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Theorem mdetunilem9 22626

Description: Lemma for mdetuni 22628. (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 = (0_g‘𝑅)
mdetuni.1r	⊢ 1 = (1_r‘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‘(1_r‘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 4513	. . . 4 ⊢ ∀𝑤 ∈ ∅ (𝑎‘𝑤) = if(𝑤 ∈ ( I ↾ 𝑁), 1 , 0 )
2		simpr 484	. . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐵) → 𝑎 ∈ 𝐵)
3		f1oi 6886	. . . . . . . 8 ⊢ ( I ↾ 𝑁):𝑁–1-1-onto→𝑁
4		f1of 6848	. . . . . . . 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 8880	. . . . . . 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 777	. . . . . . . . . . . . 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 22428	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 ∈ 𝐵 → 𝑦 ∈ (𝐾 ↑_m (𝑁 × 𝑁)))
15		elmapi 8889	. . . . . . . . . . . . . . . 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 6977	. . . . . . . . . . . . . 14 ⊢ (𝑦 ∈ 𝐵 → 𝑦 = (𝑤 ∈ (𝑁 × 𝑁) ↦ (𝑦‘𝑤)))
18	17	fveq2d 6910	. . . . . . . . . . . . 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 2737	. . . . . . . . . . . . . 14 ⊢ (𝑁 × 𝑁) = (𝑁 × 𝑁)
21		mpteq12 5234	. . . . . . . . . . . . . . 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 6910	. . . . . . . . . . . . . 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 690	. . . . . . . . . . . . 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 2829	. . . . . . . . . . . . . . . . 17 ⊢ (𝑎 = 𝑧 → (𝑎 ∈ (𝑁 ↑_m 𝑁) ↔ 𝑧 ∈ (𝑁 ↑_m 𝑁)))
26	25	anbi2d 630	. . . . . . . . . . . . . . . 16 ⊢ (𝑎 = 𝑧 → ((𝜑 ∧ 𝑎 ∈ (𝑁 ↑_m 𝑁)) ↔ (𝜑 ∧ 𝑧 ∈ (𝑁 ↑_m 𝑁))))
27		elequ2 2123	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑎 = 𝑧 → (𝑤 ∈ 𝑎 ↔ 𝑤 ∈ 𝑧))
28	27	ifbid 4549	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑎 = 𝑧 → if(𝑤 ∈ 𝑎, 1 , 0 ) = if(𝑤 ∈ 𝑧, 1 , 0 ))
29	28	mpteq2dv 5244	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑎 = 𝑧 → (𝑤 ∈ (𝑁 × 𝑁) ↦ if(𝑤 ∈ 𝑎, 1 , 0 )) = (𝑤 ∈ (𝑁 × 𝑁) ↦ if(𝑤 ∈ 𝑧, 1 , 0 )))
30	29	fveq2d 6910	. . . . . . . . . . . . . . . . 17 ⊢ (𝑎 = 𝑧 → (𝐷‘(𝑤 ∈ (𝑁 × 𝑁) ↦ if(𝑤 ∈ 𝑎, 1 , 0 ))) = (𝐷‘(𝑤 ∈ (𝑁 × 𝑁) ↦ if(𝑤 ∈ 𝑧, 1 , 0 ))))
31	30	eqeq1d 2739	. . . . . . . . . . . . . . . 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 2829	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑤 = ⟨𝑏, 𝑐⟩ → (𝑤 ∈ 𝑎 ↔ ⟨𝑏, 𝑐⟩ ∈ 𝑎))
34	33	ifbid 4549	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑤 = ⟨𝑏, 𝑐⟩ → if(𝑤 ∈ 𝑎, 1 , 0 ) = if(⟨𝑏, 𝑐⟩ ∈ 𝑎, 1 , 0 ))
35	34	mpompt 7547	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑤 ∈ (𝑁 × 𝑁) ↦ if(𝑤 ∈ 𝑎, 1 , 0 )) = (𝑏 ∈ 𝑁, 𝑐 ∈ 𝑁 ↦ if(⟨𝑏, 𝑐⟩ ∈ 𝑎, 1 , 0 ))
36		elmapi 8889	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑎 ∈ (𝑁 ↑_m 𝑁) → 𝑎:𝑁⟶𝑁)
37	36	adantl 481	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ 𝑎 ∈ (𝑁 ↑_m 𝑁)) → 𝑎:𝑁⟶𝑁)
38	37	ffnd 6737	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ 𝑎 ∈ (𝑁 ↑_m 𝑁)) → 𝑎 Fn 𝑁)
39	38	3ad2ant1 1134	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ 𝑎 ∈ (𝑁 ↑_m 𝑁)) ∧ 𝑏 ∈ 𝑁 ∧ 𝑐 ∈ 𝑁) → 𝑎 Fn 𝑁)
40		simp2 1138	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ 𝑎 ∈ (𝑁 ↑_m 𝑁)) ∧ 𝑏 ∈ 𝑁 ∧ 𝑐 ∈ 𝑁) → 𝑏 ∈ 𝑁)
41		fnopfvb 6960	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑎 Fn 𝑁 ∧ 𝑏 ∈ 𝑁) → ((𝑎‘𝑏) = 𝑐 ↔ ⟨𝑏, 𝑐⟩ ∈ 𝑎))
42	39, 40, 41	syl2anc 584	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑎 ∈ (𝑁 ↑_m 𝑁)) ∧ 𝑏 ∈ 𝑁 ∧ 𝑐 ∈ 𝑁) → ((𝑎‘𝑏) = 𝑐 ↔ ⟨𝑏, 𝑐⟩ ∈ 𝑎))
43	42	bicomd 223	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑎 ∈ (𝑁 ↑_m 𝑁)) ∧ 𝑏 ∈ 𝑁 ∧ 𝑐 ∈ 𝑁) → (⟨𝑏, 𝑐⟩ ∈ 𝑎 ↔ (𝑎‘𝑏) = 𝑐))
44	43	ifbid 4549	. . . . . . . . . . . . . . . . . . 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 7510	. . . . . . . . . . . . . . . . . 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 2789	. . . . . . . . . . . . . . . . 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 6910	. . . . . . . . . . . . . . . 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 = (0_g‘𝑅)
49		mdetuni.1r	. . . . . . . . . . . . . . . . . 18 ⊢ 1 = (1_r‘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‘(1_r‘𝐴)) = 0 )
58	11, 13, 12, 48, 49, 50, 51, 6, 52, 53, 54, 55, 56, 57	mdetunilem8 22625	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑎:𝑁⟶𝑁) → (𝐷‘(𝑏 ∈ 𝑁, 𝑐 ∈ 𝑁 ↦ if((𝑎‘𝑏) = 𝑐, 1 , 0 ))) = 0 )
59	36, 58	sylan2 593	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑎 ∈ (𝑁 ↑_m 𝑁)) → (𝐷‘(𝑏 ∈ 𝑁, 𝑐 ∈ 𝑁 ↦ if((𝑎‘𝑏) = 𝑐, 1 , 0 ))) = 0 )
60	47, 59	eqtrd 2777	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑎 ∈ (𝑁 ↑_m 𝑁)) → (𝐷‘(𝑤 ∈ (𝑁 × 𝑁) ↦ if(𝑤 ∈ 𝑎, 1 , 0 ))) = 0 )
61	32, 60	chvarvv 1998	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝑁 ↑_m 𝑁)) → (𝐷‘(𝑤 ∈ (𝑁 × 𝑁) ↦ if(𝑤 ∈ 𝑧, 1 , 0 ))) = 0 )
62	61	adantrl 716	. . . . . . . . . . . . 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 2781	. . . . . . . . . . 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 3202	. . . . . . . . 9 ⊢ (𝜑 → ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ (𝑁 ↑_m 𝑁)(∀𝑤 ∈ (𝑁 × 𝑁)(𝑦‘𝑤) = if(𝑤 ∈ 𝑧, 1 , 0 ) → (𝐷‘𝑦) = 0 ))
67		xpfi 9358	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ Fin ∧ 𝑁 ∈ Fin) → (𝑁 × 𝑁) ∈ Fin)
68	6, 6, 67	syl2anc 584	. . . . . . . . . 10 ⊢ (𝜑 → (𝑁 × 𝑁) ∈ Fin)
69		raleq 3323	. . . . . . . . . . . . 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 3221	. . . . . . . . . . 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 3680	. . . . . . . . . 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 4006	. . . . . . . . 9 ⊢ (𝑁 × 𝑁) ⊆ (𝑁 × 𝑁)
77	68	3ad2ant1 1134	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑁 × 𝑁) ⊆ (𝑁 × 𝑁) ∧ ¬ ∅ ∈ 𝑌) → (𝑁 × 𝑁) ∈ Fin)
78		sseq1 4009	. . . . . . . . . . . . . 14 ⊢ (𝑎 = ∅ → (𝑎 ⊆ (𝑁 × 𝑁) ↔ ∅ ⊆ (𝑁 × 𝑁)))
79	78	3anbi2d 1443	. . . . . . . . . . . . 13 ⊢ (𝑎 = ∅ → ((𝜑 ∧ 𝑎 ⊆ (𝑁 × 𝑁) ∧ ¬ ∅ ∈ 𝑌) ↔ (𝜑 ∧ ∅ ⊆ (𝑁 × 𝑁) ∧ ¬ ∅ ∈ 𝑌)))
80		eleq1 2829	. . . . . . . . . . . . . 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 4009	. . . . . . . . . . . . . 14 ⊢ (𝑎 = 𝑏 → (𝑎 ⊆ (𝑁 × 𝑁) ↔ 𝑏 ⊆ (𝑁 × 𝑁)))
84	83	3anbi2d 1443	. . . . . . . . . . . . 13 ⊢ (𝑎 = 𝑏 → ((𝜑 ∧ 𝑎 ⊆ (𝑁 × 𝑁) ∧ ¬ ∅ ∈ 𝑌) ↔ (𝜑 ∧ 𝑏 ⊆ (𝑁 × 𝑁) ∧ ¬ ∅ ∈ 𝑌)))
85		eleq1 2829	. . . . . . . . . . . . . 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 4009	. . . . . . . . . . . . . 14 ⊢ (𝑎 = (𝑏 ∪ {𝑐}) → (𝑎 ⊆ (𝑁 × 𝑁) ↔ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁)))
89	88	3anbi2d 1443	. . . . . . . . . . . . 13 ⊢ (𝑎 = (𝑏 ∪ {𝑐}) → ((𝜑 ∧ 𝑎 ⊆ (𝑁 × 𝑁) ∧ ¬ ∅ ∈ 𝑌) ↔ (𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ ¬ ∅ ∈ 𝑌)))
90		eleq1 2829	. . . . . . . . . . . . . 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 4009	. . . . . . . . . . . . . 14 ⊢ (𝑎 = (𝑁 × 𝑁) → (𝑎 ⊆ (𝑁 × 𝑁) ↔ (𝑁 × 𝑁) ⊆ (𝑁 × 𝑁)))
94	93	3anbi2d 1443	. . . . . . . . . . . . 13 ⊢ (𝑎 = (𝑁 × 𝑁) → ((𝜑 ∧ 𝑎 ⊆ (𝑁 × 𝑁) ∧ ¬ ∅ ∈ 𝑌) ↔ (𝜑 ∧ (𝑁 × 𝑁) ⊆ (𝑁 × 𝑁) ∧ ¬ ∅ ∈ 𝑌)))
95		eleq1 2829	. . . . . . . . . . . . . 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 1139	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ ∅ ⊆ (𝑁 × 𝑁) ∧ ¬ ∅ ∈ 𝑌) → ¬ ∅ ∈ 𝑌)
99		ssun1 4178	. . . . . . . . . . . . . . . 16 ⊢ 𝑏 ⊆ (𝑏 ∪ {𝑐})
100		sstr2 3990	. . . . . . . . . . . . . . . 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 1154	. . . . . . . . . . . . . 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 1192	. . . . . . . . . . . . . . . . . . . . . . . 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 1193	. . . . . . . . . . . . . . . . . . . . . . . 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 779	. . . . . . . . . . . . . . . . . . . . . . . 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 22428	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑎 ∈ 𝐵 → 𝑎 ∈ (𝐾 ↑_m (𝑁 × 𝑁)))
108		elmapi 8889	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 1136	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) → 𝑎:(𝑁 × 𝑁)⟶𝐾)
111	110	feqmptd 6977	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) → 𝑎 = (𝑒 ∈ (𝑁 × 𝑁) ↦ (𝑎‘𝑒)))
112	111	reseq1d 5996	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) → (𝑎 ↾ ({(1^st ‘𝑐)} × 𝑁)) = ((𝑒 ∈ (𝑁 × 𝑁) ↦ (𝑎‘𝑒)) ↾ ({(1^st ‘𝑐)} × 𝑁)))
113	52	3ad2ant1 1134	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) → 𝑅 ∈ Ring)
114		ringgrp 20235	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 ∈ ({(1^st ‘𝑐)} × 𝑁)) → 𝑅 ∈ Grp)
117	110	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) ∧ 𝑒 ∈ ({(1^st ‘𝑐)} × 𝑁)) → 𝑎:(𝑁 × 𝑁)⟶𝐾)
118		simp2 1138	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 ⊢ ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) → (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁))
119	118	unssbd 4194	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 ⊢ ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) → {𝑐} ⊆ (𝑁 × 𝑁))
120		vex 3484	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 ⊢ 𝑐 ∈ V
121	120	snss 4785	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 ⊢ (𝑐 ∈ (𝑁 × 𝑁) ↔ {𝑐} ⊆ (𝑁 × 𝑁))
122	119, 121	sylibr 234	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ⊢ ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) → 𝑐 ∈ (𝑁 × 𝑁))
123		xp1st 8046	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ⊢ (𝑐 ∈ (𝑁 × 𝑁) → (1^st ‘𝑐) ∈ 𝑁)
124	122, 123	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ⊢ ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) → (1^st ‘𝑐) ∈ 𝑁)
125	124	snssd 4809	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) → {(1^st ‘𝑐)} ⊆ 𝑁)
126		xpss1 5704	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ ({(1^st ‘𝑐)} ⊆ 𝑁 → ({(1^st ‘𝑐)} × 𝑁) ⊆ (𝑁 × 𝑁))
127	125, 126	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) → ({(1^st ‘𝑐)} × 𝑁) ⊆ (𝑁 × 𝑁))
128	127	sselda 3983	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) ∧ 𝑒 ∈ ({(1^st ‘𝑐)} × 𝑁)) → 𝑒 ∈ (𝑁 × 𝑁))
129	117, 128	ffvelcdmd 7105	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) ∧ 𝑒 ∈ ({(1^st ‘𝑐)} × 𝑁)) → (𝑎‘𝑒) ∈ 𝐾)
130	12, 49	ringidcl 20262	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ (𝑅 ∈ Ring → 1 ∈ 𝐾)
131	113, 130	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) → 1 ∈ 𝐾)
132	12, 48	ring0cl 20264	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ (𝑅 ∈ Ring → 0 ∈ 𝐾)
133	113, 132	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) → 0 ∈ 𝐾)
134	131, 133	ifcld 4572	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 ∈ ({(1^st ‘𝑐)} × 𝑁)) → if(𝑒 ∈ 𝑑, 1 , 0 ) ∈ 𝐾)
136		eqid 2737	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ (-_g‘𝑅) = (-_g‘𝑅)
137	12, 50, 136	grpnpcan 19050	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 1373	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) ∧ 𝑒 ∈ ({(1^st ‘𝑐)} × 𝑁)) → (((𝑎‘𝑒)(-_g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )) + if(𝑒 ∈ 𝑑, 1 , 0 )) = (𝑎‘𝑒))
139	138	eqcomd 2743	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) ∧ 𝑒 ∈ ({(1^st ‘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 ∈ ({(1^st ‘𝑐)} × 𝑁)) ∧ 𝑒 = 𝑐) → (𝑎‘𝑒) = (((𝑎‘𝑒)(-_g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )) + if(𝑒 ∈ 𝑑, 1 , 0 )))
141		iftrue 4531	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 4531	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (𝑒 = 𝑐 → if(𝑒 = 𝑐, if(𝑒 ∈ 𝑑, 1 , 0 ), (𝑎‘𝑒)) = if(𝑒 ∈ 𝑑, 1 , 0 ))
143	141, 142	oveq12d 7449	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 ∈ ({(1^st ‘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 2780	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) ∧ 𝑒 ∈ ({(1^st ‘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 18985	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ ((𝑅 ∈ Grp ∧ (𝑎‘𝑒) ∈ 𝐾) → ( 0 + (𝑎‘𝑒)) = (𝑎‘𝑒))
147	116, 129, 146	syl2anc 584	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) ∧ 𝑒 ∈ ({(1^st ‘𝑐)} × 𝑁)) → ( 0 + (𝑎‘𝑒)) = (𝑎‘𝑒))
148	147	eqcomd 2743	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) ∧ 𝑒 ∈ ({(1^st ‘𝑐)} × 𝑁)) → (𝑎‘𝑒) = ( 0 + (𝑎‘𝑒)))
149	148	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) ∧ 𝑒 ∈ ({(1^st ‘𝑐)} × 𝑁)) ∧ ¬ 𝑒 = 𝑐) → (𝑎‘𝑒) = ( 0 + (𝑎‘𝑒)))
150		iffalse 4534	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (¬ 𝑒 = 𝑐 → if(𝑒 = 𝑐, ((𝑎‘𝑒)(-_g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ) = 0 )
151		iffalse 4534	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (¬ 𝑒 = 𝑐 → if(𝑒 = 𝑐, if(𝑒 ∈ 𝑑, 1 , 0 ), (𝑎‘𝑒)) = (𝑎‘𝑒))
152	150, 151	oveq12d 7449	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 ∈ ({(1^st ‘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 2780	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) ∧ 𝑒 ∈ ({(1^st ‘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 813	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) ∧ 𝑒 ∈ ({(1^st ‘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 5242	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) → (𝑒 ∈ ({(1^st ‘𝑐)} × 𝑁) ↦ (𝑎‘𝑒)) = (𝑒 ∈ ({(1^st ‘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 9083	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ {(1^st ‘𝑐)} ∈ Fin
158	6	3ad2ant1 1134	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) → 𝑁 ∈ Fin)
159		xpfi 9358	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (({(1^st ‘𝑐)} ∈ Fin ∧ 𝑁 ∈ Fin) → ({(1^st ‘𝑐)} × 𝑁) ∈ Fin)
160	157, 158, 159	sylancr 587	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) → ({(1^st ‘𝑐)} × 𝑁) ∈ Fin)
161		ovex 7464	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((𝑎‘𝑒)(-_g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )) ∈ V
162	48	fvexi 6920	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ 0 ∈ V
163	161, 162	ifex 4576	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 ∈ ({(1^st ‘𝑐)} × 𝑁)) → if(𝑒 = 𝑐, ((𝑎‘𝑒)(-_g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ) ∈ V)
165	49	fvexi 6920	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ 1 ∈ V
166	165, 162	ifex 4576	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ if(𝑒 ∈ 𝑑, 1 , 0 ) ∈ V
167		fvex 6919	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝑎‘𝑒) ∈ V
168	166, 167	ifex 4576	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 ∈ ({(1^st ‘𝑐)} × 𝑁)) → if(𝑒 = 𝑐, if(𝑒 ∈ 𝑑, 1 , 0 ), (𝑎‘𝑒)) ∈ V)
170		xp1st 8046	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝑒 ∈ ({(1^st ‘𝑐)} × 𝑁) → (1^st ‘𝑒) ∈ {(1^st ‘𝑐)})
