Theorem madjusmdetlem1 33773

Description: Lemma for madjusmdet 33777. (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 22679	. . . 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 1371	. . 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 6923	. . 3 ⊢ (𝐷‘𝑈) = (𝐷‘(𝐼(((1...𝑁) minMatR1 𝑅)‘𝑀)𝐽))
12	9, 11	eqtr4di 2798	. 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 14024	. . 3 ⊢ (𝜑 → (1...𝑁) ∈ Fin)
20		crngring 20272	. . . . . 6 ⊢ (𝑅 ∈ CRing → 𝑅 ∈ Ring)
21	18, 20	syl 17	. . . . 5 ⊢ (𝜑 → 𝑅 ∈ Ring)
22	4, 5	minmar1cl 22678	. . . . 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 2848	. . 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 33771	. 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 6924	. . . . . . . . . . . . 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 725	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) ∧ 𝑖 = 𝑁) ∧ 𝑗 = 𝑁) → (𝑃‘𝑁) = 𝐼)
33	29, 32	eqtrd 2780	. . . . . . . . . . . 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 6924	. . . . . . . . . . . . 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 725	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) ∧ 𝑖 = 𝑁) ∧ 𝑗 = 𝑁) → (𝑄‘𝑁) = 𝐽)
39	35, 38	eqtrd 2780	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) ∧ 𝑖 = 𝑁) ∧ 𝑗 = 𝑁) → (𝑄‘𝑗) = 𝐽)
40	33, 39	oveq12d 7466	. . . . . . . . . . 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 725	. . . . . . . . . . . 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 725	. . . . . . . . . . . 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 725	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) ∧ 𝑖 = 𝑁) ∧ 𝑗 = 𝑁) → 𝐽 ∈ (1...𝑁))
47		eqid 2740	. . . . . . . . . . . . 13 ⊢ ((1...𝑁) minMatR1 𝑅) = ((1...𝑁) minMatR1 𝑅)
48		eqid 2740	. . . . . . . . . . . . 13 ⊢ (1r‘𝑅) = (1r‘𝑅)
49		eqid 2740	. . . . . . . . . . . . 13 ⊢ (0g‘𝑅) = (0g‘𝑅)
50	4, 5, 47, 48, 49	minmar1eval 22676	. . . . . . . . . . . 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 1379	. . . . . . . . . . 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 2740	. . . . . . . . . . . . . 14 ⊢ 𝐼 = 𝐼
53	52	iftruei 4555	. . . . . . . . . . . . 13 ⊢ if(𝐼 = 𝐼, if(𝐽 = 𝐽, (1r‘𝑅), (0g‘𝑅)), (𝐼𝑀𝐽)) = if(𝐽 = 𝐽, (1r‘𝑅), (0g‘𝑅))
54		eqid 2740	. . . . . . . . . . . . . 14 ⊢ 𝐽 = 𝐽
55	54	iftruei 4555	. . . . . . . . . . . . 13 ⊢ if(𝐽 = 𝐽, (1r‘𝑅), (0g‘𝑅)) = (1r‘𝑅)
56	53, 55	eqtri 2768	. . . . . . . . . . . 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 2785	. . . . . . . . . 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 6924	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) ∧ 𝑖 = 𝑁) ∧ ¬ 𝑗 = 𝑁) → (𝑃‘𝑖) = (𝑃‘𝑁))
61	31	ad2antrr 725	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) ∧ 𝑖 = 𝑁) ∧ ¬ 𝑗 = 𝑁) → (𝑃‘𝑁) = 𝐼)
62	60, 61	eqtrd 2780	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) ∧ 𝑖 = 𝑁) ∧ ¬ 𝑗 = 𝑁) → (𝑃‘𝑖) = 𝐼)
63	62	oveq1d 7463	. . . . . . . . . . 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 725	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) ∧ 𝑖 = 𝑁) ∧ ¬ 𝑗 = 𝑁) → 𝑀 ∈ 𝐵)
