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Theorem mdetunilem9 22962

Description: Lemma for mdetuni 22964. (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 4457	. . . 4 ⊢ ∀𝑤 ∈ ∅ (𝑎‘𝑤) = if(𝑤 ∈ ( I ↾ 𝑁), 1 , 0 )
2		simpr 490	. . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐵) → 𝑎 ∈ 𝐵)
3		f1oi 6883	. . . . . . . 8 ⊢ ( I ↾ 𝑁):𝑁⟶1-1onto𝑁
4		f1of 6843	. . . . . . . 8 ⊢ (( I ↾ 𝑁):𝑁⟶1-1onto𝑁 → ( I ↾ 𝑁):𝑁⟶𝑁)
5	3, 4	mp1i 14	. . . . . . 7 ⊢ (𝜑 → ( I ↾ 𝑁):𝑁⟶𝑁)
6		mdetuni.n	. . . . . . . 8 ⊢ (𝜑 → 𝑁 ∈ Fin)
7	6, 6	elmapd 8885	. . . . . . 7 ⊢ (𝜑 → (( I ↾ 𝑁) ∈ (𝑁 ↑_m 𝑁) ↔ ( I ↾ 𝑁):𝑁⟶𝑁))
8	5, 7	mpbird 260	. . . . . 6 ⊢ (𝜑 → ( I ↾ 𝑁) ∈ (𝑁 ↑_m 𝑁))
9	8	adantr 486	. . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐵) → ( I ↾ 𝑁) ∈ (𝑁 ↑_m 𝑁))
10		simplrl 789	. . . . . . . . . . . . 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 22764	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 ∈ 𝐵 → 𝑦 ∈ (𝐾 ↑_m (𝑁 × 𝑁)))
15		elmapi 8894	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 ∈ (𝐾 ↑_m (𝑁 × 𝑁)) → 𝑦:(𝑁 × 𝑁)⟶𝐾)
16	14, 15	syl 18	. . . . . . . . . . . . . . 15 ⊢ (𝑦 ∈ 𝐵 → 𝑦:(𝑁 × 𝑁)⟶𝐾)
17	16	feqmptd 6973	. . . . . . . . . . . . . 14 ⊢ (𝑦 ∈ 𝐵 → 𝑦 = (𝑤 ∈ (𝑁 × 𝑁) ↦ (𝑦‘𝑤)))
18	17	fveq2d 6909	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ 𝐵 → (𝐷‘𝑦) = (𝐷‘(𝑤 ∈ (𝑁 × 𝑁) ↦ (𝑦‘𝑤))))
19	10, 18	syl 18	. . . . . . . . . . . 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 2763	. . . . . . . . . . . . . 14 ⊢ (𝑁 × 𝑁) = (𝑁 × 𝑁)
21		mpteq12 5196	. . . . . . . . . . . . . . 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 6909	. . . . . . . . . . . . . 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 703	. . . . . . . . . . . . 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 487	. . . . . . . . . . . 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 2852	. . . . . . . . . . . . . . . . 17 ⊢ (𝑎 = 𝑧 → (𝑎 ∈ (𝑁 ↑_m 𝑁) ↔ 𝑧 ∈ (𝑁 ↑_m 𝑁)))
26	25	anbi2d 642	. . . . . . . . . . . . . . . 16 ⊢ (𝑎 = 𝑧 → ((𝜑 ∧ 𝑎 ∈ (𝑁 ↑_m 𝑁)) ↔ (𝜑 ∧ 𝑧 ∈ (𝑁 ↑_m 𝑁))))
27		elequ2 2162	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑎 = 𝑧 → (𝑤 ∈ 𝑎 ↔ 𝑤 ∈ 𝑧))
28	27	ifbid 4509	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑎 = 𝑧 → if(𝑤 ∈ 𝑎, 1 , 0 ) = if(𝑤 ∈ 𝑧, 1 , 0 ))
29	28	mpteq2dv 5202	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑎 = 𝑧 → (𝑤 ∈ (𝑁 × 𝑁) ↦ if(𝑤 ∈ 𝑎, 1 , 0 )) = (𝑤 ∈ (𝑁 × 𝑁) ↦ if(𝑤 ∈ 𝑧, 1 , 0 )))
30	29	fveq2d 6909	. . . . . . . . . . . . . . . . 17 ⊢ (𝑎 = 𝑧 → (𝐷‘(𝑤 ∈ (𝑁 × 𝑁) ↦ if(𝑤 ∈ 𝑎, 1 , 0 ))) = (𝐷‘(𝑤 ∈ (𝑁 × 𝑁) ↦ if(𝑤 ∈ 𝑧, 1 , 0 ))))
31	30	eqeq1d 2765	. . . . . . . . . . . . . . . 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 347	. . . . . . . . . . . . . . 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 2852	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑤 = ⟨𝑏, 𝑐⟩ → (𝑤 ∈ 𝑎 ↔ ⟨𝑏, 𝑐⟩ ∈ 𝑎))
34	33	ifbid 4509	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑤 = ⟨𝑏, 𝑐⟩ → if(𝑤 ∈ 𝑎, 1 , 0 ) = if(⟨𝑏, 𝑐⟩ ∈ 𝑎, 1 , 0 ))
35	34	mpompt 7554	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑤 ∈ (𝑁 × 𝑁) ↦ if(𝑤 ∈ 𝑎, 1 , 0 )) = (𝑏 ∈ 𝑁, 𝑐 ∈ 𝑁 ↦ if(⟨𝑏, 𝑐⟩ ∈ 𝑎, 1 , 0 ))
36		elmapi 8894	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑎 ∈ (𝑁 ↑_m 𝑁) → 𝑎:𝑁⟶𝑁)
37	36	adantl 487	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ 𝑎 ∈ (𝑁 ↑_m 𝑁)) → 𝑎:𝑁⟶𝑁)
38	37	ffnd 6729	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ 𝑎 ∈ (𝑁 ↑_m 𝑁)) → 𝑎 Fn 𝑁)
39	38	3ad2ant1 1151	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ 𝑎 ∈ (𝑁 ↑_m 𝑁)) ∧ 𝑏 ∈ 𝑁 ∧ 𝑐 ∈ 𝑁) → 𝑎 Fn 𝑁)
40		simp2 1155	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ 𝑎 ∈ (𝑁 ↑_m 𝑁)) ∧ 𝑏 ∈ 𝑁 ∧ 𝑐 ∈ 𝑁) → 𝑏 ∈ 𝑁)
41		fnopfvb 6956	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑎 Fn 𝑁 ∧ 𝑏 ∈ 𝑁) → ((𝑎‘𝑏) = 𝑐 ↔ ⟨𝑏, 𝑐⟩ ∈ 𝑎))
42	39, 40, 41	syl2anc 596	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑎 ∈ (𝑁 ↑_m 𝑁)) ∧ 𝑏 ∈ 𝑁 ∧ 𝑐 ∈ 𝑁) → ((𝑎‘𝑏) = 𝑐 ↔ ⟨𝑏, 𝑐⟩ ∈ 𝑎))
43	42	bicomd 226	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑎 ∈ (𝑁 ↑_m 𝑁)) ∧ 𝑏 ∈ 𝑁 ∧ 𝑐 ∈ 𝑁) → (⟨𝑏, 𝑐⟩ ∈ 𝑎 ↔ (𝑎‘𝑏) = 𝑐))
44	43	ifbid 4509	. . . . . . . . . . . . . . . . . . 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 7517	. . . . . . . . . . . . . . . . . 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 2810	. . . . . . . . . . . . . . . . 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 6909	. . . . . . . . . . . . . . . 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 22961	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑎:𝑁⟶𝑁) → (𝐷‘(𝑏 ∈ 𝑁, 𝑐 ∈ 𝑁 ↦ if((𝑎‘𝑏) = 𝑐, 1 , 0 ))) = 0 )
59	36, 58	sylan2 605	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑎 ∈ (𝑁 ↑_m 𝑁)) → (𝐷‘(𝑏 ∈ 𝑁, 𝑐 ∈ 𝑁 ↦ if((𝑎‘𝑏) = 𝑐, 1 , 0 ))) = 0 )
60	47, 59	eqtrd 2798	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑎 ∈ (𝑁 ↑_m 𝑁)) → (𝐷‘(𝑤 ∈ (𝑁 × 𝑁) ↦ if(𝑤 ∈ 𝑎, 1 , 0 ))) = 0 )
61	32, 60	chvarvv 2024	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝑁 ↑_m 𝑁)) → (𝐷‘(𝑤 ∈ (𝑁 × 𝑁) ↦ if(𝑤 ∈ 𝑧, 1 , 0 ))) = 0 )
62	61	adantrl 729	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ (𝑁 ↑_m 𝑁))) → (𝐷‘(𝑤 ∈ (𝑁 × 𝑁) ↦ if(𝑤 ∈ 𝑧, 1 , 0 ))) = 0 )
63	62	adantr 486	. . . . . . . . . . . 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 2802	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ (𝑁 ↑_m 𝑁))) ∧ ∀𝑤 ∈ (𝑁 × 𝑁)(𝑦‘𝑤) = if(𝑤 ∈ 𝑧, 1 , 0 )) → (𝐷‘𝑦) = 0 )
65	64	ex 418	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ (𝑁 ↑_m 𝑁))) → (∀𝑤 ∈ (𝑁 × 𝑁)(𝑦‘𝑤) = if(𝑤 ∈ 𝑧, 1 , 0 ) → (𝐷‘𝑦) = 0 ))
66	65	ralrimivva 3209	. . . . . . . . 9 ⊢ (𝜑 → ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ (𝑁 ↑_m 𝑁)(∀𝑤 ∈ (𝑁 × 𝑁)(𝑦‘𝑤) = if(𝑤 ∈ 𝑧, 1 , 0 ) → (𝐷‘𝑦) = 0 ))
67		xpfi 9336	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ Fin ∧ 𝑁 ∈ Fin) → (𝑁 × 𝑁) ∈ Fin)
68	6, 6, 67	syl2anc 596	. . . . . . . . . 10 ⊢ (𝜑 → (𝑁 × 𝑁) ∈ Fin)
69		raleq 3320	. . . . . . . . . . . . 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 344	. . . . . . . . . . . 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 3230	. . . . . . . . . . 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 3637	. . . . . . . . . 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 18	. . . . . . . . 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 260	. . . . . . . 8 ⊢ (𝜑 → (𝑁 × 𝑁) ∈ 𝑌)
76		ssid 3935	. . . . . . . . 9 ⊢ (𝑁 × 𝑁) ⊆ (𝑁 × 𝑁)
77	68	3ad2ant1 1151	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑁 × 𝑁) ⊆ (𝑁 × 𝑁) ∧ ¬ ∅ ∈ 𝑌) → (𝑁 × 𝑁) ∈ Fin)
78		sseq1 3938	. . . . . . . . . . . . . 14 ⊢ (𝑎 = ∅ → (𝑎 ⊆ (𝑁 × 𝑁) ↔ ∅ ⊆ (𝑁 × 𝑁)))
79	78	3anbi2d 1470	. . . . . . . . . . . . 13 ⊢ (𝑎 = ∅ → ((𝜑 ∧ 𝑎 ⊆ (𝑁 × 𝑁) ∧ ¬ ∅ ∈ 𝑌) ↔ (𝜑 ∧ ∅ ⊆ (𝑁 × 𝑁) ∧ ¬ ∅ ∈ 𝑌)))
80		eleq1 2852	. . . . . . . . . . . . . 14 ⊢ (𝑎 = ∅ → (𝑎 ∈ 𝑌 ↔ ∅ ∈ 𝑌))
81	80	notbid 321	. . . . . . . . . . . . 13 ⊢ (𝑎 = ∅ → (¬ 𝑎 ∈ 𝑌 ↔ ¬ ∅ ∈ 𝑌))
82	79, 81	imbi12d 347	. . . . . . . . . . . 12 ⊢ (𝑎 = ∅ → (((𝜑 ∧ 𝑎 ⊆ (𝑁 × 𝑁) ∧ ¬ ∅ ∈ 𝑌) → ¬ 𝑎 ∈ 𝑌) ↔ ((𝜑 ∧ ∅ ⊆ (𝑁 × 𝑁) ∧ ¬ ∅ ∈ 𝑌) → ¬ ∅ ∈ 𝑌)))
83		sseq1 3938	. . . . . . . . . . . . . 14 ⊢ (𝑎 = 𝑏 → (𝑎 ⊆ (𝑁 × 𝑁) ↔ 𝑏 ⊆ (𝑁 × 𝑁)))
84	83	3anbi2d 1470	. . . . . . . . . . . . 13 ⊢ (𝑎 = 𝑏 → ((𝜑 ∧ 𝑎 ⊆ (𝑁 × 𝑁) ∧ ¬ ∅ ∈ 𝑌) ↔ (𝜑 ∧ 𝑏 ⊆ (𝑁 × 𝑁) ∧ ¬ ∅ ∈ 𝑌)))
85		eleq1 2852	. . . . . . . . . . . . . 14 ⊢ (𝑎 = 𝑏 → (𝑎 ∈ 𝑌 ↔ 𝑏 ∈ 𝑌))
86	85	notbid 321	. . . . . . . . . . . . 13 ⊢ (𝑎 = 𝑏 → (¬ 𝑎 ∈ 𝑌 ↔ ¬ 𝑏 ∈ 𝑌))
87	84, 86	imbi12d 347	. . . . . . . . . . . 12 ⊢ (𝑎 = 𝑏 → (((𝜑 ∧ 𝑎 ⊆ (𝑁 × 𝑁) ∧ ¬ ∅ ∈ 𝑌) → ¬ 𝑎 ∈ 𝑌) ↔ ((𝜑 ∧ 𝑏 ⊆ (𝑁 × 𝑁) ∧ ¬ ∅ ∈ 𝑌) → ¬ 𝑏 ∈ 𝑌)))
88		sseq1 3938	. . . . . . . . . . . . . 14 ⊢ (𝑎 = (𝑏 ∪ {𝑐}) → (𝑎 ⊆ (𝑁 × 𝑁) ↔ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁)))
89	88	3anbi2d 1470	. . . . . . . . . . . . 13 ⊢ (𝑎 = (𝑏 ∪ {𝑐}) → ((𝜑 ∧ 𝑎 ⊆ (𝑁 × 𝑁) ∧ ¬ ∅ ∈ 𝑌) ↔ (𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ ¬ ∅ ∈ 𝑌)))
90		eleq1 2852	. . . . . . . . . . . . . 14 ⊢ (𝑎 = (𝑏 ∪ {𝑐}) → (𝑎 ∈ 𝑌 ↔ (𝑏 ∪ {𝑐}) ∈ 𝑌))
91	90	notbid 321	. . . . . . . . . . . . 13 ⊢ (𝑎 = (𝑏 ∪ {𝑐}) → (¬ 𝑎 ∈ 𝑌 ↔ ¬ (𝑏 ∪ {𝑐}) ∈ 𝑌))
92	89, 91	imbi12d 347	. . . . . . . . . . . 12 ⊢ (𝑎 = (𝑏 ∪ {𝑐}) → (((𝜑 ∧ 𝑎 ⊆ (𝑁 × 𝑁) ∧ ¬ ∅ ∈ 𝑌) → ¬ 𝑎 ∈ 𝑌) ↔ ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ ¬ ∅ ∈ 𝑌) → ¬ (𝑏 ∪ {𝑐}) ∈ 𝑌)))
93		sseq1 3938	. . . . . . . . . . . . . 14 ⊢ (𝑎 = (𝑁 × 𝑁) → (𝑎 ⊆ (𝑁 × 𝑁) ↔ (𝑁 × 𝑁) ⊆ (𝑁 × 𝑁)))
