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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 3462 | . . 3 ⊢ (𝐷 ∈ Fin → 𝐷 ∈ V) | |
2 | fveq2 6843 | . . . . . 6 ⊢ (𝑑 = 𝐷 → (pmSgn‘𝑑) = (pmSgn‘𝐷)) | |
3 | 2 | cnveqd 5832 | . . . . 5 ⊢ (𝑑 = 𝐷 → ◡(pmSgn‘𝑑) = ◡(pmSgn‘𝐷)) |
4 | 3 | imaeq1d 6013 | . . . 4 ⊢ (𝑑 = 𝐷 → (◡(pmSgn‘𝑑) “ {1}) = (◡(pmSgn‘𝐷) “ {1})) |
5 | df-evpm 19279 | . . . 4 ⊢ pmEven = (𝑑 ∈ V ↦ (◡(pmSgn‘𝑑) “ {1})) | |
6 | fvex 6856 | . . . . . 6 ⊢ (pmSgn‘𝐷) ∈ V | |
7 | 6 | cnvex 7863 | . . . . 5 ⊢ ◡(pmSgn‘𝐷) ∈ V |
8 | 7 | imaex 7854 | . . . 4 ⊢ (◡(pmSgn‘𝐷) “ {1}) ∈ V |
9 | 4, 5, 8 | fvmpt 6949 | . . 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 2733 | . . . 4 ⊢ (pmSgn‘𝐷) = (pmSgn‘𝐷) | |
13 | eqid 2733 | . . . 4 ⊢ ((mulGrp‘ℂfld) ↾s {1, -1}) = ((mulGrp‘ℂfld) ↾s {1, -1}) | |
14 | 11, 12, 13 | psgnghm2 21001 | . . 3 ⊢ (𝐷 ∈ Fin → (pmSgn‘𝐷) ∈ (𝑆 GrpHom ((mulGrp‘ℂfld) ↾s {1, -1}))) |
15 | cnring 20835 | . . . . . 6 ⊢ ℂfld ∈ Ring | |
16 | eqid 2733 | . . . . . . 7 ⊢ (mulGrp‘ℂfld) = (mulGrp‘ℂfld) | |
17 | 16 | ringmgp 19975 | . . . . . 6 ⊢ (ℂfld ∈ Ring → (mulGrp‘ℂfld) ∈ Mnd) |
18 | 15, 17 | ax-mp 5 | . . . . 5 ⊢ (mulGrp‘ℂfld) ∈ Mnd |
19 | ax-1cn 11114 | . . . . . 6 ⊢ 1 ∈ ℂ | |
20 | prid1g 4722 | . . . . . 6 ⊢ (1 ∈ ℂ → 1 ∈ {1, -1}) | |
21 | 19, 20 | ax-mp 5 | . . . . 5 ⊢ 1 ∈ {1, -1} |
22 | neg1cn 12272 | . . . . . 6 ⊢ -1 ∈ ℂ | |
23 | prssi 4782 | . . . . . 6 ⊢ ((1 ∈ ℂ ∧ -1 ∈ ℂ) → {1, -1} ⊆ ℂ) | |
24 | 19, 22, 23 | mp2an 691 | . . . . 5 ⊢ {1, -1} ⊆ ℂ |
25 | cnfldbas 20816 | . . . . . . 7 ⊢ ℂ = (Base‘ℂfld) | |
26 | 16, 25 | mgpbas 19907 | . . . . . 6 ⊢ ℂ = (Base‘(mulGrp‘ℂfld)) |
27 | cnfld1 20838 | . . . . . . 7 ⊢ 1 = (1r‘ℂfld) | |
28 | 16, 27 | ringidval 19920 | . . . . . 6 ⊢ 1 = (0g‘(mulGrp‘ℂfld)) |
29 | 13, 26, 28 | ress0g 18589 | . . . . 5 ⊢ (((mulGrp‘ℂfld) ∈ Mnd ∧ 1 ∈ {1, -1} ∧ {1, -1} ⊆ ℂ) → 1 = (0g‘((mulGrp‘ℂfld) ↾s {1, -1}))) |
30 | 18, 21, 24, 29 | mp3an 1462 | . . . 4 ⊢ 1 = (0g‘((mulGrp‘ℂfld) ↾s {1, -1})) |
31 | 30 | ghmker 19039 | . . 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 2834 | 1 ⊢ (𝐷 ∈ Fin → (pmEven‘𝐷) ∈ (NrmSGrp‘𝑆)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1542 ∈ wcel 2107 Vcvv 3444 ⊆ wss 3911 {csn 4587 {cpr 4589 ◡ccnv 5633 “ cima 5637 ‘cfv 6497 (class class class)co 7358 Fincfn 8886 ℂcc 11054 1c1 11057 -cneg 11391 ↾s cress 17117 0gc0g 17326 Mndcmnd 18561 NrmSGrpcnsg 18928 GrpHom cghm 19010 SymGrpcsymg 19153 pmSgncpsgn 19276 pmEvencevpm 19277 mulGrpcmgp 19901 Ringcrg 19969 ℂfldccnfld 20812 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5243 