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Theorem madjusmdetlem4 31183
Description: Lemma for madjusmdet 31184. (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 2801 . . 3 (Base‘(SymGrp‘(1...𝑁))) = (Base‘(SymGrp‘(1...𝑁)))
14 eqid 2801 . . 3 (pmSgn‘(1...𝑁)) = (pmSgn‘(1...𝑁))
15 eqid 2801 . . 3 (𝐼(((1...𝑁) minMatR1 𝑅)‘𝑀)𝐽) = (𝐼(((1...𝑁) minMatR1 𝑅)‘𝑀)𝐽)
16 fveq2 6649 . . . . 5 (𝑘 = 𝑖 → ((𝑃𝑆)‘𝑘) = ((𝑃𝑆)‘𝑖))
1716oveq1d 7154 . . . 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 6649 . . . . 5 (𝑙 = 𝑗 → ((𝑄𝑇)‘𝑙) = ((𝑄𝑇)‘𝑗))
1918oveq2d 7155 . . . 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 7232 . . 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 2801 . . . . . 6 (1...𝑁) = (1...𝑁)
22 madjusmdetlem2.p . . . . . 6 𝑃 = (𝑖 ∈ (1...𝑁) ↦ if(𝑖 = 1, 𝐼, if(𝑖𝐼, (𝑖 − 1), 𝑖)))
23 eqid 2801 . . . . . 6 (SymGrp‘(1...𝑁)) = (SymGrp‘(1...𝑁))
2421, 22, 23, 13fzto1st 30798 . . . . 5 (𝐼 ∈ (1...𝑁) → 𝑃 ∈ (Base‘(SymGrp‘(1...𝑁))))
2510, 24syl 17 . . . 4 (𝜑𝑃 ∈ (Base‘(SymGrp‘(1...𝑁))))
26 nnuz 12273 . . . . . . . . 9 ℕ = (ℤ‘1)
278, 26eleqtrdi 2903 . . . . . . . 8 (𝜑𝑁 ∈ (ℤ‘1))
28 eluzfz2 12914 . . . . . . . 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 30798 . . . . . . 7 (𝑁 ∈ (1...𝑁) → 𝑆 ∈ (Base‘(SymGrp‘(1...𝑁))))
3229, 31syl 17 . . . . . 6 (𝜑𝑆 ∈ (Base‘(SymGrp‘(1...𝑁))))
33 eqid 2801 . . . . . . 7 (invg‘(SymGrp‘(1...𝑁))) = (invg‘(SymGrp‘(1...𝑁)))
3423, 13, 33symginv 18525 . . . . . 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 13340 . . . . . . 7 (𝜑 → (1...𝑁) ∈ Fin)
3723symggrp 18523 . . . . . . 7 ((1...𝑁) ∈ Fin → (SymGrp‘(1...𝑁)) ∈ Grp)
3836, 37syl 17 . . . . . 6 (𝜑 → (SymGrp‘(1...𝑁)) ∈ Grp)
3913, 33grpinvcl 18146 . . . . . 6 (((SymGrp‘(1...𝑁)) ∈ Grp ∧ 𝑆 ∈ (Base‘(SymGrp‘(1...𝑁)))) → ((invg‘(SymGrp‘(1...𝑁)))‘𝑆) ∈ (Base‘(SymGrp‘(1...𝑁))))
4038, 32, 39syl2anc 587 . . . . 5 (𝜑 → ((invg‘(SymGrp‘(1...𝑁)))‘𝑆) ∈ (Base‘(SymGrp‘(1...𝑁))))
4135, 40eqeltrrd 2894 . . . 4 (𝜑𝑆 ∈ (Base‘(SymGrp‘(1...𝑁))))
42 eqid 2801 . . . . . 6 (+g‘(SymGrp‘(1...𝑁))) = (+g‘(SymGrp‘(1...𝑁)))
4323, 13, 42symgov 18507 . . . . 5 ((𝑃 ∈ (Base‘(SymGrp‘(1...𝑁))) ∧ 𝑆 ∈ (Base‘(SymGrp‘(1...𝑁)))) → (𝑃(+g‘(SymGrp‘(1...𝑁)))𝑆) = (𝑃𝑆))
