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Theorem dihvalb 32097
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 2438 . . . 4  |-  ( join `  K )  =  (
join `  K )
4 eqid 2438 . . . 4  |-  ( meet `  K )  =  (
meet `  K )
5 eqid 2438 . . . 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 2438 . . . 4  |-  ( (
DIsoC `  K ) `  W )  =  ( ( DIsoC `  K ) `  W )
10 eqid 2438 . . . 4  |-  ( (
DVecH `  K ) `  W )  =  ( ( DVecH `  K ) `  W )
11 eqid 2438 . . . 4  |-  ( LSubSp `  ( ( DVecH `  K
) `  W )
)  =  ( LSubSp `  ( ( DVecH `  K
) `  W )
)
12 eqid 2438 . . . 4  |-  ( LSSum `  ( ( DVecH `  K
) `  W )
)  =  ( LSSum `  ( ( DVecH `  K
) `  W )
)
131, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12dihval 32092 . . 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 3747 . . 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 2490 . 2  |-  ( ( ( ( K  e.  V  /\  W  e.  H )  /\  X  e.  B )  /\  X  .<_  W )  ->  (
I `  X )  =  ( D `  X ) )
1615anasss 630 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 360    = wceq 1653    e. wcel 1726   A.wral 2707   ifcif 3741   class class class wbr 4214   ` cfv 5456  (class class class)co 6083   iota_crio 6544   Basecbs 13471   lecple 13538   joincjn 14403   meetcmee 14404   LSSumclsm 15270   LSubSpclss 16010   Atomscatm 30123   LHypclh 30843   DVecHcdvh 31938   DIsoBcdib 31998   DIsoCcdic 32032   DIsoHcdih 32088
This theorem is referenced by:  dihopelvalbN  32098  dih1dimb  32100  dih2dimb  32104  dih2dimbALTN  32105  dihvalcq2  32107  dihlss  32110  dihord6apre  32116  dihord3  32117  dihord5b  32119  dihord5apre  32122  dih0  32140  dihwN  32149  dihglblem3N  32155  dihmeetlem2N  32159  dih1dimatlem  32189  dihjatcclem4  32281
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4322  ax-sep 4332  ax-nul 4340  ax-pr 4405
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-reu 2714  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-iun 4097  df-br 4215  df-opab 4269  df-mpt 4270  df-id 4500  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-ov 6086  df-riota 6551  df-dih 32089
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