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Mirrors > Home > MPE Home > Th. List > psgnodpm | Structured version Visualization version GIF version |
Description: A permutation which is odd (i.e. not even) has 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 |
---|---|
psgnodpm | ⊢ ((𝐷 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝐷))) → (𝑁‘𝐹) = -1) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eldif 3891 | . . 3 ⊢ (𝐹 ∈ (𝑃 ∖ (pmEven‘𝐷)) ↔ (𝐹 ∈ 𝑃 ∧ ¬ 𝐹 ∈ (pmEven‘𝐷))) | |
2 | simpr 488 | . . . . . . . 8 ⊢ ((𝐷 ∈ Fin ∧ 𝐹 ∈ 𝑃) → 𝐹 ∈ 𝑃) | |
3 | 2 | a1d 25 | . . . . . . 7 ⊢ ((𝐷 ∈ Fin ∧ 𝐹 ∈ 𝑃) → ((𝑁‘𝐹) = 1 → 𝐹 ∈ 𝑃)) |
4 | 3 | ancrd 555 | . . . . . 6 ⊢ ((𝐷 ∈ Fin ∧ 𝐹 ∈ 𝑃) → ((𝑁‘𝐹) = 1 → (𝐹 ∈ 𝑃 ∧ (𝑁‘𝐹) = 1))) |
5 | evpmss.s | . . . . . . . 8 ⊢ 𝑆 = (SymGrp‘𝐷) | |
6 | evpmss.p | . . . . . . . 8 ⊢ 𝑃 = (Base‘𝑆) | |
7 | psgnevpmb.n | . . . . . . . 8 ⊢ 𝑁 = (pmSgn‘𝐷) | |
8 | 5, 6, 7 | psgnevpmb 20276 | . . . . . . 7 ⊢ (𝐷 ∈ Fin → (𝐹 ∈ (pmEven‘𝐷) ↔ (𝐹 ∈ 𝑃 ∧ (𝑁‘𝐹) = 1))) |
9 | 8 | adantr 484 | . . . . . 6 ⊢ ((𝐷 ∈ Fin ∧ 𝐹 ∈ 𝑃) → (𝐹 ∈ (pmEven‘𝐷) ↔ (𝐹 ∈ 𝑃 ∧ (𝑁‘𝐹) = 1))) |
10 | 4, 9 | sylibrd 262 | . . . . 5 ⊢ ((𝐷 ∈ Fin ∧ 𝐹 ∈ 𝑃) → ((𝑁‘𝐹) = 1 → 𝐹 ∈ (pmEven‘𝐷))) |
11 | 10 | con3d 155 | . . . 4 ⊢ ((𝐷 ∈ Fin ∧ 𝐹 ∈ 𝑃) → (¬ 𝐹 ∈ (pmEven‘𝐷) → ¬ (𝑁‘𝐹) = 1)) |
12 | 11 | impr 458 | . . 3 ⊢ ((𝐷 ∈ Fin ∧ (𝐹 ∈ 𝑃 ∧ ¬ 𝐹 ∈ (pmEven‘𝐷))) → ¬ (𝑁‘𝐹) = 1) |
13 | 1, 12 | sylan2b 596 | . 2 ⊢ ((𝐷 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝐷))) → ¬ (𝑁‘𝐹) = 1) |
14 | eqid 2798 | . . . . . . 7 ⊢ ((mulGrp‘ℂfld) ↾s {1, -1}) = ((mulGrp‘ℂfld) ↾s {1, -1}) | |
15 | 5, 7, 14 | psgnghm2 20270 | . . . . . 6 ⊢ (𝐷 ∈ Fin → 𝑁 ∈ (𝑆 GrpHom ((mulGrp‘ℂfld) ↾s {1, -1}))) |
16 | 15 | adantr 484 | . . . . 5 ⊢ ((𝐷 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝐷))) → 𝑁 ∈ (𝑆 GrpHom ((mulGrp‘ℂfld) ↾s {1, -1}))) |
17 | 14 | cnmsgnbas 20267 | . . . . . 6 ⊢ {1, -1} = (Base‘((mulGrp‘ℂfld) ↾s {1, -1})) |
18 | 6, 17 | ghmf 18354 | . . . . 5 ⊢ (𝑁 ∈ (𝑆 GrpHom ((mulGrp‘ℂfld) ↾s {1, -1})) → 𝑁:𝑃⟶{1, -1}) |
19 | 16, 18 | syl 17 | . . . 4 ⊢ ((𝐷 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝐷))) → 𝑁:𝑃⟶{1, -1}) |
20 | eldifi 4054 | . . . . 5 ⊢ (𝐹 ∈ (𝑃 ∖ (pmEven‘𝐷)) → 𝐹 ∈ 𝑃) | |
21 | 20 | adantl 485 | . . . 4 ⊢ ((𝐷 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝐷))) → 𝐹 ∈ 𝑃) |
22 | 19, 21 | ffvelrnd 6829 | . . 3 ⊢ ((𝐷 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝐷))) → (𝑁‘𝐹) ∈ {1, -1}) |
23 | fvex 6658 | . . . 4 ⊢ (𝑁‘𝐹) ∈ V | |
24 | 23 | elpr 4548 | . . 3 ⊢ ((𝑁‘𝐹) ∈ {1, -1} ↔ ((𝑁‘𝐹) = 1 ∨ (𝑁‘𝐹) = -1)) |