4423, 13, 42symgcl 18508 . . . . 5 ((𝑃 ∈ (Base‘(SymGrp‘(1...𝑁))) ∧ 𝑆 ∈ (Base‘(SymGrp‘(1...𝑁)))) → (𝑃(+g‘(SymGrp‘(1...𝑁)))𝑆) ∈ (Base‘(SymGrp‘(1...𝑁))))
4543, 44eqeltrrd 2894 . . . 4 ((𝑃 ∈ (Base‘(SymGrp‘(1...𝑁))) ∧ 𝑆 ∈ (Base‘(SymGrp‘(1...𝑁)))) → (𝑃𝑆) ∈ (Base‘(SymGrp‘(1...𝑁))))
4625, 41, 45syl2anc 587 . . 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 30798 . . . . 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 30798 . . . . . . 7 (𝑁 ∈ (1...𝑁) → 𝑇 ∈ (Base‘(SymGrp‘(1...𝑁))))
5229, 51syl 17 . . . . . 6 (𝜑𝑇 ∈ (Base‘(SymGrp‘(1...𝑁))))
5323, 13, 33symginv 18525 . . . . . 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 18146 . . . . . 6 (((SymGrp‘(1...𝑁)) ∈ Grp ∧ 𝑇 ∈ (Base‘(SymGrp‘(1...𝑁)))) → ((invg‘(SymGrp‘(1...𝑁)))‘𝑇) ∈ (Base‘(SymGrp‘(1...𝑁))))
5638, 52, 55syl2anc 587 . . . . 5 (𝜑 → ((invg‘(SymGrp‘(1...𝑁)))‘𝑇) ∈ (Base‘(SymGrp‘(1...𝑁))))
5754, 56eqeltrrd 2894 . . . 4 (𝜑𝑇 ∈ (Base‘(SymGrp‘(1...𝑁))))
5823, 13, 42symgov 18507 . . . . 5 ((𝑄 ∈ (Base‘(SymGrp‘(1...𝑁))) ∧ 𝑇 ∈ (Base‘(SymGrp‘(1...𝑁)))) → (𝑄(+g‘(SymGrp‘(1...𝑁)))𝑇) = (𝑄𝑇))
5923, 13, 42symgcl 18508 . . . . 5 ((𝑄 ∈ (Base‘(SymGrp‘(1...𝑁))) ∧ 𝑇 ∈ (Base‘(SymGrp‘(1...𝑁)))) → (𝑄(+g‘(SymGrp‘(1...𝑁)))𝑇) ∈ (Base‘(SymGrp‘(1...𝑁))))
6058, 59eqeltrrd 2894 . . . 4 ((𝑄 ∈ (Base‘(SymGrp‘(1...𝑁))) ∧ 𝑇 ∈ (Base‘(SymGrp‘(1...𝑁)))) → (𝑄𝑇) ∈ (Base‘(SymGrp‘(1...𝑁))))
6149, 57, 60syl2anc 587 . . 3 (𝜑 → (𝑄𝑇) ∈ (Base‘(SymGrp‘(1...𝑁))))
6223, 13symgbasf1o 18498 . . . . . . 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 6593 . . . . . 6 (𝑆:(1...𝑁)–1-1-onto→(1...𝑁) → 𝑆:(1...𝑁)–1-1→(1...𝑁))
65 df-f1 6333 . . . . . . 7 (𝑆:(1...𝑁)–1-1→(1...𝑁) ↔ (𝑆:(1...𝑁)⟶(1...𝑁) ∧ Fun 𝑆))
6665simprbi 500 . . . . . 6 (𝑆:(1...𝑁)–1-1→(1...𝑁) → Fun 𝑆)
6763, 64, 663syl 18 . . . . 5 (𝜑 → Fun 𝑆)
68 f1ocnv 6606 . . . . . . 7 (𝑆:(1...𝑁)–1-1-onto→(1...𝑁) → 𝑆:(1...𝑁)–1-1-onto→(1...𝑁))
69 f1odm 6598 . . . . . . 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 2896 . . . . 5 (𝜑𝑁 ∈ dom 𝑆)
72 fvco 6740 . . . . 5 ((Fun 𝑆𝑁 ∈ dom 𝑆) → ((𝑃𝑆)‘𝑁) = (𝑃‘(𝑆𝑁)))
7367, 71, 72syl2anc 587 . . . 4 (𝜑 → ((𝑃𝑆)‘𝑁) = (𝑃‘(𝑆𝑁)))
7421, 30, 23, 13fzto1stinvn 30799 . . . . . 6 (𝑁 ∈ (1...𝑁) → (𝑆𝑁) = 1)
