madjusmdetlem3 - Mathbox for Thierry Arnoux

	Mathbox for Thierry Arnoux		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > Mathboxes > madjusmdetlem3		Structured version Visualization version GIF version

Theorem madjusmdetlem3 31182

Description: Lemma for madjusmdet 31184. (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 12269	. . . . . . . . . . 11 ⊢ ℕ = (ℤ_≥‘1)
3	1, 2	eleqtrdi 2900	. . . . . . . . . 10 ⊢ (𝜑 → 𝑁 ∈ (ℤ_≥‘1))
4		fzdif2 30540	. . . . . . . . . 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 4059	. . . . . . . . 9 ⊢ ((1...𝑁) ∖ {𝑁}) ⊆ (1...𝑁)
7	5, 6	eqsstrrdi 3970	. . . . . . . 8 ⊢ (𝜑 → (1...(𝑁 − 1)) ⊆ (1...𝑁))
8	7	adantr 484	. . . . . . 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 3916	. . . . . 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 3916	. . . . . 6 ⊢ ((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) → 𝑗 ∈ (1...𝑁))
13		ovexd 7170	. . . . . 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 7276	. . . . . 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 1368	. . . . 5 ⊢ ((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) → (𝑖𝑊𝑗) = (((𝑃 ∘ ^◡𝑆)‘𝑖)𝑈((𝑄 ∘ ^◡𝑇)‘𝑗)))
17	9, 11	ovresd 7295	. . . . 5 ⊢ ((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) → (𝑖(𝑊 ↾ ((1...(𝑁 − 1)) × (1...(𝑁 − 1))))𝑗) = (𝑖𝑊𝑗))
18		eqid 2798	. . . . . . 7 ⊢ (𝐼(subMat1‘𝑈)𝐽) = (𝐼(subMat1‘𝑈)𝐽)
19	1	adantr 484	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) → 𝑁 ∈ ℕ)
20		madjusmdet.i	. . . . . . . 8 ⊢ (𝜑 → 𝐼 ∈ (1...𝑁))
21	20	adantr 484	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) → 𝐼 ∈ (1...𝑁))
22		madjusmdet.j	. . . . . . . 8 ⊢ (𝜑 → 𝐽 ∈ (1...𝑁))
23	22	adantr 484	. . . . . . 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 2798	. . . . . . . . . 10 ⊢ (Base‘𝑅) = (Base‘𝑅)
27		madjusmdet.b	. . . . . . . . . 10 ⊢ 𝐵 = (Base‘𝐴)
28	25, 26, 27	matbas2i 21027	. . . . . . . . 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 484	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) → 𝑈 ∈ ((Base‘𝑅) ↑_m ((1...𝑁) × (1...𝑁))))
31		fz1ssnn 12933	. . . . . . . 8 ⊢ (1...𝑁) ⊆ ℕ
32	31, 10	sseldi 3913	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) → 𝑖 ∈ ℕ)
33	31, 12	sseldi 3913	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) → 𝑗 ∈ ℕ)
34		eqidd 2799	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) → if(𝑖 < 𝐼, 𝑖, (𝑖 + 1)) = if(𝑖 < 𝐼, 𝑖, (𝑖 + 1)))
35		eqidd 2799	. . . . . . 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 31150	. . . . . 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 31181	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...(𝑁 − 1))) → if(𝑖 < 𝐼, 𝑖, (𝑖 + 1)) = ((𝑃 ∘ ^◡𝑆)‘𝑖))
47	9, 46	syldan 594	. . . . . . 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 31181	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ (1...(𝑁 − 1))) → if(𝑗 < 𝐽, 𝑗, (𝑗 + 1)) = ((𝑄 ∘ ^◡𝑇)‘𝑗))
51	11, 50	syldan 594	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) → if(𝑗 < 𝐽, 𝑗, (𝑗 + 1)) = ((𝑄 ∘ ^◡𝑇)‘𝑗))
