madjusmdetlem4 - Mathbox for Thierry Arnoux

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Theorem madjusmdetlem4 33829

Description: Lemma for madjusmdet 33830. (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)
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)))

**Assertion**
Ref	Expression
madjusmdetlem4	⊢ (𝜑 → (𝐽(𝐾‘𝑀)𝐼) = ((𝑍‘(-1↑(𝐼 + 𝐽))) · (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 𝑆,𝑖,𝑗 𝑇,𝑖,𝑗 Allowed substitution hints: 𝐴(𝑖,𝑗) 𝐷(𝑖,𝑗) · (𝑖,𝑗) 𝐸(𝑖,𝑗) 𝐾(𝑖,𝑗) 𝑍(𝑖,𝑗)

Proof of Theorem madjusmdetlem4 Dummy variables 𝑘 𝑙 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		madjusmdet.b	. . 3 ⊢ 𝐵 = (Base‘𝐴)
2		madjusmdet.a	. . 3 ⊢ 𝐴 = ((1...𝑁) Mat 𝑅)
3		madjusmdet.d	. . 3 ⊢ 𝐷 = ((1...𝑁) maDet 𝑅)
4		madjusmdet.k	. . 3 ⊢ 𝐾 = ((1...𝑁) maAdju 𝑅)
5		madjusmdet.t	. . 3 ⊢ · = (._r‘𝑅)
6		madjusmdet.z	. . 3 ⊢ 𝑍 = (ℤRHom‘𝑅)
7		madjusmdet.e	. . 3 ⊢ 𝐸 = ((1...(𝑁 − 1)) maDet 𝑅)
8		madjusmdet.n	. . 3 ⊢ (𝜑 → 𝑁 ∈ ℕ)
9		madjusmdet.r	. . 3 ⊢ (𝜑 → 𝑅 ∈ CRing)
10		madjusmdet.i	. . 3 ⊢ (𝜑 → 𝐼 ∈ (1...𝑁))
11		madjusmdet.j	. . 3 ⊢ (𝜑 → 𝐽 ∈ (1...𝑁))
12		madjusmdet.m	. . 3 ⊢ (𝜑 → 𝑀 ∈ 𝐵)
13		eqid 2737	. . 3 ⊢ (Base‘(SymGrp‘(1...𝑁))) = (Base‘(SymGrp‘(1...𝑁)))
14		eqid 2737	. . 3 ⊢ (pmSgn‘(1...𝑁)) = (pmSgn‘(1...𝑁))
15		eqid 2737	. . 3 ⊢ (𝐼(((1...𝑁) minMatR1 𝑅)‘𝑀)𝐽) = (𝐼(((1...𝑁) minMatR1 𝑅)‘𝑀)𝐽)
16		fveq2 6906	. . . . 5 ⊢ (𝑘 = 𝑖 → ((𝑃 ∘ ^◡𝑆)‘𝑘) = ((𝑃 ∘ ^◡𝑆)‘𝑖))
17	16	oveq1d 7446	. . . 4 ⊢ (𝑘 = 𝑖 → (((𝑃 ∘ ^◡𝑆)‘𝑘)(𝐼(((1...𝑁) minMatR1 𝑅)‘𝑀)𝐽)((𝑄 ∘ ^◡𝑇)‘𝑙)) = (((𝑃 ∘ ^◡𝑆)‘𝑖)(𝐼(((1...𝑁) minMatR1 𝑅)‘𝑀)𝐽)((𝑄 ∘ ^◡𝑇)‘𝑙)))
18		fveq2 6906	. . . . 5 ⊢ (𝑙 = 𝑗 → ((𝑄 ∘ ^◡𝑇)‘𝑙) = ((𝑄 ∘ ^◡𝑇)‘𝑗))
19	18	oveq2d 7447	. . . 4 ⊢ (𝑙 = 𝑗 → (((𝑃 ∘ ^◡𝑆)‘𝑖)(𝐼(((1...𝑁) minMatR1 𝑅)‘𝑀)𝐽)((𝑄 ∘ ^◡𝑇)‘𝑙)) = (((𝑃 ∘ ^◡𝑆)‘𝑖)(𝐼(((1...𝑁) minMatR1 𝑅)‘𝑀)𝐽)((𝑄 ∘ ^◡𝑇)‘𝑗)))
20	17, 19	cbvmpov 7528	. . 3 ⊢ (𝑘 ∈ (1...𝑁), 𝑙 ∈ (1...𝑁) ↦ (((𝑃 ∘ ^◡𝑆)‘𝑘)(𝐼(((1...𝑁) minMatR1 𝑅)‘𝑀)𝐽)((𝑄 ∘ ^◡𝑇)‘𝑙))) = (𝑖 ∈ (1...𝑁), 𝑗 ∈ (1...𝑁) ↦ (((𝑃 ∘ ^◡𝑆)‘𝑖)(𝐼(((1...𝑁) minMatR1 𝑅)‘𝑀)𝐽)((𝑄 ∘ ^◡𝑇)‘𝑗)))
21		eqid 2737	. . . . . 6 ⊢ (1...𝑁) = (1...𝑁)
22		madjusmdetlem2.p	. . . . . 6 ⊢ 𝑃 = (𝑖 ∈ (1...𝑁) ↦ if(𝑖 = 1, 𝐼, if(𝑖 ≤ 𝐼, (𝑖 − 1), 𝑖)))
23		eqid 2737	. . . . . 6 ⊢ (SymGrp‘(1...𝑁)) = (SymGrp‘(1...𝑁))
24	21, 22, 23, 13	fzto1st 33123	. . . . 5 ⊢ (𝐼 ∈ (1...𝑁) → 𝑃 ∈ (Base‘(SymGrp‘(1...𝑁))))
25	10, 24	syl 17	. . . 4 ⊢ (𝜑 → 𝑃 ∈ (Base‘(SymGrp‘(1...𝑁))))
