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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 2740 | . . . . 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 20785 | . . . 4 ⊢ ((𝑁 ∈ Fin ∧ 𝐹 ∈ 𝑃) → (𝑆‘◡𝐹) = (𝑆‘𝐹)) |
5 | 4 | 3adant1 1129 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin ∧ 𝐹 ∈ 𝑃) → (𝑆‘◡𝐹) = (𝑆‘𝐹)) |
6 | 5 | fveq2d 6775 | . 2 ⊢ ((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin ∧ 𝐹 ∈ 𝑃) → (𝑌‘(𝑆‘◡𝐹)) = (𝑌‘(𝑆‘𝐹))) |
7 | eqid 2740 | . . . . . 6 ⊢ ((mulGrp‘ℂfld) ↾s {1, -1}) = ((mulGrp‘ℂfld) ↾s {1, -1}) | |
8 | 1, 2, 7 | psgnghm2 20784 | . . . . 5 ⊢ (𝑁 ∈ Fin → 𝑆 ∈ ((SymGrp‘𝑁) GrpHom ((mulGrp‘ℂfld) ↾s {1, -1}))) |
9 | eqid 2740 | . . . . . 6 ⊢ (Base‘((mulGrp‘ℂfld) ↾s {1, -1})) = (Base‘((mulGrp‘ℂfld) ↾s {1, -1})) | |
10 | 3, 9 | ghmf 18836 | . . . . 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 1133 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin ∧ 𝐹 ∈ 𝑃) → 𝑆:𝑃⟶(Base‘((mulGrp‘ℂfld) ↾s {1, -1}))) |
13 | eqid 2740 | . . . . . 6 ⊢ (invg‘(SymGrp‘𝑁)) = (invg‘(SymGrp‘𝑁)) | |
14 | 1, 3, 13 | symginv 19008 | . . . . 5 ⊢ (𝐹 ∈ 𝑃 → ((invg‘(SymGrp‘𝑁))‘𝐹) = ◡𝐹) |
15 | 14 | 3ad2ant3 1134 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin ∧ 𝐹 ∈ 𝑃) → ((invg‘(SymGrp‘𝑁))‘𝐹) = ◡𝐹) |
16 | 1 | symggrp 19006 | . . . . . 6 ⊢ (𝑁 ∈ Fin → (SymGrp‘𝑁) ∈ Grp) |
17 | 16 | 3ad2ant2 1133 | . . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin ∧ 𝐹 ∈ 𝑃) → (SymGrp‘𝑁) ∈ Grp) |
18 | simp3 1137 | . . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin ∧ 𝐹 ∈ 𝑃) → 𝐹 ∈ 𝑃) | |
19 | 3, 13 | grpinvcl 18625 | . . . . 5 ⊢ (((SymGrp‘𝑁) ∈ Grp ∧ 𝐹 ∈ 𝑃) → ((invg‘(SymGrp‘𝑁))‘𝐹) ∈ 𝑃) |
20 | 17, 18, 19 | syl2anc 584 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin ∧ 𝐹 ∈ 𝑃) → ((invg‘(SymGrp‘𝑁))‘𝐹) ∈ 𝑃) |
21 | 15, 20 | eqeltrrd 2842 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin ∧ 𝐹 ∈ 𝑃) → ◡𝐹 ∈ 𝑃) |
22 | fvco3 6864 | . . 3 ⊢ ((𝑆:𝑃⟶(Base‘((mulGrp‘ℂfld) ↾s {1, -1})) ∧ ◡𝐹 ∈ 𝑃) → ((𝑌 ∘ 𝑆)‘◡𝐹) = (𝑌‘(𝑆‘◡𝐹))) | |
23 | 12, 21, 22 | syl2anc 584 | . 2 ⊢ ((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin ∧ 𝐹 ∈ 𝑃) → ((𝑌 ∘ 𝑆)‘◡𝐹) = (𝑌‘(𝑆‘◡𝐹))) |
24 | fvco3 6864 | . . 3 ⊢ ((𝑆:𝑃⟶(Base‘((mulGrp‘ℂfld) ↾s {1, -1})) ∧ 𝐹 ∈ 𝑃) → ((𝑌 ∘ 𝑆)‘𝐹) = (𝑌‘(𝑆‘𝐹))) | |
25 | 12, 18, 24 | syl2anc 584 | . 2 ⊢ ((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin ∧ 𝐹 ∈ 𝑃) → ((𝑌 ∘ 𝑆)‘𝐹) = (𝑌‘(𝑆‘𝐹))) |
26 | 6, 23, 25 | 3eqtr4d 2790 | 1 ⊢ ((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin ∧ 𝐹 ∈ 𝑃) → ((𝑌 ∘ 𝑆)‘◡𝐹) = ((𝑌 ∘ 𝑆)‘𝐹)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1086 = wceq 1542 ∈ wcel 2110 {cpr 4569 ◡ccnv 5589 ∘ ccom 5594 ⟶wf 6428 ‘cfv 6432 (class class class)co 7271 Fincfn 8716 1c1 10873 -cneg 11206 Basecbs 16910 ↾s cress 16939 Grpcgrp 18575 invgcminusg 18576 GrpHom cghm 18829 SymGrpcsymg 18972 pmSgncpsgn 19095 mulGrpcmgp 19718 Ringcrg 19781 ℂfldccnfld 20595 ℤRHomczrh 20699 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2015 