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Theorem mdetunilem6 22842

Description: Lemma for mdetuni 22847. (Contributed by SO, 15-Jul-2018.)

**Hypotheses**
Ref	Expression
mdetuni.a	⊢ 𝐴 = (𝑁 Mat 𝑅)
mdetuni.b	⊢ 𝐵 = (Base‘𝐴)
mdetuni.k	⊢ 𝐾 = (Base‘𝑅)
mdetuni.0g	⊢ 0 = (0_g‘𝑅)
mdetuni.1r	⊢ 1 = (1_r‘𝑅)
mdetuni.pg	⊢ + = (+_g‘𝑅)
mdetuni.tg	⊢ · = (._r‘𝑅)
mdetuni.n	⊢ (𝜑 → 𝑁 ∈ Fin)
mdetuni.r	⊢ (𝜑 → 𝑅 ∈ Ring)
mdetuni.ff	⊢ (𝜑 → 𝐷:𝐵⟶𝐾)
mdetuni.al	⊢ (𝜑 → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝑁 ∀𝑧 ∈ 𝑁 ((𝑦 ≠ 𝑧 ∧ ∀𝑤 ∈ 𝑁 (𝑦𝑥𝑤) = (𝑧𝑥𝑤)) → (𝐷‘𝑥) = 0 ))
mdetuni.li	⊢ (𝜑 → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝑁 (((𝑥 ↾ ({𝑤} × 𝑁)) = ((𝑦 ↾ ({𝑤} × 𝑁)) ∘_f + (𝑧 ↾ ({𝑤} × 𝑁))) ∧ (𝑥 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑦 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) ∧ (𝑥 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑧 ↾ ((𝑁 ∖ {𝑤}) × 𝑁))) → (𝐷‘𝑥) = ((𝐷‘𝑦) + (𝐷‘𝑧))))
mdetuni.sc	⊢ (𝜑 → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐾 ∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝑁 (((𝑥 ↾ ({𝑤} × 𝑁)) = ((({𝑤} × 𝑁) × {𝑦}) ∘_f · (𝑧 ↾ ({𝑤} × 𝑁))) ∧ (𝑥 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑧 ↾ ((𝑁 ∖ {𝑤}) × 𝑁))) → (𝐷‘𝑥) = (𝑦 · (𝐷‘𝑧))))
mdetunilem6.ph	⊢ (𝜓 → 𝜑)
mdetunilem6.ef	⊢ (𝜓 → (𝐸 ∈ 𝑁 ∧ 𝐹 ∈ 𝑁 ∧ 𝐸 ≠ 𝐹))
mdetunilem6.gh	⊢ ((𝜓 ∧ 𝑏 ∈ 𝑁) → (𝐺 ∈ 𝐾 ∧ 𝐻 ∈ 𝐾))
mdetunilem6.i	⊢ ((𝜓 ∧ 𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁) → 𝐼 ∈ 𝐾)

**Assertion**
Ref	Expression
mdetunilem6	⊢ (𝜓 → (𝐷‘(𝑎 ∈ 𝑁, 𝑏 ∈ 𝑁 ↦ if(𝑎 = 𝐸, 𝐺, if(𝑎 = 𝐹, 𝐻, 𝐼)))) = ((inv_g‘𝑅)‘(𝐷‘(𝑎 ∈ 𝑁, 𝑏 ∈ 𝑁 ↦ if(𝑎 = 𝐸, 𝐻, if(𝑎 = 𝐹, 𝐺, 𝐼))))))

Distinct variable groups: 𝜑,𝑥,𝑦,𝑧,𝑤,𝑎,𝑏 𝑥,𝐵,𝑦,𝑧,𝑤,𝑎,𝑏 𝑥,𝐾,𝑦,𝑧,𝑤,𝑎,𝑏 𝑥,𝑁,𝑦,𝑧,𝑤,𝑎,𝑏 𝑥,𝐷,𝑦,𝑧,𝑤,𝑎,𝑏 𝑥, · ,𝑦,𝑧,𝑤 + ,𝑎,𝑏,𝑥,𝑦,𝑧,𝑤 0 ,𝑎,𝑏,𝑥,𝑦,𝑧,𝑤 1 ,𝑎,𝑏,𝑥,𝑦,𝑧,𝑤 𝑥,𝑅,𝑦,𝑧,𝑤 𝐴,𝑎,𝑏,𝑥,𝑦,𝑧,𝑤 𝑥,𝐸,𝑦,𝑧,𝑤 𝑥,𝐹,𝑦,𝑧,𝑤 𝑥,𝐺,𝑦,𝑧,𝑤 𝑥,𝐻,𝑦,𝑧,𝑤 𝜓,𝑎,𝑏,𝑥,𝑦,𝑧,𝑤 𝐸,𝑎,𝑏 𝐹,𝑎,𝑏 𝐺,𝑎 𝐻,𝑎 𝑥,𝐼,𝑦,𝑧,𝑤 Allowed substitution hints: 𝑅(𝑎, 𝑏) · (𝑎, 𝑏) 𝐺(𝑏) 𝐻(𝑏) 𝐼(𝑎, 𝑏)

**Proof of Theorem mdetunilem6**
Step	Hyp	Ref	Expression
1		mdetuni.a	. . . . 5 ⊢ 𝐴 = (𝑁 Mat 𝑅)