26		nnuz 12921	. . . . . . . . 9 ⊢ ℕ = (ℤ_≥‘1)
27	8, 26	eleqtrdi 2851	. . . . . . . 8 ⊢ (𝜑 → 𝑁 ∈ (ℤ_≥‘1))
28		eluzfz2 13572	. . . . . . . 8 ⊢ (𝑁 ∈ (ℤ_≥‘1) → 𝑁 ∈ (1...𝑁))
29	27, 28	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝑁 ∈ (1...𝑁))
30		madjusmdetlem2.s	. . . . . . . 8 ⊢ 𝑆 = (𝑖 ∈ (1...𝑁) ↦ if(𝑖 = 1, 𝑁, if(𝑖 ≤ 𝑁, (𝑖 − 1), 𝑖)))
31	21, 30, 23, 13	fzto1st 33123	. . . . . . 7 ⊢ (𝑁 ∈ (1...𝑁) → 𝑆 ∈ (Base‘(SymGrp‘(1...𝑁))))
32	29, 31	syl 17	. . . . . 6 ⊢ (𝜑 → 𝑆 ∈ (Base‘(SymGrp‘(1...𝑁))))
33		eqid 2737	. . . . . . 7 ⊢ (inv_g‘(SymGrp‘(1...𝑁))) = (inv_g‘(SymGrp‘(1...𝑁)))
34	23, 13, 33	symginv 19420	. . . . . 6 ⊢ (𝑆 ∈ (Base‘(SymGrp‘(1...𝑁))) → ((inv_g‘(SymGrp‘(1...𝑁)))‘𝑆) = ^◡𝑆)
35	32, 34	syl 17	. . . . 5 ⊢ (𝜑 → ((inv_g‘(SymGrp‘(1...𝑁)))‘𝑆) = ^◡𝑆)
36		fzfid 14014	. . . . . . 7 ⊢ (𝜑 → (1...𝑁) ∈ Fin)
37	23	symggrp 19418	. . . . . . 7 ⊢ ((1...𝑁) ∈ Fin → (SymGrp‘(1...𝑁)) ∈ Grp)
38	36, 37	syl 17	. . . . . 6 ⊢ (𝜑 → (SymGrp‘(1...𝑁)) ∈ Grp)
39	13, 33	grpinvcl 19005	. . . . . 6 ⊢ (((SymGrp‘(1...𝑁)) ∈ Grp ∧ 𝑆 ∈ (Base‘(SymGrp‘(1...𝑁)))) → ((inv_g‘(SymGrp‘(1...𝑁)))‘𝑆) ∈ (Base‘(SymGrp‘(1...𝑁))))
40	38, 32, 39	syl2anc 584	. . . . 5 ⊢ (𝜑 → ((inv_g‘(SymGrp‘(1...𝑁)))‘𝑆) ∈ (Base‘(SymGrp‘(1...𝑁))))
41	35, 40	eqeltrrd 2842	. . . 4 ⊢ (𝜑 → ^◡𝑆 ∈ (Base‘(SymGrp‘(1...𝑁))))
42		eqid 2737	. . . . . 6 ⊢ (+_g‘(SymGrp‘(1...𝑁))) = (+_g‘(SymGrp‘(1...𝑁)))
43	23, 13, 42	symgov 19401	. . . . 5 ⊢ ((𝑃 ∈ (Base‘(SymGrp‘(1...𝑁))) ∧ ^◡𝑆 ∈ (Base‘(SymGrp‘(1...𝑁)))) → (𝑃(+_g‘(SymGrp‘(1...𝑁)))^◡𝑆) = (𝑃 ∘ ^◡𝑆))
44	23, 13, 42	symgcl 19402	. . . . 5 ⊢ ((𝑃 ∈ (Base‘(SymGrp‘(1...𝑁))) ∧ ^◡𝑆 ∈ (Base‘(SymGrp‘(1...𝑁)))) → (𝑃(+_g‘(SymGrp‘(1...𝑁)))^◡𝑆) ∈ (Base‘(SymGrp‘(1...𝑁))))
45	43, 44	eqeltrrd 2842	. . . 4 ⊢ ((𝑃 ∈ (Base‘(SymGrp‘(1...𝑁))) ∧ ^◡𝑆 ∈ (Base‘(SymGrp‘(1...𝑁)))) → (𝑃 ∘ ^◡𝑆) ∈ (Base‘(SymGrp‘(1...𝑁))))
46	25, 41, 45	syl2anc 584	. . 3 ⊢ (𝜑 → (𝑃 ∘ ^◡𝑆) ∈ (Base‘(SymGrp‘(1...𝑁))))
47		madjusmdetlem4.q	. . . . . 6 ⊢ 𝑄 = (𝑗 ∈ (1...𝑁) ↦ if(𝑗 = 1, 𝐽, if(𝑗 ≤ 𝐽, (𝑗 − 1), 𝑗)))
48	21, 47, 23, 13	fzto1st 33123	. . . . 5 ⊢ (𝐽 ∈ (1...𝑁) → 𝑄 ∈ (Base‘(SymGrp‘(1...𝑁))))
49	11, 48	syl 17	. . . 4 ⊢ (𝜑 → 𝑄 ∈ (Base‘(SymGrp‘(1...𝑁))))
50		madjusmdetlem4.t	. . . . . . . 8 ⊢ 𝑇 = (𝑗 ∈ (1...𝑁) ↦ if(𝑗 = 1, 𝑁, if(𝑗 ≤ 𝑁, (𝑗 − 1), 𝑗)))
51	21, 50, 23, 13	fzto1st 33123	. . . . . . 7 ⊢ (𝑁 ∈ (1...𝑁) → 𝑇 ∈ (Base‘(SymGrp‘(1...𝑁))))
