Theorem madjusmdetlem1 34134

Description: Lemma for madjusmdet 34138. (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 22770	. . . 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 1394	. . 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 6874	. . 3 ⊢ (𝐷‘𝑈) = (𝐷‘(𝐼(((1...𝑁) minMatR1 𝑅)‘𝑀)𝐽))
12	9, 11	eqtr4di 2818	. 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 14000	. . 3 ⊢ (𝜑 → (1...𝑁) ∈ Fin)
20		crngring 20318	. . . . . 6 ⊢ (𝑅 ∈ CRing → 𝑅 ∈ Ring)
21	18, 20	syl 18	. . . . 5 ⊢ (𝜑 → 𝑅 ∈ Ring)
22	4, 5	minmar1cl 22769	. . . . 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 851	. . . 4 ⊢ (𝜑 → (𝐼(((1...𝑁) minMatR1 𝑅)‘𝑀)𝐽) ∈ 𝐵)
24	10, 23	eqeltrid 2869	. . 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 34132	. 2 ⊢ (𝜑 → (𝐷‘𝑈) = ((𝑍‘((𝑆‘𝑃) · (𝑆‘𝑄))) · (𝐷‘𝑊)))
28		simplr 780	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) ∧ 𝑖 = 𝑁) ∧ 𝑗 = 𝑁) → 𝑖 = 𝑁)
29	28	fveq2d 6875	. . . . . . . . . . . . 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 1149	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) → (𝑃‘𝑁) = 𝐼)
32	31	ad2antrr 738	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) ∧ 𝑖 = 𝑁) ∧ 𝑗 = 𝑁) → (𝑃‘𝑁) = 𝐼)
33	29, 32	eqtrd 2800	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) ∧ 𝑖 = 𝑁) ∧ 𝑗 = 𝑁) → (𝑃‘𝑖) = 𝐼)
34		simpr 489	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) ∧ 𝑖 = 𝑁) ∧ 𝑗 = 𝑁) → 𝑗 = 𝑁)
35	34	fveq2d 6875	. . . . . . . . . . . . 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 1149	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) → (𝑄‘𝑁) = 𝐽)
38	37	ad2antrr 738	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) ∧ 𝑖 = 𝑁) ∧ 𝑗 = 𝑁) → (𝑄‘𝑁) = 𝐽)
39	35, 38	eqtrd 2800	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) ∧ 𝑖 = 𝑁) ∧ 𝑗 = 𝑁) → (𝑄‘𝑗) = 𝐽)
40	33, 39	oveq12d 7418	. . . . . . . . . . 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 1149	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) → 𝑀 ∈ 𝐵)
42	41	ad2antrr 738	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) ∧ 𝑖 = 𝑁) ∧ 𝑗 = 𝑁) → 𝑀 ∈ 𝐵)
43	3	3ad2ant1 1149	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) → 𝐼 ∈ (1...𝑁))
44	43	ad2antrr 738	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) ∧ 𝑖 = 𝑁) ∧ 𝑗 = 𝑁) → 𝐼 ∈ (1...𝑁))
45	2	3ad2ant1 1149	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) → 𝐽 ∈ (1...𝑁))
46	45	ad2antrr 738	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) ∧ 𝑖 = 𝑁) ∧ 𝑗 = 𝑁) → 𝐽 ∈ (1...𝑁))
47		eqid 2765	. . . . . . . . . . . . 13 ⊢ ((1...𝑁) minMatR1 𝑅) = ((1...𝑁) minMatR1 𝑅)
48		eqid 2765	. . . . . . . . . . . . 13 ⊢ (1r‘𝑅) = (1r‘𝑅)
