madjusmdetlem1 - Mathbox for Thierry Arnoux

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Theorem madjusmdetlem1 30491

Description: Lemma for madjusmdet 30495. (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 20864	. . . 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 1439	. . 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 6449	. . 3 ⊢ (𝐷‘𝑈) = (𝐷‘(𝐼(((1...𝑁) minMatR1 𝑅)‘𝑀)𝐽))
12	9, 11	syl6eqr 2832	. 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 13091	. . 3 ⊢ (𝜑 → (1...𝑁) ∈ Fin)
20		crngring 18945	. . . . . 6 ⊢ (𝑅 ∈ CRing → 𝑅 ∈ Ring)
21	18, 20	syl 17	. . . . 5 ⊢ (𝜑 → 𝑅 ∈ Ring)
22	4, 5	minmar1cl 20863	. . . . 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 829	. . . 4 ⊢ (𝜑 → (𝐼(((1...𝑁) minMatR1 𝑅)‘𝑀)𝐽) ∈ 𝐵)
24	10, 23	syl5eqel 2863	. . 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 30489	. 2 ⊢ (𝜑 → (𝐷‘𝑈) = ((𝑍‘((𝑆‘𝑃) · (𝑆‘𝑄))) · (𝐷‘𝑊)))
28		simplr 759	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) ∧ 𝑖 = 𝑁) ∧ 𝑗 = 𝑁) → 𝑖 = 𝑁)
29	28	fveq2d 6450	. . . . . . . . . . . . 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 1124	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) → (𝑃‘𝑁) = 𝐼)
32	31	ad2antrr 716	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) ∧ 𝑖 = 𝑁) ∧ 𝑗 = 𝑁) → (𝑃‘𝑁) = 𝐼)
33	29, 32	eqtrd 2814	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) ∧ 𝑖 = 𝑁) ∧ 𝑗 = 𝑁) → (𝑃‘𝑖) = 𝐼)
34		simpr 479	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) ∧ 𝑖 = 𝑁) ∧ 𝑗 = 𝑁) → 𝑗 = 𝑁)
35	34	fveq2d 6450	. . . . . . . . . . . . 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 1124	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) → (𝑄‘𝑁) = 𝐽)
38	37	ad2antrr 716	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) ∧ 𝑖 = 𝑁) ∧ 𝑗 = 𝑁) → (𝑄‘𝑁) = 𝐽)
39	35, 38	eqtrd 2814	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) ∧ 𝑖 = 𝑁) ∧ 𝑗 = 𝑁) → (𝑄‘𝑗) = 𝐽)
40	33, 39	oveq12d 6940	. . . . . . . . . . 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 1124	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) → 𝑀 ∈ 𝐵)
42	41	ad2antrr 716	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) ∧ 𝑖 = 𝑁) ∧ 𝑗 = 𝑁) → 𝑀 ∈ 𝐵)
43	3	3ad2ant1 1124	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) → 𝐼 ∈ (1...𝑁))
44	43	ad2antrr 716	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) ∧ 𝑖 = 𝑁) ∧ 𝑗 = 𝑁) → 𝐼 ∈ (1...𝑁))
45	2	3ad2ant1 1124	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) → 𝐽 ∈ (1...𝑁))
46	45	ad2antrr 716	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) ∧ 𝑖 = 𝑁) ∧ 𝑗 = 𝑁) → 𝐽 ∈ (1...𝑁))
