madjusmdetlem4 - Mathbox for Thierry Arnoux

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Theorem madjusmdetlem4 34024

Description: Lemma for madjusmdet 34025. (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 2741	. . 3 ⊢ (Base‘(SymGrp‘(1...𝑁))) = (Base‘(SymGrp‘(1...𝑁)))
14		eqid 2741	. . 3 ⊢ (pmSgn‘(1...𝑁)) = (pmSgn‘(1...𝑁))
15		eqid 2741	. . 3 ⊢ (𝐼(((1...𝑁) minMatR1 𝑅)‘𝑀)𝐽) = (𝐼(((1...𝑁) minMatR1 𝑅)‘𝑀)𝐽)
16		fveq2 6830	. . . . 5 ⊢ (𝑘 = 𝑖 → ((𝑃 ∘ ^◡𝑆)‘𝑘) = ((𝑃 ∘ ^◡𝑆)‘𝑖))
17	16	oveq1d 7374	. . . 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 6830	. . . . 5 ⊢ (𝑙 = 𝑗 → ((𝑄 ∘ ^◡𝑇)‘𝑙) = ((𝑄 ∘ ^◡𝑇)‘𝑗))
19	18	oveq2d 7375	. . . 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 7454	. . 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 2741	. . . . . 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 2741	. . . . . 6 ⊢ (SymGrp‘(1...𝑁)) = (SymGrp‘(1...𝑁))
24	21, 22, 23, 13	fzto1st 33186	. . . . 5 ⊢ (𝐼 ∈ (1...𝑁) → 𝑃 ∈ (Base‘(SymGrp‘(1...𝑁))))
25	10, 24	syl 17	. . . 4 ⊢ (𝜑 → 𝑃 ∈ (Base‘(SymGrp‘(1...𝑁))))
26		nnuz 12822	. . . . . . . . 9 ⊢ ℕ = (ℤ_≥‘1)
27	8, 26	eleqtrdi 2851	. . . . . . . 8 ⊢ (𝜑 → 𝑁 ∈ (ℤ_≥‘1))
28		eluzfz2 13481	. . . . . . . 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 33186	. . . . . . 7 ⊢ (𝑁 ∈ (1...𝑁) → 𝑆 ∈ (Base‘(SymGrp‘(1...𝑁))))
32	29, 31	syl 17	. . . . . 6 ⊢ (𝜑 → 𝑆 ∈ (Base‘(SymGrp‘(1...𝑁))))
33		eqid 2741	. . . . . . 7 ⊢ (inv_g‘(SymGrp‘(1...𝑁))) = (inv_g‘(SymGrp‘(1...𝑁)))
34	23, 13, 33	symginv 19371	. . . . . 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 13930	. . . . . . 7 ⊢ (𝜑 → (1...𝑁) ∈ Fin)
37	23	symggrp 19369	. . . . . . 7 ⊢ ((1...𝑁) ∈ Fin → (SymGrp‘(1...𝑁)) ∈ Grp)
38	36, 37	syl 17	. . . . . 6 ⊢ (𝜑 → (SymGrp‘(1...𝑁)) ∈ Grp)
39	13, 33	grpinvcl 18958	. . . . . 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 591	. . . . 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 2741	. . . . . 6 ⊢ (+_g‘(SymGrp‘(1...𝑁))) = (+_g‘(SymGrp‘(1...𝑁)))
43	23, 13, 42	symgov 19353	. . . . 5 ⊢ ((𝑃 ∈ (Base‘(SymGrp‘(1...𝑁))) ∧ ^◡𝑆 ∈ (Base‘(SymGrp‘(1...𝑁)))) → (𝑃(+_g‘(SymGrp‘(1...𝑁)))^◡𝑆) = (𝑃 ∘ ^◡𝑆))
