Theorem madjusmdetlem1 33788

Description: Lemma for madjusmdet 33792. (Contributed by Thierry Arnoux, 22-Aug-2020.)

Hypotheses

Ref	Expression
madjusmdet.b	⊢ 𝐵 = (Base‘𝐴)
madjusmdet.a	⊢ 𝐴 = ((1...𝑁) Mat 𝑅)
madjusmdet.d	⊢ 𝐷 = ((1...𝑁) maDet 𝑅)
madjusmdet.k	⊢ 𝐾 = ((1...𝑁) maAdju 𝑅)
madjusmdet.t	⊢ · = (.r‘𝑅)
madjusmdet.z	⊢ 𝑍 = (ℤRHom‘𝑅)
madjusmdet.e	⊢ 𝐸 = ((1...(𝑁 − 1)) maDet 𝑅)
madjusmdet.n	⊢ (𝜑 → 𝑁 ∈ ℕ)
madjusmdet.r	⊢ (𝜑 → 𝑅 ∈ CRing)
madjusmdet.i	⊢ (𝜑 → 𝐼 ∈ (1...𝑁))
madjusmdet.j	⊢ (𝜑 → 𝐽 ∈ (1...𝑁))
madjusmdet.m	⊢ (𝜑 → 𝑀 ∈ 𝐵)
madjusmdetlem1.g	⊢ 𝐺 = (Base‘(SymGrp‘(1...𝑁)))
madjusmdetlem1.s	⊢ 𝑆 = (pmSgn‘(1...𝑁))
madjusmdetlem1.u	⊢ 𝑈 = (𝐼(((1...𝑁) minMatR1 𝑅)‘𝑀)𝐽)
madjusmdetlem1.w	⊢ 𝑊 = (𝑖 ∈ (1...𝑁), 𝑗 ∈ (1...𝑁) ↦ ((𝑃‘𝑖)𝑈(𝑄‘𝑗)))
madjusmdetlem1.p	⊢ (𝜑 → 𝑃 ∈ 𝐺)
madjusmdetlem1.q	⊢ (𝜑 → 𝑄 ∈ 𝐺)
madjusmdetlem1.1	⊢ (𝜑 → (𝑃‘𝑁) = 𝐼)
madjusmdetlem1.2	⊢ (𝜑 → (𝑄‘𝑁) = 𝐽)
madjusmdetlem1.3	⊢ (𝜑 → (𝐼(subMat1‘𝑈)𝐽) = (𝑁(subMat1‘𝑊)𝑁))

Assertion

Ref	Expression
madjusmdetlem1	⊢ (𝜑 → (𝐽(𝐾‘𝑀)𝐼) = ((𝑍‘((𝑆‘𝑃) · (𝑆‘𝑄))) · (𝐸‘(𝐼(subMat1‘𝑀)𝐽))))

Distinct variable groups:   𝐵,𝑖,𝑗   𝑖,𝐼,𝑗   𝑖,𝐽,𝑗   𝑖,𝑀,𝑗   𝑖,𝑁,𝑗   𝑃,𝑖,𝑗   𝑄,𝑖,𝑗   𝑅,𝑖,𝑗   𝜑,𝑖,𝑗   𝑖,𝐺,𝑗   𝑖,𝑊,𝑗   𝑈,𝑖,𝑗

Allowed substitution hints:   𝐴(𝑖,𝑗)   𝐷(𝑖,𝑗)   𝑆(𝑖,𝑗)   · (𝑖,𝑗)   𝐸(𝑖,𝑗)   𝐾(𝑖,𝑗)   𝑍(𝑖,𝑗)

Proof of Theorem madjusmdetlem1

Step	Hyp	Ref	Expression
1		madjusmdet.m	. . . 4 ⊢ (𝜑 → 𝑀 ∈ 𝐵)
2		madjusmdet.j	. . . 4 ⊢ (𝜑 → 𝐽 ∈ (1...𝑁))
3		madjusmdet.i	. . . 4 ⊢ (𝜑 → 𝐼 ∈ (1...𝑁))
4		madjusmdet.a	. . . . 5 ⊢ 𝐴 = ((1...𝑁) Mat 𝑅)
5		madjusmdet.b	. . . . 5 ⊢ 𝐵 = (Base‘𝐴)
6		madjusmdet.d	. . . . 5 ⊢ 𝐷 = ((1...𝑁) maDet 𝑅)
7		madjusmdet.k	. . . . 5 ⊢ 𝐾 = ((1...𝑁) maAdju 𝑅)
8	4, 5, 6, 7	maducoevalmin1 22674	. . . 4 ⊢ ((𝑀 ∈ 𝐵 ∧ 𝐽 ∈ (1...𝑁) ∧ 𝐼 ∈ (1...𝑁)) → (𝐽(𝐾‘𝑀)𝐼) = (𝐷‘(𝐼(((1...𝑁) minMatR1 𝑅)‘𝑀)𝐽)))
9	1, 2, 3, 8	syl3anc 1370	. . 3 ⊢ (𝜑 → (𝐽(𝐾‘𝑀)𝐼) = (𝐷‘(𝐼(((1...𝑁) minMatR1 𝑅)‘𝑀)𝐽)))
10		madjusmdetlem1.u	. . . 4 ⊢ 𝑈 = (𝐼(((1...𝑁) minMatR1 𝑅)‘𝑀)𝐽)
11	10	fveq2i 6910	. . 3 ⊢ (𝐷‘𝑈) = (𝐷‘(𝐼(((1...𝑁) minMatR1 𝑅)‘𝑀)𝐽))
12	9, 11	eqtr4di 2793	. 2 ⊢ (𝜑 → (𝐽(𝐾‘𝑀)𝐼) = (𝐷‘𝑈))
13		madjusmdetlem1.g	. . 3 ⊢ 𝐺 = (Base‘(SymGrp‘(1...𝑁)))
14		madjusmdetlem1.s	. . 3 ⊢ 𝑆 = (pmSgn‘(1...𝑁))
15		madjusmdet.z	. . 3 ⊢ 𝑍 = (ℤRHom‘𝑅)
16		madjusmdet.t	. . 3 ⊢ · = (.r‘𝑅)
17		madjusmdetlem1.w	. . 3 ⊢ 𝑊 = (𝑖 ∈ (1...𝑁), 𝑗 ∈ (1...𝑁) ↦ ((𝑃‘𝑖)𝑈(𝑄‘𝑗)))
18		madjusmdet.r	. . 3 ⊢ (𝜑 → 𝑅 ∈ CRing)
19		fzfid 14011	. . 3 ⊢ (𝜑 → (1...𝑁) ∈ Fin)
20		crngring 20263	. . . . . 6 ⊢ (𝑅 ∈ CRing → 𝑅 ∈ Ring)
21	18, 20	syl 17	. . . . 5 ⊢ (𝜑 → 𝑅 ∈ Ring)
22	4, 5	minmar1cl 22673	. . . . 5 ⊢ (((𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ (𝐼 ∈ (1...𝑁) ∧ 𝐽 ∈ (1...𝑁))) → (𝐼(((1...𝑁) minMatR1 𝑅)‘𝑀)𝐽) ∈ 𝐵)
23	21, 1, 3, 2, 22	syl22anc 839	. . . 4 ⊢ (𝜑 → (𝐼(((1...𝑁) minMatR1 𝑅)‘𝑀)𝐽) ∈ 𝐵)
24	10, 23	eqeltrid 2843	. . 3 ⊢ (𝜑 → 𝑈 ∈ 𝐵)
25		madjusmdetlem1.p	. . 3 ⊢ (𝜑 → 𝑃 ∈ 𝐺)
26		madjusmdetlem1.q	. . 3 ⊢ (𝜑 → 𝑄 ∈ 𝐺)
27	4, 5, 6, 13, 14, 15, 16, 17, 18, 19, 24, 25, 26	mdetpmtr12 33786	. 2 ⊢ (𝜑 → (𝐷‘𝑈) = ((𝑍‘((𝑆‘𝑃) · (𝑆‘𝑄))) · (𝐷‘𝑊)))
