Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > psgnevpmb | Structured version Visualization version GIF version |
Description: A class is an even permutation if it is a permutation with sign 1. (Contributed by SO, 9-Jul-2018.) |
Ref | Expression |
---|---|
evpmss.s | β’ π = (SymGrpβπ·) |
evpmss.p | β’ π = (Baseβπ) |
psgnevpmb.n | β’ π = (pmSgnβπ·) |
Ref | Expression |
---|---|
psgnevpmb | β’ (π· β Fin β (πΉ β (pmEvenβπ·) β (πΉ β π β§ (πβπΉ) = 1))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elex 3455 | . . . 4 β’ (π· β Fin β π· β V) | |
2 | fveq2 6800 | . . . . . . . 8 β’ (π = π· β (pmSgnβπ) = (pmSgnβπ·)) | |
3 | psgnevpmb.n | . . . . . . . 8 β’ π = (pmSgnβπ·) | |
4 | 2, 3 | eqtr4di 2794 | . . . . . . 7 β’ (π = π· β (pmSgnβπ) = π) |
5 | 4 | cnveqd 5793 | . . . . . 6 β’ (π = π· β β‘(pmSgnβπ) = β‘π) |
6 | 5 | imaeq1d 5974 | . . . . 5 β’ (π = π· β (β‘(pmSgnβπ) β {1}) = (β‘π β {1})) |
7 | df-evpm 19141 | . . . . 5 β’ pmEven = (π β V β¦ (β‘(pmSgnβπ) β {1})) | |
8 | 3 | fvexi 6814 | . . . . . . 7 β’ π β V |
9 | 8 | cnvex 7800 | . . . . . 6 β’ β‘π β V |
10 | 9 | imaex 7791 | . . . . 5 β’ (β‘π β {1}) β V |
11 | 6, 7, 10 | fvmpt 6903 | . . . 4 β’ (π· β V β (pmEvenβπ·) = (β‘π β {1})) |
12 | 1, 11 | syl 17 | . . 3 β’ (π· β Fin β (pmEvenβπ·) = (β‘π β {1})) |
13 | 12 | eleq2d 2822 | . 2 β’ (π· β Fin β (πΉ β (pmEvenβπ·) β πΉ β (β‘π β {1}))) |
14 | evpmss.s | . . . . 5 β’ π = (SymGrpβπ·) | |
15 | eqid 2736 | . . . . 5 β’ ((mulGrpββfld) βΎs {1, -1}) = ((mulGrpββfld) βΎs {1, -1}) | |
16 | 14, 3, 15 | psgnghm2 20827 | . . . 4 β’ (π· β Fin β π β (π GrpHom ((mulGrpββfld) βΎs {1, -1}))) |
17 | evpmss.p | . . . . 5 β’ π = (Baseβπ) | |
18 | eqid 2736 | . . . . 5 β’ (Baseβ((mulGrpββfld) βΎs {1, -1})) = (Baseβ((mulGrpββfld) βΎs {1, -1})) | |
19 | 17, 18 | ghmf 18879 | . . . 4 β’ (π β (π GrpHom ((mulGrpββfld) βΎs {1, -1})) β π:πβΆ(Baseβ((mulGrpββfld) βΎs {1, -1}))) |
20 | 16, 19 | syl 17 | . . 3 β’ (π· β Fin β π:πβΆ(Baseβ((mulGrpββfld) βΎs {1, -1}))) |
21 | ffn 6626 | . . 3 β’ (π:πβΆ(Baseβ((mulGrpββfld) βΎs {1, -1})) β π Fn π) | |
22 | fniniseg 6965 | . . 3 β’ (π Fn π β (πΉ β (β‘π β {1}) β (πΉ β π β§ (πβπΉ) = 1))) | |
23 | 20, 21, 22 | 3syl 18 | . 2 β’ (π· β Fin β (πΉ β (β‘π β {1}) β (πΉ β π β§ (πβπΉ) = 1))) |
24 | 13, 23 | bitrd 280 | 1 β’ (π· β Fin β (πΉ β (pmEvenβπ·) β (πΉ β π β§ (πβπΉ) = 1))) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β wb 205 β§ wa 397 = wceq 1539 β wcel 2104 Vcvv 3437 {csn 4565 {cpr 4567 β‘ccnv 5595 β cima 5599 Fn wfn 6449 βΆwf 6450 βcfv 6454 (class class class)co 7303 Fincfn 8760 1c1 10914 -cneg 11248 Basecbs 16953 βΎs cress 16982 GrpHom cghm 18872 SymGrpcsymg 19015 pmSgncpsgn 19138 pmEvencevpm 19139 mulGrpcmgp 19761 βfldccnfld 20638 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-10 2135 ax-11 2152 ax-12 2169 ax-ext 2707 ax-rep 5218 