44	23, 13, 42	symgcl 19354	. . . . 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 591	. . 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 33186	. . . . 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 33186	. . . . . . 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 19371	. . . . . 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 18958	. . . . . 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 591	. . . . 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 19353	. . . . 5 ⊢ ((𝑄 ∈ (Base‘(SymGrp‘(1...𝑁))) ∧ ^◡𝑇 ∈ (Base‘(SymGrp‘(1...𝑁)))) → (𝑄(+_g‘(SymGrp‘(1...𝑁)))^◡𝑇) = (𝑄 ∘ ^◡𝑇))
59	23, 13, 42	symgcl 19354	. . . . 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 591	. . 3 ⊢ (𝜑 → (𝑄 ∘ ^◡𝑇) ∈ (Base‘(SymGrp‘(1...𝑁))))
62	23, 13	symgbasf1o 19344	. . . . . . 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 6769	. . . . . 6 ⊢ (𝑆:(1...𝑁)–1-1-onto→(1...𝑁) → 𝑆:(1...𝑁)–1-1→(1...𝑁))
65		df-f1 6493	. . . . . . 7 ⊢ (𝑆:(1...𝑁)–1-1→(1...𝑁) ↔ (𝑆:(1...𝑁)⟶(1...𝑁) ∧ Fun ^◡𝑆))
66	65	simprbi 499	. . . . . 6 ⊢ (𝑆:(1...𝑁)–1-1→(1...𝑁) → Fun ^◡𝑆)
67	63, 64, 66	3syl 18	. . . . 5 ⊢ (𝜑 → Fun ^◡𝑆)
68		f1ocnv 6782	. . . . . . 7 ⊢ (𝑆:(1...𝑁)–1-1-onto→(1...𝑁) → ^◡𝑆:(1...𝑁)–1-1-onto→(1...𝑁))
69		f1odm 6774	. . . . . . 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 6928	. . . . 5 ⊢ ((Fun ^◡𝑆 ∧ 𝑁 ∈ dom ^◡𝑆) → ((𝑃 ∘ ^◡𝑆)‘𝑁) = (𝑃‘(^◡𝑆‘𝑁)))
73	67, 71, 72	syl2anc 591	. . . 4 ⊢ (𝜑 → ((𝑃 ∘ ^◡𝑆)‘𝑁) = (𝑃‘(^◡𝑆‘𝑁)))
74	21, 30, 23, 13	fzto1stinvn 33187	. . . . . 6 ⊢ (𝑁 ∈ (1...𝑁) → (^◡𝑆‘𝑁) = 1)
75	29, 74	syl 17	. . . . 5 ⊢ (𝜑 → (^◡𝑆‘𝑁) = 1)
76	75	fveq2d 6834	. . . 4 ⊢ (𝜑 → (𝑃‘(^◡𝑆‘𝑁)) = (𝑃‘1))
77	21, 22	fzto1stfv1 33184	. . . . 5 ⊢ (𝐼 ∈ (1...𝑁) → (𝑃‘1) = 𝐼)
78	10, 77	syl 17	. . . 4 ⊢ (𝜑 → (𝑃‘1) = 𝐼)
79	73, 76, 78	3eqtrd 2780	. . 3 ⊢ (𝜑 → ((𝑃 ∘ ^◡𝑆)‘𝑁) = 𝐼)
80	23, 13	symgbasf1o 19344	. . . . . . 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 6769	. . . . . 6 ⊢ (𝑇:(1...𝑁)–1-1-onto→(1...𝑁) → 𝑇:(1...𝑁)–1-1→(1...𝑁))
83		df-f1 6493	. . . . . . 7 ⊢ (𝑇:(1...𝑁)–1-1→(1...𝑁) ↔ (𝑇:(1...𝑁)⟶(1...𝑁) ∧ Fun ^◡𝑇))
84	83	simprbi 499	. . . . . 6 ⊢ (𝑇:(1...𝑁)–1-1→(1...𝑁) → Fun ^◡𝑇)
