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Theorem dibopelval3 35945
Description: Member of the partial isomorphism B. (Contributed by NM, 3-Mar-2014.)
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
dibopelval3  |-  ( ( ( K  e.  V  /\  W  e.  H
)  /\  ( X  e.  B  /\  X  .<_  W ) )  ->  ( <. F ,  S >.  e.  ( I `  X
)  <->  ( ( F  e.  T  /\  ( R `  F )  .<_  X )  /\  S  =  .0.  ) ) )
Distinct variable groups:    g, K    g, W    T, g
Allowed substitution hints:    B( g)    R( g)    S( g)    F( g)    H( g)    I( g)    .<_ ( g)    V( g)    X( g)    .0. ( g)

Proof of Theorem dibopelval3
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 2467 . . 3  |-  ( (
DIsoA `  K ) `  W )  =  ( ( DIsoA `  K ) `  W )
7 dibval3.i . . 3  |-  I  =  ( ( DIsoB `  K
) `  W )
81, 2, 3, 4, 5, 6, 7dibopelval2 35942 . 2  |-  ( ( ( K  e.  V  /\  W  e.  H
)  /\  ( X  e.  B  /\  X  .<_  W ) )  ->  ( <. F ,  S >.  e.  ( I `  X
)  <->  ( F  e.  ( ( ( DIsoA `  K ) `  W
) `  X )  /\  S  =  .0.  ) ) )
9 dibval3.r . . . 4  |-  R  =  ( ( trL `  K
) `  W )
101, 2, 3, 4, 9, 6diaelval 35830 . . 3  |-  ( ( ( K  e.  V  /\  W  e.  H
)  /\  ( X  e.  B  /\  X  .<_  W ) )  ->  ( F  e.  ( (
( DIsoA `  K ) `  W ) `  X
)  <->  ( F  e.  T  /\  ( R `
 F )  .<_  X ) ) )
1110anbi1d 704 . 2  |-  ( ( ( K  e.  V  /\  W  e.  H
)  /\  ( X  e.  B  /\  X  .<_  W ) )  ->  (
( F  e.  ( ( ( DIsoA `  K
) `  W ) `  X )  /\  S  =  .0.  )  <->  ( ( F  e.  T  /\  ( R `  F ) 
.<_  X )  /\  S  =  .0.  ) ) )
128, 11bitrd 253 1  |-  ( ( ( K  e.  V  /\  W  e.  H
)  /\  ( X  e.  B  /\  X  .<_  W ) )  ->  ( <. F ,  S >.  e.  ( I `  X
)  <->  ( ( F  e.  T  /\  ( R `  F )  .<_  X )  /\  S  =  .0.  ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1379    e. wcel 1767   <.cop 4033   class class class wbr 4447    |-> cmpt 4505    _I cid 4790    |` cres 5001   ` cfv 5586   Basecbs 14483   lecple 14555   LHypclh 34780   LTrncltrn 34897   trLctrl 34954   DIsoAcdia 35825   DIsoBcdib 35935
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-disoa 35826  df-dib 35936
This theorem is referenced by:  dihord2cN  36018  dihord11b  36019  dihopelvalbN  36035  dihopelvalcpre  36045  dihjatcclem4  36218
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