49		eqid 2765	. . . . . . . . . . . . 13 ⊢ (0g‘𝑅) = (0g‘𝑅)
50	4, 5, 47, 48, 49	minmar1eval 22767	. . . . . . . . . . . 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 1402	. . . . . . . . . . 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 2765	. . . . . . . . . . . . . 14 ⊢ 𝐼 = 𝐼
53	52	iftruei 4490	. . . . . . . . . . . . 13 ⊢ if(𝐼 = 𝐼, if(𝐽 = 𝐽, (1r‘𝑅), (0g‘𝑅)), (𝐼𝑀𝐽)) = if(𝐽 = 𝐽, (1r‘𝑅), (0g‘𝑅))
54		eqid 2765	. . . . . . . . . . . . . 14 ⊢ 𝐽 = 𝐽
55	54	iftruei 4490	. . . . . . . . . . . . 13 ⊢ if(𝐽 = 𝐽, (1r‘𝑅), (0g‘𝑅)) = (1r‘𝑅)
56	53, 55	eqtri 2788	. . . . . . . . . . . 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 2805	. . . . . . . . . 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 780	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) ∧ 𝑖 = 𝑁) ∧ ¬ 𝑗 = 𝑁) → 𝑖 = 𝑁)
60	59	fveq2d 6875	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) ∧ 𝑖 = 𝑁) ∧ ¬ 𝑗 = 𝑁) → (𝑃‘𝑖) = (𝑃‘𝑁))
61	31	ad2antrr 738	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) ∧ 𝑖 = 𝑁) ∧ ¬ 𝑗 = 𝑁) → (𝑃‘𝑁) = 𝐼)
62	60, 61	eqtrd 2800	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) ∧ 𝑖 = 𝑁) ∧ ¬ 𝑗 = 𝑁) → (𝑃‘𝑖) = 𝐼)
63	62	oveq1d 7415	. . . . . . . . . . 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 738	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) ∧ 𝑖 = 𝑁) ∧ ¬ 𝑗 = 𝑁) → 𝑀 ∈ 𝐵)
65	43	ad2antrr 738	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) ∧ 𝑖 = 𝑁) ∧ ¬ 𝑗 = 𝑁) → 𝐼 ∈ (1...𝑁))
66	45	ad2antrr 738	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) ∧ 𝑖 = 𝑁) ∧ ¬ 𝑗 = 𝑁) → 𝐽 ∈ (1...𝑁))
67	26	3ad2ant1 1149	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) → 𝑄 ∈ 𝐺)
68		simp3 1154	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) → 𝑗 ∈ (1...𝑁))
69		eqid 2765	. . . . . . . . . . . . . . 15 ⊢ (SymGrp‘(1...𝑁)) = (SymGrp‘(1...𝑁))
70	69, 13	symgfv 19441	. . . . . . . . . . . . . 14 ⊢ ((𝑄 ∈ 𝐺 ∧ 𝑗 ∈ (1...𝑁)) → (𝑄‘𝑗) ∈ (1...𝑁))
71	67, 68, 70	syl2anc 595	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) → (𝑄‘𝑗) ∈ (1...𝑁))
72	71	ad2antrr 738	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) ∧ 𝑖 = 𝑁) ∧ ¬ 𝑗 = 𝑁) → (𝑄‘𝑗) ∈ (1...𝑁))
73	4, 5, 47, 48, 49	minmar1eval 22767	. . . . . . . . . . . 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 1402	. . . . . . . . . . 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 4491	. . . . . . . . . . . 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 489	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) ∧ 𝑖 = 𝑁) ∧ (𝑄‘𝑗) = 𝐽) → (𝑄‘𝑗) = 𝐽)
78	77	fveq2d 6875	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) ∧ 𝑖 = 𝑁) ∧ (𝑄‘𝑗) = 𝐽) → (◡𝑄‘(𝑄‘𝑗)) = (◡𝑄‘𝐽))
