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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 | ⊢ 𝑆 = (SymGrp‘𝐷) |
evpmss.p | ⊢ 𝑃 = (Base‘𝑆) |
psgnevpmb.n | ⊢ 𝑁 = (pmSgn‘𝐷) |
Ref | Expression |
---|---|
psgnevpmb | ⊢ (𝐷 ∈ Fin → (𝐹 ∈ (pmEven‘𝐷) ↔ (𝐹 ∈ 𝑃 ∧ (𝑁‘𝐹) = 1))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elex 3427 | . . . 4 ⊢ (𝐷 ∈ Fin → 𝐷 ∈ V) | |
2 | fveq2 6493 | . . . . . . . 8 ⊢ (𝑑 = 𝐷 → (pmSgn‘𝑑) = (pmSgn‘𝐷)) | |
3 | psgnevpmb.n | . . . . . . . 8 ⊢ 𝑁 = (pmSgn‘𝐷) | |
4 | 2, 3 | syl6eqr 2826 | . . . . . . 7 ⊢ (𝑑 = 𝐷 → (pmSgn‘𝑑) = 𝑁) |
5 | 4 | cnveqd 5589 | . . . . . 6 ⊢ (𝑑 = 𝐷 → ◡(pmSgn‘𝑑) = ◡𝑁) |
6 | 5 | imaeq1d 5763 | . . . . 5 ⊢ (𝑑 = 𝐷 → (◡(pmSgn‘𝑑) “ {1}) = (◡𝑁 “ {1})) |
7 | df-evpm 18371 | . . . . 5 ⊢ pmEven = (𝑑 ∈ V ↦ (◡(pmSgn‘𝑑) “ {1})) | |
8 | 3 | fvexi 6507 | . . . . . . 7 ⊢ 𝑁 ∈ V |
9 | 8 | cnvex 7439 | . . . . . 6 ⊢ ◡𝑁 ∈ V |
10 | 9 | imaex 7430 | . . . . 5 ⊢ (◡𝑁 “ {1}) ∈ V |
11 | 6, 7, 10 | fvmpt 6589 | . . . 4 ⊢ (𝐷 ∈ V → (pmEven‘𝐷) = (◡𝑁 “ {1})) |
12 | 1, 11 | syl 17 | . . 3 ⊢ (𝐷 ∈ Fin → (pmEven‘𝐷) = (◡𝑁 “ {1})) |
13 | 12 | eleq2d 2845 | . 2 ⊢ (𝐷 ∈ Fin → (𝐹 ∈ (pmEven‘𝐷) ↔ 𝐹 ∈ (◡𝑁 “ {1}))) |
14 | evpmss.s | . . . . 5 ⊢ 𝑆 = (SymGrp‘𝐷) | |
15 | eqid 2772 | . . . . 5 ⊢ ((mulGrp‘ℂfld) ↾s {1, -1}) = ((mulGrp‘ℂfld) ↾s {1, -1}) | |
16 | 14, 3, 15 | psgnghm2 20417 | . . . 4 ⊢ (𝐷 ∈ Fin → 𝑁 ∈ (𝑆 GrpHom ((mulGrp‘ℂfld) ↾s {1, -1}))) |
17 | evpmss.p | . . . . 5 ⊢ 𝑃 = (Base‘𝑆) | |
18 | eqid 2772 | . . . . 5 ⊢ (Base‘((mulGrp‘ℂfld) ↾s {1, -1})) = (Base‘((mulGrp‘ℂfld) ↾s {1, -1})) | |
19 | 17, 18 | ghmf 18123 | . . . 4 ⊢ (𝑁 ∈ (𝑆 GrpHom ((mulGrp‘ℂfld) ↾s {1, -1})) → 𝑁:𝑃⟶(Base‘((mulGrp‘ℂfld) ↾s {1, -1}))) |
20 | 16, 19 | syl 17 | . . 3 ⊢ (𝐷 ∈ Fin → 𝑁:𝑃⟶(Base‘((mulGrp‘ℂfld) ↾s {1, -1}))) |
21 | ffn 6338 | . . 3 ⊢ (𝑁:𝑃⟶(Base‘((mulGrp‘ℂfld) ↾s {1, -1})) → 𝑁 Fn 𝑃) | |
22 | fniniseg 6649 | . . 3 ⊢ (𝑁 Fn 𝑃 → (𝐹 ∈ (◡𝑁 “ {1}) ↔ (𝐹 ∈ 𝑃 ∧ (𝑁‘𝐹) = 1))) | |
23 | 20, 21, 22 | 3syl 18 | . 2 ⊢ (𝐷 ∈ Fin → (𝐹 ∈ (◡𝑁 “ {1}) ↔ (𝐹 ∈ 𝑃 ∧ (𝑁‘𝐹) = 1))) |
24 | 13, 23 | bitrd 271 | 1 ⊢ (𝐷 ∈ Fin → (𝐹 ∈ (pmEven‘𝐷) ↔ (𝐹 ∈ 𝑃 ∧ (𝑁‘𝐹) = 1))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 ∧ wa 387 = wceq 1507 ∈ wcel 2048 Vcvv 3409 {csn 4435 {cpr 4437 ◡ccnv 5399 “ cima 5403 Fn wfn 6177 ⟶wf 6178 ‘cfv 6182 (class class class)co 6970 Fincfn 8298 1c1 10328 -cneg 10663 Basecbs 16329 ↾s cress 16330 GrpHom cghm 18116 SymGrpcsymg 18256 pmSgncpsgn 18368 pmEvencevpm 18369 mulGrpcmgp 18952 ℂfldccnfld 20237 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1964 ax-8 2050 ax-9 2057 ax-10 2077 ax-11 2091 ax-12 2104 ax-13 2299 ax-ext 2745 ax-rep 5043 ax-sep 5054 ax-nul 5061 ax-pow 5113 ax-pr 5180 ax-un 7273 ax-cnex 10383 ax-resscn 10384 ax-1cn 10385 ax-icn 10386 ax-addcl 10387 