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Mirrors > Home > MPE Home > Th. List > psgnran | Structured version Visualization version GIF version |
Description: The range of the permutation sign function for finite permutations. (Contributed by AV, 1-Jan-2019.) |
Ref | Expression |
---|---|
psgnran.p | ⊢ 𝑃 = (Base‘(SymGrp‘𝑁)) |
psgnran.s | ⊢ 𝑆 = (pmSgn‘𝑁) |
Ref | Expression |
---|---|
psgnran | ⊢ ((𝑁 ∈ Fin ∧ 𝑄 ∈ 𝑃) → (𝑆‘𝑄) ∈ {1, -1}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2622 | . . . . . . 7 ⊢ (SymGrp‘𝑁) = (SymGrp‘𝑁) | |
2 | psgnran.p | . . . . . . 7 ⊢ 𝑃 = (Base‘(SymGrp‘𝑁)) | |
3 | 1, 2 | sygbasnfpfi 17932 | . . . . . 6 ⊢ ((𝑁 ∈ Fin ∧ 𝑄 ∈ 𝑃) → dom (𝑄 ∖ I ) ∈ Fin) |
4 | 3 | ex 450 | . . . . 5 ⊢ (𝑁 ∈ Fin → (𝑄 ∈ 𝑃 → dom (𝑄 ∖ I ) ∈ Fin)) |
5 | 4 | pm4.71d 666 | . . . 4 ⊢ (𝑁 ∈ Fin → (𝑄 ∈ 𝑃 ↔ (𝑄 ∈ 𝑃 ∧ dom (𝑄 ∖ I ) ∈ Fin))) |
6 | psgnran.s | . . . . 5 ⊢ 𝑆 = (pmSgn‘𝑁) | |
7 | 1, 6, 2 | psgneldm 17923 | . . . 4 ⊢ (𝑄 ∈ dom 𝑆 ↔ (𝑄 ∈ 𝑃 ∧ dom (𝑄 ∖ I ) ∈ Fin)) |
8 | 5, 7 | syl6bbr 278 | . . 3 ⊢ (𝑁 ∈ Fin → (𝑄 ∈ 𝑃 ↔ 𝑄 ∈ dom 𝑆)) |
9 | eqid 2622 | . . . . 5 ⊢ ran (pmTrsp‘𝑁) = ran (pmTrsp‘𝑁) | |
10 | 1, 9, 6 | psgnvali 17928 | . . . 4 ⊢ (𝑄 ∈ dom 𝑆 → ∃𝑤 ∈ Word ran (pmTrsp‘𝑁)(𝑄 = ((SymGrp‘𝑁) Σg 𝑤) ∧ (𝑆‘𝑄) = (-1↑(#‘𝑤)))) |
11 | lencl 13324 | . . . . . . . . . 10 ⊢ (𝑤 ∈ Word ran (pmTrsp‘𝑁) → (#‘𝑤) ∈ ℕ0) | |
12 | 11 | nn0zd 11480 | . . . . . . . . 9 ⊢ (𝑤 ∈ Word ran (pmTrsp‘𝑁) → (#‘𝑤) ∈ ℤ) |
13 | m1expcl2 12882 | . . . . . . . . . 10 ⊢ ((#‘𝑤) ∈ ℤ → (-1↑(#‘𝑤)) ∈ {-1, 1}) | |
14 | prcom 4267 | . . . . . . . . . 10 ⊢ {-1, 1} = {1, -1} | |
15 | 13, 14 | syl6eleq 2711 | . . . . . . . . 9 ⊢ ((#‘𝑤) ∈ ℤ → (-1↑(#‘𝑤)) ∈ {1, -1}) |
16 | 12, 15 | syl 17 | . . . . . . . 8 ⊢ (𝑤 ∈ Word ran (pmTrsp‘𝑁) → (-1↑(#‘𝑤)) ∈ {1, -1}) |
17 | 16 | adantl 482 | . . . . . . 7 ⊢ ((𝑁 ∈ Fin ∧ 𝑤 ∈ Word ran (pmTrsp‘𝑁)) → (-1↑(#‘𝑤)) ∈ {1, -1}) |
18 | eleq1a 2696 | . . . . . . 7 ⊢ ((-1↑(#‘𝑤)) ∈ {1, -1} → ((𝑆‘𝑄) = (-1↑(#‘𝑤)) → (𝑆‘𝑄) ∈ {1, -1})) | |
19 | 17, 18 | syl 17 | . . . . . 6 ⊢ ((𝑁 ∈ Fin ∧ 𝑤 ∈ Word ran (pmTrsp‘𝑁)) → ((𝑆‘𝑄) = (-1↑(#‘𝑤)) → (𝑆‘𝑄) ∈ {1, -1})) |
20 | 19 | adantld 483 | . . . . 5 ⊢ ((𝑁 ∈ Fin ∧ 𝑤 ∈ Word ran (pmTrsp‘𝑁)) → ((𝑄 = ((SymGrp‘𝑁) Σg 𝑤) ∧ (𝑆‘𝑄) = (-1↑(#‘𝑤))) → (𝑆‘𝑄) ∈ {1, -1})) |
