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Theorem diaffval 31842
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
)
Assertion
Ref Expression
diaffval  |-  ( 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 }
) ) )
Distinct variable groups:    x, w, y,  .<_    w, B, x, y   
w, H    w, f, x, y, K
Allowed substitution hints:    B( f)    H( x, y, f)    .<_ ( f)    V( x, y, w, f)

Proof of Theorem diaffval
Dummy variable  k is distinct from all other variables.
StepHypRef Expression
1 elex 2809 . 2  |-  ( K  e.  V  ->  K  e.  _V )
2 fveq2 5541 . . . . 5  |-  ( k  =  K  ->  ( LHyp `  k )  =  ( LHyp `  K
) )
3 diaval.h . . . . 5  |-  H  =  ( LHyp `  K
)
42, 3syl6eqr 2346 . . . 4  |-  ( k  =  K  ->  ( LHyp `  k )  =  H )
5 fveq2 5541 . . . . . . 7  |-  ( k  =  K  ->  ( Base `  k )  =  ( Base `  K
) )
6 diaval.b . . . . . . 7  |-  B  =  ( Base `  K
)
75, 6syl6eqr 2346 . . . . . 6  |-  ( k  =  K  ->  ( Base `  k )  =  B )
8 fveq2 5541 . . . . . . . 8  |-  ( k  =  K  ->  ( le `  k )  =  ( le `  K
) )
9 diaval.l . . . . . . . 8  |-  .<_  =  ( le `  K )
108, 9syl6eqr 2346 . . . . . . 7  |-  ( k  =  K  ->  ( le `  k )  = 
.<_  )
1110breqd 4050 . . . . . 6  |-  ( k  =  K  ->  (
y ( le `  k ) w  <->  y  .<_  w ) )
127, 11rabeqbidv 2796 . . . . 5  |-  ( k  =  K  ->  { y  e.  ( Base `  k
)  |  y ( le `  k ) w }  =  {
y  e.  B  | 
y  .<_  w } )
13 fveq2 5541 . . . . . . 7  |-  ( k  =  K  ->  ( LTrn `  k )  =  ( LTrn `  K
) )
1413fveq1d 5543 . . . . . 6  |-  ( k  =  K  ->  (
( LTrn `  k ) `  w )  =  ( ( LTrn `  K
) `  w )
)
15 fveq2 5541 . . . . . . . . 9  |-  ( k  =  K  ->  ( trL `  k )  =  ( trL `  K
) )
1615fveq1d 5543 . . . . . . . 8  |-  ( k  =  K  ->  (
( trL `  k
) `  w )  =  ( ( trL `  K ) `  w
) )
1716fveq1d 5543 . . . . . . 7  |-  ( k  =  K  ->  (
( ( trL `  k
) `  w ) `  f )  =  ( ( ( trL `  K
) `  w ) `  f ) )
18 eqidd 2297 . . . . . . 7  |-  ( k  =  K  ->  x  =  x )
1917, 10, 18breq123d 4053 . . . . . 6  |-  ( k  =  K  ->  (
( ( ( trL `  k ) `  w
) `  f )
( le `  k
) x  <->  ( (
( trL `  K
) `  w ) `  f )  .<_  x ) )
2014, 19rabeqbidv 2796 . . . . 5  |-  ( k  =  K  ->  { f  e.  ( ( LTrn `  k ) `  w
)  |  ( ( ( trL `  k
) `  w ) `  f ) ( le
`  k ) x }  =  { f  e.  ( ( LTrn `  K ) `  w
)  |  ( ( ( trL `  K
) `  w ) `  f )  .<_  x }
)
2112, 20mpteq12dv 4114 . . . 4  |-  ( k  =  K  ->  (
x  e.  { y  e.  ( Base `  k
)  |  y ( le `  k ) w }  |->  { f  e.  ( ( LTrn `  k ) `  w
)  |  ( ( ( trL `  k
) `  w ) `  f ) ( le
`  k ) x } )  =  ( x  e.  { y  e.  B  |  y 
.<_  w }  |->  { f  e.  ( ( LTrn `  K ) `  w
)  |  ( ( ( trL `  K
) `  w ) `  f )  .<_  x }
) )
224, 21mpteq12dv 4114 . . 3  |-  ( k  =  K  ->  (
w  e.  ( LHyp `  k )  |->  ( x  e.  { y  e.  ( Base `  k
)  |  y ( le `  k ) w }  |->  { f  e.  ( ( LTrn `  k ) `  w
)  |  ( ( ( trL `  k
) `  w ) `  f ) ( le
`  k ) x } ) )  =  ( w  e.  H  |->  ( x  e.  {
y  e.  B  | 
y  .<_  w }  |->  { f  e.  ( (
LTrn `  K ) `  w )  |  ( ( ( trL `  K
) `  w ) `  f )  .<_  x }
) ) )
23 df-disoa 31841 . . 3  |-  DIsoA  =  ( k  e.  _V  |->  ( w  e.  ( LHyp `  k )  |->  ( x  e.  { y  e.  ( Base `  k
)  |  y ( le `  k ) w }  |->  { f  e.  ( ( LTrn `  k ) `  w
)  |  ( ( ( trL `  k
) `  w ) `  f ) ( le
`  k ) x } ) ) )
24 fvex 5555 . . . . 5  |-  ( LHyp `  K )  e.  _V
253, 24eqeltri 2366 . . . 4  |-  H  e. 
_V
2625mptex 5762 . . 3  |-  ( w  e.  H  |->  ( x  e.  { y  e.  B  |  y  .<_  w }  |->  { f  e.  ( ( LTrn `  K ) `  w
)  |  ( ( ( trL `  K
) `  w ) `  f )  .<_  x }
) )  e.  _V
2722, 23, 26fvmpt 5618 . 2  |-  ( 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 }
) ) )
281, 27syl 15 1  |-  ( 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 }
) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1632    e. wcel 1696   {crab 2560   _Vcvv 2801   class class class wbr 4039    e. cmpt 4093   ` cfv 5271   Basecbs 13164   lecple 13231   LHypclh 30795   LTrncltrn 30912   trLctrl 30969   DIsoAcdia 31840
This theorem is referenced by:  diafval  31843
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-disoa 31841
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