65	43	ad2antrr 725	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) ∧ 𝑖 = 𝑁) ∧ ¬ 𝑗 = 𝑁) → 𝐼 ∈ (1...𝑁))
66	45	ad2antrr 725	. . . . . . . . . . . 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 2740	. . . . . . . . . . . . . . 15 ⊢ (SymGrp‘(1...𝑁)) = (SymGrp‘(1...𝑁))
70	69, 13	symgfv 19421	. . . . . . . . . . . . . 14 ⊢ ((𝑄 ∈ 𝐺 ∧ 𝑗 ∈ (1...𝑁)) → (𝑄‘𝑗) ∈ (1...𝑁))
71	67, 68, 70	syl2anc 583	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) → (𝑄‘𝑗) ∈ (1...𝑁))
72	71	ad2antrr 725	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) ∧ 𝑖 = 𝑁) ∧ ¬ 𝑗 = 𝑁) → (𝑄‘𝑗) ∈ (1...𝑁))
73	4, 5, 47, 48, 49	minmar1eval 22676	. . . . . . . . . . . 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 1379	. . . . . . . . . . 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 4556	. . . . . . . . . . . 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 6924	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) ∧ 𝑖 = 𝑁) ∧ (𝑄‘𝑗) = 𝐽) → (◡𝑄‘(𝑄‘𝑗)) = (◡𝑄‘𝐽))
79	69, 13	symgbasf1o 19416	. . . . . . . . . . . . . . . . . . . 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 725	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) ∧ 𝑖 = 𝑁) ∧ (𝑄‘𝑗) = 𝐽) → 𝑄:(1...𝑁)–1-1-onto→(1...𝑁))
82	68	ad2antrr 725	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) ∧ 𝑖 = 𝑁) ∧ (𝑄‘𝑗) = 𝐽) → 𝑗 ∈ (1...𝑁))
83		f1ocnvfv1 7312	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑄:(1...𝑁)–1-1-onto→(1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) → (◡𝑄‘(𝑄‘𝑗)) = 𝑗)
84	81, 82, 83	syl2anc 583	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) ∧ 𝑖 = 𝑁) ∧ (𝑄‘𝑗) = 𝐽) → (◡𝑄‘(𝑄‘𝑗)) = 𝑗)
85	36	fveq2d 6924	. . . . . . . . . . . . . . . . . . . 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 12946	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ℕ = (ℤ≥‘1)
89	87, 88	eleqtrdi 2854	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘1))
90		eluzfz2 13592	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑁 ∈ (ℤ≥‘1) → 𝑁 ∈ (1...𝑁))
91	89, 90	syl 17	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → 𝑁 ∈ (1...𝑁))
92		f1ocnvfv1 7312	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑄:(1...𝑁)–1-1-onto→(1...𝑁) ∧ 𝑁 ∈ (1...𝑁)) → (◡𝑄‘(𝑄‘𝑁)) = 𝑁)
93	86, 91, 92	syl2anc 583	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → (◡𝑄‘(𝑄‘𝑁)) = 𝑁)
94	85, 93	eqtr3d 2782	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → (◡𝑄‘𝐽) = 𝑁)
95	94	3ad2ant1 1133	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) → (◡𝑄‘𝐽) = 𝑁)
96	95	ad2antrr 725	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) ∧ 𝑖 = 𝑁) ∧ (𝑄‘𝑗) = 𝐽) → (◡𝑄‘𝐽) = 𝑁)
97	78, 84, 96	3eqtr3d 2788	. . . . . . . . . . . . . . . 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 4559	. . . . . . . . . . . 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 2780	. . . . . . . . . . 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 2785	. . . . . . . . . 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 4584	. . . . . . . . 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 7483	. . . . . . . . . 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 7461	. . . . . . . . . . . . . 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 7528	. . . . . . . . . . . 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 2768	. . . . . . . . . . 11 ⊢ 𝑊 = (𝑖 ∈ (1...𝑁), 𝑗 ∈ (1...𝑁) ↦ ((𝑃‘𝑖)(𝐼(((1...𝑁) minMatR1 𝑅)‘𝑀)𝐽)(𝑄‘𝑗)))