79	69, 13	symgbasf1o 19436	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑄 ∈ 𝐺 → 𝑄:(1...𝑁)–1-1-onto→(1...𝑁))
80	67, 79	syl 18	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) → 𝑄:(1...𝑁)–1-1-onto→(1...𝑁))
81	80	ad2antrr 738	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) ∧ 𝑖 = 𝑁) ∧ (𝑄‘𝑗) = 𝐽) → 𝑄:(1...𝑁)–1-1-onto→(1...𝑁))
82	68	ad2antrr 738	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) ∧ 𝑖 = 𝑁) ∧ (𝑄‘𝑗) = 𝐽) → 𝑗 ∈ (1...𝑁))
83		f1ocnvfv1 7264	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑄:(1...𝑁)–1-1-onto→(1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) → (◡𝑄‘(𝑄‘𝑗)) = 𝑗)
84	81, 82, 83	syl2anc 595	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) ∧ 𝑖 = 𝑁) ∧ (𝑄‘𝑗) = 𝐽) → (◡𝑄‘(𝑄‘𝑗)) = 𝑗)
85	36	fveq2d 6875	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → (◡𝑄‘(𝑄‘𝑁)) = (◡𝑄‘𝐽))
86	26, 79	syl 18	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → 𝑄:(1...𝑁)–1-1-onto→(1...𝑁))
87		madjusmdet.n	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝜑 → 𝑁 ∈ ℕ)
88		nnuz 12892	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ℕ = (ℤ≥‘1)
89	87, 88	eleqtrdi 2875	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘1))
90		eluzfz2 13551	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑁 ∈ (ℤ≥‘1) → 𝑁 ∈ (1...𝑁))
91	89, 90	syl 18	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → 𝑁 ∈ (1...𝑁))
92		f1ocnvfv1 7264	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑄:(1...𝑁)–1-1-onto→(1...𝑁) ∧ 𝑁 ∈ (1...𝑁)) → (◡𝑄‘(𝑄‘𝑁)) = 𝑁)
93	86, 91, 92	syl2anc 595	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → (◡𝑄‘(𝑄‘𝑁)) = 𝑁)
94	85, 93	eqtr3d 2802	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → (◡𝑄‘𝐽) = 𝑁)
95	94	3ad2ant1 1149	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) → (◡𝑄‘𝐽) = 𝑁)
96	95	ad2antrr 738	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) ∧ 𝑖 = 𝑁) ∧ (𝑄‘𝑗) = 𝐽) → (◡𝑄‘𝐽) = 𝑁)
97	78, 84, 96	3eqtr3d 2808	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) ∧ 𝑖 = 𝑁) ∧ (𝑄‘𝑗) = 𝐽) → 𝑗 = 𝑁)
98	97	ex 417	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) ∧ 𝑖 = 𝑁) → ((𝑄‘𝑗) = 𝐽 → 𝑗 = 𝑁))
99	98	con3d 153	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) ∧ 𝑖 = 𝑁) → (¬ 𝑗 = 𝑁 → ¬ (𝑄‘𝑗) = 𝐽))
100	99	imp 411	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) ∧ 𝑖 = 𝑁) ∧ ¬ 𝑗 = 𝑁) → ¬ (𝑄‘𝑗) = 𝐽)
101	100	iffalsed 4494	. . . . . . . . . . . 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 2800	. . . . . . . . . . 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 2805	. . . . . . . . . 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 4520	. . . . . . . . 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 1153	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) → 𝑖 ∈ (1...𝑁))
