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Theorem madjusmdetlem4 29702
Description: Lemma for madjusmdet 29703. (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 2621 . . 3 (Base‘(SymGrp‘(1...𝑁))) = (Base‘(SymGrp‘(1...𝑁)))
14 eqid 2621 . . 3 (pmSgn‘(1...𝑁)) = (pmSgn‘(1...𝑁))
15 eqid 2621 . . 3 (𝐼(((1...𝑁) minMatR1 𝑅)‘𝑀)𝐽) = (𝐼(((1...𝑁) minMatR1 𝑅)‘𝑀)𝐽)
16 fveq2 6153 . . . . 5 (𝑘 = 𝑖 → ((𝑃𝑆)‘𝑘) = ((𝑃𝑆)‘𝑖))
1716oveq1d 6625 . . . 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 6153 . . . . 5 (𝑙 = 𝑗 → ((𝑄𝑇)‘𝑙) = ((𝑄𝑇)‘𝑗))
1918oveq2d 6626 . . . 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, 19cbvmpt2v 6695 . . 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 2621 . . . . . 6 (1...𝑁) = (1...𝑁)
22 madjusmdetlem2.p . . . . . 6 𝑃 = (𝑖 ∈ (1...𝑁) ↦ if(𝑖 = 1, 𝐼, if(𝑖𝐼, (𝑖 − 1), 𝑖)))
23 eqid 2621 . . . . . 6 (SymGrp‘(1...𝑁)) = (SymGrp‘(1...𝑁))
2421, 22, 23, 13fzto1st 29662 . . . . 5 (𝐼 ∈ (1...𝑁) → 𝑃 ∈ (Base‘(SymGrp‘(1...𝑁))))
2510, 24syl 17 . . . 4 (𝜑𝑃 ∈ (Base‘(SymGrp‘(1...𝑁))))
26 nnuz 11675 . . . . . . . . 9 ℕ = (ℤ‘1)
278, 26syl6eleq 2708 . . . . . . . 8 (𝜑𝑁 ∈ (ℤ‘1))
28 eluzfz2 12299 . . . . . . . 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 29662 . . . . . . 7 (𝑁 ∈ (1...𝑁) → 𝑆 ∈ (Base‘(SymGrp‘(1...𝑁))))
3229, 31syl 17 . . . . . 6 (𝜑𝑆 ∈ (Base‘(SymGrp‘(1...𝑁))))
33 eqid 2621 . . . . . . 7 (invg‘(SymGrp‘(1...𝑁))) = (invg‘(SymGrp‘(1...𝑁)))
3423, 13, 33symginv 17754 . . . . . 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 12720 . . . . . . 7 (𝜑 → (1...𝑁) ∈ Fin)
3723symggrp 17752 . . . . . . 7 ((1...𝑁) ∈ Fin → (SymGrp‘(1...𝑁)) ∈ Grp)
3836, 37syl 17 . . . . . 6 (𝜑 → (SymGrp‘(1...𝑁)) ∈ Grp)
3913, 33grpinvcl 17399 . . . . . 6 (((SymGrp‘(1...𝑁)) ∈ Grp ∧ 𝑆 ∈ (Base‘(SymGrp‘(1...𝑁)))) → ((invg‘(SymGrp‘(1...𝑁)))‘𝑆) ∈ (Base‘(SymGrp‘(1...𝑁))))
4038, 32, 39syl2anc 692 . . . . 5 (𝜑 → ((invg‘(SymGrp‘(1...𝑁)))‘𝑆) ∈ (Base‘(SymGrp‘(1...𝑁))))
4135, 40eqeltrrd 2699 . . . 4 (𝜑𝑆 ∈ (Base‘(SymGrp‘(1...𝑁))))
42 eqid 2621 . . . . . 6 (+g‘(SymGrp‘(1...𝑁))) = (+g‘(SymGrp‘(1...𝑁)))
4323, 13, 42symgov 17742 . . . . 5 ((𝑃 ∈ (Base‘(SymGrp‘(1...𝑁))) ∧ 𝑆 ∈ (Base‘(SymGrp‘(1...𝑁)))) → (𝑃(+g‘(SymGrp‘(1...𝑁)))𝑆) = (𝑃𝑆))
