madjusmdetlem4 - Mathbox for Thierry Arnoux

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Theorem madjusmdetlem4 34529

Description: Lemma for madjusmdet 34530. (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 2763	. . 3 ⊢ (Base‘(SymGrp‘(1...𝑁))) = (Base‘(SymGrp‘(1...𝑁)))
14		eqid 2763	. . 3 ⊢ (pmSgn‘(1...𝑁)) = (pmSgn‘(1...𝑁))
15		eqid 2763	. . 3 ⊢ (𝐼(((1...𝑁) minMatR1 𝑅)‘𝑀)𝐽) = (𝐼(((1...𝑁) minMatR1 𝑅)‘𝑀)𝐽)
16		fveq2 6905	. . . . 5 ⊢ (𝑘 = 𝑖 → ((𝑃 ∘ ^◡𝑆)‘𝑘) = ((𝑃 ∘ ^◡𝑆)‘𝑖))
17	16	oveq1d 7455	. . . 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 6905	. . . . 5 ⊢ (𝑙 = 𝑗 → ((𝑄 ∘ ^◡𝑇)‘𝑙) = ((𝑄 ∘ ^◡𝑇)‘𝑗))
19	18	oveq2d 7456	. . . 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 7535	. . 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 2763	. . . . . 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 2763	. . . . . 6 ⊢ (SymGrp‘(1...𝑁)) = (SymGrp‘(1...𝑁))
24	21, 22, 23, 13	fzto1st 33731	. . . . 5 ⊢ (𝐼 ∈ (1...𝑁) → 𝑃 ∈ (Base‘(SymGrp‘(1...𝑁))))
25	10, 24	syl 18	. . . 4 ⊢ (𝜑 → 𝑃 ∈ (Base‘(SymGrp‘(1...𝑁))))
26		nnuz 13028	. . . . . . . . 9 ⊢ ℕ = (ℤ_≥‘1)
27	8, 26	eleqtrdi 2874	. . . . . . . 8 ⊢ (𝜑 → 𝑁 ∈ (ℤ_≥‘1))
28		eluzfz2 13689	. . . . . . . 8 ⊢ (𝑁 ∈ (ℤ_≥‘1) → 𝑁 ∈ (1...𝑁))
29	27, 28	syl 18	. . . . . . 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 33731	. . . . . . 7 ⊢ (𝑁 ∈ (1...𝑁) → 𝑆 ∈ (Base‘(SymGrp‘(1...𝑁))))
32	29, 31	syl 18	. . . . . 6 ⊢ (𝜑 → 𝑆 ∈ (Base‘(SymGrp‘(1...𝑁))))
33		eqid 2763	. . . . . . 7 ⊢ (inv_g‘(SymGrp‘(1...𝑁))) = (inv_g‘(SymGrp‘(1...𝑁)))
34	23, 13, 33	symginv 19643	. . . . . 6 ⊢ (𝑆 ∈ (Base‘(SymGrp‘(1...𝑁))) → ((inv_g‘(SymGrp‘(1...𝑁)))‘𝑆) = ^◡𝑆)
35	32, 34	syl 18	. . . . 5 ⊢ (𝜑 → ((inv_g‘(SymGrp‘(1...𝑁)))‘𝑆) = ^◡𝑆)
36		fzfid 14140	. . . . . . 7 ⊢ (𝜑 → (1...𝑁) ∈ Fin)
37	23	symggrp 19641	. . . . . . 7 ⊢ ((1...𝑁) ∈ Fin → (SymGrp‘(1...𝑁)) ∈ Grp)
38	36, 37	syl 18	. . . . . 6 ⊢ (𝜑 → (SymGrp‘(1...𝑁)) ∈ Grp)
39	13, 33	grpinvcl 19224	. . . . . 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 596	. . . . 5 ⊢ (𝜑 → ((inv_g‘(SymGrp‘(1...𝑁)))‘𝑆) ∈ (Base‘(SymGrp‘(1...𝑁))))
