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Theorem dihfval 30688
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 30687 . . . 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 5487 . . 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 2328 . 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 4028 . . . . 5  |-  ( w  =  W  ->  (
x  .<_  w  <->  x  .<_  W ) )
12 fveq2 5485 . . . . . . 7  |-  ( w  =  W  ->  (
( DIsoB `  K ) `  w )  =  ( ( DIsoB `  K ) `  W ) )
13 dihval.d . . . . . . 7  |-  D  =  ( ( DIsoB `  K
) `  W )
1412, 13syl6eqr 2334 . . . . . 6  |-  ( w  =  W  ->  (
( DIsoB `  K ) `  w )  =  D )
1514fveq1d 5487 . . . . 5  |-  ( w  =  W  ->  (
( ( DIsoB `  K
) `  w ) `  x )  =  ( D `  x ) )
16 fveq2 5485 . . . . . . . . 9  |-  ( w  =  W  ->  (
( DVecH `  K ) `  w )  =  ( ( DVecH `  K ) `  W ) )
17 dihval.u . . . . . . . . 9  |-  U  =  ( ( DVecH `  K
) `  W )
1816, 17syl6eqr 2334 . . . . . . . 8  |-  ( w  =  W  ->  (
( DVecH `  K ) `  w )  =  U )
1918fveq2d 5489 . . . . . . 7  |-  ( w  =  W  ->  ( LSubSp `
 ( ( DVecH `  K ) `  w
) )  =  (
LSubSp `  U ) )
20 dihval.s . . . . . . 7  |-  S  =  ( LSubSp `  U )
2119, 20syl6eqr 2334 . . . . . 6  |-  ( w  =  W  ->  ( LSubSp `
 ( ( DVecH `  K ) `  w
) )  =  S )
22 breq2 4028 . . . . . . . . . 10  |-  ( w  =  W  ->  (
q  .<_  w  <->  q  .<_  W ) )
2322notbid 287 . . . . . . . . 9  |-  ( w  =  W  ->  ( -.  q  .<_  w  <->  -.  q  .<_  W ) )
24 oveq2 5827 . . . . . . . . . . 11  |-  ( w  =  W  ->  (
x  ./\  w )  =  ( x  ./\  W ) )
2524oveq2d 5835 . . . . . . . . . 10  |-  ( w  =  W  ->  (
q  .\/  ( x  ./\  w ) )  =  ( q  .\/  (
x  ./\  W )
) )
2625eqeq1d 2292 . . . . . . . . 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 5489 . . . . . . . . . . 11  |-  ( w  =  W  ->  ( LSSum `  ( ( DVecH `  K ) `  w
) )  =  (
LSSum `  U ) )
29 dihval.p . . . . . . . . . . 11  |-  .(+)  =  (
LSSum `  U )
3028, 29syl6eqr 2334 . . . . . . . . . 10  |-  ( w  =  W  ->  ( LSSum `  ( ( DVecH `  K ) `  w
) )  =  .(+)  )
31 fveq2 5485 . . . . . . . . . . . 12  |-  ( w  =  W  ->  (
( DIsoC `  K ) `  w )  =  ( ( DIsoC `  K ) `  W ) )
32 dihval.c . . . . . . . . . . . 12  |-  C  =  ( ( DIsoC `  K
) `  W )
3331, 32syl6eqr 2334 . . . . . . . . . . 11  |-  ( w  =  W  ->  (
( DIsoC `  K ) `  w )  =  C )
3433fveq1d 5487 . . . . . . . . . 10  |-  ( w  =  W  ->  (
( ( DIsoC `  K
) `  w ) `  q )  =  ( C `  q ) )
3514, 24fveq12d 5491 . . . . . . . . . 10  |-  ( w  =  W  ->  (
( ( DIsoB `  K
) `  w ) `  ( x  ./\  w
) )  =  ( D `  ( x 
./\  W ) ) )
3630, 34, 35oveq123d 5840 . . . . . . . . 9  |-  ( w  =  W  ->  (
( ( ( DIsoC `  K ) `  w
) `  q )
( LSSum `  ( ( DVecH `  K ) `  w ) ) ( ( ( DIsoB `  K
) `  w ) `  ( x  ./\  w
) ) )  =  ( ( C `  q )  .(+)  ( D `
 ( x  ./\  W ) ) ) )
3736eqeq2d 2295 . . . . . . . 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 2564 . . . . . 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 6302 . . . . 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 3588 . . . 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 4108 . . 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 2284 . . 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 5499 . . . . 5  |-  ( Base `  K )  e.  _V
452, 44eqeltri 2354 . . . 4  |-  B  e. 
_V
4645mptex 5707 . . 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 5563 . 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 2336 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 1628    e. wcel 1688   A.wral 2544   _Vcvv 2789   ifcif 3566   class class class wbr 4024    e. cmpt 4078   ` cfv 5221  (class class class)co 5819   iota_crio 6290   Basecbs 13142   lecple 13209   joincjn 14072   meetcmee 14073   LSSumclsm 14939   LSubSpclss 15683   Atomscatm 28720   LHypclh 29440   DVecHcdvh 30535   DIsoBcdib 30595   DIsoCcdic 30629   DIsoHcdih 30685
This theorem is referenced by:  dihval  30689  dihf11lem  30723
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1538  ax-5 1549  ax-17 1608  ax-9 1641  ax-8 1648  ax-13 1690  ax-14 1692  ax-6 1707  ax-7 1712  ax-11 1719  ax-12 1869  ax-ext 2265  ax-rep 4132  ax-sep 4142  ax-nul 4150  ax-pr 4213  ax-un 4511
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 941  df-tru 1315  df-ex 1534  df-nf 1537  df-sb 1636  df-eu 2148  df-mo 2149  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-ral 2549  df-rex 2550  df-reu 2551  df-rab 2553  df-v 2791  df-sbc 2993  df-csb 3083  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3457  df-if 3567  df-sn 3647  df-pr 3648  df-op 3650  df-uni 3829  df-iun 3908  df-br 4025  df-opab 4079  df-mpt 4080  df-id 4308  df-xp 4694  df-rel 4695  df-cnv 4696  df-co 4697  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701  df-fun 5223  df-fn 5224  df-f 5225  df-f1 5226  df-fo 5227  df-f1o 5228  df-fv 5229  df-ov 5822  df-iota 6252  df-riota 6299  df-dih 30686
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