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Theorem dihval 31968
Description: Value of isomorphism H for a lattice  K. Definition of isomorphism map in [Crawley] p. 122 line 3. (Contributed by NM, 3-Feb-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
dihval  |-  ( ( ( K  e.  V  /\  W  e.  H
)  /\  X  e.  B )  ->  (
I `  X )  =  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, K    u, S    W, q, u    X, q, u
Allowed substitution hints:    A( u)    B( u, q)    C( u, q)    D( u, q)    .(+) ( u, q)    S( q)    U( u, q)    H( u, q)    I( u, q)    .\/ ( u, q)    .<_ ( u, q)    ./\ ( u, q)    V( u, q)

Proof of Theorem dihval
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 dihval.b . . . 4  |-  B  =  ( Base `  K
)
2 dihval.l . . . 4  |-  .<_  =  ( le `  K )
3 dihval.j . . . 4  |-  .\/  =  ( join `  K )
4 dihval.m . . . 4  |-  ./\  =  ( meet `  K )
5 dihval.a . . . 4  |-  A  =  ( Atoms `  K )
6 dihval.h . . . 4  |-  H  =  ( LHyp `  K
)
7 dihval.i . . . 4  |-  I  =  ( ( DIsoH `  K
) `  W )
8 dihval.d . . . 4  |-  D  =  ( ( DIsoB `  K
) `  W )
9 dihval.c . . . 4  |-  C  =  ( ( DIsoC `  K
) `  W )
10 dihval.u . . . 4  |-  U  =  ( ( DVecH `  K
) `  W )
11 dihval.s . . . 4  |-  S  =  ( LSubSp `  U )
12 dihval.p . . . 4  |-  .(+)  =  (
LSSum `  U )
131, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12dihfval 31967 . . 3  |-  ( ( 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 ) ) ) ) ) ) ) )
1413fveq1d 5723 . 2  |-  ( ( K  e.  V  /\  W  e.  H )  ->  ( I `  X
)  =  ( ( 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 ) ) ) ) ) ) ) `  X
) )
15 breq1 4208 . . . 4  |-  ( x  =  X  ->  (
x  .<_  W  <->  X  .<_  W ) )
16 fveq2 5721 . . . 4  |-  ( x  =  X  ->  ( D `  x )  =  ( D `  X ) )
17 oveq1 6081 . . . . . . . . . 10  |-  ( x  =  X  ->  (
x  ./\  W )  =  ( X  ./\  W ) )
1817oveq2d 6090 . . . . . . . . 9  |-  ( x  =  X  ->  (
q  .\/  ( x  ./\ 
W ) )  =  ( q  .\/  ( X  ./\  W ) ) )
19 id 20 . . . . . . . . 9  |-  ( x  =  X  ->  x  =  X )
2018, 19eqeq12d 2450 . . . . . . . 8  |-  ( x  =  X  ->  (
( q  .\/  (
x  ./\  W )
)  =  x  <->  ( q  .\/  ( X  ./\  W
) )  =  X ) )
2120anbi2d 685 . . . . . . 7  |-  ( x  =  X  ->  (
( -.  q  .<_  W  /\  ( q  .\/  ( x  ./\  W ) )  =  x )  <-> 
( -.  q  .<_  W  /\  ( q  .\/  ( X  ./\  W ) )  =  X ) ) )
2217fveq2d 5725 . . . . . . . . 9  |-  ( x  =  X  ->  ( D `  ( x  ./\ 
W ) )  =  ( D `  ( X  ./\  W ) ) )
2322oveq2d 6090 . . . . . . . 8  |-  ( x  =  X  ->  (
( C `  q
)  .(+)  ( D `  ( x  ./\  W ) ) )  =  ( ( C `  q
)  .(+)  ( D `  ( X  ./\  W ) ) ) )
2423eqeq2d 2447 . . . . . . 7  |-  ( x  =  X  ->  (
u  =  ( ( C `  q ) 
.(+)  ( D `  ( x  ./\  W ) ) )  <->  u  =  ( ( C `  q )  .(+)  ( D `
 ( X  ./\  W ) ) ) ) )
2521, 24imbi12d 312 . . . . . 6  |-  ( x  =  X  ->  (
( ( -.  q  .<_  W  /\  ( q 
.\/  ( x  ./\  W ) )  =  x )  ->  u  =  ( ( C `  q )  .(+)  ( D `
 ( x  ./\  W ) ) ) )  <-> 
( ( -.  q  .<_  W  /\  ( q 
