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Theorem madjusmdetlem4 32411
Description: Lemma for madjusmdet 32412. (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 2736 . . 3 (Base‘(SymGrp‘(1...𝑁))) = (Base‘(SymGrp‘(1...𝑁)))
14 eqid 2736 . . 3 (pmSgn‘(1...𝑁)) = (pmSgn‘(1...𝑁))
15 eqid 2736 . . 3 (𝐼(((1...𝑁) minMatR1 𝑅)‘𝑀)𝐽) = (𝐼(((1...𝑁) minMatR1 𝑅)‘𝑀)𝐽)
16 fveq2 6842 . . . . 5 (𝑘 = 𝑖 → ((𝑃𝑆)‘𝑘) = ((𝑃𝑆)‘𝑖))
1716oveq1d 7372 . . . 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 6842 . . . . 5 (𝑙 = 𝑗 → ((𝑄𝑇)‘𝑙) = ((𝑄𝑇)‘𝑗))
1918oveq2d 7373 . . . 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 7452 . . 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 2736 . . . . . 6 (1...𝑁) = (1...𝑁)
22 madjusmdetlem2.p . . . . . 6 𝑃 = (𝑖 ∈ (1...𝑁) ↦ if(𝑖 = 1, 𝐼, if(𝑖𝐼, (𝑖 − 1), 𝑖)))
23 eqid 2736 . . . . . 6 (SymGrp‘(1...𝑁)) = (SymGrp‘(1...𝑁))
2421, 22, 23, 13fzto1st 31952 . . . . 5 (𝐼 ∈ (1...𝑁) → 𝑃 ∈ (Base‘(SymGrp‘(1...𝑁))))
2510, 24syl 17 . . . 4 (𝜑𝑃 ∈ (Base‘(SymGrp‘(1...𝑁))))
26 nnuz 12806 . . . . . . . . 9 ℕ = (ℤ‘1)
278, 26eleqtrdi 2848 . . . . . . . 8 (𝜑𝑁 ∈ (ℤ‘1))
28 eluzfz2 13449 . . . . . . . 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 31952 . . . . . . 7 (𝑁 ∈ (1...𝑁) → 𝑆 ∈ (Base‘(SymGrp‘(1...𝑁))))
3229, 31syl 17 . . . . . 6 (𝜑𝑆 ∈ (Base‘(SymGrp‘(1...𝑁))))
33 eqid 2736 . . . . . . 7 (invg‘(SymGrp‘(1...𝑁))) = (invg‘(SymGrp‘(1...𝑁)))
3423, 13, 33symginv 19184 . . . . . 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 13878 . . . . . . 7 (𝜑 → (1...𝑁) ∈ Fin)
3723symggrp 19182 . . . . . . 7 ((1...𝑁) ∈ Fin → (SymGrp‘(1...𝑁)) ∈ Grp)
3836, 37syl 17 . . . . . 6 (𝜑 → (SymGrp‘(1...𝑁)) ∈ Grp)
3913, 33grpinvcl 18798 . . . . . 6 (((SymGrp‘(1...𝑁)) ∈ Grp ∧ 𝑆 ∈ (Base‘(SymGrp‘(1...𝑁)))) → ((invg‘(SymGrp‘(1...𝑁)))‘𝑆) ∈ (Base‘(SymGrp‘(1...𝑁))))
4038, 32, 39syl2anc 584 . . . . 5 (𝜑 → ((invg‘(SymGrp‘(1...𝑁)))‘𝑆) ∈ (Base‘(SymGrp‘(1...𝑁))))
4135, 40eqeltrrd 2839 . . . 4 (𝜑𝑆 ∈ (Base‘(SymGrp‘(1...𝑁))))
42 eqid 2736 . . . . . 6 (+g‘(SymGrp‘(1...𝑁))) = (+g‘(SymGrp‘(1...𝑁)))
4323, 13, 42symgov 19165 . . . . 5 ((𝑃 ∈ (Base‘(SymGrp‘(1...𝑁))) ∧ 𝑆 ∈ (Base‘(SymGrp‘(1...𝑁)))) → (𝑃(+g‘(SymGrp‘(1...𝑁)))𝑆) = (𝑃𝑆))
