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Mirrors > Home > MPE Home > Th. List > evpmss | Structured version Visualization version GIF version |
Description: Even permutations are permutations. (Contributed by SO, 9-Jul-2018.) |
Ref | Expression |
---|---|
evpmss.s | ⊢ 𝑆 = (SymGrp‘𝐷) |
evpmss.p | ⊢ 𝑃 = (Base‘𝑆) |
Ref | Expression |
---|---|
evpmss | ⊢ (pmEven‘𝐷) ⊆ 𝑃 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fveq2 6891 | . . . . . 6 ⊢ (𝑑 = 𝐷 → (pmSgn‘𝑑) = (pmSgn‘𝐷)) | |
2 | 1 | cnveqd 5872 | . . . . 5 ⊢ (𝑑 = 𝐷 → ◡(pmSgn‘𝑑) = ◡(pmSgn‘𝐷)) |
3 | 2 | imaeq1d 6056 | . . . 4 ⊢ (𝑑 = 𝐷 → (◡(pmSgn‘𝑑) “ {1}) = (◡(pmSgn‘𝐷) “ {1})) |
4 | df-evpm 19440 | . . . 4 ⊢ pmEven = (𝑑 ∈ V ↦ (◡(pmSgn‘𝑑) “ {1})) | |
5 | fvex 6904 | . . . . . 6 ⊢ (pmSgn‘𝐷) ∈ V | |
6 | 5 | cnvex 7927 | . . . . 5 ⊢ ◡(pmSgn‘𝐷) ∈ V |
7 | 6 | imaex 7916 | . . . 4 ⊢ (◡(pmSgn‘𝐷) “ {1}) ∈ V |
8 | 3, 4, 7 | fvmpt 6999 | . . 3 ⊢ (𝐷 ∈ V → (pmEven‘𝐷) = (◡(pmSgn‘𝐷) “ {1})) |
9 | cnvimass 6079 | . . . 4 ⊢ (◡(pmSgn‘𝐷) “ {1}) ⊆ dom (pmSgn‘𝐷) | |
10 | evpmss.s | . . . . . . 7 ⊢ 𝑆 = (SymGrp‘𝐷) | |
11 | eqid 2727 | . . . . . . 7 ⊢ (pmSgn‘𝐷) = (pmSgn‘𝐷) | |
12 | eqid 2727 | . . . . . . 7 ⊢ (𝑆 ↾s dom (pmSgn‘𝐷)) = (𝑆 ↾s dom (pmSgn‘𝐷)) | |
13 | eqid 2727 | . . . . . . 7 ⊢ ((mulGrp‘ℂfld) ↾s {1, -1}) = ((mulGrp‘ℂfld) ↾s {1, -1}) | |
14 | 10, 11, 12, 13 | psgnghm 21505 | . . . . . 6 ⊢ (𝐷 ∈ V → (pmSgn‘𝐷) ∈ ((𝑆 ↾s dom (pmSgn‘𝐷)) GrpHom ((mulGrp‘ℂfld) ↾s {1, -1}))) |
15 | eqid 2727 | . . . . . . 7 ⊢ (Base‘(𝑆 ↾s dom (pmSgn‘𝐷))) = (Base‘(𝑆 ↾s dom (pmSgn‘𝐷))) | |
16 | eqid 2727 | . . . . . . 7 ⊢ (Base‘((mulGrp‘ℂfld) ↾s {1, -1})) = (Base‘((mulGrp‘ℂfld) ↾s {1, -1})) | |
17 | 15, 16 | ghmf 19167 | . . . . . 6 ⊢ ((pmSgn‘𝐷) ∈ ((𝑆 ↾s dom (pmSgn‘𝐷)) GrpHom ((mulGrp‘ℂfld) ↾s {1, -1})) → (pmSgn‘𝐷):(Base‘(𝑆 ↾s dom (pmSgn‘𝐷)))⟶(Base‘((mulGrp‘ℂfld) ↾s {1, -1}))) |
18 | fdm 6725 | . . . . . 6 ⊢ ((pmSgn‘𝐷):(Base‘(𝑆 ↾s dom (pmSgn‘𝐷)))⟶(Base‘((mulGrp‘ℂfld) ↾s {1, -1})) → dom (pmSgn‘𝐷) = (Base‘(𝑆 ↾s dom (pmSgn‘𝐷)))) | |
19 | 14, 17, 18 | 3syl 18 | . . . . 5 ⊢ (𝐷 ∈ V → dom (pmSgn‘𝐷) = (Base‘(𝑆 ↾s dom (pmSgn‘𝐷)))) |
20 | evpmss.p | . . . . . 6 ⊢ 𝑃 = (Base‘𝑆) | |
21 | 12, 20 | ressbasss 17212 | . . . . 5 ⊢ (Base‘(𝑆 ↾s dom (pmSgn‘𝐷))) ⊆ 𝑃 |
22 | 19, 21 | eqsstrdi 4032 | . . . 4 ⊢ (𝐷 ∈ V → dom (pmSgn‘𝐷) ⊆ 𝑃) |
23 | 9, 22 | sstrid 3989 | . . 3 ⊢ (𝐷 ∈ V → (◡(pmSgn‘𝐷) “ {1}) ⊆ 𝑃) |
24 | 8, 23 | eqsstrd 4016 | . 2 ⊢ (𝐷 ∈ V → (pmEven‘𝐷) ⊆ 𝑃) |
25 | fvprc 6883 | . . 3 ⊢ (¬ 𝐷 ∈ V → (pmEven‘𝐷) = ∅) | |
26 | 0ss 4392 | . . 3 ⊢ ∅ ⊆ 𝑃 | |
27 | 25, 26 | eqsstrdi 4032 | . 2 ⊢ (¬ 𝐷 ∈ V → (pmEven‘𝐷) ⊆ 𝑃) |
28 | 24, 27 | pm2.61i 182 | 1 ⊢ (pmEven‘𝐷) ⊆ 𝑃 |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 = wceq 1534 ∈ wcel 2099 Vcvv 3469 ⊆ wss 3944 ∅c0 4318 {csn 4624 {cpr 4626 ◡ccnv 5671 dom cdm 5672 “ cima 5675 ⟶wf 6538 ‘cfv 6542 (class class class)co 7414 1c1 11133 -cneg 11469 Basecbs 17173 ↾s cress 17202 GrpHom cghm 19160 SymGrpcsymg 19314 pmSgncpsgn 19437 pmEvencevpm 19438 mulGrpcmgp 20067 ℂfldccnfld 21272 |
