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Theorem mdetunilem6 22559

Description: Lemma for mdetuni 22564. (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 1142	. . . . 5 ⊢ (𝜓 → 𝐸 ∈ 𝑁)
17		mdetunilem6.gh	. . . . . . . 8 ⊢ ((𝜓 ∧ 𝑏 ∈ 𝑁) → (𝐺 ∈ 𝐾 ∧ 𝐻 ∈ 𝐾))
18	17	simprd 495	. . . . . . 7 ⊢ ((𝜓 ∧ 𝑏 ∈ 𝑁) → 𝐻 ∈ 𝐾)
19	18	3adant2 1131	. . . . . 6 ⊢ ((𝜓 ∧ 𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁) → 𝐻 ∈ 𝐾)
20	17	simpld 494	. . . . . . 7 ⊢ ((𝜓 ∧ 𝑏 ∈ 𝑁) → 𝐺 ∈ 𝐾)
21	20	3adant2 1131	. . . . . 6 ⊢ ((𝜓 ∧ 𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁) → 𝐺 ∈ 𝐾)
22		ringgrp 20171	. . . . . . . . . . 11 ⊢ (𝑅 ∈ Ring → 𝑅 ∈ Grp)
23	14, 9, 22	3syl 18	. . . . . . . . . 10 ⊢ (𝜓 → 𝑅 ∈ Grp)
24	23	adantr 480	. . . . . . . . 9 ⊢ ((𝜓 ∧ 𝑏 ∈ 𝑁) → 𝑅 ∈ Grp)
25	3, 6	grpcl 18869	. . . . . . . . 9 ⊢ ((𝑅 ∈ Grp ∧ 𝐻 ∈ 𝐾 ∧ 𝐺 ∈ 𝐾) → (𝐻 + 𝐺) ∈ 𝐾)
26	24, 18, 20, 25	syl3anc 1373	. . . . . . . 8 ⊢ ((𝜓 ∧ 𝑏 ∈ 𝑁) → (𝐻 + 𝐺) ∈ 𝐾)
27	26	3adant2 1131	. . . . . . 7 ⊢ ((𝜓 ∧ 𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁) → (𝐻 + 𝐺) ∈ 𝐾)
28		mdetunilem6.i	. . . . . . 7 ⊢ ((𝜓 ∧ 𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁) → 𝐼 ∈ 𝐾)
29	27, 28	ifcld 4524	. . . . . 6 ⊢ ((𝜓 ∧ 𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁) → if(𝑎 = 𝐹, (𝐻 + 𝐺), 𝐼) ∈ 𝐾)
30	19, 21, 29	3jca 1128	. . . . 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 22558	. . . 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 22555	. . . 4 ⊢ (𝜓 → (𝐷‘(𝑎 ∈ 𝑁, 𝑏 ∈ 𝑁 ↦ if(𝑎 = 𝐸, (𝐻 + 𝐺), if(𝑎 = 𝐹, (𝐻 + 𝐺), 𝐼)))) = 0 )
33	15	simp2d 1143	. . . . . . . 8 ⊢ (𝜓 → 𝐹 ∈ 𝑁)
34	19, 28	ifcld 4524	. . . . . . . . 9 ⊢ ((𝜓 ∧ 𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁) → if(𝑎 = 𝐸, 𝐻, 𝐼) ∈ 𝐾)
35	19, 21, 34	3jca 1128	. . . . . . . 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 22558	. . . . . . 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 1144	. . . . . . . . . . 11 ⊢ (𝜓 → 𝐸 ≠ 𝐹)
38	37	necomd 2985	. . . . . . . . . 10 ⊢ (𝜓 → 𝐹 ≠ 𝐸)
39	33, 16, 38	3jca 1128	. . . . . . . . 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 22555	. . . . . . . 8 ⊢ (𝜓 → (𝐷‘(𝑎 ∈ 𝑁, 𝑏 ∈ 𝑁 ↦ if(𝑎 = 𝐹, 𝐻, if(𝑎 = 𝐸, 𝐻, 𝐼)))) = 0 )
41	40	oveq1d 7371	. . . . . . 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 2935	. . . . . . . . . . . . . 14 ⊢ (𝜓 → ¬ 𝐸 = 𝐹)
43		eqtr2 2755	. . . . . . . . . . . . . 14 ⊢ ((𝑎 = 𝐸 ∧ 𝑎 = 𝐹) → 𝐸 = 𝐹)
44	42, 43	nsyl 140	. . . . . . . . . . . . 13 ⊢ (𝜓 → ¬ (𝑎 = 𝐸 ∧ 𝑎 = 𝐹))
45	44	3ad2ant1 1133	. . . . . . . . . . . 12 ⊢ ((𝜓 ∧ 𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁) → ¬ (𝑎 = 𝐸 ∧ 𝑎 = 𝐹))
46		ifcomnan 4534	. . . . . . . . . . . 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 17	. . . . . . . . . . 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 7433	. . . . . . . . . 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 6836	. . . . . . . . 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 17	. . . . . . . . . 10 ⊢ (𝜓 → 𝐷:𝐵⟶𝐾)
51	14, 8	syl 17	. . . . . . . . . . 11 ⊢ (𝜓 → 𝑁 ∈ Fin)
52	21, 28	ifcld 4524	. . . . . . . . . . . 12 ⊢ ((𝜓 ∧ 𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁) → if(𝑎 = 𝐹, 𝐺, 𝐼) ∈ 𝐾)
53	19, 52	ifcld 4524	. . . . . . . . . . 11 ⊢ ((𝜓 ∧ 𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁) → if(𝑎 = 𝐸, 𝐻, if(𝑎 = 𝐹, 𝐺, 𝐼)) ∈ 𝐾)
