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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > evpmsubg | Structured version Visualization version GIF version |
Description: The alternating group is a subgroup of the symmetric group. (Contributed by Thierry Arnoux, 1-Nov-2023.) |
Ref | Expression |
---|---|
evpmsubg.s | ⊢ 𝑆 = (SymGrp‘𝐷) |
evpmsubg.a | ⊢ 𝐴 = (pmEven‘𝐷) |
Ref | Expression |
---|---|
evpmsubg | ⊢ (𝐷 ∈ Fin → 𝐴 ∈ (SubGrp‘𝑆)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | evpmsubg.a | . . 3 ⊢ 𝐴 = (pmEven‘𝐷) | |
2 | 1 | evpmval 31820 | . 2 ⊢ (𝐷 ∈ Fin → 𝐴 = (◡(pmSgn‘𝐷) “ {1})) |
3 | evpmsubg.s | . . . 4 ⊢ 𝑆 = (SymGrp‘𝐷) | |
4 | eqid 2737 | . . . 4 ⊢ (pmSgn‘𝐷) = (pmSgn‘𝐷) | |
5 | eqid 2737 | . . . 4 ⊢ ((mulGrp‘ℂfld) ↾s {1, -1}) = ((mulGrp‘ℂfld) ↾s {1, -1}) | |
6 | 3, 4, 5 | psgnghm2 20938 | . . 3 ⊢ (𝐷 ∈ Fin → (pmSgn‘𝐷) ∈ (𝑆 GrpHom ((mulGrp‘ℂfld) ↾s {1, -1}))) |
7 | 5 | cnmsgngrp 20936 | . . . 4 ⊢ ((mulGrp‘ℂfld) ↾s {1, -1}) ∈ Grp |
8 | 5 | cnmsgn0g 31821 | . . . . 5 ⊢ 1 = (0g‘((mulGrp‘ℂfld) ↾s {1, -1})) |
9 | 8 | 0subg 18912 | . . . 4 ⊢ (((mulGrp‘ℂfld) ↾s {1, -1}) ∈ Grp → {1} ∈ (SubGrp‘((mulGrp‘ℂfld) ↾s {1, -1}))) |
10 | 7, 9 | ax-mp 5 | . . 3 ⊢ {1} ∈ (SubGrp‘((mulGrp‘ℂfld) ↾s {1, -1})) |
11 | ghmpreima 18989 | . . 3 ⊢ (((pmSgn‘𝐷) ∈ (𝑆 GrpHom ((mulGrp‘ℂfld) ↾s {1, -1})) ∧ {1} ∈ (SubGrp‘((mulGrp‘ℂfld) ↾s {1, -1}))) → (◡(pmSgn‘𝐷) “ {1}) ∈ (SubGrp‘𝑆)) | |
12 | 6, 10, 11 | sylancl 586 | . 2 ⊢ (𝐷 ∈ Fin → (◡(pmSgn‘𝐷) “ {1}) ∈ (SubGrp‘𝑆)) |
13 | 2, 12 | eqeltrd 2838 | 1 ⊢ (𝐷 ∈ Fin → 𝐴 ∈ (SubGrp‘𝑆)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1541 ∈ wcel 2106 {csn 4584 {cpr 4586 ◡ccnv 5630 “ cima 5634 ‘cfv 6493 (class class class)co 7351 Fincfn 8841 1c1 11010 -cneg 11344 ↾s cress 17072 Grpcgrp 18708 SubGrpcsubg 18881 GrpHom cghm 18964 SymGrpcsymg 19107 pmSgncpsgn 19230 pmEvencevpm 19231 mulGrpcmgp 19855 ℂfldccnfld 20749 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2708 ax-rep 5240 ax-sep 5254 ax-nul 5261 ax-pow 5318 ax-pr 5382 ax-un 7664 ax-cnex 11065 ax-resscn 11066 ax-1cn 11067 ax-icn 11068 ax-addcl 11069 ax-addrcl 11070 ax-mulcl 11071 ax-mulrcl 11072 ax-mulcom 11073 ax-addass 11074 ax-mulass 11075 ax-distr 11076 ax-i2m1 11077 ax-1ne0 11078 ax-1rid 11079 ax-rnegex 11080 ax-rrecex 11081 ax-cnre 11082 ax-pre-lttri 11083 ax-pre-lttrn 11084 ax-pre-ltadd 11085 ax-pre-mulgt0 11086 ax-addf 11088 ax-mulf 11089 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-xor 1510 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2887 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3351 df-reu 3352 df-rab 3406 df-v 3445 df-sbc 3738 df-csb 3854 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-pss 3927 df-nul 4281 df-if 4485 df-pw 4560 df-sn 4585 df-pr 4587 df-tp 4589 df-op 4591 df-ot 4593 df-uni 4864 df-int 4906 df-iun 4954 df-iin 4955 df-br 5104 df-opab 5166 df-mpt 5187 df-tr 5221 df-id 5529 df-eprel 5535 df-po 5543 df-so 5544 df-fr 5586 df-se 5587 df-we 5588 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6251 df-ord 6318 df-on 6319 df-lim 6320 df-suc 6321 df-iota 6445 df-fun 6495 df-fn 6496 df-f 6497 df-f1 6498 df-fo 6499 df-f1o 6500 df-fv 6501 df-isom 6502 df-riota 7307 df-ov 7354 df-oprab 7355 df-mpo 7356 df-om 7795 df-1st 7913 df-2nd 7914 df-tpos 8149 df-frecs 8204 df-wrecs 8235 df-recs 8309 df-rdg 8348 df-1o 8404 df-2o 8405 df-er 8606 df-map 8725 df-en 8842 df-dom 8843 df-sdom 8844 df-fin 8845 df-card 9833 df-pnf 11149 df-mnf 11150 df-xr 11151 df-ltxr 11152 df-le 11153 df-sub 11345 df-neg 11346 df-div 11771 df-nn 12112 df-2 12174 df-3 12175 df-4 12176 df-5 12177 df-6 12178 df-7 12179 df-8 12180 df-9 12181 df-n0 12372 df-xnn0 12444 df-z 12458 df-dec 12577 df-uz 12722 df-rp 12870 df-fz 13379 df-fzo 13522 df-seq 13861 df-exp 13922 df-hash 14185 df-word 14357 df-lsw 14405 df-concat 14413 df-s1 14438 df-substr 14487 df-pfx 14517 df-splice 14596 df-reverse 14605 df-s2 14695 df-struct 16979 df-sets 16996 df-slot 17014 df-ndx 17026 df-base 17044 df-ress 17073 df-plusg 17106 df-mulr 17107 df-starv 17108 df-tset 17112 df-ple 17113 df-ds 17115 df-unif 17116 df-0g 17283 df-gsum 17284 df-mre 17426 df-mrc 17427 df-acs 17429 df-mgm 18457 df-sgrp 18506 df-mnd 18517 df-mhm 18561 df-submnd 18562 df-efmnd 18639 df-grp 18711 df-minusg 18712 df-subg 18884 df-ghm 18965 df-gim 19008 df-oppg 19083 df-symg 19108 df-pmtr 19183 df-psgn 19232 df-evpm 19233 df-cmn 19523 df-abl 19524 df-mgp 19856 df-ur 19873 df-ring 19920 df-cring 19921 df-oppr 20002 df-dvdsr 20023 df-unit 20024 df-invr 20054 df-dvr 20065 df-drng 20140 df-cnfld 20750 |
This theorem is referenced by: (None) |
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