![]() |
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 32887 | . 2 ⊢ (𝐷 ∈ Fin → 𝐴 = (◡(pmSgn‘𝐷) “ {1})) |
3 | evpmsubg.s | . . . 4 ⊢ 𝑆 = (SymGrp‘𝐷) | |
4 | eqid 2728 | . . . 4 ⊢ (pmSgn‘𝐷) = (pmSgn‘𝐷) | |
5 | eqid 2728 | . . . 4 ⊢ ((mulGrp‘ℂfld) ↾s {1, -1}) = ((mulGrp‘ℂfld) ↾s {1, -1}) | |
6 | 3, 4, 5 | psgnghm2 21520 | . . 3 ⊢ (𝐷 ∈ Fin → (pmSgn‘𝐷) ∈ (𝑆 GrpHom ((mulGrp‘ℂfld) ↾s {1, -1}))) |
7 | 5 | cnmsgngrp 21518 | . . . 4 ⊢ ((mulGrp‘ℂfld) ↾s {1, -1}) ∈ Grp |
8 | 5 | cnmsgn0g 32888 | . . . . 5 ⊢ 1 = (0g‘((mulGrp‘ℂfld) ↾s {1, -1})) |
9 | 8 | 0subg 19113 | . . . 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 19199 | . . 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 584 | . 2 ⊢ (𝐷 ∈ Fin → (◡(pmSgn‘𝐷) “ {1}) ∈ (SubGrp‘𝑆)) |
13 | 2, 12 | eqeltrd 2829 | 1 ⊢ (𝐷 ∈ Fin → 𝐴 ∈ (SubGrp‘𝑆)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2098 {csn 4632 {cpr 4634 ◡ccnv 5681 “ cima 5685 ‘cfv 6553 (class class class)co 7426 Fincfn 8970 1c1 11147 -cneg 11483 ↾s cress 17216 Grpcgrp 18897 SubGrpcsubg 19082 GrpHom cghm 19174 SymGrpcsymg 19328 pmSgncpsgn 19451 pmEvencevpm 19452 mulGrpcmgp 20081 ℂfldccnfld 21286 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2699 ax-rep 5289 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-cnex 11202 ax-resscn 11203 ax-1cn 11204 ax-icn 11205 ax-addcl 11206 ax-addrcl 11207 ax-mulcl 11208 ax-mulrcl 11209 ax-mulcom 11210 ax-addass 11211 ax-mulass 11212 ax-distr 11213 ax-i2m1 11214 ax-1ne0 11215 ax-1rid 11216 ax-rnegex 11217 ax-rrecex 11218 ax-cnre 11219 ax-pre-lttri 11220 ax-pre-lttrn 11221 ax-pre-ltadd 11222 ax-pre-mulgt0 11223 ax-addf 11225 ax-mulf 11226 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-xor 1505 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3374 df-reu 3375 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-tp 4637 df-op 4639 df-ot 4641 df-uni 4913 df-int 4954 df-iun 5002 df-iin 5003 df-br 5153 df-opab 5215 df-mpt 5236 df-tr 5270 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-se 5638 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6310 df-ord 6377 df-on 6378 df-lim 6379 df-suc 6380 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-isom 6562 df-riota 7382 df-ov 7429 df-oprab 7430 df-mpo 7431 df-om 7877 df-1st 7999 df-2nd 8000 df-tpos 8238 df-frecs 8293 df-wrecs 8324 df-recs 8398 df-rdg 8437 df-1o 8493 df-2o 8494 df-er 8731 df-map 8853 df-en 8971 df-dom 8972 df-sdom 8973 df-fin 8974 df-card 9970 df-pnf 11288 df-mnf 11289 df-xr 11290 df-ltxr 11291 df-le 11292 df-sub 11484 df-neg 11485 df-div 11910 df-nn 12251 df-2 12313 df-3 12314 df-4 12315 df-5 12316 df-6 12317 df-7 12318 df-8 12319 df-9 12320 df-n0 12511 df-xnn0 12583 df-z 12597 df-dec 12716 df-uz 12861 df-rp 13015 df-fz 13525 df-fzo 13668 df-seq 14007 df-exp 14067 df-hash 14330 df-word 14505 df-lsw 14553 df-concat 14561 df-s1 14586 df-substr 14631 df-pfx 14661 df-splice 14740 df-reverse 14749 df-s2 14839 df-struct 17123 df-sets 17140 df-slot 17158 df-ndx 17170 df-base 17188 df-ress 17217 df-plusg 17253 df-mulr 17254 df-starv 17255 df-tset 17259 df-ple 17260 df-ds 17262 df-unif 17263 df-0g 17430 df-gsum 17431 df-mre 17573 df-mrc 17574 df-acs 17576 df-mgm 18607 df-sgrp 18686 df-mnd 18702 df-mhm 18747 df-submnd 18748 df-efmnd 18828 df-grp 18900 df-minusg 18901 df-subg 19085 df-ghm 19175 df-gim 19220 df-oppg 19304 df-symg 19329 df-pmtr 19404 df-psgn 19453 df-evpm 19454 df-cmn 19744 df-abl 19745 df-mgp 20082 df-rng 20100 df-ur 20129 df-ring 20182 df-cring 20183 df-oppr 20280 df-dvdsr 20303 df-unit 20304 df-invr 20334 df-dvr 20347 df-drng 20633 df-cnfld 21287 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |