Users' Mathboxes Mathbox for Thierry Arnoux < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  madjusmdetlem4 Structured version   Visualization version   GIF version

Theorem madjusmdetlem4 32076
Description: Lemma for madjusmdet 32077. (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 2737 . . 3 (Base‘(SymGrp‘(1...𝑁))) = (Base‘(SymGrp‘(1...𝑁)))
14 eqid 2737 . . 3 (pmSgn‘(1...𝑁)) = (pmSgn‘(1...𝑁))
15 eqid 2737 . . 3 (𝐼(((1...𝑁) minMatR1 𝑅)‘𝑀)𝐽) = (𝐼(((1...𝑁) minMatR1 𝑅)‘𝑀)𝐽)
16 fveq2 6829 . . . . 5 (𝑘 = 𝑖 → ((𝑃𝑆)‘𝑘) = ((𝑃𝑆)‘𝑖))
1716oveq1d 7356 . . . 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 6829 . . . . 5 (𝑙 = 𝑗 → ((𝑄𝑇)‘𝑙) = ((𝑄𝑇)‘𝑗))
1918oveq2d 7357 . . . 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 7436 . . 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 2737 . . . . . 6 (1...𝑁) = (1...𝑁)
22 madjusmdetlem2.p . . . . . 6 𝑃 = (𝑖 ∈ (1...𝑁) ↦ if(𝑖 = 1, 𝐼, if(𝑖𝐼, (𝑖 − 1), 𝑖)))
23 eqid 2737 . . . . . 6 (SymGrp‘(1...𝑁)) = (SymGrp‘(1...𝑁))
2421, 22, 23, 13fzto1st 31655 . . . . 5 (𝐼 ∈ (1...𝑁) → 𝑃 ∈ (Base‘(SymGrp‘(1...𝑁))))
2510, 24syl 17 . . . 4 (𝜑𝑃 ∈ (Base‘(SymGrp‘(1...𝑁))))
26 nnuz 12726 . . . . . . . . 9 ℕ = (ℤ‘1)
278, 26eleqtrdi 2848 . . . . . . . 8 (𝜑𝑁 ∈ (ℤ‘1))
28 eluzfz2 13369 . . . . . . . 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 31655 . . . . . . 7 (𝑁 ∈ (1...𝑁) → 𝑆 ∈ (Base‘(SymGrp‘(1...𝑁))))
3229, 31syl 17 . . . . . 6 (𝜑𝑆 ∈ (Base‘(SymGrp‘(1...𝑁))))
33 eqid 2737 . . . . . . 7 (invg‘(SymGrp‘(1...𝑁))) = (invg‘(SymGrp‘(1...𝑁)))
3423, 13, 33symginv 19106 . . . . . 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 13798 . . . . . . 7 (𝜑 → (1...𝑁) ∈ Fin)
3723symggrp 19104 . . . . . . 7 ((1...𝑁) ∈ Fin → (SymGrp‘(1...𝑁)) ∈ Grp)
3836, 37syl 17 . . . . . 6 (𝜑 → (SymGrp‘(1...𝑁)) ∈ Grp)
3913, 33grpinvcl 18723 . . . . . 6 (((SymGrp‘(1...𝑁)) ∈ Grp ∧ 𝑆 ∈ (Base‘(SymGrp‘(1...𝑁)))) → ((invg‘(SymGrp‘(1...𝑁)))‘𝑆) ∈ (Base‘(SymGrp‘(1...𝑁))))
4038, 32, 39syl2anc 585 . . . . 5 (𝜑 → ((invg‘(SymGrp‘(1...𝑁)))‘𝑆) ∈ (Base‘(SymGrp‘(1...𝑁))))
4135, 40eqeltrrd 2839 . . . 4 (𝜑𝑆 ∈ (Base‘(SymGrp‘(1...𝑁))))
42 eqid 2737 . . . . . 6 (+g‘(SymGrp‘(1...𝑁))) = (+g‘(SymGrp‘(1...𝑁)))
4323, 13, 42symgov 19087 . . . . 5 ((𝑃 ∈ (Base‘(SymGrp‘(1...𝑁))) ∧ 𝑆 ∈ (Base‘(SymGrp‘(1...𝑁)))) → (𝑃(+g‘(SymGrp‘(1...𝑁)))𝑆) = (𝑃𝑆))