106	105	adantr 485	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) ∧ ¬ 𝑖 = 𝑁) → 𝑖 ∈ (1...𝑁))
107	68	adantr 485	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) ∧ ¬ 𝑖 = 𝑁) → 𝑗 ∈ (1...𝑁))
108		ovexd 7435	. . . . . . . . . 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 7413	. . . . . . . . . . . . . 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 7478	. . . . . . . . . . . 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 2788	. . . . . . . . . . 11 ⊢ 𝑊 = (𝑖 ∈ (1...𝑁), 𝑗 ∈ (1...𝑁) ↦ ((𝑃‘𝑖)(𝐼(((1...𝑁) minMatR1 𝑅)‘𝑀)𝐽)(𝑄‘𝑗)))
113	112	ovmpt4g 7547	. . . . . . . . . 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 1394	. . . . . . . . 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 4520	. . . . . . . 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 7477	. . . . . . 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 2765	. . . . . . . . . 10 ⊢ (Base‘𝑅) = (Base‘𝑅)
118	25	3ad2ant1 1149	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) → 𝑃 ∈ 𝐺)
119	69, 13	symgfv 19441	. . . . . . . . . . . 12 ⊢ ((𝑃 ∈ 𝐺 ∧ 𝑖 ∈ (1...𝑁)) → (𝑃‘𝑖) ∈ (1...𝑁))
120	118, 105, 119	syl2anc 595	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) → (𝑃‘𝑖) ∈ (1...𝑁))
121	24	3ad2ant1 1149	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) → 𝑈 ∈ 𝐵)
122	4, 117, 5, 120, 71, 121	matecld 22544	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) → ((𝑃‘𝑖)𝑈(𝑄‘𝑗)) ∈ (Base‘𝑅))
123	4, 117, 5, 19, 18, 122	matbas2d 22541	. . . . . . . . 9 ⊢ (𝜑 → (𝑖 ∈ (1...𝑁), 𝑗 ∈ (1...𝑁) ↦ ((𝑃‘𝑖)𝑈(𝑄‘𝑗))) ∈ 𝐵)
124	17, 123	eqeltrid 2869	. . . . . . . 8 ⊢ (𝜑 → 𝑊 ∈ 𝐵)
125	117, 48	ringidcl 20339	. . . . . . . . 9 ⊢ (𝑅 ∈ Ring → (1r‘𝑅) ∈ (Base‘𝑅))
126	21, 125	syl 18	. . . . . . . 8 ⊢ (𝜑 → (1r‘𝑅) ∈ (Base‘𝑅))
127		eqid 2765	. . . . . . . . 9 ⊢ ((1...𝑁) matRRep 𝑅) = ((1...𝑁) matRRep 𝑅)
128	4, 5, 127, 49	marrepval 22680	. . . . . . . 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 851	. . . . . . 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 2810	. . . . . 6 ⊢ (𝜑 → (𝑁(𝑊((1...𝑁) matRRep 𝑅)(1r‘𝑅))𝑁) = 𝑊)
132	131	fveq2d 6875	. . . . 5 ⊢ (𝜑 → (𝐷‘(𝑁(𝑊((1...𝑁) matRRep 𝑅)(1r‘𝑅))𝑁)) = (𝐷‘𝑊))
133		eqid 2765	. . . . . . . . . . . 12 ⊢ ((1...𝑁) subMat 𝑅) = ((1...𝑁) subMat 𝑅)
134	4, 133, 5	submaval 22699	. . . . . . . . . . 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 1394	. . . . . . . . . 10 ⊢ (𝜑 → (𝑁(((1...𝑁) subMat 𝑅)‘𝑊)𝑁) = (𝑖 ∈ ((1...𝑁) ∖ {𝑁}), 𝑗 ∈ ((1...𝑁) ∖ {𝑁}) ↦ (𝑖𝑊𝑗)))
