2nd ‘𝑐)⟩)‘(1st ‘𝑐)))
426		vex 3470	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 ⊢ 𝑑 ∈ V
427		fvex 6894	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 ⊢ (1st ‘𝑐) ∈ V
428		fvex 6894	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 ⊢ (2nd ‘𝑐) ∈ V
429		fvsetsid 17100	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 ⊢ ((𝑑 ∈ V ∧ (1st ‘𝑐) ∈ V ∧ (2nd ‘𝑐) ∈ V) → ((𝑑 sSet ⟨(1st ‘𝑐), (2nd ‘𝑐)⟩)‘(1st ‘𝑐)) = (2nd ‘𝑐))
430	426, 427, 428, 429	mp3an 1457	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 ⊢ ((𝑑 sSet ⟨(1st ‘𝑐), (2nd ‘𝑐)⟩)‘(1st ‘𝑐)) = (2nd ‘𝑐)
431	425, 430	eqtrdi 2780	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 ⊢ (((((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) ∧ 𝑑 ∈ (𝑁 ↑m 𝑁)) ∧ 𝑤 ∈ (𝑏 ∪ {𝑐})) ∧ (1st ‘𝑤) = (1st ‘𝑐)) → ((𝑑 sSet ⟨(1st ‘𝑐), (2nd ‘𝑐)⟩)‘(1st ‘𝑤)) = (2nd ‘𝑐))
432	431	eqeq1d 2726	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 ⊢ (((((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) ∧ 𝑑 ∈ (𝑁 ↑m 𝑁)) ∧ 𝑤 ∈ (𝑏 ∪ {𝑐})) ∧ (1st ‘𝑤) = (1st ‘𝑐)) → (((𝑑 sSet ⟨(1st ‘𝑐), (2nd ‘𝑐)⟩)‘(1st ‘𝑤)) = (2nd ‘𝑤) ↔ (2nd ‘𝑐) = (2nd ‘𝑤)))
433		eqcom 2731	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 ⊢ ((2nd ‘𝑐) = (2nd ‘𝑤) ↔ (2nd ‘𝑤) = (2nd ‘𝑐))
434	432, 433	bitrdi 287	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ⊢ (((((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) ∧ 𝑑 ∈ (𝑁 ↑m 𝑁)) ∧ 𝑤 ∈ (𝑏 ∪ {𝑐})) ∧ (1st ‘𝑤) = (1st ‘𝑐)) → (((𝑑 sSet ⟨(1st ‘𝑐), (2nd ‘𝑐)⟩)‘(1st ‘𝑤)) = (2nd ‘𝑤) ↔ (2nd ‘𝑤) = (2nd ‘𝑐)))
435	407, 423, 434	3bitr2rd 308	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ⊢ (((((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) ∧ 𝑑 ∈ (𝑁 ↑m 𝑁)) ∧ 𝑤 ∈ (𝑏 ∪ {𝑐})) ∧ (1st ‘𝑤) = (1st ‘𝑐)) → ((2nd ‘𝑤) = (2nd ‘𝑐) ↔ 𝑤 ∈ (𝑑 sSet ⟨(1st ‘𝑐), (2nd ‘𝑐)⟩)))
436	122	ad3antrrr 727	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ⊢ (((((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) ∧ 𝑑 ∈ (𝑁 ↑m 𝑁)) ∧ 𝑤 ∈ (𝑏 ∪ {𝑐})) ∧ (1st ‘𝑤) = (1st ‘𝑐)) → 𝑐 ∈ (𝑁 × 𝑁))
437		xpopth 8009	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ⊢ ((𝑤 ∈ (𝑁 × 𝑁) ∧ 𝑐 ∈ (𝑁 × 𝑁)) → (((1st ‘𝑤) = (1st ‘𝑐) ∧ (2nd ‘𝑤) = (2nd ‘𝑐)) ↔ 𝑤 = 𝑐))
438	404, 436, 437	syl2anc 583	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ⊢ (((((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) ∧ 𝑑 ∈ (𝑁 ↑m 𝑁)) ∧ 𝑤 ∈ (𝑏 ∪ {𝑐})) ∧ (1st ‘𝑤) = (1st ‘𝑐)) → (((1st ‘𝑤) = (1st ‘𝑐) ∧ (2nd ‘𝑤) = (2nd ‘𝑐)) ↔ 𝑤 = 𝑐))
439	400, 435, 438	3bitr3rd 310	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ (((((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) ∧ 𝑑 ∈ (𝑁 ↑m 𝑁)) ∧ 𝑤 ∈ (𝑏 ∪ {𝑐})) ∧ (1st ‘𝑤) = (1st ‘𝑐)) → (𝑤 = 𝑐 ↔ 𝑤 ∈ (𝑑 sSet ⟨(1st ‘𝑐), (2nd ‘𝑐)⟩)))
440	439	ifbid 4543	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ (((((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) ∧ 𝑑 ∈ (𝑁 ↑m 𝑁)) ∧ 𝑤 ∈ (𝑏 ∪ {𝑐})) ∧ (1st ‘𝑤) = (1st ‘𝑐)) → if(𝑤 = 𝑐, 1 , 0 ) = if(𝑤 ∈ (𝑑 sSet ⟨(1st ‘𝑐), (2nd ‘𝑐)⟩), 1 , 0 ))
441	398, 440	eqtrd 2764	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ (((((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) ∧ 𝑑 ∈ (𝑁 ↑m 𝑁)) ∧ 𝑤 ∈ (𝑏 ∪ {𝑐})) ∧ (1st ‘𝑤) = (1st ‘𝑐)) → if((1st ‘𝑤) = (1st ‘𝑐), if(𝑤 = 𝑐, 1 , 0 ), (𝑎‘𝑤)) = if(𝑤 ∈ (𝑑 sSet ⟨(1st ‘𝑐), (2nd ‘𝑐)⟩), 1 , 0 ))
442	441	a1d 25	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (((((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) ∧ 𝑑 ∈ (𝑁 ↑m 𝑁)) ∧ 𝑤 ∈ (𝑏 ∪ {𝑐})) ∧ (1st ‘𝑤) = (1st ‘𝑐)) → ((𝑤 ∈ 𝑏 → (𝑎‘𝑤) = if(𝑤 ∈ 𝑑, 1 , 0 )) → if((1st ‘𝑤) = (1st ‘𝑐), if(𝑤 = 𝑐, 1 , 0 ), (𝑎‘𝑤)) = if(𝑤 ∈ (𝑑 sSet ⟨(1st ‘𝑐), (2nd ‘𝑐)⟩), 1 , 0 )))
443		elsni 4637	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 ⊢ (𝑤 ∈ {𝑐} → 𝑤 = 𝑐)
444	443	fveq2d 6885	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ⊢ (𝑤 ∈ {𝑐} → (1st ‘𝑤) = (1st ‘𝑐))
445	444	con3i 154	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ⊢ (¬ (1st ‘𝑤) = (1st ‘𝑐) → ¬ 𝑤 ∈ {𝑐})
446	445	adantl 481	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ ((𝑤 ∈ (𝑏 ∪ {𝑐}) ∧ ¬ (1st ‘𝑤) = (1st ‘𝑐)) → ¬ 𝑤 ∈ {𝑐})
447		elun 4140	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ⊢ (𝑤 ∈ (𝑏 ∪ {𝑐}) ↔ (𝑤 ∈ 𝑏 ∨ 𝑤 ∈ {𝑐}))
448	447	biimpi 215	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ⊢ (𝑤 ∈ (𝑏 ∪ {𝑐}) → (𝑤 ∈ 𝑏 ∨ 𝑤 ∈ {𝑐}))
449	448	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ ((𝑤 ∈ (𝑏 ∪ {𝑐}) ∧ ¬ (1st ‘𝑤) = (1st ‘𝑐)) → (𝑤 ∈ 𝑏 ∨ 𝑤 ∈ {𝑐}))
450		orel2 887	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ (¬ 𝑤 ∈ {𝑐} → ((𝑤 ∈ 𝑏 ∨ 𝑤 ∈ {𝑐}) → 𝑤 ∈ 𝑏))
451	446, 449, 450	sylc 65	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ ((𝑤 ∈ (𝑏 ∪ {𝑐}) ∧ ¬ (1st ‘𝑤) = (1st ‘𝑐)) → 𝑤 ∈ 𝑏)
452	451	adantll 711	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ (((((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) ∧ 𝑑 ∈ (𝑁 ↑m 𝑁)) ∧ 𝑤 ∈ (𝑏 ∪ {𝑐})) ∧ ¬ (1st ‘𝑤) = (1st ‘𝑐)) → 𝑤 ∈ 𝑏)
453		iffalse 4529	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ⊢ (¬ (1st ‘𝑤) = (1st ‘𝑐) → if((1st ‘𝑤) = (1st ‘𝑐), if(𝑤 = 𝑐, 1 , 0 ), if(𝑤 ∈ 𝑑, 1 , 0 )) = if(𝑤 ∈ 𝑑, 1 , 0 ))
454	453	adantl 481	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ (((((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) ∧ 𝑑 ∈ (𝑁 ↑m 𝑁)) ∧ 𝑤 ∈ (𝑏 ∪ {𝑐})) ∧ ¬ (1st ‘𝑤) = (1st ‘𝑐)) → if((1st ‘𝑤) = (1st ‘𝑐), if(𝑤 = 𝑐, 1 , 0 ), if(𝑤 ∈ 𝑑, 1 , 0 )) = if(𝑤 ∈ 𝑑, 1 , 0 ))
455		setsres 17110	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 ⊢ (𝑑 ∈ V → ((𝑑 sSet ⟨(1st ‘𝑐), (2nd ‘𝑐)⟩) ↾ (V ∖ {(1st ‘𝑐)})) = (𝑑 ↾ (V ∖ {(1st ‘𝑐)})))
456	455	eleq2d 2811	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 ⊢ (𝑑 ∈ V → (⟨(1st ‘𝑤), (2nd ‘𝑤)⟩ ∈ ((𝑑 sSet ⟨(1st ‘𝑐), (2nd ‘𝑐)⟩) ↾ (V ∖ {(1st ‘𝑐)})) ↔ ⟨(1st ‘𝑤), (2nd ‘𝑤)⟩ ∈ (𝑑 ↾ (V ∖ {(1st ‘𝑐)}))))
457	426, 456	mp1i 13	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ⊢ (((((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) ∧ 𝑑 ∈ (𝑁 ↑m 𝑁)) ∧ 𝑤 ∈ (𝑏 ∪ {𝑐})) ∧ ¬ (1st ‘𝑤) = (1st ‘𝑐)) → (⟨(1st ‘𝑤), (2nd ‘𝑤)⟩ ∈ ((𝑑 sSet ⟨(1st ‘𝑐), (2nd ‘𝑐)⟩) ↾ (V ∖ {(1st ‘𝑐)})) ↔ ⟨(1st ‘𝑤), (2nd ‘𝑤)⟩ ∈ (𝑑 ↾ (V ∖ {(1st ‘𝑐)}))))
