madjusmdetlem4 - Mathbox for Thierry Arnoux

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Theorem madjusmdetlem4 33861

Description: Lemma for madjusmdet 33862. (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 2735	. . 3 ⊢ (Base‘(SymGrp‘(1...𝑁))) = (Base‘(SymGrp‘(1...𝑁)))
14		eqid 2735	. . 3 ⊢ (pmSgn‘(1...𝑁)) = (pmSgn‘(1...𝑁))
15		eqid 2735	. . 3 ⊢ (𝐼(((1...𝑁) minMatR1 𝑅)‘𝑀)𝐽) = (𝐼(((1...𝑁) minMatR1 𝑅)‘𝑀)𝐽)
16		fveq2 6876	. . . . 5 ⊢ (𝑘 = 𝑖 → ((𝑃 ∘ ^◡𝑆)‘𝑘) = ((𝑃 ∘ ^◡𝑆)‘𝑖))
17	16	oveq1d 7420	. . . 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 6876	. . . . 5 ⊢ (𝑙 = 𝑗 → ((𝑄 ∘ ^◡𝑇)‘𝑙) = ((𝑄 ∘ ^◡𝑇)‘𝑗))
19	18	oveq2d 7421	. . . 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 7502	. . 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 2735	. . . . . 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 2735	. . . . . 6 ⊢ (SymGrp‘(1...𝑁)) = (SymGrp‘(1...𝑁))
24	21, 22, 23, 13	fzto1st 33114	. . . . 5 ⊢ (𝐼 ∈ (1...𝑁) → 𝑃 ∈ (Base‘(SymGrp‘(1...𝑁))))
25	10, 24	syl 17	. . . 4 ⊢ (𝜑 → 𝑃 ∈ (Base‘(SymGrp‘(1...𝑁))))
26		nnuz 12895	. . . . . . . . 9 ⊢ ℕ = (ℤ_≥‘1)
27	8, 26	eleqtrdi 2844	. . . . . . . 8 ⊢ (𝜑 → 𝑁 ∈ (ℤ_≥‘1))
28		eluzfz2 13549	. . . . . . . 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 33114	. . . . . . 7 ⊢ (𝑁 ∈ (1...𝑁) → 𝑆 ∈ (Base‘(SymGrp‘(1...𝑁))))
32	29, 31	syl 17	. . . . . 6 ⊢ (𝜑 → 𝑆 ∈ (Base‘(SymGrp‘(1...𝑁))))
33		eqid 2735	. . . . . . 7 ⊢ (inv_g‘(SymGrp‘(1...𝑁))) = (inv_g‘(SymGrp‘(1...𝑁)))
34	23, 13, 33	symginv 19383	. . . . . 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 13991	. . . . . . 7 ⊢ (𝜑 → (1...𝑁) ∈ Fin)
37	23	symggrp 19381	. . . . . . 7 ⊢ ((1...𝑁) ∈ Fin → (SymGrp‘(1...𝑁)) ∈ Grp)
38	36, 37	syl 17	. . . . . 6 ⊢ (𝜑 → (SymGrp‘(1...𝑁)) ∈ Grp)
39	13, 33	grpinvcl 18970	. . . . . 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 2835	. . . 4 ⊢ (𝜑 → ^◡𝑆 ∈ (Base‘(SymGrp‘(1...𝑁))))
42		eqid 2735	. . . . . 6 ⊢ (+_g‘(SymGrp‘(1...𝑁))) = (+_g‘(SymGrp‘(1...𝑁)))
43	23, 13, 42	symgov 19365	. . . . 5 ⊢ ((𝑃 ∈ (Base‘(SymGrp‘(1...𝑁))) ∧ ^◡𝑆 ∈ (Base‘(SymGrp‘(1...𝑁)))) → (𝑃(+_g‘(SymGrp‘(1...𝑁)))^◡𝑆) = (𝑃 ∘ ^◡𝑆))
