Theorem madjusmdetlem1 33863

Description: Lemma for madjusmdet 33867. (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 22595	. . . 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 1373	. . 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 6884	. . 3 ⊢ (𝐷‘𝑈) = (𝐷‘(𝐼(((1...𝑁) minMatR1 𝑅)‘𝑀)𝐽))
12	9, 11	eqtr4di 2789	. 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 13996	. . 3 ⊢ (𝜑 → (1...𝑁) ∈ Fin)
20		crngring 20210	. . . . . 6 ⊢ (𝑅 ∈ CRing → 𝑅 ∈ Ring)
21	18, 20	syl 17	. . . . 5 ⊢ (𝜑 → 𝑅 ∈ Ring)
22	4, 5	minmar1cl 22594	. . . . 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 838	. . . 4 ⊢ (𝜑 → (𝐼(((1...𝑁) minMatR1 𝑅)‘𝑀)𝐽) ∈ 𝐵)
24	10, 23	eqeltrid 2839	. . 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 33861	. 2 ⊢ (𝜑 → (𝐷‘𝑈) = ((𝑍‘((𝑆‘𝑃) · (𝑆‘𝑄))) · (𝐷‘𝑊)))
28		simplr 768	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) ∧ 𝑖 = 𝑁) ∧ 𝑗 = 𝑁) → 𝑖 = 𝑁)
29	28	fveq2d 6885	. . . . . . . . . . . . 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 1133	. . . . . . . . . . . . . 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 2771	. . . . . . . . . . . 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 6885	. . . . . . . . . . . . 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 1133	. . . . . . . . . . . . . 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 2771	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) ∧ 𝑖 = 𝑁) ∧ 𝑗 = 𝑁) → (𝑄‘𝑗) = 𝐽)
40	33, 39	oveq12d 7428	. . . . . . . . . . 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 1133	. . . . . . . . . . . . 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 1133	. . . . . . . . . . . . 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 1133	. . . . . . . . . . . . 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 2736	. . . . . . . . . . . . 13 ⊢ ((1...𝑁) minMatR1 𝑅) = ((1...𝑁) minMatR1 𝑅)
48		eqid 2736	. . . . . . . . . . . . 13 ⊢ (1r‘𝑅) = (1r‘𝑅)
49		eqid 2736	. . . . . . . . . . . . 13 ⊢ (0g‘𝑅) = (0g‘𝑅)
50	4, 5, 47, 48, 49	minmar1eval 22592	. . . . . . . . . . . 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 1381	. . . . . . . . . . 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 2736	. . . . . . . . . . . . . 14 ⊢ 𝐼 = 𝐼
53	52	iftruei 4512	. . . . . . . . . . . . 13 ⊢ if(𝐼 = 𝐼, if(𝐽 = 𝐽, (1r‘𝑅), (0g‘𝑅)), (𝐼𝑀𝐽)) = if(𝐽 = 𝐽, (1r‘𝑅), (0g‘𝑅))
54		eqid 2736	. . . . . . . . . . . . . 14 ⊢ 𝐽 = 𝐽
55	54	iftruei 4512	. . . . . . . . . . . . 13 ⊢ if(𝐽 = 𝐽, (1r‘𝑅), (0g‘𝑅)) = (1r‘𝑅)
56	53, 55	eqtri 2759	. . . . . . . . . . . 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 2776	. . . . . . . . . 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 768	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) ∧ 𝑖 = 𝑁) ∧ ¬ 𝑗 = 𝑁) → 𝑖 = 𝑁)
60	59	fveq2d 6885	. . . . . . . . . . . . 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 2771	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) ∧ 𝑖 = 𝑁) ∧ ¬ 𝑗 = 𝑁) → (𝑃‘𝑖) = 𝐼)
63	62	oveq1d 7425	. . . . . . . . . . 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 1133	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) → 𝑄 ∈ 𝐺)
68		simp3 1138	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) → 𝑗 ∈ (1...𝑁))
69		eqid 2736	. . . . . . . . . . . . . . 15 ⊢ (SymGrp‘(1...𝑁)) = (SymGrp‘(1...𝑁))
70	69, 13	symgfv 19366	. . . . . . . . . . . . . 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 22592	. . . . . . . . . . . 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 1381	. . . . . . . . . . 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 4513	. . . . . . . . . . . 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 6885	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) ∧ 𝑖 = 𝑁) ∧ (𝑄‘𝑗) = 𝐽) → (◡𝑄‘(𝑄‘𝑗)) = (◡𝑄‘𝐽))
