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Theorem dibopelval3 35132
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 2454 . . 3  |-  ( (
DIsoA `  K ) `  W )  =  ( ( DIsoA `  K ) `  W )
7 dibval3.i . . 3  |-  I  =  ( ( DIsoB `  K
) `  W )
81, 2, 3, 4, 5, 6, 7dibopelval2 35129 . 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 35017 . . 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 1370    e. wcel 1758   <.cop 3992   class class class wbr 4401    |-> cmpt 4459    _I cid 4740    |` cres 4951   ` cfv 5527   Basecbs 14293   lecple 14365   LHypclh 33967   LTrncltrn 34084   trLctrl 34141   DIsoAcdia 35012   DIsoBcdib 35122
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4512  ax-sep 4522  ax-nul 4530  ax-pow 4579  ax-pr 4640  ax-un 6483
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-reu 2806  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3397  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-nul 3747  df-if 3901  df-pw 3971  df-sn 3987  df-pr 3989  df-op 3993  df-uni 4201  df-iun 4282  df-br 4402  df-opab 4460  df-mpt 4461  df-id 4745  df-xp 4955  df-rel 4956  df-cnv 4957  df-co 4958  df-dm 4959  df-rn 4960  df-res 4961  df-ima 4962  df-iota 5490  df-fun 5529  df-fn 5530  df-f 5531  df-f1 5532  df-fo 5533  df-f1o 5534  df-fv 5535  df-disoa 35013  df-dib 35123
This theorem is referenced by:  dihord2cN  35205  dihord11b  35206  dihopelvalbN  35222  dihopelvalcpre  35232  dihjatcclem4  35405
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