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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 2823 | . . . . . . 7 ⊢ (SymGrp‘𝑁) = (SymGrp‘𝑁) | |
2 | psgnran.p | . . . . . . 7 ⊢ 𝑃 = (Base‘(SymGrp‘𝑁)) | |
3 | 1, 2 | sygbasnfpfi 18642 | . . . . . 6 ⊢ ((𝑁 ∈ Fin ∧ 𝑄 ∈ 𝑃) → dom (𝑄 ∖ I ) ∈ Fin) |
4 | 3 | ex 415 | . . . . 5 ⊢ (𝑁 ∈ Fin → (𝑄 ∈ 𝑃 → dom (𝑄 ∖ I ) ∈ Fin)) |
5 | 4 | pm4.71d 564 | . . . 4 ⊢ (𝑁 ∈ Fin → (𝑄 ∈ 𝑃 ↔ (𝑄 ∈ 𝑃 ∧ dom (𝑄 ∖ I ) ∈ Fin))) |
6 | psgnran.s | . . . . 5 ⊢ 𝑆 = (pmSgn‘𝑁) | |
7 | 1, 6, 2 | psgneldm 18633 | . . . 4 ⊢ (𝑄 ∈ dom 𝑆 ↔ (𝑄 ∈ 𝑃 ∧ dom (𝑄 ∖ I ) ∈ Fin)) |
8 | 5, 7 | syl6bbr 291 | . . 3 ⊢ (𝑁 ∈ Fin → (𝑄 ∈ 𝑃 ↔ 𝑄 ∈ dom 𝑆)) |
9 | eqid 2823 | . . . . 5 ⊢ ran (pmTrsp‘𝑁) = ran (pmTrsp‘𝑁) | |
10 | 1, 9, 6 | psgnvali 18638 | . . . 4 ⊢ (𝑄 ∈ dom 𝑆 → ∃𝑤 ∈ Word ran (pmTrsp‘𝑁)(𝑄 = ((SymGrp‘𝑁) Σg 𝑤) ∧ (𝑆‘𝑄) = (-1↑(♯‘𝑤)))) |
11 | lencl 13885 | . . . . . . . . . 10 ⊢ (𝑤 ∈ Word ran (pmTrsp‘𝑁) → (♯‘𝑤) ∈ ℕ0) | |
12 | 11 | nn0zd 12088 | . . . . . . . . 9 ⊢ (𝑤 ∈ Word ran (pmTrsp‘𝑁) → (♯‘𝑤) ∈ ℤ) |
13 | m1expcl2 13454 | . . . . . . . . . 10 ⊢ ((♯‘𝑤) ∈ ℤ → (-1↑(♯‘𝑤)) ∈ {-1, 1}) | |
14 | prcom 4670 | . . . . . . . . . 10 ⊢ {-1, 1} = {1, -1} | |
15 | 13, 14 | eleqtrdi 2925 | . . . . . . . . 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 484 | . . . . . . 7 ⊢ ((𝑁 ∈ Fin ∧ 𝑤 ∈ Word ran (pmTrsp‘𝑁)) → (-1↑(♯‘𝑤)) ∈ {1, -1}) |
18 | eleq1a 2910 | . . . . . . 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 493 | . . . . 5 ⊢ ((𝑁 ∈ Fin ∧ 𝑤 ∈ Word ran (pmTrsp‘𝑁)) → ((𝑄 = ((SymGrp‘𝑁) Σg 𝑤) ∧ (𝑆‘𝑄) = (-1↑(♯‘𝑤))) → (𝑆‘𝑄) ∈ {1, -1})) |
21 | 20 | rexlimdva 3286 | . . . 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 242 | . 2 ⊢ (𝑁 ∈ Fin → (𝑄 ∈ 𝑃 → (𝑆‘𝑄) ∈ {1, -1})) |
24 | 23 | imp 409 | 1 ⊢ ((𝑁 ∈ Fin ∧ 𝑄 ∈ 𝑃) → (𝑆‘𝑄) ∈ {1, -1}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ∃wrex 3141 ∖ cdif 3935 {cpr 4571 I cid 5461 dom cdm 5557 ran crn 5558 ‘cfv 6357 (class class class)co 7158 Fincfn 8511 1c1 10540 -cneg 10873 ℤcz 11984 ↑cexp 13432 ♯chash 13693 Word cword 13864 Basecbs 16485 Σg cgsu 16716 SymGrpcsymg 18497 pmTrspcpmtr 18571 pmSgncpsgn 18619 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-xor 1502 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-ot 4578 df-uni 4841 df-int 4879 df-iun 4923 df-iin 4924 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-se 5517 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-isom 6366 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-om 7583 df-1st 7691 df-2nd 7692 df-tpos 7894 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-1o 8104 df-2o 8105 df-oadd 8108 df-er 8291 df-map 8410 df-en 8512 df-dom 8513 df-sdom 8514 df-fin 8515 df-card 9370 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-div 11300 df-nn 11641 df-2 11703 df-3 11704 df-4 11705 df-5 11706 df-6 11707 df-7 11708 df-8 11709 df-9 11710 df-n0 11901 df-xnn0 11971 df-z 11985 df-uz 12247 df-rp 12393 df-fz 12896 df-fzo 13037 df-seq 13373 df-exp 13433 df-hash 13694 df-word 13865 df-lsw 13917 df-concat 13925 df-s1 13952 df-substr 14005 df-pfx 14035 df-splice 14114 df-reverse 14123 df-s2 14212 df-struct 16487 df-ndx 16488 df-slot 16489 df-base 16491 df-sets 16492 df-ress 16493 df-plusg 16580 df-tset 16586 df-0g 16717 df-gsum 16718 df-mre 16859 df-mrc 16860 df-acs 16862 df-mgm 17854 df-sgrp 17903 df-mnd 17914 df-mhm 17958 df-submnd 17959 df-efmnd 18036 df-grp 18108 df-minusg 18109 df-subg 18278 df-ghm 18358 df-gim 18401 df-oppg 18476 df-symg 18498 df-pmtr 18572 df-psgn 18621 |
This theorem is referenced by: zrhpsgnelbas 20740 mdetpmtr1 31090 mdetpmtr12 31092 |
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