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Theorem dihwN 31479
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 30187 . . . . 5  |-  ( W  e.  H  ->  W  e.  B )
62, 5syl 15 . . . 4  |-  ( ph  ->  W  e.  B )
71simpld 445 . . . . . 6  |-  ( ph  ->  K  e.  HL )
8 hllat 29553 . . . . . 6  |-  ( K  e.  HL  ->  K  e.  Lat )
97, 8syl 15 . . . . 5  |-  ( ph  ->  K  e.  Lat )
10 eqid 2283 . . . . . 6  |-  ( le
`  K )  =  ( le `  K
)
113, 10latref 14159 . . . . 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 2283 . . . 4  |-  ( (
DIsoB `  K ) `  W )  =  ( ( DIsoB `  K ) `  W )
163, 10, 4, 14, 15dihvalb 31427 . . 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 2283 . . . 4  |-  ( (
DIsoA `  K ) `  W )  =  ( ( DIsoA `  K ) `  W )
213, 10, 4, 18, 19, 20, 15dibval2 31334 . . 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 2283 . . . . . 6  |-  ( ( trL `  K ) `
 W )  =  ( ( trL `  K
) `  W )
243, 10, 4, 18, 23, 20diaval 31222 . . . . 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 30373 . . . . . . 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 2626 . . . . 5  |-  ( ph  ->  A. g  e.  T  ( ( ( trL `  K ) `  W
) `  g )
( le `  K
) W )
29 rabid2 2717 . . . . 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 2318 . . 3  |-  ( ph  ->  ( ( ( DIsoA `  K ) `  W
) `  W )  =  T )
3231xpeq1d 4712 . 2  |-  ( ph  ->  ( ( ( (
DIsoA `  K ) `  W ) `  W
)  X.  {  .0.  } )  =  ( T  X.  {  .0.  }
) )
3317, 22, 323eqtrd 2319 1  |-  ( ph  ->  ( I `  W
)  =  ( T  X.  {  .0.  }
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684   A.wral 2543   {crab 2547   {csn 3640   class class class wbr 4023    e. cmpt 4077    _I cid 4304    X. cxp 4687    |` cres 4691   ` cfv 5255   Basecbs 13148   lecple 13215   Latclat 14151   HLchlt 29540   LHypclh 30173   LTrncltrn 30290   trLctrl 30347   DIsoAcdia 31218   DIsoBcdib 31328   DIsoHcdih 31418
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-13 1686  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-pow 4188  ax-pr 4214  ax-un 4512
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-nel 2449  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-pw 3627  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-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-undef 6298  df-riota 6304  df-map 6774  df-poset 14080  df-plt 14092  df-lub 14108  df-glb 14109  df-meet 14111  df-p0 14145  df-p1 14146  df-lat 14152  df-oposet 29366  df-ol 29368  df-oml 29369  df-covers 29456  df-ats 29457  df-atl 29488  df-cvlat 29512  df-hlat 29541  df-lhyp 30177  df-laut 30178  df-ldil 30293  df-ltrn 30294  df-trl 30348  df-disoa 31219  df-dib 31329  df-dih 31419
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