4423, 13, 42symgcl 17743 . . . . 5 ((𝑃 ∈ (Base‘(SymGrp‘(1...𝑁))) ∧ 𝑆 ∈ (Base‘(SymGrp‘(1...𝑁)))) → (𝑃(+g‘(SymGrp‘(1...𝑁)))𝑆) ∈ (Base‘(SymGrp‘(1...𝑁))))
4543, 44eqeltrrd 2699 . . . 4 ((𝑃 ∈ (Base‘(SymGrp‘(1...𝑁))) ∧ 𝑆 ∈ (Base‘(SymGrp‘(1...𝑁)))) → (𝑃𝑆) ∈ (Base‘(SymGrp‘(1...𝑁))))
4625, 41, 45syl2anc 692 . . 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 29662 . . . . 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 29662 . . . . . . 7 (𝑁 ∈ (1...𝑁) → 𝑇 ∈ (Base‘(SymGrp‘(1...𝑁))))
5229, 51syl 17 . . . . . 6 (𝜑𝑇 ∈ (Base‘(SymGrp‘(1...𝑁))))
5323, 13, 33symginv 17754 . . . . . 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 17399 . . . . . 6 (((SymGrp‘(1...𝑁)) ∈ Grp ∧ 𝑇 ∈ (Base‘(SymGrp‘(1...𝑁)))) → ((invg‘(SymGrp‘(1...𝑁)))‘𝑇) ∈ (Base‘(SymGrp‘(1...𝑁))))
5638, 52, 55syl2anc 692 . . . . 5 (𝜑 → ((invg‘(SymGrp‘(1...𝑁)))‘𝑇) ∈ (Base‘(SymGrp‘(1...𝑁))))
5754, 56eqeltrrd 2699 . . . 4 (𝜑𝑇 ∈ (Base‘(SymGrp‘(1...𝑁))))
5823, 13, 42symgov 17742 . . . . 5 ((𝑄 ∈ (Base‘(SymGrp‘(1...𝑁))) ∧ 𝑇 ∈ (Base‘(SymGrp‘(1...𝑁)))) → (𝑄(+g‘(SymGrp‘(1...𝑁)))𝑇) = (𝑄𝑇))
5923, 13, 42symgcl 17743 . . . . 5 ((𝑄 ∈ (Base‘(SymGrp‘(1...𝑁))) ∧ 𝑇 ∈ (Base‘(SymGrp‘(1...𝑁)))) → (𝑄(+g‘(SymGrp‘(1...𝑁)))𝑇) ∈ (Base‘(SymGrp‘(1...𝑁))))
6058, 59eqeltrrd 2699 . . . 4 ((𝑄 ∈ (Base‘(SymGrp‘(1...𝑁))) ∧ 𝑇 ∈ (Base‘(SymGrp‘(1...𝑁)))) → (𝑄𝑇) ∈ (Base‘(SymGrp‘(1...𝑁))))
6149, 57, 60syl2anc 692 . . 3 (𝜑 → (𝑄𝑇) ∈ (Base‘(SymGrp‘(1...𝑁))))
6223, 13symgbasf1o 17735 . . . . . . 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 6098 . . . . . 6 (𝑆:(1...𝑁)–1-1-onto→(1...𝑁) → 𝑆:(1...𝑁)–1-1→(1...𝑁))
65 df-f1 5857 . . . . . . 7 (𝑆:(1...𝑁)–1-1→(1...𝑁) ↔ (𝑆:(1...𝑁)⟶(1...𝑁) ∧ Fun 𝑆))
6665simprbi 480 . . . . . 6 (𝑆:(1...𝑁)–1-1→(1...𝑁) → Fun 𝑆)
6763, 64, 663syl 18 . . . . 5 (𝜑 → Fun 𝑆)
68 f1ocnv 6111 . . . . . . 7 (𝑆:(1...𝑁)–1-1-onto→(1...𝑁) → 𝑆:(1...𝑁)–1-1-onto→(1...𝑁))
69 f1odm 6103 . . . . . . 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 2701 . . . . 5 (𝜑𝑁 ∈ dom 𝑆)
72 fvco 6236 . . . . 5 ((Fun 𝑆𝑁 ∈ dom 𝑆) → ((𝑃𝑆)‘𝑁) = (𝑃‘(𝑆𝑁)))
7367, 71, 72syl2anc 692 . . . 4 (𝜑 → ((𝑃𝑆)‘𝑁) = (𝑃‘(𝑆𝑁)))
7421, 30, 23, 13fzto1stinvn 29663 . . . . . 6 (𝑁 ∈ (1...𝑁) → (𝑆𝑁) = 1)
