Theorem mdetunilem9 20245

Description: Lemma for mdetuni 20247. (Contributed by SO, 15-Jul-2018.)

Hypotheses

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

Assertion

Ref	Expression
mdetunilem9	⊢ (𝜑 → 𝐷 = (𝐵 × { 0 }))

Distinct variable groups:   𝜑,𝑥,𝑦,𝑧,𝑤   𝑥,𝐵,𝑦,𝑧,𝑤   𝑥,𝐾,𝑦,𝑧,𝑤   𝑥,𝑁,𝑦,𝑧,𝑤   𝑥,𝐷,𝑦,𝑧,𝑤   𝑥, · ,𝑦,𝑧,𝑤   𝑥, + ,𝑦,𝑧,𝑤   𝑥, 0 ,𝑦,𝑧,𝑤   𝑥, 1 ,𝑦,𝑧,𝑤   𝑥,𝑅,𝑦,𝑧,𝑤   𝑥,𝐴,𝑦,𝑧,𝑤

Allowed substitution hints:   𝑌(𝑥,𝑦,𝑧,𝑤)

Proof of Theorem mdetunilem9

Dummy variables 𝑎 𝑏 𝑐 𝑑 𝑒 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ral0 4028	. . . 4 ⊢ ∀𝑤 ∈ ∅ (𝑎‘𝑤) = if(𝑤 ∈ ( I ↾ 𝑁), 1 , 0 )
2		simpr 476	. . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐵) → 𝑎 ∈ 𝐵)
3		f1oi 6086	. . . . . . . 8 ⊢ ( I ↾ 𝑁):𝑁–1-1-onto→𝑁
4		f1of 6050	. . . . . . . 8 ⊢ (( I ↾ 𝑁):𝑁–1-1-onto→𝑁 → ( I ↾ 𝑁):𝑁⟶𝑁)
5	3, 4	mp1i 13	. . . . . . 7 ⊢ (𝜑 → ( I ↾ 𝑁):𝑁⟶𝑁)
6		mdetuni.n	. . . . . . . 8 ⊢ (𝜑 → 𝑁 ∈ Fin)
7	6, 6	elmapd 7758	. . . . . . 7 ⊢ (𝜑 → (( I ↾ 𝑁) ∈ (𝑁 ↑𝑚 𝑁) ↔ ( I ↾ 𝑁):𝑁⟶𝑁))
8	5, 7	mpbird 246	. . . . . 6 ⊢ (𝜑 → ( I ↾ 𝑁) ∈ (𝑁 ↑𝑚 𝑁))
9	8	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐵) → ( I ↾ 𝑁) ∈ (𝑁 ↑𝑚 𝑁))
10		simplrl 796	. . . . . . . . . . . . 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 20047	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 ∈ 𝐵 → 𝑦 ∈ (𝐾 ↑𝑚 (𝑁 × 𝑁)))
15		elmapi 7765	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 ∈ (𝐾 ↑𝑚 (𝑁 × 𝑁)) → 𝑦:(𝑁 × 𝑁)⟶𝐾)
16	14, 15	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝑦 ∈ 𝐵 → 𝑦:(𝑁 × 𝑁)⟶𝐾)
17	16	feqmptd 6159	. . . . . . . . . . . . . 14 ⊢ (𝑦 ∈ 𝐵 → 𝑦 = (𝑤 ∈ (𝑁 × 𝑁) ↦ (𝑦‘𝑤)))
18	17	fveq2d 6107	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ 𝐵 → (𝐷‘𝑦) = (𝐷‘(𝑤 ∈ (𝑁 × 𝑁) ↦ (𝑦‘𝑤))))
19	10, 18	syl 17	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ (𝑁 ↑𝑚 𝑁))) ∧ ∀𝑤 ∈ (𝑁 × 𝑁)(𝑦‘𝑤) = if(𝑤 ∈ 𝑧, 1 , 0 )) → (𝐷‘𝑦) = (𝐷‘(𝑤 ∈ (𝑁 × 𝑁) ↦ (𝑦‘𝑤))))
20		eqid 2610	. . . . . . . . . . . . . 14 ⊢ (𝑁 × 𝑁) = (𝑁 × 𝑁)
21		mpteq12 4664	. . . . . . . . . . . . . . 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 6107	. . . . . . . . . . . . . 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 702	. . . . . . . . . . . . 13 ⊢ (∀𝑤 ∈ (𝑁 × 𝑁)(𝑦‘𝑤) = if(𝑤 ∈ 𝑧, 1 , 0 ) → (𝐷‘(𝑤 ∈ (𝑁 × 𝑁) ↦ (𝑦‘𝑤))) = (𝐷‘(𝑤 ∈ (𝑁 × 𝑁) ↦ if(𝑤 ∈ 𝑧, 1 , 0 ))))
24	23	adantl 481	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ (𝑁 ↑𝑚 𝑁))) ∧ ∀𝑤 ∈ (𝑁 × 𝑁)(𝑦‘𝑤) = if(𝑤 ∈ 𝑧, 1 , 0 )) → (𝐷‘(𝑤 ∈ (𝑁 × 𝑁) ↦ (𝑦‘𝑤))) = (𝐷‘(𝑤 ∈ (𝑁 × 𝑁) ↦ if(𝑤 ∈ 𝑧, 1 , 0 ))))
25		eleq1 2676	. . . . . . . . . . . . . . . . 17 ⊢ (𝑎 = 𝑧 → (𝑎 ∈ (𝑁 ↑𝑚 𝑁) ↔ 𝑧 ∈ (𝑁 ↑𝑚 𝑁)))
26	25	anbi2d 736	. . . . . . . . . . . . . . . 16 ⊢ (𝑎 = 𝑧 → ((𝜑 ∧ 𝑎 ∈ (𝑁 ↑𝑚 𝑁)) ↔ (𝜑 ∧ 𝑧 ∈ (𝑁 ↑𝑚 𝑁))))
27		elequ2 1991	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑎 = 𝑧 → (𝑤 ∈ 𝑎 ↔ 𝑤 ∈ 𝑧))
28	27	ifbid 4058	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑎 = 𝑧 → if(𝑤 ∈ 𝑎, 1 , 0 ) = if(𝑤 ∈ 𝑧, 1 , 0 ))
29	28	mpteq2dv 4673	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑎 = 𝑧 → (𝑤 ∈ (𝑁 × 𝑁) ↦ if(𝑤 ∈ 𝑎, 1 , 0 )) = (𝑤 ∈ (𝑁 × 𝑁) ↦ if(𝑤 ∈ 𝑧, 1 , 0 )))
30	29	fveq2d 6107	. . . . . . . . . . . . . . . . 17 ⊢ (𝑎 = 𝑧 → (𝐷‘(𝑤 ∈ (𝑁 × 𝑁) ↦ if(𝑤 ∈ 𝑎, 1 , 0 ))) = (𝐷‘(𝑤 ∈ (𝑁 × 𝑁) ↦ if(𝑤 ∈ 𝑧, 1 , 0 ))))
31	30	eqeq1d 2612	. . . . . . . . . . . . . . . 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 333	. . . . . . . . . . . . . . 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 2676	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑤 = ⟨𝑏, 𝑐⟩ → (𝑤 ∈ 𝑎 ↔ ⟨𝑏, 𝑐⟩ ∈ 𝑎))
34	33	ifbid 4058	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑤 = ⟨𝑏, 𝑐⟩ → if(𝑤 ∈ 𝑎, 1 , 0 ) = if(⟨𝑏, 𝑐⟩ ∈ 𝑎, 1 , 0 ))
35	34	mpt2mpt 6650	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑤 ∈ (𝑁 × 𝑁) ↦ if(𝑤 ∈ 𝑎, 1 , 0 )) = (𝑏 ∈ 𝑁, 𝑐 ∈ 𝑁 ↦ if(⟨𝑏, 𝑐⟩ ∈ 𝑎, 1 , 0 ))
36		elmapi 7765	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑎 ∈ (𝑁 ↑𝑚 𝑁) → 𝑎:𝑁⟶𝑁)
37	36	adantl 481	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ 𝑎 ∈ (𝑁 ↑𝑚 𝑁)) → 𝑎:𝑁⟶𝑁)
38		ffn 5958	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑎:𝑁⟶𝑁 → 𝑎 Fn 𝑁)
39	37, 38	syl 17	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ 𝑎 ∈ (𝑁 ↑𝑚 𝑁)) → 𝑎 Fn 𝑁)
40	39	3ad2ant1 1075	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ 𝑎 ∈ (𝑁 ↑𝑚 𝑁)) ∧ 𝑏 ∈ 𝑁 ∧ 𝑐 ∈ 𝑁) → 𝑎 Fn 𝑁)
41		simp2 1055	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ 𝑎 ∈ (𝑁 ↑𝑚 𝑁)) ∧ 𝑏 ∈ 𝑁 ∧ 𝑐 ∈ 𝑁) → 𝑏 ∈ 𝑁)
42		fnopfvb 6147	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑎 Fn 𝑁 ∧ 𝑏 ∈ 𝑁) → ((𝑎‘𝑏) = 𝑐 ↔ ⟨𝑏, 𝑐⟩ ∈ 𝑎))
43	40, 41, 42	syl2anc 691	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑎 ∈ (𝑁 ↑𝑚 𝑁)) ∧ 𝑏 ∈ 𝑁 ∧ 𝑐 ∈ 𝑁) → ((𝑎‘𝑏) = 𝑐 ↔ ⟨𝑏, 𝑐⟩ ∈ 𝑎))
44	43	bicomd 212	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑎 ∈ (𝑁 ↑𝑚 𝑁)) ∧ 𝑏 ∈ 𝑁 ∧ 𝑐 ∈ 𝑁) → (⟨𝑏, 𝑐⟩ ∈ 𝑎 ↔ (𝑎‘𝑏) = 𝑐))
45	44	ifbid 4058	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑎 ∈ (𝑁 ↑𝑚 𝑁)) ∧ 𝑏 ∈ 𝑁 ∧ 𝑐 ∈ 𝑁) → if(⟨𝑏, 𝑐⟩ ∈ 𝑎, 1 , 0 ) = if((𝑎‘𝑏) = 𝑐, 1 , 0 ))
46	45	mpt2eq3dva 6617	. . . . . . . . . . . . . . . . . 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 )))
47	35, 46	syl5eq 2656	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑎 ∈ (𝑁 ↑𝑚 𝑁)) → (𝑤 ∈ (𝑁 × 𝑁) ↦ if(𝑤 ∈ 𝑎, 1 , 0 )) = (𝑏 ∈ 𝑁, 𝑐 ∈ 𝑁 ↦ if((𝑎‘𝑏) = 𝑐, 1 , 0 )))
48	47	fveq2d 6107	. . . . . . . . . . . . . . . 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 ))))
49		mdetuni.0g	. . . . . . . . . . . . . . . . . 18 ⊢ 0 = (0g‘𝑅)
50		mdetuni.1r	. . . . . . . . . . . . . . . . . 18 ⊢ 1 = (1r‘𝑅)
51		mdetuni.pg	. . . . . . . . . . . . . . . . . 18 ⊢ + = (+g‘𝑅)
52		mdetuni.tg	. . . . . . . . . . . . . . . . . 18 ⊢ · = (.r‘𝑅)
53		mdetuni.r	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝑅 ∈ Ring)
54		mdetuni.ff	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝐷:𝐵⟶𝐾)
55		mdetuni.al	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝑁 ∀𝑧 ∈ 𝑁 ((𝑦 ≠ 𝑧 ∧ ∀𝑤 ∈ 𝑁 (𝑦𝑥𝑤) = (𝑧𝑥𝑤)) → (𝐷‘𝑥) = 0 ))
56		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))))
57		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))))
58		mdetunilem9.id	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (𝐷‘(1r‘𝐴)) = 0 )
59	11, 13, 12, 49, 50, 51, 52, 6, 53, 54, 55, 56, 57, 58	mdetunilem8 20244	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑎:𝑁⟶𝑁) → (𝐷‘(𝑏 ∈ 𝑁, 𝑐 ∈ 𝑁 ↦ if((𝑎‘𝑏) = 𝑐, 1 , 0 ))) = 0 )
60	36, 59	sylan2 490	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑎 ∈ (𝑁 ↑𝑚 𝑁)) → (𝐷‘(𝑏 ∈ 𝑁, 𝑐 ∈ 𝑁 ↦ if((𝑎‘𝑏) = 𝑐, 1 , 0 ))) = 0 )
61	48, 60	eqtrd 2644	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑎 ∈ (𝑁 ↑𝑚 𝑁)) → (𝐷‘(𝑤 ∈ (𝑁 × 𝑁) ↦ if(𝑤 ∈ 𝑎, 1 , 0 ))) = 0 )
62	32, 61	chvarv 2251	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝑁 ↑𝑚 𝑁)) → (𝐷‘(𝑤 ∈ (𝑁 × 𝑁) ↦ if(𝑤 ∈ 𝑧, 1 , 0 ))) = 0 )
63	62	adantrl 748	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ (𝑁 ↑𝑚 𝑁))) → (𝐷‘(𝑤 ∈ (𝑁 × 𝑁) ↦ if(𝑤 ∈ 𝑧, 1 , 0 ))) = 0 )
64	63	adantr 480	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ (𝑁 ↑𝑚 𝑁))) ∧ ∀𝑤 ∈ (𝑁 × 𝑁)(𝑦‘𝑤) = if(𝑤 ∈ 𝑧, 1 , 0 )) → (𝐷‘(𝑤 ∈ (𝑁 × 𝑁) ↦ if(𝑤 ∈ 𝑧, 1 , 0 ))) = 0 )
65	19, 24, 64	3eqtrd 2648	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ (𝑁 ↑𝑚 𝑁))) ∧ ∀𝑤 ∈ (𝑁 × 𝑁)(𝑦‘𝑤) = if(𝑤 ∈ 𝑧, 1 , 0 )) → (𝐷‘𝑦) = 0 )
66	65	ex 449	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ (𝑁 ↑𝑚 𝑁))) → (∀𝑤 ∈ (𝑁 × 𝑁)(𝑦‘𝑤) = if(𝑤 ∈ 𝑧, 1 , 0 ) → (𝐷‘𝑦) = 0 ))
67	66	ralrimivva 2954	. . . . . . . . 9 ⊢ (𝜑 → ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ (𝑁 ↑𝑚 𝑁)(∀𝑤 ∈ (𝑁 × 𝑁)(𝑦‘𝑤) = if(𝑤 ∈ 𝑧, 1 , 0 ) → (𝐷‘𝑦) = 0 ))
68		xpfi 8116	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ Fin ∧ 𝑁 ∈ Fin) → (𝑁 × 𝑁) ∈ Fin)
69	6, 6, 68	syl2anc 691	. . . . . . . . . 10 ⊢ (𝜑 → (𝑁 × 𝑁) ∈ Fin)
70		raleq 3115	. . . . . . . . . . . . 13 ⊢ (𝑥 = (𝑁 × 𝑁) → (∀𝑤 ∈ 𝑥 (𝑦‘𝑤) = if(𝑤 ∈ 𝑧, 1 , 0 ) ↔ ∀𝑤 ∈ (𝑁 × 𝑁)(𝑦‘𝑤) = if(𝑤 ∈ 𝑧, 1 , 0 )))
71	70	imbi1d 330	. . . . . . . . . . . 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 )))
72	71	2ralbidv 2972	. . . . . . . . . . 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 )))
73		mdetunilem9.y	. . . . . . . . . . 11 ⊢ 𝑌 = {𝑥 ∣ ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ (𝑁 ↑𝑚 𝑁)(∀𝑤 ∈ 𝑥 (𝑦‘𝑤) = if(𝑤 ∈ 𝑧, 1 , 0 ) → (𝐷‘𝑦) = 0 )}
74	72, 73	elab2g 3322	. . . . . . . . . 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 )))
75	69, 74	syl 17	. . . . . . . . 9 ⊢ (𝜑 → ((𝑁 × 𝑁) ∈ 𝑌 ↔ ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ (𝑁 ↑𝑚 𝑁)(∀𝑤 ∈ (𝑁 × 𝑁)(𝑦‘𝑤) = if(𝑤 ∈ 𝑧, 1 , 0 ) → (𝐷‘𝑦) = 0 )))
76	67, 75	mpbird 246	. . . . . . . 8 ⊢ (𝜑 → (𝑁 × 𝑁) ∈ 𝑌)
77		ssid 3587	. . . . . . . . 9 ⊢ (𝑁 × 𝑁) ⊆ (𝑁 × 𝑁)
78	69	3ad2ant1 1075	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑁 × 𝑁) ⊆ (𝑁 × 𝑁) ∧ ¬ ∅ ∈ 𝑌) → (𝑁 × 𝑁) ∈ Fin)
79		sseq1 3589	. . . . . . . . . . . . . 14 ⊢ (𝑎 = ∅ → (𝑎 ⊆ (𝑁 × 𝑁) ↔ ∅ ⊆ (𝑁 × 𝑁)))
80	79	3anbi2d 1396	. . . . . . . . . . . . 13 ⊢ (𝑎 = ∅ → ((𝜑 ∧ 𝑎 ⊆ (𝑁 × 𝑁) ∧ ¬ ∅ ∈ 𝑌) ↔ (𝜑 ∧ ∅ ⊆ (𝑁 × 𝑁) ∧ ¬ ∅ ∈ 𝑌)))
81		eleq1 2676	. . . . . . . . . . . . . 14 ⊢ (𝑎 = ∅ → (𝑎 ∈ 𝑌 ↔ ∅ ∈ 𝑌))
82	81	notbid 307	. . . . . . . . . . . . 13 ⊢ (𝑎 = ∅ → (¬ 𝑎 ∈ 𝑌 ↔ ¬ ∅ ∈ 𝑌))
83	80, 82	imbi12d 333	. . . . . . . . . . . 12 ⊢ (𝑎 = ∅ → (((𝜑 ∧ 𝑎 ⊆ (𝑁 × 𝑁) ∧ ¬ ∅ ∈ 𝑌) → ¬ 𝑎 ∈ 𝑌) ↔ ((𝜑 ∧ ∅ ⊆ (𝑁 × 𝑁) ∧ ¬ ∅ ∈ 𝑌) → ¬ ∅ ∈ 𝑌)))
84		sseq1 3589	. . . . . . . . . . . . . 14 ⊢ (𝑎 = 𝑏 → (𝑎 ⊆ (𝑁 × 𝑁) ↔ 𝑏 ⊆ (𝑁 × 𝑁)))
85	84	3anbi2d 1396	. . . . . . . . . . . . 13 ⊢ (𝑎 = 𝑏 → ((𝜑 ∧ 𝑎 ⊆ (𝑁 × 𝑁) ∧ ¬ ∅ ∈ 𝑌) ↔ (𝜑 ∧ 𝑏 ⊆ (𝑁 × 𝑁) ∧ ¬ ∅ ∈ 𝑌)))
86		eleq1 2676	. . . . . . . . . . . . . 14 ⊢ (𝑎 = 𝑏 → (𝑎 ∈ 𝑌 ↔ 𝑏 ∈ 𝑌))
87	86	notbid 307	. . . . . . . . . . . . 13 ⊢ (𝑎 = 𝑏 → (¬ 𝑎 ∈ 𝑌 ↔ ¬ 𝑏 ∈ 𝑌))
88	85, 87	imbi12d 333	. . . . . . . . . . . 12 ⊢ (𝑎 = 𝑏 → (((𝜑 ∧ 𝑎 ⊆ (𝑁 × 𝑁) ∧ ¬ ∅ ∈ 𝑌) → ¬ 𝑎 ∈ 𝑌) ↔ ((𝜑 ∧ 𝑏 ⊆ (𝑁 × 𝑁) ∧ ¬ ∅ ∈ 𝑌) → ¬ 𝑏 ∈ 𝑌)))
89		sseq1 3589	. . . . . . . . . . . . . 14 ⊢ (𝑎 = (𝑏 ∪ {𝑐}) → (𝑎 ⊆ (𝑁 × 𝑁) ↔ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁)))
90	89	3anbi2d 1396	. . . . . . . . . . . . 13 ⊢ (𝑎 = (𝑏 ∪ {𝑐}) → ((𝜑 ∧ 𝑎 ⊆ (𝑁 × 𝑁) ∧ ¬ ∅ ∈ 𝑌) ↔ (𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ ¬ ∅ ∈ 𝑌)))
91		eleq1 2676	. . . . . . . . . . . . . 14 ⊢ (𝑎 = (𝑏 ∪ {𝑐}) → (𝑎 ∈ 𝑌 ↔ (𝑏 ∪ {𝑐}) ∈ 𝑌))
92	91	notbid 307	. . . . . . . . . . . . 13 ⊢ (𝑎 = (𝑏 ∪ {𝑐}) → (¬ 𝑎 ∈ 𝑌 ↔ ¬ (𝑏 ∪ {𝑐}) ∈ 𝑌))
93	90, 92	imbi12d 333	. . . . . . . . . . . 12 ⊢ (𝑎 = (𝑏 ∪ {𝑐}) → (((𝜑 ∧ 𝑎 ⊆ (𝑁 × 𝑁) ∧ ¬ ∅ ∈ 𝑌) → ¬ 𝑎 ∈ 𝑌) ↔ ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ ¬ ∅ ∈ 𝑌) → ¬ (𝑏 ∪ {𝑐}) ∈ 𝑌)))
94		sseq1 3589	. . . . . . . . . . . . . 14 ⊢ (𝑎 = (𝑁 × 𝑁) → (𝑎 ⊆ (𝑁 × 𝑁) ↔ (𝑁 × 𝑁) ⊆ (𝑁 × 𝑁)))
95	94	3anbi2d 1396	. . . . . . . . . . . . 13 ⊢ (𝑎 = (𝑁 × 𝑁) → ((𝜑 ∧ 𝑎 ⊆ (𝑁 × 𝑁) ∧ ¬ ∅ ∈ 𝑌) ↔ (𝜑 ∧ (𝑁 × 𝑁) ⊆ (𝑁 × 𝑁) ∧ ¬ ∅ ∈ 𝑌)))
96		eleq1 2676	. . . . . . . . . . . . . 14 ⊢ (𝑎 = (𝑁 × 𝑁) → (𝑎 ∈ 𝑌 ↔ (𝑁 × 𝑁) ∈ 𝑌))
97	96	notbid 307	. . . . . . . . . . . . 13 ⊢ (𝑎 = (𝑁 × 𝑁) → (¬ 𝑎 ∈ 𝑌 ↔ ¬ (𝑁 × 𝑁) ∈ 𝑌))
98	95, 97	imbi12d 333	. . . . . . . . . . . 12 ⊢ (𝑎 = (𝑁 × 𝑁) → (((𝜑 ∧ 𝑎 ⊆ (𝑁 × 𝑁) ∧ ¬ ∅ ∈ 𝑌) → ¬ 𝑎 ∈ 𝑌) ↔ ((𝜑 ∧ (𝑁 × 𝑁) ⊆ (𝑁 × 𝑁) ∧ ¬ ∅ ∈ 𝑌) → ¬ (𝑁 × 𝑁) ∈ 𝑌)))
99		simp3 1056	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ ∅ ⊆ (𝑁 × 𝑁) ∧ ¬ ∅ ∈ 𝑌) → ¬ ∅ ∈ 𝑌)
100		ssun1 3738	. . . . . . . . . . . . . . . 16 ⊢ 𝑏 ⊆ (𝑏 ∪ {𝑐})
101		sstr2 3575	. . . . . . . . . . . . . . . 16 ⊢ (𝑏 ⊆ (𝑏 ∪ {𝑐}) → ((𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) → 𝑏 ⊆ (𝑁 × 𝑁)))
102	100, 101	ax-mp 5	. . . . . . . . . . . . . . 15 ⊢ ((𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) → 𝑏 ⊆ (𝑁 × 𝑁))
103	102	3anim2i 1242	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ ¬ ∅ ∈ 𝑌) → (𝜑 ∧ 𝑏 ⊆ (𝑁 × 𝑁) ∧ ¬ ∅ ∈ 𝑌))
104	103	imim1i 61	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑏 ⊆ (𝑁 × 𝑁) ∧ ¬ ∅ ∈ 𝑌) → ¬ 𝑏 ∈ 𝑌) → ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ ¬ ∅ ∈ 𝑌) → ¬ 𝑏 ∈ 𝑌))
105		simpl1 1057	. . . . . . . . . . . . . . . . . . . . . . . 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)
106		simpl2 1058	. . . . . . . . . . . . . . . . . . . . . . . 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))
107		simprll 798	. . . . . . . . . . . . . . . . . . . . . . . 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)
108	11, 12, 13	matbas2i 20047	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑎 ∈ 𝐵 → 𝑎 ∈ (𝐾 ↑𝑚 (𝑁 × 𝑁)))
109		elmapi 7765	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑎 ∈ (𝐾 ↑𝑚 (𝑁 × 𝑁)) → 𝑎:(𝑁 × 𝑁)⟶𝐾)
110	108, 109	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑎 ∈ 𝐵 → 𝑎:(𝑁 × 𝑁)⟶𝐾)
111	110	3ad2ant3 1077	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) → 𝑎:(𝑁 × 𝑁)⟶𝐾)
112	111	feqmptd 6159	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) → 𝑎 = (𝑒 ∈ (𝑁 × 𝑁) ↦ (𝑎‘𝑒)))
113	112	reseq1d 5316	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) → (𝑎 ↾ ({(1st ‘𝑐)} × 𝑁)) = ((𝑒 ∈ (𝑁 × 𝑁) ↦ (𝑎‘𝑒)) ↾ ({(1st ‘𝑐)} × 𝑁)))
114	53	3ad2ant1 1075	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) → 𝑅 ∈ Ring)
115		ringgrp 18375	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ (𝑅 ∈ Ring → 𝑅 ∈ Grp)
116	114, 115	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) → 𝑅 ∈ Grp)
117	116	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) ∧ 𝑒 ∈ ({(1st ‘𝑐)} × 𝑁)) → 𝑅 ∈ Grp)
118	111	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) ∧ 𝑒 ∈ ({(1st ‘𝑐)} × 𝑁)) → 𝑎:(𝑁 × 𝑁)⟶𝐾)
119		simp2 1055	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 ⊢ ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) → (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁))
120	119	unssbd 3753	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 ⊢ ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) → {𝑐} ⊆ (𝑁 × 𝑁))
121		vex 3176	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 ⊢ 𝑐 ∈ V
122	121	snss 4259	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 ⊢ (𝑐 ∈ (𝑁 × 𝑁) ↔ {𝑐} ⊆ (𝑁 × 𝑁))
123	120, 122	sylibr 223	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ⊢ ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) → 𝑐 ∈ (𝑁 × 𝑁))
124		xp1st 7089	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ⊢ (𝑐 ∈ (𝑁 × 𝑁) → (1st ‘𝑐) ∈ 𝑁)
125	123, 124	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ⊢ ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) → (1st ‘𝑐) ∈ 𝑁)
126	125	snssd 4281	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) → {(1st ‘𝑐)} ⊆ 𝑁)
127		xpss1 5151	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ ({(1st ‘𝑐)} ⊆ 𝑁 → ({(1st ‘𝑐)} × 𝑁) ⊆ (𝑁 × 𝑁))
128	126, 127	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) → ({(1st ‘𝑐)} × 𝑁) ⊆ (𝑁 × 𝑁))
129	128	sselda 3568	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) ∧ 𝑒 ∈ ({(1st ‘𝑐)} × 𝑁)) → 𝑒 ∈ (𝑁 × 𝑁))
130	118, 129	ffvelrnd 6268	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) ∧ 𝑒 ∈ ({(1st ‘𝑐)} × 𝑁)) → (𝑎‘𝑒) ∈ 𝐾)
131	12, 50	ringidcl 18391	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ (𝑅 ∈ Ring → 1 ∈ 𝐾)
132	114, 131	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) → 1 ∈ 𝐾)
133	12, 49	ring0cl 18392	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ (𝑅 ∈ Ring → 0 ∈ 𝐾)
134	114, 133	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) → 0 ∈ 𝐾)
135	132, 134	ifcld 4081	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) → if(𝑒 ∈ 𝑑, 1 , 0 ) ∈ 𝐾)
136	135	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) ∧ 𝑒 ∈ ({(1st ‘𝑐)} × 𝑁)) → if(𝑒 ∈ 𝑑, 1 , 0 ) ∈ 𝐾)
137		eqid 2610	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ (-g‘𝑅) = (-g‘𝑅)
138	12, 51, 137	grpnpcan 17330	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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))
139	117, 130, 136, 138	syl3anc 1318	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) ∧ 𝑒 ∈ ({(1st ‘𝑐)} × 𝑁)) → (((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )) + if(𝑒 ∈ 𝑑, 1 , 0 )) = (𝑎‘𝑒))
140	139	eqcomd 2616	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) ∧ 𝑒 ∈ ({(1st ‘𝑐)} × 𝑁)) → (𝑎‘𝑒) = (((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )) + if(𝑒 ∈ 𝑑, 1 , 0 )))
141	140	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) ∧ 𝑒 ∈ ({(1st ‘𝑐)} × 𝑁)) ∧ 𝑒 = 𝑐) → (𝑎‘𝑒) = (((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )) + if(𝑒 ∈ 𝑑, 1 , 0 )))
142		iftrue 4042	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 )))