25 | 22, 24 | sylib 221 | . 2 ⊢ ((𝐷 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝐷))) → ((𝑁‘𝐹) = 1 ∨ (𝑁‘𝐹) = -1)) |
26 | orel1 886 | . 2 ⊢ (¬ (𝑁‘𝐹) = 1 → (((𝑁‘𝐹) = 1 ∨ (𝑁‘𝐹) = -1) → (𝑁‘𝐹) = -1)) | |
27 | 13, 25, 26 | sylc 65 | 1 ⊢ ((𝐷 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝐷))) → (𝑁‘𝐹) = -1) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 399 ∨ wo 844 = wceq 1538 ∈ wcel 2111 ∖ cdif 3878 {cpr 4527 ⟶wf 6320 ‘cfv 6324 (class class class)co 7135 Fincfn 8492 1c1 10527 -cneg 10860 Basecbs 16475 ↾s cress 16476 GrpHom cghm 18347 SymGrpcsymg 18487 pmSgncpsgn 18609 pmEvencevpm 18610 mulGrpcmgp 19232 ℂfldccnfld 20091 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 ax-addf 10605 ax-mulf 10606 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-xor 1503 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-ot 4534 df-uni 4801 df-int 4839 df-iun 4883 df-iin 4884 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-se 5479 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-isom 6333 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-1st 7671 df-2nd 7672 df-tpos 7875 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-1o 8085 df-2o 8086 df-oadd 8089 df-er 8272 df-map 8391 df-en 8493 df-dom 8494 df-sdom 8495 df-fin 8496 df-card 9352 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-div 11287 df-nn 11626 df-2 11688 df-3 11689 df-4 11690 df-5 11691 df-6 11692 df-7 11693 df-8 11694 df-9 11695 df-n0 11886 df-xnn0 11956 df-z 11970 df-dec 12087 df-uz 12232 df-rp 12378 df-fz 12886 df-fzo 13029 df-seq 13365 df-exp 13426 df-hash 13687 df-word 13858 df-lsw 13906 df-concat 13914 df-s1 13941 df-substr 13994 df-pfx 14024 df-splice 14103 df-reverse 14112 df-s2 14201 df-struct 16477 df-ndx 16478 df-slot 16479 df-base 16481 df-sets 16482 df-ress 16483 df-plusg 16570 df-mulr 16571 df-starv 16572 df-tset 16576 df-ple 16577 df-ds 16579 df-unif 16580 df-0g 16707 df-gsum 16708 df-mre 16849 df-mrc 16850 df-acs 16852 df-mgm 17844 df-sgrp 17893 df-mnd 17904 df-mhm 17948 df-submnd 17949 df-efmnd 18026 df-grp 18098 df-minusg 18099 df-subg 18268 df-ghm 18348 df-gim 18391 df-oppg 18466 df-symg 18488 df-pmtr 18562 df-psgn 18611 df-evpm 18612 df-cmn 18900 df-abl 18901 df-mgp 19233 df-ur 19245 df-ring 19292 df-cring 19293 df-oppr 19369 df-dvdsr 19387 df-unit 19388 df-invr 19418 df-dvr 19429 df-drng 19497 df-cnfld 20092 |
This theorem is referenced by: zrhpsgnodpm 20281 evpmodpmf1o 20285 odpmco 30780 |
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