52	29, 51	syl 17	. . . . . 6 ⊢ (𝜑 → 𝑇 ∈ (Base‘(SymGrp‘(1...𝑁))))
53	23, 13, 33	symginv 19420	. . . . . 6 ⊢ (𝑇 ∈ (Base‘(SymGrp‘(1...𝑁))) → ((inv_g‘(SymGrp‘(1...𝑁)))‘𝑇) = ^◡𝑇)
54	52, 53	syl 17	. . . . 5 ⊢ (𝜑 → ((inv_g‘(SymGrp‘(1...𝑁)))‘𝑇) = ^◡𝑇)
55	13, 33	grpinvcl 19005	. . . . . 6 ⊢ (((SymGrp‘(1...𝑁)) ∈ Grp ∧ 𝑇 ∈ (Base‘(SymGrp‘(1...𝑁)))) → ((inv_g‘(SymGrp‘(1...𝑁)))‘𝑇) ∈ (Base‘(SymGrp‘(1...𝑁))))
56	38, 52, 55	syl2anc 584	. . . . 5 ⊢ (𝜑 → ((inv_g‘(SymGrp‘(1...𝑁)))‘𝑇) ∈ (Base‘(SymGrp‘(1...𝑁))))
57	54, 56	eqeltrrd 2842	. . . 4 ⊢ (𝜑 → ^◡𝑇 ∈ (Base‘(SymGrp‘(1...𝑁))))
58	23, 13, 42	symgov 19401	. . . . 5 ⊢ ((𝑄 ∈ (Base‘(SymGrp‘(1...𝑁))) ∧ ^◡𝑇 ∈ (Base‘(SymGrp‘(1...𝑁)))) → (𝑄(+_g‘(SymGrp‘(1...𝑁)))^◡𝑇) = (𝑄 ∘ ^◡𝑇))
59	23, 13, 42	symgcl 19402	. . . . 5 ⊢ ((𝑄 ∈ (Base‘(SymGrp‘(1...𝑁))) ∧ ^◡𝑇 ∈ (Base‘(SymGrp‘(1...𝑁)))) → (𝑄(+_g‘(SymGrp‘(1...𝑁)))^◡𝑇) ∈ (Base‘(SymGrp‘(1...𝑁))))
60	58, 59	eqeltrrd 2842	. . . 4 ⊢ ((𝑄 ∈ (Base‘(SymGrp‘(1...𝑁))) ∧ ^◡𝑇 ∈ (Base‘(SymGrp‘(1...𝑁)))) → (𝑄 ∘ ^◡𝑇) ∈ (Base‘(SymGrp‘(1...𝑁))))
61	49, 57, 60	syl2anc 584	. . 3 ⊢ (𝜑 → (𝑄 ∘ ^◡𝑇) ∈ (Base‘(SymGrp‘(1...𝑁))))
62	23, 13	symgbasf1o 19392	. . . . . . 7 ⊢ (𝑆 ∈ (Base‘(SymGrp‘(1...𝑁))) → 𝑆:(1...𝑁)–1-1-onto→(1...𝑁))
63	32, 62	syl 17	. . . . . 6 ⊢ (𝜑 → 𝑆:(1...𝑁)–1-1-onto→(1...𝑁))
64		f1of1 6847	. . . . . 6 ⊢ (𝑆:(1...𝑁)–1-1-onto→(1...𝑁) → 𝑆:(1...𝑁)–1-1→(1...𝑁))
65		df-f1 6566	. . . . . . 7 ⊢ (𝑆:(1...𝑁)–1-1→(1...𝑁) ↔ (𝑆:(1...𝑁)⟶(1...𝑁) ∧ Fun ^◡𝑆))
66	65	simprbi 496	. . . . . 6 ⊢ (𝑆:(1...𝑁)–1-1→(1...𝑁) → Fun ^◡𝑆)
67	63, 64, 66	3syl 18	. . . . 5 ⊢ (𝜑 → Fun ^◡𝑆)
68		f1ocnv 6860	. . . . . . 7 ⊢ (𝑆:(1...𝑁)–1-1-onto→(1...𝑁) → ^◡𝑆:(1...𝑁)–1-1-onto→(1...𝑁))
69		f1odm 6852	. . . . . . 7 ⊢ (^◡𝑆:(1...𝑁)–1-1-onto→(1...𝑁) → dom ^◡𝑆 = (1...𝑁))
70	63, 68, 69	3syl 18	. . . . . 6 ⊢ (𝜑 → dom ^◡𝑆 = (1...𝑁))
71	29, 70	eleqtrrd 2844	. . . . 5 ⊢ (𝜑 → 𝑁 ∈ dom ^◡𝑆)
72		fvco 7007	. . . . 5 ⊢ ((Fun ^◡𝑆 ∧ 𝑁 ∈ dom ^◡𝑆) → ((𝑃 ∘ ^◡𝑆)‘𝑁) = (𝑃‘(^◡𝑆‘𝑁)))
73	67, 71, 72	syl2anc 584	. . . 4 ⊢ (𝜑 → ((𝑃 ∘ ^◡𝑆)‘𝑁) = (𝑃‘(^◡𝑆‘𝑁)))
74	21, 30, 23, 13	fzto1stinvn 33124	. . . . . 6 ⊢ (𝑁 ∈ (1...𝑁) → (^◡𝑆‘𝑁) = 1)
75	29, 74	syl 17	. . . . 5 ⊢ (𝜑 → (^◡𝑆‘𝑁) = 1)
76	75	fveq2d 6910	. . . 4 ⊢ (𝜑 → (𝑃‘(^◡𝑆‘𝑁)) = (𝑃‘1))
77	21, 22	fzto1stfv1 33121	. . . . 5 ⊢ (𝐼 ∈ (1...𝑁) → (𝑃‘1) = 𝐼)
78	10, 77	syl 17	. . . 4 ⊢ (𝜑 → (𝑃‘1) = 𝐼)
79	73, 76, 78	3eqtrd 2781	. . 3 ⊢ (𝜑 → ((𝑃 ∘ ^◡𝑆)‘𝑁) = 𝐼)
