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Theorem mdetunilem6 21766

Description: Lemma for mdetuni 21771. (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 1141	. . . . 5 ⊢ (𝜓 → 𝐸 ∈ 𝑁)
17		mdetunilem6.gh	. . . . . . . 8 ⊢ ((𝜓 ∧ 𝑏 ∈ 𝑁) → (𝐺 ∈ 𝐾 ∧ 𝐻 ∈ 𝐾))
18	17	simprd 496	. . . . . . 7 ⊢ ((𝜓 ∧ 𝑏 ∈ 𝑁) → 𝐻 ∈ 𝐾)
19	18	3adant2 1130	. . . . . 6 ⊢ ((𝜓 ∧ 𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁) → 𝐻 ∈ 𝐾)
20	17	simpld 495	. . . . . . 7 ⊢ ((𝜓 ∧ 𝑏 ∈ 𝑁) → 𝐺 ∈ 𝐾)
21	20	3adant2 1130	. . . . . 6 ⊢ ((𝜓 ∧ 𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁) → 𝐺 ∈ 𝐾)
22		ringgrp 19788	. . . . . . . . . . 11 ⊢ (𝑅 ∈ Ring → 𝑅 ∈ Grp)
23	14, 9, 22	3syl 18	. . . . . . . . . 10 ⊢ (𝜓 → 𝑅 ∈ Grp)
24	23	adantr 481	. . . . . . . . 9 ⊢ ((𝜓 ∧ 𝑏 ∈ 𝑁) → 𝑅 ∈ Grp)
25	3, 6	grpcl 18585	. . . . . . . . 9 ⊢ ((𝑅 ∈ Grp ∧ 𝐻 ∈ 𝐾 ∧ 𝐺 ∈ 𝐾) → (𝐻 + 𝐺) ∈ 𝐾)
26	24, 18, 20, 25	syl3anc 1370	. . . . . . . 8 ⊢ ((𝜓 ∧ 𝑏 ∈ 𝑁) → (𝐻 + 𝐺) ∈ 𝐾)
27	26	3adant2 1130	. . . . . . 7 ⊢ ((𝜓 ∧ 𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁) → (𝐻 + 𝐺) ∈ 𝐾)
28		mdetunilem6.i	. . . . . . 7 ⊢ ((𝜓 ∧ 𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁) → 𝐼 ∈ 𝐾)
29	27, 28	ifcld 4505	. . . . . 6 ⊢ ((𝜓 ∧ 𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁) → if(𝑎 = 𝐹, (𝐻 + 𝐺), 𝐼) ∈ 𝐾)
30	19, 21, 29	3jca 1127	. . . . 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 21765	. . . 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 21762	. . . 4 ⊢ (𝜓 → (𝐷‘(𝑎 ∈ 𝑁, 𝑏 ∈ 𝑁 ↦ if(𝑎 = 𝐸, (𝐻 + 𝐺), if(𝑎 = 𝐹, (𝐻 + 𝐺), 𝐼)))) = 0 )
33	15	simp2d 1142	. . . . . . . 8 ⊢ (𝜓 → 𝐹 ∈ 𝑁)
34	19, 28	ifcld 4505	. . . . . . . . 9 ⊢ ((𝜓 ∧ 𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁) → if(𝑎 = 𝐸, 𝐻, 𝐼) ∈ 𝐾)
35	19, 21, 34	3jca 1127	. . . . . . . 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 21765	. . . . . . 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 1143	. . . . . . . . . . 11 ⊢ (𝜓 → 𝐸 ≠ 𝐹)
38	37	necomd 2999	. . . . . . . . . 10 ⊢ (𝜓 → 𝐹 ≠ 𝐸)
39	33, 16, 38	3jca 1127	. . . . . . . . 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 21762	. . . . . . . 8 ⊢ (𝜓 → (𝐷‘(𝑎 ∈ 𝑁, 𝑏 ∈ 𝑁 ↦ if(𝑎 = 𝐹, 𝐻, if(𝑎 = 𝐸, 𝐻, 𝐼)))) = 0 )
41	40	oveq1d 7290	. . . . . . 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 2948	. . . . . . . . . . . . . 14 ⊢ (𝜓 → ¬ 𝐸 = 𝐹)
43		eqtr2 2762	. . . . . . . . . . . . . 14 ⊢ ((𝑎 = 𝐸 ∧ 𝑎 = 𝐹) → 𝐸 = 𝐹)
44	42, 43	nsyl 140	. . . . . . . . . . . . 13 ⊢ (𝜓 → ¬ (𝑎 = 𝐸 ∧ 𝑎 = 𝐹))
45	44	3ad2ant1 1132	. . . . . . . . . . . 12 ⊢ ((𝜓 ∧ 𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁) → ¬ (𝑎 = 𝐸 ∧ 𝑎 = 𝐹))
46		ifcomnan 4515	. . . . . . . . . . . 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 7352	. . . . . . . . . 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 6778	. . . . . . . . 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 4505	. . . . . . . . . . . 12 ⊢ ((𝜓 ∧ 𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁) → if(𝑎 = 𝐹, 𝐺, 𝐼) ∈ 𝐾)
53	19, 52	ifcld 4505	. . . . . . . . . . 11 ⊢ ((𝜓 ∧ 𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁) → if(𝑎 = 𝐸, 𝐻, if(𝑎 = 𝐹, 𝐺, 𝐼)) ∈ 𝐾)
54	1, 3, 2, 51, 23, 53	matbas2d 21572	. . . . . . . . . 10 ⊢ (𝜓 → (𝑎 ∈ 𝑁, 𝑏 ∈ 𝑁 ↦ if(𝑎 = 𝐸, 𝐻, if(𝑎 = 𝐹, 𝐺, 𝐼))) ∈ 𝐵)
