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Theorem dihffval 30109
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 2735 . 2  |-  ( K  e.  V  ->  K  e.  _V )
2 fveq2 5377 . . . . 5  |-  ( k  =  K  ->  ( LHyp `  k )  =  ( LHyp `  K
) )
3 dihval.h . . . . 5  |-  H  =  ( LHyp `  K
)
42, 3syl6eqr 2303 . . . 4  |-  ( k  =  K  ->  ( LHyp `  k )  =  H )
5 fveq2 5377 . . . . . 6  |-  ( k  =  K  ->  ( Base `  k )  =  ( Base `  K
) )
6 dihval.b . . . . . 6  |-  B  =  ( Base `  K
)
75, 6syl6eqr 2303 . . . . 5  |-  ( k  =  K  ->  ( Base `  k )  =  B )
8 fveq2 5377 . . . . . . . 8  |-  ( k  =  K  ->  ( le `  k )  =  ( le `  K
) )
9 dihval.l . . . . . . . 8  |-  .<_  =  ( le `  K )
108, 9syl6eqr 2303 . . . . . . 7  |-  ( k  =  K  ->  ( le `  k )  = 
.<_  )
1110breqd 3931 . . . . . 6  |-  ( k  =  K  ->  (
x ( le `  k ) w  <->  x  .<_  w ) )
12 fveq2 5377 . . . . . . . 8  |-  ( k  =  K  ->  ( DIsoB `  k )  =  ( DIsoB `  K )
)
1312fveq1d 5379 . . . . . . 7  |-  ( k  =  K  ->  (
( DIsoB `  k ) `  w )  =  ( ( DIsoB `  K ) `  w ) )
1413fveq1d 5379 . . . . . 6  |-  ( k  =  K  ->  (
( ( DIsoB `  k
) `  w ) `  x )  =  ( ( ( DIsoB `  K
) `  w ) `  x ) )
15 fveq2 5377 . . . . . . . . 9  |-  ( k  =  K  ->  ( DVecH `  k )  =  ( DVecH `  K )
)
1615fveq1d 5379 . . . . . . . 8  |-  ( k  =  K  ->  (
( DVecH `  k ) `  w )  =  ( ( DVecH `  K ) `  w ) )
1716fveq2d 5381 . . . . . . 7  |-  ( k  =  K  ->  ( LSubSp `
 ( ( DVecH `  k ) `  w
) )  =  (
LSubSp `  ( ( DVecH `  K ) `  w
) ) )
18 fveq2 5377 . . . . . . . . 9  |-  ( k  =  K  ->  ( Atoms `  k )  =  ( Atoms `  K )
)
19 dihval.a . . . . . . . . 9  |-  A  =  ( Atoms `  K )
2018, 19syl6eqr 2303 . . . . . . . 8  |-  ( k  =  K  ->  ( Atoms `  k )  =  A )
2110breqd 3931 . . . . . . . . . . 11  |-  ( k  =  K  ->  (
q ( le `  k ) w  <->  q  .<_  w ) )
2221notbid 287 . . . . . . . . . 10  |-  ( k  =  K  ->  ( -.  q ( le `  k ) w  <->  -.  q  .<_  w ) )
23 fveq2 5377 . . . . . . . . . . . . 13  |-  ( k  =  K  ->  ( join `  k )  =  ( join `  K
) )
24 dihval.j . . . . . . . . . . . . 13  |-  .\/  =  ( join `  K )
2523, 24syl6eqr 2303 . . . . . . . . . . . 12  |-  ( k  =  K  ->  ( join `  k )  = 
.\/  )
26 eqidd 2254 . . . . . . . . . . . 12  |-  ( k  =  K  ->  q  =  q )
27 fveq2 5377 . . . . . . . . . . . . . 14  |-  ( k  =  K  ->  ( meet `  k )  =  ( meet `  K
) )
28 dihval.m . . . . . . . . . . . . . 14  |-  ./\  =  ( meet `  K )
2927, 28syl6eqr 2303 . . . . . . . . . . . . 13  |-  ( k  =  K  ->  ( meet `  k )  = 
./\  )
3029oveqd 5727 . . . . . . . . . . . 12  |-  ( k  =  K  ->  (
x ( meet `  k
) w )  =  ( x  ./\  w
) )
3125, 26, 30oveq123d 5731 . . . . . . . . . . 11  |-  ( k  =  K  ->  (
q ( join `  k
) ( x (
meet `  k )
w ) )  =  ( q  .\/  (
x  ./\  w )
) )
3231eqeq1d 2261 . . . . . . . . . 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 5381 . . . . . . . . . . 11  |-  ( k  =  K  ->  ( LSSum `  ( ( DVecH `  k ) `  w
) )  =  (
LSSum `  ( ( DVecH `  K ) `  w
) ) )
35 fveq2 5377 . . . . . . . . . . . . 13  |-  ( k  =  K  ->  ( DIsoC `  k )  =  ( DIsoC `  K )
)
3635fveq1d 5379 . . . . . . . . . . . 12  |-  ( k  =  K  ->  (
( DIsoC `  k ) `  w )  =  ( ( DIsoC `  K ) `  w ) )
3736fveq1d 5379 . . . . . . . . . . 11  |-  ( k  =  K  ->  (
( ( DIsoC `  k
) `  w ) `  q )  =  ( ( ( DIsoC `  K
) `  w ) `  q ) )
3813, 30fveq12d 5383 . . . . . . . . . . 11  |-  ( k  =  K  ->  (
( ( DIsoB `  k
) `  w ) `  ( x ( meet `  k ) w ) )  =  ( ( ( DIsoB `  K ) `  w ) `  (
x  ./\  w )
) )
3934, 37, 38oveq123d 5731 . . . . . . . . . 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 2264 . . . . . . . . 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 2699 . . . . . . 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 6193 . . . . . 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 3492 . . . . 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 3995 . . . 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 3995 . . 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 30108 . . 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 5391 . . . . 5  |-  ( LHyp `  K )  e.  _V
493, 48eqeltri 2323 . . . 4  |-  H  e. 
_V
5049mptex 5598 . . 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 5454 . 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 2509   _Vcvv 2727   ifcif 3470   class class class wbr 3920    e. cmpt 3974   ` cfv 4592  (class class class)co 5710   iota_crio 6181   Basecbs 13022   lecple 13089   joincjn 13922   meetcmee 13923   LSSumclsm 14780   LSubSpclss 15524   Atomscatm 28142   LHypclh 28862   DVecHcdvh 29957   DIsoBcdib 30017   DIsoCcdic 30051   DIsoHcdih 30107
This theorem is referenced by:  dihfval  30110
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 1926  ax-ext 2234  ax-rep 4028  ax-sep 4038  ax-nul 4046  ax-pr 4108  ax-un 4403
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 1883  df-eu 2118  df-mo 2119  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-ne 2414  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2516  df-v 2729  df-sbc 2922  df-csb 3010  df-dif 3081  df-un 3083  df-in 3085  df-ss 3089  df-nul 3363  df-if 3471  df-sn 3550  df-pr 3551  df-op 3553  df-uni 3728  df-iun 3805  df-br 3921  df-opab 3975  df-mpt 3976  df-id 4202  df-xp 4594  df-rel 4595  df-cnv 4596  df-co 4597  df-dm 4598  df-rn 4599  df-res 4600  df-ima 4601  df-fun 4602  df-fn 4603  df-f 4604  df-f1 4605  df-fo 4606  df-f1o 4607  df-fv 4608  df-ov 5713  df-iota 6143  df-riota 6190  df-dih 30108
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