2		mdetuni.b	. . . . 5 ⊢ 𝐵 = (Base‘𝐴)
3		mdetuni.k	. . . . 5 ⊢ 𝐾 = (Base‘𝑅)
4		mdetuni.0g	. . . . 5 ⊢ 0 = (0_g‘𝑅)
5		mdetuni.1r	. . . . 5 ⊢ 1 = (1_r‘𝑅)
6		mdetuni.pg	. . . . 5 ⊢ + = (+_g‘𝑅)
7		mdetuni.tg	. . . . 5 ⊢ · = (._r‘𝑅)
8		mdetuni.n	. . . . 5 ⊢ (𝜑 → 𝑁 ∈ Fin)
9		mdetuni.r	. . . . 5 ⊢ (𝜑 → 𝑅 ∈ Ring)
10		mdetuni.ff	. . . . 5 ⊢ (𝜑 → 𝐷:𝐵⟶𝐾)
11		mdetuni.al	. . . . 5 ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝑁 ∀𝑧 ∈ 𝑁 ((𝑦 ≠ 𝑧 ∧ ∀𝑤 ∈ 𝑁 (𝑦𝑥𝑤) = (𝑧𝑥𝑤)) → (𝐷‘𝑥) = 0 ))
12		mdetuni.li	. . . . 5 ⊢ (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))))
13		mdetuni.sc	. . . . 5 ⊢ (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))))
14		mdetunilem6.ph	. . . . 5 ⊢ (𝜓 → 𝜑)
15		mdetunilem6.ef	. . . . . 6 ⊢ (𝜓 → (𝐸 ∈ 𝑁 ∧ 𝐹 ∈ 𝑁 ∧ 𝐸 ≠ 𝐹))
16	15	simp1d 1160	. . . . 5 ⊢ (𝜓 → 𝐸 ∈ 𝑁)
17		mdetunilem6.gh	. . . . . . . 8 ⊢ ((𝜓 ∧ 𝑏 ∈ 𝑁) → (𝐺 ∈ 𝐾 ∧ 𝐻 ∈ 𝐾))
18	17	simprd 501	. . . . . . 7 ⊢ ((𝜓 ∧ 𝑏 ∈ 𝑁) → 𝐻 ∈ 𝐾)
19	18	3adant2 1149	. . . . . 6 ⊢ ((𝜓 ∧ 𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁) → 𝐻 ∈ 𝐾)
20	17	simpld 500	. . . . . . 7 ⊢ ((𝜓 ∧ 𝑏 ∈ 𝑁) → 𝐺 ∈ 𝐾)
21	20	3adant2 1149	. . . . . 6 ⊢ ((𝜓 ∧ 𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁) → 𝐺 ∈ 𝐾)
22		ringgrp 20380	. . . . . . . . . . 11 ⊢ (𝑅 ∈ Ring → 𝑅 ∈ Grp)
23	14, 9, 22	3syl 19	. . . . . . . . . 10 ⊢ (𝜓 → 𝑅 ∈ Grp)
24	23	adantr 486	. . . . . . . . 9 ⊢ ((𝜓 ∧ 𝑏 ∈ 𝑁) → 𝑅 ∈ Grp)
25	3, 6	grpcl 19068	. . . . . . . . 9 ⊢ ((𝑅 ∈ Grp ∧ 𝐻 ∈ 𝐾 ∧ 𝐺 ∈ 𝐾) → (𝐻 + 𝐺) ∈ 𝐾)
26	24, 18, 20, 25	syl3anc 1398	. . . . . . . 8 ⊢ ((𝜓 ∧ 𝑏 ∈ 𝑁) → (𝐻 + 𝐺) ∈ 𝐾)
27	26	3adant2 1149	. . . . . . 7 ⊢ ((𝜓 ∧ 𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁) → (𝐻 + 𝐺) ∈ 𝐾)
28		mdetunilem6.i	. . . . . . 7 ⊢ ((𝜓 ∧ 𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁) → 𝐼 ∈ 𝐾)
29	27, 28	ifcld 4529	. . . . . 6 ⊢ ((𝜓 ∧ 𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁) → if(𝑎 = 𝐹, (𝐻 + 𝐺), 𝐼) ∈ 𝐾)
30	19, 21, 29	3jca 1146	. . . . 5 ⊢ ((𝜓 ∧ 𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁) → (𝐻 ∈ 𝐾 ∧ 𝐺 ∈ 𝐾 ∧ if(𝑎 = 𝐹, (𝐻 + 𝐺), 𝐼) ∈ 𝐾))
