madjusmdetlem3 - Mathbox for Thierry Arnoux

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Theorem madjusmdetlem3 33812

Description: Lemma for madjusmdet 33814. (Contributed by Thierry Arnoux, 27-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)
madjusmdetlem2.p	⊢ 𝑃 = (𝑖 ∈ (1...𝑁) ↦ if(𝑖 = 1, 𝐼, if(𝑖 ≤ 𝐼, (𝑖 − 1), 𝑖)))
madjusmdetlem2.s	⊢ 𝑆 = (𝑖 ∈ (1...𝑁) ↦ if(𝑖 = 1, 𝑁, if(𝑖 ≤ 𝑁, (𝑖 − 1), 𝑖)))
madjusmdetlem4.q	⊢ 𝑄 = (𝑗 ∈ (1...𝑁) ↦ if(𝑗 = 1, 𝐽, if(𝑗 ≤ 𝐽, (𝑗 − 1), 𝑗)))
madjusmdetlem4.t	⊢ 𝑇 = (𝑗 ∈ (1...𝑁) ↦ if(𝑗 = 1, 𝑁, if(𝑗 ≤ 𝑁, (𝑗 − 1), 𝑗)))
madjusmdetlem3.w	⊢ 𝑊 = (𝑖 ∈ (1...𝑁), 𝑗 ∈ (1...𝑁) ↦ (((𝑃 ∘ ^◡𝑆)‘𝑖)𝑈((𝑄 ∘ ^◡𝑇)‘𝑗)))
madjusmdetlem3.u	⊢ (𝜑 → 𝑈 ∈ 𝐵)

**Assertion**
Ref	Expression
madjusmdetlem3	⊢ (𝜑 → (𝐼(subMat1‘𝑈)𝐽) = (𝑁(subMat1‘𝑊)𝑁))

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

**Proof of Theorem madjusmdetlem3**
Step	Hyp	Ref	Expression
1		madjusmdet.n	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑁 ∈ ℕ)
2		nnuz 12778	. . . . . . . . . . 11 ⊢ ℕ = (ℤ_≥‘1)
3	1, 2	eleqtrdi 2838	. . . . . . . . . 10 ⊢ (𝜑 → 𝑁 ∈ (ℤ_≥‘1))
4		fzdif2 32742	. . . . . . . . . 10 ⊢ (𝑁 ∈ (ℤ_≥‘1) → ((1...𝑁) ∖ {𝑁}) = (1...(𝑁 − 1)))
5	3, 4	syl 17	. . . . . . . . 9 ⊢ (𝜑 → ((1...𝑁) ∖ {𝑁}) = (1...(𝑁 − 1)))
6		difss 4087	. . . . . . . . 9 ⊢ ((1...𝑁) ∖ {𝑁}) ⊆ (1...𝑁)
7	5, 6	eqsstrrdi 3981	. . . . . . . 8 ⊢ (𝜑 → (1...(𝑁 − 1)) ⊆ (1...𝑁))
8	7	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) → (1...(𝑁 − 1)) ⊆ (1...𝑁))
9		simprl 770	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) → 𝑖 ∈ (1...(𝑁 − 1)))
10	8, 9	sseldd 3936	. . . . . 6 ⊢ ((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) → 𝑖 ∈ (1...𝑁))
11		simprr 772	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) → 𝑗 ∈ (1...(𝑁 − 1)))
12	8, 11	sseldd 3936	. . . . . 6 ⊢ ((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) → 𝑗 ∈ (1...𝑁))
13		ovexd 7384	. . . . . 6 ⊢ ((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) → (((𝑃 ∘ ^◡𝑆)‘𝑖)𝑈((𝑄 ∘ ^◡𝑇)‘𝑗)) ∈ V)