7529, 74syl 17 . . . . 5 (𝜑 → (𝑆𝑁) = 1)
7675fveq2d 6157 . . . 4 (𝜑 → (𝑃‘(𝑆𝑁)) = (𝑃‘1))
7721, 22fzto1stfv1 29660 . . . . 5 (𝐼 ∈ (1...𝑁) → (𝑃‘1) = 𝐼)
7810, 77syl 17 . . . 4 (𝜑 → (𝑃‘1) = 𝐼)
7973, 76, 783eqtrd 2659 . . 3 (𝜑 → ((𝑃𝑆)‘𝑁) = 𝐼)
8023, 13symgbasf1o 17735 . . . . . . 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 6098 . . . . . 6 (𝑇:(1...𝑁)–1-1-onto→(1...𝑁) → 𝑇:(1...𝑁)–1-1→(1...𝑁))
83 df-f1 5857 . . . . . . 7 (𝑇:(1...𝑁)–1-1→(1...𝑁) ↔ (𝑇:(1...𝑁)⟶(1...𝑁) ∧ Fun 𝑇))
8483simprbi 480 . . . . . 6 (𝑇:(1...𝑁)–1-1→(1...𝑁) → Fun 𝑇)
8581, 82, 843syl 18 . . . . 5 (𝜑 → Fun 𝑇)
86 f1ocnv 6111 . . . . . . 7 (𝑇:(1...𝑁)–1-1-onto→(1...𝑁) → 𝑇:(1...𝑁)–1-1-onto→(1...𝑁))
87 f1odm 6103 . . . . . . 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 2701 . . . . 5 (𝜑𝑁 ∈ dom 𝑇)
90 fvco 6236 . . . . 5 ((Fun 𝑇𝑁 ∈ dom 𝑇) → ((𝑄𝑇)‘𝑁) = (𝑄‘(𝑇𝑁)))
9185, 89, 90syl2anc 692 . . . 4 (𝜑 → ((𝑄𝑇)‘𝑁) = (𝑄‘(𝑇𝑁)))
9221, 50, 23, 13fzto1stinvn 29663 . . . . . 6 (𝑁 ∈ (1...𝑁) → (𝑇𝑁) = 1)
9329, 92syl 17 . . . . 5 (𝜑 → (𝑇𝑁) = 1)
9493fveq2d 6157 . . . 4 (𝜑 → (𝑄‘(𝑇𝑁)) = (𝑄‘1))
9521, 47fzto1stfv1 29660 . . . . 5 (𝐽 ∈ (1...𝑁) → (𝑄‘1) = 𝐽)
9611, 95syl 17 . . . 4 (𝜑 → (𝑄‘1) = 𝐽)
9791, 94, 963eqtrd 2659 . . 3 (𝜑 → ((𝑄𝑇)‘𝑁) = 𝐽)
98 crngring 18490 . . . . . 6 (𝑅 ∈ CRing → 𝑅 ∈ Ring)
999, 98syl 17 . . . . 5 (𝜑𝑅 ∈ Ring)
1002, 1minmar1cl 20389 . . . . 5 (((𝑅 ∈ Ring ∧ 𝑀𝐵) ∧ (𝐼 ∈ (1...𝑁) ∧ 𝐽 ∈ (1...𝑁))) → (𝐼(((1...𝑁) minMatR1 𝑅)‘𝑀)𝐽) ∈ 𝐵)
10199, 12, 10, 11, 100syl22anc 1324 . . . 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 29701 . . 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 29699 . 2 (𝜑 → (𝐽(𝐾𝑀)𝐼) = ((𝑍‘(((pmSgn‘(1...𝑁))‘(𝑃𝑆)) · ((pmSgn‘(1...𝑁))‘(𝑄𝑇)))) · (𝐸‘(𝐼(subMat1‘𝑀)𝐽))))
10423, 14, 13psgnco 19861 . . . . . . . 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 1323 . . . . . . 7 (𝜑 → ((pmSgn‘(1...𝑁))‘(𝑃𝑆)) = (((pmSgn‘(1...𝑁))‘𝑃) · ((pmSgn‘(1...𝑁))‘𝑆)))
10621, 22, 23, 13, 14psgnfzto1st 29664 . . . . . . . . 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 19860 . . . . . . . . . 10 (((1...𝑁) ∈ Fin ∧ 𝑆 ∈ (Base‘(SymGrp‘(1...𝑁)))) → ((pmSgn‘(1...𝑁))‘𝑆) = ((pmSgn‘(1...𝑁))‘𝑆))
10936, 32, 108syl2anc 692 . . . . . . . . 9 (𝜑 → ((pmSgn‘(1...𝑁))‘𝑆) = ((pmSgn‘(1...𝑁))‘𝑆))