171		elsni 4643	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((1^st ‘𝑒) ∈ {(1^st ‘𝑐)} → (1^st ‘𝑒) = (1^st ‘𝑐))
172		iftrue 4531	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((1^st ‘𝑒) = (1^st ‘𝑐) → if((1^st ‘𝑒) = (1^st ‘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 ∈ ({(1^st ‘𝑐)} × 𝑁) → if((1^st ‘𝑒) = (1^st ‘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 5245	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑒 ∈ ({(1^st ‘𝑐)} × 𝑁) ↦ if((1^st ‘𝑒) = (1^st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-_g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) = (𝑒 ∈ ({(1^st ‘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 ∈ ({(1^st ‘𝑐)} × 𝑁) ↦ if((1^st ‘𝑒) = (1^st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-_g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) = (𝑒 ∈ ({(1^st ‘𝑐)} × 𝑁) ↦ if(𝑒 = 𝑐, ((𝑎‘𝑒)(-_g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 )))
176		eqidd 2738	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) → (𝑒 ∈ ({(1^st ‘𝑐)} × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒 ∈ 𝑑, 1 , 0 ), (𝑎‘𝑒))) = (𝑒 ∈ ({(1^st ‘𝑐)} × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒 ∈ 𝑑, 1 , 0 ), (𝑎‘𝑒))))
177	160, 164, 169, 175, 176	offval2 7717	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) → ((𝑒 ∈ ({(1^st ‘𝑐)} × 𝑁) ↦ if((1^st ‘𝑒) = (1^st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-_g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ∘_f + (𝑒 ∈ ({(1^st ‘𝑐)} × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒 ∈ 𝑑, 1 , 0 ), (𝑎‘𝑒)))) = (𝑒 ∈ ({(1^st ‘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 2780	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) → (𝑒 ∈ ({(1^st ‘𝑐)} × 𝑁) ↦ (𝑎‘𝑒)) = ((𝑒 ∈ ({(1^st ‘𝑐)} × 𝑁) ↦ if((1^st ‘𝑒) = (1^st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-_g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ∘_f + (𝑒 ∈ ({(1^st ‘𝑐)} × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒 ∈ 𝑑, 1 , 0 ), (𝑎‘𝑒)))))
179	127	resmptd 6058	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) → ((𝑒 ∈ (𝑁 × 𝑁) ↦ (𝑎‘𝑒)) ↾ ({(1^st ‘𝑐)} × 𝑁)) = (𝑒 ∈ ({(1^st ‘𝑐)} × 𝑁) ↦ (𝑎‘𝑒)))
180	127	resmptd 6058	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) → ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1^st ‘𝑒) = (1^st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-_g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ↾ ({(1^st ‘𝑐)} × 𝑁)) = (𝑒 ∈ ({(1^st ‘𝑐)} × 𝑁) ↦ if((1^st ‘𝑒) = (1^st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-_g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))))
181	127	resmptd 6058	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) → ((𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒 ∈ 𝑑, 1 , 0 ), (𝑎‘𝑒))) ↾ ({(1^st ‘𝑐)} × 𝑁)) = (𝑒 ∈ ({(1^st ‘𝑐)} × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒 ∈ 𝑑, 1 , 0 ), (𝑎‘𝑒))))
182	180, 181	oveq12d 7449	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) → (((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1^st ‘𝑒) = (1^st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-_g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ↾ ({(1^st ‘𝑐)} × 𝑁)) ∘_f + ((𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒 ∈ 𝑑, 1 , 0 ), (𝑎‘𝑒))) ↾ ({(1^st ‘𝑐)} × 𝑁))) = ((𝑒 ∈ ({(1^st ‘𝑐)} × 𝑁) ↦ if((1^st ‘𝑒) = (1^st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-_g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ∘_f + (𝑒 ∈ ({(1^st ‘𝑐)} × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒 ∈ 𝑑, 1 , 0 ), (𝑎‘𝑒)))))
183	178, 179, 182	3eqtr4d 2787	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) → ((𝑒 ∈ (𝑁 × 𝑁) ↦ (𝑎‘𝑒)) ↾ ({(1^st ‘𝑐)} × 𝑁)) = (((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1^st ‘𝑒) = (1^st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-_g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ↾ ({(1^st ‘𝑐)} × 𝑁)) ∘_f + ((𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒 ∈ 𝑑, 1 , 0 ), (𝑎‘𝑒))) ↾ ({(1^st ‘𝑐)} × 𝑁))))
184	112, 183	eqtrd 2777	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) → (𝑎 ↾ ({(1^st ‘𝑐)} × 𝑁)) = (((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1^st ‘𝑒) = (1^st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-_g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ↾ ({(1^st ‘𝑐)} × 𝑁)) ∘_f + ((𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒 ∈ 𝑑, 1 , 0 ), (𝑎‘𝑒))) ↾ ({(1^st ‘𝑐)} × 𝑁))))
185	111	reseq1d 5996	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) → (𝑎 ↾ ((𝑁 ∖ {(1^st ‘𝑐)}) × 𝑁)) = ((𝑒 ∈ (𝑁 × 𝑁) ↦ (𝑎‘𝑒)) ↾ ((𝑁 ∖ {(1^st ‘𝑐)}) × 𝑁)))
186		xp1st 8046	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝑒 ∈ ((𝑁 ∖ {(1^st ‘𝑐)}) × 𝑁) → (1^st ‘𝑒) ∈ (𝑁 ∖ {(1^st ‘𝑐)}))
187		eldifsni 4790	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((1^st ‘𝑒) ∈ (𝑁 ∖ {(1^st ‘𝑐)}) → (1^st ‘𝑒) ≠ (1^st ‘𝑐))
188	186, 187	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝑒 ∈ ((𝑁 ∖ {(1^st ‘𝑐)}) × 𝑁) → (1^st ‘𝑒) ≠ (1^st ‘𝑐))
189	188	neneqd 2945	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑒 ∈ ((𝑁 ∖ {(1^st ‘𝑐)}) × 𝑁) → ¬ (1^st ‘𝑒) = (1^st ‘𝑐))
190	189	adantl 481	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) ∧ 𝑒 ∈ ((𝑁 ∖ {(1^st ‘𝑐)}) × 𝑁)) → ¬ (1^st ‘𝑒) = (1^st ‘𝑐))
191	190	iffalsed 4536	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) ∧ 𝑒 ∈ ((𝑁 ∖ {(1^st ‘𝑐)}) × 𝑁)) → if((1^st ‘𝑒) = (1^st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-_g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒)) = (𝑎‘𝑒))
192	191	mpteq2dva 5242	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) → (𝑒 ∈ ((𝑁 ∖ {(1^st ‘𝑐)}) × 𝑁) ↦ if((1^st ‘𝑒) = (1^st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-_g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) = (𝑒 ∈ ((𝑁 ∖ {(1^st ‘𝑐)}) × 𝑁) ↦ (𝑎‘𝑒)))
193		difss 4136	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑁 ∖ {(1^st ‘𝑐)}) ⊆ 𝑁
194		xpss1 5704	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((𝑁 ∖ {(1^st ‘𝑐)}) ⊆ 𝑁 → ((𝑁 ∖ {(1^st ‘𝑐)}) × 𝑁) ⊆ (𝑁 × 𝑁))
195	193, 194	ax-mp 5	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝑁 ∖ {(1^st ‘𝑐)}) × 𝑁) ⊆ (𝑁 × 𝑁)
196		resmpt 6055	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (((𝑁 ∖ {(1^st ‘𝑐)}) × 𝑁) ⊆ (𝑁 × 𝑁) → ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1^st ‘𝑒) = (1^st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-_g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ↾ ((𝑁 ∖ {(1^st ‘𝑐)}) × 𝑁)) = (𝑒 ∈ ((𝑁 ∖ {(1^st ‘𝑐)}) × 𝑁) ↦ if((1^st ‘𝑒) = (1^st ‘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((1^st ‘𝑒) = (1^st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-_g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ↾ ((𝑁 ∖ {(1^st ‘𝑐)}) × 𝑁)) = (𝑒 ∈ ((𝑁 ∖ {(1^st ‘𝑐)}) × 𝑁) ↦ if((1^st ‘𝑒) = (1^st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-_g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))))
198		resmpt 6055	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (((𝑁 ∖ {(1^st ‘𝑐)}) × 𝑁) ⊆ (𝑁 × 𝑁) → ((𝑒 ∈ (𝑁 × 𝑁) ↦ (𝑎‘𝑒)) ↾ ((𝑁 ∖ {(1^st ‘𝑐)}) × 𝑁)) = (𝑒 ∈ ((𝑁 ∖ {(1^st ‘𝑐)}) × 𝑁) ↦ (𝑎‘𝑒)))
199	195, 198	mp1i 13	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) → ((𝑒 ∈ (𝑁 × 𝑁) ↦ (𝑎‘𝑒)) ↾ ((𝑁 ∖ {(1^st ‘𝑐)}) × 𝑁)) = (𝑒 ∈ ((𝑁 ∖ {(1^st ‘𝑐)}) × 𝑁) ↦ (𝑎‘𝑒)))
200	192, 197, 199	3eqtr4rd 2788	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) → ((𝑒 ∈ (𝑁 × 𝑁) ↦ (𝑎‘𝑒)) ↾ ((𝑁 ∖ {(1^st ‘𝑐)}) × 𝑁)) = ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1^st ‘𝑒) = (1^st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-_g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ↾ ((𝑁 ∖ {(1^st ‘𝑐)}) × 𝑁)))
201	185, 200	eqtrd 2777	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) → (𝑎 ↾ ((𝑁 ∖ {(1^st ‘𝑐)}) × 𝑁)) = ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1^st ‘𝑒) = (1^st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-_g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ↾ ((𝑁 ∖ {(1^st ‘𝑐)}) × 𝑁)))
202		fveq2 6906	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑒 = 𝑐 → (1^st ‘𝑒) = (1^st ‘𝑐))
203	190, 202	nsyl 140	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) ∧ 𝑒 ∈ ((𝑁 ∖ {(1^st ‘𝑐)}) × 𝑁)) → ¬ 𝑒 = 𝑐)
204	203	iffalsed 4536	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) ∧ 𝑒 ∈ ((𝑁 ∖ {(1^st ‘𝑐)}) × 𝑁)) → if(𝑒 = 𝑐, if(𝑒 ∈ 𝑑, 1 , 0 ), (𝑎‘𝑒)) = (𝑎‘𝑒))
205	204	mpteq2dva 5242	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) → (𝑒 ∈ ((𝑁 ∖ {(1^st ‘𝑐)}) × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒 ∈ 𝑑, 1 , 0 ), (𝑎‘𝑒))) = (𝑒 ∈ ((𝑁 ∖ {(1^st ‘𝑐)}) × 𝑁) ↦ (𝑎‘𝑒)))
206		resmpt 6055	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (((𝑁 ∖ {(1^st ‘𝑐)}) × 𝑁) ⊆ (𝑁 × 𝑁) → ((𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒 ∈ 𝑑, 1 , 0 ), (𝑎‘𝑒))) ↾ ((𝑁 ∖ {(1^st ‘𝑐)}) × 𝑁)) = (𝑒 ∈ ((𝑁 ∖ {(1^st ‘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 ∖ {(1^st ‘𝑐)}) × 𝑁)) = (𝑒 ∈ ((𝑁 ∖ {(1^st ‘𝑐)}) × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒 ∈ 𝑑, 1 , 0 ), (𝑎‘𝑒))))
208	205, 207, 199	3eqtr4rd 2788	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) → ((𝑒 ∈ (𝑁 × 𝑁) ↦ (𝑎‘𝑒)) ↾ ((𝑁 ∖ {(1^st ‘𝑐)}) × 𝑁)) = ((𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒 ∈ 𝑑, 1 , 0 ), (𝑎‘𝑒))) ↾ ((𝑁 ∖ {(1^st ‘𝑐)}) × 𝑁)))
209	185, 208	eqtrd 2777	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) → (𝑎 ↾ ((𝑁 ∖ {(1^st ‘𝑐)}) × 𝑁)) = ((𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒 ∈ 𝑑, 1 , 0 ), (𝑎‘𝑒))) ↾ ((𝑁 ∖ {(1^st ‘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 7104	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) ∧ 𝑒 ∈ (𝑁 × 𝑁)) → (𝑎‘𝑒) ∈ 𝐾)
212	210, 211	ifcld 4572	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 7135	. . . . . . . . . . . . . . . . . . . . . . . . . . . 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 6920	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 8879	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 587	. . . . . . . . . . . . . . . . . . . . . . . . . . . 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 22427	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (𝐾 ↑_m (𝑁 × 𝑁)) = (Base‘𝐴))
221	158, 113, 220	syl2anc 584	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) → (𝐾 ↑_m (𝑁 × 𝑁)) = (Base‘𝐴))
222	221, 13	eqtr4di 2795	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) → (𝐾 ↑_m (𝑁 × 𝑁)) = 𝐵)
223	219, 222	eleqtrd 2843	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) → (𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒 ∈ 𝑑, 1 , 0 ), (𝑎‘𝑒))) ∈ 𝐵)
224		simp3 1139	. . . . . . . . . . . . . . . . . . . . . . . . . . 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 19038	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 1373	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 4572	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 4572	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) ∧ 𝑒 ∈ (𝑁 × 𝑁)) → if((1^st ‘𝑒) = (1^st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-_g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒)) ∈ 𝐾)
231	230	fmpttd 7135	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) → (𝑒 ∈ (𝑁 × 𝑁) ↦ if((1^st ‘𝑒) = (1^st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-_g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))):(𝑁 × 𝑁)⟶𝐾)
232		elmapg 8879	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((𝐾 ∈ V ∧ (𝑁 × 𝑁) ∈ Fin) → ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1^st ‘𝑒) = (1^st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-_g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ∈ (𝐾 ↑_m (𝑁 × 𝑁)) ↔ (𝑒 ∈ (𝑁 × 𝑁) ↦ if((1^st ‘𝑒) = (1^st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-_g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))):(𝑁 × 𝑁)⟶𝐾))
233	214, 216, 232	sylancr 587	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) → ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1^st ‘𝑒) = (1^st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-_g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ∈ (𝐾 ↑_m (𝑁 × 𝑁)) ↔ (𝑒 ∈ (𝑁 × 𝑁) ↦ if((1^st ‘𝑒) = (1^st ‘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((1^st ‘𝑒) = (1^st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-_g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ∈ (𝐾 ↑_m (𝑁 × 𝑁)))
235	234, 222	eleqtrd 2843	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) → (𝑒 ∈ (𝑁 × 𝑁) ↦ if((1^st ‘𝑒) = (1^st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-_g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ∈ 𝐵)
236	55	3ad2ant1 1134	. . . . . . . . . . . . . . . . . . . . . . . . . . 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 5991	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝑥 = 𝑎 → (𝑥 ↾ ({𝑤} × 𝑁)) = (𝑎 ↾ ({𝑤} × 𝑁)))
238	237	eqeq1d 2739	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝑥 = 𝑎 → ((𝑥 ↾ ({𝑤} × 𝑁)) = ((𝑦 ↾ ({𝑤} × 𝑁)) ∘_f + (𝑧 ↾ ({𝑤} × 𝑁))) ↔ (𝑎 ↾ ({𝑤} × 𝑁)) = ((𝑦 ↾ ({𝑤} × 𝑁)) ∘_f + (𝑧 ↾ ({𝑤} × 𝑁)))))
239		reseq1 5991	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝑥 = 𝑎 → (𝑥 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑎 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)))
240	239	eqeq1d 2739	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝑥 = 𝑎 → ((𝑥 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑦 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) ↔ (𝑎 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑦 ↾ ((𝑁 ∖ {𝑤}) × 𝑁))))
241	239	eqeq1d 2739	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝑥 = 𝑎 → ((𝑥 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑧 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) ↔ (𝑎 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑧 ↾ ((𝑁 ∖ {𝑤}) × 𝑁))))
242	238, 240, 241	3anbi123d 1438	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑥 = 𝑎 → (((𝑥 ↾ ({𝑤} × 𝑁)) = ((𝑦 ↾ ({𝑤} × 𝑁)) ∘_f + (𝑧 ↾ ({𝑤} × 𝑁))) ∧ (𝑥 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑦 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) ∧ (𝑥 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑧 ↾ ((𝑁 ∖ {𝑤}) × 𝑁))) ↔ ((𝑎 ↾ ({𝑤} × 𝑁)) = ((𝑦 ↾ ({𝑤} × 𝑁)) ∘_f + (𝑧 ↾ ({𝑤} × 𝑁))) ∧ (𝑎 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑦 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) ∧ (𝑎 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑧 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)))))