79	69, 13	symgbasf1o 19361	. . . . . . . . . . . . . . . . . . . 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 7274	. . . . . . . . . . . . . . . . . 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 6885	. . . . . . . . . . . . . . . . . . . 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 12900	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ℕ = (ℤ≥‘1)
89	87, 88	eleqtrdi 2845	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘1))
90		eluzfz2 13554	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑁 ∈ (ℤ≥‘1) → 𝑁 ∈ (1...𝑁))
91	89, 90	syl 17	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → 𝑁 ∈ (1...𝑁))
92		f1ocnvfv1 7274	. . . . . . . . . . . . . . . . . . . . 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 2773	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → (◡𝑄‘𝐽) = 𝑁)
95	94	3ad2ant1 1133	. . . . . . . . . . . . . . . . . 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 2779	. . . . . . . . . . . . . . . 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 4516	. . . . . . . . . . . 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 2771	. . . . . . . . . . 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 2776	. . . . . . . . . 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 4542	. . . . . . . . 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 1137	. . . . . . . . . . 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 7445	. . . . . . . . . 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 7423	. . . . . . . . . . . . . 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 7490	. . . . . . . . . . . 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 2759	. . . . . . . . . . 11 ⊢ 𝑊 = (𝑖 ∈ (1...𝑁), 𝑗 ∈ (1...𝑁) ↦ ((𝑃‘𝑖)(𝐼(((1...𝑁) minMatR1 𝑅)‘𝑀)𝐽)(𝑄‘𝑗)))
113	112	ovmpt4g 7559	. . . . . . . . . 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 1373	. . . . . . . . 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 4542	. . . . . . . 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 7489	. . . . . . 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 2736	. . . . . . . . . 10 ⊢ (Base‘𝑅) = (Base‘𝑅)
118	25	3ad2ant1 1133	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) → 𝑃 ∈ 𝐺)
119	69, 13	symgfv 19366	. . . . . . . . . . . 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 1133	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) → 𝑈 ∈ 𝐵)
122	4, 117, 5, 120, 71, 121	matecld 22369	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) → ((𝑃‘𝑖)𝑈(𝑄‘𝑗)) ∈ (Base‘𝑅))
123	4, 117, 5, 19, 18, 122	matbas2d 22366	. . . . . . . . 9 ⊢ (𝜑 → (𝑖 ∈ (1...𝑁), 𝑗 ∈ (1...𝑁) ↦ ((𝑃‘𝑖)𝑈(𝑄‘𝑗))) ∈ 𝐵)
124	17, 123	eqeltrid 2839	. . . . . . . 8 ⊢ (𝜑 → 𝑊 ∈ 𝐵)
125	117, 48	ringidcl 20230	. . . . . . . . 9 ⊢ (𝑅 ∈ Ring → (1r‘𝑅) ∈ (Base‘𝑅))
126	21, 125	syl 17	. . . . . . . 8 ⊢ (𝜑 → (1r‘𝑅) ∈ (Base‘𝑅))
127		eqid 2736	. . . . . . . . 9 ⊢ ((1...𝑁) matRRep 𝑅) = ((1...𝑁) matRRep 𝑅)
128	4, 5, 127, 49	marrepval 22505	. . . . . . . 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 838	. . . . . . 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 2781	. . . . . 6 ⊢ (𝜑 → (𝑁(𝑊((1...𝑁) matRRep 𝑅)(1r‘𝑅))𝑁) = 𝑊)
132	131	fveq2d 6885	. . . . 5 ⊢ (𝜑 → (𝐷‘(𝑁(𝑊((1...𝑁) matRRep 𝑅)(1r‘𝑅))𝑁)) = (𝐷‘𝑊))
