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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 3455 | . . 3 ⊢ (𝐷 ∈ Fin → 𝐷 ∈ V) | |
2 | fveq2 6538 | . . . . . 6 ⊢ (𝑑 = 𝐷 → (pmSgn‘𝑑) = (pmSgn‘𝐷)) | |
3 | 2 | cnveqd 5632 | . . . . 5 ⊢ (𝑑 = 𝐷 → ◡(pmSgn‘𝑑) = ◡(pmSgn‘𝐷)) |
4 | 3 | imaeq1d 5805 | . . . 4 ⊢ (𝑑 = 𝐷 → (◡(pmSgn‘𝑑) “ {1}) = (◡(pmSgn‘𝐷) “ {1})) |
5 | df-evpm 18351 | . . . 4 ⊢ pmEven = (𝑑 ∈ V ↦ (◡(pmSgn‘𝑑) “ {1})) | |
6 | fvex 6551 | . . . . . 6 ⊢ (pmSgn‘𝐷) ∈ V | |
7 | 6 | cnvex 7486 | . . . . 5 ⊢ ◡(pmSgn‘𝐷) ∈ V |
8 | 7 | imaex 7477 | . . . 4 ⊢ (◡(pmSgn‘𝐷) “ {1}) ∈ V |
9 | 4, 5, 8 | fvmpt 6635 | . . 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 2795 | . . . 4 ⊢ (pmSgn‘𝐷) = (pmSgn‘𝐷) | |
13 | eqid 2795 | . . . 4 ⊢ ((mulGrp‘ℂfld) ↾s {1, -1}) = ((mulGrp‘ℂfld) ↾s {1, -1}) | |
14 | 11, 12, 13 | psgnghm2 20407 | . . 3 ⊢ (𝐷 ∈ Fin → (pmSgn‘𝐷) ∈ (𝑆 GrpHom ((mulGrp‘ℂfld) ↾s {1, -1}))) |
15 | cnring 20249 | . . . . . 6 ⊢ ℂfld ∈ Ring | |
16 | eqid 2795 | . . . . . . 7 ⊢ (mulGrp‘ℂfld) = (mulGrp‘ℂfld) | |
17 | 16 | ringmgp 18993 | . . . . . 6 ⊢ (ℂfld ∈ Ring → (mulGrp‘ℂfld) ∈ Mnd) |
18 | 15, 17 | ax-mp 5 | . . . . 5 ⊢ (mulGrp‘ℂfld) ∈ Mnd |
19 | ax-1cn 10441 | . . . . . 6 ⊢ 1 ∈ ℂ | |
20 | prid1g 4603 | . . . . . 6 ⊢ (1 ∈ ℂ → 1 ∈ {1, -1}) | |
21 | 19, 20 | ax-mp 5 | . . . . 5 ⊢ 1 ∈ {1, -1} |
22 | neg1cn 11599 | . . . . . 6 ⊢ -1 ∈ ℂ | |
23 | prssi 4661 | . . . . . 6 ⊢ ((1 ∈ ℂ ∧ -1 ∈ ℂ) → {1, -1} ⊆ ℂ) | |
24 | 19, 22, 23 | mp2an 688 | . . . . 5 ⊢ {1, -1} ⊆ ℂ |
25 | cnfldbas 20231 | . . . . . . 7 ⊢ ℂ = (Base‘ℂfld) | |
26 | 16, 25 | mgpbas 18935 | . . . . . 6 ⊢ ℂ = (Base‘(mulGrp‘ℂfld)) |
27 | cnfld1 20252 | . . . . . . 7 ⊢ 1 = (1r‘ℂfld) | |
28 | 16, 27 | ringidval 18943 | . . . . . 6 ⊢ 1 = (0g‘(mulGrp‘ℂfld)) |
29 | 13, 26, 28 | ress0g 17758 | . . . . 5 ⊢ (((mulGrp‘ℂfld) ∈ Mnd ∧ 1 ∈ {1, -1} ∧ {1, -1} ⊆ ℂ) → 1 = (0g‘((mulGrp‘ℂfld) ↾s {1, -1}))) |
30 | 18, 21, 24, 29 | mp3an 1453 | . . . 4 ⊢ 1 = (0g‘((mulGrp‘ℂfld) ↾s {1, -1})) |
31 | 30 | ghmker 18125 | . . 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 2883 | 1 ⊢ (𝐷 ∈ Fin → (pmEven‘𝐷) ∈ (NrmSGrp‘𝑆)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1522 ∈ wcel 2081 Vcvv 3437 ⊆ wss 3859 {csn 4472 {cpr 4474 ◡ccnv 5442 “ cima 5446 ‘cfv 6225 (class class class)co 7016 Fincfn 8357 ℂcc 10381 1c1 10384 -cneg 10718 ↾s cress 16313 0gc0g 16542 Mndcmnd 17733 NrmSGrpcnsg 18028 GrpHom cghm 18096 SymGrpcsymg 18236 pmSgncpsgn 18348 pmEvencevpm 18349 mulGrpcmgp 18929 Ringcrg 18987 ℂfldccnfld 20227 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1777 ax-4 1791 ax-5 1888 ax-6 1947 ax-7 1992 ax-8 2083 ax-9 2091 ax-10 2112 ax-11 2126 ax-12 2141 ax-13 2344 ax-ext 2769 ax-rep 5081 ax-sep 5094 ax-nul 5101 ax-pow 5157 ax-pr 5221 ax-un 7319 ax-cnex 10439 