31	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 16, 30	mdetunilem5 22841	. . . 4 ⊢ (𝜓 → (𝐷‘(𝑎 ∈ 𝑁, 𝑏 ∈ 𝑁 ↦ if(𝑎 = 𝐸, (𝐻 + 𝐺), if(𝑎 = 𝐹, (𝐻 + 𝐺), 𝐼)))) = ((𝐷‘(𝑎 ∈ 𝑁, 𝑏 ∈ 𝑁 ↦ if(𝑎 = 𝐸, 𝐻, if(𝑎 = 𝐹, (𝐻 + 𝐺), 𝐼)))) + (𝐷‘(𝑎 ∈ 𝑁, 𝑏 ∈ 𝑁 ↦ if(𝑎 = 𝐸, 𝐺, if(𝑎 = 𝐹, (𝐻 + 𝐺), 𝐼))))))
32	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 26, 28	mdetunilem2 22838	. . . 4 ⊢ (𝜓 → (𝐷‘(𝑎 ∈ 𝑁, 𝑏 ∈ 𝑁 ↦ if(𝑎 = 𝐸, (𝐻 + 𝐺), if(𝑎 = 𝐹, (𝐻 + 𝐺), 𝐼)))) = 0 )
33	15	simp2d 1161	. . . . . . . 8 ⊢ (𝜓 → 𝐹 ∈ 𝑁)
34	19, 28	ifcld 4529	. . . . . . . . 9 ⊢ ((𝜓 ∧ 𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁) → if(𝑎 = 𝐸, 𝐻, 𝐼) ∈ 𝐾)
35	19, 21, 34	3jca 1146	. . . . . . . 8 ⊢ ((𝜓 ∧ 𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁) → (𝐻 ∈ 𝐾 ∧ 𝐺 ∈ 𝐾 ∧ if(𝑎 = 𝐸, 𝐻, 𝐼) ∈ 𝐾))
36	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 33, 35	mdetunilem5 22841	. . . . . . 7 ⊢ (𝜓 → (𝐷‘(𝑎 ∈ 𝑁, 𝑏 ∈ 𝑁 ↦ if(𝑎 = 𝐹, (𝐻 + 𝐺), if(𝑎 = 𝐸, 𝐻, 𝐼)))) = ((𝐷‘(𝑎 ∈ 𝑁, 𝑏 ∈ 𝑁 ↦ if(𝑎 = 𝐹, 𝐻, if(𝑎 = 𝐸, 𝐻, 𝐼)))) + (𝐷‘(𝑎 ∈ 𝑁, 𝑏 ∈ 𝑁 ↦ if(𝑎 = 𝐹, 𝐺, if(𝑎 = 𝐸, 𝐻, 𝐼))))))
37	15	simp3d 1162	. . . . . . . . . . 11 ⊢ (𝜓 → 𝐸 ≠ 𝐹)
38	37	necomd 3010	. . . . . . . . . 10 ⊢ (𝜓 → 𝐹 ≠ 𝐸)
39	33, 16, 38	3jca 1146	. . . . . . . . 9 ⊢ (𝜓 → (𝐹 ∈ 𝑁 ∧ 𝐸 ∈ 𝑁 ∧ 𝐹 ≠ 𝐸))
40	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 39, 18, 28	mdetunilem2 22838	. . . . . . . 8 ⊢ (𝜓 → (𝐷‘(𝑎 ∈ 𝑁, 𝑏 ∈ 𝑁 ↦ if(𝑎 = 𝐹, 𝐻, if(𝑎 = 𝐸, 𝐻, 𝐼)))) = 0 )
41	40	oveq1d 7429	. . . . . . 7 ⊢ (𝜓 → ((𝐷‘(𝑎 ∈ 𝑁, 𝑏 ∈ 𝑁 ↦ if(𝑎 = 𝐹, 𝐻, if(𝑎 = 𝐸, 𝐻, 𝐼)))) + (𝐷‘(𝑎 ∈ 𝑁, 𝑏 ∈ 𝑁 ↦ if(𝑎 = 𝐹, 𝐺, if(𝑎 = 𝐸, 𝐻, 𝐼))))) = ( 0 + (𝐷‘(𝑎 ∈ 𝑁, 𝑏 ∈ 𝑁 ↦ if(𝑎 = 𝐹, 𝐺, if(𝑎 = 𝐸, 𝐻, 𝐼))))))
42	37	neneqd 2960	. . . . . . . . . . . . . 14 ⊢ (𝜓 → ¬ 𝐸 = 𝐹)
43		eqtr2 2781	. . . . . . . . . . . . . 14 ⊢ ((𝑎 = 𝐸 ∧ 𝑎 = 𝐹) → 𝐸 = 𝐹)
44	42, 43	nsyl 141	. . . . . . . . . . . . 13 ⊢ (𝜓 → ¬ (𝑎 = 𝐸 ∧ 𝑎 = 𝐹))
45	44	3ad2ant1 1151	. . . . . . . . . . . 12 ⊢ ((𝜓 ∧ 𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁) → ¬ (𝑎 = 𝐸 ∧ 𝑎 = 𝐹))