94	93	3anbi2d 1470	. . . . . . . . . . . . 13 ⊢ (𝑎 = (𝑁 × 𝑁) → ((𝜑 ∧ 𝑎 ⊆ (𝑁 × 𝑁) ∧ ¬ ∅ ∈ 𝑌) ↔ (𝜑 ∧ (𝑁 × 𝑁) ⊆ (𝑁 × 𝑁) ∧ ¬ ∅ ∈ 𝑌)))
95		eleq1 2852	. . . . . . . . . . . . . 14 ⊢ (𝑎 = (𝑁 × 𝑁) → (𝑎 ∈ 𝑌 ↔ (𝑁 × 𝑁) ∈ 𝑌))
96	95	notbid 321	. . . . . . . . . . . . 13 ⊢ (𝑎 = (𝑁 × 𝑁) → (¬ 𝑎 ∈ 𝑌 ↔ ¬ (𝑁 × 𝑁) ∈ 𝑌))
97	94, 96	imbi12d 347	. . . . . . . . . . . 12 ⊢ (𝑎 = (𝑁 × 𝑁) → (((𝜑 ∧ 𝑎 ⊆ (𝑁 × 𝑁) ∧ ¬ ∅ ∈ 𝑌) → ¬ 𝑎 ∈ 𝑌) ↔ ((𝜑 ∧ (𝑁 × 𝑁) ⊆ (𝑁 × 𝑁) ∧ ¬ ∅ ∈ 𝑌) → ¬ (𝑁 × 𝑁) ∈ 𝑌)))
98		simp3 1156	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ ∅ ⊆ (𝑁 × 𝑁) ∧ ¬ ∅ ∈ 𝑌) → ¬ ∅ ∈ 𝑌)
99		ssun1 4179	. . . . . . . . . . . . . . . 16 ⊢ 𝑏 ⊆ (𝑏 ∪ {𝑐})
100		sstr2 3920	. . . . . . . . . . . . . . . 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 1171	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ ¬ ∅ ∈ 𝑌) → (𝜑 ∧ 𝑏 ⊆ (𝑁 × 𝑁) ∧ ¬ ∅ ∈ 𝑌))
103	102	imim1i 64	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑏 ⊆ (𝑁 × 𝑁) ∧ ¬ ∅ ∈ 𝑌) → ¬ 𝑏 ∈ 𝑌) → ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ ¬ ∅ ∈ 𝑌) → ¬ 𝑏 ∈ 𝑌))
104		simpl1 1210	. . . . . . . . . . . . . . . . . . . . . . . 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 1211	. . . . . . . . . . . . . . . . . . . . . . . 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 791	. . . . . . . . . . . . . . . . . . . . . . . 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 22764	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑎 ∈ 𝐵 → 𝑎 ∈ (𝐾 ↑_m (𝑁 × 𝑁)))
108		elmapi 8894	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑎 ∈ (𝐾 ↑_m (𝑁 × 𝑁)) → 𝑎:(𝑁 × 𝑁)⟶𝐾)
109	107, 108	syl 18	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑎 ∈ 𝐵 → 𝑎:(𝑁 × 𝑁)⟶𝐾)
110	109	3ad2ant3 1153	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) → 𝑎:(𝑁 × 𝑁)⟶𝐾)
111	110	feqmptd 6973	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) → 𝑎 = (𝑒 ∈ (𝑁 × 𝑁) ↦ (𝑎‘𝑒)))
112	111	reseq1d 5979	. . . . . . . . . . . . . . . . . . . . . . . . . 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 1151	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) → 𝑅 ∈ Ring)
114		ringgrp 20491	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ (𝑅 ∈ Ring → 𝑅 ∈ Grp)
115	113, 114	syl 18	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) → 𝑅 ∈ Grp)
116	115	adantr 486	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) ∧ 𝑒 ∈ ({(1^st ‘𝑐)} × 𝑁)) → 𝑅 ∈ Grp)
117	110	adantr 486	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) ∧ 𝑒 ∈ ({(1^st ‘𝑐)} × 𝑁)) → 𝑎:(𝑁 × 𝑁)⟶𝐾)
118		simp2 1155	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 ⊢ ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) → (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁))
119	118	unssbd 4195	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 ⊢ ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) → {𝑐} ⊆ (𝑁 × 𝑁))
120		vex 3458	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 ⊢ 𝑐 ∈ V
121	120	snss 4745	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 ⊢ (𝑐 ∈ (𝑁 × 𝑁) ↔ {𝑐} ⊆ (𝑁 × 𝑁))
122	119, 121	sylibr 237	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ⊢ ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) → 𝑐 ∈ (𝑁 × 𝑁))
123		xp1st 8054	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ⊢ (𝑐 ∈ (𝑁 × 𝑁) → (1^st ‘𝑐) ∈ 𝑁)
124	122, 123	syl 18	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ⊢ ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) → (1^st ‘𝑐) ∈ 𝑁)
125	124	snssd 4747	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) → {(1^st ‘𝑐)} ⊆ 𝑁)
126		xpss1 5674	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ ({(1^st ‘𝑐)} ⊆ 𝑁 → ({(1^st ‘𝑐)} × 𝑁) ⊆ (𝑁 × 𝑁))
127	125, 126	syl 18	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) → ({(1^st ‘𝑐)} × 𝑁) ⊆ (𝑁 × 𝑁))
128	127	sselda 3913	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) ∧ 𝑒 ∈ ({(1^st ‘𝑐)} × 𝑁)) → 𝑒 ∈ (𝑁 × 𝑁))
129	117, 128	ffvelcdmd 7106	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) ∧ 𝑒 ∈ ({(1^st ‘𝑐)} × 𝑁)) → (𝑎‘𝑒) ∈ 𝐾)
130	12, 49	ringidcl 20521	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ (𝑅 ∈ Ring → 1 ∈ 𝐾)
131	113, 130	syl 18	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) → 1 ∈ 𝐾)
132	12, 48	ring0cl 20523	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ (𝑅 ∈ Ring → 0 ∈ 𝐾)
133	113, 132	syl 18	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) → 0 ∈ 𝐾)
134	131, 133	ifcld 4532	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) → if(𝑒 ∈ 𝑑, 1 , 0 ) ∈ 𝐾)
135	134	adantr 486	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) ∧ 𝑒 ∈ ({(1^st ‘𝑐)} × 𝑁)) → if(𝑒 ∈ 𝑑, 1 , 0 ) ∈ 𝐾)
136		eqid 2763	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ (-_g‘𝑅) = (-_g‘𝑅)
137	12, 50, 136	grpnpcan 19268	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 1399	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 2769	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 486	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 4491	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 4491	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (𝑒 = 𝑐 → if(𝑒 = 𝑐, if(𝑒 ∈ 𝑑, 1 , 0 ), (𝑎‘𝑒)) = if(𝑒 ∈ 𝑑, 1 , 0 ))
143	141, 142	oveq12d 7458	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 487	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 2801	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 19204	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ ((𝑅 ∈ Grp ∧ (𝑎‘𝑒) ∈ 𝐾) → ( 0 + (𝑎‘𝑒)) = (𝑎‘𝑒))
147	116, 129, 146	syl2anc 596	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) ∧ 𝑒 ∈ ({(1^st ‘𝑐)} × 𝑁)) → ( 0 + (𝑎‘𝑒)) = (𝑎‘𝑒))
148	147	eqcomd 2769	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) ∧ 𝑒 ∈ ({(1^st ‘𝑐)} × 𝑁)) → (𝑎‘𝑒) = ( 0 + (𝑎‘𝑒)))
149	148	adantr 486	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) ∧ 𝑒 ∈ ({(1^st ‘𝑐)} × 𝑁)) ∧ ¬ 𝑒 = 𝑐) → (𝑎‘𝑒) = ( 0 + (𝑎‘𝑒)))
150		iffalse 4494	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (¬ 𝑒 = 𝑐 → if(𝑒 = 𝑐, ((𝑎‘𝑒)(-_g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ) = 0 )
151		iffalse 4494	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (¬ 𝑒 = 𝑐 → if(𝑒 = 𝑐, if(𝑒 ∈ 𝑑, 1 , 0 ), (𝑎‘𝑒)) = (𝑎‘𝑒))
152	150, 151	oveq12d 7458	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 487	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 2801	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 825	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 5201	. . . . . . . . . . . . . . . . . . . . . . . . . . . 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 9096	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ {(1^st ‘𝑐)} ∈ Fin
158	6	3ad2ant1 1151	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) → 𝑁 ∈ Fin)
159		xpfi 9336	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (({(1^st ‘𝑐)} ∈ Fin ∧ 𝑁 ∈ Fin) → ({(1^st ‘𝑐)} × 𝑁) ∈ Fin)
160	157, 158, 159	sylancr 599	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) → ({(1^st ‘𝑐)} × 𝑁) ∈ Fin)
161		ovex 7473	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((𝑎‘𝑒)(-_g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )) ∈ V
162	48	fvexi 6919	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ 0 ∈ V
163	161, 162	ifex 4536	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 6919	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ 1 ∈ V
166	165, 162	ifex 4536	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ if(𝑒 ∈ 𝑑, 1 , 0 ) ∈ V
167		fvex 6918	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝑎‘𝑒) ∈ V
168	166, 167	ifex 4536	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 8054	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝑒 ∈ ({(1^st ‘𝑐)} × 𝑁) → (1^st ‘𝑒) ∈ {(1^st ‘𝑐)})
171		elsni 4601	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((1^st ‘𝑒) ∈ {(1^st ‘𝑐)} → (1^st ‘𝑒) = (1^st ‘𝑐))
