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Theorem dihffval 30550
Description: The isomorphism H for a lattice  K. Definition of isomorphism map in [Crawley] p. 122 line 3. (Contributed by NM, 28-Jan-2014.)
Hypotheses
Ref Expression
dihval.b  |-  B  =  ( Base `  K
)
dihval.l  |-  .<_  =  ( le `  K )
dihval.j  |-  .\/  =  ( join `  K )
dihval.m  |-  ./\  =  ( meet `  K )
dihval.a  |-  A  =  ( Atoms `  K )
dihval.h  |-  H  =  ( LHyp `  K
)
Assertion
Ref Expression
dihffval  |-  ( K  e.  V  ->  ( DIsoH `  K )  =  ( w  e.  H  |->  ( x  e.  B  |->  if ( x  .<_  w ,  ( ( (
DIsoB `  K ) `  w ) `  x
) ,  ( iota_ u  e.  ( LSubSp `  (
( DVecH `  K ) `  w ) ) A. q  e.  A  (
( -.  q  .<_  w  /\  ( q  .\/  ( x  ./\  w ) )  =  x )  ->  u  =  ( ( ( ( DIsoC `  K ) `  w
) `  q )
( LSSum `  ( ( DVecH `  K ) `  w ) ) ( ( ( DIsoB `  K
) `  w ) `  ( x  ./\  w
) ) ) ) ) ) ) ) )
Distinct variable groups:    A, q    w, H    u, q, w, x, K
Allowed substitution hints:    A( x, w, u)    B( x, w, u, q)    H( x, u, q)    .\/ ( x, w, u, q)    .<_ ( x, w, u, q)    ./\ (
x, w, u, q)    V( x, w, u, q)

Proof of Theorem dihffval
StepHypRef Expression
1 elex 2748 . 2  |-  ( K  e.  V  ->  K  e.  _V )
2 fveq2 5423 . . . . 5  |-  ( k  =  K  ->  ( LHyp `  k )  =  ( LHyp `  K
) )
3 dihval.h . . . . 5  |-  H  =  ( LHyp `  K
)
42, 3syl6eqr 2306 . . . 4  |-  ( k  =  K  ->  ( LHyp `  k )  =  H )
5 fveq2 5423 . . . . . 6  |-  ( k  =  K  ->  ( Base `  k )  =  ( Base `  K
) )
6 dihval.b . . . . . 6  |-  B  =  ( Base `  K
)
75, 6syl6eqr 2306 . . . . 5  |-  ( k  =  K  ->  ( Base `  k )  =  B )
8 fveq2 5423 . . . . . . . 8  |-  ( k  =  K  ->  ( le `  k )  =  ( le `  K
) )
9 dihval.l . . . . . . . 8  |-  .<_  =  ( le `  K )
108, 9syl6eqr 2306 . . . . . . 7  |-  ( k  =  K  ->  ( le `  k )  = 
.<_  )
1110breqd 3974 . . . . . 6  |-  ( k  =  K  ->  (
x ( le `  k ) w  <->  x  .<_  w ) )
12 fveq2 5423 . . . . . . . 8  |-  ( k  =  K  ->  ( DIsoB `  k )  =  ( DIsoB `  K )
)
1312fveq1d 5425 . . . . . . 7  |-  ( k  =  K  ->  (
( DIsoB `  k ) `  w )  =  ( ( DIsoB `  K ) `  w ) )
1413fveq1d 5425 . . . . . 6  |-  ( k  =  K  ->  (
( ( DIsoB `  k
) `  w ) `  x )  =  ( ( ( DIsoB `  K
) `  w ) `  x ) )
15 fveq2 5423 . . . . . . . . 9  |-  ( k  =  K  ->  ( DVecH `  k )  =  ( DVecH `  K )
)
1615fveq1d 5425 . . . . . . . 8  |-  ( k  =  K  ->  (
( DVecH `  k ) `  w )  =  ( ( DVecH `  K ) `  w ) )
1716fveq2d 5427 . . . . . . 7  |-  ( k  =  K  ->  ( LSubSp `
 ( ( DVecH `  k ) `  w
) )  =  (
