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Theorem dihfval 30551
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 30550 . . . 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 5425 . . 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 2300 . 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 3967 . . . . 5  |-  ( w  =  W  ->  (
x  .<_  w  <->  x  .<_  W ) )
12 fveq2 5423 . . . . . . 7  |-  ( w  =  W  ->  (
( DIsoB `  K ) `  w )  =  ( ( DIsoB `  K ) `  W ) )
13 dihval.d . . . . . . 7  |-  D  =  ( ( DIsoB `  K
) `  W )
1412, 13syl6eqr 2306 . . . . . 6  |-  ( w  =  W  ->  (
( DIsoB `  K ) `  w )  =  D )
1514fveq1d 5425 . . . . 5  |-  ( w  =  W  ->  (
( ( DIsoB `  K
) `  w ) `  x )  =  ( D `  x ) )
16 fveq2 5423 . . . . . . . . 9  |-  ( w  =  W  ->  (
( DVecH `  K ) `  w )  =  ( ( DVecH `  K ) `  W ) )
17 dihval.u . . . . . . . . 9  |-  U  =  ( ( DVecH `  K
) `  W )
1816, 17syl6eqr 2306 . . . . . . . 8  |-  ( w  =  W  ->  (
( DVecH `  K ) `  w )  =  U )
1918fveq2d 5427 . . . . . . 7  |-  ( w  =  W  ->  ( LSubSp `
 ( ( DVecH `  K ) `  w
) )  =  (
LSubSp `  U ) )
20 dihval.s . . . . . . 7  |-  S  =  ( LSubSp `  U )
2119, 20syl6eqr 2306 . . . . . 6  |-  ( w  =  W  ->  ( LSubSp `
 ( ( DVecH `  K ) `  w
) )  =  S )
22 breq2 3967 . . . . . . . . . 10  |-  ( w  =  W  ->  (
q  .<_  w  <->  q  .<_  W ) )
2322notbid 287 . . . . . . . . 9  |-  ( w  =  W  ->  ( -.  q  .<_  w  <->  -.  q  .<_  W ) )
24 oveq2 5765 . . . . . . . . . . 11  |-  ( w  =  W  ->  (
x  ./\  w )  =  ( x  ./\  W ) )
2524oveq2d 5773 . . . . . . . . . 10  |-  ( w  =  W  ->  (
q  .\/  ( x  ./\  w ) )  =  ( q  .\/  (
x  ./\  W )
) )
2625eqeq1d 2264 . . . . . . . . 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 5427 . . . . . . . . . . 11  |-  ( w  =  W  ->  ( LSSum `  ( ( DVecH `  K ) `  w
) )  =  (
LSSum `  U ) )
29 dihval.p . . . . . . . . . . 11  |-  .(+)  =  (
LSSum `  U )
3028, 29syl6eqr 2306 . . . . . . . . . 10  |-  ( w  =  W  ->  ( LSSum `  ( ( DVecH `  K ) `  w
) )  =  .(+)  )
31 fveq2 5423 . . . . . . . . . . . 12  |-  ( w  =  W  ->  (
( DIsoC `  K ) `  w )  =  ( ( DIsoC `  K ) `  W ) )
32 dihval.c . . . . . . . . . . . 12  |-  C  =  ( ( DIsoC `  K
) `  W )
3331, 32syl6eqr 2306 . . . . . . . . . . 11  |-  ( w  =  W  ->  (
( DIsoC `  K ) `  w )  =  C )
3433fveq1d 5425 . . . . . . . . . 10  |-  ( w  =  W  ->  (
( ( DIsoC `  K
) `  w ) `  q )  =  ( C `  q ) )
3514, 24fveq12d 5429 . . . . . . . . . 10  |-  ( w  =  W  ->  (
( ( DIsoB `  K
) `  w ) `  ( x  ./\  w
) )  =  ( D `  ( x 
./\  W ) ) )
3630, 34, 35oveq123d 5778 . . . . . . . . 9  |-  ( w  =  W  ->  (
( ( ( DIsoC `  K ) `  w
) `  q )
( LSSum `  ( ( DVecH `  K ) `  w ) ) ( ( ( DIsoB `  K
) `  w ) `  ( x  ./\  w
) ) )  =  ( ( C `  q )  .(+)  ( D `
 ( x  ./\  W ) ) ) )
3736eqeq2d 2267 . . . . . . . 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 2534 . . . . . 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 6240 . . . . 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 3528 . . . 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 4047 . . 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 2256 . . 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 5437 . . . . 5  |-  ( Base `  K )  e.  _V
452, 44eqeltri 2326 . . . 4  |-  B  e. 
_V
4645mptex 5645 . . 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 5501 . 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 2308 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 2516   _Vcvv 2740   ifcif 3506   class class class wbr 3963    e. cmpt 4017   ` cfv 4638  (class class class)co 5757   iota_crio 6228   Basecbs 13075   lecple 13142   joincjn 14005   meetcmee 14006   LSSumclsm 14872   LSubSpclss 15616   Atomscatm 28583   LHypclh 29303   DVecHcdvh 30398   DIsoBcdib 30458   DIsoCcdic 30492   DIsoHcdih 30548
This theorem is referenced by:  dihval  30552  dihf11lem  30586
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 2237  ax-rep 4071  ax-sep 4081  ax-nul 4089  ax-pr 4152  ax-un 4449
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 2121  df-mo 2122  df-clab 2243  df-cleq 2249  df-clel 2252  df-nfc 2381  df-ne 2421  df-ral 2520  df-rex 2521  df-reu 2522  df-rab 2523  df-v 2742  df-sbc 2936  df-csb 3024  df-dif 3097  df-un 3099  df-in 3101  df-ss 3108  df-nul 3398  df-if 3507  df-sn 3587  df-pr 3588  df-op 3590  df-uni 3769  df-iun 3848  df-br 3964  df-opab 4018  df-mpt 4019  df-id 4246  df-xp 4640  df-rel 4641  df-cnv 4642  df-co 4643  df-dm 4644  df-rn 4645  df-res 4646  df-ima 4647  df-fun 4648  df-fn 4649  df-f 4650  df-f1 4651  df-fo 4652  df-f1o 4653  df-fv 4654  df-ov 5760  df-iota 6190  df-riota 6237  df-dih 30549
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