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Theorem dihfval 30588
Description: 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
)
dihval.i  |-  I  =  ( ( DIsoH `  K
) `  W )
dihval.d  |-  D  =  ( ( DIsoB `  K
) `  W )
dihval.c  |-  C  =  ( ( DIsoC `  K
) `  W )
dihval.u  |-  U  =  ( ( DVecH `  K
) `  W )
dihval.s  |-  S  =  ( LSubSp `  U )
dihval.p  |-  .(+)  =  (
LSSum `  U )
Assertion
Ref Expression
dihfval  |-  ( ( K  e.  V  /\  W  e.  H )  ->  I  =  ( x  e.  B  |->  if ( x  .<_  W , 
( D `  x
) ,  ( iota_ u  e.  S A. q  e.  A  ( ( -.  q  .<_  W  /\  ( q  .\/  (
x  ./\  W )
)  =  x )  ->  u  =  ( ( C `  q
)  .(+)  ( D `  ( x  ./\  W ) ) ) ) ) ) ) )
Distinct variable groups:    A, q    u, q, x, K    x, B    u, S    W, q, u, x
Allowed substitution hints:    A( x, u)    B( u, q)    C( x, u, q)    D( x, u, q)    .(+) ( x, u, q)    S( x, q)    U( x, u, q)    H( x, u, q)    I( x, u, q)    .\/ ( x, u, q)    .<_ ( x, u, q)    ./\ (
x, u, q)    V( x, u, q)

Proof of Theorem dihfval
StepHypRef Expression
1 dihval.i . . 3  |-  I  =  ( ( DIsoH `  K
) `  W )
2 dihval.b . . . . 5  |-  B  =  ( Base `  K
)
3 dihval.l . . . . 5  |-  .<_  =  ( le `  K )
4 dihval.j . . . . 5  |-  .\/  =  ( join `  K )
5 dihval.m . . . . 5  |-  ./\  =  ( meet `  K )
6 dihval.a . . . . 5  |-  A  =  ( Atoms `  K )
7 dihval.h . . . . 5  |-  H  =  ( LHyp `  K
)
82, 3, 4, 5, 6, 7dihffval 30587 . . . 4  |-  ( 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
) ) ) ) ) ) ) ) )
98fveq1d 5460 . . 3  |-  ( K  e.  V  ->  (
( DIsoH `  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
) ) ) ) ) ) ) ) `
 W ) )
101, 9syl5eq 2302 . 2  |-  ( K  e.  V  ->  I  =  ( ( 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
) ) ) ) ) ) ) ) `
 W ) )
11 breq2 4001 . . . . 5  |-  ( w  =  W  ->  (
x  .<_  w  <->  x  .<_  W ) )
12 fveq2 5458 . . . . . . 7  |-  ( w  =  W  ->  (
( DIsoB `  K ) `  w )  =  ( ( DIsoB `  K ) `  W ) )
13 dihval.d . . . . . . 7  |-  D  =  ( ( DIsoB `  K
) `  W )
1412, 13syl6eqr 2308 . . . . . 6  |-  ( w  =  W  ->  (
( DIsoB `  K ) `  w )  =  D )
1514fveq1d 5460 . . . . 5  |-  ( w  =  W  ->  (
( ( DIsoB `  K
) `  w ) `  x )  =  ( D `  x ) )
16 fveq2 5458 . . . . . . . . 9  |-  ( w  =  W  ->  (
( DVecH `  K ) `  w )  =  ( ( DVecH `  K ) `  W ) )
17 dihval.u . . . . . . . . 9  |-  U  =  ( ( DVecH `  K
) `  W )
1816, 17syl6eqr 2308 . . . . . . . 8  |-  ( w  =  W  ->  (
( DVecH `  K ) `  w )  =  U )
1918fveq2d 5462 . . . . . . 7  |-  ( w  =  W  ->  ( LSubSp `
 ( ( DVecH `  K ) `  w
) )  =  (
LSubSp `  U ) )