44	23, 13, 42	symgcl 19366	. . . . 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 2835	. . . 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 33114	. . . . 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 33114	. . . . . . 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 19383	. . . . . 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 18970	. . . . . 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 2835	. . . 4 ⊢ (𝜑 → ^◡𝑇 ∈ (Base‘(SymGrp‘(1...𝑁))))
58	23, 13, 42	symgov 19365	. . . . 5 ⊢ ((𝑄 ∈ (Base‘(SymGrp‘(1...𝑁))) ∧ ^◡𝑇 ∈ (Base‘(SymGrp‘(1...𝑁)))) → (𝑄(+_g‘(SymGrp‘(1...𝑁)))^◡𝑇) = (𝑄 ∘ ^◡𝑇))
59	23, 13, 42	symgcl 19366	. . . . 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 2835	. . . 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 19356	. . . . . . 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 6817	. . . . . 6 ⊢ (𝑆:(1...𝑁)–1-1-onto→(1...𝑁) → 𝑆:(1...𝑁)–1-1→(1...𝑁))
65		df-f1 6536	. . . . . . 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 6830	. . . . . . 7 ⊢ (𝑆:(1...𝑁)–1-1-onto→(1...𝑁) → ^◡𝑆:(1...𝑁)–1-1-onto→(1...𝑁))
69		f1odm 6822	. . . . . . 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 2837	. . . . 5 ⊢ (𝜑 → 𝑁 ∈ dom ^◡𝑆)
72		fvco 6977	. . . . 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 33115	. . . . . 6 ⊢ (𝑁 ∈ (1...𝑁) → (^◡𝑆‘𝑁) = 1)
75	29, 74	syl 17	. . . . 5 ⊢ (𝜑 → (^◡𝑆‘𝑁) = 1)
76	75	fveq2d 6880	. . . 4 ⊢ (𝜑 → (𝑃‘(^◡𝑆‘𝑁)) = (𝑃‘1))
77	21, 22	fzto1stfv1 33112	. . . . 5 ⊢ (𝐼 ∈ (1...𝑁) → (𝑃‘1) = 𝐼)
78	10, 77	syl 17	. . . 4 ⊢ (𝜑 → (𝑃‘1) = 𝐼)
79	73, 76, 78	3eqtrd 2774	. . 3 ⊢ (𝜑 → ((𝑃 ∘ ^◡𝑆)‘𝑁) = 𝐼)
80	23, 13	symgbasf1o 19356	. . . . . . 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 6817	. . . . . 6 ⊢ (𝑇:(1...𝑁)–1-1-onto→(1...𝑁) → 𝑇:(1...𝑁)–1-1→(1...𝑁))
83		df-f1 6536	. . . . . . 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 6830	. . . . . . 7 ⊢ (𝑇:(1...𝑁)–1-1-onto→(1...𝑁) → ^◡𝑇:(1...𝑁)–1-1-onto→(1...𝑁))
87		f1odm 6822	. . . . . . 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 2837	. . . . 5 ⊢ (𝜑 → 𝑁 ∈ dom ^◡𝑇)
90		fvco 6977	. . . . 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 33115	. . . . . 6 ⊢ (𝑁 ∈ (1...𝑁) → (^◡𝑇‘𝑁) = 1)
93	29, 92	syl 17	. . . . 5 ⊢ (𝜑 → (^◡𝑇‘𝑁) = 1)
94	93	fveq2d 6880	. . . 4 ⊢ (𝜑 → (𝑄‘(^◡𝑇‘𝑁)) = (𝑄‘1))
95	21, 47	fzto1stfv1 33112	. . . . 5 ⊢ (𝐽 ∈ (1...𝑁) → (𝑄‘1) = 𝐽)
96	11, 95	syl 17	. . . 4 ⊢ (𝜑 → (𝑄‘1) = 𝐽)
97	91, 94, 96	3eqtrd 2774	. . 3 ⊢ (𝜑 → ((𝑄 ∘ ^◡𝑇)‘𝑁) = 𝐽)