7529, 74syl 17 . . . . 5 (𝜑 → (𝑆𝑁) = 1)
7675fveq2d 6653 . . . 4 (𝜑 → (𝑃‘(𝑆𝑁)) = (𝑃‘1))
7721, 22fzto1stfv1 30796 . . . . 5 (𝐼 ∈ (1...𝑁) → (𝑃‘1) = 𝐼)
7810, 77syl 17 . . . 4 (𝜑 → (𝑃‘1) = 𝐼)
7973, 76, 783eqtrd 2840 . . 3 (𝜑 → ((𝑃𝑆)‘𝑁) = 𝐼)
8023, 13symgbasf1o 18498 . . . . . . 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 6593 . . . . . 6 (𝑇:(1...𝑁)–1-1-onto→(1...𝑁) → 𝑇:(1...𝑁)–1-1→(1...𝑁))
83 df-f1 6333 . . . . . . 7 (𝑇:(1...𝑁)–1-1→(1...𝑁) ↔ (𝑇:(1...𝑁)⟶(1...𝑁) ∧ Fun 𝑇))
8483simprbi 500 . . . . . 6 (𝑇:(1...𝑁)–1-1→(1...𝑁) → Fun 𝑇)
8581, 82, 843syl 18 . . . . 5 (𝜑 → Fun 𝑇)
86 f1ocnv 6606 . . . . . . 7 (𝑇:(1...𝑁)–1-1-onto→(1...𝑁) → 𝑇:(1...𝑁)–1-1-onto→(1...𝑁))
87 f1odm 6598 . . . . . . 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 2896 . . . . 5 (𝜑𝑁 ∈ dom 𝑇)
90 fvco 6740 . . . . 5 ((Fun 𝑇𝑁 ∈ dom 𝑇) → ((𝑄𝑇)‘𝑁) = (𝑄‘(𝑇𝑁)))
9185, 89, 90syl2anc 587 . . . 4 (𝜑 → ((𝑄𝑇)‘𝑁) = (𝑄‘(𝑇𝑁)))
9221, 50, 23, 13fzto1stinvn 30799 . . . . . 6 (𝑁 ∈ (1...𝑁) → (𝑇𝑁) = 1)
9329, 92syl 17 . . . . 5 (𝜑 → (𝑇𝑁) = 1)
9493fveq2d 6653 . . . 4 (𝜑 → (𝑄‘(𝑇𝑁)) = (𝑄‘1))
9521, 47fzto1stfv1 30796 . . . . 5 (𝐽 ∈ (1...𝑁) → (𝑄‘1) = 𝐽)
9611, 95syl 17 . . . 4 (𝜑 → (𝑄‘1) = 𝐽)
9791, 94, 963eqtrd 2840 . . 3 (𝜑 → ((𝑄𝑇)‘𝑁) = 𝐽)
98 crngring 19305 . . . . . 6 (𝑅 ∈ CRing → 𝑅 ∈ Ring)
999, 98syl 17 . . . . 5 (𝜑𝑅 ∈ Ring)
1002, 1minmar1cl 21259 . . . . 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 31182 . . 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 31180 . 2 (𝜑 → (𝐽(𝐾𝑀)𝐼) = ((𝑍‘(((pmSgn‘(1...𝑁))‘(𝑃𝑆)) · ((pmSgn‘(1...𝑁))‘(𝑄𝑇)))) · (𝐸‘(𝐼(subMat1‘𝑀)𝐽))))
10423, 14, 13psgnco 20275 . . . . . . . 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 1368 . . . . . . 7 (𝜑 → ((pmSgn‘(1...𝑁))‘(𝑃𝑆)) = (((pmSgn‘(1...𝑁))‘𝑃) · ((pmSgn‘(1...𝑁))‘𝑆)))
10621, 22, 23, 13, 14psgnfzto1st 30800 . . . . . . . . 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 20274 . . . . . . . . . 10 (((1...𝑁) ∈ Fin ∧ 𝑆 ∈ (Base‘(SymGrp‘(1...𝑁)))) → ((pmSgn‘(1...𝑁))‘𝑆) = ((pmSgn‘(1...𝑁))‘𝑆))
10936, 32, 108syl2anc 587 . . . . . . . . 9 (𝜑 → ((pmSgn‘(1...𝑁))‘𝑆) = ((pmSgn‘(1...𝑁))‘𝑆))