46		ifcomnan 4539	. . . . . . . . . . . 12 ⊢ (¬ (𝑎 = 𝐸 ∧ 𝑎 = 𝐹) → if(𝑎 = 𝐸, 𝐻, if(𝑎 = 𝐹, 𝐺, 𝐼)) = if(𝑎 = 𝐹, 𝐺, if(𝑎 = 𝐸, 𝐻, 𝐼)))
47	45, 46	syl 18	. . . . . . . . . . 11 ⊢ ((𝜓 ∧ 𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁) → if(𝑎 = 𝐸, 𝐻, if(𝑎 = 𝐹, 𝐺, 𝐼)) = if(𝑎 = 𝐹, 𝐺, if(𝑎 = 𝐸, 𝐻, 𝐼)))
48	47	mpoeq3dva 7491	. . . . . . . . . 10 ⊢ (𝜓 → (𝑎 ∈ 𝑁, 𝑏 ∈ 𝑁 ↦ if(𝑎 = 𝐸, 𝐻, if(𝑎 = 𝐹, 𝐺, 𝐼))) = (𝑎 ∈ 𝑁, 𝑏 ∈ 𝑁 ↦ if(𝑎 = 𝐹, 𝐺, if(𝑎 = 𝐸, 𝐻, 𝐼))))
49	48	fveq2d 6883	. . . . . . . . 9 ⊢ (𝜓 → (𝐷‘(𝑎 ∈ 𝑁, 𝑏 ∈ 𝑁 ↦ if(𝑎 = 𝐸, 𝐻, if(𝑎 = 𝐹, 𝐺, 𝐼)))) = (𝐷‘(𝑎 ∈ 𝑁, 𝑏 ∈ 𝑁 ↦ if(𝑎 = 𝐹, 𝐺, if(𝑎 = 𝐸, 𝐻, 𝐼)))))
50	14, 10	syl 18	. . . . . . . . . 10 ⊢ (𝜓 → 𝐷:𝐵⟶𝐾)
51	14, 8	syl 18	. . . . . . . . . . 11 ⊢ (𝜓 → 𝑁 ∈ Fin)
52	21, 28	ifcld 4529	. . . . . . . . . . . 12 ⊢ ((𝜓 ∧ 𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁) → if(𝑎 = 𝐹, 𝐺, 𝐼) ∈ 𝐾)
53	19, 52	ifcld 4529	. . . . . . . . . . 11 ⊢ ((𝜓 ∧ 𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁) → if(𝑎 = 𝐸, 𝐻, if(𝑎 = 𝐹, 𝐺, 𝐼)) ∈ 𝐾)
54	1, 3, 2, 51, 23, 53	matbas2d 22648	. . . . . . . . . 10 ⊢ (𝜓 → (𝑎 ∈ 𝑁, 𝑏 ∈ 𝑁 ↦ if(𝑎 = 𝐸, 𝐻, if(𝑎 = 𝐹, 𝐺, 𝐼))) ∈ 𝐵)
55	50, 54	ffvelcdmd 7079	. . . . . . . . 9 ⊢ (𝜓 → (𝐷‘(𝑎 ∈ 𝑁, 𝑏 ∈ 𝑁 ↦ if(𝑎 = 𝐸, 𝐻, if(𝑎 = 𝐹, 𝐺, 𝐼)))) ∈ 𝐾)
56	49, 55	eqeltrrd 2861	. . . . . . . 8 ⊢ (𝜓 → (𝐷‘(𝑎 ∈ 𝑁, 𝑏 ∈ 𝑁 ↦ if(𝑎 = 𝐹, 𝐺, if(𝑎 = 𝐸, 𝐻, 𝐼)))) ∈ 𝐾)
57	3, 6, 4	grplid 19094	. . . . . . . 8 ⊢ ((𝑅 ∈ Grp ∧ (𝐷‘(𝑎 ∈ 𝑁, 𝑏 ∈ 𝑁 ↦ if(𝑎 = 𝐹, 𝐺, if(𝑎 = 𝐸, 𝐻, 𝐼)))) ∈ 𝐾) → ( 0 + (𝐷‘(𝑎 ∈ 𝑁, 𝑏 ∈ 𝑁 ↦ if(𝑎 = 𝐹, 𝐺, if(𝑎 = 𝐸, 𝐻, 𝐼))))) = (𝐷‘(𝑎 ∈ 𝑁, 𝑏 ∈ 𝑁 ↦ if(𝑎 = 𝐹, 𝐺, if(𝑎 = 𝐸, 𝐻, 𝐼)))))
58	23, 56, 57	syl2anc 596	. . . . . . 7 ⊢ (𝜓 → ( 0 + (𝐷‘(𝑎 ∈ 𝑁, 𝑏 ∈ 𝑁 ↦ if(𝑎 = 𝐹, 𝐺, if(𝑎 = 𝐸, 𝐻, 𝐼))))) = (𝐷‘(𝑎 ∈ 𝑁, 𝑏 ∈ 𝑁 ↦ if(𝑎 = 𝐹, 𝐺, if(𝑎 = 𝐸, 𝐻, 𝐼)))))
59	36, 41, 58	3eqtrd 2799	. . . . . 6 ⊢ (𝜓 → (𝐷‘(𝑎 ∈ 𝑁, 𝑏 ∈ 𝑁 ↦ if(𝑎 = 𝐹, (𝐻 + 𝐺), if(𝑎 = 𝐸, 𝐻, 𝐼)))) = (𝐷‘(𝑎 ∈ 𝑁, 𝑏 ∈ 𝑁 ↦ if(𝑎 = 𝐹, 𝐺, if(𝑎 = 𝐸, 𝐻, 𝐼)))))
60		ifcomnan 4539	. . . . . . . . 9 ⊢ (¬ (𝑎 = 𝐸 ∧ 𝑎 = 𝐹) → if(𝑎 = 𝐸, 𝐻, if(𝑎 = 𝐹, (𝐻 + 𝐺), 𝐼)) = if(𝑎 = 𝐹, (𝐻 + 𝐺), if(𝑎 = 𝐸, 𝐻, 𝐼)))