458		fvex 6894	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 ⊢ (1st ‘𝑤) ∈ V
459	458	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 ⊢ (¬ (1st ‘𝑤) = (1st ‘𝑐) → (1st ‘𝑤) ∈ V)
460		neqne 2940	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 ⊢ (¬ (1st ‘𝑤) = (1st ‘𝑐) → (1st ‘𝑤) ≠ (1st ‘𝑐))
461		eldifsn 4782	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 ⊢ ((1st ‘𝑤) ∈ (V ∖ {(1st ‘𝑐)}) ↔ ((1st ‘𝑤) ∈ V ∧ (1st ‘𝑤) ≠ (1st ‘𝑐)))
462	459, 460, 461	sylanbrc 582	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 ⊢ (¬ (1st ‘𝑤) = (1st ‘𝑐) → (1st ‘𝑤) ∈ (V ∖ {(1st ‘𝑐)}))
463		fvex 6894	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 ⊢ (2nd ‘𝑤) ∈ V
464	463	opres 5981	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 ⊢ ((1st ‘𝑤) ∈ (V ∖ {(1st ‘𝑐)}) → (⟨(1st ‘𝑤), (2nd ‘𝑤)⟩ ∈ ((𝑑 sSet ⟨(1st ‘𝑐), (2nd ‘𝑐)⟩) ↾ (V ∖ {(1st ‘𝑐)})) ↔ ⟨(1st ‘𝑤), (2nd ‘𝑤)⟩ ∈ (𝑑 sSet ⟨(1st ‘𝑐), (2nd ‘𝑐)⟩)))
465	464	adantl 481	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 ⊢ ((𝑤 ∈ (𝑁 × 𝑁) ∧ (1st ‘𝑤) ∈ (V ∖ {(1st ‘𝑐)})) → (⟨(1st ‘𝑤), (2nd ‘𝑤)⟩ ∈ ((𝑑 sSet ⟨(1st ‘𝑐), (2nd ‘𝑐)⟩) ↾ (V ∖ {(1st ‘𝑐)})) ↔ ⟨(1st ‘𝑤), (2nd ‘𝑤)⟩ ∈ (𝑑 sSet ⟨(1st ‘𝑐), (2nd ‘𝑐)⟩)))
466		1st2nd2 8007	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 ⊢ (𝑤 ∈ (𝑁 × 𝑁) → 𝑤 = ⟨(1st ‘𝑤), (2nd ‘𝑤)⟩)
467	466	eleq1d 2810	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 ⊢ (𝑤 ∈ (𝑁 × 𝑁) → (𝑤 ∈ (𝑑 sSet ⟨(1st ‘𝑐), (2nd ‘𝑐)⟩) ↔ ⟨(1st ‘𝑤), (2nd ‘𝑤)⟩ ∈ (𝑑 sSet ⟨(1st ‘𝑐), (2nd ‘𝑐)⟩)))
468	467	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 ⊢ ((𝑤 ∈ (𝑁 × 𝑁) ∧ (1st ‘𝑤) ∈ (V ∖ {(1st ‘𝑐)})) → (𝑤 ∈ (𝑑 sSet ⟨(1st ‘𝑐), (2nd ‘𝑐)⟩) ↔ ⟨(1st ‘𝑤), (2nd ‘𝑤)⟩ ∈ (𝑑 sSet ⟨(1st ‘𝑐), (2nd ‘𝑐)⟩)))
469	465, 468	bitr4d 282	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 ⊢ ((𝑤 ∈ (𝑁 × 𝑁) ∧ (1st ‘𝑤) ∈ (V ∖ {(1st ‘𝑐)})) → (⟨(1st ‘𝑤), (2nd ‘𝑤)⟩ ∈ ((𝑑 sSet ⟨(1st ‘𝑐), (2nd ‘𝑐)⟩) ↾ (V ∖ {(1st ‘𝑐)})) ↔ 𝑤 ∈ (𝑑 sSet ⟨(1st ‘𝑐), (2nd ‘𝑐)⟩)))
470	403, 462, 469	syl2an 595	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ⊢ (((((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) ∧ 𝑑 ∈ (𝑁 ↑m 𝑁)) ∧ 𝑤 ∈ (𝑏 ∪ {𝑐})) ∧ ¬ (1st ‘𝑤) = (1st ‘𝑐)) → (⟨(1st ‘𝑤), (2nd ‘𝑤)⟩ ∈ ((𝑑 sSet ⟨(1st ‘𝑐), (2nd ‘𝑐)⟩) ↾ (V ∖ {(1st ‘𝑐)})) ↔ 𝑤 ∈ (𝑑 sSet ⟨(1st ‘𝑐), (2nd ‘𝑐)⟩)))
471	463	opres 5981	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 ⊢ ((1st ‘𝑤) ∈ (V ∖ {(1st ‘𝑐)}) → (⟨(1st ‘𝑤), (2nd ‘𝑤)⟩ ∈ (𝑑 ↾ (V ∖ {(1st ‘𝑐)})) ↔ ⟨(1st ‘𝑤), (2nd ‘𝑤)⟩ ∈ 𝑑))
472	471	adantl 481	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 ⊢ ((𝑤 ∈ (𝑁 × 𝑁) ∧ (1st ‘𝑤) ∈ (V ∖ {(1st ‘𝑐)})) → (⟨(1st ‘𝑤), (2nd ‘𝑤)⟩ ∈ (𝑑 ↾ (V ∖ {(1st ‘𝑐)})) ↔ ⟨(1st ‘𝑤), (2nd ‘𝑤)⟩ ∈ 𝑑))
473	466	eleq1d 2810	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 ⊢ (𝑤 ∈ (𝑁 × 𝑁) → (𝑤 ∈ 𝑑 ↔ ⟨(1st ‘𝑤), (2nd ‘𝑤)⟩ ∈ 𝑑))
474	473	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 ⊢ ((𝑤 ∈ (𝑁 × 𝑁) ∧ (1st ‘𝑤) ∈ (V ∖ {(1st ‘𝑐)})) → (𝑤 ∈ 𝑑 ↔ ⟨(1st ‘𝑤), (2nd ‘𝑤)⟩ ∈ 𝑑))
475	472, 474	bitr4d 282	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 ⊢ ((𝑤 ∈ (𝑁 × 𝑁) ∧ (1st ‘𝑤) ∈ (V ∖ {(1st ‘𝑐)})) → (⟨(1st ‘𝑤), (2nd ‘𝑤)⟩ ∈ (𝑑 ↾ (V ∖ {(1st ‘𝑐)})) ↔ 𝑤 ∈ 𝑑))
476	403, 462, 475	syl2an 595	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ⊢ (((((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) ∧ 𝑑 ∈ (𝑁 ↑m 𝑁)) ∧ 𝑤 ∈ (𝑏 ∪ {𝑐})) ∧ ¬ (1st ‘𝑤) = (1st ‘𝑐)) → (⟨(1st ‘𝑤), (2nd ‘𝑤)⟩ ∈ (𝑑 ↾ (V ∖ {(1st ‘𝑐)})) ↔ 𝑤 ∈ 𝑑))
477	457, 470, 476	3bitr3rd 310	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ⊢ (((((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) ∧ 𝑑 ∈ (𝑁 ↑m 𝑁)) ∧ 𝑤 ∈ (𝑏 ∪ {𝑐})) ∧ ¬ (1st ‘𝑤) = (1st ‘𝑐)) → (𝑤 ∈ 𝑑 ↔ 𝑤 ∈ (𝑑 sSet ⟨(1st ‘𝑐), (2nd ‘𝑐)⟩)))
478	477	ifbid 4543	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ (((((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) ∧ 𝑑 ∈ (𝑁 ↑m 𝑁)) ∧ 𝑤 ∈ (𝑏 ∪ {𝑐})) ∧ ¬ (1st ‘𝑤) = (1st ‘𝑐)) → if(𝑤 ∈ 𝑑, 1 , 0 ) = if(𝑤 ∈ (𝑑 sSet ⟨(1st ‘𝑐), (2nd ‘𝑐)⟩), 1 , 0 ))
479	454, 478	eqtrd 2764	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ (((((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) ∧ 𝑑 ∈ (𝑁 ↑m 𝑁)) ∧ 𝑤 ∈ (𝑏 ∪ {𝑐})) ∧ ¬ (1st ‘𝑤) = (1st ‘𝑐)) → if((1st ‘𝑤) = (1st ‘𝑐), if(𝑤 = 𝑐, 1 , 0 ), if(𝑤 ∈ 𝑑, 1 , 0 )) = if(𝑤 ∈ (𝑑 sSet ⟨(1st ‘𝑐), (2nd ‘𝑐)⟩), 1 , 0 ))
480		ifeq2 4525	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ ((𝑎‘𝑤) = if(𝑤 ∈ 𝑑, 1 , 0 ) → if((1st ‘𝑤) = (1st ‘𝑐), if(𝑤 = 𝑐, 1 , 0 ), (𝑎‘𝑤)) = if((1st ‘𝑤) = (1st ‘𝑐), if(𝑤 = 𝑐, 1 , 0 ), if(𝑤 ∈ 𝑑, 1 , 0 )))
481	480	eqeq1d 2726	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ ((𝑎‘𝑤) = if(𝑤 ∈ 𝑑, 1 , 0 ) → (if((1st ‘𝑤) = (1st ‘𝑐), if(𝑤 = 𝑐, 1 , 0 ), (𝑎‘𝑤)) = if(𝑤 ∈ (𝑑 sSet ⟨(1st ‘𝑐), (2nd ‘𝑐)⟩), 1 , 0 ) ↔ if((1st ‘𝑤) = (1st ‘𝑐), if(𝑤 = 𝑐, 1 , 0 ), if(𝑤 ∈ 𝑑, 1 , 0 )) = if(𝑤 ∈ (𝑑 sSet ⟨(1st ‘𝑐), (2nd ‘𝑐)⟩), 1 , 0 )))
482	479, 481	syl5ibrcom 246	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ (((((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) ∧ 𝑑 ∈ (𝑁 ↑m 𝑁)) ∧ 𝑤 ∈ (𝑏 ∪ {𝑐})) ∧ ¬ (1st ‘𝑤) = (1st ‘𝑐)) → ((𝑎‘𝑤) = if(𝑤 ∈ 𝑑, 1 , 0 ) → if((1st ‘𝑤) = (1st ‘𝑐), if(𝑤 = 𝑐, 1 , 0 ), (𝑎‘𝑤)) = if(𝑤 ∈ (𝑑 sSet ⟨(1st ‘𝑐), (2nd ‘𝑐)⟩), 1 , 0 )))
483	452, 482	embantd 59	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (((((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) ∧ 𝑑 ∈ (𝑁 ↑m 𝑁)) ∧ 𝑤 ∈ (𝑏 ∪ {𝑐})) ∧ ¬ (1st ‘𝑤) = (1st ‘𝑐)) → ((𝑤 ∈ 𝑏 → (𝑎‘𝑤) = if(𝑤 ∈ 𝑑, 1 , 0 )) → if((1st ‘𝑤) = (1st ‘𝑐), if(𝑤 = 𝑐, 1 , 0 ), (𝑎‘𝑤)) = if(𝑤 ∈ (𝑑 sSet ⟨(1st ‘𝑐), (2nd ‘𝑐)⟩), 1 , 0 )))
484	442, 483	pm2.61dan 810	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ ((((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) ∧ 𝑑 ∈ (𝑁 ↑m 𝑁)) ∧ 𝑤 ∈ (𝑏 ∪ {𝑐})) → ((𝑤 ∈ 𝑏 → (𝑎‘𝑤) = if(𝑤 ∈ 𝑑, 1 , 0 )) → if((1st ‘𝑤) = (1st ‘𝑐), if(𝑤 = 𝑐, 1 , 0 ), (𝑎‘𝑤)) = if(𝑤 ∈ (𝑑 sSet ⟨(1st ‘𝑐), (2nd ‘𝑐)⟩), 1 , 0 )))