54	1, 3, 2, 51, 23, 53	matbas2d 22365	. . . . . . . . . 10 ⊢ (𝜓 → (𝑎 ∈ 𝑁, 𝑏 ∈ 𝑁 ↦ if(𝑎 = 𝐸, 𝐻, if(𝑎 = 𝐹, 𝐺, 𝐼))) ∈ 𝐵)
55	50, 54	ffvelcdmd 7028	. . . . . . . . 9 ⊢ (𝜓 → (𝐷‘(𝑎 ∈ 𝑁, 𝑏 ∈ 𝑁 ↦ if(𝑎 = 𝐸, 𝐻, if(𝑎 = 𝐹, 𝐺, 𝐼)))) ∈ 𝐾)
56	49, 55	eqeltrrd 2835	. . . . . . . 8 ⊢ (𝜓 → (𝐷‘(𝑎 ∈ 𝑁, 𝑏 ∈ 𝑁 ↦ if(𝑎 = 𝐹, 𝐺, if(𝑎 = 𝐸, 𝐻, 𝐼)))) ∈ 𝐾)
57	3, 6, 4	grplid 18895	. . . . . . . 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 584	. . . . . . 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 2773	. . . . . 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 4534	. . . . . . . . 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 17	. . . . . . . 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 7433	. . . . . . 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 6836	. . . . . 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 2779	. . . . 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 4524	. . . . . . . . 9 ⊢ ((𝜓 ∧ 𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁) → if(𝑎 = 𝐸, 𝐺, 𝐼) ∈ 𝐾)
66	19, 21, 65	3jca 1128	. . . . . . . 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 22558	. . . . . . 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 22555	. . . . . . . 8 ⊢ (𝜓 → (𝐷‘(𝑎 ∈ 𝑁, 𝑏 ∈ 𝑁 ↦ if(𝑎 = 𝐹, 𝐺, if(𝑎 = 𝐸, 𝐺, 𝐼)))) = 0 )
69	68	oveq2d 7372	. . . . . . 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 4534	. . . . . . . . . . . 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 17	. . . . . . . . . . 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 7433	. . . . . . . . . 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 6836	. . . . . . . . 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 4524	. . . . . . . . . . . 12 ⊢ ((𝜓 ∧ 𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁) → if(𝑎 = 𝐹, 𝐻, 𝐼) ∈ 𝐾)
75	21, 74	ifcld 4524	. . . . . . . . . . 11 ⊢ ((𝜓 ∧ 𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁) → if(𝑎 = 𝐸, 𝐺, if(𝑎 = 𝐹, 𝐻, 𝐼)) ∈ 𝐾)
76	1, 3, 2, 51, 23, 75	matbas2d 22365	. . . . . . . . . 10 ⊢ (𝜓 → (𝑎 ∈ 𝑁, 𝑏 ∈ 𝑁 ↦ if(𝑎 = 𝐸, 𝐺, if(𝑎 = 𝐹, 𝐻, 𝐼))) ∈ 𝐵)
77	50, 76	ffvelcdmd 7028	. . . . . . . . 9 ⊢ (𝜓 → (𝐷‘(𝑎 ∈ 𝑁, 𝑏 ∈ 𝑁 ↦ if(𝑎 = 𝐸, 𝐺, if(𝑎 = 𝐹, 𝐻, 𝐼)))) ∈ 𝐾)
78	73, 77	eqeltrrd 2835	. . . . . . . 8 ⊢ (𝜓 → (𝐷‘(𝑎 ∈ 𝑁, 𝑏 ∈ 𝑁 ↦ if(𝑎 = 𝐹, 𝐻, if(𝑎 = 𝐸, 𝐺, 𝐼)))) ∈ 𝐾)
79	3, 6, 4	grprid 18896	. . . . . . . 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 584	. . . . . . 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 2773	. . . . . 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 4534	. . . . . . . . 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 17	. . . . . . . 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 7433	. . . . . . 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 6836	. . . . . 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 2779	. . . . 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 7374	. . . 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 2778	. . 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 2734	. . . . 5 ⊢ (inv_g‘𝑅) = (inv_g‘𝑅)
90	3, 6, 4, 89	grpinvid1 18919	. . . 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 1373	. . 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 257	. 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 2740	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

