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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 5875 | . . . . 5 ⊢ (𝑑 = 𝐷 → ◡(pmSgn‘𝑑) = ◡(pmSgn‘𝐷)) |
3 | 2 | imaeq1d 6058 | . . . 4 ⊢ (𝑑 = 𝐷 → (◡(pmSgn‘𝑑) “ {1}) = (◡(pmSgn‘𝐷) “ {1})) |
4 | df-evpm 19405 | . . . 4 ⊢ pmEven = (𝑑 ∈ V ↦ (◡(pmSgn‘𝑑) “ {1})) | |
5 | fvex 6904 | . . . . . 6 ⊢ (pmSgn‘𝐷) ∈ V | |
6 | 5 | cnvex 7920 | . . . . 5 ⊢ ◡(pmSgn‘𝐷) ∈ V |
7 | 6 | imaex 7911 | . . . 4 ⊢ (◡(pmSgn‘𝐷) “ {1}) ∈ V |
8 | 3, 4, 7 | fvmpt 6998 | . . 3 ⊢ (𝐷 ∈ V → (pmEven‘𝐷) = (◡(pmSgn‘𝐷) “ {1})) |
9 | cnvimass 6080 | . . . 4 ⊢ (◡(pmSgn‘𝐷) “ {1}) ⊆ dom (pmSgn‘𝐷) | |
10 | evpmss.s | . . . . . . 7 ⊢ 𝑆 = (SymGrp‘𝐷) | |
11 | eqid 2731 | . . . . . . 7 ⊢ (pmSgn‘𝐷) = (pmSgn‘𝐷) | |
12 | eqid 2731 | . . . . . . 7 ⊢ (𝑆 ↾s dom (pmSgn‘𝐷)) = (𝑆 ↾s dom (pmSgn‘𝐷)) | |
13 | eqid 2731 | . . . . . . 7 ⊢ ((mulGrp‘ℂfld) ↾s {1, -1}) = ((mulGrp‘ℂfld) ↾s {1, -1}) | |
14 | 10, 11, 12, 13 | psgnghm 21356 | . . . . . 6 ⊢ (𝐷 ∈ V → (pmSgn‘𝐷) ∈ ((𝑆 ↾s dom (pmSgn‘𝐷)) GrpHom ((mulGrp‘ℂfld) ↾s {1, -1}))) |
15 | eqid 2731 | . . . . . . 7 ⊢ (Base‘(𝑆 ↾s dom (pmSgn‘𝐷))) = (Base‘(𝑆 ↾s dom (pmSgn‘𝐷))) | |
16 | eqid 2731 | . . . . . . 7 ⊢ (Base‘((mulGrp‘ℂfld) ↾s {1, -1})) = (Base‘((mulGrp‘ℂfld) ↾s {1, -1})) | |
17 | 15, 16 | ghmf 19138 | . . . . . 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 6726 | . . . . . 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 17190 | . . . . 5 ⊢ (Base‘(𝑆 ↾s dom (pmSgn‘𝐷))) ⊆ 𝑃 |
22 | 19, 21 | eqsstrdi 4036 | . . . 4 ⊢ (𝐷 ∈ V → dom (pmSgn‘𝐷) ⊆ 𝑃) |
23 | 9, 22 | sstrid 3993 | . . 3 ⊢ (𝐷 ∈ V → (◡(pmSgn‘𝐷) “ {1}) ⊆ 𝑃) |
24 | 8, 23 | eqsstrd 4020 | . 2 ⊢ (𝐷 ∈ V → (pmEven‘𝐷) ⊆ 𝑃) |
25 | fvprc 6883 | . . 3 ⊢ (¬ 𝐷 ∈ V → (pmEven‘𝐷) = ∅) | |
26 | 0ss 4396 | . . 3 ⊢ ∅ ⊆ 𝑃 | |
27 | 25, 26 | eqsstrdi 4036 | . 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 1540 ∈ wcel 2105 Vcvv 3473 ⊆ wss 3948 ∅c0 4322 {csn 4628 {cpr 4630 ◡ccnv 5675 dom cdm 5676 “ cima 5679 ⟶wf 6539 ‘cfv 6543 (class class class)co 7412 1c1 11117 -cneg 11452 Basecbs 17151 ↾s cress 17180 GrpHom cghm 19131 SymGrpcsymg 19279 pmSgncpsgn 19402 pmEvencevpm 19403 mulGrpcmgp 20032 ℂfldccnfld 21148 |
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 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7729 ax-cnex 11172 ax-resscn 11173 ax-1cn 11174 ax-icn 11175 ax-addcl 11176 ax-addrcl 11177 ax-mulcl 11178 ax-mulrcl 11179 ax-mulcom 11180 ax-addass 11181 ax-mulass 11182 ax-distr 11183 ax-i2m1 11184 ax-1ne0 11185 ax-1rid 11186 ax-rnegex 11187 ax-rrecex 11188 ax-cnre 11189 ax-pre-lttri 11190 ax-pre-lttrn 11191 ax-pre-ltadd 11192 ax-pre-mulgt0 11193 ax-addf 11195 ax-mulf 11196 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1087 df-3an 1088 df-xor 1509 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3375 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-tp 4633 df-op 4635 df-ot 4637 df-uni 4909 df-int 4951 df-iun 4999 df-iin 5000 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-se 5632 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-isom 6552 df-riota 7368 df-ov 7415 df-oprab 7416 df-mpo 7417 df-om 7860 df-1st 7979 df-2nd 7980 df-tpos 8217 df-frecs 8272 df-wrecs 8303 df-recs 8377 df-rdg 8416 df-1o 8472 df-2o 8473 df-er 8709 df-map 8828 df-en 8946 df-dom 8947 df-sdom 8948 df-fin 8949 df-card 9940 df-pnf 11257 df-mnf 11258 df-xr 11259 df-ltxr 11260 df-le 11261 df-sub 11453 df-neg 11454 df-div 11879 df-nn 12220 df-2 12282 df-3 12283 df-4 12284 df-5 12285 df-6 12286 df-7 12287 df-8 12288 df-9 12289 df-n0 12480 df-xnn0 12552 df-z 12566 df-dec 12685 df-uz 12830 df-rp 12982 df-fz 13492 df-fzo 13635 df-seq 13974 df-exp 14035 df-hash 14298 df-word 14472 df-lsw 14520 df-concat 14528 df-s1 14553 df-substr 14598 df-pfx 14628 df-splice 14707 df-reverse 14716 df-s2 14806 df-struct 17087 df-sets 17104 df-slot 17122 df-ndx 17134 df-base 17152 df-ress 17181 df-plusg 17217 df-mulr 17218 df-starv 17219 df-tset 17223 df-ple 17224 df-ds 17226 df-unif 17227 df-0g 17394 df-gsum 17395 df-mre 17537 df-mrc 17538 df-acs 17540 df-mgm 18568 df-sgrp 18647 df-mnd 18663 df-mhm 18708 df-submnd 18709 df-efmnd 18789 df-grp 18861 df-minusg 18862 df-subg 19043 df-ghm 19132 df-gim 19177 df-oppg 19255 df-symg 19280 df-pmtr 19355 df-psgn 19404 df-evpm 19405 df-cmn 19695 df-abl 19696 df-mgp 20033 df-rng 20051 df-ur 20080 df-ring 20133 df-cring 20134 df-oppr 20229 df-dvdsr 20252 df-unit 20253 df-invr 20283 df-dvr 20296 df-drng 20506 df-cnfld 21149 |
This theorem is referenced by: zrhpsgnevpm 21367 evpmodpmf1o 21372 mdetralt 22343 cyc3genpm 32596 |
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