Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  dibelval3 Structured version   Unicode version

Theorem dibelval3 31946
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 2437 . . . 4  |-  ( (
DIsoA `  K ) `  W )  =  ( ( DIsoA `  K ) `  W )
7 dibval3.i . . . 4  |-  I  =  ( ( DIsoB `  K
) `  W )
81, 2, 3, 4, 5, 6, 7dibval2 31943 . . 3  |-  ( ( ( K  e.  V  /\  W  e.  H
)  /\  ( X  e.  B  /\  X  .<_  W ) )  ->  (
I `  X )  =  ( ( ( ( DIsoA `  K ) `  W ) `  X
)  X.  {  .0.  } ) )
98eleq2d 2504 . 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 31832 . . . . . 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 687 . . . . 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 776 . . . . . . . 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 3830 . . . . . . . . 9  |-  ( s  e.  {  .0.  }  <->  s  =  .0.  )
1514anbi1i 678 . . . . . . . 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 242 . . . . . . 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 1593 . . . . . 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 5743 . . . . . . . . . 10  |-  ( (
LTrn `  K ) `  W )  e.  _V
194, 18eqeltri 2507 . . . . . . . . 9  |-  T  e. 
_V
2019mptex 5967 . . . . . . . 8  |-  ( g  e.  T  |->  (  _I  |`  B ) )  e. 
_V
215, 20eqeltri 2507 . . . . . . 7  |-  .0.  e.  _V
22 opeq2 3986 . . . . . . . . 9  |-  ( s  =  .0.  ->  <. f ,  s >.  =  <. f ,  .0.  >. )
2322eqeq2d 2448 . . . . . . . 8  |-  ( s  =  .0.  ->  ( Y  =  <. f ,  s >.  <->  Y  =  <. f ,  .0.  >. )
)
2423anbi2d 686 . . . . . . 7  |-  ( s  =  .0.  ->  (
( f  e.  ( ( ( DIsoA `  K
) `  W ) `  X )  /\  Y  =  <. f ,  s
>. )  <->  ( f  e.  ( ( ( DIsoA `  K ) `  W
) `  X )  /\  Y  =  <. f ,  .0.  >. )
) )
2521, 24ceqsexv 2992 . . . . . 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 242 . . . . 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 632 . . . . . 6  |-  ( ( ( f  e.  T  /\  Y  =  <. f ,  .0.  >. )  /\  ( R `  f
)  .<_  X )  <->  ( f  e.  T  /\  ( Y  =  <. f ,  .0.  >.  /\  ( R `  f )  .<_  X ) ) )
28 an32 775 . . . . . 6  |-  ( ( ( f  e.  T  /\  Y  =  <. f ,  .0.  >. )  /\  ( R `  f
)  .<_  X )  <->  ( (
f  e.  T  /\  ( R `  f ) 
.<_  X )  /\  Y  =  <. f ,  .0.  >.
) )
2927, 28bitr3i 244 . . . . 5  |-  ( ( f  e.  T  /\  ( Y  =  <. f ,  .0.  >.  /\  ( R `  f )  .<_  X ) )  <->  ( (
f  e.  T  /\  ( R `  f ) 
.<_  X )  /\  Y  =  <. f ,  .0.  >.
) )
3012, 26, 293bitr4g 281 . . . 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 1637 . . 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 4896 . . 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 2712 . . 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 281 . 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 246 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 178    /\ wa 360   E.wex 1551    = wceq 1653    e. wcel 1726   E.wrex 2707   _Vcvv 2957   {csn 3815   <.cop 3818   class class class wbr 4213    e. cmpt 4267    _I cid 4494    X. cxp 4877    |` cres 4881   ` cfv 5455   Basecbs 13470   lecple 13537   LHypclh 30782   LTrncltrn 30899   trLctrl 30956   DIsoAcdia 31827   DIsoBcdib 31937
This theorem is referenced by:  cdlemn11pre  32009  dihord2pre  32024
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2418  ax-rep 4321  ax-sep 4331  ax-nul 4339  ax-pow 4378  ax-pr 4404  ax-un 4702
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2286  df-mo 2287  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ne 2602  df-ral 2711  df-rex 2712  df-reu 2713  df-rab 2715  df-v 2959  df-sbc 3163  df-csb 3253  df-dif 3324  df-un 3326  df-in 3328  df-ss 3335  df-nul 3630  df-if 3741  df-pw 3802  df-sn 3821  df-pr 3822  df-op 3824  df-uni 4017  df-iun 4096  df-br 4214  df-opab 4268  df-mpt 4269  df-id 4499  df-xp 4885  df-rel 4886  df-cnv 4887  df-co 4888  df-dm 4889  df-rn 4890  df-res 4891  df-ima 4892  df-iota 5419  df-fun 5457  df-fn 5458  df-f 5459  df-f1 5460  df-fo 5461  df-f1o 5462  df-fv 5463  df-disoa 31828  df-dib 31938
  Copyright terms: Public domain W3C validator