Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  dihval Structured version   Unicode version

Theorem dihval 37356
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 37355 . . 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 5850 . 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 4442 . . . 4  |-  ( x  =  X  ->  (
x  .<_  W  <->  X  .<_  W ) )
16 fveq2 5848 . . . 4  |-  ( x  =  X  ->  ( D `  x )  =  ( D `  X ) )
17 oveq1 6277 . . . . . . . . . 10  |-  ( x  =  X  ->  (
x  ./\  W )  =  ( X  ./\  W ) )
1817oveq2d 6286 . . . . . . . . 9  |-  ( x  =  X  ->  (
q  .\/  ( x  ./\ 
W ) )  =  ( q  .\/  ( X  ./\  W ) ) )
19 id 22 . . . . . . . . 9  |-  ( x  =  X  ->  x  =  X )
2018, 19eqeq12d 2476 . . . . . . . 8  |-  ( x  =  X  ->  (
( q  .\/  (
x  ./\  W )
)  =  x  <->  ( q  .\/  ( X  ./\  W
) )  =  X ) )
2120anbi2d 701 . . . . . . 7  |-  ( x  =  X  ->  (
( -.  q  .<_  W  /\  ( q  .\/  ( x  ./\  W ) )  =  x )  <-> 
( -.  q  .<_  W  /\  ( q  .\/  ( X  ./\  W ) )  =  X ) ) )
2217fveq2d 5852 . . . . . . . . 9  |-  ( x  =  X  ->  ( D `  ( x  ./\ 
W ) )  =  ( D `  ( X  ./\  W ) ) )
2322oveq2d 6286 . . . . . . . 8  |-  ( x  =  X  ->  (
( C `  q
)  .(+)  ( D `  ( x  ./\  W ) ) )  =  ( ( C `  q
)  .(+)  ( D `  ( X  ./\  W ) ) ) )
2423eqeq2d 2468 . . . . . . 7  |-  ( x  =  X  ->  (
u  =  ( ( C `  q ) 
.(+)  ( D `  ( x  ./\  W ) ) )  <->  u  =  ( ( C `  q )  .(+)  ( D `
 ( X  ./\  W ) ) ) ) )
2521, 24imbi12d 318 . . . . . 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 2893 . . . . 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 6234 . . . 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 3956 . . 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 2454 . . 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 5858 . . . 4  |-  ( D `
 X )  e. 
_V
31 riotaex 6236 . . . 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 3997 . . 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 5931 . 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 2515 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 setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 367    = wceq 1398    e. wcel 1823   A.wral 2804   ifcif 3929   class class class wbr 4439    |-> cmpt 4497   ` cfv 5570   iota_crio 6231  (class class class)co 6270   Basecbs 14716   lecple 14791   joincjn 15772   meetcmee 15773   LSSumclsm 16853   LSubSpclss 17773   Atomscatm 35385   LHypclh 36105   DVecHcdvh 37202   DIsoBcdib 37262   DIsoCcdic 37296   DIsoHcdih 37352
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pr 4676
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-reu 2811  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-id 4784  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-riota 6232  df-ov 6273  df-dih 37353
This theorem is referenced by:  dihvalc  37357  dihvalb  37361
  Copyright terms: Public domain W3C validator