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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 6672 | . . . . . 6 ⊢ (𝑑 = 𝐷 → (pmSgn‘𝑑) = (pmSgn‘𝐷)) | |
2 | 1 | cnveqd 5748 | . . . . 5 ⊢ (𝑑 = 𝐷 → ◡(pmSgn‘𝑑) = ◡(pmSgn‘𝐷)) |
3 | 2 | imaeq1d 5930 | . . . 4 ⊢ (𝑑 = 𝐷 → (◡(pmSgn‘𝑑) “ {1}) = (◡(pmSgn‘𝐷) “ {1})) |
4 | df-evpm 18622 | . . . 4 ⊢ pmEven = (𝑑 ∈ V ↦ (◡(pmSgn‘𝑑) “ {1})) | |
5 | fvex 6685 | . . . . . 6 ⊢ (pmSgn‘𝐷) ∈ V | |
6 | 5 | cnvex 7632 | . . . . 5 ⊢ ◡(pmSgn‘𝐷) ∈ V |
7 | 6 | imaex 7623 | . . . 4 ⊢ (◡(pmSgn‘𝐷) “ {1}) ∈ V |
8 | 3, 4, 7 | fvmpt 6770 | . . 3 ⊢ (𝐷 ∈ V → (pmEven‘𝐷) = (◡(pmSgn‘𝐷) “ {1})) |
9 | cnvimass 5951 | . . . 4 ⊢ (◡(pmSgn‘𝐷) “ {1}) ⊆ dom (pmSgn‘𝐷) | |
10 | evpmss.s | . . . . . . 7 ⊢ 𝑆 = (SymGrp‘𝐷) | |
11 | eqid 2823 | . . . . . . 7 ⊢ (pmSgn‘𝐷) = (pmSgn‘𝐷) | |
12 | eqid 2823 | . . . . . . 7 ⊢ (𝑆 ↾s dom (pmSgn‘𝐷)) = (𝑆 ↾s dom (pmSgn‘𝐷)) | |
13 | eqid 2823 | . . . . . . 7 ⊢ ((mulGrp‘ℂfld) ↾s {1, -1}) = ((mulGrp‘ℂfld) ↾s {1, -1}) | |
14 | 10, 11, 12, 13 | psgnghm 20726 | . . . . . 6 ⊢ (𝐷 ∈ V → (pmSgn‘𝐷) ∈ ((𝑆 ↾s dom (pmSgn‘𝐷)) GrpHom ((mulGrp‘ℂfld) ↾s {1, -1}))) |
15 | eqid 2823 | . . . . . . 7 ⊢ (Base‘(𝑆 ↾s dom (pmSgn‘𝐷))) = (Base‘(𝑆 ↾s dom (pmSgn‘𝐷))) | |
16 | eqid 2823 | . . . . . . 7 ⊢ (Base‘((mulGrp‘ℂfld) ↾s {1, -1})) = (Base‘((mulGrp‘ℂfld) ↾s {1, -1})) | |
17 | 15, 16 | ghmf 18364 | . . . . . 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 6524 | . . . . . 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 16558 | . . . . 5 ⊢ (Base‘(𝑆 ↾s dom (pmSgn‘𝐷))) ⊆ 𝑃 |
22 | 19, 21 | eqsstrdi 4023 | . . . 4 ⊢ (𝐷 ∈ V → dom (pmSgn‘𝐷) ⊆ 𝑃) |
23 | 9, 22 | sstrid 3980 | . . 3 ⊢ (𝐷 ∈ V → (◡(pmSgn‘𝐷) “ {1}) ⊆ 𝑃) |
24 | 8, 23 | eqsstrd 4007 | . 2 ⊢ (𝐷 ∈ V → (pmEven‘𝐷) ⊆ 𝑃) |
25 | fvprc 6665 | . . 3 ⊢ (¬ 𝐷 ∈ V → (pmEven‘𝐷) = ∅) | |
26 | 0ss 4352 | . . 3 ⊢ ∅ ⊆ 𝑃 | |
27 | 25, 26 | eqsstrdi 4023 | . 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 1537 ∈ wcel 2114 Vcvv 3496 ⊆ wss 3938 ∅c0 4293 {csn 4569 {cpr 4571 ◡ccnv 5556 dom cdm 5557 “ cima 5560 ⟶wf 6353 ‘cfv 6357 (class class class)co 7158 1c1 10540 -cneg 10873 Basecbs 16485 ↾s cress 16486 GrpHom cghm 18357 SymGrpcsymg 18497 pmSgncpsgn 18619 pmEvencevpm 18620 mulGrpcmgp 19241 ℂfldccnfld 20547 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616 ax-addf 10618 ax-mulf 10619 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-xor 1502 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-ot 4578 df-uni 4841 df-int 4879 df-iun 4923 df-iin 4924 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-se 5517 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-isom 6366 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-om 7583 df-1st 7691 df-2nd 7692 df-tpos 7894 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-1o 8104 df-2o 8105 df-oadd 8108 df-er 8291 df-map 8410 df-en 8512 df-dom 8513 df-sdom 8514 df-fin 8515 df-card 9370 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-div 11300 df-nn 11641 df-2 11703 df-3 11704 df-4 11705 df-5 11706 df-6 11707 df-7 11708 df-8 11709 df-9 11710 df-n0 11901 df-xnn0 11971 df-z 11985 df-dec 12102 df-uz 12247 df-rp 12393 df-fz 12896 df-fzo 13037 df-seq 13373 df-exp 13433 df-hash 13694 df-word 13865 df-lsw 13917 df-concat 13925 df-s1 13952 df-substr 14005 df-pfx 14035 df-splice 14114 df-reverse 14123 df-s2 14212 df-struct 16487 df-ndx 16488 df-slot 16489 df-base 16491 df-sets 16492 df-ress 16493 df-plusg 16580 df-mulr 16581 df-starv 16582 df-tset 16586 df-ple 16587 df-ds 16589 df-unif 16590 df-0g 16717 df-gsum 16718 df-mre 16859 df-mrc 16860 df-acs 16862 df-mgm 17854 df-sgrp 17903 df-mnd 17914 df-mhm 17958 df-submnd 17959 df-efmnd 18036 df-grp 18108 df-minusg 18109 df-subg 18278 df-ghm 18358 df-gim 18401 df-oppg 18476 df-symg 18498 df-pmtr 18572 df-psgn 18621 df-evpm 18622 df-cmn 18910 df-abl 18911 df-mgp 19242 df-ur 19254 df-ring 19301 df-cring 19302 df-oppr 19375 df-dvdsr 19393 df-unit 19394 df-invr 19424 df-dvr 19435 df-drng 19506 df-cnfld 20548 |
This theorem is referenced by: zrhpsgnevpm 20737 evpmodpmf1o 20742 mdetralt 21219 cyc3genpm 30796 |
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