14		madjusmdetlem3.w	. . . . . . 7 ⊢ 𝑊 = (𝑖 ∈ (1...𝑁), 𝑗 ∈ (1...𝑁) ↦ (((𝑃 ∘ ^◡𝑆)‘𝑖)𝑈((𝑄 ∘ ^◡𝑇)‘𝑗)))
15	14	ovmpt4g 7496	. . . . . 6 ⊢ ((𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁) ∧ (((𝑃 ∘ ^◡𝑆)‘𝑖)𝑈((𝑄 ∘ ^◡𝑇)‘𝑗)) ∈ V) → (𝑖𝑊𝑗) = (((𝑃 ∘ ^◡𝑆)‘𝑖)𝑈((𝑄 ∘ ^◡𝑇)‘𝑗)))
16	10, 12, 13, 15	syl3anc 1373	. . . . 5 ⊢ ((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) → (𝑖𝑊𝑗) = (((𝑃 ∘ ^◡𝑆)‘𝑖)𝑈((𝑄 ∘ ^◡𝑇)‘𝑗)))
17	9, 11	ovresd 7516	. . . . 5 ⊢ ((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) → (𝑖(𝑊 ↾ ((1...(𝑁 − 1)) × (1...(𝑁 − 1))))𝑗) = (𝑖𝑊𝑗))
18		eqid 2729	. . . . . . 7 ⊢ (𝐼(subMat1‘𝑈)𝐽) = (𝐼(subMat1‘𝑈)𝐽)
19	1	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) → 𝑁 ∈ ℕ)
20		madjusmdet.i	. . . . . . . 8 ⊢ (𝜑 → 𝐼 ∈ (1...𝑁))
21	20	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) → 𝐼 ∈ (1...𝑁))
22		madjusmdet.j	. . . . . . . 8 ⊢ (𝜑 → 𝐽 ∈ (1...𝑁))
23	22	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) → 𝐽 ∈ (1...𝑁))
24		madjusmdetlem3.u	. . . . . . . . 9 ⊢ (𝜑 → 𝑈 ∈ 𝐵)
25		madjusmdet.a	. . . . . . . . . 10 ⊢ 𝐴 = ((1...𝑁) Mat 𝑅)
26		eqid 2729	. . . . . . . . . 10 ⊢ (Base‘𝑅) = (Base‘𝑅)
27		madjusmdet.b	. . . . . . . . . 10 ⊢ 𝐵 = (Base‘𝐴)
28	25, 26, 27	matbas2i 22307	. . . . . . . . 9 ⊢ (𝑈 ∈ 𝐵 → 𝑈 ∈ ((Base‘𝑅) ↑_m ((1...𝑁) × (1...𝑁))))
29	24, 28	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝑈 ∈ ((Base‘𝑅) ↑_m ((1...𝑁) × (1...𝑁))))
30	29	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) → 𝑈 ∈ ((Base‘𝑅) ↑_m ((1...𝑁) × (1...𝑁))))
31		fz1ssnn 13458	. . . . . . . 8 ⊢ (1...𝑁) ⊆ ℕ
32	31, 10	sselid 3933	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) → 𝑖 ∈ ℕ)
33	31, 12	sselid 3933	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) → 𝑗 ∈ ℕ)
34		eqidd 2730	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) → if(𝑖 < 𝐼, 𝑖, (𝑖 + 1)) = if(𝑖 < 𝐼, 𝑖, (𝑖 + 1)))
35		eqidd 2730	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) → if(𝑗 < 𝐽, 𝑗, (𝑗 + 1)) = if(𝑗 < 𝐽, 𝑗, (𝑗 + 1)))