28		simplr 769	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) ∧ 𝑖 = 𝑁) ∧ 𝑗 = 𝑁) → 𝑖 = 𝑁)
29	28	fveq2d 6911	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) ∧ 𝑖 = 𝑁) ∧ 𝑗 = 𝑁) → (𝑃‘𝑖) = (𝑃‘𝑁))
30		madjusmdetlem1.1	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝑃‘𝑁) = 𝐼)
31	30	3ad2ant1 1132	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) → (𝑃‘𝑁) = 𝐼)
32	31	ad2antrr 726	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) ∧ 𝑖 = 𝑁) ∧ 𝑗 = 𝑁) → (𝑃‘𝑁) = 𝐼)
33	29, 32	eqtrd 2775	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) ∧ 𝑖 = 𝑁) ∧ 𝑗 = 𝑁) → (𝑃‘𝑖) = 𝐼)
34		simpr 484	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) ∧ 𝑖 = 𝑁) ∧ 𝑗 = 𝑁) → 𝑗 = 𝑁)
35	34	fveq2d 6911	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) ∧ 𝑖 = 𝑁) ∧ 𝑗 = 𝑁) → (𝑄‘𝑗) = (𝑄‘𝑁))
36		madjusmdetlem1.2	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝑄‘𝑁) = 𝐽)
37	36	3ad2ant1 1132	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) → (𝑄‘𝑁) = 𝐽)
38	37	ad2antrr 726	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) ∧ 𝑖 = 𝑁) ∧ 𝑗 = 𝑁) → (𝑄‘𝑁) = 𝐽)
39	35, 38	eqtrd 2775	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) ∧ 𝑖 = 𝑁) ∧ 𝑗 = 𝑁) → (𝑄‘𝑗) = 𝐽)
40	33, 39	oveq12d 7449	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) ∧ 𝑖 = 𝑁) ∧ 𝑗 = 𝑁) → ((𝑃‘𝑖)(𝐼(((1...𝑁) minMatR1 𝑅)‘𝑀)𝐽)(𝑄‘𝑗)) = (𝐼(𝐼(((1...𝑁) minMatR1 𝑅)‘𝑀)𝐽)𝐽))
41	1	3ad2ant1 1132	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) → 𝑀 ∈ 𝐵)
42	41	ad2antrr 726	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) ∧ 𝑖 = 𝑁) ∧ 𝑗 = 𝑁) → 𝑀 ∈ 𝐵)
43	3	3ad2ant1 1132	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) → 𝐼 ∈ (1...𝑁))
44	43	ad2antrr 726	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) ∧ 𝑖 = 𝑁) ∧ 𝑗 = 𝑁) → 𝐼 ∈ (1...𝑁))
45	2	3ad2ant1 1132	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) → 𝐽 ∈ (1...𝑁))
46	45	ad2antrr 726	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) ∧ 𝑖 = 𝑁) ∧ 𝑗 = 𝑁) → 𝐽 ∈ (1...𝑁))
47		eqid 2735	. . . . . . . . . . . . 13 ⊢ ((1...𝑁) minMatR1 𝑅) = ((1...𝑁) minMatR1 𝑅)
48		eqid 2735	. . . . . . . . . . . . 13 ⊢ (1r‘𝑅) = (1r‘𝑅)
49		eqid 2735	. . . . . . . . . . . . 13 ⊢ (0g‘𝑅) = (0g‘𝑅)
50	4, 5, 47, 48, 49	minmar1eval 22671	. . . . . . . . . . . 12 ⊢ ((𝑀 ∈ 𝐵 ∧ (𝐼 ∈ (1...𝑁) ∧ 𝐽 ∈ (1...𝑁)) ∧ (𝐼 ∈ (1...𝑁) ∧ 𝐽 ∈ (1...𝑁))) → (𝐼(𝐼(((1...𝑁) minMatR1 𝑅)‘𝑀)𝐽)𝐽) = if(𝐼 = 𝐼, if(𝐽 = 𝐽, (1r‘𝑅), (0g‘𝑅)), (𝐼𝑀𝐽)))
51	42, 44, 46, 44, 46, 50	syl122anc 1378	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) ∧ 𝑖 = 𝑁) ∧ 𝑗 = 𝑁) → (𝐼(𝐼(((1...𝑁) minMatR1 𝑅)‘𝑀)𝐽)𝐽) = if(𝐼 = 𝐼, if(𝐽 = 𝐽, (1r‘𝑅), (0g‘𝑅)), (𝐼𝑀𝐽)))
52		eqid 2735	. . . . . . . . . . . . . 14 ⊢ 𝐼 = 𝐼
53	52	iftruei 4538	. . . . . . . . . . . . 13 ⊢ if(𝐼 = 𝐼, if(𝐽 = 𝐽, (1r‘𝑅), (0g‘𝑅)), (𝐼𝑀𝐽)) = if(𝐽 = 𝐽, (1r‘𝑅), (0g‘𝑅))
54		eqid 2735	. . . . . . . . . . . . . 14 ⊢ 𝐽 = 𝐽
55	54	iftruei 4538	. . . . . . . . . . . . 13 ⊢ if(𝐽 = 𝐽, (1r‘𝑅), (0g‘𝑅)) = (1r‘𝑅)
56	53, 55	eqtri 2763	. . . . . . . . . . . 12 ⊢ if(𝐼 = 𝐼, if(𝐽 = 𝐽, (1r‘𝑅), (0g‘𝑅)), (𝐼𝑀𝐽)) = (1r‘𝑅)
