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Theorem dihffval 31713
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
Dummy variable  k is distinct from all other variables.
StepHypRef Expression
1 elex 2924 . 2  |-  ( K  e.  V  ->  K  e.  _V )
2 fveq2 5687 . . . . 5  |-  ( k  =  K  ->  ( LHyp `  k )  =  ( LHyp `  K
) )
3 dihval.h . . . . 5  |-  H  =  ( LHyp `  K
)
42, 3syl6eqr 2454 . . . 4  |-  ( k  =  K  ->  ( LHyp `  k )  =  H )
5 fveq2 5687 . . . . . 6  |-  ( k  =  K  ->  ( Base `  k )  =  ( Base `  K
) )
6 dihval.b . . . . . 6  |-  B  =  ( Base `  K
)
75, 6syl6eqr 2454 . . . . 5  |-  ( k  =  K  ->  ( Base `  k )  =  B )
8 fveq2 5687 . . . . . . . 8  |-  ( k  =  K  ->  ( le `  k )  =  ( le `  K
) )
9 dihval.l . . . . . . . 8  |-  .<_  =  ( le `  K )
108, 9syl6eqr 2454 . . . . . . 7  |-  ( k  =  K  ->  ( le `  k )  = 
.<_  )
1110breqd 4183 . . . . . 6  |-  ( k  =  K  ->  (
x ( le `  k ) w  <->  x  .<_  w ) )
12 fveq2 5687 . . . . . . . 8  |-  ( k  =  K  ->  ( DIsoB `  k )  =  ( DIsoB `  K )
)
1312fveq1d 5689 . . . . . . 7  |-  ( k  =  K  ->  (
( DIsoB `  k ) `  w )  =  ( ( DIsoB `  K ) `  w ) )
1413fveq1d 5689 . . . . . 6  |-  ( k  =  K  ->  (
( ( DIsoB `  k
) `  w ) `  x )  =  ( ( ( DIsoB `  K
) `  w ) `  x ) )
15 fveq2 5687 . . . . . . . . 9  |-  ( k  =  K  ->  ( DVecH `  k )  =  ( DVecH `  K )
)
1615fveq1d 5689 . . . . . . . 8  |-  ( k  =  K  ->  (
( DVecH `  k ) `  w )  =  ( ( DVecH `  K ) `  w ) )
1716fveq2d 5691 . . . . . . 7  |-  ( k  =  K  ->  ( LSubSp `
 ( ( DVecH `  k ) `  w
) )  =  (
LSubSp `  ( ( DVecH `  K ) `  w
) ) )
18 fveq2 5687 . . . . . . . . 9  |-  ( k  =  K  ->  ( Atoms `  k )  =  ( Atoms `  K )
)
19 dihval.a . . . . . . . . 9  |-  A  =  ( Atoms `  K )
2018, 19syl6eqr 2454 . . . . . . . 8  |-  ( k  =  K  ->  ( Atoms `  k )  =  A )
2110breqd 4183 . . . . . . . . . . 11  |-  ( k  =  K  ->  (
q ( le `  k ) w  <->  q  .<_  w ) )
2221notbid 286 . . . . . . . . . 10  |-  ( k  =  K  ->  ( -.  q ( le `  k ) w  <->  -.  q  .<_  w ) )
23 fveq2 5687 . . . . . . . . . . . . 13  |-  ( k  =  K  ->  ( join `  k )  =  ( join `  K
) )
24 dihval.j . . . . . . . . . . . . 13  |-  .\/  =  ( join `  K )
2523, 24syl6eqr 2454 . . . . . . . . . . . 12  |-  ( k  =  K  ->  ( join `  k )  = 
.\/  )
26 eqidd 2405 . . . . . . . . . . . 12  |-  ( k  =  K  ->  q  =  q )
27 fveq2 5687 . . . . . . . . . . . . . 14  |-  ( k  =  K  ->  ( meet `  k )  =  ( meet `  K
) )
28 dihval.m . . . . . . . . . . . . . 14  |-  ./\  =  ( meet `  K )
2927, 28syl6eqr 2454 . . . . . . . . . . . . 13  |-  ( k  =  K  ->  ( meet `  k )  = 
./\  )
3029oveqd 6057 . . . . . . . . . . . 12  |-  ( k  =  K  ->  (
x ( meet `  k
) w )  =  ( x  ./\  w
) )
3125, 26, 30oveq123d 6061 . . . . . . . . . . 11  |-  ( k  =  K  ->  (
q ( join `  k
) ( x (
meet `  k )
w ) )  =  ( q  .\/  (
x  ./\  w )
) )
3231eqeq1d 2412 . . . . . . . . . 10  |-  ( k  =  K  ->  (
( q ( join `  k ) ( x ( meet `  k
) w ) )  =  x  <->  ( q  .\/  ( x  ./\  w
) )  =  x ) )
3322, 32anbi12d 692 . . . . . . . . 9  |-  ( k  =  K  ->  (
( -.  q ( le `  k ) w  /\  ( q ( join `  k
) ( x (
meet `  k )
w ) )  =  x )  <->  ( -.  q  .<_  w  /\  (
q  .\/  ( x  ./\  w ) )  =  x ) ) )
3416fveq2d 5691 . . . . . . . . . . 11  |-  ( k  =  K  ->  ( LSSum `  ( ( DVecH `  k ) `  w
) )  =  (
LSSum `  ( ( DVecH `  K ) `  w
) ) )
35 fveq2 5687 . . . . . . . . . . . . 13  |-  ( k  =  K  ->  ( DIsoC `  k )  =  ( DIsoC `  K )
)
3635fveq1d 5689 . . . . . . . . . . . 12  |-  ( k  =  K  ->  (
( DIsoC `  k ) `  w )  =  ( ( DIsoC `  K ) `  w ) )
3736fveq1d 5689 . . . . . . . . . . 11  |-  ( k  =  K  ->  (
( ( DIsoC `  k
) `  w ) `  q )  =  ( ( ( DIsoC `  K
) `  w ) `  q ) )
3813, 30fveq12d 5693 . . . . . . . . . . 11  |-  ( k  =  K  ->  (
( ( DIsoB `  k
) `  w ) `  ( x ( meet `  k ) w ) )  =  ( ( ( DIsoB `  K ) `  w ) `  (
x  ./\  w )
) )
3934, 37, 38oveq123d 6061 . . . . . . . . . 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 2415 . . . . . . . . 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 312 . . . . . . . 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 2876 . . . . . . 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 6511 . . . . . 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 3721 . . . . 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 4247 . . . 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 4247 . . 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 31712 . . 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 5701 . . . . 5  |-  ( LHyp `  K )  e.  _V
493, 48eqeltri 2474 . . . 4  |-  H  e. 
_V
5049mptex 5925 . . 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 5765 . 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 16 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 3    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1721   A.wral 2666   _Vcvv 2916   ifcif 3699   class class class wbr 4172    e. cmpt 4226   ` cfv 5413  (class class class)co 6040   iota_crio 6501   Basecbs 13424   lecple 13491   joincjn 14356   meetcmee 14357   LSSumclsm 15223   LSubSpclss 15963   Atomscatm 29746   LHypclh 30466   DVecHcdvh 31561   DIsoBcdib 31621   DIsoCcdic 31655   DIsoHcdih 31711
This theorem is referenced by:  dihfval  31714
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pr 4363
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-reu 2673  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-id 4458  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-riota 6508  df-dih 31712
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