Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  diatrl Structured version   Unicode version

Theorem diatrl 31842
Description: Trace of a member of the partial isomorphism A. (Contributed by NM, 17-Jan-2014.)
Hypotheses
Ref Expression
diatrl.b  |-  B  =  ( Base `  K
)
diatrl.l  |-  .<_  =  ( le `  K )
diatrl.h  |-  H  =  ( LHyp `  K
)
diatrl.t  |-  T  =  ( ( LTrn `  K
) `  W )
diatrl.r  |-  R  =  ( ( trL `  K
) `  W )
diatrl.i  |-  I  =  ( ( DIsoA `  K
) `  W )
Assertion
Ref Expression
diatrl  |-  ( ( ( K  e.  V  /\  W  e.  H
)  /\  ( X  e.  B  /\  X  .<_  W )  /\  F  e.  ( I `  X
) )  ->  ( R `  F )  .<_  X )

Proof of Theorem diatrl
StepHypRef Expression
1 diatrl.b . . . 4  |-  B  =  ( Base `  K
)
2 diatrl.l . . . 4  |-  .<_  =  ( le `  K )
3 diatrl.h . . . 4  |-  H  =  ( LHyp `  K
)
4 diatrl.t . . . 4  |-  T  =  ( ( LTrn `  K
) `  W )
5 diatrl.r . . . 4  |-  R  =  ( ( trL `  K
) `  W )
6 diatrl.i . . . 4  |-  I  =  ( ( DIsoA `  K
) `  W )
71, 2, 3, 4, 5, 6diaelval 31831 . . 3  |-  ( ( ( K  e.  V  /\  W  e.  H
)  /\  ( X  e.  B  /\  X  .<_  W ) )  ->  ( F  e.  ( I `  X )  <->  ( F  e.  T  /\  ( R `  F )  .<_  X ) ) )
8 simpr 448 . . 3  |-  ( ( F  e.  T  /\  ( R `  F ) 
.<_  X )  ->  ( R `  F )  .<_  X )
97, 8syl6bi 220 . 2  |-  ( ( ( K  e.  V  /\  W  e.  H
)  /\  ( X  e.  B  /\  X  .<_  W ) )  ->  ( F  e.  ( I `  X )  ->  ( R `  F )  .<_  X ) )
1093impia 1150 1  |-  ( ( ( K  e.  V  /\  W  e.  H
)  /\  ( X  e.  B  /\  X  .<_  W )  /\  F  e.  ( I `  X
) )  ->  ( R `  F )  .<_  X )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725   class class class wbr 4212   ` cfv 5454   Basecbs 13469   lecple 13536   LHypclh 30781   LTrncltrn 30898   trLctrl 30955   DIsoAcdia 31826
This theorem is referenced by:  dialss  31844  dibelval1st2N  31949  diblss  31968
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pr 4403
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-reu 2712  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-disoa 31827
  Copyright terms: Public domain W3C validator