4423, 13, 42symgcl 19088 . . . . 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 585 . . 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 31655 . . . . 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 31655 . . . . . . 7 (𝑁 ∈ (1...𝑁) → 𝑇 ∈ (Base‘(SymGrp‘(1...𝑁))))
5229, 51syl 17 . . . . . 6 (𝜑𝑇 ∈ (Base‘(SymGrp‘(1...𝑁))))
5323, 13, 33symginv 19106 . . . . . 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 18723 . . . . . 6 (((SymGrp‘(1...𝑁)) ∈ Grp ∧ 𝑇 ∈ (Base‘(SymGrp‘(1...𝑁)))) → ((invg‘(SymGrp‘(1...𝑁)))‘𝑇) ∈ (Base‘(SymGrp‘(1...𝑁))))
5638, 52, 55syl2anc 585 . . . . 5 (𝜑 → ((invg‘(SymGrp‘(1...𝑁)))‘𝑇) ∈ (Base‘(SymGrp‘(1...𝑁))))
5754, 56eqeltrrd 2839 . . . 4 (𝜑𝑇 ∈ (Base‘(SymGrp‘(1...𝑁))))
5823, 13, 42symgov 19087 . . . . 5 ((𝑄 ∈ (Base‘(SymGrp‘(1...𝑁))) ∧ 𝑇 ∈ (Base‘(SymGrp‘(1...𝑁)))) → (𝑄(+g‘(SymGrp‘(1...𝑁)))𝑇) = (𝑄𝑇))
5923, 13, 42symgcl 19088 . . . . 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 585 . . 3 (𝜑 → (𝑄𝑇) ∈ (Base‘(SymGrp‘(1...𝑁))))
6223, 13symgbasf1o 19078 . . . . . . 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 6770 . . . . . 6 (𝑆:(1...𝑁)–1-1-onto→(1...𝑁) → 𝑆:(1...𝑁)–1-1→(1...𝑁))
65 df-f1 6488 . . . . . . 7 (𝑆:(1...𝑁)–1-1→(1...𝑁) ↔ (𝑆:(1...𝑁)⟶(1...𝑁) ∧ Fun 𝑆))
6665simprbi 498 . . . . . 6 (𝑆:(1...𝑁)–1-1→(1...𝑁) → Fun 𝑆)
6763, 64, 663syl 18 . . . . 5 (𝜑 → Fun 𝑆)
68 f1ocnv 6783 . . . . . . 7 (𝑆:(1...𝑁)–1-1-onto→(1...𝑁) → 𝑆:(1...𝑁)–1-1-onto→(1...𝑁))
69 f1odm 6775 . . . . . . 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 6926 . . . . 5 ((Fun 𝑆𝑁 ∈ dom 𝑆) → ((𝑃𝑆)‘𝑁) = (𝑃‘(𝑆𝑁)))
7367, 71, 72syl2anc 585 . . . 4 (𝜑 → ((𝑃𝑆)‘𝑁) = (𝑃‘(𝑆𝑁)))
7421, 30, 23, 13fzto1stinvn 31656 . . . . . 6 (𝑁 ∈ (1...𝑁) → (𝑆𝑁) = 1)
7529, 74syl 17 . . . . 5 (𝜑 → (𝑆𝑁) = 1)
7675fveq2d 6833 . . . 4 (𝜑 → (𝑃‘(𝑆𝑁)) = (𝑃‘1))
7721, 22fzto1stfv1 31653 . . . . 5 (𝐼 ∈ (1...𝑁) → (𝑃‘1) = 𝐼)
7810, 77syl 17 . . . 4 (𝜑 → (𝑃‘1) = 𝐼)
7973, 76, 783eqtrd 2781 . . 3 (𝜑 → ((𝑃𝑆)‘𝑁) = 𝐼)
8023, 13symgbasf1o 19078 . . . . . . 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 6770 . . . . . 6 (𝑇:(1...𝑁)–1-1-onto→(1...𝑁) → 𝑇:(1...𝑁)–1-1→(1...𝑁))
83 df-f1 6488 . . . . . . 7 (𝑇:(1...𝑁)–1-1→(1...𝑁) ↔ (𝑇:(1...𝑁)⟶(1...𝑁) ∧ Fun 𝑇))
8483simprbi 498 . . . . . 6 (𝑇:(1...𝑁)–1-1→(1...𝑁) → Fun 𝑇)
8581, 82, 843syl 18 . . . . 5 (𝜑 → Fun 𝑇)
86 f1ocnv 6783 . . . . . . 7 (𝑇:(1...𝑁)–1-1-onto→(1...𝑁) → 𝑇:(1...𝑁)–1-1-onto→(1...𝑁))
87 f1odm 6775 . . . . . . 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 6926 . . . . 5 ((Fun 𝑇𝑁 ∈ dom 𝑇) → ((𝑄𝑇)‘𝑁) = (𝑄‘(𝑇𝑁)))
9185, 89, 90syl2anc 585 . . . 4 (𝜑 → ((𝑄𝑇)‘𝑁) = (𝑄‘(𝑇𝑁)))
