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Theorem dihval 30701
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 30700 . . 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 5488 . 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 4027 . . . 4  |-  ( x  =  X  ->  (
x  .<_  W  <->  X  .<_  W ) )
16 fveq2 5486 . . . 4  |-  ( x  =  X  ->  ( D `  x )  =  ( D `  X ) )
17 oveq1 5827 . . . . . . . . . 10  |-  ( x  =  X  ->  (
x  ./\  W )  =  ( X  ./\  W ) )
1817oveq2d 5836 . . . . . . . . 9  |-  ( x  =  X  ->  (
q  .\/  ( x  ./\ 
W ) )  =  ( q  .\/  ( X  ./\  W ) ) )
19 id 19 . . . . . . . . 9  |-  ( x  =  X  ->  x  =  X )
2018, 19eqeq12d 2298 . . . . . . . 8  |-  ( x  =  X  ->  (
( q  .\/  (
x  ./\  W )
)  =  x  <->  ( q  .\/  ( X  ./\  W
) )  =  X ) )
2120anbi2d 684 . . . . . . 7  |-  ( x  =  X  ->  (
( -.  q  .<_  W  /\  ( q  .\/  ( x  ./\  W ) )  =  x )  <-> 
( -.  q  .<_  W  /\  ( q  .\/  ( X  ./\  W ) )  =  X ) ) )
2217fveq2d 5490 . . . . . . . . 9  |-  ( x  =  X  ->  ( D `  ( x  ./\ 
W ) )  =  ( D `  ( X  ./\  W ) ) )
2322oveq2d 5836 . . . . . . . 8  |-  ( x  =  X  ->  (
( C `  q
)  .(+)  ( D `  ( x  ./\  W ) ) )  =  ( ( C `  q
)  .(+)  ( D `  ( X  ./\  W ) ) ) )
2423eqeq2d 2295 . . . . . . 7  |-  ( x  =  X  ->  (
u  =  ( ( C `  q ) 
.(+)  ( D `  ( x  ./\  W ) ) )  <->  u  =  ( ( C `  q )  .(+)  ( D `
 ( X  ./\  W ) ) ) ) )
2521, 24imbi12d 311 . . . . . 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 2564 . . . . 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 6302 . . . 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 3588 . . 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 2284 . . 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 5500 . . . 4  |-  ( D `
 X )  e. 
_V
31 riotaex 6304 . . . 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 3624 . . 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 5564 . 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 2336 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 358    = wceq 1623    e. wcel 1685   A.wral 2544   ifcif 3566   class class class wbr 4024    e. cmpt 4078   ` cfv 5221  (class class class)co 5820   iota_crio 6291   Basecbs 13144   lecple 13211   joincjn 14074   meetcmee 14075   LSSumclsm 14941   LSubSpclss 15685   Atomscatm 28732   LHypclh 29452   DVecHcdvh 30547   DIsoBcdib 30607   DIsoCcdic 30641   DIsoHcdih 30697
This theorem is referenced by:  dihvalc  30702  dihvalb  30706
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1636  ax-8 1644  ax-13 1687  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1868  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 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1631  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 5823  df-iota 6253  df-riota 6300  df-dih 30698
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