172		iftrue 4491	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 19	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 5203	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 2764	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 7734	. . . . . . . . . . . . . . . . . . . . . . . . . . . 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 2801	. . . . . . . . . . . . . . . . . . . . . . . . . . 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 6043	. . . . . . . . . . . . . . . . . . . . . . . . . . 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 6043	. . . . . . . . . . . . . . . . . . . . . . . . . . . 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 6043	. . . . . . . . . . . . . . . . . . . . . . . . . . . 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 7458	. . . . . . . . . . . . . . . . . . . . . . . . . . 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 2808	. . . . . . . . . . . . . . . . . . . . . . . . . 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 2798	. . . . . . . . . . . . . . . . . . . . . . . . 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 5979	. . . . . . . . . . . . . . . . . . . . . . . . . 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 8054	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝑒 ∈ ((𝑁 ∖ {(1^st ‘𝑐)}) × 𝑁) → (1^st ‘𝑒) ∈ (𝑁 ∖ {(1^st ‘𝑐)}))
187		eldifsni 4753	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((1^st ‘𝑒) ∈ (𝑁 ∖ {(1^st ‘𝑐)}) → (1^st ‘𝑒) ≠ (1^st ‘𝑐))
188	186, 187	syl 18	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝑒 ∈ ((𝑁 ∖ {(1^st ‘𝑐)}) × 𝑁) → (1^st ‘𝑒) ≠ (1^st ‘𝑐))
189	188	neneqd 2964	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑒 ∈ ((𝑁 ∖ {(1^st ‘𝑐)}) × 𝑁) → ¬ (1^st ‘𝑒) = (1^st ‘𝑐))
190	189	adantl 487	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) ∧ 𝑒 ∈ ((𝑁 ∖ {(1^st ‘𝑐)}) × 𝑁)) → ¬ (1^st ‘𝑒) = (1^st ‘𝑐))
191	190	iffalsed 4496	. . . . . . . . . . . . . . . . . . . . . . . . . . . 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 5201	. . . . . . . . . . . . . . . . . . . . . . . . . . 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 4075	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑁 ∖ {(1^st ‘𝑐)}) ⊆ 𝑁
194		xpss1 5674	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 6040	. . . . . . . . . . . . . . . . . . . . . . . . . . . 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 14	. . . . . . . . . . . . . . . . . . . . . . . . . . 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 6040	. . . . . . . . . . . . . . . . . . . . . . . . . . . 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 14	. . . . . . . . . . . . . . . . . . . . . . . . . . 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 2809	. . . . . . . . . . . . . . . . . . . . . . . . . 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 2798	. . . . . . . . . . . . . . . . . . . . . . . . 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 6905	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑒 = 𝑐 → (1^st ‘𝑒) = (1^st ‘𝑐))
203	190, 202	nsyl 141	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) ∧ 𝑒 ∈ ((𝑁 ∖ {(1^st ‘𝑐)}) × 𝑁)) → ¬ 𝑒 = 𝑐)
204	203	iffalsed 4496	. . . . . . . . . . . . . . . . . . . . . . . . . . . 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 5201	. . . . . . . . . . . . . . . . . . . . . . . . . . 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 6040	. . . . . . . . . . . . . . . . . . . . . . . . . . . 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 14	. . . . . . . . . . . . . . . . . . . . . . . . . . 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 2809	. . . . . . . . . . . . . . . . . . . . . . . . . 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 2798	. . . . . . . . . . . . . . . . . . . . . . . . 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 486	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) ∧ 𝑒 ∈ (𝑁 × 𝑁)) → if(𝑒 ∈ 𝑑, 1 , 0 ) ∈ 𝐾)
211	110	ffvelcdmda 7105	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) ∧ 𝑒 ∈ (𝑁 × 𝑁)) → (𝑎‘𝑒) ∈ 𝐾)
212	210, 211	ifcld 4532	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 7136	. . . . . . . . . . . . . . . . . . . . . . . . . . . 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 6919	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ 𝐾 ∈ V
215	67	anidms 577	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑁 ∈ Fin → (𝑁 × 𝑁) ∈ Fin)
216	158, 215	syl 18	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) → (𝑁 × 𝑁) ∈ Fin)
217		elmapg 8884	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 599	. . . . . . . . . . . . . . . . . . . . . . . . . . . 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 260	. . . . . . . . . . . . . . . . . . . . . . . . . . 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 22763	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (𝐾 ↑_m (𝑁 × 𝑁)) = (Base‘𝐴))
221	158, 113, 220	syl2anc 596	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) → (𝐾 ↑_m (𝑁 × 𝑁)) = (Base‘𝐴))
222	221, 13	eqtr4di 2816	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) → (𝐾 ↑_m (𝑁 × 𝑁)) = 𝐵)
223	219, 222	eleqtrd 2866	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) → (𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒 ∈ 𝑑, 1 , 0 ), (𝑎‘𝑒))) ∈ 𝐵)
224		simp3 1156	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) → 𝑎 ∈ 𝐵)
225	115	adantr 486	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) ∧ 𝑒 ∈ (𝑁 × 𝑁)) → 𝑅 ∈ Grp)
226	12, 136	grpsubcl 19256	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 1399	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) ∧ 𝑒 ∈ (𝑁 × 𝑁)) → ((𝑎‘𝑒)(-_g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )) ∈ 𝐾)
228	133	adantr 486	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) ∧ 𝑒 ∈ (𝑁 × 𝑁)) → 0 ∈ 𝐾)
229	227, 228	ifcld 4532	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 4532	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 7136	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 8884	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 599	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 260	. . . . . . . . . . . . . . . . . . . . . . . . . . . 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 2866	. . . . . . . . . . . . . . . . . . . . . . . . . . 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 1151	. . . . . . . . . . . . . . . . . . . . . . . . . . 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 5974	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝑥 = 𝑎 → (𝑥 ↾ ({𝑤} × 𝑁)) = (𝑎 ↾ ({𝑤} × 𝑁)))
238	237	eqeq1d 2765	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 5974	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝑥 = 𝑎 → (𝑥 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑎 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)))
240	239	eqeq1d 2765	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝑥 = 𝑎 → ((𝑥 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑦 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) ↔ (𝑎 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑦 ↾ ((𝑁 ∖ {𝑤}) × 𝑁))))
241	239	eqeq1d 2765	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝑥 = 𝑎 → ((𝑥 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑧 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) ↔ (𝑎 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑧 ↾ ((𝑁 ∖ {𝑤}) × 𝑁))))
242	238, 240, 241	3anbi123d 1465	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 6914	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑥 = 𝑎 → ((𝐷‘𝑥) = ((𝐷‘𝑦) + (𝐷‘𝑧)) ↔ (𝐷‘𝑎) = ((𝐷‘𝑦) + (𝐷‘𝑧))))
244	242, 243	imbi12d 347	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 3230	. . . . . . . . . . . . . . . . . . . . . . . . . . . 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 5974	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 7455	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 2774	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 5974	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 2774	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 1466	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 6905	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 7455	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 2774	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 347	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 3230	. . . . . . . . . . . . . . . . . . . . . . . . . . . 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 3591	. . . . . . . . . . . . . . . . . . . . . . . . . . 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 851	. . . . . . . . . . . . . . . . . . . . . . . . . 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 5974	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 7456	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 2774	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 5974	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 2774	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 1467	. . . . . . . . . . . . . . . . . . . . . . . . . . . 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 6905	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 7456	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 2774	. . . . . . . . . . . . . . . . . . . . . . . . . . . 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 347	. . . . . . . . . . . . . . . . . . . . . . . . . . 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 4594	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝑤 = (1^st ‘𝑐) → {𝑤} = {(1^st ‘𝑐)})