98		crngring 20205	. . . . . 6 ⊢ (𝑅 ∈ CRing → 𝑅 ∈ Ring)
99	9, 98	syl 17	. . . . 5 ⊢ (𝜑 → 𝑅 ∈ Ring)
100	2, 1	minmar1cl 22589	. . . . 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 838	. . . 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 33860	. . 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 33858	. 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 21543	. . . . . . . 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 33116	. . . . . . . . 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 21542	. . . . . . . . . 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 33116	. . . . . . . . . 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 2770	. . . . . . . 8 ⊢ (𝜑 → ((pmSgn‘(1...𝑁))‘^◡𝑆) = (-1↑(𝑁 + 1)))
113	107, 112	oveq12d 7423	. . . . . . 7 ⊢ (𝜑 → (((pmSgn‘(1...𝑁))‘𝑃) · ((pmSgn‘(1...𝑁))‘^◡𝑆)) = ((-1↑(𝐼 + 1)) · (-1↑(𝑁 + 1))))
114	105, 113	eqtrd 2770	. . . . . 6 ⊢ (𝜑 → ((pmSgn‘(1...𝑁))‘(𝑃 ∘ ^◡𝑆)) = ((-1↑(𝐼 + 1)) · (-1↑(𝑁 + 1))))
115	23, 14, 13	psgnco 21543	. . . . . . . 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 33116	. . . . . . . . 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 21542	. . . . . . . . . 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 33116	. . . . . . . . . 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 2770	. . . . . . . 8 ⊢ (𝜑 → ((pmSgn‘(1...𝑁))‘^◡𝑇) = (-1↑(𝑁 + 1)))
124	118, 123	oveq12d 7423	. . . . . . 7 ⊢ (𝜑 → (((pmSgn‘(1...𝑁))‘𝑄) · ((pmSgn‘(1...𝑁))‘^◡𝑇)) = ((-1↑(𝐽 + 1)) · (-1↑(𝑁 + 1))))
125	116, 124	eqtrd 2770	. . . . . 6 ⊢ (𝜑 → ((pmSgn‘(1...𝑁))‘(𝑄 ∘ ^◡𝑇)) = ((-1↑(𝐽 + 1)) · (-1↑(𝑁 + 1))))
126	114, 125	oveq12d 7423	. . . . 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 11230	. . . . . . . . 9 ⊢ (𝜑 → 1 ∈ ℂ)
128	127	negcld 11581	. . . . . . . 8 ⊢ (𝜑 → -1 ∈ ℂ)
129		fz1ssnn 13572	. . . . . . . . . . 11 ⊢ (1...𝑁) ⊆ ℕ
130	129, 10	sselid 3956	. . . . . . . . . 10 ⊢ (𝜑 → 𝐼 ∈ ℕ)
131	130	nnnn0d 12562	. . . . . . . . 9 ⊢ (𝜑 → 𝐼 ∈ ℕ₀)
132		1nn0 12517	. . . . . . . . . 10 ⊢ 1 ∈ ℕ₀
133	132	a1i 11	. . . . . . . . 9 ⊢ (𝜑 → 1 ∈ ℕ₀)
134	131, 133	nn0addcld 12566	. . . . . . . 8 ⊢ (𝜑 → (𝐼 + 1) ∈ ℕ₀)
135	128, 134	expcld 14164	. . . . . . 7 ⊢ (𝜑 → (-1↑(𝐼 + 1)) ∈ ℂ)
136	8	nnnn0d 12562	. . . . . . . . 9 ⊢ (𝜑 → 𝑁 ∈ ℕ₀)