11021, 30, 23, 13, 14psgnfzto1st 29664 . . . . . . . . . 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 2655 . . . . . . . 8 (𝜑 → ((pmSgn‘(1...𝑁))‘𝑆) = (-1↑(𝑁 + 1)))
113107, 112oveq12d 6628 . . . . . . 7 (𝜑 → (((pmSgn‘(1...𝑁))‘𝑃) · ((pmSgn‘(1...𝑁))‘𝑆)) = ((-1↑(𝐼 + 1)) · (-1↑(𝑁 + 1))))
114105, 113eqtrd 2655 . . . . . 6 (𝜑 → ((pmSgn‘(1...𝑁))‘(𝑃𝑆)) = ((-1↑(𝐼 + 1)) · (-1↑(𝑁 + 1))))
11523, 14, 13psgnco 19861 . . . . . . . 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 1323 . . . . . . 7 (𝜑 → ((pmSgn‘(1...𝑁))‘(𝑄𝑇)) = (((pmSgn‘(1...𝑁))‘𝑄) · ((pmSgn‘(1...𝑁))‘𝑇)))
11721, 47, 23, 13, 14psgnfzto1st 29664 . . . . . . . . 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 19860 . . . . . . . . . 10 (((1...𝑁) ∈ Fin ∧ 𝑇 ∈ (Base‘(SymGrp‘(1...𝑁)))) → ((pmSgn‘(1...𝑁))‘𝑇) = ((pmSgn‘(1...𝑁))‘𝑇))
12036, 52, 119syl2anc 692 . . . . . . . . 9 (𝜑 → ((pmSgn‘(1...𝑁))‘𝑇) = ((pmSgn‘(1...𝑁))‘𝑇))
12121, 50, 23, 13, 14psgnfzto1st 29664 . . . . . . . . . 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 2655 . . . . . . . 8 (𝜑 → ((pmSgn‘(1...𝑁))‘𝑇) = (-1↑(𝑁 + 1)))
124118, 123oveq12d 6628 . . . . . . 7 (𝜑 → (((pmSgn‘(1...𝑁))‘𝑄) · ((pmSgn‘(1...𝑁))‘𝑇)) = ((-1↑(𝐽 + 1)) · (-1↑(𝑁 + 1))))
125116, 124eqtrd 2655 . . . . . 6 (𝜑 → ((pmSgn‘(1...𝑁))‘(𝑄𝑇)) = ((-1↑(𝐽 + 1)) · (-1↑(𝑁 + 1))))
126114, 125oveq12d 6628 . . . . 5 (𝜑 → (((pmSgn‘(1...𝑁))‘(𝑃𝑆)) · ((pmSgn‘(1...𝑁))‘(𝑄𝑇))) = (((-1↑(𝐼 + 1)) · (-1↑(𝑁 + 1))) · ((-1↑(𝐽 + 1)) · (-1↑(𝑁 + 1)))))
127 1cnd 10008 . . . . . . . . 9 (𝜑 → 1 ∈ ℂ)
128127negcld 10331 . . . . . . . 8 (𝜑 → -1 ∈ ℂ)
129 fz1ssnn 12322 . . . . . . . . . . 11 (1...𝑁) ⊆ ℕ
130129, 10sseldi 3585 . . . . . . . . . 10 (𝜑𝐼 ∈ ℕ)
131130nnnn0d 11303 . . . . . . . . 9 (𝜑𝐼 ∈ ℕ0)
132 1nn0 11260 . . . . . . . . . 10 1 ∈ ℕ0
133132a1i 11 . . . . . . . . 9 (𝜑 → 1 ∈ ℕ0)
134131, 133nn0addcld 11307 . . . . . . . 8 (𝜑 → (𝐼 + 1) ∈ ℕ0)
135128, 134expcld 12956 . . . . . . 7 (𝜑 → (-1↑(𝐼 + 1)) ∈ ℂ)
1368nnnn0d 11303 . . . . . . . . 9 (𝜑𝑁 ∈ ℕ0)
137136, 133nn0addcld 11307 . . . . . . . 8 (𝜑 → (𝑁 + 1) ∈ ℕ0)
138128, 137expcld 12956 . . . . . . 7 (𝜑 → (-1↑(𝑁 + 1)) ∈ ℂ)
139129, 11sseldi 3585 . . . . . . . . . 10 (𝜑𝐽 ∈ ℕ)