4423, 13, 42symgcl 19166 . . . . 5 ((𝑃 ∈ (Base‘(SymGrp‘(1...𝑁))) ∧ 𝑆 ∈ (Base‘(SymGrp‘(1...𝑁)))) → (𝑃(+g‘(SymGrp‘(1...𝑁)))𝑆) ∈ (Base‘(SymGrp‘(1...𝑁))))
4543, 44eqeltrrd 2839 . . . 4 ((𝑃 ∈ (Base‘(SymGrp‘(1...𝑁))) ∧ 𝑆 ∈ (Base‘(SymGrp‘(1...𝑁)))) → (𝑃𝑆) ∈ (Base‘(SymGrp‘(1...𝑁))))
4625, 41, 45syl2anc 584 . . 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 31952 . . . . 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 31952 . . . . . . 7 (𝑁 ∈ (1...𝑁) → 𝑇 ∈ (Base‘(SymGrp‘(1...𝑁))))
5229, 51syl 17 . . . . . 6 (𝜑𝑇 ∈ (Base‘(SymGrp‘(1...𝑁))))
5323, 13, 33symginv 19184 . . . . . 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 18798 . . . . . 6 (((SymGrp‘(1...𝑁)) ∈ Grp ∧ 𝑇 ∈ (Base‘(SymGrp‘(1...𝑁)))) → ((invg‘(SymGrp‘(1...𝑁)))‘𝑇) ∈ (Base‘(SymGrp‘(1...𝑁))))
5638, 52, 55syl2anc 584 . . . . 5 (𝜑 → ((invg‘(SymGrp‘(1...𝑁)))‘𝑇) ∈ (Base‘(SymGrp‘(1...𝑁))))
5754, 56eqeltrrd 2839 . . . 4 (𝜑𝑇 ∈ (Base‘(SymGrp‘(1...𝑁))))
5823, 13, 42symgov 19165 . . . . 5 ((𝑄 ∈ (Base‘(SymGrp‘(1...𝑁))) ∧ 𝑇 ∈ (Base‘(SymGrp‘(1...𝑁)))) → (𝑄(+g‘(SymGrp‘(1...𝑁)))𝑇) = (𝑄𝑇))
5923, 13, 42symgcl 19166 . . . . 5 ((𝑄 ∈ (Base‘(SymGrp‘(1...𝑁))) ∧ 𝑇 ∈ (Base‘(SymGrp‘(1...𝑁)))) → (𝑄(+g‘(SymGrp‘(1...𝑁)))𝑇) ∈ (Base‘(SymGrp‘(1...𝑁))))
6058, 59eqeltrrd 2839 . . . 4 ((𝑄 ∈ (Base‘(SymGrp‘(1...𝑁))) ∧ 𝑇 ∈ (Base‘(SymGrp‘(1...𝑁)))) → (𝑄𝑇) ∈ (Base‘(SymGrp‘(1...𝑁))))
6149, 57, 60syl2anc 584 . . 3 (𝜑 → (𝑄𝑇) ∈ (Base‘(SymGrp‘(1...𝑁))))
6223, 13symgbasf1o 19156 . . . . . . 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 6783 . . . . . 6 (𝑆:(1...𝑁)–1-1-onto→(1...𝑁) → 𝑆:(1...𝑁)–1-1→(1...𝑁))
65 df-f1 6501 . . . . . . 7 (𝑆:(1...𝑁)–1-1→(1...𝑁) ↔ (𝑆:(1...𝑁)⟶(1...𝑁) ∧ Fun 𝑆))
6665simprbi 497 . . . . . 6 (𝑆:(1...𝑁)–1-1→(1...𝑁) → Fun 𝑆)
6763, 64, 663syl 18 . . . . 5 (𝜑 → Fun 𝑆)
68 f1ocnv 6796 . . . . . . 7 (𝑆:(1...𝑁)–1-1-onto→(1...𝑁) → 𝑆:(1...𝑁)–1-1-onto→(1...𝑁))
69 f1odm 6788 . . . . . . 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 2841 . . . . 5 (𝜑𝑁 ∈ dom 𝑆)
72 fvco 6939 . . . . 5 ((Fun 𝑆𝑁 ∈ dom 𝑆) → ((𝑃𝑆)‘𝑁) = (𝑃‘(𝑆𝑁)))