11021, 30, 23, 13, 14psgnfzto1st 30800 . . . . . . . . . 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 2836 . . . . . . . 8 (𝜑 → ((pmSgn‘(1...𝑁))‘𝑆) = (-1↑(𝑁 + 1)))
113107, 112oveq12d 7157 . . . . . . 7 (𝜑 → (((pmSgn‘(1...𝑁))‘𝑃) · ((pmSgn‘(1...𝑁))‘𝑆)) = ((-1↑(𝐼 + 1)) · (-1↑(𝑁 + 1))))
114105, 113eqtrd 2836 . . . . . 6 (𝜑 → ((pmSgn‘(1...𝑁))‘(𝑃𝑆)) = ((-1↑(𝐼 + 1)) · (-1↑(𝑁 + 1))))
11523, 14, 13psgnco 20275 . . . . . . . 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 1368 . . . . . . 7 (𝜑 → ((pmSgn‘(1...𝑁))‘(𝑄𝑇)) = (((pmSgn‘(1...𝑁))‘𝑄) · ((pmSgn‘(1...𝑁))‘𝑇)))
11721, 47, 23, 13, 14psgnfzto1st 30800 . . . . . . . . 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 20274 . . . . . . . . . 10 (((1...𝑁) ∈ Fin ∧ 𝑇 ∈ (Base‘(SymGrp‘(1...𝑁)))) → ((pmSgn‘(1...𝑁))‘𝑇) = ((pmSgn‘(1...𝑁))‘𝑇))
12036, 52, 119syl2anc 587 . . . . . . . . 9 (𝜑 → ((pmSgn‘(1...𝑁))‘𝑇) = ((pmSgn‘(1...𝑁))‘𝑇))
12121, 50, 23, 13, 14psgnfzto1st 30800 . . . . . . . . . 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 2836 . . . . . . . 8 (𝜑 → ((pmSgn‘(1...𝑁))‘𝑇) = (-1↑(𝑁 + 1)))
124118, 123oveq12d 7157 . . . . . . 7 (𝜑 → (((pmSgn‘(1...𝑁))‘𝑄) · ((pmSgn‘(1...𝑁))‘𝑇)) = ((-1↑(𝐽 + 1)) · (-1↑(𝑁 + 1))))
125116, 124eqtrd 2836 . . . . . 6 (𝜑 → ((pmSgn‘(1...𝑁))‘(𝑄𝑇)) = ((-1↑(𝐽 + 1)) · (-1↑(𝑁 + 1))))
126114, 125oveq12d 7157 . . . . 5 (𝜑 → (((pmSgn‘(1...𝑁))‘(𝑃𝑆)) · ((pmSgn‘(1...𝑁))‘(𝑄𝑇))) = (((-1↑(𝐼 + 1)) · (-1↑(𝑁 + 1))) · ((-1↑(𝐽 + 1)) · (-1↑(𝑁 + 1)))))
127 1cnd 10629 . . . . . . . . 9 (𝜑 → 1 ∈ ℂ)
128127negcld 10977 . . . . . . . 8 (𝜑 → -1 ∈ ℂ)
129 fz1ssnn 12937 . . . . . . . . . . 11 (1...𝑁) ⊆ ℕ
130129, 10sseldi 3916 . . . . . . . . . 10 (𝜑𝐼 ∈ ℕ)
131130nnnn0d 11947 . . . . . . . . 9 (𝜑𝐼 ∈ ℕ0)
132 1nn0 11905 . . . . . . . . . 10 1 ∈ ℕ0
133132a1i 11 . . . . . . . . 9 (𝜑 → 1 ∈ ℕ0)
134131, 133nn0addcld 11951 . . . . . . . 8 (𝜑 → (𝐼 + 1) ∈ ℕ0)
135128, 134expcld 13510 . . . . . . 7 (𝜑 → (-1↑(𝐼 + 1)) ∈ ℂ)
1368nnnn0d 11947 . . . . . . . . 9 (𝜑𝑁 ∈ ℕ0)
137136, 133nn0addcld 11951 . . . . . . . 8 (𝜑 → (𝑁 + 1) ∈ ℕ0)
138128, 137expcld 13510 . . . . . . 7 (𝜑 → (-1↑(𝑁 + 1)) ∈ ℂ)
139129, 11sseldi 3916 . . . . . . . . . 10 (𝜑𝐽 ∈ ℕ)
140139nnnn0d 11947 . . . . . . . . 9 (𝜑𝐽 ∈ ℕ0)
141140, 133nn0addcld 11951 . . . . . . . 8 (𝜑 → (𝐽 + 1) ∈ ℕ0)
