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Theorem madjusmdetlem4 30694
Description: Lemma for madjusmdet 30695. (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 · = (.r𝑅)
madjusmdet.z 𝑍 = (ℤRHom‘𝑅)
madjusmdet.e 𝐸 = ((1...(𝑁 − 1)) maDet 𝑅)
madjusmdet.n (𝜑𝑁 ∈ ℕ)
madjusmdet.r (𝜑𝑅 ∈ CRing)
madjusmdet.i (𝜑𝐼 ∈ (1...𝑁))
madjusmdet.j (𝜑𝐽 ∈ (1...𝑁))
madjusmdet.m (𝜑𝑀𝐵)
madjusmdetlem2.p 𝑃 = (𝑖 ∈ (1...𝑁) ↦ if(𝑖 = 1, 𝐼, if(𝑖𝐼, (𝑖 − 1), 𝑖)))
madjusmdetlem2.s 𝑆 = (𝑖 ∈ (1...𝑁) ↦ if(𝑖 = 1, 𝑁, if(𝑖𝑁, (𝑖 − 1), 𝑖)))
madjusmdetlem4.q 𝑄 = (𝑗 ∈ (1...𝑁) ↦ if(𝑗 = 1, 𝐽, if(𝑗𝐽, (𝑗 − 1), 𝑗)))
madjusmdetlem4.t 𝑇 = (𝑗 ∈ (1...𝑁) ↦ if(𝑗 = 1, 𝑁, if(𝑗𝑁, (𝑗 − 1), 𝑗)))
Assertion
Ref Expression
madjusmdetlem4 (𝜑 → (𝐽(𝐾𝑀)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.
StepHypRef 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 · = (.r𝑅)
6 madjusmdet.z . . 3 𝑍 = (ℤRHom‘𝑅)
7 madjusmdet.e . . 3 𝐸 = ((1...(𝑁 − 1)) maDet 𝑅)
8 madjusmdet.n . . 3 (𝜑𝑁 ∈ ℕ)
9 madjusmdet.r . . 3 (𝜑𝑅 ∈ CRing)
10 madjusmdet.i . . 3 (𝜑𝐼 ∈ (1...𝑁))
11 madjusmdet.j . . 3 (𝜑𝐽 ∈ (1...𝑁))
12 madjusmdet.m . . 3 (𝜑𝑀𝐵)
13 eqid 2772 . . 3 (Base‘(SymGrp‘(1...𝑁))) = (Base‘(SymGrp‘(1...𝑁)))
14 eqid 2772 . . 3 (pmSgn‘(1...𝑁)) = (pmSgn‘(1...𝑁))
15 eqid 2772 . . 3 (𝐼(((1...𝑁) minMatR1 𝑅)‘𝑀)𝐽) = (𝐼(((1...𝑁) minMatR1 𝑅)‘𝑀)𝐽)
16 fveq2 6493 . . . . 5 (𝑘 = 𝑖 → ((𝑃𝑆)‘𝑘) = ((𝑃𝑆)‘𝑖))
1716oveq1d 6985 . . . 4 (𝑘 = 𝑖 → (((𝑃𝑆)‘𝑘)(𝐼(((1...𝑁) minMatR1 𝑅)‘𝑀)𝐽)((𝑄𝑇)‘𝑙)) = (((𝑃𝑆)‘𝑖)(𝐼(((1...𝑁) minMatR1 𝑅)‘𝑀)𝐽)((𝑄𝑇)‘𝑙)))
18 fveq2 6493 . . . . 5 (𝑙 = 𝑗 → ((𝑄𝑇)‘𝑙) = ((𝑄𝑇)‘𝑗))
1918oveq2d 6986 . . . 4 (𝑙 = 𝑗 → (((𝑃𝑆)‘𝑖)(𝐼(((1...𝑁) minMatR1 𝑅)‘𝑀)𝐽)((𝑄𝑇)‘𝑙)) = (((𝑃𝑆)‘𝑖)(𝐼(((1...𝑁) minMatR1 𝑅)‘𝑀)𝐽)((𝑄𝑇)‘𝑗)))
2017, 19cbvmpov 7059 . . 3 (𝑘 ∈ (1...𝑁), 𝑙 ∈ (1...𝑁) ↦ (((𝑃𝑆)‘𝑘)(𝐼(((1...𝑁) minMatR1 𝑅)‘𝑀)𝐽)((𝑄𝑇)‘𝑙))) = (𝑖 ∈ (1...𝑁), 𝑗 ∈ (1...𝑁) ↦ (((𝑃𝑆)‘𝑖)(𝐼(((1...𝑁) minMatR1 𝑅)‘𝑀)𝐽)((𝑄𝑇)‘𝑗)))
21 eqid 2772 . . . . . 6 (1...𝑁) = (1...𝑁)
22 madjusmdetlem2.p . . . . . 6 𝑃 = (𝑖 ∈ (1...𝑁) ↦ if(𝑖 = 1, 𝐼, if(𝑖𝐼, (𝑖 − 1), 𝑖)))
23 eqid 2772 . . . . . 6 (SymGrp‘(1...𝑁)) = (SymGrp‘(1...𝑁))
2421, 22, 23, 13fzto1st 30651 . . . . 5 (𝐼 ∈ (1...𝑁) → 𝑃 ∈ (Base‘(SymGrp‘(1...𝑁))))
2510, 24syl 17 . . . 4 (𝜑𝑃 ∈ (Base‘(SymGrp‘(1...𝑁))))
26 nnuz 12088 . . . . . . . . 9 ℕ = (ℤ‘1)
278, 26syl6eleq 2870 . . . . . . . 8 (𝜑𝑁 ∈ (ℤ‘1))
28 eluzfz2 12724 . . . . . . . 8 (𝑁 ∈ (ℤ‘1) → 𝑁 ∈ (1...𝑁))