485		fveqeq2 6890	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ (𝑒 = 𝑤 → ((1st ‘𝑒) = (1st ‘𝑐) ↔ (1st ‘𝑤) = (1st ‘𝑐)))
486		equequ1 2020	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ⊢ (𝑒 = 𝑤 → (𝑒 = 𝑐 ↔ 𝑤 = 𝑐))
487	486	ifbid 4543	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ (𝑒 = 𝑤 → if(𝑒 = 𝑐, 1 , 0 ) = if(𝑤 = 𝑐, 1 , 0 ))
488		fveq2 6881	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ (𝑒 = 𝑤 → (𝑎‘𝑒) = (𝑎‘𝑤))
489	485, 487, 488	ifbieq12d 4548	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ (𝑒 = 𝑤 → if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒)) = if((1st ‘𝑤) = (1st ‘𝑐), if(𝑤 = 𝑐, 1 , 0 ), (𝑎‘𝑤)))
490		eqid 2724	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ (𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒))) = (𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒)))
491	165, 162	ifex 4570	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ if(𝑤 = 𝑐, 1 , 0 ) ∈ V
492		fvex 6894	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ (𝑎‘𝑤) ∈ V
493	491, 492	ifex 4570	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ if((1st ‘𝑤) = (1st ‘𝑐), if(𝑤 = 𝑐, 1 , 0 ), (𝑎‘𝑤)) ∈ V
494	489, 490, 493	fvmpt 6988	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ (𝑤 ∈ (𝑁 × 𝑁) → ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒)))‘𝑤) = if((1st ‘𝑤) = (1st ‘𝑐), if(𝑤 = 𝑐, 1 , 0 ), (𝑎‘𝑤)))
495	494	eqeq1d 2726	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (𝑤 ∈ (𝑁 × 𝑁) → (((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒)))‘𝑤) = if(𝑤 ∈ (𝑑 sSet ⟨(1st ‘𝑐), (2nd ‘𝑐)⟩), 1 , 0 ) ↔ if((1st ‘𝑤) = (1st ‘𝑐), if(𝑤 = 𝑐, 1 , 0 ), (𝑎‘𝑤)) = if(𝑤 ∈ (𝑑 sSet ⟨(1st ‘𝑐), (2nd ‘𝑐)⟩), 1 , 0 )))
496	403, 495	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ ((((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) ∧ 𝑑 ∈ (𝑁 ↑m 𝑁)) ∧ 𝑤 ∈ (𝑏 ∪ {𝑐})) → (((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒)))‘𝑤) = if(𝑤 ∈ (𝑑 sSet ⟨(1st ‘𝑐), (2nd ‘𝑐)⟩), 1 , 0 ) ↔ if((1st ‘𝑤) = (1st ‘𝑐), if(𝑤 = 𝑐, 1 , 0 ), (𝑎‘𝑤)) = if(𝑤 ∈ (𝑑 sSet ⟨(1st ‘𝑐), (2nd ‘𝑐)⟩), 1 , 0 )))
497	484, 496	sylibrd 259	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) ∧ 𝑑 ∈ (𝑁 ↑m 𝑁)) ∧ 𝑤 ∈ (𝑏 ∪ {𝑐})) → ((𝑤 ∈ 𝑏 → (𝑎‘𝑤) = if(𝑤 ∈ 𝑑, 1 , 0 )) → ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒)))‘𝑤) = if(𝑤 ∈ (𝑑 sSet ⟨(1st ‘𝑐), (2nd ‘𝑐)⟩), 1 , 0 )))
498	497	ralimdva 3159	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) ∧ 𝑑 ∈ (𝑁 ↑m 𝑁)) → (∀𝑤 ∈ (𝑏 ∪ {𝑐})(𝑤 ∈ 𝑏 → (𝑎‘𝑤) = if(𝑤 ∈ 𝑑, 1 , 0 )) → ∀𝑤 ∈ (𝑏 ∪ {𝑐})((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒)))‘𝑤) = if(𝑤 ∈ (𝑑 sSet ⟨(1st ‘𝑐), (2nd ‘𝑐)⟩), 1 , 0 )))
499	396, 498	biimtrid 241	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) ∧ 𝑑 ∈ (𝑁 ↑m 𝑁)) → (∀𝑤 ∈ 𝑏 (𝑎‘𝑤) = if(𝑤 ∈ 𝑑, 1 , 0 ) → ∀𝑤 ∈ (𝑏 ∪ {𝑐})((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒)))‘𝑤) = if(𝑤 ∈ (𝑑 sSet ⟨(1st ‘𝑐), (2nd ‘𝑐)⟩), 1 , 0 )))
500	499	impr 454	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) ∧ (𝑑 ∈ (𝑁 ↑m 𝑁) ∧ ∀𝑤 ∈ 𝑏 (𝑎‘𝑤) = if(𝑤 ∈ 𝑑, 1 , 0 ))) → ∀𝑤 ∈ (𝑏 ∪ {𝑐})((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒)))‘𝑤) = if(𝑤 ∈ (𝑑 sSet ⟨(1st ‘𝑐), (2nd ‘𝑐)⟩), 1 , 0 ))
501	500	3adantr1 1166	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) ∧ ((𝑏 ∪ {𝑐}) ∈ 𝑌 ∧ 𝑑 ∈ (𝑁 ↑m 𝑁) ∧ ∀𝑤 ∈ 𝑏 (𝑎‘𝑤) = if(𝑤 ∈ 𝑑, 1 , 0 ))) → ∀𝑤 ∈ (𝑏 ∪ {𝑐})((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒)))‘𝑤) = if(𝑤 ∈ (𝑑 sSet ⟨(1st ‘𝑐), (2nd ‘𝑐)⟩), 1 , 0 ))
502	348	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) ∧ ((𝑏 ∪ {𝑐}) ∈ 𝑌 ∧ 𝑑 ∈ (𝑁 ↑m 𝑁) ∧ ∀𝑤 ∈ 𝑏 (𝑎‘𝑤) = if(𝑤 ∈ 𝑑, 1 , 0 ))) → (𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒))) ∈ 𝐵)
503		simpr2 1192	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) ∧ ((𝑏 ∪ {𝑐}) ∈ 𝑌 ∧ 𝑑 ∈ (𝑁 ↑m 𝑁) ∧ ∀𝑤 ∈ 𝑏 (𝑎‘𝑤) = if(𝑤 ∈ 𝑑, 1 , 0 ))) → 𝑑 ∈ (𝑁 ↑m 𝑁))
504	503, 409	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) ∧ ((𝑏 ∪ {𝑐}) ∈ 𝑌 ∧ 𝑑 ∈ (𝑁 ↑m 𝑁) ∧ ∀𝑤 ∈ 𝑏 (𝑎‘𝑤) = if(𝑤 ∈ 𝑑, 1 , 0 ))) → 𝑑:𝑁⟶𝑁)
505	124	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) ∧ ((𝑏 ∪ {𝑐}) ∈ 𝑌 ∧ 𝑑 ∈ (𝑁 ↑m 𝑁) ∧ ∀𝑤 ∈ 𝑏 (𝑎‘𝑤) = if(𝑤 ∈ 𝑑, 1 , 0 ))) → (1st ‘𝑐) ∈ 𝑁)
506	413	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) ∧ ((𝑏 ∪ {𝑐}) ∈ 𝑌 ∧ 𝑑 ∈ (𝑁 ↑m 𝑁) ∧ ∀𝑤 ∈ 𝑏 (𝑎‘𝑤) = if(𝑤 ∈ 𝑑, 1 , 0 ))) → (2nd ‘𝑐) ∈ 𝑁)
507	503, 504, 505, 506, 415	syl211anc 1373	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) ∧ ((𝑏 ∪ {𝑐}) ∈ 𝑌 ∧ 𝑑 ∈ (𝑁 ↑m 𝑁) ∧ ∀𝑤 ∈ 𝑏 (𝑎‘𝑤) = if(𝑤 ∈ 𝑑, 1 , 0 ))) → (𝑑 sSet ⟨(1st ‘𝑐), (2nd ‘𝑐)⟩):𝑁⟶𝑁)
508	158, 158	elmapd 8830	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) → ((𝑑 sSet ⟨(1st ‘𝑐), (2nd ‘𝑐)⟩) ∈ (𝑁 ↑m 𝑁) ↔ (𝑑 sSet ⟨(1st ‘𝑐), (2nd ‘𝑐)⟩):𝑁⟶𝑁))
509	508	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) ∧ ((𝑏 ∪ {𝑐}) ∈ 𝑌 ∧ 𝑑 ∈ (𝑁 ↑m 𝑁) ∧ ∀𝑤 ∈ 𝑏 (𝑎‘𝑤) = if(𝑤 ∈ 𝑑, 1 , 0 ))) → ((𝑑 sSet ⟨(1st ‘𝑐), (2nd ‘𝑐)⟩) ∈ (𝑁 ↑m 𝑁) ↔ (𝑑 sSet ⟨(1st ‘𝑐), (2nd ‘𝑐)⟩):𝑁⟶𝑁))
510	507, 509	mpbird 257	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) ∧ ((𝑏 ∪ {𝑐}) ∈ 𝑌 ∧ 𝑑 ∈ (𝑁 ↑m 𝑁) ∧ ∀𝑤 ∈ 𝑏 (𝑎‘𝑤) = if(𝑤 ∈ 𝑑, 1 , 0 ))) → (𝑑 sSet ⟨(1st ‘𝑐), (2nd ‘𝑐)⟩) ∈ (𝑁 ↑m 𝑁))
511		simpr1 1191	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) ∧ ((𝑏 ∪ {𝑐}) ∈ 𝑌 ∧ 𝑑 ∈ (𝑁 ↑m 𝑁) ∧ ∀𝑤 ∈ 𝑏 (𝑎‘𝑤) = if(𝑤 ∈ 𝑑, 1 , 0 ))) → (𝑏 ∪ {𝑐}) ∈ 𝑌)
512		raleq 3314	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (𝑥 = (𝑏 ∪ {𝑐}) → (∀𝑤 ∈ 𝑥 (𝑦‘𝑤) = if(𝑤 ∈ 𝑧, 1 , 0 ) ↔ ∀𝑤 ∈ (𝑏 ∪ {𝑐})(𝑦‘𝑤) = if(𝑤 ∈ 𝑧, 1 , 0 )))
513	512	imbi1d 341	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (𝑥 = (𝑏 ∪ {𝑐}) → ((∀𝑤 ∈ 𝑥 (𝑦‘𝑤) = if(𝑤 ∈ 𝑧, 1 , 0 ) → (𝐷‘𝑦) = 0 ) ↔ (∀𝑤 ∈ (𝑏 ∪ {𝑐})(𝑦‘𝑤) = if(𝑤 ∈ 𝑧, 1 , 0 ) → (𝐷‘𝑦) = 0 )))
514	513	2ralbidv 3210	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝑥 = (𝑏 ∪ {𝑐}) → (∀𝑦 ∈ 𝐵 ∀𝑧 ∈ (𝑁 ↑m 𝑁)(∀𝑤 ∈ 𝑥 (𝑦‘𝑤) = if(𝑤 ∈ 𝑧, 1 , 0 ) → (𝐷‘𝑦) = 0 ) ↔ ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ (𝑁 ↑m 𝑁)(∀𝑤 ∈ (𝑏 ∪ {𝑐})(𝑦‘𝑤) = if(𝑤 ∈ 𝑧, 1 , 0 ) → (𝐷‘𝑦) = 0 )))