47		eqid 2778	. . . . . . . . . . . . 13 ⊢ ((1...𝑁) minMatR1 𝑅) = ((1...𝑁) minMatR1 𝑅)
48		eqid 2778	. . . . . . . . . . . . 13 ⊢ (1_r‘𝑅) = (1_r‘𝑅)
49		eqid 2778	. . . . . . . . . . . . 13 ⊢ (0_g‘𝑅) = (0_g‘𝑅)
50	4, 5, 47, 48, 49	minmar1eval 20860	. . . . . . . . . . . 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, (1_r‘𝑅), (0_g‘𝑅)), (𝐼𝑀𝐽)))
51	42, 44, 46, 44, 46, 50	syl122anc 1447	. . . . . . . . . . 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, (1_r‘𝑅), (0_g‘𝑅)), (𝐼𝑀𝐽)))
52		eqid 2778	. . . . . . . . . . . . . 14 ⊢ 𝐼 = 𝐼
53	52	iftruei 4314	. . . . . . . . . . . . 13 ⊢ if(𝐼 = 𝐼, if(𝐽 = 𝐽, (1_r‘𝑅), (0_g‘𝑅)), (𝐼𝑀𝐽)) = if(𝐽 = 𝐽, (1_r‘𝑅), (0_g‘𝑅))
54		eqid 2778	. . . . . . . . . . . . . 14 ⊢ 𝐽 = 𝐽
55	54	iftruei 4314	. . . . . . . . . . . . 13 ⊢ if(𝐽 = 𝐽, (1_r‘𝑅), (0_g‘𝑅)) = (1_r‘𝑅)
56	53, 55	eqtri 2802	. . . . . . . . . . . 12 ⊢ if(𝐼 = 𝐼, if(𝐽 = 𝐽, (1_r‘𝑅), (0_g‘𝑅)), (𝐼𝑀𝐽)) = (1_r‘𝑅)
57	56	a1i 11	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) ∧ 𝑖 = 𝑁) ∧ 𝑗 = 𝑁) → if(𝐼 = 𝐼, if(𝐽 = 𝐽, (1_r‘𝑅), (0_g‘𝑅)), (𝐼𝑀𝐽)) = (1_r‘𝑅))
58	40, 51, 57	3eqtrrd 2819	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) ∧ 𝑖 = 𝑁) ∧ 𝑗 = 𝑁) → (1_r‘𝑅) = ((𝑃‘𝑖)(𝐼(((1...𝑁) minMatR1 𝑅)‘𝑀)𝐽)(𝑄‘𝑗)))
59		simplr 759	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) ∧ 𝑖 = 𝑁) ∧ ¬ 𝑗 = 𝑁) → 𝑖 = 𝑁)
60	59	fveq2d 6450	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) ∧ 𝑖 = 𝑁) ∧ ¬ 𝑗 = 𝑁) → (𝑃‘𝑖) = (𝑃‘𝑁))
61	31	ad2antrr 716	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) ∧ 𝑖 = 𝑁) ∧ ¬ 𝑗 = 𝑁) → (𝑃‘𝑁) = 𝐼)
62	60, 61	eqtrd 2814	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) ∧ 𝑖 = 𝑁) ∧ ¬ 𝑗 = 𝑁) → (𝑃‘𝑖) = 𝐼)
63	62	oveq1d 6937	. . . . . . . . . . 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 716	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) ∧ 𝑖 = 𝑁) ∧ ¬ 𝑗 = 𝑁) → 𝑀 ∈ 𝐵)
65	43	ad2antrr 716	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) ∧ 𝑖 = 𝑁) ∧ ¬ 𝑗 = 𝑁) → 𝐼 ∈ (1...𝑁))
66	45	ad2antrr 716	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) ∧ 𝑖 = 𝑁) ∧ ¬ 𝑗 = 𝑁) → 𝐽 ∈ (1...𝑁))
67	26	3ad2ant1 1124	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) → 𝑄 ∈ 𝐺)
68		simp3 1129	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) → 𝑗 ∈ (1...𝑁))
69		eqid 2778	. . . . . . . . . . . . . . 15 ⊢ (SymGrp‘(1...𝑁)) = (SymGrp‘(1...𝑁))
70	69, 13	symgfv 18190	. . . . . . . . . . . . . 14 ⊢ ((𝑄 ∈ 𝐺 ∧ 𝑗 ∈ (1...𝑁)) → (𝑄‘𝑗) ∈ (1...𝑁))
71	67, 68, 70	syl2anc 579	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) → (𝑄‘𝑗) ∈ (1...𝑁))