41	35, 40	eqeltrrd 2865	. . . 4 ⊢ (𝜑 → ^◡𝑆 ∈ (Base‘(SymGrp‘(1...𝑁))))
42		eqid 2763	. . . . . 6 ⊢ (+_g‘(SymGrp‘(1...𝑁))) = (+_g‘(SymGrp‘(1...𝑁)))
43	23, 13, 42	symgov 19625	. . . . 5 ⊢ ((𝑃 ∈ (Base‘(SymGrp‘(1...𝑁))) ∧ ^◡𝑆 ∈ (Base‘(SymGrp‘(1...𝑁)))) → (𝑃(+_g‘(SymGrp‘(1...𝑁)))^◡𝑆) = (𝑃 ∘ ^◡𝑆))
44	23, 13, 42	symgcl 19626	. . . . 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 2865	. . . 4 ⊢ ((𝑃 ∈ (Base‘(SymGrp‘(1...𝑁))) ∧ ^◡𝑆 ∈ (Base‘(SymGrp‘(1...𝑁)))) → (𝑃 ∘ ^◡𝑆) ∈ (Base‘(SymGrp‘(1...𝑁))))
46	25, 41, 45	syl2anc 596	. . 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 33731	. . . . 5 ⊢ (𝐽 ∈ (1...𝑁) → 𝑄 ∈ (Base‘(SymGrp‘(1...𝑁))))
49	11, 48	syl 18	. . . 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 33731	. . . . . . 7 ⊢ (𝑁 ∈ (1...𝑁) → 𝑇 ∈ (Base‘(SymGrp‘(1...𝑁))))
52	29, 51	syl 18	. . . . . 6 ⊢ (𝜑 → 𝑇 ∈ (Base‘(SymGrp‘(1...𝑁))))
53	23, 13, 33	symginv 19643	. . . . . 6 ⊢ (𝑇 ∈ (Base‘(SymGrp‘(1...𝑁))) → ((inv_g‘(SymGrp‘(1...𝑁)))‘𝑇) = ^◡𝑇)
54	52, 53	syl 18	. . . . 5 ⊢ (𝜑 → ((inv_g‘(SymGrp‘(1...𝑁)))‘𝑇) = ^◡𝑇)
55	13, 33	grpinvcl 19224	. . . . . 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 596	. . . . 5 ⊢ (𝜑 → ((inv_g‘(SymGrp‘(1...𝑁)))‘𝑇) ∈ (Base‘(SymGrp‘(1...𝑁))))
57	54, 56	eqeltrrd 2865	. . . 4 ⊢ (𝜑 → ^◡𝑇 ∈ (Base‘(SymGrp‘(1...𝑁))))
58	23, 13, 42	symgov 19625	. . . . 5 ⊢ ((𝑄 ∈ (Base‘(SymGrp‘(1...𝑁))) ∧ ^◡𝑇 ∈ (Base‘(SymGrp‘(1...𝑁)))) → (𝑄(+_g‘(SymGrp‘(1...𝑁)))^◡𝑇) = (𝑄 ∘ ^◡𝑇))
59	23, 13, 42	symgcl 19626	. . . . 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 2865	. . . 4 ⊢ ((𝑄 ∈ (Base‘(SymGrp‘(1...𝑁))) ∧ ^◡𝑇 ∈ (Base‘(SymGrp‘(1...𝑁)))) → (𝑄 ∘ ^◡𝑇) ∈ (Base‘(SymGrp‘(1...𝑁))))
61	49, 57, 60	syl2anc 596	. . 3 ⊢ (𝜑 → (𝑄 ∘ ^◡𝑇) ∈ (Base‘(SymGrp‘(1...𝑁))))
62	23, 13	symgbasf1o 19616	. . . . . . 7 ⊢ (𝑆 ∈ (Base‘(SymGrp‘(1...𝑁))) → 𝑆:(1...𝑁)⟶1-1onto(1...𝑁))
63	32, 62	syl 18	. . . . . 6 ⊢ (𝜑 → 𝑆:(1...𝑁)⟶1-1onto(1...𝑁))
64		f1of1 6842	. . . . . 6 ⊢ (𝑆:(1...𝑁)⟶1-1onto(1...𝑁) → 𝑆:(1...𝑁)⟶1-1(1...𝑁))
65		df-f1 6560	. . . . . . 7 ⊢ (𝑆:(1...𝑁)⟶1-1(1...𝑁) ↔ (𝑆:(1...𝑁)⟶(1...𝑁) ∧ Fun ^◡𝑆))
66	65	simprbi 503	. . . . . 6 ⊢ (𝑆:(1...𝑁)⟶1-1(1...𝑁) → Fun ^◡𝑆)