143		iftrue 4042	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (𝑒 = 𝑐 → if(𝑒 = 𝑐, if(𝑒 ∈ 𝑑, 1 , 0 ), (𝑎‘𝑒)) = if(𝑒 ∈ 𝑑, 1 , 0 ))
144	142, 143	oveq12d 6567	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 )))
145	144	adantl 481	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) ∧ 𝑒 ∈ ({(1st ‘𝑐)} × 𝑁)) ∧ 𝑒 = 𝑐) → (if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ) + if(𝑒 = 𝑐, if(𝑒 ∈ 𝑑, 1 , 0 ), (𝑎‘𝑒))) = (((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )) + if(𝑒 ∈ 𝑑, 1 , 0 )))
146	141, 145	eqtr4d 2647	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) ∧ 𝑒 ∈ ({(1st ‘𝑐)} × 𝑁)) ∧ 𝑒 = 𝑐) → (𝑎‘𝑒) = (if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ) + if(𝑒 = 𝑐, if(𝑒 ∈ 𝑑, 1 , 0 ), (𝑎‘𝑒))))
147	12, 51, 49	grplid 17275	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ ((𝑅 ∈ Grp ∧ (𝑎‘𝑒) ∈ 𝐾) → ( 0 + (𝑎‘𝑒)) = (𝑎‘𝑒))
148	117, 130, 147	syl2anc 691	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) ∧ 𝑒 ∈ ({(1st ‘𝑐)} × 𝑁)) → ( 0 + (𝑎‘𝑒)) = (𝑎‘𝑒))
149	148	eqcomd 2616	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) ∧ 𝑒 ∈ ({(1st ‘𝑐)} × 𝑁)) → (𝑎‘𝑒) = ( 0 + (𝑎‘𝑒)))
150	149	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) ∧ 𝑒 ∈ ({(1st ‘𝑐)} × 𝑁)) ∧ ¬ 𝑒 = 𝑐) → (𝑎‘𝑒) = ( 0 + (𝑎‘𝑒)))
151		iffalse 4045	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (¬ 𝑒 = 𝑐 → if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ) = 0 )
152		iffalse 4045	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (¬ 𝑒 = 𝑐 → if(𝑒 = 𝑐, if(𝑒 ∈ 𝑑, 1 , 0 ), (𝑎‘𝑒)) = (𝑎‘𝑒))
153	151, 152	oveq12d 6567	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)))
154	153	adantl 481	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) ∧ 𝑒 ∈ ({(1st ‘𝑐)} × 𝑁)) ∧ ¬ 𝑒 = 𝑐) → (if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ) + if(𝑒 = 𝑐, if(𝑒 ∈ 𝑑, 1 , 0 ), (𝑎‘𝑒))) = ( 0 + (𝑎‘𝑒)))
155	150, 154	eqtr4d 2647	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) ∧ 𝑒 ∈ ({(1st ‘𝑐)} × 𝑁)) ∧ ¬ 𝑒 = 𝑐) → (𝑎‘𝑒) = (if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ) + if(𝑒 = 𝑐, if(𝑒 ∈ 𝑑, 1 , 0 ), (𝑎‘𝑒))))
156	146, 155	pm2.61dan 828	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) ∧ 𝑒 ∈ ({(1st ‘𝑐)} × 𝑁)) → (𝑎‘𝑒) = (if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ) + if(𝑒 = 𝑐, if(𝑒 ∈ 𝑑, 1 , 0 ), (𝑎‘𝑒))))
157	156	mpteq2dva 4672	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) → (𝑒 ∈ ({(1st ‘𝑐)} × 𝑁) ↦ (𝑎‘𝑒)) = (𝑒 ∈ ({(1st ‘𝑐)} × 𝑁) ↦ (if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ) + if(𝑒 = 𝑐, if(𝑒 ∈ 𝑑, 1 , 0 ), (𝑎‘𝑒)))))
158		snfi 7923	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ {(1st ‘𝑐)} ∈ Fin
159	6	3ad2ant1 1075	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) → 𝑁 ∈ Fin)
160		xpfi 8116	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (({(1st ‘𝑐)} ∈ Fin ∧ 𝑁 ∈ Fin) → ({(1st ‘𝑐)} × 𝑁) ∈ Fin)
161	158, 159, 160	sylancr 694	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) → ({(1st ‘𝑐)} × 𝑁) ∈ Fin)
162		ovex 6577	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )) ∈ V
163		fvex 6113	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (0g‘𝑅) ∈ V
164	49, 163	eqeltri 2684	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ 0 ∈ V
165	162, 164	ifex 4106	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ) ∈ V
166	165	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) ∧ 𝑒 ∈ ({(1st ‘𝑐)} × 𝑁)) → if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ) ∈ V)
167		fvex 6113	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (1r‘𝑅) ∈ V
168	50, 167	eqeltri 2684	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ 1 ∈ V
169	168, 164	ifex 4106	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ if(𝑒 ∈ 𝑑, 1 , 0 ) ∈ V
170		fvex 6113	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝑎‘𝑒) ∈ V
171	169, 170	ifex 4106	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ if(𝑒 = 𝑐, if(𝑒 ∈ 𝑑, 1 , 0 ), (𝑎‘𝑒)) ∈ V
172	171	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) ∧ 𝑒 ∈ ({(1st ‘𝑐)} × 𝑁)) → if(𝑒 = 𝑐, if(𝑒 ∈ 𝑑, 1 , 0 ), (𝑎‘𝑒)) ∈ V)
173		xp1st 7089	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝑒 ∈ ({(1st ‘𝑐)} × 𝑁) → (1st ‘𝑒) ∈ {(1st ‘𝑐)})
174		elsni 4142	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((1st ‘𝑒) ∈ {(1st ‘𝑐)} → (1st ‘𝑒) = (1st ‘𝑐))
175		iftrue 4042	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((1st ‘𝑒) = (1st ‘𝑐) → if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒)) = if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ))
176	173, 174, 175	3syl 18	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝑒 ∈ ({(1st ‘𝑐)} × 𝑁) → if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒)) = if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ))
177	176	mpteq2ia 4668	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑒 ∈ ({(1st ‘𝑐)} × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) = (𝑒 ∈ ({(1st ‘𝑐)} × 𝑁) ↦ if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ))
178	177	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) → (𝑒 ∈ ({(1st ‘𝑐)} × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) = (𝑒 ∈ ({(1st ‘𝑐)} × 𝑁) ↦ if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 )))
179		eqidd 2611	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) → (𝑒 ∈ ({(1st ‘𝑐)} × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒 ∈ 𝑑, 1 , 0 ), (𝑎‘𝑒))) = (𝑒 ∈ ({(1st ‘𝑐)} × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒 ∈ 𝑑, 1 , 0 ), (𝑎‘𝑒))))
180	161, 166, 172, 178, 179	offval2 6812	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) → ((𝑒 ∈ ({(1st ‘𝑐)} × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ∘𝑓 + (𝑒 ∈ ({(1st ‘𝑐)} × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒 ∈ 𝑑, 1 , 0 ), (𝑎‘𝑒)))) = (𝑒 ∈ ({(1st ‘𝑐)} × 𝑁) ↦ (if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ) + if(𝑒 = 𝑐, if(𝑒 ∈ 𝑑, 1 , 0 ), (𝑎‘𝑒)))))
181	157, 180	eqtr4d 2647	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) → (𝑒 ∈ ({(1st ‘𝑐)} × 𝑁) ↦ (𝑎‘𝑒)) = ((𝑒 ∈ ({(1st ‘𝑐)} × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ∘𝑓 + (𝑒 ∈ ({(1st ‘𝑐)} × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒 ∈ 𝑑, 1 , 0 ), (𝑎‘𝑒)))))
182	128	resmptd 5371	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) → ((𝑒 ∈ (𝑁 × 𝑁) ↦ (𝑎‘𝑒)) ↾ ({(1st ‘𝑐)} × 𝑁)) = (𝑒 ∈ ({(1st ‘𝑐)} × 𝑁) ↦ (𝑎‘𝑒)))
183	128	resmptd 5371	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) → ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ↾ ({(1st ‘𝑐)} × 𝑁)) = (𝑒 ∈ ({(1st ‘𝑐)} × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))))
184	128	resmptd 5371	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) → ((𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒 ∈ 𝑑, 1 , 0 ), (𝑎‘𝑒))) ↾ ({(1st ‘𝑐)} × 𝑁)) = (𝑒 ∈ ({(1st ‘𝑐)} × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒 ∈ 𝑑, 1 , 0 ), (𝑎‘𝑒))))
185	183, 184	oveq12d 6567	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) → (((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ↾ ({(1st ‘𝑐)} × 𝑁)) ∘𝑓 + ((𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒 ∈ 𝑑, 1 , 0 ), (𝑎‘𝑒))) ↾ ({(1st ‘𝑐)} × 𝑁))) = ((𝑒 ∈ ({(1st ‘𝑐)} × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ∘𝑓 + (𝑒 ∈ ({(1st ‘𝑐)} × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒 ∈ 𝑑, 1 , 0 ), (𝑎‘𝑒)))))
186	181, 182, 185	3eqtr4d 2654	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) → ((𝑒 ∈ (𝑁 × 𝑁) ↦ (𝑎‘𝑒)) ↾ ({(1st ‘𝑐)} × 𝑁)) = (((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ↾ ({(1st ‘𝑐)} × 𝑁)) ∘𝑓 + ((𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒 ∈ 𝑑, 1 , 0 ), (𝑎‘𝑒))) ↾ ({(1st ‘𝑐)} × 𝑁))))
187	113, 186	eqtrd 2644	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) → (𝑎 ↾ ({(1st ‘𝑐)} × 𝑁)) = (((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ↾ ({(1st ‘𝑐)} × 𝑁)) ∘𝑓 + ((𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒 ∈ 𝑑, 1 , 0 ), (𝑎‘𝑒))) ↾ ({(1st ‘𝑐)} × 𝑁))))
188	112	reseq1d 5316	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) → (𝑎 ↾ ((𝑁 ∖ {(1st ‘𝑐)}) × 𝑁)) = ((𝑒 ∈ (𝑁 × 𝑁) ↦ (𝑎‘𝑒)) ↾ ((𝑁 ∖ {(1st ‘𝑐)}) × 𝑁)))
189		xp1st 7089	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝑒 ∈ ((𝑁 ∖ {(1st ‘𝑐)}) × 𝑁) → (1st ‘𝑒) ∈ (𝑁 ∖ {(1st ‘𝑐)}))
190		eldifsni 4261	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((1st ‘𝑒) ∈ (𝑁 ∖ {(1st ‘𝑐)}) → (1st ‘𝑒) ≠ (1st ‘𝑐))
191	189, 190	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝑒 ∈ ((𝑁 ∖ {(1st ‘𝑐)}) × 𝑁) → (1st ‘𝑒) ≠ (1st ‘𝑐))
192	191	neneqd 2787	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑒 ∈ ((𝑁 ∖ {(1st ‘𝑐)}) × 𝑁) → ¬ (1st ‘𝑒) = (1st ‘𝑐))
193	192	adantl 481	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) ∧ 𝑒 ∈ ((𝑁 ∖ {(1st ‘𝑐)}) × 𝑁)) → ¬ (1st ‘𝑒) = (1st ‘𝑐))
194	193	iffalsed 4047	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) ∧ 𝑒 ∈ ((𝑁 ∖ {(1st ‘𝑐)}) × 𝑁)) → if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒)) = (𝑎‘𝑒))
195	194	mpteq2dva 4672	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) → (𝑒 ∈ ((𝑁 ∖ {(1st ‘𝑐)}) × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) = (𝑒 ∈ ((𝑁 ∖ {(1st ‘𝑐)}) × 𝑁) ↦ (𝑎‘𝑒)))
196		difss 3699	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑁 ∖ {(1st ‘𝑐)}) ⊆ 𝑁
197		xpss1 5151	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((𝑁 ∖ {(1st ‘𝑐)}) ⊆ 𝑁 → ((𝑁 ∖ {(1st ‘𝑐)}) × 𝑁) ⊆ (𝑁 × 𝑁))
198	196, 197	ax-mp 5	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝑁 ∖ {(1st ‘𝑐)}) × 𝑁) ⊆ (𝑁 × 𝑁)
199		resmpt 5369	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (((𝑁 ∖ {(1st ‘𝑐)}) × 𝑁) ⊆ (𝑁 × 𝑁) → ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ↾ ((𝑁 ∖ {(1st ‘𝑐)}) × 𝑁)) = (𝑒 ∈ ((𝑁 ∖ {(1st ‘𝑐)}) × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))))
200	198, 199	mp1i 13	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) → ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ↾ ((𝑁 ∖ {(1st ‘𝑐)}) × 𝑁)) = (𝑒 ∈ ((𝑁 ∖ {(1st ‘𝑐)}) × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))))
201		resmpt 5369	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (((𝑁 ∖ {(1st ‘𝑐)}) × 𝑁) ⊆ (𝑁 × 𝑁) → ((𝑒 ∈ (𝑁 × 𝑁) ↦ (𝑎‘𝑒)) ↾ ((𝑁 ∖ {(1st ‘𝑐)}) × 𝑁)) = (𝑒 ∈ ((𝑁 ∖ {(1st ‘𝑐)}) × 𝑁) ↦ (𝑎‘𝑒)))
202	198, 201	mp1i 13	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) → ((𝑒 ∈ (𝑁 × 𝑁) ↦ (𝑎‘𝑒)) ↾ ((𝑁 ∖ {(1st ‘𝑐)}) × 𝑁)) = (𝑒 ∈ ((𝑁 ∖ {(1st ‘𝑐)}) × 𝑁) ↦ (𝑎‘𝑒)))
203	195, 200, 202	3eqtr4rd 2655	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) → ((𝑒 ∈ (𝑁 × 𝑁) ↦ (𝑎‘𝑒)) ↾ ((𝑁 ∖ {(1st ‘𝑐)}) × 𝑁)) = ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ↾ ((𝑁 ∖ {(1st ‘𝑐)}) × 𝑁)))
204	188, 203	eqtrd 2644	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) → (𝑎 ↾ ((𝑁 ∖ {(1st ‘𝑐)}) × 𝑁)) = ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ↾ ((𝑁 ∖ {(1st ‘𝑐)}) × 𝑁)))
205		fveq2 6103	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑒 = 𝑐 → (1st ‘𝑒) = (1st ‘𝑐))
206	193, 205	nsyl 134	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) ∧ 𝑒 ∈ ((𝑁 ∖ {(1st ‘𝑐)}) × 𝑁)) → ¬ 𝑒 = 𝑐)
207	206	iffalsed 4047	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) ∧ 𝑒 ∈ ((𝑁 ∖ {(1st ‘𝑐)}) × 𝑁)) → if(𝑒 = 𝑐, if(𝑒 ∈ 𝑑, 1 , 0 ), (𝑎‘𝑒)) = (𝑎‘𝑒))
208	207	mpteq2dva 4672	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) → (𝑒 ∈ ((𝑁 ∖ {(1st ‘𝑐)}) × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒 ∈ 𝑑, 1 , 0 ), (𝑎‘𝑒))) = (𝑒 ∈ ((𝑁 ∖ {(1st ‘𝑐)}) × 𝑁) ↦ (𝑎‘𝑒)))
209		resmpt 5369	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (((𝑁 ∖ {(1st ‘𝑐)}) × 𝑁) ⊆ (𝑁 × 𝑁) → ((𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒 ∈ 𝑑, 1 , 0 ), (𝑎‘𝑒))) ↾ ((𝑁 ∖ {(1st ‘𝑐)}) × 𝑁)) = (𝑒 ∈ ((𝑁 ∖ {(1st ‘𝑐)}) × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒 ∈ 𝑑, 1 , 0 ), (𝑎‘𝑒))))
210	198, 209	mp1i 13	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) → ((𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒 ∈ 𝑑, 1 , 0 ), (𝑎‘𝑒))) ↾ ((𝑁 ∖ {(1st ‘𝑐)}) × 𝑁)) = (𝑒 ∈ ((𝑁 ∖ {(1st ‘𝑐)}) × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒 ∈ 𝑑, 1 , 0 ), (𝑎‘𝑒))))
211	208, 210, 202	3eqtr4rd 2655	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) → ((𝑒 ∈ (𝑁 × 𝑁) ↦ (𝑎‘𝑒)) ↾ ((𝑁 ∖ {(1st ‘𝑐)}) × 𝑁)) = ((𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒 ∈ 𝑑, 1 , 0 ), (𝑎‘𝑒))) ↾ ((𝑁 ∖ {(1st ‘𝑐)}) × 𝑁)))
212	188, 211	eqtrd 2644	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) → (𝑎 ↾ ((𝑁 ∖ {(1st ‘𝑐)}) × 𝑁)) = ((𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒 ∈ 𝑑, 1 , 0 ), (𝑎‘𝑒))) ↾ ((𝑁 ∖ {(1st ‘𝑐)}) × 𝑁)))
213	135	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) ∧ 𝑒 ∈ (𝑁 × 𝑁)) → if(𝑒 ∈ 𝑑, 1 , 0 ) ∈ 𝐾)
214	111	ffvelrnda 6267	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) ∧ 𝑒 ∈ (𝑁 × 𝑁)) → (𝑎‘𝑒) ∈ 𝐾)
215	213, 214	ifcld 4081	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) ∧ 𝑒 ∈ (𝑁 × 𝑁)) → if(𝑒 = 𝑐, if(𝑒 ∈ 𝑑, 1 , 0 ), (𝑎‘𝑒)) ∈ 𝐾)
216		eqid 2610	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (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)))
217	215, 216	fmptd 6292	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) → (𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒 ∈ 𝑑, 1 , 0 ), (𝑎‘𝑒))):(𝑁 × 𝑁)⟶𝐾)
218		fvex 6113	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (Base‘𝑅) ∈ V
219	12, 218	eqeltri 2684	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ 𝐾 ∈ V
220	68	anidms 675	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑁 ∈ Fin → (𝑁 × 𝑁) ∈ Fin)
221	159, 220	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) → (𝑁 × 𝑁) ∈ Fin)
222		elmapg 7757	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 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))
223	219, 221, 222	sylancr 694	. . . . . . . . . . . . . . . . . . . . . . . . . . . 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))
224	217, 223	mpbird 246	. . . . . . . . . . . . . . . . . . . . . . . . . . 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)))
225	11, 12	matbas2 20046	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (𝐾 ↑𝑚 (𝑁 × 𝑁)) = (Base‘𝐴))
226	159, 114, 225	syl2anc 691	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) → (𝐾 ↑𝑚 (𝑁 × 𝑁)) = (Base‘𝐴))
227	226, 13	syl6eqr 2662	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) → (𝐾 ↑𝑚 (𝑁 × 𝑁)) = 𝐵)
228	224, 227	eleqtrd 2690	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) → (𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒 ∈ 𝑑, 1 , 0 ), (𝑎‘𝑒))) ∈ 𝐵)
229		simp3 1056	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) → 𝑎 ∈ 𝐵)
230	116	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) ∧ 𝑒 ∈ (𝑁 × 𝑁)) → 𝑅 ∈ Grp)
231	12, 137	grpsubcl 17318	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ ((𝑅 ∈ Grp ∧ (𝑎‘𝑒) ∈ 𝐾 ∧ if(𝑒 ∈ 𝑑, 1 , 0 ) ∈ 𝐾) → ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )) ∈ 𝐾)
232	230, 214, 213, 231	syl3anc 1318	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) ∧ 𝑒 ∈ (𝑁 × 𝑁)) → ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )) ∈ 𝐾)
233	134	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) ∧ 𝑒 ∈ (𝑁 × 𝑁)) → 0 ∈ 𝐾)
234	232, 233	ifcld 4081	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)
235	234, 214	ifcld 4081	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) ∧ 𝑒 ∈ (𝑁 × 𝑁)) → if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒)) ∈ 𝐾)
236		eqid 2610	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) = (𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒)))
237	235, 236	fmptd 6292	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) → (𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))):(𝑁 × 𝑁)⟶𝐾)
238		elmapg 7757	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((𝐾 ∈ V ∧ (𝑁 × 𝑁) ∈ Fin) → ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ∈ (𝐾 ↑𝑚 (𝑁 × 𝑁)) ↔ (𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))):(𝑁 × 𝑁)⟶𝐾))
239	219, 221, 238	sylancr 694	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) → ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ∈ (𝐾 ↑𝑚 (𝑁 × 𝑁)) ↔ (𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))):(𝑁 × 𝑁)⟶𝐾))
240	237, 239	mpbird 246	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) → (𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ∈ (𝐾 ↑𝑚 (𝑁 × 𝑁)))