LSubSp `  ( ( DVecH `  K ) `  w
) ) )
18 fveq2 5423 . . . . . . . . 9  |-  ( k  =  K  ->  ( Atoms `  k )  =  ( Atoms `  K )
)
19 dihval.a . . . . . . . . 9  |-  A  =  ( Atoms `  K )
2018, 19syl6eqr 2306 . . . . . . . 8  |-  ( k  =  K  ->  ( Atoms `  k )  =  A )
2110breqd 3974 . . . . . . . . . . 11  |-  ( k  =  K  ->  (
q ( le `  k ) w  <->  q  .<_  w ) )
2221notbid 287 . . . . . . . . . 10  |-  ( k  =  K  ->  ( -.  q ( le `  k ) w  <->  -.  q  .<_  w ) )
23 fveq2 5423 . . . . . . . . . . . . 13  |-  ( k  =  K  ->  ( join `  k )  =  ( join `  K
) )
24 dihval.j . . . . . . . . . . . . 13  |-  .\/  =  ( join `  K )
2523, 24syl6eqr 2306 . . . . . . . . . . . 12  |-  ( k  =  K  ->  ( join `  k )  = 
.\/  )
26 eqidd 2257 . . . . . . . . . . . 12  |-  ( k  =  K  ->  q  =  q )
27 fveq2 5423 . . . . . . . . . . . . . 14  |-  ( k  =  K  ->  ( meet `  k )  =  ( meet `  K
) )
28 dihval.m . . . . . . . . . . . . . 14  |-  ./\  =  ( meet `  K )
2927, 28syl6eqr 2306 . . . . . . . . . . . . 13  |-  ( k  =  K  ->  ( meet `  k )  = 
./\  )
3029oveqd 5774 . . . . . . . . . . . 12  |-  ( k  =  K  ->  (
x ( meet `  k
) w )  =  ( x  ./\  w
) )
3125, 26, 30oveq123d 5778 . . . . . . . . . . 11  |-  ( k  =  K  ->  (
q ( join `  k
) ( x (
meet `  k )
w ) )  =  ( q  .\/  (
x  ./\  w )
) )
3231eqeq1d 2264 . . . . . . . . . 10  |-  ( k  =  K  ->  (
( q ( join `  k ) ( x ( meet `  k
) w ) )  =  x  <->  ( q  .\/  ( x  ./\  w
) )  =  x ) )
3322, 32anbi12d 694 . . . . . . . . 9  |-  ( k  =  K  ->  (
( -.  q ( le `  k ) w  /\  ( q ( join `  k
) ( x (
meet `  k )
w ) )  =  x )  <->  ( -.  q  .<_  w  /\  (
q  .\/  ( x  ./\  w ) )  =  x ) ) )
3416fveq2d 5427 . . . . . . . . . . 11  |-  ( k  =  K  ->  ( LSSum `  ( ( DVecH `  k ) `  w
) )  =  (
LSSum `  ( ( DVecH `  K ) `  w
) ) )
35 fveq2 5423 . . . . . . . . . . . . 13  |-  ( k  =  K  ->  ( DIsoC `  k )  =  ( DIsoC `  K )
)
3635fveq1d 5425 . . . . . . . . . . . 12  |-  ( k  =  K  ->  (
( DIsoC `  k ) `  w )  =  ( ( DIsoC `  K ) `  w ) )
3736fveq1d 5425 . . . . . . . . . . 11  |-  ( k  =  K  ->  (
( ( DIsoC `  k
) `  w ) `  q )  =  ( ( ( DIsoC `  K
) `  w ) `  q ) )
3813, 30fveq12d 5429 . . . . . . . . . . 11  |-  ( k  =  K  ->  (
( ( DIsoB `  k
) `  w ) `  ( x ( meet `  k ) w ) )  =  ( ( ( DIsoB `  K ) `  w ) `  (
x  ./\  w )
) )
3934, 37, 38oveq123d 5778 . . . . . . . . . 10  |-  ( k  =  K  ->  (
( ( ( DIsoC `  k ) `  w
) `  q )
( LSSum `  ( ( DVecH `  k ) `  w ) ) ( ( ( DIsoB `  k
) `  w ) `  ( x ( meet `  k ) w ) ) )  =  ( ( ( ( DIsoC `  K ) `  w