137	136, 133	nn0addcld 12566	. . . . . . . 8 ⊢ (𝜑 → (𝑁 + 1) ∈ ℕ₀)
138	128, 137	expcld 14164	. . . . . . 7 ⊢ (𝜑 → (-1↑(𝑁 + 1)) ∈ ℂ)
139	129, 11	sselid 3956	. . . . . . . . . 10 ⊢ (𝜑 → 𝐽 ∈ ℕ)
140	139	nnnn0d 12562	. . . . . . . . 9 ⊢ (𝜑 → 𝐽 ∈ ℕ₀)
141	140, 133	nn0addcld 12566	. . . . . . . 8 ⊢ (𝜑 → (𝐽 + 1) ∈ ℕ₀)
142	128, 141	expcld 14164	. . . . . . 7 ⊢ (𝜑 → (-1↑(𝐽 + 1)) ∈ ℂ)
143	135, 138, 142, 138	mul4d 11447	. . . . . 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 14166	. . . . . . . 8 ⊢ (𝜑 → (-1↑((𝐼 + 1) + (𝐽 + 1))) = ((-1↑(𝐼 + 1)) · (-1↑(𝐽 + 1))))
145	130	nncnd 12256	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐼 ∈ ℂ)
146	139	nncnd 12256	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐽 ∈ ℂ)
147	145, 127, 146, 127	add4d 11464	. . . . . . . . . . 11 ⊢ (𝜑 → ((𝐼 + 1) + (𝐽 + 1)) = ((𝐼 + 𝐽) + (1 + 1)))
148		1p1e2 12365	. . . . . . . . . . . 12 ⊢ (1 + 1) = 2
149	148	oveq2i 7416	. . . . . . . . . . 11 ⊢ ((𝐼 + 𝐽) + (1 + 1)) = ((𝐼 + 𝐽) + 2)
150	147, 149	eqtrdi 2786	. . . . . . . . . 10 ⊢ (𝜑 → ((𝐼 + 1) + (𝐽 + 1)) = ((𝐼 + 𝐽) + 2))
151	150	oveq2d 7421	. . . . . . . . 9 ⊢ (𝜑 → (-1↑((𝐼 + 1) + (𝐽 + 1))) = (-1↑((𝐼 + 𝐽) + 2)))
152		2nn0 12518	. . . . . . . . . . . 12 ⊢ 2 ∈ ℕ₀
153	152	a1i 11	. . . . . . . . . . 11 ⊢ (𝜑 → 2 ∈ ℕ₀)
154	131, 140	nn0addcld 12566	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐼 + 𝐽) ∈ ℕ₀)
155	128, 153, 154	expaddd 14166	. . . . . . . . . 10 ⊢ (𝜑 → (-1↑((𝐼 + 𝐽) + 2)) = ((-1↑(𝐼 + 𝐽)) · (-1↑2)))
156		neg1sqe1 14214	. . . . . . . . . . 11 ⊢ (-1↑2) = 1
157	156	oveq2i 7416	. . . . . . . . . 10 ⊢ ((-1↑(𝐼 + 𝐽)) · (-1↑2)) = ((-1↑(𝐼 + 𝐽)) · 1)
158	155, 157	eqtrdi 2786	. . . . . . . . 9 ⊢ (𝜑 → (-1↑((𝐼 + 𝐽) + 2)) = ((-1↑(𝐼 + 𝐽)) · 1))
159	128, 154	expcld 14164	. . . . . . . . . 10 ⊢ (𝜑 → (-1↑(𝐼 + 𝐽)) ∈ ℂ)
160	159	mulridd 11252	. . . . . . . . 9 ⊢ (𝜑 → ((-1↑(𝐼 + 𝐽)) · 1) = (-1↑(𝐼 + 𝐽)))
161	151, 158, 160	3eqtrd 2774	. . . . . . . 8 ⊢ (𝜑 → (-1↑((𝐼 + 1) + (𝐽 + 1))) = (-1↑(𝐼 + 𝐽)))
162	144, 161	eqtr3d 2772	. . . . . . 7 ⊢ (𝜑 → ((-1↑(𝐼 + 1)) · (-1↑(𝐽 + 1))) = (-1↑(𝐼 + 𝐽)))
163	137	nn0zd 12614	. . . . . . . 8 ⊢ (𝜑 → (𝑁 + 1) ∈ ℤ)
164		m1expcl2 14103	. . . . . . . 8 ⊢ ((𝑁 + 1) ∈ ℤ → (-1↑(𝑁 + 1)) ∈ {-1, 1})
