MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  tposeq Unicode version

Theorem tposeq 6252
Description: Equality theorem for transposition. (Contributed by Mario Carneiro, 10-Sep-2015.)
Assertion
Ref Expression
tposeq  |-  ( F  =  G  -> tpos  F  = tpos 
G )

Proof of Theorem tposeq
StepHypRef Expression
1 eqimss 3243 . . 3  |-  ( F  =  G  ->  F  C_  G )
2 tposss 6251 . . 3  |-  ( F 
C_  G  -> tpos  F  C_ tpos  G )
31, 2syl 15 . 2  |-  ( F  =  G  -> tpos  F  C_ tpos  G )
4 eqimss2 3244 . . 3  |-  ( F  =  G  ->  G  C_  F )
5 tposss 6251 . . 3  |-  ( G 
C_  F  -> tpos  G  C_ tpos  F )
64, 5syl 15 . 2  |-  ( F  =  G  -> tpos  G  C_ tpos  F )
73, 6eqssd 3209 1  |-  ( F  =  G  -> tpos  F  = tpos 
G )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1632    C_ wss 3165  tpos ctpos 6249
This theorem is referenced by:  tposeqd  6253  tposeqi  6283  oppcval  13632  oppchomfval  13633  oppccofval  13635  oppcmon  13657  oppgval  14836  oppgplusfval  14837  oppglsm  14969  opprval  15422  opprmulfval  15423
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-br 4040  df-opab 4094  df-mpt 4095  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-res 4717  df-tpos 6250
  Copyright terms: Public domain W3C validator