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1086 = wceq 1541 ∈ wcel 2113 ≠ wne 2930 ∀wral 3049 ∖ cdif 3896 ifcif 4477 {csn 4578 × cxp 5620 ↾ cres 5624 ⟶wf 6486 ‘cfv 6490 (class class class)co 7356 ∈ cmpo 7358 ∘_f cof 7618 Fincfn 8881 Basecbs 17134 +_gcplusg 17175 ._rcmulr 17176 0_gc0g 17357 Grpcgrp 18861 inv_gcminusg 18862 1_rcur 20114 Ringcrg 20166 Mat cmat 22349

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2115 ax-9 2123 ax-10 2146 ax-11 2162 ax-12 2182 ax-ext 2706 ax-rep 5222 ax-sep 5239 ax-nul 5249 ax-pow 5308 ax-pr 5375 ax-un 7678 ax-cnex 11080 ax-resscn 11081 ax-1cn 11082 ax-icn 11083 ax-addcl 11084 ax-addrcl 11085 ax-mulcl 11086 ax-mulrcl 11087 ax-mulcom 11088 ax-addass 11089 ax-mulass 11090 ax-distr 11091 ax-i2m1 11092 ax-1ne0 11093 ax-1rid 11094 ax-rnegex 11095 ax-rrecex 11096 ax-cnre 11097 ax-pre-lttri 11098 ax-pre-lttrn 11099 ax-pre-ltadd 11100 ax-pre-mulgt0 11101

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2537 df-eu 2567 df-clab 2713 df-cleq 2726 df-clel 2809 df-nfc 2883 df-ne 2931 df-nel 3035 df-ral 3050 df-rex 3059 df-rmo 3348 df-reu 3349 df-rab 3398 df-v 3440 df-sbc 3739 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-pss 3919 df-nul 4284 df-if 4478 df-pw 4554 df-sn 4579 df-pr 4581 df-tp 4583 df-op 4585 df-ot 4587 df-uni 4862 df-iun 4946 df-br 5097 df-opab 5159 df-mpt 5178 df-tr 5204 df-id 5517 df-eprel 5522 df-po 5530 df-so 5531 df-fr 5575 df-we 5577 df-xp 5628 df-rel 5629 df-cnv 5630 df-co 5631 df-dm 5632 df-rn 5633 df-res 5634 df-ima 5635 df-pred 6257 df-ord 6318 df-on 6319 df-lim 6320 df-suc 6321 df-iota 6446 df-fun 6492 df-fn 6493 df-f 6494 df-f1 6495 df-fo 6496 df-f1o 6497 df-fv 6498 df-riota 7313 df-ov 7359 df-oprab 7360 df-mpo 7361 df-of 7620 df-om 7807 df-1st 7931 df-2nd 7932 df-supp 8101 df-frecs 8221 df-wrecs 8252 df-recs 8301 df-rdg 8339 df-1o 8395 df-er 8633 df-map 8763 df-ixp 8834 df-en 8882 df-dom 8883 df-sdom 8884 df-fin 8885 df-fsupp 9263 df-sup 9343 df-pnf 11166 df-mnf 11167 df-xr 11168 df-ltxr 11169 df-le 11170 df-sub 11364 df-neg 11365 df-nn 12144 df-2 12206 df-3 12207 df-4 12208 df-5 12209 df-6 12210 df-7 12211 df-8 12212 df-9 12213 df-n0 12400 df-z 12487 df-dec 12606 df-uz 12750 df-fz 13422 df-struct 17072 df-sets 17089 df-slot 17107 df-ndx 17119 df-base 17135 df-ress 17156 df-plusg 17188 df-mulr 17189 df-sca 17191 df-vsca 17192 df-ip 17193 df-tset 17194 df-ple 17195 df-ds 17197 df-hom 17199 df-cco 17200 df-0g 17359 df-prds 17365 df-pws 17367 df-mgm 18563 df-sgrp 18642 df-mnd 18658 df-grp 18864 df-minusg 18865 df-ring 20168 df-sra 21123 df-rgmod 21124 df-dsmm 21685 df-frlm 21700 df-mat 22350

This theorem is referenced by: mdetunilem7 22560

W3C validator