madjusmdetlem3 - Mathbox for Thierry Arnoux

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Theorem madjusmdetlem3 33988

Description: Lemma for madjusmdet 33990. (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 12794	. . . . . . . . . . 11 ⊢ ℕ = (ℤ_≥‘1)
3	1, 2	eleqtrdi 2847	. . . . . . . . . 10 ⊢ (𝜑 → 𝑁 ∈ (ℤ_≥‘1))
4		fzdif2 32872	. . . . . . . . . 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 4089	. . . . . . . . 9 ⊢ ((1...𝑁) ∖ {𝑁}) ⊆ (1...𝑁)
7	5, 6	eqsstrrdi 3980	. . . . . . . 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 771	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) → 𝑖 ∈ (1...(𝑁 − 1)))
10	8, 9	sseldd 3935	. . . . . 6 ⊢ ((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) → 𝑖 ∈ (1...𝑁))
11		simprr 773	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) → 𝑗 ∈ (1...(𝑁 − 1)))
12	8, 11	sseldd 3935	. . . . . 6 ⊢ ((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) → 𝑗 ∈ (1...𝑁))
13		ovexd 7395	. . . . . 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 7507	. . . . . 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 1374	. . . . 5 ⊢ ((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) → (𝑖𝑊𝑗) = (((𝑃 ∘ ^◡𝑆)‘𝑖)𝑈((𝑄 ∘ ^◡𝑇)‘𝑗)))
17	9, 11	ovresd 7527	. . . . 5 ⊢ ((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) → (𝑖(𝑊 ↾ ((1...(𝑁 − 1)) × (1...(𝑁 − 1))))𝑗) = (𝑖𝑊𝑗))
18		eqid 2737	. . . . . . 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 2737	. . . . . . . . . 10 ⊢ (Base‘𝑅) = (Base‘𝑅)
27		madjusmdet.b	. . . . . . . . . 10 ⊢ 𝐵 = (Base‘𝐴)
28	25, 26, 27	matbas2i 22370	. . . . . . . . 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 13475	. . . . . . . 8 ⊢ (1...𝑁) ⊆ ℕ
32	31, 10	sselid 3932	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) → 𝑖 ∈ ℕ)
33	31, 12	sselid 3932	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) → 𝑗 ∈ ℕ)
34		eqidd 2738	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) → if(𝑖 < 𝐼, 𝑖, (𝑖 + 1)) = if(𝑖 < 𝐼, 𝑖, (𝑖 + 1)))
35		eqidd 2738	. . . . . . 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 33956	. . . . . 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 33987	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...(𝑁 − 1))) → if(𝑖 < 𝐼, 𝑖, (𝑖 + 1)) = ((𝑃 ∘ ^◡𝑆)‘𝑖))
47	9, 46	syldan 592	. . . . . . 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 33987	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ (1...(𝑁 − 1))) → if(𝑗 < 𝐽, 𝑗, (𝑗 + 1)) = ((𝑄 ∘ ^◡𝑇)‘𝑗))
51	11, 50	syldan 592	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) → if(𝑗 < 𝐽, 𝑗, (𝑗 + 1)) = ((𝑄 ∘ ^◡𝑇)‘𝑗))
52	47, 51	oveq12d 7378	. . . . . 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 2772	. . . . 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 2783	. . . 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 3180	. . 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 2737	. . . . 5 ⊢ (Base‘((1...(𝑁 − 1)) Mat 𝑅)) = (Base‘((1...(𝑁 − 1)) Mat 𝑅))