) `  q )
( LSSum `  ( ( DVecH `  K ) `  w ) ) ( ( ( DIsoB `  K
) `  w ) `  ( x  ./\  w
) ) ) )
4039eqeq2d 2267 . . . . . . . . 9  |-  ( k  =  K  ->  (
u  =  ( ( ( ( DIsoC `  k
) `  w ) `  q ) ( LSSum `  ( ( DVecH `  k
) `  w )
) ( ( (
DIsoB `  k ) `  w ) `  (
x ( meet `  k
) w ) ) )  <->  u  =  (
( ( ( DIsoC `  K ) `  w
) `  q )
( LSSum `  ( ( DVecH `  K ) `  w ) ) ( ( ( DIsoB `  K
) `  w ) `  ( x  ./\  w
) ) ) ) )
4133, 40imbi12d 313 . . . . . . . 8  |-  ( k  =  K  ->  (
( ( -.  q
( le `  k
) w  /\  (
q ( join `  k
) ( x (
meet `  k )
w ) )  =  x )  ->  u  =  ( ( ( ( DIsoC `  k ) `  w ) `  q
) ( LSSum `  (
( DVecH `  k ) `  w ) ) ( ( ( DIsoB `  k
) `  w ) `  ( x ( meet `  k ) w ) ) ) )  <->  ( ( -.  q  .<_  w  /\  ( q  .\/  (
x  ./\  w )
)  =  x )  ->  u  =  ( ( ( ( DIsoC `  K ) `  w
) `  q )
( LSSum `  ( ( DVecH `  K ) `  w ) ) ( ( ( DIsoB `  K
) `  w ) `  ( x  ./\  w
) ) ) ) ) )
4220, 41raleqbidv 2708 . . . . . . 7  |-  ( k  =  K  ->  ( A. q  e.  ( Atoms `  k ) ( ( -.  q ( le `  k ) w  /\  ( q ( join `  k
) ( x (
meet `  k )
w ) )  =  x )  ->  u  =  ( ( ( ( DIsoC `  k ) `  w ) `  q
) ( LSSum `  (
( DVecH `  k ) `  w ) ) ( ( ( DIsoB `  k
) `  w ) `  ( x ( meet `  k ) w ) ) ) )  <->  A. q  e.  A  ( ( -.  q  .<_  w  /\  ( q  .\/  (
x  ./\  w )
)  =  x )  ->  u  =  ( ( ( ( DIsoC `  K ) `  w
) `  q )
( LSSum `  ( ( DVecH `  K ) `  w ) ) ( ( ( DIsoB `  K
) `  w ) `  ( x  ./\  w
) ) ) ) ) )
4317, 42riotaeqbidv 6240 . . . . . 6  |-  ( k  =  K  ->  ( iota_ u  e.  ( LSubSp `  ( ( DVecH `  k
) `  w )
) A. q  e.  ( Atoms `  k )
( ( -.  q
( le `  k
) w  /\  (
q ( join `  k
) ( x (
meet `  k )
w ) )  =  x )  ->  u  =  ( ( ( ( DIsoC `  k ) `  w ) `  q
) ( LSSum `  (
( DVecH `  k ) `  w ) ) ( ( ( DIsoB `  k
) `  w ) `  ( x ( meet `  k ) w ) ) ) ) )  =  ( iota_ u  e.  ( LSubSp `  ( ( DVecH `  K ) `  w ) ) A. q  e.  A  (
( -.  q  .<_  w  /\  ( q  .\/  ( x  ./\  w ) )  =  x )  ->  u  =  ( ( ( ( DIsoC `  K ) `  w
) `  q )
( LSSum `  ( ( DVecH `  K ) `  w ) ) ( ( ( DIsoB `  K
) `  w ) `  ( x  ./\  w
) ) ) ) ) )
4411, 14, 43ifbieq12d 3528 . . . . 5  |-  ( k  =  K  ->  if ( x ( le
`  k ) w ,  ( ( (
DIsoB `  k ) `  w ) `  x
) ,  ( iota_ u  e.  ( LSubSp `  (
( DVecH `  k ) `  w ) ) A. q  e.  ( Atoms `  k ) ( ( -.  q ( le
`  k ) w  /\  ( q (
join `  k )
( x ( meet `  k ) w ) )  =  x )  ->  u  =  ( ( ( ( DIsoC `  k ) `  w
) `  q )
( LSSum `  ( ( DVecH `  k ) `  w ) ) ( ( ( DIsoB `  k
) `  w ) `  ( x ( meet `  k ) w ) ) ) ) ) )  =  if ( x  .<_  w , 