52	47, 51	oveq12d 7153	. . . . . 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 2833	. . . . 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 2844	. . . 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 3156	. . 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 2798	. . . . 5 ⊢ (Base‘((1...(𝑁 − 1)) Mat 𝑅)) = (Base‘((1...(𝑁 − 1)) Mat 𝑅))
57	25, 27, 56, 18, 1, 20, 22, 24	smatcl 31155	. . . 4 ⊢ (𝜑 → (𝐼(subMat1‘𝑈)𝐽) ∈ (Base‘((1...(𝑁 − 1)) Mat 𝑅)))
58		fzfid 13336	. . . . . . . 8 ⊢ (𝜑 → (1...𝑁) ∈ Fin)
59		eqid 2798	. . . . . . . . . . . . . 14 ⊢ (1...𝑁) = (1...𝑁)
60		eqid 2798	. . . . . . . . . . . . . 14 ⊢ (SymGrp‘(1...𝑁)) = (SymGrp‘(1...𝑁))
61		eqid 2798	. . . . . . . . . . . . . 14 ⊢ (Base‘(SymGrp‘(1...𝑁))) = (Base‘(SymGrp‘(1...𝑁)))
62	59, 44, 60, 61	fzto1st 30795	. . . . . . . . . . . . 13 ⊢ (𝐼 ∈ (1...𝑁) → 𝑃 ∈ (Base‘(SymGrp‘(1...𝑁))))
63	20, 62	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑃 ∈ (Base‘(SymGrp‘(1...𝑁))))
64		eluzfz2 12910	. . . . . . . . . . . . . . . 16 ⊢ (𝑁 ∈ (ℤ_≥‘1) → 𝑁 ∈ (1...𝑁))
65	3, 64	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑁 ∈ (1...𝑁))
66	59, 45, 60, 61	fzto1st 30795	. . . . . . . . . . . . . . 15 ⊢ (𝑁 ∈ (1...𝑁) → 𝑆 ∈ (Base‘(SymGrp‘(1...𝑁))))
67	65, 66	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑆 ∈ (Base‘(SymGrp‘(1...𝑁))))
68		eqid 2798	. . . . . . . . . . . . . . 15 ⊢ (inv_g‘(SymGrp‘(1...𝑁))) = (inv_g‘(SymGrp‘(1...𝑁)))
69	60, 61, 68	symginv 18522	. . . . . . . . . . . . . 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 18520	. . . . . . . . . . . . . . 15 ⊢ ((1...𝑁) ∈ Fin → (SymGrp‘(1...𝑁)) ∈ Grp)
72	58, 71	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (SymGrp‘(1...𝑁)) ∈ Grp)
73	61, 68	grpinvcl 18143	. . . . . . . . . . . . . 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 587	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((inv_g‘(SymGrp‘(1...𝑁)))‘𝑆) ∈ (Base‘(SymGrp‘(1...𝑁))))
75	70, 74	eqeltrrd 2891	. . . . . . . . . . . 12 ⊢ (𝜑 → ^◡𝑆 ∈ (Base‘(SymGrp‘(1...𝑁))))
76		eqid 2798	. . . . . . . . . . . . . 14 ⊢ (+_g‘(SymGrp‘(1...𝑁))) = (+_g‘(SymGrp‘(1...𝑁)))
77	60, 61, 76	symgov 18504	. . . . . . . . . . . . 13 ⊢ ((𝑃 ∈ (Base‘(SymGrp‘(1...𝑁))) ∧ ^◡𝑆 ∈ (Base‘(SymGrp‘(1...𝑁)))) → (𝑃(+_g‘(SymGrp‘(1...𝑁)))^◡𝑆) = (𝑃 ∘ ^◡𝑆))
78	60, 61, 76	symgcl 18505	. . . . . . . . . . . . 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 2891	. . . . . . . . . . . 12 ⊢ ((𝑃 ∈ (Base‘(SymGrp‘(1...𝑁))) ∧ ^◡𝑆 ∈ (Base‘(SymGrp‘(1...𝑁)))) → (𝑃 ∘ ^◡𝑆) ∈ (Base‘(SymGrp‘(1...𝑁))))
80	63, 75, 79	syl2anc 587	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑃 ∘ ^◡𝑆) ∈ (Base‘(SymGrp‘(1...𝑁))))
81	80	3ad2ant1 1130	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) → (𝑃 ∘ ^◡𝑆) ∈ (Base‘(SymGrp‘(1...𝑁))))
82		simp2 1134	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) → 𝑖 ∈ (1...𝑁))
83	60, 61	symgfv 18500	. . . . . . . . . 10 ⊢ (((𝑃 ∘ ^◡𝑆) ∈ (Base‘(SymGrp‘(1...𝑁))) ∧ 𝑖 ∈ (1...𝑁)) → ((𝑃 ∘ ^◡𝑆)‘𝑖) ∈ (1...𝑁))
84	81, 82, 83	syl2anc 587	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) → ((𝑃 ∘ ^◡𝑆)‘𝑖) ∈ (1...𝑁))