61	45, 60	syl 18	. . . . . . . 8 ⊢ ((𝜓 ∧ 𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁) → if(𝑎 = 𝐸, 𝐻, if(𝑎 = 𝐹, (𝐻 + 𝐺), 𝐼)) = if(𝑎 = 𝐹, (𝐻 + 𝐺), if(𝑎 = 𝐸, 𝐻, 𝐼)))
62	61	mpoeq3dva 7491	. . . . . . 7 ⊢ (𝜓 → (𝑎 ∈ 𝑁, 𝑏 ∈ 𝑁 ↦ if(𝑎 = 𝐸, 𝐻, if(𝑎 = 𝐹, (𝐻 + 𝐺), 𝐼))) = (𝑎 ∈ 𝑁, 𝑏 ∈ 𝑁 ↦ if(𝑎 = 𝐹, (𝐻 + 𝐺), if(𝑎 = 𝐸, 𝐻, 𝐼))))
63	62	fveq2d 6883	. . . . . 6 ⊢ (𝜓 → (𝐷‘(𝑎 ∈ 𝑁, 𝑏 ∈ 𝑁 ↦ if(𝑎 = 𝐸, 𝐻, if(𝑎 = 𝐹, (𝐻 + 𝐺), 𝐼)))) = (𝐷‘(𝑎 ∈ 𝑁, 𝑏 ∈ 𝑁 ↦ if(𝑎 = 𝐹, (𝐻 + 𝐺), if(𝑎 = 𝐸, 𝐻, 𝐼)))))
64	59, 63, 49	3eqtr4d 2805	. . . . 5 ⊢ (𝜓 → (𝐷‘(𝑎 ∈ 𝑁, 𝑏 ∈ 𝑁 ↦ if(𝑎 = 𝐸, 𝐻, if(𝑎 = 𝐹, (𝐻 + 𝐺), 𝐼)))) = (𝐷‘(𝑎 ∈ 𝑁, 𝑏 ∈ 𝑁 ↦ if(𝑎 = 𝐸, 𝐻, if(𝑎 = 𝐹, 𝐺, 𝐼)))))
65	21, 28	ifcld 4529	. . . . . . . . 9 ⊢ ((𝜓 ∧ 𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁) → if(𝑎 = 𝐸, 𝐺, 𝐼) ∈ 𝐾)
66	19, 21, 65	3jca 1146	. . . . . . . 8 ⊢ ((𝜓 ∧ 𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁) → (𝐻 ∈ 𝐾 ∧ 𝐺 ∈ 𝐾 ∧ if(𝑎 = 𝐸, 𝐺, 𝐼) ∈ 𝐾))
67	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 33, 66	mdetunilem5 22841	. . . . . . 7 ⊢ (𝜓 → (𝐷‘(𝑎 ∈ 𝑁, 𝑏 ∈ 𝑁 ↦ if(𝑎 = 𝐹, (𝐻 + 𝐺), if(𝑎 = 𝐸, 𝐺, 𝐼)))) = ((𝐷‘(𝑎 ∈ 𝑁, 𝑏 ∈ 𝑁 ↦ if(𝑎 = 𝐹, 𝐻, if(𝑎 = 𝐸, 𝐺, 𝐼)))) + (𝐷‘(𝑎 ∈ 𝑁, 𝑏 ∈ 𝑁 ↦ if(𝑎 = 𝐹, 𝐺, if(𝑎 = 𝐸, 𝐺, 𝐼))))))
68	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 39, 20, 28	mdetunilem2 22838	. . . . . . . 8 ⊢ (𝜓 → (𝐷‘(𝑎 ∈ 𝑁, 𝑏 ∈ 𝑁 ↦ if(𝑎 = 𝐹, 𝐺, if(𝑎 = 𝐸, 𝐺, 𝐼)))) = 0 )
69	68	oveq2d 7430	. . . . . . 7 ⊢ (𝜓 → ((𝐷‘(𝑎 ∈ 𝑁, 𝑏 ∈ 𝑁 ↦ if(𝑎 = 𝐹, 𝐻, if(𝑎 = 𝐸, 𝐺, 𝐼)))) + (𝐷‘(𝑎 ∈ 𝑁, 𝑏 ∈ 𝑁 ↦ if(𝑎 = 𝐹, 𝐺, if(𝑎 = 𝐸, 𝐺, 𝐼))))) = ((𝐷‘(𝑎 ∈ 𝑁, 𝑏 ∈ 𝑁 ↦ if(𝑎 = 𝐹, 𝐻, if(𝑎 = 𝐸, 𝐺, 𝐼)))) + 0 ))
70		ifcomnan 4539	. . . . . . . . . . . 12 ⊢ (¬ (𝑎 = 𝐸 ∧ 𝑎 = 𝐹) → if(𝑎 = 𝐸, 𝐺, if(𝑎 = 𝐹, 𝐻, 𝐼)) = if(𝑎 = 𝐹, 𝐻, if(𝑎 = 𝐸, 𝐺, 𝐼)))
71	45, 70	syl 18	. . . . . . . . . . 11 ⊢ ((𝜓 ∧ 𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁) → if(𝑎 = 𝐸, 𝐺, if(𝑎 = 𝐹, 𝐻, 𝐼)) = if(𝑎 = 𝐹, 𝐻, if(𝑎 = 𝐸, 𝐺, 𝐼)))
72	71	mpoeq3dva 7491	. . . . . . . . . 10 ⊢ (𝜓 → (𝑎 ∈ 𝑁, 𝑏 ∈ 𝑁 ↦ if(𝑎 = 𝐸, 𝐺, if(𝑎 = 𝐹, 𝐻, 𝐼))) = (𝑎 ∈ 𝑁, 𝑏 ∈ 𝑁 ↦ if(𝑎 = 𝐹, 𝐻, if(𝑎 = 𝐸, 𝐺, 𝐼))))
73	72	fveq2d 6883	. . . . . . . . 9 ⊢ (𝜓 → (𝐷‘(𝑎 ∈ 𝑁, 𝑏 ∈ 𝑁 ↦ if(𝑎 = 𝐸, 𝐺, if(𝑎 = 𝐹, 𝐻, 𝐼)))) = (𝐷‘(𝑎 ∈ 𝑁, 𝑏 ∈ 𝑁 ↦ if(𝑎 = 𝐹, 𝐻, if(𝑎 = 𝐸, 𝐺, 𝐼)))))