36	18, 19, 19, 21, 23, 30, 32, 33, 34, 35	smatlem 33780	. . . . . 6 ⊢ ((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) → (𝑖(𝐼(subMat1‘𝑈)𝐽)𝑗) = (if(𝑖 < 𝐼, 𝑖, (𝑖 + 1))𝑈if(𝑗 < 𝐽, 𝑗, (𝑗 + 1))))
37		madjusmdet.d	. . . . . . . . 9 ⊢ 𝐷 = ((1...𝑁) maDet 𝑅)
38		madjusmdet.k	. . . . . . . . 9 ⊢ 𝐾 = ((1...𝑁) maAdju 𝑅)
39		madjusmdet.t	. . . . . . . . 9 ⊢ · = (._r‘𝑅)
40		madjusmdet.z	. . . . . . . . 9 ⊢ 𝑍 = (ℤRHom‘𝑅)
41		madjusmdet.e	. . . . . . . . 9 ⊢ 𝐸 = ((1...(𝑁 − 1)) maDet 𝑅)
42		madjusmdet.r	. . . . . . . . 9 ⊢ (𝜑 → 𝑅 ∈ CRing)
43		madjusmdet.m	. . . . . . . . 9 ⊢ (𝜑 → 𝑀 ∈ 𝐵)
44		madjusmdetlem2.p	. . . . . . . . 9 ⊢ 𝑃 = (𝑖 ∈ (1...𝑁) ↦ if(𝑖 = 1, 𝐼, if(𝑖 ≤ 𝐼, (𝑖 − 1), 𝑖)))
45		madjusmdetlem2.s	. . . . . . . . 9 ⊢ 𝑆 = (𝑖 ∈ (1...𝑁) ↦ if(𝑖 = 1, 𝑁, if(𝑖 ≤ 𝑁, (𝑖 − 1), 𝑖)))
46	27, 25, 37, 38, 39, 40, 41, 1, 42, 20, 20, 43, 44, 45	madjusmdetlem2 33811	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...(𝑁 − 1))) → if(𝑖 < 𝐼, 𝑖, (𝑖 + 1)) = ((𝑃 ∘ ^◡𝑆)‘𝑖))
47	9, 46	syldan 591	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) → if(𝑖 < 𝐼, 𝑖, (𝑖 + 1)) = ((𝑃 ∘ ^◡𝑆)‘𝑖))
48		madjusmdetlem4.q	. . . . . . . . 9 ⊢ 𝑄 = (𝑗 ∈ (1...𝑁) ↦ if(𝑗 = 1, 𝐽, if(𝑗 ≤ 𝐽, (𝑗 − 1), 𝑗)))
49		madjusmdetlem4.t	. . . . . . . . 9 ⊢ 𝑇 = (𝑗 ∈ (1...𝑁) ↦ if(𝑗 = 1, 𝑁, if(𝑗 ≤ 𝑁, (𝑗 − 1), 𝑗)))
50	27, 25, 37, 38, 39, 40, 41, 1, 42, 22, 22, 43, 48, 49	madjusmdetlem2 33811	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ (1...(𝑁 − 1))) → if(𝑗 < 𝐽, 𝑗, (𝑗 + 1)) = ((𝑄 ∘ ^◡𝑇)‘𝑗))
51	11, 50	syldan 591	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) → if(𝑗 < 𝐽, 𝑗, (𝑗 + 1)) = ((𝑄 ∘ ^◡𝑇)‘𝑗))
52	47, 51	oveq12d 7367	. . . . . 6 ⊢ ((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) → (if(𝑖 < 𝐼, 𝑖, (𝑖 + 1))𝑈if(𝑗 < 𝐽, 𝑗, (𝑗 + 1))) = (((𝑃 ∘ ^◡𝑆)‘𝑖)𝑈((𝑄 ∘ ^◡𝑇)‘𝑗)))
53	36, 52	eqtrd 2764	. . . . 5 ⊢ ((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) → (𝑖(𝐼(subMat1‘𝑈)𝐽)𝑗) = (((𝑃 ∘ ^◡𝑆)‘𝑖)𝑈((𝑄 ∘ ^◡𝑇)‘𝑗)))
54	16, 17, 53	3eqtr4rd 2775	. . . 4 ⊢ ((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) → (𝑖(𝐼(subMat1‘𝑈)𝐽)𝑗) = (𝑖(𝑊 ↾ ((1...(𝑁 − 1)) × (1...(𝑁 − 1))))𝑗))