85	59, 48, 60, 61	fzto1st 30795	. . . . . . . . . . . . 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 30795	. . . . . . . . . . . . . . 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 18522	. . . . . . . . . . . . . 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 18143	. . . . . . . . . . . . . 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 587	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((inv_g‘(SymGrp‘(1...𝑁)))‘𝑇) ∈ (Base‘(SymGrp‘(1...𝑁))))
93	90, 92	eqeltrrd 2891	. . . . . . . . . . . 12 ⊢ (𝜑 → ^◡𝑇 ∈ (Base‘(SymGrp‘(1...𝑁))))
94	60, 61, 76	symgov 18504	. . . . . . . . . . . . 13 ⊢ ((𝑄 ∈ (Base‘(SymGrp‘(1...𝑁))) ∧ ^◡𝑇 ∈ (Base‘(SymGrp‘(1...𝑁)))) → (𝑄(+_g‘(SymGrp‘(1...𝑁)))^◡𝑇) = (𝑄 ∘ ^◡𝑇))
95	60, 61, 76	symgcl 18505	. . . . . . . . . . . . 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 2891	. . . . . . . . . . . 12 ⊢ ((𝑄 ∈ (Base‘(SymGrp‘(1...𝑁))) ∧ ^◡𝑇 ∈ (Base‘(SymGrp‘(1...𝑁)))) → (𝑄 ∘ ^◡𝑇) ∈ (Base‘(SymGrp‘(1...𝑁))))
97	86, 93, 96	syl2anc 587	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑄 ∘ ^◡𝑇) ∈ (Base‘(SymGrp‘(1...𝑁))))
98	97	3ad2ant1 1130	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) → (𝑄 ∘ ^◡𝑇) ∈ (Base‘(SymGrp‘(1...𝑁))))
99		simp3 1135	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) → 𝑗 ∈ (1...𝑁))
100	60, 61	symgfv 18500	. . . . . . . . . 10 ⊢ (((𝑄 ∘ ^◡𝑇) ∈ (Base‘(SymGrp‘(1...𝑁))) ∧ 𝑗 ∈ (1...𝑁)) → ((𝑄 ∘ ^◡𝑇)‘𝑗) ∈ (1...𝑁))
101	98, 99, 100	syl2anc 587	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) → ((𝑄 ∘ ^◡𝑇)‘𝑗) ∈ (1...𝑁))
102	24	3ad2ant1 1130	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) → 𝑈 ∈ 𝐵)
103	25, 26, 27, 84, 101, 102	matecld 21031	. . . . . . . 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 21028	. . . . . . 7 ⊢ (𝜑 → (𝑖 ∈ (1...𝑁), 𝑗 ∈ (1...𝑁) ↦ (((𝑃 ∘ ^◡𝑆)‘𝑖)𝑈((𝑄 ∘ ^◡𝑇)‘𝑗))) ∈ 𝐵)
105	14, 104	eqeltrid 2894	. . . . . 6 ⊢ (𝜑 → 𝑊 ∈ 𝐵)
106	25, 27	submatres 31159	. . . . . 6 ⊢ ((𝑁 ∈ ℕ ∧ 𝑊 ∈ 𝐵) → (𝑁(subMat1‘𝑊)𝑁) = (𝑊 ↾ ((1...(𝑁 − 1)) × (1...(𝑁 − 1)))))
107	1, 105, 106	syl2anc 587	. . . . 5 ⊢ (𝜑 → (𝑁(subMat1‘𝑊)𝑁) = (𝑊 ↾ ((1...(𝑁 − 1)) × (1...(𝑁 − 1)))))
108		eqid 2798	. . . . . 6 ⊢ (𝑁(subMat1‘𝑊)𝑁) = (𝑁(subMat1‘𝑊)𝑁)
109	25, 27, 56, 108, 1, 65, 65, 105	smatcl 31155	. . . . 5 ⊢ (𝜑 → (𝑁(subMat1‘𝑊)𝑁) ∈ (Base‘((1...(𝑁 − 1)) Mat 𝑅)))
110	107, 109	eqeltrrd 2891	. . . 4 ⊢ (𝜑 → (𝑊 ↾ ((1...(𝑁 − 1)) × (1...(𝑁 − 1)))) ∈ (Base‘((1...(𝑁 − 1)) Mat 𝑅)))
111		eqid 2798	. . . . 5 ⊢ ((1...(𝑁 − 1)) Mat 𝑅) = ((1...(𝑁 − 1)) Mat 𝑅)
112	111, 56	eqmat 21029	. . . 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 587	. . 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 260	. 2 ⊢ (𝜑 → (𝐼(subMat1‘𝑈)𝐽) = (𝑊 ↾ ((1...(𝑁 − 1)) × (1...(𝑁 − 1)))))
115	114, 107	eqtr4d 2836	1 ⊢ (𝜑 → (𝐼(subMat1‘𝑈)𝐽) = (𝑁(subMat1‘𝑊)𝑁))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 ∧ w3a 1084 = wceq 1538 ∈ wcel 2111 ∀wral 3106 Vcvv 3441 ∖ cdif 3878 ⊆ wss 3881 ifcif 4425 {csn 4525 class class class wbr 5030 ↦ cmpt 5110 × cxp 5517 ^◡ccnv 5518 ↾ cres 5521 ∘ ccom 5523 ‘cfv 6324 (class class class)co 7135 ∈ cmpo 7137 ↑_m cmap 8389 Fincfn 8492 1c1 10527 + caddc 10529 < clt 10664 ≤ cle 10665 − cmin 10859 ℕcn 11625 ℤ_≥cuz 12231 ...cfz 12885 Basecbs 16475 +_gcplusg 16557 ._rcmulr 16558 Grpcgrp 18095 inv_gcminusg 18096 SymGrpcsymg 18487 CRingccrg 19291 ℤRHomczrh 20193 Mat cmat 21012 maDet cmdat 21189 maAdju cmadu 21237 subMat1csmat 31146