57	25, 27, 56, 18, 1, 20, 22, 24	smatcl 33961	. . . 4 ⊢ (𝜑 → (𝐼(subMat1‘𝑈)𝐽) ∈ (Base‘((1...(𝑁 − 1)) Mat 𝑅)))
58		fzfid 13900	. . . . . . . 8 ⊢ (𝜑 → (1...𝑁) ∈ Fin)
59		eqid 2737	. . . . . . . . . . . . . 14 ⊢ (1...𝑁) = (1...𝑁)
60		eqid 2737	. . . . . . . . . . . . . 14 ⊢ (SymGrp‘(1...𝑁)) = (SymGrp‘(1...𝑁))
61		eqid 2737	. . . . . . . . . . . . . 14 ⊢ (Base‘(SymGrp‘(1...𝑁))) = (Base‘(SymGrp‘(1...𝑁)))
62	59, 44, 60, 61	fzto1st 33187	. . . . . . . . . . . . 13 ⊢ (𝐼 ∈ (1...𝑁) → 𝑃 ∈ (Base‘(SymGrp‘(1...𝑁))))
63	20, 62	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑃 ∈ (Base‘(SymGrp‘(1...𝑁))))
64		eluzfz2 13452	. . . . . . . . . . . . . . . 16 ⊢ (𝑁 ∈ (ℤ_≥‘1) → 𝑁 ∈ (1...𝑁))
65	3, 64	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑁 ∈ (1...𝑁))
66	59, 45, 60, 61	fzto1st 33187	. . . . . . . . . . . . . . 15 ⊢ (𝑁 ∈ (1...𝑁) → 𝑆 ∈ (Base‘(SymGrp‘(1...𝑁))))
67	65, 66	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑆 ∈ (Base‘(SymGrp‘(1...𝑁))))
68		eqid 2737	. . . . . . . . . . . . . . 15 ⊢ (inv_g‘(SymGrp‘(1...𝑁))) = (inv_g‘(SymGrp‘(1...𝑁)))
69	60, 61, 68	symginv 19335	. . . . . . . . . . . . . 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 19333	. . . . . . . . . . . . . . 15 ⊢ ((1...𝑁) ∈ Fin → (SymGrp‘(1...𝑁)) ∈ Grp)
72	58, 71	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (SymGrp‘(1...𝑁)) ∈ Grp)
73	61, 68	grpinvcl 18921	. . . . . . . . . . . . . 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 585	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((inv_g‘(SymGrp‘(1...𝑁)))‘𝑆) ∈ (Base‘(SymGrp‘(1...𝑁))))
75	70, 74	eqeltrrd 2838	. . . . . . . . . . . 12 ⊢ (𝜑 → ^◡𝑆 ∈ (Base‘(SymGrp‘(1...𝑁))))
76		eqid 2737	. . . . . . . . . . . . . 14 ⊢ (+_g‘(SymGrp‘(1...𝑁))) = (+_g‘(SymGrp‘(1...𝑁)))
77	60, 61, 76	symgov 19317	. . . . . . . . . . . . 13 ⊢ ((𝑃 ∈ (Base‘(SymGrp‘(1...𝑁))) ∧ ^◡𝑆 ∈ (Base‘(SymGrp‘(1...𝑁)))) → (𝑃(+_g‘(SymGrp‘(1...𝑁)))^◡𝑆) = (𝑃 ∘ ^◡𝑆))
78	60, 61, 76	symgcl 19318	. . . . . . . . . . . . 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 2838	. . . . . . . . . . . 12 ⊢ ((𝑃 ∈ (Base‘(SymGrp‘(1...𝑁))) ∧ ^◡𝑆 ∈ (Base‘(SymGrp‘(1...𝑁)))) → (𝑃 ∘ ^◡𝑆) ∈ (Base‘(SymGrp‘(1...𝑁))))
80	63, 75, 79	syl2anc 585	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑃 ∘ ^◡𝑆) ∈ (Base‘(SymGrp‘(1...𝑁))))
81	80	3ad2ant1 1134	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) → (𝑃 ∘ ^◡𝑆) ∈ (Base‘(SymGrp‘(1...𝑁))))
82		simp2 1138	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) → 𝑖 ∈ (1...𝑁))
83	60, 61	symgfv 19313	. . . . . . . . . 10 ⊢ (((𝑃 ∘ ^◡𝑆) ∈ (Base‘(SymGrp‘(1...𝑁))) ∧ 𝑖 ∈ (1...𝑁)) → ((𝑃 ∘ ^◡𝑆)‘𝑖) ∈ (1...𝑁))