142128, 141expcld 13510 . . . . . . 7 (𝜑 → (-1↑(𝐽 + 1)) ∈ ℂ)
143135, 138, 142, 138mul4d 10845 . . . . . 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 13512 . . . . . . . 8 (𝜑 → (-1↑((𝐼 + 1) + (𝐽 + 1))) = ((-1↑(𝐼 + 1)) · (-1↑(𝐽 + 1))))
145130nncnd 11645 . . . . . . . . . . . 12 (𝜑𝐼 ∈ ℂ)
146139nncnd 11645 . . . . . . . . . . . 12 (𝜑𝐽 ∈ ℂ)
147145, 127, 146, 127add4d 10861 . . . . . . . . . . 11 (𝜑 → ((𝐼 + 1) + (𝐽 + 1)) = ((𝐼 + 𝐽) + (1 + 1)))
148 1p1e2 11754 . . . . . . . . . . . 12 (1 + 1) = 2
149148oveq2i 7150 . . . . . . . . . . 11 ((𝐼 + 𝐽) + (1 + 1)) = ((𝐼 + 𝐽) + 2)
150147, 149eqtrdi 2852 . . . . . . . . . 10 (𝜑 → ((𝐼 + 1) + (𝐽 + 1)) = ((𝐼 + 𝐽) + 2))
151150oveq2d 7155 . . . . . . . . 9 (𝜑 → (-1↑((𝐼 + 1) + (𝐽 + 1))) = (-1↑((𝐼 + 𝐽) + 2)))
152 2nn0 11906 . . . . . . . . . . . 12 2 ∈ ℕ0
153152a1i 11 . . . . . . . . . . 11 (𝜑 → 2 ∈ ℕ0)
154131, 140nn0addcld 11951 . . . . . . . . . . 11 (𝜑 → (𝐼 + 𝐽) ∈ ℕ0)
155128, 153, 154expaddd 13512 . . . . . . . . . 10 (𝜑 → (-1↑((𝐼 + 𝐽) + 2)) = ((-1↑(𝐼 + 𝐽)) · (-1↑2)))
156 neg1sqe1 13559 . . . . . . . . . . 11 (-1↑2) = 1
157156oveq2i 7150 . . . . . . . . . 10 ((-1↑(𝐼 + 𝐽)) · (-1↑2)) = ((-1↑(𝐼 + 𝐽)) · 1)
158155, 157eqtrdi 2852 . . . . . . . . 9 (𝜑 → (-1↑((𝐼 + 𝐽) + 2)) = ((-1↑(𝐼 + 𝐽)) · 1))
159128, 154expcld 13510 . . . . . . . . . 10 (𝜑 → (-1↑(𝐼 + 𝐽)) ∈ ℂ)
160159mulid1d 10651 . . . . . . . . 9 (𝜑 → ((-1↑(𝐼 + 𝐽)) · 1) = (-1↑(𝐼 + 𝐽)))
161151, 158, 1603eqtrd 2840 . . . . . . . 8 (𝜑 → (-1↑((𝐼 + 1) + (𝐽 + 1))) = (-1↑(𝐼 + 𝐽)))
162144, 161eqtr3d 2838 . . . . . . 7 (𝜑 → ((-1↑(𝐼 + 1)) · (-1↑(𝐽 + 1))) = (-1↑(𝐼 + 𝐽)))
163137nn0zd 12077 . . . . . . . 8 (𝜑 → (𝑁 + 1) ∈ ℤ)
164 m1expcl2 13451 . . . . . . . 8 ((𝑁 + 1) ∈ ℤ → (-1↑(𝑁 + 1)) ∈ {-1, 1})
165 1neg1t1neg1 30502 . . . . . . . 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 7157 . . . . . 6 (𝜑 → (((-1↑(𝐼 + 1)) · (-1↑(𝐽 + 1))) · ((-1↑(𝑁 + 1)) · (-1↑(𝑁 + 1)))) = ((-1↑(𝐼 + 𝐽)) · 1))
168143, 167, 1603eqtrd 2840 . . . . 5 (𝜑 → (((-1↑(𝐼 + 1)) · (-1↑(𝑁 + 1))) · ((-1↑(𝐽 + 1)) · (-1↑(𝑁 + 1)))) = (-1↑(𝐼 + 𝐽)))
169126, 168eqtrd 2836 . . . 4 (𝜑 → (((pmSgn‘(1...𝑁))‘(𝑃𝑆)) · ((pmSgn‘(1...𝑁))‘(𝑄𝑇))) = (-1↑(𝐼 + 𝐽)))