67	63, 64, 66	3syl 19	. . . . 5 ⊢ (𝜑 → Fun ^◡𝑆)
68		f1ocnv 6856	. . . . . . 7 ⊢ (𝑆:(1...𝑁)⟶1-1onto(1...𝑁) → ^◡𝑆:(1...𝑁)⟶1-1onto(1...𝑁))
69		f1odm 6847	. . . . . . 7 ⊢ (^◡𝑆:(1...𝑁)⟶1-1onto(1...𝑁) → dom ^◡𝑆 = (1...𝑁))
70	63, 68, 69	3syl 19	. . . . . 6 ⊢ (𝜑 → dom ^◡𝑆 = (1...𝑁))
71	29, 70	eleqtrrd 2867	. . . . 5 ⊢ (𝜑 → 𝑁 ∈ dom ^◡𝑆)
72		fvco 7003	. . . . 5 ⊢ ((Fun ^◡𝑆 ∧ 𝑁 ∈ dom ^◡𝑆) → ((𝑃 ∘ ^◡𝑆)‘𝑁) = (𝑃‘(^◡𝑆‘𝑁)))
73	67, 71, 72	syl2anc 596	. . . 4 ⊢ (𝜑 → ((𝑃 ∘ ^◡𝑆)‘𝑁) = (𝑃‘(^◡𝑆‘𝑁)))
74	21, 30, 23, 13	fzto1stinvn 33732	. . . . . 6 ⊢ (𝑁 ∈ (1...𝑁) → (^◡𝑆‘𝑁) = 1)
75	29, 74	syl 18	. . . . 5 ⊢ (𝜑 → (^◡𝑆‘𝑁) = 1)
76	75	fveq2d 6909	. . . 4 ⊢ (𝜑 → (𝑃‘(^◡𝑆‘𝑁)) = (𝑃‘1))
77	21, 22	fzto1stfv1 33729	. . . . 5 ⊢ (𝐼 ∈ (1...𝑁) → (𝑃‘1) = 𝐼)
78	10, 77	syl 18	. . . 4 ⊢ (𝜑 → (𝑃‘1) = 𝐼)
79	73, 76, 78	3eqtrd 2802	. . 3 ⊢ (𝜑 → ((𝑃 ∘ ^◡𝑆)‘𝑁) = 𝐼)
80	23, 13	symgbasf1o 19616	. . . . . . 7 ⊢ (𝑇 ∈ (Base‘(SymGrp‘(1...𝑁))) → 𝑇:(1...𝑁)⟶1-1onto(1...𝑁))
81	52, 80	syl 18	. . . . . 6 ⊢ (𝜑 → 𝑇:(1...𝑁)⟶1-1onto(1...𝑁))
82		f1of1 6842	. . . . . 6 ⊢ (𝑇:(1...𝑁)⟶1-1onto(1...𝑁) → 𝑇:(1...𝑁)⟶1-1(1...𝑁))
83		df-f1 6560	. . . . . . 7 ⊢ (𝑇:(1...𝑁)⟶1-1(1...𝑁) ↔ (𝑇:(1...𝑁)⟶(1...𝑁) ∧ Fun ^◡𝑇))
84	83	simprbi 503	. . . . . 6 ⊢ (𝑇:(1...𝑁)⟶1-1(1...𝑁) → Fun ^◡𝑇)
85	81, 82, 84	3syl 19	. . . . 5 ⊢ (𝜑 → Fun ^◡𝑇)
86		f1ocnv 6856	. . . . . . 7 ⊢ (𝑇:(1...𝑁)⟶1-1onto(1...𝑁) → ^◡𝑇:(1...𝑁)⟶1-1onto(1...𝑁))
87		f1odm 6847	. . . . . . 7 ⊢ (^◡𝑇:(1...𝑁)⟶1-1onto(1...𝑁) → dom ^◡𝑇 = (1...𝑁))
88	81, 86, 87	3syl 19	. . . . . 6 ⊢ (𝜑 → dom ^◡𝑇 = (1...𝑁))
89	29, 88	eleqtrrd 2867	. . . . 5 ⊢ (𝜑 → 𝑁 ∈ dom ^◡𝑇)
90		fvco 7003	. . . . 5 ⊢ ((Fun ^◡𝑇 ∧ 𝑁 ∈ dom ^◡𝑇) → ((𝑄 ∘ ^◡𝑇)‘𝑁) = (𝑄‘(^◡𝑇‘𝑁)))
91	85, 89, 90	syl2anc 596	. . . 4 ⊢ (𝜑 → ((𝑄 ∘ ^◡𝑇)‘𝑁) = (𝑄‘(^◡𝑇‘𝑁)))
92	21, 50, 23, 13	fzto1stinvn 33732	. . . . . 6 ⊢ (𝑁 ∈ (1...𝑁) → (^◡𝑇‘𝑁) = 1)
93	29, 92	syl 18	. . . . 5 ⊢ (𝜑 → (^◡𝑇‘𝑁) = 1)
94	93	fveq2d 6909	. . . 4 ⊢ (𝜑 → (𝑄‘(^◡𝑇‘𝑁)) = (𝑄‘1))
95	21, 47	fzto1stfv1 33729	. . . . 5 ⊢ (𝐽 ∈ (1...𝑁) → (𝑄‘1) = 𝐽)
96	11, 95	syl 18	. . . 4 ⊢ (𝜑 → (𝑄‘1) = 𝐽)
97	91, 94, 96	3eqtrd 2802	. . 3 ⊢ (𝜑 → ((𝑄 ∘ ^◡𝑇)‘𝑁) = 𝐽)