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2711 ax-rep 5214 ax-sep 5227 ax-nul 5234 ax-pow 5292 ax-pr 5356 ax-un 7582 ax-cnex 10928 ax-resscn 10929 ax-1cn 10930 ax-icn 10931 ax-addcl 10932 ax-addrcl 10933 ax-mulcl 10934 ax-mulrcl 10935 ax-mulcom 10936 ax-addass 10937 ax-mulass 10938 ax-distr 10939 ax-i2m1 10940 ax-1ne0 10941 ax-1rid 10942 ax-rnegex 10943 ax-rrecex 10944 ax-cnre 10945 ax-pre-lttri 10946 ax-pre-lttrn 10947 ax-pre-ltadd 10948 ax-pre-mulgt0 10949 ax-addf 10951 ax-mulf 10952 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-xor 1507 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2072 df-mo 2542 df-eu 2571 df-clab 2718 df-cleq 2732 df-clel 2818 df-nfc 2891 df-ne 2946 df-nel 3052 df-ral 3071 df-rex 3072 df-reu 3073 df-rmo 3074 df-rab 3075 df-v 3433 df-sbc 3721 df-csb 3838 df-dif 3895 df-un 3897 df-in 3899 df-ss 3909 df-pss 3911 df-nul 4263 df-if 4466 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-ot 4576 df-uni 4846 df-int 4886 df-iun 4932 df-iin 4933 df-br 5080 df-opab 5142 df-mpt 5163 df-tr 5197 df-id 5490 df-eprel 5496 df-po 5504 df-so 5505 df-fr 5545 df-se 5546 df-we 5547 df-xp 5596 df-rel 5597 df-cnv 5598 df-co 5599 df-dm 5600 df-rn 5601 df-res 5602 df-ima 5603 df-pred 6201 df-ord 6268 df-on 6269 df-lim 6270 df-suc 6271 df-iota 6390 df-fun 6434 df-fn 6435 df-f 6436 df-f1 6437 df-fo 6438 df-f1o 6439 df-fv 6440 df-isom 6441 df-riota 7228 df-ov 7274 df-oprab 7275 df-mpo 7276 df-om 7707 df-1st 7824 df-2nd 7825 df-tpos 8033 df-frecs 8088 df-wrecs 8119 df-recs 8193 df-rdg 8232 df-1o 8288 df-2o 8289 df-er 8481 df-map 8600 df-en 8717 df-dom 8718 df-sdom 8719 df-fin 8720 df-card 9698 df-pnf 11012 df-mnf 11013 df-xr 11014 df-ltxr 11015 df-le 11016 df-sub 11207 df-neg 11208 df-div 11633 df-nn 11974 df-2 12036 df-3 12037 df-4 12038 df-5 12039 df-6 12040 df-7 12041 df-8 12042 df-9 12043 df-n0 12234 df-xnn0 12306 df-z 12320 df-dec 12437 df-uz 12582 df-rp 12730 df-fz 13239 df-fzo 13382 df-seq 13720 df-exp 13781 df-hash 14043 df-word 14216 df-lsw 14264 df-concat 14272 df-s1 14299 df-substr 14352 df-pfx 14382 df-splice 14461 df-reverse 14470 df-s2 14559 df-struct 16846 df-sets 16863 df-slot 16881 df-ndx 16893 df-base 16911 df-ress 16940 df-plusg 16973 df-mulr 16974 df-starv 16975 df-tset 16979 df-ple 16980 df-ds 16982 df-unif 16983 df-0g 17150 df-gsum 17151 df-mre 17293 df-mrc 17294 df-acs 17296 df-mgm 18324 df-sgrp 18373 df-mnd 18384 df-mhm 18428 df-submnd 18429 df-efmnd 18506 df-grp 18578 df-minusg 18579 df-subg 18750 df-ghm 18830 df-gim 18873 df-oppg 18948 df-symg 18973 df-pmtr 19048 df-psgn 19097 df-cmn 19386 df-abl 19387 df-mgp 19719 df-ur 19736 df-ring 19783 df-cring 19784 df-oppr 19860 df-dvdsr 19881 df-unit 19882 df-invr 19912 df-dvr 19923 df-drng 19991 df-cnfld 20596 |
This theorem is referenced by: mdetleib2 21735 |
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