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Theorem diafval 34673
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 34672 . . . 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 5691 . . 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 2485 . 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 4294 . . . . 5  |-  ( w  =  W  ->  (
y  .<_  w  <->  y  .<_  W ) )
98rabbidv 2962 . . . 4  |-  ( w  =  W  ->  { y  e.  B  |  y 
.<_  w }  =  {
y  e.  B  | 
y  .<_  W } )
10 fveq2 5689 . . . . . 6  |-  ( w  =  W  ->  (
( LTrn `  K ) `  w )  =  ( ( LTrn `  K
) `  W )
)
11 diaval.t . . . . . 6  |-  T  =  ( ( LTrn `  K
) `  W )
1210, 11syl6eqr 2491 . . . . 5  |-  ( w  =  W  ->  (
( LTrn `  K ) `  w )  =  T )
13 fveq2 5689 . . . . . . . 8  |-  ( w  =  W  ->  (
( trL `  K
) `  w )  =  ( ( trL `  K ) `  W
) )
14 diaval.r . . . . . . . 8  |-  R  =  ( ( trL `  K
) `  W )
1513, 14syl6eqr 2491 . . . . . . 7  |-  ( w  =  W  ->  (
( trL `  K
) `  w )  =  R )
1615fveq1d 5691 . . . . . 6  |-  ( w  =  W  ->  (
( ( trL `  K
) `  w ) `  f )  =  ( R `  f ) )
1716breq1d 4300 . . . . 5  |-  ( w  =  W  ->  (
( ( ( trL `  K ) `  w
) `  f )  .<_  x  <->  ( R `  f )  .<_  x ) )
1812, 17rabeqbidv 2965 . . . 4  |-  ( w  =  W  ->  { f  e.  ( ( LTrn `  K ) `  w
)  |  ( ( ( trL `  K
) `  w ) `  f )  .<_  x }  =  { f  e.  T  |  ( R `  f )  .<_  x }
)
199, 18mpteq12dv 4368 . . 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 2441 . . 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 5699 . . . . . 6  |-  ( Base `  K )  e.  _V
222, 21eqeltri 2511 . . . . 5  |-  B  e. 
_V
2322rabex 4441 . . . 4  |-  { y  e.  B  |  y 
.<_  W }  e.  _V
2423mptex 5946 . . 3  |-  ( x  e.  { y  e.  B  |  y  .<_  W }  |->  { f  e.  T  |  ( R `  f ) 
.<_  x } )  e. 
_V
2519, 20, 24fvmpt 5772 . 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 } ) )
267, 25sylan9eq 2493 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 1369    e. wcel 1756   {crab 2717   _Vcvv 2970   class class class wbr 4290    e. cmpt 4348   ` cfv 5416   Basecbs 14172   lecple 14243   LHypclh 33625   LTrncltrn 33742   trLctrl 33799   DIsoAcdia 34670
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2422  ax-rep 4401  ax-sep 4411  ax-nul 4419  ax-pr 4529
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-ral 2718  df-rex 2719  df-reu 2720  df-rab 2722  df-v 2972  df-sbc 3185  df-csb 3287  df-dif 3329  df-un 3331  df-in 3333  df-ss 3340  df-nul 3636  df-if 3790  df-sn 3876  df-pr 3878  df-op 3882  df-uni 4090  df-iun 4171  df-br 4291  df-opab 4349  df-mpt 4350  df-id 4634  df-xp 4844  df-rel 4845  df-cnv 4846  df-co 4847  df-dm 4848  df-rn 4849  df-res 4850  df-ima 4851  df-iota 5379  df-fun 5418  df-fn 5419  df-f 5420  df-f1 5421  df-fo 5422  df-f1o 5423  df-fv 5424  df-disoa 34671
This theorem is referenced by:  diaval  34674  diafn  34676
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