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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 2795 | . . 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 20406 | . 2 ⊢ (𝐷 ∈ Fin → 𝑁 ∈ ((𝑆 ↾s dom 𝑁) GrpHom 𝑈)) |
6 | eqid 2795 | . . . . . . . 8 ⊢ (Base‘𝑆) = (Base‘𝑆) | |
7 | 1, 6 | sygbasnfpfi 18371 | . . . . . . 7 ⊢ ((𝐷 ∈ Fin ∧ 𝑥 ∈ (Base‘𝑆)) → dom (𝑥 ∖ I ) ∈ Fin) |
8 | 7 | ralrimiva 3149 | . . . . . 6 ⊢ (𝐷 ∈ Fin → ∀𝑥 ∈ (Base‘𝑆)dom (𝑥 ∖ I ) ∈ Fin) |
9 | rabid2 3340 | . . . . . 6 ⊢ ((Base‘𝑆) = {𝑥 ∈ (Base‘𝑆) ∣ dom (𝑥 ∖ I ) ∈ Fin} ↔ ∀𝑥 ∈ (Base‘𝑆)dom (𝑥 ∖ I ) ∈ Fin) | |
10 | 8, 9 | sylibr 235 | . . . . 5 ⊢ (𝐷 ∈ Fin → (Base‘𝑆) = {𝑥 ∈ (Base‘𝑆) ∣ dom (𝑥 ∖ I ) ∈ Fin}) |
11 | eqid 2795 | . . . . . . 7 ⊢ {𝑥 ∈ (Base‘𝑆) ∣ dom (𝑥 ∖ I ) ∈ Fin} = {𝑥 ∈ (Base‘𝑆) ∣ dom (𝑥 ∖ I ) ∈ Fin} | |
12 | 1, 6, 11, 2 | psgnfn 18360 | . . . . . 6 ⊢ 𝑁 Fn {𝑥 ∈ (Base‘𝑆) ∣ dom (𝑥 ∖ I ) ∈ Fin} |
13 | fndm 6325 | . . . . . 6 ⊢ (𝑁 Fn {𝑥 ∈ (Base‘𝑆) ∣ dom (𝑥 ∖ I ) ∈ Fin} → dom 𝑁 = {𝑥 ∈ (Base‘𝑆) ∣ dom (𝑥 ∖ I ) ∈ Fin}) | |
14 | 12, 13 | ax-mp 5 | . . . . 5 ⊢ dom 𝑁 = {𝑥 ∈ (Base‘𝑆) ∣ dom (𝑥 ∖ I ) ∈ Fin} |
15 | 10, 14 | syl6eqr 2849 | . . . 4 ⊢ (𝐷 ∈ Fin → (Base‘𝑆) = dom 𝑁) |
16 | eqimss 3944 | . . . 4 ⊢ ((Base‘𝑆) = dom 𝑁 → (Base‘𝑆) ⊆ dom 𝑁) | |
17 | 1 | fvexi 6552 | . . . . 5 ⊢ 𝑆 ∈ V |
18 | 2 | fvexi 6552 | . . . . . 6 ⊢ 𝑁 ∈ V |
19 | 18 | dmex 7472 | . . . . 5 ⊢ dom 𝑁 ∈ V |
20 | 3, 6 | ressid2 16381 | . . . . 5 ⊢ (((Base‘𝑆) ⊆ dom 𝑁 ∧ 𝑆 ∈ V ∧ dom 𝑁 ∈ V) → (𝑆 ↾s dom 𝑁) = 𝑆) |
21 | 17, 19, 20 | mp3an23 1445 | . . . 4 ⊢ ((Base‘𝑆) ⊆ dom 𝑁 → (𝑆 ↾s dom 𝑁) = 𝑆) |
22 | 15, 16, 21 | 3syl 18 | . . 3 ⊢ (𝐷 ∈ Fin → (𝑆 ↾s dom 𝑁) = 𝑆) |
23 | 22 | oveq1d 7031 | . 2 ⊢ (𝐷 ∈ Fin → ((𝑆 ↾s dom 𝑁) GrpHom 𝑈) = (𝑆 GrpHom 𝑈)) |
24 | 5, 23 | eleqtrd 2885 | 1 ⊢ (𝐷 ∈ Fin → 𝑁 ∈ (𝑆 GrpHom 𝑈)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1522 ∈ wcel 2081 ∀wral 3105 {crab 3109 Vcvv 3437 ∖ cdif 3856 ⊆ wss 3859 {cpr 4474 I cid 5347 dom cdm 5443 Fn wfn 6220 ‘cfv 6225 (class class class)co 7016 Fincfn 8357 1c1 10384 -cneg 10718 Basecbs 16312 ↾s cress 16313 GrpHom cghm 18096 SymGrpcsymg 18236 pmSgncpsgn 18348 mulGrpcmgp 18929 ℂfldccnfld 20227 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1777 ax-4 1791 ax-5 1888 ax-6 1947 ax-7 1992 ax-8 2083 ax-9 2091 ax-10 2112 ax-11 2126 ax-12 2141 ax-13 2344 ax-ext 2769 ax-rep 5081 ax-sep 5094 ax-nul 5101 ax-pow 5157 ax-pr 5221 ax-un 7319 ax-cnex 10439 ax-resscn 10440 ax-1cn 10441 ax-icn 10442 ax-addcl 10443 ax-addrcl 10444 ax-mulcl 10445 ax-mulrcl 10446 ax-mulcom 10447 ax-addass 10448 ax-mulass 10449 ax-distr 10450 ax-i2m1 10451 ax-1ne0 10452 ax-1rid 10453 ax-rnegex 10454 ax-rrecex 10455 ax-cnre 10456 