Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  dibval3N Unicode version

Theorem dibval3N 30709
Description: Value of the partial isomorphism B for a lattice  K. (Contributed by NM, 24-Feb-2014.) (New usage is discouraged.)
Hypotheses
Ref Expression
dibval3.b  |-  B  =  ( Base `  K
)
dibval3.l  |-  .<_  =  ( le `  K )
dibval3.h  |-  H  =  ( LHyp `  K
)
dibval3.t  |-  T  =  ( ( LTrn `  K
) `  W )
dibval3.r  |-  R  =  ( ( trL `  K
) `  W )
dibval3.o  |-  .0.  =  ( g  e.  T  |->  (  _I  |`  B ) )
dibval3.i  |-  I  =  ( ( DIsoB `  K
) `  W )
Assertion
Ref Expression
dibval3N  |-  ( ( ( K  e.  V  /\  W  e.  H
)  /\  ( X  e.  B  /\  X  .<_  W ) )  ->  (
I `  X )  =  ( { f  e.  T  |  ( R `  f ) 
.<_  X }  X.  {  .0.  } ) )
Distinct variable groups:    f, K    g, K    T, f    f, W   
g, W    f, X
Allowed substitution hints:    B( f, g)    R( f, g)    T( g)    H( f, g)    I( f, g)    .<_ ( f, g)    V( f, g)    X( g)    .0. ( f, g)

Proof of Theorem dibval3N
StepHypRef Expression
1 dibval3.b . . 3  |-  B  =  ( Base `  K
)
2 dibval3.l . . 3  |-  .<_  =  ( le `  K )
3 dibval3.h . . 3  |-  H  =  ( LHyp `  K
)
4 dibval3.t . . 3  |-  T  =  ( ( LTrn `  K
) `  W )
5 dibval3.o . . 3  |-  .0.  =  ( g  e.  T  |->  (  _I  |`  B ) )
6 eqid 2283 . . 3  |-  ( (
DIsoA `  K ) `  W )  =  ( ( DIsoA `  K ) `  W )
7 dibval3.i . . 3  |-  I  =  ( ( DIsoB `  K
) `  W )
81, 2, 3, 4, 5, 6, 7dibval2 30707 . 2  |-  ( ( ( K  e.  V  /\  W  e.  H
)  /\  ( X  e.  B  /\  X  .<_  W ) )  ->  (
I `  X )  =  ( ( ( ( DIsoA `  K ) `  W ) `  X
)  X.  {  .0.  } ) )
9 dibval3.r . . . 4  |-  R  =  ( ( trL `  K
) `  W )
101, 2, 3, 4, 9, 6diaval 30595 . . 3  |-  ( ( ( K  e.  V  /\  W  e.  H
)  /\  ( X  e.  B  /\  X  .<_  W ) )  ->  (
( ( DIsoA `  K
) `  W ) `  X )  =  {
f  e.  T  | 
( R `  f
)  .<_  X } )
1110xpeq1d 4712 . 2  |-  ( ( ( K  e.  V  /\  W  e.  H
)  /\  ( X  e.  B  /\  X  .<_  W ) )  ->  (
( ( ( DIsoA `  K ) `  W
) `  X )  X.  {  .0.  } )  =  ( { f  e.  T  |  ( R `  f ) 
.<_  X }  X.  {  .0.  } ) )
128, 11eqtrd 2315 1  |-  ( ( ( K  e.  V  /\  W  e.  H
)  /\  ( X  e.  B  /\  X  .<_  W ) )  ->  (
I `  X )  =  ( { f  e.  T  |  ( R `  f ) 
.<_  X }  X.  {  .0.  } ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684   {crab 2547   {csn 3640   class class class wbr 4023    e. cmpt 4077    _I cid 4304    X. cxp 4687    |` cres 4691   ` cfv 5255   Basecbs 13148   lecple 13215   LHypclh 29546   LTrncltrn 29663   trLctrl 29720   DIsoAcdia 30591   DIsoBcdib 30701
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-disoa 30592  df-dib 30702
  Copyright terms: Public domain W3C validator