165		1neg1t1neg1 32715	. . . . . . . 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 7423	. . . . . 6 ⊢ (𝜑 → (((-1↑(𝐼 + 1)) · (-1↑(𝐽 + 1))) · ((-1↑(𝑁 + 1)) · (-1↑(𝑁 + 1)))) = ((-1↑(𝐼 + 𝐽)) · 1))
168	143, 167, 160	3eqtrd 2774	. . . . 5 ⊢ (𝜑 → (((-1↑(𝐼 + 1)) · (-1↑(𝑁 + 1))) · ((-1↑(𝐽 + 1)) · (-1↑(𝑁 + 1)))) = (-1↑(𝐼 + 𝐽)))
169	126, 168	eqtrd 2770	. . . 4 ⊢ (𝜑 → (((pmSgn‘(1...𝑁))‘(𝑃 ∘ ^◡𝑆)) · ((pmSgn‘(1...𝑁))‘(𝑄 ∘ ^◡𝑇))) = (-1↑(𝐼 + 𝐽)))
170	169	fveq2d 6880	. . 3 ⊢ (𝜑 → (𝑍‘(((pmSgn‘(1...𝑁))‘(𝑃 ∘ ^◡𝑆)) · ((pmSgn‘(1...𝑁))‘(𝑄 ∘ ^◡𝑇)))) = (𝑍‘(-1↑(𝐼 + 𝐽))))
171	170	oveq1d 7420	. 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 2770	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 4500 {cpr 4603 class class class wbr 5119 ↦ cmpt 5201 ^◡ccnv 5653 dom cdm 5654 ∘ ccom 5658 Fun wfun 6525 ⟶wf 6527 –1-1→wf1 6528 –1-1-onto→wf1o 6530 ‘cfv 6531 (class class class)co 7405 ∈ cmpo 7407 Fincfn 8959 1c1 11130 + caddc 11132 · cmul 11134 ≤ cle 11270 − cmin 11466 -cneg 11467 ℕcn 12240 2c2 12295 ℕ₀cn0 12501 ℤcz 12588 ℤ_≥cuz 12852 ...cfz 13524 ↑cexp 14079 Basecbs 17228 +_gcplusg 17271 ._rcmulr 17272 Grpcgrp 18916 inv_gcminusg 18917 SymGrpcsymg 19350 pmSgncpsgn 19470 Ringcrg 20193 CRingccrg 20194 ℤRHomczrh 21460 Mat cmat 22345 maDet cmdat 22522 maAdju cmadu 22570 minMatR1 cminmar1 22571 subMat1csmat 33824

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 2707 ax-rep 5249 ax-sep 5266 ax-nul 5276 ax-pow 5335 ax-pr 5402 ax-un 7729 ax-cnex 11185 ax-resscn 11186 ax-1cn 11187 ax-icn 11188 ax-addcl 11189 ax-addrcl 11190 ax-mulcl 11191 ax-mulrcl 11192 ax-mulcom 11193 ax-addass 11194 ax-mulass 11195 ax-distr 11196 ax-i2m1 11197 ax-1ne0 11198 ax-1rid 11199 ax-rnegex 11200 ax-rrecex 11201 ax-cnre 11202 ax-pre-lttri 11203 ax-pre-lttrn 11204 ax-pre-ltadd 11205 ax-pre-mulgt0 11206 ax-addf 11208 ax-mulf 11209

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-xor 1512 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2727 df-clel 2809 df-nfc 2885 df-ne 2933 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3359 df-reu 3360 df-rab 3416 df-v 3461 df-sbc 3766 df-csb 3875 df-dif 3929 df-un 3931 df-in 3933 df-ss 3943 df-pss 3946 df-nul 4309 df-if 4501 df-pw 4577 df-sn 4602 df-pr 4604 df-tp 4606 df-op 4608 df-ot 4610 df-uni 4884 df-int 4923 df-iun 4969 df-iin 4970 df-br 5120 df-opab 5182 