136		fzdif2 33047	. . . . . . . . . . . 12 ⊢ (𝑁 ∈ (ℤ≥‘1) → ((1...𝑁) ∖ {𝑁}) = (1...(𝑁 − 1)))
137	89, 136	syl 18	. . . . . . . . . . 11 ⊢ (𝜑 → ((1...𝑁) ∖ {𝑁}) = (1...(𝑁 − 1)))
138		mpoeq12 7473	. . . . . . . . . . 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 595	. . . . . . . . . 10 ⊢ (𝜑 → (𝑖 ∈ ((1...𝑁) ∖ {𝑁}), 𝑗 ∈ ((1...𝑁) ∖ {𝑁}) ↦ (𝑖𝑊𝑗)) = (𝑖 ∈ (1...(𝑁 − 1)), 𝑗 ∈ (1...(𝑁 − 1)) ↦ (𝑖𝑊𝑗)))
140	135, 139	eqtrd 2800	. . . . . . . . 9 ⊢ (𝜑 → (𝑁(((1...𝑁) subMat 𝑅)‘𝑊)𝑁) = (𝑖 ∈ (1...(𝑁 − 1)), 𝑗 ∈ (1...(𝑁 − 1)) ↦ (𝑖𝑊𝑗)))
141		difssd 4093	. . . . . . . . . . 11 ⊢ (𝜑 → ((1...𝑁) ∖ {𝑁}) ⊆ (1...𝑁))
142	137, 141	eqsstrrd 3974	. . . . . . . . . 10 ⊢ (𝜑 → (1...(𝑁 − 1)) ⊆ (1...𝑁))
143	4, 5	submabas 22696	. . . . . . . . . 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 595	. . . . . . . . 9 ⊢ (𝜑 → (𝑖 ∈ (1...(𝑁 − 1)), 𝑗 ∈ (1...(𝑁 − 1)) ↦ (𝑖𝑊𝑗)) ∈ (Base‘((1...(𝑁 − 1)) Mat 𝑅)))
145	140, 144	eqeltrd 2865	. . . . . . . 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 2765	. . . . . . . . 9 ⊢ ((1...(𝑁 − 1)) Mat 𝑅) = ((1...(𝑁 − 1)) Mat 𝑅)
148		eqid 2765	. . . . . . . . 9 ⊢ (Base‘((1...(𝑁 − 1)) Mat 𝑅)) = (Base‘((1...(𝑁 − 1)) Mat 𝑅))
149	146, 147, 148, 117	mdetcl 22714	. . . . . . . 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 595	. . . . . . 7 ⊢ (𝜑 → (𝐸‘(𝑁(((1...𝑁) subMat 𝑅)‘𝑊)𝑁)) ∈ (Base‘𝑅))
151	117, 16, 48	ringlidm 20343	. . . . . . 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 595	. . . . . 6 ⊢ (𝜑 → ((1r‘𝑅) · (𝐸‘(𝑁(((1...𝑁) subMat 𝑅)‘𝑊)𝑁))) = (𝐸‘(𝑁(((1...𝑁) subMat 𝑅)‘𝑊)𝑁)))
153	4	fveq2i 6874	. . . . . . . . . . 11 ⊢ (Base‘𝐴) = (Base‘((1...𝑁) Mat 𝑅))
154	5, 153	eqtri 2788	. . . . . . . . . 10 ⊢ 𝐵 = (Base‘((1...𝑁) Mat 𝑅))
155	124, 154	eleqtrdi 2875	. . . . . . . . 9 ⊢ (𝜑 → 𝑊 ∈ (Base‘((1...𝑁) Mat 𝑅)))
156		smadiadetr 22793	. . . . . . . . 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 851	. . . . . . . 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 6872	. . . . . . . . 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 7413	. . . . . . . . 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 2783	. . . . . . . 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 237	. . . . . . 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 7415	. . . . . . . . . 10 ⊢ (𝜑 → (((1...𝑁) ∖ {𝑁}) maDet 𝑅) = ((1...(𝑁 − 1)) maDet 𝑅))
163	162, 146	eqtr4di 2818	. . . . . . . . 9 ⊢ (𝜑 → (((1...𝑁) ∖ {𝑁}) maDet 𝑅) = 𝐸)
164	163	fveq1d 6873	. . . . . . . 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 7416	. . . . . . 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 2800	. . . . . 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 34112	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ ∧ 𝑊 ∈ 𝐵) → (𝑁(subMat1‘𝑊)𝑁) = (𝑁(((1...𝑁) subMat 𝑅)‘𝑊)𝑁))