515	514, 72	elab2g 3662	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((𝑏 ∪ {𝑐}) ∈ 𝑌 → ((𝑏 ∪ {𝑐}) ∈ 𝑌 ↔ ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ (𝑁 ↑m 𝑁)(∀𝑤 ∈ (𝑏 ∪ {𝑐})(𝑦‘𝑤) = if(𝑤 ∈ 𝑧, 1 , 0 ) → (𝐷‘𝑦) = 0 )))
516	515	ibi 267	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((𝑏 ∪ {𝑐}) ∈ 𝑌 → ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ (𝑁 ↑m 𝑁)(∀𝑤 ∈ (𝑏 ∪ {𝑐})(𝑦‘𝑤) = if(𝑤 ∈ 𝑧, 1 , 0 ) → (𝐷‘𝑦) = 0 ))
517	511, 516	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) ∧ ((𝑏 ∪ {𝑐}) ∈ 𝑌 ∧ 𝑑 ∈ (𝑁 ↑m 𝑁) ∧ ∀𝑤 ∈ 𝑏 (𝑎‘𝑤) = if(𝑤 ∈ 𝑑, 1 , 0 ))) → ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ (𝑁 ↑m 𝑁)(∀𝑤 ∈ (𝑏 ∪ {𝑐})(𝑦‘𝑤) = if(𝑤 ∈ 𝑧, 1 , 0 ) → (𝐷‘𝑦) = 0 ))
518		fveq1 6880	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (𝑦 = (𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒))) → (𝑦‘𝑤) = ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒)))‘𝑤))
519	518	eqeq1d 2726	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝑦 = (𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒))) → ((𝑦‘𝑤) = if(𝑤 ∈ 𝑧, 1 , 0 ) ↔ ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒)))‘𝑤) = if(𝑤 ∈ 𝑧, 1 , 0 )))
520	519	ralbidv 3169	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝑦 = (𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒))) → (∀𝑤 ∈ (𝑏 ∪ {𝑐})(𝑦‘𝑤) = if(𝑤 ∈ 𝑧, 1 , 0 ) ↔ ∀𝑤 ∈ (𝑏 ∪ {𝑐})((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒)))‘𝑤) = if(𝑤 ∈ 𝑧, 1 , 0 )))
521		fveqeq2 6890	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝑦 = (𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒))) → ((𝐷‘𝑦) = 0 ↔ (𝐷‘(𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒)))) = 0 ))
522	520, 521	imbi12d 344	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑦 = (𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒))) → ((∀𝑤 ∈ (𝑏 ∪ {𝑐})(𝑦‘𝑤) = if(𝑤 ∈ 𝑧, 1 , 0 ) → (𝐷‘𝑦) = 0 ) ↔ (∀𝑤 ∈ (𝑏 ∪ {𝑐})((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒)))‘𝑤) = if(𝑤 ∈ 𝑧, 1 , 0 ) → (𝐷‘(𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒)))) = 0 )))
523		eleq2 2814	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (𝑧 = (𝑑 sSet ⟨(1st ‘𝑐), (2nd ‘𝑐)⟩) → (𝑤 ∈ 𝑧 ↔ 𝑤 ∈ (𝑑 sSet ⟨(1st ‘𝑐), (2nd ‘𝑐)⟩)))
524	523	ifbid 4543	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (𝑧 = (𝑑 sSet ⟨(1st ‘𝑐), (2nd ‘𝑐)⟩) → if(𝑤 ∈ 𝑧, 1 , 0 ) = if(𝑤 ∈ (𝑑 sSet ⟨(1st ‘𝑐), (2nd ‘𝑐)⟩), 1 , 0 ))
525	524	eqeq2d 2735	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝑧 = (𝑑 sSet ⟨(1st ‘𝑐), (2nd ‘𝑐)⟩) → (((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒)))‘𝑤) = if(𝑤 ∈ 𝑧, 1 , 0 ) ↔ ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒)))‘𝑤) = if(𝑤 ∈ (𝑑 sSet ⟨(1st ‘𝑐), (2nd ‘𝑐)⟩), 1 , 0 )))
526	525	ralbidv 3169	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝑧 = (𝑑 sSet ⟨(1st ‘𝑐), (2nd ‘𝑐)⟩) → (∀𝑤 ∈ (𝑏 ∪ {𝑐})((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒)))‘𝑤) = if(𝑤 ∈ 𝑧, 1 , 0 ) ↔ ∀𝑤 ∈ (𝑏 ∪ {𝑐})((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒)))‘𝑤) = if(𝑤 ∈ (𝑑 sSet ⟨(1st ‘𝑐), (2nd ‘𝑐)⟩), 1 , 0 )))
527	526	imbi1d 341	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑧 = (𝑑 sSet ⟨(1st ‘𝑐), (2nd ‘𝑐)⟩) → ((∀𝑤 ∈ (𝑏 ∪ {𝑐})((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒)))‘𝑤) = if(𝑤 ∈ 𝑧, 1 , 0 ) → (𝐷‘(𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒)))) = 0 ) ↔ (∀𝑤 ∈ (𝑏 ∪ {𝑐})((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒)))‘𝑤) = if(𝑤 ∈ (𝑑 sSet ⟨(1st ‘𝑐), (2nd ‘𝑐)⟩), 1 , 0 ) → (𝐷‘(𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒)))) = 0 )))
528	522, 527	rspc2va 3615	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒))) ∈ 𝐵 ∧ (𝑑 sSet ⟨(1st ‘𝑐), (2nd ‘𝑐)⟩) ∈ (𝑁 ↑m 𝑁)) ∧ ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ (𝑁 ↑m 𝑁)(∀𝑤 ∈ (𝑏 ∪ {𝑐})(𝑦‘𝑤) = if(𝑤 ∈ 𝑧, 1 , 0 ) → (𝐷‘𝑦) = 0 )) → (∀𝑤 ∈ (𝑏 ∪ {𝑐})((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒)))‘𝑤) = if(𝑤 ∈ (𝑑 sSet ⟨(1st ‘𝑐), (2nd ‘𝑐)⟩), 1 , 0 ) → (𝐷‘(𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒)))) = 0 ))
529	502, 510, 517, 528	syl21anc 835	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) ∧ ((𝑏 ∪ {𝑐}) ∈ 𝑌 ∧ 𝑑 ∈ (𝑁 ↑m 𝑁) ∧ ∀𝑤 ∈ 𝑏 (𝑎‘𝑤) = if(𝑤 ∈ 𝑑, 1 , 0 ))) → (∀𝑤 ∈ (𝑏 ∪ {𝑐})((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒)))‘𝑤) = if(𝑤 ∈ (𝑑 sSet ⟨(1st ‘𝑐), (2nd ‘𝑐)⟩), 1 , 0 ) → (𝐷‘(𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒)))) = 0 ))
530	501, 529	mpd 15	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) ∧ ((𝑏 ∪ {𝑐}) ∈ 𝑌 ∧ 𝑑 ∈ (𝑁 ↑m 𝑁) ∧ ∀𝑤 ∈ 𝑏 (𝑎‘𝑤) = if(𝑤 ∈ 𝑑, 1 , 0 ))) → (𝐷‘(𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒)))) = 0 )
531	530	oveq2d 7417	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) ∧ ((𝑏 ∪ {𝑐}) ∈ 𝑌 ∧ 𝑑 ∈ (𝑁 ↑m 𝑁) ∧ ∀𝑤 ∈ 𝑏 (𝑎‘𝑤) = if(𝑤 ∈ 𝑑, 1 , 0 ))) → (((𝑎‘𝑐)(-g‘𝑅)if(𝑐 ∈ 𝑑, 1 , 0 )) · (𝐷‘(𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒))))) = (((𝑎‘𝑐)(-g‘𝑅)if(𝑐 ∈ 𝑑, 1 , 0 )) · 0 ))
532	118	unssad 4179	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) → 𝑏 ⊆ (𝑁 × 𝑁))
533	532	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) ∧ ((𝑏 ∪ {𝑐}) ∈ 𝑌 ∧ 𝑑 ∈ (𝑁 ↑m 𝑁) ∧ ∀𝑤 ∈ 𝑏 (𝑎‘𝑤) = if(𝑤 ∈ 𝑑, 1 , 0 ))) → 𝑏 ⊆ (𝑁 × 𝑁))
534		simpr3 1193	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) ∧ ((𝑏 ∪ {𝑐}) ∈ 𝑌 ∧ 𝑑 ∈ (𝑁 ↑m 𝑁) ∧ ∀𝑤 ∈ 𝑏 (𝑎‘𝑤) = if(𝑤 ∈ 𝑑, 1 , 0 ))) → ∀𝑤 ∈ 𝑏 (𝑎‘𝑤) = if(𝑤 ∈ 𝑑, 1 , 0 ))
535		ssel2 3969	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ ((𝑏 ⊆ (𝑁 × 𝑁) ∧ 𝑤 ∈ 𝑏) → 𝑤 ∈ (𝑁 × 𝑁))
536	535	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (((𝑏 ⊆ (𝑁 × 𝑁) ∧ 𝑤 ∈ 𝑏) ∧ (𝑎‘𝑤) = if(𝑤 ∈ 𝑑, 1 , 0 )) → 𝑤 ∈ (𝑁 × 𝑁))
537		elequ1 2105	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ (𝑒 = 𝑤 → (𝑒 ∈ 𝑑 ↔ 𝑤 ∈ 𝑑))
538	537	ifbid 4543	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ (𝑒 = 𝑤 → if(𝑒 ∈ 𝑑, 1 , 0 ) = if(𝑤 ∈ 𝑑, 1 , 0 ))
