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Theorem dibelval3 31959
Description: Member of the partial isomorphism B. (Contributed by NM, 26-Feb-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
dibelval3  |-  ( ( ( K  e.  V  /\  W  e.  H
)  /\  ( X  e.  B  /\  X  .<_  W ) )  ->  ( Y  e.  ( I `  X )  <->  E. f  e.  T  ( Y  =  <. f ,  .0.  >.  /\  ( R `  f
)  .<_  X ) ) )
Distinct variable groups:    f, K    g, K    T, f    f, W   
g, W    f, X    .<_ , f    B, f    f, H    .0. , f    T, g    f, V   
f, Y
Allowed substitution hints:    B( g)    R( f, g)    H( g)    I(
f, g)    .<_ ( g)    V( g)    X( g)    Y( g)    .0. ( g)

Proof of Theorem dibelval3
Dummy variable  s is distinct from all other variables.
StepHypRef Expression
1 dibval3.b . . . 4  |-  B  =  ( Base `  K
)
2 dibval3.l . . . 4  |-  .<_  =  ( le `  K )
3 dibval3.h . . . 4  |-  H  =  ( LHyp `  K
)
4 dibval3.t . . . 4  |-  T  =  ( ( LTrn `  K
) `  W )
5 dibval3.o . . . 4  |-  .0.  =  ( g  e.  T  |->  (  _I  |`  B ) )
6 eqid 2296 . . . 4  |-  ( (
DIsoA `  K ) `  W )  =  ( ( DIsoA `  K ) `  W )
7 dibval3.i . . . 4  |-  I  =  ( ( DIsoB `  K
) `  W )
81, 2, 3, 4, 5, 6, 7dibval2 31956 . . 3  |-  ( ( ( K  e.  V  /\  W  e.  H
)  /\  ( X  e.  B  /\  X  .<_  W ) )  ->  (
I `  X )  =  ( ( ( ( DIsoA `  K ) `  W ) `  X
)  X.  {  .0.  } ) )
98eleq2d 2363 . 2  |-  ( ( ( K  e.  V  /\  W  e.  H
)  /\  ( X  e.  B  /\  X  .<_  W ) )  ->  ( Y  e.  ( I `  X )  <->  Y  e.  ( ( ( (
DIsoA `  K ) `  W ) `  X
)  X.  {  .0.  } ) ) )
10 dibval3.r . . . . . . 7  |-  R  =  ( ( trL `  K
) `  W )
111, 2, 3, 4, 10, 6diaelval 31845 . . . . . 6  |-  ( ( ( K  e.  V  /\  W  e.  H
)  /\  ( X  e.  B  /\  X  .<_  W ) )  ->  (
f  e.  ( ( ( DIsoA `  K ) `  W ) `  X
)  <->  ( f  e.  T  /\  ( R `
 f )  .<_  X ) ) )
1211anbi1d 685 . . . . 5  |-  ( ( ( K  e.  V  /\  W  e.  H
)  /\  ( X  e.  B  /\  X  .<_  W ) )  ->  (
( f  e.  ( ( ( DIsoA `  K
) `  W ) `  X )  /\  Y  =  <. f ,  .0.  >.
)  <->  ( ( f  e.  T  /\  ( R `  f )  .<_  X )  /\  Y  =  <. f ,  .0.  >.
) ) )
13 an13 774 . . . . . . . 8  |-  ( ( Y  =  <. f ,  s >.  /\  (
f  e.  ( ( ( DIsoA `  K ) `  W ) `  X
)  /\  s  e.  {  .0.  } ) )  <-> 
( s  e.  {  .0.  }  /\  ( f  e.  ( ( (
DIsoA `  K ) `  W ) `  X
)  /\  Y  =  <. f ,  s >.
) ) )
14 elsn 3668 . . . . . . . . 9  |-  ( s  e.  {  .0.  }  <->  s  =  .0.  )
1514anbi1i 676 . . . . . . . 8  |-  ( ( s  e.  {  .0.  }  /\  ( f  e.  ( ( ( DIsoA `  K ) `  W
) `  X )  /\  Y  =  <. f ,  s >. )
)  <->  ( s  =  .0.  /\  ( f  e.  ( ( (
DIsoA `  K ) `  W ) `  X
)  /\  Y  =  <. f ,  s >.
) ) )
1613, 15bitri 240 . . . . . . 7  |-  ( ( Y  =  <. f ,  s >.  /\  (
f  e.  ( ( ( DIsoA `  K ) `  W ) `  X
)  /\  s  e.  {  .0.  } ) )  <-> 
( s  =  .0. 
/\  ( f  e.  ( ( ( DIsoA `  K ) `  W
) `  X )  /\  Y  =  <. f ,  s >. )
) )
1716exbii 1572 . . . . . 6  |-  ( E. s ( Y  = 
<. f ,  s >.  /\  ( f  e.  ( ( ( DIsoA `  K
) `  W ) `  X )  /\  s  e.  {  .0.  } ) )  <->  E. s ( s  =  .0.  /\  (
f  e.  ( ( ( DIsoA `  K ) `  W ) `  X
)  /\  Y  =  <. f ,  s >.
) ) )
18 fvex 5555 . . . . . . . . . 10  |-  ( (
LTrn `  K ) `  W )  e.  _V
194, 18eqeltri 2366 . . . . . . . . 9  |-  T  e. 
_V
2019mptex 5762 . . . . . . . 8  |-  ( g  e.  T  |->  (  _I  |`  B ) )  e. 