57	56	a1i 11	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) ∧ 𝑖 = 𝑁) ∧ 𝑗 = 𝑁) → if(𝐼 = 𝐼, if(𝐽 = 𝐽, (1r‘𝑅), (0g‘𝑅)), (𝐼𝑀𝐽)) = (1r‘𝑅))
58	40, 51, 57	3eqtrrd 2780	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) ∧ 𝑖 = 𝑁) ∧ 𝑗 = 𝑁) → (1r‘𝑅) = ((𝑃‘𝑖)(𝐼(((1...𝑁) minMatR1 𝑅)‘𝑀)𝐽)(𝑄‘𝑗)))
59		simplr 769	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) ∧ 𝑖 = 𝑁) ∧ ¬ 𝑗 = 𝑁) → 𝑖 = 𝑁)
60	59	fveq2d 6911	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) ∧ 𝑖 = 𝑁) ∧ ¬ 𝑗 = 𝑁) → (𝑃‘𝑖) = (𝑃‘𝑁))
61	31	ad2antrr 726	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) ∧ 𝑖 = 𝑁) ∧ ¬ 𝑗 = 𝑁) → (𝑃‘𝑁) = 𝐼)
62	60, 61	eqtrd 2775	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) ∧ 𝑖 = 𝑁) ∧ ¬ 𝑗 = 𝑁) → (𝑃‘𝑖) = 𝐼)
63	62	oveq1d 7446	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) ∧ 𝑖 = 𝑁) ∧ ¬ 𝑗 = 𝑁) → ((𝑃‘𝑖)(𝐼(((1...𝑁) minMatR1 𝑅)‘𝑀)𝐽)(𝑄‘𝑗)) = (𝐼(𝐼(((1...𝑁) minMatR1 𝑅)‘𝑀)𝐽)(𝑄‘𝑗)))
64	41	ad2antrr 726	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) ∧ 𝑖 = 𝑁) ∧ ¬ 𝑗 = 𝑁) → 𝑀 ∈ 𝐵)
65	43	ad2antrr 726	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) ∧ 𝑖 = 𝑁) ∧ ¬ 𝑗 = 𝑁) → 𝐼 ∈ (1...𝑁))
66	45	ad2antrr 726	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) ∧ 𝑖 = 𝑁) ∧ ¬ 𝑗 = 𝑁) → 𝐽 ∈ (1...𝑁))
67	26	3ad2ant1 1132	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) → 𝑄 ∈ 𝐺)
68		simp3 1137	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) → 𝑗 ∈ (1...𝑁))
69		eqid 2735	. . . . . . . . . . . . . . 15 ⊢ (SymGrp‘(1...𝑁)) = (SymGrp‘(1...𝑁))
70	69, 13	symgfv 19412	. . . . . . . . . . . . . 14 ⊢ ((𝑄 ∈ 𝐺 ∧ 𝑗 ∈ (1...𝑁)) → (𝑄‘𝑗) ∈ (1...𝑁))
71	67, 68, 70	syl2anc 584	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) → (𝑄‘𝑗) ∈ (1...𝑁))
72	71	ad2antrr 726	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) ∧ 𝑖 = 𝑁) ∧ ¬ 𝑗 = 𝑁) → (𝑄‘𝑗) ∈ (1...𝑁))
73	4, 5, 47, 48, 49	minmar1eval 22671	. . . . . . . . . . . 12 ⊢ ((𝑀 ∈ 𝐵 ∧ (𝐼 ∈ (1...𝑁) ∧ 𝐽 ∈ (1...𝑁)) ∧ (𝐼 ∈ (1...𝑁) ∧ (𝑄‘𝑗) ∈ (1...𝑁))) → (𝐼(𝐼(((1...𝑁) minMatR1 𝑅)‘𝑀)𝐽)(𝑄‘𝑗)) = if(𝐼 = 𝐼, if((𝑄‘𝑗) = 𝐽, (1r‘𝑅), (0g‘𝑅)), (𝐼𝑀(𝑄‘𝑗))))
74	64, 65, 66, 65, 72, 73	syl122anc 1378	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) ∧ 𝑖 = 𝑁) ∧ ¬ 𝑗 = 𝑁) → (𝐼(𝐼(((1...𝑁) minMatR1 𝑅)‘𝑀)𝐽)(𝑄‘𝑗)) = if(𝐼 = 𝐼, if((𝑄‘𝑗) = 𝐽, (1r‘𝑅), (0g‘𝑅)), (𝐼𝑀(𝑄‘𝑗))))
75	52	a1i 11	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) ∧ 𝑖 = 𝑁) ∧ ¬ 𝑗 = 𝑁) → 𝐼 = 𝐼)
76	75	iftrued 4539	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) ∧ 𝑖 = 𝑁) ∧ ¬ 𝑗 = 𝑁) → if(𝐼 = 𝐼, if((𝑄‘𝑗) = 𝐽, (1r‘𝑅), (0g‘𝑅)), (𝐼𝑀(𝑄‘𝑗))) = if((𝑄‘𝑗) = 𝐽, (1r‘𝑅), (0g‘𝑅)))
77		simpr 484	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) ∧ 𝑖 = 𝑁) ∧ (𝑄‘𝑗) = 𝐽) → (𝑄‘𝑗) = 𝐽)
78	77	fveq2d 6911	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) ∧ 𝑖 = 𝑁) ∧ (𝑄‘𝑗) = 𝐽) → (◡𝑄‘(𝑄‘𝑗)) = (◡𝑄‘𝐽))
79	69, 13	symgbasf1o 19407	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑄 ∈ 𝐺 → 𝑄:(1...𝑁)–1-1-onto→(1...𝑁))
80	67, 79	syl 17	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) → 𝑄:(1...𝑁)–1-1-onto→(1...𝑁))
81	80	ad2antrr 726	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) ∧ 𝑖 = 𝑁) ∧ (𝑄‘𝑗) = 𝐽) → 𝑄:(1...𝑁)–1-1-onto→(1...𝑁))
