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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > altgnsg | Structured version Visualization version GIF version |
Description: The alternating group (pmEven‘𝐷) is a normal subgroup of the symmetric group. (Contributed by Thierry Arnoux, 18-Sep-2023.) |
Ref | Expression |
---|---|
evpmid.1 | ⊢ 𝑆 = (SymGrp‘𝐷) |
Ref | Expression |
---|---|
altgnsg | ⊢ (𝐷 ∈ Fin → (pmEven‘𝐷) ∈ (NrmSGrp‘𝑆)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elex 3487 | . . 3 ⊢ (𝐷 ∈ Fin → 𝐷 ∈ V) | |
2 | fveq2 6884 | . . . . . 6 ⊢ (𝑑 = 𝐷 → (pmSgn‘𝑑) = (pmSgn‘𝐷)) | |
3 | 2 | cnveqd 5868 | . . . . 5 ⊢ (𝑑 = 𝐷 → ◡(pmSgn‘𝑑) = ◡(pmSgn‘𝐷)) |
4 | 3 | imaeq1d 6051 | . . . 4 ⊢ (𝑑 = 𝐷 → (◡(pmSgn‘𝑑) “ {1}) = (◡(pmSgn‘𝐷) “ {1})) |
5 | df-evpm 19409 | . . . 4 ⊢ pmEven = (𝑑 ∈ V ↦ (◡(pmSgn‘𝑑) “ {1})) | |
6 | fvex 6897 | . . . . . 6 ⊢ (pmSgn‘𝐷) ∈ V | |
7 | 6 | cnvex 7912 | . . . . 5 ⊢ ◡(pmSgn‘𝐷) ∈ V |
8 | 7 | imaex 7903 | . . . 4 ⊢ (◡(pmSgn‘𝐷) “ {1}) ∈ V |
9 | 4, 5, 8 | fvmpt 6991 | . . 3 ⊢ (𝐷 ∈ V → (pmEven‘𝐷) = (◡(pmSgn‘𝐷) “ {1})) |
10 | 1, 9 | syl 17 | . 2 ⊢ (𝐷 ∈ Fin → (pmEven‘𝐷) = (◡(pmSgn‘𝐷) “ {1})) |
11 | evpmid.1 | . . . 4 ⊢ 𝑆 = (SymGrp‘𝐷) | |
12 | eqid 2726 | . . . 4 ⊢ (pmSgn‘𝐷) = (pmSgn‘𝐷) | |
13 | eqid 2726 | . . . 4 ⊢ ((mulGrp‘ℂfld) ↾s {1, -1}) = ((mulGrp‘ℂfld) ↾s {1, -1}) | |
14 | 11, 12, 13 | psgnghm2 21469 | . . 3 ⊢ (𝐷 ∈ Fin → (pmSgn‘𝐷) ∈ (𝑆 GrpHom ((mulGrp‘ℂfld) ↾s {1, -1}))) |
15 | cnring 21274 | . . . . . 6 ⊢ ℂfld ∈ Ring | |
16 | eqid 2726 | . . . . . . 7 ⊢ (mulGrp‘ℂfld) = (mulGrp‘ℂfld) | |
17 | 16 | ringmgp 20141 | . . . . . 6 ⊢ (ℂfld ∈ Ring → (mulGrp‘ℂfld) ∈ Mnd) |
18 | 15, 17 | ax-mp 5 | . . . . 5 ⊢ (mulGrp‘ℂfld) ∈ Mnd |
19 | ax-1cn 11167 | . . . . . 6 ⊢ 1 ∈ ℂ | |
20 | prid1g 4759 | . . . . . 6 ⊢ (1 ∈ ℂ → 1 ∈ {1, -1}) | |
21 | 19, 20 | ax-mp 5 | . . . . 5 ⊢ 1 ∈ {1, -1} |
22 | neg1cn 12327 | . . . . . 6 ⊢ -1 ∈ ℂ | |
23 | prssi 4819 | . . . . . 6 ⊢ ((1 ∈ ℂ ∧ -1 ∈ ℂ) → {1, -1} ⊆ ℂ) | |
24 | 19, 22, 23 | mp2an 689 | . . . . 5 ⊢ {1, -1} ⊆ ℂ |
25 | cnfldbas 21239 | . . . . . . 7 ⊢ ℂ = (Base‘ℂfld) | |
26 | 16, 25 | mgpbas 20042 | . . . . . 6 ⊢ ℂ = (Base‘(mulGrp‘ℂfld)) |
27 | cnfld1 21277 | . . . . . . 7 ⊢ 1 = (1r‘ℂfld) | |
28 | 16, 27 | ringidval 20085 | . . . . . 6 ⊢ 1 = (0g‘(mulGrp‘ℂfld)) |
29 | 13, 26, 28 | ress0g 18692 | . . . . 5 ⊢ (((mulGrp‘ℂfld) ∈ Mnd ∧ 1 ∈ {1, -1} ∧ {1, -1} ⊆ ℂ) → 1 = (0g‘((mulGrp‘ℂfld) ↾s {1, -1}))) |
30 | 18, 21, 24, 29 | mp3an 1457 | . . . 4 ⊢ 1 = (0g‘((mulGrp‘ℂfld) ↾s {1, -1})) |
31 | 30 | ghmker 19164 | . . 3 ⊢ ((pmSgn‘𝐷) ∈ (𝑆 GrpHom ((mulGrp‘ℂfld) ↾s {1, -1})) → (◡(pmSgn‘𝐷) “ {1}) ∈ (NrmSGrp‘𝑆)) |
32 | 14, 31 | syl 17 | . 2 ⊢ (𝐷 ∈ Fin → (◡(pmSgn‘𝐷) “ {1}) ∈ (NrmSGrp‘𝑆)) |