9221, 50, 23, 13fzto1stinvn 31656 . . . . . 6 (𝑁 ∈ (1...𝑁) → (𝑇𝑁) = 1)
9329, 92syl 17 . . . . 5 (𝜑 → (𝑇𝑁) = 1)
9493fveq2d 6833 . . . 4 (𝜑 → (𝑄‘(𝑇𝑁)) = (𝑄‘1))
9521, 47fzto1stfv1 31653 . . . . 5 (𝐽 ∈ (1...𝑁) → (𝑄‘1) = 𝐽)
9611, 95syl 17 . . . 4 (𝜑 → (𝑄‘1) = 𝐽)
9791, 94, 963eqtrd 2781 . . 3 (𝜑 → ((𝑄𝑇)‘𝑁) = 𝐽)
98 crngring 19889 . . . . . 6 (𝑅 ∈ CRing → 𝑅 ∈ Ring)
999, 98syl 17 . . . . 5 (𝜑𝑅 ∈ Ring)
1002, 1minmar1cl 21905 . . . . 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 32075 . . 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 32073 . 2 (𝜑 → (𝐽(𝐾𝑀)𝐼) = ((𝑍‘(((pmSgn‘(1...𝑁))‘(𝑃𝑆)) · ((pmSgn‘(1...𝑁))‘(𝑄𝑇)))) · (𝐸‘(𝐼(subMat1‘𝑀)𝐽))))
10423, 14, 13psgnco 20893 . . . . . . . 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 31657 . . . . . . . . 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 20892 . . . . . . . . . 10 (((1...𝑁) ∈ Fin ∧ 𝑆 ∈ (Base‘(SymGrp‘(1...𝑁)))) → ((pmSgn‘(1...𝑁))‘𝑆) = ((pmSgn‘(1...𝑁))‘𝑆))
10936, 32, 108syl2anc 585 . . . . . . . . 9 (𝜑 → ((pmSgn‘(1...𝑁))‘𝑆) = ((pmSgn‘(1...𝑁))‘𝑆))
11021, 30, 23, 13, 14psgnfzto1st 31657 . . . . . . . . . 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 2777 . . . . . . . 8 (𝜑 → ((pmSgn‘(1...𝑁))‘𝑆) = (-1↑(𝑁 + 1)))
113107, 112oveq12d 7359 . . . . . . 7 (𝜑 → (((pmSgn‘(1...𝑁))‘𝑃) · ((pmSgn‘(1...𝑁))‘𝑆)) = ((-1↑(𝐼 + 1)) · (-1↑(𝑁 + 1))))
114105, 113eqtrd 2777 . . . . . 6 (𝜑 → ((pmSgn‘(1...𝑁))‘(𝑃𝑆)) = ((-1↑(𝐼 + 1)) · (-1↑(𝑁 + 1))))
11523, 14, 13psgnco 20893 . . . . . . . 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 31657 . . . . . . . . 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 20892 . . . . . . . . . 10 (((1...𝑁) ∈ Fin ∧ 𝑇 ∈ (Base‘(SymGrp‘(1...𝑁)))) → ((pmSgn‘(1...𝑁))‘𝑇) = ((pmSgn‘(1...𝑁))‘𝑇))
12036, 52, 119syl2anc 585 . . . . . . . . 9 (𝜑 → ((pmSgn‘(1...𝑁))‘𝑇) = ((pmSgn‘(1...𝑁))‘𝑇))
12121, 50, 23, 13, 14psgnfzto1st 31657 . . . . . . . . . 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 2777 . . . . . . . 8 (𝜑 → ((pmSgn‘(1...𝑁))‘𝑇) = (-1↑(𝑁 + 1)))
124118, 123oveq12d 7359 . . . . . . 7 (𝜑 → (((pmSgn‘(1...𝑁))‘𝑄) · ((pmSgn‘(1...𝑁))‘𝑇)) = ((-1↑(𝐽 + 1)) · (-1↑(𝑁 + 1))))
125116, 124eqtrd 2777 . . . . . 6 (𝜑 → ((pmSgn‘(1...𝑁))‘(𝑄𝑇)) = ((-1↑(𝐽 + 1)) · (-1↑(𝑁 + 1))))
126114, 125oveq12d 7359 . . . . 5 (𝜑 → (((pmSgn‘(1...𝑁))‘(𝑃𝑆)) · ((pmSgn‘(1...𝑁))‘(𝑄𝑇))) = (((-1↑(𝐼 + 1)) · (-1↑(𝑁 + 1))) · ((-1↑(𝐽 + 1)) · (-1↑(𝑁 + 1)))))
127 1cnd 11075 . . . . . . . . 9 (𝜑 → 1 ∈ ℂ)