82	68	ad2antrr 726	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) ∧ 𝑖 = 𝑁) ∧ (𝑄‘𝑗) = 𝐽) → 𝑗 ∈ (1...𝑁))
83		f1ocnvfv1 7296	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑄:(1...𝑁)–1-1-onto→(1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) → (◡𝑄‘(𝑄‘𝑗)) = 𝑗)
84	81, 82, 83	syl2anc 584	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) ∧ 𝑖 = 𝑁) ∧ (𝑄‘𝑗) = 𝐽) → (◡𝑄‘(𝑄‘𝑗)) = 𝑗)
85	36	fveq2d 6911	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → (◡𝑄‘(𝑄‘𝑁)) = (◡𝑄‘𝐽))
86	26, 79	syl 17	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → 𝑄:(1...𝑁)–1-1-onto→(1...𝑁))
87		madjusmdet.n	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝜑 → 𝑁 ∈ ℕ)
88		nnuz 12919	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ℕ = (ℤ≥‘1)
89	87, 88	eleqtrdi 2849	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘1))
90		eluzfz2 13569	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑁 ∈ (ℤ≥‘1) → 𝑁 ∈ (1...𝑁))
91	89, 90	syl 17	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → 𝑁 ∈ (1...𝑁))
92		f1ocnvfv1 7296	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑄:(1...𝑁)–1-1-onto→(1...𝑁) ∧ 𝑁 ∈ (1...𝑁)) → (◡𝑄‘(𝑄‘𝑁)) = 𝑁)
93	86, 91, 92	syl2anc 584	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → (◡𝑄‘(𝑄‘𝑁)) = 𝑁)
94	85, 93	eqtr3d 2777	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → (◡𝑄‘𝐽) = 𝑁)
95	94	3ad2ant1 1132	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) → (◡𝑄‘𝐽) = 𝑁)
96	95	ad2antrr 726	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) ∧ 𝑖 = 𝑁) ∧ (𝑄‘𝑗) = 𝐽) → (◡𝑄‘𝐽) = 𝑁)
97	78, 84, 96	3eqtr3d 2783	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) ∧ 𝑖 = 𝑁) ∧ (𝑄‘𝑗) = 𝐽) → 𝑗 = 𝑁)
98	97	ex 412	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) ∧ 𝑖 = 𝑁) → ((𝑄‘𝑗) = 𝐽 → 𝑗 = 𝑁))
99	98	con3d 152	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) ∧ 𝑖 = 𝑁) → (¬ 𝑗 = 𝑁 → ¬ (𝑄‘𝑗) = 𝐽))
100	99	imp 406	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) ∧ 𝑖 = 𝑁) ∧ ¬ 𝑗 = 𝑁) → ¬ (𝑄‘𝑗) = 𝐽)
101	100	iffalsed 4542	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) ∧ 𝑖 = 𝑁) ∧ ¬ 𝑗 = 𝑁) → if((𝑄‘𝑗) = 𝐽, (1r‘𝑅), (0g‘𝑅)) = (0g‘𝑅))
102	76, 101	eqtrd 2775	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) ∧ 𝑖 = 𝑁) ∧ ¬ 𝑗 = 𝑁) → if(𝐼 = 𝐼, if((𝑄‘𝑗) = 𝐽, (1r‘𝑅), (0g‘𝑅)), (𝐼𝑀(𝑄‘𝑗))) = (0g‘𝑅))
103	63, 74, 102	3eqtrrd 2780	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) ∧ 𝑖 = 𝑁) ∧ ¬ 𝑗 = 𝑁) → (0g‘𝑅) = ((𝑃‘𝑖)(𝐼(((1...𝑁) minMatR1 𝑅)‘𝑀)𝐽)(𝑄‘𝑗)))
104	58, 103	ifeqda 4567	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) ∧ 𝑖 = 𝑁) → if(𝑗 = 𝑁, (1r‘𝑅), (0g‘𝑅)) = ((𝑃‘𝑖)(𝐼(((1...𝑁) minMatR1 𝑅)‘𝑀)𝐽)(𝑄‘𝑗)))
105		simp2 1136	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) → 𝑖 ∈ (1...𝑁))
106	105	adantr 480	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) ∧ ¬ 𝑖 = 𝑁) → 𝑖 ∈ (1...𝑁))
107	68	adantr 480	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) ∧ ¬ 𝑖 = 𝑁) → 𝑗 ∈ (1...𝑁))
