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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 6669 | . . . . . 6 ⊢ (𝑑 = 𝐷 → (pmSgn‘𝑑) = (pmSgn‘𝐷)) | |
2 | 1 | cnveqd 5745 | . . . . 5 ⊢ (𝑑 = 𝐷 → ◡(pmSgn‘𝑑) = ◡(pmSgn‘𝐷)) |
3 | 2 | imaeq1d 5927 | . . . 4 ⊢ (𝑑 = 𝐷 → (◡(pmSgn‘𝑑) “ {1}) = (◡(pmSgn‘𝐷) “ {1})) |
4 | df-evpm 18619 | . . . 4 ⊢ pmEven = (𝑑 ∈ V ↦ (◡(pmSgn‘𝑑) “ {1})) | |
5 | fvex 6682 | . . . . . 6 ⊢ (pmSgn‘𝐷) ∈ V | |
6 | 5 | cnvex 7629 | . . . . 5 ⊢ ◡(pmSgn‘𝐷) ∈ V |
7 | 6 | imaex 7620 | . . . 4 ⊢ (◡(pmSgn‘𝐷) “ {1}) ∈ V |
8 | 3, 4, 7 | fvmpt 6767 | . . 3 ⊢ (𝐷 ∈ V → (pmEven‘𝐷) = (◡(pmSgn‘𝐷) “ {1})) |
9 | cnvimass 5948 | . . . 4 ⊢ (◡(pmSgn‘𝐷) “ {1}) ⊆ dom (pmSgn‘𝐷) | |
10 | evpmss.s | . . . . . . 7 ⊢ 𝑆 = (SymGrp‘𝐷) | |
11 | eqid 2821 | . . . . . . 7 ⊢ (pmSgn‘𝐷) = (pmSgn‘𝐷) | |
12 | eqid 2821 | . . . . . . 7 ⊢ (𝑆 ↾s dom (pmSgn‘𝐷)) = (𝑆 ↾s dom (pmSgn‘𝐷)) | |
13 | eqid 2821 | . . . . . . 7 ⊢ ((mulGrp‘ℂfld) ↾s {1, -1}) = ((mulGrp‘ℂfld) ↾s {1, -1}) | |
14 | 10, 11, 12, 13 | psgnghm 20723 | . . . . . 6 ⊢ (𝐷 ∈ V → (pmSgn‘𝐷) ∈ ((𝑆 ↾s dom (pmSgn‘𝐷)) GrpHom ((mulGrp‘ℂfld) ↾s {1, -1}))) |
15 | eqid 2821 | . . . . . . 7 ⊢ (Base‘(𝑆 ↾s dom (pmSgn‘𝐷))) = (Base‘(𝑆 ↾s dom (pmSgn‘𝐷))) | |
16 | eqid 2821 | . . . . . . 7 ⊢ (Base‘((mulGrp‘ℂfld) ↾s {1, -1})) = (Base‘((mulGrp‘ℂfld) ↾s {1, -1})) | |
17 | 15, 16 | ghmf 18361 | . . . . . 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 6521 | . . . . . 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 16555 | . . . . 5 ⊢ (Base‘(𝑆 ↾s dom (pmSgn‘𝐷))) ⊆ 𝑃 |
22 | 19, 21 | eqsstrdi 4020 | . . . 4 ⊢ (𝐷 ∈ V → dom (pmSgn‘𝐷) ⊆ 𝑃) |
23 | 9, 22 | sstrid 3977 | . . 3 ⊢ (𝐷 ∈ V → (◡(pmSgn‘𝐷) “ {1}) ⊆ 𝑃) |
24 | 8, 23 | eqsstrd 4004 | . 2 ⊢ (𝐷 ∈ V → (pmEven‘𝐷) ⊆ 𝑃) |
25 | fvprc 6662 | . . 3 ⊢ (¬ 𝐷 ∈ V → (pmEven‘𝐷) = ∅) | |
26 | 0ss 4349 | . . 3 ⊢ ∅ ⊆ 𝑃 | |
27 | 25, 26 | eqsstrdi 4020 | . 2 ⊢ (¬ 𝐷 ∈ V → (pmEven‘𝐷) ⊆ 𝑃) |
28 | 24, 27 | pm2.61i 184 | 1 ⊢ (pmEven‘𝐷) ⊆ 𝑃 |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 = wceq 1533 ∈ wcel 2110 Vcvv 3494 ⊆ wss 3935 ∅c0 4290 {csn 4566 {cpr 4568 ◡ccnv 5553 dom cdm 5554 “ cima 5557 ⟶wf 6350 ‘cfv 6354 (class class class)co 7155 1c1 10537 -cneg 10870 Basecbs 16482 ↾s cress 16483 GrpHom cghm 18354 SymGrpcsymg 18494 pmSgncpsgn 18616 pmEvencevpm 18617 mulGrpcmgp 19238 ℂfldccnfld 20544 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-rep 5189 ax-sep 5202 ax-nul 5209 ax-pow 5265 ax-pr 5329 ax-un 7460 ax-cnex 10592 ax-resscn 10593 ax-1cn 10594 