74	19, 28	ifcld 4529	. . . . . . . . . . . 12 ⊢ ((𝜓 ∧ 𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁) → if(𝑎 = 𝐹, 𝐻, 𝐼) ∈ 𝐾)
75	21, 74	ifcld 4529	. . . . . . . . . . 11 ⊢ ((𝜓 ∧ 𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁) → if(𝑎 = 𝐸, 𝐺, if(𝑎 = 𝐹, 𝐻, 𝐼)) ∈ 𝐾)
76	1, 3, 2, 51, 23, 75	matbas2d 22648	. . . . . . . . . 10 ⊢ (𝜓 → (𝑎 ∈ 𝑁, 𝑏 ∈ 𝑁 ↦ if(𝑎 = 𝐸, 𝐺, if(𝑎 = 𝐹, 𝐻, 𝐼))) ∈ 𝐵)
77	50, 76	ffvelcdmd 7079	. . . . . . . . 9 ⊢ (𝜓 → (𝐷‘(𝑎 ∈ 𝑁, 𝑏 ∈ 𝑁 ↦ if(𝑎 = 𝐸, 𝐺, if(𝑎 = 𝐹, 𝐻, 𝐼)))) ∈ 𝐾)
78	73, 77	eqeltrrd 2861	. . . . . . . 8 ⊢ (𝜓 → (𝐷‘(𝑎 ∈ 𝑁, 𝑏 ∈ 𝑁 ↦ if(𝑎 = 𝐹, 𝐻, if(𝑎 = 𝐸, 𝐺, 𝐼)))) ∈ 𝐾)
79	3, 6, 4	grprid 19095	. . . . . . . 8 ⊢ ((𝑅 ∈ Grp ∧ (𝐷‘(𝑎 ∈ 𝑁, 𝑏 ∈ 𝑁 ↦ if(𝑎 = 𝐹, 𝐻, if(𝑎 = 𝐸, 𝐺, 𝐼)))) ∈ 𝐾) → ((𝐷‘(𝑎 ∈ 𝑁, 𝑏 ∈ 𝑁 ↦ if(𝑎 = 𝐹, 𝐻, if(𝑎 = 𝐸, 𝐺, 𝐼)))) + 0 ) = (𝐷‘(𝑎 ∈ 𝑁, 𝑏 ∈ 𝑁 ↦ if(𝑎 = 𝐹, 𝐻, if(𝑎 = 𝐸, 𝐺, 𝐼)))))
80	23, 78, 79	syl2anc 596	. . . . . . 7 ⊢ (𝜓 → ((𝐷‘(𝑎 ∈ 𝑁, 𝑏 ∈ 𝑁 ↦ if(𝑎 = 𝐹, 𝐻, if(𝑎 = 𝐸, 𝐺, 𝐼)))) + 0 ) = (𝐷‘(𝑎 ∈ 𝑁, 𝑏 ∈ 𝑁 ↦ if(𝑎 = 𝐹, 𝐻, if(𝑎 = 𝐸, 𝐺, 𝐼)))))
81	67, 69, 80	3eqtrd 2799	. . . . . 6 ⊢ (𝜓 → (𝐷‘(𝑎 ∈ 𝑁, 𝑏 ∈ 𝑁 ↦ if(𝑎 = 𝐹, (𝐻 + 𝐺), if(𝑎 = 𝐸, 𝐺, 𝐼)))) = (𝐷‘(𝑎 ∈ 𝑁, 𝑏 ∈ 𝑁 ↦ if(𝑎 = 𝐹, 𝐻, if(𝑎 = 𝐸, 𝐺, 𝐼)))))
82		ifcomnan 4539	. . . . . . . . 9 ⊢ (¬ (𝑎 = 𝐸 ∧ 𝑎 = 𝐹) → if(𝑎 = 𝐸, 𝐺, if(𝑎 = 𝐹, (𝐻 + 𝐺), 𝐼)) = if(𝑎 = 𝐹, (𝐻 + 𝐺), if(𝑎 = 𝐸, 𝐺, 𝐼)))
83	45, 82	syl 18	. . . . . . . 8 ⊢ ((𝜓 ∧ 𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁) → if(𝑎 = 𝐸, 𝐺, if(𝑎 = 𝐹, (𝐻 + 𝐺), 𝐼)) = if(𝑎 = 𝐹, (𝐻 + 𝐺), if(𝑎 = 𝐸, 𝐺, 𝐼)))
84	83	mpoeq3dva 7491	. . . . . . 7 ⊢ (𝜓 → (𝑎 ∈ 𝑁, 𝑏 ∈ 𝑁 ↦ if(𝑎 = 𝐸, 𝐺, if(𝑎 = 𝐹, (𝐻 + 𝐺), 𝐼))) = (𝑎 ∈ 𝑁, 𝑏 ∈ 𝑁 ↦ if(𝑎 = 𝐹, (𝐻 + 𝐺), if(𝑎 = 𝐸, 𝐺, 𝐼))))
85	84	fveq2d 6883	. . . . . 6 ⊢ (𝜓 → (𝐷‘(𝑎 ∈ 𝑁, 𝑏 ∈ 𝑁 ↦ if(𝑎 = 𝐸, 𝐺, if(𝑎 = 𝐹, (𝐻 + 𝐺), 𝐼)))) = (𝐷‘(𝑎 ∈ 𝑁, 𝑏 ∈ 𝑁 ↦ if(𝑎 = 𝐹, (𝐻 + 𝐺), if(𝑎 = 𝐸, 𝐺, 𝐼)))))
86	81, 85, 73	3eqtr4d 2805	. . . . 5 ⊢ (𝜓 → (𝐷‘(𝑎 ∈ 𝑁, 𝑏 ∈ 𝑁 ↦ if(𝑎 = 𝐸, 𝐺, if(𝑎 = 𝐹, (𝐻 + 𝐺), 𝐼)))) = (𝐷‘(𝑎 ∈ 𝑁, 𝑏 ∈ 𝑁 ↦ if(𝑎 = 𝐸, 𝐺, if(𝑎 = 𝐹, 𝐻, 𝐼)))))