241	240, 227	eleqtrd 2690	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) → (𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ∈ 𝐵)
242	56	3ad2ant1 1075	. . . . . . . . . . . . . . . . . . . . . . . . . . 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))))
243		reseq1 5311	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝑥 = 𝑎 → (𝑥 ↾ ({𝑤} × 𝑁)) = (𝑎 ↾ ({𝑤} × 𝑁)))
244	243	eqeq1d 2612	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝑥 = 𝑎 → ((𝑥 ↾ ({𝑤} × 𝑁)) = ((𝑦 ↾ ({𝑤} × 𝑁)) ∘𝑓 + (𝑧 ↾ ({𝑤} × 𝑁))) ↔ (𝑎 ↾ ({𝑤} × 𝑁)) = ((𝑦 ↾ ({𝑤} × 𝑁)) ∘𝑓 + (𝑧 ↾ ({𝑤} × 𝑁)))))
245		reseq1 5311	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝑥 = 𝑎 → (𝑥 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑎 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)))
246	245	eqeq1d 2612	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝑥 = 𝑎 → ((𝑥 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑦 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) ↔ (𝑎 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑦 ↾ ((𝑁 ∖ {𝑤}) × 𝑁))))
247	245	eqeq1d 2612	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝑥 = 𝑎 → ((𝑥 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑧 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) ↔ (𝑎 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑧 ↾ ((𝑁 ∖ {𝑤}) × 𝑁))))
248	244, 246, 247	3anbi123d 1391	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑥 = 𝑎 → (((𝑥 ↾ ({𝑤} × 𝑁)) = ((𝑦 ↾ ({𝑤} × 𝑁)) ∘𝑓 + (𝑧 ↾ ({𝑤} × 𝑁))) ∧ (𝑥 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑦 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) ∧ (𝑥 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑧 ↾ ((𝑁 ∖ {𝑤}) × 𝑁))) ↔ ((𝑎 ↾ ({𝑤} × 𝑁)) = ((𝑦 ↾ ({𝑤} × 𝑁)) ∘𝑓 + (𝑧 ↾ ({𝑤} × 𝑁))) ∧ (𝑎 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑦 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) ∧ (𝑎 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑧 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)))))
249		fveq2 6103	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝑥 = 𝑎 → (𝐷‘𝑥) = (𝐷‘𝑎))
250	249	eqeq1d 2612	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑥 = 𝑎 → ((𝐷‘𝑥) = ((𝐷‘𝑦) + (𝐷‘𝑧)) ↔ (𝐷‘𝑎) = ((𝐷‘𝑦) + (𝐷‘𝑧))))
251	248, 250	imbi12d 333	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)))))
252	251	2ralbidv 2972	. . . . . . . . . . . . . . . . . . . . . . . . . . . 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)))))
253		reseq1 5311	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (𝑦 = (𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) → (𝑦 ↾ ({𝑤} × 𝑁)) = ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ↾ ({𝑤} × 𝑁)))
254	253	oveq1d 6564	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝑦 = (𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) → ((𝑦 ↾ ({𝑤} × 𝑁)) ∘𝑓 + (𝑧 ↾ ({𝑤} × 𝑁))) = (((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ↾ ({𝑤} × 𝑁)) ∘𝑓 + (𝑧 ↾ ({𝑤} × 𝑁))))
255	254	eqeq2d 2620	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝑦 = (𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) → ((𝑎 ↾ ({𝑤} × 𝑁)) = ((𝑦 ↾ ({𝑤} × 𝑁)) ∘𝑓 + (𝑧 ↾ ({𝑤} × 𝑁))) ↔ (𝑎 ↾ ({𝑤} × 𝑁)) = (((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ↾ ({𝑤} × 𝑁)) ∘𝑓 + (𝑧 ↾ ({𝑤} × 𝑁)))))
256		reseq1 5311	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝑦 = (𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) → (𝑦 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ↾ ((𝑁 ∖ {𝑤}) × 𝑁)))
257	256	eqeq2d 2620	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝑦 = (𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) → ((𝑎 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑦 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) ↔ (𝑎 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ↾ ((𝑁 ∖ {𝑤}) × 𝑁))))
258	255, 257	3anbi12d 1392	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑦 = (𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) → (((𝑎 ↾ ({𝑤} × 𝑁)) = ((𝑦 ↾ ({𝑤} × 𝑁)) ∘𝑓 + (𝑧 ↾ ({𝑤} × 𝑁))) ∧ (𝑎 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑦 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) ∧ (𝑎 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑧 ↾ ((𝑁 ∖ {𝑤}) × 𝑁))) ↔ ((𝑎 ↾ ({𝑤} × 𝑁)) = (((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ↾ ({𝑤} × 𝑁)) ∘𝑓 + (𝑧 ↾ ({𝑤} × 𝑁))) ∧ (𝑎 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) ∧ (𝑎 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑧 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)))))
259		fveq2 6103	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝑦 = (𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) → (𝐷‘𝑦) = (𝐷‘(𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒)))))
260	259	oveq1d 6564	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝑦 = (𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) → ((𝐷‘𝑦) + (𝐷‘𝑧)) = ((𝐷‘(𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒)))) + (𝐷‘𝑧)))
261	260	eqeq2d 2620	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑦 = (𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) → ((𝐷‘𝑎) = ((𝐷‘𝑦) + (𝐷‘𝑧)) ↔ (𝐷‘𝑎) = ((𝐷‘(𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒)))) + (𝐷‘𝑧))))
262	258, 261	imbi12d 333	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑦 = (𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) → ((((𝑎 ↾ ({𝑤} × 𝑁)) = ((𝑦 ↾ ({𝑤} × 𝑁)) ∘𝑓 + (𝑧 ↾ ({𝑤} × 𝑁))) ∧ (𝑎 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑦 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) ∧ (𝑎 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑧 ↾ ((𝑁 ∖ {𝑤}) × 𝑁))) → (𝐷‘𝑎) = ((𝐷‘𝑦) + (𝐷‘𝑧))) ↔ (((𝑎 ↾ ({𝑤} × 𝑁)) = (((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ↾ ({𝑤} × 𝑁)) ∘𝑓 + (𝑧 ↾ ({𝑤} × 𝑁))) ∧ (𝑎 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) ∧ (𝑎 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑧 ↾ ((𝑁 ∖ {𝑤}) × 𝑁))) → (𝐷‘𝑎) = ((𝐷‘(𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒)))) + (𝐷‘𝑧)))))
263	262	2ralbidv 2972	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑦 = (𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) → (∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝑁 (((𝑎 ↾ ({𝑤} × 𝑁)) = ((𝑦 ↾ ({𝑤} × 𝑁)) ∘𝑓 + (𝑧 ↾ ({𝑤} × 𝑁))) ∧ (𝑎 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑦 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) ∧ (𝑎 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑧 ↾ ((𝑁 ∖ {𝑤}) × 𝑁))) → (𝐷‘𝑎) = ((𝐷‘𝑦) + (𝐷‘𝑧))) ↔ ∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝑁 (((𝑎 ↾ ({𝑤} × 𝑁)) = (((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ↾ ({𝑤} × 𝑁)) ∘𝑓 + (𝑧 ↾ ({𝑤} × 𝑁))) ∧ (𝑎 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) ∧ (𝑎 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑧 ↾ ((𝑁 ∖ {𝑤}) × 𝑁))) → (𝐷‘𝑎) = ((𝐷‘(𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒)))) + (𝐷‘𝑧)))))
264	252, 263	rspc2va 3294	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((𝑎 ∈ 𝐵 ∧ (𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ∈ 𝐵) ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝑁 (((𝑥 ↾ ({𝑤} × 𝑁)) = ((𝑦 ↾ ({𝑤} × 𝑁)) ∘𝑓 + (𝑧 ↾ ({𝑤} × 𝑁))) ∧ (𝑥 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑦 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) ∧ (𝑥 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑧 ↾ ((𝑁 ∖ {𝑤}) × 𝑁))) → (𝐷‘𝑥) = ((𝐷‘𝑦) + (𝐷‘𝑧)))) → ∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝑁 (((𝑎 ↾ ({𝑤} × 𝑁)) = (((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ↾ ({𝑤} × 𝑁)) ∘𝑓 + (𝑧 ↾ ({𝑤} × 𝑁))) ∧ (𝑎 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) ∧ (𝑎 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑧 ↾ ((𝑁 ∖ {𝑤}) × 𝑁))) → (𝐷‘𝑎) = ((𝐷‘(𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒)))) + (𝐷‘𝑧))))
265	229, 241, 242, 264	syl21anc 1317	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) → ∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝑁 (((𝑎 ↾ ({𝑤} × 𝑁)) = (((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ↾ ({𝑤} × 𝑁)) ∘𝑓 + (𝑧 ↾ ({𝑤} × 𝑁))) ∧ (𝑎 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) ∧ (𝑎 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑧 ↾ ((𝑁 ∖ {𝑤}) × 𝑁))) → (𝐷‘𝑎) = ((𝐷‘(𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒)))) + (𝐷‘𝑧))))
266		reseq1 5311	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)))
267	266	oveq2d 6565	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑧 = (𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒 ∈ 𝑑, 1 , 0 ), (𝑎‘𝑒))) → (((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ↾ ({𝑤} × 𝑁)) ∘𝑓 + (𝑧 ↾ ({𝑤} × 𝑁))) = (((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ↾ ({𝑤} × 𝑁)) ∘𝑓 + ((𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒 ∈ 𝑑, 1 , 0 ), (𝑎‘𝑒))) ↾ ({𝑤} × 𝑁))))
268	267	eqeq2d 2620	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑧 = (𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒 ∈ 𝑑, 1 , 0 ), (𝑎‘𝑒))) → ((𝑎 ↾ ({𝑤} × 𝑁)) = (((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ↾ ({𝑤} × 𝑁)) ∘𝑓 + (𝑧 ↾ ({𝑤} × 𝑁))) ↔ (𝑎 ↾ ({𝑤} × 𝑁)) = (((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ↾ ({𝑤} × 𝑁)) ∘𝑓 + ((𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒 ∈ 𝑑, 1 , 0 ), (𝑎‘𝑒))) ↾ ({𝑤} × 𝑁)))))
269		reseq1 5311	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)))
270	269	eqeq2d 2620	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 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))))
271	268, 270	3anbi13d 1393	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑧 = (𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒 ∈ 𝑑, 1 , 0 ), (𝑎‘𝑒))) → (((𝑎 ↾ ({𝑤} × 𝑁)) = (((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ↾ ({𝑤} × 𝑁)) ∘𝑓 + (𝑧 ↾ ({𝑤} × 𝑁))) ∧ (𝑎 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) ∧ (𝑎 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑧 ↾ ((𝑁 ∖ {𝑤}) × 𝑁))) ↔ ((𝑎 ↾ ({𝑤} × 𝑁)) = (((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ↾ ({𝑤} × 𝑁)) ∘𝑓 + ((𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒 ∈ 𝑑, 1 , 0 ), (𝑎‘𝑒))) ↾ ({𝑤} × 𝑁))) ∧ (𝑎 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) ∧ (𝑎 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = ((𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒 ∈ 𝑑, 1 , 0 ), (𝑎‘𝑒))) ↾ ((𝑁 ∖ {𝑤}) × 𝑁)))))
272		fveq2 6103	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)))))
273	272	oveq2d 6565	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑧 = (𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒 ∈ 𝑑, 1 , 0 ), (𝑎‘𝑒))) → ((𝐷‘(𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒)))) + (𝐷‘𝑧)) = ((𝐷‘(𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒)))) + (𝐷‘(𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒 ∈ 𝑑, 1 , 0 ), (𝑎‘𝑒))))))
274	273	eqeq2d 2620	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑧 = (𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒 ∈ 𝑑, 1 , 0 ), (𝑎‘𝑒))) → ((𝐷‘𝑎) = ((𝐷‘(𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒)))) + (𝐷‘𝑧)) ↔ (𝐷‘𝑎) = ((𝐷‘(𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒)))) + (𝐷‘(𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒 ∈ 𝑑, 1 , 0 ), (𝑎‘𝑒)))))))
275	271, 274	imbi12d 333	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑧 = (𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒 ∈ 𝑑, 1 , 0 ), (𝑎‘𝑒))) → ((((𝑎 ↾ ({𝑤} × 𝑁)) = (((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ↾ ({𝑤} × 𝑁)) ∘𝑓 + (𝑧 ↾ ({𝑤} × 𝑁))) ∧ (𝑎 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) ∧ (𝑎 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑧 ↾ ((𝑁 ∖ {𝑤}) × 𝑁))) → (𝐷‘𝑎) = ((𝐷‘(𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒)))) + (𝐷‘𝑧))) ↔ (((𝑎 ↾ ({𝑤} × 𝑁)) = (((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ↾ ({𝑤} × 𝑁)) ∘𝑓 + ((𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒 ∈ 𝑑, 1 , 0 ), (𝑎‘𝑒))) ↾ ({𝑤} × 𝑁))) ∧ (𝑎 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) ∧ (𝑎 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = ((𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒 ∈ 𝑑, 1 , 0 ), (𝑎‘𝑒))) ↾ ((𝑁 ∖ {𝑤}) × 𝑁))) → (𝐷‘𝑎) = ((𝐷‘(𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒)))) + (𝐷‘(𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒 ∈ 𝑑, 1 , 0 ), (𝑎‘𝑒))))))))
276		sneq 4135	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝑤 = (1st ‘𝑐) → {𝑤} = {(1st ‘𝑐)})
277	276	xpeq1d 5062	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝑤 = (1st ‘𝑐) → ({𝑤} × 𝑁) = ({(1st ‘𝑐)} × 𝑁))
278	277	reseq2d 5317	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑤 = (1st ‘𝑐) → (𝑎 ↾ ({𝑤} × 𝑁)) = (𝑎 ↾ ({(1st ‘𝑐)} × 𝑁)))
279	277	reseq2d 5317	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝑤 = (1st ‘𝑐) → ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ↾ ({𝑤} × 𝑁)) = ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ↾ ({(1st ‘𝑐)} × 𝑁)))
280	277	reseq2d 5317	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝑤 = (1st ‘𝑐) → ((𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒 ∈ 𝑑, 1 , 0 ), (𝑎‘𝑒))) ↾ ({𝑤} × 𝑁)) = ((𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒 ∈ 𝑑, 1 , 0 ), (𝑎‘𝑒))) ↾ ({(1st ‘𝑐)} × 𝑁)))
281	279, 280	oveq12d 6567	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑤 = (1st ‘𝑐) → (((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ↾ ({𝑤} × 𝑁)) ∘𝑓 + ((𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒 ∈ 𝑑, 1 , 0 ), (𝑎‘𝑒))) ↾ ({𝑤} × 𝑁))) = (((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ↾ ({(1st ‘𝑐)} × 𝑁)) ∘𝑓 + ((𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒 ∈ 𝑑, 1 , 0 ), (𝑎‘𝑒))) ↾ ({(1st ‘𝑐)} × 𝑁))))
282	278, 281	eqeq12d 2625	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑤 = (1st ‘𝑐) → ((𝑎 ↾ ({𝑤} × 𝑁)) = (((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ↾ ({𝑤} × 𝑁)) ∘𝑓 + ((𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒 ∈ 𝑑, 1 , 0 ), (𝑎‘𝑒))) ↾ ({𝑤} × 𝑁))) ↔ (𝑎 ↾ ({(1st ‘𝑐)} × 𝑁)) = (((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ↾ ({(1st ‘𝑐)} × 𝑁)) ∘𝑓 + ((𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒 ∈ 𝑑, 1 , 0 ), (𝑎‘𝑒))) ↾ ({(1st ‘𝑐)} × 𝑁)))))
283	276	difeq2d 3690	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝑤 = (1st ‘𝑐) → (𝑁 ∖ {𝑤}) = (𝑁 ∖ {(1st ‘𝑐)}))
284	283	xpeq1d 5062	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝑤 = (1st ‘𝑐) → ((𝑁 ∖ {𝑤}) × 𝑁) = ((𝑁 ∖ {(1st ‘𝑐)}) × 𝑁))
285	284	reseq2d 5317	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑤 = (1st ‘𝑐) → (𝑎 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑎 ↾ ((𝑁 ∖ {(1st ‘𝑐)}) × 𝑁)))
286	284	reseq2d 5317	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑤 = (1st ‘𝑐) → ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ↾ ((𝑁 ∖ {(1st ‘𝑐)}) × 𝑁)))
287	285, 286	eqeq12d 2625	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑤 = (1st ‘𝑐) → ((𝑎 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) ↔ (𝑎 ↾ ((𝑁 ∖ {(1st ‘𝑐)}) × 𝑁)) = ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ↾ ((𝑁 ∖ {(1st ‘𝑐)}) × 𝑁))))
288	284	reseq2d 5317	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑤 = (1st ‘𝑐) → ((𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒 ∈ 𝑑, 1 , 0 ), (𝑎‘𝑒))) ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = ((𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒 ∈ 𝑑, 1 , 0 ), (𝑎‘𝑒))) ↾ ((𝑁 ∖ {(1st ‘𝑐)}) × 𝑁)))
289	285, 288	eqeq12d 2625	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑤 = (1st ‘𝑐) → ((𝑎 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = ((𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒 ∈ 𝑑, 1 , 0 ), (𝑎‘𝑒))) ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) ↔ (𝑎 ↾ ((𝑁 ∖ {(1st ‘𝑐)}) × 𝑁)) = ((𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒 ∈ 𝑑, 1 , 0 ), (𝑎‘𝑒))) ↾ ((𝑁 ∖ {(1st ‘𝑐)}) × 𝑁))))