270	269	xpeq1d 5684	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝑤 = (1^st ‘𝑐) → ({𝑤} × 𝑁) = ({(1^st ‘𝑐)} × 𝑁))
271	270	reseq2d 5980	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑤 = (1^st ‘𝑐) → (𝑎 ↾ ({𝑤} × 𝑁)) = (𝑎 ↾ ({(1^st ‘𝑐)} × 𝑁)))
272	270	reseq2d 5980	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 5980	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 7458	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 2779	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 4066	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝑤 = (1^st ‘𝑐) → (𝑁 ∖ {𝑤}) = (𝑁 ∖ {(1^st ‘𝑐)}))
277	276	xpeq1d 5684	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝑤 = (1^st ‘𝑐) → ((𝑁 ∖ {𝑤}) × 𝑁) = ((𝑁 ∖ {(1^st ‘𝑐)}) × 𝑁))
278	277	reseq2d 5980	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑤 = (1^st ‘𝑐) → (𝑎 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑎 ↾ ((𝑁 ∖ {(1^st ‘𝑐)}) × 𝑁)))
279	277	reseq2d 5980	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 2779	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 5980	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 2779	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 1465	. . . . . . . . . . . . . . . . . . . . . . . . . . . 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 344	. . . . . . . . . . . . . . . . . . . . . . . . . . 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 3591	. . . . . . . . . . . . . . . . . . . . . . . . . 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 851	. . . . . . . . . . . . . . . . . . . . . . . . 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 1494	. . . . . . . . . . . . . . . . . . . . . . . 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 1399	. . . . . . . . . . . . . . . . . . . . . . 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 6905	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (𝑒 = 𝑐 → (𝑎‘𝑒) = (𝑎‘𝑐))
290		elequ1 2154	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ (𝑒 = 𝑐 → (𝑒 ∈ 𝑑 ↔ 𝑐 ∈ 𝑑))
291	290	ifbid 4509	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (𝑒 = 𝑐 → if(𝑒 ∈ 𝑑, 1 , 0 ) = if(𝑐 ∈ 𝑑, 1 , 0 ))
292	289, 291	oveq12d 7458	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (𝑒 = 𝑐 → ((𝑎‘𝑒)(-_g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )) = ((𝑎‘𝑐)(-_g‘𝑅)if(𝑐 ∈ 𝑑, 1 , 0 )))
293	292	adantl 487	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 7106	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) → (𝑎‘𝑐) ∈ 𝐾)
295	131, 133	ifcld 4532	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) → if(𝑐 ∈ 𝑑, 1 , 0 ) ∈ 𝐾)
296	12, 136	grpsubcl 19256	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 1399	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) → ((𝑎‘𝑐)(-_g‘𝑅)if(𝑐 ∈ 𝑑, 1 , 0 )) ∈ 𝐾)
298	12, 51, 49	ringridm 20526	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 596	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 739	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 2801	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 487	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 4491	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (𝑒 = 𝑐 → if(𝑒 = 𝑐, 1 , 0 ) = 1 )
304	303	oveq2d 7456	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 487	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 2808	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 20552	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 596	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) → (((𝑎‘𝑐)(-_g‘𝑅)if(𝑐 ∈ 𝑑, 1 , 0 )) · 0 ) = 0 )
309	308	eqcomd 2769	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) → 0 = (((𝑎‘𝑐)(-_g‘𝑅)if(𝑐 ∈ 𝑑, 1 , 0 )) · 0 ))
310	309	ad2antrr 739	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 487	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 4494	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (¬ 𝑒 = 𝑐 → if(𝑒 = 𝑐, 1 , 0 ) = 0 )
313	312	oveq2d 7456	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 487	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 2808	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 825	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 487	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) ∧ 𝑒 ∈ ({(1^st ‘𝑐)} × 𝑁)) → (1^st ‘𝑒) ∈ {(1^st ‘𝑐)})
318	317, 171	syl 18	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) ∧ 𝑒 ∈ ({(1^st ‘𝑐)} × 𝑁)) → (1^st ‘𝑒) = (1^st ‘𝑐))
319	318	iftrued 4493	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 4493	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 7456	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 2808	. . . . . . . . . . . . . . . . . . . . . . . . . . . 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 5201	. . . . . . . . . . . . . . . . . . . . . . . . . . 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 7475	. . . . . . . . . . . . . . . . . . . . . . . . . . . 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 4536	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ if(𝑒 = 𝑐, 1 , 0 ) ∈ V
326	325, 167	ifex 4536	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 5717	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 6043	. . . . . . . . . . . . . . . . . . . . . . . . . . . 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 7734	. . . . . . . . . . . . . . . . . . . . . . . . . . 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 2808	. . . . . . . . . . . . . . . . . . . . . . . . . 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 4494	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 4494	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 2801	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 18	. . . . . . . . . . . . . . . . . . . . . . . . . . . 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 5201	. . . . . . . . . . . . . . . . . . . . . . . . . . 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 6040	. . . . . . . . . . . . . . . . . . . . . . . . . . . 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 14	. . . . . . . . . . . . . . . . . . . . . . . . . . 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 2808	. . . . . . . . . . . . . . . . . . . . . . . . . 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 4532	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) → if(𝑒 = 𝑐, 1 , 0 ) ∈ 𝐾)
342	341	adantr 486	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) ∧ 𝑒 ∈ (𝑁 × 𝑁)) → if(𝑒 = 𝑐, 1 , 0 ) ∈ 𝐾)
343	342, 211	ifcld 4532	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 7136	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 8884	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 599	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 260	. . . . . . . . . . . . . . . . . . . . . . . . . . . 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 2866	. . . . . . . . . . . . . . . . . . . . . . . . . . 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 1151	. . . . . . . . . . . . . . . . . . . . . . . . . . . 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 5974	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 2765	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 5974	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 2765	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 644	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 6914	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 347	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 3230	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 4594	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ (𝑦 = ((𝑎‘𝑐)(-_g‘𝑅)if(𝑐 ∈ 𝑑, 1 , 0 )) → {𝑦} = {((𝑎‘𝑐)(-_g‘𝑅)if(𝑐 ∈ 𝑑, 1 , 0 ))})