72	71	ad2antrr 716	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) ∧ 𝑖 = 𝑁) ∧ ¬ 𝑗 = 𝑁) → (𝑄‘𝑗) ∈ (1...𝑁))
73	4, 5, 47, 48, 49	minmar1eval 20860	. . . . . . . . . . . 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, (1_r‘𝑅), (0_g‘𝑅)), (𝐼𝑀(𝑄‘𝑗))))
74	64, 65, 66, 65, 72, 73	syl122anc 1447	. . . . . . . . . . 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, (1_r‘𝑅), (0_g‘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 4315	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) ∧ 𝑖 = 𝑁) ∧ ¬ 𝑗 = 𝑁) → if(𝐼 = 𝐼, if((𝑄‘𝑗) = 𝐽, (1_r‘𝑅), (0_g‘𝑅)), (𝐼𝑀(𝑄‘𝑗))) = if((𝑄‘𝑗) = 𝐽, (1_r‘𝑅), (0_g‘𝑅)))
77		simpr 479	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) ∧ 𝑖 = 𝑁) ∧ (𝑄‘𝑗) = 𝐽) → (𝑄‘𝑗) = 𝐽)
78	77	fveq2d 6450	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) ∧ 𝑖 = 𝑁) ∧ (𝑄‘𝑗) = 𝐽) → (^◡𝑄‘(𝑄‘𝑗)) = (^◡𝑄‘𝐽))
79	69, 13	symgbasf1o 18186	. . . . . . . . . . . . . . . . . . . 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 716	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) ∧ 𝑖 = 𝑁) ∧ (𝑄‘𝑗) = 𝐽) → 𝑄:(1...𝑁)–1-1-onto→(1...𝑁))
82	68	ad2antrr 716	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) ∧ 𝑖 = 𝑁) ∧ (𝑄‘𝑗) = 𝐽) → 𝑗 ∈ (1...𝑁))
83		f1ocnvfv1 6804	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑄:(1...𝑁)–1-1-onto→(1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) → (^◡𝑄‘(𝑄‘𝑗)) = 𝑗)
84	81, 82, 83	syl2anc 579	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) ∧ 𝑖 = 𝑁) ∧ (𝑄‘𝑗) = 𝐽) → (^◡𝑄‘(𝑄‘𝑗)) = 𝑗)
85	36	fveq2d 6450	. . . . . . . . . . . . . . . . . . . 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 12029	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ℕ = (ℤ_≥‘1)
89	87, 88	syl6eleq 2869	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → 𝑁 ∈ (ℤ_≥‘1))
90		eluzfz2 12666	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑁 ∈ (ℤ_≥‘1) → 𝑁 ∈ (1...𝑁))
91	89, 90	syl 17	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → 𝑁 ∈ (1...𝑁))
92		f1ocnvfv1 6804	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑄:(1...𝑁)–1-1-onto→(1...𝑁) ∧ 𝑁 ∈ (1...𝑁)) → (^◡𝑄‘(𝑄‘𝑁)) = 𝑁)
93	86, 91, 92	syl2anc 579	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → (^◡𝑄‘(𝑄‘𝑁)) = 𝑁)
94	85, 93	eqtr3d 2816	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → (^◡𝑄‘𝐽) = 𝑁)
95	94	3ad2ant1 1124	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) → (^◡𝑄‘𝐽) = 𝑁)
96	95	ad2antrr 716	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) ∧ 𝑖 = 𝑁) ∧ (𝑄‘𝑗) = 𝐽) → (^◡𝑄‘𝐽) = 𝑁)
97	78, 84, 96	3eqtr3d 2822	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) ∧ 𝑖 = 𝑁) ∧ (𝑄‘𝑗) = 𝐽) → 𝑗 = 𝑁)