55	54	ralrimivva 3172	. . 3 ⊢ (𝜑 → ∀𝑖 ∈ (1...(𝑁 − 1))∀𝑗 ∈ (1...(𝑁 − 1))(𝑖(𝐼(subMat1‘𝑈)𝐽)𝑗) = (𝑖(𝑊 ↾ ((1...(𝑁 − 1)) × (1...(𝑁 − 1))))𝑗))
56		eqid 2729	. . . . 5 ⊢ (Base‘((1...(𝑁 − 1)) Mat 𝑅)) = (Base‘((1...(𝑁 − 1)) Mat 𝑅))
57	25, 27, 56, 18, 1, 20, 22, 24	smatcl 33785	. . . 4 ⊢ (𝜑 → (𝐼(subMat1‘𝑈)𝐽) ∈ (Base‘((1...(𝑁 − 1)) Mat 𝑅)))
58		fzfid 13880	. . . . . . . 8 ⊢ (𝜑 → (1...𝑁) ∈ Fin)
59		eqid 2729	. . . . . . . . . . . . . 14 ⊢ (1...𝑁) = (1...𝑁)
60		eqid 2729	. . . . . . . . . . . . . 14 ⊢ (SymGrp‘(1...𝑁)) = (SymGrp‘(1...𝑁))
61		eqid 2729	. . . . . . . . . . . . . 14 ⊢ (Base‘(SymGrp‘(1...𝑁))) = (Base‘(SymGrp‘(1...𝑁)))
62	59, 44, 60, 61	fzto1st 33054	. . . . . . . . . . . . 13 ⊢ (𝐼 ∈ (1...𝑁) → 𝑃 ∈ (Base‘(SymGrp‘(1...𝑁))))
63	20, 62	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑃 ∈ (Base‘(SymGrp‘(1...𝑁))))
64		eluzfz2 13435	. . . . . . . . . . . . . . . 16 ⊢ (𝑁 ∈ (ℤ_≥‘1) → 𝑁 ∈ (1...𝑁))
65	3, 64	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑁 ∈ (1...𝑁))
66	59, 45, 60, 61	fzto1st 33054	. . . . . . . . . . . . . . 15 ⊢ (𝑁 ∈ (1...𝑁) → 𝑆 ∈ (Base‘(SymGrp‘(1...𝑁))))
67	65, 66	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑆 ∈ (Base‘(SymGrp‘(1...𝑁))))
68		eqid 2729	. . . . . . . . . . . . . . 15 ⊢ (inv_g‘(SymGrp‘(1...𝑁))) = (inv_g‘(SymGrp‘(1...𝑁)))
69	60, 61, 68	symginv 19281	. . . . . . . . . . . . . 14 ⊢ (𝑆 ∈ (Base‘(SymGrp‘(1...𝑁))) → ((inv_g‘(SymGrp‘(1...𝑁)))‘𝑆) = ^◡𝑆)
70	67, 69	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((inv_g‘(SymGrp‘(1...𝑁)))‘𝑆) = ^◡𝑆)
71	60	symggrp 19279	. . . . . . . . . . . . . . 15 ⊢ ((1...𝑁) ∈ Fin → (SymGrp‘(1...𝑁)) ∈ Grp)
72	58, 71	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (SymGrp‘(1...𝑁)) ∈ Grp)
73	61, 68	grpinvcl 18866	. . . . . . . . . . . . . 14 ⊢ (((SymGrp‘(1...𝑁)) ∈ Grp ∧ 𝑆 ∈ (Base‘(SymGrp‘(1...𝑁)))) → ((inv_g‘(SymGrp‘(1...𝑁)))‘𝑆) ∈ (Base‘(SymGrp‘(1...𝑁))))