2927, 28syl 17 . . . . . . 7 (𝜑𝑁 ∈ (1...𝑁))
30 madjusmdetlem2.s . . . . . . . 8 𝑆 = (𝑖 ∈ (1...𝑁) ↦ if(𝑖 = 1, 𝑁, if(𝑖𝑁, (𝑖 − 1), 𝑖)))
3121, 30, 23, 13fzto1st 30651 . . . . . . 7 (𝑁 ∈ (1...𝑁) → 𝑆 ∈ (Base‘(SymGrp‘(1...𝑁))))
3229, 31syl 17 . . . . . 6 (𝜑𝑆 ∈ (Base‘(SymGrp‘(1...𝑁))))
33 eqid 2772 . . . . . . 7 (invg‘(SymGrp‘(1...𝑁))) = (invg‘(SymGrp‘(1...𝑁)))
3423, 13, 33symginv 18281 . . . . . 6 (𝑆 ∈ (Base‘(SymGrp‘(1...𝑁))) → ((invg‘(SymGrp‘(1...𝑁)))‘𝑆) = 𝑆)
3532, 34syl 17 . . . . 5 (𝜑 → ((invg‘(SymGrp‘(1...𝑁)))‘𝑆) = 𝑆)
36 fzfid 13149 . . . . . . 7 (𝜑 → (1...𝑁) ∈ Fin)
3723symggrp 18279 . . . . . . 7 ((1...𝑁) ∈ Fin → (SymGrp‘(1...𝑁)) ∈ Grp)
3836, 37syl 17 . . . . . 6 (𝜑 → (SymGrp‘(1...𝑁)) ∈ Grp)
3913, 33grpinvcl 17928 . . . . . 6 (((SymGrp‘(1...𝑁)) ∈ Grp ∧ 𝑆 ∈ (Base‘(SymGrp‘(1...𝑁)))) → ((invg‘(SymGrp‘(1...𝑁)))‘𝑆) ∈ (Base‘(SymGrp‘(1...𝑁))))
4038, 32, 39syl2anc 576 . . . . 5 (𝜑 → ((invg‘(SymGrp‘(1...𝑁)))‘𝑆) ∈ (Base‘(SymGrp‘(1...𝑁))))
4135, 40eqeltrrd 2861 . . . 4 (𝜑𝑆 ∈ (Base‘(SymGrp‘(1...𝑁))))
42 eqid 2772 . . . . . 6 (+g‘(SymGrp‘(1...𝑁))) = (+g‘(SymGrp‘(1...𝑁)))
4323, 13, 42symgov 18269 . . . . 5 ((𝑃 ∈ (Base‘(SymGrp‘(1...𝑁))) ∧ 𝑆 ∈ (Base‘(SymGrp‘(1...𝑁)))) → (𝑃(+g‘(SymGrp‘(1...𝑁)))𝑆) = (𝑃𝑆))
4423, 13, 42symgcl 18270 . . . . 5 ((𝑃 ∈ (Base‘(SymGrp‘(1...𝑁))) ∧ 𝑆 ∈ (Base‘(SymGrp‘(1...𝑁)))) → (𝑃(+g‘(SymGrp‘(1...𝑁)))𝑆) ∈ (Base‘(SymGrp‘(1...𝑁))))
4543, 44eqeltrrd 2861 . . . 4 ((𝑃 ∈ (Base‘(SymGrp‘(1...𝑁))) ∧ 𝑆 ∈ (Base‘(SymGrp‘(1...𝑁)))) → (𝑃𝑆) ∈ (Base‘(SymGrp‘(1...𝑁))))
4625, 41, 45syl2anc 576 . . 3 (𝜑 → (𝑃𝑆) ∈ (Base‘(SymGrp‘(1...𝑁))))
47 madjusmdetlem4.q . . . . . 6 𝑄 = (𝑗 ∈ (1...𝑁) ↦ if(𝑗 = 1, 𝐽, if(𝑗𝐽, (𝑗 − 1), 𝑗)))
4821, 47, 23, 13fzto1st 30651 . . . . 5 (𝐽 ∈ (1...𝑁) → 𝑄 ∈ (Base‘(SymGrp‘(1...𝑁))))
4911, 48syl 17 . . . 4 (𝜑𝑄 ∈ (Base‘(SymGrp‘(1...𝑁))))
50 madjusmdetlem4.t . . . . . . . 8 𝑇 = (𝑗 ∈ (1...𝑁) ↦ if(𝑗 = 1, 𝑁, if(𝑗𝑁, (𝑗 − 1), 𝑗)))
5121, 50, 23, 13fzto1st 30651 . . . . . . 7 (𝑁 ∈ (1...𝑁) → 𝑇 ∈ (Base‘(SymGrp‘(1...𝑁))))
5229, 51syl 17 . . . . . 6 (𝜑𝑇 ∈ (Base‘(SymGrp‘(1...𝑁))))
5323, 13, 33symginv 18281 . . . . . 6 (𝑇 ∈ (Base‘(SymGrp‘(1...𝑁))) → ((invg‘(SymGrp‘(1...𝑁)))‘𝑇) = 𝑇)
5452, 53syl 17 . . . . 5 (𝜑 → ((invg‘(SymGrp‘(1...𝑁)))‘𝑇) = 𝑇)
5513, 33grpinvcl 17928 . . . . . 6 (((SymGrp‘(1...𝑁)) ∈ Grp ∧ 𝑇 ∈ (Base‘(SymGrp‘(1...𝑁)))) → ((invg‘(SymGrp‘(1...𝑁)))‘𝑇) ∈ (Base‘(SymGrp‘(1...𝑁))))