7367, 71, 72syl2anc 584 . . . 4 (𝜑 → ((𝑃𝑆)‘𝑁) = (𝑃‘(𝑆𝑁)))
7421, 30, 23, 13fzto1stinvn 31953 . . . . . 6 (𝑁 ∈ (1...𝑁) → (𝑆𝑁) = 1)
7529, 74syl 17 . . . . 5 (𝜑 → (𝑆𝑁) = 1)
7675fveq2d 6846 . . . 4 (𝜑 → (𝑃‘(𝑆𝑁)) = (𝑃‘1))
7721, 22fzto1stfv1 31950 . . . . 5 (𝐼 ∈ (1...𝑁) → (𝑃‘1) = 𝐼)
7810, 77syl 17 . . . 4 (𝜑 → (𝑃‘1) = 𝐼)
7973, 76, 783eqtrd 2780 . . 3 (𝜑 → ((𝑃𝑆)‘𝑁) = 𝐼)
8023, 13symgbasf1o 19156 . . . . . . 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 6783 . . . . . 6 (𝑇:(1...𝑁)–1-1-onto→(1...𝑁) → 𝑇:(1...𝑁)–1-1→(1...𝑁))
83 df-f1 6501 . . . . . . 7 (𝑇:(1...𝑁)–1-1→(1...𝑁) ↔ (𝑇:(1...𝑁)⟶(1...𝑁) ∧ Fun 𝑇))
8483simprbi 497 . . . . . 6 (𝑇:(1...𝑁)–1-1→(1...𝑁) → Fun 𝑇)
8581, 82, 843syl 18 . . . . 5 (𝜑 → Fun 𝑇)
86 f1ocnv 6796 . . . . . . 7 (𝑇:(1...𝑁)–1-1-onto→(1...𝑁) → 𝑇:(1...𝑁)–1-1-onto→(1...𝑁))
87 f1odm 6788 . . . . . . 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 2841 . . . . 5 (𝜑𝑁 ∈ dom 𝑇)
90 fvco 6939 . . . . 5 ((Fun 𝑇𝑁 ∈ dom 𝑇) → ((𝑄𝑇)‘𝑁) = (𝑄‘(𝑇𝑁)))
9185, 89, 90syl2anc 584 . . . 4 (𝜑 → ((𝑄𝑇)‘𝑁) = (𝑄‘(𝑇𝑁)))
9221, 50, 23, 13fzto1stinvn 31953 . . . . . 6 (𝑁 ∈ (1...𝑁) → (𝑇𝑁) = 1)
9329, 92syl 17 . . . . 5 (𝜑 → (𝑇𝑁) = 1)
9493fveq2d 6846 . . . 4 (𝜑 → (𝑄‘(𝑇𝑁)) = (𝑄‘1))
9521, 47fzto1stfv1 31950 . . . . 5 (𝐽 ∈ (1...𝑁) → (𝑄‘1) = 𝐽)
9611, 95syl 17 . . . 4 (𝜑 → (𝑄‘1) = 𝐽)
9791, 94, 963eqtrd 2780 . . 3 (𝜑 → ((𝑄𝑇)‘𝑁) = 𝐽)
98 crngring 19976 . . . . . 6 (𝑅 ∈ CRing → 𝑅 ∈ Ring)
999, 98syl 17 . . . . 5 (𝜑𝑅 ∈ Ring)
1002, 1minmar1cl 22000 . . . . 5 (((𝑅 ∈ Ring ∧ 𝑀𝐵) ∧ (𝐼 ∈ (1...𝑁) ∧ 𝐽 ∈ (1...𝑁))) → (𝐼(((1...𝑁) minMatR1 𝑅)‘𝑀)𝐽) ∈ 𝐵)
10199, 12, 10, 11, 100syl22anc 837 . . . 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 32410 . . 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 32408 . 2 (𝜑 → (𝐽(𝐾𝑀)𝐼) = ((𝑍‘(((pmSgn‘(1...𝑁))‘(𝑃𝑆)) · ((pmSgn‘(1...𝑁))‘(𝑄𝑇)))) · (𝐸‘(𝐼(subMat1‘𝑀)𝐽))))
10423, 14, 13psgnco 20987 . . . . . . . 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 1371 . . . . . . 7 (𝜑 → ((pmSgn‘(1...𝑁))‘(𝑃𝑆)) = (((pmSgn‘(1...𝑁))‘𝑃) · ((pmSgn‘(1...𝑁))‘𝑆)))