98	97	ex 403	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) ∧ 𝑖 = 𝑁) → ((𝑄‘𝑗) = 𝐽 → 𝑗 = 𝑁))
99	98	con3d 150	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) ∧ 𝑖 = 𝑁) → (¬ 𝑗 = 𝑁 → ¬ (𝑄‘𝑗) = 𝐽))
100	99	imp 397	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) ∧ 𝑖 = 𝑁) ∧ ¬ 𝑗 = 𝑁) → ¬ (𝑄‘𝑗) = 𝐽)
101	100	iffalsed 4318	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) ∧ 𝑖 = 𝑁) ∧ ¬ 𝑗 = 𝑁) → if((𝑄‘𝑗) = 𝐽, (1_r‘𝑅), (0_g‘𝑅)) = (0_g‘𝑅))
102	76, 101	eqtrd 2814	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) ∧ 𝑖 = 𝑁) ∧ ¬ 𝑗 = 𝑁) → if(𝐼 = 𝐼, if((𝑄‘𝑗) = 𝐽, (1_r‘𝑅), (0_g‘𝑅)), (𝐼𝑀(𝑄‘𝑗))) = (0_g‘𝑅))
103	63, 74, 102	3eqtrrd 2819	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) ∧ 𝑖 = 𝑁) ∧ ¬ 𝑗 = 𝑁) → (0_g‘𝑅) = ((𝑃‘𝑖)(𝐼(((1...𝑁) minMatR1 𝑅)‘𝑀)𝐽)(𝑄‘𝑗)))
104	58, 103	ifeqda 4342	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) ∧ 𝑖 = 𝑁) → if(𝑗 = 𝑁, (1_r‘𝑅), (0_g‘𝑅)) = ((𝑃‘𝑖)(𝐼(((1...𝑁) minMatR1 𝑅)‘𝑀)𝐽)(𝑄‘𝑗)))
105		simp2 1128	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) → 𝑖 ∈ (1...𝑁))
106	105	adantr 474	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) ∧ ¬ 𝑖 = 𝑁) → 𝑖 ∈ (1...𝑁))
107	68	adantr 474	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) ∧ ¬ 𝑖 = 𝑁) → 𝑗 ∈ (1...𝑁))
108		ovexd 6956	. . . . . . . . . 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 6935	. . . . . . . . . . . . . 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	mpt2eq3ia 6997	. . . . . . . . . . . 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 2802	. . . . . . . . . . 11 ⊢ 𝑊 = (𝑖 ∈ (1...𝑁), 𝑗 ∈ (1...𝑁) ↦ ((𝑃‘𝑖)(𝐼(((1...𝑁) minMatR1 𝑅)‘𝑀)𝐽)(𝑄‘𝑗)))
113	112	ovmpt4g 7060	. . . . . . . . . 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 1439	. . . . . . . . 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 4342	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) → if(𝑖 = 𝑁, if(𝑗 = 𝑁, (1_r‘𝑅), (0_g‘𝑅)), (𝑖𝑊𝑗)) = ((𝑃‘𝑖)(𝐼(((1...𝑁) minMatR1 𝑅)‘𝑀)𝐽)(𝑄‘𝑗)))
116	115	mpt2eq3dva 6996	. . . . . . 7 ⊢ (𝜑 → (𝑖 ∈ (1...𝑁), 𝑗 ∈ (1...𝑁) ↦ if(𝑖 = 𝑁, if(𝑗 = 𝑁, (1_r‘𝑅), (0_g‘𝑅)), (𝑖𝑊𝑗))) = (𝑖 ∈ (1...𝑁), 𝑗 ∈ (1...𝑁) ↦ ((𝑃‘𝑖)(𝐼(((1...𝑁) minMatR1 𝑅)‘𝑀)𝐽)(𝑄‘𝑗))))
117		eqid 2778	. . . . . . . . . 10 ⊢ (Base‘𝑅) = (Base‘𝑅)
118	25	3ad2ant1 1124	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) → 𝑃 ∈ 𝐺)
119	69, 13	symgfv 18190	. . . . . . . . . . . 12 ⊢ ((𝑃 ∈ 𝐺 ∧ 𝑖 ∈ (1...𝑁)) → (𝑃‘𝑖) ∈ (1...𝑁))
120	118, 105, 119	syl2anc 579	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) → (𝑃‘𝑖) ∈ (1...𝑁))