108		ovexd 7466	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) ∧ ¬ 𝑖 = 𝑁) → ((𝑃‘𝑖)(𝐼(((1...𝑁) minMatR1 𝑅)‘𝑀)𝐽)(𝑄‘𝑗)) ∈ V)
109	10	oveqi 7444	. . . . . . . . . . . . . 14 ⊢ ((𝑃‘𝑖)𝑈(𝑄‘𝑗)) = ((𝑃‘𝑖)(𝐼(((1...𝑁) minMatR1 𝑅)‘𝑀)𝐽)(𝑄‘𝑗))
110	109	a1i 11	. . . . . . . . . . . . 13 ⊢ ((𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) → ((𝑃‘𝑖)𝑈(𝑄‘𝑗)) = ((𝑃‘𝑖)(𝐼(((1...𝑁) minMatR1 𝑅)‘𝑀)𝐽)(𝑄‘𝑗)))
111	110	mpoeq3ia 7511	. . . . . . . . . . . 12 ⊢ (𝑖 ∈ (1...𝑁), 𝑗 ∈ (1...𝑁) ↦ ((𝑃‘𝑖)𝑈(𝑄‘𝑗))) = (𝑖 ∈ (1...𝑁), 𝑗 ∈ (1...𝑁) ↦ ((𝑃‘𝑖)(𝐼(((1...𝑁) minMatR1 𝑅)‘𝑀)𝐽)(𝑄‘𝑗)))
112	17, 111	eqtri 2763	. . . . . . . . . . 11 ⊢ 𝑊 = (𝑖 ∈ (1...𝑁), 𝑗 ∈ (1...𝑁) ↦ ((𝑃‘𝑖)(𝐼(((1...𝑁) minMatR1 𝑅)‘𝑀)𝐽)(𝑄‘𝑗)))
113	112	ovmpt4g 7580	. . . . . . . . . 10 ⊢ ((𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁) ∧ ((𝑃‘𝑖)(𝐼(((1...𝑁) minMatR1 𝑅)‘𝑀)𝐽)(𝑄‘𝑗)) ∈ V) → (𝑖𝑊𝑗) = ((𝑃‘𝑖)(𝐼(((1...𝑁) minMatR1 𝑅)‘𝑀)𝐽)(𝑄‘𝑗)))
114	106, 107, 108, 113	syl3anc 1370	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) ∧ ¬ 𝑖 = 𝑁) → (𝑖𝑊𝑗) = ((𝑃‘𝑖)(𝐼(((1...𝑁) minMatR1 𝑅)‘𝑀)𝐽)(𝑄‘𝑗)))
115	104, 114	ifeqda 4567	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) → if(𝑖 = 𝑁, if(𝑗 = 𝑁, (1r‘𝑅), (0g‘𝑅)), (𝑖𝑊𝑗)) = ((𝑃‘𝑖)(𝐼(((1...𝑁) minMatR1 𝑅)‘𝑀)𝐽)(𝑄‘𝑗)))
116	115	mpoeq3dva 7510	. . . . . . 7 ⊢ (𝜑 → (𝑖 ∈ (1...𝑁), 𝑗 ∈ (1...𝑁) ↦ if(𝑖 = 𝑁, if(𝑗 = 𝑁, (1r‘𝑅), (0g‘𝑅)), (𝑖𝑊𝑗))) = (𝑖 ∈ (1...𝑁), 𝑗 ∈ (1...𝑁) ↦ ((𝑃‘𝑖)(𝐼(((1...𝑁) minMatR1 𝑅)‘𝑀)𝐽)(𝑄‘𝑗))))
117		eqid 2735	. . . . . . . . . 10 ⊢ (Base‘𝑅) = (Base‘𝑅)
118	25	3ad2ant1 1132	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) → 𝑃 ∈ 𝐺)
119	69, 13	symgfv 19412	. . . . . . . . . . . 12 ⊢ ((𝑃 ∈ 𝐺 ∧ 𝑖 ∈ (1...𝑁)) → (𝑃‘𝑖) ∈ (1...𝑁))
120	118, 105, 119	syl2anc 584	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) → (𝑃‘𝑖) ∈ (1...𝑁))
121	24	3ad2ant1 1132	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) → 𝑈 ∈ 𝐵)
122	4, 117, 5, 120, 71, 121	matecld 22448	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) → ((𝑃‘𝑖)𝑈(𝑄‘𝑗)) ∈ (Base‘𝑅))
123	4, 117, 5, 19, 18, 122	matbas2d 22445	. . . . . . . . 9 ⊢ (𝜑 → (𝑖 ∈ (1...𝑁), 𝑗 ∈ (1...𝑁) ↦ ((𝑃‘𝑖)𝑈(𝑄‘𝑗))) ∈ 𝐵)
124	17, 123	eqeltrid 2843	. . . . . . . 8 ⊢ (𝜑 → 𝑊 ∈ 𝐵)
125	117, 48	ringidcl 20280	. . . . . . . . 9 ⊢ (𝑅 ∈ Ring → (1r‘𝑅) ∈ (Base‘𝑅))
126	21, 125	syl 17	. . . . . . . 8 ⊢ (𝜑 → (1r‘𝑅) ∈ (Base‘𝑅))
127		eqid 2735	. . . . . . . . 9 ⊢ ((1...𝑁) matRRep 𝑅) = ((1...𝑁) matRRep 𝑅)
128	4, 5, 127, 49	marrepval 22584	. . . . . . . 8 ⊢ (((𝑊 ∈ 𝐵 ∧ (1r‘𝑅) ∈ (Base‘𝑅)) ∧ (𝑁 ∈ (1...𝑁) ∧ 𝑁 ∈ (1...𝑁))) → (𝑁(𝑊((1...𝑁) matRRep 𝑅)(1r‘𝑅))𝑁) = (𝑖 ∈ (1...𝑁), 𝑗 ∈ (1...𝑁) ↦ if(𝑖 = 𝑁, if(𝑗 = 𝑁, (1r‘𝑅), (0g‘𝑅)), (𝑖𝑊𝑗))))
129	124, 126, 91, 91, 128	syl22anc 839	. . . . . . 7 ⊢ (𝜑 → (𝑁(𝑊((1...𝑁) matRRep 𝑅)(1r‘𝑅))𝑁) = (𝑖 ∈ (1...𝑁), 𝑗 ∈ (1...𝑁) ↦ if(𝑖 = 𝑁, if(𝑗 = 𝑁, (1r‘𝑅), (0g‘𝑅)), (𝑖𝑊𝑗))))
130	112	a1i 11	. . . . . . 7 ⊢ (𝜑 → 𝑊 = (𝑖 ∈ (1...𝑁), 𝑗 ∈ (1...𝑁) ↦ ((𝑃‘𝑖)(𝐼(((1...𝑁) minMatR1 𝑅)‘𝑀)𝐽)(𝑄‘𝑗))))
131	116, 129, 130	3eqtr4d 2785	. . . . . 6 ⊢ (𝜑 → (𝑁(𝑊((1...𝑁) matRRep 𝑅)(1r‘𝑅))𝑁) = 𝑊)
132	131	fveq2d 6911	. . . . 5 ⊢ (𝜑 → (𝐷‘(𝑁(𝑊((1...𝑁) matRRep 𝑅)(1r‘𝑅))𝑁)) = (𝐷‘𝑊))
