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Theorem diaffval 34698
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 3000 . 2  |-  ( K  e.  V  ->  K  e.  _V )
2 fveq2 5710 . . . . 5  |-  ( k  =  K  ->  ( LHyp `  k )  =  ( LHyp `  K
) )
3 diaval.h . . . . 5  |-  H  =  ( LHyp `  K
)
42, 3syl6eqr 2493 . . . 4  |-  ( k  =  K  ->  ( LHyp `  k )  =  H )
5 fveq2 5710 . . . . . . 7  |-  ( k  =  K  ->  ( Base `  k )  =  ( Base `  K
) )
6 diaval.b . . . . . . 7  |-  B  =  ( Base `  K
)
75, 6syl6eqr 2493 . . . . . 6  |-  ( k  =  K  ->  ( Base `  k )  =  B )
8 fveq2 5710 . . . . . . . 8  |-  ( k  =  K  ->  ( le `  k )  =  ( le `  K
) )
9 diaval.l . . . . . . . 8  |-  .<_  =  ( le `  K )
108, 9syl6eqr 2493 . . . . . . 7  |-  ( k  =  K  ->  ( le `  k )  = 
.<_  )
1110breqd 4322 . . . . . 6  |-  ( k  =  K  ->  (
y ( le `  k ) w  <->  y  .<_  w ) )
127, 11rabeqbidv 2986 . . . . 5  |-  ( k  =  K  ->  { y  e.  ( Base `  k
)  |  y ( le `  k ) w }  =  {
y  e.  B  | 
y  .<_  w } )
13 fveq2 5710 . . . . . . 7  |-  ( k  =  K  ->  ( LTrn `  k )  =  ( LTrn `  K
) )
1413fveq1d 5712 . . . . . 6  |-  ( k  =  K  ->  (
( LTrn `  k ) `  w )  =  ( ( LTrn `  K
) `  w )
)
15 fveq2 5710 . . . . . . . . 9  |-  ( k  =  K  ->  ( trL `  k )  =  ( trL `  K
) )
1615fveq1d 5712 . . . . . . . 8  |-  ( k  =  K  ->  (
( trL `  k
) `  w )  =  ( ( trL `  K ) `  w
) )
1716fveq1d 5712 . . . . . . 7  |-  ( k  =  K  ->  (
( ( trL `  k
) `  w ) `  f )  =  ( ( ( trL `  K
) `  w ) `  f ) )
18 eqidd 2444 . . . . . . 7  |-  ( k  =  K  ->  x  =  x )
1917, 10, 18breq123d 4325 . . . . . 6  |-  ( k  =  K  ->  (
( ( ( trL `  k ) `  w
) `  f )
( le `  k
) x  <->  ( (
( trL `  K
) `  w ) `  f )  .<_  x ) )
2014, 19rabeqbidv 2986 . . . . 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 4389 . . . 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 4389 . . 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 34697 . . 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 5720 . . . . 5  |-  ( LHyp `  K )  e.  _V
253, 24eqeltri 2513 . . . 4  |-  H  e. 
_V
2625mptex 5967 . . 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 5793 . 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 16 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 setvar class
Syntax hints:    -> wi 4    = wceq 1369    e. wcel 1756   {crab 2738   _Vcvv 2991   class class class wbr 4311    e. cmpt 4369   ` cfv 5437   Basecbs 14193   lecple 14264   LHypclh 33651   LTrncltrn 33768   trLctrl 33825   DIsoAcdia 34696
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 2423  ax-rep 4422  ax-sep 4432  ax-nul 4440  ax-pr 4550
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 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2622  df-ral 2739  df-rex 2740  df-reu 2741  df-rab 2743  df-v 2993  df-sbc 3206  df-csb 3308  df-dif 3350  df-un 3352  df-in 3354  df-ss 3361  df-nul 3657  df-if 3811  df-sn 3897  df-pr 3899  df-op 3903  df-uni 4111  df-iun 4192  df-br 4312  df-opab 4370  df-mpt 4371  df-id 4655  df-xp 4865  df-rel 4866  df-cnv 4867  df-co 4868  df-dm 4869  df-rn 4870  df-res 4871  df-ima 4872  df-iota 5400  df-fun 5439  df-fn 5440  df-f 5441  df-f1 5442  df-fo 5443  df-f1o 5444  df-fv 5445  df-disoa 34697
This theorem is referenced by:  diafval  34699
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