ax-icn 10595 ax-addcl 10596 ax-addrcl 10597 ax-mulcl 10598 ax-mulrcl 10599 ax-mulcom 10600 ax-addass 10601 ax-mulass 10602 ax-distr 10603 ax-i2m1 10604 ax-1ne0 10605 ax-1rid 10606 ax-rnegex 10607 ax-rrecex 10608 ax-cnre 10609 ax-pre-lttri 10610 ax-pre-lttrn 10611 ax-pre-ltadd 10612 ax-pre-mulgt0 10613 ax-addf 10615 ax-mulf 10616 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-xor 1501 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4567 df-pr 4569 df-tp 4571 df-op 4573 df-ot 4575 df-uni 4838 df-int 4876 df-iun 4920 df-iin 4921 df-br 5066 df-opab 5128 df-mpt 5146 df-tr 5172 df-id 5459 df-eprel 5464 df-po 5473 df-so 5474 df-fr 5513 df-se 5514 df-we 5515 df-xp 5560 df-rel 5561 df-cnv 5562 df-co 5563 df-dm 5564 df-rn 5565 df-res 5566 df-ima 5567 df-pred 6147 df-ord 6193 df-on 6194 df-lim 6195 df-suc 6196 df-iota 6313 df-fun 6356 df-fn 6357 df-f 6358 df-f1 6359 df-fo 6360 df-f1o 6361 df-fv 6362 df-isom 6363 df-riota 7113 df-ov 7158 df-oprab 7159 df-mpo 7160 df-om 7580 df-1st 7688 df-2nd 7689 df-tpos 7891 df-wrecs 7946 df-recs 8007 df-rdg 8045 df-1o 8101 df-2o 8102 df-oadd 8105 df-er 8288 df-map 8407 df-en 8509 df-dom 8510 df-sdom 8511 df-fin 8512 df-card 9367 df-pnf 10676 df-mnf 10677 df-xr 10678 df-ltxr 10679 df-le 10680 df-sub 10871 df-neg 10872 df-div 11297 df-nn 11638 df-2 11699 df-3 11700 df-4 11701 df-5 11702 df-6 11703 df-7 11704 df-8 11705 df-9 11706 df-n0 11897 df-xnn0 11967 df-z 11981 df-dec 12098 df-uz 12243 df-rp 12389 df-fz 12892 df-fzo 13033 df-seq 13369 df-exp 13429 df-hash 13690 df-word 13861 df-lsw 13914 df-concat 13922 df-s1 13949 df-substr 14002 df-pfx 14032 df-splice 14111 df-reverse 14120 df-s2 14209 df-struct 16484 df-ndx 16485 df-slot 16486 df-base 16488 df-sets 16489 df-ress 16490 df-plusg 16577 df-mulr 16578 df-starv 16579 df-tset 16583 df-ple 16584 df-ds 16586 df-unif 16587 df-0g 16714 df-gsum 16715 df-mre 16856 df-mrc 16857 df-acs 16859 df-mgm 17851 df-sgrp 17900 df-mnd 17911 df-mhm 17955 df-submnd 17956 df-efmnd 18033 df-grp 18105 df-minusg 18106 df-subg 18275 df-ghm 18355 df-gim 18398 df-oppg 18473 df-symg 18495 df-pmtr 18569 df-psgn 18618 df-evpm 18619 df-cmn 18907 df-abl 18908 df-mgp 19239 df-ur 19251 df-ring 19298 df-cring 19299 df-oppr 19372 df-dvdsr 19390 df-unit 19391 df-invr 19421 df-dvr 19432 df-drng 19503 df-cnfld 20545 |
This theorem is referenced by: zrhpsgnevpm 20734 evpmodpmf1o 20739 mdetralt 21216 cyc3genpm 30794 |
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