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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 5394 | . . . . 5 ⊢ {⟨1, 2⟩, ⟨2, 1⟩} ∈ V | |
2 | 1 | snid 4627 | . . . 4 ⊢ {⟨1, 2⟩, ⟨2, 1⟩} ∈ {{⟨1, 2⟩, ⟨2, 1⟩}} |
3 | psgnprfval.0 | . . . . . . 7 ⊢ 𝐷 = {1, 2} | |
4 | 3 | fveq2i 6850 | . . . . . 6 ⊢ (pmTrsp‘𝐷) = (pmTrsp‘{1, 2}) |
5 | 4 | rneqi 5897 | . . . . 5 ⊢ ran (pmTrsp‘𝐷) = ran (pmTrsp‘{1, 2}) |
6 | pmtrprfvalrn 19277 | . . . . 5 ⊢ ran (pmTrsp‘{1, 2}) = {{⟨1, 2⟩, ⟨2, 1⟩}} | |
7 | 5, 6 | eqtri 2765 | . . . 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 19299 | . 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 1542 ∈ wcel 2107 {csn 4591 {cpr 4593 ⟨cop 4597 ran crn 5639 ‘cfv 6501 1c1 11059 -cneg 11393 2c2 12215 Basecbs 17090 SymGrpcsymg 19155 pmTrspcpmtr 19230 pmSgncpsgn 19278 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2708 ax-rep 5247 ax-sep 5261 ax-nul 5268 ax-pow 5325 ax-pr 5389 ax-un 7677 ax-cnex 11114 ax-resscn 11115 ax-1cn 11116 ax-icn 11117 ax-addcl 11118 ax-addrcl 11119 ax-mulcl 11120 ax-mulrcl 11121 ax-mulcom 11122 ax-addass 11123 ax-mulass 11124 ax-distr 11125 ax-i2m1 11126 ax-1ne0 11127 ax-1rid 11128 ax-rnegex 11129 ax-rrecex 11130 ax-cnre 11131 ax-pre-lttri 11132 ax-pre-lttrn 11133 ax-pre-ltadd 11134 ax-pre-mulgt0 11135 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-xor 1511 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-nel 3051 df-ral 3066 df-rex 3075 df-rmo 3356 df-reu 3357 df-rab 3411 df-v 3450 df-sbc 3745 df-csb 3861 df-dif 3918 df-un 3920 df-in 3922 df-ss 3932 df-pss 3934 df-nul 4288 df-if 4492 df-pw 4567 df-sn 4592 df-pr 4594 df-tp 4596 df-op 4598 df-ot 4600 df-uni 4871 df-int 4913 df-iun 4961 df-iin 4962 df-br 5111 df-opab 5173 df-mpt 5194 df-tr 5228 df-id 5536 df-eprel 5542 df-po 5550 df-so 5551 df-fr 5593 df-se 5594 df-we 5595 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-pred 6258 df-ord 6325 df-on 6326 df-lim 6327 df-suc 6328 df-iota 6453 df-fun 6503 df-fn 6504 df-f 6505 df-f1 6506 df-fo 6507 df-f1o 6508 df-fv 6509 df-isom 6510 df-riota 7318 df-ov 7365 df-oprab 7366 df-mpo 7367 df-om 7808 df-1st 7926 df-2nd 7927 df-tpos 8162 df-frecs 8217 df-wrecs 8248 df-recs 8322 df-rdg 8361 df-1o 8417 df-2o 8418 df-oadd 8421 df-er 8655 df-map 8774 df-en 8891 df-dom 8892 df-sdom 8893 df-fin 8894 df-dju 9844 df-card 9882 df-pnf 11198 df-mnf 11199 df-xr 11200 df-ltxr 11201 df-le 11202 df-sub 11394 df-neg 11395 df-div 11820 df-nn 12161 df-2 12223 df-3 12224 df-4 12225 df-5 12226 df-6 12227 df-7 12228 df-8 12229 df-9 12230 df-n0 12421 df-xnn0 12493 df-z 12507 df-uz 12771 df-rp 12923 df-fz 13432 df-fzo 13575 df-seq 13914 df-exp 13975 df-hash 14238 df-word 14410 df-lsw 14458 df-concat 14466 df-s1 14491 df-substr 14536 df-pfx 14566 df-splice 14645 df-reverse 14654 df-s2 14744 df-struct 17026 df-sets 17043 df-slot 17061 df-ndx 17073 df-base 17091 df-ress 17120 df-plusg 17153 df-tset 17159 df-0g 17330 df-gsum 17331 df-mre 17473 df-mrc 17474 df-acs 17476 df-mgm 18504 df-sgrp 18553 df-mnd 18564 df-mhm 18608 df-submnd 18609 df-efmnd 18686 df-grp 18758 df-minusg 18759 df-subg 18932 df-ghm 19013 df-gim 19056 df-oppg 19131 df-symg 19156 df-pmtr 19231 df-psgn 19280 |
This theorem is referenced by: m2detleiblem1 21989 m2detleiblem6 21991 |
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