10621, 22, 23, 13, 14psgnfzto1st 31954 . . . . . . . . 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 20986 . . . . . . . . . 10 (((1...𝑁) ∈ Fin ∧ 𝑆 ∈ (Base‘(SymGrp‘(1...𝑁)))) → ((pmSgn‘(1...𝑁))‘𝑆) = ((pmSgn‘(1...𝑁))‘𝑆))
10936, 32, 108syl2anc 584 . . . . . . . . 9 (𝜑 → ((pmSgn‘(1...𝑁))‘𝑆) = ((pmSgn‘(1...𝑁))‘𝑆))
11021, 30, 23, 13, 14psgnfzto1st 31954 . . . . . . . . . 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 2776 . . . . . . . 8 (𝜑 → ((pmSgn‘(1...𝑁))‘𝑆) = (-1↑(𝑁 + 1)))
113107, 112oveq12d 7375 . . . . . . 7 (𝜑 → (((pmSgn‘(1...𝑁))‘𝑃) · ((pmSgn‘(1...𝑁))‘𝑆)) = ((-1↑(𝐼 + 1)) · (-1↑(𝑁 + 1))))
114105, 113eqtrd 2776 . . . . . 6 (𝜑 → ((pmSgn‘(1...𝑁))‘(𝑃𝑆)) = ((-1↑(𝐼 + 1)) · (-1↑(𝑁 + 1))))
11523, 14, 13psgnco 20987 . . . . . . . 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 1371 . . . . . . 7 (𝜑 → ((pmSgn‘(1...𝑁))‘(𝑄𝑇)) = (((pmSgn‘(1...𝑁))‘𝑄) · ((pmSgn‘(1...𝑁))‘𝑇)))
11721, 47, 23, 13, 14psgnfzto1st 31954 . . . . . . . . 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 20986 . . . . . . . . . 10 (((1...𝑁) ∈ Fin ∧ 𝑇 ∈ (Base‘(SymGrp‘(1...𝑁)))) → ((pmSgn‘(1...𝑁))‘𝑇) = ((pmSgn‘(1...𝑁))‘𝑇))
12036, 52, 119syl2anc 584 . . . . . . . . 9 (𝜑 → ((pmSgn‘(1...𝑁))‘𝑇) = ((pmSgn‘(1...𝑁))‘𝑇))
12121, 50, 23, 13, 14psgnfzto1st 31954 . . . . . . . . . 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 2776 . . . . . . . 8 (𝜑 → ((pmSgn‘(1...𝑁))‘𝑇) = (-1↑(𝑁 + 1)))
124118, 123oveq12d 7375 . . . . . . 7 (𝜑 → (((pmSgn‘(1...𝑁))‘𝑄) · ((pmSgn‘(1...𝑁))‘𝑇)) = ((-1↑(𝐽 + 1)) · (-1↑(𝑁 + 1))))
125116, 124eqtrd 2776 . . . . . 6 (𝜑 → ((pmSgn‘(1...𝑁))‘(𝑄𝑇)) = ((-1↑(𝐽 + 1)) · (-1↑(𝑁 + 1))))
126114, 125oveq12d 7375 . . . . 5 (𝜑 → (((pmSgn‘(1...𝑁))‘(𝑃𝑆)) · ((pmSgn‘(1...𝑁))‘(𝑄𝑇))) = (((-1↑(𝐼 + 1)) · (-1↑(𝑁 + 1))) · ((-1↑(𝐽 + 1)) · (-1↑(𝑁 + 1)))))
127 1cnd 11150 . . . . . . . . 9 (𝜑 → 1 ∈ ℂ)
128127negcld 11499 . . . . . . . 8 (𝜑 → -1 ∈ ℂ)
129 fz1ssnn 13472 . . . . . . . . . . 11 (1...𝑁) ⊆ ℕ
130129, 10sselid 3942 . . . . . . . . . 10 (𝜑𝐼 ∈ ℕ)
131130nnnn0d 12473 . . . . . . . . 9 (𝜑𝐼 ∈ ℕ0)
132 1nn0 12429 . . . . . . . . . 10 1 ∈ ℕ0
133132a1i 11 . . . . . . . . 9 (𝜑 → 1 ∈ ℕ0)
134131, 133nn0addcld 12477 . . . . . . . 8 (𝜑 → (𝐼 + 1) ∈ ℕ0)
