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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 6846 | . . . . . 6 ⊢ (𝑑 = 𝐷 → (pmSgn‘𝑑) = (pmSgn‘𝐷)) | |
2 | 1 | cnveqd 5835 | . . . . 5 ⊢ (𝑑 = 𝐷 → ◡(pmSgn‘𝑑) = ◡(pmSgn‘𝐷)) |
3 | 2 | imaeq1d 6016 | . . . 4 ⊢ (𝑑 = 𝐷 → (◡(pmSgn‘𝑑) “ {1}) = (◡(pmSgn‘𝐷) “ {1})) |
4 | df-evpm 19282 | . . . 4 ⊢ pmEven = (𝑑 ∈ V ↦ (◡(pmSgn‘𝑑) “ {1})) | |
5 | fvex 6859 | . . . . . 6 ⊢ (pmSgn‘𝐷) ∈ V | |
6 | 5 | cnvex 7866 | . . . . 5 ⊢ ◡(pmSgn‘𝐷) ∈ V |
7 | 6 | imaex 7857 | . . . 4 ⊢ (◡(pmSgn‘𝐷) “ {1}) ∈ V |
8 | 3, 4, 7 | fvmpt 6952 | . . 3 ⊢ (𝐷 ∈ V → (pmEven‘𝐷) = (◡(pmSgn‘𝐷) “ {1})) |
9 | cnvimass 6037 | . . . 4 ⊢ (◡(pmSgn‘𝐷) “ {1}) ⊆ dom (pmSgn‘𝐷) | |
10 | evpmss.s | . . . . . . 7 ⊢ 𝑆 = (SymGrp‘𝐷) | |
11 | eqid 2733 | . . . . . . 7 ⊢ (pmSgn‘𝐷) = (pmSgn‘𝐷) | |
12 | eqid 2733 | . . . . . . 7 ⊢ (𝑆 ↾s dom (pmSgn‘𝐷)) = (𝑆 ↾s dom (pmSgn‘𝐷)) | |
13 | eqid 2733 | . . . . . . 7 ⊢ ((mulGrp‘ℂfld) ↾s {1, -1}) = ((mulGrp‘ℂfld) ↾s {1, -1}) | |
14 | 10, 11, 12, 13 | psgnghm 21007 | . . . . . 6 ⊢ (𝐷 ∈ V → (pmSgn‘𝐷) ∈ ((𝑆 ↾s dom (pmSgn‘𝐷)) GrpHom ((mulGrp‘ℂfld) ↾s {1, -1}))) |
15 | eqid 2733 | . . . . . . 7 ⊢ (Base‘(𝑆 ↾s dom (pmSgn‘𝐷))) = (Base‘(𝑆 ↾s dom (pmSgn‘𝐷))) | |
16 | eqid 2733 | . . . . . . 7 ⊢ (Base‘((mulGrp‘ℂfld) ↾s {1, -1})) = (Base‘((mulGrp‘ℂfld) ↾s {1, -1})) | |
17 | 15, 16 | ghmf 19020 | . . . . . 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 6681 | . . . . . 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 17129 | . . . . 5 ⊢ (Base‘(𝑆 ↾s dom (pmSgn‘𝐷))) ⊆ 𝑃 |
22 | 19, 21 | eqsstrdi 4002 | . . . 4 ⊢ (𝐷 ∈ V → dom (pmSgn‘𝐷) ⊆ 𝑃) |
23 | 9, 22 | sstrid 3959 | . . 3 ⊢ (𝐷 ∈ V → (◡(pmSgn‘𝐷) “ {1}) ⊆ 𝑃) |
24 | 8, 23 | eqsstrd 3986 | . 2 ⊢ (𝐷 ∈ V → (pmEven‘𝐷) ⊆ 𝑃) |
25 | fvprc 6838 | . . 3 ⊢ (¬ 𝐷 ∈ V → (pmEven‘𝐷) = ∅) | |
26 | 0ss 4360 | . . 3 ⊢ ∅ ⊆ 𝑃 | |
27 | 25, 26 | eqsstrdi 4002 | . 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 1542 ∈ wcel 2107 Vcvv 3447 ⊆ wss 3914 ∅c0 4286 {csn 4590 {cpr 4592 ◡ccnv 5636 dom cdm 5637 “ cima 5640 ⟶wf 6496 ‘cfv 6500 (class class class)co 7361 1c1 11060 -cneg 11394 Basecbs 17091 ↾s cress 17120 GrpHom cghm 19013 SymGrpcsymg 19156 pmSgncpsgn 19279 pmEvencevpm 19280 mulGrpcmgp 19904 ℂfldccnfld 20819 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5246 ax-sep 5260 ax-nul 5267 ax-pow 5324 ax-pr 5388 ax-un 7676 ax-cnex 11115 ax-resscn 11116 ax-1cn 11117 ax-icn 11118 ax-addcl 11119 ax-addrcl 11120 ax-mulcl 11121 ax-mulrcl 11122 ax-mulcom 11123 ax-addass 11124 ax-mulass 11125 ax-distr 