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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 2726 | . . 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 21572 | . 2 ⊢ (𝐷 ∈ Fin → 𝑁 ∈ ((𝑆 ↾s dom 𝑁) GrpHom 𝑈)) |
6 | eqid 2726 | . . . . . . . 8 ⊢ (Base‘𝑆) = (Base‘𝑆) | |
7 | 1, 6 | sygbasnfpfi 19506 | . . . . . . 7 ⊢ ((𝐷 ∈ Fin ∧ 𝑥 ∈ (Base‘𝑆)) → dom (𝑥 ∖ I ) ∈ Fin) |
8 | 7 | ralrimiva 3136 | . . . . . 6 ⊢ (𝐷 ∈ Fin → ∀𝑥 ∈ (Base‘𝑆)dom (𝑥 ∖ I ) ∈ Fin) |
9 | rabid2 3453 | . . . . . 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 2726 | . . . . . . 7 ⊢ {𝑥 ∈ (Base‘𝑆) ∣ dom (𝑥 ∖ I ) ∈ Fin} = {𝑥 ∈ (Base‘𝑆) ∣ dom (𝑥 ∖ I ) ∈ Fin} | |
12 | 1, 6, 11, 2 | psgnfn 19495 | . . . . . 6 ⊢ 𝑁 Fn {𝑥 ∈ (Base‘𝑆) ∣ dom (𝑥 ∖ I ) ∈ Fin} |
13 | 12 | fndmi 6656 | . . . . 5 ⊢ dom 𝑁 = {𝑥 ∈ (Base‘𝑆) ∣ dom (𝑥 ∖ I ) ∈ Fin} |
14 | 10, 13 | eqtr4di 2784 | . . . 4 ⊢ (𝐷 ∈ Fin → (Base‘𝑆) = dom 𝑁) |
15 | eqimss 4037 | . . . 4 ⊢ ((Base‘𝑆) = dom 𝑁 → (Base‘𝑆) ⊆ dom 𝑁) | |
16 | 1 | fvexi 6907 | . . . . 5 ⊢ 𝑆 ∈ V |
17 | 2 | fvexi 6907 | . . . . . 6 ⊢ 𝑁 ∈ V |
18 | 17 | dmex 7914 | . . . . 5 ⊢ dom 𝑁 ∈ V |
19 | 3, 6 | ressid2 17241 | . . . . 5 ⊢ (((Base‘𝑆) ⊆ dom 𝑁 ∧ 𝑆 ∈ V ∧ dom 𝑁 ∈ V) → (𝑆 ↾s dom 𝑁) = 𝑆) |
20 | 16, 18, 19 | mp3an23 1450 | . . . 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 7431 | . 2 ⊢ (𝐷 ∈ Fin → ((𝑆 ↾s dom 𝑁) GrpHom 𝑈) = (𝑆 GrpHom 𝑈)) |
23 | 5, 22 | eleqtrd 2828 | 1 ⊢ (𝐷 ∈ Fin → 𝑁 ∈ (𝑆 GrpHom 𝑈)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1534 ∈ wcel 2099 ∀wral 3051 {crab 3419 Vcvv 3462 ∖ cdif 3943 ⊆ wss 3946 {cpr 4625 I cid 5571 dom cdm 5674 ‘cfv 6546 (class class class)co 7416 Fincfn 8966 1c1 11150 -cneg 11486 Basecbs 17208 ↾s cress 17237 GrpHom cghm 19202 SymGrpcsymg 19360 pmSgncpsgn 19483 mulGrpcmgp 20113 ℂfldccnfld 21339 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-rep 5282 ax-sep 5296 ax-nul 5303 ax-pow 5361 ax-pr 5425 ax-un 7738 ax-cnex 11205 ax-resscn 11206 ax-1cn 11207 ax-icn 11208 ax-addcl 11209 ax-addrcl 11210 ax-mulcl 11211 ax-mulrcl 11212 ax-mulcom 11213 ax-addass 11214 ax-mulass 11215 ax-distr 11216 ax-i2m1 11217 ax-1ne0 11218 ax-1rid 11219 ax-rnegex 11220 ax-rrecex 11221 ax-cnre 11222 ax-pre-lttri 11223 ax-pre-lttrn 11224 ax-pre-ltadd 11225 ax-pre-mulgt0 11226 ax-addf 11228 ax-mulf 11229 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-xor 1506 