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 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603

This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-ot 4534 df-uni 4801 df-int 4839 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-1st 7671 df-2nd 7672 df-supp 7814 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-1o 8085 df-2o 8086 df-oadd 8089 df-er 8272 df-map 8391 df-ixp 8445 df-en 8493 df-dom 8494 df-sdom 8495 df-fin 8496 df-fsupp 8818 df-sup 8890 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-nn 11626 df-2 11688 df-3 11689 df-4 11690 df-5 11691 df-6 11692 df-7 11693 df-8 11694 df-9 11695 df-n0 11886 df-z 11970 df-dec 12087 df-uz 12232 df-rp 12378 df-fz 12886 df-fzo 13029 df-struct 16477 df-ndx 16478 df-slot 16479 df-base 16481 df-sets 16482 df-ress 16483 df-plusg 16570 df-mulr 16571 df-sca 16573 df-vsca 16574 df-ip 16575 df-tset 16576 df-ple 16577 df-ds 16579 df-hom 16581 df-cco 16582 df-0g 16707 df-prds 16713 df-pws 16715 df-mgm 17844 df-sgrp 17893 df-mnd 17904 df-submnd 17949 df-efmnd 18026 df-grp 18098 df-minusg 18099 df-symg 18488 df-pmtr 18562 df-sra 19937 df-rgmod 19938 df-dsmm 20421 df-frlm 20436 df-mat 21013 df-subma 21182 df-smat 31147

This theorem is referenced by: madjusmdetlem4 31183

W3C validator