539	486, 538, 488	ifbieq12d 4548	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (𝑒 = 𝑤 → if(𝑒 = 𝑐, if(𝑒 ∈ 𝑑, 1 , 0 ), (𝑎‘𝑒)) = if(𝑤 = 𝑐, if(𝑤 ∈ 𝑑, 1 , 0 ), (𝑎‘𝑤)))
540		eqid 2724	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒 ∈ 𝑑, 1 , 0 ), (𝑎‘𝑒))) = (𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒 ∈ 𝑑, 1 , 0 ), (𝑎‘𝑒)))
541	165, 162	ifex 4570	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ if(𝑤 ∈ 𝑑, 1 , 0 ) ∈ V
542	541, 492	ifex 4570	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ if(𝑤 = 𝑐, if(𝑤 ∈ 𝑑, 1 , 0 ), (𝑎‘𝑤)) ∈ V
543	539, 540, 542	fvmpt 6988	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (𝑤 ∈ (𝑁 × 𝑁) → ((𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒 ∈ 𝑑, 1 , 0 ), (𝑎‘𝑒)))‘𝑤) = if(𝑤 = 𝑐, if(𝑤 ∈ 𝑑, 1 , 0 ), (𝑎‘𝑤)))
544	536, 543	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (((𝑏 ⊆ (𝑁 × 𝑁) ∧ 𝑤 ∈ 𝑏) ∧ (𝑎‘𝑤) = if(𝑤 ∈ 𝑑, 1 , 0 )) → ((𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒 ∈ 𝑑, 1 , 0 ), (𝑎‘𝑒)))‘𝑤) = if(𝑤 = 𝑐, if(𝑤 ∈ 𝑑, 1 , 0 ), (𝑎‘𝑤)))
545		ifeq2 4525	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ ((𝑎‘𝑤) = if(𝑤 ∈ 𝑑, 1 , 0 ) → if(𝑤 = 𝑐, if(𝑤 ∈ 𝑑, 1 , 0 ), (𝑎‘𝑤)) = if(𝑤 = 𝑐, if(𝑤 ∈ 𝑑, 1 , 0 ), if(𝑤 ∈ 𝑑, 1 , 0 )))
546	545	adantl 481	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (((𝑏 ⊆ (𝑁 × 𝑁) ∧ 𝑤 ∈ 𝑏) ∧ (𝑎‘𝑤) = if(𝑤 ∈ 𝑑, 1 , 0 )) → if(𝑤 = 𝑐, if(𝑤 ∈ 𝑑, 1 , 0 ), (𝑎‘𝑤)) = if(𝑤 = 𝑐, if(𝑤 ∈ 𝑑, 1 , 0 ), if(𝑤 ∈ 𝑑, 1 , 0 )))
547		ifid 4560	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ if(𝑤 = 𝑐, if(𝑤 ∈ 𝑑, 1 , 0 ), if(𝑤 ∈ 𝑑, 1 , 0 )) = if(𝑤 ∈ 𝑑, 1 , 0 )
548	546, 547	eqtrdi 2780	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (((𝑏 ⊆ (𝑁 × 𝑁) ∧ 𝑤 ∈ 𝑏) ∧ (𝑎‘𝑤) = if(𝑤 ∈ 𝑑, 1 , 0 )) → if(𝑤 = 𝑐, if(𝑤 ∈ 𝑑, 1 , 0 ), (𝑎‘𝑤)) = if(𝑤 ∈ 𝑑, 1 , 0 ))
549	544, 548	eqtrd 2764	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (((𝑏 ⊆ (𝑁 × 𝑁) ∧ 𝑤 ∈ 𝑏) ∧ (𝑎‘𝑤) = if(𝑤 ∈ 𝑑, 1 , 0 )) → ((𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒 ∈ 𝑑, 1 , 0 ), (𝑎‘𝑒)))‘𝑤) = if(𝑤 ∈ 𝑑, 1 , 0 ))
550	549	ex 412	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((𝑏 ⊆ (𝑁 × 𝑁) ∧ 𝑤 ∈ 𝑏) → ((𝑎‘𝑤) = if(𝑤 ∈ 𝑑, 1 , 0 ) → ((𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒 ∈ 𝑑, 1 , 0 ), (𝑎‘𝑒)))‘𝑤) = if(𝑤 ∈ 𝑑, 1 , 0 )))
551	550	ralimdva 3159	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑏 ⊆ (𝑁 × 𝑁) → (∀𝑤 ∈ 𝑏 (𝑎‘𝑤) = if(𝑤 ∈ 𝑑, 1 , 0 ) → ∀𝑤 ∈ 𝑏 ((𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒 ∈ 𝑑, 1 , 0 ), (𝑎‘𝑒)))‘𝑤) = if(𝑤 ∈ 𝑑, 1 , 0 )))
552	533, 534, 551	sylc 65	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) ∧ ((𝑏 ∪ {𝑐}) ∈ 𝑌 ∧ 𝑑 ∈ (𝑁 ↑m 𝑁) ∧ ∀𝑤 ∈ 𝑏 (𝑎‘𝑤) = if(𝑤 ∈ 𝑑, 1 , 0 ))) → ∀𝑤 ∈ 𝑏 ((𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒 ∈ 𝑑, 1 , 0 ), (𝑎‘𝑒)))‘𝑤) = if(𝑤 ∈ 𝑑, 1 , 0 ))
553	142, 291	eqtrd 2764	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝑒 = 𝑐 → if(𝑒 = 𝑐, if(𝑒 ∈ 𝑑, 1 , 0 ), (𝑎‘𝑒)) = if(𝑐 ∈ 𝑑, 1 , 0 ))
554	165, 162	ifex 4570	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ if(𝑐 ∈ 𝑑, 1 , 0 ) ∈ V
555	553, 540, 554	fvmpt 6988	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑐 ∈ (𝑁 × 𝑁) → ((𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒 ∈ 𝑑, 1 , 0 ), (𝑎‘𝑒)))‘𝑐) = if(𝑐 ∈ 𝑑, 1 , 0 ))
556	122, 555	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) → ((𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒 ∈ 𝑑, 1 , 0 ), (𝑎‘𝑒)))‘𝑐) = if(𝑐 ∈ 𝑑, 1 , 0 ))
557	556	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) ∧ ((𝑏 ∪ {𝑐}) ∈ 𝑌 ∧ 𝑑 ∈ (𝑁 ↑m 𝑁) ∧ ∀𝑤 ∈ 𝑏 (𝑎‘𝑤) = if(𝑤 ∈ 𝑑, 1 , 0 ))) → ((𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒 ∈ 𝑑, 1 , 0 ), (𝑎‘𝑒)))‘𝑐) = if(𝑐 ∈ 𝑑, 1 , 0 ))
558		fveq2 6881	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝑤 = 𝑐 → ((𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒 ∈ 𝑑, 1 , 0 ), (𝑎‘𝑒)))‘𝑤) = ((𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒 ∈ 𝑑, 1 , 0 ), (𝑎‘𝑒)))‘𝑐))
559		elequ1 2105	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝑤 = 𝑐 → (𝑤 ∈ 𝑑 ↔ 𝑐 ∈ 𝑑))
560	559	ifbid 4543	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝑤 = 𝑐 → if(𝑤 ∈ 𝑑, 1 , 0 ) = if(𝑐 ∈ 𝑑, 1 , 0 ))
561	558, 560	eqeq12d 2740	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑤 = 𝑐 → (((𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒 ∈ 𝑑, 1 , 0 ), (𝑎‘𝑒)))‘𝑤) = if(𝑤 ∈ 𝑑, 1 , 0 ) ↔ ((𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒 ∈ 𝑑, 1 , 0 ), (𝑎‘𝑒)))‘𝑐) = if(𝑐 ∈ 𝑑, 1 , 0 )))
562	561	ralunsn 4886	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑐 ∈ V → (∀𝑤 ∈ (𝑏 ∪ {𝑐})((𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒 ∈ 𝑑, 1 , 0 ), (𝑎‘𝑒)))‘𝑤) = if(𝑤 ∈ 𝑑, 1 , 0 ) ↔ (∀𝑤 ∈ 𝑏 ((𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒 ∈ 𝑑, 1 , 0 ), (𝑎‘𝑒)))‘𝑤) = if(𝑤 ∈ 𝑑, 1 , 0 ) ∧ ((𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒 ∈ 𝑑, 1 , 0 ), (𝑎‘𝑒)))‘𝑐) = if(𝑐 ∈ 𝑑, 1 , 0 ))))
563	562	elv 3472	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (∀𝑤 ∈ (𝑏 ∪ {𝑐})((𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒 ∈ 𝑑, 1 , 0 ), (𝑎‘𝑒)))‘𝑤) = if(𝑤 ∈ 𝑑, 1 , 0 ) ↔ (∀𝑤 ∈ 𝑏 ((𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒 ∈ 𝑑, 1 , 0 ), (𝑎‘𝑒)))‘𝑤) = if(𝑤 ∈ 𝑑, 1 , 0 ) ∧ ((𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒 ∈ 𝑑, 1 , 0 ), (𝑎‘𝑒)))‘𝑐) = if(𝑐 ∈ 𝑑, 1 , 0 )))
564	552, 557, 563	sylanbrc 582	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) ∧ ((𝑏 ∪ {𝑐}) ∈ 𝑌 ∧ 𝑑 ∈ (𝑁 ↑m 𝑁) ∧ ∀𝑤 ∈ 𝑏 (𝑎‘𝑤) = if(𝑤 ∈ 𝑑, 1 , 0 ))) → ∀𝑤 ∈ (𝑏 ∪ {𝑐})((𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒 ∈ 𝑑, 1 , 0 ), (𝑎‘𝑒)))‘𝑤) = if(𝑤 ∈ 𝑑, 1 , 0 ))
565	223	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) ∧ ((𝑏 ∪ {𝑐}) ∈ 𝑌 ∧ 𝑑 ∈ (𝑁 ↑m 𝑁) ∧ ∀𝑤 ∈ 𝑏 (𝑎‘𝑤) = if(𝑤 ∈ 𝑑, 1 , 0 ))) → (𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒 ∈ 𝑑, 1 , 0 ), (𝑎‘𝑒))) ∈ 𝐵)