55	50, 54	ffvelrnd 6962	. . . . . . . . 9 ⊢ (𝜓 → (𝐷‘(𝑎 ∈ 𝑁, 𝑏 ∈ 𝑁 ↦ if(𝑎 = 𝐸, 𝐻, if(𝑎 = 𝐹, 𝐺, 𝐼)))) ∈ 𝐾)
56	49, 55	eqeltrrd 2840	. . . . . . . 8 ⊢ (𝜓 → (𝐷‘(𝑎 ∈ 𝑁, 𝑏 ∈ 𝑁 ↦ if(𝑎 = 𝐹, 𝐺, if(𝑎 = 𝐸, 𝐻, 𝐼)))) ∈ 𝐾)
57	3, 6, 4	grplid 18609	. . . . . . . 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 2782	. . . . . 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 4515	. . . . . . . . 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 7352	. . . . . . 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 6778	. . . . . 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 2788	. . . . 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 4505	. . . . . . . . 9 ⊢ ((𝜓 ∧ 𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁) → if(𝑎 = 𝐸, 𝐺, 𝐼) ∈ 𝐾)
66	19, 21, 65	3jca 1127	. . . . . . . 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 21765	. . . . . . 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 21762	. . . . . . . 8 ⊢ (𝜓 → (𝐷‘(𝑎 ∈ 𝑁, 𝑏 ∈ 𝑁 ↦ if(𝑎 = 𝐹, 𝐺, if(𝑎 = 𝐸, 𝐺, 𝐼)))) = 0 )
69	68	oveq2d 7291	. . . . . . 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 4515	. . . . . . . . . . . 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 7352	. . . . . . . . . 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 6778	. . . . . . . . 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 4505	. . . . . . . . . . . 12 ⊢ ((𝜓 ∧ 𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁) → if(𝑎 = 𝐹, 𝐻, 𝐼) ∈ 𝐾)
75	21, 74	ifcld 4505	. . . . . . . . . . 11 ⊢ ((𝜓 ∧ 𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁) → if(𝑎 = 𝐸, 𝐺, if(𝑎 = 𝐹, 𝐻, 𝐼)) ∈ 𝐾)
76	1, 3, 2, 51, 23, 75	matbas2d 21572	. . . . . . . . . 10 ⊢ (𝜓 → (𝑎 ∈ 𝑁, 𝑏 ∈ 𝑁 ↦ if(𝑎 = 𝐸, 𝐺, if(𝑎 = 𝐹, 𝐻, 𝐼))) ∈ 𝐵)
77	50, 76	ffvelrnd 6962	. . . . . . . . 9 ⊢ (𝜓 → (𝐷‘(𝑎 ∈ 𝑁, 𝑏 ∈ 𝑁 ↦ if(𝑎 = 𝐸, 𝐺, if(𝑎 = 𝐹, 𝐻, 𝐼)))) ∈ 𝐾)
78	73, 77	eqeltrrd 2840	. . . . . . . 8 ⊢ (𝜓 → (𝐷‘(𝑎 ∈ 𝑁, 𝑏 ∈ 𝑁 ↦ if(𝑎 = 𝐹, 𝐻, if(𝑎 = 𝐸, 𝐺, 𝐼)))) ∈ 𝐾)
79	3, 6, 4	grprid 18610	. . . . . . . 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 2782	. . . . . 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 4515	. . . . . . . . 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 7352	. . . . . . 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 6778	. . . . . 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 2788	. . . . 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 7293	. . . 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 2787	. . 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 2738	. . . . 5 ⊢ (inv_g‘𝑅) = (inv_g‘𝑅)
90	3, 6, 4, 89	grpinvid1 18630	. . . 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 1370	. . 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 256	. 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 2744	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 205 ∧ wa 396 ∧ w3a 1086 = wceq 1539 ∈ wcel 2106 ≠ wne 2943 ∀wral 3064 ∖ cdif 3884 ifcif 4459 {csn 4561 × cxp 5587 ↾ cres 5591 ⟶wf 6429 ‘cfv 6433 (class class class)co 7275 ∈ cmpo 7277 ∘_f cof 7531 Fincfn 8733 Basecbs 16912 +_gcplusg 16962 ._rcmulr 16963 0_gc0g 17150 Grpcgrp 18577 inv_gcminusg 18578 1_rcur 19737 Ringcrg 19783 Mat cmat 21554

