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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 3501 | . . . 4 ⊢ (𝐷 ∈ Fin → 𝐷 ∈ V) | |
| 2 | fveq2 6906 | . . . . . . . 8 ⊢ (𝑑 = 𝐷 → (pmSgn‘𝑑) = (pmSgn‘𝐷)) | |
| 3 | psgnevpmb.n | . . . . . . . 8 ⊢ 𝑁 = (pmSgn‘𝐷) | |
| 4 | 2, 3 | eqtr4di 2795 | . . . . . . 7 ⊢ (𝑑 = 𝐷 → (pmSgn‘𝑑) = 𝑁) |
| 5 | 4 | cnveqd 5886 | . . . . . 6 ⊢ (𝑑 = 𝐷 → ◡(pmSgn‘𝑑) = ◡𝑁) |
| 6 | 5 | imaeq1d 6077 | . . . . 5 ⊢ (𝑑 = 𝐷 → (◡(pmSgn‘𝑑) “ {1}) = (◡𝑁 “ {1})) |
| 7 | df-evpm 19510 | . . . . 5 ⊢ pmEven = (𝑑 ∈ V ↦ (◡(pmSgn‘𝑑) “ {1})) | |
| 8 | 3 | fvexi 6920 | . . . . . . 7 ⊢ 𝑁 ∈ V |
| 9 | 8 | cnvex 7947 | . . . . . 6 ⊢ ◡𝑁 ∈ V |
| 10 | 9 | imaex 7936 | . . . . 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 2827 | . 2 ⊢ (𝐷 ∈ Fin → (𝐹 ∈ (pmEven‘𝐷) ↔ 𝐹 ∈ (◡𝑁 “ {1}))) |
| 14 | evpmss.s | . . . 4 ⊢ 𝑆 = (SymGrp‘𝐷) | |
| 15 | eqid 2737 | . . . 4 ⊢ ((mulGrp‘ℂfld) ↾s {1, -1}) = ((mulGrp‘ℂfld) ↾s {1, -1}) | |
| 16 | 14, 3, 15 | psgnghm2 21599 | . . 3 ⊢ (𝐷 ∈ Fin → 𝑁 ∈ (𝑆 GrpHom ((mulGrp‘ℂfld) ↾s {1, -1}))) |
| 17 | evpmss.p | . . . 4 ⊢ 𝑃 = (Base‘𝑆) | |
| 18 | eqid 2737 | . . . 4 ⊢ (Base‘((mulGrp‘ℂfld) ↾s {1, -1})) = (Base‘((mulGrp‘ℂfld) ↾s {1, -1})) | |
| 19 | 17, 18 | ghmf 19238 | . . 3 ⊢ (𝑁 ∈ (𝑆 GrpHom ((mulGrp‘ℂfld) ↾s {1, -1})) → 𝑁:𝑃⟶(Base‘((mulGrp‘ℂfld) ↾s {1, -1}))) |
| 20 | ffn 6736 | . . 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 1540 ∈ wcel 2108 Vcvv 3480 {csn 4626 {cpr 4628 ◡ccnv 5684 “ cima 5688 Fn wfn 6556 ⟶wf 6557 ‘cfv 6561 (class class class)co 7431 Fincfn 8985 1c1 11156 -cneg 11493 Basecbs 17247 ↾s cress 17274 GrpHom cghm 19230 SymGrpcsymg 19386 pmSgncpsgn 19507 pmEvencevpm 19508 mulGrpcmgp 20137 ℂfldccnfld 21364 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 ax-addf 11234 ax-mulf 11235 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-xor 1512 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-tp 4631 df-op 4633 df-ot 4635 df-uni 4908 df-int 4947 df-iun 4993 df-iin 4994 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-se 5638 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-isom 6570 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8014 df-2nd 8015 df-tpos 8251 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-2o 8507 df-er 8745 df-map 8868 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-card 9979 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-div 11921 df-nn 12267 df-2 12329 df-3 12330 df-4 12331 df-5 12332 df-6 12333 df-7 12334 df-8 12335 df-9 12336 df-n0 12527 df-xnn0 12600 df-z 12614 df-dec 12734 df-uz 12879 df-rp 13035 df-fz 13548 df-fzo 13695 df-seq 14043 df-exp 14103 df-hash 14370 df-word 14553 df-lsw 14601 df-concat 14609 df-s1 14634 df-substr 14679 df-pfx 14709 df-splice 14788 df-reverse 14797 df-s2 14887 df-struct 17184 df-sets 17201 df-slot 17219 df-ndx 17231 df-base 17248 df-ress 17275 df-plusg 17310 df-mulr 17311 df-starv 17312 df-tset 17316 df-ple 17317 df-ds 17319 df-unif 17320 df-0g 17486 df-gsum 17487 df-mre 17629 df-mrc 17630 df-acs 17632 df-mgm 18653 df-sgrp 18732 df-mnd 18748 df-mhm 18796 df-submnd 18797 df-efmnd 18882 df-grp 18954 df-minusg 18955 df-subg 19141 df-ghm 19231 df-gim 19277 df-oppg 19364 df-symg 19387 df-pmtr 19460 df-psgn 19509 df-evpm 19510 df-cmn 19800 df-abl 19801 df-mgp 20138 df-rng 20150 df-ur 20179 df-ring 20232 df-cring 20233 df-oppr 20334 df-dvdsr 20357 df-unit 20358 df-invr 20388 df-dvr 20401 df-drng 20731 df-cnfld 21365 |
| This theorem is referenced by: psgnodpm 21606 psgnevpm 21607 evpmodpmf1o 21614 mdet0pr 22598 odpmco 33106 evpmid 33168 cyc3evpm 33170 |
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