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Theorem diaffval 37170
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 3043 . 2  |-  ( K  e.  V  ->  K  e.  _V )
2 fveq2 5774 . . . . 5  |-  ( k  =  K  ->  ( LHyp `  k )  =  ( LHyp `  K
) )
3 diaval.h . . . . 5  |-  H  =  ( LHyp `  K
)
42, 3syl6eqr 2441 . . . 4  |-  ( k  =  K  ->  ( LHyp `  k )  =  H )
5 fveq2 5774 . . . . . . 7  |-  ( k  =  K  ->  ( Base `  k )  =  ( Base `  K
) )
6 diaval.b . . . . . . 7  |-  B  =  ( Base `  K
)
75, 6syl6eqr 2441 . . . . . 6  |-  ( k  =  K  ->  ( Base `  k )  =  B )
8 fveq2 5774 . . . . . . . 8  |-  ( k  =  K  ->  ( le `  k )  =  ( le `  K
) )
9 diaval.l . . . . . . . 8  |-  .<_  =  ( le `  K )
108, 9syl6eqr 2441 . . . . . . 7  |-  ( k  =  K  ->  ( le `  k )  = 
.<_  )
1110breqd 4378 . . . . . 6  |-  ( k  =  K  ->  (
y ( le `  k ) w  <->  y  .<_  w ) )
127, 11rabeqbidv 3029 . . . . 5  |-  ( k  =  K  ->  { y  e.  ( Base `  k
)  |  y ( le `  k ) w }  =  {
y  e.  B  | 
y  .<_  w } )
13 fveq2 5774 . . . . . . 7  |-  ( k  =  K  ->  ( LTrn `  k )  =  ( LTrn `  K
) )
1413fveq1d 5776 . . . . . 6  |-  ( k  =  K  ->  (
( LTrn `  k ) `  w )  =  ( ( LTrn `  K
) `  w )
)
15 fveq2 5774 . . . . . . . . 9  |-  ( k  =  K  ->  ( trL `  k )  =  ( trL `  K
) )
1615fveq1d 5776 . . . . . . . 8  |-  ( k  =  K  ->  (
( trL `  k
) `  w )  =  ( ( trL `  K ) `  w
) )
1716fveq1d 5776 . . . . . . 7  |-  ( k  =  K  ->  (
( ( trL `  k
) `  w ) `  f )  =  ( ( ( trL `  K
) `  w ) `  f ) )
18 eqidd 2383 . . . . . . 7  |-  ( k  =  K  ->  x  =  x )
1917, 10, 18breq123d 4381 . . . . . 6  |-  ( k  =  K  ->  (
( ( ( trL `  k ) `  w
) `  f )
( le `  k
) x  <->  ( (
( trL `  K
) `  w ) `  f )  .<_  x ) )
2014, 19rabeqbidv 3029 . . . . 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 4445 . . . 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 4445 . . 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 37169 . . 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 5784 . . . . 5  |-  ( LHyp `  K )  e.  _V
253, 24eqeltri 2466 . . . 4  |-  H  e. 
_V
2625mptex 6044 . . 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 5857 . 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 1399    e. wcel 1826   {crab 2736   _Vcvv 3034   class class class wbr 4367    |-> cmpt 4425   ` cfv 5496   Basecbs 14634   lecple 14709   LHypclh 36121   LTrncltrn 36238   trLctrl 36296   DIsoAcdia 37168
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-9 1830  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006  ax-ext 2360  ax-rep 4478  ax-sep 4488  ax-nul 4496  ax-pr 4601
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1402  df-ex 1621  df-nf 1625  df-sb 1748  df-eu 2222  df-mo 2223  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2532  df-ne 2579  df-ral 2737  df-rex 2738  df-reu 2739  df-rab 2741  df-v 3036  df-sbc 3253  df-csb 3349  df-dif 3392  df-un 3394  df-in 3396  df-ss 3403  df-nul 3712  df-if 3858  df-sn 3945  df-pr 3947  df-op 3951  df-uni 4164  df-iun 4245  df-br 4368  df-opab 4426  df-mpt 4427  df-id 4709  df-xp 4919  df-rel 4920  df-cnv 4921  df-co 4922  df-dm 4923  df-rn 4924  df-res 4925  df-ima 4926  df-iota 5460  df-fun 5498  df-fn 5499  df-f 5500  df-f1 5501  df-fo 5502  df-f1o 5503  df-fv 5504  df-disoa 37169
This theorem is referenced by:  diafval  37171
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