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Theorem dihfval 31760
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
Dummy variable  w is distinct from all other variables.
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 31759 . . . 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 5716 . . 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 2474 . 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 4203 . . . . 5  |-  ( w  =  W  ->  (
x  .<_  w  <->  x  .<_  W ) )
12 fveq2 5714 . . . . . . 7  |-  ( w  =  W  ->  (
( DIsoB `  K ) `  w )  =  ( ( DIsoB `  K ) `  W ) )
13 dihval.d . . . . . . 7  |-  D  =  ( ( DIsoB `  K
) `  W )
1412, 13syl6eqr 2480 . . . . . 6  |-  ( w  =  W  ->  (
( DIsoB `  K ) `  w )  =  D )
1514fveq1d 5716 . . . . 5  |-  ( w  =  W  ->  (
( ( DIsoB `  K
) `  w ) `  x )  =  ( D `  x ) )
16 fveq2 5714 . . . . . . . . 9  |-  ( w  =  W  ->  (
( DVecH `  K ) `  w )  =  ( ( DVecH `  K ) `  W ) )
17 dihval.u . . . . . . . . 9  |-  U  =  ( ( DVecH `  K
) `  W )
1816, 17syl6eqr 2480 . . . . . . . 8  |-  ( w  =  W  ->  (
( DVecH `  K ) `  w )  =  U )
1918fveq2d 5718 . . . . . . 7  |-  ( w  =  W  ->  ( LSubSp `
 ( ( DVecH `  K ) `  w
) )  =  (
LSubSp `  U ) )
20 dihval.s . . . . . . 7  |-  S  =  ( LSubSp `  U )
2119, 20syl6eqr 2480 . . . . . 6  |-  ( w  =  W  ->  ( LSubSp `
 ( ( DVecH `  K ) `  w
) )  =  S )
22 breq2 4203 . . . . . . . . . 10  |-  ( w  =  W  ->  (
q  .<_  w  <->  q  .<_  W ) )
2322notbid 286 . . . . . . . . 9  |-  ( w  =  W  ->  ( -.  q  .<_  w  <->  -.  q  .<_  W ) )
24 oveq2 6075 . . . . . . . . . . 11  |-  ( w  =  W  ->  (
x  ./\  w )  =  ( x  ./\  W ) )
2524oveq2d 6083 . . . . . . . . . 10  |-  ( w  =  W  ->  (
q  .\/  ( x  ./\  w ) )  =  ( q  .\/  (
x  ./\  W )
) )
2625eqeq1d 2438 . . . . . . . . 9  |-  ( w  =  W  ->  (
( q  .\/  (
x  ./\  w )
)  =  x  <->  ( q  .\/  ( x  ./\  W
) )  =  x ) )
2723, 26anbi12d 692 . . . . . . . 8  |-  ( w  =  W  ->  (
( -.  q  .<_  w  /\  ( q  .\/  ( x  ./\  w ) )  =  x )  <-> 
( -.  q  .<_  W  /\  ( q  .\/  ( x  ./\  W ) )  =  x ) ) )
2818fveq2d 5718 . . . . . . . . . . 11  |-  ( w  =  W  ->  ( LSSum `  ( ( DVecH `  K ) `  w
) )  =  (
LSSum `  U ) )
29 dihval.p . . . . . . . . . . 11  |-  .(+)  =  (
LSSum `  U )
3028, 29syl6eqr 2480 . . . . . . . . . 10  |-  ( w  =  W  ->  ( LSSum `  ( ( DVecH `  K ) `  w
) )  =  .(+)  )
31 fveq2 5714 . . . . . . . . . . . 12  |-  ( w  =  W  ->  (
( DIsoC `  K ) `  w )  =  ( ( DIsoC `  K ) `  W ) )
32 dihval.c . . . . . . . . . . . 12  |-  C  =  ( ( DIsoC `  K
) `  W )
3331, 32syl6eqr 2480 . . . . . . . . . . 11  |-  ( w  =  W  ->  (
( DIsoC `  K ) `  w )  =  C )
3433fveq1d 5716 . . . . . . . . . 10  |-  ( w  =  W  ->  (
( ( DIsoC `  K
) `  w ) `  q )  =  ( C `  q ) )
3514, 24fveq12d 5720 . . . . . . . . . 10  |-  ( w  =  W  ->  (
( ( DIsoB `  K
) `  w ) `  ( x  ./\  w
) )  =  ( D `  ( x 
./\  W ) ) )
3630, 34, 35oveq123d 6088 . . . . . . . . 9  |-  ( w  =  W  ->  (
( ( ( DIsoC `  K ) `  w
) `  q )
( LSSum `  ( ( DVecH `  K ) `  w ) ) ( ( ( DIsoB `  K
) `  w ) `  ( x  ./\  w
) ) )  =  ( ( C `  q )  .(+)  ( D `
 ( x  ./\  W ) ) ) )
3736eqeq2d 2441 . . . . . . . 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 312 . . . . . . 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 2712 . . . . . 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 6538 . . . . 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 3748 . . . 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 4283 . . 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 2430 . . 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 5728 . . . . 5  |-  ( Base `  K )  e.  _V
452, 44eqeltri 2500 . . . 4  |-  B  e. 
_V
4645mptex 5952 . . 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 5792 . 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 2482 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 3    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725   A.wral 2692   _Vcvv 2943   ifcif 3726   class class class wbr 4199    e. cmpt 4253   ` cfv 5440  (class class class)co 6067   iota_crio 6528   Basecbs 13452   lecple 13519   joincjn 14384   meetcmee 14385   LSSumclsm 15251   LSubSpclss 15991   Atomscatm 29792   LHypclh 30512   DVecHcdvh 31607   DIsoBcdib 31667   DIsoCcdic 31701   DIsoHcdih 31757
This theorem is referenced by:  dihval  31761  dihf11lem  31795
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2411  ax-rep 4307  ax-sep 4317  ax-nul 4325  ax-pr 4390
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2417  df-cleq 2423  df-clel 2426  df-nfc 2555  df-ne 2595  df-ral 2697  df-rex 2698  df-reu 2699  df-rab 2701  df-v 2945  df-sbc 3149  df-csb 3239  df-dif 3310  df-un 3312  df-in 3314  df-ss 3321  df-nul 3616  df-if 3727  df-sn 3807  df-pr 3808  df-op 3810  df-uni 4003  df-iun 4082  df-br 4200  df-opab 4254  df-mpt 4255  df-id 4485  df-xp 4870  df-rel 4871  df-cnv 4872  df-co 4873  df-dm 4874  df-rn 4875  df-res 4876  df-ima 4877  df-iota 5404  df-fun 5442  df-fn 5443  df-f 5444  df-f1 5445  df-fo 5446  df-f1o 5447  df-fv 5448  df-ov 6070  df-riota 6535  df-dih 31758
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