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

Theorem dihvalc 32031
Description: Value of isomorphism H for a lattice  K when  -.  X  .<_  W. (Contributed by NM, 4-Mar-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
dihvalc  |-  ( ( ( K  e.  V  /\  W  e.  H
)  /\  ( X  e.  B  /\  -.  X  .<_  W ) )  -> 
( I `  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 dihvalc
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, 12dihval 32030 . . 3  |-  ( ( ( 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 ) ) ) ) ) ) )
14 iffalse 3746 . . 3  |-  ( -.  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 ) ) ) ) ) )  =  ( iota_ u  e.  S A. q  e.  A  ( ( -.  q  .<_  W  /\  ( q  .\/  ( X  ./\  W ) )  =  X )  ->  u  =  ( ( C `  q )  .(+)  ( D `  ( X  ./\  W ) ) ) ) ) )
1513, 14sylan9eq 2488 . 2  |-  ( ( ( ( K  e.  V  /\  W  e.  H )  /\  X  e.  B )  /\  -.  X  .<_  W )  -> 
( I `  X
)  =  ( iota_ u  e.  S A. q  e.  A  ( ( -.  q  .<_  W  /\  ( q  .\/  ( X  ./\  W ) )  =  X )  ->  u  =  ( ( C `  q )  .(+)  ( D `  ( X  ./\  W ) ) ) ) ) )
1615anasss 629 1  |-  ( ( ( K  e.  V  /\  W  e.  H
)  /\  ( X  e.  B  /\  -.  X  .<_  W ) )  -> 
( I `  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 2705   ifcif 3739   class class class wbr 4212   ` cfv 5454  (class class class)co 6081   iota_crio 6542   Basecbs 13469   lecple 13536   joincjn 14401   meetcmee 14402   LSSumclsm 15268   LSubSpclss 16008   Atomscatm 30061   LHypclh 30781   DVecHcdvh 31876   DIsoBcdib 31936   DIsoCcdic 31970   DIsoHcdih 32026
This theorem is referenced by:  dihlsscpre  32032  dihvalcqpre  32033
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 4320  ax-sep 4330  ax-nul 4338  ax-pr 4403
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 2710  df-rex 2711  df-reu 2712  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-riota 6549  df-dih 32027
  Copyright terms: Public domain W3C validator