84	81, 82, 83	syl2anc 585	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) → ((𝑃 ∘ ^◡𝑆)‘𝑖) ∈ (1...𝑁))
85	59, 48, 60, 61	fzto1st 33187	. . . . . . . . . . . . 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 33187	. . . . . . . . . . . . . . 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 19335	. . . . . . . . . . . . . 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 18921	. . . . . . . . . . . . . 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 585	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((inv_g‘(SymGrp‘(1...𝑁)))‘𝑇) ∈ (Base‘(SymGrp‘(1...𝑁))))
93	90, 92	eqeltrrd 2838	. . . . . . . . . . . 12 ⊢ (𝜑 → ^◡𝑇 ∈ (Base‘(SymGrp‘(1...𝑁))))
94	60, 61, 76	symgov 19317	. . . . . . . . . . . . 13 ⊢ ((𝑄 ∈ (Base‘(SymGrp‘(1...𝑁))) ∧ ^◡𝑇 ∈ (Base‘(SymGrp‘(1...𝑁)))) → (𝑄(+_g‘(SymGrp‘(1...𝑁)))^◡𝑇) = (𝑄 ∘ ^◡𝑇))
95	60, 61, 76	symgcl 19318	. . . . . . . . . . . . 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 2838	. . . . . . . . . . . 12 ⊢ ((𝑄 ∈ (Base‘(SymGrp‘(1...𝑁))) ∧ ^◡𝑇 ∈ (Base‘(SymGrp‘(1...𝑁)))) → (𝑄 ∘ ^◡𝑇) ∈ (Base‘(SymGrp‘(1...𝑁))))
97	86, 93, 96	syl2anc 585	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑄 ∘ ^◡𝑇) ∈ (Base‘(SymGrp‘(1...𝑁))))
98	97	3ad2ant1 1134	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) → (𝑄 ∘ ^◡𝑇) ∈ (Base‘(SymGrp‘(1...𝑁))))
99		simp3 1139	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) → 𝑗 ∈ (1...𝑁))
100	60, 61	symgfv 19313	. . . . . . . . . 10 ⊢ (((𝑄 ∘ ^◡𝑇) ∈ (Base‘(SymGrp‘(1...𝑁))) ∧ 𝑗 ∈ (1...𝑁)) → ((𝑄 ∘ ^◡𝑇)‘𝑗) ∈ (1...𝑁))
101	98, 99, 100	syl2anc 585	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) → ((𝑄 ∘ ^◡𝑇)‘𝑗) ∈ (1...𝑁))
102	24	3ad2ant1 1134	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) → 𝑈 ∈ 𝐵)
103	25, 26, 27, 84, 101, 102	matecld 22374	. . . . . . . 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 22371	. . . . . . 7 ⊢ (𝜑 → (𝑖 ∈ (1...𝑁), 𝑗 ∈ (1...𝑁) ↦ (((𝑃 ∘ ^◡𝑆)‘𝑖)𝑈((𝑄 ∘ ^◡𝑇)‘𝑗))) ∈ 𝐵)
105	14, 104	eqeltrid 2841	. . . . . 6 ⊢ (𝜑 → 𝑊 ∈ 𝐵)
106	25, 27	submatres 33965	. . . . . 6 ⊢ ((𝑁 ∈ ℕ ∧ 𝑊 ∈ 𝐵) → (𝑁(subMat1‘𝑊)𝑁) = (𝑊 ↾ ((1...(𝑁 − 1)) × (1...(𝑁 − 1)))))
107	1, 105, 106	syl2anc 585	. . . . 5 ⊢ (𝜑 → (𝑁(subMat1‘𝑊)𝑁) = (𝑊 ↾ ((1...(𝑁 − 1)) × (1...(𝑁 − 1)))))
108		eqid 2737	. . . . . 6 ⊢ (𝑁(subMat1‘𝑊)𝑁) = (𝑁(subMat1‘𝑊)𝑁)
109	25, 27, 56, 108, 1, 65, 65, 105	smatcl 33961	. . . . 5 ⊢ (𝜑 → (𝑁(subMat1‘𝑊)𝑁) ∈ (Base‘((1...(𝑁 − 1)) Mat 𝑅)))
110	107, 109	eqeltrrd 2838	. . . 4 ⊢ (𝜑 → (𝑊 ↾ ((1...(𝑁 − 1)) × (1...(𝑁 − 1)))) ∈ (Base‘((1...(𝑁 − 1)) Mat 𝑅)))
111		eqid 2737	. . . . 5 ⊢ ((1...(𝑁 − 1)) Mat 𝑅) = ((1...(𝑁 − 1)) Mat 𝑅)