20 dihval.s . . . . . . 7  |-  S  =  ( LSubSp `  U )
2119, 20syl6eqr 2308 . . . . . 6  |-  ( w  =  W  ->  ( LSubSp `
 ( ( DVecH `  K ) `  w
) )  =  S )
22 breq2 4001 . . . . . . . . . 10  |-  ( w  =  W  ->  (
q  .<_  w  <->  q  .<_  W ) )
2322notbid 287 . . . . . . . . 9  |-  ( w  =  W  ->  ( -.  q  .<_  w  <->  -.  q  .<_  W ) )
24 oveq2 5800 . . . . . . . . . . 11  |-  ( w  =  W  ->  (
x  ./\  w )  =  ( x  ./\  W ) )
2524oveq2d 5808 . . . . . . . . . 10  |-  ( w  =  W  ->  (
q  .\/  ( x  ./\  w ) )  =  ( q  .\/  (
x  ./\  W )
) )
2625eqeq1d 2266 . . . . . . . . 9  |-  ( w  =  W  ->  (
( q  .\/  (
x  ./\  w )
)  =  x  <->  ( q  .\/  ( x  ./\  W
) )  =  x ) )
2723, 26anbi12d 694 . . . . . . . 8  |-  ( w  =  W  ->  (
( -.  q  .<_  w  /\  ( q  .\/  ( x  ./\  w ) )  =  x )  <-> 
( -.  q  .<_  W  /\  ( q  .\/  ( x  ./\  W ) )  =  x ) ) )
2818fveq2d 5462 . . . . . . . . . . 11  |-  ( w  =  W  ->  ( LSSum `  ( ( DVecH `  K ) `  w
) )  =  (
LSSum `  U ) )
29 dihval.p . . . . . . . . . . 11  |-  .(+)  =  (
LSSum `  U )
3028, 29syl6eqr 2308 . . . . . . . . . 10  |-  ( w  =  W  ->  ( LSSum `  ( ( DVecH `  K ) `  w
) )  =  .(+)  )
31 fveq2 5458 . . . . . . . . . . . 12  |-  ( w  =  W  ->  (
( DIsoC `  K ) `  w )  =  ( ( DIsoC `  K ) `  W ) )
32 dihval.c . . . . . . . . . . . 12  |-  C  =  ( ( DIsoC `  K
) `  W )
3331, 32syl6eqr 2308 . . . . . . . . . . 11  |-  ( w  =  W  ->  (
( DIsoC `  K ) `  w )  =  C )
3433fveq1d 5460 . . . . . . . . . 10  |-  ( w  =  W  ->  (
( ( DIsoC `  K
) `  w ) `  q )  =  ( C `  q ) )
3514, 24fveq12d 5464 . . . . . . . . . 10  |-  ( w  =  W  ->  (
( ( DIsoB `  K
) `  w ) `  ( x  ./\  w
) )  =  ( D `  ( x 
./\  W ) ) )
3630, 34, 35oveq123d 5813 . . . . . . . . 9  |-  ( w  =  W  ->  (
( ( ( DIsoC `  K ) `  w
) `  q )
( LSSum `  ( ( DVecH `  K ) `  w ) ) ( ( ( DIsoB `  K
) `  w ) `  ( x  ./\  w
) ) )  =  ( ( C `  q )  .(+)  ( D `
 ( x  ./\  W ) ) ) )
3736eqeq2d 2269 . . . . . . . 8  |-  ( w  =  W  ->  (
u  =  ( ( ( ( DIsoC `  K
) `  w ) `  q ) ( LSSum `  ( ( DVecH `  K
) `  w )
) ( ( (
DIsoB `  K ) `  w ) `  (
x  ./\  w )
) )  <->  u  =  ( ( C `  q )  .(+)  ( D `
 ( x  ./\  W ) ) ) ) )
3827, 37imbi12d 313 . . . . . . 7  |-  ( w  =  W  ->  (
( ( -.  q  .<_  w  /\  ( q 
.\/  ( x  ./\  w ) )  =  x )  ->  u  =  ( ( ( ( DIsoC `  K ) `  w ) `  q
) ( LSSum `  (
( DVecH `  K ) `  w ) ) ( ( ( DIsoB `  K
) `  w ) `  ( x  ./\  w
) ) ) )  <-> 
( ( -.  q  .<_  W  /\  ( q 
.\/  ( x  ./\  W ) )  =  x )  ->  u  =  ( ( C `  q )  .(+)  ( D `
 ( x  ./\  W ) ) ) ) ) )
3938ralbidv 2538 . . . . . 6  |-  ( w  =  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
