Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  dibelval1st1 Structured version   Unicode version

Theorem dibelval1st1 31949
Description: Membership in value of the partial isomorphism B for a lattice  K. (Contributed by NM, 13-Feb-2014.)
Hypotheses
Ref Expression
dibelval1st1.b  |-  B  =  ( Base `  K
)
dibelval1st1.l  |-  .<_  =  ( le `  K )
dibelval1st1.h  |-  H  =  ( LHyp `  K
)
dibelval1st1.t  |-  T  =  ( ( LTrn `  K
) `  W )
dibelval1st1.i  |-  I  =  ( ( DIsoB `  K
) `  W )
Assertion
Ref Expression
dibelval1st1  |-  ( ( ( K  e.  V  /\  W  e.  H
)  /\  ( X  e.  B  /\  X  .<_  W )  /\  Y  e.  ( I `  X
) )  ->  ( 1st `  Y )  e.  T )

Proof of Theorem dibelval1st1
StepHypRef Expression
1 dibelval1st1.b . . 3  |-  B  =  ( Base `  K
)
2 dibelval1st1.l . . 3  |-  .<_  =  ( le `  K )
3 dibelval1st1.h . . 3  |-  H  =  ( LHyp `  K
)
4 eqid 2437 . . 3  |-  ( (
DIsoA `  K ) `  W )  =  ( ( DIsoA `  K ) `  W )
5 dibelval1st1.i . . 3  |-  I  =  ( ( DIsoB `  K
) `  W )
61, 2, 3, 4, 5dibelval1st 31948 . 2  |-  ( ( ( K  e.  V  /\  W  e.  H
)  /\  ( X  e.  B  /\  X  .<_  W )  /\  Y  e.  ( I `  X
) )  ->  ( 1st `  Y )  e.  ( ( ( DIsoA `  K ) `  W
) `  X )
)
7 dibelval1st1.t . . 3  |-  T  =  ( ( LTrn `  K
) `  W )
81, 2, 3, 7, 4diael 31842 . 2  |-  ( ( ( K  e.  V  /\  W  e.  H
)  /\  ( X  e.  B  /\  X  .<_  W )  /\  ( 1st `  Y )  e.  ( ( ( DIsoA `  K
) `  W ) `  X ) )  -> 
( 1st `  Y
)  e.  T )
96, 8syld3an3 1230 1  |-  ( ( ( K  e.  V  /\  W  e.  H
)  /\  ( X  e.  B  /\  X  .<_  W )  /\  Y  e.  ( I `  X
) )  ->  ( 1st `  Y )  e.  T )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1726   class class class wbr 4213   ` cfv 5455   1stc1st 6348   Basecbs 13470   lecple 13537   LHypclh 30782   LTrncltrn 30899   DIsoAcdia 31827   DIsoBcdib 31937
This theorem is referenced by:  diblss  31969
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2418  ax-rep 4321  ax-sep 4331  ax-nul 4339  ax-pow 4378  ax-pr 4404  ax-un 4702
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2286  df-mo 2287  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ne 2602  df-ral 2711  df-rex 2712  df-reu 2713  df-rab 2715  df-v 2959  df-sbc 3163  df-csb 3253  df-dif 3324  df-un 3326  df-in 3328  df-ss 3335  df-nul 3630  df-if 3741  df-pw 3802  df-sn 3821  df-pr 3822  df-op 3824  df-uni 4017  df-iun 4096  df-br 4214  df-opab 4268  df-mpt 4269  df-id 4499  df-xp 4885  df-rel 4886  df-cnv 4887  df-co 4888  df-dm 4889  df-rn 4890  df-res 4891  df-ima 4892  df-iota 5419  df-fun 5457  df-fn 5458  df-f 5459  df-f1 5460  df-fo 5461  df-f1o 5462  df-fv 5463  df-1st 6350  df-disoa 31828  df-dib 31938
  Copyright terms: Public domain W3C validator