135128, 134expcld 14051 . . . . . . 7 (𝜑 → (-1↑(𝐼 + 1)) ∈ ℂ)
1368nnnn0d 12473 . . . . . . . . 9 (𝜑𝑁 ∈ ℕ0)
137136, 133nn0addcld 12477 . . . . . . . 8 (𝜑 → (𝑁 + 1) ∈ ℕ0)
138128, 137expcld 14051 . . . . . . 7 (𝜑 → (-1↑(𝑁 + 1)) ∈ ℂ)
139129, 11sselid 3942 . . . . . . . . . 10 (𝜑𝐽 ∈ ℕ)
140139nnnn0d 12473 . . . . . . . . 9 (𝜑𝐽 ∈ ℕ0)
141140, 133nn0addcld 12477 . . . . . . . 8 (𝜑 → (𝐽 + 1) ∈ ℕ0)
142128, 141expcld 14051 . . . . . . 7 (𝜑 → (-1↑(𝐽 + 1)) ∈ ℂ)
143135, 138, 142, 138mul4d 11367 . . . . . 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 14053 . . . . . . . 8 (𝜑 → (-1↑((𝐼 + 1) + (𝐽 + 1))) = ((-1↑(𝐼 + 1)) · (-1↑(𝐽 + 1))))
145130nncnd 12169 . . . . . . . . . . . 12 (𝜑𝐼 ∈ ℂ)
146139nncnd 12169 . . . . . . . . . . . 12 (𝜑𝐽 ∈ ℂ)
147145, 127, 146, 127add4d 11383 . . . . . . . . . . 11 (𝜑 → ((𝐼 + 1) + (𝐽 + 1)) = ((𝐼 + 𝐽) + (1 + 1)))
148 1p1e2 12278 . . . . . . . . . . . 12 (1 + 1) = 2
149148oveq2i 7368 . . . . . . . . . . 11 ((𝐼 + 𝐽) + (1 + 1)) = ((𝐼 + 𝐽) + 2)
150147, 149eqtrdi 2792 . . . . . . . . . 10 (𝜑 → ((𝐼 + 1) + (𝐽 + 1)) = ((𝐼 + 𝐽) + 2))
151150oveq2d 7373 . . . . . . . . 9 (𝜑 → (-1↑((𝐼 + 1) + (𝐽 + 1))) = (-1↑((𝐼 + 𝐽) + 2)))
152 2nn0 12430 . . . . . . . . . . . 12 2 ∈ ℕ0
153152a1i 11 . . . . . . . . . . 11 (𝜑 → 2 ∈ ℕ0)
154131, 140nn0addcld 12477 . . . . . . . . . . 11 (𝜑 → (𝐼 + 𝐽) ∈ ℕ0)
155128, 153, 154expaddd 14053 . . . . . . . . . 10 (𝜑 → (-1↑((𝐼 + 𝐽) + 2)) = ((-1↑(𝐼 + 𝐽)) · (-1↑2)))
156 neg1sqe1 14100 . . . . . . . . . . 11 (-1↑2) = 1
157156oveq2i 7368 . . . . . . . . . 10 ((-1↑(𝐼 + 𝐽)) · (-1↑2)) = ((-1↑(𝐼 + 𝐽)) · 1)
158155, 157eqtrdi 2792 . . . . . . . . 9 (𝜑 → (-1↑((𝐼 + 𝐽) + 2)) = ((-1↑(𝐼 + 𝐽)) · 1))
159128, 154expcld 14051 . . . . . . . . . 10 (𝜑 → (-1↑(𝐼 + 𝐽)) ∈ ℂ)
160159mulid1d 11172 . . . . . . . . 9 (𝜑 → ((-1↑(𝐼 + 𝐽)) · 1) = (-1↑(𝐼 + 𝐽)))
161151, 158, 1603eqtrd 2780 . . . . . . . 8 (𝜑 → (-1↑((𝐼 + 1) + (𝐽 + 1))) = (-1↑(𝐼 + 𝐽)))
162144, 161eqtr3d 2778 . . . . . . 7 (𝜑 → ((-1↑(𝐼 + 1)) · (-1↑(𝐽 + 1))) = (-1↑(𝐼 + 𝐽)))
163137nn0zd 12525 . . . . . . . 8 (𝜑 → (𝑁 + 1) ∈ ℤ)
164 m1expcl2 13991 . . . . . . . 8 ((𝑁 + 1) ∈ ℤ → (-1↑(𝑁 + 1)) ∈ {-1, 1})