290	282, 287, 289	3anbi123d 1391	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑤 = (1st ‘𝑐) → (((𝑎 ↾ ({𝑤} × 𝑁)) = (((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ↾ ({𝑤} × 𝑁)) ∘𝑓 + ((𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒 ∈ 𝑑, 1 , 0 ), (𝑎‘𝑒))) ↾ ({𝑤} × 𝑁))) ∧ (𝑎 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) ∧ (𝑎 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = ((𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒 ∈ 𝑑, 1 , 0 ), (𝑎‘𝑒))) ↾ ((𝑁 ∖ {𝑤}) × 𝑁))) ↔ ((𝑎 ↾ ({(1st ‘𝑐)} × 𝑁)) = (((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ↾ ({(1st ‘𝑐)} × 𝑁)) ∘𝑓 + ((𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒 ∈ 𝑑, 1 , 0 ), (𝑎‘𝑒))) ↾ ({(1st ‘𝑐)} × 𝑁))) ∧ (𝑎 ↾ ((𝑁 ∖ {(1st ‘𝑐)}) × 𝑁)) = ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ↾ ((𝑁 ∖ {(1st ‘𝑐)}) × 𝑁)) ∧ (𝑎 ↾ ((𝑁 ∖ {(1st ‘𝑐)}) × 𝑁)) = ((𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒 ∈ 𝑑, 1 , 0 ), (𝑎‘𝑒))) ↾ ((𝑁 ∖ {(1st ‘𝑐)}) × 𝑁)))))
291	290	imbi1d 330	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑤 = (1st ‘𝑐) → ((((𝑎 ↾ ({𝑤} × 𝑁)) = (((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ↾ ({𝑤} × 𝑁)) ∘𝑓 + ((𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒 ∈ 𝑑, 1 , 0 ), (𝑎‘𝑒))) ↾ ({𝑤} × 𝑁))) ∧ (𝑎 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) ∧ (𝑎 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = ((𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒 ∈ 𝑑, 1 , 0 ), (𝑎‘𝑒))) ↾ ((𝑁 ∖ {𝑤}) × 𝑁))) → (𝐷‘𝑎) = ((𝐷‘(𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒)))) + (𝐷‘(𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒 ∈ 𝑑, 1 , 0 ), (𝑎‘𝑒)))))) ↔ (((𝑎 ↾ ({(1st ‘𝑐)} × 𝑁)) = (((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ↾ ({(1st ‘𝑐)} × 𝑁)) ∘𝑓 + ((𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒 ∈ 𝑑, 1 , 0 ), (𝑎‘𝑒))) ↾ ({(1st ‘𝑐)} × 𝑁))) ∧ (𝑎 ↾ ((𝑁 ∖ {(1st ‘𝑐)}) × 𝑁)) = ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ↾ ((𝑁 ∖ {(1st ‘𝑐)}) × 𝑁)) ∧ (𝑎 ↾ ((𝑁 ∖ {(1st ‘𝑐)}) × 𝑁)) = ((𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒 ∈ 𝑑, 1 , 0 ), (𝑎‘𝑒))) ↾ ((𝑁 ∖ {(1st ‘𝑐)}) × 𝑁))) → (𝐷‘𝑎) = ((𝐷‘(𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒)))) + (𝐷‘(𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒 ∈ 𝑑, 1 , 0 ), (𝑎‘𝑒))))))))
292	275, 291	rspc2va 3294	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((((𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒 ∈ 𝑑, 1 , 0 ), (𝑎‘𝑒))) ∈ 𝐵 ∧ (1st ‘𝑐) ∈ 𝑁) ∧ ∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝑁 (((𝑎 ↾ ({𝑤} × 𝑁)) = (((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ↾ ({𝑤} × 𝑁)) ∘𝑓 + (𝑧 ↾ ({𝑤} × 𝑁))) ∧ (𝑎 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) ∧ (𝑎 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑧 ↾ ((𝑁 ∖ {𝑤}) × 𝑁))) → (𝐷‘𝑎) = ((𝐷‘(𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒)))) + (𝐷‘𝑧)))) → (((𝑎 ↾ ({(1st ‘𝑐)} × 𝑁)) = (((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ↾ ({(1st ‘𝑐)} × 𝑁)) ∘𝑓 + ((𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒 ∈ 𝑑, 1 , 0 ), (𝑎‘𝑒))) ↾ ({(1st ‘𝑐)} × 𝑁))) ∧ (𝑎 ↾ ((𝑁 ∖ {(1st ‘𝑐)}) × 𝑁)) = ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ↾ ((𝑁 ∖ {(1st ‘𝑐)}) × 𝑁)) ∧ (𝑎 ↾ ((𝑁 ∖ {(1st ‘𝑐)}) × 𝑁)) = ((𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒 ∈ 𝑑, 1 , 0 ), (𝑎‘𝑒))) ↾ ((𝑁 ∖ {(1st ‘𝑐)}) × 𝑁))) → (𝐷‘𝑎) = ((𝐷‘(𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒)))) + (𝐷‘(𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒 ∈ 𝑑, 1 , 0 ), (𝑎‘𝑒)))))))
293	228, 125, 265, 292	syl21anc 1317	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) → (((𝑎 ↾ ({(1st ‘𝑐)} × 𝑁)) = (((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ↾ ({(1st ‘𝑐)} × 𝑁)) ∘𝑓 + ((𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒 ∈ 𝑑, 1 , 0 ), (𝑎‘𝑒))) ↾ ({(1st ‘𝑐)} × 𝑁))) ∧ (𝑎 ↾ ((𝑁 ∖ {(1st ‘𝑐)}) × 𝑁)) = ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ↾ ((𝑁 ∖ {(1st ‘𝑐)}) × 𝑁)) ∧ (𝑎 ↾ ((𝑁 ∖ {(1st ‘𝑐)}) × 𝑁)) = ((𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒 ∈ 𝑑, 1 , 0 ), (𝑎‘𝑒))) ↾ ((𝑁 ∖ {(1st ‘𝑐)}) × 𝑁))) → (𝐷‘𝑎) = ((𝐷‘(𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒)))) + (𝐷‘(𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒 ∈ 𝑑, 1 , 0 ), (𝑎‘𝑒)))))))
294	187, 204, 212, 293	mp3and 1419	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) → (𝐷‘𝑎) = ((𝐷‘(𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒)))) + (𝐷‘(𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒 ∈ 𝑑, 1 , 0 ), (𝑎‘𝑒))))))
295	105, 106, 107, 294	syl3anc 1318	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ (𝑏 ∪ {𝑐}) ∈ 𝑌) ∧ ((𝑎 ∈ 𝐵 ∧ 𝑑 ∈ (𝑁 ↑𝑚 𝑁)) ∧ ∀𝑤 ∈ 𝑏 (𝑎‘𝑤) = if(𝑤 ∈ 𝑑, 1 , 0 ))) → (𝐷‘𝑎) = ((𝐷‘(𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒)))) + (𝐷‘(𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒 ∈ 𝑑, 1 , 0 ), (𝑎‘𝑒))))))
296		fveq2 6103	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (𝑒 = 𝑐 → (𝑎‘𝑒) = (𝑎‘𝑐))
297		elequ1 1984	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ (𝑒 = 𝑐 → (𝑒 ∈ 𝑑 ↔ 𝑐 ∈ 𝑑))
298	297	ifbid 4058	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (𝑒 = 𝑐 → if(𝑒 ∈ 𝑑, 1 , 0 ) = if(𝑐 ∈ 𝑑, 1 , 0 ))
299	296, 298	oveq12d 6567	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (𝑒 = 𝑐 → ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )) = ((𝑎‘𝑐)(-g‘𝑅)if(𝑐 ∈ 𝑑, 1 , 0 )))
300	299	adantl 481	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) ∧ 𝑒 ∈ ({(1st ‘𝑐)} × 𝑁)) ∧ 𝑒 = 𝑐) → ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )) = ((𝑎‘𝑐)(-g‘𝑅)if(𝑐 ∈ 𝑑, 1 , 0 )))
301	111, 123	ffvelrnd 6268	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) → (𝑎‘𝑐) ∈ 𝐾)
302	132, 134	ifcld 4081	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) → if(𝑐 ∈ 𝑑, 1 , 0 ) ∈ 𝐾)
303	12, 137	grpsubcl 17318	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ ((𝑅 ∈ Grp ∧ (𝑎‘𝑐) ∈ 𝐾 ∧ if(𝑐 ∈ 𝑑, 1 , 0 ) ∈ 𝐾) → ((𝑎‘𝑐)(-g‘𝑅)if(𝑐 ∈ 𝑑, 1 , 0 )) ∈ 𝐾)
304	116, 301, 302, 303	syl3anc 1318	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) → ((𝑎‘𝑐)(-g‘𝑅)if(𝑐 ∈ 𝑑, 1 , 0 )) ∈ 𝐾)
305	12, 52, 50	ringridm 18395	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 )))
306	114, 304, 305	syl2anc 691	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 )))
307	306	ad2antrr 758	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) ∧ 𝑒 ∈ ({(1st ‘𝑐)} × 𝑁)) ∧ 𝑒 = 𝑐) → (((𝑎‘𝑐)(-g‘𝑅)if(𝑐 ∈ 𝑑, 1 , 0 )) · 1 ) = ((𝑎‘𝑐)(-g‘𝑅)if(𝑐 ∈ 𝑑, 1 , 0 )))
308	300, 307	eqtr4d 2647	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) ∧ 𝑒 ∈ ({(1st ‘𝑐)} × 𝑁)) ∧ 𝑒 = 𝑐) → ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )) = (((𝑎‘𝑐)(-g‘𝑅)if(𝑐 ∈ 𝑑, 1 , 0 )) · 1 ))
309	142	adantl 481	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) ∧ 𝑒 ∈ ({(1st ‘𝑐)} × 𝑁)) ∧ 𝑒 = 𝑐) → if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ) = ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )))
310		iftrue 4042	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (𝑒 = 𝑐 → if(𝑒 = 𝑐, 1 , 0 ) = 1 )
311	310	oveq2d 6565	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 ))
312	311	adantl 481	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) ∧ 𝑒 ∈ ({(1st ‘𝑐)} × 𝑁)) ∧ 𝑒 = 𝑐) → (((𝑎‘𝑐)(-g‘𝑅)if(𝑐 ∈ 𝑑, 1 , 0 )) · if(𝑒 = 𝑐, 1 , 0 )) = (((𝑎‘𝑐)(-g‘𝑅)if(𝑐 ∈ 𝑑, 1 , 0 )) · 1 ))
313	308, 309, 312	3eqtr4d 2654	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) ∧ 𝑒 ∈ ({(1st ‘𝑐)} × 𝑁)) ∧ 𝑒 = 𝑐) → if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ) = (((𝑎‘𝑐)(-g‘𝑅)if(𝑐 ∈ 𝑑, 1 , 0 )) · if(𝑒 = 𝑐, 1 , 0 )))
314	12, 52, 49	ringrz 18411	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ ((𝑅 ∈ Ring ∧ ((𝑎‘𝑐)(-g‘𝑅)if(𝑐 ∈ 𝑑, 1 , 0 )) ∈ 𝐾) → (((𝑎‘𝑐)(-g‘𝑅)if(𝑐 ∈ 𝑑, 1 , 0 )) · 0 ) = 0 )
315	114, 304, 314	syl2anc 691	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) → (((𝑎‘𝑐)(-g‘𝑅)if(𝑐 ∈ 𝑑, 1 , 0 )) · 0 ) = 0 )
316	315	eqcomd 2616	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) → 0 = (((𝑎‘𝑐)(-g‘𝑅)if(𝑐 ∈ 𝑑, 1 , 0 )) · 0 ))
317	316	ad2antrr 758	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) ∧ 𝑒 ∈ ({(1st ‘𝑐)} × 𝑁)) ∧ ¬ 𝑒 = 𝑐) → 0 = (((𝑎‘𝑐)(-g‘𝑅)if(𝑐 ∈ 𝑑, 1 , 0 )) · 0 ))
318	151	adantl 481	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) ∧ 𝑒 ∈ ({(1st ‘𝑐)} × 𝑁)) ∧ ¬ 𝑒 = 𝑐) → if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ) = 0 )
319		iffalse 4045	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (¬ 𝑒 = 𝑐 → if(𝑒 = 𝑐, 1 , 0 ) = 0 )
320	319	oveq2d 6565	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 ))
321	320	adantl 481	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) ∧ 𝑒 ∈ ({(1st ‘𝑐)} × 𝑁)) ∧ ¬ 𝑒 = 𝑐) → (((𝑎‘𝑐)(-g‘𝑅)if(𝑐 ∈ 𝑑, 1 , 0 )) · if(𝑒 = 𝑐, 1 , 0 )) = (((𝑎‘𝑐)(-g‘𝑅)if(𝑐 ∈ 𝑑, 1 , 0 )) · 0 ))
322	317, 318, 321	3eqtr4d 2654	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) ∧ 𝑒 ∈ ({(1st ‘𝑐)} × 𝑁)) ∧ ¬ 𝑒 = 𝑐) → if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ) = (((𝑎‘𝑐)(-g‘𝑅)if(𝑐 ∈ 𝑑, 1 , 0 )) · if(𝑒 = 𝑐, 1 , 0 )))
323	313, 322	pm2.61dan 828	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) ∧ 𝑒 ∈ ({(1st ‘𝑐)} × 𝑁)) → if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ) = (((𝑎‘𝑐)(-g‘𝑅)if(𝑐 ∈ 𝑑, 1 , 0 )) · if(𝑒 = 𝑐, 1 , 0 )))
324	173	adantl 481	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) ∧ 𝑒 ∈ ({(1st ‘𝑐)} × 𝑁)) → (1st ‘𝑒) ∈ {(1st ‘𝑐)})
325	324, 174	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) ∧ 𝑒 ∈ ({(1st ‘𝑐)} × 𝑁)) → (1st ‘𝑒) = (1st ‘𝑐))
326	325	iftrued 4044	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) ∧ 𝑒 ∈ ({(1st ‘𝑐)} × 𝑁)) → if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒)) = if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ))
327	325	iftrued 4044	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) ∧ 𝑒 ∈ ({(1st ‘𝑐)} × 𝑁)) → if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒)) = if(𝑒 = 𝑐, 1 , 0 ))
328	327	oveq2d 6565	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) ∧ 𝑒 ∈ ({(1st ‘𝑐)} × 𝑁)) → (((𝑎‘𝑐)(-g‘𝑅)if(𝑐 ∈ 𝑑, 1 , 0 )) · if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒))) = (((𝑎‘𝑐)(-g‘𝑅)if(𝑐 ∈ 𝑑, 1 , 0 )) · if(𝑒 = 𝑐, 1 , 0 )))
329	323, 326, 328	3eqtr4d 2654	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) ∧ 𝑒 ∈ ({(1st ‘𝑐)} × 𝑁)) → if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒)) = (((𝑎‘𝑐)(-g‘𝑅)if(𝑐 ∈ 𝑑, 1 , 0 )) · if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒))))
330	329	mpteq2dva 4672	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) → (𝑒 ∈ ({(1st ‘𝑐)} × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) = (𝑒 ∈ ({(1st ‘𝑐)} × 𝑁) ↦ (((𝑎‘𝑐)(-g‘𝑅)if(𝑐 ∈ 𝑑, 1 , 0 )) · if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒)))))
331		ovex 6577	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((𝑎‘𝑐)(-g‘𝑅)if(𝑐 ∈ 𝑑, 1 , 0 )) ∈ V
332	331	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) ∧ 𝑒 ∈ ({(1st ‘𝑐)} × 𝑁)) → ((𝑎‘𝑐)(-g‘𝑅)if(𝑐 ∈ 𝑑, 1 , 0 )) ∈ V)
333	168, 164	ifex 4106	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ if(𝑒 = 𝑐, 1 , 0 ) ∈ V
334	333, 170	ifex 4106	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒)) ∈ V
335	334	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) ∧ 𝑒 ∈ ({(1st ‘𝑐)} × 𝑁)) → if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒)) ∈ V)
336		fconstmpt 5085	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (({(1st ‘𝑐)} × 𝑁) × {((𝑎‘𝑐)(-g‘𝑅)if(𝑐 ∈ 𝑑, 1 , 0 ))}) = (𝑒 ∈ ({(1st ‘𝑐)} × 𝑁) ↦ ((𝑎‘𝑐)(-g‘𝑅)if(𝑐 ∈ 𝑑, 1 , 0 )))
337	336	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) → (({(1st ‘𝑐)} × 𝑁) × {((𝑎‘𝑐)(-g‘𝑅)if(𝑐 ∈ 𝑑, 1 , 0 ))}) = (𝑒 ∈ ({(1st ‘𝑐)} × 𝑁) ↦ ((𝑎‘𝑐)(-g‘𝑅)if(𝑐 ∈ 𝑑, 1 , 0 ))))
338	128	resmptd 5371	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) → ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒))) ↾ ({(1st ‘𝑐)} × 𝑁)) = (𝑒 ∈ ({(1st ‘𝑐)} × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒))))
339	161, 332, 335, 337, 338	offval2 6812	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) → ((({(1st ‘𝑐)} × 𝑁) × {((𝑎‘𝑐)(-g‘𝑅)if(𝑐 ∈ 𝑑, 1 , 0 ))}) ∘𝑓 · ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒))) ↾ ({(1st ‘𝑐)} × 𝑁))) = (𝑒 ∈ ({(1st ‘𝑐)} × 𝑁) ↦ (((𝑎‘𝑐)(-g‘𝑅)if(𝑐 ∈ 𝑑, 1 , 0 )) · if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒)))))
340	330, 183, 339	3eqtr4d 2654	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) → ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ↾ ({(1st ‘𝑐)} × 𝑁)) = ((({(1st ‘𝑐)} × 𝑁) × {((𝑎‘𝑐)(-g‘𝑅)if(𝑐 ∈ 𝑑, 1 , 0 ))}) ∘𝑓 · ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒))) ↾ ({(1st ‘𝑐)} × 𝑁))))
341		iffalse 4045	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (¬ (1st ‘𝑒) = (1st ‘𝑐) → if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒)) = (𝑎‘𝑒))
342		iffalse 4045	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (¬ (1st ‘𝑒) = (1st ‘𝑐) → if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒)) = (𝑎‘𝑒))
343	341, 342	eqtr4d 2647	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (¬ (1st ‘𝑒) = (1st ‘𝑐) → if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒)) = if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒)))
344	193, 343	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) ∧ 𝑒 ∈ ((𝑁 ∖ {(1st ‘𝑐)}) × 𝑁)) → if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒)) = if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒)))
345	344	mpteq2dva 4672	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) → (𝑒 ∈ ((𝑁 ∖ {(1st ‘𝑐)}) × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) = (𝑒 ∈ ((𝑁 ∖ {(1st ‘𝑐)}) × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒))))
346		resmpt 5369	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (((𝑁 ∖ {(1st ‘𝑐)}) × 𝑁) ⊆ (𝑁 × 𝑁) → ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒))) ↾ ((𝑁 ∖ {(1st ‘𝑐)}) × 𝑁)) = (𝑒 ∈ ((𝑁 ∖ {(1st ‘𝑐)}) × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒))))
347	198, 346	mp1i 13	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) → ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒))) ↾ ((𝑁 ∖ {(1st ‘𝑐)}) × 𝑁)) = (𝑒 ∈ ((𝑁 ∖ {(1st ‘𝑐)}) × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒))))
348	345, 200, 347	3eqtr4d 2654	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) → ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ↾ ((𝑁 ∖ {(1st ‘𝑐)}) × 𝑁)) = ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒))) ↾ ((𝑁 ∖ {(1st ‘𝑐)}) × 𝑁)))
349	132, 134	ifcld 4081	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) → if(𝑒 = 𝑐, 1 , 0 ) ∈ 𝐾)
350	349	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) ∧ 𝑒 ∈ (𝑁 × 𝑁)) → if(𝑒 = 𝑐, 1 , 0 ) ∈ 𝐾)
351	350, 214	ifcld 4081	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) ∧ 𝑒 ∈ (𝑁 × 𝑁)) → if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒)) ∈ 𝐾)
352		eqid 2610	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒))) = (𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒)))
353	351, 352	fmptd 6292	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) → (𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒))):(𝑁 × 𝑁)⟶𝐾)
354		elmapg 7757	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((𝐾 ∈ V ∧ (𝑁 × 𝑁) ∈ Fin) → ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒))) ∈ (𝐾 ↑𝑚 (𝑁 × 𝑁)) ↔ (𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒))):(𝑁 × 𝑁)⟶𝐾))
355	219, 221, 354	sylancr 694	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) → ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒))) ∈ (𝐾 ↑𝑚 (𝑁 × 𝑁)) ↔ (𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒))):(𝑁 × 𝑁)⟶𝐾))
356	353, 355	mpbird 246	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) → (𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒))) ∈ (𝐾 ↑𝑚 (𝑁 × 𝑁)))
357	356, 227	eleqtrd 2690	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) → (𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒))) ∈ 𝐵)
358	57	3ad2ant1 1075	. . . . . . . . . . . . . . . . . . . . . . . . . . . 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))))
359		reseq1 5311	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (𝑥 = (𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) → (𝑥 ↾ ({𝑤} × 𝑁)) = ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ↾ ({𝑤} × 𝑁)))
