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Mirrors > Home > MPE Home > Th. List > zrhpsgninv | Structured version Visualization version GIF version |
Description: The embedded sign of a permutation equals the embedded sign of the inverse of the permutation. (Contributed by SO, 9-Jul-2018.) |
Ref | Expression |
---|---|
zrhpsgninv.p | ⊢ 𝑃 = (Base‘(SymGrp‘𝑁)) |
zrhpsgninv.y | ⊢ 𝑌 = (ℤRHom‘𝑅) |
zrhpsgninv.s | ⊢ 𝑆 = (pmSgn‘𝑁) |
Ref | Expression |
---|---|
zrhpsgninv | ⊢ ((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin ∧ 𝐹 ∈ 𝑃) → ((𝑌 ∘ 𝑆)‘◡𝐹) = ((𝑌 ∘ 𝑆)‘𝐹)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2821 | . . . . 5 ⊢ (SymGrp‘𝑁) = (SymGrp‘𝑁) | |
2 | zrhpsgninv.s | . . . . 5 ⊢ 𝑆 = (pmSgn‘𝑁) | |
3 | zrhpsgninv.p | . . . . 5 ⊢ 𝑃 = (Base‘(SymGrp‘𝑁)) | |
4 | 1, 2, 3 | psgninv 20720 | . . . 4 ⊢ ((𝑁 ∈ Fin ∧ 𝐹 ∈ 𝑃) → (𝑆‘◡𝐹) = (𝑆‘𝐹)) |
5 | 4 | 3adant1 1126 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin ∧ 𝐹 ∈ 𝑃) → (𝑆‘◡𝐹) = (𝑆‘𝐹)) |
6 | 5 | fveq2d 6669 | . 2 ⊢ ((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin ∧ 𝐹 ∈ 𝑃) → (𝑌‘(𝑆‘◡𝐹)) = (𝑌‘(𝑆‘𝐹))) |
7 | eqid 2821 | . . . . . 6 ⊢ ((mulGrp‘ℂfld) ↾s {1, -1}) = ((mulGrp‘ℂfld) ↾s {1, -1}) | |
8 | 1, 2, 7 | psgnghm2 20719 | . . . . 5 ⊢ (𝑁 ∈ Fin → 𝑆 ∈ ((SymGrp‘𝑁) GrpHom ((mulGrp‘ℂfld) ↾s {1, -1}))) |
9 | eqid 2821 | . . . . . 6 ⊢ (Base‘((mulGrp‘ℂfld) ↾s {1, -1})) = (Base‘((mulGrp‘ℂfld) ↾s {1, -1})) | |
10 | 3, 9 | ghmf 18356 | . . . . 5 ⊢ (𝑆 ∈ ((SymGrp‘𝑁) GrpHom ((mulGrp‘ℂfld) ↾s {1, -1})) → 𝑆:𝑃⟶(Base‘((mulGrp‘ℂfld) ↾s {1, -1}))) |
11 | 8, 10 | syl 17 | . . . 4 ⊢ (𝑁 ∈ Fin → 𝑆:𝑃⟶(Base‘((mulGrp‘ℂfld) ↾s {1, -1}))) |
12 | 11 | 3ad2ant2 1130 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin ∧ 𝐹 ∈ 𝑃) → 𝑆:𝑃⟶(Base‘((mulGrp‘ℂfld) ↾s {1, -1}))) |
13 | eqid 2821 | . . . . . 6 ⊢ (invg‘(SymGrp‘𝑁)) = (invg‘(SymGrp‘𝑁)) | |
14 | 1, 3, 13 | symginv 18524 | . . . . 5 ⊢ (𝐹 ∈ 𝑃 → ((invg‘(SymGrp‘𝑁))‘𝐹) = ◡𝐹) |
15 | 14 | 3ad2ant3 1131 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin ∧ 𝐹 ∈ 𝑃) → ((invg‘(SymGrp‘𝑁))‘𝐹) = ◡𝐹) |
16 | 1 | symggrp 18522 | . . . . . 6 ⊢ (𝑁 ∈ Fin → (SymGrp‘𝑁) ∈ Grp) |
17 | 16 | 3ad2ant2 1130 | . . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin ∧ 𝐹 ∈ 𝑃) → (SymGrp‘𝑁) ∈ Grp) |
18 | simp3 1134 | . . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin ∧ 𝐹 ∈ 𝑃) → 𝐹 ∈ 𝑃) | |
19 | 3, 13 | grpinvcl 18145 | . . . . 5 ⊢ (((SymGrp‘𝑁) ∈ Grp ∧ 𝐹 ∈ 𝑃) → ((invg‘(SymGrp‘𝑁))‘𝐹) ∈ 𝑃) |
20 | 17, 18, 19 | syl2anc 586 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin ∧ 𝐹 ∈ 𝑃) → ((invg‘(SymGrp‘𝑁))‘𝐹) ∈ 𝑃) |
21 | 15, 20 | eqeltrrd 2914 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin ∧ 𝐹 ∈ 𝑃) → ◡𝐹 ∈ 𝑃) |
22 | fvco3 6755 | . . 3 ⊢ ((𝑆:𝑃⟶(Base‘((mulGrp‘ℂfld) ↾s {1, -1})) ∧ ◡𝐹 ∈ 𝑃) → ((𝑌 ∘ 𝑆)‘◡𝐹) = (𝑌‘(𝑆‘◡𝐹))) | |
23 | 12, 21, 22 | syl2anc 586 | . 2 ⊢ ((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin ∧ 𝐹 ∈ 𝑃) → ((𝑌 ∘ 𝑆)‘◡𝐹) = (𝑌‘(𝑆‘◡𝐹))) |
24 | fvco3 6755 | . . 3 ⊢ ((𝑆:𝑃⟶(Base‘((mulGrp‘ℂfld) ↾s {1, -1})) ∧ 𝐹 ∈ 𝑃) → ((𝑌 ∘ 𝑆)‘𝐹) = (𝑌‘(𝑆‘𝐹))) | |