243		fveqeq2 6915	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 3221	. . . . . . . . . . . . . . . . . . . . . . . . . . . 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 5991	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (𝑦 = (𝑒 ∈ (𝑁 × 𝑁) ↦ if((1^st ‘𝑒) = (1^st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-_g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) → (𝑦 ↾ ({𝑤} × 𝑁)) = ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1^st ‘𝑒) = (1^st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-_g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ↾ ({𝑤} × 𝑁)))
247	246	oveq1d 7446	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝑦 = (𝑒 ∈ (𝑁 × 𝑁) ↦ if((1^st ‘𝑒) = (1^st ‘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((1^st ‘𝑒) = (1^st ‘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 2748	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝑦 = (𝑒 ∈ (𝑁 × 𝑁) ↦ if((1^st ‘𝑒) = (1^st ‘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((1^st ‘𝑒) = (1^st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-_g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ↾ ({𝑤} × 𝑁)) ∘_f + (𝑧 ↾ ({𝑤} × 𝑁)))))
249		reseq1 5991	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝑦 = (𝑒 ∈ (𝑁 × 𝑁) ↦ if((1^st ‘𝑒) = (1^st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-_g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) → (𝑦 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1^st ‘𝑒) = (1^st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-_g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ↾ ((𝑁 ∖ {𝑤}) × 𝑁)))
250	249	eqeq2d 2748	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝑦 = (𝑒 ∈ (𝑁 × 𝑁) ↦ if((1^st ‘𝑒) = (1^st ‘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((1^st ‘𝑒) = (1^st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-_g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ↾ ((𝑁 ∖ {𝑤}) × 𝑁))))
251	248, 250	3anbi12d 1439	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑦 = (𝑒 ∈ (𝑁 × 𝑁) ↦ if((1^st ‘𝑒) = (1^st ‘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((1^st ‘𝑒) = (1^st ‘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((1^st ‘𝑒) = (1^st ‘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 6906	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝑦 = (𝑒 ∈ (𝑁 × 𝑁) ↦ if((1^st ‘𝑒) = (1^st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-_g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) → (𝐷‘𝑦) = (𝐷‘(𝑒 ∈ (𝑁 × 𝑁) ↦ if((1^st ‘𝑒) = (1^st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-_g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒)))))
253	252	oveq1d 7446	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝑦 = (𝑒 ∈ (𝑁 × 𝑁) ↦ if((1^st ‘𝑒) = (1^st ‘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((1^st ‘𝑒) = (1^st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-_g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒)))) + (𝐷‘𝑧)))
254	253	eqeq2d 2748	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑦 = (𝑒 ∈ (𝑁 × 𝑁) ↦ if((1^st ‘𝑒) = (1^st ‘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((1^st ‘𝑒) = (1^st ‘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((1^st ‘𝑒) = (1^st ‘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((1^st ‘𝑒) = (1^st ‘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((1^st ‘𝑒) = (1^st ‘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((1^st ‘𝑒) = (1^st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-_g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒)))) + (𝐷‘𝑧)))))
256	255	2ralbidv 3221	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑦 = (𝑒 ∈ (𝑁 × 𝑁) ↦ if((1^st ‘𝑒) = (1^st ‘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((1^st ‘𝑒) = (1^st ‘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((1^st ‘𝑒) = (1^st ‘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((1^st ‘𝑒) = (1^st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-_g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒)))) + (𝐷‘𝑧)))))
257	245, 256	rspc2va 3634	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((𝑎 ∈ 𝐵 ∧ (𝑒 ∈ (𝑁 × 𝑁) ↦ if((1^st ‘𝑒) = (1^st ‘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((1^st ‘𝑒) = (1^st ‘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((1^st ‘𝑒) = (1^st ‘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((1^st ‘𝑒) = (1^st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-_g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒)))) + (𝐷‘𝑧))))
258	224, 235, 236, 257	syl21anc 838	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) → ∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝑁 (((𝑎 ↾ ({𝑤} × 𝑁)) = (((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1^st ‘𝑒) = (1^st ‘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((1^st ‘𝑒) = (1^st ‘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((1^st ‘𝑒) = (1^st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-_g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒)))) + (𝐷‘𝑧))))
259		reseq1 5991	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 7447	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑧 = (𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒 ∈ 𝑑, 1 , 0 ), (𝑎‘𝑒))) → (((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1^st ‘𝑒) = (1^st ‘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((1^st ‘𝑒) = (1^st ‘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 2748	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑧 = (𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒 ∈ 𝑑, 1 , 0 ), (𝑎‘𝑒))) → ((𝑎 ↾ ({𝑤} × 𝑁)) = (((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1^st ‘𝑒) = (1^st ‘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((1^st ‘𝑒) = (1^st ‘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 5991	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 2748	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 1440	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑧 = (𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒 ∈ 𝑑, 1 , 0 ), (𝑎‘𝑒))) → (((𝑎 ↾ ({𝑤} × 𝑁)) = (((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1^st ‘𝑒) = (1^st ‘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((1^st ‘𝑒) = (1^st ‘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((1^st ‘𝑒) = (1^st ‘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((1^st ‘𝑒) = (1^st ‘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 6906	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 7447	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑧 = (𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒 ∈ 𝑑, 1 , 0 ), (𝑎‘𝑒))) → ((𝐷‘(𝑒 ∈ (𝑁 × 𝑁) ↦ if((1^st ‘𝑒) = (1^st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-_g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒)))) + (𝐷‘𝑧)) = ((𝐷‘(𝑒 ∈ (𝑁 × 𝑁) ↦ if((1^st ‘𝑒) = (1^st ‘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 2748	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑧 = (𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒 ∈ 𝑑, 1 , 0 ), (𝑎‘𝑒))) → ((𝐷‘𝑎) = ((𝐷‘(𝑒 ∈ (𝑁 × 𝑁) ↦ if((1^st ‘𝑒) = (1^st ‘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((1^st ‘𝑒) = (1^st ‘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((1^st ‘𝑒) = (1^st ‘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((1^st ‘𝑒) = (1^st ‘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((1^st ‘𝑒) = (1^st ‘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((1^st ‘𝑒) = (1^st ‘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((1^st ‘𝑒) = (1^st ‘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((1^st ‘𝑒) = (1^st ‘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 4636	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝑤 = (1^st ‘𝑐) → {𝑤} = {(1^st ‘𝑐)})
270	269	xpeq1d 5714	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝑤 = (1^st ‘𝑐) → ({𝑤} × 𝑁) = ({(1^st ‘𝑐)} × 𝑁))
271	270	reseq2d 5997	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑤 = (1^st ‘𝑐) → (𝑎 ↾ ({𝑤} × 𝑁)) = (𝑎 ↾ ({(1^st ‘𝑐)} × 𝑁)))
272	270	reseq2d 5997	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝑤 = (1^st ‘𝑐) → ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1^st ‘𝑒) = (1^st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-_g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ↾ ({𝑤} × 𝑁)) = ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1^st ‘𝑒) = (1^st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-_g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ↾ ({(1^st ‘𝑐)} × 𝑁)))
273	270	reseq2d 5997	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝑤 = (1^st ‘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))) ↾ ({(1^st ‘𝑐)} × 𝑁)))
274	272, 273	oveq12d 7449	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑤 = (1^st ‘𝑐) → (((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1^st ‘𝑒) = (1^st ‘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((1^st ‘𝑒) = (1^st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-_g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ↾ ({(1^st ‘𝑐)} × 𝑁)) ∘_f + ((𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒 ∈ 𝑑, 1 , 0 ), (𝑎‘𝑒))) ↾ ({(1^st ‘𝑐)} × 𝑁))))
275	271, 274	eqeq12d 2753	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑤 = (1^st ‘𝑐) → ((𝑎 ↾ ({𝑤} × 𝑁)) = (((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1^st ‘𝑒) = (1^st ‘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 ↾ ({(1^st ‘𝑐)} × 𝑁)) = (((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1^st ‘𝑒) = (1^st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-_g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ↾ ({(1^st ‘𝑐)} × 𝑁)) ∘_f + ((𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒 ∈ 𝑑, 1 , 0 ), (𝑎‘𝑒))) ↾ ({(1^st ‘𝑐)} × 𝑁)))))
276	269	difeq2d 4126	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝑤 = (1^st ‘𝑐) → (𝑁 ∖ {𝑤}) = (𝑁 ∖ {(1^st ‘𝑐)}))
277	276	xpeq1d 5714	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝑤 = (1^st ‘𝑐) → ((𝑁 ∖ {𝑤}) × 𝑁) = ((𝑁 ∖ {(1^st ‘𝑐)}) × 𝑁))
278	277	reseq2d 5997	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑤 = (1^st ‘𝑐) → (𝑎 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑎 ↾ ((𝑁 ∖ {(1^st ‘𝑐)}) × 𝑁)))
279	277	reseq2d 5997	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑤 = (1^st ‘𝑐) → ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1^st ‘𝑒) = (1^st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-_g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1^st ‘𝑒) = (1^st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-_g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ↾ ((𝑁 ∖ {(1^st ‘𝑐)}) × 𝑁)))
280	278, 279	eqeq12d 2753	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑤 = (1^st ‘𝑐) → ((𝑎 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1^st ‘𝑒) = (1^st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-_g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) ↔ (𝑎 ↾ ((𝑁 ∖ {(1^st ‘𝑐)}) × 𝑁)) = ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1^st ‘𝑒) = (1^st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-_g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ↾ ((𝑁 ∖ {(1^st ‘𝑐)}) × 𝑁))))
281	277	reseq2d 5997	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑤 = (1^st ‘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 ∖ {(1^st ‘𝑐)}) × 𝑁)))
282	278, 281	eqeq12d 2753	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑤 = (1^st ‘𝑐) → ((𝑎 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = ((𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒 ∈ 𝑑, 1 , 0 ), (𝑎‘𝑒))) ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) ↔ (𝑎 ↾ ((𝑁 ∖ {(1^st ‘𝑐)}) × 𝑁)) = ((𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒 ∈ 𝑑, 1 , 0 ), (𝑎‘𝑒))) ↾ ((𝑁 ∖ {(1^st ‘𝑐)}) × 𝑁))))
283	275, 280, 282	3anbi123d 1438	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑤 = (1^st ‘𝑐) → (((𝑎 ↾ ({𝑤} × 𝑁)) = (((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1^st ‘𝑒) = (1^st ‘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((1^st ‘𝑒) = (1^st ‘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 ↾ ({(1^st ‘𝑐)} × 𝑁)) = (((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1^st ‘𝑒) = (1^st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-_g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ↾ ({(1^st ‘𝑐)} × 𝑁)) ∘_f + ((𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒 ∈ 𝑑, 1 , 0 ), (𝑎‘𝑒))) ↾ ({(1^st ‘𝑐)} × 𝑁))) ∧ (𝑎 ↾ ((𝑁 ∖ {(1^st ‘𝑐)}) × 𝑁)) = ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1^st ‘𝑒) = (1^st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-_g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ↾ ((𝑁 ∖ {(1^st ‘𝑐)}) × 𝑁)) ∧ (𝑎 ↾ ((𝑁 ∖ {(1^st ‘𝑐)}) × 𝑁)) = ((𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒 ∈ 𝑑, 1 , 0 ), (𝑎‘𝑒))) ↾ ((𝑁 ∖ {(1^st ‘𝑐)}) × 𝑁)))))