98		crngring 20499	. . . . . 6 ⊢ (𝑅 ∈ CRing → 𝑅 ∈ Ring)
99	9, 98	syl 18	. . . . 5 ⊢ (𝜑 → 𝑅 ∈ Ring)
100	2, 1	minmar1cl 22993	. . . . 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 852	. . . 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 34528	. . 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 34526	. 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 21916	. . . . . . . 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 1399	. . . . . . 7 ⊢ (𝜑 → ((pmSgn‘(1...𝑁))‘(𝑃 ∘ ^◡𝑆)) = (((pmSgn‘(1...𝑁))‘𝑃) · ((pmSgn‘(1...𝑁))‘^◡𝑆)))
106	21, 22, 23, 13, 14	psgnfzto1st 33733	. . . . . . . . 9 ⊢ (𝐼 ∈ (1...𝑁) → ((pmSgn‘(1...𝑁))‘𝑃) = ( -1↑(𝐼 + 1)))
107	10, 106	syl 18	. . . . . . . 8 ⊢ (𝜑 → ((pmSgn‘(1...𝑁))‘𝑃) = ( -1↑(𝐼 + 1)))
108	23, 14, 13	psgninv 21915	. . . . . . . . . 10 ⊢ (((1...𝑁) ∈ Fin ∧ 𝑆 ∈ (Base‘(SymGrp‘(1...𝑁)))) → ((pmSgn‘(1...𝑁))‘^◡𝑆) = ((pmSgn‘(1...𝑁))‘𝑆))
109	36, 32, 108	syl2anc 596	. . . . . . . . 9 ⊢ (𝜑 → ((pmSgn‘(1...𝑁))‘^◡𝑆) = ((pmSgn‘(1...𝑁))‘𝑆))
110	21, 30, 23, 13, 14	psgnfzto1st 33733	. . . . . . . . . 10 ⊢ (𝑁 ∈ (1...𝑁) → ((pmSgn‘(1...𝑁))‘𝑆) = ( -1↑(𝑁 + 1)))
111	29, 110	syl 18	. . . . . . . . 9 ⊢ (𝜑 → ((pmSgn‘(1...𝑁))‘𝑆) = ( -1↑(𝑁 + 1)))
112	109, 111	eqtrd 2798	. . . . . . . 8 ⊢ (𝜑 → ((pmSgn‘(1...𝑁))‘^◡𝑆) = ( -1↑(𝑁 + 1)))
113	107, 112	oveq12d 7458	. . . . . . 7 ⊢ (𝜑 → (((pmSgn‘(1...𝑁))‘𝑃) · ((pmSgn‘(1...𝑁))‘^◡𝑆)) = (( -1↑(𝐼 + 1)) · ( -1↑(𝑁 + 1))))
114	105, 113	eqtrd 2798	. . . . . 6 ⊢ (𝜑 → ((pmSgn‘(1...𝑁))‘(𝑃 ∘ ^◡𝑆)) = (( -1↑(𝐼 + 1)) · ( -1↑(𝑁 + 1))))
115	23, 14, 13	psgnco 21916	. . . . . . . 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 1399	. . . . . . 7 ⊢ (𝜑 → ((pmSgn‘(1...𝑁))‘(𝑄 ∘ ^◡𝑇)) = (((pmSgn‘(1...𝑁))‘𝑄) · ((pmSgn‘(1...𝑁))‘^◡𝑇)))
117	21, 47, 23, 13, 14	psgnfzto1st 33733	. . . . . . . . 9 ⊢ (𝐽 ∈ (1...𝑁) → ((pmSgn‘(1...𝑁))‘𝑄) = ( -1↑(𝐽 + 1)))
118	11, 117	syl 18	. . . . . . . 8 ⊢ (𝜑 → ((pmSgn‘(1...𝑁))‘𝑄) = ( -1↑(𝐽 + 1)))
119	23, 14, 13	psgninv 21915	. . . . . . . . . 10 ⊢ (((1...𝑁) ∈ Fin ∧ 𝑇 ∈ (Base‘(SymGrp‘(1...𝑁)))) → ((pmSgn‘(1...𝑁))‘^◡𝑇) = ((pmSgn‘(1...𝑁))‘𝑇))