5638, 52, 55syl2anc 576 . . . . 5 (𝜑 → ((invg‘(SymGrp‘(1...𝑁)))‘𝑇) ∈ (Base‘(SymGrp‘(1...𝑁))))
5754, 56eqeltrrd 2861 . . . 4 (𝜑𝑇 ∈ (Base‘(SymGrp‘(1...𝑁))))
5823, 13, 42symgov 18269 . . . . 5 ((𝑄 ∈ (Base‘(SymGrp‘(1...𝑁))) ∧ 𝑇 ∈ (Base‘(SymGrp‘(1...𝑁)))) → (𝑄(+g‘(SymGrp‘(1...𝑁)))𝑇) = (𝑄𝑇))
5923, 13, 42symgcl 18270 . . . . 5 ((𝑄 ∈ (Base‘(SymGrp‘(1...𝑁))) ∧ 𝑇 ∈ (Base‘(SymGrp‘(1...𝑁)))) → (𝑄(+g‘(SymGrp‘(1...𝑁)))𝑇) ∈ (Base‘(SymGrp‘(1...𝑁))))
6058, 59eqeltrrd 2861 . . . 4 ((𝑄 ∈ (Base‘(SymGrp‘(1...𝑁))) ∧ 𝑇 ∈ (Base‘(SymGrp‘(1...𝑁)))) → (𝑄𝑇) ∈ (Base‘(SymGrp‘(1...𝑁))))
6149, 57, 60syl2anc 576 . . 3 (𝜑 → (𝑄𝑇) ∈ (Base‘(SymGrp‘(1...𝑁))))
6223, 13symgbasf1o 18262 . . . . . . 7 (𝑆 ∈ (Base‘(SymGrp‘(1...𝑁))) → 𝑆:(1...𝑁)–1-1-onto→(1...𝑁))
6332, 62syl 17 . . . . . 6 (𝜑𝑆:(1...𝑁)–1-1-onto→(1...𝑁))
64 f1of1 6437 . . . . . 6 (𝑆:(1...𝑁)–1-1-onto→(1...𝑁) → 𝑆:(1...𝑁)–1-1→(1...𝑁))
65 df-f1 6187 . . . . . . 7 (𝑆:(1...𝑁)–1-1→(1...𝑁) ↔ (𝑆:(1...𝑁)⟶(1...𝑁) ∧ Fun 𝑆))
6665simprbi 489 . . . . . 6 (𝑆:(1...𝑁)–1-1→(1...𝑁) → Fun 𝑆)
6763, 64, 663syl 18 . . . . 5 (𝜑 → Fun 𝑆)
68 f1ocnv 6450 . . . . . . 7 (𝑆:(1...𝑁)–1-1-onto→(1...𝑁) → 𝑆:(1...𝑁)–1-1-onto→(1...𝑁))
69 f1odm 6442 . . . . . . 7 (𝑆:(1...𝑁)–1-1-onto→(1...𝑁) → dom 𝑆 = (1...𝑁))
7063, 68, 693syl 18 . . . . . 6 (𝜑 → dom 𝑆 = (1...𝑁))
7129, 70eleqtrrd 2863 . . . . 5 (𝜑𝑁 ∈ dom 𝑆)
72 fvco 6581 . . . . 5 ((Fun 𝑆𝑁 ∈ dom 𝑆) → ((𝑃𝑆)‘𝑁) = (𝑃‘(𝑆𝑁)))
7367, 71, 72syl2anc 576 . . . 4 (𝜑 → ((𝑃𝑆)‘𝑁) = (𝑃‘(𝑆𝑁)))
7421, 30, 23, 13fzto1stinvn 30652 . . . . . 6 (𝑁 ∈ (1...𝑁) → (𝑆𝑁) = 1)
7529, 74syl 17 . . . . 5 (𝜑 → (𝑆𝑁) = 1)
7675fveq2d 6497 . . . 4 (𝜑 → (𝑃‘(𝑆𝑁)) = (𝑃‘1))
7721, 22fzto1stfv1 30649 . . . . 5 (𝐼 ∈ (1...𝑁) → (𝑃‘1) = 𝐼)
7810, 77syl 17 . . . 4 (𝜑 → (𝑃‘1) = 𝐼)
7973, 76, 783eqtrd 2812 . . 3 (𝜑 → ((𝑃𝑆)‘𝑁) = 𝐼)
8023, 13symgbasf1o 18262 . . . . . . 7 (𝑇 ∈ (Base‘(SymGrp‘(1...𝑁))) → 𝑇:(1...𝑁)–1-1-onto→(1...𝑁))
8152, 80syl 17 . . . . . 6 (𝜑𝑇:(1...𝑁)–1-1-onto→(1...𝑁))
82 f1of1 6437 . . . . . 6 (𝑇:(1...𝑁)–1-1-onto→(1...𝑁) → 𝑇:(1...𝑁)–1-1→(1...𝑁))
83 df-f1 6187 . . . . . . 7 (𝑇:(1...𝑁)–1-1→(1...𝑁) ↔ (𝑇:(1...𝑁)⟶(1...𝑁) ∧ Fun 𝑇))
8483simprbi 489 . . . . . 6 (𝑇:(1...𝑁)–1-1→(1...𝑁) → Fun 𝑇)
8581, 82, 843syl 18 . . . . 5 (𝜑 → Fun 𝑇)
86 f1ocnv 6450 . . . . . . 7 (𝑇:(1...𝑁)–1-1-onto→(1...𝑁) → 𝑇:(1...𝑁)–1-1-onto→(1...𝑁))
87 f1odm 6442 . . . . . . 7 (𝑇:(1...𝑁)–1-1-onto→(1...𝑁) → dom 𝑇 = (1...𝑁))
