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Mirrors > Home > MPE Home > Th. List > psgnodpmr | Structured version Visualization version GIF version |
Description: If a permutation has sign -1 it is odd (not even). (Contributed by SO, 9-Jul-2018.) |
Ref | Expression |
---|---|
evpmss.s | ⊢ 𝑆 = (SymGrp‘𝐷) |
evpmss.p | ⊢ 𝑃 = (Base‘𝑆) |
psgnevpmb.n | ⊢ 𝑁 = (pmSgn‘𝐷) |
Ref | Expression |
---|---|
psgnodpmr | ⊢ ((𝐷 ∈ Fin ∧ 𝐹 ∈ 𝑃 ∧ (𝑁‘𝐹) = -1) → 𝐹 ∈ (𝑃 ∖ (pmEven‘𝐷))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simp2 1128 | . 2 ⊢ ((𝐷 ∈ Fin ∧ 𝐹 ∈ 𝑃 ∧ (𝑁‘𝐹) = -1) → 𝐹 ∈ 𝑃) | |
2 | evpmss.s | . . . . . . . 8 ⊢ 𝑆 = (SymGrp‘𝐷) | |
3 | evpmss.p | . . . . . . . 8 ⊢ 𝑃 = (Base‘𝑆) | |
4 | psgnevpmb.n | . . . . . . . 8 ⊢ 𝑁 = (pmSgn‘𝐷) | |
5 | 2, 3, 4 | psgnevpm 20341 | . . . . . . 7 ⊢ ((𝐷 ∈ Fin ∧ 𝐹 ∈ (pmEven‘𝐷)) → (𝑁‘𝐹) = 1) |
6 | 5 | ex 403 | . . . . . 6 ⊢ (𝐷 ∈ Fin → (𝐹 ∈ (pmEven‘𝐷) → (𝑁‘𝐹) = 1)) |
7 | 6 | adantr 474 | . . . . 5 ⊢ ((𝐷 ∈ Fin ∧ 𝐹 ∈ 𝑃) → (𝐹 ∈ (pmEven‘𝐷) → (𝑁‘𝐹) = 1)) |
8 | neg1rr 11502 | . . . . . . 7 ⊢ -1 ∈ ℝ | |
9 | neg1lt0 11504 | . . . . . . . 8 ⊢ -1 < 0 | |
10 | 0lt1 10900 | . . . . . . . 8 ⊢ 0 < 1 | |
11 | 0re 10380 | . . . . . . . . 9 ⊢ 0 ∈ ℝ | |
12 | 1re 10378 | . . . . . . . . 9 ⊢ 1 ∈ ℝ | |
13 | 8, 11, 12 | lttri 10504 | . . . . . . . 8 ⊢ ((-1 < 0 ∧ 0 < 1) → -1 < 1) |
14 | 9, 10, 13 | mp2an 682 | . . . . . . 7 ⊢ -1 < 1 |
15 | 8, 14 | gtneii 10490 | . . . . . 6 ⊢ 1 ≠ -1 |
16 | neeq1 3031 | . . . . . 6 ⊢ ((𝑁‘𝐹) = 1 → ((𝑁‘𝐹) ≠ -1 ↔ 1 ≠ -1)) | |
17 | 15, 16 | mpbiri 250 | . . . . 5 ⊢ ((𝑁‘𝐹) = 1 → (𝑁‘𝐹) ≠ -1) |
18 | 7, 17 | syl6 35 | . . . 4 ⊢ ((𝐷 ∈ Fin ∧ 𝐹 ∈ 𝑃) → (𝐹 ∈ (pmEven‘𝐷) → (𝑁‘𝐹) ≠ -1)) |
19 | 18 | necon2bd 2985 | . . 3 ⊢ ((𝐷 ∈ Fin ∧ 𝐹 ∈ 𝑃) → ((𝑁‘𝐹) = -1 → ¬ 𝐹 ∈ (pmEven‘𝐷))) |
20 | 19 | 3impia 1106 | . 2 ⊢ ((𝐷 ∈ Fin ∧ 𝐹 ∈ 𝑃 ∧ (𝑁‘𝐹) = -1) → ¬ 𝐹 ∈ (pmEven‘𝐷)) |
21 | 1, 20 | eldifd 3803 | 1 ⊢ ((𝐷 ∈ Fin ∧ 𝐹 ∈ 𝑃 ∧ (𝑁‘𝐹) = -1) → 𝐹 ∈ (𝑃 ∖ (pmEven‘𝐷))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 386 ∧ w3a 1071 = wceq 1601 ∈ wcel 2107 ≠ wne 2969 ∖ cdif 3789 class class class wbr 4888 ‘cfv 6137 Fincfn 8243 0cc0 10274 1c1 10275 < clt 10413 -cneg 10609 Basecbs 16266 SymGrpcsymg 18191 pmSgncpsgn 18303 pmEvencevpm 18304 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-8 2109 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-rep 5008 ax-sep 5019 ax-nul 5027 ax-pow 5079 ax-pr 5140 ax-un 7228 ax-cnex 10330 ax-resscn 10331 ax-1cn 10332 ax-icn 10333 ax-addcl 10334 ax-addrcl 10335 ax-mulcl 10336 ax-mulrcl 10337 ax-mulcom 10338 ax-addass 10339 ax-mulass 10340 ax-distr 10341 