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 2164 ax-ext 2698 ax-rep 5279 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7734 ax-cnex 11188 ax-resscn 11189 ax-1cn 11190 ax-icn 11191 ax-addcl 11192 ax-addrcl 11193 ax-mulcl 11194 ax-mulrcl 11195 ax-mulcom 11196 ax-addass 11197 ax-mulass 11198 ax-distr 11199 ax-i2m1 11200 ax-1ne0 11201 ax-1rid 11202 ax-rnegex 11203 ax-rrecex 11204 ax-cnre 11205 ax-pre-lttri 11206 ax-pre-lttrn 11207 ax-pre-ltadd 11208 ax-pre-mulgt0 11209 ax-addf 11211 ax-mulf 11212 |
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 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2936 df-nel 3042 df-ral 3057 df-rex 3066 df-rmo 3371 df-reu 3372 df-rab 3428 df-v 3471 df-sbc 3775 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3963 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-tp 4629 df-op 4631 df-ot 4633 df-uni 4904 df-int 4945 df-iun 4993 df-iin 4994 df-br 5143 df-opab 5205 df-mpt 5226 df-tr 5260 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-se 5628 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-isom 6551 df-riota 7370 df-ov 7417 df-oprab 7418 df-mpo 7419 df-om 7865 df-1st 7987 df-2nd 7988 df-tpos 8225 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-rdg 8424 df-1o 8480 df-2o 8481 df-er 8718 df-map 8840 df-en 8958 df-dom 8959 df-sdom 8960 df-fin 8961 df-card 9956 df-pnf 11274 df-mnf 11275 df-xr 11276 df-ltxr 11277 df-le 11278 df-sub 11470 df-neg 11471 df-div 11896 df-nn 12237 df-2 12299 df-3 12300 df-4 12301 df-5 12302 df-6 12303 df-7 12304 df-8 12305 df-9 12306 df-n0 12497 df-xnn0 12569 df-z 12583 df-dec 12702 df-uz 12847 df-rp 13001 df-fz 13511 df-fzo 13654 df-seq 13993 df-exp 14053 df-hash 14316 df-word 14491 df-lsw 14539 df-concat 14547 df-s1 14572 df-substr 14617 df-pfx 14647 df-splice 14726 df-reverse 14735 df-s2 14825 df-struct 17109 df-sets 17126 df-slot 17144 df-ndx 17156 df-base 17174 df-ress 17203 df-plusg 17239 df-mulr 17240 df-starv 17241 df-tset 17245 df-ple 17246 df-ds 17248 df-unif 17249 df-0g 17416 df-gsum 17417 df-mre 17559 df-mrc 17560 df-acs 17562 df-mgm 18593 df-sgrp 18672 df-mnd 18688 df-mhm 18733 df-submnd 18734 df-efmnd 18814 df-grp 18886 df-minusg 18887 df-subg 19071 df-ghm 19161 df-gim 19206 df-oppg 19290 df-symg 19315 df-pmtr 19390 df-psgn 19439 df-evpm 19440 df-cmn 19730 df-abl 19731 df-mgp 20068 df-rng 20086 df-ur 20115 df-ring 20168 df-cring 20169 df-oppr 20266 df-dvdsr 20289 df-unit 20290 df-invr 20320 df-dvr 20333 df-drng 20619 df-cnfld 21273 |
This theorem is referenced by: zrhpsgnevpm 21516 evpmodpmf1o 21521 mdetralt 22503 cyc3genpm 32867 |
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