128127negcld 11424 . . . . . . . 8 (𝜑 → -1 ∈ ℂ)
129 fz1ssnn 13392 . . . . . . . . . . 11 (1...𝑁) ⊆ ℕ
130129, 10sselid 3933 . . . . . . . . . 10 (𝜑𝐼 ∈ ℕ)
131130nnnn0d 12398 . . . . . . . . 9 (𝜑𝐼 ∈ ℕ0)
132 1nn0 12354 . . . . . . . . . 10 1 ∈ ℕ0
133132a1i 11 . . . . . . . . 9 (𝜑 → 1 ∈ ℕ0)
134131, 133nn0addcld 12402 . . . . . . . 8 (𝜑 → (𝐼 + 1) ∈ ℕ0)
135128, 134expcld 13969 . . . . . . 7 (𝜑 → (-1↑(𝐼 + 1)) ∈ ℂ)
1368nnnn0d 12398 . . . . . . . . 9 (𝜑𝑁 ∈ ℕ0)
137136, 133nn0addcld 12402 . . . . . . . 8 (𝜑 → (𝑁 + 1) ∈ ℕ0)
138128, 137expcld 13969 . . . . . . 7 (𝜑 → (-1↑(𝑁 + 1)) ∈ ℂ)
139129, 11sselid 3933 . . . . . . . . . 10 (𝜑𝐽 ∈ ℕ)
140139nnnn0d 12398 . . . . . . . . 9 (𝜑𝐽 ∈ ℕ0)
141140, 133nn0addcld 12402 . . . . . . . 8 (𝜑 → (𝐽 + 1) ∈ ℕ0)
142128, 141expcld 13969 . . . . . . 7 (𝜑 → (-1↑(𝐽 + 1)) ∈ ℂ)
143135, 138, 142, 138mul4d 11292 . . . . . 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 13971 . . . . . . . 8 (𝜑 → (-1↑((𝐼 + 1) + (𝐽 + 1))) = ((-1↑(𝐼 + 1)) · (-1↑(𝐽 + 1))))
145130nncnd 12094 . . . . . . . . . . . 12 (𝜑𝐼 ∈ ℂ)
146139nncnd 12094 . . . . . . . . . . . 12 (𝜑𝐽 ∈ ℂ)
147145, 127, 146, 127add4d 11308 . . . . . . . . . . 11 (𝜑 → ((𝐼 + 1) + (𝐽 + 1)) = ((𝐼 + 𝐽) + (1 + 1)))
148 1p1e2 12203 . . . . . . . . . . . 12 (1 + 1) = 2
149148oveq2i 7352 . . . . . . . . . . 11 ((𝐼 + 𝐽) + (1 + 1)) = ((𝐼 + 𝐽) + 2)
150147, 149eqtrdi 2793 . . . . . . . . . 10 (𝜑 → ((𝐼 + 1) + (𝐽 + 1)) = ((𝐼 + 𝐽) + 2))
151150oveq2d 7357 . . . . . . . . 9 (𝜑 → (-1↑((𝐼 + 1) + (𝐽 + 1))) = (-1↑((𝐼 + 𝐽) + 2)))
152 2nn0 12355 . . . . . . . . . . . 12 2 ∈ ℕ0
153152a1i 11 . . . . . . . . . . 11 (𝜑 → 2 ∈ ℕ0)
154131, 140nn0addcld 12402 . . . . . . . . . . 11 (𝜑 → (𝐼 + 𝐽) ∈ ℕ0)
155128, 153, 154expaddd 13971 . . . . . . . . . 10 (𝜑 → (-1↑((𝐼 + 𝐽) + 2)) = ((-1↑(𝐼 + 𝐽)) · (-1↑2)))
156 neg1sqe1 14018 . . . . . . . . . . 11 (-1↑2) = 1
157156oveq2i 7352 . . . . . . . . . 10 ((-1↑(𝐼 + 𝐽)) · (-1↑2)) = ((-1↑(𝐼 + 𝐽)) · 1)
158155, 157eqtrdi 2793 . . . . . . . . 9 (𝜑 → (-1↑((𝐼 + 𝐽) + 2)) = ((-1↑(𝐼 + 𝐽)) · 1))
159128, 154expcld 13969 . . . . . . . . . 10 (𝜑 → (-1↑(𝐼 + 𝐽)) ∈ ℂ)
160159mulid1d 11097 . . . . . . . . 9 (𝜑 → ((-1↑(𝐼 + 𝐽)) · 1) = (-1↑(𝐼 + 𝐽)))
161151, 158, 1603eqtrd 2781 . . . . . . . 8 (𝜑 → (-1↑((𝐼 + 1) + (𝐽 + 1))) = (-1↑(𝐼 + 𝐽)))
162144, 161eqtr3d 2779 . . . . . . 7 (𝜑 → ((-1↑(𝐼 + 1)) · (-1↑(𝐽 + 1))) = (-1↑(𝐼 + 𝐽)))
163137nn0zd 12529 . . . . . . . 8 (𝜑 → (𝑁 + 1) ∈ ℤ)
164 m1expcl2 13909 . . . . . . . 8 ((𝑁 + 1) ∈ ℤ → (-1↑(𝑁 + 1)) ∈ {-1, 1})