85	81, 82, 84	3syl 18	. . . . 5 ⊢ (𝜑 → Fun ^◡𝑇)
86		f1ocnv 6782	. . . . . . 7 ⊢ (𝑇:(1...𝑁)–1-1-onto→(1...𝑁) → ^◡𝑇:(1...𝑁)–1-1-onto→(1...𝑁))
87		f1odm 6774	. . . . . . 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 6928	. . . . 5 ⊢ ((Fun ^◡𝑇 ∧ 𝑁 ∈ dom ^◡𝑇) → ((𝑄 ∘ ^◡𝑇)‘𝑁) = (𝑄‘(^◡𝑇‘𝑁)))
91	85, 89, 90	syl2anc 591	. . . 4 ⊢ (𝜑 → ((𝑄 ∘ ^◡𝑇)‘𝑁) = (𝑄‘(^◡𝑇‘𝑁)))
92	21, 50, 23, 13	fzto1stinvn 33187	. . . . . 6 ⊢ (𝑁 ∈ (1...𝑁) → (^◡𝑇‘𝑁) = 1)
93	29, 92	syl 17	. . . . 5 ⊢ (𝜑 → (^◡𝑇‘𝑁) = 1)
94	93	fveq2d 6834	. . . 4 ⊢ (𝜑 → (𝑄‘(^◡𝑇‘𝑁)) = (𝑄‘1))
95	21, 47	fzto1stfv1 33184	. . . . 5 ⊢ (𝐽 ∈ (1...𝑁) → (𝑄‘1) = 𝐽)
96	11, 95	syl 17	. . . 4 ⊢ (𝜑 → (𝑄‘1) = 𝐽)
97	91, 94, 96	3eqtrd 2780	. . 3 ⊢ (𝜑 → ((𝑄 ∘ ^◡𝑇)‘𝑁) = 𝐽)
98		crngring 20220	. . . . . 6 ⊢ (𝑅 ∈ CRing → 𝑅 ∈ Ring)
99	9, 98	syl 17	. . . . 5 ⊢ (𝜑 → 𝑅 ∈ Ring)
100	2, 1	minmar1cl 22637	. . . . 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 845	. . . 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 34023	. . 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 34021	. 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 21561	. . . . . . . 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 1380	. . . . . . 7 ⊢ (𝜑 → ((pmSgn‘(1...𝑁))‘(𝑃 ∘ ^◡𝑆)) = (((pmSgn‘(1...𝑁))‘𝑃) · ((pmSgn‘(1...𝑁))‘^◡𝑆)))
106	21, 22, 23, 13, 14	psgnfzto1st 33188	. . . . . . . . 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 21560	. . . . . . . . . 10 ⊢ (((1...𝑁) ∈ Fin ∧ 𝑆 ∈ (Base‘(SymGrp‘(1...𝑁)))) → ((pmSgn‘(1...𝑁))‘^◡𝑆) = ((pmSgn‘(1...𝑁))‘𝑆))
109	36, 32, 108	syl2anc 591	. . . . . . . . 9 ⊢ (𝜑 → ((pmSgn‘(1...𝑁))‘^◡𝑆) = ((pmSgn‘(1...𝑁))‘𝑆))
110	21, 30, 23, 13, 14	psgnfzto1st 33188	. . . . . . . . . 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 2776	. . . . . . . 8 ⊢ (𝜑 → ((pmSgn‘(1...𝑁))‘^◡𝑆) = (-1↑(𝑁 + 1)))
113	107, 112	oveq12d 7377	. . . . . . 7 ⊢ (𝜑 → (((pmSgn‘(1...𝑁))‘𝑃) · ((pmSgn‘(1...𝑁))‘^◡𝑆)) = ((-1↑(𝐼 + 1)) · (-1↑(𝑁 + 1))))
114	105, 113	eqtrd 2776	. . . . . 6 ⊢ (𝜑 → ((pmSgn‘(1...𝑁))‘(𝑃 ∘ ^◡𝑆)) = ((-1↑(𝐼 + 1)) · (-1↑(𝑁 + 1))))
