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Mirrors > Home > MPE Home > Th. List > psgnprfval2 | Structured version Visualization version GIF version |
Description: The permutation sign of the transposition for a pair. (Contributed by AV, 10-Dec-2018.) |
Ref | Expression |
---|---|
psgnprfval.0 | ⊢ 𝐷 = {1, 2} |
psgnprfval.g | ⊢ 𝐺 = (SymGrp‘𝐷) |
psgnprfval.b | ⊢ 𝐵 = (Base‘𝐺) |
psgnprfval.t | ⊢ 𝑇 = ran (pmTrsp‘𝐷) |
psgnprfval.n | ⊢ 𝑁 = (pmSgn‘𝐷) |
Ref | Expression |
---|---|
psgnprfval2 | ⊢ (𝑁‘{〈1, 2〉, 〈2, 1〉}) = -1 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | prex 5452 | . . . . 5 ⊢ {〈1, 2〉, 〈2, 1〉} ∈ V | |
2 | 1 | snid 4684 | . . . 4 ⊢ {〈1, 2〉, 〈2, 1〉} ∈ {{〈1, 2〉, 〈2, 1〉}} |
3 | psgnprfval.0 | . . . . . . 7 ⊢ 𝐷 = {1, 2} | |
4 | 3 | fveq2i 6923 | . . . . . 6 ⊢ (pmTrsp‘𝐷) = (pmTrsp‘{1, 2}) |
5 | 4 | rneqi 5962 | . . . . 5 ⊢ ran (pmTrsp‘𝐷) = ran (pmTrsp‘{1, 2}) |
6 | pmtrprfvalrn 19530 | . . . . 5 ⊢ ran (pmTrsp‘{1, 2}) = {{〈1, 2〉, 〈2, 1〉}} | |
7 | 5, 6 | eqtri 2768 | . . . 4 ⊢ ran (pmTrsp‘𝐷) = {{〈1, 2〉, 〈2, 1〉}} |
8 | 2, 7 | eleqtrri 2843 | . . 3 ⊢ {〈1, 2〉, 〈2, 1〉} ∈ ran (pmTrsp‘𝐷) |
9 | psgnprfval.t | . . 3 ⊢ 𝑇 = ran (pmTrsp‘𝐷) | |
10 | 8, 9 | eleqtrri 2843 | . 2 ⊢ {〈1, 2〉, 〈2, 1〉} ∈ 𝑇 |
11 | psgnprfval.g | . . 3 ⊢ 𝐺 = (SymGrp‘𝐷) | |
12 | psgnprfval.n | . . 3 ⊢ 𝑁 = (pmSgn‘𝐷) | |
13 | 11, 9, 12 | psgnpmtr 19552 | . 2 ⊢ ({〈1, 2〉, 〈2, 1〉} ∈ 𝑇 → (𝑁‘{〈1, 2〉, 〈2, 1〉}) = -1) |
14 | 10, 13 | ax-mp 5 | 1 ⊢ (𝑁‘{〈1, 2〉, 〈2, 1〉}) = -1 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1537 ∈ wcel 2108 {csn 4648 {cpr 4650 〈cop 4654 ran crn 5701 ‘cfv 6573 1c1 11185 -cneg 11521 2c2 12348 Basecbs 17258 SymGrpcsymg 19410 pmTrspcpmtr 19483 pmSgncpsgn 19531 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2158 ax-12 2178 ax-ext 2711 ax-rep 5303 ax-sep 5317 ax-nul 5324 ax-pow 5383 ax-pr 5447 ax-un 7770 ax-cnex 11240 ax-resscn 11241 ax-1cn 11242 ax-icn 11243 ax-addcl 11244 ax-addrcl 11245 ax-mulcl 11246 ax-mulrcl 11247 ax-mulcom 11248 ax-addass 11249 ax-mulass 11250 ax-distr 11251 ax-i2m1 11252 ax-1ne0 11253 ax-1rid 11254 ax-rnegex 11255 ax-rrecex 11256 ax-cnre 11257 ax-pre-lttri 11258 ax-pre-lttrn 11259 ax-pre-ltadd 11260 ax-pre-mulgt0 11261 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3or 1088 df-3an 1089 df-xor 1509 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2543 df-eu 2572 df-clab 2718 df-cleq 2732 df-clel 2819 df-nfc 2895 df-ne 2947 df-nel 3053 df-ral 3068 df-rex 3077 df-rmo 3388 df-reu 3389 df-rab 3444 df-v 3490 df-sbc 3805 df-csb 3922 df-dif 3979 df-un 3981 df-in 3983 df-ss 3993 df-pss 3996 df-nul 4353 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-tp 4653 df-op 4655 df-ot 4657 df-uni 4932 df-int 4971 df-iun 5017 df-iin 5018 df-br 5167 df-opab 5229 df-mpt 5250 df-tr 5284 df-id 5593 df-eprel 5599 df-po 5607 df-so 5608 df-fr 5652 df-se 5653 df-we 5654 df-xp 5706 df-rel 5707 df-cnv 5708 df-co 5709 df-dm 5710 df-rn 5711 df-res 5712 df-ima 5713 df-pred 6332 df-ord 6398 df-on 6399 df-lim 6400 df-suc 6401 df-iota 6525 df-fun 6575 df-fn 6576 df-f 6577 df-f1 6578 df-fo 6579 df-f1o 6580 df-fv 6581 df-isom 6582 df-riota 7404 df-ov 7451 df-oprab 7452 df-mpo 7453 df-om 7904 df-1st 8030 df-2nd 8031 df-tpos 8267 df-frecs 8322 df-wrecs 8353 df-recs 8427 df-rdg 8466 df-1o 8522 df-2o 8523 df-oadd 8526 df-er 8763 df-map 8886 df-en 9004 df-dom 9005 df-sdom 9006 df-fin 9007 df-dju 9970 df-card 10008 df-pnf 11326 df-mnf 11327 df-xr 11328 df-ltxr 11329 df-le 11330 df-sub 11522 df-neg 11523 df-div 11948 df-nn 12294 df-2 12356 df-3 12357 df-4 12358 df-5 12359 df-6 12360 df-7 12361 df-8 12362 df-9 12363 df-n0 12554 df-xnn0 12626 df-z 12640 df-uz 12904 df-rp 13058 df-fz 13568 df-fzo 13712 df-seq 14053 df-exp 14113 df-hash 14380 df-word 14563 df-lsw 14611 df-concat 14619 df-s1 14644 df-substr 14689 df-pfx 14719 df-splice 14798 df-reverse 14807 df-s2 14897 df-struct 17194 df-sets 17211 df-slot 17229 df-ndx 17241 df-base 17259 df-ress 17288 df-plusg 17324 df-tset 17330 df-0g 17501 df-gsum 17502 df-mre 17644 df-mrc 17645 df-acs 17647 df-mgm 18678 df-sgrp 18757 df-mnd 18773 df-mhm 18818 df-submnd 18819 df-efmnd 18904 df-grp 18976 df-minusg 18977 df-subg 19163 df-ghm 19253 df-gim 19299 df-oppg 19386 df-symg 19411 df-pmtr 19484 df-psgn 19533 |
This theorem is referenced by: m2detleiblem1 22651 m2detleiblem6 22653 |
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