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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 2610 | . . . . . . 7 ⊢ (SymGrp‘𝑁) = (SymGrp‘𝑁) | |
2 | psgnran.p | . . . . . . 7 ⊢ 𝑃 = (Base‘(SymGrp‘𝑁)) | |
3 | 1, 2 | sygbasnfpfi 17755 | . . . . . 6 ⊢ ((𝑁 ∈ Fin ∧ 𝑄 ∈ 𝑃) → dom (𝑄 ∖ I ) ∈ Fin) |
4 | 3 | ex 449 | . . . . 5 ⊢ (𝑁 ∈ Fin → (𝑄 ∈ 𝑃 → dom (𝑄 ∖ I ) ∈ Fin)) |
5 | 4 | pm4.71d 664 | . . . 4 ⊢ (𝑁 ∈ Fin → (𝑄 ∈ 𝑃 ↔ (𝑄 ∈ 𝑃 ∧ dom (𝑄 ∖ I ) ∈ Fin))) |
6 | psgnran.s | . . . . 5 ⊢ 𝑆 = (pmSgn‘𝑁) | |
7 | 1, 6, 2 | psgneldm 17746 | . . . 4 ⊢ (𝑄 ∈ dom 𝑆 ↔ (𝑄 ∈ 𝑃 ∧ dom (𝑄 ∖ I ) ∈ Fin)) |
8 | 5, 7 | syl6bbr 277 | . . 3 ⊢ (𝑁 ∈ Fin → (𝑄 ∈ 𝑃 ↔ 𝑄 ∈ dom 𝑆)) |
9 | eqid 2610 | . . . . 5 ⊢ ran (pmTrsp‘𝑁) = ran (pmTrsp‘𝑁) | |
10 | 1, 9, 6 | psgnvali 17751 | . . . 4 ⊢ (𝑄 ∈ dom 𝑆 → ∃𝑤 ∈ Word ran (pmTrsp‘𝑁)(𝑄 = ((SymGrp‘𝑁) Σg 𝑤) ∧ (𝑆‘𝑄) = (-1↑(#‘𝑤)))) |
11 | lencl 13179 | . . . . . . . . . 10 ⊢ (𝑤 ∈ Word ran (pmTrsp‘𝑁) → (#‘𝑤) ∈ ℕ0) | |
12 | 11 | nn0zd 11356 | . . . . . . . . 9 ⊢ (𝑤 ∈ Word ran (pmTrsp‘𝑁) → (#‘𝑤) ∈ ℤ) |
13 | m1expcl2 12744 | . . . . . . . . . 10 ⊢ ((#‘𝑤) ∈ ℤ → (-1↑(#‘𝑤)) ∈ {-1, 1}) | |
14 | prcom 4211 | . . . . . . . . . 10 ⊢ {-1, 1} = {1, -1} | |
15 | 13, 14 | syl6eleq 2698 | . . . . . . . . 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 481 | . . . . . . 7 ⊢ ((𝑁 ∈ Fin ∧ 𝑤 ∈ Word ran (pmTrsp‘𝑁)) → (-1↑(#‘𝑤)) ∈ {1, -1}) |
18 | eleq1a 2683 | . . . . . . 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 482 | . . . . 5 ⊢ ((𝑁 ∈ Fin ∧ 𝑤 ∈ Word ran (pmTrsp‘𝑁)) → ((𝑄 = ((SymGrp‘𝑁) Σg 𝑤) ∧ (𝑆‘𝑄) = (-1↑(#‘𝑤))) → (𝑆‘𝑄) ∈ {1, -1})) |
21 | 20 | rexlimdva 3013 | . . . 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 33 | . . 3 ⊢ (𝑁 ∈ Fin → (𝑄 ∈ dom 𝑆 → (𝑆‘𝑄) ∈ {1, -1})) |
23 | 8, 22 | sylbid 229 | . 2 ⊢ (𝑁 ∈ Fin → (𝑄 ∈ 𝑃 → (𝑆‘𝑄) ∈ {1, -1})) |
24 | 23 | imp 444 | 1 ⊢ ((𝑁 ∈ Fin ∧ 𝑄 ∈ 𝑃) → (𝑆‘𝑄) ∈ {1, -1}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ∃wrex 2897 ∖ cdif 3537 {cpr 4127 I cid 4948 dom cdm 5038 ran crn 5039 ‘cfv 5804 (class class class)co 6549 Fincfn 7841 1c1 9816 -cneg 10146 ℤcz 11254 ↑cexp 12722 #chash 12979 Word cword 13146 Basecbs 15695 Σg cgsu 15924 SymGrpcsymg 17620 pmTrspcpmtr 17684 pmSgncpsgn 17732 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-xor 1457 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-ot 4134 df-uni 4373 df-int 4411 df-iun 4457 df-iin 4458 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-se 4998 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-isom 5813 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-tpos 7239 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-2o 7448 df-oadd 7451 df-er 7629 df-map 7746 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-card 8648 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-div 10564 df-nn 10898 df-2 10956 df-3 10957 df-4 10958 df-5 10959 df-6 10960 df-7 10961 df-8 10962 df-9 10963 df-n0 11170 df-xnn0 11241 df-z 11255 df-uz 11564 df-rp 11709 df-fz 12198 df-fzo 12335 df-seq 12664 df-exp 12723 df-hash 12980 df-word 13154 df-lsw 13155 df-concat 13156 df-s1 13157 df-substr 13158 df-splice 13159 df-reverse 13160 df-s2 13444 df-struct 15697 df-ndx 15698 df-slot 15699 df-base 15700 df-sets 15701 df-ress 15702 df-plusg 15781 df-tset 15787 df-0g 15925 df-gsum 15926 df-mre 16069 df-mrc 16070 df-acs 16072 df-mgm 17065 df-sgrp 17107 df-mnd 17118 df-mhm 17158 df-submnd 17159 df-grp 17248 df-minusg 17249 df-subg 17414 df-ghm 17481 df-gim 17524 df-oppg 17599 df-symg 17621 df-pmtr 17685 df-psgn 17734 |
This theorem is referenced by: zrhpsgnelbas 19759 mdetpmtr1 29217 mdetpmtr12 29219 |
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