74	72, 67, 73	syl2anc 584	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((inv_g‘(SymGrp‘(1...𝑁)))‘𝑆) ∈ (Base‘(SymGrp‘(1...𝑁))))
75	70, 74	eqeltrrd 2829	. . . . . . . . . . . 12 ⊢ (𝜑 → ^◡𝑆 ∈ (Base‘(SymGrp‘(1...𝑁))))
76		eqid 2729	. . . . . . . . . . . . . 14 ⊢ (+_g‘(SymGrp‘(1...𝑁))) = (+_g‘(SymGrp‘(1...𝑁)))
77	60, 61, 76	symgov 19263	. . . . . . . . . . . . 13 ⊢ ((𝑃 ∈ (Base‘(SymGrp‘(1...𝑁))) ∧ ^◡𝑆 ∈ (Base‘(SymGrp‘(1...𝑁)))) → (𝑃(+_g‘(SymGrp‘(1...𝑁)))^◡𝑆) = (𝑃 ∘ ^◡𝑆))
78	60, 61, 76	symgcl 19264	. . . . . . . . . . . . 13 ⊢ ((𝑃 ∈ (Base‘(SymGrp‘(1...𝑁))) ∧ ^◡𝑆 ∈ (Base‘(SymGrp‘(1...𝑁)))) → (𝑃(+_g‘(SymGrp‘(1...𝑁)))^◡𝑆) ∈ (Base‘(SymGrp‘(1...𝑁))))
79	77, 78	eqeltrrd 2829	. . . . . . . . . . . 12 ⊢ ((𝑃 ∈ (Base‘(SymGrp‘(1...𝑁))) ∧ ^◡𝑆 ∈ (Base‘(SymGrp‘(1...𝑁)))) → (𝑃 ∘ ^◡𝑆) ∈ (Base‘(SymGrp‘(1...𝑁))))
80	63, 75, 79	syl2anc 584	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑃 ∘ ^◡𝑆) ∈ (Base‘(SymGrp‘(1...𝑁))))
81	80	3ad2ant1 1133	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) → (𝑃 ∘ ^◡𝑆) ∈ (Base‘(SymGrp‘(1...𝑁))))
82		simp2 1137	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) → 𝑖 ∈ (1...𝑁))
83	60, 61	symgfv 19259	. . . . . . . . . 10 ⊢ (((𝑃 ∘ ^◡𝑆) ∈ (Base‘(SymGrp‘(1...𝑁))) ∧ 𝑖 ∈ (1...𝑁)) → ((𝑃 ∘ ^◡𝑆)‘𝑖) ∈ (1...𝑁))
84	81, 82, 83	syl2anc 584	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) → ((𝑃 ∘ ^◡𝑆)‘𝑖) ∈ (1...𝑁))
85	59, 48, 60, 61	fzto1st 33054	. . . . . . . . . . . . 13 ⊢ (𝐽 ∈ (1...𝑁) → 𝑄 ∈ (Base‘(SymGrp‘(1...𝑁))))
86	22, 85	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑄 ∈ (Base‘(SymGrp‘(1...𝑁))))
87	59, 49, 60, 61	fzto1st 33054	. . . . . . . . . . . . . . 15 ⊢ (𝑁 ∈ (1...𝑁) → 𝑇 ∈ (Base‘(SymGrp‘(1...𝑁))))
88	65, 87	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑇 ∈ (Base‘(SymGrp‘(1...𝑁))))
89	60, 61, 68	symginv 19281	. . . . . . . . . . . . . 14 ⊢ (𝑇 ∈ (Base‘(SymGrp‘(1...𝑁))) → ((inv_g‘(SymGrp‘(1...𝑁)))‘𝑇) = ^◡𝑇)
90	88, 89	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((inv_g‘(SymGrp‘(1...𝑁)))‘𝑇) = ^◡𝑇)