80	23, 13	symgbasf1o 19392	. . . . . . 7 ⊢ (𝑇 ∈ (Base‘(SymGrp‘(1...𝑁))) → 𝑇:(1...𝑁)–1-1-onto→(1...𝑁))
81	52, 80	syl 17	. . . . . 6 ⊢ (𝜑 → 𝑇:(1...𝑁)–1-1-onto→(1...𝑁))
82		f1of1 6847	. . . . . 6 ⊢ (𝑇:(1...𝑁)–1-1-onto→(1...𝑁) → 𝑇:(1...𝑁)–1-1→(1...𝑁))
83		df-f1 6566	. . . . . . 7 ⊢ (𝑇:(1...𝑁)–1-1→(1...𝑁) ↔ (𝑇:(1...𝑁)⟶(1...𝑁) ∧ Fun ^◡𝑇))
84	83	simprbi 496	. . . . . 6 ⊢ (𝑇:(1...𝑁)–1-1→(1...𝑁) → Fun ^◡𝑇)
85	81, 82, 84	3syl 18	. . . . 5 ⊢ (𝜑 → Fun ^◡𝑇)
86		f1ocnv 6860	. . . . . . 7 ⊢ (𝑇:(1...𝑁)–1-1-onto→(1...𝑁) → ^◡𝑇:(1...𝑁)–1-1-onto→(1...𝑁))
87		f1odm 6852	. . . . . . 7 ⊢ (^◡𝑇:(1...𝑁)–1-1-onto→(1...𝑁) → dom ^◡𝑇 = (1...𝑁))
88	81, 86, 87	3syl 18	. . . . . 6 ⊢ (𝜑 → dom ^◡𝑇 = (1...𝑁))
89	29, 88	eleqtrrd 2844	. . . . 5 ⊢ (𝜑 → 𝑁 ∈ dom ^◡𝑇)
90		fvco 7007	. . . . 5 ⊢ ((Fun ^◡𝑇 ∧ 𝑁 ∈ dom ^◡𝑇) → ((𝑄 ∘ ^◡𝑇)‘𝑁) = (𝑄‘(^◡𝑇‘𝑁)))
91	85, 89, 90	syl2anc 584	. . . 4 ⊢ (𝜑 → ((𝑄 ∘ ^◡𝑇)‘𝑁) = (𝑄‘(^◡𝑇‘𝑁)))
92	21, 50, 23, 13	fzto1stinvn 33124	. . . . . 6 ⊢ (𝑁 ∈ (1...𝑁) → (^◡𝑇‘𝑁) = 1)
93	29, 92	syl 17	. . . . 5 ⊢ (𝜑 → (^◡𝑇‘𝑁) = 1)
94	93	fveq2d 6910	. . . 4 ⊢ (𝜑 → (𝑄‘(^◡𝑇‘𝑁)) = (𝑄‘1))
95	21, 47	fzto1stfv1 33121	. . . . 5 ⊢ (𝐽 ∈ (1...𝑁) → (𝑄‘1) = 𝐽)
96	11, 95	syl 17	. . . 4 ⊢ (𝜑 → (𝑄‘1) = 𝐽)
97	91, 94, 96	3eqtrd 2781	. . 3 ⊢ (𝜑 → ((𝑄 ∘ ^◡𝑇)‘𝑁) = 𝐽)
98		crngring 20242	. . . . . 6 ⊢ (𝑅 ∈ CRing → 𝑅 ∈ Ring)
99	9, 98	syl 17	. . . . 5 ⊢ (𝜑 → 𝑅 ∈ Ring)
100	2, 1	minmar1cl 22657	. . . . 5 ⊢ (((𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ (𝐼 ∈ (1...𝑁) ∧ 𝐽 ∈ (1...𝑁))) → (𝐼(((1...𝑁) minMatR1 𝑅)‘𝑀)𝐽) ∈ 𝐵)
101	99, 12, 10, 11, 100	syl22anc 839	. . . 4 ⊢ (𝜑 → (𝐼(((1...𝑁) minMatR1 𝑅)‘𝑀)𝐽) ∈ 𝐵)
102	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 22, 30, 47, 50, 20, 101	madjusmdetlem3 33828	. . 3 ⊢ (𝜑 → (𝐼(subMat1‘(𝐼(((1...𝑁) minMatR1 𝑅)‘𝑀)𝐽))𝐽) = (𝑁(subMat1‘(𝑘 ∈ (1...𝑁), 𝑙 ∈ (1...𝑁) ↦ (((𝑃 ∘ ^◡𝑆)‘𝑘)(𝐼(((1...𝑁) minMatR1 𝑅)‘𝑀)𝐽)((𝑄 ∘ ^◡𝑇)‘𝑙))))𝑁))
103	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 20, 46, 61, 79, 97, 102	madjusmdetlem1 33826	. 2 ⊢ (𝜑 → (𝐽(𝐾‘𝑀)𝐼) = ((𝑍‘(((pmSgn‘(1...𝑁))‘(𝑃 ∘ ^◡𝑆)) · ((pmSgn‘(1...𝑁))‘(𝑄 ∘ ^◡𝑇)))) · (𝐸‘(𝐼(subMat1‘𝑀)𝐽))))
104	23, 14, 13	psgnco 21601	. . . . . . . 8 ⊢ (((1...𝑁) ∈ Fin ∧ 𝑃 ∈ (Base‘(SymGrp‘(1...𝑁))) ∧ ^◡𝑆 ∈ (Base‘(SymGrp‘(1...𝑁)))) → ((pmSgn‘(1...𝑁))‘(𝑃 ∘ ^◡𝑆)) = (((pmSgn‘(1...𝑁))‘𝑃) · ((pmSgn‘(1...𝑁))‘^◡𝑆)))