33 | 10, 32 | eqeltrd 2827 | 1 ⊢ (𝐷 ∈ Fin → (pmEven‘𝐷) ∈ (NrmSGrp‘𝑆)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2098 Vcvv 3468 ⊆ wss 3943 {csn 4623 {cpr 4625 ◡ccnv 5668 “ cima 5672 ‘cfv 6536 (class class class)co 7404 Fincfn 8938 ℂcc 11107 1c1 11110 -cneg 11446 ↾s cress 17179 0gc0g 17391 Mndcmnd 18664 NrmSGrpcnsg 19045 GrpHom cghm 19135 SymGrpcsymg 19283 pmSgncpsgn 19406 pmEvencevpm 19407 mulGrpcmgp 20036 Ringcrg 20135 ℂfldccnfld 21235 |
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 2163 ax-ext 2697 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7721 ax-cnex 11165 ax-resscn 11166 ax-1cn 11167 ax-icn 11168 ax-addcl 11169 ax-addrcl 11170 ax-mulcl 11171 ax-mulrcl 11172 ax-mulcom 11173 ax-addass 11174 ax-mulass 11175 ax-distr 11176 ax-i2m1 11177 ax-1ne0 11178 ax-1rid 11179 ax-rnegex 11180 ax-rrecex 11181 ax-cnre 11182 ax-pre-lttri 11183 ax-pre-lttrn 11184 ax-pre-ltadd 11185 ax-pre-mulgt0 11186 ax-addf 11188 ax-mulf 11189 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 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 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-rmo 3370 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-tp 4628 df-op 4630 df-ot 4632 df-uni 4903 df-int 4944 df-iun 4992 df-iin 4993 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-se 5625 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6293 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6488 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-isom 6545 df-riota 7360 df-ov 7407 df-oprab 7408 df-mpo 7409 df-om 7852 df-1st 7971 df-2nd 7972 df-tpos 8209 df-frecs 8264 df-wrecs 8295 df-recs 8369 df-rdg 8408 df-1o 8464 df-2o 8465 df-er 8702 df-map 8821 df-en 8939 df-dom 8940 df-sdom 8941 df-fin 8942 df-card 9933 df-pnf 11251 df-mnf 11252 df-xr 11253 df-ltxr 11254 df-le 11255 df-sub 11447 df-neg 11448 df-div 11873 df-nn 12214 df-2 12276 df-3 12277 df-4 12278 df-5 12279 df-6 12280 df-7 12281 df-8 12282 df-9 12283 df-n0 12474 df-xnn0 12546 df-z 12560 df-dec 12679 df-uz 12824 df-rp 12978 df-fz 13488 df-fzo 13631 df-seq 13970 df-exp 14030 df-hash 14293 df-word 14468 df-lsw 14516 df-concat 14524 df-s1 14549 df-substr 14594 df-pfx 14624 df-splice 14703 df-reverse 14712 df-s2 14802 df-struct 17086 df-sets 17103 df-slot 17121 df-ndx 17133 df-base 17151 df-ress 17180 df-plusg 17216 df-mulr 17217 df-starv 17218 df-tset 17222 df-ple 17223 df-ds 17225 df-unif 17226 df-0g 17393 df-gsum 17394 df-mre 17536 df-mrc 17537 df-acs 17539 df-mgm 18570 df-sgrp 18649 df-mnd 18665 df-mhm 18710 df-submnd 18711 df-efmnd 18791 df-grp 18863 df-minusg 18864 df-sbg 18865 df-subg 19047 df-nsg 19048 df-ghm 19136 df-gim 19181 df-oppg 19259 df-symg 19284 df-pmtr 19359 df-psgn 19408 df-evpm 19409 df-cmn 19699 df-abl 19700 df-mgp 20037 df-rng 20055 df-ur 20084 df-ring 20137 df-cring 20138 df-oppr 20233 df-dvdsr 20256 df-unit 20257 df-invr 20287 df-dvr 20300 df-drng 20586 df-cnfld 21236 |
This theorem is referenced by: cyc3genpm 32814 |
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