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Mirrors > Home > MPE Home > Th. List > psgnghm2 | Structured version Visualization version GIF version |
Description: The sign is a homomorphism from the finite symmetric group to the numeric signs. (Contributed by Stefan O'Rear, 28-Aug-2015.) |
Ref | Expression |
---|---|
psgnghm2.s | ⊢ 𝑆 = (SymGrp‘𝐷) |
psgnghm2.n | ⊢ 𝑁 = (pmSgn‘𝐷) |
psgnghm2.u | ⊢ 𝑈 = ((mulGrp‘ℂfld) ↾s {1, -1}) |
Ref | Expression |
---|---|
psgnghm2 | ⊢ (𝐷 ∈ Fin → 𝑁 ∈ (𝑆 GrpHom 𝑈)) |
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‘ℂfld) ↾s {1, -1}) | |
5 | 1, 2, 3, 4 | psgnghm 20496 | . 2 ⊢ (𝐷 ∈ Fin → 𝑁 ∈ ((𝑆 ↾s dom 𝑁) GrpHom 𝑈)) |
6 | eqid 2736 | . . . . . . . 8 ⊢ (Base‘𝑆) = (Base‘𝑆) | |
7 | 1, 6 | sygbasnfpfi 18858 | . . . . . . 7 ⊢ ((𝐷 ∈ Fin ∧ 𝑥 ∈ (Base‘𝑆)) → dom (𝑥 ∖ I ) ∈ Fin) |
8 | 7 | ralrimiva 3095 | . . . . . 6 ⊢ (𝐷 ∈ Fin → ∀𝑥 ∈ (Base‘𝑆)dom (𝑥 ∖ I ) ∈ Fin) |
9 | rabid2 3283 | . . . . . 6 ⊢ ((Base‘𝑆) = {𝑥 ∈ (Base‘𝑆) ∣ dom (𝑥 ∖ I ) ∈ Fin} ↔ ∀𝑥 ∈ (Base‘𝑆)dom (𝑥 ∖ I ) ∈ Fin) | |
10 | 8, 9 | sylibr 237 | . . . . 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 18847 | . . . . . 6 ⊢ 𝑁 Fn {𝑥 ∈ (Base‘𝑆) ∣ dom (𝑥 ∖ I ) ∈ Fin} |
13 | 12 | fndmi 6460 | . . . . 5 ⊢ dom 𝑁 = {𝑥 ∈ (Base‘𝑆) ∣ dom (𝑥 ∖ I ) ∈ Fin} |
14 | 10, 13 | eqtr4di 2789 | . . . 4 ⊢ (𝐷 ∈ Fin → (Base‘𝑆) = dom 𝑁) |
15 | eqimss 3943 | . . . 4 ⊢ ((Base‘𝑆) = dom 𝑁 → (Base‘𝑆) ⊆ dom 𝑁) | |
16 | 1 | fvexi 6709 | . . . . 5 ⊢ 𝑆 ∈ V |
17 | 2 | fvexi 6709 | . . . . . 6 ⊢ 𝑁 ∈ V |
18 | 17 | dmex 7667 | . . . . 5 ⊢ dom 𝑁 ∈ V |
19 | 3, 6 | ressid2 16736 | . . . . 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 7206 | . 2 ⊢ (𝐷 ∈ Fin → ((𝑆 ↾s dom 𝑁) GrpHom 𝑈) = (𝑆 GrpHom 𝑈)) |
23 | 5, 22 | eleqtrd 2833 | 1 ⊢ (𝐷 ∈ Fin → 𝑁 ∈ (𝑆 GrpHom 𝑈)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1543 ∈ wcel 2112 ∀wral 3051 {crab 3055 Vcvv 3398 ∖ cdif 3850 ⊆ wss 3853 {cpr 4529 I cid 5439 dom cdm 5536 ‘cfv 6358 (class class class)co 7191 Fincfn 8604 1c1 10695 -cneg 11028 Basecbs 16666 ↾s cress 16667 GrpHom cghm 18573 SymGrpcsymg 18713 pmSgncpsgn 18835 mulGrpcmgp 19458 ℂfldccnfld 20317 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-rep 5164 ax-sep 5177 ax-nul 5184 ax-pow 5243 ax-pr 5307 ax-un 7501 ax-cnex 10750 ax-resscn 10751 ax-1cn 10752 ax-icn 10753 ax-addcl 10754 ax-addrcl 10755 ax-mulcl 10756 ax-mulrcl 10757 ax-mulcom 10758 ax-addass 10759 ax-mulass 10760 ax-distr 10761 ax-i2m1 10762 ax-1ne0 10763 ax-1rid 10764 ax-rnegex 10765 ax-rrecex 10766 ax-cnre 10767 ax-pre-lttri 10768 