105	36, 25, 41, 104	syl3anc 1373	. . . . . . 7 ⊢ (𝜑 → ((pmSgn‘(1...𝑁))‘(𝑃 ∘ ^◡𝑆)) = (((pmSgn‘(1...𝑁))‘𝑃) · ((pmSgn‘(1...𝑁))‘^◡𝑆)))
106	21, 22, 23, 13, 14	psgnfzto1st 33125	. . . . . . . . 9 ⊢ (𝐼 ∈ (1...𝑁) → ((pmSgn‘(1...𝑁))‘𝑃) = (-1↑(𝐼 + 1)))
107	10, 106	syl 17	. . . . . . . 8 ⊢ (𝜑 → ((pmSgn‘(1...𝑁))‘𝑃) = (-1↑(𝐼 + 1)))
108	23, 14, 13	psgninv 21600	. . . . . . . . . 10 ⊢ (((1...𝑁) ∈ Fin ∧ 𝑆 ∈ (Base‘(SymGrp‘(1...𝑁)))) → ((pmSgn‘(1...𝑁))‘^◡𝑆) = ((pmSgn‘(1...𝑁))‘𝑆))
109	36, 32, 108	syl2anc 584	. . . . . . . . 9 ⊢ (𝜑 → ((pmSgn‘(1...𝑁))‘^◡𝑆) = ((pmSgn‘(1...𝑁))‘𝑆))
110	21, 30, 23, 13, 14	psgnfzto1st 33125	. . . . . . . . . 10 ⊢ (𝑁 ∈ (1...𝑁) → ((pmSgn‘(1...𝑁))‘𝑆) = (-1↑(𝑁 + 1)))
111	29, 110	syl 17	. . . . . . . . 9 ⊢ (𝜑 → ((pmSgn‘(1...𝑁))‘𝑆) = (-1↑(𝑁 + 1)))
112	109, 111	eqtrd 2777	. . . . . . . 8 ⊢ (𝜑 → ((pmSgn‘(1...𝑁))‘^◡𝑆) = (-1↑(𝑁 + 1)))
113	107, 112	oveq12d 7449	. . . . . . 7 ⊢ (𝜑 → (((pmSgn‘(1...𝑁))‘𝑃) · ((pmSgn‘(1...𝑁))‘^◡𝑆)) = ((-1↑(𝐼 + 1)) · (-1↑(𝑁 + 1))))
114	105, 113	eqtrd 2777	. . . . . 6 ⊢ (𝜑 → ((pmSgn‘(1...𝑁))‘(𝑃 ∘ ^◡𝑆)) = ((-1↑(𝐼 + 1)) · (-1↑(𝑁 + 1))))
115	23, 14, 13	psgnco 21601	. . . . . . . 8 ⊢ (((1...𝑁) ∈ Fin ∧ 𝑄 ∈ (Base‘(SymGrp‘(1...𝑁))) ∧ ^◡𝑇 ∈ (Base‘(SymGrp‘(1...𝑁)))) → ((pmSgn‘(1...𝑁))‘(𝑄 ∘ ^◡𝑇)) = (((pmSgn‘(1...𝑁))‘𝑄) · ((pmSgn‘(1...𝑁))‘^◡𝑇)))
116	36, 49, 57, 115	syl3anc 1373	. . . . . . 7 ⊢ (𝜑 → ((pmSgn‘(1...𝑁))‘(𝑄 ∘ ^◡𝑇)) = (((pmSgn‘(1...𝑁))‘𝑄) · ((pmSgn‘(1...𝑁))‘^◡𝑇)))
117	21, 47, 23, 13, 14	psgnfzto1st 33125	. . . . . . . . 9 ⊢ (𝐽 ∈ (1...𝑁) → ((pmSgn‘(1...𝑁))‘𝑄) = (-1↑(𝐽 + 1)))
118	11, 117	syl 17	. . . . . . . 8 ⊢ (𝜑 → ((pmSgn‘(1...𝑁))‘𝑄) = (-1↑(𝐽 + 1)))
119	23, 14, 13	psgninv 21600	. . . . . . . . . 10 ⊢ (((1...𝑁) ∈ Fin ∧ 𝑇 ∈ (Base‘(SymGrp‘(1...𝑁)))) → ((pmSgn‘(1...𝑁))‘^◡𝑇) = ((pmSgn‘(1...𝑁))‘𝑇))
120	36, 52, 119	syl2anc 584	. . . . . . . . 9 ⊢ (𝜑 → ((pmSgn‘(1...𝑁))‘^◡𝑇) = ((pmSgn‘(1...𝑁))‘𝑇))
121	21, 50, 23, 13, 14	psgnfzto1st 33125	. . . . . . . . . 10 ⊢ (𝑁 ∈ (1...𝑁) → ((pmSgn‘(1...𝑁))‘𝑇) = (-1↑(𝑁 + 1)))
122	29, 121	syl 17	. . . . . . . . 9 ⊢ (𝜑 → ((pmSgn‘(1...𝑁))‘𝑇) = (-1↑(𝑁 + 1)))
123	120, 122	eqtrd 2777	. . . . . . . 8 ⊢ (𝜑 → ((pmSgn‘(1...𝑁))‘^◡𝑇) = (-1↑(𝑁 + 1)))
124	118, 123	oveq12d 7449	. . . . . . 7 ⊢ (𝜑 → (((pmSgn‘(1...𝑁))‘𝑄) · ((pmSgn‘(1...𝑁))‘^◡𝑇)) = ((-1↑(𝐽 + 1)) · (-1↑(𝑁 + 1))))
125	116, 124	eqtrd 2777	. . . . . 6 ⊢ (𝜑 → ((pmSgn‘(1...𝑁))‘(𝑄 ∘ ^◡𝑇)) = ((-1↑(𝐽 + 1)) · (-1↑(𝑁 + 1))))
