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 3416 | . . 3 ⊢ (𝐷 ∈ Fin → 𝐷 ∈ V) | |
2 | fveq2 6674 | . . . . . 6 ⊢ (𝑑 = 𝐷 → (pmSgn‘𝑑) = (pmSgn‘𝐷)) | |
3 | 2 | cnveqd 5718 | . . . . 5 ⊢ (𝑑 = 𝐷 → ◡(pmSgn‘𝑑) = ◡(pmSgn‘𝐷)) |
4 | 3 | imaeq1d 5902 | . . . 4 ⊢ (𝑑 = 𝐷 → (◡(pmSgn‘𝑑) “ {1}) = (◡(pmSgn‘𝐷) “ {1})) |
5 | df-evpm 18738 | . . . 4 ⊢ pmEven = (𝑑 ∈ V ↦ (◡(pmSgn‘𝑑) “ {1})) | |
6 | fvex 6687 | . . . . . 6 ⊢ (pmSgn‘𝐷) ∈ V | |
7 | 6 | cnvex 7656 | . . . . 5 ⊢ ◡(pmSgn‘𝐷) ∈ V |
8 | 7 | imaex 7647 | . . . 4 ⊢ (◡(pmSgn‘𝐷) “ {1}) ∈ V |
9 | 4, 5, 8 | fvmpt 6775 | . . 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 2738 | . . . 4 ⊢ (pmSgn‘𝐷) = (pmSgn‘𝐷) | |
13 | eqid 2738 | . . . 4 ⊢ ((mulGrp‘ℂfld) ↾s {1, -1}) = ((mulGrp‘ℂfld) ↾s {1, -1}) | |
14 | 11, 12, 13 | psgnghm2 20397 | . . 3 ⊢ (𝐷 ∈ Fin → (pmSgn‘𝐷) ∈ (𝑆 GrpHom ((mulGrp‘ℂfld) ↾s {1, -1}))) |
15 | cnring 20239 | . . . . . 6 ⊢ ℂfld ∈ Ring | |
16 | eqid 2738 | . . . . . . 7 ⊢ (mulGrp‘ℂfld) = (mulGrp‘ℂfld) | |
17 | 16 | ringmgp 19422 | . . . . . 6 ⊢ (ℂfld ∈ Ring → (mulGrp‘ℂfld) ∈ Mnd) |
18 | 15, 17 | ax-mp 5 | . . . . 5 ⊢ (mulGrp‘ℂfld) ∈ Mnd |
19 | ax-1cn 10673 | . . . . . 6 ⊢ 1 ∈ ℂ | |
20 | prid1g 4651 | . . . . . 6 ⊢ (1 ∈ ℂ → 1 ∈ {1, -1}) | |
21 | 19, 20 | ax-mp 5 | . . . . 5 ⊢ 1 ∈ {1, -1} |
22 | neg1cn 11830 | . . . . . 6 ⊢ -1 ∈ ℂ | |
23 | prssi 4709 | . . . . . 6 ⊢ ((1 ∈ ℂ ∧ -1 ∈ ℂ) → {1, -1} ⊆ ℂ) | |
24 | 19, 22, 23 | mp2an 692 | . . . . 5 ⊢ {1, -1} ⊆ ℂ |
25 | cnfldbas 20221 | . . . . . . 7 ⊢ ℂ = (Base‘ℂfld) | |
26 | 16, 25 | mgpbas 19364 | . . . . . 6 ⊢ ℂ = (Base‘(mulGrp‘ℂfld)) |
27 | cnfld1 20242 | . . . . . . 7 ⊢ 1 = (1r‘ℂfld) | |
28 | 16, 27 | ringidval 19372 | . . . . . 6 ⊢ 1 = (0g‘(mulGrp‘ℂfld)) |
29 | 13, 26, 28 | ress0g 18055 | . . . . 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 18502 | . . 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 2833 | 1 ⊢ (𝐷 ∈ Fin → (pmEven‘𝐷) ∈ (NrmSGrp‘𝑆)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1542 ∈ wcel 2114 Vcvv 3398 ⊆ wss 3843 {csn 4516 {cpr 4518 ◡ccnv 5524 “ cima 5528 ‘cfv 6339 (class class class)co 7170 Fincfn 8555 ℂcc 10613 1c1 10616 -cneg 10949 ↾s cress 16587 0gc0g 16816 Mndcmnd 18027 NrmSGrpcnsg 18392 GrpHom cghm 18473 SymGrpcsymg 18613 pmSgncpsgn 18735 pmEvencevpm 18736 mulGrpcmgp 19358 Ringcrg 19416 ℂfldccnfld 20217 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2710 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5232 ax-pr 5296 ax-un 7479 ax-cnex 10671 ax-resscn 10672 ax-1cn 10673 ax-icn 10674 ax-addcl 10675 ax-addrcl 10676 ax-mulcl 10677 ax-mulrcl 10678 ax-mulcom 10679 ax-addass 10680 ax-mulass 10681 