120	36, 52, 119	syl2anc 596	. . . . . . . . 9 ⊢ (𝜑 → ((pmSgn‘(1...𝑁))‘^◡𝑇) = ((pmSgn‘(1...𝑁))‘𝑇))
121	21, 50, 23, 13, 14	psgnfzto1st 33733	. . . . . . . . . 10 ⊢ (𝑁 ∈ (1...𝑁) → ((pmSgn‘(1...𝑁))‘𝑇) = ( -1↑(𝑁 + 1)))
122	29, 121	syl 18	. . . . . . . . 9 ⊢ (𝜑 → ((pmSgn‘(1...𝑁))‘𝑇) = ( -1↑(𝑁 + 1)))
123	120, 122	eqtrd 2798	. . . . . . . 8 ⊢ (𝜑 → ((pmSgn‘(1...𝑁))‘^◡𝑇) = ( -1↑(𝑁 + 1)))
124	118, 123	oveq12d 7458	. . . . . . 7 ⊢ (𝜑 → (((pmSgn‘(1...𝑁))‘𝑄) · ((pmSgn‘(1...𝑁))‘^◡𝑇)) = (( -1↑(𝐽 + 1)) · ( -1↑(𝑁 + 1))))
125	116, 124	eqtrd 2798	. . . . . 6 ⊢ (𝜑 → ((pmSgn‘(1...𝑁))‘(𝑄 ∘ ^◡𝑇)) = (( -1↑(𝐽 + 1)) · ( -1↑(𝑁 + 1))))
126	114, 125	oveq12d 7458	. . . . 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 11326	. . . . . . . . 9 ⊢ (𝜑 → 1 ∈ ℂ)
128	127	negcld 11680	. . . . . . . 8 ⊢ (𝜑 → -1 ∈ ℂ)
129		fz1ssnn 13713	. . . . . . . . . . 11 ⊢ (1...𝑁) ⊆ ℕ
130	129, 10	sselid 3911	. . . . . . . . . 10 ⊢ (𝜑 → 𝐼 ∈ ℕ)
131	130	nnnn0d 12691	. . . . . . . . 9 ⊢ (𝜑 → 𝐼 ∈ ℕ₀)
132		1nn0 12646	. . . . . . . . . 10 ⊢ 1 ∈ ℕ₀
133	132	a1i 11	. . . . . . . . 9 ⊢ (𝜑 → 1 ∈ ℕ₀)
134	131, 133	nn0addcld 12695	. . . . . . . 8 ⊢ (𝜑 → (𝐼 + 1) ∈ ℕ₀)
135	128, 134	expcld 14313	. . . . . . 7 ⊢ (𝜑 → ( -1↑(𝐼 + 1)) ∈ ℂ)
136	8	nnnn0d 12691	. . . . . . . . 9 ⊢ (𝜑 → 𝑁 ∈ ℕ₀)
137	136, 133	nn0addcld 12695	. . . . . . . 8 ⊢ (𝜑 → (𝑁 + 1) ∈ ℕ₀)
138	128, 137	expcld 14313	. . . . . . 7 ⊢ (𝜑 → ( -1↑(𝑁 + 1)) ∈ ℂ)
139	129, 11	sselid 3911	. . . . . . . . . 10 ⊢ (𝜑 → 𝐽 ∈ ℕ)
140	139	nnnn0d 12691	. . . . . . . . 9 ⊢ (𝜑 → 𝐽 ∈ ℕ₀)
141	140, 133	nn0addcld 12695	. . . . . . . 8 ⊢ (𝜑 → (𝐽 + 1) ∈ ℕ₀)
142	128, 141	expcld 14313	. . . . . . 7 ⊢ (𝜑 → ( -1↑(𝐽 + 1)) ∈ ℂ)
143	135, 138, 142, 138	mul4d 11546	. . . . . 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 14315	. . . . . . . 8 ⊢ (𝜑 → ( -1↑((𝐼 + 1) + (𝐽 + 1))) = (( -1↑(𝐼 + 1)) · ( -1↑(𝐽 + 1))))
145	130	nncnd 12375	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐼 ∈ ℂ)
146	139	nncnd 12375	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐽 ∈ ℂ)
147	145, 127, 146, 127	add4d 11563	. . . . . . . . . . 11 ⊢ (𝜑 → ((𝐼 + 1) + (𝐽 + 1)) = ((𝐼 + 𝐽) + (1 + 1)))
148		1p1e2 12490	. . . . . . . . . . . 12 ⊢ (1 + 1) = 2