91	61, 68	grpinvcl 18866	. . . . . . . . . . . . . 14 ⊢ (((SymGrp‘(1...𝑁)) ∈ Grp ∧ 𝑇 ∈ (Base‘(SymGrp‘(1...𝑁)))) → ((inv_g‘(SymGrp‘(1...𝑁)))‘𝑇) ∈ (Base‘(SymGrp‘(1...𝑁))))
92	72, 88, 91	syl2anc 584	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((inv_g‘(SymGrp‘(1...𝑁)))‘𝑇) ∈ (Base‘(SymGrp‘(1...𝑁))))
93	90, 92	eqeltrrd 2829	. . . . . . . . . . . 12 ⊢ (𝜑 → ^◡𝑇 ∈ (Base‘(SymGrp‘(1...𝑁))))
94	60, 61, 76	symgov 19263	. . . . . . . . . . . . 13 ⊢ ((𝑄 ∈ (Base‘(SymGrp‘(1...𝑁))) ∧ ^◡𝑇 ∈ (Base‘(SymGrp‘(1...𝑁)))) → (𝑄(+_g‘(SymGrp‘(1...𝑁)))^◡𝑇) = (𝑄 ∘ ^◡𝑇))
95	60, 61, 76	symgcl 19264	. . . . . . . . . . . . 13 ⊢ ((𝑄 ∈ (Base‘(SymGrp‘(1...𝑁))) ∧ ^◡𝑇 ∈ (Base‘(SymGrp‘(1...𝑁)))) → (𝑄(+_g‘(SymGrp‘(1...𝑁)))^◡𝑇) ∈ (Base‘(SymGrp‘(1...𝑁))))
96	94, 95	eqeltrrd 2829	. . . . . . . . . . . 12 ⊢ ((𝑄 ∈ (Base‘(SymGrp‘(1...𝑁))) ∧ ^◡𝑇 ∈ (Base‘(SymGrp‘(1...𝑁)))) → (𝑄 ∘ ^◡𝑇) ∈ (Base‘(SymGrp‘(1...𝑁))))
97	86, 93, 96	syl2anc 584	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑄 ∘ ^◡𝑇) ∈ (Base‘(SymGrp‘(1...𝑁))))
98	97	3ad2ant1 1133	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) → (𝑄 ∘ ^◡𝑇) ∈ (Base‘(SymGrp‘(1...𝑁))))
99		simp3 1138	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) → 𝑗 ∈ (1...𝑁))
100	60, 61	symgfv 19259	. . . . . . . . . 10 ⊢ (((𝑄 ∘ ^◡𝑇) ∈ (Base‘(SymGrp‘(1...𝑁))) ∧ 𝑗 ∈ (1...𝑁)) → ((𝑄 ∘ ^◡𝑇)‘𝑗) ∈ (1...𝑁))
101	98, 99, 100	syl2anc 584	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) → ((𝑄 ∘ ^◡𝑇)‘𝑗) ∈ (1...𝑁))
102	24	3ad2ant1 1133	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) → 𝑈 ∈ 𝐵)
103	25, 26, 27, 84, 101, 102	matecld 22311	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) → (((𝑃 ∘ ^◡𝑆)‘𝑖)𝑈((𝑄 ∘ ^◡𝑇)‘𝑗)) ∈ (Base‘𝑅))
104	25, 26, 27, 58, 42, 103	matbas2d 22308	. . . . . . 7 ⊢ (𝜑 → (𝑖 ∈ (1...𝑁), 𝑗 ∈ (1...𝑁) ↦ (((𝑃 ∘ ^◡𝑆)‘𝑖)𝑈((𝑄 ∘ ^◡𝑇)‘𝑗))) ∈ 𝐵)
105	14, 104	eqeltrid 2832	. . . . . 6 ⊢ (𝜑 → 𝑊 ∈ 𝐵)
106	25, 27	submatres 33789	. . . . . 6 ⊢ ((𝑁 ∈ ℕ ∧ 𝑊 ∈ 𝐵) → (𝑁(subMat1‘𝑊)𝑁) = (𝑊 ↾ ((1...(𝑁 − 1)) × (1...(𝑁 − 1)))))