566		fveq1 6880	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝑦 = (𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒 ∈ 𝑑, 1 , 0 ), (𝑎‘𝑒))) → (𝑦‘𝑤) = ((𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒 ∈ 𝑑, 1 , 0 ), (𝑎‘𝑒)))‘𝑤))
567	566	eqeq1d 2726	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝑦 = (𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒 ∈ 𝑑, 1 , 0 ), (𝑎‘𝑒))) → ((𝑦‘𝑤) = if(𝑤 ∈ 𝑧, 1 , 0 ) ↔ ((𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒 ∈ 𝑑, 1 , 0 ), (𝑎‘𝑒)))‘𝑤) = if(𝑤 ∈ 𝑧, 1 , 0 )))
568	567	ralbidv 3169	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑦 = (𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒 ∈ 𝑑, 1 , 0 ), (𝑎‘𝑒))) → (∀𝑤 ∈ (𝑏 ∪ {𝑐})(𝑦‘𝑤) = if(𝑤 ∈ 𝑧, 1 , 0 ) ↔ ∀𝑤 ∈ (𝑏 ∪ {𝑐})((𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒 ∈ 𝑑, 1 , 0 ), (𝑎‘𝑒)))‘𝑤) = if(𝑤 ∈ 𝑧, 1 , 0 )))
569		fveqeq2 6890	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑦 = (𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒 ∈ 𝑑, 1 , 0 ), (𝑎‘𝑒))) → ((𝐷‘𝑦) = 0 ↔ (𝐷‘(𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒 ∈ 𝑑, 1 , 0 ), (𝑎‘𝑒)))) = 0 ))
570	568, 569	imbi12d 344	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑦 = (𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒 ∈ 𝑑, 1 , 0 ), (𝑎‘𝑒))) → ((∀𝑤 ∈ (𝑏 ∪ {𝑐})(𝑦‘𝑤) = if(𝑤 ∈ 𝑧, 1 , 0 ) → (𝐷‘𝑦) = 0 ) ↔ (∀𝑤 ∈ (𝑏 ∪ {𝑐})((𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒 ∈ 𝑑, 1 , 0 ), (𝑎‘𝑒)))‘𝑤) = if(𝑤 ∈ 𝑧, 1 , 0 ) → (𝐷‘(𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒 ∈ 𝑑, 1 , 0 ), (𝑎‘𝑒)))) = 0 )))
571		elequ2 2113	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (𝑧 = 𝑑 → (𝑤 ∈ 𝑧 ↔ 𝑤 ∈ 𝑑))
572	571	ifbid 4543	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝑧 = 𝑑 → if(𝑤 ∈ 𝑧, 1 , 0 ) = if(𝑤 ∈ 𝑑, 1 , 0 ))
573	572	eqeq2d 2735	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝑧 = 𝑑 → (((𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒 ∈ 𝑑, 1 , 0 ), (𝑎‘𝑒)))‘𝑤) = if(𝑤 ∈ 𝑧, 1 , 0 ) ↔ ((𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒 ∈ 𝑑, 1 , 0 ), (𝑎‘𝑒)))‘𝑤) = if(𝑤 ∈ 𝑑, 1 , 0 )))
574	573	ralbidv 3169	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑧 = 𝑑 → (∀𝑤 ∈ (𝑏 ∪ {𝑐})((𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒 ∈ 𝑑, 1 , 0 ), (𝑎‘𝑒)))‘𝑤) = if(𝑤 ∈ 𝑧, 1 , 0 ) ↔ ∀𝑤 ∈ (𝑏 ∪ {𝑐})((𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒 ∈ 𝑑, 1 , 0 ), (𝑎‘𝑒)))‘𝑤) = if(𝑤 ∈ 𝑑, 1 , 0 )))
575	574	imbi1d 341	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑧 = 𝑑 → ((∀𝑤 ∈ (𝑏 ∪ {𝑐})((𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒 ∈ 𝑑, 1 , 0 ), (𝑎‘𝑒)))‘𝑤) = if(𝑤 ∈ 𝑧, 1 , 0 ) → (𝐷‘(𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒 ∈ 𝑑, 1 , 0 ), (𝑎‘𝑒)))) = 0 ) ↔ (∀𝑤 ∈ (𝑏 ∪ {𝑐})((𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒 ∈ 𝑑, 1 , 0 ), (𝑎‘𝑒)))‘𝑤) = if(𝑤 ∈ 𝑑, 1 , 0 ) → (𝐷‘(𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒 ∈ 𝑑, 1 , 0 ), (𝑎‘𝑒)))) = 0 )))
576	570, 575	rspc2va 3615	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((((𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒 ∈ 𝑑, 1 , 0 ), (𝑎‘𝑒))) ∈ 𝐵 ∧ 𝑑 ∈ (𝑁 ↑m 𝑁)) ∧ ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ (𝑁 ↑m 𝑁)(∀𝑤 ∈ (𝑏 ∪ {𝑐})(𝑦‘𝑤) = if(𝑤 ∈ 𝑧, 1 , 0 ) → (𝐷‘𝑦) = 0 )) → (∀𝑤 ∈ (𝑏 ∪ {𝑐})((𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒 ∈ 𝑑, 1 , 0 ), (𝑎‘𝑒)))‘𝑤) = if(𝑤 ∈ 𝑑, 1 , 0 ) → (𝐷‘(𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒 ∈ 𝑑, 1 , 0 ), (𝑎‘𝑒)))) = 0 ))
577	565, 503, 517, 576	syl21anc 835	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) ∧ ((𝑏 ∪ {𝑐}) ∈ 𝑌 ∧ 𝑑 ∈ (𝑁 ↑m 𝑁) ∧ ∀𝑤 ∈ 𝑏 (𝑎‘𝑤) = if(𝑤 ∈ 𝑑, 1 , 0 ))) → (∀𝑤 ∈ (𝑏 ∪ {𝑐})((𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒 ∈ 𝑑, 1 , 0 ), (𝑎‘𝑒)))‘𝑤) = if(𝑤 ∈ 𝑑, 1 , 0 ) → (𝐷‘(𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒 ∈ 𝑑, 1 , 0 ), (𝑎‘𝑒)))) = 0 ))
578	564, 577	mpd 15	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) ∧ ((𝑏 ∪ {𝑐}) ∈ 𝑌 ∧ 𝑑 ∈ (𝑁 ↑m 𝑁) ∧ ∀𝑤 ∈ 𝑏 (𝑎‘𝑤) = if(𝑤 ∈ 𝑑, 1 , 0 ))) → (𝐷‘(𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒 ∈ 𝑑, 1 , 0 ), (𝑎‘𝑒)))) = 0 )
579	531, 578	oveq12d 7419	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) ∧ ((𝑏 ∪ {𝑐}) ∈ 𝑌 ∧ 𝑑 ∈ (𝑁 ↑m 𝑁) ∧ ∀𝑤 ∈ 𝑏 (𝑎‘𝑤) = if(𝑤 ∈ 𝑑, 1 , 0 ))) → ((((𝑎‘𝑐)(-g‘𝑅)if(𝑐 ∈ 𝑑, 1 , 0 )) · (𝐷‘(𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒))))) + (𝐷‘(𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒 ∈ 𝑑, 1 , 0 ), (𝑎‘𝑒))))) = ((((𝑎‘𝑐)(-g‘𝑅)if(𝑐 ∈ 𝑑, 1 , 0 )) · 0 ) + 0 ))
580	308	oveq1d 7416	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) → ((((𝑎‘𝑐)(-g‘𝑅)if(𝑐 ∈ 𝑑, 1 , 0 )) · 0 ) + 0 ) = ( 0 + 0 ))
581	12, 50, 48	grplid 18887	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝑅 ∈ Grp ∧ 0 ∈ 𝐾) → ( 0 + 0 ) = 0 )
582	115, 133, 581	syl2anc 583	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) → ( 0 + 0 ) = 0 )
583	580, 582	eqtrd 2764	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) → ((((𝑎‘𝑐)(-g‘𝑅)if(𝑐 ∈ 𝑑, 1 , 0 )) · 0 ) + 0 ) = 0 )
584	583	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) ∧ ((𝑏 ∪ {𝑐}) ∈ 𝑌 ∧ 𝑑 ∈ (𝑁 ↑m 𝑁) ∧ ∀𝑤 ∈ 𝑏 (𝑎‘𝑤) = if(𝑤 ∈ 𝑑, 1 , 0 ))) → ((((𝑎‘𝑐)(-g‘𝑅)if(𝑐 ∈ 𝑑, 1 , 0 )) · 0 ) + 0 ) = 0 )
585	579, 584	eqtrd 2764	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) ∧ ((𝑏 ∪ {𝑐}) ∈ 𝑌 ∧ 𝑑 ∈ (𝑁 ↑m 𝑁) ∧ ∀𝑤 ∈ 𝑏 (𝑎‘𝑤) = if(𝑤 ∈ 𝑑, 1 , 0 ))) → ((((𝑎‘𝑐)(-g‘𝑅)if(𝑐 ∈ 𝑑, 1 , 0 )) · (𝐷‘(𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒))))) + (𝐷‘(𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒 ∈ 𝑑, 1 , 0 ), (𝑎‘𝑒))))) = 0 )
586	104, 105, 106, 392, 393, 394, 585	syl33anc 1382	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ (𝑏 ∪ {𝑐}) ∈ 𝑌) ∧ ((𝑎 ∈ 𝐵 ∧ 𝑑 ∈ (𝑁 ↑m 𝑁)) ∧ ∀𝑤 ∈ 𝑏 (𝑎‘𝑤) = if(𝑤 ∈ 𝑑, 1 , 0 ))) → ((((𝑎‘𝑐)(-g‘𝑅)if(𝑐 ∈ 𝑑, 1 , 0 )) · (𝐷‘(𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒))))) + (𝐷‘(𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒 ∈ 𝑑, 1 , 0 ), (𝑎‘𝑒))))) = 0 )
