psgnghm2 - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > psgnghm2		Structured version Visualization version GIF version

Theorem psgnghm2 21536

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 2736	. . 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 21535	. 2 ⊢ (𝐷 ∈ Fin → 𝑁 ∈ ((𝑆 ↾_s dom 𝑁) GrpHom 𝑈))
6		eqid 2736	. . . . . . . 8 ⊢ (Base‘𝑆) = (Base‘𝑆)
7	1, 6	sygbasnfpfi 19441	. . . . . . 7 ⊢ ((𝐷 ∈ Fin ∧ 𝑥 ∈ (Base‘𝑆)) → dom (𝑥 ∖ I ) ∈ Fin)
8	7	ralrimiva 3128	. . . . . 6 ⊢ (𝐷 ∈ Fin → ∀𝑥 ∈ (Base‘𝑆)dom (𝑥 ∖ I ) ∈ Fin)
9		rabid2 3432	. . . . . 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 2736	. . . . . . 7 ⊢ {𝑥 ∈ (Base‘𝑆) ∣ dom (𝑥 ∖ I ) ∈ Fin} = {𝑥 ∈ (Base‘𝑆) ∣ dom (𝑥 ∖ I ) ∈ Fin}
12	1, 6, 11, 2	psgnfn 19430	. . . . . 6 ⊢ 𝑁 Fn {𝑥 ∈ (Base‘𝑆) ∣ dom (𝑥 ∖ I ) ∈ Fin}
13	12	fndmi 6596	. . . . 5 ⊢ dom 𝑁 = {𝑥 ∈ (Base‘𝑆) ∣ dom (𝑥 ∖ I ) ∈ Fin}
14	10, 13	eqtr4di 2789	. . . 4 ⊢ (𝐷 ∈ Fin → (Base‘𝑆) = dom 𝑁)
15		eqimss 3992	. . . 4 ⊢ ((Base‘𝑆) = dom 𝑁 → (Base‘𝑆) ⊆ dom 𝑁)
16	1	fvexi 6848	. . . . 5 ⊢ 𝑆 ∈ V
17	2	fvexi 6848	. . . . . 6 ⊢ 𝑁 ∈ V
18	17	dmex 7851	. . . . 5 ⊢ dom 𝑁 ∈ V
19	3, 6	ressid2 17161	. . . . 5 ⊢ (((Base‘𝑆) ⊆ dom 𝑁 ∧ 𝑆 ∈ V ∧ dom 𝑁 ∈ V) → (𝑆 ↾_s dom 𝑁) = 𝑆)
20	16, 18, 19	mp3an23 1455	. . . 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 7373	. 2 ⊢ (𝐷 ∈ Fin → ((𝑆 ↾_s dom 𝑁) GrpHom 𝑈) = (𝑆 GrpHom 𝑈))
23	5, 22	eleqtrd 2838	1 ⊢ (𝐷 ∈ Fin → 𝑁 ∈ (𝑆 GrpHom 𝑈))

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1541 ∈ wcel 2113 ∀wral 3051 {crab 3399 Vcvv 3440 ∖ cdif 3898 ⊆ wss 3901 {cpr 4582 I cid 5518 dom cdm 5624 ‘cfv 6492 (class class class)co 7358 Fincfn 8883 1c1 11027 -cneg 11365 Basecbs 17136 ↾_s cress 17157 GrpHom cghm 19141 SymGrpcsymg 19298 pmSgncpsgn 19418 mulGrpcmgp 20075 ℂ_fldccnfld 21309

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2115 ax-9 2123 ax-10 2146 ax-11 2162 ax-12 2184 ax-ext 2708 ax-rep 5224 ax-sep 5241 ax-nul 5251 ax-pow 5310 ax-pr 5377 ax-un 7680 ax-cnex 11082 ax-resscn 11083 ax-1cn 11084 ax-icn 11085 ax-addcl 11086 ax-addrcl 11087 ax-mulcl 11088 ax-mulrcl 11089 ax-mulcom 11090 ax-addass 11091 ax-mulass 11092 ax-distr 11093 ax-i2m1 11094 ax-1ne0 11095 ax-1rid 11096 ax-rnegex 11097 ax-rrecex 11098 ax-cnre 11099 ax-pre-lttri 11100 ax-pre-lttrn 11101 ax-pre-ltadd 11102 ax-pre-mulgt0 11103 ax-addf 11105 ax-mulf 11106