284	283	imbi1d 341	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑤 = (1^st ‘𝑐) → ((((𝑎 ↾ ({𝑤} × 𝑁)) = (((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1^st ‘𝑒) = (1^st ‘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((1^st ‘𝑒) = (1^st ‘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((1^st ‘𝑒) = (1^st ‘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 ↾ ({(1^st ‘𝑐)} × 𝑁)) = (((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1^st ‘𝑒) = (1^st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-_g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ↾ ({(1^st ‘𝑐)} × 𝑁)) ∘_f + ((𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒 ∈ 𝑑, 1 , 0 ), (𝑎‘𝑒))) ↾ ({(1^st ‘𝑐)} × 𝑁))) ∧ (𝑎 ↾ ((𝑁 ∖ {(1^st ‘𝑐)}) × 𝑁)) = ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1^st ‘𝑒) = (1^st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-_g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ↾ ((𝑁 ∖ {(1^st ‘𝑐)}) × 𝑁)) ∧ (𝑎 ↾ ((𝑁 ∖ {(1^st ‘𝑐)}) × 𝑁)) = ((𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒 ∈ 𝑑, 1 , 0 ), (𝑎‘𝑒))) ↾ ((𝑁 ∖ {(1^st ‘𝑐)}) × 𝑁))) → (𝐷‘𝑎) = ((𝐷‘(𝑒 ∈ (𝑁 × 𝑁) ↦ if((1^st ‘𝑒) = (1^st ‘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 3634	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((((𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒 ∈ 𝑑, 1 , 0 ), (𝑎‘𝑒))) ∈ 𝐵 ∧ (1^st ‘𝑐) ∈ 𝑁) ∧ ∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝑁 (((𝑎 ↾ ({𝑤} × 𝑁)) = (((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1^st ‘𝑒) = (1^st ‘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((1^st ‘𝑒) = (1^st ‘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((1^st ‘𝑒) = (1^st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-_g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒)))) + (𝐷‘𝑧)))) → (((𝑎 ↾ ({(1^st ‘𝑐)} × 𝑁)) = (((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1^st ‘𝑒) = (1^st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-_g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ↾ ({(1^st ‘𝑐)} × 𝑁)) ∘_f + ((𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒 ∈ 𝑑, 1 , 0 ), (𝑎‘𝑒))) ↾ ({(1^st ‘𝑐)} × 𝑁))) ∧ (𝑎 ↾ ((𝑁 ∖ {(1^st ‘𝑐)}) × 𝑁)) = ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1^st ‘𝑒) = (1^st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-_g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ↾ ((𝑁 ∖ {(1^st ‘𝑐)}) × 𝑁)) ∧ (𝑎 ↾ ((𝑁 ∖ {(1^st ‘𝑐)}) × 𝑁)) = ((𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒 ∈ 𝑑, 1 , 0 ), (𝑎‘𝑒))) ↾ ((𝑁 ∖ {(1^st ‘𝑐)}) × 𝑁))) → (𝐷‘𝑎) = ((𝐷‘(𝑒 ∈ (𝑁 × 𝑁) ↦ if((1^st ‘𝑒) = (1^st ‘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 838	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) → (((𝑎 ↾ ({(1^st ‘𝑐)} × 𝑁)) = (((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1^st ‘𝑒) = (1^st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-_g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ↾ ({(1^st ‘𝑐)} × 𝑁)) ∘_f + ((𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒 ∈ 𝑑, 1 , 0 ), (𝑎‘𝑒))) ↾ ({(1^st ‘𝑐)} × 𝑁))) ∧ (𝑎 ↾ ((𝑁 ∖ {(1^st ‘𝑐)}) × 𝑁)) = ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1^st ‘𝑒) = (1^st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-_g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ↾ ((𝑁 ∖ {(1^st ‘𝑐)}) × 𝑁)) ∧ (𝑎 ↾ ((𝑁 ∖ {(1^st ‘𝑐)}) × 𝑁)) = ((𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒 ∈ 𝑑, 1 , 0 ), (𝑎‘𝑒))) ↾ ((𝑁 ∖ {(1^st ‘𝑐)}) × 𝑁))) → (𝐷‘𝑎) = ((𝐷‘(𝑒 ∈ (𝑁 × 𝑁) ↦ if((1^st ‘𝑒) = (1^st ‘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 1466	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) → (𝐷‘𝑎) = ((𝐷‘(𝑒 ∈ (𝑁 × 𝑁) ↦ if((1^st ‘𝑒) = (1^st ‘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 1373	. . . . . . . . . . . . . . . . . . . . . . 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((1^st ‘𝑒) = (1^st ‘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 6906	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (𝑒 = 𝑐 → (𝑎‘𝑒) = (𝑎‘𝑐))
290		elequ1 2115	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ (𝑒 = 𝑐 → (𝑒 ∈ 𝑑 ↔ 𝑐 ∈ 𝑑))
291	290	ifbid 4549	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (𝑒 = 𝑐 → if(𝑒 ∈ 𝑑, 1 , 0 ) = if(𝑐 ∈ 𝑑, 1 , 0 ))
292	289, 291	oveq12d 7449	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 ∈ ({(1^st ‘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 7105	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) → (𝑎‘𝑐) ∈ 𝐾)
295	131, 133	ifcld 4572	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) → if(𝑐 ∈ 𝑑, 1 , 0 ) ∈ 𝐾)
296	12, 136	grpsubcl 19038	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 1373	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) → ((𝑎‘𝑐)(-_g‘𝑅)if(𝑐 ∈ 𝑑, 1 , 0 )) ∈ 𝐾)
298	12, 51, 49	ringridm 20267	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 584	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 726	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) ∧ 𝑒 ∈ ({(1^st ‘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 2780	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) ∧ 𝑒 ∈ ({(1^st ‘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 ∈ ({(1^st ‘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 4531	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (𝑒 = 𝑐 → if(𝑒 = 𝑐, 1 , 0 ) = 1 )
304	303	oveq2d 7447	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 ∈ ({(1^st ‘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 2787	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) ∧ 𝑒 ∈ ({(1^st ‘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 20291	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 584	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) → (((𝑎‘𝑐)(-_g‘𝑅)if(𝑐 ∈ 𝑑, 1 , 0 )) · 0 ) = 0 )
309	308	eqcomd 2743	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) → 0 = (((𝑎‘𝑐)(-_g‘𝑅)if(𝑐 ∈ 𝑑, 1 , 0 )) · 0 ))
310	309	ad2antrr 726	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) ∧ 𝑒 ∈ ({(1^st ‘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 ∈ ({(1^st ‘𝑐)} × 𝑁)) ∧ ¬ 𝑒 = 𝑐) → if(𝑒 = 𝑐, ((𝑎‘𝑒)(-_g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ) = 0 )
312		iffalse 4534	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (¬ 𝑒 = 𝑐 → if(𝑒 = 𝑐, 1 , 0 ) = 0 )
313	312	oveq2d 7447	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 ∈ ({(1^st ‘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 2787	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) ∧ 𝑒 ∈ ({(1^st ‘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 813	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) ∧ 𝑒 ∈ ({(1^st ‘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 ∈ ({(1^st ‘𝑐)} × 𝑁)) → (1^st ‘𝑒) ∈ {(1^st ‘𝑐)})
318	317, 171	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) ∧ 𝑒 ∈ ({(1^st ‘𝑐)} × 𝑁)) → (1^st ‘𝑒) = (1^st ‘𝑐))
319	318	iftrued 4533	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) ∧ 𝑒 ∈ ({(1^st ‘𝑐)} × 𝑁)) → if((1^st ‘𝑒) = (1^st ‘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 4533	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) ∧ 𝑒 ∈ ({(1^st ‘𝑐)} × 𝑁)) → if((1^st ‘𝑒) = (1^st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒)) = if(𝑒 = 𝑐, 1 , 0 ))
321	320	oveq2d 7447	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) ∧ 𝑒 ∈ ({(1^st ‘𝑐)} × 𝑁)) → (((𝑎‘𝑐)(-_g‘𝑅)if(𝑐 ∈ 𝑑, 1 , 0 )) · if((1^st ‘𝑒) = (1^st ‘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 2787	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) ∧ 𝑒 ∈ ({(1^st ‘𝑐)} × 𝑁)) → if((1^st ‘𝑒) = (1^st ‘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((1^st ‘𝑒) = (1^st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒))))
323	322	mpteq2dva 5242	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) → (𝑒 ∈ ({(1^st ‘𝑐)} × 𝑁) ↦ if((1^st ‘𝑒) = (1^st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-_g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) = (𝑒 ∈ ({(1^st ‘𝑐)} × 𝑁) ↦ (((𝑎‘𝑐)(-_g‘𝑅)if(𝑐 ∈ 𝑑, 1 , 0 )) · if((1^st ‘𝑒) = (1^st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒)))))
324		ovexd 7466	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) ∧ 𝑒 ∈ ({(1^st ‘𝑐)} × 𝑁)) → ((𝑎‘𝑐)(-_g‘𝑅)if(𝑐 ∈ 𝑑, 1 , 0 )) ∈ V)
325	165, 162	ifex 4576	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ if(𝑒 = 𝑐, 1 , 0 ) ∈ V
326	325, 167	ifex 4576	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ if((1^st ‘𝑒) = (1^st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒)) ∈ V
327	326	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) ∧ 𝑒 ∈ ({(1^st ‘𝑐)} × 𝑁)) → if((1^st ‘𝑒) = (1^st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒)) ∈ V)
328		fconstmpt 5747	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (({(1^st ‘𝑐)} × 𝑁) × {((𝑎‘𝑐)(-_g‘𝑅)if(𝑐 ∈ 𝑑, 1 , 0 ))}) = (𝑒 ∈ ({(1^st ‘𝑐)} × 𝑁) ↦ ((𝑎‘𝑐)(-_g‘𝑅)if(𝑐 ∈ 𝑑, 1 , 0 )))
329	328	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) → (({(1^st ‘𝑐)} × 𝑁) × {((𝑎‘𝑐)(-_g‘𝑅)if(𝑐 ∈ 𝑑, 1 , 0 ))}) = (𝑒 ∈ ({(1^st ‘𝑐)} × 𝑁) ↦ ((𝑎‘𝑐)(-_g‘𝑅)if(𝑐 ∈ 𝑑, 1 , 0 ))))
330	127	resmptd 6058	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) → ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1^st ‘𝑒) = (1^st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒))) ↾ ({(1^st ‘𝑐)} × 𝑁)) = (𝑒 ∈ ({(1^st ‘𝑐)} × 𝑁) ↦ if((1^st ‘𝑒) = (1^st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒))))
331	160, 324, 327, 329, 330	offval2 7717	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) → ((({(1^st ‘𝑐)} × 𝑁) × {((𝑎‘𝑐)(-_g‘𝑅)if(𝑐 ∈ 𝑑, 1 , 0 ))}) ∘_f · ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1^st ‘𝑒) = (1^st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒))) ↾ ({(1^st ‘𝑐)} × 𝑁))) = (𝑒 ∈ ({(1^st ‘𝑐)} × 𝑁) ↦ (((𝑎‘𝑐)(-_g‘𝑅)if(𝑐 ∈ 𝑑, 1 , 0 )) · if((1^st ‘𝑒) = (1^st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒)))))
332	323, 180, 331	3eqtr4d 2787	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) → ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1^st ‘𝑒) = (1^st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-_g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ↾ ({(1^st ‘𝑐)} × 𝑁)) = ((({(1^st ‘𝑐)} × 𝑁) × {((𝑎‘𝑐)(-_g‘𝑅)if(𝑐 ∈ 𝑑, 1 , 0 ))}) ∘_f · ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1^st ‘𝑒) = (1^st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒))) ↾ ({(1^st ‘𝑐)} × 𝑁))))
333		iffalse 4534	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (¬ (1^st ‘𝑒) = (1^st ‘𝑐) → if((1^st ‘𝑒) = (1^st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-_g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒)) = (𝑎‘𝑒))
334		iffalse 4534	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (¬ (1^st ‘𝑒) = (1^st ‘𝑐) → if((1^st ‘𝑒) = (1^st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒)) = (𝑎‘𝑒))
335	333, 334	eqtr4d 2780	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (¬ (1^st ‘𝑒) = (1^st ‘𝑐) → if((1^st ‘𝑒) = (1^st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-_g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒)) = if((1^st ‘𝑒) = (1^st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒)))
336	190, 335	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) ∧ 𝑒 ∈ ((𝑁 ∖ {(1^st ‘𝑐)}) × 𝑁)) → if((1^st ‘𝑒) = (1^st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-_g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒)) = if((1^st ‘𝑒) = (1^st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒)))
337	336	mpteq2dva 5242	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) → (𝑒 ∈ ((𝑁 ∖ {(1^st ‘𝑐)}) × 𝑁) ↦ if((1^st ‘𝑒) = (1^st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-_g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) = (𝑒 ∈ ((𝑁 ∖ {(1^st ‘𝑐)}) × 𝑁) ↦ if((1^st ‘𝑒) = (1^st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒))))
338		resmpt 6055	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (((𝑁 ∖ {(1^st ‘𝑐)}) × 𝑁) ⊆ (𝑁 × 𝑁) → ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1^st ‘𝑒) = (1^st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒))) ↾ ((𝑁 ∖ {(1^st ‘𝑐)}) × 𝑁)) = (𝑒 ∈ ((𝑁 ∖ {(1^st ‘𝑐)}) × 𝑁) ↦ if((1^st ‘𝑒) = (1^st ‘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((1^st ‘𝑒) = (1^st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒))) ↾ ((𝑁 ∖ {(1^st ‘𝑐)}) × 𝑁)) = (𝑒 ∈ ((𝑁 ∖ {(1^st ‘𝑐)}) × 𝑁) ↦ if((1^st ‘𝑒) = (1^st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒))))
340	337, 197, 339	3eqtr4d 2787	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) → ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1^st ‘𝑒) = (1^st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-_g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ↾ ((𝑁 ∖ {(1^st ‘𝑐)}) × 𝑁)) = ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1^st ‘𝑒) = (1^st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒))) ↾ ((𝑁 ∖ {(1^st ‘𝑐)}) × 𝑁)))
341	131, 133	ifcld 4572	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 4572	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) ∧ 𝑒 ∈ (𝑁 × 𝑁)) → if((1^st ‘𝑒) = (1^st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒)) ∈ 𝐾)
344	343	fmpttd 7135	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) → (𝑒 ∈ (𝑁 × 𝑁) ↦ if((1^st ‘𝑒) = (1^st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒))):(𝑁 × 𝑁)⟶𝐾)
345		elmapg 8879	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((𝐾 ∈ V ∧ (𝑁 × 𝑁) ∈ Fin) → ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1^st ‘𝑒) = (1^st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒))) ∈ (𝐾 ↑_m (𝑁 × 𝑁)) ↔ (𝑒 ∈ (𝑁 × 𝑁) ↦ if((1^st ‘𝑒) = (1^st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒))):(𝑁 × 𝑁)⟶𝐾))
346	214, 216, 345	sylancr 587	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) → ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1^st ‘𝑒) = (1^st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒))) ∈ (𝐾 ↑_m (𝑁 × 𝑁)) ↔ (𝑒 ∈ (𝑁 × 𝑁) ↦ if((1^st ‘𝑒) = (1^st ‘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((1^st ‘𝑒) = (1^st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒))) ∈ (𝐾 ↑_m (𝑁 × 𝑁)))
348	347, 222	eleqtrd 2843	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) → (𝑒 ∈ (𝑁 × 𝑁) ↦ if((1^st ‘𝑒) = (1^st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒))) ∈ 𝐵)
349	56	3ad2ant1 1134	. . . . . . . . . . . . . . . . . . . . . . . . . . . 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 5991	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (𝑥 = (𝑒 ∈ (𝑁 × 𝑁) ↦ if((1^st ‘𝑒) = (1^st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-_g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) → (𝑥 ↾ ({𝑤} × 𝑁)) = ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1^st ‘𝑒) = (1^st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-_g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ↾ ({𝑤} × 𝑁)))
351	350	eqeq1d 2739	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝑥 = (𝑒 ∈ (𝑁 × 𝑁) ↦ if((1^st ‘𝑒) = (1^st ‘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((1^st ‘𝑒) = (1^st ‘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 5991	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (𝑥 = (𝑒 ∈ (𝑁 × 𝑁) ↦ if((1^st ‘𝑒) = (1^st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-_g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) → (𝑥 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1^st ‘𝑒) = (1^st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-_g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ↾ ((𝑁 ∖ {𝑤}) × 𝑁)))