149	148	oveq2i 7451	. . . . . . . . . . 11 ⊢ ((𝐼 + 𝐽) + (1 + 1)) = ((𝐼 + 𝐽) + 2)
150	147, 149	eqtrdi 2814	. . . . . . . . . 10 ⊢ (𝜑 → ((𝐼 + 1) + (𝐽 + 1)) = ((𝐼 + 𝐽) + 2))
151	150	oveq2d 7456	. . . . . . . . 9 ⊢ (𝜑 → ( -1↑((𝐼 + 1) + (𝐽 + 1))) = ( -1↑((𝐼 + 𝐽) + 2)))
152		2nn0 12647	. . . . . . . . . . . 12 ⊢ 2 ∈ ℕ₀
153	152	a1i 11	. . . . . . . . . . 11 ⊢ (𝜑 → 2 ∈ ℕ₀)
154	131, 140	nn0addcld 12695	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐼 + 𝐽) ∈ ℕ₀)
155	128, 153, 154	expaddd 14315	. . . . . . . . . 10 ⊢ (𝜑 → ( -1↑((𝐼 + 𝐽) + 2)) = (( -1↑(𝐼 + 𝐽)) · ( -1↑2)))
156		neg1sqe1 14363	. . . . . . . . . . 11 ⊢ ( -1↑2) = 1
157	156	oveq2i 7451	. . . . . . . . . 10 ⊢ (( -1↑(𝐼 + 𝐽)) · ( -1↑2)) = (( -1↑(𝐼 + 𝐽)) · 1)
158	155, 157	eqtrdi 2814	. . . . . . . . 9 ⊢ (𝜑 → ( -1↑((𝐼 + 𝐽) + 2)) = (( -1↑(𝐼 + 𝐽)) · 1))
159	128, 154	expcld 14313	. . . . . . . . . 10 ⊢ (𝜑 → ( -1↑(𝐼 + 𝐽)) ∈ ℂ)
160	159	mulridd 11350	. . . . . . . . 9 ⊢ (𝜑 → (( -1↑(𝐼 + 𝐽)) · 1) = ( -1↑(𝐼 + 𝐽)))
161	151, 158, 160	3eqtrd 2802	. . . . . . . 8 ⊢ (𝜑 → ( -1↑((𝐼 + 1) + (𝐽 + 1))) = ( -1↑(𝐼 + 𝐽)))
162	144, 161	eqtr3d 2800	. . . . . . 7 ⊢ (𝜑 → (( -1↑(𝐼 + 1)) · ( -1↑(𝐽 + 1))) = ( -1↑(𝐼 + 𝐽)))
163	137	nn0zd 12742	. . . . . . . 8 ⊢ (𝜑 → (𝑁 + 1) ∈ ℤ)
164		m1expcl2 14252	. . . . . . . 8 ⊢ ((𝑁 + 1) ∈ ℤ → ( -1↑(𝑁 + 1)) ∈ { -1, 1})
165		1neg1t1neg1 33397	. . . . . . . 8 ⊢ (( -1↑(𝑁 + 1)) ∈ { -1, 1} → (( -1↑(𝑁 + 1)) · ( -1↑(𝑁 + 1))) = 1)
166	163, 164, 165	3syl 19	. . . . . . 7 ⊢ (𝜑 → (( -1↑(𝑁 + 1)) · ( -1↑(𝑁 + 1))) = 1)
167	162, 166	oveq12d 7458	. . . . . 6 ⊢ (𝜑 → ((( -1↑(𝐼 + 1)) · ( -1↑(𝐽 + 1))) · (( -1↑(𝑁 + 1)) · ( -1↑(𝑁 + 1)))) = (( -1↑(𝐼 + 𝐽)) · 1))
168	143, 167, 160	3eqtrd 2802	. . . . 5 ⊢ (𝜑 → ((( -1↑(𝐼 + 1)) · ( -1↑(𝑁 + 1))) · (( -1↑(𝐽 + 1)) · ( -1↑(𝑁 + 1)))) = ( -1↑(𝐼 + 𝐽)))
169	126, 168	eqtrd 2798	. . . 4 ⊢ (𝜑 → (((pmSgn‘(1...𝑁))‘(𝑃 ∘ ^◡𝑆)) · ((pmSgn‘(1...𝑁))‘(𝑄 ∘ ^◡𝑇))) = ( -1↑(𝐼 + 𝐽)))
170	169	fveq2d 6909	. . 3 ⊢ (𝜑 → (𝑍‘(((pmSgn‘(1...𝑁))‘(𝑃 ∘ ^◡𝑆)) · ((pmSgn‘(1...𝑁))‘(𝑄 ∘ ^◡𝑇)))) = (𝑍‘( -1↑(𝐼 + 𝐽))))
