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Theorem dihffval 30670
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 2771 . 2  |-  ( K  e.  V  ->  K  e.  _V )
2 fveq2 5458 . . . . 5  |-  ( k  =  K  ->  ( LHyp `  k )  =  ( LHyp `  K
) )
3 dihval.h . . . . 5  |-  H  =  ( LHyp `  K
)
42, 3syl6eqr 2308 . . . 4  |-  ( k  =  K  ->  ( LHyp `  k )  =  H )
5 fveq2 5458 . . . . . 6  |-  ( k  =  K  ->  ( Base `  k )  =  ( Base `  K
) )
6 dihval.b . . . . . 6  |-  B  =  ( Base `  K
)
75, 6syl6eqr 2308 . . . . 5  |-  ( k  =  K  ->  ( Base `  k )  =  B )
8 fveq2 5458 . . . . . . . 8  |-  ( k  =  K  ->  ( le `  k )  =  ( le `  K
) )
9 dihval.l . . . . . . . 8  |-  .<_  =  ( le `  K )
108, 9syl6eqr 2308 . . . . . . 7  |-  ( k  =  K  ->  ( le `  k )  = 
.<_  )
1110breqd 4008 . . . . . 6  |-  ( k  =  K  ->  (
x ( le `  k ) w  <->  x  .<_  w ) )
12 fveq2 5458 . . . . . . . 8  |-  ( k  =  K  ->  ( DIsoB `  k )  =  ( DIsoB `  K )
)
1312fveq1d 5460 . . . . . . 7  |-  ( k  =  K  ->  (
( DIsoB `  k ) `  w )  =  ( ( DIsoB `  K ) `  w ) )
1413fveq1d 5460 . . . . . 6  |-  ( k  =  K  ->  (
( ( DIsoB `  k
) `  w ) `  x )  =  ( ( ( DIsoB `  K
) `  w ) `  x ) )
15 fveq2 5458 . . . . . . . . 9  |-  ( k  =  K  ->  ( DVecH `  k )  =  ( DVecH `  K )
)
1615fveq1d 5460 . . . . . . . 8  |-  ( k  =  K  ->  (
( DVecH `  k ) `  w )  =  ( ( DVecH `  K ) `  w ) )
1716fveq2d 5462 . . . . . . 7  |-  ( k  =  K  ->  ( LSubSp `
 ( ( DVecH `  k ) `  w
) )  =  (
LSubSp `  ( ( DVecH `  K ) `  w
) ) )
18 fveq2 5458 . . . . . . . . 9  |-  ( k  =  K  ->  ( Atoms `  k )  =  ( Atoms `  K )
)
19 dihval.a . . . . . . . . 9  |-  A  =  ( Atoms `  K )
2018, 19syl6eqr 2308 . . . . . . . 8  |-  ( k  =  K  ->  ( Atoms `  k )  =  A )
2110breqd 4008 . . . . . . . . . . 11  |-  ( k  =  K  ->  (
q ( le `  k ) w  <->  q  .<_  w ) )
2221notbid 287 . . . . . . . . . 10  |-  ( k  =  K  ->  ( -.  q ( le `  k ) w  <->  -.  q  .<_  w ) )
23 fveq2 5458 . . . . . . . . . . . . 13  |-  ( k  =  K  ->  ( join `  k )  =  ( join `  K
) )
24 dihval.j . . . . . . . . . . . . 13  |-  .\/  =  ( join `  K )
2523, 24syl6eqr 2308 . . . . . . . . . . . 12  |-  ( k  =  K  ->  ( join `  k )  = 
.\/  )
26 eqidd 2259 . . . . . . . . . . . 12  |-  ( k  =  K  ->  q  =  q )
27 fveq2 5458 . . . . . . . . . . . . . 14  |-  ( k  =  K  ->  ( meet `  k )  =  ( meet `  K
) )
28 dihval.m . . . . . . . . . . . . . 14  |-  ./\  =  ( meet `  K )
2927, 28syl6eqr 2308 . . . . . . . . . . . . 13  |-  ( k  =  K  ->  ( meet `  k )  = 
./\  )
3029oveqd 5809 . . . . . . . . . . . 12  |-  ( k  =  K  ->  (
x ( meet `  k
) w )  =  ( x  ./\  w
) )
3125, 26, 30oveq123d 5813 . . . . . . . . . . 11  |-  ( k  =  K  ->  (
q ( join `  k
) ( x (
meet `  k )
w ) )  =  ( q  .\/  (
x  ./\  w )
) )