107	1, 105, 106	syl2anc 584	. . . . 5 ⊢ (𝜑 → (𝑁(subMat1‘𝑊)𝑁) = (𝑊 ↾ ((1...(𝑁 − 1)) × (1...(𝑁 − 1)))))
108		eqid 2729	. . . . . 6 ⊢ (𝑁(subMat1‘𝑊)𝑁) = (𝑁(subMat1‘𝑊)𝑁)
109	25, 27, 56, 108, 1, 65, 65, 105	smatcl 33785	. . . . 5 ⊢ (𝜑 → (𝑁(subMat1‘𝑊)𝑁) ∈ (Base‘((1...(𝑁 − 1)) Mat 𝑅)))
110	107, 109	eqeltrrd 2829	. . . 4 ⊢ (𝜑 → (𝑊 ↾ ((1...(𝑁 − 1)) × (1...(𝑁 − 1)))) ∈ (Base‘((1...(𝑁 − 1)) Mat 𝑅)))
111		eqid 2729	. . . . 5 ⊢ ((1...(𝑁 − 1)) Mat 𝑅) = ((1...(𝑁 − 1)) Mat 𝑅)
112	111, 56	eqmat 22309	. . . 4 ⊢ (((𝐼(subMat1‘𝑈)𝐽) ∈ (Base‘((1...(𝑁 − 1)) Mat 𝑅)) ∧ (𝑊 ↾ ((1...(𝑁 − 1)) × (1...(𝑁 − 1)))) ∈ (Base‘((1...(𝑁 − 1)) Mat 𝑅))) → ((𝐼(subMat1‘𝑈)𝐽) = (𝑊 ↾ ((1...(𝑁 − 1)) × (1...(𝑁 − 1)))) ↔ ∀𝑖 ∈ (1...(𝑁 − 1))∀𝑗 ∈ (1...(𝑁 − 1))(𝑖(𝐼(subMat1‘𝑈)𝐽)𝑗) = (𝑖(𝑊 ↾ ((1...(𝑁 − 1)) × (1...(𝑁 − 1))))𝑗)))
113	57, 110, 112	syl2anc 584	. . 3 ⊢ (𝜑 → ((𝐼(subMat1‘𝑈)𝐽) = (𝑊 ↾ ((1...(𝑁 − 1)) × (1...(𝑁 − 1)))) ↔ ∀𝑖 ∈ (1...(𝑁 − 1))∀𝑗 ∈ (1...(𝑁 − 1))(𝑖(𝐼(subMat1‘𝑈)𝐽)𝑗) = (𝑖(𝑊 ↾ ((1...(𝑁 − 1)) × (1...(𝑁 − 1))))𝑗)))
114	55, 113	mpbird 257	. 2 ⊢ (𝜑 → (𝐼(subMat1‘𝑈)𝐽) = (𝑊 ↾ ((1...(𝑁 − 1)) × (1...(𝑁 − 1)))))
115	114, 107	eqtr4d 2767	1 ⊢ (𝜑 → (𝐼(subMat1‘𝑈)𝐽) = (𝑁(subMat1‘𝑊)𝑁))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1086 = wceq 1540 ∈ wcel 2109 ∀wral 3044 Vcvv 3436 ∖ cdif 3900 ⊆ wss 3903 ifcif 4476 {csn 4577 class class class wbr 5092 ↦ cmpt 5173 × cxp 5617 ^◡ccnv 5618 ↾ cres 5621 ∘ ccom 5623 ‘cfv 6482 (class class class)co 7349 ∈ cmpo 7351 ↑_m cmap 8753 Fincfn 8872 1c1 11010 + caddc 11012 < clt 11149 ≤ cle 11150 − cmin 11347 ℕcn 12128 ℤ_≥cuz 12735 ...cfz 13410 Basecbs 17120 +_gcplusg 17161 ._rcmulr 17162 Grpcgrp 18812 inv_gcminusg 18813 SymGrpcsymg 19248 CRingccrg 20119 ℤRHomczrh 21406 Mat cmat 22292 maDet cmdat 22469 maAdju cmadu 22517 subMat1csmat 33776

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 2701 ax-rep 5218 ax-sep 5235 ax-nul 5245 ax-pow 5304 ax-pr 5371 ax-un 7671 ax-cnex 11065 ax-resscn 11066 ax-1cn 11067 ax-icn 11068 ax-addcl 11069 ax-addrcl 11070 ax-mulcl 11071 ax-mulrcl 11072 ax-mulcom 11073 ax-addass 11074 ax-mulass 11075 ax-distr 11076 ax-i2m1 11077 ax-1ne0 11078 ax-1rid 11079 ax-rnegex 11080 ax-rrecex 11081 ax-cnre 11082 ax-pre-lttri 11083 ax-pre-lttrn 11084 ax-pre-ltadd 11085 ax-pre-mulgt0 11086