8881, 86, 873syl 18 . . . . . 6 (𝜑 → dom 𝑇 = (1...𝑁))
8929, 88eleqtrrd 2863 . . . . 5 (𝜑𝑁 ∈ dom 𝑇)
90 fvco 6581 . . . . 5 ((Fun 𝑇𝑁 ∈ dom 𝑇) → ((𝑄𝑇)‘𝑁) = (𝑄‘(𝑇𝑁)))
9185, 89, 90syl2anc 576 . . . 4 (𝜑 → ((𝑄𝑇)‘𝑁) = (𝑄‘(𝑇𝑁)))
9221, 50, 23, 13fzto1stinvn 30652 . . . . . 6 (𝑁 ∈ (1...𝑁) → (𝑇𝑁) = 1)
9329, 92syl 17 . . . . 5 (𝜑 → (𝑇𝑁) = 1)
9493fveq2d 6497 . . . 4 (𝜑 → (𝑄‘(𝑇𝑁)) = (𝑄‘1))
9521, 47fzto1stfv1 30649 . . . . 5 (𝐽 ∈ (1...𝑁) → (𝑄‘1) = 𝐽)
9611, 95syl 17 . . . 4 (𝜑 → (𝑄‘1) = 𝐽)
9791, 94, 963eqtrd 2812 . . 3 (𝜑 → ((𝑄𝑇)‘𝑁) = 𝐽)
98 crngring 19021 . . . . . 6 (𝑅 ∈ CRing → 𝑅 ∈ Ring)
999, 98syl 17 . . . . 5 (𝜑𝑅 ∈ Ring)
1002, 1minmar1cl 20954 . . . . 5 (((𝑅 ∈ Ring ∧ 𝑀𝐵) ∧ (𝐼 ∈ (1...𝑁) ∧ 𝐽 ∈ (1...𝑁))) → (𝐼(((1...𝑁) minMatR1 𝑅)‘𝑀)𝐽) ∈ 𝐵)
10199, 12, 10, 11, 100syl22anc 826 . . . 4 (𝜑 → (𝐼(((1...𝑁) minMatR1 𝑅)‘𝑀)𝐽) ∈ 𝐵)
1021, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 22, 30, 47, 50, 20, 101madjusmdetlem3 30693 . . 3 (𝜑 → (𝐼(subMat1‘(𝐼(((1...𝑁) minMatR1 𝑅)‘𝑀)𝐽))𝐽) = (𝑁(subMat1‘(𝑘 ∈ (1...𝑁), 𝑙 ∈ (1...𝑁) ↦ (((𝑃𝑆)‘𝑘)(𝐼(((1...𝑁) minMatR1 𝑅)‘𝑀)𝐽)((𝑄𝑇)‘𝑙))))𝑁))
1031, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 20, 46, 61, 79, 97, 102madjusmdetlem1 30691 . 2 (𝜑 → (𝐽(𝐾𝑀)𝐼) = ((𝑍‘(((pmSgn‘(1...𝑁))‘(𝑃𝑆)) · ((pmSgn‘(1...𝑁))‘(𝑄𝑇)))) · (𝐸‘(𝐼(subMat1‘𝑀)𝐽))))
10423, 14, 13psgnco 20419 . . . . . . . 8 (((1...𝑁) ∈ Fin ∧ 𝑃 ∈ (Base‘(SymGrp‘(1...𝑁))) ∧ 𝑆 ∈ (Base‘(SymGrp‘(1...𝑁)))) → ((pmSgn‘(1...𝑁))‘(𝑃𝑆)) = (((pmSgn‘(1...𝑁))‘𝑃) · ((pmSgn‘(1...𝑁))‘𝑆)))
10536, 25, 41, 104syl3anc 1351 . . . . . . 7 (𝜑 → ((pmSgn‘(1...𝑁))‘(𝑃𝑆)) = (((pmSgn‘(1...𝑁))‘𝑃) · ((pmSgn‘(1...𝑁))‘𝑆)))
10621, 22, 23, 13, 14psgnfzto1st 30653 . . . . . . . . 9 (𝐼 ∈ (1...𝑁) → ((pmSgn‘(1...𝑁))‘𝑃) = (-1↑(𝐼 + 1)))
10710, 106syl 17 . . . . . . . 8 (𝜑 → ((pmSgn‘(1...𝑁))‘𝑃) = (-1↑(𝐼 + 1)))
10823, 14, 13psgninv 20418 . . . . . . . . . 10 (((1...𝑁) ∈ Fin ∧ 𝑆 ∈ (Base‘(SymGrp‘(1...𝑁)))) → ((pmSgn‘(1...𝑁))‘𝑆) = ((pmSgn‘(1...𝑁))‘𝑆))
10936, 32, 108syl2anc 576 . . . . . . . . 9 (𝜑 → ((pmSgn‘(1...𝑁))‘𝑆) = ((pmSgn‘(1...𝑁))‘𝑆))
11021, 30, 23, 13, 14psgnfzto1st 30653 . . . . . . . . . 10 (𝑁 ∈ (1...𝑁) → ((pmSgn‘(1...𝑁))‘𝑆) = (-1↑(𝑁 + 1)))
11129, 110syl 17 . . . . . . . . 9 (𝜑 → ((pmSgn‘(1...𝑁))‘𝑆) = (-1↑(𝑁 + 1)))
112109, 111eqtrd 2808 . . . . . . . 8 (𝜑 → ((pmSgn‘(1...𝑁))‘𝑆) = (-1↑(𝑁 + 1)))