ax-pre-lttrn 10769 ax-pre-ltadd 10770 ax-pre-mulgt0 10771 ax-addf 10773 ax-mulf 10774 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-xor 1508 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ne 2933 df-nel 3037 df-ral 3056 df-rex 3057 df-reu 3058 df-rmo 3059 df-rab 3060 df-v 3400 df-sbc 3684 df-csb 3799 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-pss 3872 df-nul 4224 df-if 4426 df-pw 4501 df-sn 4528 df-pr 4530 df-tp 4532 df-op 4534 df-ot 4536 df-uni 4806 df-int 4846 df-iun 4892 df-iin 4893 df-br 5040 df-opab 5102 df-mpt 5121 df-tr 5147 df-id 5440 df-eprel 5445 df-po 5453 df-so 5454 df-fr 5494 df-se 5495 df-we 5496 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 df-res 5548 df-ima 5549 df-pred 6140 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6316 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-isom 6367 df-riota 7148 df-ov 7194 df-oprab 7195 df-mpo 7196 df-om 7623 df-1st 7739 df-2nd 7740 df-tpos 7946 df-wrecs 8025 df-recs 8086 df-rdg 8124 df-1o 8180 df-2o 8181 df-er 8369 df-map 8488 df-en 8605 df-dom 8606 df-sdom 8607 df-fin 8608 df-card 9520 df-pnf 10834 df-mnf 10835 df-xr 10836 df-ltxr 10837 df-le 10838 df-sub 11029 df-neg 11030 df-div 11455 df-nn 11796 df-2 11858 df-3 11859 df-4 11860 df-5 11861 df-6 11862 df-7 11863 df-8 11864 df-9 11865 df-n0 12056 df-xnn0 12128 df-z 12142 df-dec 12259 df-uz 12404 df-rp 12552 df-fz 13061 df-fzo 13204 df-seq 13540 df-exp 13601 df-hash 13862 df-word 14035 df-lsw 14083 df-concat 14091 df-s1 14118 df-substr 14171 df-pfx 14201 df-splice 14280 df-reverse 14289 df-s2 14378 df-struct 16668 df-ndx 16669 df-slot 16670 df-base 16672 df-sets 16673 df-ress 16674 df-plusg 16762 df-mulr 16763 df-starv 16764 df-tset 16768 df-ple 16769 df-ds 16771 df-unif 16772 df-0g 16900 df-gsum 16901 df-mre 17043 df-mrc 17044 df-acs 17046 df-mgm 18068 df-sgrp 18117 df-mnd 18128 df-mhm 18172 df-submnd 18173 df-efmnd 18250 df-grp 18322 df-minusg 18323 df-subg 18494 df-ghm 18574 df-gim 18617 df-oppg 18692 df-symg 18714 df-pmtr 18788 df-psgn 18837 df-cmn 19126 df-abl 19127 df-mgp 19459 df-ur 19471 df-ring 19518 df-cring 19519 df-oppr 19595 df-dvdsr 19613 df-unit 19614 df-invr 19644 df-dvr 19655 df-drng 19723 df-cnfld 20318 |
This theorem is referenced by: psgninv 20498 psgnco 20499 zrhpsgnmhm 20500 zrhpsgninv 20501 psgnevpmb 20503 psgnodpm 20504 zrhpsgnevpm 20507 zrhpsgnodpm 20508 evpmodpmf1o 20512 mdetralt 21459 psgnid 31037 evpmsubg 31087 altgnsg 31089 |
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