121	24	3ad2ant1 1124	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) → 𝑈 ∈ 𝐵)
122	4, 117, 5, 120, 71, 121	matecld 20636	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) → ((𝑃‘𝑖)𝑈(𝑄‘𝑗)) ∈ (Base‘𝑅))
123	4, 117, 5, 19, 18, 122	matbas2d 20633	. . . . . . . . 9 ⊢ (𝜑 → (𝑖 ∈ (1...𝑁), 𝑗 ∈ (1...𝑁) ↦ ((𝑃‘𝑖)𝑈(𝑄‘𝑗))) ∈ 𝐵)
124	17, 123	syl5eqel 2863	. . . . . . . 8 ⊢ (𝜑 → 𝑊 ∈ 𝐵)
125	117, 48	ringidcl 18955	. . . . . . . . 9 ⊢ (𝑅 ∈ Ring → (1_r‘𝑅) ∈ (Base‘𝑅))
126	21, 125	syl 17	. . . . . . . 8 ⊢ (𝜑 → (1_r‘𝑅) ∈ (Base‘𝑅))
127		eqid 2778	. . . . . . . . 9 ⊢ ((1...𝑁) matRRep 𝑅) = ((1...𝑁) matRRep 𝑅)
128	4, 5, 127, 49	marrepval 20773	. . . . . . . 8 ⊢ (((𝑊 ∈ 𝐵 ∧ (1_r‘𝑅) ∈ (Base‘𝑅)) ∧ (𝑁 ∈ (1...𝑁) ∧ 𝑁 ∈ (1...𝑁))) → (𝑁(𝑊((1...𝑁) matRRep 𝑅)(1_r‘𝑅))𝑁) = (𝑖 ∈ (1...𝑁), 𝑗 ∈ (1...𝑁) ↦ if(𝑖 = 𝑁, if(𝑗 = 𝑁, (1_r‘𝑅), (0_g‘𝑅)), (𝑖𝑊𝑗))))
129	124, 126, 91, 91, 128	syl22anc 829	. . . . . . 7 ⊢ (𝜑 → (𝑁(𝑊((1...𝑁) matRRep 𝑅)(1_r‘𝑅))𝑁) = (𝑖 ∈ (1...𝑁), 𝑗 ∈ (1...𝑁) ↦ if(𝑖 = 𝑁, if(𝑗 = 𝑁, (1_r‘𝑅), (0_g‘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 2824	. . . . . 6 ⊢ (𝜑 → (𝑁(𝑊((1...𝑁) matRRep 𝑅)(1_r‘𝑅))𝑁) = 𝑊)
132	131	fveq2d 6450	. . . . 5 ⊢ (𝜑 → (𝐷‘(𝑁(𝑊((1...𝑁) matRRep 𝑅)(1_r‘𝑅))𝑁)) = (𝐷‘𝑊))
133		eqid 2778	. . . . . . . . . . . 12 ⊢ ((1...𝑁) subMat 𝑅) = ((1...𝑁) subMat 𝑅)
134	4, 133, 5	submaval 20792	. . . . . . . . . . 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 1439	. . . . . . . . . 10 ⊢ (𝜑 → (𝑁(((1...𝑁) subMat 𝑅)‘𝑊)𝑁) = (𝑖 ∈ ((1...𝑁) ∖ {𝑁}), 𝑗 ∈ ((1...𝑁) ∖ {𝑁}) ↦ (𝑖𝑊𝑗)))
136		fzdif2 30115	. . . . . . . . . . . 12 ⊢ (𝑁 ∈ (ℤ_≥‘1) → ((1...𝑁) ∖ {𝑁}) = (1...(𝑁 − 1)))
137	89, 136	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → ((1...𝑁) ∖ {𝑁}) = (1...(𝑁 − 1)))
138		mpt2eq12 6992	. . . . . . . . . . 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 579	. . . . . . . . . 10 ⊢ (𝜑 → (𝑖 ∈ ((1...𝑁) ∖ {𝑁}), 𝑗 ∈ ((1...𝑁) ∖ {𝑁}) ↦ (𝑖𝑊𝑗)) = (𝑖 ∈ (1...(𝑁 − 1)), 𝑗 ∈ (1...(𝑁 − 1)) ↦ (𝑖𝑊𝑗)))
140	135, 139	eqtrd 2814	. . . . . . . . 9 ⊢ (𝜑 → (𝑁(((1...𝑁) subMat 𝑅)‘𝑊)𝑁) = (𝑖 ∈ (1...(𝑁 − 1)), 𝑗 ∈ (1...(𝑁 − 1)) ↦ (𝑖𝑊𝑗)))
141		difssd 3961	. . . . . . . . . . 11 ⊢ (𝜑 → ((1...𝑁) ∖ {𝑁}) ⊆ (1...𝑁))
142	137, 141	eqsstr3d 3859	. . . . . . . . . 10 ⊢ (𝜑 → (1...(𝑁 − 1)) ⊆ (1...𝑁))
143	4, 5	submabas 20789	. . . . . . . . . 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 579	. . . . . . . . 9 ⊢ (𝜑 → (𝑖 ∈ (1...(𝑁 − 1)), 𝑗 ∈ (1...(𝑁 − 1)) ↦ (𝑖𝑊𝑗)) ∈ (Base‘((1...(𝑁 − 1)) Mat 𝑅)))
145	140, 144	eqeltrd 2859	. . . . . . . 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 2778	. . . . . . . . 9 ⊢ ((1...(𝑁 − 1)) Mat 𝑅) = ((1...(𝑁 − 1)) Mat 𝑅)