133		eqid 2736	. . . . . . . . . . . 12 ⊢ ((1...𝑁) subMat 𝑅) = ((1...𝑁) subMat 𝑅)
134	4, 133, 5	submaval 22524	. . . . . . . . . . 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 1373	. . . . . . . . . 10 ⊢ (𝜑 → (𝑁(((1...𝑁) subMat 𝑅)‘𝑊)𝑁) = (𝑖 ∈ ((1...𝑁) ∖ {𝑁}), 𝑗 ∈ ((1...𝑁) ∖ {𝑁}) ↦ (𝑖𝑊𝑗)))
136		fzdif2 32772	. . . . . . . . . . . 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 7485	. . . . . . . . . . 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 2771	. . . . . . . . 9 ⊢ (𝜑 → (𝑁(((1...𝑁) subMat 𝑅)‘𝑊)𝑁) = (𝑖 ∈ (1...(𝑁 − 1)), 𝑗 ∈ (1...(𝑁 − 1)) ↦ (𝑖𝑊𝑗)))
141		difssd 4117	. . . . . . . . . . 11 ⊢ (𝜑 → ((1...𝑁) ∖ {𝑁}) ⊆ (1...𝑁))
142	137, 141	eqsstrrd 3999	. . . . . . . . . 10 ⊢ (𝜑 → (1...(𝑁 − 1)) ⊆ (1...𝑁))
143	4, 5	submabas 22521	. . . . . . . . . 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 2835	. . . . . . . 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 2736	. . . . . . . . 9 ⊢ ((1...(𝑁 − 1)) Mat 𝑅) = ((1...(𝑁 − 1)) Mat 𝑅)
148		eqid 2736	. . . . . . . . 9 ⊢ (Base‘((1...(𝑁 − 1)) Mat 𝑅)) = (Base‘((1...(𝑁 − 1)) Mat 𝑅))
149	146, 147, 148, 117	mdetcl 22539	. . . . . . . 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 20234	. . . . . . 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 6884	. . . . . . . . . . 11 ⊢ (Base‘𝐴) = (Base‘((1...𝑁) Mat 𝑅))
154	5, 153	eqtri 2759	. . . . . . . . . 10 ⊢ 𝐵 = (Base‘((1...𝑁) Mat 𝑅))
155	124, 154	eleqtrdi 2845	. . . . . . . . 9 ⊢ (𝜑 → 𝑊 ∈ (Base‘((1...𝑁) Mat 𝑅)))
156		smadiadetr 22618	. . . . . . . . 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 838	. . . . . . . 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 6882	. . . . . . . . 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 7423	. . . . . . . . 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 2754	. . . . . . . 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 7425	. . . . . . . . . 10 ⊢ (𝜑 → (((1...𝑁) ∖ {𝑁}) maDet 𝑅) = ((1...(𝑁 − 1)) maDet 𝑅))
163	162, 146	eqtr4di 2789	. . . . . . . . 9 ⊢ (𝜑 → (((1...𝑁) ∖ {𝑁}) maDet 𝑅) = 𝐸)
164	163	fveq1d 6883	. . . . . . . 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 7426	. . . . . . 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 2771	. . . . . 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 33841	. . . . . . . 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 6885	. . . . . 6 ⊢ (𝜑 → (𝐸‘(𝑁(subMat1‘𝑊)𝑁)) = (𝐸‘(𝑁(((1...𝑁) subMat 𝑅)‘𝑊)𝑁)))
170	152, 166, 169	3eqtr4d 2781	. . . . 5 ⊢ (𝜑 → (𝐷‘(𝑁(𝑊((1...𝑁) matRRep 𝑅)(1r‘𝑅))𝑁)) = (𝐸‘(𝑁(subMat1‘𝑊)𝑁)))
171	132, 170	eqtr3d 2773	. . . 4 ⊢ (𝜑 → (𝐷‘𝑊) = (𝐸‘(𝑁(subMat1‘𝑊)𝑁)))
172	4, 5, 87, 3, 2, 21, 1, 10	submatminr1 33846	. . . . . 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 2771	. . . . 5 ⊢ (𝜑 → (𝐼(subMat1‘𝑀)𝐽) = (𝑁(subMat1‘𝑊)𝑁))
175	174	fveq2d 6885	. . . 4 ⊢ (𝜑 → (𝐸‘(𝐼(subMat1‘𝑀)𝐽)) = (𝐸‘(𝑁(subMat1‘𝑊)𝑁)))
176	171, 175	eqtr4d 2774	. . 3 ⊢ (𝜑 → (𝐷‘𝑊) = (𝐸‘(𝐼(subMat1‘𝑀)𝐽)))
177	176	oveq2d 7426	. 2 ⊢ (𝜑 → ((𝑍‘((𝑆‘𝑃) · (𝑆‘𝑄))) · (𝐷‘𝑊)) = ((𝑍‘((𝑆‘𝑃) · (𝑆‘𝑄))) · (𝐸‘(𝐼(subMat1‘𝑀)𝐽))))
178	12, 27, 177	3eqtrd 2775	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 1540   ∈ wcel 2109  Vcvv 3464   ∖ cdif 3928   ⊆ wss 3931  ifcif 4505  {csn 4606  ^◡ccnv 5658  –1-1-onto→wf1o 6535  ‘cfv 6536  (class class class)co 7410   ∈ cmpo 7412  1c1 11135   · cmul 11139   − cmin 11471  ℕcn 12245  ℤ_≥cuz 12857  ...cfz 13529  Basecbs 17233  ._rcmulr 17277  0_gc0g 17458  SymGrpcsymg 19355  pmSgncpsgn 19475  1_rcur 20146  Ringcrg 20198  CRingccrg 20199  ℤRHomczrh 21465   Mat cmat 22350   matRRep cmarrep 22499   subMat csubma 22519   maDet cmdat 22527   maAdju cmadu 22575   minMatR1 cminmar1 22576  subMat1csmat 33829