3231eqeq1d 2266 . . . . . . . . . 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 5462 . . . . . . . . . . 11  |-  ( k  =  K  ->  ( LSSum `  ( ( DVecH `  k ) `  w
) )  =  (
LSSum `  ( ( DVecH `  K ) `  w
) ) )
35 fveq2 5458 . . . . . . . . . . . . 13  |-  ( k  =  K  ->  ( DIsoC `  k )  =  ( DIsoC `  K )
)
3635fveq1d 5460 . . . . . . . . . . . 12  |-  ( k  =  K  ->  (
( DIsoC `  k ) `  w )  =  ( ( DIsoC `  K ) `  w ) )
3736fveq1d 5460 . . . . . . . . . . 11  |-  ( k  =  K  ->  (
( ( DIsoC `  k
) `  w ) `  q )  =  ( ( ( DIsoC `  K
) `  w ) `  q ) )
3813, 30fveq12d 5464 . . . . . . . . . . 11  |-  ( k  =  K  ->  (
( ( DIsoB `  k
) `  w ) `  ( x ( meet `  k ) w ) )  =  ( ( ( DIsoB `  K ) `  w ) `  (
x  ./\  w )
) )
3934, 37, 38oveq123d 5813 . . . . . . . . . 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 2269 . . . . . . . . 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 2723 . . . . . . 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 6275 . . . . . 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 3561 . . . . 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 4072 . . . 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 4072 . . 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 30669 . . 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 5472 . . . . 5  |-  ( LHyp `  K )  e.  _V
493, 48eqeltri 2328 . . . 4  |-  H  e. 
_V
5049mptex 5680 . . 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 5536 . 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 2518   _Vcvv 2763   ifcif 3539   class class class wbr 3997    e. cmpt 4051   ` cfv 4673  (class class class)co 5792   iota_crio 6263   Basecbs 13111   lecple 13178   joincjn 14041   meetcmee 14042   LSSumclsm 14908   LSubSpclss 15652   Atomscatm 28703   LHypclh 29423   DVecHcdvh 30518   DIsoBcdib 30578   DIsoCcdic 30612   DIsoHcdih 30668
This theorem is referenced by:  dihfval  30671
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 2239  ax-rep 4105  ax-sep 4115  ax-nul 4123  ax-pr 4186  ax-un 4484
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 2122  df-mo 2123  df-clab 2245  df-cleq 2251  df-clel 2254  df-nfc 2383  df-ne 2423  df-ral 2523  df-rex 2524  df-reu 2525  df-rab 2527  df-v 2765  df-sbc 2967  df-csb 3057  df-dif 3130  df-un 3132  df-in 3134  df-ss 3141  df-nul 3431  df-if 3540  df-sn 3620  df-pr 3621  df-op 3623  df-uni 3802  df-iun 3881  df-br 3998  df-opab 4052  df-mpt 4053  df-id 4281  df-xp 4675  df-rel 4676  df-cnv 4677  df-co 4678  df-dm 4679  df-rn 4680  df-res 4681  df-ima 4682  df-fun 4683  df-fn 4684  df-f 4685  df-f1 4686  df-fo 4687  df-f1o 4688  df-fv 4689  df-ov 5795  df-iota 6225  df-riota 6272  df-dih 30669
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