113107, 112oveq12d 6988 . . . . . . 7 (𝜑 → (((pmSgn‘(1...𝑁))‘𝑃) · ((pmSgn‘(1...𝑁))‘𝑆)) = ((-1↑(𝐼 + 1)) · (-1↑(𝑁 + 1))))
114105, 113eqtrd 2808 . . . . . 6 (𝜑 → ((pmSgn‘(1...𝑁))‘(𝑃𝑆)) = ((-1↑(𝐼 + 1)) · (-1↑(𝑁 + 1))))
11523, 14, 13psgnco 20419 . . . . . . . 8 (((1...𝑁) ∈ Fin ∧ 𝑄 ∈ (Base‘(SymGrp‘(1...𝑁))) ∧ 𝑇 ∈ (Base‘(SymGrp‘(1...𝑁)))) → ((pmSgn‘(1...𝑁))‘(𝑄𝑇)) = (((pmSgn‘(1...𝑁))‘𝑄) · ((pmSgn‘(1...𝑁))‘𝑇)))
11636, 49, 57, 115syl3anc 1351 . . . . . . 7 (𝜑 → ((pmSgn‘(1...𝑁))‘(𝑄𝑇)) = (((pmSgn‘(1...𝑁))‘𝑄) · ((pmSgn‘(1...𝑁))‘𝑇)))
11721, 47, 23, 13, 14psgnfzto1st 30653 . . . . . . . . 9 (𝐽 ∈ (1...𝑁) → ((pmSgn‘(1...𝑁))‘𝑄) = (-1↑(𝐽 + 1)))
11811, 117syl 17 . . . . . . . 8 (𝜑 → ((pmSgn‘(1...𝑁))‘𝑄) = (-1↑(𝐽 + 1)))
11923, 14, 13psgninv 20418 . . . . . . . . . 10 (((1...𝑁) ∈ Fin ∧ 𝑇 ∈ (Base‘(SymGrp‘(1...𝑁)))) → ((pmSgn‘(1...𝑁))‘𝑇) = ((pmSgn‘(1...𝑁))‘𝑇))
12036, 52, 119syl2anc 576 . . . . . . . . 9 (𝜑 → ((pmSgn‘(1...𝑁))‘𝑇) = ((pmSgn‘(1...𝑁))‘𝑇))
12121, 50, 23, 13, 14psgnfzto1st 30653 . . . . . . . . . 10 (𝑁 ∈ (1...𝑁) → ((pmSgn‘(1...𝑁))‘𝑇) = (-1↑(𝑁 + 1)))
12229, 121syl 17 . . . . . . . . 9 (𝜑 → ((pmSgn‘(1...𝑁))‘𝑇) = (-1↑(𝑁 + 1)))
123120, 122eqtrd 2808 . . . . . . . 8 (𝜑 → ((pmSgn‘(1...𝑁))‘𝑇) = (-1↑(𝑁 + 1)))
124118, 123oveq12d 6988 . . . . . . 7 (𝜑 → (((pmSgn‘(1...𝑁))‘𝑄) · ((pmSgn‘(1...𝑁))‘𝑇)) = ((-1↑(𝐽 + 1)) · (-1↑(𝑁 + 1))))
125116, 124eqtrd 2808 . . . . . 6 (𝜑 → ((pmSgn‘(1...𝑁))‘(𝑄𝑇)) = ((-1↑(𝐽 + 1)) · (-1↑(𝑁 + 1))))
126114, 125oveq12d 6988 . . . . 5 (𝜑 → (((pmSgn‘(1...𝑁))‘(𝑃𝑆)) · ((pmSgn‘(1...𝑁))‘(𝑄𝑇))) = (((-1↑(𝐼 + 1)) · (-1↑(𝑁 + 1))) · ((-1↑(𝐽 + 1)) · (-1↑(𝑁 + 1)))))
127 1cnd 10426 . . . . . . . . 9 (𝜑 → 1 ∈ ℂ)
128127negcld 10777 . . . . . . . 8 (𝜑 → -1 ∈ ℂ)
129 fz1ssnn 12747 . . . . . . . . . . 11 (1...𝑁) ⊆ ℕ
130129, 10sseldi 3852 . . . . . . . . . 10 (𝜑𝐼 ∈ ℕ)
131130nnnn0d 11760 . . . . . . . . 9 (𝜑𝐼 ∈ ℕ0)
132 1nn0 11718 . . . . . . . . . 10 1 ∈ ℕ0
133132a1i 11 . . . . . . . . 9 (𝜑 → 1 ∈ ℕ0)
134131, 133nn0addcld 11764 . . . . . . . 8 (𝜑 → (𝐼 + 1) ∈ ℕ0)
135128, 134expcld 13318 . . . . . . 7 (𝜑 → (-1↑(𝐼 + 1)) ∈ ℂ)
1368nnnn0d 11760 . . . . . . . . 9 (𝜑𝑁 ∈ ℕ0)
137136, 133nn0addcld 11764 . . . . . . . 8 (𝜑 → (𝑁 + 1) ∈ ℕ0)
138128, 137expcld 13318 . . . . . . 7 (𝜑 → (-1↑(𝑁 + 1)) ∈ ℂ)
139129, 11sseldi 3852 . . . . . . . . . 10 (𝜑𝐽 ∈ ℕ)
140139nnnn0d 11760 . . . . . . . . 9 (𝜑𝐽 ∈ ℕ0)
141140, 133nn0addcld 11764 . . . . . . . 8 (𝜑 → (𝐽 + 1) ∈ ℕ0)