115	23, 14, 13	psgnco 21561	. . . . . . . 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 1380	. . . . . . 7 ⊢ (𝜑 → ((pmSgn‘(1...𝑁))‘(𝑄 ∘ ^◡𝑇)) = (((pmSgn‘(1...𝑁))‘𝑄) · ((pmSgn‘(1...𝑁))‘^◡𝑇)))
117	21, 47, 23, 13, 14	psgnfzto1st 33188	. . . . . . . . 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 21560	. . . . . . . . . 10 ⊢ (((1...𝑁) ∈ Fin ∧ 𝑇 ∈ (Base‘(SymGrp‘(1...𝑁)))) → ((pmSgn‘(1...𝑁))‘^◡𝑇) = ((pmSgn‘(1...𝑁))‘𝑇))
120	36, 52, 119	syl2anc 591	. . . . . . . . 9 ⊢ (𝜑 → ((pmSgn‘(1...𝑁))‘^◡𝑇) = ((pmSgn‘(1...𝑁))‘𝑇))
121	21, 50, 23, 13, 14	psgnfzto1st 33188	. . . . . . . . . 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 2776	. . . . . . . 8 ⊢ (𝜑 → ((pmSgn‘(1...𝑁))‘^◡𝑇) = (-1↑(𝑁 + 1)))
124	118, 123	oveq12d 7377	. . . . . . 7 ⊢ (𝜑 → (((pmSgn‘(1...𝑁))‘𝑄) · ((pmSgn‘(1...𝑁))‘^◡𝑇)) = ((-1↑(𝐽 + 1)) · (-1↑(𝑁 + 1))))
125	116, 124	eqtrd 2776	. . . . . 6 ⊢ (𝜑 → ((pmSgn‘(1...𝑁))‘(𝑄 ∘ ^◡𝑇)) = ((-1↑(𝐽 + 1)) · (-1↑(𝑁 + 1))))
126	114, 125	oveq12d 7377	. . . . 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 11135	. . . . . . . . 9 ⊢ (𝜑 → 1 ∈ ℂ)
128	127	negcld 11488	. . . . . . . 8 ⊢ (𝜑 → -1 ∈ ℂ)
129		fz1ssnn 13504	. . . . . . . . . . 11 ⊢ (1...𝑁) ⊆ ℕ
130	129, 10	sselid 3914	. . . . . . . . . 10 ⊢ (𝜑 → 𝐼 ∈ ℕ)
131	130	nnnn0d 12493	. . . . . . . . 9 ⊢ (𝜑 → 𝐼 ∈ ℕ₀)
132		1nn0 12448	. . . . . . . . . 10 ⊢ 1 ∈ ℕ₀
133	132	a1i 11	. . . . . . . . 9 ⊢ (𝜑 → 1 ∈ ℕ₀)
134	131, 133	nn0addcld 12497	. . . . . . . 8 ⊢ (𝜑 → (𝐼 + 1) ∈ ℕ₀)
135	128, 134	expcld 14103	. . . . . . 7 ⊢ (𝜑 → (-1↑(𝐼 + 1)) ∈ ℂ)
136	8	nnnn0d 12493	. . . . . . . . 9 ⊢ (𝜑 → 𝑁 ∈ ℕ₀)
137	136, 133	nn0addcld 12497	. . . . . . . 8 ⊢ (𝜑 → (𝑁 + 1) ∈ ℕ₀)
138	128, 137	expcld 14103	. . . . . . 7 ⊢ (𝜑 → (-1↑(𝑁 + 1)) ∈ ℂ)
139	129, 11	sselid 3914	. . . . . . . . . 10 ⊢ (𝜑 → 𝐽 ∈ ℕ)
140	139	nnnn0d 12493	. . . . . . . . 9 ⊢ (𝜑 → 𝐽 ∈ ℕ₀)
141	140, 133	nn0addcld 12497	. . . . . . . 8 ⊢ (𝜑 → (𝐽 + 1) ∈ ℕ₀)
142	128, 141	expcld 14103	. . . . . . 7 ⊢ (𝜑 → (-1↑(𝐽 + 1)) ∈ ℂ)
143	135, 138, 142, 138	mul4d 11354	. . . . . 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 14105	. . . . . . . 8 ⊢ (𝜑 → (-1↑((𝐼 + 1) + (𝐽 + 1))) = ((-1↑(𝐼 + 1)) · (-1↑(𝐽 + 1))))
145	130	nncnd 12185	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐼 ∈ ℂ)
