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Mirrors > Home > MPE Home > Th. List > psgnfieu | Structured version Visualization version GIF version |
Description: A permutation of a finite set has exactly one parity. (Contributed by AV, 13-Jan-2019.) |
Ref | Expression |
---|---|
psgnfitr.g | ⊢ 𝐺 = (SymGrp‘𝑁) |
psgnfitr.p | ⊢ 𝐵 = (Base‘𝐺) |
psgnfitr.t | ⊢ 𝑇 = ran (pmTrsp‘𝑁) |
Ref | Expression |
---|---|
psgnfieu | ⊢ ((𝑁 ∈ Fin ∧ 𝑄 ∈ 𝐵) → ∃!𝑠∃𝑤 ∈ Word 𝑇(𝑄 = (𝐺 Σg 𝑤) ∧ 𝑠 = (-1↑(♯‘𝑤)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpr 483 | . . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑄 ∈ 𝐵) → 𝑄 ∈ 𝐵) | |
2 | psgnfitr.g | . . . 4 ⊢ 𝐺 = (SymGrp‘𝑁) | |
3 | psgnfitr.p | . . . 4 ⊢ 𝐵 = (Base‘𝐺) | |
4 | 2, 3 | sygbasnfpfi 19481 | . . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑄 ∈ 𝐵) → dom (𝑄 ∖ I ) ∈ Fin) |
5 | eqid 2728 | . . . 4 ⊢ (pmSgn‘𝑁) = (pmSgn‘𝑁) | |
6 | 2, 5, 3 | psgneldm 19472 | . . 3 ⊢ (𝑄 ∈ dom (pmSgn‘𝑁) ↔ (𝑄 ∈ 𝐵 ∧ dom (𝑄 ∖ I ) ∈ Fin)) |
7 | 1, 4, 6 | sylanbrc 581 | . 2 ⊢ ((𝑁 ∈ Fin ∧ 𝑄 ∈ 𝐵) → 𝑄 ∈ dom (pmSgn‘𝑁)) |
8 | psgnfitr.t | . . 3 ⊢ 𝑇 = ran (pmTrsp‘𝑁) | |
9 | 2, 8, 5 | psgneu 19475 | . 2 ⊢ (𝑄 ∈ dom (pmSgn‘𝑁) → ∃!𝑠∃𝑤 ∈ Word 𝑇(𝑄 = (𝐺 Σg 𝑤) ∧ 𝑠 = (-1↑(♯‘𝑤)))) |
10 | 7, 9 | syl 17 | 1 ⊢ ((𝑁 ∈ Fin ∧ 𝑄 ∈ 𝐵) → ∃!𝑠∃𝑤 ∈ Word 𝑇(𝑄 = (𝐺 Σg 𝑤) ∧ 𝑠 = (-1↑(♯‘𝑤)))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 = wceq 1533 ∈ wcel 2098 ∃!weu 2557 ∃wrex 3067 ∖ cdif 3946 I cid 5579 dom cdm 5682 ran crn 5683 ‘cfv 6553 (class class class)co 7426 Fincfn 8972 1c1 11149 -cneg 11485 ↑cexp 14068 ♯chash 14331 Word cword 14506 Basecbs 17189 Σg cgsu 17431 SymGrpcsymg 19335 pmTrspcpmtr 19410 pmSgncpsgn 19458 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2699 ax-rep 5289 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7748 ax-cnex 11204 ax-resscn 11205 ax-1cn 11206 ax-icn 11207 ax-addcl 11208 ax-addrcl 11209 ax-mulcl 11210 ax-mulrcl 11211 ax-mulcom 11212 ax-addass 11213 ax-mulass 11214 ax-distr 11215 ax-i2m1 11216 ax-1ne0 11217 ax-1rid 11218 ax-rnegex 11219 ax-rrecex 11220 ax-cnre 11221 ax-pre-lttri 11222 ax-pre-lttrn 11223 ax-pre-ltadd 11224 ax-pre-mulgt0 11225 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-xor 1505 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3374 df-reu 3375 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-tp 4637 df-op 4639 df-ot 4641 df-uni 4913 df-int 4954 df-iun 5002 df-iin 5003 df-br 5153 df-opab 5215 df-mpt 5236 df-tr 5270 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-se 5638 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6310 df-ord 6377 df-on 6378 df-lim 6379 df-suc 6380 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-isom 6562 df-riota 7382 df-ov 7429 df-oprab 7430 df-mpo 7431 df-om 7879 df-1st 8001 df-2nd 8002 df-tpos 8240 df-frecs 8295 df-wrecs 8326 df-recs 8400 df-rdg 8439 df-1o 8495 df-2o 8496 df-er 8733 df-map 8855 df-en 8973 df-dom 8974 df-sdom 8975 df-fin 8976 df-card 9972 df-pnf 11290 df-mnf 11291 df-xr 11292 df-ltxr 11293 df-le 11294 df-sub 11486 df-neg 11487 df-div 11912 df-nn 12253 df-2 12315 df-3 12316 df-4 12317 df-5 12318 df-6 12319 df-7 12320 df-8 12321 df-9 12322 df-n0 12513 df-xnn0 12585 df-z 12599 df-uz 12863 df-rp 13017 df-fz 13527 df-fzo 13670 df-seq 14009 df-exp 14069 df-hash 14332 df-word 14507 df-lsw 14555 df-concat 14563 df-s1 14588 df-substr 14633 df-pfx 14663 df-splice 14742 df-reverse 14751 df-s2 14841 df-struct 17125 df-sets 17142 df-slot 17160 df-ndx 17172 df-base 17190 df-ress 17219 df-plusg 17255 df-tset 17261 df-0g 17432 df-gsum 17433 df-mre 17575 df-mrc 17576 df-acs 17578 df-mgm 18609 df-sgrp 18688 df-mnd 18704 df-mhm 18749 df-submnd 18750 df-efmnd 18835 df-grp 18907 df-minusg 18908 df-subg 19092 df-ghm 19182 df-gim 19227 df-oppg 19311 df-symg 19336 df-pmtr 19411 df-psgn 19460 |
This theorem is referenced by: (None) |
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