171	170	oveq1d 7455	. 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 2798	1 ⊢ (𝜑 → (𝐽(𝐾‘𝑀)𝐼) = ((𝑍‘( -1↑(𝐼 + 𝐽))) · (𝐸‘(𝐼(subMat1‘𝑀)𝐽))))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ∧ wa 401 = wceq 1570 ∈ wcel 2147 ifcif 4485 {cpr 4590 class class class wbr 5106 ↦ cmpt 5189 ^◡ccnv 5654 dom cdm 5655 ∘ ccom 5659 Fun wfun 6549 ⟶wf 6551 ⟶1-1wf1 6552 ⟶1-1ontowf1o 6554 ‘cfv 6555 (class class class)co 7440 ∈ cmpo 7442 Fincfn 8998 1c1 11225 + caddc 11227 · cmul 11229 ≤ cle 11368 − cmin 11565 -cneg 11566 ℕcn 12359 2c2 12421 ℕ₀cn0 12630 ℤcz 12717 ℤ_≥cuz 12989 ...cfz 13663 ↑cexp 14228 Basecbs 17413 +_gcplusg 17454 ._rcmulr 17455 Grpcgrp 19170 inv_gcminusg 19171 SymGrpcsymg 19610 pmSgncpsgn 19730 Ringcrg 20486 CRingccrg 20487 ℤRHomczrh 21832 Mat cmat 22749 maDet cmdat 22926 maAdju cmadu 22974 minMatR1 cminmar1 22975 subMat1csmat 34492

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1945 ax-6 2002 ax-7 2043 ax-8 2149 ax-9 2157 ax-10 2180 ax-11 2197 ax-12 2216 ax-ext 2735 ax-rep 5235 ax-sep 5252 ax-nul 5263 ax-pow 5331 ax-pr 5395 ax-un 7772 ax-cnex 11280 ax-resscn 11281 ax-1cn 11282 ax-icn 11283 ax-addcl 11284 ax-addrcl 11285 ax-mulcl 11286 ax-mulrcl 11287 ax-mulcom 11288 ax-addass 11289 ax-mulass 11290 ax-distr 11291 ax-i2m1 11292 ax-1ne0 11293 ax-1rid 11294 ax-rnegex 11295 ax-rrecex 11296 ax-cnre 11297 ax-pre-lttri 11298 ax-pre-lttrn 11299 ax-pre-ltadd 11300 ax-pre-mulgt0 11301 ax-addf 11303 ax-mulf 11304

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-xor 1542 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2102 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2839 df-nfc 2913 df-ne 2960 df-nel 3066 df-ral 3081 df-rex 3091 df-rmo 3369 df-reu 3370 df-rab 3417 df-v 3456 df-sbc 3743 df-csb 3852 df-ss 3901 df-pss 4017 df-dif 4050 df-in 4094 df-un 4154 df-nul 4282 df-if 4486 df-pw 4562 df-sn 4589 df-pr 4591 df-tp 4593 df-op 4839 df-ot 4862 df-uni 4871 df-int 4911 df-iun 4956 df-iin 4957 df-br 5107 df-opab 5171 df-mpt 5190 df-tr 5216 df-id 5550 df-eprel 5555 df-po 5563 df-so 5564 df-fr 5608 df-se 5609 df-we 5610 df-xp 5661 df-rel 5662 df-cnv 5663 df-co 5664 df-dm 5665 