25 | 12, 18, 24 | syl2anc 586 | . 2 ⊢ ((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin ∧ 𝐹 ∈ 𝑃) → ((𝑌 ∘ 𝑆)‘𝐹) = (𝑌‘(𝑆‘𝐹))) |
26 | 6, 23, 25 | 3eqtr4d 2866 | 1 ⊢ ((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin ∧ 𝐹 ∈ 𝑃) → ((𝑌 ∘ 𝑆)‘◡𝐹) = ((𝑌 ∘ 𝑆)‘𝐹)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1083 = wceq 1533 ∈ wcel 2110 {cpr 4563 ◡ccnv 5549 ∘ ccom 5554 ⟶wf 6346 ‘cfv 6350 (class class class)co 7150 Fincfn 8503 1c1 10532 -cneg 10865 Basecbs 16477 ↾s cress 16478 Grpcgrp 18097 invgcminusg 18098 GrpHom cghm 18349 SymGrpcsymg 18489 pmSgncpsgn 18611 mulGrpcmgp 19233 Ringcrg 19291 ℂfldccnfld 20539 ℤRHomczrh 20641 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 ax-rep 5183 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5322 ax-un 7455 ax-cnex 10587 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 ax-pre-mulgt0 10608 ax-addf 10610 ax-mulf 10611 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-xor 1501 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3497 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4562 df-pr 4564 df-tp 4566 df-op 4568 df-ot 4570 df-uni 4833 df-int 4870 df-iun 4914 df-iin 4915 df-br 5060 df-opab 5122 df-mpt 5140 df-tr 5166 df-id 5455 df-eprel 5460 df-po 5469 df-so 5470 df-fr 5509 df-se 5510 df-we 5511 df-xp 5556 df-rel 5557 df-cnv 5558 df-co 5559 df-dm 5560 df-rn 5561 df-res 5562 df-ima 5563 df-pred 6143 df-ord 6189 df-on 6190 df-lim 6191 df-suc 6192 df-iota 6309 df-fun 6352 df-fn 6353 df-f 6354 df-f1 6355 df-fo 6356 df-f1o 6357 df-fv 6358 df-isom 6359 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-om 7575 df-1st 7683 df-2nd 7684 df-tpos 7886 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-1o 8096 df-2o 8097 df-oadd 8100 df-er 8283 df-map 8402 df-en 8504 df-dom 8505 df-sdom 8506 df-fin 8507 df-card 9362 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-sub 10866 df-neg 10867 df-div 11292 df-nn 11633 df-2 11694 df-3 11695 df-4 11696 df-5 11697 df-6 11698 df-7 11699 df-8 11700 df-9 11701 df-n0 11892 df-xnn0 11962 df-z 11976 df-dec 12093 df-uz 12238 df-rp 12384 df-fz 12887 df-fzo 13028 df-seq 13364 df-exp 13424 df-hash 13685 df-word 13856 df-lsw 13909 df-concat 13917 df-s1 13944 df-substr 13997 df-pfx 14027 df-splice 14106 df-reverse 14115 df-s2 14204 df-struct 16479 df-ndx 16480 df-slot 16481 df-base 16483 df-sets 16484 df-ress 16485 df-plusg 16572 df-mulr 16573 df-starv 16574 df-tset 16578 df-ple 16579 df-ds 16581 df-unif 16582 df-0g 16709 df-gsum 16710 df-mre 16851 df-mrc 16852 df-acs 16854 df-mgm 17846 df-sgrp 17895 df-mnd 17906 df-mhm 17950 df-submnd 17951 df-efmnd 18028 df-grp 18100 df-minusg 18101 df-subg 18270 df-ghm 18350 df-gim 18393 df-oppg 18468 df-symg 18490 df-pmtr 18564 df-psgn 18613 df-cmn 18902 df-abl 18903 df-mgp 19234 df-ur 19246 df-ring 19293 df-cring 19294 df-oppr 19367 df-dvdsr 19385 df-unit 19386 df-invr 19416 df-dvr 19427 df-drng 19498 df-cnfld 20540 |
This theorem is referenced by: mdetleib2 21191 |
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