170169fveq2d 6653 . . 3 (𝜑 → (𝑍‘(((pmSgn‘(1...𝑁))‘(𝑃𝑆)) · ((pmSgn‘(1...𝑁))‘(𝑄𝑇)))) = (𝑍‘(-1↑(𝐼 + 𝐽))))
171170oveq1d 7154 . 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 2836 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 399   = wceq 1538  wcel 2112  ifcif 4428  {cpr 4530   class class class wbr 5033  cmpt 5113  ccnv 5522  dom cdm 5523  ccom 5527  Fun wfun 6322  wf 6324  1-1wf1 6325  1-1-ontowf1o 6327  cfv 6328  (class class class)co 7139  cmpo 7141  Fincfn 8496  1c1 10531   + caddc 10533   · cmul 10535  cle 10669  cmin 10863  -cneg 10864  cn 11629  2c2 11684  0cn0 11889  cz 11973  cuz 12235  ...cfz 12889  cexp 13429  Basecbs 16478  +gcplusg 16560  .rcmulr 16561  Grpcgrp 18098  invgcminusg 18099  SymGrpcsymg 18490  pmSgncpsgn 18612  Ringcrg 19293  CRingccrg 19294  ℤRHomczrh 20196   Mat cmat 21015   maDet cmdat 21192   maAdju cmadu 21240   minMatR1 cminmar1 21241  subMat1csmat 31146
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 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-rep 5157  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298  ax-un 7445  ax-cnex 10586  ax-resscn 10587  ax-1cn 10588  ax-icn 10589  ax-addcl 10590  ax-addrcl 10591  ax-mulcl 10592  ax-mulrcl 10593  ax-mulcom 10594  ax-addass 10595  ax-mulass 10596  ax-distr 10597  ax-i2m1 10598  ax-1ne0 10599  ax-1rid 10600  ax-rnegex 10601  ax-rrecex 10602  ax-cnre 10603  ax-pre-lttri 10604  ax-pre-lttrn 10605  ax-pre-ltadd 10606  ax-pre-mulgt0 10607  ax-addf 10609  ax-mulf 10610
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-xor 1503  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-nel 3095  df-ral 3114  df-rex 3115  df-reu 3116  df-rmo 3117  df-rab 3118  df-v 3446  df-sbc 3724  df-csb 3832  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-pss 3903  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-tp 4533  df-op 4535  df-ot 4537  df-uni 4804  df-int 4842  df-iun 4886  df-iin 4887  df-br 5034  df-opab 5096  df-mpt 5114  df-tr 5140  df-id 5428  df-eprel 5433  df-po 5442  df-so 5443  df-fr 5482  df-se 5483  df-we 5484  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-pred 6120  