140139nnnn0d 11303 . . . . . . . . 9 (𝜑𝐽 ∈ ℕ0)
141140, 133nn0addcld 11307 . . . . . . . 8 (𝜑 → (𝐽 + 1) ∈ ℕ0)
142128, 141expcld 12956 . . . . . . 7 (𝜑 → (-1↑(𝐽 + 1)) ∈ ℂ)
143135, 138, 142, 138mul4d 10200 . . . . . 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 12958 . . . . . . . 8 (𝜑 → (-1↑((𝐼 + 1) + (𝐽 + 1))) = ((-1↑(𝐼 + 1)) · (-1↑(𝐽 + 1))))
145130nncnd 10988 . . . . . . . . . . . 12 (𝜑𝐼 ∈ ℂ)
146139nncnd 10988 . . . . . . . . . . . 12 (𝜑𝐽 ∈ ℂ)
147145, 127, 146, 127add4d 10216 . . . . . . . . . . 11 (𝜑 → ((𝐼 + 1) + (𝐽 + 1)) = ((𝐼 + 𝐽) + (1 + 1)))
148 1p1e2 11086 . . . . . . . . . . . 12 (1 + 1) = 2
149148oveq2i 6621 . . . . . . . . . . 11 ((𝐼 + 𝐽) + (1 + 1)) = ((𝐼 + 𝐽) + 2)
150147, 149syl6eq 2671 . . . . . . . . . 10 (𝜑 → ((𝐼 + 1) + (𝐽 + 1)) = ((𝐼 + 𝐽) + 2))
151150oveq2d 6626 . . . . . . . . 9 (𝜑 → (-1↑((𝐼 + 1) + (𝐽 + 1))) = (-1↑((𝐼 + 𝐽) + 2)))
152 2nn0 11261 . . . . . . . . . . . 12 2 ∈ ℕ0
153152a1i 11 . . . . . . . . . . 11 (𝜑 → 2 ∈ ℕ0)
154131, 140nn0addcld 11307 . . . . . . . . . . 11 (𝜑 → (𝐼 + 𝐽) ∈ ℕ0)
155128, 153, 154expaddd 12958 . . . . . . . . . 10 (𝜑 → (-1↑((𝐼 + 𝐽) + 2)) = ((-1↑(𝐼 + 𝐽)) · (-1↑2)))
156 neg1sqe1 12907 . . . . . . . . . . 11 (-1↑2) = 1
157156oveq2i 6621 . . . . . . . . . 10 ((-1↑(𝐼 + 𝐽)) · (-1↑2)) = ((-1↑(𝐼 + 𝐽)) · 1)
158155, 157syl6eq 2671 . . . . . . . . 9 (𝜑 → (-1↑((𝐼 + 𝐽) + 2)) = ((-1↑(𝐼 + 𝐽)) · 1))
159128, 154expcld 12956 . . . . . . . . . 10 (𝜑 → (-1↑(𝐼 + 𝐽)) ∈ ℂ)
160159mulid1d 10009 . . . . . . . . 9 (𝜑 → ((-1↑(𝐼 + 𝐽)) · 1) = (-1↑(𝐼 + 𝐽)))
161151, 158, 1603eqtrd 2659 . . . . . . . 8 (𝜑 → (-1↑((𝐼 + 1) + (𝐽 + 1))) = (-1↑(𝐼 + 𝐽)))
162144, 161eqtr3d 2657 . . . . . . 7 (𝜑 → ((-1↑(𝐼 + 1)) · (-1↑(𝐽 + 1))) = (-1↑(𝐼 + 𝐽)))
163137nn0zd 11432 . . . . . . . 8 (𝜑 → (𝑁 + 1) ∈ ℤ)
164 m1expcl2 12830 . . . . . . . 8 ((𝑁 + 1) ∈ ℤ → (-1↑(𝑁 + 1)) ∈ {-1, 1})
165 1neg1t1neg1 29380 . . . . . . . 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 6628 . . . . . 6 (𝜑 → (((-1↑(𝐼 + 1)) · (-1↑(𝐽 + 1))) · ((-1↑(𝑁 + 1)) · (-1↑(𝑁 + 1)))) = ((-1↑(𝐼 + 𝐽)) · 1))
168143, 167, 1603eqtrd 2659 . . . . 5 (𝜑 → (((-1↑(𝐼 + 1)) · (-1↑(𝑁 + 1))) · ((-1↑(𝐽 + 1)) · (-1↑(𝑁 + 1)))) = (-1↑(𝐼 + 𝐽)))