168	87, 124, 167	syl2anc 595	. . . . . . 7 ⊢ (𝜑 → (𝑁(subMat1‘𝑊)𝑁) = (𝑁(((1...𝑁) subMat 𝑅)‘𝑊)𝑁))
169	168	fveq2d 6875	. . . . . 6 ⊢ (𝜑 → (𝐸‘(𝑁(subMat1‘𝑊)𝑁)) = (𝐸‘(𝑁(((1...𝑁) subMat 𝑅)‘𝑊)𝑁)))
170	152, 166, 169	3eqtr4d 2810	. . . . 5 ⊢ (𝜑 → (𝐷‘(𝑁(𝑊((1...𝑁) matRRep 𝑅)(1r‘𝑅))𝑁)) = (𝐸‘(𝑁(subMat1‘𝑊)𝑁)))
171	132, 170	eqtr3d 2802	. . . 4 ⊢ (𝜑 → (𝐷‘𝑊) = (𝐸‘(𝑁(subMat1‘𝑊)𝑁)))
172	4, 5, 87, 3, 2, 21, 1, 10	submatminr1 34117	. . . . . 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 2800	. . . . 5 ⊢ (𝜑 → (𝐼(subMat1‘𝑀)𝐽) = (𝑁(subMat1‘𝑊)𝑁))
175	174	fveq2d 6875	. . . 4 ⊢ (𝜑 → (𝐸‘(𝐼(subMat1‘𝑀)𝐽)) = (𝐸‘(𝑁(subMat1‘𝑊)𝑁)))
176	171, 175	eqtr4d 2803	. . 3 ⊢ (𝜑 → (𝐷‘𝑊) = (𝐸‘(𝐼(subMat1‘𝑀)𝐽)))
177	176	oveq2d 7416	. 2 ⊢ (𝜑 → ((𝑍‘((𝑆‘𝑃) · (𝑆‘𝑄))) · (𝐷‘𝑊)) = ((𝑍‘((𝑆‘𝑃) · (𝑆‘𝑄))) · (𝐸‘(𝐼(subMat1‘𝑀)𝐽))))
178	12, 27, 177	3eqtrd 2804	1 ⊢ (𝜑 → (𝐽(𝐾‘𝑀)𝐼) = ((𝑍‘((𝑆‘𝑃) · (𝑆‘𝑄))) · (𝐸‘(𝐼(subMat1‘𝑀)𝐽))))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 400   ∧ w3a 1101   = wceq 1563   ∈ wcel 2145  Vcvv 3457   ∖ cdif 3904   ⊆ wss 3907  ifcif 4483  {csn 4585  ^◡ccnv 5651  –1-1-onto→wf1o 6524  ‘cfv 6525  (class class class)co 7400   ∈ cmpo 7402  1c1 11089   · cmul 11093   − cmin 11429  ℕcn 12224  ℤ_≥cuz 12853  ...cfz 13526  Basecbs 17259  ._rcmulr 17301  0_gc0g 17482  SymGrpcsymg 19430  pmSgncpsgn 19550  1_rcur 20254  Ringcrg 20306  CRingccrg 20307  ℤRHomczrh 21609   Mat cmat 22525   matRRep cmarrep 22674   subMat csubma 22694   maDet cmdat 22702   maAdju cmadu 22750   minMatR1 cminmar1 22751  subMat1csmat 34100

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2737  ax-rep 5232  ax-sep 5251  ax-nul 5261  ax-pow 5327  ax-pr 5395  ax-un 7722  ax-cnex 11144  ax-resscn 11145  ax-1cn 11146  ax-icn 11147  ax-addcl 11148  ax-addrcl 11149  ax-mulcl 11150  ax-mulrcl 11151  ax-mulcom 11152  ax-addass 11153  ax-mulass 11154  ax-distr 11155  ax-i2m1 11156  ax-1ne0 11157  ax-1rid 11158  ax-rnegex 11159  ax-rrecex 11160  ax-cnre 11161  ax-pre-lttri 11162  ax-pre-lttrn 11163  ax-pre-ltadd 11164  ax-pre-mulgt0 11165  ax-addf 11167  ax-mulf 11168