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-xor 1513 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2539 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2811 df-nfc 2885 df-ne 2933 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3350 df-reu 3351 df-rab 3400 df-v 3442 df-sbc 3741 df-csb 3850 df-dif 3904 df-un 3906 df-in 3908 df-ss 3918 df-pss 3921 df-nul 4286 df-if 4480 df-pw 4556 df-sn 4581 df-pr 4583 df-tp 4585 df-op 4587 df-ot 4589 df-uni 4864 df-int 4903 df-iun 4948 df-iin 4949 df-br 5099 df-opab 5161 df-mpt 5180 df-tr 5206 df-id 5519 df-eprel 5524 df-po 5532 df-so 5533 df-fr 5577 df-se 5578 df-we 5579 df-xp 5630 df-rel 5631 df-cnv 5632 df-co 5633 df-dm 5634 df-rn 5635 df-res 5636 df-ima 5637 df-pred 6259 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6494 df-fn 6495 df-f 6496 df-f1 6497 df-fo 6498 df-f1o 6499 df-fv 6500 df-isom 6501 df-riota 7315 df-ov 7361 df-oprab 7362 df-mpo 7363 df-om 7809 df-1st 7933 df-2nd 7934 df-tpos 8168 df-frecs 8223 df-wrecs 8254 df-recs 8303 df-rdg 8341 df-1o 8397 df-2o 8398 df-er 8635 df-map 8765 df-en 8884 df-dom 8885 df-sdom 8886 df-fin 8887 df-card 9851 df-pnf 11168 df-mnf 11169 df-xr 11170 df-ltxr 11171 df-le 11172 df-sub 11366 df-neg 11367 df-div 11795 df-nn 12146 df-2 12208 df-3 12209 df-4 12210 df-5 12211 df-6 12212 df-7 12213 df-8 12214 df-9 12215 df-n0 12402 df-xnn0 12475 df-z 12489 df-dec 12608 df-uz 12752 df-rp 12906 df-fz 13424 df-fzo 13571 df-seq 13925 df-exp 13985 df-hash 14254 df-word 14437 df-lsw 14486 df-concat 14494 df-s1 14520 df-substr 14565 df-pfx 14595 df-splice 14673 df-reverse 14682 df-s2 14771 df-struct 17074 df-sets 17091 df-slot 17109 df-ndx 17121 df-base 17137 df-ress 17158 df-plusg 17190 df-mulr 17191 df-starv 17192 df-tset 17196 df-ple 17197 df-ds 17199 df-unif 17200 df-0g 17361 df-gsum 17362 df-mre 17505 df-mrc 17506 df-acs 17508 df-mgm 18565 df-sgrp 18644 df-mnd 18660 df-mhm 18708 df-submnd 18709 df-efmnd 18794 df-grp 18866 df-minusg 18867 df-subg 19053 df-ghm 19142 df-gim 19188 df-oppg 19275 df-symg 19299 df-pmtr 19371 df-psgn 19420 df-cmn 19711 df-abl 19712 df-mgp 20076 df-rng 20088 df-ur 20117 df-ring 20170 df-cring 20171 df-oppr 20273 df-dvdsr 20293 df-unit 20294 df-invr 20324 df-dvr 20337 df-drng 20664 df-cnfld 21310

This theorem is referenced by: psgninv 21537 psgnco 21538 zrhpsgnmhm 21539 zrhpsgninv 21540 psgnevpmb 21542 psgnodpm 21543 zrhpsgnevpm 21546 zrhpsgnodpm 21547 evpmodpmf1o 21551 mdetralt 22552 psgnid 33179 evpmsubg 33229 altgnsg 33231

W3C validator