df-ord 6166  df-on 6167  df-lim 6168  df-suc 6169  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-isom 6337  df-riota 7097  df-ov 7142  df-oprab 7143  df-mpo 7144  df-of 7393  df-om 7565  df-1st 7675  df-2nd 7676  df-supp 7818  df-tpos 7879  df-wrecs 7934  df-recs 7995  df-rdg 8033  df-1o 8089  df-2o 8090  df-oadd 8093  df-er 8276  df-map 8395  df-pm 8396  df-ixp 8449  df-en 8497  df-dom 8498  df-sdom 8499  df-fin 8500  df-fsupp 8822  df-sup 8894  df-oi 8962  df-card 9356  df-pnf 10670  df-mnf 10671  df-xr 10672  df-ltxr 10673  df-le 10674  df-sub 10865  df-neg 10866  df-div 11291  df-nn 11630  df-2 11692  df-3 11693  df-4 11694  df-5 11695  df-6 11696  df-7 11697  df-8 11698  df-9 11699  df-n0 11890  df-xnn0 11960  df-z 11974  df-dec 12091  df-uz 12236  df-rp 12382  df-fz 12890  df-fzo 13033  df-seq 13369  df-exp 13430  df-hash 13691  df-word 13862  df-lsw 13910  df-concat 13918  df-s1 13945  df-substr 13998  df-pfx 14028  df-splice 14107  df-reverse 14116  df-s2 14205  df-struct 16480  df-ndx 16481  df-slot 16482  df-base 16484  df-sets 16485  df-ress 16486  df-plusg 16573  df-mulr 16574  df-starv 16575  df-sca 16576  df-vsca 16577  df-ip 16578  df-tset 16579  df-ple 16580  df-ds 16582  df-unif 16583  df-hom 16584  df-cco 16585  df-0g 16710  df-gsum 16711  df-prds 16716  df-pws 16718  df-mre 16852  df-mrc 16853  df-acs 16855  df-mgm 17847  df-sgrp 17896  df-mnd 17907  df-mhm 17951  df-submnd 17952  df-efmnd 18029  df-grp 18101  df-minusg 18102  df-mulg 18220  df-subg 18271  df-ghm 18351  df-gim 18394  df-cntz 18442  df-oppg 18469  df-symg 18491  df-pmtr 18565  df-psgn 18614  df-cmn 18903  df-abl 18904  df-mgp 19236  df-ur 19248  df-ring 19295  df-cring 19296  df-oppr 19372  df-dvdsr 19390  df-unit 19391  df-invr 19421  df-dvr 19432  df-rnghom 19466  df-drng 19500  df-subrg 19529  df-sra 19940  df-rgmod 19941  df-cnfld 20095  df-zring 20167  df-zrh 20200  df-dsmm 20424  df-frlm 20439  df-mat 21016  df-marrep 21166  df-subma 21185  df-mdet 21193  df-madu 21242  df-minmar1 21243  df-smat 31147
This theorem is referenced by:  madjusmdet  31184
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