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-rep 5209 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7588 ax-cnex 10927 ax-resscn 10928 ax-1cn 10929 ax-icn 10930 ax-addcl 10931 ax-addrcl 10932 ax-mulcl 10933 ax-mulrcl 10934 ax-mulcom 10935 ax-addass 10936 ax-mulass 10937 ax-distr 10938 ax-i2m1 10939 ax-1ne0 10940 ax-1rid 10941 ax-rnegex 10942 ax-rrecex 10943 ax-cnre 10944 ax-pre-lttri 10945 ax-pre-lttrn 10946 ax-pre-ltadd 10947 ax-pre-mulgt0 10948

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-rmo 3071 df-reu 3072 df-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-pss 3906 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-tp 4566 df-op 4568 df-ot 4570 df-uni 4840 df-iun 4926 df-br 5075 df-opab 5137 df-mpt 5158 df-tr 5192 df-id 5489 df-eprel 5495 df-po 5503 df-so 5504 df-fr 5544 df-we 5546 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-pred 6202 df-ord 6269 df-on 6270 df-lim 6271 df-suc 6272 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-f1 6438 df-fo 6439 df-f1o 6440 df-fv 6441 df-riota 7232 df-ov 7278 df-oprab 7279 df-mpo 7280 df-of 7533 df-om 7713 df-1st 7831 df-2nd 7832 df-supp 7978 df-frecs 8097 df-wrecs 8128 df-recs 8202 df-rdg 8241 df-1o 8297 df-er 8498 df-map 8617 df-ixp 8686 df-en 8734 df-dom 8735 df-sdom 8736 df-fin 8737 df-fsupp 9129 df-sup 9201 df-pnf 11011 df-mnf 11012 df-xr 11013 df-ltxr 11014 df-le 11015 df-sub 11207 df-neg 11208 df-nn 11974 df-2 12036 df-3 12037 df-4 12038 df-5 12039 df-6 12040 df-7 12041 df-8 12042 df-9 12043 df-n0 12234 df-z 12320 df-dec 12438 df-uz 12583 df-fz 13240 df-struct 16848 df-sets 16865 df-slot 16883 df-ndx 16895 df-base 16913 df-ress 16942 df-plusg 16975 df-mulr 16976 df-sca 16978 df-vsca 16979 df-ip 16980 df-tset 16981 df-ple 16982 df-ds 16984 df-hom 16986 df-cco 16987 df-0g 17152 df-prds 17158 df-pws 17160 df-mgm 18326 df-sgrp 18375 df-mnd 18386 df-grp 18580 df-minusg 18581 df-ring 19785 df-sra 20434 df-rgmod 20435 df-dsmm 20939 df-frlm 20954 df-mat 21555

This theorem is referenced by: mdetunilem7 21767

W3C validator