( ( ( DIsoB `  K ) `  w
) `  x ) ,  ( iota_ u  e.  ( LSubSp `  ( ( DVecH `  K ) `  w ) ) A. q  e.  A  (
( -.  q  .<_  w  /\  ( q  .\/  ( x  ./\  w ) )  =  x )  ->  u  =  ( ( ( ( DIsoC `  K ) `  w
) `  q )
( LSSum `  ( ( DVecH `  K ) `  w ) ) ( ( ( DIsoB `  K
) `  w ) `  ( x  ./\  w
) ) ) ) ) ) )
457, 44mpteq12dv 4038 . . . 4  |-  ( k  =  K  ->  (
x  e.  ( Base `  k )  |->  if ( x ( le `  k ) w ,  ( ( ( DIsoB `  k ) `  w
) `  x ) ,  ( iota_ u  e.  ( LSubSp `  ( ( DVecH `  k ) `  w ) ) A. q  e.  ( Atoms `  k ) ( ( -.  q ( le
`  k ) w  /\  ( q (
join `  k )
( x ( meet `  k ) w ) )  =  x )  ->  u  =  ( ( ( ( DIsoC `  k ) `  w
) `  q )
( LSSum `  ( ( DVecH `  k ) `  w ) ) ( ( ( DIsoB `  k
) `  w ) `  ( x ( meet `  k ) w ) ) ) ) ) ) )  =  ( x  e.  B  |->  if ( x  .<_  w ,  ( ( ( DIsoB `  K ) `  w
) `  x ) ,  ( iota_ u  e.  ( LSubSp `  ( ( DVecH `  K ) `  w ) ) A. q  e.  A  (
( -.  q  .<_  w  /\  ( q  .\/  ( x  ./\  w ) )  =  x )  ->  u  =  ( ( ( ( DIsoC `  K ) `  w
) `  q )
( LSSum `  ( ( DVecH `  K ) `  w ) ) ( ( ( DIsoB `  K
) `  w ) `  ( x  ./\  w
) ) ) ) ) ) ) )
464, 45mpteq12dv 4038 . . 3  |-  ( k  =  K  ->  (
w  e.  ( LHyp `  k )  |->  ( x  e.  ( Base `  k
)  |->  if ( x ( le `  k
) w ,  ( ( ( DIsoB `  k
) `  w ) `  x ) ,  (
iota_ u  e.  ( LSubSp `
 ( ( DVecH `  k ) `  w
) ) A. q  e.  ( Atoms `  k )
( ( -.  q
( le `  k
) w  /\  (
q ( join `  k
) ( x (
meet `  k )
w ) )  =  x )  ->  u  =  ( ( ( ( DIsoC `  k ) `  w ) `  q
) ( LSSum `  (
( DVecH `  k ) `  w ) ) ( ( ( DIsoB `  k
) `  w ) `  ( x ( meet `  k ) w ) ) ) ) ) ) ) )  =  ( w  e.  H  |->  ( x  e.  B  |->  if ( x  .<_  w ,  ( ( (
DIsoB `  K ) `  w ) `  x
) ,  ( iota_ u  e.  ( LSubSp `  (
( DVecH `  K ) `  w ) ) A. q  e.  A  (
( -.  q  .<_  w  /\  ( q  .\/  ( x  ./\  w ) )  =  x )  ->  u  =  ( ( ( ( DIsoC `  K ) `  w
) `  q )
( LSSum `  ( ( DVecH `  K ) `  w ) ) ( ( ( DIsoB `  K
) `  w ) `  ( x  ./\  w
) ) ) ) ) ) ) ) )
47 df-dih 30549 . . 3  |-  DIsoH  =  ( k  e.  _V  |->  ( w  e.  ( LHyp `  k )  |->  ( x  e.  ( Base `  k
)  |->  if ( x ( le `  k
) w ,  ( ( ( DIsoB `  k
) `  w ) `  x ) ,  (
iota_ u  e.  ( LSubSp `
 ( ( DVecH `  k ) `  w
) ) A. q  e.  ( Atoms `  k )
( ( -.  q
( le `  k
) w  /\  (
q ( join `  k
) ( x (
meet `  k )
w ) )  =  x )  ->  u  =  ( ( ( ( DIsoC `  k ) `  w ) `  q
) ( LSSum `  (
( DVecH `  k ) `  w ) ) ( ( ( DIsoB `  k
) `  w ) `  ( x ( meet `  k ) w ) ) ) ) ) ) ) ) )
48 fvex 5437 . . . . 5  |-  ( LHyp `  K )  e.  _V
493, 48eqeltri 2326 . . . 4  |-  H  e. 