353	352	eqeq1d 2739	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝑥 = (𝑒 ∈ (𝑁 × 𝑁) ↦ if((1^st ‘𝑒) = (1^st ‘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((1^st ‘𝑒) = (1^st ‘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 632	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝑥 = (𝑒 ∈ (𝑁 × 𝑁) ↦ if((1^st ‘𝑒) = (1^st ‘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((1^st ‘𝑒) = (1^st ‘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((1^st ‘𝑒) = (1^st ‘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 6915	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝑥 = (𝑒 ∈ (𝑁 × 𝑁) ↦ if((1^st ‘𝑒) = (1^st ‘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((1^st ‘𝑒) = (1^st ‘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((1^st ‘𝑒) = (1^st ‘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((1^st ‘𝑒) = (1^st ‘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((1^st ‘𝑒) = (1^st ‘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((1^st ‘𝑒) = (1^st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-_g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒)))) = (𝑦 · (𝐷‘𝑧)))))
357	356	2ralbidv 3221	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑥 = (𝑒 ∈ (𝑁 × 𝑁) ↦ if((1^st ‘𝑒) = (1^st ‘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((1^st ‘𝑒) = (1^st ‘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((1^st ‘𝑒) = (1^st ‘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((1^st ‘𝑒) = (1^st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-_g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒)))) = (𝑦 · (𝐷‘𝑧)))))
358		sneq 4636	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ (𝑦 = ((𝑎‘𝑐)(-_g‘𝑅)if(𝑐 ∈ 𝑑, 1 , 0 )) → {𝑦} = {((𝑎‘𝑐)(-_g‘𝑅)if(𝑐 ∈ 𝑑, 1 , 0 ))})
359	358	xpeq2d 5715	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 7446	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 2748	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝑦 = ((𝑎‘𝑐)(-_g‘𝑅)if(𝑐 ∈ 𝑑, 1 , 0 )) → (((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1^st ‘𝑒) = (1^st ‘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((1^st ‘𝑒) = (1^st ‘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 631	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝑦 = ((𝑎‘𝑐)(-_g‘𝑅)if(𝑐 ∈ 𝑑, 1 , 0 )) → ((((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1^st ‘𝑒) = (1^st ‘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((1^st ‘𝑒) = (1^st ‘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((1^st ‘𝑒) = (1^st ‘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((1^st ‘𝑒) = (1^st ‘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 7438	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 2748	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝑦 = ((𝑎‘𝑐)(-_g‘𝑅)if(𝑐 ∈ 𝑑, 1 , 0 )) → ((𝐷‘(𝑒 ∈ (𝑁 × 𝑁) ↦ if((1^st ‘𝑒) = (1^st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-_g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒)))) = (𝑦 · (𝐷‘𝑧)) ↔ (𝐷‘(𝑒 ∈ (𝑁 × 𝑁) ↦ if((1^st ‘𝑒) = (1^st ‘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((1^st ‘𝑒) = (1^st ‘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((1^st ‘𝑒) = (1^st ‘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((1^st ‘𝑒) = (1^st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-_g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒)))) = (𝑦 · (𝐷‘𝑧))) ↔ ((((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1^st ‘𝑒) = (1^st ‘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((1^st ‘𝑒) = (1^st ‘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((1^st ‘𝑒) = (1^st ‘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 3221	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑦 = ((𝑎‘𝑐)(-_g‘𝑅)if(𝑐 ∈ 𝑑, 1 , 0 )) → (∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝑁 ((((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1^st ‘𝑒) = (1^st ‘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((1^st ‘𝑒) = (1^st ‘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((1^st ‘𝑒) = (1^st ‘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((1^st ‘𝑒) = (1^st ‘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((1^st ‘𝑒) = (1^st ‘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((1^st ‘𝑒) = (1^st ‘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 3634	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1^st ‘𝑒) = (1^st ‘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((1^st ‘𝑒) = (1^st ‘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((1^st ‘𝑒) = (1^st ‘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((1^st ‘𝑒) = (1^st ‘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 838	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) → ∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝑁 ((((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1^st ‘𝑒) = (1^st ‘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((1^st ‘𝑒) = (1^st ‘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((1^st ‘𝑒) = (1^st ‘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 5991	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝑧 = (𝑒 ∈ (𝑁 × 𝑁) ↦ if((1^st ‘𝑒) = (1^st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒))) → (𝑧 ↾ ({𝑤} × 𝑁)) = ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1^st ‘𝑒) = (1^st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒))) ↾ ({𝑤} × 𝑁)))
370	369	oveq2d 7447	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝑧 = (𝑒 ∈ (𝑁 × 𝑁) ↦ if((1^st ‘𝑒) = (1^st ‘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((1^st ‘𝑒) = (1^st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒))) ↾ ({𝑤} × 𝑁))))
371	370	eqeq2d 2748	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑧 = (𝑒 ∈ (𝑁 × 𝑁) ↦ if((1^st ‘𝑒) = (1^st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒))) → (((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1^st ‘𝑒) = (1^st ‘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((1^st ‘𝑒) = (1^st ‘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((1^st ‘𝑒) = (1^st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒))) ↾ ({𝑤} × 𝑁)))))
372		reseq1 5991	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝑧 = (𝑒 ∈ (𝑁 × 𝑁) ↦ if((1^st ‘𝑒) = (1^st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒))) → (𝑧 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1^st ‘𝑒) = (1^st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒))) ↾ ((𝑁 ∖ {𝑤}) × 𝑁)))
373	372	eqeq2d 2748	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑧 = (𝑒 ∈ (𝑁 × 𝑁) ↦ if((1^st ‘𝑒) = (1^st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒))) → (((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1^st ‘𝑒) = (1^st ‘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((1^st ‘𝑒) = (1^st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-_g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1^st ‘𝑒) = (1^st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒))) ↾ ((𝑁 ∖ {𝑤}) × 𝑁))))
374	371, 373	anbi12d 632	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑧 = (𝑒 ∈ (𝑁 × 𝑁) ↦ if((1^st ‘𝑒) = (1^st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒))) → ((((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1^st ‘𝑒) = (1^st ‘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((1^st ‘𝑒) = (1^st ‘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((1^st ‘𝑒) = (1^st ‘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((1^st ‘𝑒) = (1^st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒))) ↾ ({𝑤} × 𝑁))) ∧ ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1^st ‘𝑒) = (1^st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-_g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1^st ‘𝑒) = (1^st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒))) ↾ ((𝑁 ∖ {𝑤}) × 𝑁)))))
375		fveq2 6906	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝑧 = (𝑒 ∈ (𝑁 × 𝑁) ↦ if((1^st ‘𝑒) = (1^st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒))) → (𝐷‘𝑧) = (𝐷‘(𝑒 ∈ (𝑁 × 𝑁) ↦ if((1^st ‘𝑒) = (1^st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒)))))
376	375	oveq2d 7447	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑧 = (𝑒 ∈ (𝑁 × 𝑁) ↦ if((1^st ‘𝑒) = (1^st ‘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((1^st ‘𝑒) = (1^st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒))))))
377	376	eqeq2d 2748	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑧 = (𝑒 ∈ (𝑁 × 𝑁) ↦ if((1^st ‘𝑒) = (1^st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒))) → ((𝐷‘(𝑒 ∈ (𝑁 × 𝑁) ↦ if((1^st ‘𝑒) = (1^st ‘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((1^st ‘𝑒) = (1^st ‘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((1^st ‘𝑒) = (1^st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒)))))))
378	374, 377	imbi12d 344	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑧 = (𝑒 ∈ (𝑁 × 𝑁) ↦ if((1^st ‘𝑒) = (1^st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒))) → (((((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1^st ‘𝑒) = (1^st ‘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((1^st ‘𝑒) = (1^st ‘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((1^st ‘𝑒) = (1^st ‘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((1^st ‘𝑒) = (1^st ‘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((1^st ‘𝑒) = (1^st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒))) ↾ ({𝑤} × 𝑁))) ∧ ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1^st ‘𝑒) = (1^st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-_g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1^st ‘𝑒) = (1^st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒))) ↾ ((𝑁 ∖ {𝑤}) × 𝑁))) → (𝐷‘(𝑒 ∈ (𝑁 × 𝑁) ↦ if((1^st ‘𝑒) = (1^st ‘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((1^st ‘𝑒) = (1^st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒))))))))
379	270	xpeq1d 5714	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝑤 = (1^st ‘𝑐) → (({𝑤} × 𝑁) × {((𝑎‘𝑐)(-_g‘𝑅)if(𝑐 ∈ 𝑑, 1 , 0 ))}) = (({(1^st ‘𝑐)} × 𝑁) × {((𝑎‘𝑐)(-_g‘𝑅)if(𝑐 ∈ 𝑑, 1 , 0 ))}))
380	270	reseq2d 5997	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝑤 = (1^st ‘𝑐) → ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1^st ‘𝑒) = (1^st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒))) ↾ ({𝑤} × 𝑁)) = ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1^st ‘𝑒) = (1^st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒))) ↾ ({(1^st ‘𝑐)} × 𝑁)))
381	379, 380	oveq12d 7449	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝑤 = (1^st ‘𝑐) → ((({𝑤} × 𝑁) × {((𝑎‘𝑐)(-_g‘𝑅)if(𝑐 ∈ 𝑑, 1 , 0 ))}) ∘_f · ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1^st ‘𝑒) = (1^st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒))) ↾ ({𝑤} × 𝑁))) = ((({(1^st ‘𝑐)} × 𝑁) × {((𝑎‘𝑐)(-_g‘𝑅)if(𝑐 ∈ 𝑑, 1 , 0 ))}) ∘_f · ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1^st ‘𝑒) = (1^st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒))) ↾ ({(1^st ‘𝑐)} × 𝑁))))
382	272, 381	eqeq12d 2753	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑤 = (1^st ‘𝑐) → (((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1^st ‘𝑒) = (1^st ‘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((1^st ‘𝑒) = (1^st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒))) ↾ ({𝑤} × 𝑁))) ↔ ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1^st ‘𝑒) = (1^st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-_g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ↾ ({(1^st ‘𝑐)} × 𝑁)) = ((({(1^st ‘𝑐)} × 𝑁) × {((𝑎‘𝑐)(-_g‘𝑅)if(𝑐 ∈ 𝑑, 1 , 0 ))}) ∘_f · ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1^st ‘𝑒) = (1^st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒))) ↾ ({(1^st ‘𝑐)} × 𝑁)))))
383	277	reseq2d 5997	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝑤 = (1^st ‘𝑐) → ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1^st ‘𝑒) = (1^st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒))) ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1^st ‘𝑒) = (1^st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒))) ↾ ((𝑁 ∖ {(1^st ‘𝑐)}) × 𝑁)))
384	279, 383	eqeq12d 2753	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑤 = (1^st ‘𝑐) → (((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1^st ‘𝑒) = (1^st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-_g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1^st ‘𝑒) = (1^st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒))) ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) ↔ ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1^st ‘𝑒) = (1^st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-_g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ↾ ((𝑁 ∖ {(1^st ‘𝑐)}) × 𝑁)) = ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1^st ‘𝑒) = (1^st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒))) ↾ ((𝑁 ∖ {(1^st ‘𝑐)}) × 𝑁))))
385	382, 384	anbi12d 632	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑤 = (1^st ‘𝑐) → ((((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1^st ‘𝑒) = (1^st ‘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((1^st ‘𝑒) = (1^st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒))) ↾ ({𝑤} × 𝑁))) ∧ ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1^st ‘𝑒) = (1^st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-_g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1^st ‘𝑒) = (1^st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒))) ↾ ((𝑁 ∖ {𝑤}) × 𝑁))) ↔ (((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1^st ‘𝑒) = (1^st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-_g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ↾ ({(1^st ‘𝑐)} × 𝑁)) = ((({(1^st ‘𝑐)} × 𝑁) × {((𝑎‘𝑐)(-_g‘𝑅)if(𝑐 ∈ 𝑑, 1 , 0 ))}) ∘_f · ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1^st ‘𝑒) = (1^st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒))) ↾ ({(1^st ‘𝑐)} × 𝑁))) ∧ ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1^st ‘𝑒) = (1^st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-_g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ↾ ((𝑁 ∖ {(1^st ‘𝑐)}) × 𝑁)) = ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1^st ‘𝑒) = (1^st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒))) ↾ ((𝑁 ∖ {(1^st ‘𝑐)}) × 𝑁)))))
