psgnghm2 - Metamath Proof Explorer

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Theorem psgnghm2 21622

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 2740	. . 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 21621	. 2 ⊢ (𝐷 ∈ Fin → 𝑁 ∈ ((𝑆 ↾_s dom 𝑁) GrpHom 𝑈))
6		eqid 2740	. . . . . . . 8 ⊢ (Base‘𝑆) = (Base‘𝑆)
7	1, 6	sygbasnfpfi 19554	. . . . . . 7 ⊢ ((𝐷 ∈ Fin ∧ 𝑥 ∈ (Base‘𝑆)) → dom (𝑥 ∖ I ) ∈ Fin)
8	7	ralrimiva 3152	. . . . . 6 ⊢ (𝐷 ∈ Fin → ∀𝑥 ∈ (Base‘𝑆)dom (𝑥 ∖ I ) ∈ Fin)
9		rabid2 3478	. . . . . 6 ⊢ ((Base‘𝑆) = {𝑥 ∈ (Base‘𝑆) ∣ dom (𝑥 ∖ I ) ∈ Fin} ↔ ∀𝑥 ∈ (Base‘𝑆)dom (𝑥 ∖ I ) ∈ Fin)
10	8, 9	sylibr 234	. . . . 5 ⊢ (𝐷 ∈ Fin → (Base‘𝑆) = {𝑥 ∈ (Base‘𝑆) ∣ dom (𝑥 ∖ I ) ∈ Fin})
11		eqid 2740	. . . . . . 7 ⊢ {𝑥 ∈ (Base‘𝑆) ∣ dom (𝑥 ∖ I ) ∈ Fin} = {𝑥 ∈ (Base‘𝑆) ∣ dom (𝑥 ∖ I ) ∈ Fin}
12	1, 6, 11, 2	psgnfn 19543	. . . . . 6 ⊢ 𝑁 Fn {𝑥 ∈ (Base‘𝑆) ∣ dom (𝑥 ∖ I ) ∈ Fin}
13	12	fndmi 6683	. . . . 5 ⊢ dom 𝑁 = {𝑥 ∈ (Base‘𝑆) ∣ dom (𝑥 ∖ I ) ∈ Fin}
14	10, 13	eqtr4di 2798	. . . 4 ⊢ (𝐷 ∈ Fin → (Base‘𝑆) = dom 𝑁)
15		eqimss 4067	. . . 4 ⊢ ((Base‘𝑆) = dom 𝑁 → (Base‘𝑆) ⊆ dom 𝑁)
16	1	fvexi 6934	. . . . 5 ⊢ 𝑆 ∈ V
17	2	fvexi 6934	. . . . . 6 ⊢ 𝑁 ∈ V
18	17	dmex 7949	. . . . 5 ⊢ dom 𝑁 ∈ V
19	3, 6	ressid2 17291	. . . . 5 ⊢ (((Base‘𝑆) ⊆ dom 𝑁 ∧ 𝑆 ∈ V ∧ dom 𝑁 ∈ V) → (𝑆 ↾_s dom 𝑁) = 𝑆)
20	16, 18, 19	mp3an23 1453	. . . 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 7463	. 2 ⊢ (𝐷 ∈ Fin → ((𝑆 ↾_s dom 𝑁) GrpHom 𝑈) = (𝑆 GrpHom 𝑈))
23	5, 22	eleqtrd 2846	1 ⊢ (𝐷 ∈ Fin → 𝑁 ∈ (𝑆 GrpHom 𝑈))

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2108 ∀wral 3067 {crab 3443 Vcvv 3488 ∖ cdif 3973 ⊆ wss 3976 {cpr 4650 I cid 5592 dom cdm 5700 ‘cfv 6573 (class class class)co 7448 Fincfn 9003 1c1 11185 -cneg 11521 Basecbs 17258 ↾_s cress 17287 GrpHom cghm 19252 SymGrpcsymg 19410 pmSgncpsgn 19531 mulGrpcmgp 20161 ℂ_fldccnfld 21387

