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Mirrors > Home > MPE Home > Th. List > psgnevpmb | Structured version Visualization version GIF version |
Description: A class is an even permutation if it is a permutation with sign 1. (Contributed by SO, 9-Jul-2018.) |
Ref | Expression |
---|---|
evpmss.s | ⊢ 𝑆 = (SymGrp‘𝐷) |
evpmss.p | ⊢ 𝑃 = (Base‘𝑆) |
psgnevpmb.n | ⊢ 𝑁 = (pmSgn‘𝐷) |
Ref | Expression |
---|---|
psgnevpmb | ⊢ (𝐷 ∈ Fin → (𝐹 ∈ (pmEven‘𝐷) ↔ (𝐹 ∈ 𝑃 ∧ (𝑁‘𝐹) = 1))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elex 3499 | . . . 4 ⊢ (𝐷 ∈ Fin → 𝐷 ∈ V) | |
2 | fveq2 6907 | . . . . . . . 8 ⊢ (𝑑 = 𝐷 → (pmSgn‘𝑑) = (pmSgn‘𝐷)) | |
3 | psgnevpmb.n | . . . . . . . 8 ⊢ 𝑁 = (pmSgn‘𝐷) | |
4 | 2, 3 | eqtr4di 2793 | . . . . . . 7 ⊢ (𝑑 = 𝐷 → (pmSgn‘𝑑) = 𝑁) |
5 | 4 | cnveqd 5889 | . . . . . 6 ⊢ (𝑑 = 𝐷 → ◡(pmSgn‘𝑑) = ◡𝑁) |
6 | 5 | imaeq1d 6079 | . . . . 5 ⊢ (𝑑 = 𝐷 → (◡(pmSgn‘𝑑) “ {1}) = (◡𝑁 “ {1})) |
7 | df-evpm 19525 | . . . . 5 ⊢ pmEven = (𝑑 ∈ V ↦ (◡(pmSgn‘𝑑) “ {1})) | |
8 | 3 | fvexi 6921 | . . . . . . 7 ⊢ 𝑁 ∈ V |
9 | 8 | cnvex 7948 | . . . . . 6 ⊢ ◡𝑁 ∈ V |
10 | 9 | imaex 7937 | . . . . 5 ⊢ (◡𝑁 “ {1}) ∈ V |
11 | 6, 7, 10 | fvmpt 7016 | . . . 4 ⊢ (𝐷 ∈ V → (pmEven‘𝐷) = (◡𝑁 “ {1})) |
12 | 1, 11 | syl 17 | . . 3 ⊢ (𝐷 ∈ Fin → (pmEven‘𝐷) = (◡𝑁 “ {1})) |
13 | 12 | eleq2d 2825 | . 2 ⊢ (𝐷 ∈ Fin → (𝐹 ∈ (pmEven‘𝐷) ↔ 𝐹 ∈ (◡𝑁 “ {1}))) |
14 | evpmss.s | . . . 4 ⊢ 𝑆 = (SymGrp‘𝐷) | |
15 | eqid 2735 | . . . 4 ⊢ ((mulGrp‘ℂfld) ↾s {1, -1}) = ((mulGrp‘ℂfld) ↾s {1, -1}) | |
16 | 14, 3, 15 | psgnghm2 21617 | . . 3 ⊢ (𝐷 ∈ Fin → 𝑁 ∈ (𝑆 GrpHom ((mulGrp‘ℂfld) ↾s {1, -1}))) |
17 | evpmss.p | . . . 4 ⊢ 𝑃 = (Base‘𝑆) | |
18 | eqid 2735 | . . . 4 ⊢ (Base‘((mulGrp‘ℂfld) ↾s {1, -1})) = (Base‘((mulGrp‘ℂfld) ↾s {1, -1})) | |
19 | 17, 18 | ghmf 19251 | . . 3 ⊢ (𝑁 ∈ (𝑆 GrpHom ((mulGrp‘ℂfld) ↾s {1, -1})) → 𝑁:𝑃⟶(Base‘((mulGrp‘ℂfld) ↾s {1, -1}))) |
20 | ffn 6737 | . . 3 ⊢ (𝑁:𝑃⟶(Base‘((mulGrp‘ℂfld) ↾s {1, -1})) → 𝑁 Fn 𝑃) | |
21 | fniniseg 7080 | . . 3 ⊢ (𝑁 Fn 𝑃 → (𝐹 ∈ (◡𝑁 “ {1}) ↔ (𝐹 ∈ 𝑃 ∧ (𝑁‘𝐹) = 1))) | |
22 | 16, 19, 20, 21 | 4syl 19 | . 2 ⊢ (𝐷 ∈ Fin → (𝐹 ∈ (◡𝑁 “ {1}) ↔ (𝐹 ∈ 𝑃 ∧ (𝑁‘𝐹) = 1))) |
