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Theorem diafval 31221
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 31220 . . . 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 5527 . . 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 2327 . 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 4027 . . . . 5  |-  ( w  =  W  ->  (
y  .<_  w  <->  y  .<_  W ) )
98rabbidv 2780 . . . 4  |-  ( w  =  W  ->  { y  e.  B  |  y 
.<_  w }  =  {
y  e.  B  | 
y  .<_  W } )
10 fveq2 5525 . . . . . 6  |-  ( w  =  W  ->  (
( LTrn `  K ) `  w )  =  ( ( LTrn `  K
) `  W )
)
11 diaval.t . . . . . 6  |-  T  =  ( ( LTrn `  K
) `  W )
1210, 11syl6eqr 2333 . . . . 5  |-  ( w  =  W  ->  (
( LTrn `  K ) `  w )  =  T )
13 fveq2 5525 . . . . . . . 8  |-  ( w  =  W  ->  (
( trL `  K
) `  w )  =  ( ( trL `  K ) `  W
) )
14 diaval.r . . . . . . . 8  |-  R  =  ( ( trL `  K
) `  W )
1513, 14syl6eqr 2333 . . . . . . 7  |-  ( w  =  W  ->  (
( trL `  K
) `  w )  =  R )
1615fveq1d 5527 . . . . . 6  |-  ( w  =  W  ->  (
( ( trL `  K
) `  w ) `  f )  =  ( R `  f ) )
1716breq1d 4033 . . . . 5  |-  ( w  =  W  ->  (
( ( ( trL `  K ) `  w
) `  f )  .<_  x  <->  ( R `  f )  .<_  x ) )
1812, 17rabeqbidv 2783 . . . 4  |-  ( w  =  W  ->  { f  e.  ( ( LTrn `  K ) `  w
)  |  ( ( ( trL `  K
) `  w ) `  f )  .<_  x }  =  { f  e.  T  |  ( R `  f )  .<_  x }
)
199, 18mpteq12dv 4098 . . 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 2283 . . 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 5539 . . . . . 6  |-  ( Base `  K )  e.  _V
222, 21eqeltri 2353 . . . . 5  |-  B  e. 
_V
2322rabex 4165 . . . 4  |-  { y  e.  B  |  y 
.<_  W }  e.  _V
2423mptex 5746 . . 3  |-  ( x  e.  { y  e.  B  |  y  .<_  W }  |->  { f  e.  T  |  ( R `  f ) 
.<_  x } )  e. 
_V
2519, 20, 24fvmpt 5602 . 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 2335 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 set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684   {crab 2547   _Vcvv 2788   class class class wbr 4023    e. cmpt 4077   ` cfv 5255   Basecbs 13148   lecple 13215   LHypclh 30173   LTrncltrn 30290   trLctrl 30347   DIsoAcdia 31218
This theorem is referenced by:  diaval  31222  diafn  31224
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-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-pr 4214
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-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
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