169126, 168eqtrd 2655 . . . 4 (𝜑 → (((pmSgn‘(1...𝑁))‘(𝑃𝑆)) · ((pmSgn‘(1...𝑁))‘(𝑄𝑇))) = (-1↑(𝐼 + 𝐽)))
170169fveq2d 6157 . . 3 (𝜑 → (𝑍‘(((pmSgn‘(1...𝑁))‘(𝑃𝑆)) · ((pmSgn‘(1...𝑁))‘(𝑄𝑇)))) = (𝑍‘(-1↑(𝐼 + 𝐽))))
171170oveq1d 6625 . 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 2655 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 384   = wceq 1480  wcel 1987  ifcif 4063  {cpr 4155   class class class wbr 4618  cmpt 4678  ccnv 5078  dom cdm 5079  ccom 5083  Fun wfun 5846  wf 5848  1-1wf1 5849  1-1-ontowf1o 5851  cfv 5852  (class class class)co 6610  cmpt2 6612  Fincfn 7907  1c1 9889   + caddc 9891   · cmul 9893  cle 10027  cmin 10218  -cneg 10219  cn 10972  2c2 11022  0cn0 11244  cz 11329  cuz 11639  ...cfz 12276  cexp 12808  Basecbs 15792  +gcplusg 15873  .rcmulr 15874  Grpcgrp 17354  invgcminusg 17355  SymGrpcsymg 17729  pmSgncpsgn 17841  Ringcrg 18479  CRingccrg 18480  ℤRHomczrh 19780   Mat cmat 20145   maDet cmdat 20322   maAdju cmadu 20370   minMatR1 cminmar1 20371  subMat1csmat 29665
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4736  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6909  ax-inf2 8490  ax-cnex 9944  ax-resscn 9945  ax-1cn 9946  ax-icn 9947  ax-addcl 9948  ax-addrcl 9949  ax-mulcl 9950  ax-mulrcl 9951  ax-mulcom 9952  ax-addass 9953  ax-mulass 9954  ax-distr 9955  ax-i2m1 9956  ax-1ne0 9957  ax-1rid 9958  ax-rnegex 9959  ax-rrecex 9960  ax-cnre 9961  ax-pre-lttri 9962  ax-pre-lttrn 9963  ax-pre-ltadd 9964  ax-pre-mulgt0 9965  ax-addf 9967  ax-mulf 9968
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-xor 1462  df-tru 1483  df-fal 1486  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2912  df-rex 2913  df-reu 2914  df-rmo 2915  df-rab 2916  df-v 3191  df-sbc 3422  df-csb 3519  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-pss 3575  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-tp 4158  df-op 4160  df-ot 4162  df-uni 4408  df-int 4446  df-iun 4492  df-iin 4493  df-br 4619  df-opab 4679  df-mpt 4680  df-tr 4718  df-eprel 4990  df-id 4994  df-po 5000  df-so 5001  df-fr 5038  df-se 5039  df-we 5040  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-pred 5644  df-ord 