360	359	eqeq1d 2612	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝑥 = (𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) → ((𝑥 ↾ ({𝑤} × 𝑁)) = ((({𝑤} × 𝑁) × {𝑦}) ∘𝑓 · (𝑧 ↾ ({𝑤} × 𝑁))) ↔ ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ↾ ({𝑤} × 𝑁)) = ((({𝑤} × 𝑁) × {𝑦}) ∘𝑓 · (𝑧 ↾ ({𝑤} × 𝑁)))))
361		reseq1 5311	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (𝑥 = (𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) → (𝑥 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ↾ ((𝑁 ∖ {𝑤}) × 𝑁)))
362	361	eqeq1d 2612	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝑥 = (𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) → ((𝑥 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑧 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) ↔ ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑧 ↾ ((𝑁 ∖ {𝑤}) × 𝑁))))
363	360, 362	anbi12d 743	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝑥 = (𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) → (((𝑥 ↾ ({𝑤} × 𝑁)) = ((({𝑤} × 𝑁) × {𝑦}) ∘𝑓 · (𝑧 ↾ ({𝑤} × 𝑁))) ∧ (𝑥 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑧 ↾ ((𝑁 ∖ {𝑤}) × 𝑁))) ↔ (((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ↾ ({𝑤} × 𝑁)) = ((({𝑤} × 𝑁) × {𝑦}) ∘𝑓 · (𝑧 ↾ ({𝑤} × 𝑁))) ∧ ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑧 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)))))
364		fveq2 6103	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝑥 = (𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) → (𝐷‘𝑥) = (𝐷‘(𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒)))))
365	364	eqeq1d 2612	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝑥 = (𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) → ((𝐷‘𝑥) = (𝑦 · (𝐷‘𝑧)) ↔ (𝐷‘(𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒)))) = (𝑦 · (𝐷‘𝑧))))
366	363, 365	imbi12d 333	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑥 = (𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) → ((((𝑥 ↾ ({𝑤} × 𝑁)) = ((({𝑤} × 𝑁) × {𝑦}) ∘𝑓 · (𝑧 ↾ ({𝑤} × 𝑁))) ∧ (𝑥 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑧 ↾ ((𝑁 ∖ {𝑤}) × 𝑁))) → (𝐷‘𝑥) = (𝑦 · (𝐷‘𝑧))) ↔ ((((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ↾ ({𝑤} × 𝑁)) = ((({𝑤} × 𝑁) × {𝑦}) ∘𝑓 · (𝑧 ↾ ({𝑤} × 𝑁))) ∧ ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑧 ↾ ((𝑁 ∖ {𝑤}) × 𝑁))) → (𝐷‘(𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒)))) = (𝑦 · (𝐷‘𝑧)))))
367	366	2ralbidv 2972	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑥 = (𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) → (∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝑁 (((𝑥 ↾ ({𝑤} × 𝑁)) = ((({𝑤} × 𝑁) × {𝑦}) ∘𝑓 · (𝑧 ↾ ({𝑤} × 𝑁))) ∧ (𝑥 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑧 ↾ ((𝑁 ∖ {𝑤}) × 𝑁))) → (𝐷‘𝑥) = (𝑦 · (𝐷‘𝑧))) ↔ ∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝑁 ((((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ↾ ({𝑤} × 𝑁)) = ((({𝑤} × 𝑁) × {𝑦}) ∘𝑓 · (𝑧 ↾ ({𝑤} × 𝑁))) ∧ ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑧 ↾ ((𝑁 ∖ {𝑤}) × 𝑁))) → (𝐷‘(𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒)))) = (𝑦 · (𝐷‘𝑧)))))
368		sneq 4135	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ (𝑦 = ((𝑎‘𝑐)(-g‘𝑅)if(𝑐 ∈ 𝑑, 1 , 0 )) → {𝑦} = {((𝑎‘𝑐)(-g‘𝑅)if(𝑐 ∈ 𝑑, 1 , 0 ))})
369	368	xpeq2d 5063	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 ))}))
370	369	oveq1d 6564	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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))))
371	370	eqeq2d 2620	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝑦 = ((𝑎‘𝑐)(-g‘𝑅)if(𝑐 ∈ 𝑑, 1 , 0 )) → (((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ↾ ({𝑤} × 𝑁)) = ((({𝑤} × 𝑁) × {𝑦}) ∘𝑓 · (𝑧 ↾ ({𝑤} × 𝑁))) ↔ ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ↾ ({𝑤} × 𝑁)) = ((({𝑤} × 𝑁) × {((𝑎‘𝑐)(-g‘𝑅)if(𝑐 ∈ 𝑑, 1 , 0 ))}) ∘𝑓 · (𝑧 ↾ ({𝑤} × 𝑁)))))
372	371	anbi1d 737	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝑦 = ((𝑎‘𝑐)(-g‘𝑅)if(𝑐 ∈ 𝑑, 1 , 0 )) → ((((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ↾ ({𝑤} × 𝑁)) = ((({𝑤} × 𝑁) × {𝑦}) ∘𝑓 · (𝑧 ↾ ({𝑤} × 𝑁))) ∧ ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑧 ↾ ((𝑁 ∖ {𝑤}) × 𝑁))) ↔ (((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ↾ ({𝑤} × 𝑁)) = ((({𝑤} × 𝑁) × {((𝑎‘𝑐)(-g‘𝑅)if(𝑐 ∈ 𝑑, 1 , 0 ))}) ∘𝑓 · (𝑧 ↾ ({𝑤} × 𝑁))) ∧ ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑧 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)))))
373		oveq1 6556	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)))
374	373	eqeq2d 2620	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝑦 = ((𝑎‘𝑐)(-g‘𝑅)if(𝑐 ∈ 𝑑, 1 , 0 )) → ((𝐷‘(𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒)))) = (𝑦 · (𝐷‘𝑧)) ↔ (𝐷‘(𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒)))) = (((𝑎‘𝑐)(-g‘𝑅)if(𝑐 ∈ 𝑑, 1 , 0 )) · (𝐷‘𝑧))))
375	372, 374	imbi12d 333	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑦 = ((𝑎‘𝑐)(-g‘𝑅)if(𝑐 ∈ 𝑑, 1 , 0 )) → (((((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ↾ ({𝑤} × 𝑁)) = ((({𝑤} × 𝑁) × {𝑦}) ∘𝑓 · (𝑧 ↾ ({𝑤} × 𝑁))) ∧ ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑧 ↾ ((𝑁 ∖ {𝑤}) × 𝑁))) → (𝐷‘(𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒)))) = (𝑦 · (𝐷‘𝑧))) ↔ ((((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ↾ ({𝑤} × 𝑁)) = ((({𝑤} × 𝑁) × {((𝑎‘𝑐)(-g‘𝑅)if(𝑐 ∈ 𝑑, 1 , 0 ))}) ∘𝑓 · (𝑧 ↾ ({𝑤} × 𝑁))) ∧ ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑧 ↾ ((𝑁 ∖ {𝑤}) × 𝑁))) → (𝐷‘(𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒)))) = (((𝑎‘𝑐)(-g‘𝑅)if(𝑐 ∈ 𝑑, 1 , 0 )) · (𝐷‘𝑧)))))
376	375	2ralbidv 2972	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑦 = ((𝑎‘𝑐)(-g‘𝑅)if(𝑐 ∈ 𝑑, 1 , 0 )) → (∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝑁 ((((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ↾ ({𝑤} × 𝑁)) = ((({𝑤} × 𝑁) × {𝑦}) ∘𝑓 · (𝑧 ↾ ({𝑤} × 𝑁))) ∧ ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑧 ↾ ((𝑁 ∖ {𝑤}) × 𝑁))) → (𝐷‘(𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒)))) = (𝑦 · (𝐷‘𝑧))) ↔ ∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝑁 ((((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ↾ ({𝑤} × 𝑁)) = ((({𝑤} × 𝑁) × {((𝑎‘𝑐)(-g‘𝑅)if(𝑐 ∈ 𝑑, 1 , 0 ))}) ∘𝑓 · (𝑧 ↾ ({𝑤} × 𝑁))) ∧ ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑧 ↾ ((𝑁 ∖ {𝑤}) × 𝑁))) → (𝐷‘(𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒)))) = (((𝑎‘𝑐)(-g‘𝑅)if(𝑐 ∈ 𝑑, 1 , 0 )) · (𝐷‘𝑧)))))
377	367, 376	rspc2va 3294	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ∈ 𝐵 ∧ ((𝑎‘𝑐)(-g‘𝑅)if(𝑐 ∈ 𝑑, 1 , 0 )) ∈ 𝐾) ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐾 ∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝑁 (((𝑥 ↾ ({𝑤} × 𝑁)) = ((({𝑤} × 𝑁) × {𝑦}) ∘𝑓 · (𝑧 ↾ ({𝑤} × 𝑁))) ∧ (𝑥 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑧 ↾ ((𝑁 ∖ {𝑤}) × 𝑁))) → (𝐷‘𝑥) = (𝑦 · (𝐷‘𝑧)))) → ∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝑁 ((((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ↾ ({𝑤} × 𝑁)) = ((({𝑤} × 𝑁) × {((𝑎‘𝑐)(-g‘𝑅)if(𝑐 ∈ 𝑑, 1 , 0 ))}) ∘𝑓 · (𝑧 ↾ ({𝑤} × 𝑁))) ∧ ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑧 ↾ ((𝑁 ∖ {𝑤}) × 𝑁))) → (𝐷‘(𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒)))) = (((𝑎‘𝑐)(-g‘𝑅)if(𝑐 ∈ 𝑑, 1 , 0 )) · (𝐷‘𝑧))))
378	241, 304, 358, 377	syl21anc 1317	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) → ∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝑁 ((((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ↾ ({𝑤} × 𝑁)) = ((({𝑤} × 𝑁) × {((𝑎‘𝑐)(-g‘𝑅)if(𝑐 ∈ 𝑑, 1 , 0 ))}) ∘𝑓 · (𝑧 ↾ ({𝑤} × 𝑁))) ∧ ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑧 ↾ ((𝑁 ∖ {𝑤}) × 𝑁))) → (𝐷‘(𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒)))) = (((𝑎‘𝑐)(-g‘𝑅)if(𝑐 ∈ 𝑑, 1 , 0 )) · (𝐷‘𝑧))))
379		reseq1 5311	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝑧 = (𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒))) → (𝑧 ↾ ({𝑤} × 𝑁)) = ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒))) ↾ ({𝑤} × 𝑁)))
380	379	oveq2d 6565	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝑧 = (𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒))) → ((({𝑤} × 𝑁) × {((𝑎‘𝑐)(-g‘𝑅)if(𝑐 ∈ 𝑑, 1 , 0 ))}) ∘𝑓 · (𝑧 ↾ ({𝑤} × 𝑁))) = ((({𝑤} × 𝑁) × {((𝑎‘𝑐)(-g‘𝑅)if(𝑐 ∈ 𝑑, 1 , 0 ))}) ∘𝑓 · ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒))) ↾ ({𝑤} × 𝑁))))
381	380	eqeq2d 2620	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑧 = (𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒))) → (((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ↾ ({𝑤} × 𝑁)) = ((({𝑤} × 𝑁) × {((𝑎‘𝑐)(-g‘𝑅)if(𝑐 ∈ 𝑑, 1 , 0 ))}) ∘𝑓 · (𝑧 ↾ ({𝑤} × 𝑁))) ↔ ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ↾ ({𝑤} × 𝑁)) = ((({𝑤} × 𝑁) × {((𝑎‘𝑐)(-g‘𝑅)if(𝑐 ∈ 𝑑, 1 , 0 ))}) ∘𝑓 · ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒))) ↾ ({𝑤} × 𝑁)))))
382		reseq1 5311	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝑧 = (𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒))) → (𝑧 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒))) ↾ ((𝑁 ∖ {𝑤}) × 𝑁)))
383	382	eqeq2d 2620	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑧 = (𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒))) → (((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑧 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) ↔ ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒))) ↾ ((𝑁 ∖ {𝑤}) × 𝑁))))
384	381, 383	anbi12d 743	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑧 = (𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒))) → ((((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ↾ ({𝑤} × 𝑁)) = ((({𝑤} × 𝑁) × {((𝑎‘𝑐)(-g‘𝑅)if(𝑐 ∈ 𝑑, 1 , 0 ))}) ∘𝑓 · (𝑧 ↾ ({𝑤} × 𝑁))) ∧ ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑧 ↾ ((𝑁 ∖ {𝑤}) × 𝑁))) ↔ (((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ↾ ({𝑤} × 𝑁)) = ((({𝑤} × 𝑁) × {((𝑎‘𝑐)(-g‘𝑅)if(𝑐 ∈ 𝑑, 1 , 0 ))}) ∘𝑓 · ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒))) ↾ ({𝑤} × 𝑁))) ∧ ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒))) ↾ ((𝑁 ∖ {𝑤}) × 𝑁)))))
385		fveq2 6103	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝑧 = (𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒))) → (𝐷‘𝑧) = (𝐷‘(𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒)))))
386	385	oveq2d 6565	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑧 = (𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒))) → (((𝑎‘𝑐)(-g‘𝑅)if(𝑐 ∈ 𝑑, 1 , 0 )) · (𝐷‘𝑧)) = (((𝑎‘𝑐)(-g‘𝑅)if(𝑐 ∈ 𝑑, 1 , 0 )) · (𝐷‘(𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒))))))
387	386	eqeq2d 2620	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑧 = (𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒))) → ((𝐷‘(𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒)))) = (((𝑎‘𝑐)(-g‘𝑅)if(𝑐 ∈ 𝑑, 1 , 0 )) · (𝐷‘𝑧)) ↔ (𝐷‘(𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒)))) = (((𝑎‘𝑐)(-g‘𝑅)if(𝑐 ∈ 𝑑, 1 , 0 )) · (𝐷‘(𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒)))))))
388	384, 387	imbi12d 333	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑧 = (𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒))) → (((((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ↾ ({𝑤} × 𝑁)) = ((({𝑤} × 𝑁) × {((𝑎‘𝑐)(-g‘𝑅)if(𝑐 ∈ 𝑑, 1 , 0 ))}) ∘𝑓 · (𝑧 ↾ ({𝑤} × 𝑁))) ∧ ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑧 ↾ ((𝑁 ∖ {𝑤}) × 𝑁))) → (𝐷‘(𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒)))) = (((𝑎‘𝑐)(-g‘𝑅)if(𝑐 ∈ 𝑑, 1 , 0 )) · (𝐷‘𝑧))) ↔ ((((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ↾ ({𝑤} × 𝑁)) = ((({𝑤} × 𝑁) × {((𝑎‘𝑐)(-g‘𝑅)if(𝑐 ∈ 𝑑, 1 , 0 ))}) ∘𝑓 · ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒))) ↾ ({𝑤} × 𝑁))) ∧ ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒))) ↾ ((𝑁 ∖ {𝑤}) × 𝑁))) → (𝐷‘(𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒)))) = (((𝑎‘𝑐)(-g‘𝑅)if(𝑐 ∈ 𝑑, 1 , 0 )) · (𝐷‘(𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒))))))))
389	277	xpeq1d 5062	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝑤 = (1st ‘𝑐) → (({𝑤} × 𝑁) × {((𝑎‘𝑐)(-g‘𝑅)if(𝑐 ∈ 𝑑, 1 , 0 ))}) = (({(1st ‘𝑐)} × 𝑁) × {((𝑎‘𝑐)(-g‘𝑅)if(𝑐 ∈ 𝑑, 1 , 0 ))}))
390	277	reseq2d 5317	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝑤 = (1st ‘𝑐) → ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒))) ↾ ({𝑤} × 𝑁)) = ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒))) ↾ ({(1st ‘𝑐)} × 𝑁)))
391	389, 390	oveq12d 6567	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝑤 = (1st ‘𝑐) → ((({𝑤} × 𝑁) × {((𝑎‘𝑐)(-g‘𝑅)if(𝑐 ∈ 𝑑, 1 , 0 ))}) ∘𝑓 · ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒))) ↾ ({𝑤} × 𝑁))) = ((({(1st ‘𝑐)} × 𝑁) × {((𝑎‘𝑐)(-g‘𝑅)if(𝑐 ∈ 𝑑, 1 , 0 ))}) ∘𝑓 · ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒))) ↾ ({(1st ‘𝑐)} × 𝑁))))
392	279, 391	eqeq12d 2625	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑤 = (1st ‘𝑐) → (((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ↾ ({𝑤} × 𝑁)) = ((({𝑤} × 𝑁) × {((𝑎‘𝑐)(-g‘𝑅)if(𝑐 ∈ 𝑑, 1 , 0 ))}) ∘𝑓 · ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒))) ↾ ({𝑤} × 𝑁))) ↔ ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ↾ ({(1st ‘𝑐)} × 𝑁)) = ((({(1st ‘𝑐)} × 𝑁) × {((𝑎‘𝑐)(-g‘𝑅)if(𝑐 ∈ 𝑑, 1 , 0 ))}) ∘𝑓 · ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒))) ↾ ({(1st ‘𝑐)} × 𝑁)))))
393	284	reseq2d 5317	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝑤 = (1st ‘𝑐) → ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒))) ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒))) ↾ ((𝑁 ∖ {(1st ‘𝑐)}) × 𝑁)))
394	286, 393	eqeq12d 2625	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑤 = (1st ‘𝑐) → (((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒))) ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) ↔ ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ↾ ((𝑁 ∖ {(1st ‘𝑐)}) × 𝑁)) = ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒))) ↾ ((𝑁 ∖ {(1st ‘𝑐)}) × 𝑁))))
395	392, 394	anbi12d 743	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑤 = (1st ‘𝑐) → ((((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ↾ ({𝑤} × 𝑁)) = ((({𝑤} × 𝑁) × {((𝑎‘𝑐)(-g‘𝑅)if(𝑐 ∈ 𝑑, 1 , 0 ))}) ∘𝑓 · ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒))) ↾ ({𝑤} × 𝑁))) ∧ ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒))) ↾ ((𝑁 ∖ {𝑤}) × 𝑁))) ↔ (((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ↾ ({(1st ‘𝑐)} × 𝑁)) = ((({(1st ‘𝑐)} × 𝑁) × {((𝑎‘𝑐)(-g‘𝑅)if(𝑐 ∈ 𝑑, 1 , 0 ))}) ∘𝑓 · ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒))) ↾ ({(1st ‘𝑐)} × 𝑁))) ∧ ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ↾ ((𝑁 ∖ {(1st ‘𝑐)}) × 𝑁)) = ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒))) ↾ ((𝑁 ∖ {(1st ‘𝑐)}) × 𝑁)))))
396	395	imbi1d 330	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑤 = (1st ‘𝑐) → (((((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ↾ ({𝑤} × 𝑁)) = ((({𝑤} × 𝑁) × {((𝑎‘𝑐)(-g‘𝑅)if(𝑐 ∈ 𝑑, 1 , 0 ))}) ∘𝑓 · ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒))) ↾ ({𝑤} × 𝑁))) ∧ ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒))) ↾ ((𝑁 ∖ {𝑤}) × 𝑁))) → (𝐷‘(𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒)))) = (((𝑎‘𝑐)(-g‘𝑅)if(𝑐 ∈ 𝑑, 1 , 0 )) · (𝐷‘(𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒)))))) ↔ ((((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ↾ ({(1st ‘𝑐)} × 𝑁)) = ((({(1st ‘𝑐)} × 𝑁) × {((𝑎‘𝑐)(-g‘𝑅)if(𝑐 ∈ 𝑑, 1 , 0 ))}) ∘𝑓 · ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒))) ↾ ({(1st ‘𝑐)} × 𝑁))) ∧ ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ↾ ((𝑁 ∖ {(1st ‘𝑐)}) × 𝑁)) = ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒))) ↾ ((𝑁 ∖ {(1st ‘𝑐)}) × 𝑁))) → (𝐷‘(𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒)))) = (((𝑎‘𝑐)(-g‘𝑅)if(𝑐 ∈ 𝑑, 1 , 0 )) · (𝐷‘(𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒))))))))