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-rmo 3343 df-reu 3344 df-rab 3395 df-v 3438 df-sbc 3743 df-csb 3852 df-dif 3906 df-un 3908 df-in 3910 df-ss 3920 df-pss 3923 df-nul 4285 df-if 4477 df-pw 4553 df-sn 4578 df-pr 4580 df-tp 4582 df-op 4584 df-ot 4586 df-uni 4859 df-iun 4943 df-br 5093 df-opab 5155 df-mpt 5174 df-tr 5200 df-id 5514 df-eprel 5519 df-po 5527 df-so 5528 df-fr 5572 df-we 5574 df-xp 5625 df-rel 5626 df-cnv 5627 df-co 5628 df-dm 5629 df-rn 5630 df-res 5631 df-ima 5632 df-pred 6249 df-ord 6310 df-on 6311 df-lim 6312 df-suc 6313 df-iota 6438 df-fun 6484 df-fn 6485 df-f 6486 df-f1 6487 df-fo 6488 df-f1o 6489 df-fv 6490 df-riota 7306 df-ov 7352 df-oprab 7353 df-mpo 7354 df-om 7800 df-1st 7924 df-2nd 7925 df-supp 8094 df-frecs 8214 df-wrecs 8245 df-recs 8294 df-rdg 8332 df-1o 8388 df-2o 8389 df-er 8625 df-map 8755 df-ixp 8825 df-en 8873 df-dom 8874 df-sdom 8875 df-fin 8876 df-fsupp 9252 df-sup 9332 df-pnf 11151 df-mnf 11152 df-xr 11153 df-ltxr 11154 df-le 11155 df-sub 11349 df-neg 11350 df-nn 12129 df-2 12191 df-3 12192 df-4 12193 df-5 12194 df-6 12195 df-7 12196 df-8 12197 df-9 12198 df-n0 12385 df-z 12472 df-dec 12592 df-uz 12736 df-rp 12894 df-fz 13411 df-fzo 13558 df-struct 17058 df-sets 17075 df-slot 17093 df-ndx 17105 df-base 17121 df-ress 17142 df-plusg 17174 df-mulr 17175 df-sca 17177 df-vsca 17178 df-ip 17179 df-tset 17180 df-ple 17181 df-ds 17183 df-hom 17185 df-cco 17186 df-0g 17345 df-prds 17351 df-pws 17353 df-mgm 18514 df-sgrp 18593 df-mnd 18609 df-submnd 18658 df-efmnd 18743 df-grp 18815 df-minusg 18816 df-symg 19249 df-pmtr 19321 df-sra 21077 df-rgmod 21078 df-dsmm 21639 df-frlm 21654 df-mat 22293 df-subma 22462 df-smat 33777

This theorem is referenced by: madjusmdetlem4 33813

W3C validator