ax-resscn 10440 ax-1cn 10441 ax-icn 10442 ax-addcl 10443 ax-addrcl 10444 ax-mulcl 10445 ax-mulrcl 10446 ax-mulcom 10447 ax-addass 10448 ax-mulass 10449 ax-distr 10450 ax-i2m1 10451 ax-1ne0 10452 ax-1rid 10453 ax-rnegex 10454 ax-rrecex 10455 ax-cnre 10456 ax-pre-lttri 10457 ax-pre-lttrn 10458 ax-pre-ltadd 10459 ax-pre-mulgt0 10460 ax-addf 10462 ax-mulf 10463 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3or 1081 df-3an 1082 df-xor 1497 df-tru 1525 df-ex 1762 df-nf 1766 df-sb 2043 df-mo 2576 df-eu 2612 df-clab 2776 df-cleq 2788 df-clel 2863 df-nfc 2935 df-ne 2985 df-nel 3091 df-ral 3110 df-rex 3111 df-reu 3112 df-rmo 3113 df-rab 3114 df-v 3439 df-sbc 3707 df-csb 3812 df-dif 3862 df-un 3864 df-in 3866 df-ss 3874 df-pss 3876 df-nul 4212 df-if 4382 df-pw 4455 df-sn 4473 df-pr 4475 df-tp 4477 df-op 4479 df-ot 4481 df-uni 4746 df-int 4783 df-iun 4827 df-iin 4828 df-br 4963 df-opab 5025 df-mpt 5042 df-tr 5064 df-id 5348 df-eprel 5353 df-po 5362 df-so 5363 df-fr 5402 df-se 5403 df-we 5404 df-xp 5449 df-rel 5450 df-cnv 5451 df-co 5452 df-dm 5453 df-rn 5454 df-res 5455 df-ima 5456 df-pred 6023 df-ord 6069 df-on 6070 df-lim 6071 df-suc 6072 df-iota 6189 df-fun 6227 df-fn 6228 df-f 6229 df-f1 6230 df-fo 6231 df-f1o 6232 df-fv 6233 df-isom 6234 df-riota 6977 df-ov 7019 df-oprab 7020 df-mpo 7021 df-om 7437 df-1st 7545 df-2nd 7546 df-tpos 7743 df-wrecs 7798 df-recs 7860 df-rdg 7898 df-1o 7953 df-2o 7954 df-oadd 7957 df-er 8139 df-map 8258 df-en 8358 df-dom 8359 df-sdom 8360 df-fin 8361 df-card 9214 df-pnf 10523 df-mnf 10524 df-xr 10525 df-ltxr 10526 df-le 10527 df-sub 10719 df-neg 10720 df-div 11146 df-nn 11487 df-2 11548 df-3 11549 df-4 11550 df-5 11551 df-6 11552 df-7 11553 df-8 11554 df-9 11555 df-n0 11746 df-xnn0 11816 df-z 11830 df-dec 11948 df-uz 12094 df-rp 12240 df-fz 12743 df-fzo 12884 df-seq 13220 df-exp 13280 df-hash 13541 df-word 13708 df-lsw 13761 df-concat 13769 df-s1 13794 df-substr 13839 df-pfx 13869 df-splice 13948 df-reverse 13957 df-s2 14046 df-struct 16314 df-ndx 16315 df-slot 16316 df-base 16318 df-sets 16319 df-ress 16320 df-plusg 16407 df-mulr 16408 df-starv 16409 df-tset 16413 df-ple 16414 df-ds 16416 df-unif 16417 df-0g 16544 df-gsum 16545 df-mre 16686 df-mrc 16687 df-acs 16689 df-mgm 17681 df-sgrp 17723 df-mnd 17734 df-mhm 17774 df-submnd 17775 df-grp 17864 df-minusg 17865 df-sbg 17866 df-subg 18030 df-nsg 18031 df-ghm 18097 df-gim 18140 df-oppg 18215 df-symg 18237 df-pmtr 18301 df-psgn 18350 df-evpm 18351 df-cmn 18635 df-abl 18636 df-mgp 18930 df-ur 18942 df-ring 18989 df-cring 18990 df-oppr 19063 df-dvdsr 19081 df-unit 19082 df-invr 19112 df-dvr 19123 df-drng 19194 df-cnfld 20228 |
This theorem is referenced by: cyc3genpm 30432 |
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