359	358	xpeq2d 5685	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 7455	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 2774	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 643	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 7447	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 2774	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 347	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 3230	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 3591	. . . . . . . . . . . . . . . . . . . . . . . . . . . 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 851	. . . . . . . . . . . . . . . . . . . . . . . . . . 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 5974	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 7456	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 2774	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 5974	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 2774	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 644	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 6905	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 7456	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 2774	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 347	. . . . . . . . . . . . . . . . . . . . . . . . . . . 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 5684	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 5980	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 7458	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 2779	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 5980	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 2779	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 644	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 344	. . . . . . . . . . . . . . . . . . . . . . . . . . . 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 3591	. . . . . . . . . . . . . . . . . . . . . . . . . . 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 851	. . . . . . . . . . . . . . . . . . . . . . . . . 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 712	. . . . . . . . . . . . . . . . . . . . . . . . 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 7455	. . . . . . . . . . . . . . . . . . . . . . . 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 1399	. . . . . . . . . . . . . . . . . . . . . . 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 1212	. . . . . . . . . . . . . . . . . . . . . . . 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 792	. . . . . . . . . . . . . . . . . . . . . . . 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 785	. . . . . . . . . . . . . . . . . . . . . . . 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 3986	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 4491	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 487	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 538	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 487	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 5673	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 ⊢ Rel (𝑁 × 𝑁)
402		simpl2 1211	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 ⊢ (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) ∧ 𝑑 ∈ (𝑁 ↑_m 𝑁)) → (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁))
403	402	sselda 3913	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 ⊢ ((((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) ∧ 𝑑 ∈ (𝑁 ↑_m 𝑁)) ∧ 𝑤 ∈ (𝑏 ∪ {𝑐})) → 𝑤 ∈ (𝑁 × 𝑁))
404	403	adantr 486	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 8071	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 ⊢ ((Rel (𝑁 × 𝑁) ∧ 𝑤 ∈ (𝑁 × 𝑁)) → 𝑤 = ⟨(1^st ‘𝑤), (2^nd ‘𝑤)⟩)
406	401, 404, 405	sylancr 599	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 2849	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 490	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 ⊢ (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) ∧ 𝑑 ∈ (𝑁 ↑_m 𝑁)) → 𝑑 ∈ (𝑁 ↑_m 𝑁))
409		elmapi 8894	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 ⊢ (𝑑 ∈ (𝑁 ↑_m 𝑁) → 𝑑:𝑁⟶𝑁)
410	409	adantl 487	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 ⊢ (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) ∧ 𝑑 ∈ (𝑁 ↑_m 𝑁)) → 𝑑:𝑁⟶𝑁)
411	124	adantr 486	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 ⊢ (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) ∧ 𝑑 ∈ (𝑁 ↑_m 𝑁)) → (1^st ‘𝑐) ∈ 𝑁)
412		xp2nd 8055	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 ⊢ (𝑐 ∈ (𝑁 × 𝑁) → (2^nd ‘𝑐) ∈ 𝑁)
413	122, 412	syl 18	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 ⊢ ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) → (2^nd ‘𝑐) ∈ 𝑁)
414	413	adantr 486	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 ⊢ (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) ∧ 𝑑 ∈ (𝑁 ↑_m 𝑁)) → (2^nd ‘𝑐) ∈ 𝑁)
415		fsets 17373	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 1404	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 6729	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 739	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 8054	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 ⊢ (𝑤 ∈ (𝑁 × 𝑁) → (1^st ‘𝑤) ∈ 𝑁)
420	403, 419	syl 18	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 ⊢ ((((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) ∧ 𝑑 ∈ (𝑁 ↑_m 𝑁)) ∧ 𝑤 ∈ (𝑏 ∪ {𝑐})) → (1^st ‘𝑤) ∈ 𝑁)
421	420	adantr 486	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 6956	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 596	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 6905	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 487	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 3458	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 ⊢ 𝑑 ∈ V
427		fvex 6918	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 ⊢ (1^st ‘𝑐) ∈ V
428		fvex 6918	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 ⊢ (2^nd ‘𝑐) ∈ V
429		fvsetsid 17372	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 1491	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 ⊢ ((𝑑 sSet ⟨(1^st ‘𝑐), (2^nd ‘𝑐)⟩)‘(1^st ‘𝑐)) = (2^nd ‘𝑐)
431	425, 430	eqtrdi 2814	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 2765	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 2770	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 ⊢ ((2^nd ‘𝑐) = (2^nd ‘𝑤) ↔ (2^nd ‘𝑤) = (2^nd ‘𝑐))
434	432, 433	bitrdi 290	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 311	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 743	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 8063	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 596	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 313	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 4509	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 2798	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 26	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 4601	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 ⊢ (𝑤 ∈ {𝑐} → 𝑤 = 𝑐)
444	443	fveq2d 6909	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ⊢ (𝑤 ∈ {𝑐} → (1^st ‘𝑤) = (1^st ‘𝑐))
445	444	con3i 155	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ⊢ (¬ (1^st ‘𝑤) = (1^st ‘𝑐) → ¬ 𝑤 ∈ {𝑐})
446	445	adantl 487	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ ((𝑤 ∈ (𝑏 ∪ {𝑐}) ∧ ¬ (1^st ‘𝑤) = (1^st ‘𝑐)) → ¬ 𝑤 ∈ {𝑐})
447		elun 4155	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ⊢ (𝑤 ∈ (𝑏 ∪ {𝑐}) ↔ (𝑤 ∈ 𝑏 ∨ 𝑤 ∈ {𝑐}))
448	447	birani 509	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ ((𝑤 ∈ (𝑏 ∪ {𝑐}) ∧ ¬ (1^st ‘𝑤) = (1^st ‘𝑐)) → (𝑤 ∈ 𝑏 ∨ 𝑤 ∈ {𝑐}))
449		orel2 904	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ (¬ 𝑤 ∈ {𝑐} → ((𝑤 ∈ 𝑏 ∨ 𝑤 ∈ {𝑐}) → 𝑤 ∈ 𝑏))
450	446, 448, 449	sylc 66	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ ((𝑤 ∈ (𝑏 ∪ {𝑐}) ∧ ¬ (1^st ‘𝑤) = (1^st ‘𝑐)) → 𝑤 ∈ 𝑏)
451	450	adantll 727	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ (((((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) ∧ 𝑑 ∈ (𝑁 ↑_m 𝑁)) ∧ 𝑤 ∈ (𝑏 ∪ {𝑐})) ∧ ¬ (1^st ‘𝑤) = (1^st ‘𝑐)) → 𝑤 ∈ 𝑏)
