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Mirrors > Home > MPE Home > Th. List > psgnpmtr | Structured version Visualization version GIF version |
Description: All transpositions are odd. (Contributed by Stefan O'Rear, 29-Aug-2015.) |
Ref | Expression |
---|---|
psgnval.g | ⊢ 𝐺 = (SymGrp‘𝐷) |
psgnval.t | ⊢ 𝑇 = ran (pmTrsp‘𝐷) |
psgnval.n | ⊢ 𝑁 = (pmSgn‘𝐷) |
Ref | Expression |
---|---|
psgnpmtr | ⊢ (𝑃 ∈ 𝑇 → (𝑁‘𝑃) = -1) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | psgnval.t | . . . . . 6 ⊢ 𝑇 = ran (pmTrsp‘𝐷) | |
2 | psgnval.g | . . . . . 6 ⊢ 𝐺 = (SymGrp‘𝐷) | |
3 | eqid 2732 | . . . . . 6 ⊢ (Base‘𝐺) = (Base‘𝐺) | |
4 | 1, 2, 3 | symgtrf 19378 | . . . . 5 ⊢ 𝑇 ⊆ (Base‘𝐺) |
5 | 4 | sseli 3978 | . . . 4 ⊢ (𝑃 ∈ 𝑇 → 𝑃 ∈ (Base‘𝐺)) |
6 | 3 | gsumws1 18755 | . . . 4 ⊢ (𝑃 ∈ (Base‘𝐺) → (𝐺 Σg ⟨“𝑃”⟩) = 𝑃) |
7 | 5, 6 | syl 17 | . . 3 ⊢ (𝑃 ∈ 𝑇 → (𝐺 Σg ⟨“𝑃”⟩) = 𝑃) |
8 | 7 | fveq2d 6895 | . 2 ⊢ (𝑃 ∈ 𝑇 → (𝑁‘(𝐺 Σg ⟨“𝑃”⟩)) = (𝑁‘𝑃)) |
9 | 2, 3 | elbasfv 17154 | . . . . 5 ⊢ (𝑃 ∈ (Base‘𝐺) → 𝐷 ∈ V) |
10 | 5, 9 | syl 17 | . . . 4 ⊢ (𝑃 ∈ 𝑇 → 𝐷 ∈ V) |
11 | s1cl 14556 | . . . 4 ⊢ (𝑃 ∈ 𝑇 → ⟨“𝑃”⟩ ∈ Word 𝑇) | |
12 | psgnval.n | . . . . 5 ⊢ 𝑁 = (pmSgn‘𝐷) | |
13 | 2, 1, 12 | psgnvalii 19418 | . . . 4 ⊢ ((𝐷 ∈ V ∧ ⟨“𝑃”⟩ ∈ Word 𝑇) → (𝑁‘(𝐺 Σg ⟨“𝑃”⟩)) = (-1↑(♯‘⟨“𝑃”⟩))) |
14 | 10, 11, 13 | syl2anc 584 | . . 3 ⊢ (𝑃 ∈ 𝑇 → (𝑁‘(𝐺 Σg ⟨“𝑃”⟩)) = (-1↑(♯‘⟨“𝑃”⟩))) |
15 | s1len 14560 | . . . . 5 ⊢ (♯‘⟨“𝑃”⟩) = 1 | |
16 | 15 | oveq2i 7422 | . . . 4 ⊢ (-1↑(♯‘⟨“𝑃”⟩)) = (-1↑1) |
17 | neg1cn 12330 | . . . . 5 ⊢ -1 ∈ ℂ | |
18 | exp1 14037 | . . . . 5 ⊢ (-1 ∈ ℂ → (-1↑1) = -1) | |
19 | 17, 18 | ax-mp 5 | . . . 4 ⊢ (-1↑1) = -1 |
20 | 16, 19 | eqtri 2760 | . . 3 ⊢ (-1↑(♯‘⟨“𝑃”⟩)) = -1 |
21 | 14, 20 | eqtrdi 2788 | . 2 ⊢ (𝑃 ∈ 𝑇 → (𝑁‘(𝐺 Σg ⟨“𝑃”⟩)) = -1) |
22 | 8, 21 | eqtr3d 2774 | 1 ⊢ (𝑃 ∈ 𝑇 → (𝑁‘𝑃) = -1) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1541 ∈ wcel 2106 Vcvv 3474 ran crn 5677 ‘cfv 6543 (class class class)co 7411 ℂcc 11110 1c1 11113 -cneg 11449 ↑cexp 14031 ♯chash 14294 Word cword 14468 ⟨“cs1 14549 Basecbs 17148 Σg cgsu 17390 SymGrpcsymg 19275 pmTrspcpmtr 19350 pmSgncpsgn 19398 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7727 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-xor 1510 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-tp 4633 df-op 4635 df-ot 4637 df-uni 4909 df-int 4951 df-iun 4999 df-iin 5000 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-se 5632 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-isom 6552 df-riota 7367 df-ov 7414 df-oprab 7415 df-mpo 7416 df-om 7858 df-1st 7977 df-2nd 7978 df-tpos 8213 df-frecs 8268 df-wrecs 8299 df-recs 8373 df-rdg 8412 df-1o 8468 df-2o 8469 df-er 8705 df-map 8824 df-en 8942 df-dom 8943 df-sdom 8944 df-fin 8945 df-card 9936 df-pnf 11254 df-mnf 11255 df-xr 11256 df-ltxr 11257 df-le 11258 df-sub 11450 df-neg 11451 df-div 11876 df-nn 12217 df-2 12279 df-3 12280 df-4 12281 df-5 12282 df-6 12283 df-7 12284 df-8 12285 df-9 12286 df-n0 12477 df-xnn0 12549 df-z 12563 df-uz 12827 df-rp 12979 df-fz 13489 df-fzo 13632 df-seq 13971 df-exp 14032 df-hash 14295 df-word 14469 df-lsw 14517 df-concat 14525 df-s1 14550 df-substr 14595 df-pfx 14625 df-splice 14704 df-reverse 14713 df-s2 14803 df-struct 17084 df-sets 17101 df-slot 17119 df-ndx 17131 df-base 17149 df-ress 17178 df-plusg 17214 df-tset 17220 df-0g 17391 df-gsum 17392 df-mre 17534 df-mrc 17535 df-acs 17537 df-mgm 18565 df-sgrp 18644 df-mnd 18660 df-mhm 18705 df-submnd 18706 df-efmnd 18786 df-grp 18858 df-minusg 18859 df-subg 19039 df-ghm 19128 df-gim 19173 df-oppg 19251 df-symg 19276 df-pmtr 19351 df-psgn 19400 |
This theorem is referenced by: psgnprfval2 19432 pmtrodpm 21369 mdetralt 22330 psgnfzto1st 32522 cyc3evpm 32567 |
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