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Mirrors > Home > MPE Home > Th. List > psgnco | Structured version Visualization version GIF version |
Description: Multiplicativity of the permutation sign function. (Contributed by SO, 9-Jul-2018.) |
Ref | Expression |
---|---|
psgninv.s | โข ๐ = (SymGrpโ๐ท) |
psgninv.n | โข ๐ = (pmSgnโ๐ท) |
psgninv.p | โข ๐ = (Baseโ๐) |
Ref | Expression |
---|---|
psgnco | โข ((๐ท โ Fin โง ๐น โ ๐ โง ๐บ โ ๐) โ (๐โ(๐น โ ๐บ)) = ((๐โ๐น) ยท (๐โ๐บ))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | psgninv.s | . . . . 5 โข ๐ = (SymGrpโ๐ท) | |
2 | psgninv.p | . . . . 5 โข ๐ = (Baseโ๐) | |
3 | eqid 2733 | . . . . 5 โข (+gโ๐) = (+gโ๐) | |
4 | 1, 2, 3 | symgov 19170 | . . . 4 โข ((๐น โ ๐ โง ๐บ โ ๐) โ (๐น(+gโ๐)๐บ) = (๐น โ ๐บ)) |
5 | 4 | 3adant1 1131 | . . 3 โข ((๐ท โ Fin โง ๐น โ ๐ โง ๐บ โ ๐) โ (๐น(+gโ๐)๐บ) = (๐น โ ๐บ)) |
6 | 5 | fveq2d 6847 | . 2 โข ((๐ท โ Fin โง ๐น โ ๐ โง ๐บ โ ๐) โ (๐โ(๐น(+gโ๐)๐บ)) = (๐โ(๐น โ ๐บ))) |
7 | psgninv.n | . . . 4 โข ๐ = (pmSgnโ๐ท) | |
8 | eqid 2733 | . . . 4 โข ((mulGrpโโfld) โพs {1, -1}) = ((mulGrpโโfld) โพs {1, -1}) | |
9 | 1, 7, 8 | psgnghm2 21001 | . . 3 โข (๐ท โ Fin โ ๐ โ (๐ GrpHom ((mulGrpโโfld) โพs {1, -1}))) |
10 | prex 5390 | . . . . 5 โข {1, -1} โ V | |
11 | eqid 2733 | . . . . . . 7 โข (mulGrpโโfld) = (mulGrpโโfld) | |
12 | cnfldmul 20818 | . . . . . . 7 โข ยท = (.rโโfld) | |
13 | 11, 12 | mgpplusg 19905 | . . . . . 6 โข ยท = (+gโ(mulGrpโโfld)) |
14 | 8, 13 | ressplusg 17176 | . . . . 5 โข ({1, -1} โ V โ ยท = (+gโ((mulGrpโโfld) โพs {1, -1}))) |
15 | 10, 14 | ax-mp 5 | . . . 4 โข ยท = (+gโ((mulGrpโโfld) โพs {1, -1})) |
16 | 2, 3, 15 | ghmlin 19018 | . . 3 โข ((๐ โ (๐ GrpHom ((mulGrpโโfld) โพs {1, -1})) โง ๐น โ ๐ โง ๐บ โ ๐) โ (๐โ(๐น(+gโ๐)๐บ)) = ((๐โ๐น) ยท (๐โ๐บ))) |
17 | 9, 16 | syl3an1 1164 | . 2 โข ((๐ท โ Fin โง ๐น โ ๐ โง ๐บ โ ๐) โ (๐โ(๐น(+gโ๐)๐บ)) = ((๐โ๐น) ยท (๐โ๐บ))) |
18 | 6, 17 | eqtr3d 2775 | 1 โข ((๐ท โ Fin โง ๐น โ ๐ โง ๐บ โ ๐) โ (๐โ(๐น โ ๐บ)) = ((๐โ๐น) ยท (๐โ๐บ))) |
Colors of variables: wff setvar class |
Syntax hints: โ wi 4 โง w3a 1088 = wceq 1542 โ wcel 2107 Vcvv 3444 {cpr 4589 โ ccom 5638 โcfv 6497 (class class class)co 7358 Fincfn 8886 1c1 11057 ยท cmul 11061 -cneg 11391 Basecbs 17088 โพs cress 17117 +gcplusg 17138 GrpHom cghm 19010 SymGrpcsymg 19153 pmSgncpsgn 19276 mulGrpcmgp 19901 โfldccnfld 20812 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5243 ax-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 ax-cnex 11112 ax-resscn 11113 ax-1cn 11114 ax-icn 11115 ax-addcl 