397	388, 396	rspc2va 3294	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒))) ∈ 𝐵 ∧ (1st ‘𝑐) ∈ 𝑁) ∧ ∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝑁 ((((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ↾ ({𝑤} × 𝑁)) = ((({𝑤} × 𝑁) × {((𝑎‘𝑐)(-g‘𝑅)if(𝑐 ∈ 𝑑, 1 , 0 ))}) ∘𝑓 · (𝑧 ↾ ({𝑤} × 𝑁))) ∧ ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑧 ↾ ((𝑁 ∖ {𝑤}) × 𝑁))) → (𝐷‘(𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒)))) = (((𝑎‘𝑐)(-g‘𝑅)if(𝑐 ∈ 𝑑, 1 , 0 )) · (𝐷‘𝑧)))) → ((((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ↾ ({(1st ‘𝑐)} × 𝑁)) = ((({(1st ‘𝑐)} × 𝑁) × {((𝑎‘𝑐)(-g‘𝑅)if(𝑐 ∈ 𝑑, 1 , 0 ))}) ∘𝑓 · ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒))) ↾ ({(1st ‘𝑐)} × 𝑁))) ∧ ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ↾ ((𝑁 ∖ {(1st ‘𝑐)}) × 𝑁)) = ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒))) ↾ ((𝑁 ∖ {(1st ‘𝑐)}) × 𝑁))) → (𝐷‘(𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒)))) = (((𝑎‘𝑐)(-g‘𝑅)if(𝑐 ∈ 𝑑, 1 , 0 )) · (𝐷‘(𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒)))))))
398	357, 125, 378, 397	syl21anc 1317	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) → ((((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ↾ ({(1st ‘𝑐)} × 𝑁)) = ((({(1st ‘𝑐)} × 𝑁) × {((𝑎‘𝑐)(-g‘𝑅)if(𝑐 ∈ 𝑑, 1 , 0 ))}) ∘𝑓 · ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒))) ↾ ({(1st ‘𝑐)} × 𝑁))) ∧ ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒))) ↾ ((𝑁 ∖ {(1st ‘𝑐)}) × 𝑁)) = ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒))) ↾ ((𝑁 ∖ {(1st ‘𝑐)}) × 𝑁))) → (𝐷‘(𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒)))) = (((𝑎‘𝑐)(-g‘𝑅)if(𝑐 ∈ 𝑑, 1 , 0 )) · (𝐷‘(𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒)))))))
399	340, 348, 398	mp2and 711	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) → (𝐷‘(𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒)))) = (((𝑎‘𝑐)(-g‘𝑅)if(𝑐 ∈ 𝑑, 1 , 0 )) · (𝐷‘(𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒))))))
400	399	oveq1d 6564	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) → ((𝐷‘(𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒)))) + (𝐷‘(𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒 ∈ 𝑑, 1 , 0 ), (𝑎‘𝑒))))) = ((((𝑎‘𝑐)(-g‘𝑅)if(𝑐 ∈ 𝑑, 1 , 0 )) · (𝐷‘(𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒))))) + (𝐷‘(𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒 ∈ 𝑑, 1 , 0 ), (𝑎‘𝑒))))))
401	105, 106, 107, 400	syl3anc 1318	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ (𝑏 ∪ {𝑐}) ∈ 𝑌) ∧ ((𝑎 ∈ 𝐵 ∧ 𝑑 ∈ (𝑁 ↑𝑚 𝑁)) ∧ ∀𝑤 ∈ 𝑏 (𝑎‘𝑤) = if(𝑤 ∈ 𝑑, 1 , 0 ))) → ((𝐷‘(𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, ((𝑎‘𝑒)(-g‘𝑅)if(𝑒 ∈ 𝑑, 1 , 0 )), 0 ), (𝑎‘𝑒)))) + (𝐷‘(𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒 ∈ 𝑑, 1 , 0 ), (𝑎‘𝑒))))) = ((((𝑎‘𝑐)(-g‘𝑅)if(𝑐 ∈ 𝑑, 1 , 0 )) · (𝐷‘(𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒))))) + (𝐷‘(𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒 ∈ 𝑑, 1 , 0 ), (𝑎‘𝑒))))))
402		simpl3 1059	. . . . . . . . . . . . . . . . . . . . . . . 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)
403		simprlr 799	. . . . . . . . . . . . . . . . . . . . . . . 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))
404		simprr 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 ))) → ∀𝑤 ∈ 𝑏 (𝑎‘𝑤) = if(𝑤 ∈ 𝑑, 1 , 0 ))
405		ralss 3631	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 ))))
406	100, 405	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 )))
407		iftrue 4042	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ ((1st ‘𝑤) = (1st ‘𝑐) → if((1st ‘𝑤) = (1st ‘𝑐), if(𝑤 = 𝑐, 1 , 0 ), (𝑎‘𝑤)) = if(𝑤 = 𝑐, 1 , 0 ))
408	407	adantl 481	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ (((((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) ∧ 𝑑 ∈ (𝑁 ↑𝑚 𝑁)) ∧ 𝑤 ∈ (𝑏 ∪ {𝑐})) ∧ (1st ‘𝑤) = (1st ‘𝑐)) → if((1st ‘𝑤) = (1st ‘𝑐), if(𝑤 = 𝑐, 1 , 0 ), (𝑎‘𝑤)) = if(𝑤 = 𝑐, 1 , 0 ))
409		ibar 524	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ⊢ ((1st ‘𝑤) = (1st ‘𝑐) → ((2nd ‘𝑤) = (2nd ‘𝑐) ↔ ((1st ‘𝑤) = (1st ‘𝑐) ∧ (2nd ‘𝑤) = (2nd ‘𝑐))))
410	409	adantl 481	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ⊢ (((((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) ∧ 𝑑 ∈ (𝑁 ↑𝑚 𝑁)) ∧ 𝑤 ∈ (𝑏 ∪ {𝑐})) ∧ (1st ‘𝑤) = (1st ‘𝑐)) → ((2nd ‘𝑤) = (2nd ‘𝑐) ↔ ((1st ‘𝑤) = (1st ‘𝑐) ∧ (2nd ‘𝑤) = (2nd ‘𝑐))))
411		relxp 5150	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 ⊢ Rel (𝑁 × 𝑁)
412		simpl2 1058	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 ⊢ (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) ∧ 𝑑 ∈ (𝑁 ↑𝑚 𝑁)) → (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁))
413	412	sselda 3568	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 ⊢ ((((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) ∧ 𝑑 ∈ (𝑁 ↑𝑚 𝑁)) ∧ 𝑤 ∈ (𝑏 ∪ {𝑐})) → 𝑤 ∈ (𝑁 × 𝑁))
414	413	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 ⊢ (((((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) ∧ 𝑑 ∈ (𝑁 ↑𝑚 𝑁)) ∧ 𝑤 ∈ (𝑏 ∪ {𝑐})) ∧ (1st ‘𝑤) = (1st ‘𝑐)) → 𝑤 ∈ (𝑁 × 𝑁))
415		1st2nd 7105	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 ⊢ ((Rel (𝑁 × 𝑁) ∧ 𝑤 ∈ (𝑁 × 𝑁)) → 𝑤 = ⟨(1st ‘𝑤), (2nd ‘𝑤)⟩)
416	411, 414, 415	sylancr 694	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 ⊢ (((((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) ∧ 𝑑 ∈ (𝑁 ↑𝑚 𝑁)) ∧ 𝑤 ∈ (𝑏 ∪ {𝑐})) ∧ (1st ‘𝑤) = (1st ‘𝑐)) → 𝑤 = ⟨(1st ‘𝑤), (2nd ‘𝑤)⟩)
417	416	eleq1d 2672	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ⊢ (((((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) ∧ 𝑑 ∈ (𝑁 ↑𝑚 𝑁)) ∧ 𝑤 ∈ (𝑏 ∪ {𝑐})) ∧ (1st ‘𝑤) = (1st ‘𝑐)) → (𝑤 ∈ (𝑑 sSet ⟨(1st ‘𝑐), (2nd ‘𝑐)⟩) ↔ ⟨(1st ‘𝑤), (2nd ‘𝑤)⟩ ∈ (𝑑 sSet ⟨(1st ‘𝑐), (2nd ‘𝑐)⟩)))
418		simpr 476	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 ⊢ (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) ∧ 𝑑 ∈ (𝑁 ↑𝑚 𝑁)) → 𝑑 ∈ (𝑁 ↑𝑚 𝑁))
419		elmapi 7765	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 ⊢ (𝑑 ∈ (𝑁 ↑𝑚 𝑁) → 𝑑:𝑁⟶𝑁)
420	419	adantl 481	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 ⊢ (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) ∧ 𝑑 ∈ (𝑁 ↑𝑚 𝑁)) → 𝑑:𝑁⟶𝑁)
421	125	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 ⊢ (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) ∧ 𝑑 ∈ (𝑁 ↑𝑚 𝑁)) → (1st ‘𝑐) ∈ 𝑁)
422		xp2nd 7090	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 ⊢ (𝑐 ∈ (𝑁 × 𝑁) → (2nd ‘𝑐) ∈ 𝑁)
423	123, 422	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 ⊢ ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) → (2nd ‘𝑐) ∈ 𝑁)
424	423	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 ⊢ (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) ∧ 𝑑 ∈ (𝑁 ↑𝑚 𝑁)) → (2nd ‘𝑐) ∈ 𝑁)
425		fsets 15723	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 ⊢ (((𝑑 ∈ (𝑁 ↑𝑚 𝑁) ∧ 𝑑:𝑁⟶𝑁) ∧ (1st ‘𝑐) ∈ 𝑁 ∧ (2nd ‘𝑐) ∈ 𝑁) → (𝑑 sSet ⟨(1st ‘𝑐), (2nd ‘𝑐)⟩):𝑁⟶𝑁)
426	418, 420, 421, 424, 425	syl211anc 1324	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 ⊢ (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) ∧ 𝑑 ∈ (𝑁 ↑𝑚 𝑁)) → (𝑑 sSet ⟨(1st ‘𝑐), (2nd ‘𝑐)⟩):𝑁⟶𝑁)
427		ffn 5958	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 ⊢ ((𝑑 sSet ⟨(1st ‘𝑐), (2nd ‘𝑐)⟩):𝑁⟶𝑁 → (𝑑 sSet ⟨(1st ‘𝑐), (2nd ‘𝑐)⟩) Fn 𝑁)
428	426, 427	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 ⊢ (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) ∧ 𝑑 ∈ (𝑁 ↑𝑚 𝑁)) → (𝑑 sSet ⟨(1st ‘𝑐), (2nd ‘𝑐)⟩) Fn 𝑁)
429	428	ad2antrr 758	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 ⊢ (((((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) ∧ 𝑑 ∈ (𝑁 ↑𝑚 𝑁)) ∧ 𝑤 ∈ (𝑏 ∪ {𝑐})) ∧ (1st ‘𝑤) = (1st ‘𝑐)) → (𝑑 sSet ⟨(1st ‘𝑐), (2nd ‘𝑐)⟩) Fn 𝑁)
430		xp1st 7089	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 ⊢ (𝑤 ∈ (𝑁 × 𝑁) → (1st ‘𝑤) ∈ 𝑁)
431	413, 430	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 ⊢ ((((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) ∧ 𝑑 ∈ (𝑁 ↑𝑚 𝑁)) ∧ 𝑤 ∈ (𝑏 ∪ {𝑐})) → (1st ‘𝑤) ∈ 𝑁)
432	431	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 ⊢ (((((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) ∧ 𝑑 ∈ (𝑁 ↑𝑚 𝑁)) ∧ 𝑤 ∈ (𝑏 ∪ {𝑐})) ∧ (1st ‘𝑤) = (1st ‘𝑐)) → (1st ‘𝑤) ∈ 𝑁)
433		fnopfvb 6147	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 ⊢ (((𝑑 sSet ⟨(1st ‘𝑐), (2nd ‘𝑐)⟩) Fn 𝑁 ∧ (1st ‘𝑤) ∈ 𝑁) → (((𝑑 sSet ⟨(1st ‘𝑐), (2nd ‘𝑐)⟩)‘(1st ‘𝑤)) = (2nd ‘𝑤) ↔ ⟨(1st ‘𝑤), (2nd ‘𝑤)⟩ ∈ (𝑑 sSet ⟨(1st ‘𝑐), (2nd ‘𝑐)⟩)))
434	429, 432, 433	syl2anc 691	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ⊢ (((((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) ∧ 𝑑 ∈ (𝑁 ↑𝑚 𝑁)) ∧ 𝑤 ∈ (𝑏 ∪ {𝑐})) ∧ (1st ‘𝑤) = (1st ‘𝑐)) → (((𝑑 sSet ⟨(1st ‘𝑐), (2nd ‘𝑐)⟩)‘(1st ‘𝑤)) = (2nd ‘𝑤) ↔ ⟨(1st ‘𝑤), (2nd ‘𝑤)⟩ ∈ (𝑑 sSet ⟨(1st ‘𝑐), (2nd ‘𝑐)⟩)))
435		fveq2 6103	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 ⊢ ((1st ‘𝑤) = (1st ‘𝑐) → ((𝑑 sSet ⟨(1st ‘𝑐), (2nd ‘𝑐)⟩)‘(1st ‘𝑤)) = ((𝑑 sSet ⟨(1st ‘𝑐), (2nd ‘𝑐)⟩)‘(1st ‘𝑐)))
436	435	adantl 481	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 ⊢ (((((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) ∧ 𝑑 ∈ (𝑁 ↑𝑚 𝑁)) ∧ 𝑤 ∈ (𝑏 ∪ {𝑐})) ∧ (1st ‘𝑤) = (1st ‘𝑐)) → ((𝑑 sSet ⟨(1st ‘𝑐), (2nd ‘𝑐)⟩)‘(1st ‘𝑤)) = ((𝑑 sSet ⟨(1st ‘𝑐), (2nd ‘𝑐)⟩)‘(1st ‘𝑐)))
437		vex 3176	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 ⊢ 𝑑 ∈ V
438		fvex 6113	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 ⊢ (1st ‘𝑐) ∈ V
439		fvex 6113	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 ⊢ (2nd ‘𝑐) ∈ V
440		fvsetsid 15722	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 ⊢ ((𝑑 ∈ V ∧ (1st ‘𝑐) ∈ V ∧ (2nd ‘𝑐) ∈ V) → ((𝑑 sSet ⟨(1st ‘𝑐), (2nd ‘𝑐)⟩)‘(1st ‘𝑐)) = (2nd ‘𝑐))
441	437, 438, 439, 440	mp3an 1416	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 ⊢ ((𝑑 sSet ⟨(1st ‘𝑐), (2nd ‘𝑐)⟩)‘(1st ‘𝑐)) = (2nd ‘𝑐)
442	436, 441	syl6eq 2660	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 ⊢ (((((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) ∧ 𝑑 ∈ (𝑁 ↑𝑚 𝑁)) ∧ 𝑤 ∈ (𝑏 ∪ {𝑐})) ∧ (1st ‘𝑤) = (1st ‘𝑐)) → ((𝑑 sSet ⟨(1st ‘𝑐), (2nd ‘𝑐)⟩)‘(1st ‘𝑤)) = (2nd ‘𝑐))
443	442	eqeq1d 2612	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 ⊢ (((((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) ∧ 𝑑 ∈ (𝑁 ↑𝑚 𝑁)) ∧ 𝑤 ∈ (𝑏 ∪ {𝑐})) ∧ (1st ‘𝑤) = (1st ‘𝑐)) → (((𝑑 sSet ⟨(1st ‘𝑐), (2nd ‘𝑐)⟩)‘(1st ‘𝑤)) = (2nd ‘𝑤) ↔ (2nd ‘𝑐) = (2nd ‘𝑤)))
444		eqcom 2617	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 ⊢ ((2nd ‘𝑐) = (2nd ‘𝑤) ↔ (2nd ‘𝑤) = (2nd ‘𝑐))
445	443, 444	syl6bb 275	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ⊢ (((((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) ∧ 𝑑 ∈ (𝑁 ↑𝑚 𝑁)) ∧ 𝑤 ∈ (𝑏 ∪ {𝑐})) ∧ (1st ‘𝑤) = (1st ‘𝑐)) → (((𝑑 sSet ⟨(1st ‘𝑐), (2nd ‘𝑐)⟩)‘(1st ‘𝑤)) = (2nd ‘𝑤) ↔ (2nd ‘𝑤) = (2nd ‘𝑐)))
446	417, 434, 445	3bitr2rd 296	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ⊢ (((((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) ∧ 𝑑 ∈ (𝑁 ↑𝑚 𝑁)) ∧ 𝑤 ∈ (𝑏 ∪ {𝑐})) ∧ (1st ‘𝑤) = (1st ‘𝑐)) → ((2nd ‘𝑤) = (2nd ‘𝑐) ↔ 𝑤 ∈ (𝑑 sSet ⟨(1st ‘𝑐), (2nd ‘𝑐)⟩)))
447	123	ad3antrrr 762	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ⊢ (((((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) ∧ 𝑑 ∈ (𝑁 ↑𝑚 𝑁)) ∧ 𝑤 ∈ (𝑏 ∪ {𝑐})) ∧ (1st ‘𝑤) = (1st ‘𝑐)) → 𝑐 ∈ (𝑁 × 𝑁))
448		xpopth 7098	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ⊢ ((𝑤 ∈ (𝑁 × 𝑁) ∧ 𝑐 ∈ (𝑁 × 𝑁)) → (((1st ‘𝑤) = (1st ‘𝑐) ∧ (2nd ‘𝑤) = (2nd ‘𝑐)) ↔ 𝑤 = 𝑐))
449	414, 447, 448	syl2anc 691	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ⊢ (((((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) ∧ 𝑑 ∈ (𝑁 ↑𝑚 𝑁)) ∧ 𝑤 ∈ (𝑏 ∪ {𝑐})) ∧ (1st ‘𝑤) = (1st ‘𝑐)) → (((1st ‘𝑤) = (1st ‘𝑐) ∧ (2nd ‘𝑤) = (2nd ‘𝑐)) ↔ 𝑤 = 𝑐))
450	410, 446, 449	3bitr3rd 298	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ (((((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) ∧ 𝑑 ∈ (𝑁 ↑𝑚 𝑁)) ∧ 𝑤 ∈ (𝑏 ∪ {𝑐})) ∧ (1st ‘𝑤) = (1st ‘𝑐)) → (𝑤 = 𝑐 ↔ 𝑤 ∈ (𝑑 sSet ⟨(1st ‘𝑐), (2nd ‘𝑐)⟩)))
451	450	ifbid 4058	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ (((((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) ∧ 𝑑 ∈ (𝑁 ↑𝑚 𝑁)) ∧ 𝑤 ∈ (𝑏 ∪ {𝑐})) ∧ (1st ‘𝑤) = (1st ‘𝑐)) → if(𝑤 = 𝑐, 1 , 0 ) = if(𝑤 ∈ (𝑑 sSet ⟨(1st ‘𝑐), (2nd ‘𝑐)⟩), 1 , 0 ))
452	408, 451	eqtrd 2644	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ (((((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) ∧ 𝑑 ∈ (𝑁 ↑𝑚 𝑁)) ∧ 𝑤 ∈ (𝑏 ∪ {𝑐})) ∧ (1st ‘𝑤) = (1st ‘𝑐)) → if((1st ‘𝑤) = (1st ‘𝑐), if(𝑤 = 𝑐, 1 , 0 ), (𝑎‘𝑤)) = if(𝑤 ∈ (𝑑 sSet ⟨(1st ‘𝑐), (2nd ‘𝑐)⟩), 1 , 0 ))
453	452	a1d 25	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (((((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) ∧ 𝑑 ∈ (𝑁 ↑𝑚 𝑁)) ∧ 𝑤 ∈ (𝑏 ∪ {𝑐})) ∧ (1st ‘𝑤) = (1st ‘𝑐)) → ((𝑤 ∈ 𝑏 → (𝑎‘𝑤) = if(𝑤 ∈ 𝑑, 1 , 0 )) → if((1st ‘𝑤) = (1st ‘𝑐), if(𝑤 = 𝑐, 1 , 0 ), (𝑎‘𝑤)) = if(𝑤 ∈ (𝑑 sSet ⟨(1st ‘𝑐), (2nd ‘𝑐)⟩), 1 , 0 )))
454		elsni 4142	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 ⊢ (𝑤 ∈ {𝑐} → 𝑤 = 𝑐)
455	454	fveq2d 6107	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ⊢ (𝑤 ∈ {𝑐} → (1st ‘𝑤) = (1st ‘𝑐))
456	455	con3i 149	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ⊢ (¬ (1st ‘𝑤) = (1st ‘𝑐) → ¬ 𝑤 ∈ {𝑐})
457	456	adantl 481	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ ((𝑤 ∈ (𝑏 ∪ {𝑐}) ∧ ¬ (1st ‘𝑤) = (1st ‘𝑐)) → ¬ 𝑤 ∈ {𝑐})
458		elun 3715	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ⊢ (𝑤 ∈ (𝑏 ∪ {𝑐}) ↔ (𝑤 ∈ 𝑏 ∨ 𝑤 ∈ {𝑐}))
459	458	biimpi 205	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ⊢ (𝑤 ∈ (𝑏 ∪ {𝑐}) → (𝑤 ∈ 𝑏 ∨ 𝑤 ∈ {𝑐}))
460	459	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ ((𝑤 ∈ (𝑏 ∪ {𝑐}) ∧ ¬ (1st ‘𝑤) = (1st ‘𝑐)) → (𝑤 ∈ 𝑏 ∨ 𝑤 ∈ {𝑐}))
461		orel2 397	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ (¬ 𝑤 ∈ {𝑐} → ((𝑤 ∈ 𝑏 ∨ 𝑤 ∈ {𝑐}) → 𝑤 ∈ 𝑏))
462	457, 460, 461	sylc 63	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ ((𝑤 ∈ (𝑏 ∪ {𝑐}) ∧ ¬ (1st ‘𝑤) = (1st ‘𝑐)) → 𝑤 ∈ 𝑏)
463	462	adantll 746	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ (((((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) ∧ 𝑑 ∈ (𝑁 ↑𝑚 𝑁)) ∧ 𝑤 ∈ (𝑏 ∪ {𝑐})) ∧ ¬ (1st ‘𝑤) = (1st ‘𝑐)) → 𝑤 ∈ 𝑏)
464		iffalse 4045	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ⊢ (¬ (1st ‘𝑤) = (1st ‘𝑐) → if((1st ‘𝑤) = (1st ‘𝑐), if(𝑤 = 𝑐, 1 , 0 ), if(𝑤 ∈ 𝑑, 1 , 0 )) = if(𝑤 ∈ 𝑑, 1 , 0 ))
465	464	adantl 481	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ (((((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) ∧ 𝑑 ∈ (𝑁 ↑𝑚 𝑁)) ∧ 𝑤 ∈ (𝑏 ∪ {𝑐})) ∧ ¬ (1st ‘𝑤) = (1st ‘𝑐)) → if((1st ‘𝑤) = (1st ‘𝑐), if(𝑤 = 𝑐, 1 , 0 ), if(𝑤 ∈ 𝑑, 1 , 0 )) = if(𝑤 ∈ 𝑑, 1 , 0 ))
466		setsres 15729	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 ⊢ (𝑑 ∈ V → ((𝑑 sSet ⟨(1st ‘𝑐), (2nd ‘𝑐)⟩) ↾ (V ∖ {(1st ‘𝑐)})) = (𝑑 ↾ (V ∖ {(1st ‘𝑐)})))
467	466	eleq2d 2673	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 ⊢ (𝑑 ∈ V → (⟨(1st ‘𝑤), (2nd ‘𝑤)⟩ ∈ ((𝑑 sSet ⟨(1st ‘𝑐), (2nd ‘𝑐)⟩) ↾ (V ∖ {(1st ‘𝑐)})) ↔ ⟨(1st ‘𝑤), (2nd ‘𝑤)⟩ ∈ (𝑑 ↾ (V ∖ {(1st ‘𝑐)}))))
468	437, 467	mp1i 13	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ⊢ (((((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) ∧ 𝑑 ∈ (𝑁 ↑𝑚 𝑁)) ∧ 𝑤 ∈ (𝑏 ∪ {𝑐})) ∧ ¬ (1st ‘𝑤) = (1st ‘𝑐)) → (⟨(1st ‘𝑤), (2nd ‘𝑤)⟩ ∈ ((𝑑 sSet ⟨(1st ‘𝑐), (2nd ‘𝑐)⟩) ↾ (V ∖ {(1st ‘𝑐)})) ↔ ⟨(1st ‘𝑤), (2nd ‘𝑤)⟩ ∈ (𝑑 ↾ (V ∖ {(1st ‘𝑐)}))))
469		fvex 6113	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 ⊢ (1st ‘𝑤) ∈ V
470	469	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 ⊢ (¬ (1st ‘𝑤) = (1st ‘𝑐) → (1st ‘𝑤) ∈ V)
471		df-ne 2782	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 ⊢ ((1st ‘𝑤) ≠ (1st ‘𝑐) ↔ ¬ (1st ‘𝑤) = (1st ‘𝑐))
472	471	biimpri 217	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 ⊢ (¬ (1st ‘𝑤) = (1st ‘𝑐) → (1st ‘𝑤) ≠ (1st ‘𝑐))
473		eldifsn 4260	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 ⊢ ((1st ‘𝑤) ∈ (V ∖ {(1st ‘𝑐)}) ↔ ((1st ‘𝑤) ∈ V ∧ (1st ‘𝑤) ≠ (1st ‘𝑐)))
474	470, 472, 473	sylanbrc 695	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 ⊢ (¬ (1st ‘𝑤) = (1st ‘𝑐) → (1st ‘𝑤) ∈ (V ∖ {(1st ‘𝑐)}))
475		fvex 6113	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 ⊢ (2nd ‘𝑤) ∈ V
476	475	opres 5326	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 ⊢ ((1st ‘𝑤) ∈ (V ∖ {(1st ‘𝑐)}) → (⟨(1st ‘𝑤), (2nd ‘𝑤)⟩ ∈ ((𝑑 sSet ⟨(1st ‘𝑐), (2nd ‘𝑐)⟩) ↾ (V ∖ {(1st ‘𝑐)})) ↔ ⟨(1st ‘𝑤), (2nd ‘𝑤)⟩ ∈ (𝑑 sSet ⟨(1st ‘𝑐), (2nd ‘𝑐)⟩)))
477	476	adantl 481	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 ⊢ ((𝑤 ∈ (𝑁 × 𝑁) ∧ (1st ‘𝑤) ∈ (V ∖ {(1st ‘𝑐)})) → (⟨(1st ‘𝑤), (2nd ‘𝑤)⟩ ∈ ((𝑑 sSet ⟨(1st ‘𝑐), (2nd ‘𝑐)⟩) ↾ (V ∖ {(1st ‘𝑐)})) ↔ ⟨(1st ‘𝑤), (2nd ‘𝑤)⟩ ∈ (𝑑 sSet ⟨(1st ‘𝑐), (2nd ‘𝑐)⟩)))
478		1st2nd2 7096	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 ⊢ (𝑤 ∈ (𝑁 × 𝑁) → 𝑤 = ⟨(1st ‘𝑤), (2nd ‘𝑤)⟩)