ax-i2m1 10342 ax-1ne0 10343 ax-1rid 10344 ax-rnegex 10345 ax-rrecex 10346 ax-cnre 10347 ax-pre-lttri 10348 ax-pre-lttrn 10349 ax-pre-ltadd 10350 ax-pre-mulgt0 10351 ax-addf 10353 ax-mulf 10354 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3or 1072 df-3an 1073 df-xor 1583 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ne 2970 df-nel 3076 df-ral 3095 df-rex 3096 df-reu 3097 df-rmo 3098 df-rab 3099 df-v 3400 df-sbc 3653 df-csb 3752 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-pss 3808 df-nul 4142 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-tp 4403 df-op 4405 df-ot 4407 df-uni 4674 df-int 4713 df-iun 4757 df-iin 4758 df-br 4889 df-opab 4951 df-mpt 4968 df-tr 4990 df-id 5263 df-eprel 5268 df-po 5276 df-so 5277 df-fr 5316 df-se 5317 df-we 5318 df-xp 5363 df-rel 5364 df-cnv 5365 df-co 5366 df-dm 5367 df-rn 5368 df-res 5369 df-ima 5370 df-pred 5935 df-ord 5981 df-on 5982 df-lim 5983 df-suc 5984 df-iota 6101 df-fun 6139 df-fn 6140 df-f 6141 df-f1 6142 df-fo 6143 df-f1o 6144 df-fv 6145 df-isom 6146 df-riota 6885 df-ov 6927 df-oprab 6928 df-mpt2 6929 df-om 7346 df-1st 7447 df-2nd 7448 df-tpos 7636 df-wrecs 7691 df-recs 7753 df-rdg 7791 df-1o 7845 df-2o 7846 df-oadd 7849 df-er 8028 df-map 8144 df-en 8244 df-dom 8245 df-sdom 8246 df-fin 8247 df-card 9100 df-pnf 10415 df-mnf 10416 df-xr 10417 df-ltxr 10418 df-le 10419 df-sub 10610 df-neg 10611 df-div 11036 df-nn 11380 df-2 11443 df-3 11444 df-4 11445 df-5 11446 df-6 11447 df-7 11448 df-8 11449 df-9 11450 df-n0 11648 df-xnn0 11720 df-z 11734 df-dec 11851 df-uz 11998 df-rp 12143 df-fz 12649 df-fzo 12790 df-seq 13125 df-exp 13184 df-hash 13442 df-word 13606 df-lsw 13659 df-concat 13667 df-s1 13692 df-substr 13737 df-pfx 13786 df-splice 13893 df-reverse 13911 df-s2 14005 df-struct 16268 df-ndx 16269 df-slot 16270 df-base 16272 df-sets 16273 df-ress 16274 df-plusg 16362 df-mulr 16363 df-starv 16364 df-tset 16368 df-ple 16369 df-ds 16371 df-unif 16372 df-0g 16499 df-gsum 16500 df-mre 16643 df-mrc 16644 df-acs 16646 df-mgm 17639 df-sgrp 17681 df-mnd 17692 df-mhm 17732 df-submnd 17733 df-grp 17823 df-minusg 17824 df-subg 17986 df-ghm 18053 df-gim 18096 df-oppg 18170 df-symg 18192 df-pmtr 18256 df-psgn 18305 df-evpm 18306 df-cmn 18592 df-abl 18593 df-mgp 18888 df-ur 18900 df-ring 18947 df-cring 18948 df-oppr 19021 df-dvdsr 19039 df-unit 19040 df-invr 19070 df-dvr 19081 df-drng 19152 df-cnfld 20154 |
This theorem is referenced by: evpmodpmf1o 20349 pmtrodpm 20350 mdetralt 20830 |
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