11126 ax-i2m1 11127 ax-1ne0 11128 ax-1rid 11129 ax-rnegex 11130 ax-rrecex 11131 ax-cnre 11132 ax-pre-lttri 11133 ax-pre-lttrn 11134 ax-pre-ltadd 11135 ax-pre-mulgt0 11136 ax-addf 11138 ax-mulf 11139 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-xor 1511 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3352 df-reu 3353 df-rab 3407 df-v 3449 df-sbc 3744 df-csb 3860 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-pss 3933 df-nul 4287 df-if 4491 df-pw 4566 df-sn 4591 df-pr 4593 df-tp 4595 df-op 4597 df-ot 4599 df-uni 4870 df-int 4912 df-iun 4960 df-iin 4961 df-br 5110 df-opab 5172 df-mpt 5193 df-tr 5227 df-id 5535 df-eprel 5541 df-po 5549 df-so 5550 df-fr 5592 df-se 5593 df-we 5594 df-xp 5643 df-rel 5644 df-cnv 5645 df-co 5646 df-dm 5647 df-rn 5648 df-res 5649 df-ima 5650 df-pred 6257 df-ord 6324 df-on 6325 df-lim 6326 df-suc 6327 df-iota 6452 df-fun 6502 df-fn 6503 df-f 6504 df-f1 6505 df-fo 6506 df-f1o 6507 df-fv 6508 df-isom 6509 df-riota 7317 df-ov 7364 df-oprab 7365 df-mpo 7366 df-om 7807 df-1st 7925 df-2nd 7926 df-tpos 8161 df-frecs 8216 df-wrecs 8247 df-recs 8321 df-rdg 8360 df-1o 8416 df-2o 8417 df-er 8654 df-map 8773 df-en 8890 df-dom 8891 df-sdom 8892 df-fin 8893 df-card 9883 df-pnf 11199 df-mnf 11200 df-xr 11201 df-ltxr 11202 df-le 11203 df-sub 11395 df-neg 11396 df-div 11821 df-nn 12162 df-2 12224 df-3 12225 df-4 12226 df-5 12227 df-6 12228 df-7 12229 df-8 12230 df-9 12231 df-n0 12422 df-xnn0 12494 df-z 12508 df-dec 12627 df-uz 12772 df-rp 12924 df-fz 13434 df-fzo 13577 df-seq 13916 df-exp 13977 df-hash 14240 df-word 14412 df-lsw 14460 df-concat 14468 df-s1 14493 df-substr 14538 df-pfx 14568 df-splice 14647 df-reverse 14656 df-s2 14746 df-struct 17027 df-sets 17044 df-slot 17062 df-ndx 17074 df-base 17092 df-ress 17121 df-plusg 17154 df-mulr 17155 df-starv 17156 df-tset 17160 df-ple 17161 df-ds 17163 df-unif 17164 df-0g 17331 df-gsum 17332 df-mre 17474 df-mrc 17475 df-acs 17477 df-mgm 18505 df-sgrp 18554 df-mnd 18565 df-mhm 18609 df-submnd 18610 df-efmnd 18687 df-grp 18759 df-minusg 18760 df-subg 18933 df-ghm 19014 df-gim 19057 df-oppg 19132 df-symg 19157 df-pmtr 19232 df-psgn 19281 df-evpm 19282 df-cmn 19572 df-abl 19573 df-mgp 19905 df-ur 19922 df-ring 19974 df-cring 19975 df-oppr 20057 df-dvdsr 20078 df-unit 20079 df-invr 20109 df-dvr 20120 df-drng 20221 df-cnfld 20820 |
This theorem is referenced by: zrhpsgnevpm 21018 evpmodpmf1o 21023 mdetralt 21980 cyc3genpm 32057 |
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