479	478	eleq1d 2672	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 ⊢ (𝑤 ∈ (𝑁 × 𝑁) → (𝑤 ∈ (𝑑 sSet ⟨(1st ‘𝑐), (2nd ‘𝑐)⟩) ↔ ⟨(1st ‘𝑤), (2nd ‘𝑤)⟩ ∈ (𝑑 sSet ⟨(1st ‘𝑐), (2nd ‘𝑐)⟩)))
480	479	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 ⊢ ((𝑤 ∈ (𝑁 × 𝑁) ∧ (1st ‘𝑤) ∈ (V ∖ {(1st ‘𝑐)})) → (𝑤 ∈ (𝑑 sSet ⟨(1st ‘𝑐), (2nd ‘𝑐)⟩) ↔ ⟨(1st ‘𝑤), (2nd ‘𝑤)⟩ ∈ (𝑑 sSet ⟨(1st ‘𝑐), (2nd ‘𝑐)⟩)))
481	477, 480	bitr4d 270	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 ⊢ ((𝑤 ∈ (𝑁 × 𝑁) ∧ (1st ‘𝑤) ∈ (V ∖ {(1st ‘𝑐)})) → (⟨(1st ‘𝑤), (2nd ‘𝑤)⟩ ∈ ((𝑑 sSet ⟨(1st ‘𝑐), (2nd ‘𝑐)⟩) ↾ (V ∖ {(1st ‘𝑐)})) ↔ 𝑤 ∈ (𝑑 sSet ⟨(1st ‘𝑐), (2nd ‘𝑐)⟩)))
482	413, 474, 481	syl2an 493	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ⊢ (((((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) ∧ 𝑑 ∈ (𝑁 ↑𝑚 𝑁)) ∧ 𝑤 ∈ (𝑏 ∪ {𝑐})) ∧ ¬ (1st ‘𝑤) = (1st ‘𝑐)) → (⟨(1st ‘𝑤), (2nd ‘𝑤)⟩ ∈ ((𝑑 sSet ⟨(1st ‘𝑐), (2nd ‘𝑐)⟩) ↾ (V ∖ {(1st ‘𝑐)})) ↔ 𝑤 ∈ (𝑑 sSet ⟨(1st ‘𝑐), (2nd ‘𝑐)⟩)))
483	475	opres 5326	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 ⊢ ((1st ‘𝑤) ∈ (V ∖ {(1st ‘𝑐)}) → (⟨(1st ‘𝑤), (2nd ‘𝑤)⟩ ∈ (𝑑 ↾ (V ∖ {(1st ‘𝑐)})) ↔ ⟨(1st ‘𝑤), (2nd ‘𝑤)⟩ ∈ 𝑑))
484	483	adantl 481	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 ⊢ ((𝑤 ∈ (𝑁 × 𝑁) ∧ (1st ‘𝑤) ∈ (V ∖ {(1st ‘𝑐)})) → (⟨(1st ‘𝑤), (2nd ‘𝑤)⟩ ∈ (𝑑 ↾ (V ∖ {(1st ‘𝑐)})) ↔ ⟨(1st ‘𝑤), (2nd ‘𝑤)⟩ ∈ 𝑑))
485	478	eleq1d 2672	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 ⊢ (𝑤 ∈ (𝑁 × 𝑁) → (𝑤 ∈ 𝑑 ↔ ⟨(1st ‘𝑤), (2nd ‘𝑤)⟩ ∈ 𝑑))
486	485	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 ⊢ ((𝑤 ∈ (𝑁 × 𝑁) ∧ (1st ‘𝑤) ∈ (V ∖ {(1st ‘𝑐)})) → (𝑤 ∈ 𝑑 ↔ ⟨(1st ‘𝑤), (2nd ‘𝑤)⟩ ∈ 𝑑))
487	484, 486	bitr4d 270	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 ⊢ ((𝑤 ∈ (𝑁 × 𝑁) ∧ (1st ‘𝑤) ∈ (V ∖ {(1st ‘𝑐)})) → (⟨(1st ‘𝑤), (2nd ‘𝑤)⟩ ∈ (𝑑 ↾ (V ∖ {(1st ‘𝑐)})) ↔ 𝑤 ∈ 𝑑))
488	413, 474, 487	syl2an 493	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ⊢ (((((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) ∧ 𝑑 ∈ (𝑁 ↑𝑚 𝑁)) ∧ 𝑤 ∈ (𝑏 ∪ {𝑐})) ∧ ¬ (1st ‘𝑤) = (1st ‘𝑐)) → (⟨(1st ‘𝑤), (2nd ‘𝑤)⟩ ∈ (𝑑 ↾ (V ∖ {(1st ‘𝑐)})) ↔ 𝑤 ∈ 𝑑))
489	468, 482, 488	3bitr3rd 298	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ⊢ (((((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) ∧ 𝑑 ∈ (𝑁 ↑𝑚 𝑁)) ∧ 𝑤 ∈ (𝑏 ∪ {𝑐})) ∧ ¬ (1st ‘𝑤) = (1st ‘𝑐)) → (𝑤 ∈ 𝑑 ↔ 𝑤 ∈ (𝑑 sSet ⟨(1st ‘𝑐), (2nd ‘𝑐)⟩)))
490	489	ifbid 4058	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ (((((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) ∧ 𝑑 ∈ (𝑁 ↑𝑚 𝑁)) ∧ 𝑤 ∈ (𝑏 ∪ {𝑐})) ∧ ¬ (1st ‘𝑤) = (1st ‘𝑐)) → if(𝑤 ∈ 𝑑, 1 , 0 ) = if(𝑤 ∈ (𝑑 sSet ⟨(1st ‘𝑐), (2nd ‘𝑐)⟩), 1 , 0 ))
491	465, 490	eqtrd 2644	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ (((((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) ∧ 𝑑 ∈ (𝑁 ↑𝑚 𝑁)) ∧ 𝑤 ∈ (𝑏 ∪ {𝑐})) ∧ ¬ (1st ‘𝑤) = (1st ‘𝑐)) → if((1st ‘𝑤) = (1st ‘𝑐), if(𝑤 = 𝑐, 1 , 0 ), if(𝑤 ∈ 𝑑, 1 , 0 )) = if(𝑤 ∈ (𝑑 sSet ⟨(1st ‘𝑐), (2nd ‘𝑐)⟩), 1 , 0 ))
492		ifeq2 4041	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ ((𝑎‘𝑤) = if(𝑤 ∈ 𝑑, 1 , 0 ) → if((1st ‘𝑤) = (1st ‘𝑐), if(𝑤 = 𝑐, 1 , 0 ), (𝑎‘𝑤)) = if((1st ‘𝑤) = (1st ‘𝑐), if(𝑤 = 𝑐, 1 , 0 ), if(𝑤 ∈ 𝑑, 1 , 0 )))
493	492	eqeq1d 2612	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ ((𝑎‘𝑤) = if(𝑤 ∈ 𝑑, 1 , 0 ) → (if((1st ‘𝑤) = (1st ‘𝑐), if(𝑤 = 𝑐, 1 , 0 ), (𝑎‘𝑤)) = if(𝑤 ∈ (𝑑 sSet ⟨(1st ‘𝑐), (2nd ‘𝑐)⟩), 1 , 0 ) ↔ if((1st ‘𝑤) = (1st ‘𝑐), if(𝑤 = 𝑐, 1 , 0 ), if(𝑤 ∈ 𝑑, 1 , 0 )) = if(𝑤 ∈ (𝑑 sSet ⟨(1st ‘𝑐), (2nd ‘𝑐)⟩), 1 , 0 )))
494	491, 493	syl5ibrcom 236	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ (((((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) ∧ 𝑑 ∈ (𝑁 ↑𝑚 𝑁)) ∧ 𝑤 ∈ (𝑏 ∪ {𝑐})) ∧ ¬ (1st ‘𝑤) = (1st ‘𝑐)) → ((𝑎‘𝑤) = if(𝑤 ∈ 𝑑, 1 , 0 ) → if((1st ‘𝑤) = (1st ‘𝑐), if(𝑤 = 𝑐, 1 , 0 ), (𝑎‘𝑤)) = if(𝑤 ∈ (𝑑 sSet ⟨(1st ‘𝑐), (2nd ‘𝑐)⟩), 1 , 0 )))
495	463, 494	embantd 57	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (((((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) ∧ 𝑑 ∈ (𝑁 ↑𝑚 𝑁)) ∧ 𝑤 ∈ (𝑏 ∪ {𝑐})) ∧ ¬ (1st ‘𝑤) = (1st ‘𝑐)) → ((𝑤 ∈ 𝑏 → (𝑎‘𝑤) = if(𝑤 ∈ 𝑑, 1 , 0 )) → if((1st ‘𝑤) = (1st ‘𝑐), if(𝑤 = 𝑐, 1 , 0 ), (𝑎‘𝑤)) = if(𝑤 ∈ (𝑑 sSet ⟨(1st ‘𝑐), (2nd ‘𝑐)⟩), 1 , 0 )))
496	453, 495	pm2.61dan 828	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ ((((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) ∧ 𝑑 ∈ (𝑁 ↑𝑚 𝑁)) ∧ 𝑤 ∈ (𝑏 ∪ {𝑐})) → ((𝑤 ∈ 𝑏 → (𝑎‘𝑤) = if(𝑤 ∈ 𝑑, 1 , 0 )) → if((1st ‘𝑤) = (1st ‘𝑐), if(𝑤 = 𝑐, 1 , 0 ), (𝑎‘𝑤)) = if(𝑤 ∈ (𝑑 sSet ⟨(1st ‘𝑐), (2nd ‘𝑐)⟩), 1 , 0 )))
497		fveq2 6103	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ⊢ (𝑒 = 𝑤 → (1st ‘𝑒) = (1st ‘𝑤))
498	497	eqeq1d 2612	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ (𝑒 = 𝑤 → ((1st ‘𝑒) = (1st ‘𝑐) ↔ (1st ‘𝑤) = (1st ‘𝑐)))
499		equequ1 1939	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ⊢ (𝑒 = 𝑤 → (𝑒 = 𝑐 ↔ 𝑤 = 𝑐))
500	499	ifbid 4058	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ (𝑒 = 𝑤 → if(𝑒 = 𝑐, 1 , 0 ) = if(𝑤 = 𝑐, 1 , 0 ))
501		fveq2 6103	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ (𝑒 = 𝑤 → (𝑎‘𝑒) = (𝑎‘𝑤))
502	498, 500, 501	ifbieq12d 4063	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ (𝑒 = 𝑤 → if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒)) = if((1st ‘𝑤) = (1st ‘𝑐), if(𝑤 = 𝑐, 1 , 0 ), (𝑎‘𝑤)))
503	168, 164	ifex 4106	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ if(𝑤 = 𝑐, 1 , 0 ) ∈ V
504		fvex 6113	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ (𝑎‘𝑤) ∈ V
505	503, 504	ifex 4106	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ if((1st ‘𝑤) = (1st ‘𝑐), if(𝑤 = 𝑐, 1 , 0 ), (𝑎‘𝑤)) ∈ V
506	502, 352, 505	fvmpt 6191	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ (𝑤 ∈ (𝑁 × 𝑁) → ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒)))‘𝑤) = if((1st ‘𝑤) = (1st ‘𝑐), if(𝑤 = 𝑐, 1 , 0 ), (𝑎‘𝑤)))
507	506	eqeq1d 2612	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (𝑤 ∈ (𝑁 × 𝑁) → (((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒)))‘𝑤) = if(𝑤 ∈ (𝑑 sSet ⟨(1st ‘𝑐), (2nd ‘𝑐)⟩), 1 , 0 ) ↔ if((1st ‘𝑤) = (1st ‘𝑐), if(𝑤 = 𝑐, 1 , 0 ), (𝑎‘𝑤)) = if(𝑤 ∈ (𝑑 sSet ⟨(1st ‘𝑐), (2nd ‘𝑐)⟩), 1 , 0 )))
508	413, 507	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ ((((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) ∧ 𝑑 ∈ (𝑁 ↑𝑚 𝑁)) ∧ 𝑤 ∈ (𝑏 ∪ {𝑐})) → (((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒)))‘𝑤) = if(𝑤 ∈ (𝑑 sSet ⟨(1st ‘𝑐), (2nd ‘𝑐)⟩), 1 , 0 ) ↔ if((1st ‘𝑤) = (1st ‘𝑐), if(𝑤 = 𝑐, 1 , 0 ), (𝑎‘𝑤)) = if(𝑤 ∈ (𝑑 sSet ⟨(1st ‘𝑐), (2nd ‘𝑐)⟩), 1 , 0 )))
509	496, 508	sylibrd 248	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) ∧ 𝑑 ∈ (𝑁 ↑𝑚 𝑁)) ∧ 𝑤 ∈ (𝑏 ∪ {𝑐})) → ((𝑤 ∈ 𝑏 → (𝑎‘𝑤) = if(𝑤 ∈ 𝑑, 1 , 0 )) → ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒)))‘𝑤) = if(𝑤 ∈ (𝑑 sSet ⟨(1st ‘𝑐), (2nd ‘𝑐)⟩), 1 , 0 )))
510	509	ralimdva 2945	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) ∧ 𝑑 ∈ (𝑁 ↑𝑚 𝑁)) → (∀𝑤 ∈ (𝑏 ∪ {𝑐})(𝑤 ∈ 𝑏 → (𝑎‘𝑤) = if(𝑤 ∈ 𝑑, 1 , 0 )) → ∀𝑤 ∈ (𝑏 ∪ {𝑐})((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒)))‘𝑤) = if(𝑤 ∈ (𝑑 sSet ⟨(1st ‘𝑐), (2nd ‘𝑐)⟩), 1 , 0 )))
511	406, 510	syl5bi 231	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) ∧ 𝑑 ∈ (𝑁 ↑𝑚 𝑁)) → (∀𝑤 ∈ 𝑏 (𝑎‘𝑤) = if(𝑤 ∈ 𝑑, 1 , 0 ) → ∀𝑤 ∈ (𝑏 ∪ {𝑐})((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒)))‘𝑤) = if(𝑤 ∈ (𝑑 sSet ⟨(1st ‘𝑐), (2nd ‘𝑐)⟩), 1 , 0 )))
512	511	impr 647	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) ∧ (𝑑 ∈ (𝑁 ↑𝑚 𝑁) ∧ ∀𝑤 ∈ 𝑏 (𝑎‘𝑤) = if(𝑤 ∈ 𝑑, 1 , 0 ))) → ∀𝑤 ∈ (𝑏 ∪ {𝑐})((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒)))‘𝑤) = if(𝑤 ∈ (𝑑 sSet ⟨(1st ‘𝑐), (2nd ‘𝑐)⟩), 1 , 0 ))
513	512	3adantr1 1213	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) ∧ ((𝑏 ∪ {𝑐}) ∈ 𝑌 ∧ 𝑑 ∈ (𝑁 ↑𝑚 𝑁) ∧ ∀𝑤 ∈ 𝑏 (𝑎‘𝑤) = if(𝑤 ∈ 𝑑, 1 , 0 ))) → ∀𝑤 ∈ (𝑏 ∪ {𝑐})((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒)))‘𝑤) = if(𝑤 ∈ (𝑑 sSet ⟨(1st ‘𝑐), (2nd ‘𝑐)⟩), 1 , 0 ))
514	357	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) ∧ ((𝑏 ∪ {𝑐}) ∈ 𝑌 ∧ 𝑑 ∈ (𝑁 ↑𝑚 𝑁) ∧ ∀𝑤 ∈ 𝑏 (𝑎‘𝑤) = if(𝑤 ∈ 𝑑, 1 , 0 ))) → (𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒))) ∈ 𝐵)
515		simpr2 1061	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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))
516	515, 419	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) ∧ ((𝑏 ∪ {𝑐}) ∈ 𝑌 ∧ 𝑑 ∈ (𝑁 ↑𝑚 𝑁) ∧ ∀𝑤 ∈ 𝑏 (𝑎‘𝑤) = if(𝑤 ∈ 𝑑, 1 , 0 ))) → 𝑑:𝑁⟶𝑁)
517	125	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) ∧ ((𝑏 ∪ {𝑐}) ∈ 𝑌 ∧ 𝑑 ∈ (𝑁 ↑𝑚 𝑁) ∧ ∀𝑤 ∈ 𝑏 (𝑎‘𝑤) = if(𝑤 ∈ 𝑑, 1 , 0 ))) → (1st ‘𝑐) ∈ 𝑁)
518	423	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) ∧ ((𝑏 ∪ {𝑐}) ∈ 𝑌 ∧ 𝑑 ∈ (𝑁 ↑𝑚 𝑁) ∧ ∀𝑤 ∈ 𝑏 (𝑎‘𝑤) = if(𝑤 ∈ 𝑑, 1 , 0 ))) → (2nd ‘𝑐) ∈ 𝑁)
519	515, 516, 517, 518, 425	syl211anc 1324	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) ∧ ((𝑏 ∪ {𝑐}) ∈ 𝑌 ∧ 𝑑 ∈ (𝑁 ↑𝑚 𝑁) ∧ ∀𝑤 ∈ 𝑏 (𝑎‘𝑤) = if(𝑤 ∈ 𝑑, 1 , 0 ))) → (𝑑 sSet ⟨(1st ‘𝑐), (2nd ‘𝑐)⟩):𝑁⟶𝑁)
520	159, 159	elmapd 7758	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) → ((𝑑 sSet ⟨(1st ‘𝑐), (2nd ‘𝑐)⟩) ∈ (𝑁 ↑𝑚 𝑁) ↔ (𝑑 sSet ⟨(1st ‘𝑐), (2nd ‘𝑐)⟩):𝑁⟶𝑁))
521	520	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) ∧ ((𝑏 ∪ {𝑐}) ∈ 𝑌 ∧ 𝑑 ∈ (𝑁 ↑𝑚 𝑁) ∧ ∀𝑤 ∈ 𝑏 (𝑎‘𝑤) = if(𝑤 ∈ 𝑑, 1 , 0 ))) → ((𝑑 sSet ⟨(1st ‘𝑐), (2nd ‘𝑐)⟩) ∈ (𝑁 ↑𝑚 𝑁) ↔ (𝑑 sSet ⟨(1st ‘𝑐), (2nd ‘𝑐)⟩):𝑁⟶𝑁))
522	519, 521	mpbird 246	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) ∧ ((𝑏 ∪ {𝑐}) ∈ 𝑌 ∧ 𝑑 ∈ (𝑁 ↑𝑚 𝑁) ∧ ∀𝑤 ∈ 𝑏 (𝑎‘𝑤) = if(𝑤 ∈ 𝑑, 1 , 0 ))) → (𝑑 sSet ⟨(1st ‘𝑐), (2nd ‘𝑐)⟩) ∈ (𝑁 ↑𝑚 𝑁))
523		simpr1 1060	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)
524		raleq 3115	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (𝑥 = (𝑏 ∪ {𝑐}) → (∀𝑤 ∈ 𝑥 (𝑦‘𝑤) = if(𝑤 ∈ 𝑧, 1 , 0 ) ↔ ∀𝑤 ∈ (𝑏 ∪ {𝑐})(𝑦‘𝑤) = if(𝑤 ∈ 𝑧, 1 , 0 )))
525	524	imbi1d 330	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 )))
526	525	2ralbidv 2972	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 )))
527	526, 73	elab2g 3322	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 )))
528	527	ibi 255	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((𝑏 ∪ {𝑐}) ∈ 𝑌 → ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ (𝑁 ↑𝑚 𝑁)(∀𝑤 ∈ (𝑏 ∪ {𝑐})(𝑦‘𝑤) = if(𝑤 ∈ 𝑧, 1 , 0 ) → (𝐷‘𝑦) = 0 ))
529	523, 528	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) ∧ ((𝑏 ∪ {𝑐}) ∈ 𝑌 ∧ 𝑑 ∈ (𝑁 ↑𝑚 𝑁) ∧ ∀𝑤 ∈ 𝑏 (𝑎‘𝑤) = if(𝑤 ∈ 𝑑, 1 , 0 ))) → ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ (𝑁 ↑𝑚 𝑁)(∀𝑤 ∈ (𝑏 ∪ {𝑐})(𝑦‘𝑤) = if(𝑤 ∈ 𝑧, 1 , 0 ) → (𝐷‘𝑦) = 0 ))
530		fveq1 6102	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (𝑦 = (𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒))) → (𝑦‘𝑤) = ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒)))‘𝑤))
531	530	eqeq1d 2612	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝑦 = (𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒))) → ((𝑦‘𝑤) = if(𝑤 ∈ 𝑧, 1 , 0 ) ↔ ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒)))‘𝑤) = if(𝑤 ∈ 𝑧, 1 , 0 )))
532	531	ralbidv 2969	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝑦 = (𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒))) → (∀𝑤 ∈ (𝑏 ∪ {𝑐})(𝑦‘𝑤) = if(𝑤 ∈ 𝑧, 1 , 0 ) ↔ ∀𝑤 ∈ (𝑏 ∪ {𝑐})((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒)))‘𝑤) = if(𝑤 ∈ 𝑧, 1 , 0 )))
533		fveq2 6103	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝑦 = (𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒))) → (𝐷‘𝑦) = (𝐷‘(𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒)))))
534	533	eqeq1d 2612	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝑦 = (𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒))) → ((𝐷‘𝑦) = 0 ↔ (𝐷‘(𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒)))) = 0 ))
535	532, 534	imbi12d 333	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑦 = (𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒))) → ((∀𝑤 ∈ (𝑏 ∪ {𝑐})(𝑦‘𝑤) = if(𝑤 ∈ 𝑧, 1 , 0 ) → (𝐷‘𝑦) = 0 ) ↔ (∀𝑤 ∈ (𝑏 ∪ {𝑐})((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒)))‘𝑤) = if(𝑤 ∈ 𝑧, 1 , 0 ) → (𝐷‘(𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒)))) = 0 )))
536		eleq2 2677	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (𝑧 = (𝑑 sSet ⟨(1st ‘𝑐), (2nd ‘𝑐)⟩) → (𝑤 ∈ 𝑧 ↔ 𝑤 ∈ (𝑑 sSet ⟨(1st ‘𝑐), (2nd ‘𝑐)⟩)))
537	536	ifbid 4058	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (𝑧 = (𝑑 sSet ⟨(1st ‘𝑐), (2nd ‘𝑐)⟩) → if(𝑤 ∈ 𝑧, 1 , 0 ) = if(𝑤 ∈ (𝑑 sSet ⟨(1st ‘𝑐), (2nd ‘𝑐)⟩), 1 , 0 ))
538	537	eqeq2d 2620	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝑧 = (𝑑 sSet ⟨(1st ‘𝑐), (2nd ‘𝑐)⟩) → (((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒)))‘𝑤) = if(𝑤 ∈ 𝑧, 1 , 0 ) ↔ ((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒)))‘𝑤) = if(𝑤 ∈ (𝑑 sSet ⟨(1st ‘𝑐), (2nd ‘𝑐)⟩), 1 , 0 )))
539	538	ralbidv 2969	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝑧 = (𝑑 sSet ⟨(1st ‘𝑐), (2nd ‘𝑐)⟩) → (∀𝑤 ∈ (𝑏 ∪ {𝑐})((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒)))‘𝑤) = if(𝑤 ∈ 𝑧, 1 , 0 ) ↔ ∀𝑤 ∈ (𝑏 ∪ {𝑐})((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒)))‘𝑤) = if(𝑤 ∈ (𝑑 sSet ⟨(1st ‘𝑐), (2nd ‘𝑐)⟩), 1 , 0 )))
540	539	imbi1d 330	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑧 = (𝑑 sSet ⟨(1st ‘𝑐), (2nd ‘𝑐)⟩) → ((∀𝑤 ∈ (𝑏 ∪ {𝑐})((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒)))‘𝑤) = if(𝑤 ∈ 𝑧, 1 , 0 ) → (𝐷‘(𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒)))) = 0 ) ↔ (∀𝑤 ∈ (𝑏 ∪ {𝑐})((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒)))‘𝑤) = if(𝑤 ∈ (𝑑 sSet ⟨(1st ‘𝑐), (2nd ‘𝑐)⟩), 1 , 0 ) → (𝐷‘(𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒)))) = 0 )))
541	535, 540	rspc2va 3294	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒))) ∈ 𝐵 ∧ (𝑑 sSet ⟨(1st ‘𝑐), (2nd ‘𝑐)⟩) ∈ (𝑁 ↑𝑚 𝑁)) ∧ ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ (𝑁 ↑𝑚 𝑁)(∀𝑤 ∈ (𝑏 ∪ {𝑐})(𝑦‘𝑤) = if(𝑤 ∈ 𝑧, 1 , 0 ) → (𝐷‘𝑦) = 0 )) → (∀𝑤 ∈ (𝑏 ∪ {𝑐})((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒)))‘𝑤) = if(𝑤 ∈ (𝑑 sSet ⟨(1st ‘𝑐), (2nd ‘𝑐)⟩), 1 , 0 ) → (𝐷‘(𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒)))) = 0 ))
542	514, 522, 529, 541	syl21anc 1317	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) ∧ ((𝑏 ∪ {𝑐}) ∈ 𝑌 ∧ 𝑑 ∈ (𝑁 ↑𝑚 𝑁) ∧ ∀𝑤 ∈ 𝑏 (𝑎‘𝑤) = if(𝑤 ∈ 𝑑, 1 , 0 ))) → (∀𝑤 ∈ (𝑏 ∪ {𝑐})((𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒)))‘𝑤) = if(𝑤 ∈ (𝑑 sSet ⟨(1st ‘𝑐), (2nd ‘𝑐)⟩), 1 , 0 ) → (𝐷‘(𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒)))) = 0 ))
543	513, 542	mpd 15	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) ∧ ((𝑏 ∪ {𝑐}) ∈ 𝑌 ∧ 𝑑 ∈ (𝑁 ↑𝑚 𝑁) ∧ ∀𝑤 ∈ 𝑏 (𝑎‘𝑤) = if(𝑤 ∈ 𝑑, 1 , 0 ))) → (𝐷‘(𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒)))) = 0 )
544	543	oveq2d 6565	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) ∧ ((𝑏 ∪ {𝑐}) ∈ 𝑌 ∧ 𝑑 ∈ (𝑁 ↑𝑚 𝑁) ∧ ∀𝑤 ∈ 𝑏 (𝑎‘𝑤) = if(𝑤 ∈ 𝑑, 1 , 0 ))) → (((𝑎‘𝑐)(-g‘𝑅)if(𝑐 ∈ 𝑑, 1 , 0 )) · (𝐷‘(𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒))))) = (((𝑎‘𝑐)(-g‘𝑅)if(𝑐 ∈ 𝑑, 1 , 0 )) · 0 ))
545	119	unssad 3752	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) → 𝑏 ⊆ (𝑁 × 𝑁))
546	545	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) ∧ ((𝑏 ∪ {𝑐}) ∈ 𝑌 ∧ 𝑑 ∈ (𝑁 ↑𝑚 𝑁) ∧ ∀𝑤 ∈ 𝑏 (𝑎‘𝑤) = if(𝑤 ∈ 𝑑, 1 , 0 ))) → 𝑏 ⊆ (𝑁 × 𝑁))
547		simpr3 1062	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 ))
548		ssel2 3563	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ ((𝑏 ⊆ (𝑁 × 𝑁) ∧ 𝑤 ∈ 𝑏) → 𝑤 ∈ (𝑁 × 𝑁))
549	548	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (((𝑏 ⊆ (𝑁 × 𝑁) ∧ 𝑤 ∈ 𝑏) ∧ (𝑎‘𝑤) = if(𝑤 ∈ 𝑑, 1 , 0 )) → 𝑤 ∈ (𝑁 × 𝑁))
550		elequ1 1984	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ (𝑒 = 𝑤 → (𝑒 ∈ 𝑑 ↔ 𝑤 ∈ 𝑑))
551	550	ifbid 4058	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ (𝑒 = 𝑤 → if(𝑒 ∈ 𝑑, 1 , 0 ) = if(𝑤 ∈ 𝑑, 1 , 0 ))
552	499, 551, 501	ifbieq12d 4063	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (𝑒 = 𝑤 → if(𝑒 = 𝑐, if(𝑒 ∈ 𝑑, 1 , 0 ), (𝑎‘𝑒)) = if(𝑤 = 𝑐, if(𝑤 ∈ 𝑑, 1 , 0 ), (𝑎‘𝑤)))
553	168, 164	ifex 4106	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ if(𝑤 ∈ 𝑑, 1 , 0 ) ∈ V
554	553, 504	ifex 4106	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ if(𝑤 = 𝑐, if(𝑤 ∈ 𝑑, 1 , 0 ), (𝑎‘𝑤)) ∈ V
555	552, 216, 554	fvmpt 6191	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)))