146	139	nncnd 12185	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐽 ∈ ℂ)
147	145, 127, 146, 127	add4d 11371	. . . . . . . . . . 11 ⊢ (𝜑 → ((𝐼 + 1) + (𝐽 + 1)) = ((𝐼 + 𝐽) + (1 + 1)))
148		1p1e2 12296	. . . . . . . . . . . 12 ⊢ (1 + 1) = 2
149	148	oveq2i 7370	. . . . . . . . . . 11 ⊢ ((𝐼 + 𝐽) + (1 + 1)) = ((𝐼 + 𝐽) + 2)
150	147, 149	eqtrdi 2792	. . . . . . . . . 10 ⊢ (𝜑 → ((𝐼 + 1) + (𝐽 + 1)) = ((𝐼 + 𝐽) + 2))
151	150	oveq2d 7375	. . . . . . . . 9 ⊢ (𝜑 → (-1↑((𝐼 + 1) + (𝐽 + 1))) = (-1↑((𝐼 + 𝐽) + 2)))
152		2nn0 12449	. . . . . . . . . . . 12 ⊢ 2 ∈ ℕ₀
153	152	a1i 11	. . . . . . . . . . 11 ⊢ (𝜑 → 2 ∈ ℕ₀)
154	131, 140	nn0addcld 12497	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐼 + 𝐽) ∈ ℕ₀)
155	128, 153, 154	expaddd 14105	. . . . . . . . . 10 ⊢ (𝜑 → (-1↑((𝐼 + 𝐽) + 2)) = ((-1↑(𝐼 + 𝐽)) · (-1↑2)))
156		neg1sqe1 14153	. . . . . . . . . . 11 ⊢ (-1↑2) = 1
157	156	oveq2i 7370	. . . . . . . . . 10 ⊢ ((-1↑(𝐼 + 𝐽)) · (-1↑2)) = ((-1↑(𝐼 + 𝐽)) · 1)
158	155, 157	eqtrdi 2792	. . . . . . . . 9 ⊢ (𝜑 → (-1↑((𝐼 + 𝐽) + 2)) = ((-1↑(𝐼 + 𝐽)) · 1))
159	128, 154	expcld 14103	. . . . . . . . . 10 ⊢ (𝜑 → (-1↑(𝐼 + 𝐽)) ∈ ℂ)
160	159	mulridd 11158	. . . . . . . . 9 ⊢ (𝜑 → ((-1↑(𝐼 + 𝐽)) · 1) = (-1↑(𝐼 + 𝐽)))
161	151, 158, 160	3eqtrd 2780	. . . . . . . 8 ⊢ (𝜑 → (-1↑((𝐼 + 1) + (𝐽 + 1))) = (-1↑(𝐼 + 𝐽)))
162	144, 161	eqtr3d 2778	. . . . . . 7 ⊢ (𝜑 → ((-1↑(𝐼 + 1)) · (-1↑(𝐽 + 1))) = (-1↑(𝐼 + 𝐽)))
163	137	nn0zd 12544	. . . . . . . 8 ⊢ (𝜑 → (𝑁 + 1) ∈ ℤ)
164		m1expcl2 14042	. . . . . . . 8 ⊢ ((𝑁 + 1) ∈ ℤ → (-1↑(𝑁 + 1)) ∈ {-1, 1})
165		1neg1t1neg1 32832	. . . . . . . 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 7377	. . . . . 6 ⊢ (𝜑 → (((-1↑(𝐼 + 1)) · (-1↑(𝐽 + 1))) · ((-1↑(𝑁 + 1)) · (-1↑(𝑁 + 1)))) = ((-1↑(𝐼 + 𝐽)) · 1))
168	143, 167, 160	3eqtrd 2780	. . . . 5 ⊢ (𝜑 → (((-1↑(𝐼 + 1)) · (-1↑(𝑁 + 1))) · ((-1↑(𝐽 + 1)) · (-1↑(𝑁 + 1)))) = (-1↑(𝐼 + 𝐽)))
169	126, 168	eqtrd 2776	. . . 4 ⊢ (𝜑 → (((pmSgn‘(1...𝑁))‘(𝑃 ∘ ^◡𝑆)) · ((pmSgn‘(1...𝑁))‘(𝑄 ∘ ^◡𝑇))) = (-1↑(𝐼 + 𝐽)))
170	169	fveq2d 6834	. . 3 ⊢ (𝜑 → (𝑍‘(((pmSgn‘(1...𝑁))‘(𝑃 ∘ ^◡𝑆)) · ((pmSgn‘(1...𝑁))‘(𝑄 ∘ ^◡𝑇)))) = (𝑍‘(-1↑(𝐼 + 𝐽))))