df-rn 5666 df-res 5667 df-ima 5668 df-pred 6319 df-ord 6380 df-on 6381 df-lim 6382 df-suc 6383 df-iota 6509 df-fun 6557 df-fn 6558 df-f 6559 df-f1 6560 df-fo 6561 df-f1o 6562 df-fv 6563 df-isom 6564 df-riota 7397 df-ov 7443 df-oprab 7444 df-mpo 7445 df-of 7714 df-om 7899 df-1st 8022 df-2nd 8023 df-supp 8202 df-tpos 8268 df-frecs 8324 df-wrecs 8355 df-recs 8404 df-rdg 8443 df-1o 8501 df-2o 8502 df-er 8742 df-map 8874 df-pm 8875 df-ixp 8951 df-en 8999 df-dom 9000 df-sdom 9001 df-fin 9002 df-fsupp 9379 df-sup 9458 df-oi 9528 df-card 10044 df-pnf 11369 df-mnf 11370 df-xr 11371 df-ltxr 11372 df-le 11373 df-sub 11567 df-neg 11568 df-div 11998 df-nn 12360 df-2 12429 df-3 12430 df-4 12431 df-5 12432 df-6 12433 df-7 12434 df-8 12435 df-9 12436 df-n0 12631 df-xnn0 12704 df-z 12718 df-dec 12839 df-uz 12990 df-rp 13145 df-fz 13664 df-fzo 13813 df-seq 14169 df-exp 14229 df-hash 14499 df-word 14683 df-lsw 14732 df-concat 14740 df-s1 14767 df-substr 14814 df-pfx 14846 df-splice 14924 df-reverse 14933 df-s2 15024 df-struct 17351 df-sets 17368 df-slot 17386 df-ndx 17398 df-base 17414 df-ress 17435 df-plusg 17467 df-mulr 17468 df-starv 17469 df-sca 17470 df-vsca 17471 df-ip 17472 df-tset 17473 df-ple 17474 df-ds 17476 df-unif 17477 df-hom 17478 df-cco 17479 df-0g 17638 df-gsum 17639 df-prds 17644 df-pws 17646 df-mre 17782 df-mrc 17783 df-acs 17785 df-mgm 18842 df-sgrp 18934 df-mnd 18950 df-mhm 19004 df-submnd 19005 df-efmnd 19091 df-grp 19173 df-minusg 19174 df-mulg 19304 df-subg 19359 df-ghm 19454 df-gim 19499 df-cntz 19558 df-oppg 19587 df-symg 19611 df-pmtr 19683 df-psgn 19732 df-cmn 20023 df-abl 20024 df-mgp 20388 df-rng 20402 df-ur 20435 df-ring 20488 df-cring 20489 df-oppr 20594 df-dvdsr 20614 df-unit 20615 df-invr 20645 df-dvr 20658 df-rhm 20729 df-subrng 20825 df-subrg 20849 df-drng 21009 df-sra 21475 df-rgmod 21476 df-cnfld 21706 df-zring 21780 df-zrh 21836 df-dsmm 22065 df-frlm 22080 df-mat 22750 df-marrep 22900 df-subma 22919 df-mdet 22927 df-madu 22976 df-minmar1 22977 df-smat 34493

This theorem is used by: madjusmdet 34530

W3C validator