587	288, 391, 586	3eqtrd 2768	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ (𝑏 ∪ {𝑐}) ∈ 𝑌) ∧ ((𝑎 ∈ 𝐵 ∧ 𝑑 ∈ (𝑁 ↑m 𝑁)) ∧ ∀𝑤 ∈ 𝑏 (𝑎‘𝑤) = if(𝑤 ∈ 𝑑, 1 , 0 ))) → (𝐷‘𝑎) = 0 )
588	587	expr 456	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ (𝑏 ∪ {𝑐}) ∈ 𝑌) ∧ (𝑎 ∈ 𝐵 ∧ 𝑑 ∈ (𝑁 ↑m 𝑁))) → (∀𝑤 ∈ 𝑏 (𝑎‘𝑤) = if(𝑤 ∈ 𝑑, 1 , 0 ) → (𝐷‘𝑎) = 0 ))
589	588	ralrimivva 3192	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ (𝑏 ∪ {𝑐}) ∈ 𝑌) → ∀𝑎 ∈ 𝐵 ∀𝑑 ∈ (𝑁 ↑m 𝑁)(∀𝑤 ∈ 𝑏 (𝑎‘𝑤) = if(𝑤 ∈ 𝑑, 1 , 0 ) → (𝐷‘𝑎) = 0 ))
590		fveq1 6880	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑎 = 𝑦 → (𝑎‘𝑤) = (𝑦‘𝑤))
591	590	eqeq1d 2726	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑎 = 𝑦 → ((𝑎‘𝑤) = if(𝑤 ∈ 𝑑, 1 , 0 ) ↔ (𝑦‘𝑤) = if(𝑤 ∈ 𝑑, 1 , 0 )))
592	591	ralbidv 3169	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑎 = 𝑦 → (∀𝑤 ∈ 𝑏 (𝑎‘𝑤) = if(𝑤 ∈ 𝑑, 1 , 0 ) ↔ ∀𝑤 ∈ 𝑏 (𝑦‘𝑤) = if(𝑤 ∈ 𝑑, 1 , 0 )))
593		fveqeq2 6890	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑎 = 𝑦 → ((𝐷‘𝑎) = 0 ↔ (𝐷‘𝑦) = 0 ))
594	592, 593	imbi12d 344	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑎 = 𝑦 → ((∀𝑤 ∈ 𝑏 (𝑎‘𝑤) = if(𝑤 ∈ 𝑑, 1 , 0 ) → (𝐷‘𝑎) = 0 ) ↔ (∀𝑤 ∈ 𝑏 (𝑦‘𝑤) = if(𝑤 ∈ 𝑑, 1 , 0 ) → (𝐷‘𝑦) = 0 )))
595		elequ2 2113	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑑 = 𝑧 → (𝑤 ∈ 𝑑 ↔ 𝑤 ∈ 𝑧))
596	595	ifbid 4543	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑑 = 𝑧 → if(𝑤 ∈ 𝑑, 1 , 0 ) = if(𝑤 ∈ 𝑧, 1 , 0 ))
597	596	eqeq2d 2735	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑑 = 𝑧 → ((𝑦‘𝑤) = if(𝑤 ∈ 𝑑, 1 , 0 ) ↔ (𝑦‘𝑤) = if(𝑤 ∈ 𝑧, 1 , 0 )))
598	597	ralbidv 3169	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑑 = 𝑧 → (∀𝑤 ∈ 𝑏 (𝑦‘𝑤) = if(𝑤 ∈ 𝑑, 1 , 0 ) ↔ ∀𝑤 ∈ 𝑏 (𝑦‘𝑤) = if(𝑤 ∈ 𝑧, 1 , 0 )))
599	598	imbi1d 341	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑑 = 𝑧 → ((∀𝑤 ∈ 𝑏 (𝑦‘𝑤) = if(𝑤 ∈ 𝑑, 1 , 0 ) → (𝐷‘𝑦) = 0 ) ↔ (∀𝑤 ∈ 𝑏 (𝑦‘𝑤) = if(𝑤 ∈ 𝑧, 1 , 0 ) → (𝐷‘𝑦) = 0 )))
600	594, 599	cbvral2vw 3230	. . . . . . . . . . . . . . . . . . . 20 ⊢ (∀𝑎 ∈ 𝐵 ∀𝑑 ∈ (𝑁 ↑m 𝑁)(∀𝑤 ∈ 𝑏 (𝑎‘𝑤) = if(𝑤 ∈ 𝑑, 1 , 0 ) → (𝐷‘𝑎) = 0 ) ↔ ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ (𝑁 ↑m 𝑁)(∀𝑤 ∈ 𝑏 (𝑦‘𝑤) = if(𝑤 ∈ 𝑧, 1 , 0 ) → (𝐷‘𝑦) = 0 ))
601	589, 600	sylib 217	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ (𝑏 ∪ {𝑐}) ∈ 𝑌) → ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ (𝑁 ↑m 𝑁)(∀𝑤 ∈ 𝑏 (𝑦‘𝑤) = if(𝑤 ∈ 𝑧, 1 , 0 ) → (𝐷‘𝑦) = 0 ))
602		vex 3470	. . . . . . . . . . . . . . . . . . . 20 ⊢ 𝑏 ∈ V
603		raleq 3314	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑥 = 𝑏 → (∀𝑤 ∈ 𝑥 (𝑦‘𝑤) = if(𝑤 ∈ 𝑧, 1 , 0 ) ↔ ∀𝑤 ∈ 𝑏 (𝑦‘𝑤) = if(𝑤 ∈ 𝑧, 1 , 0 )))
604	603	imbi1d 341	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑥 = 𝑏 → ((∀𝑤 ∈ 𝑥 (𝑦‘𝑤) = if(𝑤 ∈ 𝑧, 1 , 0 ) → (𝐷‘𝑦) = 0 ) ↔ (∀𝑤 ∈ 𝑏 (𝑦‘𝑤) = if(𝑤 ∈ 𝑧, 1 , 0 ) → (𝐷‘𝑦) = 0 )))
605	604	2ralbidv 3210	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑥 = 𝑏 → (∀𝑦 ∈ 𝐵 ∀𝑧 ∈ (𝑁 ↑m 𝑁)(∀𝑤 ∈ 𝑥 (𝑦‘𝑤) = if(𝑤 ∈ 𝑧, 1 , 0 ) → (𝐷‘𝑦) = 0 ) ↔ ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ (𝑁 ↑m 𝑁)(∀𝑤 ∈ 𝑏 (𝑦‘𝑤) = if(𝑤 ∈ 𝑧, 1 , 0 ) → (𝐷‘𝑦) = 0 )))
606	602, 605, 72	elab2 3664	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑏 ∈ 𝑌 ↔ ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ (𝑁 ↑m 𝑁)(∀𝑤 ∈ 𝑏 (𝑦‘𝑤) = if(𝑤 ∈ 𝑧, 1 , 0 ) → (𝐷‘𝑦) = 0 ))
607	601, 606	sylibr 233	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ (𝑏 ∪ {𝑐}) ∈ 𝑌) → 𝑏 ∈ 𝑌)
608	607	3expia 1118	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁)) → ((𝑏 ∪ {𝑐}) ∈ 𝑌 → 𝑏 ∈ 𝑌))
609	608	con3d 152	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁)) → (¬ 𝑏 ∈ 𝑌 → ¬ (𝑏 ∪ {𝑐}) ∈ 𝑌))
610	609	3adant3 1129	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ ¬ ∅ ∈ 𝑌) → (¬ 𝑏 ∈ 𝑌 → ¬ (𝑏 ∪ {𝑐}) ∈ 𝑌))
611	610	a1i 11	. . . . . . . . . . . . . 14 ⊢ ((𝑏 ∈ Fin ∧ ¬ 𝑐 ∈ 𝑏) → ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ ¬ ∅ ∈ 𝑌) → (¬ 𝑏 ∈ 𝑌 → ¬ (𝑏 ∪ {𝑐}) ∈ 𝑌)))
612	611	a2d 29	. . . . . . . . . . . . 13 ⊢ ((𝑏 ∈ Fin ∧ ¬ 𝑐 ∈ 𝑏) → (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ ¬ ∅ ∈ 𝑌) → ¬ 𝑏 ∈ 𝑌) → ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ ¬ ∅ ∈ 𝑌) → ¬ (𝑏 ∪ {𝑐}) ∈ 𝑌)))
613	103, 612	syl5 34	. . . . . . . . . . . 12 ⊢ ((𝑏 ∈ Fin ∧ ¬ 𝑐 ∈ 𝑏) → (((𝜑 ∧ 𝑏 ⊆ (𝑁 × 𝑁) ∧ ¬ ∅ ∈ 𝑌) → ¬ 𝑏 ∈ 𝑌) → ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ ¬ ∅ ∈ 𝑌) → ¬ (𝑏 ∪ {𝑐}) ∈ 𝑌)))
614	82, 87, 92, 97, 98, 613	findcard2s 9161	. . . . . . . . . . 11 ⊢ ((𝑁 × 𝑁) ∈ Fin → ((𝜑 ∧ (𝑁 × 𝑁) ⊆ (𝑁 × 𝑁) ∧ ¬ ∅ ∈ 𝑌) → ¬ (𝑁 × 𝑁) ∈ 𝑌))
615	77, 614	mpcom 38	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑁 × 𝑁) ⊆ (𝑁 × 𝑁) ∧ ¬ ∅ ∈ 𝑌) → ¬ (𝑁 × 𝑁) ∈ 𝑌)
616	615	3exp 1116	. . . . . . . . 9 ⊢ (𝜑 → ((𝑁 × 𝑁) ⊆ (𝑁 × 𝑁) → (¬ ∅ ∈ 𝑌 → ¬ (𝑁 × 𝑁) ∈ 𝑌)))
617	76, 616	mpi 20	. . . . . . . 8 ⊢ (𝜑 → (¬ ∅ ∈ 𝑌 → ¬ (𝑁 × 𝑁) ∈ 𝑌))
618	75, 617	mt4d 117	. . . . . . 7 ⊢ (𝜑 → ∅ ∈ 𝑌)
619	618	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐵) → ∅ ∈ 𝑌)
620		0ex 5297	. . . . . . 7 ⊢ ∅ ∈ V
621		raleq 3314	. . . . . . . . 9 ⊢ (𝑥 = ∅ → (∀𝑤 ∈ 𝑥 (𝑦‘𝑤) = if(𝑤 ∈ 𝑧, 1 , 0 ) ↔ ∀𝑤 ∈ ∅ (𝑦‘𝑤) = if(𝑤 ∈ 𝑧, 1 , 0 )))
622	621	imbi1d 341	. . . . . . . 8 ⊢ (𝑥 = ∅ → ((∀𝑤 ∈ 𝑥 (𝑦‘𝑤) = if(𝑤 ∈ 𝑧, 1 , 0 ) → (𝐷‘𝑦) = 0 ) ↔ (∀𝑤 ∈ ∅ (𝑦‘𝑤) = if(𝑤 ∈ 𝑧, 1 , 0 ) → (𝐷‘𝑦) = 0 )))
623	622	2ralbidv 3210	. . . . . . 7 ⊢ (𝑥 = ∅ → (∀𝑦 ∈ 𝐵 ∀𝑧 ∈ (𝑁 ↑m 𝑁)(∀𝑤 ∈ 𝑥 (𝑦‘𝑤) = if(𝑤 ∈ 𝑧, 1 , 0 ) → (𝐷‘𝑦) = 0 ) ↔ ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ (𝑁 ↑m 𝑁)(∀𝑤 ∈ ∅ (𝑦‘𝑤) = if(𝑤 ∈ 𝑧, 1 , 0 ) → (𝐷‘𝑦) = 0 )))
624	620, 623, 72	elab2 3664	. . . . . 6 ⊢ (∅ ∈ 𝑌 ↔ ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ (𝑁 ↑m 𝑁)(∀𝑤 ∈ ∅ (𝑦‘𝑤) = if(𝑤 ∈ 𝑧, 1 , 0 ) → (𝐷‘𝑦) = 0 ))
625	619, 624	sylib 217	. . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐵) → ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ (𝑁 ↑m 𝑁)(∀𝑤 ∈ ∅ (𝑦‘𝑤) = if(𝑤 ∈ 𝑧, 1 , 0 ) → (𝐷‘𝑦) = 0 ))
626		fveq1 6880	. . . . . . . . 9 ⊢ (𝑦 = 𝑎 → (𝑦‘𝑤) = (𝑎‘𝑤))