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2708  ax-rep 5254  ax-sep 5271  ax-nul 5281  ax-pow 5340  ax-pr 5407  ax-un 7734  ax-cnex 11190  ax-resscn 11191  ax-1cn 11192  ax-icn 11193  ax-addcl 11194  ax-addrcl 11195  ax-mulcl 11196  ax-mulrcl 11197  ax-mulcom 11198  ax-addass 11199  ax-mulass 11200  ax-distr 11201  ax-i2m1 11202  ax-1ne0 11203  ax-1rid 11204  ax-rnegex 11205  ax-rrecex 11206  ax-cnre 11207  ax-pre-lttri 11208  ax-pre-lttrn 11209  ax-pre-ltadd 11210  ax-pre-mulgt0 11211  ax-addf 11213  ax-mulf 11214

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-xor 1512  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2810  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3062  df-rmo 3364  df-reu 3365  df-rab 3421  df-v 3466  df-sbc 3771  df-csb 3880  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-pss 3951  df-nul 4314  df-if 4506  df-pw 4582  df-sn 4607  df-pr 4609  df-tp 4611  df-op 4613  df-ot 4615  df-uni 4889  df-int 4928  df-iun 4974  df-iin 4975  df-br 5125  df-opab 5187  df-mpt 5207  df-tr 5235  df-id 5553  df-eprel 5558  df-po 5566  df-so 5567  df-fr 5611  df-se 5612  df-we 5613  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-pred 6295  df-ord 6360  df-on 6361  df-lim 6362  df-suc 6363  df-iota 6489  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-isom 6545  df-riota 7367  df-ov 7413  df-oprab 7414  df-mpo 7415  df-of 7676  df-om 7867  df-1st 7993  df-2nd 7994  df-supp 8165  df-tpos 8230  df-frecs 8285  df-wrecs 8316  df-recs 8390  df-rdg 8429  df-1o 8485  df-2o 8486  df-er 8724  df-map 8847  df-pm 8848  df-ixp 8917  df-en 8965  df-dom 8966  df-sdom 8967  df-fin 8968  df-fsupp 9379  df-sup 9459  df-oi 9529  df-card 9958  df-pnf 11276  df-mnf 11277  df-xr 11278  df-ltxr 11279  df-le 11280  df-sub 11473  df-neg 11474  df-div 11900  df-nn 12246  df-2 12308  df-3 12309  df-4 12310  df-5 12311  df-6 12312  df-7 12313  df-8 12314  df-9 12315  df-n0 12507  df-xnn0 12580  df-z 12594  df-dec 12714  df-uz 12858  df-rp 13014  df-fz 13530  df-fzo 13677  df-seq 14025  df-exp 14085  df-hash 14354  df-word 14537  df-lsw 14586  df-concat 14594  df-s1 14619  df-substr 14664  df-pfx 14694  df-splice 14773  df-reverse 14782  df-s2 14872  df-struct 17171  df-sets 17188  df-slot 17206  df-ndx 17218  df-base 17234  df-ress 17257  df-plusg 17289  df-mulr 17290  df-starv 17291  df-sca 17292  df-vsca 17293  df-ip 17294  df-tset 17295  df-ple 17296  df-ds 17298  df-unif 17299  df-hom 17300  df-cco 17301  df-0g 17460  df-gsum 17461  df-prds 17466  df-pws 17468  df-mre 17603  df-mrc 17604  df-acs 17606  df-mgm 18623  df-sgrp 18702  df-mnd 18718  df-mhm 18766  df-submnd 18767  df-efmnd 18852  df-grp 18924  df-minusg 18925  df-mulg 19056  df-subg 19111  df-ghm 19201  df-gim 19247  df-cntz 19305  df-oppg 19334  df-symg 19356  df-pmtr 19428  df-psgn 19477  df-cmn 19768  df-abl 19769  df-mgp 20106  df-rng 20118  df-ur 20147  df-ring 20200  df-cring 20201  df-oppr 20302  df-dvdsr 20322  df-unit 20323  df-invr 20353  df-dvr 20366  df-rhm 20437  df-subrng 20511  df-subrg 20535  df-drng 20696  df-sra 21136  df-rgmod 21137  df-cnfld 21321  df-zring 21413  df-zrh 21469  df-dsmm 21697  df-frlm 21712  df-mat 22351  df-marrep 22501  df-subma 22520  df-mdet 22528  df-madu 22577  df-minmar1 22578  df-smat 33830

This theorem is referenced by:  madjusmdetlem4  33866