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Theorem diafval 36859
Description: The partial isomorphism A for a lattice  K. (Contributed by NM, 15-Oct-2013.)
Hypotheses
Ref Expression
diaval.b  |-  B  =  ( Base `  K
)
diaval.l  |-  .<_  =  ( le `  K )
diaval.h  |-  H  =  ( LHyp `  K
)
diaval.t  |-  T  =  ( ( LTrn `  K
) `  W )
diaval.r  |-  R  =  ( ( trL `  K
) `  W )
diaval.i  |-  I  =  ( ( DIsoA `  K
) `  W )
Assertion
Ref Expression
diafval  |-  ( ( K  e.  V  /\  W  e.  H )  ->  I  =  ( x  e.  { y  e.  B  |  y  .<_  W }  |->  { f  e.  T  |  ( R `  f ) 
.<_  x } ) )
Distinct variable groups:    x, y,  .<_    x, B, y    x, f, y, K    x, R    T, f, x    f, W, x, y
Allowed substitution hints:    B( f)    R( y, f)    T( y)    H( x, y, f)    I( x, y, f)    .<_ ( f)    V( x, y, f)

Proof of Theorem diafval
Dummy variable  w is distinct from all other variables.
StepHypRef Expression
1 diaval.i . . 3  |-  I  =  ( ( DIsoA `  K
) `  W )
2 diaval.b . . . . 5  |-  B  =  ( Base `  K
)
3 diaval.l . . . . 5  |-  .<_  =  ( le `  K )
4 diaval.h . . . . 5  |-  H  =  ( LHyp `  K
)
52, 3, 4diaffval 36858 . . . 4  |-  ( K  e.  V  ->  ( DIsoA `  K )  =  ( w  e.  H  |->  ( x  e.  {
y  e.  B  | 
y  .<_  w }  |->  { f  e.  ( (
LTrn `  K ) `  w )  |  ( ( ( trL `  K
) `  w ) `  f )  .<_  x }
) ) )
65fveq1d 5874 . . 3  |-  ( K  e.  V  ->  (
( DIsoA `  K ) `  W )  =  ( ( w  e.  H  |->  ( x  e.  {
y  e.  B  | 
y  .<_  w }  |->  { f  e.  ( (
LTrn `  K ) `  w )  |  ( ( ( trL `  K
) `  w ) `  f )  .<_  x }
) ) `  W
) )
71, 6syl5eq 2510 . 2  |-  ( K  e.  V  ->  I  =  ( ( w  e.  H  |->  ( x  e.  { y  e.  B  |  y  .<_  w }  |->  { f  e.  ( ( LTrn `  K ) `  w
)  |  ( ( ( trL `  K
) `  w ) `  f )  .<_  x }
) ) `  W
) )
8 breq2 4460 . . . . 5  |-  ( w  =  W  ->  (
y  .<_  w  <->  y  .<_  W ) )
98rabbidv 3101 . . . 4  |-  ( w  =  W  ->  { y  e.  B  |  y 
.<_  w }  =  {
y  e.  B  | 
y  .<_  W } )
10 fveq2 5872 . . . . . 6  |-  ( w  =  W  ->  (
( LTrn `  K ) `  w )  =  ( ( LTrn `  K
) `  W )
)
11 diaval.t . . . . . 6  |-  T  =  ( ( LTrn `  K
) `  W )
1210, 11syl6eqr 2516 . . . . 5  |-  ( w  =  W  ->  (
( LTrn `  K ) `  w )  =  T )
13 fveq2 5872 . . . . . . . 8  |-  ( w  =  W  ->  (
( trL `  K
) `  w )  =  ( ( trL `  K ) `  W
) )
14 diaval.r . . . . . . . 8  |-  R  =  ( ( trL `  K
) `  W )
1513, 14syl6eqr 2516 . . . . . . 7  |-  ( w  =  W  ->  (
( trL `  K
) `  w )  =  R )
1615fveq1d 5874 . . . . . 6  |-  ( w  =  W  ->  (
( ( trL `  K
) `  w ) `  f )  =  ( R `  f ) )
1716breq1d 4466 . . . . 5  |-  ( w  =  W  ->  (
( ( ( trL `  K ) `  w
) `  f )  .<_  x  <->  ( R `  f )  .<_  x ) )
1812, 17rabeqbidv 3104 . . . 4  |-  ( w  =  W  ->  { f  e.  ( ( LTrn `  K ) `  w
)  |  ( ( ( trL `  K
) `  w ) `  f )  .<_  x }  =  { f  e.  T  |  ( R `  f )  .<_  x }
)
199, 18mpteq12dv 4535 . . 3  |-  ( w  =  W  ->  (
x  e.  { y  e.  B  |  y 
.<_  w }  |->  { f  e.  ( ( LTrn `  K ) `  w
)  |  ( ( ( trL `  K
) `  w ) `  f )  .<_  x }
)  =  ( x  e.  { y  e.  B  |  y  .<_  W }  |->  { f  e.  T  |  ( R `  f ) 
.<_  x } ) )
20 eqid 2457 . . 3  |-  ( w  e.  H  |->  ( x  e.  { y  e.  B  |  y  .<_  w }  |->  { f  e.  ( ( LTrn `  K ) `  w
)  |  ( ( ( trL `  K
) `  w ) `  f )  .<_  x }
) )  =  ( w  e.  H  |->  ( x  e.  { y  e.  B  |  y 
.<_  w }  |->  { f  e.  ( ( LTrn `  K ) `  w
)  |  ( ( ( trL `  K
) `  w ) `  f )  .<_  x }
) )
21 fvex 5882 . . . . 5  |-  ( Base `  K )  e.  _V
222, 21eqeltri 2541 . . . 4  |-  B  e. 
_V
2322mptrabex 6145 . . 3  |-  ( x  e.  { y  e.  B  |  y  .<_  W }  |->  { f  e.  T  |  ( R `  f ) 
.<_  x } )  e. 
_V
2419, 20, 23fvmpt 5956 . 2  |-  ( W  e.  H  ->  (
( w  e.  H  |->  ( x  e.  {
y  e.  B  | 
y  .<_  w }  |->  { f  e.  ( (
LTrn `  K ) `  w )  |  ( ( ( trL `  K
) `  w ) `  f )  .<_  x }
) ) `  W
)  =  ( x  e.  { y  e.  B  |  y  .<_  W }  |->  { f  e.  T  |  ( R `  f ) 
.<_  x } ) )
257, 24sylan9eq 2518 1  |-  ( ( K  e.  V  /\  W  e.  H )  ->  I  =  ( x  e.  { y  e.  B  |  y  .<_  W }  |->  { f  e.  T  |  ( R `  f ) 
.<_  x } ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1395    e. wcel 1819   {crab 2811   _Vcvv 3109   class class class wbr 4456    |-> cmpt 4515   ` cfv 5594   Basecbs 14643   lecple 14718   LHypclh 35809   LTrncltrn 35926   trLctrl 35984   DIsoAcdia 36856
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-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-pr 4695
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-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
This theorem is referenced by:  diaval  36860  diafn  36862
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