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Theorem dibval3N 31944
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 2436 . . 3  |-  ( (
DIsoA `  K ) `  W )  =  ( ( DIsoA `  K ) `  W )
7 dibval3.i . . 3  |-  I  =  ( ( DIsoB `  K
) `  W )
81, 2, 3, 4, 5, 6, 7dibval2 31942 . 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 31830 . . 3  |-  ( ( ( K  e.  V  /\  W  e.  H
)  /\  ( X  e.  B  /\  X  .<_  W ) )  ->  (
( ( DIsoA `  K
) `  W ) `  X )  =  {
f  e.  T  | 
( R `  f
)  .<_  X } )
1110xpeq1d 4901 . 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 2468 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 359    = wceq 1652    e. wcel 1725   {crab 2709   {csn 3814   class class class wbr 4212    e. cmpt 4266    _I cid 4493    X. cxp 4876    |` cres 4880   ` cfv 5454   Basecbs 13469   lecple 13536   LHypclh 30781   LTrncltrn 30898   trLctrl 30955   DIsoAcdia 31826   DIsoBcdib 31936
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-13 1727  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-pow 4377  ax-pr 4403  ax-un 4701
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-pw 3801  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  df-dib 31937
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