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Theorem diaffval 31220
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 2796 . 2  |-  ( K  e.  V  ->  K  e.  _V )
2 fveq2 5525 . . . . 5  |-  ( k  =  K  ->  ( LHyp `  k )  =  ( LHyp `  K
) )
3 diaval.h . . . . 5  |-  H  =  ( LHyp `  K
)
42, 3syl6eqr 2333 . . . 4  |-  ( k  =  K  ->  ( LHyp `  k )  =  H )
5 fveq2 5525 . . . . . . 7  |-  ( k  =  K  ->  ( Base `  k )  =  ( Base `  K
) )
6 diaval.b . . . . . . 7  |-  B  =  ( Base `  K
)
75, 6syl6eqr 2333 . . . . . 6  |-  ( k  =  K  ->  ( Base `  k )  =  B )
8 fveq2 5525 . . . . . . . 8  |-  ( k  =  K  ->  ( le `  k )  =  ( le `  K
) )
9 diaval.l . . . . . . . 8  |-  .<_  =  ( le `  K )
108, 9syl6eqr 2333 . . . . . . 7  |-  ( k  =  K  ->  ( le `  k )  = 
.<_  )
1110breqd 4034 . . . . . 6  |-  ( k  =  K  ->  (
y ( le `  k ) w  <->  y  .<_  w ) )
127, 11rabeqbidv 2783 . . . . 5  |-  ( k  =  K  ->  { y  e.  ( Base `  k
)  |  y ( le `  k ) w }  =  {
y  e.  B  | 
y  .<_  w } )
13 fveq2 5525 . . . . . . 7  |-  ( k  =  K  ->  ( LTrn `  k )  =  ( LTrn `  K
) )
1413fveq1d 5527 . . . . . 6  |-  ( k  =  K  ->  (
( LTrn `  k ) `  w )  =  ( ( LTrn `  K
) `  w )
)
15 fveq2 5525 . . . . . . . . 9  |-  ( k  =  K  ->  ( trL `  k )  =  ( trL `  K
) )
1615fveq1d 5527 . . . . . . . 8  |-  ( k  =  K  ->  (
( trL `  k
) `  w )  =  ( ( trL `  K ) `  w
) )
1716fveq1d 5527 . . . . . . 7  |-  ( k  =  K  ->  (
( ( trL `  k
) `  w ) `  f )  =  ( ( ( trL `  K
) `  w ) `  f ) )
18 eqidd 2284 . . . . . . 7  |-  ( k  =  K  ->  x  =  x )
1917, 10, 18breq123d 4037 . . . . . 6  |-  ( k  =  K  ->  (
( ( ( trL `  k ) `  w
) `  f )
( le `  k
) x  <->  ( (
( trL `  K
) `  w ) `  f )  .<_  x ) )
2014, 19rabeqbidv 2783 . . . . 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 4098 . . . 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 4098 . . 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 31219 . . 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 5539 . . . . 5  |-  ( LHyp `  K )  e.  _V
253, 24eqeltri 2353 . . . 4  |-  H  e. 
_V
2625mptex 5746 . . 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 5602 . 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 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:  diafval  31221
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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