87	64, 86	oveq12d 7432	. . . 4 ⊢ (𝜓 → ((𝐷‘(𝑎 ∈ 𝑁, 𝑏 ∈ 𝑁 ↦ if(𝑎 = 𝐸, 𝐻, if(𝑎 = 𝐹, (𝐻 + 𝐺), 𝐼)))) + (𝐷‘(𝑎 ∈ 𝑁, 𝑏 ∈ 𝑁 ↦ if(𝑎 = 𝐸, 𝐺, if(𝑎 = 𝐹, (𝐻 + 𝐺), 𝐼))))) = ((𝐷‘(𝑎 ∈ 𝑁, 𝑏 ∈ 𝑁 ↦ if(𝑎 = 𝐸, 𝐻, if(𝑎 = 𝐹, 𝐺, 𝐼)))) + (𝐷‘(𝑎 ∈ 𝑁, 𝑏 ∈ 𝑁 ↦ if(𝑎 = 𝐸, 𝐺, if(𝑎 = 𝐹, 𝐻, 𝐼))))))
88	31, 32, 87	3eqtr3rd 2804	. . 3 ⊢ (𝜓 → ((𝐷‘(𝑎 ∈ 𝑁, 𝑏 ∈ 𝑁 ↦ if(𝑎 = 𝐸, 𝐻, if(𝑎 = 𝐹, 𝐺, 𝐼)))) + (𝐷‘(𝑎 ∈ 𝑁, 𝑏 ∈ 𝑁 ↦ if(𝑎 = 𝐸, 𝐺, if(𝑎 = 𝐹, 𝐻, 𝐼))))) = 0 )
89		eqid 2760	. . . . 5 ⊢ (inv_g‘𝑅) = (inv_g‘𝑅)
90	3, 6, 4, 89	grpinvid1 19118	. . . 4 ⊢ ((𝑅 ∈ Grp ∧ (𝐷‘(𝑎 ∈ 𝑁, 𝑏 ∈ 𝑁 ↦ if(𝑎 = 𝐸, 𝐻, if(𝑎 = 𝐹, 𝐺, 𝐼)))) ∈ 𝐾 ∧ (𝐷‘(𝑎 ∈ 𝑁, 𝑏 ∈ 𝑁 ↦ if(𝑎 = 𝐸, 𝐺, if(𝑎 = 𝐹, 𝐻, 𝐼)))) ∈ 𝐾) → (((inv_g‘𝑅)‘(𝐷‘(𝑎 ∈ 𝑁, 𝑏 ∈ 𝑁 ↦ if(𝑎 = 𝐸, 𝐻, if(𝑎 = 𝐹, 𝐺, 𝐼))))) = (𝐷‘(𝑎 ∈ 𝑁, 𝑏 ∈ 𝑁 ↦ if(𝑎 = 𝐸, 𝐺, if(𝑎 = 𝐹, 𝐻, 𝐼)))) ↔ ((𝐷‘(𝑎 ∈ 𝑁, 𝑏 ∈ 𝑁 ↦ if(𝑎 = 𝐸, 𝐻, if(𝑎 = 𝐹, 𝐺, 𝐼)))) + (𝐷‘(𝑎 ∈ 𝑁, 𝑏 ∈ 𝑁 ↦ if(𝑎 = 𝐸, 𝐺, if(𝑎 = 𝐹, 𝐻, 𝐼))))) = 0 ))
91	23, 55, 77, 90	syl3anc 1398	. . 3 ⊢ (𝜓 → (((inv_g‘𝑅)‘(𝐷‘(𝑎 ∈ 𝑁, 𝑏 ∈ 𝑁 ↦ if(𝑎 = 𝐸, 𝐻, if(𝑎 = 𝐹, 𝐺, 𝐼))))) = (𝐷‘(𝑎 ∈ 𝑁, 𝑏 ∈ 𝑁 ↦ if(𝑎 = 𝐸, 𝐺, if(𝑎 = 𝐹, 𝐻, 𝐼)))) ↔ ((𝐷‘(𝑎 ∈ 𝑁, 𝑏 ∈ 𝑁 ↦ if(𝑎 = 𝐸, 𝐻, if(𝑎 = 𝐹, 𝐺, 𝐼)))) + (𝐷‘(𝑎 ∈ 𝑁, 𝑏 ∈ 𝑁 ↦ if(𝑎 = 𝐸, 𝐺, if(𝑎 = 𝐹, 𝐻, 𝐼))))) = 0 ))
92	88, 91	mpbird 260	. 2 ⊢ (𝜓 → ((inv_g‘𝑅)‘(𝐷‘(𝑎 ∈ 𝑁, 𝑏 ∈ 𝑁 ↦ if(𝑎 = 𝐸, 𝐻, if(𝑎 = 𝐹, 𝐺, 𝐼))))) = (𝐷‘(𝑎 ∈ 𝑁, 𝑏 ∈ 𝑁 ↦ if(𝑎 = 𝐸, 𝐺, if(𝑎 = 𝐹, 𝐻, 𝐼)))))
93	92	eqcomd 2766	1 ⊢ (𝜓 → (𝐷‘(𝑎 ∈ 𝑁, 𝑏 ∈ 𝑁 ↦ if(𝑎 = 𝐸, 𝐺, if(𝑎 = 𝐹, 𝐻, 𝐼)))) = ((inv_g‘𝑅)‘(𝐷‘(𝑎 ∈ 𝑁, 𝑏 ∈ 𝑁 ↦ if(𝑎 = 𝐸, 𝐻, if(𝑎 = 𝐹, 𝐺, 𝐼))))))

Colors of variables: wff setvar class

This proof depends on syntax axioms: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 401 ∧ w3a 1103 = wceq 1570 ∈ wcel 2145 ≠ wne 2955 ∀wral 3076 ∖ cdif 3896 ifcif 4482 {csn 4584 × cxp 5653 ↾ cres 5657 ⟶wf 6529 ‘cfv 6533 (class class class)co 7414 ∈ cmpo 7416 ∘_f cof 7677 Fincfn 8955 Basecbs 17304 +_gcplusg 17345 ._rcmulr 17346 0_gc0g 17527 Grpcgrp 19060 inv_gcminusg 19061 1_rcur 20323 Ringcrg 20375 Mat cmat 22632

