psgnghm2 - Metamath Proof Explorer

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Theorem psgnghm2 20698

Description: The sign is a homomorphism from the finite symmetric group to the numeric signs. (Contributed by Stefan O'Rear, 28-Aug-2015.)

**Hypotheses**
Ref	Expression
psgnghm2.s	⊢ 𝑆 = (SymGrp‘𝐷)
psgnghm2.n	⊢ 𝑁 = (pmSgn‘𝐷)
psgnghm2.u	⊢ 𝑈 = ((mulGrp‘ℂ_fld) ↾_s {1, -1})

**Assertion**
Ref	Expression
psgnghm2	⊢ (𝐷 ∈ Fin → 𝑁 ∈ (𝑆 GrpHom 𝑈))

Proof of Theorem psgnghm2 Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		psgnghm2.s	. . 3 ⊢ 𝑆 = (SymGrp‘𝐷)
2		psgnghm2.n	. . 3 ⊢ 𝑁 = (pmSgn‘𝐷)
3		eqid 2738	. . 3 ⊢ (𝑆 ↾_s dom 𝑁) = (𝑆 ↾_s dom 𝑁)
4		psgnghm2.u	. . 3 ⊢ 𝑈 = ((mulGrp‘ℂ_fld) ↾_s {1, -1})
5	1, 2, 3, 4	psgnghm 20697	. 2 ⊢ (𝐷 ∈ Fin → 𝑁 ∈ ((𝑆 ↾_s dom 𝑁) GrpHom 𝑈))
6		eqid 2738	. . . . . . . 8 ⊢ (Base‘𝑆) = (Base‘𝑆)
7	1, 6	sygbasnfpfi 19035	. . . . . . 7 ⊢ ((𝐷 ∈ Fin ∧ 𝑥 ∈ (Base‘𝑆)) → dom (𝑥 ∖ I ) ∈ Fin)
8	7	ralrimiva 3107	. . . . . 6 ⊢ (𝐷 ∈ Fin → ∀𝑥 ∈ (Base‘𝑆)dom (𝑥 ∖ I ) ∈ Fin)
9		rabid2 3307	. . . . . 6 ⊢ ((Base‘𝑆) = {𝑥 ∈ (Base‘𝑆) ∣ dom (𝑥 ∖ I ) ∈ Fin} ↔ ∀𝑥 ∈ (Base‘𝑆)dom (𝑥 ∖ I ) ∈ Fin)
10	8, 9	sylibr 233	. . . . 5 ⊢ (𝐷 ∈ Fin → (Base‘𝑆) = {𝑥 ∈ (Base‘𝑆) ∣ dom (𝑥 ∖ I ) ∈ Fin})
11		eqid 2738	. . . . . . 7 ⊢ {𝑥 ∈ (Base‘𝑆) ∣ dom (𝑥 ∖ I ) ∈ Fin} = {𝑥 ∈ (Base‘𝑆) ∣ dom (𝑥 ∖ I ) ∈ Fin}
12	1, 6, 11, 2	psgnfn 19024	. . . . . 6 ⊢ 𝑁 Fn {𝑥 ∈ (Base‘𝑆) ∣ dom (𝑥 ∖ I ) ∈ Fin}
13	12	fndmi 6521	. . . . 5 ⊢ dom 𝑁 = {𝑥 ∈ (Base‘𝑆) ∣ dom (𝑥 ∖ I ) ∈ Fin}
14	10, 13	eqtr4di 2797	. . . 4 ⊢ (𝐷 ∈ Fin → (Base‘𝑆) = dom 𝑁)
15		eqimss 3973	. . . 4 ⊢ ((Base‘𝑆) = dom 𝑁 → (Base‘𝑆) ⊆ dom 𝑁)
16	1	fvexi 6770	. . . . 5 ⊢ 𝑆 ∈ V
17	2	fvexi 6770	. . . . . 6 ⊢ 𝑁 ∈ V
18	17	dmex 7732	. . . . 5 ⊢ dom 𝑁 ∈ V
19	3, 6	ressid2 16871	. . . . 5 ⊢ (((Base‘𝑆) ⊆ dom 𝑁 ∧ 𝑆 ∈ V ∧ dom 𝑁 ∈ V) → (𝑆 ↾_s dom 𝑁) = 𝑆)
20	16, 18, 19	mp3an23 1451	. . . 4 ⊢ ((Base‘𝑆) ⊆ dom 𝑁 → (𝑆 ↾_s dom 𝑁) = 𝑆)
21	14, 15, 20	3syl 18	. . 3 ⊢ (𝐷 ∈ Fin → (𝑆 ↾_s dom 𝑁) = 𝑆)
22	21	oveq1d 7270	. 2 ⊢ (𝐷 ∈ Fin → ((𝑆 ↾_s dom 𝑁) GrpHom 𝑈) = (𝑆 GrpHom 𝑈))
23	5, 22	eleqtrd 2841	1 ⊢ (𝐷 ∈ Fin → 𝑁 ∈ (𝑆 GrpHom 𝑈))