142128, 141expcld 13318 . . . . . . 7 (𝜑 → (-1↑(𝐽 + 1)) ∈ ℂ)
143135, 138, 142, 138mul4d 10644 . . . . . 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)))))
144128, 141, 134expaddd 13320 . . . . . . . 8 (𝜑 → (-1↑((𝐼 + 1) + (𝐽 + 1))) = ((-1↑(𝐼 + 1)) · (-1↑(𝐽 + 1))))
145130nncnd 11449 . . . . . . . . . . . 12 (𝜑𝐼 ∈ ℂ)
146139nncnd 11449 . . . . . . . . . . . 12 (𝜑𝐽 ∈ ℂ)
147145, 127, 146, 127add4d 10660 . . . . . . . . . . 11 (𝜑 → ((𝐼 + 1) + (𝐽 + 1)) = ((𝐼 + 𝐽) + (1 + 1)))
148 1p1e2 11565 . . . . . . . . . . . 12 (1 + 1) = 2
149148oveq2i 6981 . . . . . . . . . . 11 ((𝐼 + 𝐽) + (1 + 1)) = ((𝐼 + 𝐽) + 2)
150147, 149syl6eq 2824 . . . . . . . . . 10 (𝜑 → ((𝐼 + 1) + (𝐽 + 1)) = ((𝐼 + 𝐽) + 2))
151150oveq2d 6986 . . . . . . . . 9 (𝜑 → (-1↑((𝐼 + 1) + (𝐽 + 1))) = (-1↑((𝐼 + 𝐽) + 2)))
152 2nn0 11719 . . . . . . . . . . . 12 2 ∈ ℕ0
153152a1i 11 . . . . . . . . . . 11 (𝜑 → 2 ∈ ℕ0)
154131, 140nn0addcld 11764 . . . . . . . . . . 11 (𝜑 → (𝐼 + 𝐽) ∈ ℕ0)
155128, 153, 154expaddd 13320 . . . . . . . . . 10 (𝜑 → (-1↑((𝐼 + 𝐽) + 2)) = ((-1↑(𝐼 + 𝐽)) · (-1↑2)))
156 neg1sqe1 13367 . . . . . . . . . . 11 (-1↑2) = 1
157156oveq2i 6981 . . . . . . . . . 10 ((-1↑(𝐼 + 𝐽)) · (-1↑2)) = ((-1↑(𝐼 + 𝐽)) · 1)
158155, 157syl6eq 2824 . . . . . . . . 9 (𝜑 → (-1↑((𝐼 + 𝐽) + 2)) = ((-1↑(𝐼 + 𝐽)) · 1))
159128, 154expcld 13318 . . . . . . . . . 10 (𝜑 → (-1↑(𝐼 + 𝐽)) ∈ ℂ)
160159mulid1d 10449 . . . . . . . . 9 (𝜑 → ((-1↑(𝐼 + 𝐽)) · 1) = (-1↑(𝐼 + 𝐽)))
161151, 158, 1603eqtrd 2812 . . . . . . . 8 (𝜑 → (-1↑((𝐼 + 1) + (𝐽 + 1))) = (-1↑(𝐼 + 𝐽)))
162144, 161eqtr3d 2810 . . . . . . 7 (𝜑 → ((-1↑(𝐼 + 1)) · (-1↑(𝐽 + 1))) = (-1↑(𝐼 + 𝐽)))
163137nn0zd 11891 . . . . . . . 8 (𝜑 → (𝑁 + 1) ∈ ℤ)
164 m1expcl2 13259 . . . . . . . 8 ((𝑁 + 1) ∈ ℤ → (-1↑(𝑁 + 1)) ∈ {-1, 1})
165 1neg1t1neg1 30214 . . . . . . . 8 ((-1↑(𝑁 + 1)) ∈ {-1, 1} → ((-1↑(𝑁 + 1)) · (-1↑(𝑁 + 1))) = 1)
166163, 164, 1653syl 18 . . . . . . 7 (𝜑 → ((-1↑(𝑁 + 1)) · (-1↑(𝑁 + 1))) = 1)
167162, 166oveq12d 6988 . . . . . 6 (𝜑 → (((-1↑(𝐼 + 1)) · (-1↑(𝐽 + 1))) · ((-1↑(𝑁 + 1)) · (-1↑(𝑁 + 1)))) = ((-1↑(𝐼 + 𝐽)) · 1))
168143, 167, 1603eqtrd 2812 . . . . 5 (𝜑 → (((-1↑(𝐼 + 1)) · (-1↑(𝑁 + 1))) · ((-1↑(𝐽 + 1)) · (-1↑(𝑁 + 1)))) = (-1↑(𝐼 + 𝐽)))
169126, 168eqtrd 2808 . . . 4 (𝜑 → (((pmSgn‘(1...𝑁))‘(𝑃𝑆)) · ((pmSgn‘(1...𝑁))‘(𝑄𝑇))) = (-1↑(𝐼 + 𝐽)))