452		iffalse 4494	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 ))
453	452	adantl 487	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 ))
454		setsres 17382	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 ⊢ (𝑑 ∈ V → ((𝑑 sSet ⟨(1^st ‘𝑐), (2^nd ‘𝑐)⟩) ↾ (V ∖ {(1^st ‘𝑐)})) = (𝑑 ↾ (V ∖ {(1^st ‘𝑐)})))
455	454	eleq2d 2850	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)}))))
456	426, 455	mp1i 14	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)}))))
457		fvex 6918	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 ⊢ (1^st ‘𝑤) ∈ V
458	457	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 ⊢ (¬ (1^st ‘𝑤) = (1^st ‘𝑐) → (1^st ‘𝑤) ∈ V)
459		neqne 2967	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 ⊢ (¬ (1^st ‘𝑤) = (1^st ‘𝑐) → (1^st ‘𝑤) ≠ (1^st ‘𝑐))
460		eldifsn 4748	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 ⊢ ((1^st ‘𝑤) ∈ (V ∖ {(1^st ‘𝑐)}) ↔ ((1^st ‘𝑤) ∈ V ∧ (1^st ‘𝑤) ≠ (1^st ‘𝑐)))
461	458, 459, 460	sylanbrc 595	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 ⊢ (¬ (1^st ‘𝑤) = (1^st ‘𝑐) → (1^st ‘𝑤) ∈ (V ∖ {(1^st ‘𝑐)}))
462		fvex 6918	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 ⊢ (2^nd ‘𝑤) ∈ V
463	462	opres 5990	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)⟩)))
464	463	adantl 487	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)⟩)))
465		1st2nd2 8061	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 ⊢ (𝑤 ∈ (𝑁 × 𝑁) → 𝑤 = ⟨(1^st ‘𝑤), (2^nd ‘𝑤)⟩)
466	465	eleq1d 2849	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)⟩)))
467	466	adantr 486	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)⟩)))
468	464, 467	bitr4d 285	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)⟩)))
469	403, 461, 468	syl2an 608	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)⟩)))
470	462	opres 5990	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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))
471	470	adantl 487	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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))
472	465	eleq1d 2849	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 ⊢ (𝑤 ∈ (𝑁 × 𝑁) → (𝑤 ∈ 𝑑 ↔ ⟨(1^st ‘𝑤), (2^nd ‘𝑤)⟩ ∈ 𝑑))
473	472	adantr 486	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 ⊢ ((𝑤 ∈ (𝑁 × 𝑁) ∧ (1^st ‘𝑤) ∈ (V ∖ {(1^st ‘𝑐)})) → (𝑤 ∈ 𝑑 ↔ ⟨(1^st ‘𝑤), (2^nd ‘𝑤)⟩ ∈ 𝑑))
474	471, 473	bitr4d 285	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 ⊢ ((𝑤 ∈ (𝑁 × 𝑁) ∧ (1^st ‘𝑤) ∈ (V ∖ {(1^st ‘𝑐)})) → (⟨(1^st ‘𝑤), (2^nd ‘𝑤)⟩ ∈ (𝑑 ↾ (V ∖ {(1^st ‘𝑐)})) ↔ 𝑤 ∈ 𝑑))
475	403, 461, 474	syl2an 608	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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))
476	456, 469, 475	3bitr3rd 313	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)⟩)))
477	476	ifbid 4509	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 ))
478	453, 477	eqtrd 2798	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 ))
479		ifeq2 4490	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 )))
480	479	eqeq1d 2765	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 )))
481	478, 480	syl5ibrcom 250	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 )))
482	451, 481	embantd 60	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 )))
483	442, 482	pm2.61dan 825	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 )))
484		fveqeq2 6914	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ (𝑒 = 𝑤 → ((1^st ‘𝑒) = (1^st ‘𝑐) ↔ (1^st ‘𝑤) = (1^st ‘𝑐)))
485		equequ1 2060	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ⊢ (𝑒 = 𝑤 → (𝑒 = 𝑐 ↔ 𝑤 = 𝑐))
486	485	ifbid 4509	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ (𝑒 = 𝑤 → if(𝑒 = 𝑐, 1 , 0 ) = if(𝑤 = 𝑐, 1 , 0 ))
487		fveq2 6905	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ (𝑒 = 𝑤 → (𝑎‘𝑒) = (𝑎‘𝑤))
488	484, 486, 487	ifbieq12d 4514	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)))
489		eqid 2763	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)))
490	165, 162	ifex 4536	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ if(𝑤 = 𝑐, 1 , 0 ) ∈ V
491		fvex 6918	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ (𝑎‘𝑤) ∈ V
492	490, 491	ifex 4536	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ if((1^st ‘𝑤) = (1^st ‘𝑐), if(𝑤 = 𝑐, 1 , 0 ), (𝑎‘𝑤)) ∈ V
493	488, 489, 492	fvmpt 7013	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)))
494	493	eqeq1d 2765	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 )))
495	403, 494	syl 18	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 )))
496	483, 495	sylibrd 262	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 )))
497	496	ralimdva 3178	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 )))
498	396, 497	biimtrid 245	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 )))
499	498	impr 460	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 ))
500	499	3adantr1 1188	. . . . . . . . . . . . . . . . . . . . . . . . . . . 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 ))
501	348	adantr 486	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)
502		simpr2 1214	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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))
503	502, 409	syl 18	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)
504	124	adantr 486	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)
505	413	adantr 486	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)
506	502, 503, 504, 505, 415	syl211anc 1404	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)
507	158, 158	elmapd 8885	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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))
508	507	adantr 486	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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))
509	506, 508	mpbird 260	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 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))
510		simpr1 1213	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)
511		raleq 3320	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (𝑥 = (𝑏 ∪ {𝑐}) → (∀𝑤 ∈ 𝑥 (𝑦‘𝑤) = if(𝑤 ∈ 𝑧, 1 , 0 ) ↔ ∀𝑤 ∈ (𝑏 ∪ {𝑐})(𝑦‘𝑤) = if(𝑤 ∈ 𝑧, 1 , 0 )))
512	511	imbi1d 344	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 )))
513	512	2ralbidv 3230	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 )))
514	513, 72	elab2g 3637	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 )))
515	514	ibi 270	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((𝑏 ∪ {𝑐}) ∈ 𝑌 → ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ (𝑁 ↑_m 𝑁)(∀𝑤 ∈ (𝑏 ∪ {𝑐})(𝑦‘𝑤) = if(𝑤 ∈ 𝑧, 1 , 0 ) → (𝐷‘𝑦) = 0 ))
516	510, 515	syl 18	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 ))
517		fveq1 6904	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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))
518	517	eqeq1d 2765	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 )))
519	518	ralbidv 3189	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 )))
520		fveqeq2 6914	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 ))
521	519, 520	imbi12d 347	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 )))
522		eleq2 2853	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (𝑧 = (𝑑 sSet ⟨(1^st ‘𝑐), (2^nd ‘𝑐)⟩) → (𝑤 ∈ 𝑧 ↔ 𝑤 ∈ (𝑑 sSet ⟨(1^st ‘𝑐), (2^nd ‘𝑐)⟩)))
523	522	ifbid 4509	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 ))
524	523	eqeq2d 2774	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 )))
525	524	ralbidv 3189	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 )))
526	525	imbi1d 344	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 )))
527	521, 526	rspc2va 3591	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 ))
528	501, 509, 516, 527	syl21anc 851	. . . . . . . . . . . . . . . . . . . . . . . . . . . 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 ))
529	500, 528	mpd 16	. . . . . . . . . . . . . . . . . . . . . . . . . . 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 )
530	529	oveq2d 7456	. . . . . . . . . . . . . . . . . . . . . . . . . 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 ))
531	118	unssad 4194	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) → 𝑏 ⊆ (𝑁 × 𝑁))
