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Theorem dihvalb 34257
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 2402 . . . 4  |-  ( join `  K )  =  (
join `  K )
4 eqid 2402 . . . 4  |-  ( meet `  K )  =  (
meet `  K )
5 eqid 2402 . . . 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 2402 . . . 4  |-  ( (
DIsoC `  K ) `  W )  =  ( ( DIsoC `  K ) `  W )
10 eqid 2402 . . . 4  |-  ( (
DVecH `  K ) `  W )  =  ( ( DVecH `  K ) `  W )
11 eqid 2402 . . . 4  |-  ( LSubSp `  ( ( DVecH `  K
) `  W )
)  =  ( LSubSp `  ( ( DVecH `  K
) `  W )
)
12 eqid 2402 . . . 4  |-  ( LSSum `  ( ( DVecH `  K
) `  W )
)  =  ( LSSum `  ( ( DVecH `  K
) `  W )
)
131, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12dihval 34252 . . 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 3891 . . 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 2463 . 2  |-  ( ( ( ( K  e.  V  /\  W  e.  H )  /\  X  e.  B )  /\  X  .<_  W )  ->  (
I `  X )  =  ( D `  X ) )
1615anasss 645 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 367    = wceq 1405    e. wcel 1842   A.wral 2754   ifcif 3885   class class class wbr 4395   ` cfv 5569   iota_crio 6239  (class class class)co 6278   Basecbs 14841   lecple 14916   joincjn 15897   meetcmee 15898   LSSumclsm 16978   LSubSpclss 17898   Atomscatm 32281   LHypclh 33001   DVecHcdvh 34098   DIsoBcdib 34158   DIsoCcdic 34192   DIsoHcdih 34248
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4507  ax-sep 4517  ax-nul 4525  ax-pr 4630
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2759  df-rex 2760  df-reu 2761  df-rab 2763  df-v 3061  df-sbc 3278  df-csb 3374  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-nul 3739  df-if 3886  df-sn 3973  df-pr 3975  df-op 3979  df-uni 4192  df-iun 4273  df-br 4396  df-opab 4454  df-mpt 4455  df-id 4738  df-xp 4829  df-rel 4830  df-cnv 4831  df-co 4832  df-dm 4833  df-rn 4834  df-res 4835  df-ima 4836  df-iota 5533  df-fun 5571  df-fn 5572  df-f 5573  df-f1 5574  df-fo 5575  df-f1o 5576  df-fv 5577  df-riota 6240  df-ov 6281  df-dih 34249
This theorem is referenced by:  dihopelvalbN  34258  dih1dimb  34260  dih2dimb  34264  dih2dimbALTN  34265  dihvalcq2  34267  dihlss  34270  dihord6apre  34276  dihord3  34277  dihord5b  34279  dihord5apre  34282  dih0  34300  dihwN  34309  dihglblem3N  34315  dihmeetlem2N  34319  dih1dimatlem  34349  dihjatcclem4  34441
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