112	111, 56	eqmat 22372	. . . 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 585	. . 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 2775	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 1087 = wceq 1542 ∈ wcel 2114 ∀wral 3052 Vcvv 3441 ∖ cdif 3899 ⊆ wss 3902 ifcif 4480 {csn 4581 class class class wbr 5099 ↦ cmpt 5180 × cxp 5623 ^◡ccnv 5624 ↾ cres 5627 ∘ ccom 5629 ‘cfv 6493 (class class class)co 7360 ∈ cmpo 7362 ↑_m cmap 8767 Fincfn 8887 1c1 11031 + caddc 11033 < clt 11170 ≤ cle 11171 − cmin 11368 ℕcn 12149 ℤ_≥cuz 12755 ...cfz 13427 Basecbs 17140 +_gcplusg 17181 ._rcmulr 17182 Grpcgrp 18867 inv_gcminusg 18868 SymGrpcsymg 19302 CRingccrg 20173 ℤRHomczrh 21458 Mat cmat 22355 maDet cmdat 22532 maAdju cmadu 22580 subMat1csmat 33952

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-rep 5225 ax-sep 5242 ax-nul 5252 ax-pow 5311 ax-pr 5378 ax-un 7682 ax-cnex 11086 ax-resscn 11087 ax-1cn 11088 ax-icn 11089 ax-addcl 11090 ax-addrcl 11091 ax-mulcl 11092 ax-mulrcl 11093 ax-mulcom 11094 ax-addass 11095 ax-mulass 11096 ax-distr 11097 ax-i2m1 11098 ax-1ne0 11099 ax-1rid 11100 ax-rnegex 11101 ax-rrecex 11102 ax-cnre 11103 ax-pre-lttri 11104 ax-pre-lttrn 11105 ax-pre-ltadd 11106 ax-pre-mulgt0 11107

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3062 df-rmo 3351 df-reu 3352 df-rab 3401 df-v 3443 df-sbc 3742 df-csb 3851 df-dif 3905 df-un 3907 df-in 3909 df-ss 3919 df-pss 3922 df-nul 4287 df-if 4481 df-pw 4557 df-sn 4582 df-pr 4584 df-tp 4586 df-op 4588 df-ot 4590 df-uni 4865 df-iun 4949 df-br 5100 df-opab 5162 df-mpt 5181 df-tr 5207 df-id 5520 df-eprel 5525 df-po 5533 df-so 5534 df-fr 5578 df-we 5580 df-xp 5631 df-rel 5632 df-cnv 5633 df-co 5634 df-dm 5635 df-rn 5636 df-res 5637 df-ima 5638 df-pred 6260 df-ord 6321 df-on 6322 df-lim 6323 df-suc 6324 df-iota 6449 df-fun 6495 df-fn 6496 df-f 6497 df-f1 6498 df-fo 6499 df-f1o 6500 df-fv 6501 df-riota 7317 df-ov 7363 df-oprab 7364 df-mpo 7365 df-om 7811 df-1st 7935 df-2nd 7936 df-supp 8105 df-frecs 8225 df-wrecs 8256 df-recs 8305 df-rdg 8343 df-1o 8399 df-2o 8400 df-er 8637 df-map 8769 df-ixp 8840 df-en 8888 df-dom 8889 df-sdom 8890 df-fin 8891 df-fsupp 9269 df-sup 9349 df-pnf 11172 df-mnf 11173 df-xr 11174 df-ltxr 11175 df-le 11176 df-sub 11370 df-neg 11371 df-nn 12150 df-2 12212 df-3 12213 df-4 12214 df-5 12215 df-6 12216 df-7 12217 df-8 12218 df-9 12219 df-n0 12406 df-z 12493 df-dec 12612 df-uz 12756 df-rp 12910 df-fz 13428 df-fzo 13575 df-struct 17078 df-sets 17095 df-slot 17113 df-ndx 17125 df-base 17141 df-ress 17162 df-plusg 17194 df-mulr 17195 df-sca 17197 df-vsca 17198 df-ip 17199 df-tset 17200 df-ple 17201 df-ds 17203 df-hom 17205 df-cco 17206 df-0g 17365 df-prds 17371 df-pws 17373 df-mgm 18569 df-sgrp 18648 df-mnd 18664 df-submnd 18713 df-efmnd 18798 df-grp 18870 df-minusg 18871 df-symg 19303 df-pmtr 19375 df-sra 21129 df-rgmod 21130 df-dsmm 21691 df-frlm 21706 df-mat 22356 df-subma 22525 df-smat 33953

This theorem is referenced by: madjusmdetlem4 33989

W3C validator