170169fveq2d 6497 . . 3 (𝜑 → (𝑍‘(((pmSgn‘(1...𝑁))‘(𝑃𝑆)) · ((pmSgn‘(1...𝑁))‘(𝑄𝑇)))) = (𝑍‘(-1↑(𝐼 + 𝐽))))
171170oveq1d 6985 . 2 (𝜑 → ((𝑍‘(((pmSgn‘(1...𝑁))‘(𝑃𝑆)) · ((pmSgn‘(1...𝑁))‘(𝑄𝑇)))) · (𝐸‘(𝐼(subMat1‘𝑀)𝐽))) = ((𝑍‘(-1↑(𝐼 + 𝐽))) · (𝐸‘(𝐼(subMat1‘𝑀)𝐽))))
172103, 171eqtrd 2808 1 (𝜑 → (𝐽(𝐾𝑀)𝐼) = ((𝑍‘(-1↑(𝐼 + 𝐽))) · (𝐸‘(𝐼(subMat1‘𝑀)𝐽))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 387   = wceq 1507  wcel 2048  ifcif 4344  {cpr 4437   class class class wbr 4923  cmpt 5002  ccnv 5399  dom cdm 5400  ccom 5404  Fun wfun 6176  wf 6178  1-1wf1 6179  1-1-ontowf1o 6181  cfv 6182  (class class class)co 6970  cmpo 6972  Fincfn 8298  1c1 10328   + caddc 10330   · cmul 10332  cle 10467  cmin 10662  -cneg 10663  cn 11431  2c2 11488  0cn0 11700  cz 11786  cuz 12051  ...cfz 12701  cexp 13237  Basecbs 16329  +gcplusg 16411  .rcmulr 16412  Grpcgrp 17881  invgcminusg 17882  SymGrpcsymg 18256  pmSgncpsgn 18368  Ringcrg 19010  CRingccrg 19011  ℤRHomczrh 20339   Mat cmat 20710   maDet cmdat 20887   maAdju cmadu 20935   minMatR1 cminmar1 20936  subMat1csmat 30657
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1964  ax-8 2050  ax-9 2057  ax-10 2077  ax-11 2091  ax-12 2104  ax-13 2299  ax-ext 2745  ax-rep 5043  ax-sep 5054  ax-nul 5061  ax-pow 5113  ax-pr 5180  ax-un 7273  ax-cnex 10383  ax-resscn 10384  ax-1cn 10385  ax-icn 10386  ax-addcl 10387  ax-addrcl 10388  ax-mulcl 10389  ax-mulrcl 10390  ax-mulcom 10391  ax-addass 10392  ax-mulass 10393  ax-distr 10394  ax-i2m1 10395  ax-1ne0 10396  ax-1rid 10397  ax-rnegex 10398  ax-rrecex 10399  ax-cnre 10400  ax-pre-lttri 10401  ax-pre-lttrn 10402  ax-pre-ltadd 10403  ax-pre-mulgt0 10404  ax-addf 10406  ax-mulf 10407
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3or 1069  df-3an 1070  df-xor 1489  df-tru 1510  df-fal 1520  df-ex 1743  df-nf 1747  df-sb 2014  df-mo 2544  df-eu 2580  df-clab 2754  df-cleq 2765  df-clel 2840  df-nfc 2912  df-ne 2962  df-nel 3068  df-ral 3087  df-rex 3088  df-reu 3089  df-rmo 3090  df-rab 3091  df-v 3411  df-sbc 3678  df-csb 3783  df-dif 3828  df-un 3830  df-in 3832  df-ss 3839  df-pss 3841  df-nul 4174  df-if 4345  df-pw 4418  df-sn 4436  df-pr 4438  df-tp 4440  df-op 4442  df-ot 4444  df-uni 4707  df-int 4744  df-iun 4788  df-iin 4789  df-br 4924  df-opab 4986  df-mpt 5003  df-tr 5025  df-id 5305  df-eprel 5310  df-po 5319  df-so 5320  df-fr 