126	114, 125	oveq12d 7449	. . . . 5 ⊢ (𝜑 → (((pmSgn‘(1...𝑁))‘(𝑃 ∘ ^◡𝑆)) · ((pmSgn‘(1...𝑁))‘(𝑄 ∘ ^◡𝑇))) = (((-1↑(𝐼 + 1)) · (-1↑(𝑁 + 1))) · ((-1↑(𝐽 + 1)) · (-1↑(𝑁 + 1)))))
127		1cnd 11256	. . . . . . . . 9 ⊢ (𝜑 → 1 ∈ ℂ)
128	127	negcld 11607	. . . . . . . 8 ⊢ (𝜑 → -1 ∈ ℂ)
129		fz1ssnn 13595	. . . . . . . . . . 11 ⊢ (1...𝑁) ⊆ ℕ
130	129, 10	sselid 3981	. . . . . . . . . 10 ⊢ (𝜑 → 𝐼 ∈ ℕ)
131	130	nnnn0d 12587	. . . . . . . . 9 ⊢ (𝜑 → 𝐼 ∈ ℕ₀)
132		1nn0 12542	. . . . . . . . . 10 ⊢ 1 ∈ ℕ₀
133	132	a1i 11	. . . . . . . . 9 ⊢ (𝜑 → 1 ∈ ℕ₀)
134	131, 133	nn0addcld 12591	. . . . . . . 8 ⊢ (𝜑 → (𝐼 + 1) ∈ ℕ₀)
135	128, 134	expcld 14186	. . . . . . 7 ⊢ (𝜑 → (-1↑(𝐼 + 1)) ∈ ℂ)
136	8	nnnn0d 12587	. . . . . . . . 9 ⊢ (𝜑 → 𝑁 ∈ ℕ₀)
137	136, 133	nn0addcld 12591	. . . . . . . 8 ⊢ (𝜑 → (𝑁 + 1) ∈ ℕ₀)
138	128, 137	expcld 14186	. . . . . . 7 ⊢ (𝜑 → (-1↑(𝑁 + 1)) ∈ ℂ)
139	129, 11	sselid 3981	. . . . . . . . . 10 ⊢ (𝜑 → 𝐽 ∈ ℕ)
140	139	nnnn0d 12587	. . . . . . . . 9 ⊢ (𝜑 → 𝐽 ∈ ℕ₀)
141	140, 133	nn0addcld 12591	. . . . . . . 8 ⊢ (𝜑 → (𝐽 + 1) ∈ ℕ₀)
142	128, 141	expcld 14186	. . . . . . 7 ⊢ (𝜑 → (-1↑(𝐽 + 1)) ∈ ℂ)
143	135, 138, 142, 138	mul4d 11473	. . . . . 6 ⊢ (𝜑 → (((-1↑(𝐼 + 1)) · (-1↑(𝑁 + 1))) · ((-1↑(𝐽 + 1)) · (-1↑(𝑁 + 1)))) = (((-1↑(𝐼 + 1)) · (-1↑(𝐽 + 1))) · ((-1↑(𝑁 + 1)) · (-1↑(𝑁 + 1)))))
144	128, 141, 134	expaddd 14188	. . . . . . . 8 ⊢ (𝜑 → (-1↑((𝐼 + 1) + (𝐽 + 1))) = ((-1↑(𝐼 + 1)) · (-1↑(𝐽 + 1))))
145	130	nncnd 12282	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐼 ∈ ℂ)
146	139	nncnd 12282	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐽 ∈ ℂ)
147	145, 127, 146, 127	add4d 11490	. . . . . . . . . . 11 ⊢ (𝜑 → ((𝐼 + 1) + (𝐽 + 1)) = ((𝐼 + 𝐽) + (1 + 1)))
148		1p1e2 12391	. . . . . . . . . . . 12 ⊢ (1 + 1) = 2
149	148	oveq2i 7442	. . . . . . . . . . 11 ⊢ ((𝐼 + 𝐽) + (1 + 1)) = ((𝐼 + 𝐽) + 2)
150	147, 149	eqtrdi 2793	. . . . . . . . . 10 ⊢ (𝜑 → ((𝐼 + 1) + (𝐽 + 1)) = ((𝐼 + 𝐽) + 2))
151	150	oveq2d 7447	. . . . . . . . 9 ⊢ (𝜑 → (-1↑((𝐼 + 1) + (𝐽 + 1))) = (-1↑((𝐼 + 𝐽) + 2)))
152		2nn0 12543	. . . . . . . . . . . 12 ⊢ 2 ∈ ℕ₀
153	152	a1i 11	. . . . . . . . . . 11 ⊢ (𝜑 → 2 ∈ ℕ₀)
154	131, 140	nn0addcld 12591	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐼 + 𝐽) ∈ ℕ₀)
155	128, 153, 154	expaddd 14188	. . . . . . . . . 10 ⊢ (𝜑 → (-1↑((𝐼 + 𝐽) + 2)) = ((-1↑(𝐼 + 𝐽)) · (-1↑2)))
