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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 2728 | . . . . 5 โข (+gโ๐) = (+gโ๐) | |
4 | 1, 2, 3 | symgov 19338 | . . . 4 โข ((๐น โ ๐ โง ๐บ โ ๐) โ (๐น(+gโ๐)๐บ) = (๐น โ ๐บ)) |
5 | 4 | 3adant1 1128 | . . 3 โข ((๐ท โ Fin โง ๐น โ ๐ โง ๐บ โ ๐) โ (๐น(+gโ๐)๐บ) = (๐น โ ๐บ)) |
6 | 5 | fveq2d 6901 | . 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 2728 | . . . 4 โข ((mulGrpโโfld) โพs {1, -1}) = ((mulGrpโโfld) โพs {1, -1}) | |
9 | 1, 7, 8 | psgnghm2 21513 | . . 3 โข (๐ท โ Fin โ ๐ โ (๐ GrpHom ((mulGrpโโfld) โพs {1, -1}))) |
10 | prex 5434 | . . . . 5 โข {1, -1} โ V | |
11 | eqid 2728 | . . . . . . 7 โข (mulGrpโโfld) = (mulGrpโโfld) | |
12 | cnfldmul 21287 | . . . . . . 7 โข ยท = (.rโโfld) | |
13 | 11, 12 | mgpplusg 20078 | . . . . . 6 โข ยท = (+gโ(mulGrpโโfld)) |
14 | 8, 13 | ressplusg 17271 | . . . . 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 19175 | . . 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 1161 | . 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 2770 | 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 1085 = wceq 1534 โ wcel 2099 Vcvv 3471 {cpr 4631 โ ccom 5682 โcfv 6548 (class class class)co 7420 Fincfn 8964 1c1 11140 ยท cmul 11144 -cneg 11476 Basecbs 17180 โพs cress 17209 +gcplusg 17233 GrpHom cghm 19167 SymGrpcsymg 19321 pmSgncpsgn 19444 mulGrpcmgp 20074 โfldccnfld 21279 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 ax-un 7740 ax-cnex 11195 ax-resscn 11196 ax-1cn 11197 ax-icn 11198 ax-addcl 11199 ax-addrcl 11200 ax-mulcl 11201 ax-mulrcl 11202 ax-mulcom 11203 ax-addass 11204 ax-mulass 11205 ax-distr 11206 ax-i2m1 11207 ax-1ne0 11208 ax-1rid 11209 ax-rnegex 11210 ax-rrecex 11211 ax-cnre 11212 ax-pre-lttri 11213 ax-pre-lttrn 11214 ax-pre-ltadd 11215 ax-pre-mulgt0 11216 ax-addf 11218 ax-mulf 11219 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-xor 1506 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3373 df-reu 3374 df-rab 3430 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-tp 4634 df-op 4636 df-ot 4638 df-uni 4909 df-int 4950 df-iun 4998 df-iin 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-se 5634 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6305 df-ord 6372 df-on 6373 df-lim 6374 df-suc 6375 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-fv 6556 df-isom 6557 df-riota 7376 df-ov 7423 df-oprab 7424 df-mpo 7425 df-om 7871 df-1st 7993 df-2nd 7994 df-tpos 8232 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-1o 8487 df-2o 8488 df-er 8725 df-map 8847 df-en 8965 df-dom 8966 df-sdom 8967 df-fin 8968 df-card 9963 df-pnf 11281 df-mnf 11282 df-xr 11283 df-ltxr 11284 df-le 11285 df-sub 11477 df-neg 11478 df-div 11903 df-nn 12244 df-2 12306 df-3 12307 df-4 12308 df-5 12309 df-6 12310 df-7 12311 df-8 12312 df-9 12313 df-n0 12504 df-xnn0 12576 df-z 12590 df-dec 12709 df-uz 12854 df-rp 13008 df-fz 13518 df-fzo 13661 df-seq 14000 df-exp 14060 df-hash 14323 df-word 14498 df-lsw 14546 df-concat 14554 df-s1 14579 df-substr 14624 df-pfx 14654 df-splice 14733 df-reverse 14742 df-s2 14832 df-struct 17116 df-sets 17133 df-slot 17151 df-ndx 17163 df-base 17181 df-ress 17210 df-plusg 17246 df-mulr 17247 df-starv 17248 df-tset 17252 df-ple 17253 df-ds 17255 df-unif 17256 df-0g 17423 df-gsum 17424 df-mre 17566 df-mrc 17567 df-acs 17569 df-mgm 18600 df-sgrp 18679 df-mnd 18695 df-mhm 18740 df-submnd 18741 df-efmnd 18821 df-grp 18893 df-minusg 18894 df-subg 19078 df-ghm 19168 df-gim 19213 df-oppg 19297 df-symg 19322 df-pmtr 19397 df-psgn 19446 df-cmn 19737 df-abl 19738 df-mgp 20075 df-rng 20093 df-ur 20122 df-ring 20175 df-cring 20176 df-oppr 20273 df-dvdsr 20296 df-unit 20297 df-invr 20327 df-dvr 20340 df-drng 20626 df-cnfld 21280 |
This theorem is referenced by: odpmco 32822 psgnfzto1st 32839 cyc3evpm 32884 mdetpmtr1 33424 madjusmdetlem4 33431 |
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