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2158 ax-12 2178 ax-ext 2711 ax-rep 5303 ax-sep 5317 ax-nul 5324 ax-pow 5383 ax-pr 5447 ax-un 7770 ax-cnex 11240 ax-resscn 11241 ax-1cn 11242 ax-icn 11243 ax-addcl 11244 ax-addrcl 11245 ax-mulcl 11246 ax-mulrcl 11247 ax-mulcom 11248 ax-addass 11249 ax-mulass 11250 ax-distr 11251 ax-i2m1 11252 ax-1ne0 11253 ax-1rid 11254 ax-rnegex 11255 ax-rrecex 11256 ax-cnre 11257 ax-pre-lttri 11258 ax-pre-lttrn 11259 ax-pre-ltadd 11260 ax-pre-mulgt0 11261 ax-addf 11263 ax-mulf 11264

This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3or 1088 df-3an 1089 df-xor 1509 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2543 df-eu 2572 df-clab 2718 df-cleq 2732 df-clel 2819 df-nfc 2895 df-ne 2947 df-nel 3053 df-ral 3068 df-rex 3077 df-rmo 3388 df-reu 3389 df-rab 3444 df-v 3490 df-sbc 3805 df-csb 3922 df-dif 3979 df-un 3981 df-in 3983 df-ss 3993 df-pss 3996 df-nul 4353 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-tp 4653 df-op 4655 df-ot 4657 df-uni 4932 df-int 4971 df-iun 5017 df-iin 5018 df-br 5167 df-opab 5229 df-mpt 5250 df-tr 5284 df-id 5593 df-eprel 5599 df-po 5607 df-so 5608 df-fr 5652 df-se 5653 df-we 5654 df-xp 5706 df-rel 5707 df-cnv 5708 df-co 5709 df-dm 5710 df-rn 5711 df-res 5712 df-ima 5713 df-pred 6332 df-ord 6398 df-on 6399 df-lim 6400 df-suc 6401 df-iota 6525 df-fun 6575 df-fn 6576 df-f 6577 df-f1 6578 df-fo 6579 df-f1o 6580 df-fv 6581 df-isom 6582 df-riota 7404 df-ov 7451 df-oprab 7452 df-mpo 7453 df-om 7904 df-1st 8030 df-2nd 8031 df-tpos 8267 df-frecs 8322 df-wrecs 8353 df-recs 8427 df-rdg 8466 df-1o 8522 df-2o 8523 df-er 8763 df-map 8886 df-en 9004 df-dom 9005 df-sdom 9006 df-fin 9007 df-card 10008 df-pnf 11326 df-mnf 11327 df-xr 11328 df-ltxr 11329 df-le 11330 df-sub 11522 df-neg 11523 df-div 11948 df-nn 12294 df-2 12356 df-3 12357 df-4 12358 df-5 12359 df-6 12360 df-7 12361 df-8 12362 df-9 12363 df-n0 12554 df-xnn0 12626 df-z 12640 df-dec 12759 df-uz 12904 df-rp 13058 df-fz 13568 df-fzo 13712 df-seq 14053 df-exp 14113 df-hash 14380 df-word 14563 df-lsw 14611 df-concat 14619 df-s1 14644 df-substr 14689 df-pfx 14719 df-splice 14798 df-reverse 14807 df-s2 14897 df-struct 17194 df-sets 17211 df-slot 17229 df-ndx 17241 df-base 17259 df-ress 17288 df-plusg 17324 df-mulr 17325 df-starv 17326 df-tset 17330 df-ple 17331 df-ds 17333 df-unif 17334 df-0g 17501 df-gsum 17502 df-mre 17644 df-mrc 17645 df-acs 17647 df-mgm 18678 df-sgrp 18757 df-mnd 18773 df-mhm 18818 df-submnd 18819 df-efmnd 18904 df-grp 18976 df-minusg 18977 df-subg 19163 df-ghm 19253 df-gim 19299 df-oppg 19386 df-symg 19411 df-pmtr 19484 df-psgn 19533 df-cmn 19824 df-abl 19825 df-mgp 20162 df-rng 20180 df-ur 20209 df-ring 20262 df-cring 20263 df-oppr 20360 df-dvdsr 20383 df-unit 20384 df-invr 20414 df-dvr 20427 df-drng 20753 df-cnfld 21388

This theorem is referenced by: psgninv 21623 psgnco 21624 zrhpsgnmhm 21625 zrhpsgninv 21626 psgnevpmb 21628 psgnodpm 21629 zrhpsgnevpm 21632 zrhpsgnodpm 21633 evpmodpmf1o 21637 mdetralt 22635 psgnid 33090 evpmsubg 33140 altgnsg 33142

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