5690  df-on 5691  df-lim 5692  df-suc 5693  df-iota 5815  df-fun 5854  df-fn 5855  df-f 5856  df-f1 5857  df-fo 5858  df-f1o 5859  df-fv 5860  df-isom 5861  df-riota 6571  df-ov 6613  df-oprab 6614  df-mpt2 6615  df-of 6857  df-om 7020  df-1st 7120  df-2nd 7121  df-supp 7248  df-tpos 7304  df-wrecs 7359  df-recs 7420  df-rdg 7458  df-1o 7512  df-2o 7513  df-oadd 7516  df-er 7694  df-map 7811  df-pm 7812  df-ixp 7861  df-en 7908  df-dom 7909  df-sdom 7910  df-fin 7911  df-fsupp 8228  df-sup 8300  df-oi 8367  df-card 8717  df-pnf 10028  df-mnf 10029  df-xr 10030  df-ltxr 10031  df-le 10032  df-sub 10220  df-neg 10221  df-div 10637  df-nn 10973  df-2 11031  df-3 11032  df-4 11033  df-5 11034  df-6 11035  df-7 11036  df-8 11037  df-9 11038  df-n0 11245  df-xnn0 11316  df-z 11330  df-dec 11446  df-uz 11640  df-rp 11785  df-fz 12277  df-fzo 12415  df-seq 12750  df-exp 12809  df-hash 13066  df-word 13246  df-lsw 13247  df-concat 13248  df-s1 13249  df-substr 13250  df-splice 13251  df-reverse 13252  df-s2 13538  df-struct 15794  df-ndx 15795  df-slot 15796  df-base 15797  df-sets 15798  df-ress 15799  df-plusg 15886  df-mulr 15887  df-starv 15888  df-sca 15889  df-vsca 15890  df-ip 15891  df-tset 15892  df-ple 15893  df-ds 15896  df-unif 15897  df-hom 15898  df-cco 15899  df-0g 16034  df-gsum 16035  df-prds 16040  df-pws 16042  df-mre 16178  df-mrc 16179  df-acs 16181  df-mgm 17174  df-sgrp 17216  df-mnd 17227  df-mhm 17267  df-submnd 17268  df-grp 17357  df-minusg 17358  df-mulg 17473  df-subg 17523  df-ghm 17590  df-gim 17633  df-cntz 17682  df-oppg 17708  df-symg 17730  df-pmtr 17794  df-psgn 17843  df-cmn 18127  df-abl 18128  df-mgp 18422  df-ur 18434  df-ring 18481  df-cring 18482  df-oppr 18555  df-dvdsr 18573  df-unit 18574  df-invr 18604  df-dvr 18615  df-rnghom 18647  df-drng 18681  df-subrg 18710  df-sra 19104  df-rgmod 19105  df-cnfld 19679  df-zring 19751  df-zrh 19784  df-dsmm 20008  df-frlm 20023  df-mat 20146  df-marrep 20296  df-subma 20315  df-mdet 20323  df-madu 20372  df-minmar1 20373  df-smat 29666
This theorem is referenced by:  madjusmdet  29703
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