21 | 20 | rexlimdva 3031 | . . . 4 ⊢ (𝑁 ∈ Fin → (∃𝑤 ∈ Word ran (pmTrsp‘𝑁)(𝑄 = ((SymGrp‘𝑁) Σg 𝑤) ∧ (𝑆‘𝑄) = (-1↑(#‘𝑤))) → (𝑆‘𝑄) ∈ {1, -1})) |
22 | 10, 21 | syl5 34 | . . 3 ⊢ (𝑁 ∈ Fin → (𝑄 ∈ dom 𝑆 → (𝑆‘𝑄) ∈ {1, -1})) |
23 | 8, 22 | sylbid 230 | . 2 ⊢ (𝑁 ∈ Fin → (𝑄 ∈ 𝑃 → (𝑆‘𝑄) ∈ {1, -1})) |
24 | 23 | imp 445 | 1 ⊢ ((𝑁 ∈ Fin ∧ 𝑄 ∈ 𝑃) → (𝑆‘𝑄) ∈ {1, -1}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 384 = wceq 1483 ∈ wcel 1990 ∃wrex 2913 ∖ cdif 3571 {cpr 4179 I cid 5023 dom cdm 5114 ran crn 5115 ‘cfv 5888 (class class class)co 6650 Fincfn 7955 1c1 9937 -cneg 10267 ℤcz 11377 ↑cexp 12860 #chash 13117 Word cword 13291 Basecbs 15857 Σg cgsu 16101 SymGrpcsymg 17797 pmTrspcpmtr 17861 pmSgncpsgn 17909 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-xor 1465 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-ot 4186 df-uni 4437 df-int 4476 df-iun 4522 df-iin 4523 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-se 5074 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-isom 5897 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-tpos 7352 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-2o 7561 df-oadd 7564 df-er 7742 df-map 7859 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-card 8765 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-div 10685 df-nn 11021 df-2 11079 df-3 11080 df-4 11081 df-5 11082 df-6 11083 df-7 11084 df-8 11085 df-9 11086 df-n0 11293 df-xnn0 11364 df-z 11378 df-uz 11688 df-rp 11833 df-fz 12327 df-fzo 12466 df-seq 12802 df-exp 12861 df-hash 13118 df-word 13299 df-lsw 13300 df-concat 13301 df-s1 13302 df-substr 13303 df-splice 13304 df-reverse 13305 df-s2 13593 df-struct 15859 df-ndx 15860 df-slot 15861 df-base 15863 df-sets 15864 df-ress 15865 df-plusg 15954 df-tset 15960 df-0g 16102 df-gsum 16103 df-mre 16246 df-mrc 16247 df-acs 16249 df-mgm 17242 df-sgrp 17284 df-mnd 17295 df-mhm 17335 df-submnd 17336 df-grp 17425 df-minusg 17426 df-subg 17591 df-ghm 17658 df-gim 17701 df-oppg 17776 df-symg 17798 df-pmtr 17862 df-psgn 17911 |
This theorem is referenced by: zrhpsgnelbas 19940 mdetpmtr1 29889 mdetpmtr12 29891 |
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