23 | 13, 22 | bitrd 279 | 1 ⊢ (𝐷 ∈ Fin → (𝐹 ∈ (pmEven‘𝐷) ↔ (𝐹 ∈ 𝑃 ∧ (𝑁‘𝐹) = 1))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1537 ∈ wcel 2106 Vcvv 3478 {csn 4631 {cpr 4633 ◡ccnv 5688 “ cima 5692 Fn wfn 6558 ⟶wf 6559 ‘cfv 6563 (class class class)co 7431 Fincfn 8984 1c1 11154 -cneg 11491 Basecbs 17245 ↾s cress 17274 GrpHom cghm 19243 SymGrpcsymg 19401 pmSgncpsgn 19522 pmEvencevpm 19523 mulGrpcmgp 20152 ℂfldccnfld 21382 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 ax-addf 11232 ax-mulf 11233 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-xor 1509 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-tp 4636 df-op 4638 df-ot 4640 df-uni 4913 df-int 4952 df-iun 4998 df-iin 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-se 5642 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-isom 6572 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8013 df-2nd 8014 df-tpos 8250 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-1o 8505 df-2o 8506 df-er 8744 df-map 8867 df-en 8985 df-dom 8986 df-sdom 8987 df-fin 8988 df-card 9977 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-div 11919 df-nn 12265 df-2 12327 df-3 12328 df-4 12329 df-5 12330 df-6 12331 df-7 12332 df-8 12333 df-9 12334 df-n0 12525 df-xnn0 12598 df-z 12612 df-dec 12732 df-uz 12877 df-rp 13033 df-fz 13545 df-fzo 13692 df-seq 14040 df-exp 14100 df-hash 14367 df-word 14550 df-lsw 14598 df-concat 14606 df-s1 14631 df-substr 14676 df-pfx 14706 df-splice 14785 df-reverse 14794 df-s2 14884 df-struct 17181 df-sets 17198 df-slot 17216 df-ndx 17228 df-base 17246 df-ress 17275 df-plusg 17311 df-mulr 17312 df-starv 17313 df-tset 17317 df-ple 17318 df-ds 17320 df-unif 17321 df-0g 17488 df-gsum 17489 df-mre 17631 df-mrc 17632 df-acs 17634 df-mgm 18666 df-sgrp 18745 df-mnd 18761 df-mhm 18809 df-submnd 18810 df-efmnd 18895 df-grp 18967 df-minusg 18968 df-subg 19154 df-ghm 19244 df-gim 19290 df-oppg 19377 df-symg 19402 df-pmtr 19475 df-psgn 19524 df-evpm 19525 df-cmn 19815 df-abl 19816 df-mgp 20153 df-rng 20171 df-ur 20200 df-ring 20253 df-cring 20254 df-oppr 20351 df-dvdsr 20374 df-unit 20375 df-invr 20405 df-dvr 20418 df-drng 20748 df-cnfld 21383 |
This theorem is referenced by: psgnodpm 21624 psgnevpm 21625 evpmodpmf1o 21632 mdet0pr 22614 odpmco 33089 evpmid 33151 cyc3evpm 33153 |
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