) ) ) )  <->  A. q  e.  A  ( ( -.  q  .<_  W  /\  ( q 
.\/  ( x  ./\  W ) )  =  x )  ->  u  =  ( ( C `  q )  .(+)  ( D `
 ( x  ./\  W ) ) ) ) ) )
4021, 39riotaeqbidv 6275 . . . . 5  |-  ( w  =  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
) ) ) ) )  =  ( iota_ u  e.  S A. q  e.  A  ( ( -.  q  .<_  W  /\  ( q  .\/  (
x  ./\  W )
)  =  x )  ->  u  =  ( ( C `  q
)  .(+)  ( D `  ( x  ./\  W ) ) ) ) ) )
4111, 15, 40ifbieq12d 3561 . . . 4  |-  ( w  =  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
) ) ) ) ) )  =  if ( x  .<_  W , 
( D `  x
) ,  ( iota_ u  e.  S A. q  e.  A  ( ( -.  q  .<_  W  /\  ( q  .\/  (
x  ./\  W )
)  =  x )  ->  u  =  ( ( C `  q
)  .(+)  ( D `  ( x  ./\  W ) ) ) ) ) ) )
4241mpteq2dv 4081 . . 3  |-  ( w  =  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
) ) ) ) ) ) )  =  ( x  e.  B  |->  if ( x  .<_  W ,  ( D `  x ) ,  (
iota_ u  e.  S A. q  e.  A  ( ( -.  q  .<_  W  /\  ( q 
.\/  ( x  ./\  W ) )  =  x )  ->  u  =  ( ( C `  q )  .(+)  ( D `
 ( x  ./\  W ) ) ) ) ) ) ) )
43 eqid 2258 . . 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
) ) ) ) ) ) ) )  =  ( 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
) ) ) ) ) ) ) )
44 fvex 5472 . . . . 5  |-  ( Base `  K )  e.  _V
452, 44eqeltri 2328 . . . 4  |-  B  e. 
_V
4645mptex 5680 . . 3  |-  ( x  e.  B  |->  if ( x  .<_  W , 
( D `  x
) ,  ( iota_ u  e.  S A. q  e.  A  ( ( -.  q  .<_  W  /\  ( q  .\/  (
x  ./\  W )
)  =  x )  ->  u  =  ( ( C `  q
)  .(+)  ( D `  ( x  ./\  W ) ) ) ) ) ) )  e.  _V
4742, 43, 46fvmpt 5536 . 2  |-  ( W  e.  H  ->  (
( 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
) ) ) ) ) ) ) ) `
 W )  =  ( x  e.  B  |->  if ( x  .<_  W ,  ( D `  x ) ,  (
iota_ u  e.  S A. q  e.  A  ( ( -.  q  .<_  W  /\  ( q 
.\/  ( x  ./\  W ) )  =  x )  ->  u  =  ( ( C `  q )  .(+)  ( D `
 ( x  ./\  W ) ) ) ) ) ) ) )
4810, 47sylan9eq 2310 1  |-  ( ( K  e.  V  /\  W  e.  H )  ->  I  =  ( x  e.  B  |->  if ( x  .<_  W , 
( D `  x
) ,  ( iota_ u  e.  S A. q  e.  A  ( ( -.  q  .<_  W  /\  ( q  .\/  (
x  ./\  W )
)  =  x )  ->  u  =  ( ( C `  q
)  .(+)  ( D `  ( 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 13110   lecple 13177   joincjn 14040   meetcmee 14041   LSSumclsm 14907   LSubSpclss 15651   Atomscatm 28620   LHypclh 29340   DVecHcdvh 30435   DIsoBcdib 30495   DIsoCcdic 30529   DIsoHcdih 30585
This theorem is referenced by:  dihval  30589  dihf11lem  30623
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 30586
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