165 1neg1t1neg1 31654 . . . . . . . 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 7375 . . . . . 6 (𝜑 → (((-1↑(𝐼 + 1)) · (-1↑(𝐽 + 1))) · ((-1↑(𝑁 + 1)) · (-1↑(𝑁 + 1)))) = ((-1↑(𝐼 + 𝐽)) · 1))
168143, 167, 1603eqtrd 2780 . . . . 5 (𝜑 → (((-1↑(𝐼 + 1)) · (-1↑(𝑁 + 1))) · ((-1↑(𝐽 + 1)) · (-1↑(𝑁 + 1)))) = (-1↑(𝐼 + 𝐽)))
169126, 168eqtrd 2776 . . . 4 (𝜑 → (((pmSgn‘(1...𝑁))‘(𝑃𝑆)) · ((pmSgn‘(1...𝑁))‘(𝑄𝑇))) = (-1↑(𝐼 + 𝐽)))
170169fveq2d 6846 . . 3 (𝜑 → (𝑍‘(((pmSgn‘(1...𝑁))‘(𝑃𝑆)) · ((pmSgn‘(1...𝑁))‘(𝑄𝑇)))) = (𝑍‘(-1↑(𝐼 + 𝐽))))
171170oveq1d 7372 . 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 2776 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 396   = wceq 1541  wcel 2106  ifcif 4486  {cpr 4588   class class class wbr 5105  cmpt 5188  ccnv 5632  dom cdm 5633  ccom 5637  Fun wfun 6490  wf 6492  1-1wf1 6493  1-1-ontowf1o 6495  cfv 6496  (class class class)co 7357  cmpo 7359  Fincfn 8883  1c1 11052   + caddc 11054   · cmul 11056  cle 11190  cmin 11385  -cneg 11386  cn 12153  2c2 12208  0cn0 12413  cz 12499  cuz 12763  ...cfz 13424  cexp 13967  Basecbs 17083  +gcplusg 17133  .rcmulr 17134  Grpcgrp 18748  invgcminusg 18749  SymGrpcsymg 19148  pmSgncpsgn 19271  Ringcrg 19964  CRingccrg 19965  ℤRHomczrh 20900   Mat cmat 21754   maDet cmdat 21933   maAdju cmadu 21981   minMatR1 cminmar1 21982  subMat1csmat 32374
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-cnex 11107  ax-resscn 11108  ax-1cn 11109  ax-icn 11110  ax-addcl 11111  ax-addrcl 11112  ax-mulcl 11113  ax-mulrcl 11114  ax-mulcom 11115  ax-addass 11116  ax-mulass 11117  ax-distr 11118  ax-i2m1 11119  ax-1ne0 11120  ax-1rid 11121  ax-rnegex 11122  ax-rrecex 11123  ax-cnre 11124  ax-pre-lttri 11125  ax-pre-lttrn 11126  ax-pre-ltadd 11127  ax-pre-mulgt0 11128  ax-addf 11130  ax-mulf 11131
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-xor 1510  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-rmo 3353  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-tp 4591  df-op 4593  df-ot 4595  df-uni 4866  df-int 4908  df-iun 4956  df-iin 4957  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-se 