11116 ax-addrcl 11117 ax-mulcl 11118 ax-mulrcl 11119 ax-mulcom 11120 ax-addass 11121 ax-mulass 11122 ax-distr 11123 ax-i2m1 11124 ax-1ne0 11125 ax-1rid 11126 ax-rnegex 11127 ax-rrecex 11128 ax-cnre 11129 ax-pre-lttri 11130 ax-pre-lttrn 11131 ax-pre-ltadd 11132 ax-pre-mulgt0 11133 ax-addf 11135 ax-mulf 11136 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-xor 1511 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3352 df-reu 3353 df-rab 3407 df-v 3446 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3930 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-tp 4592 df-op 4594 df-ot 4596 df-uni 4867 df-int 4909 df-iun 4957 df-iin 4958 df-br 5107 df-opab 5169 df-mpt 5190 df-tr 5224 df-id 5532 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5589 df-se 5590 df-we 5591 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-pred 6254 df-ord 6321 df-on 6322 df-lim 6323 df-suc 6324 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 df-isom 6506 df-riota 7314 df-ov 7361 df-oprab 7362 df-mpo 7363 df-om 7804 df-1st 7922 df-2nd 7923 df-tpos 8158 df-frecs 8213 df-wrecs 8244 df-recs 8318 df-rdg 8357 df-1o 8413 df-2o 8414 df-er 8651 df-map 8770 df-en 8887 df-dom 8888 df-sdom 8889 df-fin 8890 df-card 9880 df-pnf 11196 df-mnf 11197 df-xr 11198 df-ltxr 11199 df-le 11200 df-sub 11392 df-neg 11393 df-div 11818 df-nn 12159 df-2 12221 df-3 12222 df-4 12223 df-5 12224 df-6 12225 df-7 12226 df-8 12227 df-9 12228 df-n0 12419 df-xnn0 12491 df-z 12505 df-dec 12624 df-uz 12769 df-rp 12921 df-fz 13431 df-fzo 13574 df-seq 13913 df-exp 13974 df-hash 14237 df-word 14409 df-lsw 14457 df-concat 14465 df-s1 14490 df-substr 14535 df-pfx 14565 df-splice 14644 df-reverse 14653 df-s2 14743 df-struct 17024 df-sets 17041 df-slot 17059 df-ndx 17071 df-base 17089 df-ress 17118 df-plusg 17151 df-mulr 17152 df-starv 17153 df-tset 17157 df-ple 17158 df-ds 17160 df-unif 17161 df-0g 17328 df-gsum 17329 df-mre 17471 df-mrc 17472 df-acs 17474 df-mgm 18502 df-sgrp 18551 df-mnd 18562 df-mhm 18606 df-submnd 18607 df-efmnd 18684 df-grp 18756 df-minusg 18757 df-subg 18930 df-ghm 19011 df-gim 19054 df-oppg 19129 df-symg 19154 df-pmtr 19229 df-psgn 19278 df-cmn 19569 df-abl 19570 df-mgp 19902 df-ur 19919 df-ring 19971 df-cring 19972 df-oppr 20054 df-dvdsr 20075 df-unit 20076 df-invr 20106 df-dvr 20117 df-drng 20199 df-cnfld 20813 |
This theorem is referenced by: odpmco 31986 psgnfzto1st 32003 cyc3evpm 32048 mdetpmtr1 32461 madjusmdetlem4 32468 |
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