ax-distr 10682 ax-i2m1 10683 ax-1ne0 10684 ax-1rid 10685 ax-rnegex 10686 ax-rrecex 10687 ax-cnre 10688 ax-pre-lttri 10689 ax-pre-lttrn 10690 ax-pre-ltadd 10691 ax-pre-mulgt0 10692 ax-addf 10694 ax-mulf 10695 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3or 1089 df-3an 1090 df-xor 1507 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2540 df-eu 2570 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-ne 2935 df-nel 3039 df-ral 3058 df-rex 3059 df-reu 3060 df-rmo 3061 df-rab 3062 df-v 3400 df-sbc 3681 df-csb 3791 df-dif 3846 df-un 3848 df-in 3850 df-ss 3860 df-pss 3862 df-nul 4212 df-if 4415 df-pw 4490 df-sn 4517 df-pr 4519 df-tp 4521 df-op 4523 df-ot 4525 df-uni 4797 df-int 4837 df-iun 4883 df-iin 4884 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5429 df-eprel 5434 df-po 5442 df-so 5443 df-fr 5483 df-se 5484 df-we 5485 df-xp 5531 df-rel 5532 df-cnv 5533 df-co 5534 df-dm 5535 df-rn 5536 df-res 5537 df-ima 5538 df-pred 6129 df-ord 6175 df-on 6176 df-lim 6177 df-suc 6178 df-iota 6297 df-fun 6341 df-fn 6342 df-f 6343 df-f1 6344 df-fo 6345 df-f1o 6346 df-fv 6347 df-isom 6348 df-riota 7127 df-ov 7173 df-oprab 7174 df-mpo 7175 df-om 7600 df-1st 7714 df-2nd 7715 df-tpos 7921 df-wrecs 7976 df-recs 8037 df-rdg 8075 df-1o 8131 df-2o 8132 df-er 8320 df-map 8439 df-en 8556 df-dom 8557 df-sdom 8558 df-fin 8559 df-card 9441 df-pnf 10755 df-mnf 10756 df-xr 10757 df-ltxr 10758 df-le 10759 df-sub 10950 df-neg 10951 df-div 11376 df-nn 11717 df-2 11779 df-3 11780 df-4 11781 df-5 11782 df-6 11783 df-7 11784 df-8 11785 df-9 11786 df-n0 11977 df-xnn0 12049 df-z 12063 df-dec 12180 df-uz 12325 df-rp 12473 df-fz 12982 df-fzo 13125 df-seq 13461 df-exp 13522 df-hash 13783 df-word 13956 df-lsw 14004 df-concat 14012 df-s1 14039 df-substr 14092 df-pfx 14122 df-splice 14201 df-reverse 14210 df-s2 14299 df-struct 16588 df-ndx 16589 df-slot 16590 df-base 16592 df-sets 16593 df-ress 16594 df-plusg 16681 df-mulr 16682 df-starv 16683 df-tset 16687 df-ple 16688 df-ds 16690 df-unif 16691 df-0g 16818 df-gsum 16819 df-mre 16960 df-mrc 16961 df-acs 16963 df-mgm 17968 df-sgrp 18017 df-mnd 18028 df-mhm 18072 df-submnd 18073 df-efmnd 18150 df-grp 18222 df-minusg 18223 df-sbg 18224 df-subg 18394 df-nsg 18395 df-ghm 18474 df-gim 18517 df-oppg 18592 df-symg 18614 df-pmtr 18688 df-psgn 18737 df-evpm 18738 df-cmn 19026 df-abl 19027 df-mgp 19359 df-ur 19371 df-ring 19418 df-cring 19419 df-oppr 19495 df-dvdsr 19513 df-unit 19514 df-invr 19544 df-dvr 19555 df-drng 19623 df-cnfld 20218 |
This theorem is referenced by: cyc3genpm 30996 |
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