386	385	imbi1d 341	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑤 = (1^st ‘𝑐) → (((((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1^st ‘𝑒) = (1^st ‘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((1^st ‘𝑒) = (1^st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒))) ↾ ({𝑤} × 𝑁))) ∧ ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1^st ‘𝑒) = (1^st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-_g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1^st ‘𝑒) = (1^st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒))) ↾ ((𝑁 ∖ {𝑤}) × 𝑁))) → (𝐷‘(𝑒 ∈ (𝑁 × 𝑁) ↦ if((1^st ‘𝑒) = (1^st ‘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((1^st ‘𝑒) = (1^st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒)))))) ↔ ((((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1^st ‘𝑒) = (1^st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-_g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ↾ ({(1^st ‘𝑐)} × 𝑁)) = ((({(1^st ‘𝑐)} × 𝑁) × {((𝑎‘𝑐)(-_g‘𝑅)if(𝑐 ∈ 𝑑, 1 , 0 ))}) ∘_f · ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1^st ‘𝑒) = (1^st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒))) ↾ ({(1^st ‘𝑐)} × 𝑁))) ∧ ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1^st ‘𝑒) = (1^st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-_g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ↾ ((𝑁 ∖ {(1^st ‘𝑐)}) × 𝑁)) = ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1^st ‘𝑒) = (1^st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒))) ↾ ((𝑁 ∖ {(1^st ‘𝑐)}) × 𝑁))) → (𝐷‘(𝑒 ∈ (𝑁 × 𝑁) ↦ if((1^st ‘𝑒) = (1^st ‘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((1^st ‘𝑒) = (1^st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒))))))))
387	378, 386	rspc2va 3634	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1^st ‘𝑒) = (1^st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒))) ∈ 𝐵 ∧ (1^st ‘𝑐) ∈ 𝑁) ∧ ∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝑁 ((((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1^st ‘𝑒) = (1^st ‘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((1^st ‘𝑒) = (1^st ‘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((1^st ‘𝑒) = (1^st ‘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((1^st ‘𝑒) = (1^st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-_g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ↾ ({(1^st ‘𝑐)} × 𝑁)) = ((({(1^st ‘𝑐)} × 𝑁) × {((𝑎‘𝑐)(-_g‘𝑅)if(𝑐 ∈ 𝑑, 1 , 0 ))}) ∘_f · ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1^st ‘𝑒) = (1^st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒))) ↾ ({(1^st ‘𝑐)} × 𝑁))) ∧ ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1^st ‘𝑒) = (1^st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-_g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ↾ ((𝑁 ∖ {(1^st ‘𝑐)}) × 𝑁)) = ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1^st ‘𝑒) = (1^st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒))) ↾ ((𝑁 ∖ {(1^st ‘𝑐)}) × 𝑁))) → (𝐷‘(𝑒 ∈ (𝑁 × 𝑁) ↦ if((1^st ‘𝑒) = (1^st ‘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((1^st ‘𝑒) = (1^st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒)))))))
388	348, 124, 368, 387	syl21anc 838	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) → ((((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1^st ‘𝑒) = (1^st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-_g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ↾ ({(1^st ‘𝑐)} × 𝑁)) = ((({(1^st ‘𝑐)} × 𝑁) × {((𝑎‘𝑐)(-_g‘𝑅)if(𝑐 ∈ 𝑑, 1 , 0 ))}) ∘_f · ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1^st ‘𝑒) = (1^st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒))) ↾ ({(1^st ‘𝑐)} × 𝑁))) ∧ ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1^st ‘𝑒) = (1^st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-_g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ↾ ((𝑁 ∖ {(1^st ‘𝑐)}) × 𝑁)) = ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1^st ‘𝑒) = (1^st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒))) ↾ ((𝑁 ∖ {(1^st ‘𝑐)}) × 𝑁))) → (𝐷‘(𝑒 ∈ (𝑁 × 𝑁) ↦ if((1^st ‘𝑒) = (1^st ‘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((1^st ‘𝑒) = (1^st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒)))))))
389	332, 340, 388	mp2and 699	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) → (𝐷‘(𝑒 ∈ (𝑁 × 𝑁) ↦ if((1^st ‘𝑒) = (1^st ‘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((1^st ‘𝑒) = (1^st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒))))))
390	389	oveq1d 7446	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) → ((𝐷‘(𝑒 ∈ (𝑁 × 𝑁) ↦ if((1^st ‘𝑒) = (1^st ‘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((1^st ‘𝑒) = (1^st ‘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 1373	. . . . . . . . . . . . . . . . . . . . . . 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((1^st ‘𝑒) = (1^st ‘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((1^st ‘𝑒) = (1^st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒))))) + (𝐷‘(𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒 ∈ 𝑑, 1 , 0 ), (𝑎‘𝑒))))))
392		simpl3 1194	. . . . . . . . . . . . . . . . . . . . . . . 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 780	. . . . . . . . . . . . . . . . . . . . . . . 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 773	. . . . . . . . . . . . . . . . . . . . . . . 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 4058	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 4531	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ ((1^st ‘𝑤) = (1^st ‘𝑐) → if((1^st ‘𝑤) = (1^st ‘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})) ∧ (1^st ‘𝑤) = (1^st ‘𝑐)) → if((1^st ‘𝑤) = (1^st ‘𝑐), if(𝑤 = 𝑐, 1 , 0 ), (𝑎‘𝑤)) = if(𝑤 = 𝑐, 1 , 0 ))
399		ibar 528	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ⊢ ((1^st ‘𝑤) = (1^st ‘𝑐) → ((2^nd ‘𝑤) = (2^nd ‘𝑐) ↔ ((1^st ‘𝑤) = (1^st ‘𝑐) ∧ (2^nd ‘𝑤) = (2^nd ‘𝑐))))
400	399	adantl 481	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ⊢ (((((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) ∧ 𝑑 ∈ (𝑁 ↑_m 𝑁)) ∧ 𝑤 ∈ (𝑏 ∪ {𝑐})) ∧ (1^st ‘𝑤) = (1^st ‘𝑐)) → ((2^nd ‘𝑤) = (2^nd ‘𝑐) ↔ ((1^st ‘𝑤) = (1^st ‘𝑐) ∧ (2^nd ‘𝑤) = (2^nd ‘𝑐))))
401		relxp 5703	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 ⊢ Rel (𝑁 × 𝑁)
402		simpl2 1193	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 ⊢ (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) ∧ 𝑑 ∈ (𝑁 ↑_m 𝑁)) → (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁))
403	402	sselda 3983	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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})) ∧ (1^st ‘𝑤) = (1^st ‘𝑐)) → 𝑤 ∈ (𝑁 × 𝑁))
405		1st2nd 8064	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 ⊢ ((Rel (𝑁 × 𝑁) ∧ 𝑤 ∈ (𝑁 × 𝑁)) → 𝑤 = ⟨(1^st ‘𝑤), (2^nd ‘𝑤)⟩)
406	401, 404, 405	sylancr 587	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 ⊢ (((((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) ∧ 𝑑 ∈ (𝑁 ↑_m 𝑁)) ∧ 𝑤 ∈ (𝑏 ∪ {𝑐})) ∧ (1^st ‘𝑤) = (1^st ‘𝑐)) → 𝑤 = ⟨(1^st ‘𝑤), (2^nd ‘𝑤)⟩)
407	406	eleq1d 2826	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ⊢ (((((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) ∧ 𝑑 ∈ (𝑁 ↑_m 𝑁)) ∧ 𝑤 ∈ (𝑏 ∪ {𝑐})) ∧ (1^st ‘𝑤) = (1^st ‘𝑐)) → (𝑤 ∈ (𝑑 sSet ⟨(1^st ‘𝑐), (2^nd ‘𝑐)⟩) ↔ ⟨(1^st ‘𝑤), (2^nd ‘𝑤)⟩ ∈ (𝑑 sSet ⟨(1^st ‘𝑐), (2^nd ‘𝑐)⟩)))
408		simpr 484	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 ⊢ (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) ∧ 𝑑 ∈ (𝑁 ↑_m 𝑁)) → 𝑑 ∈ (𝑁 ↑_m 𝑁))
409		elmapi 8889	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)) → (1^st ‘𝑐) ∈ 𝑁)
412		xp2nd 8047	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 ⊢ (𝑐 ∈ (𝑁 × 𝑁) → (2^nd ‘𝑐) ∈ 𝑁)
413	122, 412	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 ⊢ ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) → (2^nd ‘𝑐) ∈ 𝑁)
414	413	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 ⊢ (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) ∧ 𝑑 ∈ (𝑁 ↑_m 𝑁)) → (2^nd ‘𝑐) ∈ 𝑁)
415		fsets 17206	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 ⊢ (((𝑑 ∈ (𝑁 ↑_m 𝑁) ∧ 𝑑:𝑁⟶𝑁) ∧ (1^st ‘𝑐) ∈ 𝑁 ∧ (2^nd ‘𝑐) ∈ 𝑁) → (𝑑 sSet ⟨(1^st ‘𝑐), (2^nd ‘𝑐)⟩):𝑁⟶𝑁)
416	408, 410, 411, 414, 415	syl211anc 1378	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 ⊢ (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) ∧ 𝑑 ∈ (𝑁 ↑_m 𝑁)) → (𝑑 sSet ⟨(1^st ‘𝑐), (2^nd ‘𝑐)⟩):𝑁⟶𝑁)
417	416	ffnd 6737	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 ⊢ (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) ∧ 𝑑 ∈ (𝑁 ↑_m 𝑁)) → (𝑑 sSet ⟨(1^st ‘𝑐), (2^nd ‘𝑐)⟩) Fn 𝑁)
418	417	ad2antrr 726	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 ⊢ (((((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) ∧ 𝑑 ∈ (𝑁 ↑_m 𝑁)) ∧ 𝑤 ∈ (𝑏 ∪ {𝑐})) ∧ (1^st ‘𝑤) = (1^st ‘𝑐)) → (𝑑 sSet ⟨(1^st ‘𝑐), (2^nd ‘𝑐)⟩) Fn 𝑁)
419		xp1st 8046	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 ⊢ (𝑤 ∈ (𝑁 × 𝑁) → (1^st ‘𝑤) ∈ 𝑁)
420	403, 419	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 ⊢ ((((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) ∧ 𝑑 ∈ (𝑁 ↑_m 𝑁)) ∧ 𝑤 ∈ (𝑏 ∪ {𝑐})) → (1^st ‘𝑤) ∈ 𝑁)
421	420	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 ⊢ (((((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) ∧ 𝑑 ∈ (𝑁 ↑_m 𝑁)) ∧ 𝑤 ∈ (𝑏 ∪ {𝑐})) ∧ (1^st ‘𝑤) = (1^st ‘𝑐)) → (1^st ‘𝑤) ∈ 𝑁)
422		fnopfvb 6960	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 ⊢ (((𝑑 sSet ⟨(1^st ‘𝑐), (2^nd ‘𝑐)⟩) Fn 𝑁 ∧ (1^st ‘𝑤) ∈ 𝑁) → (((𝑑 sSet ⟨(1^st ‘𝑐), (2^nd ‘𝑐)⟩)‘(1^st ‘𝑤)) = (2^nd ‘𝑤) ↔ ⟨(1^st ‘𝑤), (2^nd ‘𝑤)⟩ ∈ (𝑑 sSet ⟨(1^st ‘𝑐), (2^nd ‘𝑐)⟩)))
423	418, 421, 422	syl2anc 584	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ⊢ (((((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) ∧ 𝑑 ∈ (𝑁 ↑_m 𝑁)) ∧ 𝑤 ∈ (𝑏 ∪ {𝑐})) ∧ (1^st ‘𝑤) = (1^st ‘𝑐)) → (((𝑑 sSet ⟨(1^st ‘𝑐), (2^nd ‘𝑐)⟩)‘(1^st ‘𝑤)) = (2^nd ‘𝑤) ↔ ⟨(1^st ‘𝑤), (2^nd ‘𝑤)⟩ ∈ (𝑑 sSet ⟨(1^st ‘𝑐), (2^nd ‘𝑐)⟩)))
424		fveq2 6906	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 ⊢ ((1^st ‘𝑤) = (1^st ‘𝑐) → ((𝑑 sSet ⟨(1^st ‘𝑐), (2^nd ‘𝑐)⟩)‘(1^st ‘𝑤)) = ((𝑑 sSet ⟨(1^st ‘𝑐), (2^nd ‘𝑐)⟩)‘(1^st ‘𝑐)))
425	424	adantl 481	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 ⊢ (((((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) ∧ 𝑑 ∈ (𝑁 ↑_m 𝑁)) ∧ 𝑤 ∈ (𝑏 ∪ {𝑐})) ∧ (1^st ‘𝑤) = (1^st ‘𝑐)) → ((𝑑 sSet ⟨(1^st ‘𝑐), (2^nd ‘𝑐)⟩)‘(1^st ‘𝑤)) = ((𝑑 sSet ⟨(1^st ‘𝑐), (2^nd ‘𝑐)⟩)‘(1^st ‘𝑐)))
426		vex 3484	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 ⊢ 𝑑 ∈ V
427		fvex 6919	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 ⊢ (1^st ‘𝑐) ∈ V
428		fvex 6919	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 ⊢ (2^nd ‘𝑐) ∈ V
429		fvsetsid 17205	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 ⊢ ((𝑑 ∈ V ∧ (1^st ‘𝑐) ∈ V ∧ (2^nd ‘𝑐) ∈ V) → ((𝑑 sSet ⟨(1^st ‘𝑐), (2^nd ‘𝑐)⟩)‘(1^st ‘𝑐)) = (2^nd ‘𝑐))
430	426, 427, 428, 429	mp3an 1463	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 ⊢ ((𝑑 sSet ⟨(1^st ‘𝑐), (2^nd ‘𝑐)⟩)‘(1^st ‘𝑐)) = (2^nd ‘𝑐)
431	425, 430	eqtrdi 2793	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 ⊢ (((((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) ∧ 𝑑 ∈ (𝑁 ↑_m 𝑁)) ∧ 𝑤 ∈ (𝑏 ∪ {𝑐})) ∧ (1^st ‘𝑤) = (1^st ‘𝑐)) → ((𝑑 sSet ⟨(1^st ‘𝑐), (2^nd ‘𝑐)⟩)‘(1^st ‘𝑤)) = (2^nd ‘𝑐))
432	431	eqeq1d 2739	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 ⊢ (((((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) ∧ 𝑑 ∈ (𝑁 ↑_m 𝑁)) ∧ 𝑤 ∈ (𝑏 ∪ {𝑐})) ∧ (1^st ‘𝑤) = (1^st ‘𝑐)) → (((𝑑 sSet ⟨(1^st ‘𝑐), (2^nd ‘𝑐)⟩)‘(1^st ‘𝑤)) = (2^nd ‘𝑤) ↔ (2^nd ‘𝑐) = (2^nd ‘𝑤)))
433		eqcom 2744	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 ⊢ ((2^nd ‘𝑐) = (2^nd ‘𝑤) ↔ (2^nd ‘𝑤) = (2^nd ‘𝑐))
434	432, 433	bitrdi 287	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ⊢ (((((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) ∧ 𝑑 ∈ (𝑁 ↑_m 𝑁)) ∧ 𝑤 ∈ (𝑏 ∪ {𝑐})) ∧ (1^st ‘𝑤) = (1^st ‘𝑐)) → (((𝑑 sSet ⟨(1^st ‘𝑐), (2^nd ‘𝑐)⟩)‘(1^st ‘𝑤)) = (2^nd ‘𝑤) ↔ (2^nd ‘𝑤) = (2^nd ‘𝑐)))
435	407, 423, 434	3bitr2rd 308	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ⊢ (((((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) ∧ 𝑑 ∈ (𝑁 ↑_m 𝑁)) ∧ 𝑤 ∈ (𝑏 ∪ {𝑐})) ∧ (1^st ‘𝑤) = (1^st ‘𝑐)) → ((2^nd ‘𝑤) = (2^nd ‘𝑐) ↔ 𝑤 ∈ (𝑑 sSet ⟨(1^st ‘𝑐), (2^nd ‘𝑐)⟩)))
436	122	ad3antrrr 730	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ⊢ (((((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) ∧ 𝑑 ∈ (𝑁 ↑_m 𝑁)) ∧ 𝑤 ∈ (𝑏 ∪ {𝑐})) ∧ (1^st ‘𝑤) = (1^st ‘𝑐)) → 𝑐 ∈ (𝑁 × 𝑁))
437		xpopth 8055	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ⊢ ((𝑤 ∈ (𝑁 × 𝑁) ∧ 𝑐 ∈ (𝑁 × 𝑁)) → (((1^st ‘𝑤) = (1^st ‘𝑐) ∧ (2^nd ‘𝑤) = (2^nd ‘𝑐)) ↔ 𝑤 = 𝑐))
438	404, 436, 437	syl2anc 584	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ⊢ (((((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) ∧ 𝑑 ∈ (𝑁 ↑_m 𝑁)) ∧ 𝑤 ∈ (𝑏 ∪ {𝑐})) ∧ (1^st ‘𝑤) = (1^st ‘𝑐)) → (((1^st ‘𝑤) = (1^st ‘𝑐) ∧ (2^nd ‘𝑤) = (2^nd ‘𝑐)) ↔ 𝑤 = 𝑐))
439	400, 435, 438	3bitr3rd 310	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ (((((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) ∧ 𝑑 ∈ (𝑁 ↑_m 𝑁)) ∧ 𝑤 ∈ (𝑏 ∪ {𝑐})) ∧ (1^st ‘𝑤) = (1^st ‘𝑐)) → (𝑤 = 𝑐 ↔ 𝑤 ∈ (𝑑 sSet ⟨(1^st ‘𝑐), (2^nd ‘𝑐)⟩)))
440	439	ifbid 4549	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ (((((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) ∧ 𝑑 ∈ (𝑁 ↑_m 𝑁)) ∧ 𝑤 ∈ (𝑏 ∪ {𝑐})) ∧ (1^st ‘𝑤) = (1^st ‘𝑐)) → if(𝑤 = 𝑐, 1 , 0 ) = if(𝑤 ∈ (𝑑 sSet ⟨(1^st ‘𝑐), (2^nd ‘𝑐)⟩), 1 , 0 ))
441	398, 440	eqtrd 2777	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ (((((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) ∧ 𝑑 ∈ (𝑁 ↑_m 𝑁)) ∧ 𝑤 ∈ (𝑏 ∪ {𝑐})) ∧ (1^st ‘𝑤) = (1^st ‘𝑐)) → if((1^st ‘𝑤) = (1^st ‘𝑐), if(𝑤 = 𝑐, 1 , 0 ), (𝑎‘𝑤)) = if(𝑤 ∈ (𝑑 sSet ⟨(1^st ‘𝑐), (2^nd ‘𝑐)⟩), 1 , 0 ))