627	626	eqeq1d 2726	. . . . . . . 8 ⊢ (𝑦 = 𝑎 → ((𝑦‘𝑤) = if(𝑤 ∈ 𝑧, 1 , 0 ) ↔ (𝑎‘𝑤) = if(𝑤 ∈ 𝑧, 1 , 0 )))
628	627	ralbidv 3169	. . . . . . 7 ⊢ (𝑦 = 𝑎 → (∀𝑤 ∈ ∅ (𝑦‘𝑤) = if(𝑤 ∈ 𝑧, 1 , 0 ) ↔ ∀𝑤 ∈ ∅ (𝑎‘𝑤) = if(𝑤 ∈ 𝑧, 1 , 0 )))
629		fveqeq2 6890	. . . . . . 7 ⊢ (𝑦 = 𝑎 → ((𝐷‘𝑦) = 0 ↔ (𝐷‘𝑎) = 0 ))
630	628, 629	imbi12d 344	. . . . . 6 ⊢ (𝑦 = 𝑎 → ((∀𝑤 ∈ ∅ (𝑦‘𝑤) = if(𝑤 ∈ 𝑧, 1 , 0 ) → (𝐷‘𝑦) = 0 ) ↔ (∀𝑤 ∈ ∅ (𝑎‘𝑤) = if(𝑤 ∈ 𝑧, 1 , 0 ) → (𝐷‘𝑎) = 0 )))
631		eleq2 2814	. . . . . . . . . 10 ⊢ (𝑧 = ( I ↾ 𝑁) → (𝑤 ∈ 𝑧 ↔ 𝑤 ∈ ( I ↾ 𝑁)))
632	631	ifbid 4543	. . . . . . . . 9 ⊢ (𝑧 = ( I ↾ 𝑁) → if(𝑤 ∈ 𝑧, 1 , 0 ) = if(𝑤 ∈ ( I ↾ 𝑁), 1 , 0 ))
633	632	eqeq2d 2735	. . . . . . . 8 ⊢ (𝑧 = ( I ↾ 𝑁) → ((𝑎‘𝑤) = if(𝑤 ∈ 𝑧, 1 , 0 ) ↔ (𝑎‘𝑤) = if(𝑤 ∈ ( I ↾ 𝑁), 1 , 0 )))
634	633	ralbidv 3169	. . . . . . 7 ⊢ (𝑧 = ( I ↾ 𝑁) → (∀𝑤 ∈ ∅ (𝑎‘𝑤) = if(𝑤 ∈ 𝑧, 1 , 0 ) ↔ ∀𝑤 ∈ ∅ (𝑎‘𝑤) = if(𝑤 ∈ ( I ↾ 𝑁), 1 , 0 )))
635	634	imbi1d 341	. . . . . 6 ⊢ (𝑧 = ( I ↾ 𝑁) → ((∀𝑤 ∈ ∅ (𝑎‘𝑤) = if(𝑤 ∈ 𝑧, 1 , 0 ) → (𝐷‘𝑎) = 0 ) ↔ (∀𝑤 ∈ ∅ (𝑎‘𝑤) = if(𝑤 ∈ ( I ↾ 𝑁), 1 , 0 ) → (𝐷‘𝑎) = 0 )))
636	630, 635	rspc2va 3615	. . . . 5 ⊢ (((𝑎 ∈ 𝐵 ∧ ( I ↾ 𝑁) ∈ (𝑁 ↑m 𝑁)) ∧ ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ (𝑁 ↑m 𝑁)(∀𝑤 ∈ ∅ (𝑦‘𝑤) = if(𝑤 ∈ 𝑧, 1 , 0 ) → (𝐷‘𝑦) = 0 )) → (∀𝑤 ∈ ∅ (𝑎‘𝑤) = if(𝑤 ∈ ( I ↾ 𝑁), 1 , 0 ) → (𝐷‘𝑎) = 0 ))
637	2, 9, 625, 636	syl21anc 835	. . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐵) → (∀𝑤 ∈ ∅ (𝑎‘𝑤) = if(𝑤 ∈ ( I ↾ 𝑁), 1 , 0 ) → (𝐷‘𝑎) = 0 ))
638	1, 637	mpi 20	. . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐵) → (𝐷‘𝑎) = 0 )
639	638	mpteq2dva 5238	. 2 ⊢ (𝜑 → (𝑎 ∈ 𝐵 ↦ (𝐷‘𝑎)) = (𝑎 ∈ 𝐵 ↦ 0 ))
640	53	feqmptd 6950	. 2 ⊢ (𝜑 → 𝐷 = (𝑎 ∈ 𝐵 ↦ (𝐷‘𝑎)))
641		fconstmpt 5728	. . 3 ⊢ (𝐵 × { 0 }) = (𝑎 ∈ 𝐵 ↦ 0 )
642	641	a1i 11	. 2 ⊢ (𝜑 → (𝐵 × { 0 }) = (𝑎 ∈ 𝐵 ↦ 0 ))
643	639, 640, 642	3eqtr4d 2774	1 ⊢ (𝜑 → 𝐷 = (𝐵 × { 0 }))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 395   ∨ wo 844   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098  {cab 2701   ≠ wne 2932  ∀wral 3053  Vcvv 3466   ∖ cdif 3937   ∪ cun 3938   ⊆ wss 3940  ∅c0 4314  ifcif 4520  {csn 4620  ⟨cop 4626   ↦ cmpt 5221   I cid 5563   × cxp 5664   ↾ cres 5668  Rel wrel 5671   Fn wfn 6528  ⟶wf 6529  –1-1-onto→wf1o 6532  ‘cfv 6533  (class class class)co 7401   ∈ cmpo 7403   ∘_f cof 7661  1^st c1st 7966  2^nd c2nd 7967   ↑_m cmap 8816  Fincfn 8935   sSet csts 17095  Basecbs 17143  +_gcplusg 17196  ._rcmulr 17197  0_gc0g 17384  Grpcgrp 18853  -_gcsg 18855  1_rcur 20076  Ringcrg 20128   Mat cmat 22229

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-rep 5275  ax-sep 5289  ax-nul 5296  ax-pow 5353  ax-pr 5417  ax-un 7718  ax-cnex 11162  ax-resscn 11163  ax-1cn 11164  ax-icn 11165  ax-addcl 11166  ax-addrcl 11167  ax-mulcl 11168  ax-mulrcl 11169  ax-mulcom 11170  ax-addass 11171  ax-mulass 11172  ax-distr 11173  ax-i2m1 11174  ax-1ne0 11175  ax-1rid 11176  ax-rnegex 11177  ax-rrecex 11178  ax-cnre 11179  ax-pre-lttri 11180  ax-pre-lttrn 11181  ax-pre-ltadd 11182  ax-pre-mulgt0 11183  ax-addf 11185  ax-mulf 11186

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-xor 1505  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-nel 3039  df-ral 3054  df-rex 3063  df-rmo 3368  df-reu 3369  df-rab 3425  df-v 3468  df-sbc 3770  df-csb 3886  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-pss 3959  df-nul 4315  df-if 4521  df-pw 4596  df-sn 4621  df-pr 4623  df-tp 4625  df-op 4627  df-ot 4629  df-uni 4900  df-int 4941  df-iun 4989  df-iin 4990  df-br 5139  df-opab 5201  df-mpt 5222  df-tr 5256  df-id 5564  df-eprel 5570  df-po 5578  df-so 5579  df-fr 5621  df-se 5622  df-we 5623  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-res 5678  df-ima 5679  df-pred 6290  df-ord 6357  df-on 6358  df-lim 6359  df-suc 6360  df-iota 6485  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-isom 6542  df-riota 7357  df-ov 7404  df-oprab 7405  df-mpo 7406  df-of 7663  df-om 7849  df-1st 7968  df-2nd 7969  df-supp 8141  df-tpos 8206  df-frecs 8261  df-wrecs 8292  df-recs 8366  df-rdg 8405  df-1o 8461  df-2o 8462  df-er 8699  df-map 8818  df-ixp 8888  df-en 8936  df-dom 8937  df-sdom 8938  df-fin 8939  df-fsupp 9358  df-sup 9433  df-oi 9501  df-card 9930  df-pnf 11247  df-mnf 11248  df-xr 11249  df-ltxr 11250  df-le 11251  df-sub 11443  df-neg 11444  df-div 11869  df-nn 12210  df-2 12272  df-3 12273  df-4 12274  df-5 12275  df-6 12276  df-7 12277  df-8 12278  df-9 12279  df-n0 12470  df-xnn0 12542  df-z 12556  df-dec 12675  df-uz 12820  df-rp 12972  df-fz 13482  df-fzo 13625  df-seq 13964  df-exp 14025  df-hash 14288  df-word 14462  df-lsw 14510  df-concat 14518  df-s1 14543  df-substr 14588  df-pfx 14618  df-splice 14697  df-reverse 14706  df-s2 14796  df-struct 17079  df-sets 17096  df-slot 17114  df-ndx 17126  df-base 17144  df-ress 17173  df-plusg 17209  df-mulr 17210  df-starv 17211  df-sca 17212  df-vsca 17213  df-ip 17214  df-tset 17215  df-ple 17216  df-ds 17218  df-unif 17219  df-hom 17220  df-cco 17221  df-0g 17386  df-gsum 17387  df-prds 17392  df-pws 17394  df-mre 17529  df-mrc 17530  df-acs 17532  df-mgm 18563  df-sgrp 18642  df-mnd 18658  df-mhm 18703  df-submnd 18704  df-efmnd 18784  df-grp 18856  df-minusg 18857  df-sbg 18858  df-mulg 18986  df-subg 19040  df-ghm 19129  df-gim 19174  df-cntz 19223  df-oppg 19252  df-symg 19277  df-pmtr 19352  df-psgn 19401  df-evpm 19402  df-cmn 19692  df-abl 19693  df-mgp 20030  df-rng 20048  df-ur 20077  df-ring 20130  df-cring 20131  df-oppr 20226  df-dvdsr 20249  df-unit 20250  df-invr 20280  df-dvr 20293  df-rhm 20364  df-subrng 20436  df-subrg 20461  df-drng 20579  df-lmod 20698  df-lss 20769  df-sra 21011  df-rgmod 21012  df-cnfld 21229  df-zring 21302  df-zrh 21358  df-dsmm 21595  df-frlm 21610  df-mamu 22208  df-mat 22230

This theorem is referenced by:  mdetuni0  22445