171	170	oveq1d 7374	. 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 2776	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 397 = wceq 1548 ∈ wcel 2121 ifcif 4456 {cpr 4559 class class class wbr 5074 ↦ cmpt 5155 ^◡ccnv 5619 dom cdm 5620 ∘ ccom 5624 Fun wfun 6482 ⟶wf 6484 –1-1→wf1 6485 –1-1-onto→wf1o 6487 ‘cfv 6488 (class class class)co 7359 ∈ cmpo 7361 Fincfn 8887 1c1 11035 + caddc 11037 · cmul 11039 ≤ cle 11176 − cmin 11373 -cneg 11374 ℕcn 12169 2c2 12231 ℕ₀cn0 12432 ℤcz 12519 ℤ_≥cuz 12783 ...cfz 13456 ↑cexp 14018 Basecbs 17174 +_gcplusg 17215 ._rcmulr 17216 Grpcgrp 18904 inv_gcminusg 18905 SymGrpcsymg 19338 pmSgncpsgn 19458 Ringcrg 20208 CRingccrg 20209 ℤRHomczrh 21477 Mat cmat 22393 maDet cmdat 22570 maAdju cmadu 22618 minMatR1 cminmar1 22619 subMat1csmat 33987

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1975 ax-7 2016 ax-8 2123 ax-9 2131 ax-10 2154 ax-11 2170 ax-12 2191 ax-ext 2713 ax-rep 5201 ax-sep 5220 ax-nul 5230 ax-pow 5296 ax-pr 5364 ax-un 7681 ax-cnex 11090 ax-resscn 11091 ax-1cn 11092 ax-icn 11093 ax-addcl 11094 ax-addrcl 11095 ax-mulcl 11096 ax-mulrcl 11097 ax-mulcom 11098 ax-addass 11099 ax-mulass 11100 ax-distr 11101 ax-i2m1 11102 ax-1ne0 11103 ax-1rid 11104 ax-rnegex 11105 ax-rrecex 11106 ax-cnre 11107 ax-pre-lttri 11108 ax-pre-lttrn 11109 ax-pre-ltadd 11110 ax-pre-mulgt0 11111 ax-addf 11113 ax-mulf 11114

This theorem depends on definitions: df-bi 209 df-an 398 df-or 855 df-3or 1094 df-3an 1095 df-xor 1520 df-tru 1551 df-fal 1561 df-ex 1788 df-nf 1792 df-sb 2075 df-mo 2545 df-eu 2575 df-clab 2720 df-cleq 2733 df-clel 2816 df-nfc 2890 df-ne 2937 df-nel 3041 df-ral 3056 df-rex 3066 df-rmo 3346 df-reu 3347 df-rab 3394 df-v 3435 df-sbc 3725 df-csb 3833 df-dif 3887 df-un 3889 df-in 3891 df-ss 3901 df-pss 3904 df-nul 4264 df-if 4457 df-pw 4533 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-ot 4566 df-uni 4841 df-int 4880 df-iun 4925 df-iin 4926 df-br 5075 df-opab 5137 df-mpt 5156 df-tr 5182 df-id 5515 df-eprel 5520 df-po 5528 df-so 5529 df-fr 5573 df-se 5574 df-we 5575 df-xp 5626 df-rel 5627 df-cnv 5628 