5589  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-isom 6505  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-of 7617  df-om 7803  df-1st 7921  df-2nd 7922  df-supp 8093  df-tpos 8157  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-1o 8412  df-2o 8413  df-er 8648  df-map 8767  df-pm 8768  df-ixp 8836  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-fsupp 9306  df-sup 9378  df-oi 9446  df-card 9875  df-pnf 11191  df-mnf 11192  df-xr 11193  df-ltxr 11194  df-le 11195  df-sub 11387  df-neg 11388  df-div 11813  df-nn 12154  df-2 12216  df-3 12217  df-4 12218  df-5 12219  df-6 12220  df-7 12221  df-8 12222  df-9 12223  df-n0 12414  df-xnn0 12486  df-z 12500  df-dec 12619  df-uz 12764  df-rp 12916  df-fz 13425  df-fzo 13568  df-seq 13907  df-exp 13968  df-hash 14231  df-word 14403  df-lsw 14451  df-concat 14459  df-s1 14484  df-substr 14529  df-pfx 14559  df-splice 14638  df-reverse 14647  df-s2 14737  df-struct 17019  df-sets 17036  df-slot 17054  df-ndx 17066  df-base 17084  df-ress 17113  df-plusg 17146  df-mulr 17147  df-starv 17148  df-sca 17149  df-vsca 17150  df-ip 17151  df-tset 17152  df-ple 17153  df-ds 17155  df-unif 17156  df-hom 17157  df-cco 17158  df-0g 17323  df-gsum 17324  df-prds 17329  df-pws 17331  df-mre 17466  df-mrc 17467  df-acs 17469  df-mgm 18497  df-sgrp 18546  df-mnd 18557  df-mhm 18601  df-submnd 18602  df-efmnd 18679  df-grp 18751  df-minusg 18752  df-mulg 18873  df-subg 18925  df-ghm 19006  df-gim 19049  df-cntz 19097  df-oppg 19124  df-symg 19149  df-pmtr 19224  df-psgn 19273  df-cmn 19564  df-abl 19565  df-mgp 19897  df-ur 19914  df-ring 19966  df-cring 19967  df-oppr 20049  df-dvdsr 20070  df-unit 20071  df-invr 20101  df-dvr 20112  df-rnghom 20146  df-drng 20187  df-subrg 20220  df-sra 20633  df-rgmod 20634  df-cnfld 20797  df-zring 20870  df-zrh 20904  df-dsmm 21138  df-frlm 21153  df-mat 21755  df-marrep 21907  df-subma 21926  df-mdet 21934  df-madu 21983  df-minmar1 21984  df-smat 32375
This theorem is referenced by:  madjusmdet  32412
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