_V
5049mptex 5645 . . 3  |-  ( w  e.  H  |->  ( x  e.  B  |->  if ( x  .<_  w , 
( ( ( DIsoB `  K ) `  w
) `  x ) ,  ( iota_ u  e.  ( LSubSp `  ( ( DVecH `  K ) `  w ) ) A. q  e.  A  (
( -.  q  .<_  w  /\  ( q  .\/  ( x  ./\  w ) )  =  x )  ->  u  =  ( ( ( ( DIsoC `  K ) `  w
) `  q )
( LSSum `  ( ( DVecH `  K ) `  w ) ) ( ( ( DIsoB `  K
) `  w ) `  ( x  ./\  w
) ) ) ) ) ) ) )  e.  _V
5146, 47, 50fvmpt 5501 . 2  |-  ( K  e.  _V  ->  ( DIsoH `  K )  =  ( w  e.  H  |->  ( x  e.  B  |->  if ( x  .<_  w ,  ( ( (
DIsoB `  K ) `  w ) `  x
) ,  ( iota_ u  e.  ( LSubSp `  (
( DVecH `  K ) `  w ) ) A. q  e.  A  (
( -.  q  .<_  w  /\  ( q  .\/  ( x  ./\  w ) )  =  x )  ->  u  =  ( ( ( ( DIsoC `  K ) `  w
) `  q )
( LSSum `  ( ( DVecH `  K ) `  w ) ) ( ( ( DIsoB `  K
) `  w ) `  ( x  ./\  w
) ) ) ) ) ) ) ) )
521, 51syl 17 1  |-  ( K  e.  V  ->  ( DIsoH `  K )  =  ( w  e.  H  |->  ( x  e.  B  |->  if ( x  .<_  w ,  ( ( (
DIsoB `  K ) `  w ) `  x
) ,  ( iota_ u  e.  ( LSubSp `  (
( DVecH `  K ) `  w ) ) A. q  e.  A  (
( -.  q  .<_  w  /\  ( q  .\/  ( x  ./\  w ) )  =  x )  ->  u  =  ( ( ( ( DIsoC `  K ) `  w
) `  q )
( LSSum `  ( ( DVecH `  K ) `  w ) ) ( ( ( DIsoB `  K
) `  w ) `  ( x  ./\  w
) ) ) ) ) ) ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 5    -> wi 6    /\ wa 360    = wceq 1619    e. wcel 1621   A.wral 2516   _Vcvv 2740   ifcif 3506   class class class wbr 3963    e. cmpt 4017   ` cfv 4638  (class class class)co 5757   iota_crio 6228   Basecbs 13075   lecple 13142   joincjn 14005   meetcmee 14006   LSSumclsm 14872   LSubSpclss 15616   Atomscatm 28583   LHypclh 29303   DVecHcdvh 30398   DIsoBcdib 30458   DIsoCcdic 30492   DIsoHcdih 30548
This theorem is referenced by:  dihfval  30551
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2237  ax-rep 4071  ax-sep 4081  ax-nul 4089  ax-pr 4152  ax-un 4449
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-eu 2121  df-mo 2122  df-clab 2243  df-cleq 2249  df-clel 2252  df-nfc 2381  df-ne 2421  df-ral 2520  df-rex 2521  df-reu 2522  df-rab 2523  df-v 2742  df-sbc 2936  df-csb 3024  df-dif 3097  df-un 3099  df-in 3101  df-ss 3108  df-nul 3398  df-if 3507  df-sn 3587  df-pr 3588  df-op 3590  df-uni 3769  df-iun 3848  df-br 3964  df-opab 4018  df-mpt 4019  df-id 4246  df-xp 4640  df-rel 4641  df-cnv 4642  df-co 4643  df-dm 4644  df-rn 4645  df-res 4646  df-ima 4647  df-fun 4648  df-fn 4649  df-f 4650  df-f1 4651  df-fo 4652  df-f1o 4653  df-fv 4654  df-ov 5760  df-iota 6190  df-riota 6237  df-dih 30549
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