442	441	a1d 25	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (((((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) ∧ 𝑑 ∈ (𝑁 ↑_m 𝑁)) ∧ 𝑤 ∈ (𝑏 ∪ {𝑐})) ∧ (1^st ‘𝑤) = (1^st ‘𝑐)) → ((𝑤 ∈ 𝑏 → (𝑎‘𝑤) = if(𝑤 ∈ 𝑑, 1 , 0 )) → if((1^st ‘𝑤) = (1^st ‘𝑐), if(𝑤 = 𝑐, 1 , 0 ), (𝑎‘𝑤)) = if(𝑤 ∈ (𝑑 sSet ⟨(1^st ‘𝑐), (2^nd ‘𝑐)⟩), 1 , 0 )))
443		elsni 4643	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 ⊢ (𝑤 ∈ {𝑐} → 𝑤 = 𝑐)
444	443	fveq2d 6910	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ⊢ (𝑤 ∈ {𝑐} → (1^st ‘𝑤) = (1^st ‘𝑐))
445	444	con3i 154	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ⊢ (¬ (1^st ‘𝑤) = (1^st ‘𝑐) → ¬ 𝑤 ∈ {𝑐})
446	445	adantl 481	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ ((𝑤 ∈ (𝑏 ∪ {𝑐}) ∧ ¬ (1^st ‘𝑤) = (1^st ‘𝑐)) → ¬ 𝑤 ∈ {𝑐})
447		elun 4153	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ⊢ (𝑤 ∈ (𝑏 ∪ {𝑐}) ↔ (𝑤 ∈ 𝑏 ∨ 𝑤 ∈ {𝑐}))
448	447	biimpi 216	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ⊢ (𝑤 ∈ (𝑏 ∪ {𝑐}) → (𝑤 ∈ 𝑏 ∨ 𝑤 ∈ {𝑐}))
449	448	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ ((𝑤 ∈ (𝑏 ∪ {𝑐}) ∧ ¬ (1^st ‘𝑤) = (1^st ‘𝑐)) → (𝑤 ∈ 𝑏 ∨ 𝑤 ∈ {𝑐}))
450		orel2 891	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ (¬ 𝑤 ∈ {𝑐} → ((𝑤 ∈ 𝑏 ∨ 𝑤 ∈ {𝑐}) → 𝑤 ∈ 𝑏))
451	446, 449, 450	sylc 65	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ ((𝑤 ∈ (𝑏 ∪ {𝑐}) ∧ ¬ (1^st ‘𝑤) = (1^st ‘𝑐)) → 𝑤 ∈ 𝑏)
452	451	adantll 714	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ (((((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) ∧ 𝑑 ∈ (𝑁 ↑_m 𝑁)) ∧ 𝑤 ∈ (𝑏 ∪ {𝑐})) ∧ ¬ (1^st ‘𝑤) = (1^st ‘𝑐)) → 𝑤 ∈ 𝑏)
453		iffalse 4534	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ⊢ (¬ (1^st ‘𝑤) = (1^st ‘𝑐) → if((1^st ‘𝑤) = (1^st ‘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})) ∧ ¬ (1^st ‘𝑤) = (1^st ‘𝑐)) → if((1^st ‘𝑤) = (1^st ‘𝑐), if(𝑤 = 𝑐, 1 , 0 ), if(𝑤 ∈ 𝑑, 1 , 0 )) = if(𝑤 ∈ 𝑑, 1 , 0 ))
455		setsres 17215	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 ⊢ (𝑑 ∈ V → ((𝑑 sSet ⟨(1^st ‘𝑐), (2^nd ‘𝑐)⟩) ↾ (V ∖ {(1^st ‘𝑐)})) = (𝑑 ↾ (V ∖ {(1^st ‘𝑐)})))
456	455	eleq2d 2827	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 ⊢ (𝑑 ∈ V → (⟨(1^st ‘𝑤), (2^nd ‘𝑤)⟩ ∈ ((𝑑 sSet ⟨(1^st ‘𝑐), (2^nd ‘𝑐)⟩) ↾ (V ∖ {(1^st ‘𝑐)})) ↔ ⟨(1^st ‘𝑤), (2^nd ‘𝑤)⟩ ∈ (𝑑 ↾ (V ∖ {(1^st ‘𝑐)}))))
457	426, 456	mp1i 13	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ⊢ (((((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) ∧ 𝑑 ∈ (𝑁 ↑_m 𝑁)) ∧ 𝑤 ∈ (𝑏 ∪ {𝑐})) ∧ ¬ (1^st ‘𝑤) = (1^st ‘𝑐)) → (⟨(1^st ‘𝑤), (2^nd ‘𝑤)⟩ ∈ ((𝑑 sSet ⟨(1^st ‘𝑐), (2^nd ‘𝑐)⟩) ↾ (V ∖ {(1^st ‘𝑐)})) ↔ ⟨(1^st ‘𝑤), (2^nd ‘𝑤)⟩ ∈ (𝑑 ↾ (V ∖ {(1^st ‘𝑐)}))))
458		fvex 6919	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 ⊢ (1^st ‘𝑤) ∈ V
459	458	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 ⊢ (¬ (1^st ‘𝑤) = (1^st ‘𝑐) → (1^st ‘𝑤) ∈ V)
460		neqne 2948	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 ⊢ (¬ (1^st ‘𝑤) = (1^st ‘𝑐) → (1^st ‘𝑤) ≠ (1^st ‘𝑐))
461		eldifsn 4786	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 ⊢ ((1^st ‘𝑤) ∈ (V ∖ {(1^st ‘𝑐)}) ↔ ((1^st ‘𝑤) ∈ V ∧ (1^st ‘𝑤) ≠ (1^st ‘𝑐)))
462	459, 460, 461	sylanbrc 583	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 ⊢ (¬ (1^st ‘𝑤) = (1^st ‘𝑐) → (1^st ‘𝑤) ∈ (V ∖ {(1^st ‘𝑐)}))
463		fvex 6919	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 ⊢ (2^nd ‘𝑤) ∈ V
464	463	opres 6007	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 ⊢ ((1^st ‘𝑤) ∈ (V ∖ {(1^st ‘𝑐)}) → (⟨(1^st ‘𝑤), (2^nd ‘𝑤)⟩ ∈ ((𝑑 sSet ⟨(1^st ‘𝑐), (2^nd ‘𝑐)⟩) ↾ (V ∖ {(1^st ‘𝑐)})) ↔ ⟨(1^st ‘𝑤), (2^nd ‘𝑤)⟩ ∈ (𝑑 sSet ⟨(1^st ‘𝑐), (2^nd ‘𝑐)⟩)))
465	464	adantl 481	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 ⊢ ((𝑤 ∈ (𝑁 × 𝑁) ∧ (1^st ‘𝑤) ∈ (V ∖ {(1^st ‘𝑐)})) → (⟨(1^st ‘𝑤), (2^nd ‘𝑤)⟩ ∈ ((𝑑 sSet ⟨(1^st ‘𝑐), (2^nd ‘𝑐)⟩) ↾ (V ∖ {(1^st ‘𝑐)})) ↔ ⟨(1^st ‘𝑤), (2^nd ‘𝑤)⟩ ∈ (𝑑 sSet ⟨(1^st ‘𝑐), (2^nd ‘𝑐)⟩)))
466		1st2nd2 8053	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 ⊢ (𝑤 ∈ (𝑁 × 𝑁) → 𝑤 = ⟨(1^st ‘𝑤), (2^nd ‘𝑤)⟩)
467	466	eleq1d 2826	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 ⊢ (𝑤 ∈ (𝑁 × 𝑁) → (𝑤 ∈ (𝑑 sSet ⟨(1^st ‘𝑐), (2^nd ‘𝑐)⟩) ↔ ⟨(1^st ‘𝑤), (2^nd ‘𝑤)⟩ ∈ (𝑑 sSet ⟨(1^st ‘𝑐), (2^nd ‘𝑐)⟩)))
468	467	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 ⊢ ((𝑤 ∈ (𝑁 × 𝑁) ∧ (1^st ‘𝑤) ∈ (V ∖ {(1^st ‘𝑐)})) → (𝑤 ∈ (𝑑 sSet ⟨(1^st ‘𝑐), (2^nd ‘𝑐)⟩) ↔ ⟨(1^st ‘𝑤), (2^nd ‘𝑤)⟩ ∈ (𝑑 sSet ⟨(1^st ‘𝑐), (2^nd ‘𝑐)⟩)))
469	465, 468	bitr4d 282	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 ⊢ ((𝑤 ∈ (𝑁 × 𝑁) ∧ (1^st ‘𝑤) ∈ (V ∖ {(1^st ‘𝑐)})) → (⟨(1^st ‘𝑤), (2^nd ‘𝑤)⟩ ∈ ((𝑑 sSet ⟨(1^st ‘𝑐), (2^nd ‘𝑐)⟩) ↾ (V ∖ {(1^st ‘𝑐)})) ↔ 𝑤 ∈ (𝑑 sSet ⟨(1^st ‘𝑐), (2^nd ‘𝑐)⟩)))
470	403, 462, 469	syl2an 596	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ⊢ (((((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) ∧ 𝑑 ∈ (𝑁 ↑_m 𝑁)) ∧ 𝑤 ∈ (𝑏 ∪ {𝑐})) ∧ ¬ (1^st ‘𝑤) = (1^st ‘𝑐)) → (⟨(1^st ‘𝑤), (2^nd ‘𝑤)⟩ ∈ ((𝑑 sSet ⟨(1^st ‘𝑐), (2^nd ‘𝑐)⟩) ↾ (V ∖ {(1^st ‘𝑐)})) ↔ 𝑤 ∈ (𝑑 sSet ⟨(1^st ‘𝑐), (2^nd ‘𝑐)⟩)))
471	463	opres 6007	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 ⊢ ((1^st ‘𝑤) ∈ (V ∖ {(1^st ‘𝑐)}) → (⟨(1^st ‘𝑤), (2^nd ‘𝑤)⟩ ∈ (𝑑 ↾ (V ∖ {(1^st ‘𝑐)})) ↔ ⟨(1^st ‘𝑤), (2^nd ‘𝑤)⟩ ∈ 𝑑))
472	471	adantl 481	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 ⊢ ((𝑤 ∈ (𝑁 × 𝑁) ∧ (1^st ‘𝑤) ∈ (V ∖ {(1^st ‘𝑐)})) → (⟨(1^st ‘𝑤), (2^nd ‘𝑤)⟩ ∈ (𝑑 ↾ (V ∖ {(1^st ‘𝑐)})) ↔ ⟨(1^st ‘𝑤), (2^nd ‘𝑤)⟩ ∈ 𝑑))
473	466	eleq1d 2826	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 ⊢ (𝑤 ∈ (𝑁 × 𝑁) → (𝑤 ∈ 𝑑 ↔ ⟨(1^st ‘𝑤), (2^nd ‘𝑤)⟩ ∈ 𝑑))
474	473	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 ⊢ ((𝑤 ∈ (𝑁 × 𝑁) ∧ (1^st ‘𝑤) ∈ (V ∖ {(1^st ‘𝑐)})) → (𝑤 ∈ 𝑑 ↔ ⟨(1^st ‘𝑤), (2^nd ‘𝑤)⟩ ∈ 𝑑))
475	472, 474	bitr4d 282	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 ⊢ ((𝑤 ∈ (𝑁 × 𝑁) ∧ (1^st ‘𝑤) ∈ (V ∖ {(1^st ‘𝑐)})) → (⟨(1^st ‘𝑤), (2^nd ‘𝑤)⟩ ∈ (𝑑 ↾ (V ∖ {(1^st ‘𝑐)})) ↔ 𝑤 ∈ 𝑑))
476	403, 462, 475	syl2an 596	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ⊢ (((((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) ∧ 𝑑 ∈ (𝑁 ↑_m 𝑁)) ∧ 𝑤 ∈ (𝑏 ∪ {𝑐})) ∧ ¬ (1^st ‘𝑤) = (1^st ‘𝑐)) → (⟨(1^st ‘𝑤), (2^nd ‘𝑤)⟩ ∈ (𝑑 ↾ (V ∖ {(1^st ‘𝑐)})) ↔ 𝑤 ∈ 𝑑))
477	457, 470, 476	3bitr3rd 310	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ⊢ (((((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) ∧ 𝑑 ∈ (𝑁 ↑_m 𝑁)) ∧ 𝑤 ∈ (𝑏 ∪ {𝑐})) ∧ ¬ (1^st ‘𝑤) = (1^st ‘𝑐)) → (𝑤 ∈ 𝑑 ↔ 𝑤 ∈ (𝑑 sSet ⟨(1^st ‘𝑐), (2^nd ‘𝑐)⟩)))
478	477	ifbid 4549	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ (((((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) ∧ 𝑑 ∈ (𝑁 ↑_m 𝑁)) ∧ 𝑤 ∈ (𝑏 ∪ {𝑐})) ∧ ¬ (1^st ‘𝑤) = (1^st ‘𝑐)) → if(𝑤 ∈ 𝑑, 1 , 0 ) = if(𝑤 ∈ (𝑑 sSet ⟨(1^st ‘𝑐), (2^nd ‘𝑐)⟩), 1 , 0 ))
479	454, 478	eqtrd 2777	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ (((((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) ∧ 𝑑 ∈ (𝑁 ↑_m 𝑁)) ∧ 𝑤 ∈ (𝑏 ∪ {𝑐})) ∧ ¬ (1^st ‘𝑤) = (1^st ‘𝑐)) → if((1^st ‘𝑤) = (1^st ‘𝑐), if(𝑤 = 𝑐, 1 , 0 ), if(𝑤 ∈ 𝑑, 1 , 0 )) = if(𝑤 ∈ (𝑑 sSet ⟨(1^st ‘𝑐), (2^nd ‘𝑐)⟩), 1 , 0 ))
480		ifeq2 4530	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ ((𝑎‘𝑤) = if(𝑤 ∈ 𝑑, 1 , 0 ) → if((1^st ‘𝑤) = (1^st ‘𝑐), if(𝑤 = 𝑐, 1 , 0 ), (𝑎‘𝑤)) = if((1^st ‘𝑤) = (1^st ‘𝑐), if(𝑤 = 𝑐, 1 , 0 ), if(𝑤 ∈ 𝑑, 1 , 0 )))
481	480	eqeq1d 2739	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ ((𝑎‘𝑤) = if(𝑤 ∈ 𝑑, 1 , 0 ) → (if((1^st ‘𝑤) = (1^st ‘𝑐), if(𝑤 = 𝑐, 1 , 0 ), (𝑎‘𝑤)) = if(𝑤 ∈ (𝑑 sSet ⟨(1^st ‘𝑐), (2^nd ‘𝑐)⟩), 1 , 0 ) ↔ if((1^st ‘𝑤) = (1^st ‘𝑐), if(𝑤 = 𝑐, 1 , 0 ), if(𝑤 ∈ 𝑑, 1 , 0 )) = if(𝑤 ∈ (𝑑 sSet ⟨(1^st ‘𝑐), (2^nd ‘𝑐)⟩), 1 , 0 )))
482	479, 481	syl5ibrcom 247	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ (((((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) ∧ 𝑑 ∈ (𝑁 ↑_m 𝑁)) ∧ 𝑤 ∈ (𝑏 ∪ {𝑐})) ∧ ¬ (1^st ‘𝑤) = (1^st ‘𝑐)) → ((𝑎‘𝑤) = if(𝑤 ∈ 𝑑, 1 , 0 ) → if((1^st ‘𝑤) = (1^st ‘𝑐), if(𝑤 = 𝑐, 1 , 0 ), (𝑎‘𝑤)) = if(𝑤 ∈ (𝑑 sSet ⟨(1^st ‘𝑐), (2^nd ‘𝑐)⟩), 1 , 0 )))
483	452, 482	embantd 59	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (((((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) ∧ 𝑑 ∈ (𝑁 ↑_m 𝑁)) ∧ 𝑤 ∈ (𝑏 ∪ {𝑐})) ∧ ¬ (1^st ‘𝑤) = (1^st ‘𝑐)) → ((𝑤 ∈ 𝑏 → (𝑎‘𝑤) = if(𝑤 ∈ 𝑑, 1 , 0 )) → if((1^st ‘𝑤) = (1^st ‘𝑐), if(𝑤 = 𝑐, 1 , 0 ), (𝑎‘𝑤)) = if(𝑤 ∈ (𝑑 sSet ⟨(1^st ‘𝑐), (2^nd ‘𝑐)⟩), 1 , 0 )))
484	442, 483	pm2.61dan 813	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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((1^st ‘𝑤) = (1^st ‘𝑐), if(𝑤 = 𝑐, 1 , 0 ), (𝑎‘𝑤)) = if(𝑤 ∈ (𝑑 sSet ⟨(1^st ‘𝑐), (2^nd ‘𝑐)⟩), 1 , 0 )))
485		fveqeq2 6915	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ (𝑒 = 𝑤 → ((1^st ‘𝑒) = (1^st ‘𝑐) ↔ (1^st ‘𝑤) = (1^st ‘𝑐)))
486		equequ1 2024	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ⊢ (𝑒 = 𝑤 → (𝑒 = 𝑐 ↔ 𝑤 = 𝑐))
487	486	ifbid 4549	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ (𝑒 = 𝑤 → if(𝑒 = 𝑐, 1 , 0 ) = if(𝑤 = 𝑐, 1 , 0 ))
488		fveq2 6906	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ (𝑒 = 𝑤 → (𝑎‘𝑒) = (𝑎‘𝑤))
489	485, 487, 488	ifbieq12d 4554	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ (𝑒 = 𝑤 → if((1^st ‘𝑒) = (1^st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒)) = if((1^st ‘𝑤) = (1^st ‘𝑐), if(𝑤 = 𝑐, 1 , 0 ), (𝑎‘𝑤)))
490		eqid 2737	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ (𝑒 ∈ (𝑁 × 𝑁) ↦ if((1^st ‘𝑒) = (1^st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒))) = (𝑒 ∈ (𝑁 × 𝑁) ↦ if((1^st ‘𝑒) = (1^st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒)))
491	165, 162	ifex 4576	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ if(𝑤 = 𝑐, 1 , 0 ) ∈ V
492		fvex 6919	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ (𝑎‘𝑤) ∈ V
493	491, 492	ifex 4576	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ if((1^st ‘𝑤) = (1^st ‘𝑐), if(𝑤 = 𝑐, 1 , 0 ), (𝑎‘𝑤)) ∈ V
494	489, 490, 493	fvmpt 7016	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ (𝑤 ∈ (𝑁 × 𝑁) → ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1^st ‘𝑒) = (1^st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒)))‘𝑤) = if((1^st ‘𝑤) = (1^st ‘𝑐), if(𝑤 = 𝑐, 1 , 0 ), (𝑎‘𝑤)))
495	494	eqeq1d 2739	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (𝑤 ∈ (𝑁 × 𝑁) → (((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1^st ‘𝑒) = (1^st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒)))‘𝑤) = if(𝑤 ∈ (𝑑 sSet ⟨(1^st ‘𝑐), (2^nd ‘𝑐)⟩), 1 , 0 ) ↔ if((1^st ‘𝑤) = (1^st ‘𝑐), if(𝑤 = 𝑐, 1 , 0 ), (𝑎‘𝑤)) = if(𝑤 ∈ (𝑑 sSet ⟨(1^st ‘𝑐), (2^nd ‘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((1^st ‘𝑒) = (1^st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒)))‘𝑤) = if(𝑤 ∈ (𝑑 sSet ⟨(1^st ‘𝑐), (2^nd ‘𝑐)⟩), 1 , 0 ) ↔ if((1^st ‘𝑤) = (1^st ‘𝑐), if(𝑤 = 𝑐, 1 , 0 ), (𝑎‘𝑤)) = if(𝑤 ∈ (𝑑 sSet ⟨(1^st ‘𝑐), (2^nd ‘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((1^st ‘𝑒) = (1^st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒)))‘𝑤) = if(𝑤 ∈ (𝑑 sSet ⟨(1^st ‘𝑐), (2^nd ‘𝑐)⟩), 1 , 0 )))
498	497	ralimdva 3167	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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((1^st ‘𝑒) = (1^st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒)))‘𝑤) = if(𝑤 ∈ (𝑑 sSet ⟨(1^st ‘𝑐), (2^nd ‘𝑐)⟩), 1 , 0 )))
499	396, 498	biimtrid 242	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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((1^st ‘𝑒) = (1^st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒)))‘𝑤) = if(𝑤 ∈ (𝑑 sSet ⟨(1^st ‘𝑐), (2^nd ‘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((1^st ‘𝑒) = (1^st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒)))‘𝑤) = if(𝑤 ∈ (𝑑 sSet ⟨(1^st ‘𝑐), (2^nd ‘𝑐)⟩), 1 , 0 ))
501	500	3adantr1 1170	. . . . . . . . . . . . . . . . . . . . . . . . . . . 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((1^st ‘𝑒) = (1^st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒)))‘𝑤) = if(𝑤 ∈ (𝑑 sSet ⟨(1^st ‘𝑐), (2^nd ‘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((1^st ‘𝑒) = (1^st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒))) ∈ 𝐵)
503		simpr2 1196	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 ))) → (1^st ‘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 ))) → (2^nd ‘𝑐) ∈ 𝑁)
507	503, 504, 505, 506, 415	syl211anc 1378	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 ⟨(1^st ‘𝑐), (2^nd ‘𝑐)⟩):𝑁⟶𝑁)
508	158, 158	elmapd 8880	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) → ((𝑑 sSet ⟨(1^st ‘𝑐), (2^nd ‘𝑐)⟩) ∈ (𝑁 ↑_m 𝑁) ↔ (𝑑 sSet ⟨(1^st ‘𝑐), (2^nd ‘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 ⟨(1^st ‘𝑐), (2^nd ‘𝑐)⟩) ∈ (𝑁 ↑_m 𝑁) ↔ (𝑑 sSet ⟨(1^st ‘𝑐), (2^nd ‘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 ⟨(1^st ‘𝑐), (2^nd ‘𝑐)⟩) ∈ (𝑁 ↑_m 𝑁))
511		simpr1 1195	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 3323	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 3221	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 3680	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 6905	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (𝑦 = (𝑒 ∈ (𝑁 × 𝑁) ↦ if((1^st ‘𝑒) = (1^st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒))) → (𝑦‘𝑤) = ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1^st ‘𝑒) = (1^st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒)))‘𝑤))
519	518	eqeq1d 2739	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝑦 = (𝑒 ∈ (𝑁 × 𝑁) ↦ if((1^st ‘𝑒) = (1^st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒))) → ((𝑦‘𝑤) = if(𝑤 ∈ 𝑧, 1 , 0 ) ↔ ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1^st ‘𝑒) = (1^st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒)))‘𝑤) = if(𝑤 ∈ 𝑧, 1 , 0 )))
520	519	ralbidv 3178	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝑦 = (𝑒 ∈ (𝑁 × 𝑁) ↦ if((1^st ‘𝑒) = (1^st ‘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((1^st ‘𝑒) = (1^st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒)))‘𝑤) = if(𝑤 ∈ 𝑧, 1 , 0 )))
521		fveqeq2 6915	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝑦 = (𝑒 ∈ (𝑁 × 𝑁) ↦ if((1^st ‘𝑒) = (1^st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒))) → ((𝐷‘𝑦) = 0 ↔ (𝐷‘(𝑒 ∈ (𝑁 × 𝑁) ↦ if((1^st ‘𝑒) = (1^st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒)))) = 0 ))