113	112	ovmpt4g 7597	. . . . . . . . . 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 1371	. . . . . . . . 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 4584	. . . . . . . 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 7527	. . . . . . 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 2740	. . . . . . . . . 10 ⊢ (Base‘𝑅) = (Base‘𝑅)
118	25	3ad2ant1 1133	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) → 𝑃 ∈ 𝐺)
119	69, 13	symgfv 19421	. . . . . . . . . . . 12 ⊢ ((𝑃 ∈ 𝐺 ∧ 𝑖 ∈ (1...𝑁)) → (𝑃‘𝑖) ∈ (1...𝑁))
120	118, 105, 119	syl2anc 583	. . . . . . . . . . 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 22453	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) → ((𝑃‘𝑖)𝑈(𝑄‘𝑗)) ∈ (Base‘𝑅))
123	4, 117, 5, 19, 18, 122	matbas2d 22450	. . . . . . . . 9 ⊢ (𝜑 → (𝑖 ∈ (1...𝑁), 𝑗 ∈ (1...𝑁) ↦ ((𝑃‘𝑖)𝑈(𝑄‘𝑗))) ∈ 𝐵)
124	17, 123	eqeltrid 2848	. . . . . . . 8 ⊢ (𝜑 → 𝑊 ∈ 𝐵)
125	117, 48	ringidcl 20289	. . . . . . . . 9 ⊢ (𝑅 ∈ Ring → (1r‘𝑅) ∈ (Base‘𝑅))
126	21, 125	syl 17	. . . . . . . 8 ⊢ (𝜑 → (1r‘𝑅) ∈ (Base‘𝑅))
127		eqid 2740	. . . . . . . . 9 ⊢ ((1...𝑁) matRRep 𝑅) = ((1...𝑁) matRRep 𝑅)
128	4, 5, 127, 49	marrepval 22589	. . . . . . . 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 2790	. . . . . 6 ⊢ (𝜑 → (𝑁(𝑊((1...𝑁) matRRep 𝑅)(1r‘𝑅))𝑁) = 𝑊)
132	131	fveq2d 6924	. . . . 5 ⊢ (𝜑 → (𝐷‘(𝑁(𝑊((1...𝑁) matRRep 𝑅)(1r‘𝑅))𝑁)) = (𝐷‘𝑊))
133		eqid 2740	. . . . . . . . . . . 12 ⊢ ((1...𝑁) subMat 𝑅) = ((1...𝑁) subMat 𝑅)
134	4, 133, 5	submaval 22608	. . . . . . . . . . 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 1371	. . . . . . . . . 10 ⊢ (𝜑 → (𝑁(((1...𝑁) subMat 𝑅)‘𝑊)𝑁) = (𝑖 ∈ ((1...𝑁) ∖ {𝑁}), 𝑗 ∈ ((1...𝑁) ∖ {𝑁}) ↦ (𝑖𝑊𝑗)))
136		fzdif2 32796	. . . . . . . . . . . 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 7523	. . . . . . . . . . 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 583	. . . . . . . . . 10 ⊢ (𝜑 → (𝑖 ∈ ((1...𝑁) ∖ {𝑁}), 𝑗 ∈ ((1...𝑁) ∖ {𝑁}) ↦ (𝑖𝑊𝑗)) = (𝑖 ∈ (1...(𝑁 − 1)), 𝑗 ∈ (1...(𝑁 − 1)) ↦ (𝑖𝑊𝑗)))
140	135, 139	eqtrd 2780	. . . . . . . . 9 ⊢ (𝜑 → (𝑁(((1...𝑁) subMat 𝑅)‘𝑊)𝑁) = (𝑖 ∈ (1...(𝑁 − 1)), 𝑗 ∈ (1...(𝑁 − 1)) ↦ (𝑖𝑊𝑗)))
141		difssd 4160	. . . . . . . . . . 11 ⊢ (𝜑 → ((1...𝑁) ∖ {𝑁}) ⊆ (1...𝑁))
142	137, 141	eqsstrrd 4048	. . . . . . . . . 10 ⊢ (𝜑 → (1...(𝑁 − 1)) ⊆ (1...𝑁))
143	4, 5	submabas 22605	. . . . . . . . . 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 583	. . . . . . . . 9 ⊢ (𝜑 → (𝑖 ∈ (1...(𝑁 − 1)), 𝑗 ∈ (1...(𝑁 − 1)) ↦ (𝑖𝑊𝑗)) ∈ (Base‘((1...(𝑁 − 1)) Mat 𝑅)))
145	140, 144	eqeltrd 2844	. . . . . . . 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 2740	. . . . . . . . 9 ⊢ ((1...(𝑁 − 1)) Mat 𝑅) = ((1...(𝑁 − 1)) Mat 𝑅)
148		eqid 2740	. . . . . . . . 9 ⊢ (Base‘((1...(𝑁 − 1)) Mat 𝑅)) = (Base‘((1...(𝑁 − 1)) Mat 𝑅))