df-co 5629 df-dm 5630 df-rn 5631 df-res 5632 df-ima 5633 df-pred 6255 df-ord 6316 df-on 6317 df-lim 6318 df-suc 6319 df-iota 6444 df-fun 6490 df-fn 6491 df-f 6492 df-f1 6493 df-fo 6494 df-f1o 6495 df-fv 6496 df-isom 6497 df-riota 7316 df-ov 7362 df-oprab 7363 df-mpo 7364 df-of 7623 df-om 7810 df-1st 7933 df-2nd 7934 df-supp 8103 df-tpos 8168 df-frecs 8224 df-wrecs 8255 df-recs 8304 df-rdg 8343 df-1o 8399 df-2o 8400 df-er 8637 df-map 8769 df-pm 8770 df-ixp 8840 df-en 8888 df-dom 8889 df-sdom 8890 df-fin 8891 df-fsupp 9269 df-sup 9349 df-oi 9419 df-card 9858 df-pnf 11177 df-mnf 11178 df-xr 11179 df-ltxr 11180 df-le 11181 df-sub 11375 df-neg 11376 df-div 11804 df-nn 12170 df-2 12239 df-3 12240 df-4 12241 df-5 12242 df-6 12243 df-7 12244 df-8 12245 df-9 12246 df-n0 12433 df-xnn0 12506 df-z 12520 df-dec 12640 df-uz 12784 df-rp 12938 df-fz 13457 df-fzo 13604 df-seq 13959 df-exp 14019 df-hash 14288 df-word 14471 df-lsw 14520 df-concat 14528 df-s1 14554 df-substr 14599 df-pfx 14629 df-splice 14707 df-reverse 14716 df-s2 14805 df-struct 17112 df-sets 17129 df-slot 17147 df-ndx 17159 df-base 17175 df-ress 17196 df-plusg 17228 df-mulr 17229 df-starv 17230 df-sca 17231 df-vsca 17232 df-ip 17233 df-tset 17234 df-ple 17235 df-ds 17237 df-unif 17238 df-hom 17239 df-cco 17240 df-0g 17399 df-gsum 17400 df-prds 17405 df-pws 17407 df-mre 17543 df-mrc 17544 df-acs 17546 df-mgm 18603 df-sgrp 18682 df-mnd 18698 df-mhm 18746 df-submnd 18747 df-efmnd 18832 df-grp 18907 df-minusg 18908 df-mulg 19039 df-subg 19094 df-ghm 19183 df-gim 19228 df-cntz 19286 df-oppg 19315 df-symg 19339 df-pmtr 19411 df-psgn 19460 df-cmn 19751 df-abl 19752 df-mgp 20116 df-rng 20128 df-ur 20157 df-ring 20210 df-cring 20211 df-oppr 20311 df-dvdsr 20331 df-unit 20332 df-invr 20362 df-dvr 20375 df-rhm 20446 df-subrng 20521 df-subrg 20545 df-drng 20706 df-sra 21166 df-rgmod 21167 df-cnfld 21351 df-zring 21425 df-zrh 21481 df-dsmm 21710 df-frlm 21725 df-mat 22394 df-marrep 22544 df-subma 22563 df-mdet 22571 df-madu 22620 df-minmar1 22621 df-smat 33988

This theorem is referenced by: madjusmdet 34025

W3C validator