532	531	adantr 486	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 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))
533		simpr3 1215	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 ))
534		ssel2 3908	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ ((𝑏 ⊆ (𝑁 × 𝑁) ∧ 𝑤 ∈ 𝑏) → 𝑤 ∈ (𝑁 × 𝑁))
535	534	adantr 486	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (((𝑏 ⊆ (𝑁 × 𝑁) ∧ 𝑤 ∈ 𝑏) ∧ (𝑎‘𝑤) = if(𝑤 ∈ 𝑑, 1 , 0 )) → 𝑤 ∈ (𝑁 × 𝑁))
536		elequ1 2154	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ (𝑒 = 𝑤 → (𝑒 ∈ 𝑑 ↔ 𝑤 ∈ 𝑑))
537	536	ifbid 4509	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ (𝑒 = 𝑤 → if(𝑒 ∈ 𝑑, 1 , 0 ) = if(𝑤 ∈ 𝑑, 1 , 0 ))
538	485, 537, 487	ifbieq12d 4514	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (𝑒 = 𝑤 → if(𝑒 = 𝑐, if(𝑒 ∈ 𝑑, 1 , 0 ), (𝑎‘𝑒)) = if(𝑤 = 𝑐, if(𝑤 ∈ 𝑑, 1 , 0 ), (𝑎‘𝑤)))
539		eqid 2763	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)))
540	165, 162	ifex 4536	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ if(𝑤 ∈ 𝑑, 1 , 0 ) ∈ V
541	540, 491	ifex 4536	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ if(𝑤 = 𝑐, if(𝑤 ∈ 𝑑, 1 , 0 ), (𝑎‘𝑤)) ∈ V
542	538, 539, 541	fvmpt 7013	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)))
543	535, 542	syl 18	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)))
544		ifeq2 4490	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 )))
545	544	adantl 487	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 )))
546		ifid 4526	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ if(𝑤 = 𝑐, if(𝑤 ∈ 𝑑, 1 , 0 ), if(𝑤 ∈ 𝑑, 1 , 0 )) = if(𝑤 ∈ 𝑑, 1 , 0 )
547	545, 546	eqtrdi 2814	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 ))
548	543, 547	eqtrd 2798	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 ))
549	548	ex 418	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 )))
550	549	ralimdva 3178	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 )))
551	532, 533, 550	sylc 66	. . . . . . . . . . . . . . . . . . . . . . . . . . . 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 ))
552	142, 291	eqtrd 2798	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝑒 = 𝑐 → if(𝑒 = 𝑐, if(𝑒 ∈ 𝑑, 1 , 0 ), (𝑎‘𝑒)) = if(𝑐 ∈ 𝑑, 1 , 0 ))
553	165, 162	ifex 4536	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ if(𝑐 ∈ 𝑑, 1 , 0 ) ∈ V
554	552, 539, 553	fvmpt 7013	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑐 ∈ (𝑁 × 𝑁) → ((𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒 ∈ 𝑑, 1 , 0 ), (𝑎‘𝑒)))‘𝑐) = if(𝑐 ∈ 𝑑, 1 , 0 ))
555	122, 554	syl 18	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 ))
556	555	adantr 486	. . . . . . . . . . . . . . . . . . . . . . . . . . . 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 ))
557		fveq2 6905	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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))
558		elequ1 2154	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝑤 = 𝑐 → (𝑤 ∈ 𝑑 ↔ 𝑐 ∈ 𝑑))
559	558	ifbid 4509	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝑤 = 𝑐 → if(𝑤 ∈ 𝑑, 1 , 0 ) = if(𝑐 ∈ 𝑑, 1 , 0 ))
560	557, 559	eqeq12d 2779	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 )))
561	560	ralunsn 4828	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 ))))
562	561	elv 3459	. . . . . . . . . . . . . . . . . . . . . . . . . . . 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 )))
563	551, 556, 562	sylanbrc 595	. . . . . . . . . . . . . . . . . . . . . . . . . . 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 ))
564	223	adantr 486	. . . . . . . . . . . . . . . . . . . . . . . . . . . 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)
565		fveq1 6904	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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))
566	565	eqeq1d 2765	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 )))
567	566	ralbidv 3189	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 )))
568		fveqeq2 6914	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 ))
569	567, 568	imbi12d 347	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 )))
570		elequ2 2162	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (𝑧 = 𝑑 → (𝑤 ∈ 𝑧 ↔ 𝑤 ∈ 𝑑))
571	570	ifbid 4509	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝑧 = 𝑑 → if(𝑤 ∈ 𝑧, 1 , 0 ) = if(𝑤 ∈ 𝑑, 1 , 0 ))
572	571	eqeq2d 2774	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 )))
573	572	ralbidv 3189	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 )))
574	573	imbi1d 344	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 )))
575	569, 574	rspc2va 3591	. . . . . . . . . . . . . . . . . . . . . . . . . . . 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 ))
576	564, 502, 516, 575	syl21anc 851	. . . . . . . . . . . . . . . . . . . . . . . . . . 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 ))
577	563, 576	mpd 16	. . . . . . . . . . . . . . . . . . . . . . . . . 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 )
578	530, 577	oveq12d 7458	. . . . . . . . . . . . . . . . . . . . . . . . 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 ))
579	308	oveq1d 7455	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) → ((((𝑎‘𝑐)(-_g‘𝑅)if(𝑐 ∈ 𝑑, 1 , 0 )) · 0 ) + 0 ) = ( 0 + 0 ))
580	12, 50, 48	grplid 19204	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝑅 ∈ Grp ∧ 0 ∈ 𝐾) → ( 0 + 0 ) = 0 )
581	115, 133, 580	syl2anc 596	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) → ( 0 + 0 ) = 0 )
582	579, 581	eqtrd 2798	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) → ((((𝑎‘𝑐)(-_g‘𝑅)if(𝑐 ∈ 𝑑, 1 , 0 )) · 0 ) + 0 ) = 0 )
583	582	adantr 486	. . . . . . . . . . . . . . . . . . . . . . . . 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 )
584	578, 583	eqtrd 2798	. . . . . . . . . . . . . . . . . . . . . . . 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 )
585	104, 105, 106, 392, 393, 394, 584	syl33anc 1413	. . . . . . . . . . . . . . . . . . . . . . 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 )
586	288, 391, 585	3eqtrd 2802	. . . . . . . . . . . . . . . . . . . . . 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 )
587	586	expr 462	. . . . . . . . . . . . . . . . . . . . 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 ))
588	587	ralrimivva 3209	. . . . . . . . . . . . . . . . . . . 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 ))
589		fveq1 6904	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑎 = 𝑦 → (𝑎‘𝑤) = (𝑦‘𝑤))
590	589	eqeq1d 2765	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑎 = 𝑦 → ((𝑎‘𝑤) = if(𝑤 ∈ 𝑑, 1 , 0 ) ↔ (𝑦‘𝑤) = if(𝑤 ∈ 𝑑, 1 , 0 )))
591	590	ralbidv 3189	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑎 = 𝑦 → (∀𝑤 ∈ 𝑏 (𝑎‘𝑤) = if(𝑤 ∈ 𝑑, 1 , 0 ) ↔ ∀𝑤 ∈ 𝑏 (𝑦‘𝑤) = if(𝑤 ∈ 𝑑, 1 , 0 )))
592		fveqeq2 6914	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑎 = 𝑦 → ((𝐷‘𝑎) = 0 ↔ (𝐷‘𝑦) = 0 ))
593	591, 592	imbi12d 347	. . . . . . . . . . . . . . . . . . . . 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 )))
594		elequ2 2162	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑑 = 𝑧 → (𝑤 ∈ 𝑑 ↔ 𝑤 ∈ 𝑧))
595	594	ifbid 4509	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑑 = 𝑧 → if(𝑤 ∈ 𝑑, 1 , 0 ) = if(𝑤 ∈ 𝑧, 1 , 0 ))
596	595	eqeq2d 2774	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑑 = 𝑧 → ((𝑦‘𝑤) = if(𝑤 ∈ 𝑑, 1 , 0 ) ↔ (𝑦‘𝑤) = if(𝑤 ∈ 𝑧, 1 , 0 )))
597	596	ralbidv 3189	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑑 = 𝑧 → (∀𝑤 ∈ 𝑏 (𝑦‘𝑤) = if(𝑤 ∈ 𝑑, 1 , 0 ) ↔ ∀𝑤 ∈ 𝑏 (𝑦‘𝑤) = if(𝑤 ∈ 𝑧, 1 , 0 )))
598	597	imbi1d 344	. . . . . . . . . . . . . . . . . . . . 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 )))
599	593, 598	cbvral2vw 3248	. . . . . . . . . . . . . . . . . . . 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 ))
600	588, 599	sylib 221	. . . . . . . . . . . . . . . . . . 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 ))
601		vex 3458	. . . . . . . . . . . . . . . . . . . 20 ⊢ 𝑏 ∈ V
602		raleq 3320	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑥 = 𝑏 → (∀𝑤 ∈ 𝑥 (𝑦‘𝑤) = if(𝑤 ∈ 𝑧, 1 , 0 ) ↔ ∀𝑤 ∈ 𝑏 (𝑦‘𝑤) = if(𝑤 ∈ 𝑧, 1 , 0 )))
603	602	imbi1d 344	. . . . . . . . . . . . . . . . . . . . 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 )))