165 1neg1t1neg1 31357 . . . . . . . 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 7359 . . . . . 6 (𝜑 → (((-1↑(𝐼 + 1)) · (-1↑(𝐽 + 1))) · ((-1↑(𝑁 + 1)) · (-1↑(𝑁 + 1)))) = ((-1↑(𝐼 + 𝐽)) · 1))
168143, 167, 1603eqtrd 2781 . . . . 5 (𝜑 → (((-1↑(𝐼 + 1)) · (-1↑(𝑁 + 1))) · ((-1↑(𝐽 + 1)) · (-1↑(𝑁 + 1)))) = (-1↑(𝐼 + 𝐽)))
169126, 168eqtrd 2777 . . . 4 (𝜑 → (((pmSgn‘(1...𝑁))‘(𝑃𝑆)) · ((pmSgn‘(1...𝑁))‘(𝑄𝑇))) = (-1↑(𝐼 + 𝐽)))
170169fveq2d 6833 . . 3 (𝜑 → (𝑍‘(((pmSgn‘(1...𝑁))‘(𝑃𝑆)) · ((pmSgn‘(1...𝑁))‘(𝑄𝑇)))) = (𝑍‘(-1↑(𝐼 + 𝐽))))
171170oveq1d 7356 . 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 2777 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 397   = wceq 1541  wcel 2106  ifcif 4477  {cpr 4579   class class class wbr 5096  cmpt 5179  ccnv 5623  dom cdm 5624  ccom 5628  Fun wfun 6477  wf 6479  1-1wf1 6480  1-1-ontowf1o 6482  cfv 6483  (class class class)co 7341  cmpo 7343  Fincfn 8808  1c1 10977   + caddc 10979   · cmul 10981  cle 11115  cmin 11310  -cneg 11311  cn 12078  2c2 12133  0cn0 12338  cz 12424  cuz 12687  ...cfz 13344  cexp 13887  Basecbs 17009  +gcplusg 17059  .rcmulr 17060  Grpcgrp 18673  invgcminusg 18674  SymGrpcsymg 19070  pmSgncpsgn 19193  Ringcrg 19877  CRingccrg 19878  ℤRHomczrh 20806   Mat cmat 21659   maDet cmdat 21838   maAdju cmadu 21886   minMatR1 cminmar1 21887  subMat1csmat 32039
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 2708  ax-rep 5233  ax-sep 5247  ax-nul 5254  ax-pow 5312  ax-pr 5376  ax-un 7654  ax-cnex 11032  ax-resscn 11033  ax-1cn 11034  ax-icn 11035  ax-addcl 11036  ax-addrcl 11037  ax-mulcl 11038  ax-mulrcl 11039  ax-mulcom 11040  ax-addass 11041  ax-mulass 11042  ax-distr 11043  ax-i2m1 11044  ax-1ne0 11045  ax-1rid 11046  ax-rnegex 11047  ax-rrecex 11048  ax-cnre 11049  ax-pre-lttri 11050  ax-pre-lttrn 11051  ax-pre-ltadd 11052  ax-pre-mulgt0 11053  ax-addf 11055  ax-mulf 11056
This theorem depends on definitions:  df-bi 206  df-an 398  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 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2887  df-ne 2942  df-nel 3048  df-ral 3063  df-rex 3072  df-rmo 3350  df-reu 3351  df-rab 3405  df-v 3444  df-sbc 3731  df-csb 3847  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-pss 3920  df-nul 4274  df-if 4478  df-pw 4553  df-sn 4578  df-pr 4580  df-tp 4582  df-op 4584  df-ot 4586  df-uni 4857  df-int 4899  df-iun 4947  