ax-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 ax-cnex 11112 ax-resscn 11113 ax-1cn 11114 ax-icn 11115 ax-addcl 11116 ax-addrcl 11117 ax-mulcl 11118 ax-mulrcl 11119 ax-mulcom 11120 ax-addass 11121 ax-mulass 11122 ax-distr 11123 ax-i2m1 11124 ax-1ne0 11125 ax-1rid 11126 ax-rnegex 11127 ax-rrecex 11128 ax-cnre 11129 ax-pre-lttri 11130 ax-pre-lttrn 11131 ax-pre-ltadd 11132 ax-pre-mulgt0 11133 ax-addf 11135 ax-mulf 11136 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-xor 1511 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3352 df-reu 3353 df-rab 3407 df-v 3446 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3930 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-tp 4592 df-op 4594 df-ot 4596 df-uni 4867 df-int 4909 df-iun 4957 df-iin 4958 df-br 5107 df-opab 5169 df-mpt 5190 df-tr 5224 df-id 5532 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5589 df-se 5590 df-we 5591 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-pred 6254 df-ord 6321 df-on 6322 df-lim 6323 df-suc 6324 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 df-isom 6506 df-riota 7314 df-ov 7361 df-oprab 7362 df-mpo 7363 df-om 7804 df-1st 7922 df-2nd 7923 df-tpos 8158 df-frecs 8213 df-wrecs 8244 df-recs 8318 df-rdg 8357 df-1o 8413 df-2o 8414 df-er 8651 df-map 8770 df-en 8887 df-dom 8888 df-sdom 8889 df-fin 8890 df-card 9880 df-pnf 11196 df-mnf 11197 df-xr 11198 df-ltxr 11199 df-le 11200 df-sub 11392 df-neg 11393 df-div 11818 df-nn 12159 df-2 12221 df-3 12222 df-4 12223 df-5 12224 df-6 12225 df-7 12226 df-8 12227 df-9 12228 df-n0 12419 df-xnn0 12491 df-z 12505 df-dec 12624 df-uz 12769 df-rp 12921 df-fz 13431 df-fzo 13574 df-seq 13913 df-exp 13974 df-hash 14237 df-word 14409 df-lsw 14457 df-concat 14465 df-s1 14490 df-substr 14535 df-pfx 14565 df-splice 14644 df-reverse 14653 df-s2 14743 df-struct 17024 df-sets 17041 df-slot 17059 df-ndx 17071 df-base 17089 df-ress 17118 df-plusg 17151 df-mulr 17152 df-starv 17153 df-tset 17157 df-ple 17158 df-ds 17160 df-unif 17161 df-0g 17328 df-gsum 17329 df-mre 17471 df-mrc 17472 df-acs 17474 df-mgm 18502 df-sgrp 18551 df-mnd 18562 df-mhm 18606 df-submnd 18607 df-efmnd 18684 df-grp 18756 df-minusg 18757 df-sbg 18758 df-subg 18930 df-nsg 18931 df-ghm 19011 df-gim 19054 df-oppg 19129 df-symg 19154 df-pmtr 19229 df-psgn 19278 df-evpm 19279 df-cmn 19569 df-abl 19570 df-mgp 19902 df-ur 19919 df-ring 19971 df-cring 19972 df-oppr 20054 df-dvdsr 20075 df-unit 20076 df-invr 20106 df-dvr 20117 df-drng 20199 df-cnfld 20813 |
This theorem is referenced by: cyc3genpm 32050 |
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