.\/  ( X  ./\  W ) )  =  X )  ->  u  =  ( ( C `  q )  .(+)  ( D `
 ( X  ./\  W ) ) ) ) ) )
2625ralbidv 2718 . . . . 5  |-  ( x  =  X  ->  ( A. q  e.  A  ( ( -.  q  .<_  W  /\  ( q 
.\/  ( x  ./\  W ) )  =  x )  ->  u  =  ( ( C `  q )  .(+)  ( D `
 ( x  ./\  W ) ) ) )  <->  A. q  e.  A  ( ( -.  q  .<_  W  /\  ( q 
.\/  ( X  ./\  W ) )  =  X )  ->  u  =  ( ( C `  q )  .(+)  ( D `
 ( X  ./\  W ) ) ) ) ) )
2726riotabidv 6544 . . . 4  |-  ( x  =  X  ->  ( iota_ u  e.  S A. q  e.  A  (
( -.  q  .<_  W  /\  ( q  .\/  ( x  ./\  W ) )  =  x )  ->  u  =  ( ( C `  q
)  .(+)  ( D `  ( x  ./\  W ) ) ) ) )  =  ( iota_ u  e.  S A. q  e.  A  ( ( -.  q  .<_  W  /\  ( q  .\/  ( X  ./\  W ) )  =  X )  ->  u  =  ( ( C `  q )  .(+)  ( D `  ( X  ./\  W ) ) ) ) ) )
2815, 16, 27ifbieq12d 3754 . . 3  |-  ( x  =  X  ->  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 ) ) ) ) ) )  =  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 ) ) ) ) ) ) )
29 eqid 2436 . . 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 ) ) ) ) ) ) )  =  ( 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 ) ) ) ) ) ) )
30 fvex 5735 . . . 4  |-  ( D `
 X )  e. 
_V
31 riotaex 6546 . . . 4  |-  ( iota_ u  e.  S A. q  e.  A  ( ( -.  q  .<_  W  /\  ( q  .\/  ( X  ./\  W ) )  =  X )  ->  u  =  ( ( C `  q )  .(+)  ( D `  ( X  ./\  W ) ) ) ) )  e. 
_V
3230, 31ifex 3790 . . 3  |-  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
3328, 29, 32fvmpt 5799 . 2  |-  ( X  e.  B  ->  (
( 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 ) ) ) ) ) ) ) `  X )  =  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 ) ) ) ) ) ) )
3414, 33sylan9eq 2488 1  |-  ( ( ( K  e.  V  /\  W  e.  H
)  /\  X  e.  B )  ->  (
I `  X )  =  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 2698   ifcif 3732   class class class wbr 4205    e. cmpt 4259   ` cfv 5447  (class class class)co 6074   iota_crio 6535   Basecbs 13462   lecple 13529   joincjn 14394   meetcmee 14395   LSSumclsm 15261   LSubSpclss 16001   Atomscatm 29999   LHypclh 30719   DVecHcdvh 31814   DIsoBcdib 31874   DIsoCcdic 31908   DIsoHcdih 31964
This theorem is referenced by:  dihvalc  31969  dihvalb  31973
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 2417  ax-rep 4313  ax-sep 4323  ax-nul 4331  ax-pr 4396
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 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2703  df-rex 2704  df-reu 2705  df-rab 2707  df-v 2951  df-sbc 3155  df-csb 3245  df-dif 3316  df-un 3318  df-in 3320  df-ss 3327  df-nul 3622  df-if 3733  df-sn 3813  df-pr 3814  df-op 3816  df-uni 4009  df-iun 4088  df-br 4206  df-opab 4260  df-mpt 4261  df-id 4491  df-xp 4877  df-rel 4878  df-cnv 4879  df-co 4880  df-dm 4881  df-rn 4882  df-res 4883  df-ima 4884  df-iota 5411  df-fun 5449  df-fn 5450  df-f 5451  df-f1 5452  df-fo 5453  df-f1o 5454  df-fv 5455  df-ov 6077  df-riota 6542  df-dih 31965
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