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Theorem dibopelval3 36976
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 2457 . . 3  |-  ( (
DIsoA `  K ) `  W )  =  ( ( DIsoA `  K ) `  W )
7 dibval3.i . . 3  |-  I  =  ( ( DIsoB `  K
) `  W )
81, 2, 3, 4, 5, 6, 7dibopelval2 36973 . 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 36861 . . 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 1395    e. wcel 1819   <.cop 4038   class class class wbr 4456    |-> cmpt 4515    _I cid 4799    |` cres 5010   ` cfv 5594   Basecbs 14643   lecple 14718   LHypclh 35809   LTrncltrn 35926   trLctrl 35984   DIsoAcdia 36856   DIsoBcdib 36966
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-id 4804  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-disoa 36857  df-dib 36967
This theorem is referenced by:  dihord2cN  37049  dihord11b  37050  dihopelvalbN  37066  dihopelvalcpre  37076  dihjatcclem4  37249
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