Mathbox for Thierry Arnoux |
< Previous
Next >
Nearby theorems |
||
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 30806 | . 2 ⊢ (𝐷 ∈ Fin → 𝐴 = (◡(pmSgn‘𝐷) “ {1})) |
3 | evpmsubg.s | . . . 4 ⊢ 𝑆 = (SymGrp‘𝐷) | |
4 | eqid 2820 | . . . 4 ⊢ (pmSgn‘𝐷) = (pmSgn‘𝐷) | |
5 | eqid 2820 | . . . 4 ⊢ ((mulGrp‘ℂfld) ↾s {1, -1}) = ((mulGrp‘ℂfld) ↾s {1, -1}) | |
6 | 3, 4, 5 | psgnghm2 20718 | . . 3 ⊢ (𝐷 ∈ Fin → (pmSgn‘𝐷) ∈ (𝑆 GrpHom ((mulGrp‘ℂfld) ↾s {1, -1}))) |
7 | 5 | cnmsgngrp 20716 | . . . 4 ⊢ ((mulGrp‘ℂfld) ↾s {1, -1}) ∈ Grp |
8 | 5 | cnmsgn0g 30807 | . . . . 5 ⊢ 1 = (0g‘((mulGrp‘ℂfld) ↾s {1, -1})) |
9 | 8 | 0subg 18297 | . . . 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 18373 | . . 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 588 | . 2 ⊢ (𝐷 ∈ Fin → (◡(pmSgn‘𝐷) “ {1}) ∈ (SubGrp‘𝑆)) |
13 | 2, 12 | eqeltrd 2912 | 1 ⊢ (𝐷 ∈ Fin → 𝐴 ∈ (SubGrp‘𝑆)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1536 ∈ wcel 2113 {csn 4560 {cpr 4562 ◡ccnv 5547 “ cima 5551 ‘cfv 6348 (class class class)co 7149 Fincfn 8502 1c1 10531 -cneg 10864 ↾s cress 16477 Grpcgrp 18096 SubGrpcsubg 18266 GrpHom cghm 18348 SymGrpcsymg 18488 pmSgncpsgn 18610 pmEvencevpm 18611 mulGrpcmgp 19232 ℂfldccnfld 20538 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2792 ax-rep 5183 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5323 ax-un 7454 ax-cnex 10586 ax-resscn 10587 ax-1cn 10588 ax-icn 10589 ax-addcl 10590 ax-addrcl 10591 ax-mulcl 10592 ax-mulrcl 10593 ax-mulcom 10594 ax-addass 10595 ax-mulass 10596 ax-distr 10597 ax-i2m1 10598 ax-1ne0 10599 ax-1rid 10600 ax-rnegex 10601 ax-rrecex 10602 ax-cnre 10603 ax-pre-lttri 10604 ax-pre-lttrn 10605 ax-pre-ltadd 10606 ax-pre-mulgt0 10607 ax-addf 10609 ax-mulf 10610 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1083 df-3an 1084 df-xor 1501 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2799 df-cleq 2813 df-clel 2892 df-nfc 2962 df-ne 3016 df-nel 3123 df-ral 3142 df-rex 3143 df-reu 3144 df-rmo 3145 df-rab 3146 df-v 3493 df-sbc 3769 df-csb 3877 df-dif 3932 df-un 3934 df-in 3936 df-ss 3945 df-pss 3947 df-nul 4285 df-if 4461 df-pw 4534 df-sn 4561 df-pr 4563 df-tp 4565 df-op 4567 df-ot 4569 df-uni 4832 df-int 4870 df-iun 4914 df-iin 4915 df-br 5060 df-opab 5122 df-mpt 5140 df-tr 5166 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-se 5508 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-isom 6357 df-riota 7107 df-ov 7152 df-oprab 7153 df-mpo 7154 df-om 7574 df-1st 7682 df-2nd 7683 df-tpos 7885 df-wrecs 7940 df-recs 8001 df-rdg 8039 df-1o 8095 df-2o 8096 df-oadd 8099 df-er 8282 df-map 8401 df-en 8503 df-dom 8504 df-sdom 8505 df-fin 8506 df-card 9361 df-pnf 10670 df-mnf 10671 df-xr 10672 df-ltxr 10673 df-le 10674 df-sub 10865 df-neg 10866 df-div 11291 df-nn 11632 df-2 11694 df-3 11695 df-4 11696 df-5 11697 df-6 11698 df-7 11699 df-8 11700 df-9 11701 df-n0 11892 df-xnn0 11962 df-z 11976 df-dec 12093 df-uz 12238 df-rp 12384 df-fz 12890 df-fzo 13031 df-seq 13367 df-exp 13427 df-hash 13688 df-word 13859 df-lsw 13908 df-concat 13916 df-s1 13943 df-substr 13996 df-pfx 14026 df-splice 14105 df-reverse 14114 df-s2 14203 df-struct 16478 df-ndx 16479 df-slot 16480 df-base 16482 df-sets 16483 df-ress 16484 df-plusg 16571 df-mulr 16572 df-starv 16573 df-tset 16577 df-ple 16578 df-ds 16580 df-unif 16581 df-0g 16708 df-gsum 16709 df-mre 16850 df-mrc 16851 df-acs 16853 df-mgm 17845 df-sgrp 17894 df-mnd 17905 df-mhm 17949 df-submnd 17950 df-efmnd 18027 df-grp 18099 df-minusg 18100 df-subg 18269 df-ghm 18349 df-gim 18392 df-oppg 18467 df-symg 18489 df-pmtr 18563 df-psgn 18612 df-evpm 18613 df-cmn 18901 df-abl 18902 df-mgp 19233 df-ur 19245 df-ring 19292 df-cring 19293 df-oppr 19366 df-dvdsr 19384 df-unit 19385 df-invr 19415 df-dvr 19426 df-drng 19497 df-cnfld 20539 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |