psgnghm2 - Metamath Proof Explorer

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Theorem psgnghm2 20786

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 20785	. 2 ⊢ (𝐷 ∈ Fin → 𝑁 ∈ ((𝑆 ↾_s dom 𝑁) GrpHom 𝑈))
6		eqid 2738	. . . . . . . 8 ⊢ (Base‘𝑆) = (Base‘𝑆)
7	1, 6	sygbasnfpfi 19120	. . . . . . 7 ⊢ ((𝐷 ∈ Fin ∧ 𝑥 ∈ (Base‘𝑆)) → dom (𝑥 ∖ I ) ∈ Fin)
8	7	ralrimiva 3103	. . . . . 6 ⊢ (𝐷 ∈ Fin → ∀𝑥 ∈ (Base‘𝑆)dom (𝑥 ∖ I ) ∈ Fin)
9		rabid2 3314	. . . . . 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 19109	. . . . . 6 ⊢ 𝑁 Fn {𝑥 ∈ (Base‘𝑆) ∣ dom (𝑥 ∖ I ) ∈ Fin}
13	12	fndmi 6537	. . . . 5 ⊢ dom 𝑁 = {𝑥 ∈ (Base‘𝑆) ∣ dom (𝑥 ∖ I ) ∈ Fin}
14	10, 13	eqtr4di 2796	. . . 4 ⊢ (𝐷 ∈ Fin → (Base‘𝑆) = dom 𝑁)
15		eqimss 3977	. . . 4 ⊢ ((Base‘𝑆) = dom 𝑁 → (Base‘𝑆) ⊆ dom 𝑁)
16	1	fvexi 6788	. . . . 5 ⊢ 𝑆 ∈ V
17	2	fvexi 6788	. . . . . 6 ⊢ 𝑁 ∈ V
18	17	dmex 7758	. . . . 5 ⊢ dom 𝑁 ∈ V
19	3, 6	ressid2 16945	. . . . 5 ⊢ (((Base‘𝑆) ⊆ dom 𝑁 ∧ 𝑆 ∈ V ∧ dom 𝑁 ∈ V) → (𝑆 ↾_s dom 𝑁) = 𝑆)
20	16, 18, 19	mp3an23 1452	. . . 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 7290	. 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 2106 ∀wral 3064 {crab 3068 Vcvv 3432 ∖ cdif 3884 ⊆ wss 3887 {cpr 4563 I cid 5488 dom cdm 5589 ‘cfv 6433 (class class class)co 7275 Fincfn 8733 1c1 10872 -cneg 11206 Basecbs 16912 ↾_s cress 16941 GrpHom cghm 18831 SymGrpcsymg 18974 pmSgncpsgn 19097 mulGrpcmgp 19720 ℂ_fldccnfld 20597

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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-rep 5209 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7588 ax-cnex 10927 ax-resscn 10928 ax-1cn 10929 ax-icn 10930 ax-addcl 10931 ax-addrcl 10932 ax-mulcl 10933 ax-mulrcl 10934 ax-mulcom 10935 ax-addass 10936 ax-mulass 10937 ax-distr 10938 ax-i2m1 10939 ax-1ne0 10940 ax-1rid 10941 ax-rnegex 10942 ax-rrecex 10943 ax-cnre 10944 ax-pre-lttri 10945 ax-pre-lttrn 10946 ax-pre-ltadd 10947 ax-pre-mulgt0 10948 ax-addf 10950 ax-mulf 10951

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-xor 1507 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-rmo 3071 df-reu 3072 df-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-pss 3906 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-tp 4566 df-op 4568 df-ot 4570 df-uni 4840 df-int 4880 df-iun 4926 df-iin 4927 df-br 5075 df-opab 5137 df-mpt 5158 df-tr 5192 df-id 5489 df-eprel 5495 df-po 5503 df-so 5504 df-fr 5544 df-se 5545 df-we 5546 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-pred 6202 df-ord 6269 df-on 6270 df-lim 6271 df-suc 6272 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-f1 6438 df-fo 6439 df-f1o 6440 df-fv 6441 df-isom 6442 df-riota 7232 df-ov 7278 df-oprab 7279 df-mpo 7280 df-om 7713 df-1st 7831 df-2nd 7832 df-tpos 8042 df-frecs 8097 df-wrecs 8128 df-recs 8202 df-rdg 8241 df-1o 8297 df-2o 8298 df-er 8498 df-map 8617 df-en 8734 df-dom 8735 df-sdom 8736 df-fin 8737 df-card 9697 df-pnf 11011 df-mnf 11012 df-xr 11013 df-ltxr 11014 df-le 11015 df-sub 11207 df-neg 11208 df-div 11633 df-nn 11974 df-2 12036 df-3 12037 df-4 12038 df-5 12039 df-6 12040 df-7 12041 df-8 12042 df-9 12043 df-n0 12234 df-xnn0 12306 df-z 12320 df-dec 12438 df-uz 12583 df-rp 12731 df-fz 13240 df-fzo 13383 df-seq 13722 df-exp 13783 df-hash 14045 df-word 14218 df-lsw 14266 df-concat 14274 df-s1 14301 df-substr 14354 df-pfx 14384 df-splice 14463 df-reverse 14472 df-s2 14561 df-struct 16848 df-sets 16865 df-slot 16883 df-ndx 16895 df-base 16913 df-ress 16942 df-plusg 16975 df-mulr 16976 df-starv 16977 df-tset 16981 df-ple 16982 df-ds 16984 df-unif 16985 df-0g 17152 df-gsum 17153 df-mre 17295 df-mrc 17296 df-acs 17298 df-mgm 18326 df-sgrp 18375 df-mnd 18386 df-mhm 18430 df-submnd 18431 df-efmnd 18508 df-grp 18580 df-minusg 18581 df-subg 18752 df-ghm 18832 df-gim 18875 df-oppg 18950 df-symg 18975 df-pmtr 19050 df-psgn 19099 df-cmn 19388 df-abl 19389 df-mgp 19721 df-ur 19738 df-ring 19785 df-cring 19786 df-oppr 19862 df-dvdsr 19883 df-unit 19884 df-invr 19914 df-dvr 19925 df-drng 19993 df-cnfld 20598

This theorem is referenced by: psgninv 20787 psgnco 20788 zrhpsgnmhm 20789 zrhpsgninv 20790 psgnevpmb 20792 psgnodpm 20793 zrhpsgnevpm 20796 zrhpsgnodpm 20797 evpmodpmf1o 20801 mdetralt 21757 psgnid 31364 evpmsubg 31414 altgnsg 31416

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