ax-pre-lttri 10457 ax-pre-lttrn 10458 ax-pre-ltadd 10459 ax-pre-mulgt0 10460 ax-addf 10462 ax-mulf 10463 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3or 1081 df-3an 1082 df-xor 1497 df-tru 1525 df-ex 1762 df-nf 1766 df-sb 2043 df-mo 2576 df-eu 2612 df-clab 2776 df-cleq 2788 df-clel 2863 df-nfc 2935 df-ne 2985 df-nel 3091 df-ral 3110 df-rex 3111 df-reu 3112 df-rmo 3113 df-rab 3114 df-v 3439 df-sbc 3707 df-csb 3812 df-dif 3862 df-un 3864 df-in 3866 df-ss 3874 df-pss 3876 df-nul 4212 df-if 4382 df-pw 4455 df-sn 4473 df-pr 4475 df-tp 4477 df-op 4479 df-ot 4481 df-uni 4746 df-int 4783 df-iun 4827 df-iin 4828 df-br 4963 df-opab 5025 df-mpt 5042 df-tr 5064 df-id 5348 df-eprel 5353 df-po 5362 df-so 5363 df-fr 5402 df-se 5403 df-we 5404 df-xp 5449 df-rel 5450 df-cnv 5451 df-co 5452 df-dm 5453 df-rn 5454 df-res 5455 df-ima 5456 df-pred 6023 df-ord 6069 df-on 6070 df-lim 6071 df-suc 6072 df-iota 6189 df-fun 6227 df-fn 6228 df-f 6229 df-f1 6230 df-fo 6231 df-f1o 6232 df-fv 6233 df-isom 6234 df-riota 6977 df-ov 7019 df-oprab 7020 df-mpo 7021 df-om 7437 df-1st 7545 df-2nd 7546 df-tpos 7743 df-wrecs 7798 df-recs 7860 df-rdg 7898 df-1o 7953 df-2o 7954 df-oadd 7957 df-er 8139 df-map 8258 df-en 8358 df-dom 8359 df-sdom 8360 df-fin 8361 df-card 9214 df-pnf 10523 df-mnf 10524 df-xr 10525 df-ltxr 10526 df-le 10527 df-sub 10719 df-neg 10720 df-div 11146 df-nn 11487 df-2 11548 df-3 11549 df-4 11550 df-5 11551 df-6 11552 df-7 11553 df-8 11554 df-9 11555 df-n0 11746 df-xnn0 11816 df-z 11830 df-dec 11948 df-uz 12094 df-rp 12240 df-fz 12743 df-fzo 12884 df-seq 13220 df-exp 13280 df-hash 13541 df-word 13708 df-lsw 13761 df-concat 13769 df-s1 13794 df-substr 13839 df-pfx 13869 df-splice 13948 df-reverse 13957 df-s2 14046 df-struct 16314 df-ndx 16315 df-slot 16316 df-base 16318 df-sets 16319 df-ress 16320 df-plusg 16407 df-mulr 16408 df-starv 16409 df-tset 16413 df-ple 16414 df-ds 16416 df-unif 16417 df-0g 16544 df-gsum 16545 df-mre 16686 df-mrc 16687 df-acs 16689 df-mgm 17681 df-sgrp 17723 df-mnd 17734 df-mhm 17774 df-submnd 17775 df-grp 17864 df-minusg 17865 df-subg 18030 df-ghm 18097 df-gim 18140 df-oppg 18215 df-symg 18237 df-pmtr 18301 df-psgn 18350 df-cmn 18635 df-abl 18636 df-mgp 18930 df-ur 18942 df-ring 18989 df-cring 18990 df-oppr 19063 df-dvdsr 19081 df-unit 19082 df-invr 19112 df-dvr 19123 df-drng 19194 df-cnfld 20228 |
This theorem is referenced by: psgninv 20408 psgnco 20409 zrhpsgnmhm 20410 zrhpsgninv 20411 psgnevpmb 20413 psgnodpm 20414 zrhpsgnevpm 20417 zrhpsgnodpm 20418 evpmodpmf1o 20422 mdetralt 20901 evpmsubg 30426 psgnid 30427 altgnsg 30429 |
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