556	549, 555	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (((𝑏 ⊆ (𝑁 × 𝑁) ∧ 𝑤 ∈ 𝑏) ∧ (𝑎‘𝑤) = if(𝑤 ∈ 𝑑, 1 , 0 )) → ((𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒 ∈ 𝑑, 1 , 0 ), (𝑎‘𝑒)))‘𝑤) = if(𝑤 = 𝑐, if(𝑤 ∈ 𝑑, 1 , 0 ), (𝑎‘𝑤)))
557		ifeq2 4041	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 )))
558	557	adantl 481	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (((𝑏 ⊆ (𝑁 × 𝑁) ∧ 𝑤 ∈ 𝑏) ∧ (𝑎‘𝑤) = if(𝑤 ∈ 𝑑, 1 , 0 )) → if(𝑤 = 𝑐, if(𝑤 ∈ 𝑑, 1 , 0 ), (𝑎‘𝑤)) = if(𝑤 = 𝑐, if(𝑤 ∈ 𝑑, 1 , 0 ), if(𝑤 ∈ 𝑑, 1 , 0 )))
559		ifid 4075	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ if(𝑤 = 𝑐, if(𝑤 ∈ 𝑑, 1 , 0 ), if(𝑤 ∈ 𝑑, 1 , 0 )) = if(𝑤 ∈ 𝑑, 1 , 0 )
560	558, 559	syl6eq 2660	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 ))
561	556, 560	eqtrd 2644	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 ))
562	561	ex 449	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 )))
563	562	ralimdva 2945	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 )))
564	546, 547, 563	sylc 63	. . . . . . . . . . . . . . . . . . . . . . . . . . . 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 ))
565	143, 298	eqtrd 2644	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝑒 = 𝑐 → if(𝑒 = 𝑐, if(𝑒 ∈ 𝑑, 1 , 0 ), (𝑎‘𝑒)) = if(𝑐 ∈ 𝑑, 1 , 0 ))
566	168, 164	ifex 4106	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ if(𝑐 ∈ 𝑑, 1 , 0 ) ∈ V
567	565, 216, 566	fvmpt 6191	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑐 ∈ (𝑁 × 𝑁) → ((𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒 ∈ 𝑑, 1 , 0 ), (𝑎‘𝑒)))‘𝑐) = if(𝑐 ∈ 𝑑, 1 , 0 ))
568	123, 567	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) → ((𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒 ∈ 𝑑, 1 , 0 ), (𝑎‘𝑒)))‘𝑐) = if(𝑐 ∈ 𝑑, 1 , 0 ))
569	568	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) ∧ ((𝑏 ∪ {𝑐}) ∈ 𝑌 ∧ 𝑑 ∈ (𝑁 ↑𝑚 𝑁) ∧ ∀𝑤 ∈ 𝑏 (𝑎‘𝑤) = if(𝑤 ∈ 𝑑, 1 , 0 ))) → ((𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒 ∈ 𝑑, 1 , 0 ), (𝑎‘𝑒)))‘𝑐) = if(𝑐 ∈ 𝑑, 1 , 0 ))
570		fveq2 6103	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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))
571		elequ1 1984	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝑤 = 𝑐 → (𝑤 ∈ 𝑑 ↔ 𝑐 ∈ 𝑑))
572	571	ifbid 4058	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝑤 = 𝑐 → if(𝑤 ∈ 𝑑, 1 , 0 ) = if(𝑐 ∈ 𝑑, 1 , 0 ))
573	570, 572	eqeq12d 2625	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 )))
574	573	ralunsn 4360	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 ))))
575	121, 574	ax-mp 5	. . . . . . . . . . . . . . . . . . . . . . . . . . . 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 )))
576	564, 569, 575	sylanbrc 695	. . . . . . . . . . . . . . . . . . . . . . . . . . 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 ))
577	228	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) ∧ ((𝑏 ∪ {𝑐}) ∈ 𝑌 ∧ 𝑑 ∈ (𝑁 ↑𝑚 𝑁) ∧ ∀𝑤 ∈ 𝑏 (𝑎‘𝑤) = if(𝑤 ∈ 𝑑, 1 , 0 ))) → (𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒 ∈ 𝑑, 1 , 0 ), (𝑎‘𝑒))) ∈ 𝐵)
578		fveq1 6102	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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))
579	578	eqeq1d 2612	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 )))
580	579	ralbidv 2969	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 )))
581		fveq2 6103	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝑦 = (𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒 ∈ 𝑑, 1 , 0 ), (𝑎‘𝑒))) → (𝐷‘𝑦) = (𝐷‘(𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒 ∈ 𝑑, 1 , 0 ), (𝑎‘𝑒)))))
582	581	eqeq1d 2612	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 ))
583	580, 582	imbi12d 333	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 )))
584		elequ2 1991	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (𝑧 = 𝑑 → (𝑤 ∈ 𝑧 ↔ 𝑤 ∈ 𝑑))
585	584	ifbid 4058	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝑧 = 𝑑 → if(𝑤 ∈ 𝑧, 1 , 0 ) = if(𝑤 ∈ 𝑑, 1 , 0 ))
586	585	eqeq2d 2620	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 )))
587	586	ralbidv 2969	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 )))
588	587	imbi1d 330	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 )))
589	583, 588	rspc2va 3294	. . . . . . . . . . . . . . . . . . . . . . . . . . . 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 ))
590	577, 515, 529, 589	syl21anc 1317	. . . . . . . . . . . . . . . . . . . . . . . . . . 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 ))
591	576, 590	mpd 15	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) ∧ ((𝑏 ∪ {𝑐}) ∈ 𝑌 ∧ 𝑑 ∈ (𝑁 ↑𝑚 𝑁) ∧ ∀𝑤 ∈ 𝑏 (𝑎‘𝑤) = if(𝑤 ∈ 𝑑, 1 , 0 ))) → (𝐷‘(𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒 ∈ 𝑑, 1 , 0 ), (𝑎‘𝑒)))) = 0 )
592	544, 591	oveq12d 6567	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) ∧ ((𝑏 ∪ {𝑐}) ∈ 𝑌 ∧ 𝑑 ∈ (𝑁 ↑𝑚 𝑁) ∧ ∀𝑤 ∈ 𝑏 (𝑎‘𝑤) = if(𝑤 ∈ 𝑑, 1 , 0 ))) → ((((𝑎‘𝑐)(-g‘𝑅)if(𝑐 ∈ 𝑑, 1 , 0 )) · (𝐷‘(𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒))))) + (𝐷‘(𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒 ∈ 𝑑, 1 , 0 ), (𝑎‘𝑒))))) = ((((𝑎‘𝑐)(-g‘𝑅)if(𝑐 ∈ 𝑑, 1 , 0 )) · 0 ) + 0 ))
593	315	oveq1d 6564	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) → ((((𝑎‘𝑐)(-g‘𝑅)if(𝑐 ∈ 𝑑, 1 , 0 )) · 0 ) + 0 ) = ( 0 + 0 ))
594	12, 51, 49	grplid 17275	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝑅 ∈ Grp ∧ 0 ∈ 𝐾) → ( 0 + 0 ) = 0 )
595	116, 134, 594	syl2anc 691	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) → ( 0 + 0 ) = 0 )
596	593, 595	eqtrd 2644	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) → ((((𝑎‘𝑐)(-g‘𝑅)if(𝑐 ∈ 𝑑, 1 , 0 )) · 0 ) + 0 ) = 0 )
597	596	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) ∧ ((𝑏 ∪ {𝑐}) ∈ 𝑌 ∧ 𝑑 ∈ (𝑁 ↑𝑚 𝑁) ∧ ∀𝑤 ∈ 𝑏 (𝑎‘𝑤) = if(𝑤 ∈ 𝑑, 1 , 0 ))) → ((((𝑎‘𝑐)(-g‘𝑅)if(𝑐 ∈ 𝑑, 1 , 0 )) · 0 ) + 0 ) = 0 )
598	592, 597	eqtrd 2644	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ 𝑎 ∈ 𝐵) ∧ ((𝑏 ∪ {𝑐}) ∈ 𝑌 ∧ 𝑑 ∈ (𝑁 ↑𝑚 𝑁) ∧ ∀𝑤 ∈ 𝑏 (𝑎‘𝑤) = if(𝑤 ∈ 𝑑, 1 , 0 ))) → ((((𝑎‘𝑐)(-g‘𝑅)if(𝑐 ∈ 𝑑, 1 , 0 )) · (𝐷‘(𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒))))) + (𝐷‘(𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒 ∈ 𝑑, 1 , 0 ), (𝑎‘𝑒))))) = 0 )
599	105, 106, 107, 402, 403, 404, 598	syl33anc 1333	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ (𝑏 ∪ {𝑐}) ∈ 𝑌) ∧ ((𝑎 ∈ 𝐵 ∧ 𝑑 ∈ (𝑁 ↑𝑚 𝑁)) ∧ ∀𝑤 ∈ 𝑏 (𝑎‘𝑤) = if(𝑤 ∈ 𝑑, 1 , 0 ))) → ((((𝑎‘𝑐)(-g‘𝑅)if(𝑐 ∈ 𝑑, 1 , 0 )) · (𝐷‘(𝑒 ∈ (𝑁 × 𝑁) ↦ if((1st ‘𝑒) = (1st ‘𝑐), if(𝑒 = 𝑐, 1 , 0 ), (𝑎‘𝑒))))) + (𝐷‘(𝑒 ∈ (𝑁 × 𝑁) ↦ if(𝑒 = 𝑐, if(𝑒 ∈ 𝑑, 1 , 0 ), (𝑎‘𝑒))))) = 0 )
600	295, 401, 599	3eqtrd 2648	. . . . . . . . . . . . . . . . . . . . . 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 )
601	600	expr 641	. . . . . . . . . . . . . . . . . . . . 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 ))
602	601	ralrimivva 2954	. . . . . . . . . . . . . . . . . . . 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 ))
603		fveq1 6102	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑎 = 𝑦 → (𝑎‘𝑤) = (𝑦‘𝑤))
604	603	eqeq1d 2612	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑎 = 𝑦 → ((𝑎‘𝑤) = if(𝑤 ∈ 𝑑, 1 , 0 ) ↔ (𝑦‘𝑤) = if(𝑤 ∈ 𝑑, 1 , 0 )))
605	604	ralbidv 2969	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑎 = 𝑦 → (∀𝑤 ∈ 𝑏 (𝑎‘𝑤) = if(𝑤 ∈ 𝑑, 1 , 0 ) ↔ ∀𝑤 ∈ 𝑏 (𝑦‘𝑤) = if(𝑤 ∈ 𝑑, 1 , 0 )))
606		fveq2 6103	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑎 = 𝑦 → (𝐷‘𝑎) = (𝐷‘𝑦))
607	606	eqeq1d 2612	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑎 = 𝑦 → ((𝐷‘𝑎) = 0 ↔ (𝐷‘𝑦) = 0 ))
608	605, 607	imbi12d 333	. . . . . . . . . . . . . . . . . . . . 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 )))
609		elequ2 1991	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑑 = 𝑧 → (𝑤 ∈ 𝑑 ↔ 𝑤 ∈ 𝑧))
610	609	ifbid 4058	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑑 = 𝑧 → if(𝑤 ∈ 𝑑, 1 , 0 ) = if(𝑤 ∈ 𝑧, 1 , 0 ))
611	610	eqeq2d 2620	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑑 = 𝑧 → ((𝑦‘𝑤) = if(𝑤 ∈ 𝑑, 1 , 0 ) ↔ (𝑦‘𝑤) = if(𝑤 ∈ 𝑧, 1 , 0 )))
612	611	ralbidv 2969	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑑 = 𝑧 → (∀𝑤 ∈ 𝑏 (𝑦‘𝑤) = if(𝑤 ∈ 𝑑, 1 , 0 ) ↔ ∀𝑤 ∈ 𝑏 (𝑦‘𝑤) = if(𝑤 ∈ 𝑧, 1 , 0 )))
613	612	imbi1d 330	. . . . . . . . . . . . . . . . . . . . 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 )))
614	608, 613	cbvral2v 3155	. . . . . . . . . . . . . . . . . . . 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 ))
615	602, 614	sylib 207	. . . . . . . . . . . . . . . . . . 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 ))
616		vex 3176	. . . . . . . . . . . . . . . . . . . 20 ⊢ 𝑏 ∈ V
617		raleq 3115	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑥 = 𝑏 → (∀𝑤 ∈ 𝑥 (𝑦‘𝑤) = if(𝑤 ∈ 𝑧, 1 , 0 ) ↔ ∀𝑤 ∈ 𝑏 (𝑦‘𝑤) = if(𝑤 ∈ 𝑧, 1 , 0 )))
618	617	imbi1d 330	. . . . . . . . . . . . . . . . . . . . 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 )))
619	618	2ralbidv 2972	. . . . . . . . . . . . . . . . . . . 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 )))
620	616, 619, 73	elab2 3323	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑏 ∈ 𝑌 ↔ ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ (𝑁 ↑𝑚 𝑁)(∀𝑤 ∈ 𝑏 (𝑦‘𝑤) = if(𝑤 ∈ 𝑧, 1 , 0 ) → (𝐷‘𝑦) = 0 ))
621	615, 620	sylibr 223	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ (𝑏 ∪ {𝑐}) ∈ 𝑌) → 𝑏 ∈ 𝑌)
622	621	3expia 1259	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁)) → ((𝑏 ∪ {𝑐}) ∈ 𝑌 → 𝑏 ∈ 𝑌))
623	622	con3d 147	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁)) → (¬ 𝑏 ∈ 𝑌 → ¬ (𝑏 ∪ {𝑐}) ∈ 𝑌))
624	623	3adant3 1074	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ ¬ ∅ ∈ 𝑌) → (¬ 𝑏 ∈ 𝑌 → ¬ (𝑏 ∪ {𝑐}) ∈ 𝑌))
625	624	a1i 11	. . . . . . . . . . . . . 14 ⊢ ((𝑏 ∈ Fin ∧ ¬ 𝑐 ∈ 𝑏) → ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ ¬ ∅ ∈ 𝑌) → (¬ 𝑏 ∈ 𝑌 → ¬ (𝑏 ∪ {𝑐}) ∈ 𝑌)))
626	625	a2d 29	. . . . . . . . . . . . 13 ⊢ ((𝑏 ∈ Fin ∧ ¬ 𝑐 ∈ 𝑏) → (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ ¬ ∅ ∈ 𝑌) → ¬ 𝑏 ∈ 𝑌) → ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ ¬ ∅ ∈ 𝑌) → ¬ (𝑏 ∪ {𝑐}) ∈ 𝑌)))
627	104, 626	syl5 33	. . . . . . . . . . . 12 ⊢ ((𝑏 ∈ Fin ∧ ¬ 𝑐 ∈ 𝑏) → (((𝜑 ∧ 𝑏 ⊆ (𝑁 × 𝑁) ∧ ¬ ∅ ∈ 𝑌) → ¬ 𝑏 ∈ 𝑌) → ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑁 × 𝑁) ∧ ¬ ∅ ∈ 𝑌) → ¬ (𝑏 ∪ {𝑐}) ∈ 𝑌)))
628	83, 88, 93, 98, 99, 627	findcard2s 8086	. . . . . . . . . . 11 ⊢ ((𝑁 × 𝑁) ∈ Fin → ((𝜑 ∧ (𝑁 × 𝑁) ⊆ (𝑁 × 𝑁) ∧ ¬ ∅ ∈ 𝑌) → ¬ (𝑁 × 𝑁) ∈ 𝑌))
629	78, 628	mpcom 37	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑁 × 𝑁) ⊆ (𝑁 × 𝑁) ∧ ¬ ∅ ∈ 𝑌) → ¬ (𝑁 × 𝑁) ∈ 𝑌)
630	629	3exp 1256	. . . . . . . . 9 ⊢ (𝜑 → ((𝑁 × 𝑁) ⊆ (𝑁 × 𝑁) → (¬ ∅ ∈ 𝑌 → ¬ (𝑁 × 𝑁) ∈ 𝑌)))
631	77, 630	mpi 20	. . . . . . . 8 ⊢ (𝜑 → (¬ ∅ ∈ 𝑌 → ¬ (𝑁 × 𝑁) ∈ 𝑌))
632	76, 631	mt4d 151	. . . . . . 7 ⊢ (𝜑 → ∅ ∈ 𝑌)
633	632	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐵) → ∅ ∈ 𝑌)
634		0ex 4718	. . . . . . 7 ⊢ ∅ ∈ V
635		raleq 3115	. . . . . . . . 9 ⊢ (𝑥 = ∅ → (∀𝑤 ∈ 𝑥 (𝑦‘𝑤) = if(𝑤 ∈ 𝑧, 1 , 0 ) ↔ ∀𝑤 ∈ ∅ (𝑦‘𝑤) = if(𝑤 ∈ 𝑧, 1 , 0 )))
636	635	imbi1d 330	. . . . . . . 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 )))
637	636	2ralbidv 2972	. . . . . . 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 )))
638	634, 637, 73	elab2 3323	. . . . . 6 ⊢ (∅ ∈ 𝑌 ↔ ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ (𝑁 ↑𝑚 𝑁)(∀𝑤 ∈ ∅ (𝑦‘𝑤) = if(𝑤 ∈ 𝑧, 1 , 0 ) → (𝐷‘𝑦) = 0 ))
639	633, 638	sylib 207	. . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐵) → ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ (𝑁 ↑𝑚 𝑁)(∀𝑤 ∈ ∅ (𝑦‘𝑤) = if(𝑤 ∈ 𝑧, 1 , 0 ) → (𝐷‘𝑦) = 0 ))
640		fveq1 6102	. . . . . . . . 9 ⊢ (𝑦 = 𝑎 → (𝑦‘𝑤) = (𝑎‘𝑤))
641	640	eqeq1d 2612	. . . . . . . 8 ⊢ (𝑦 = 𝑎 → ((𝑦‘𝑤) = if(𝑤 ∈ 𝑧, 1 , 0 ) ↔ (𝑎‘𝑤) = if(𝑤 ∈ 𝑧, 1 , 0 )))
642	641	ralbidv 2969	. . . . . . 7 ⊢ (𝑦 = 𝑎 → (∀𝑤 ∈ ∅ (𝑦‘𝑤) = if(𝑤 ∈ 𝑧, 1 , 0 ) ↔ ∀𝑤 ∈ ∅ (𝑎‘𝑤) = if(𝑤 ∈ 𝑧, 1 , 0 )))
643		fveq2 6103	. . . . . . . 8 ⊢ (𝑦 = 𝑎 → (𝐷‘𝑦) = (𝐷‘𝑎))
644	643	eqeq1d 2612	. . . . . . 7 ⊢ (𝑦 = 𝑎 → ((𝐷‘𝑦) = 0 ↔ (𝐷‘𝑎) = 0 ))
645	642, 644	imbi12d 333	. . . . . 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 )))
646		eleq2 2677	. . . . . . . . . 10 ⊢ (𝑧 = ( I ↾ 𝑁) → (𝑤 ∈ 𝑧 ↔ 𝑤 ∈ ( I ↾ 𝑁)))
647	646	ifbid 4058	. . . . . . . . 9 ⊢ (𝑧 = ( I ↾ 𝑁) → if(𝑤 ∈ 𝑧, 1 , 0 ) = if(𝑤 ∈ ( I ↾ 𝑁), 1 , 0 ))
648	647	eqeq2d 2620	. . . . . . . 8 ⊢ (𝑧 = ( I ↾ 𝑁) → ((𝑎‘𝑤) = if(𝑤 ∈ 𝑧, 1 , 0 ) ↔ (𝑎‘𝑤) = if(𝑤 ∈ ( I ↾ 𝑁), 1 , 0 )))
649	648	ralbidv 2969	. . . . . . 7 ⊢ (𝑧 = ( I ↾ 𝑁) → (∀𝑤 ∈ ∅ (𝑎‘𝑤) = if(𝑤 ∈ 𝑧, 1 , 0 ) ↔ ∀𝑤 ∈ ∅ (𝑎‘𝑤) = if(𝑤 ∈ ( I ↾ 𝑁), 1 , 0 )))
650	649	imbi1d 330	. . . . . 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 )))
651	645, 650	rspc2va 3294	. . . . 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 ))
652	2, 9, 639, 651	syl21anc 1317	. . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐵) → (∀𝑤 ∈ ∅ (𝑎‘𝑤) = if(𝑤 ∈ ( I ↾ 𝑁), 1 , 0 ) → (𝐷‘𝑎) = 0 ))
653	1, 652	mpi 20	. . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐵) → (𝐷‘𝑎) = 0 )
654	653	mpteq2dva 4672	. 2 ⊢ (𝜑 → (𝑎 ∈ 𝐵 ↦ (𝐷‘𝑎)) = (𝑎 ∈ 𝐵 ↦ 0 ))
655	54	feqmptd 6159	. 2 ⊢ (𝜑 → 𝐷 = (𝑎 ∈ 𝐵 ↦ (𝐷‘𝑎)))
656		fconstmpt 5085	. . 3 ⊢ (𝐵 × { 0 }) = (𝑎 ∈ 𝐵 ↦ 0 )
657	656	a1i 11	. 2 ⊢ (𝜑 → (𝐵 × { 0 }) = (𝑎 ∈ 𝐵 ↦ 0 ))
658	654, 655, 657	3eqtr4d 2654	1 ⊢ (𝜑 → 𝐷 = (𝐵 × { 0 }))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 195   ∨ wo 382   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  {cab 2596   ≠ wne 2780  ∀wral 2896  Vcvv 3173   ∖ cdif 3537   ∪ cun 3538   ⊆ wss 3540  ∅c0 3874  ifcif 4036  {csn 4125  ⟨cop 4131   ↦ cmpt 4643   I cid 4948   × cxp 5036   ↾ cres 5040  Rel wrel 5043   Fn wfn 5799  ⟶wf 5800  –1-1-onto→wf1o 5803  ‘cfv 5804  (class class class)co 6549   ↦ cmpt2 6551   ∘_𝑓 cof 6793  1^st c1st 7057  2^nd c2nd 7058   ↑_𝑚 cmap 7744  Fincfn 7841   sSet csts 15693  Basecbs 15695  +_gcplusg 15768  ._rcmulr 15769  0_gc0g 15923  Grpcgrp 17245  -_gcsg 17247  1_rcur 18324  Ringcrg 18370   Mat cmat 20032

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-inf2 8421  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892  ax-addf 9894  ax-mulf 9895

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-xor 1457  df-tru 1478  df-fal 1481  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-ot 4134  df-uni 4373  df-int 4411  df-iun 4457  df-iin 4458  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-se 4998  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-isom 5813  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-of 6795  df-om 6958  df-1st 7059  df-2nd 7060  df-supp 7183  df-tpos 7239  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-2o 7448  df-oadd 7451  df-er 7629  df-map 7746  df-ixp 7795  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-fsupp 8159  df-sup 8231  df-oi 8298  df-card 8648  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-div 10564  df-nn 10898  df-2 10956  df-3 10957  df-4 10958  df-5 10959  df-6 10960  df-7 10961  df-8 10962  df-9 10963  df-n0 11170  df-xnn0 11241  df-z 11255  df-dec 11370  df-uz 11564  df-rp 11709  df-fz 12198  df-fzo 12335  df-seq 12664  df-exp 12723  df-hash 12980  df-word 13154  df-lsw 13155  df-concat 13156  df-s1 13157  df-substr 13158  df-splice 13159  df-reverse 13160  df-s2 13444  df-struct 15697  df-ndx 15698  df-slot 15699  df-base 15700  df-sets 15701  df-ress 15702  df-plusg 15781  df-mulr 15782  df-starv 15783  df-sca 15784  df-vsca 15785  df-ip 15786  df-tset 15787  df-ple 15788  df-ds 15791  df-unif 15792  df-hom 15793  df-cco 15794  df-0g 15925  df-gsum 15926  df-prds 15931  df-pws 15933  df-mre 16069  df-mrc 16070  df-acs 16072  df-mgm 17065  df-sgrp 17107  df-mnd 17118  df-mhm 17158  df-submnd 17159  df-grp 17248  df-minusg 17249  df-sbg 17250  df-mulg 17364  df-subg 17414  df-ghm 17481  df-gim 17524  df-cntz 17573  df-oppg 17599  df-symg 17621  df-pmtr 17685  df-psgn 17734  df-evpm 17735  df-cmn 18018  df-abl 18019  df-mgp 18313  df-ur 18325  df-ring 18372  df-cring 18373  df-oppr 18446  df-dvdsr 18464  df-unit 18465  df-invr 18495  df-dvr 18506  df-rnghom 18538  df-drng 18572  df-subrg 18601  df-lmod 18688  df-lss 18754  df-sra 18993  df-rgmod 18994  df-cnfld 19568  df-zring 19638  df-zrh 19671  df-dsmm 19895  df-frlm 19910  df-mamu 20009  df-mat 20033

This theorem is referenced by:  mdetuni0  20246