149	146, 147, 148, 117	mdetcl 22623	. . . . . . . 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 583	. . . . . . 7 ⊢ (𝜑 → (𝐸‘(𝑁(((1...𝑁) subMat 𝑅)‘𝑊)𝑁)) ∈ (Base‘𝑅))
151	117, 16, 48	ringlidm 20292	. . . . . . 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 583	. . . . . 6 ⊢ (𝜑 → ((1r‘𝑅) · (𝐸‘(𝑁(((1...𝑁) subMat 𝑅)‘𝑊)𝑁))) = (𝐸‘(𝑁(((1...𝑁) subMat 𝑅)‘𝑊)𝑁)))
153	4	fveq2i 6923	. . . . . . . . . . 11 ⊢ (Base‘𝐴) = (Base‘((1...𝑁) Mat 𝑅))
154	5, 153	eqtri 2768	. . . . . . . . . 10 ⊢ 𝐵 = (Base‘((1...𝑁) Mat 𝑅))
155	124, 154	eleqtrdi 2854	. . . . . . . . 9 ⊢ (𝜑 → 𝑊 ∈ (Base‘((1...𝑁) Mat 𝑅)))
156		smadiadetr 22702	. . . . . . . . 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 6921	. . . . . . . . 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 7461	. . . . . . . . 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 2758	. . . . . . . 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 7463	. . . . . . . . . 10 ⊢ (𝜑 → (((1...𝑁) ∖ {𝑁}) maDet 𝑅) = ((1...(𝑁 − 1)) maDet 𝑅))
163	162, 146	eqtr4di 2798	. . . . . . . . 9 ⊢ (𝜑 → (((1...𝑁) ∖ {𝑁}) maDet 𝑅) = 𝐸)
164	163	fveq1d 6922	. . . . . . . 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 7464	. . . . . . 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 2780	. . . . . 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 33751	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ ∧ 𝑊 ∈ 𝐵) → (𝑁(subMat1‘𝑊)𝑁) = (𝑁(((1...𝑁) subMat 𝑅)‘𝑊)𝑁))
168	87, 124, 167	syl2anc 583	. . . . . . 7 ⊢ (𝜑 → (𝑁(subMat1‘𝑊)𝑁) = (𝑁(((1...𝑁) subMat 𝑅)‘𝑊)𝑁))
169	168	fveq2d 6924	. . . . . 6 ⊢ (𝜑 → (𝐸‘(𝑁(subMat1‘𝑊)𝑁)) = (𝐸‘(𝑁(((1...𝑁) subMat 𝑅)‘𝑊)𝑁)))
170	152, 166, 169	3eqtr4d 2790	. . . . 5 ⊢ (𝜑 → (𝐷‘(𝑁(𝑊((1...𝑁) matRRep 𝑅)(1r‘𝑅))𝑁)) = (𝐸‘(𝑁(subMat1‘𝑊)𝑁)))
171	132, 170	eqtr3d 2782	. . . 4 ⊢ (𝜑 → (𝐷‘𝑊) = (𝐸‘(𝑁(subMat1‘𝑊)𝑁)))
172	4, 5, 87, 3, 2, 21, 1, 10	submatminr1 33756	. . . . . 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 2780	. . . . 5 ⊢ (𝜑 → (𝐼(subMat1‘𝑀)𝐽) = (𝑁(subMat1‘𝑊)𝑁))
175	174	fveq2d 6924	. . . 4 ⊢ (𝜑 → (𝐸‘(𝐼(subMat1‘𝑀)𝐽)) = (𝐸‘(𝑁(subMat1‘𝑊)𝑁)))
176	171, 175	eqtr4d 2783	. . 3 ⊢ (𝜑 → (𝐷‘𝑊) = (𝐸‘(𝐼(subMat1‘𝑀)𝐽)))
177	176	oveq2d 7464	. 2 ⊢ (𝜑 → ((𝑍‘((𝑆‘𝑃) · (𝑆‘𝑄))) · (𝐷‘𝑊)) = ((𝑍‘((𝑆‘𝑃) · (𝑆‘𝑄))) · (𝐸‘(𝐼(subMat1‘𝑀)𝐽))))
178	12, 27, 177	3eqtrd 2784	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 1087   = wceq 1537   ∈ wcel 2108  Vcvv 3488   ∖ cdif 3973   ⊆ wss 3976  ifcif 4548  {csn 4648  ^◡ccnv 5699  –1-1-onto→wf1o 6572  ‘cfv 6573  (class class class)co 7448   ∈ cmpo 7450  1c1 11185   · cmul 11189   − cmin 11520  ℕcn 12293  ℤ_≥cuz 12903  ...cfz 13567  Basecbs 17258  ._rcmulr 17312  0_gc0g 17499  SymGrpcsymg 19410  pmSgncpsgn 19531  1_rcur 20208  Ringcrg 20260  CRingccrg 20261  ℤRHomczrh 21533   Mat cmat 22432   matRRep cmarrep 22583   subMat csubma 22603   maDet cmdat 22611   maAdju cmadu 22659   minMatR1 cminmar1 22660  subMat1csmat 33739