148		eqid 2778	. . . . . . . . 9 ⊢ (Base‘((1...(𝑁 − 1)) Mat 𝑅)) = (Base‘((1...(𝑁 − 1)) Mat 𝑅))
149	146, 147, 148, 117	mdetcl 20807	. . . . . . . 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 579	. . . . . . 7 ⊢ (𝜑 → (𝐸‘(𝑁(((1...𝑁) subMat 𝑅)‘𝑊)𝑁)) ∈ (Base‘𝑅))
151	117, 16, 48	ringlidm 18958	. . . . . . 7 ⊢ ((𝑅 ∈ Ring ∧ (𝐸‘(𝑁(((1...𝑁) subMat 𝑅)‘𝑊)𝑁)) ∈ (Base‘𝑅)) → ((1_r‘𝑅) · (𝐸‘(𝑁(((1...𝑁) subMat 𝑅)‘𝑊)𝑁))) = (𝐸‘(𝑁(((1...𝑁) subMat 𝑅)‘𝑊)𝑁)))
152	21, 150, 151	syl2anc 579	. . . . . 6 ⊢ (𝜑 → ((1_r‘𝑅) · (𝐸‘(𝑁(((1...𝑁) subMat 𝑅)‘𝑊)𝑁))) = (𝐸‘(𝑁(((1...𝑁) subMat 𝑅)‘𝑊)𝑁)))
153	4	fveq2i 6449	. . . . . . . . . . 11 ⊢ (Base‘𝐴) = (Base‘((1...𝑁) Mat 𝑅))
154	5, 153	eqtri 2802	. . . . . . . . . 10 ⊢ 𝐵 = (Base‘((1...𝑁) Mat 𝑅))
155	124, 154	syl6eleq 2869	. . . . . . . . 9 ⊢ (𝜑 → 𝑊 ∈ (Base‘((1...𝑁) Mat 𝑅)))
156		smadiadetr 20887	. . . . . . . . 9 ⊢ (((𝑅 ∈ CRing ∧ 𝑊 ∈ (Base‘((1...𝑁) Mat 𝑅))) ∧ (𝑁 ∈ (1...𝑁) ∧ (1_r‘𝑅) ∈ (Base‘𝑅))) → (((1...𝑁) maDet 𝑅)‘(𝑁(𝑊((1...𝑁) matRRep 𝑅)(1_r‘𝑅))𝑁)) = ((1_r‘𝑅)(._r‘𝑅)((((1...𝑁) ∖ {𝑁}) maDet 𝑅)‘(𝑁(((1...𝑁) subMat 𝑅)‘𝑊)𝑁))))
157	18, 155, 91, 126, 156	syl22anc 829	. . . . . . . 8 ⊢ (𝜑 → (((1...𝑁) maDet 𝑅)‘(𝑁(𝑊((1...𝑁) matRRep 𝑅)(1_r‘𝑅))𝑁)) = ((1_r‘𝑅)(._r‘𝑅)((((1...𝑁) ∖ {𝑁}) maDet 𝑅)‘(𝑁(((1...𝑁) subMat 𝑅)‘𝑊)𝑁))))
158	6	fveq1i 6447	. . . . . . . . 9 ⊢ (𝐷‘(𝑁(𝑊((1...𝑁) matRRep 𝑅)(1_r‘𝑅))𝑁)) = (((1...𝑁) maDet 𝑅)‘(𝑁(𝑊((1...𝑁) matRRep 𝑅)(1_r‘𝑅))𝑁))
159	16	oveqi 6935	. . . . . . . . 9 ⊢ ((1_r‘𝑅) · ((((1...𝑁) ∖ {𝑁}) maDet 𝑅)‘(𝑁(((1...𝑁) subMat 𝑅)‘𝑊)𝑁))) = ((1_r‘𝑅)(._r‘𝑅)((((1...𝑁) ∖ {𝑁}) maDet 𝑅)‘(𝑁(((1...𝑁) subMat 𝑅)‘𝑊)𝑁)))
160	158, 159	eqeq12i 2792	. . . . . . . 8 ⊢ ((𝐷‘(𝑁(𝑊((1...𝑁) matRRep 𝑅)(1_r‘𝑅))𝑁)) = ((1_r‘𝑅) · ((((1...𝑁) ∖ {𝑁}) maDet 𝑅)‘(𝑁(((1...𝑁) subMat 𝑅)‘𝑊)𝑁))) ↔ (((1...𝑁) maDet 𝑅)‘(𝑁(𝑊((1...𝑁) matRRep 𝑅)(1_r‘𝑅))𝑁)) = ((1_r‘𝑅)(._r‘𝑅)((((1...𝑁) ∖ {𝑁}) maDet 𝑅)‘(𝑁(((1...𝑁) subMat 𝑅)‘𝑊)𝑁))))
161	157, 160	sylibr 226	. . . . . . 7 ⊢ (𝜑 → (𝐷‘(𝑁(𝑊((1...𝑁) matRRep 𝑅)(1_r‘𝑅))𝑁)) = ((1_r‘𝑅) · ((((1...𝑁) ∖ {𝑁}) maDet 𝑅)‘(𝑁(((1...𝑁) subMat 𝑅)‘𝑊)𝑁))))
162	137	oveq1d 6937	. . . . . . . . . 10 ⊢ (𝜑 → (((1...𝑁) ∖ {𝑁}) maDet 𝑅) = ((1...(𝑁 − 1)) maDet 𝑅))
163	162, 146	syl6eqr 2832	. . . . . . . . 9 ⊢ (𝜑 → (((1...𝑁) ∖ {𝑁}) maDet 𝑅) = 𝐸)
164	163	fveq1d 6448	. . . . . . . 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 6938	. . . . . . 7 ⊢ (𝜑 → ((1_r‘𝑅) · ((((1...𝑁) ∖ {𝑁}) maDet 𝑅)‘(𝑁(((1...𝑁) subMat 𝑅)‘𝑊)𝑁))) = ((1_r‘𝑅) · (𝐸‘(𝑁(((1...𝑁) subMat 𝑅)‘𝑊)𝑁))))