156		neg1sqe1 14235	. . . . . . . . . . 11 ⊢ (-1↑2) = 1
157	156	oveq2i 7442	. . . . . . . . . 10 ⊢ ((-1↑(𝐼 + 𝐽)) · (-1↑2)) = ((-1↑(𝐼 + 𝐽)) · 1)
158	155, 157	eqtrdi 2793	. . . . . . . . 9 ⊢ (𝜑 → (-1↑((𝐼 + 𝐽) + 2)) = ((-1↑(𝐼 + 𝐽)) · 1))
159	128, 154	expcld 14186	. . . . . . . . . 10 ⊢ (𝜑 → (-1↑(𝐼 + 𝐽)) ∈ ℂ)
160	159	mulridd 11278	. . . . . . . . 9 ⊢ (𝜑 → ((-1↑(𝐼 + 𝐽)) · 1) = (-1↑(𝐼 + 𝐽)))
161	151, 158, 160	3eqtrd 2781	. . . . . . . 8 ⊢ (𝜑 → (-1↑((𝐼 + 1) + (𝐽 + 1))) = (-1↑(𝐼 + 𝐽)))
162	144, 161	eqtr3d 2779	. . . . . . 7 ⊢ (𝜑 → ((-1↑(𝐼 + 1)) · (-1↑(𝐽 + 1))) = (-1↑(𝐼 + 𝐽)))
163	137	nn0zd 12639	. . . . . . . 8 ⊢ (𝜑 → (𝑁 + 1) ∈ ℤ)
164		m1expcl2 14126	. . . . . . . 8 ⊢ ((𝑁 + 1) ∈ ℤ → (-1↑(𝑁 + 1)) ∈ {-1, 1})
165		1neg1t1neg1 32748	. . . . . . . 8 ⊢ ((-1↑(𝑁 + 1)) ∈ {-1, 1} → ((-1↑(𝑁 + 1)) · (-1↑(𝑁 + 1))) = 1)
166	163, 164, 165	3syl 18	. . . . . . 7 ⊢ (𝜑 → ((-1↑(𝑁 + 1)) · (-1↑(𝑁 + 1))) = 1)
167	162, 166	oveq12d 7449	. . . . . 6 ⊢ (𝜑 → (((-1↑(𝐼 + 1)) · (-1↑(𝐽 + 1))) · ((-1↑(𝑁 + 1)) · (-1↑(𝑁 + 1)))) = ((-1↑(𝐼 + 𝐽)) · 1))
168	143, 167, 160	3eqtrd 2781	. . . . 5 ⊢ (𝜑 → (((-1↑(𝐼 + 1)) · (-1↑(𝑁 + 1))) · ((-1↑(𝐽 + 1)) · (-1↑(𝑁 + 1)))) = (-1↑(𝐼 + 𝐽)))
169	126, 168	eqtrd 2777	. . . 4 ⊢ (𝜑 → (((pmSgn‘(1...𝑁))‘(𝑃 ∘ ^◡𝑆)) · ((pmSgn‘(1...𝑁))‘(𝑄 ∘ ^◡𝑇))) = (-1↑(𝐼 + 𝐽)))
170	169	fveq2d 6910	. . 3 ⊢ (𝜑 → (𝑍‘(((pmSgn‘(1...𝑁))‘(𝑃 ∘ ^◡𝑆)) · ((pmSgn‘(1...𝑁))‘(𝑄 ∘ ^◡𝑇)))) = (𝑍‘(-1↑(𝐼 + 𝐽))))
171	170	oveq1d 7446	. 2 ⊢ (𝜑 → ((𝑍‘(((pmSgn‘(1...𝑁))‘(𝑃 ∘ ^◡𝑆)) · ((pmSgn‘(1...𝑁))‘(𝑄 ∘ ^◡𝑇)))) · (𝐸‘(𝐼(subMat1‘𝑀)𝐽))) = ((𝑍‘(-1↑(𝐼 + 𝐽))) · (𝐸‘(𝐼(subMat1‘𝑀)𝐽))))
172	103, 171	eqtrd 2777	1 ⊢ (𝜑 → (𝐽(𝐾‘𝑀)𝐼) = ((𝑍‘(-1↑(𝐼 + 𝐽))) · (𝐸‘(𝐼(subMat1‘𝑀)𝐽))))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2108 ifcif 4525 {cpr 4628 class class class wbr 5143 ↦ cmpt 5225 ^◡ccnv 5684 dom cdm 5685 ∘ ccom 5689 Fun wfun 6555 ⟶wf 6557 –1-1→wf1 6558 –1-1-onto→wf1o 6560 ‘cfv 6561 (class class class)co 7431 ∈ cmpo 7433 Fincfn 8985 1c1 11156 + caddc 11158 · cmul 11160 ≤ cle 11296 − cmin 11492 -cneg 11493 ℕcn 12266 2c2 12321 ℕ₀cn0 12526 ℤcz 12613 ℤ_≥cuz 12878 ...cfz 13547 ↑cexp 14102 Basecbs 17247 +_gcplusg 17297 ._rcmulr 17298 Grpcgrp 18951 inv_gcminusg 18952 SymGrpcsymg 19386 pmSgncpsgn 19507 Ringcrg 20230 CRingccrg 20231 ℤRHomczrh 21510 Mat cmat 22411 maDet cmdat 22590 maAdju cmadu 22638 minMatR1 cminmar1 22639 subMat1csmat 33792