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-rep 5303  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770  ax-cnex 11240  ax-resscn 11241  ax-1cn 11242  ax-icn 11243  ax-addcl 11244  ax-addrcl 11245  ax-mulcl 11246  ax-mulrcl 11247  ax-mulcom 11248  ax-addass 11249  ax-mulass 11250  ax-distr 11251  ax-i2m1 11252  ax-1ne0 11253  ax-1rid 11254  ax-rnegex 11255  ax-rrecex 11256  ax-cnre 11257  ax-pre-lttri 11258  ax-pre-lttrn 11259  ax-pre-ltadd 11260  ax-pre-mulgt0 11261  ax-addf 11263  ax-mulf 11264

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-xor 1509  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-nel 3053  df-ral 3068  df-rex 3077  df-rmo 3388  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-pss 3996  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-tp 4653  df-op 4655  df-ot 4657  df-uni 4932  df-int 4971  df-iun 5017  df-iin 5018  df-br 5167  df-opab 5229  df-mpt 5250  df-tr 5284  df-id 5593  df-eprel 5599  df-po 5607  df-so 5608  df-fr 5652  df-se 5653  df-we 5654  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-pred 6332  df-ord 6398  df-on 6399  df-lim 6400  df-suc 6401  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-isom 6582  df-riota 7404  df-ov 7451  df-oprab 7452  df-mpo 7453  df-of 7714  df-om 7904  df-1st 8030  df-2nd 8031  df-supp 8202  df-tpos 8267  df-frecs 8322  df-wrecs 8353  df-recs 8427  df-rdg 8466  df-1o 8522  df-2o 8523  df-er 8763  df-map 8886  df-pm 8887  df-ixp 8956  df-en 9004  df-dom 9005  df-sdom 9006  df-fin 9007  df-fsupp 9432  df-sup 9511  df-oi 9579  df-card 10008  df-pnf 11326  df-mnf 11327  df-xr 11328  df-ltxr 11329  df-le 11330  df-sub 11522  df-neg 11523  df-div 11948  df-nn 12294  df-2 12356  df-3 12357  df-4 12358  df-5 12359  df-6 12360  df-7 12361  df-8 12362  df-9 12363  df-n0 12554  df-xnn0 12626  df-z 12640  df-dec 12759  df-uz 12904  df-rp 13058  df-fz 13568  df-fzo 13712  df-seq 14053  df-exp 14113  df-hash 14380  df-word 14563  df-lsw 14611  df-concat 14619  df-s1 14644  df-substr 14689  df-pfx 14719  df-splice 14798  df-reverse 14807  df-s2 14897  df-struct 17194  df-sets 17211  df-slot 17229  df-ndx 17241  df-base 17259  df-ress 17288  df-plusg 17324  df-mulr 17325  df-starv 17326  df-sca 17327  df-vsca 17328  df-ip 17329  df-tset 17330  df-ple 17331  df-ds 17333  df-unif 17334  df-hom 17335  df-cco 17336  df-0g 17501  df-gsum 17502  df-prds 17507  df-pws 17509  df-mre 17644  df-mrc 17645  df-acs 17647  df-mgm 18678  df-sgrp 18757  df-mnd 18773  df-mhm 18818  df-submnd 18819  df-efmnd 18904  df-grp 18976  df-minusg 18977  df-mulg 19108  df-subg 19163  df-ghm 19253  df-gim 19299  df-cntz 19357  df-oppg 19386  df-symg 19411  df-pmtr 19484  df-psgn 19533  df-cmn 19824  df-abl 19825  df-mgp 20162  df-rng 20180  df-ur 20209  df-ring 20262  df-cring 20263  df-oppr 20360  df-dvdsr 20383  df-unit 20384  df-invr 20414  df-dvr 20427  df-rhm 20498  df-subrng 20572  df-subrg 20597  df-drng 20753  df-sra 21195  df-rgmod 21196  df-cnfld 21388  df-zring 21481  df-zrh 21537  df-dsmm 21775  df-frlm 21790  df-mat 22433  df-marrep 22585  df-subma 22604  df-mdet 22612  df-madu 22661  df-minmar1 22662  df-smat 33740

This theorem is referenced by:  madjusmdetlem4  33776