522	520, 521	imbi12d 344	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑦 = (𝑒 ∈ (𝑁 × 𝑁) ↦ if((1^st ‘𝑒) = (1^st ‘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((1^st ‘𝑒) = (1^st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒)))‘𝑤) = if(𝑤 ∈ 𝑧, 1 , 0 ) → (𝐷‘(𝑒 ∈ (𝑁 × 𝑁) ↦ if((1^st ‘𝑒) = (1^st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒)))) = 0 )))
523		eleq2 2830	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (𝑧 = (𝑑 sSet ⟨(1^st ‘𝑐), (2^nd ‘𝑐)⟩) → (𝑤 ∈ 𝑧 ↔ 𝑤 ∈ (𝑑 sSet ⟨(1^st ‘𝑐), (2^nd ‘𝑐)⟩)))
524	523	ifbid 4549	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (𝑧 = (𝑑 sSet ⟨(1^st ‘𝑐), (2^nd ‘𝑐)⟩) → if(𝑤 ∈ 𝑧, 1 , 0 ) = if(𝑤 ∈ (𝑑 sSet ⟨(1^st ‘𝑐), (2^nd ‘𝑐)⟩), 1 , 0 ))
525	524	eqeq2d 2748	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝑧 = (𝑑 sSet ⟨(1^st ‘𝑐), (2^nd ‘𝑐)⟩) → (((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1^st ‘𝑒) = (1^st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒)))‘𝑤) = if(𝑤 ∈ 𝑧, 1 , 0 ) ↔ ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1^st ‘𝑒) = (1^st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒)))‘𝑤) = if(𝑤 ∈ (𝑑 sSet ⟨(1^st ‘𝑐), (2^nd ‘𝑐)⟩), 1 , 0 )))
526	525	ralbidv 3178	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝑧 = (𝑑 sSet ⟨(1^st ‘𝑐), (2^nd ‘𝑐)⟩) → (∀𝑤 ∈ (𝑏 ∪ {𝑐})((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1^st ‘𝑒) = (1^st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒)))‘𝑤) = if(𝑤 ∈ 𝑧, 1 , 0 ) ↔ ∀𝑤 ∈ (𝑏 ∪ {𝑐})((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1^st ‘𝑒) = (1^st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒)))‘𝑤) = if(𝑤 ∈ (𝑑 sSet ⟨(1^st ‘𝑐), (2^nd ‘𝑐)⟩), 1 , 0 )))
527	526	imbi1d 341	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑧 = (𝑑 sSet ⟨(1^st ‘𝑐), (2^nd ‘𝑐)⟩) → ((∀𝑤 ∈ (𝑏 ∪ {𝑐})((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1^st ‘𝑒) = (1^st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒)))‘𝑤) = if(𝑤 ∈ 𝑧, 1 , 0 ) → (𝐷‘(𝑒 ∈ (𝑁 × 𝑁) ↦ if((1^st ‘𝑒) = (1^st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒)))) = 0 ) ↔ (∀𝑤 ∈ (𝑏 ∪ {𝑐})((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1^st ‘𝑒) = (1^st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒)))‘𝑤) = if(𝑤 ∈ (𝑑 sSet ⟨(1^st ‘𝑐), (2^nd ‘𝑐)⟩), 1 , 0 ) → (𝐷‘(𝑒 ∈ (𝑁 × 𝑁) ↦ if((1^st ‘𝑒) = (1^st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒)))) = 0 )))
528	522, 527	rspc2va 3634	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1^st ‘𝑒) = (1^st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒))) ∈ 𝐵 ∧ (𝑑 sSet ⟨(1^st ‘𝑐), (2^nd ‘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((1^st ‘𝑒) = (1^st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒)))‘𝑤) = if(𝑤 ∈ (𝑑 sSet ⟨(1^st ‘𝑐), (2^nd ‘𝑐)⟩), 1 , 0 ) → (𝐷‘(𝑒 ∈ (𝑁 × 𝑁) ↦ if((1^st ‘𝑒) = (1^st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒)))) = 0 ))
529	502, 510, 517, 528	syl21anc 838	. . . . . . . . . . . . . . . . . . . . . . . . . . . 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((1^st ‘𝑒) = (1^st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒)))‘𝑤) = if(𝑤 ∈ (𝑑 sSet ⟨(1^st ‘𝑐), (2^nd ‘𝑐)⟩), 1 , 0 ) → (𝐷‘(𝑒 ∈ (𝑁 × 𝑁) ↦ if((1^st ‘𝑒) = (1^st ‘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((1^st ‘𝑒) = (1^st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒)))) = 0 )
531	530	oveq2d 7447	. . . . . . . . . . . . . . . . . . . . . . . . . 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((1^st ‘𝑒) = (1^st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒))))) = (((𝑎‘𝑐)(-_g‘𝑅)if(𝑐 ∈ 𝑑, 1 , 0 )) · 0 ))
532	118	unssad 4193	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 1197	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 3978	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 2115	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ (𝑒 = 𝑤 → (𝑒 ∈ 𝑑 ↔ 𝑤 ∈ 𝑑))
538	537	ifbid 4549	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ (𝑒 = 𝑤 → if(𝑒 ∈ 𝑑, 1 , 0 ) = if(𝑤 ∈ 𝑑, 1 , 0 ))
539	486, 538, 488	ifbieq12d 4554	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 2737	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 4576	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ if(𝑤 ∈ 𝑑, 1 , 0 ) ∈ V
542	541, 492	ifex 4576	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ if(𝑤 = 𝑐, if(𝑤 ∈ 𝑑, 1 , 0 ), (𝑎‘𝑤)) ∈ V
543	539, 540, 542	fvmpt 7016	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 4530	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 4566	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ if(𝑤 = 𝑐, if(𝑤 ∈ 𝑑, 1 , 0 ), if(𝑤 ∈ 𝑑, 1 , 0 )) = if(𝑤 ∈ 𝑑, 1 , 0 )
548	546, 547	eqtrdi 2793	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 2777	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 3167	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 2777	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝑒 = 𝑐 → if(𝑒 = 𝑐, if(𝑒 ∈ 𝑑, 1 , 0 ), (𝑎‘𝑒)) = if(𝑐 ∈ 𝑑, 1 , 0 ))
554	165, 162	ifex 4576	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ if(𝑐 ∈ 𝑑, 1 , 0 ) ∈ V
555	553, 540, 554	fvmpt 7016	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 6906	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 2115	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝑤 = 𝑐 → (𝑤 ∈ 𝑑 ↔ 𝑐 ∈ 𝑑))
560	559	ifbid 4549	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝑤 = 𝑐 → if(𝑤 ∈ 𝑑, 1 , 0 ) = if(𝑐 ∈ 𝑑, 1 , 0 ))
561	558, 560	eqeq12d 2753	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 4894	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 3485	. . . . . . . . . . . . . . . . . . . . . . . . . . . 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 583	. . . . . . . . . . . . . . . . . . . . . . . . . . 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 6905	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 2739	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 3178	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 6915	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 2123	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (𝑧 = 𝑑 → (𝑤 ∈ 𝑧 ↔ 𝑤 ∈ 𝑑))
572	571	ifbid 4549	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝑧 = 𝑑 → if(𝑤 ∈ 𝑧, 1 , 0 ) = if(𝑤 ∈ 𝑑, 1 , 0 ))
573	572	eqeq2d 2748	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 3178	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 3634	. . . . . . . . . . . . . . . . . . . . . . . . . . . 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 838	. . . . . . . . . . . . . . . . . . . . . . . . . . 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 7449	. . . . . . . . . . . . . . . . . . . . . . . . 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((1^st ‘𝑒) = (1^st ‘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 7446	. . . . . . . . . . . . . . . . . . . . . . . . . . 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 18985	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝑅 ∈ Grp ∧ 0 ∈ 𝐾) → ( 0 + 0 ) = 0 )
582	115, 133, 581	syl2anc 584	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) → ( 0 + 0 ) = 0 )
583	580, 582	eqtrd 2777	. . . . . . . . . . . . . . . . . . . . . . . . . 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 2777	. . . . . . . . . . . . . . . . . . . . . . . 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((1^st ‘𝑒) = (1^st ‘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 1387	. . . . . . . . . . . . . . . . . . . . . . 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((1^st ‘𝑒) = (1^st ‘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 2781	. . . . . . . . . . . . . . . . . . . . . 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 3202	. . . . . . . . . . . . . . . . . . . 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 6905	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑎 = 𝑦 → (𝑎‘𝑤) = (𝑦‘𝑤))
591	590	eqeq1d 2739	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑎 = 𝑦 → ((𝑎‘𝑤) = if(𝑤 ∈ 𝑑, 1 , 0 ) ↔ (𝑦‘𝑤) = if(𝑤 ∈ 𝑑, 1 , 0 )))
592	591	ralbidv 3178	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑎 = 𝑦 → (∀𝑤 ∈ 𝑏 (𝑎‘𝑤) = if(𝑤 ∈ 𝑑, 1 , 0 ) ↔ ∀𝑤 ∈ 𝑏 (𝑦‘𝑤) = if(𝑤 ∈ 𝑑, 1 , 0 )))
593		fveqeq2 6915	. . . . . . . . . . . . . . . . . . . . . 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 2123	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑑 = 𝑧 → (𝑤 ∈ 𝑑 ↔ 𝑤 ∈ 𝑧))
596	595	ifbid 4549	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑑 = 𝑧 → if(𝑤 ∈ 𝑑, 1 , 0 ) = if(𝑤 ∈ 𝑧, 1 , 0 ))
597	596	eqeq2d 2748	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑑 = 𝑧 → ((𝑦‘𝑤) = if(𝑤 ∈ 𝑑, 1 , 0 ) ↔ (𝑦‘𝑤) = if(𝑤 ∈ 𝑧, 1 , 0 )))
598	597	ralbidv 3178	. . . . . . . . . . . . . . . . . . . . . 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 3241	. . . . . . . . . . . . . . . . . . . 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 218	. . . . . . . . . . . . . . . . . . 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 3484	. . . . . . . . . . . . . . . . . . . 20 ⊢ 𝑏 ∈ V
603		raleq 3323	. . . . . . . . . . . . . . . . . . . . . 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 3221	. . . . . . . . . . . . . . . . . . . 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 3682	. . . . . . . . . . . . . . . . . . 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 234	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ (𝑏 ∪ {𝑐}) ∈ 𝑌) → 𝑏 ∈ 𝑌)
608	607	3expia 1122	. . . . . . . . . . . . . . . . 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 1133	. . . . . . . . . . . . . . 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 9205	. . . . . . . . . . 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 1120	. . . . . . . . 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 5307	. . . . . . 7 ⊢ ∅ ∈ V
621		raleq 3323	. . . . . . . . 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 3221	. . . . . . 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 3682	. . . . . 6 ⊢ (∅ ∈ 𝑌 ↔ ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ (𝑁 ↑_m 𝑁)(∀𝑤 ∈ ∅ (𝑦‘𝑤) = if(𝑤 ∈ 𝑧, 1 , 0 ) → (𝐷‘𝑦) = 0 ))
625	619, 624	sylib 218	. . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐵) → ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ (𝑁 ↑_m 𝑁)(∀𝑤 ∈ ∅ (𝑦‘𝑤) = if(𝑤 ∈ 𝑧, 1 , 0 ) → (𝐷‘𝑦) = 0 ))
626		fveq1 6905	. . . . . . . . 9 ⊢ (𝑦 = 𝑎 → (𝑦‘𝑤) = (𝑎‘𝑤))
627	626	eqeq1d 2739	. . . . . . . 8 ⊢ (𝑦 = 𝑎 → ((𝑦‘𝑤) = if(𝑤 ∈ 𝑧, 1 , 0 ) ↔ (𝑎‘𝑤) = if(𝑤 ∈ 𝑧, 1 , 0 )))
628	627	ralbidv 3178	. . . . . . 7 ⊢ (𝑦 = 𝑎 → (∀𝑤 ∈ ∅ (𝑦‘𝑤) = if(𝑤 ∈ 𝑧, 1 , 0 ) ↔ ∀𝑤 ∈ ∅ (𝑎‘𝑤) = if(𝑤 ∈ 𝑧, 1 , 0 )))
629		fveqeq2 6915	. . . . . . 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 2830	. . . . . . . . . 10 ⊢ (𝑧 = ( I ↾ 𝑁) → (𝑤 ∈ 𝑧 ↔ 𝑤 ∈ ( I ↾ 𝑁)))
632	631	ifbid 4549	. . . . . . . . 9 ⊢ (𝑧 = ( I ↾ 𝑁) → if(𝑤 ∈ 𝑧, 1 , 0 ) = if(𝑤 ∈ ( I ↾ 𝑁), 1 , 0 ))
633	632	eqeq2d 2748	. . . . . . . 8 ⊢ (𝑧 = ( I ↾ 𝑁) → ((𝑎‘𝑤) = if(𝑤 ∈ 𝑧, 1 , 0 ) ↔ (𝑎‘𝑤) = if(𝑤 ∈ ( I ↾ 𝑁), 1 , 0 )))
634	633	ralbidv 3178	. . . . . . 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 3634	. . . . 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 838	. . . 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 5242	. 2 ⊢ (𝜑 → (𝑎 ∈ 𝐵 ↦ (𝐷‘𝑎)) = (𝑎 ∈ 𝐵 ↦ 0 ))
640	53	feqmptd 6977	. 2 ⊢ (𝜑 → 𝐷 = (𝑎 ∈ 𝐵 ↦ (𝐷‘𝑎)))
641		fconstmpt 5747	. . 3 ⊢ (𝐵 × { 0 }) = (𝑎 ∈ 𝐵 ↦ 0 )
642	641	a1i 11	. 2 ⊢ (𝜑 → (𝐵 × { 0 }) = (𝑎 ∈ 𝐵 ↦ 0 ))
643	639, 640, 642	3eqtr4d 2787	1 ⊢ (𝜑 → 𝐷 = (𝐵 × { 0 }))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 ∨ wo 848 ∧ w3a 1087 = wceq 1540 ∈ wcel 2108 {cab 2714 ≠ wne 2940 ∀wral 3061 Vcvv 3480 ∖ cdif 3948 ∪ cun 3949 ⊆ wss 3951 ∅c0 4333 ifcif 4525 {csn 4626 ⟨cop 4632 ↦ cmpt 5225 I cid 5577 × cxp 5683 ↾ cres 5687 Rel wrel 5690 Fn wfn 6556 ⟶wf 6557 –1-1-onto→wf1o 6560 ‘cfv 6561 (class class class)co 7431 ∈ cmpo 7433 ∘_f cof 7695 1^st c1st 8012 2^nd c2nd 8013 ↑_m cmap 8866 Fincfn 8985 sSet csts 17200 Basecbs 17247 +_gcplusg 17297 ._rcmulr 17298 0_gc0g 17484 Grpcgrp 18951 -_gcsg 18953 1_rcur 20178 Ringcrg 20230 Mat cmat 22411

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 ax-addf 11234 ax-mulf 11235

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-xor 1512 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-tp 4631 df-op 4633 df-ot 4635 df-uni 4908 df-int 4947 df-iun 4993 df-iin 4994 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-se 5638 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-isom 6570 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-of 7697 df-om 7888 df-1st 8014 df-2nd 8015 df-supp 8186 df-tpos 8251 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-2o 8507 df-er 8745 df-map 8868 df-ixp 8938 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-fsupp 9402 df-sup 9482 df-oi 9550 df-card 9979 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-div 11921 df-nn 12267 df-2 12329 df-3 12330 df-4 12331 df-5 12332 df-6 12333 df-7 12334 df-8 12335 df-9 12336 df-n0 12527 df-xnn0 12600 df-z 12614 df-dec 12734 df-uz 12879 df-rp 13035 df-fz 13548 df-fzo 13695 df-seq 14043 df-exp 14103 df-hash 14370 df-word 14553 df-lsw 14601 df-concat 14609 df-s1 14634 df-substr 14679 df-pfx 14709 df-splice 14788 df-reverse 14797 df-s2 14887 df-struct 17184 df-sets 17201 df-slot 17219 df-ndx 17231 df-base 17248 df-ress 17275 df-plusg 17310 df-mulr 17311 df-starv 17312 df-sca 17313 df-vsca 17314 df-ip 17315 df-tset 17316 df-ple 17317 df-ds 17319 df-unif 17320 df-hom 17321 df-cco 17322 df-0g 17486 df-gsum 17487 df-prds 17492 df-pws 17494 df-mre 17629 df-mrc 17630 df-acs 17632 df-mgm 18653 df-sgrp 18732 df-mnd 18748 df-mhm 18796 df-submnd 18797 df-efmnd 18882 df-grp 18954 df-minusg 18955 df-sbg 18956 df-mulg 19086 df-subg 19141 df-ghm 19231 df-gim 19277 df-cntz 19335 df-oppg 19364 df-symg 19387 df-pmtr 19460 df-psgn 19509 df-evpm 19510 df-cmn 19800 df-abl 19801 df-mgp 20138 df-rng 20150 df-ur 20179 df-ring 20232 df-cring 20233 df-oppr 20334 df-dvdsr 20357 df-unit 20358 df-invr 20388 df-dvr 20401 df-rhm 20472 df-subrng 20546 df-subrg 20570 df-drng 20731 df-lmod 20860 df-lss 20930 df-sra 21172 df-rgmod 21173 df-cnfld 21365 df-zring 21458 df-zrh 21514 df-dsmm 21752 df-frlm 21767 df-mamu 22395 df-mat 22412

This theorem is referenced by: mdetuni0 22627

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