604	603	2ralbidv 3230	. . . . . . . . . . . . . . . . . . . 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 )))
605	601, 604, 72	elab2 3639	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑏 ∈ 𝑌 ↔ ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ (𝑁 ↑_m 𝑁)(∀𝑤 ∈ 𝑏 (𝑦‘𝑤) = if(𝑤 ∈ 𝑧, 1 , 0 ) → (𝐷‘𝑦) = 0 ))
606	600, 605	sylibr 237	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ (𝑏 ∪ {𝑐}) ∈ 𝑌) → 𝑏 ∈ 𝑌)
607	606	3expia 1139	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁)) → ((𝑏 ∪ {𝑐}) ∈ 𝑌 → 𝑏 ∈ 𝑌))
608	607	con3d 153	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁)) → (¬ 𝑏 ∈ 𝑌 → ¬ (𝑏 ∪ {𝑐}) ∈ 𝑌))
609	608	3adant3 1150	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ ¬ ∅ ∈ 𝑌) → (¬ 𝑏 ∈ 𝑌 → ¬ (𝑏 ∪ {𝑐}) ∈ 𝑌))
610	609	a1i 11	. . . . . . . . . . . . . 14 ⊢ ((𝑏 ∈ Fin ∧ ¬ 𝑐 ∈ 𝑏) → ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ ¬ ∅ ∈ 𝑌) → (¬ 𝑏 ∈ 𝑌 → ¬ (𝑏 ∪ {𝑐}) ∈ 𝑌)))
611	610	a2d 30	. . . . . . . . . . . . 13 ⊢ ((𝑏 ∈ Fin ∧ ¬ 𝑐 ∈ 𝑏) → (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ ¬ ∅ ∈ 𝑌) → ¬ 𝑏 ∈ 𝑌) → ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ ¬ ∅ ∈ 𝑌) → ¬ (𝑏 ∪ {𝑐}) ∈ 𝑌)))
612	103, 611	syl5 35	. . . . . . . . . . . 12 ⊢ ((𝑏 ∈ Fin ∧ ¬ 𝑐 ∈ 𝑏) → (((𝜑 ∧ 𝑏 ⊆ (𝑁 × 𝑁) ∧ ¬ ∅ ∈ 𝑌) → ¬ 𝑏 ∈ 𝑌) → ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ ¬ ∅ ∈ 𝑌) → ¬ (𝑏 ∪ {𝑐}) ∈ 𝑌)))
613	82, 87, 92, 97, 98, 612	findcard2s 9206	. . . . . . . . . . 11 ⊢ ((𝑁 × 𝑁) ∈ Fin → ((𝜑 ∧ (𝑁 × 𝑁) ⊆ (𝑁 × 𝑁) ∧ ¬ ∅ ∈ 𝑌) → ¬ (𝑁 × 𝑁) ∈ 𝑌))
614	77, 613	mpcom 39	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑁 × 𝑁) ⊆ (𝑁 × 𝑁) ∧ ¬ ∅ ∈ 𝑌) → ¬ (𝑁 × 𝑁) ∈ 𝑌)
615	614	3exp 1137	. . . . . . . . 9 ⊢ (𝜑 → ((𝑁 × 𝑁) ⊆ (𝑁 × 𝑁) → (¬ ∅ ∈ 𝑌 → ¬ (𝑁 × 𝑁) ∈ 𝑌)))
616	76, 615	mpi 21	. . . . . . . 8 ⊢ (𝜑 → (¬ ∅ ∈ 𝑌 → ¬ (𝑁 × 𝑁) ∈ 𝑌))
617	75, 616	mt4d 118	. . . . . . 7 ⊢ (𝜑 → ∅ ∈ 𝑌)
618	617	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐵) → ∅ ∈ 𝑌)
619		0ex 5264	. . . . . . 7 ⊢ ∅ ∈ V
620		raleq 3320	. . . . . . . . 9 ⊢ (𝑥 = ∅ → (∀𝑤 ∈ 𝑥 (𝑦‘𝑤) = if(𝑤 ∈ 𝑧, 1 , 0 ) ↔ ∀𝑤 ∈ ∅ (𝑦‘𝑤) = if(𝑤 ∈ 𝑧, 1 , 0 )))
621	620	imbi1d 344	. . . . . . . 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 )))
622	621	2ralbidv 3230	. . . . . . 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 )))
623	619, 622, 72	elab2 3639	. . . . . 6 ⊢ (∅ ∈ 𝑌 ↔ ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ (𝑁 ↑_m 𝑁)(∀𝑤 ∈ ∅ (𝑦‘𝑤) = if(𝑤 ∈ 𝑧, 1 , 0 ) → (𝐷‘𝑦) = 0 ))
624	618, 623	sylib 221	. . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐵) → ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ (𝑁 ↑_m 𝑁)(∀𝑤 ∈ ∅ (𝑦‘𝑤) = if(𝑤 ∈ 𝑧, 1 , 0 ) → (𝐷‘𝑦) = 0 ))
625		fveq1 6904	. . . . . . . . 9 ⊢ (𝑦 = 𝑎 → (𝑦‘𝑤) = (𝑎‘𝑤))
626	625	eqeq1d 2765	. . . . . . . 8 ⊢ (𝑦 = 𝑎 → ((𝑦‘𝑤) = if(𝑤 ∈ 𝑧, 1 , 0 ) ↔ (𝑎‘𝑤) = if(𝑤 ∈ 𝑧, 1 , 0 )))
627	626	ralbidv 3189	. . . . . . 7 ⊢ (𝑦 = 𝑎 → (∀𝑤 ∈ ∅ (𝑦‘𝑤) = if(𝑤 ∈ 𝑧, 1 , 0 ) ↔ ∀𝑤 ∈ ∅ (𝑎‘𝑤) = if(𝑤 ∈ 𝑧, 1 , 0 )))
628		fveqeq2 6914	. . . . . . 7 ⊢ (𝑦 = 𝑎 → ((𝐷‘𝑦) = 0 ↔ (𝐷‘𝑎) = 0 ))
629	627, 628	imbi12d 347	. . . . . 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 )))
630		eleq2 2853	. . . . . . . . . 10 ⊢ (𝑧 = ( I ↾ 𝑁) → (𝑤 ∈ 𝑧 ↔ 𝑤 ∈ ( I ↾ 𝑁)))
631	630	ifbid 4509	. . . . . . . . 9 ⊢ (𝑧 = ( I ↾ 𝑁) → if(𝑤 ∈ 𝑧, 1 , 0 ) = if(𝑤 ∈ ( I ↾ 𝑁), 1 , 0 ))
632	631	eqeq2d 2774	. . . . . . . 8 ⊢ (𝑧 = ( I ↾ 𝑁) → ((𝑎‘𝑤) = if(𝑤 ∈ 𝑧, 1 , 0 ) ↔ (𝑎‘𝑤) = if(𝑤 ∈ ( I ↾ 𝑁), 1 , 0 )))
633	632	ralbidv 3189	. . . . . . 7 ⊢ (𝑧 = ( I ↾ 𝑁) → (∀𝑤 ∈ ∅ (𝑎‘𝑤) = if(𝑤 ∈ 𝑧, 1 , 0 ) ↔ ∀𝑤 ∈ ∅ (𝑎‘𝑤) = if(𝑤 ∈ ( I ↾ 𝑁), 1 , 0 )))
634	633	imbi1d 344	. . . . . 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 )))
635	629, 634	rspc2va 3591	. . . . 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 ))
636	2, 9, 624, 635	syl21anc 851	. . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐵) → (∀𝑤 ∈ ∅ (𝑎‘𝑤) = if(𝑤 ∈ ( I ↾ 𝑁), 1 , 0 ) → (𝐷‘𝑎) = 0 ))
637	1, 636	mpi 21	. . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐵) → (𝐷‘𝑎) = 0 )
638	637	mpteq2dva 5201	. 2 ⊢ (𝜑 → (𝑎 ∈ 𝐵 ↦ (𝐷‘𝑎)) = (𝑎 ∈ 𝐵 ↦ 0 ))
639	53	feqmptd 6973	. 2 ⊢ (𝜑 → 𝐷 = (𝑎 ∈ 𝐵 ↦ (𝐷‘𝑎)))
640		fconstmpt 5717	. . 3 ⊢ (𝐵 × { 0 }) = (𝑎 ∈ 𝐵 ↦ 0 )
641	640	a1i 11	. 2 ⊢ (𝜑 → (𝐵 × { 0 }) = (𝑎 ∈ 𝐵 ↦ 0 ))
642	638, 639, 641	3eqtr4d 2808	1 ⊢ (𝜑 → 𝐷 = (𝐵 × { 0 }))

Colors of variables: wff setvar class

This proof depends on syntax axioms: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 401 ∨ wo 861 ∧ w3a 1103 = wceq 1570 ∈ wcel 2147 {cab 2741 ≠ wne 2959 ∀wral 3080 Vcvv 3454 ⊆ wss 3900 ∖ cdif 4048 ∪ cun 4152 ∅c0 4281 ifcif 4485 {csn 4588 ⟨cop 4838 ↦ cmpt 5189 I cid 5549 × cxp 5653 ↾ cres 5657 Rel wrel 5660 Fn wfn 6550 ⟶wf 6551 ⟶1-1ontowf1o 6554 ‘cfv 6555 (class class class)co 7440 ∈ cmpo 7442 ∘_f cof 7712 1^st c1st 8020 2^nd c2nd 8021 ↑_m cmap 8872 Fincfn 8998 sSet csts 17367 Basecbs 17413 +_gcplusg 17454 ._rcmulr 17455 0_gc0g 17636 Grpcgrp 19170 -_gcsg 19172 1_rcur 20434 Ringcrg 20486 Mat cmat 22749

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1945 ax-6 2002 ax-7 2043 ax-8 2149 ax-9 2157 ax-10 2180 ax-11 2197 ax-12 2216 ax-ext 2735 ax-rep 5235 ax-sep 5252 ax-nul 5263 ax-pow 5331 ax-pr 5395 ax-un 7772 ax-cnex 11280 ax-resscn 11281 ax-1cn 11282 ax-icn 11283 ax-addcl 11284 ax-addrcl 11285 ax-mulcl 11286 ax-mulrcl 11287 ax-mulcom 11288 ax-addass 11289 ax-mulass 11290 ax-distr 11291 ax-i2m1 11292 ax-1ne0 11293 ax-1rid 11294 ax-rnegex 11295 ax-rrecex 11296 ax-cnre 11297 ax-pre-lttri 11298 ax-pre-lttrn 11299 ax-pre-ltadd 11300 ax-pre-mulgt0 11301 ax-addf 11303 ax-mulf 11304

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-xor 1542 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2102 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2839 df-nfc 2913 df-ne 2960 df-nel 3066 df-ral 3081 df-rex 3091 df-rmo 3369 df-reu 3370 df-rab 3417 df-v 3456 df-sbc 3743 df-csb 3852 df-ss 3901 df-pss 4017 df-dif 4050 df-in 4094 df-un 4154 df-nul 4282 df-if 4486 df-pw 4562 df-sn 4589 df-pr 4591 df-tp 4593 df-op 4839 df-ot 4862 df-uni 4871 df-int 4911 df-iun 4956 df-iin 4957 df-br 5107 df-opab 5171 df-mpt 5190 df-tr 5216 df-id 5550 df-eprel 5555 df-po 5563 df-so 5564 df-fr 5608 df-se 5609 df-we 5610 df-xp 5661 df-rel 5662 df-cnv 5663 df-co 5664 df-dm 5665 df-rn 5666 df-res 5667 df-ima 5668 df-pred 6319 df-ord 6380 df-on 6381 df-lim 6382 df-suc 6383 df-iota 6509 df-fun 6557 df-fn 6558 df-f 6559 df-f1 6560 df-fo 6561 df-f1o 6562 df-fv 6563 df-isom 6564 df-riota 7397 df-ov 7443 df-oprab 7444 df-mpo 7445 df-of 7714 df-om 7899 df-1st 8022 df-2nd 8023 df-supp 8202 df-tpos 8268 df-frecs 8324 df-wrecs 8355 df-recs 8404 df-rdg 8443 df-1o 8501 df-2o 8502 df-er 8742 df-map 8874 df-ixp 8951 df-en 8999 df-dom 9000 df-sdom 9001 df-fin 9002 df-fsupp 9379 df-sup 9458 df-oi 9528 df-card 10044 df-pnf 11369 df-mnf 11370 df-xr 11371 df-ltxr 11372 df-le 11373 df-sub 11567 df-neg 11568 df-div 11998 df-nn 12360 df-2 12429 df-3 12430 df-4 12431 df-5 12432 df-6 12433 df-7 12434 df-8 12435 df-9 12436 df-n0 12631 df-xnn0 12704 df-z 12718 df-dec 12839 df-uz 12990 df-rp 13145 df-fz 13664 df-fzo 13813 df-seq 14169 df-exp 14229 df-hash 14499 df-word 14683 df-lsw 14732 df-concat 14740 df-s1 14767 df-substr 14814 df-pfx 14846 df-splice 14924 df-reverse 14933 df-s2 15024 df-struct 17351 df-sets 17368 df-slot 17386 df-ndx 17398 df-base 17414 df-ress 17435 df-plusg 17467 df-mulr 17468 df-starv 17469 df-sca 17470 df-vsca 17471 df-ip 17472 df-tset 17473 df-ple 17474 df-ds 17476 df-unif 17477 df-hom 17478 df-cco 17479 df-0g 17638 df-gsum 17639 df-prds 17644 df-pws 17646 df-mre 17782 df-mrc 17783 df-acs 17785 df-mgm 18842 df-sgrp 18934 df-mnd 18950 df-mhm 19004 df-submnd 19005 df-efmnd 19091 df-grp 19173 df-minusg 19174 df-sbg 19175 df-mulg 19304 df-subg 19359 df-ghm 19454 df-gim 19499 df-cntz 19558 df-oppg 19587 df-symg 19611 df-pmtr 19683 df-psgn 19732 df-evpm 19733 df-cmn 20023 df-abl 20024 df-mgp 20388 df-rng 20402 df-ur 20435 df-ring 20488 df-cring 20489 df-oppr 20594 df-dvdsr 20614 df-unit 20615 df-invr 20645 df-dvr 20658 df-rhm 20729 df-subrng 20825 df-subrg 20849 df-drng 21009 df-lmod 21164 df-lss 21234 df-sra 21475 df-rgmod 21476 df-cnfld 21706 df-zring 21780 df-zrh 21836 df-dsmm 22065 df-frlm 22080 df-mamu 22733 df-mat 22750

This theorem is used by: mdetuni0 22963

W3C validator