ax-addrcl 10388 ax-mulcl 10389 ax-mulrcl 10390 ax-mulcom 10391 ax-addass 10392 ax-mulass 10393 ax-distr 10394 ax-i2m1 10395 ax-1ne0 10396 ax-1rid 10397 ax-rnegex 10398 ax-rrecex 10399 ax-cnre 10400 ax-pre-lttri 10401 ax-pre-lttrn 10402 ax-pre-ltadd 10403 ax-pre-mulgt0 10404 ax-addf 10406 ax-mulf 10407 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3or 1069 df-3an 1070 df-xor 1489 df-tru 1510 df-ex 1743 df-nf 1747 df-sb 2014 df-mo 2544 df-eu 2580 df-clab 2754 df-cleq 2765 df-clel 2840 df-nfc 2912 df-ne 2962 df-nel 3068 df-ral 3087 df-rex 3088 df-reu 3089 df-rmo 3090 df-rab 3091 df-v 3411 df-sbc 3678 df-csb 3783 df-dif 3828 df-un 3830 df-in 3832 df-ss 3839 df-pss 3841 df-nul 4174 df-if 4345 df-pw 4418 df-sn 4436 df-pr 4438 df-tp 4440 df-op 4442 df-ot 4444 df-uni 4707 df-int 4744 df-iun 4788 df-iin 4789 df-br 4924 df-opab 4986 df-mpt 5003 df-tr 5025 df-id 5305 df-eprel 5310 df-po 5319 df-so 5320 df-fr 5359 df-se 5360 df-we 5361 df-xp 5406 df-rel 5407 df-cnv 5408 df-co 5409 df-dm 5410 df-rn 5411 df-res 5412 df-ima 5413 df-pred 5980 df-ord 6026 df-on 6027 df-lim 6028 df-suc 6029 df-iota 6146 df-fun 6184 df-fn 6185 df-f 6186 df-f1 6187 df-fo 6188 df-f1o 6189 df-fv 6190 df-isom 6191 df-riota 6931 df-ov 6973 df-oprab 6974 df-mpo 6975 df-om 7391 df-1st 7494 df-2nd 7495 df-tpos 7688 df-wrecs 7743 df-recs 7805 df-rdg 7843 df-1o 7897 df-2o 7898 df-oadd 7901 df-er 8081 df-map 8200 df-en 8299 df-dom 8300 df-sdom 8301 df-fin 8302 df-card 9154 df-pnf 10468 df-mnf 10469 df-xr 10470 df-ltxr 10471 df-le 10472 df-sub 10664 df-neg 10665 df-div 11091 df-nn 11432 df-2 11496 df-3 11497 df-4 11498 df-5 11499 df-6 11500 df-7 11501 df-8 11502 df-9 11503 df-n0 11701 df-xnn0 11773 df-z 11787 df-dec 11905 df-uz 12052 df-rp 12198 df-fz 12702 df-fzo 12843 df-seq 13178 df-exp 13238 df-hash 13499 df-word 13663 df-lsw 13716 df-concat 13724 df-s1 13749 df-substr 13794 df-pfx 13843 df-splice 13950 df-reverse 13968 df-s2 14062 df-struct 16331 df-ndx 16332 df-slot 16333 df-base 16335 df-sets 16336 df-ress 16337 df-plusg 16424 df-mulr 16425 df-starv 16426 df-tset 16430 df-ple 16431 df-ds 16433 df-unif 16434 df-0g 16561 df-gsum 16562 df-mre 16705 df-mrc 16706 df-acs 16708 df-mgm 17700 df-sgrp 17742 df-mnd 17753 df-mhm 17793 df-submnd 17794 df-grp 17884 df-minusg 17885 df-subg 18050 df-ghm 18117 df-gim 18160 df-oppg 18235 df-symg 18257 df-pmtr 18321 df-psgn 18370 df-evpm 18371 df-cmn 18658 df-abl 18659 df-mgp 18953 df-ur 18965 df-ring 19012 df-cring 19013 df-oppr 19086 df-dvdsr 19104 df-unit 19105 df-invr 19135 df-dvr 19146 df-drng 19217 df-cnfld 20238 |
This theorem is referenced by: psgnodpm 20424 psgnevpm 20425 evpmodpmf1o 20432 mdet0pr 20895 |
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