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Theorem dihvalb 36034
Description: Value of isomorphism H for a lattice  K when  X  .<_  W. (Contributed by NM, 4-Mar-2014.)
Hypotheses
Ref Expression
dihvalb.b  |-  B  =  ( Base `  K
)
dihvalb.l  |-  .<_  =  ( le `  K )
dihvalb.h  |-  H  =  ( LHyp `  K
)
dihvalb.i  |-  I  =  ( ( DIsoH `  K
) `  W )
dihvalb.d  |-  D  =  ( ( DIsoB `  K
) `  W )
Assertion
Ref Expression
dihvalb  |-  ( ( ( K  e.  V  /\  W  e.  H
)  /\  ( X  e.  B  /\  X  .<_  W ) )  ->  (
I `  X )  =  ( D `  X ) )

Proof of Theorem dihvalb
Dummy variables  u  q are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 dihvalb.b . . . 4  |-  B  =  ( Base `  K
)
2 dihvalb.l . . . 4  |-  .<_  =  ( le `  K )
3 eqid 2467 . . . 4  |-  ( join `  K )  =  (
join `  K )
4 eqid 2467 . . . 4  |-  ( meet `  K )  =  (
meet `  K )
5 eqid 2467 . . . 4  |-  ( Atoms `  K )  =  (
Atoms `  K )
6 dihvalb.h . . . 4  |-  H  =  ( LHyp `  K
)
7 dihvalb.i . . . 4  |-  I  =  ( ( DIsoH `  K
) `  W )
8 dihvalb.d . . . 4  |-  D  =  ( ( DIsoB `  K
) `  W )
9 eqid 2467 . . . 4  |-  ( (
DIsoC `  K ) `  W )  =  ( ( DIsoC `  K ) `  W )
10 eqid 2467 . . . 4  |-  ( (
DVecH `  K ) `  W )  =  ( ( DVecH `  K ) `  W )
11 eqid 2467 . . . 4  |-  ( LSubSp `  ( ( DVecH `  K
) `  W )
)  =  ( LSubSp `  ( ( DVecH `  K
) `  W )
)
12 eqid 2467 . . . 4  |-  ( LSSum `  ( ( DVecH `  K
) `  W )
)  =  ( LSSum `  ( ( DVecH `  K
) `  W )
)
131, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12dihval 36029 . . 3  |-  ( ( ( K  e.  V  /\  W  e.  H
)  /\  X  e.  B )  ->  (
I `  X )  =  if ( X  .<_  W ,  ( D `  X ) ,  (
iota_ u  e.  ( LSubSp `
 ( ( DVecH `  K ) `  W
) ) A. q  e.  ( Atoms `  K )
( ( -.  q  .<_  W  /\  ( q ( join `  K
) ( X (
meet `  K ) W ) )  =  X )  ->  u  =  ( ( ( ( DIsoC `  K ) `  W ) `  q
) ( LSSum `  (
( DVecH `  K ) `  W ) ) ( D `  ( X ( meet `  K
) W ) ) ) ) ) ) )
14 iftrue 3945 . . 3  |-  ( X 
.<_  W  ->  if ( X  .<_  W ,  ( D `  X ) ,  ( iota_ u  e.  ( LSubSp `  ( ( DVecH `  K ) `  W ) ) A. q  e.  ( Atoms `  K ) ( ( -.  q  .<_  W  /\  ( q ( join `  K ) ( X ( meet `  K
) W ) )  =  X )  ->  u  =  ( (
( ( DIsoC `  K
) `  W ) `  q ) ( LSSum `  ( ( DVecH `  K
) `  W )
) ( D `  ( X ( meet `  K
) W ) ) ) ) ) )  =  ( D `  X ) )
1513, 14sylan9eq 2528 . 2  |-  ( ( ( ( K  e.  V  /\  W  e.  H )  /\  X  e.  B )  /\  X  .<_  W )  ->  (
I `  X )  =  ( D `  X ) )
1615anasss 647 1  |-  ( ( ( K  e.  V  /\  W  e.  H
)  /\  ( X  e.  B  /\  X  .<_  W ) )  ->  (
I `  X )  =  ( D `  X ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 369    = wceq 1379    e. wcel 1767   A.wral 2814   ifcif 3939   class class class wbr 4447   ` cfv 5586   iota_crio 6242  (class class class)co 6282   Basecbs 14486   lecple 14558   joincjn 15427   meetcmee 15428   LSSumclsm 16450   LSubSpclss 17361   Atomscatm 34060   LHypclh 34780   DVecHcdvh 35875   DIsoBcdib 35935   DIsoCcdic 35969   DIsoHcdih 36025
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pr 4686
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-riota 6243  df-ov 6285  df-dih 36026
This theorem is referenced by:  dihopelvalbN  36035  dih1dimb  36037  dih2dimb  36041  dih2dimbALTN  36042  dihvalcq2  36044  dihlss  36047  dihord6apre  36053  dihord3  36054  dihord5b  36056  dihord5apre  36059  dih0  36077  dihwN  36086  dihglblem3N  36092  dihmeetlem2N  36096  dih1dimatlem  36126  dihjatcclem4  36218
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