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-xor 1535  df-tru 1566  df-fal 1576  df-ex 1803  df-nf 1807  df-sb 2094  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-nel 3065  df-ral 3080  df-rex 3090  df-rmo 3370  df-reu 3371  df-rab 3418  df-v 3459  df-sbc 3748  df-csb 3856  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-pss 3927  df-nul 4289  df-if 4484  df-pw 4560  df-sn 4586  df-pr 4588  df-tp 4590  df-op 4592  df-ot 4594  df-uni 4869  df-int 4909  df-iun 4954  df-iin 4955  df-br 5106  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5547  df-eprel 5552  df-po 5560  df-so 5561  df-fr 5605  df-se 5606  df-we 5607  df-xp 5658  df-rel 5659  df-cnv 5660  df-co 5661  df-dm 5662  df-rn 5663  df-res 5664  df-ima 5665  df-pred 6292  df-ord 6353  df-on 6354  df-lim 6355  df-suc 6356  df-iota 6481  df-fun 6527  df-fn 6528  df-f 6529  df-f1 6530  df-fo 6531  df-f1o 6532  df-fv 6533  df-isom 6534  df-riota 7357  df-ov 7403  df-oprab 7404  df-mpo 7405  df-of 7664  df-om 7851  df-1st 7974  df-2nd 7975  df-supp 8145  df-tpos 8210  df-frecs 8266  df-wrecs 8297  df-recs 8346  df-rdg 8385  df-1o 8441  df-2o 8442  df-er 8682  df-map 8814  df-pm 8815  df-ixp 8884  df-en 8932  df-dom 8933  df-sdom 8934  df-fin 8935  df-fsupp 9310  df-sup 9390  df-oi 9460  df-card 9913  df-pnf 11233  df-mnf 11234  df-xr 11235  df-ltxr 11236  df-le 11237  df-sub 11431  df-neg 11432  df-div 11860  df-nn 12225  df-2 12294  df-3 12295  df-4 12296  df-5 12297  df-6 12298  df-7 12299  df-8 12300  df-9 12301  df-n0 12496  df-xnn0 12569  df-z 12583  df-dec 12703  df-uz 12854  df-rp 13008  df-fz 13527  df-fzo 13674  df-seq 14029  df-exp 14089  df-hash 14358  df-word 14541  df-lsw 14590  df-concat 14598  df-s1 14624  df-substr 14669  df-pfx 14699  df-splice 14777  df-reverse 14786  df-s2 14875  df-struct 17197  df-sets 17214  df-slot 17232  df-ndx 17244  df-base 17260  df-ress 17281  df-plusg 17313  df-mulr 17314  df-starv 17315  df-sca 17316  df-vsca 17317  df-ip 17318  df-tset 17319  df-ple 17320  df-ds 17322  df-unif 17323  df-hom 17324  df-cco 17325  df-0g 17484  df-gsum 17485  df-prds 17490  df-pws 17492  df-mre 17628  df-mrc 17629  df-acs 17631  df-mgm 18688  df-sgrp 18767  df-mnd 18783  df-mhm 18831  df-submnd 18832  df-efmnd 18918  df-grp 18993  df-minusg 18994  df-mulg 19125  df-subg 19180  df-ghm 19275  df-gim 19320  df-cntz 19378  df-oppg 19407  df-symg 19431  df-pmtr 19503  df-psgn 19552  df-cmn 19843  df-abl 19844  df-mgp 20208  df-rng 20222  df-ur 20255  df-ring 20308  df-cring 20309  df-oppr 20410  df-dvdsr 20430  df-unit 20431  df-invr 20461  df-dvr 20474  df-rhm 20545  df-subrng 20622  df-subrg 20646  df-drng 20806  df-sra 21263  df-rgmod 21264  df-cnfld 21483  df-zring 21557  df-zrh 21613  df-dsmm 21842  df-frlm 21857  df-mat 22526  df-marrep 22676  df-subma 22695  df-mdet 22703  df-madu 22752  df-minmar1 22753  df-smat 34101

This theorem is referenced by:  madjusmdetlem4  34137