df-iin 4948  df-br 5097  df-opab 5159  df-mpt 5180  df-tr 5214  df-id 5522  df-eprel 5528  df-po 5536  df-so 5537  df-fr 5579  df-se 5580  df-we 5581  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-pred 6242  df-ord 6309  df-on 6310  df-lim 6311  df-suc 6312  df-iota 6435  df-fun 6485  df-fn 6486  df-f 6487  df-f1 6488  df-fo 6489  df-f1o 6490  df-fv 6491  df-isom 6492  df-riota 7297  df-ov 7344  df-oprab 7345  df-mpo 7346  df-of 7599  df-om 7785  df-1st 7903  df-2nd 7904  df-supp 8052  df-tpos 8116  df-frecs 8171  df-wrecs 8202  df-recs 8276  df-rdg 8315  df-1o 8371  df-2o 8372  df-er 8573  df-map 8692  df-pm 8693  df-ixp 8761  df-en 8809  df-dom 8810  df-sdom 8811  df-fin 8812  df-fsupp 9231  df-sup 9303  df-oi 9371  df-card 9800  df-pnf 11116  df-mnf 11117  df-xr 11118  df-ltxr 11119  df-le 11120  df-sub 11312  df-neg 11313  df-div 11738  df-nn 12079  df-2 12141  df-3 12142  df-4 12143  df-5 12144  df-6 12145  df-7 12146  df-8 12147  df-9 12148  df-n0 12339  df-xnn0 12411  df-z 12425  df-dec 12543  df-uz 12688  df-rp 12836  df-fz 13345  df-fzo 13488  df-seq 13827  df-exp 13888  df-hash 14150  df-word 14322  df-lsw 14370  df-concat 14378  df-s1 14403  df-substr 14452  df-pfx 14482  df-splice 14561  df-reverse 14570  df-s2 14660  df-struct 16945  df-sets 16962  df-slot 16980  df-ndx 16992  df-base 17010  df-ress 17039  df-plusg 17072  df-mulr 17073  df-starv 17074  df-sca 17075  df-vsca 17076  df-ip 17077  df-tset 17078  df-ple 17079  df-ds 17081  df-unif 17082  df-hom 17083  df-cco 17084  df-0g 17249  df-gsum 17250  df-prds 17255  df-pws 17257  df-mre 17392  df-mrc 17393  df-acs 17395  df-mgm 18423  df-sgrp 18472  df-mnd 18483  df-mhm 18527  df-submnd 18528  df-efmnd 18604  df-grp 18676  df-minusg 18677  df-mulg 18797  df-subg 18848  df-ghm 18928  df-gim 18971  df-cntz 19019  df-oppg 19046  df-symg 19071  df-pmtr 19146  df-psgn 19195  df-cmn 19483  df-abl 19484  df-mgp 19815  df-ur 19832  df-ring 19879  df-cring 19880  df-oppr 19956  df-dvdsr 19977  df-unit 19978  df-invr 20008  df-dvr 20019  df-rnghom 20053  df-drng 20094  df-subrg 20126  df-sra 20539  df-rgmod 20540  df-cnfld 20703  df-zring 20776  df-zrh 20810  df-dsmm 21044  df-frlm 21059  df-mat 21660  df-marrep 21812  df-subma 21831  df-mdet 21839  df-madu 21888  df-minmar1 21889  df-smat 32040
This theorem is referenced by:  madjusmdet  32077
  Copyright terms: Public domain W3C validator