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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 5389 | . . . . 5 ⊢ {〈1, 2〉, 〈2, 1〉} ∈ V | |
2 | 1 | snid 4622 | . . . 4 ⊢ {〈1, 2〉, 〈2, 1〉} ∈ {{〈1, 2〉, 〈2, 1〉}} |
3 | psgnprfval.0 | . . . . . . 7 ⊢ 𝐷 = {1, 2} | |
4 | 3 | fveq2i 6845 | . . . . . 6 ⊢ (pmTrsp‘𝐷) = (pmTrsp‘{1, 2}) |
5 | 4 | rneqi 5892 | . . . . 5 ⊢ ran (pmTrsp‘𝐷) = ran (pmTrsp‘{1, 2}) |
6 | pmtrprfvalrn 19268 | . . . . 5 ⊢ ran (pmTrsp‘{1, 2}) = {{〈1, 2〉, 〈2, 1〉}} | |
7 | 5, 6 | eqtri 2764 | . . . 4 ⊢ ran (pmTrsp‘𝐷) = {{〈1, 2〉, 〈2, 1〉}} |
8 | 2, 7 | eleqtrri 2837 | . . 3 ⊢ {〈1, 2〉, 〈2, 1〉} ∈ ran (pmTrsp‘𝐷) |
9 | psgnprfval.t | . . 3 ⊢ 𝑇 = ran (pmTrsp‘𝐷) | |
10 | 8, 9 | eleqtrri 2837 | . 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 19290 | . 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 1541 ∈ wcel 2106 {csn 4586 {cpr 4588 〈cop 4592 ran crn 5634 ‘cfv 6496 1c1 11051 -cneg 11385 2c2 12207 Basecbs 17082 SymGrpcsymg 19146 pmTrspcpmtr 19221 pmSgncpsgn 19269 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-rep 5242 ax-sep 5256 ax-nul 5263 ax-pow 5320 ax-pr 5384 ax-un 7671 ax-cnex 11106 ax-resscn 11107 ax-1cn 11108 ax-icn 11109 ax-addcl 11110 ax-addrcl 11111 ax-mulcl 11112 ax-mulrcl 11113 ax-mulcom 11114 ax-addass 11115 ax-mulass 11116 ax-distr 11117 ax-i2m1 11118 ax-1ne0 11119 ax-1rid 11120 ax-rnegex 11121 ax-rrecex 11122 ax-cnre 11123 ax-pre-lttri 11124 ax-pre-lttrn 11125 ax-pre-ltadd 11126 ax-pre-mulgt0 11127 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-xor 1510 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3065 df-rex 3074 df-rmo 3353 df-reu 3354 df-rab 3408 df-v 3447 df-sbc 3740 df-csb 3856 df-dif 3913 df-un 3915 df-in 3917 df-ss 3927 df-pss 3929 df-nul 4283 df-if 4487 df-pw 4562 df-sn 4587 df-pr 4589 df-tp 4591 df-op 4593 df-ot 4595 df-uni 4866 df-int 4908 df-iun 4956 df-iin 4957 df-br 5106 df-opab 5168 df-mpt 5189 df-tr 5223 df-id 5531 df-eprel 5537 df-po 5545 df-so 5546 df-fr 5588 df-se 5589 df-we 5590 df-xp 5639 df-rel 5640 df-cnv 5641 df-co 5642 df-dm 5643 df-rn 5644 df-res 5645 df-ima 5646 df-pred 6253 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6498 df-fn 6499 df-f 6500 df-f1 6501 df-fo 6502 df-f1o 6503 df-fv 6504 df-isom 6505 df-riota 7312 df-ov 7359 df-oprab 7360 df-mpo 7361 df-om 7802 df-1st 7920 df-2nd 7921 df-tpos 8156 df-frecs 8211 df-wrecs 8242 df-recs 8316 df-rdg 8355 df-1o 8411 df-2o 8412 df-oadd 8415 df-er 8647 df-map 8766 df-en 8883 df-dom 8884 df-sdom 8885 df-fin 8886 df-dju 9836 df-card 9874 df-pnf 11190 df-mnf 11191 df-xr 11192 df-ltxr 11193 df-le 11194 df-sub 11386 df-neg 11387 df-div 11812 df-nn 12153 df-2 12215 df-3 12216 df-4 12217 df-5 12218 df-6 12219 df-7 12220 df-8 12221 df-9 12222 df-n0 12413 df-xnn0 12485 df-z 12499 df-uz 12763 df-rp 12915 df-fz 13424 df-fzo 13567 df-seq 13906 df-exp 13967 df-hash 14230 df-word 14402 df-lsw 14450 df-concat 14458 df-s1 14483 df-substr 14528 df-pfx 14558 df-splice 14637 df-reverse 14646 df-s2 14736 df-struct 17018 df-sets 17035 df-slot 17053 df-ndx 17065 df-base 17083 df-ress 17112 df-plusg 17145 df-tset 17151 df-0g 17322 df-gsum 17323 df-mre 17465 df-mrc 17466 df-acs 17468 df-mgm 18496 df-sgrp 18545 df-mnd 18556 df-mhm 18600 df-submnd 18601 df-efmnd 18678 df-grp 18750 df-minusg 18751 df-subg 18923 df-ghm 19004 df-gim 19047 df-oppg 19122 df-symg 19147 df-pmtr 19222 df-psgn 19271 |
This theorem is referenced by: m2detleiblem1 21971 m2detleiblem6 21973 |
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