133		eqid 2735	. . . . . . . . . . . 12 ⊢ ((1...𝑁) subMat 𝑅) = ((1...𝑁) subMat 𝑅)
134	4, 133, 5	submaval 22603	. . . . . . . . . . 11 ⊢ ((𝑊 ∈ 𝐵 ∧ 𝑁 ∈ (1...𝑁) ∧ 𝑁 ∈ (1...𝑁)) → (𝑁(((1...𝑁) subMat 𝑅)‘𝑊)𝑁) = (𝑖 ∈ ((1...𝑁) ∖ {𝑁}), 𝑗 ∈ ((1...𝑁) ∖ {𝑁}) ↦ (𝑖𝑊𝑗)))
135	124, 91, 91, 134	syl3anc 1370	. . . . . . . . . 10 ⊢ (𝜑 → (𝑁(((1...𝑁) subMat 𝑅)‘𝑊)𝑁) = (𝑖 ∈ ((1...𝑁) ∖ {𝑁}), 𝑗 ∈ ((1...𝑁) ∖ {𝑁}) ↦ (𝑖𝑊𝑗)))
136		fzdif2 32799	. . . . . . . . . . . 12 ⊢ (𝑁 ∈ (ℤ≥‘1) → ((1...𝑁) ∖ {𝑁}) = (1...(𝑁 − 1)))
137	89, 136	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → ((1...𝑁) ∖ {𝑁}) = (1...(𝑁 − 1)))
138		mpoeq12 7506	. . . . . . . . . . 11 ⊢ ((((1...𝑁) ∖ {𝑁}) = (1...(𝑁 − 1)) ∧ ((1...𝑁) ∖ {𝑁}) = (1...(𝑁 − 1))) → (𝑖 ∈ ((1...𝑁) ∖ {𝑁}), 𝑗 ∈ ((1...𝑁) ∖ {𝑁}) ↦ (𝑖𝑊𝑗)) = (𝑖 ∈ (1...(𝑁 − 1)), 𝑗 ∈ (1...(𝑁 − 1)) ↦ (𝑖𝑊𝑗)))
139	137, 137, 138	syl2anc 584	. . . . . . . . . 10 ⊢ (𝜑 → (𝑖 ∈ ((1...𝑁) ∖ {𝑁}), 𝑗 ∈ ((1...𝑁) ∖ {𝑁}) ↦ (𝑖𝑊𝑗)) = (𝑖 ∈ (1...(𝑁 − 1)), 𝑗 ∈ (1...(𝑁 − 1)) ↦ (𝑖𝑊𝑗)))
140	135, 139	eqtrd 2775	. . . . . . . . 9 ⊢ (𝜑 → (𝑁(((1...𝑁) subMat 𝑅)‘𝑊)𝑁) = (𝑖 ∈ (1...(𝑁 − 1)), 𝑗 ∈ (1...(𝑁 − 1)) ↦ (𝑖𝑊𝑗)))
141		difssd 4147	. . . . . . . . . . 11 ⊢ (𝜑 → ((1...𝑁) ∖ {𝑁}) ⊆ (1...𝑁))
142	137, 141	eqsstrrd 4035	. . . . . . . . . 10 ⊢ (𝜑 → (1...(𝑁 − 1)) ⊆ (1...𝑁))
143	4, 5	submabas 22600	. . . . . . . . . 10 ⊢ ((𝑊 ∈ 𝐵 ∧ (1...(𝑁 − 1)) ⊆ (1...𝑁)) → (𝑖 ∈ (1...(𝑁 − 1)), 𝑗 ∈ (1...(𝑁 − 1)) ↦ (𝑖𝑊𝑗)) ∈ (Base‘((1...(𝑁 − 1)) Mat 𝑅)))
144	124, 142, 143	syl2anc 584	. . . . . . . . 9 ⊢ (𝜑 → (𝑖 ∈ (1...(𝑁 − 1)), 𝑗 ∈ (1...(𝑁 − 1)) ↦ (𝑖𝑊𝑗)) ∈ (Base‘((1...(𝑁 − 1)) Mat 𝑅)))
145	140, 144	eqeltrd 2839	. . . . . . . 8 ⊢ (𝜑 → (𝑁(((1...𝑁) subMat 𝑅)‘𝑊)𝑁) ∈ (Base‘((1...(𝑁 − 1)) Mat 𝑅)))
146		madjusmdet.e	. . . . . . . . 9 ⊢ 𝐸 = ((1...(𝑁 − 1)) maDet 𝑅)
147		eqid 2735	. . . . . . . . 9 ⊢ ((1...(𝑁 − 1)) Mat 𝑅) = ((1...(𝑁 − 1)) Mat 𝑅)
148		eqid 2735	. . . . . . . . 9 ⊢ (Base‘((1...(𝑁 − 1)) Mat 𝑅)) = (Base‘((1...(𝑁 − 1)) Mat 𝑅))
149	146, 147, 148, 117	mdetcl 22618	. . . . . . . 8 ⊢ ((𝑅 ∈ CRing ∧ (𝑁(((1...𝑁) subMat 𝑅)‘𝑊)𝑁) ∈ (Base‘((1...(𝑁 − 1)) Mat 𝑅))) → (𝐸‘(𝑁(((1...𝑁) subMat 𝑅)‘𝑊)𝑁)) ∈ (Base‘𝑅))
150	18, 145, 149	syl2anc 584	. . . . . . 7 ⊢ (𝜑 → (𝐸‘(𝑁(((1...𝑁) subMat 𝑅)‘𝑊)𝑁)) ∈ (Base‘𝑅))
151	117, 16, 48	ringlidm 20283	. . . . . . 7 ⊢ ((𝑅 ∈ Ring ∧ (𝐸‘(𝑁(((1...𝑁) subMat 𝑅)‘𝑊)𝑁)) ∈ (Base‘𝑅)) → ((1r‘𝑅) · (𝐸‘(𝑁(((1...𝑁) subMat 𝑅)‘𝑊)𝑁))) = (𝐸‘(𝑁(((1...𝑁) subMat 𝑅)‘𝑊)𝑁)))
152	21, 150, 151	syl2anc 584	. . . . . 6 ⊢ (𝜑 → ((1r‘𝑅) · (𝐸‘(𝑁(((1...𝑁) subMat 𝑅)‘𝑊)𝑁))) = (𝐸‘(𝑁(((1...𝑁) subMat 𝑅)‘𝑊)𝑁)))
153	4	fveq2i 6910	. . . . . . . . . . 11 ⊢ (Base‘𝐴) = (Base‘((1...𝑁) Mat 𝑅))
154	5, 153	eqtri 2763	. . . . . . . . . 10 ⊢ 𝐵 = (Base‘((1...𝑁) Mat 𝑅))
155	124, 154	eleqtrdi 2849	. . . . . . . . 9 ⊢ (𝜑 → 𝑊 ∈ (Base‘((1...𝑁) Mat 𝑅)))
156		smadiadetr 22697	. . . . . . . . 9 ⊢ (((𝑅 ∈ CRing ∧ 𝑊 ∈ (Base‘((1...𝑁) Mat 𝑅))) ∧ (𝑁 ∈ (1...𝑁) ∧ (1r‘𝑅) ∈ (Base‘𝑅))) → (((1...𝑁) maDet 𝑅)‘(𝑁(𝑊((1...𝑁) matRRep 𝑅)(1r‘𝑅))𝑁)) = ((1r‘𝑅)(.r‘𝑅)((((1...𝑁) ∖ {𝑁}) maDet 𝑅)‘(𝑁(((1...𝑁) subMat 𝑅)‘𝑊)𝑁))))