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 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 ax-addf 11234 ax-mulf 11235

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-xor 1512 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-tp 4631 df-op 4633 df-ot 4635 df-uni 4908 df-int 4947 df-iun 4993 df-iin 4994 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-se 5638 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-isom 6570 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-of 7697 df-om 7888 df-1st 8014 df-2nd 8015 df-supp 8186 df-tpos 8251 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-2o 8507 df-er 8745 df-map 8868 df-pm 8869 df-ixp 8938 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-fsupp 9402 df-sup 9482 df-oi 9550 df-card 9979 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-div 11921 df-nn 12267 df-2 12329 df-3 12330 df-4 12331 df-5 12332 df-6 12333 df-7 12334 df-8 12335 df-9 12336 df-n0 12527 df-xnn0 12600 df-z 12614 df-dec 12734 df-uz 12879 df-rp 13035 df-fz 13548 df-fzo 13695 df-seq 14043 df-exp 14103 df-hash 14370 df-word 14553 df-lsw 14601 df-concat 14609 df-s1 14634 df-substr 14679 df-pfx 14709 df-splice 14788 df-reverse 14797 df-s2 14887 df-struct 17184 df-sets 17201 df-slot 17219 df-ndx 17231 df-base 17248 df-ress 17275 df-plusg 17310 df-mulr 17311 df-starv 17312 df-sca 17313 df-vsca 17314 df-ip 17315 df-tset 17316 df-ple 17317 df-ds 17319 df-unif 17320 df-hom 17321 df-cco 17322 df-0g 17486 df-gsum 17487 df-prds 17492 df-pws 17494 df-mre 17629 df-mrc 17630 df-acs 17632 df-mgm 18653 df-sgrp 18732 df-mnd 18748 df-mhm 18796 df-submnd 18797 df-efmnd 18882 df-grp 18954 df-minusg 18955 df-mulg 19086 df-subg 19141 df-ghm 19231 df-gim 19277 df-cntz 19335 df-oppg 19364 df-symg 19387 df-pmtr 19460 df-psgn 19509 df-cmn 19800 df-abl 19801 df-mgp 20138 df-rng 20150 df-ur 20179 df-ring 20232 df-cring 20233 df-oppr 20334 df-dvdsr 20357 df-unit 20358 df-invr 20388 df-dvr 20401 df-rhm 20472 df-subrng 20546 df-subrg 20570 df-drng 20731 df-sra 21172 df-rgmod 21173 df-cnfld 21365 df-zring 21458 df-zrh 21514 df-dsmm 21752 df-frlm 21767 df-mat 22412 df-marrep 22564 df-subma 22583 df-mdet 22591 df-madu 22640 df-minmar1 22641 df-smat 33793

This theorem is referenced by: madjusmdet 33830

W3C validator