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Theorem dihwN 32101
Description: Value of isomorphism H at the fiducial hyperplane  W. (Contributed by NM, 25-Aug-2014.) (New usage is discouraged.)
Hypotheses
Ref Expression
dihw.b  |-  B  =  ( Base `  K
)
dihw.h  |-  H  =  ( LHyp `  K
)
dihw.t  |-  T  =  ( ( LTrn `  K
) `  W )
dihw.o  |-  .0.  =  ( f  e.  T  |->  (  _I  |`  B ) )
dihw.i  |-  I  =  ( ( DIsoH `  K
) `  W )
dihw.k  |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )
Assertion
Ref Expression
dihwN  |-  ( ph  ->  ( I `  W
)  =  ( T  X.  {  .0.  }
) )
Distinct variable groups:    f, K    f, W
Allowed substitution hints:    ph( f)    B( f)    T( f)    H( f)    I( f)    .0. ( f)

Proof of Theorem dihwN
Dummy variable  g is distinct from all other variables.
StepHypRef Expression
1 dihw.k . . 3  |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )
21simprd 449 . . . . 5  |-  ( ph  ->  W  e.  H )
3 dihw.b . . . . . 6  |-  B  =  ( Base `  K
)
4 dihw.h . . . . . 6  |-  H  =  ( LHyp `  K
)
53, 4lhpbase 30809 . . . . 5  |-  ( W  e.  H  ->  W  e.  B )
62, 5syl 15 . . . 4  |-  ( ph  ->  W  e.  B )
71simpld 445 . . . . . 6  |-  ( ph  ->  K  e.  HL )
8 hllat 30175 . . . . . 6  |-  ( K  e.  HL  ->  K  e.  Lat )
97, 8syl 15 . . . . 5  |-  ( ph  ->  K  e.  Lat )
10 eqid 2296 . . . . . 6  |-  ( le
`  K )  =  ( le `  K
)
113, 10latref 14175 . . . . 5  |-  ( ( K  e.  Lat  /\  W  e.  B )  ->  W ( le `  K ) W )
129, 6, 11syl2anc 642 . . . 4  |-  ( ph  ->  W ( le `  K ) W )
136, 12jca 518 . . 3  |-  ( ph  ->  ( W  e.  B  /\  W ( le `  K ) W ) )
14 dihw.i . . . 4  |-  I  =  ( ( DIsoH `  K
) `  W )
15 eqid 2296 . . . 4  |-  ( (
DIsoB `  K ) `  W )  =  ( ( DIsoB `  K ) `  W )
163, 10, 4, 14, 15dihvalb 32049 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( W  e.  B  /\  W ( le `  K ) W ) )  -> 
( I `  W
)  =  ( ( ( DIsoB `  K ) `  W ) `  W
) )
171, 13, 16syl2anc 642 . 2  |-  ( ph  ->  ( I `  W
)  =  ( ( ( DIsoB `  K ) `  W ) `  W
) )
18 dihw.t . . . 4  |-  T  =  ( ( LTrn `  K
) `  W )
19 dihw.o . . . 4  |-  .0.  =  ( f  e.  T  |->  (  _I  |`  B ) )
20 eqid 2296 . . . 4  |-  ( (
DIsoA `  K ) `  W )  =  ( ( DIsoA `  K ) `  W )
213, 10, 4, 18, 19, 20, 15dibval2 31956 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( W  e.  B  /\  W ( le `  K ) W ) )  -> 
( ( ( DIsoB `  K ) `  W
) `  W )  =  ( ( ( ( DIsoA `  K ) `  W ) `  W
)  X.  {  .0.  } ) )
221, 13, 21syl2anc 642 . 2  |-  ( ph  ->  ( ( ( DIsoB `  K ) `  W
) `  W )  =  ( ( ( ( DIsoA `  K ) `  W ) `  W
)  X.  {  .0.  } ) )
23 eqid 2296 . . . . . 6  |-  ( ( trL `  K ) `
 W )  =  ( ( trL `  K
) `  W )
243, 10, 4, 18, 23, 20diaval 31844 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( W  e.  B  /\  W ( le `  K ) W ) )  -> 
( ( ( DIsoA `  K ) `  W
) `  W )  =  { g  e.  T  |  ( ( ( trL `  K ) `
 W ) `  g ) ( le
`  K ) W } )
251, 13, 24syl2anc 642 . . . 4  |-  ( ph  ->  ( ( ( DIsoA `  K ) `  W
) `  W )  =  { g  e.  T  |  ( ( ( trL `  K ) `
 W ) `  g ) ( le
`  K ) W } )
2610, 4, 18, 23trlle 30995 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  g  e.  T
)  ->  ( (
( trL `  K
) `  W ) `  g ) ( le
`  K ) W )
271, 26sylan 457 . . . . . 6  |-  ( (
ph  /\  g  e.  T )  ->  (
( ( trL `  K
) `  W ) `  g ) ( le
`  K ) W )
2827ralrimiva 2639 . . . . 5  |-  ( ph  ->  A. g  e.  T  ( ( ( trL `  K ) `  W
) `  g )
( le `  K
) W )
29 rabid2 2730 . . . . 5  |-  ( T  =  { g  e.  T  |  ( ( ( trL `  K
) `  W ) `  g ) ( le
`  K ) W }  <->  A. g  e.  T  ( ( ( trL `  K ) `  W
) `  g )
( le `  K
) W )
3028, 29sylibr 203 . . . 4  |-  ( ph  ->  T  =  { g  e.  T  |  ( ( ( trL `  K
) `  W ) `  g ) ( le
`  K ) W } )
3125, 30eqtr4d 2331 . . 3  |-  ( ph  ->  ( ( ( DIsoA `  K ) `  W
) `  W )  =  T )
3231xpeq1d 4728 . 2  |-  ( ph  ->  ( ( ( (
DIsoA `  K ) `  W ) `  W
)  X.  {  .0.  } )  =  ( T  X.  {  .0.  }
) )
3317, 22, 323eqtrd 2332 1  |-  ( ph  ->  ( I `  W
)  =  ( T  X.  {  .0.  }
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1632    e. wcel 1696   A.wral 2556   {crab 2560   {csn 3653   class class class wbr 4039    e. cmpt 4093    _I cid 4320    X. cxp 4703    |` cres 4707   ` cfv 5271   Basecbs 13164   lecple 13231   Latclat 14167   HLchlt 30162   LHypclh 30795   LTrncltrn 30912   trLctrl 30969   DIsoAcdia 31840   DIsoBcdib 31950   DIsoHcdih 32040
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-13 1698  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-pow 4204  ax-pr 4230  ax-un 4528
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-nel 2462  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-pw 3640  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-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-undef 6314  df-riota 6320  df-map 6790  df-poset 14096  df-plt 14108  df-lub 14124  df-glb 14125  df-meet 14127  df-p0 14161  df-p1 14162  df-lat 14168  df-oposet 29988  df-ol 29990  df-oml 29991  df-covers 30078  df-ats 30079  df-atl 30110  df-cvlat 30134  df-hlat 30163  df-lhyp 30799  df-laut 30800  df-ldil 30915  df-ltrn 30916  df-trl 30970  df-disoa 31841  df-dib 31951  df-dih 32041
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