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Theorem dihvalb 31427
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 2283 . . . 4  |-  ( join `  K )  =  (
join `  K )
4 eqid 2283 . . . 4  |-  ( meet `  K )  =  (
meet `  K )
5 eqid 2283 . . . 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 2283 . . . 4  |-  ( (
DIsoC `  K ) `  W )  =  ( ( DIsoC `  K ) `  W )
10 eqid 2283 . . . 4  |-  ( (
DVecH `  K ) `  W )  =  ( ( DVecH `  K ) `  W )
11 eqid 2283 . . . 4  |-  ( LSubSp `  ( ( DVecH `  K
) `  W )
)  =  ( LSubSp `  ( ( DVecH `  K
) `  W )
)
12 eqid 2283 . . . 4  |-  ( LSSum `  ( ( DVecH `  K
) `  W )
)  =  ( LSSum `  ( ( DVecH `  K
) `  W )
)
131, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12dihval 31422 . . 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 3571 . . 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 2335 . 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 1623    e. wcel 1684   A.wral 2543   ifcif 3565   class class class wbr 4023   ` cfv 5255  (class class class)co 5858   iota_crio 6297   Basecbs 13148   lecple 13215   joincjn 14078   meetcmee 14079   LSSumclsm 14945   LSubSpclss 15689   Atomscatm 29453   LHypclh 30173   DVecHcdvh 31268   DIsoBcdib 31328   DIsoCcdic 31362   DIsoHcdih 31418
This theorem is referenced by:  dihopelvalbN  31428  dih1dimb  31430  dih2dimb  31434  dih2dimbALTN  31435  dihvalcq2  31437  dihlss  31440  dihord6apre  31446  dihord3  31447  dihord5b  31449  dihord5apre  31452  dih0  31470  dihwN  31479  dihglblem3N  31485  dihmeetlem2N  31489  dih1dimatlem  31519  dihjatcclem4  31611
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-riota 6304  df-dih 31419
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