![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 5429 | . . . . 5 ⊢ {⟨1, 2⟩, ⟨2, 1⟩} ∈ V | |
2 | 1 | snid 4661 | . . . 4 ⊢ {⟨1, 2⟩, ⟨2, 1⟩} ∈ {{⟨1, 2⟩, ⟨2, 1⟩}} |
3 | psgnprfval.0 | . . . . . . 7 ⊢ 𝐷 = {1, 2} | |
4 | 3 | fveq2i 6893 | . . . . . 6 ⊢ (pmTrsp‘𝐷) = (pmTrsp‘{1, 2}) |
5 | 4 | rneqi 5934 | . . . . 5 ⊢ ran (pmTrsp‘𝐷) = ran (pmTrsp‘{1, 2}) |
6 | pmtrprfvalrn 19442 | . . . . 5 ⊢ ran (pmTrsp‘{1, 2}) = {{⟨1, 2⟩, ⟨2, 1⟩}} | |
7 | 5, 6 | eqtri 2753 | . . . 4 ⊢ ran (pmTrsp‘𝐷) = {{⟨1, 2⟩, ⟨2, 1⟩}} |
8 | 2, 7 | eleqtrri 2824 | . . 3 ⊢ {⟨1, 2⟩, ⟨2, 1⟩} ∈ ran (pmTrsp‘𝐷) |
9 | psgnprfval.t | . . 3 ⊢ 𝑇 = ran (pmTrsp‘𝐷) | |
10 | 8, 9 | eleqtrri 2824 | . 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 19464 | . 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 1533 ∈ wcel 2098 {csn 4625 {cpr 4627 ⟨cop 4631 ran crn 5674 ‘cfv 6543 1c1 11134 -cneg 11470 2c2 12292 Basecbs 17174 SymGrpcsymg 19320 pmTrspcpmtr 19395 pmSgncpsgn 19443 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-rep 5281 ax-sep 5295 ax-nul 5302 ax-pow 5360 ax-pr 5424 ax-un 7735 ax-cnex 11189 ax-resscn 11190 ax-1cn 11191 ax-icn 11192 ax-addcl 11193 ax-addrcl 11194 ax-mulcl 11195 ax-mulrcl 11196 ax-mulcom 11197 ax-addass 11198 ax-mulass 11199 ax-distr 11200 ax-i2m1 11201 ax-1ne0 11202 ax-1rid 11203 ax-rnegex 11204 ax-rrecex 11205 ax-cnre 11206 ax-pre-lttri 11207 ax-pre-lttrn 11208 ax-pre-ltadd 11209 ax-pre-mulgt0 11210 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-xor 1505 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3465 df-sbc 3771 df-csb 3887 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-pss 3961 df-nul 4320 df-if 4526 df-pw 4601 df-sn 4626 df-pr 4628 df-tp 4630 df-op 4632 df-ot 4634 df-uni 4905 df-int 4946 df-iun 4994 df-iin 4995 df-br 5145 df-opab 5207 df-mpt 5228 df-tr 5262 df-id 5571 df-eprel 5577 df-po 5585 df-so 5586 df-fr 5628 df-se 5629 df-we 5630 df-xp 5679 df-rel 5680 df-cnv 5681 df-co 5682 df-dm 5683 df-rn 5684 df-res 5685 df-ima 5686 df-pred 6301 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-isom 6552 df-riota 7369 df-ov 7416 df-oprab 7417 df-mpo 7418 df-om 7866 df-1st 7987 df-2nd 7988 df-tpos 8225 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-rdg 8424 df-1o 8480 df-2o 8481 df-oadd 8484 df-er 8718 df-map 8840 df-en 8958 df-dom 8959 df-sdom 8960 df-fin 8961 df-dju 9919 df-card 9957 df-pnf 11275 df-mnf 11276 df-xr 11277 df-ltxr 11278 df-le 11279 df-sub 11471 df-neg 11472 df-div 11897 df-nn 12238 df-2 12300 df-3 12301 df-4 12302 df-5 12303 df-6 12304 df-7 12305 df-8 12306 df-9 12307 df-n0 12498 df-xnn0 12570 df-z 12584 df-uz 12848 df-rp 13002 df-fz 13512 df-fzo 13655 df-seq 13994 df-exp 14054 df-hash 14317 df-word 14492 df-lsw 14540 df-concat 14548 df-s1 14573 df-substr 14618 df-pfx 14648 df-splice 14727 df-reverse 14736 df-s2 14826 df-struct 17110 df-sets 17127 df-slot 17145 df-ndx 17157 df-base 17175 df-ress 17204 df-plusg 17240 df-tset 17246 df-0g 17417 df-gsum 17418 df-mre 17560 df-mrc 17561 df-acs 17563 df-mgm 18594 df-sgrp 18673 df-mnd 18689 df-mhm 18734 df-submnd 18735 df-efmnd 18820 df-grp 18892 df-minusg 18893 df-subg 19077 df-ghm 19167 df-gim 19212 df-oppg 19296 df-symg 19321 df-pmtr 19396 df-psgn 19445 |
This theorem is referenced by: m2detleiblem1 22539 m2detleiblem6 22541 |
Copyright terms: Public domain | W3C validator |