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Theorem dibelval3 31337
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 2283 . . . 4  |-  ( (
DIsoA `  K ) `  W )  =  ( ( DIsoA `  K ) `  W )
7 dibval3.i . . . 4  |-  I  =  ( ( DIsoB `  K
) `  W )
81, 2, 3, 4, 5, 6, 7dibval2 31334 . . 3  |-  ( ( ( K  e.  V  /\  W  e.  H
)  /\  ( X  e.  B  /\  X  .<_  W ) )  ->  (
I `  X )  =  ( ( ( ( DIsoA `  K ) `  W ) `  X
)  X.  {  .0.  } ) )
98eleq2d 2350 . 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 31223 . . . . . 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 3655 . . . . . . . . 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 1569 . . . . . 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 5539 . . . . . . . . . 10  |-  ( (
LTrn `  K ) `  W )  e.  _V
194, 18eqeltri 2353 . . . . . . . . 9  |-  T  e. 
_V
2019mptex 5746 . . . . . . . 8  |-  ( g  e.  T  |->  (  _I  |`  B ) )  e. 
_V
215, 20eqeltri 2353 . . . . . . 7  |-  .0.  e.  _V
22 opeq2 3797 . . . . . . . . 9  |-  ( s  =  .0.  ->  <. f ,  s >.  =  <. f ,  .0.  >. )
2322eqeq2d 2294 . . . . . . . 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 2823 . . . . . 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 1612 . . 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 4706 . . 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 2549 . . 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 1528    = wceq 1623    e. wcel 1684   E.wrex 2544   _Vcvv 2788   {csn 3640   <.cop 3643   class class class wbr 4023    e. cmpt 4077    _I cid 4304    X. cxp 4687    |` cres 4691   ` cfv 5255   Basecbs 13148   lecple 13215   LHypclh 30173   LTrncltrn 30290   trLctrl 30347   DIsoAcdia 31218   DIsoBcdib 31328
This theorem is referenced by:  cdlemn11pre  31400  dihord2pre  31415
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-disoa 31219  df-dib 31329
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