ax-sep 5232 ax-nul 5239 ax-pow 5297 ax-pr 5361 ax-un 7616 ax-cnex 10969 ax-resscn 10970 ax-1cn 10971 ax-icn 10972 ax-addcl 10973 ax-addrcl 10974 ax-mulcl 10975 ax-mulrcl 10976 ax-mulcom 10977 ax-addass 10978 ax-mulass 10979 ax-distr 10980 ax-i2m1 10981 ax-1ne0 10982 ax-1rid 10983 ax-rnegex 10984 ax-rrecex 10985 ax-cnre 10986 ax-pre-lttri 10987 ax-pre-lttrn 10988 ax-pre-ltadd 10989 ax-pre-mulgt0 10990 ax-addf 10992 ax-mulf 10993 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 846 df-3or 1088 df-3an 1089 df-xor 1508 df-tru 1542 df-fal 1552 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2887 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3285 df-reu 3286 df-rab 3287 df-v 3439 df-sbc 3722 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 4566 df-pr 4568 df-tp 4570 df-op 4572 df-ot 4574 df-uni 4845 df-int 4887 df-iun 4933 df-iin 4934 df-br 5082 df-opab 5144 df-mpt 5165 df-tr 5199 df-id 5496 df-eprel 5502 df-po 5510 df-so 5511 df-fr 5551 df-se 5552 df-we 5553 df-xp 5602 df-rel 5603 df-cnv 5604 df-co 5605 df-dm 5606 df-rn 5607 df-res 5608 df-ima 5609 df-pred 6213 df-ord 6280 df-on 6281 df-lim 6282 df-suc 6283 df-iota 6406 df-fun 6456 df-fn 6457 df-f 6458 df-f1 6459 df-fo 6460 df-f1o 6461 df-fv 6462 df-isom 6463 df-riota 7260 df-ov 7306 df-oprab 7307 df-mpo 7308 df-om 7741 df-1st 7859 df-2nd 7860 df-tpos 8069 df-frecs 8124 df-wrecs 8155 df-recs 8229 df-rdg 8268 df-1o 8324 df-2o 8325 df-er 8525 df-map 8644 df-en 8761 df-dom 8762 df-sdom 8763 df-fin 8764 df-card 9737 df-pnf 11053 df-mnf 11054 df-xr 11055 df-ltxr 11056 df-le 11057 df-sub 11249 df-neg 11250 df-div 11675 df-nn 12016 df-2 12078 df-3 12079 df-4 12080 df-5 12081 df-6 12082 df-7 12083 df-8 12084 df-9 12085 df-n0 12276 df-xnn0 12348 df-z 12362 df-dec 12480 df-uz 12625 df-rp 12773 df-fz 13282 df-fzo 13425 df-seq 13764 df-exp 13825 df-hash 14087 df-word 14259 df-lsw 14307 df-concat 14315 df-s1 14342 df-substr 14395 df-pfx 14425 df-splice 14504 df-reverse 14513 df-s2 14602 df-struct 16889 df-sets 16906 df-slot 16924 df-ndx 16936 df-base 16954 df-ress 16983 df-plusg 17016 df-mulr 17017 df-starv 17018 df-tset 17022 df-ple 17023 df-ds 17025 df-unif 17026 df-0g 17193 df-gsum 17194 df-mre 17336 df-mrc 17337 df-acs 17339 df-mgm 18367 df-sgrp 18416 df-mnd 18427 df-mhm 18471 df-submnd 18472 df-efmnd 18549 df-grp 18621 df-minusg 18622 df-subg 18793 df-ghm 18873 df-gim 18916 df-oppg 18991 df-symg 19016 df-pmtr 19091 df-psgn 19140 df-evpm 19141 df-cmn 19429 df-abl 19430 df-mgp 19762 df-ur 19779 df-ring 19826 df-cring 19827 df-oppr 19903 df-dvdsr 19924 df-unit 19925 df-invr 19955 df-dvr 19966 df-drng 20034 df-cnfld 20639 |
This theorem is referenced by: psgnodpm 20834 psgnevpm 20835 evpmodpmf1o 20842 mdet0pr 21782 odpmco 31396 evpmid 31456 cyc3evpm 31458 |
Copyright terms: Public domain | W3C validator |