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2213 ax-ext 2732 ax-rep 5232 ax-sep 5251 ax-nul 5263 ax-pow 5330 ax-pr 5398 ax-un 7737 ax-cnex 11183 ax-resscn 11184 ax-1cn 11185 ax-icn 11186 ax-addcl 11187 ax-addrcl 11188 ax-mulcl 11189 ax-mulrcl 11190 ax-mulcom 11191 ax-addass 11192 ax-mulass 11193 ax-distr 11194 ax-i2m1 11195 ax-1ne0 11196 ax-1rid 11197 ax-rnegex 11198 ax-rrecex 11199 ax-cnre 11200 ax-pre-lttri 11201 ax-pre-lttrn 11202 ax-pre-ltadd 11203 ax-pre-mulgt0 11204

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ne 2956 df-nel 3062 df-ral 3077 df-rex 3087 df-rmo 3365 df-reu 3366 df-rab 3413 df-v 3452 df-sbc 3740 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-pss 3919 df-nul 4280 df-if 4483 df-pw 4559 df-sn 4585 df-pr 4587 df-tp 4589 df-op 4591 df-ot 4593 df-uni 4868 df-iun 4953 df-br 5104 df-opab 5168 df-mpt 5187 df-tr 5213 df-id 5550 df-eprel 5555 df-po 5563 df-so 5564 df-fr 5608 df-we 5610 df-xp 5661 df-rel 5662 df-cnv 5663 df-co 5664 df-dm 5665 df-rn 5666 df-res 5667 df-ima 5668 df-pred 6299 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6489 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-riota 7371 df-ov 7417 df-oprab 7418 df-mpo 7419 df-of 7679 df-om 7864 df-1st 7987 df-2nd 7988 df-supp 8160 df-frecs 8281 df-wrecs 8312 df-recs 8361 df-rdg 8400 df-1o 8458 df-er 8699 df-map 8831 df-ixp 8908 df-en 8956 df-dom 8957 df-sdom 8958 df-fin 8959 df-fsupp 9335 df-sup 9415 df-pnf 11272 df-mnf 11273 df-xr 11274 df-ltxr 11275 df-le 11276 df-sub 11470 df-neg 11471 df-nn 12261 df-2 12330 df-3 12331 df-4 12332 df-5 12333 df-6 12334 df-7 12335 df-8 12336 df-9 12337 df-n0 12532 df-z 12619 df-dec 12740 df-uz 12891 df-fz 13565 df-struct 17242 df-sets 17259 df-slot 17277 df-ndx 17289 df-base 17305 df-ress 17326 df-plusg 17358 df-mulr 17359 df-sca 17361 df-vsca 17362 df-ip 17363 df-tset 17364 df-ple 17365 df-ds 17367 df-hom 17369 df-cco 17370 df-0g 17529 df-prds 17535 df-pws 17537 df-mgm 18733 df-sgrp 18824 df-mnd 18840 df-grp 19063 df-minusg 19064 df-ring 20377 df-sra 21360 df-rgmod 21361 df-dsmm 21948 df-frlm 21963 df-mat 22633

This theorem is used by: mdetunilem7 22843

W3C validator