166	161, 165	eqtrd 2814	. . . . . 6 ⊢ (𝜑 → (𝐷‘(𝑁(𝑊((1...𝑁) matRRep 𝑅)(1_r‘𝑅))𝑁)) = ((1_r‘𝑅) · (𝐸‘(𝑁(((1...𝑁) subMat 𝑅)‘𝑊)𝑁))))
167	4, 5	submat1n 30469	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ ∧ 𝑊 ∈ 𝐵) → (𝑁(subMat1‘𝑊)𝑁) = (𝑁(((1...𝑁) subMat 𝑅)‘𝑊)𝑁))
168	87, 124, 167	syl2anc 579	. . . . . . 7 ⊢ (𝜑 → (𝑁(subMat1‘𝑊)𝑁) = (𝑁(((1...𝑁) subMat 𝑅)‘𝑊)𝑁))
169	168	fveq2d 6450	. . . . . 6 ⊢ (𝜑 → (𝐸‘(𝑁(subMat1‘𝑊)𝑁)) = (𝐸‘(𝑁(((1...𝑁) subMat 𝑅)‘𝑊)𝑁)))
170	152, 166, 169	3eqtr4d 2824	. . . . 5 ⊢ (𝜑 → (𝐷‘(𝑁(𝑊((1...𝑁) matRRep 𝑅)(1_r‘𝑅))𝑁)) = (𝐸‘(𝑁(subMat1‘𝑊)𝑁)))
171	132, 170	eqtr3d 2816	. . . 4 ⊢ (𝜑 → (𝐷‘𝑊) = (𝐸‘(𝑁(subMat1‘𝑊)𝑁)))
172	4, 5, 87, 3, 2, 21, 1, 10	submatminr1 30474	. . . . . 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 2814	. . . . 5 ⊢ (𝜑 → (𝐼(subMat1‘𝑀)𝐽) = (𝑁(subMat1‘𝑊)𝑁))
175	174	fveq2d 6450	. . . 4 ⊢ (𝜑 → (𝐸‘(𝐼(subMat1‘𝑀)𝐽)) = (𝐸‘(𝑁(subMat1‘𝑊)𝑁)))
176	171, 175	eqtr4d 2817	. . 3 ⊢ (𝜑 → (𝐷‘𝑊) = (𝐸‘(𝐼(subMat1‘𝑀)𝐽)))
177	176	oveq2d 6938	. 2 ⊢ (𝜑 → ((𝑍‘((𝑆‘𝑃) · (𝑆‘𝑄))) · (𝐷‘𝑊)) = ((𝑍‘((𝑆‘𝑃) · (𝑆‘𝑄))) · (𝐸‘(𝐼(subMat1‘𝑀)𝐽))))
178	12, 27, 177	3eqtrd 2818	1 ⊢ (𝜑 → (𝐽(𝐾‘𝑀)𝐼) = ((𝑍‘((𝑆‘𝑃) · (𝑆‘𝑄))) · (𝐸‘(𝐼(subMat1‘𝑀)𝐽))))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 386 ∧ w3a 1071 = wceq 1601 ∈ wcel 2107 Vcvv 3398 ∖ cdif 3789 ⊆ wss 3792 ifcif 4307 {csn 4398 ^◡ccnv 5354 –1-1-onto→wf1o 6134 ‘cfv 6135 (class class class)co 6922 ↦ cmpt2 6924 1c1 10273 · cmul 10277 − cmin 10606 ℕcn 11374 ℤ_≥cuz 11992 ...cfz 12643 Basecbs 16255 ._rcmulr 16339 0_gc0g 16486 SymGrpcsymg 18180 pmSgncpsgn 18292 1_rcur 18888 Ringcrg 18934 CRingccrg 18935 ℤRHomczrh 20244 Mat cmat 20617 matRRep cmarrep 20767 subMat csubma 20787 maDet cmdat 20795 maAdju cmadu 20843 minMatR1 cminmar1 20844 subMat1csmat 30457

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-8 2109 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-rep 5006 ax-sep 5017 ax-nul 5025 ax-pow 5077 ax-pr 5138 ax-un 7226 ax-inf2 8835 ax-cnex 10328 ax-resscn 10329 ax-1cn 10330 ax-icn 10331 ax-addcl 10332 ax-addrcl 10333 ax-mulcl 10334 ax-mulrcl 10335 ax-mulcom 10336 ax-addass 10337 ax-mulass 10338 ax-distr 10339 ax-i2m1 10340 ax-1ne0 10341 ax-1rid 10342 ax-rnegex 10343 ax-rrecex 10344 ax-cnre 10345 ax-pre-lttri 10346 ax-pre-lttrn 10347 ax-pre-ltadd 10348 ax-pre-mulgt0 10349 ax-addf 10351 ax-mulf 10352