157	18, 155, 91, 126, 156	syl22anc 839	. . . . . . . 8 ⊢ (𝜑 → (((1...𝑁) maDet 𝑅)‘(𝑁(𝑊((1...𝑁) matRRep 𝑅)(1r‘𝑅))𝑁)) = ((1r‘𝑅)(.r‘𝑅)((((1...𝑁) ∖ {𝑁}) maDet 𝑅)‘(𝑁(((1...𝑁) subMat 𝑅)‘𝑊)𝑁))))
158	6	fveq1i 6908	. . . . . . . . 9 ⊢ (𝐷‘(𝑁(𝑊((1...𝑁) matRRep 𝑅)(1r‘𝑅))𝑁)) = (((1...𝑁) maDet 𝑅)‘(𝑁(𝑊((1...𝑁) matRRep 𝑅)(1r‘𝑅))𝑁))
159	16	oveqi 7444	. . . . . . . . 9 ⊢ ((1r‘𝑅) · ((((1...𝑁) ∖ {𝑁}) maDet 𝑅)‘(𝑁(((1...𝑁) subMat 𝑅)‘𝑊)𝑁))) = ((1r‘𝑅)(.r‘𝑅)((((1...𝑁) ∖ {𝑁}) maDet 𝑅)‘(𝑁(((1...𝑁) subMat 𝑅)‘𝑊)𝑁)))
160	158, 159	eqeq12i 2753	. . . . . . . 8 ⊢ ((𝐷‘(𝑁(𝑊((1...𝑁) matRRep 𝑅)(1r‘𝑅))𝑁)) = ((1r‘𝑅) · ((((1...𝑁) ∖ {𝑁}) maDet 𝑅)‘(𝑁(((1...𝑁) subMat 𝑅)‘𝑊)𝑁))) ↔ (((1...𝑁) maDet 𝑅)‘(𝑁(𝑊((1...𝑁) matRRep 𝑅)(1r‘𝑅))𝑁)) = ((1r‘𝑅)(.r‘𝑅)((((1...𝑁) ∖ {𝑁}) maDet 𝑅)‘(𝑁(((1...𝑁) subMat 𝑅)‘𝑊)𝑁))))
161	157, 160	sylibr 234	. . . . . . 7 ⊢ (𝜑 → (𝐷‘(𝑁(𝑊((1...𝑁) matRRep 𝑅)(1r‘𝑅))𝑁)) = ((1r‘𝑅) · ((((1...𝑁) ∖ {𝑁}) maDet 𝑅)‘(𝑁(((1...𝑁) subMat 𝑅)‘𝑊)𝑁))))
162	137	oveq1d 7446	. . . . . . . . . 10 ⊢ (𝜑 → (((1...𝑁) ∖ {𝑁}) maDet 𝑅) = ((1...(𝑁 − 1)) maDet 𝑅))
163	162, 146	eqtr4di 2793	. . . . . . . . 9 ⊢ (𝜑 → (((1...𝑁) ∖ {𝑁}) maDet 𝑅) = 𝐸)
164	163	fveq1d 6909	. . . . . . . 8 ⊢ (𝜑 → ((((1...𝑁) ∖ {𝑁}) maDet 𝑅)‘(𝑁(((1...𝑁) subMat 𝑅)‘𝑊)𝑁)) = (𝐸‘(𝑁(((1...𝑁) subMat 𝑅)‘𝑊)𝑁)))
165	164	oveq2d 7447	. . . . . . 7 ⊢ (𝜑 → ((1r‘𝑅) · ((((1...𝑁) ∖ {𝑁}) maDet 𝑅)‘(𝑁(((1...𝑁) subMat 𝑅)‘𝑊)𝑁))) = ((1r‘𝑅) · (𝐸‘(𝑁(((1...𝑁) subMat 𝑅)‘𝑊)𝑁))))
166	161, 165	eqtrd 2775	. . . . . 6 ⊢ (𝜑 → (𝐷‘(𝑁(𝑊((1...𝑁) matRRep 𝑅)(1r‘𝑅))𝑁)) = ((1r‘𝑅) · (𝐸‘(𝑁(((1...𝑁) subMat 𝑅)‘𝑊)𝑁))))
167	4, 5	submat1n 33766	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ ∧ 𝑊 ∈ 𝐵) → (𝑁(subMat1‘𝑊)𝑁) = (𝑁(((1...𝑁) subMat 𝑅)‘𝑊)𝑁))
168	87, 124, 167	syl2anc 584	. . . . . . 7 ⊢ (𝜑 → (𝑁(subMat1‘𝑊)𝑁) = (𝑁(((1...𝑁) subMat 𝑅)‘𝑊)𝑁))
169	168	fveq2d 6911	. . . . . 6 ⊢ (𝜑 → (𝐸‘(𝑁(subMat1‘𝑊)𝑁)) = (𝐸‘(𝑁(((1...𝑁) subMat 𝑅)‘𝑊)𝑁)))
170	152, 166, 169	3eqtr4d 2785	. . . . 5 ⊢ (𝜑 → (𝐷‘(𝑁(𝑊((1...𝑁) matRRep 𝑅)(1r‘𝑅))𝑁)) = (𝐸‘(𝑁(subMat1‘𝑊)𝑁)))
171	132, 170	eqtr3d 2777	. . . 4 ⊢ (𝜑 → (𝐷‘𝑊) = (𝐸‘(𝑁(subMat1‘𝑊)𝑁)))
172	4, 5, 87, 3, 2, 21, 1, 10	submatminr1 33771	. . . . . 6 ⊢ (𝜑 → (𝐼(subMat1‘𝑀)𝐽) = (𝐼(subMat1‘𝑈)𝐽))
173		madjusmdetlem1.3	. . . . . 6 ⊢ (𝜑 → (𝐼(subMat1‘𝑈)𝐽) = (𝑁(subMat1‘𝑊)𝑁))
174	172, 173	eqtrd 2775	. . . . 5 ⊢ (𝜑 → (𝐼(subMat1‘𝑀)𝐽) = (𝑁(subMat1‘𝑊)𝑁))
175	174	fveq2d 6911	. . . 4 ⊢ (𝜑 → (𝐸‘(𝐼(subMat1‘𝑀)𝐽)) = (𝐸‘(𝑁(subMat1‘𝑊)𝑁)))
176	171, 175	eqtr4d 2778	. . 3 ⊢ (𝜑 → (𝐷‘𝑊) = (𝐸‘(𝐼(subMat1‘𝑀)𝐽)))
177	176	oveq2d 7447	. 2 ⊢ (𝜑 → ((𝑍‘((𝑆‘𝑃) · (𝑆‘𝑄))) · (𝐷‘𝑊)) = ((𝑍‘((𝑆‘𝑃) · (𝑆‘𝑄))) · (𝐸‘(𝐼(subMat1‘𝑀)𝐽))))
178	12, 27, 177	3eqtrd 2779	1 ⊢ (𝜑 → (𝐽(𝐾‘𝑀)𝐼) = ((𝑍‘((𝑆‘𝑃) · (𝑆‘𝑄))) · (𝐸‘(𝐼(subMat1‘𝑀)𝐽))))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   ∧ w3a 1086   = wceq 1537   ∈ wcel 2106  Vcvv 3478   ∖ cdif 3960   ⊆ wss 3963  ifcif 4531  {csn 4631  ^◡ccnv 5688  –1-1-onto→wf1o 6562  ‘cfv 6563  (class class class)co 7431   ∈ cmpo 7433  1c1 11154   · cmul 11158   − cmin 11490  ℕcn 12264  ℤ_≥cuz 12876  ...cfz 13544  Basecbs 17245  ._rcmulr 17299  0_gc0g 17486  SymGrpcsymg 19401  pmSgncpsgn 19522  1_rcur 20199  Ringcrg 20251  CRingccrg 20252  ℤRHomczrh 21528   Mat cmat 22427   matRRep cmarrep 22578   subMat csubma 22598   maDet cmdat 22606   maAdju cmadu 22654   minMatR1 cminmar1 22655  subMat1csmat 33754