5359  df-se 5360  df-we 5361  df-xp 5406  df-rel 5407  df-cnv 5408  df-co 5409  df-dm 5410  df-rn 5411  df-res 5412  df-ima 5413  df-pred 5980  df-ord 6026  df-on 6027  df-lim 6028  df-suc 6029  df-iota 6146  df-fun 6184  df-fn 6185  df-f 6186  df-f1 6187  df-fo 6188  df-f1o 6189  df-fv 6190  df-isom 6191  df-riota 6931  df-ov 6973  df-oprab 6974  df-mpo 6975  df-of 7221  df-om 7391  df-1st 7494  df-2nd 7495  df-supp 7627  df-tpos 7688  df-wrecs 7743  df-recs 7805  df-rdg 7843  df-1o 7897  df-2o 7898  df-oadd 7901  df-er 8081  df-map 8200  df-pm 8201  df-ixp 8252  df-en 8299  df-dom 8300  df-sdom 8301  df-fin 8302  df-fsupp 8621  df-sup 8693  df-oi 8761  df-card 9154  df-pnf 10468  df-mnf 10469  df-xr 10470  df-ltxr 10471  df-le 10472  df-sub 10664  df-neg 10665  df-div 11091  df-nn 11432  df-2 11496  df-3 11497  df-4 11498  df-5 11499  df-6 11500  df-7 11501  df-8 11502  df-9 11503  df-n0 11701  df-xnn0 11773  df-z 11787  df-dec 11905  df-uz 12052  df-rp 12198  df-fz 12702  df-fzo 12843  df-seq 13178  df-exp 13238  df-hash 13499  df-word 13663  df-lsw 13716  df-concat 13724  df-s1 13749  df-substr 13794  df-pfx 13843  df-splice 13950  df-reverse 13968  df-s2 14062  df-struct 16331  df-ndx 16332  df-slot 16333  df-base 16335  df-sets 16336  df-ress 16337  df-plusg 16424  df-mulr 16425  df-starv 16426  df-sca 16427  df-vsca 16428  df-ip 16429  df-tset 16430  df-ple 16431  df-ds 16433  df-unif 16434  df-hom 16435  df-cco 16436  df-0g 16561  df-gsum 16562  df-prds 16567  df-pws 16569  df-mre 16705  df-mrc 16706  df-acs 16708  df-mgm 17700  df-sgrp 17742  df-mnd 17753  df-mhm 17793  df-submnd 17794  df-grp 17884  df-minusg 17885  df-mulg 18002  df-subg 18050  df-ghm 18117  df-gim 18160  df-cntz 18208  df-oppg 18235  df-symg 18257  df-pmtr 18321  df-psgn 18370  df-cmn 18658  df-abl 18659  df-mgp 18953  df-ur 18965  df-ring 19012  df-cring 19013  df-oppr 19086  df-dvdsr 19104  df-unit 19105  df-invr 19135  df-dvr 19146  df-rnghom 19180  df-drng 19217  df-subrg 19246  df-sra 19656  df-rgmod 19657  df-cnfld 20238  df-zring 20310  df-zrh 20343  df-dsmm 20568  df-frlm 20583  df-mat 20711  df-marrep 20861  df-subma 20880  df-mdet 20888  df-madu 20937  df-minmar1 20938  df-smat 30658
This theorem is referenced by:  madjusmdet  30695
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