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1539 ∈ wcel 2108 ∀wral 3063 {crab 3067 Vcvv 3422 ∖ cdif 3880 ⊆ wss 3883 {cpr 4560 I cid 5479 dom cdm 5580 ‘cfv 6418 (class class class)co 7255 Fincfn 8691 1c1 10803 -cneg 11136 Basecbs 16840 ↾_s cress 16867 GrpHom cghm 18746 SymGrpcsymg 18889 pmSgncpsgn 19012 mulGrpcmgp 19635 ℂ_fldccnfld 20510

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1799 ax-4 1813 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2156 ax-12 2173 ax-ext 2709 ax-rep 5205 ax-sep 5218 ax-nul 5225 ax-pow 5283 ax-pr 5347 ax-un 7566 ax-cnex 10858 ax-resscn 10859 ax-1cn 10860 ax-icn 10861 ax-addcl 10862 ax-addrcl 10863 ax-mulcl 10864 ax-mulrcl 10865 ax-mulcom 10866 ax-addass 10867 ax-mulass 10868 ax-distr 10869 ax-i2m1 10870 ax-1ne0 10871 ax-1rid 10872 ax-rnegex 10873 ax-rrecex 10874 ax-cnre 10875 ax-pre-lttri 10876 ax-pre-lttrn 10877 ax-pre-ltadd 10878 ax-pre-mulgt0 10879 ax-addf 10881 ax-mulf 10882

This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3or 1086 df-3an 1087 df-xor 1504 df-tru 1542 df-fal 1552 df-ex 1784 df-nf 1788 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2817 df-nfc 2888 df-ne 2943 df-nel 3049 df-ral 3068 df-rex 3069 df-reu 3070 df-rmo 3071 df-rab 3072 df-v 3424 df-sbc 3712 df-csb 3829 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-pss 3902 df-nul 4254 df-if 4457 df-pw 4532 df-sn 4559 df-pr 4561 df-tp 4563 df-op 4565 df-ot 4567 df-uni 4837 df-int 4877 df-iun 4923 df-iin 4924 df-br 5071 df-opab 5133 df-mpt 5154 df-tr 5188 df-id 5480 df-eprel 5486 df-po 5494 df-so 5495 df-fr 5535 df-se 5536 df-we 5537 df-xp 5586 df-rel 5587 df-cnv 5588 df-co 5589 df-dm 5590 df-rn 5591 df-res 5592 df-ima 5593 df-pred 6191 df-ord 6254 df-on 6255 df-lim 6256 df-suc 6257 df-iota 6376 df-fun 6420 df-fn 6421 df-f 6422 df-f1 6423 df-fo 6424 df-f1o 6425 df-fv 6426 df-isom 6427 df-riota 7212 df-ov 7258 df-oprab 7259 df-mpo 7260 df-om 7688 df-1st 7804 df-2nd 7805 df-tpos 8013 df-frecs 8068 df-wrecs 8099 df-recs 8173 df-rdg 8212 df-1o 8267 df-2o 8268 df-er 8456 df-map 8575 df-en 8692 df-dom 8693 df-sdom 8694 df-fin 8695 df-card 9628 df-pnf 10942 df-mnf 10943 df-xr 10944 df-ltxr 10945 df-le 10946 df-sub 11137 df-neg 11138 df-div 11563 df-nn 11904 df-2 11966 df-3 11967 df-4 11968 df-5 11969 df-6 11970 df-7 11971 df-8 11972 df-9 11973 df-n0 12164 df-xnn0 12236 df-z 12250 df-dec 12367 df-uz 12512 df-rp 12660 df-fz 13169 df-fzo 13312 df-seq 13650 df-exp 13711 df-hash 13973 df-word 14146 df-lsw 14194 df-concat 14202 df-s1 14229 df-substr 14282 df-pfx 14312 df-splice 14391 df-reverse 14400 df-s2 14489 df-struct 16776 df-sets 16793 df-slot 16811 df-ndx 16823 df-base 16841 df-ress 16868 df-plusg 16901 df-mulr 16902 df-starv 16903 df-tset 16907 df-ple 16908 df-ds 16910 df-unif 16911 df-0g 17069 df-gsum 17070 df-mre 17212 df-mrc 17213 df-acs 17215 df-mgm 18241 df-sgrp 18290 df-mnd 18301 df-mhm 18345 df-submnd 18346 df-efmnd 18423 df-grp 18495 df-minusg 18496 df-subg 18667 df-ghm 18747 df-gim 18790 df-oppg 18865 df-symg 18890 df-pmtr 18965 df-psgn 19014 df-cmn 19303 df-abl 19304 df-mgp 19636 df-ur 19653 df-ring 19700 df-cring 19701 df-oppr 19777 df-dvdsr 19798 df-unit 19799 df-invr 19829 df-dvr 19840 df-drng 19908 df-cnfld 20511

This theorem is referenced by: psgninv 20699 psgnco 20700 zrhpsgnmhm 20701 zrhpsgninv 20702 psgnevpmb 20704 psgnodpm 20705 zrhpsgnevpm 20708 zrhpsgnodpm 20709 evpmodpmf1o 20713 mdetralt 21665 psgnid 31266 evpmsubg 31316 altgnsg 31318

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