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3464 df-sbc 3776 df-csb 3892 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-pss 3966 df-nul 4323 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-tp 4628 df-op 4630 df-ot 4632 df-uni 4906 df-int 4947 df-iun 4995 df-iin 4996 df-br 5146 df-opab 5208 df-mpt 5229 df-tr 5263 df-id 5572 df-eprel 5578 df-po 5586 df-so 5587 df-fr 5629 df-se 5630 df-we 5631 df-xp 5680 df-rel 5681 df-cnv 5682 df-co 5683 df-dm 5684 df-rn 5685 df-res 5686 df-ima 5687 df-pred 6304 df-ord 6371 df-on 6372 df-lim 6373 df-suc 6374 df-iota 6498 df-fun 6548 df-fn 6549 df-f 6550 df-f1 6551 df-fo 6552 df-f1o 6553 df-fv 6554 df-isom 6555 df-riota 7372 df-ov 7419 df-oprab 7420 df-mpo 7421 df-om 7869 df-1st 7995 df-2nd 7996 df-tpos 8233 df-frecs 8288 df-wrecs 8319 df-recs 8393 df-rdg 8432 df-1o 8488 df-2o 8489 df-er 8726 df-map 8849 df-en 8967 df-dom 8968 df-sdom 8969 df-fin 8970 df-card 9975 df-pnf 11291 df-mnf 11292 df-xr 11293 df-ltxr 11294 df-le 11295 df-sub 11487 df-neg 11488 df-div 11913 df-nn 12259 df-2 12321 df-3 12322 df-4 12323 df-5 12324 df-6 12325 df-7 12326 df-8 12327 df-9 12328 df-n0 12519 df-xnn0 12591 df-z 12605 df-dec 12724 df-uz 12869 df-rp 13023 df-fz 13533 df-fzo 13676 df-seq 14016 df-exp 14076 df-hash 14343 df-word 14518 df-lsw 14566 df-concat 14574 df-s1 14599 df-substr 14644 df-pfx 14674 df-splice 14753 df-reverse 14762 df-s2 14852 df-struct 17144 df-sets 17161 df-slot 17179 df-ndx 17191 df-base 17209 df-ress 17238 df-plusg 17274 df-mulr 17275 df-starv 17276 df-tset 17280 df-ple 17281 df-ds 17283 df-unif 17284 df-0g 17451 df-gsum 17452 df-mre 17594 df-mrc 17595 df-acs 17597 df-mgm 18628 df-sgrp 18707 df-mnd 18723 df-mhm 18768 df-submnd 18769 df-efmnd 18854 df-grp 18926 df-minusg 18927 df-subg 19113 df-ghm 19203 df-gim 19249 df-oppg 19336 df-symg 19361 df-pmtr 19436 df-psgn 19485 df-cmn 19776 df-abl 19777 df-mgp 20114 df-rng 20132 df-ur 20161 df-ring 20214 df-cring 20215 df-oppr 20312 df-dvdsr 20335 df-unit 20336 df-invr 20366 df-dvr 20379 df-drng 20705 df-cnfld 21340 |
This theorem is referenced by: psgninv 21574 psgnco 21575 zrhpsgnmhm 21576 zrhpsgninv 21577 psgnevpmb 21579 psgnodpm 21580 zrhpsgnevpm 21583 zrhpsgnodpm 21584 evpmodpmf1o 21588 mdetralt 22598 psgnid 32979 evpmsubg 33029 altgnsg 33031 |
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