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-cnex 11209  ax-resscn 11210  ax-1cn 11211  ax-icn 11212  ax-addcl 11213  ax-addrcl 11214  ax-mulcl 11215  ax-mulrcl 11216  ax-mulcom 11217  ax-addass 11218  ax-mulass 11219  ax-distr 11220  ax-i2m1 11221  ax-1ne0 11222  ax-1rid 11223  ax-rnegex 11224  ax-rrecex 11225  ax-cnre 11226  ax-pre-lttri 11227  ax-pre-lttrn 11228  ax-pre-ltadd 11229  ax-pre-mulgt0 11230  ax-addf 11232  ax-mulf 11233

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-xor 1509  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-tp 4636  df-op 4638  df-ot 4640  df-uni 4913  df-int 4952  df-iun 4998  df-iin 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-se 5642  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-isom 6572  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-of 7697  df-om 7888  df-1st 8013  df-2nd 8014  df-supp 8185  df-tpos 8250  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-1o 8505  df-2o 8506  df-er 8744  df-map 8867  df-pm 8868  df-ixp 8937  df-en 8985  df-dom 8986  df-sdom 8987  df-fin 8988  df-fsupp 9400  df-sup 9480  df-oi 9548  df-card 9977  df-pnf 11295  df-mnf 11296  df-xr 11297  df-ltxr 11298  df-le 11299  df-sub 11492  df-neg 11493  df-div 11919  df-nn 12265  df-2 12327  df-3 12328  df-4 12329  df-5 12330  df-6 12331  df-7 12332  df-8 12333  df-9 12334  df-n0 12525  df-xnn0 12598  df-z 12612  df-dec 12732  df-uz 12877  df-rp 13033  df-fz 13545  df-fzo 13692  df-seq 14040  df-exp 14100  df-hash 14367  df-word 14550  df-lsw 14598  df-concat 14606  df-s1 14631  df-substr 14676  df-pfx 14706  df-splice 14785  df-reverse 14794  df-s2 14884  df-struct 17181  df-sets 17198  df-slot 17216  df-ndx 17228  df-base 17246  df-ress 17275  df-plusg 17311  df-mulr 17312  df-starv 17313  df-sca 17314  df-vsca 17315  df-ip 17316  df-tset 17317  df-ple 17318  df-ds 17320  df-unif 17321  df-hom 17322  df-cco 17323  df-0g 17488  df-gsum 17489  df-prds 17494  df-pws 17496  df-mre 17631  df-mrc 17632  df-acs 17634  df-mgm 18666  df-sgrp 18745  df-mnd 18761  df-mhm 18809  df-submnd 18810  df-efmnd 18895  df-grp 18967  df-minusg 18968  df-mulg 19099  df-subg 19154  df-ghm 19244  df-gim 19290  df-cntz 19348  df-oppg 19377  df-symg 19402  df-pmtr 19475  df-psgn 19524  df-cmn 19815  df-abl 19816  df-mgp 20153  df-rng 20171  df-ur 20200  df-ring 20253  df-cring 20254  df-oppr 20351  df-dvdsr 20374  df-unit 20375  df-invr 20405  df-dvr 20418  df-rhm 20489  df-subrng 20563  df-subrg 20587  df-drng 20748  df-sra 21190  df-rgmod 21191  df-cnfld 21383  df-zring 21476  df-zrh 21532  df-dsmm 21770  df-frlm 21785  df-mat 22428  df-marrep 22580  df-subma 22599  df-mdet 22607  df-madu 22656  df-minmar1 22657  df-smat 33755

This theorem is referenced by:  madjusmdetlem4  33791