df-mpt 5202 df-tr 5230 df-id 5548 df-eprel 5553 df-po 5561 df-so 5562 df-fr 5606 df-se 5607 df-we 5608 df-xp 5660 df-rel 5661 df-cnv 5662 df-co 5663 df-dm 5664 df-rn 5665 df-res 5666 df-ima 5667 df-pred 6290 df-ord 6355 df-on 6356 df-lim 6357 df-suc 6358 df-iota 6484 df-fun 6533 df-fn 6534 df-f 6535 df-f1 6536 df-fo 6537 df-f1o 6538 df-fv 6539 df-isom 6540 df-riota 7362 df-ov 7408 df-oprab 7409 df-mpo 7410 df-of 7671 df-om 7862 df-1st 7988 df-2nd 7989 df-supp 8160 df-tpos 8225 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-rdg 8424 df-1o 8480 df-2o 8481 df-er 8719 df-map 8842 df-pm 8843 df-ixp 8912 df-en 8960 df-dom 8961 df-sdom 8962 df-fin 8963 df-fsupp 9374 df-sup 9454 df-oi 9524 df-card 9953 df-pnf 11271 df-mnf 11272 df-xr 11273 df-ltxr 11274 df-le 11275 df-sub 11468 df-neg 11469 df-div 11895 df-nn 12241 df-2 12303 df-3 12304 df-4 12305 df-5 12306 df-6 12307 df-7 12308 df-8 12309 df-9 12310 df-n0 12502 df-xnn0 12575 df-z 12589 df-dec 12709 df-uz 12853 df-rp 13009 df-fz 13525 df-fzo 13672 df-seq 14020 df-exp 14080 df-hash 14349 df-word 14532 df-lsw 14581 df-concat 14589 df-s1 14614 df-substr 14659 df-pfx 14689 df-splice 14768 df-reverse 14777 df-s2 14867 df-struct 17166 df-sets 17183 df-slot 17201 df-ndx 17213 df-base 17229 df-ress 17252 df-plusg 17284 df-mulr 17285 df-starv 17286 df-sca 17287 df-vsca 17288 df-ip 17289 df-tset 17290 df-ple 17291 df-ds 17293 df-unif 17294 df-hom 17295 df-cco 17296 df-0g 17455 df-gsum 17456 df-prds 17461 df-pws 17463 df-mre 17598 df-mrc 17599 df-acs 17601 df-mgm 18618 df-sgrp 18697 df-mnd 18713 df-mhm 18761 df-submnd 18762 df-efmnd 18847 df-grp 18919 df-minusg 18920 df-mulg 19051 df-subg 19106 df-ghm 19196 df-gim 19242 df-cntz 19300 df-oppg 19329 df-symg 19351 df-pmtr 19423 df-psgn 19472 df-cmn 19763 df-abl 19764 df-mgp 20101 df-rng 20113 df-ur 20142 df-ring 20195 df-cring 20196 df-oppr 20297 df-dvdsr 20317 df-unit 20318 df-invr 20348 df-dvr 20361 df-rhm 20432 df-subrng 20506 df-subrg 20530 df-drng 20691 df-sra 21131 df-rgmod 21132 df-cnfld 21316 df-zring 21408 df-zrh 21464 df-dsmm 21692 df-frlm 21707 df-mat 22346 df-marrep 22496 df-subma 22515 df-mdet 22523 df-madu 22572 df-minmar1 22573 df-smat 33825

This theorem is referenced by: madjusmdet 33862

W3C validator