_V
215, 20eqeltri 2366 . . . . . . 7  |-  .0.  e.  _V
22 opeq2 3813 . . . . . . . . 9  |-  ( s  =  .0.  ->  <. f ,  s >.  =  <. f ,  .0.  >. )
2322eqeq2d 2307 . . . . . . . 8  |-  ( s  =  .0.  ->  ( Y  =  <. f ,  s >.  <->  Y  =  <. f ,  .0.  >. )
)
2423anbi2d 684 . . . . . . 7  |-  ( s  =  .0.  ->  (
( f  e.  ( ( ( DIsoA `  K
) `  W ) `  X )  /\  Y  =  <. f ,  s
>. )  <->  ( f  e.  ( ( ( DIsoA `  K ) `  W
) `  X )  /\  Y  =  <. f ,  .0.  >. )
) )
2521, 24ceqsexv 2836 . . . . . 6  |-  ( E. s ( s  =  .0.  /\  ( f  e.  ( ( (
DIsoA `  K ) `  W ) `  X
)  /\  Y  =  <. f ,  s >.
) )  <->  ( f  e.  ( ( ( DIsoA `  K ) `  W
) `  X )  /\  Y  =  <. f ,  .0.  >. )
)
2617, 25bitri 240 . . . . 5  |-  ( E. s ( Y  = 
<. f ,  s >.  /\  ( f  e.  ( ( ( DIsoA `  K
) `  W ) `  X )  /\  s  e.  {  .0.  } ) )  <->  ( f  e.  ( ( ( DIsoA `  K ) `  W
) `  X )  /\  Y  =  <. f ,  .0.  >. )
)
27 anass 630 . . . . . 6  |-  ( ( ( f  e.  T  /\  Y  =  <. f ,  .0.  >. )  /\  ( R `  f
)  .<_  X )  <->  ( f  e.  T  /\  ( Y  =  <. f ,  .0.  >.  /\  ( R `  f )  .<_  X ) ) )
28 an32 773 . . . . . 6  |-  ( ( ( f  e.  T  /\  Y  =  <. f ,  .0.  >. )  /\  ( R `  f
)  .<_  X )  <->  ( (
f  e.  T  /\  ( R `  f ) 
.<_  X )  /\  Y  =  <. f ,  .0.  >.
) )
2927, 28bitr3i 242 . . . . 5  |-  ( ( f  e.  T  /\  ( Y  =  <. f ,  .0.  >.  /\  ( R `  f )  .<_  X ) )  <->  ( (
f  e.  T  /\  ( R `  f ) 
.<_  X )  /\  Y  =  <. f ,  .0.  >.
) )
3012, 26, 293bitr4g 279 . . . 4  |-  ( ( ( K  e.  V  /\  W  e.  H
)  /\  ( X  e.  B  /\  X  .<_  W ) )  ->  ( E. s ( Y  = 
<. f ,  s >.  /\  ( f  e.  ( ( ( DIsoA `  K
) `  W ) `  X )  /\  s  e.  {  .0.  } ) )  <->  ( f  e.  T  /\  ( Y  =  <. f ,  .0.  >.  /\  ( R `  f
)  .<_  X ) ) ) )
3130exbidv 1616 . . 3  |-  ( ( ( K  e.  V  /\  W  e.  H
)  /\  ( X  e.  B  /\  X  .<_  W ) )  ->  ( E. f E. s ( Y  =  <. f ,  s >.  /\  (
f  e.  ( ( ( DIsoA `  K ) `  W ) `  X
)  /\  s  e.  {  .0.  } ) )  <->  E. f ( f  e.  T  /\  ( Y  =  <. f ,  .0.  >.  /\  ( R `  f
)  .<_  X ) ) ) )
32 elxp 4722 . . 3  |-  ( Y  e.  ( ( ( ( DIsoA `  K ) `  W ) `  X
)  X.  {  .0.  } )  <->  E. f E. s
( Y  =  <. f ,  s >.  /\  (
f  e.  ( ( ( DIsoA `  K ) `  W ) `  X
)  /\  s  e.  {  .0.  } ) ) )
33 df-rex 2562 . . 3  |-  ( E. f  e.  T  ( Y  =  <. f ,  .0.  >.  /\  ( R `  f )  .<_  X )  <->  E. f ( f  e.  T  /\  ( Y  =  <. f ,  .0.  >.  /\  ( R `  f
)  .<_  X ) ) )
3431, 32, 333bitr4g 279 . 2  |-  ( ( ( K  e.  V  /\  W  e.  H
)  /\  ( X  e.  B  /\  X  .<_  W ) )  ->  ( Y  e.  ( (
( ( DIsoA `  K
) `  W ) `  X )  X.  {  .0.  } )  <->  E. f  e.  T  ( Y  =  <. f ,  .0.  >.  /\  ( R `  f
)  .<_  X ) ) )
359, 34bitrd 244 1  |-  ( ( ( K  e.  V  /\  W  e.  H
)  /\  ( X  e.  B  /\  X  .<_  W ) )  ->  ( Y  e.  ( I `  X )  <->  E. f  e.  T  ( Y  =  <. f ,  .0.  >.  /\  ( R `  f
)  .<_  X ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358   E.wex 1531    = wceq 1632    e. wcel 1696   E.wrex 2557   _Vcvv 2801   {csn 3653   <.cop 3656   class class class wbr 4039    e. cmpt 4093    _I cid 4320    X. cxp 4703    |` cres 4707   ` cfv 5271   Basecbs 13164   lecple 13231   LHypclh 30795   LTrncltrn 30912   trLctrl 30969   DIsoAcdia 31840   DIsoBcdib 31950
This theorem is referenced by:  cdlemn11pre  32022  dihord2pre  32037
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-disoa 31841  df-dib 31951
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