This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3or 1072 df-3an 1073 df-xor 1583 df-tru 1605 df-fal 1615 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ne 2970 df-nel 3076 df-ral 3095 df-rex 3096 df-reu 3097 df-rmo 3098 df-rab 3099 df-v 3400 df-sbc 3653 df-csb 3752 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-pss 3808 df-nul 4142 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-tp 4403 df-op 4405 df-ot 4407 df-uni 4672 df-int 4711 df-iun 4755 df-iin 4756 df-br 4887 df-opab 4949 df-mpt 4966 df-tr 4988 df-id 5261 df-eprel 5266 df-po 5274 df-so 5275 df-fr 5314 df-se 5315 df-we 5316 df-xp 5361 df-rel 5362 df-cnv 5363 df-co 5364 df-dm 5365 df-rn 5366 df-res 5367 df-ima 5368 df-pred 5933 df-ord 5979 df-on 5980 df-lim 5981 df-suc 5982 df-iota 6099 df-fun 6137 df-fn 6138 df-f 6139 df-f1 6140 df-fo 6141 df-f1o 6142 df-fv 6143 df-isom 6144 df-riota 6883 df-ov 6925 df-oprab 6926 df-mpt2 6927 df-of 7174 df-om 7344 df-1st 7445 df-2nd 7446 df-supp 7577 df-tpos 7634 df-wrecs 7689 df-recs 7751 df-rdg 7789 df-1o 7843 df-2o 7844 df-oadd 7847 df-er 8026 df-map 8142 df-pm 8143 df-ixp 8195 df-en 8242 df-dom 8243 df-sdom 8244 df-fin 8245 df-fsupp 8564 df-sup 8636 df-oi 8704 df-card 9098 df-pnf 10413 df-mnf 10414 df-xr 10415 df-ltxr 10416 df-le 10417 df-sub 10608 df-neg 10609 df-div 11033 df-nn 11375 df-2 11438 df-3 11439 df-4 11440 df-5 11441 df-6 11442 df-7 11443 df-8 11444 df-9 11445 df-n0 11643 df-xnn0 11715 df-z 11729 df-dec 11846 df-uz 11993 df-rp 12138 df-fz 12644 df-fzo 12785 df-seq 13120 df-exp 13179 df-hash 13436 df-word 13600 df-lsw 13653 df-concat 13661 df-s1 13686 df-substr 13731 df-pfx 13780 df-splice 13887 df-reverse 13905 df-s2 13999 df-struct 16257 df-ndx 16258 df-slot 16259 df-base 16261 df-sets 16262 df-ress 16263 df-plusg 16351 df-mulr 16352 df-starv 16353 df-sca 16354 df-vsca 16355 df-ip 16356 df-tset 16357 df-ple 16358 df-ds 16360 df-unif 16361 df-hom 16362 df-cco 16363 df-0g 16488 df-gsum 16489 df-prds 16494 df-pws 16496 df-mre 16632 df-mrc 16633 df-acs 16635 df-mgm 17628 df-sgrp 17670 df-mnd 17681 df-mhm 17721 df-submnd 17722 df-grp 17812 df-minusg 17813 df-mulg 17928 df-subg 17975 df-ghm 18042 df-gim 18085 df-cntz 18133 df-oppg 18159 df-symg 18181 df-pmtr 18245 df-psgn 18294 df-cmn 18581 df-abl 18582 df-mgp 18877 df-ur 18889 df-ring 18936 df-cring 18937 df-oppr 19010 df-dvdsr 19028 df-unit 19029 df-invr 19059 df-dvr 19070 df-rnghom 19104 df-drng 19141 df-subrg 19170 df-sra 19569 df-rgmod 19570 df-cnfld 20143 df-zring 20215 df-zrh 20248 df-dsmm 20475 df-frlm 20490 df-mat 20618 df-marrep 20769 df-subma 20788 df-mdet 20796 df-madu 20845 df-minmar1 20846 df-smat 30458

This theorem is referenced by: madjusmdetlem4 30494

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