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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 2733 | . . . . . 6 ⊢ (Base‘𝐺) = (Base‘𝐺) | |
4 | 1, 2, 3 | symgtrf 19321 | . . . . 5 ⊢ 𝑇 ⊆ (Base‘𝐺) |
5 | 4 | sseli 3976 | . . . 4 ⊢ (𝑃 ∈ 𝑇 → 𝑃 ∈ (Base‘𝐺)) |
6 | 3 | gsumws1 18706 | . . . 4 ⊢ (𝑃 ∈ (Base‘𝐺) → (𝐺 Σg 〈“𝑃”〉) = 𝑃) |
7 | 5, 6 | syl 17 | . . 3 ⊢ (𝑃 ∈ 𝑇 → (𝐺 Σg 〈“𝑃”〉) = 𝑃) |
8 | 7 | fveq2d 6885 | . 2 ⊢ (𝑃 ∈ 𝑇 → (𝑁‘(𝐺 Σg 〈“𝑃”〉)) = (𝑁‘𝑃)) |
9 | 2, 3 | elbasfv 17137 | . . . . 5 ⊢ (𝑃 ∈ (Base‘𝐺) → 𝐷 ∈ V) |
10 | 5, 9 | syl 17 | . . . 4 ⊢ (𝑃 ∈ 𝑇 → 𝐷 ∈ V) |
11 | s1cl 14539 | . . . 4 ⊢ (𝑃 ∈ 𝑇 → 〈“𝑃”〉 ∈ Word 𝑇) | |
12 | psgnval.n | . . . . 5 ⊢ 𝑁 = (pmSgn‘𝐷) | |
13 | 2, 1, 12 | psgnvalii 19361 | . . . 4 ⊢ ((𝐷 ∈ V ∧ 〈“𝑃”〉 ∈ Word 𝑇) → (𝑁‘(𝐺 Σg 〈“𝑃”〉)) = (-1↑(♯‘〈“𝑃”〉))) |
14 | 10, 11, 13 | syl2anc 585 | . . 3 ⊢ (𝑃 ∈ 𝑇 → (𝑁‘(𝐺 Σg 〈“𝑃”〉)) = (-1↑(♯‘〈“𝑃”〉))) |
15 | s1len 14543 | . . . . 5 ⊢ (♯‘〈“𝑃”〉) = 1 | |
16 | 15 | oveq2i 7407 | . . . 4 ⊢ (-1↑(♯‘〈“𝑃”〉)) = (-1↑1) |
17 | neg1cn 12313 | . . . . 5 ⊢ -1 ∈ ℂ | |
18 | exp1 14020 | . . . . 5 ⊢ (-1 ∈ ℂ → (-1↑1) = -1) | |
19 | 17, 18 | ax-mp 5 | . . . 4 ⊢ (-1↑1) = -1 |
20 | 16, 19 | eqtri 2761 | . . 3 ⊢ (-1↑(♯‘〈“𝑃”〉)) = -1 |
21 | 14, 20 | eqtrdi 2789 | . 2 ⊢ (𝑃 ∈ 𝑇 → (𝑁‘(𝐺 Σg 〈“𝑃”〉)) = -1) |
22 | 8, 21 | eqtr3d 2775 | 1 ⊢ (𝑃 ∈ 𝑇 → (𝑁‘𝑃) = -1) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1542 ∈ wcel 2107 Vcvv 3475 ran crn 5673 ‘cfv 6535 (class class class)co 7396 ℂcc 11095 1c1 11098 -cneg 11432 ↑cexp 14014 ♯chash 14277 Word cword 14451 〈“cs1 14532 Basecbs 17131 Σg cgsu 17373 SymGrpcsymg 19218 pmTrspcpmtr 19293 pmSgncpsgn 19341 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5281 ax-sep 5295 ax-nul 5302 ax-pow 5359 ax-pr 5423 ax-un 7712 ax-cnex 11153 ax-resscn 11154 ax-1cn 11155 ax-icn 11156 ax-addcl 11157 ax-addrcl 11158 ax-mulcl 11159 ax-mulrcl 11160 ax-mulcom 11161 ax-addass 11162 ax-mulass 11163 ax-distr 11164 ax-i2m1 11165 ax-1ne0 11166 ax-1rid 11167 ax-rnegex 11168 ax-rrecex 11169 ax-cnre 11170 ax-pre-lttri 11171 ax-pre-lttrn 11172 ax-pre-ltadd 11173 ax-pre-mulgt0 11174 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-xor 1511 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3776 df-csb 3892 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-pss 3965 df-nul 4321 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-tp 4629 df-op 4631 df-ot 4633 df-uni 4905 df-int 4947 df-iun 4995 df-iin 4996 df-br 5145 df-opab 5207 df-mpt 5228 df-tr 5262 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-se 5628 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6292 df-ord 6359 df-on 6360 df-lim 6361 df-suc 6362 df-iota 6487 df-fun 6537 df-fn 6538 df-f 6539 df-f1 6540 df-fo 6541 df-f1o 6542 df-fv 6543 df-isom 6544 df-riota 7352 df-ov 7399 df-oprab 7400 df-mpo 7401 df-om 7843 df-1st 7962 df-2nd 7963 df-tpos 8198 df-frecs 8253 df-wrecs 8284 df-recs 8358 df-rdg 8397 df-1o 8453 df-2o 8454 df-er 8691 df-map 8810 df-en 8928 df-dom 8929 df-sdom 8930 df-fin 8931 df-card 9921 df-pnf 11237 df-mnf 11238 df-xr 11239 df-ltxr 11240 df-le 11241 df-sub 11433 df-neg 11434 df-div 11859 df-nn 12200 df-2 12262 df-3 12263 df-4 12264 df-5 12265 df-6 12266 df-7 12267 df-8 12268 df-9 12269 df-n0 12460 df-xnn0 12532 df-z 12546 df-uz 12810 df-rp 12962 df-fz 13472 df-fzo 13615 df-seq 13954 df-exp 14015 df-hash 14278 df-word 14452 df-lsw 14500 df-concat 14508 df-s1 14533 df-substr 14578 df-pfx 14608 df-splice 14687 df-reverse 14696 df-s2 14786 df-struct 17067 df-sets 17084 df-slot 17102 df-ndx 17114 df-base 17132 df-ress 17161 df-plusg 17197 df-tset 17203 df-0g 17374 df-gsum 17375 df-mre 17517 df-mrc 17518 df-acs 17520 df-mgm 18548 df-sgrp 18597 df-mnd 18613 df-mhm 18658 df-submnd 18659 df-efmnd 18737 df-grp 18809 df-minusg 18810 df-subg 18988 df-ghm 19075 df-gim 19118 df-oppg 19194 df-symg 19219 df-pmtr 19294 df-psgn 19343 |
This theorem is referenced by: psgnprfval2 19375 pmtrodpm 21123 mdetralt 22079 psgnfzto1st 32235 cyc3evpm 32280 |
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