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Theorem dihvalb 32049
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 2296 . . . 4  |-  ( join `  K )  =  (
join `  K )
4 eqid 2296 . . . 4  |-  ( meet `  K )  =  (
meet `  K )
5 eqid 2296 . . . 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 2296 . . . 4  |-  ( (
DIsoC `  K ) `  W )  =  ( ( DIsoC `  K ) `  W )
10 eqid 2296 . . . 4  |-  ( (
DVecH `  K ) `  W )  =  ( ( DVecH `  K ) `  W )
11 eqid 2296 . . . 4  |-  ( LSubSp `  ( ( DVecH `  K
) `  W )
)  =  ( LSubSp `  ( ( DVecH `  K
) `  W )
)
12 eqid 2296 . . . 4  |-  ( LSSum `  ( ( DVecH `  K
) `  W )
)  =  ( LSSum `  ( ( DVecH `  K
) `  W )
)
131, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12dihval 32044 . . 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 3584 . . 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 2348 . 2  |-  ( ( ( ( K  e.  V  /\  W  e.  H )  /\  X  e.  B )  /\  X  .<_  W )  ->  (
I `  X )  =  ( D `  X ) )
1615anasss 628 1  |-  ( ( ( K  e.  V  /\  W  e.  H
)  /\  ( X  e.  B  /\  X  .<_  W ) )  ->  (
I `  X )  =  ( D `  X ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 358    = wceq 1632    e. wcel 1696   A.wral 2556   ifcif 3578   class class class wbr 4039   ` cfv 5271  (class class class)co 5874   iota_crio 6313   Basecbs 13164   lecple 13231   joincjn 14094   meetcmee 14095   LSSumclsm 14961   LSubSpclss 15705   Atomscatm 30075   LHypclh 30795   DVecHcdvh 31890   DIsoBcdib 31950   DIsoCcdic 31984   DIsoHcdih 32040
This theorem is referenced by:  dihopelvalbN  32050  dih1dimb  32052  dih2dimb  32056  dih2dimbALTN  32057  dihvalcq2  32059  dihlss  32062  dihord6apre  32068  dihord3  32069  dihord5b  32071  dihord5apre  32074  dih0  32092  dihwN  32101  dihglblem3N  32107  dihmeetlem2N  32111  dih1dimatlem  32141  dihjatcclem4  32233
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-riota 6320  df-dih 32041
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