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Theorem dihwN 37159
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 463 . . . . 5  |-  ( ph  ->  W  e.  H )
3 dihw.b . . . . . 6  |-  B  =  ( Base `  K
)
4 dihw.h . . . . . 6  |-  H  =  ( LHyp `  K
)
53, 4lhpbase 35865 . . . . 5  |-  ( W  e.  H  ->  W  e.  B )
62, 5syl 16 . . . 4  |-  ( ph  ->  W  e.  B )
71simpld 459 . . . . . 6  |-  ( ph  ->  K  e.  HL )
8 hllat 35231 . . . . . 6  |-  ( K  e.  HL  ->  K  e.  Lat )
97, 8syl 16 . . . . 5  |-  ( ph  ->  K  e.  Lat )
10 eqid 2457 . . . . . 6  |-  ( le
`  K )  =  ( le `  K
)
113, 10latref 15810 . . . . 5  |-  ( ( K  e.  Lat  /\  W  e.  B )  ->  W ( le `  K ) W )
129, 6, 11syl2anc 661 . . . 4  |-  ( ph  ->  W ( le `  K ) W )
136, 12jca 532 . . 3  |-  ( ph  ->  ( W  e.  B  /\  W ( le `  K ) W ) )
14 dihw.i . . . 4  |-  I  =  ( ( DIsoH `  K
) `  W )
15 eqid 2457 . . . 4  |-  ( (
DIsoB `  K ) `  W )  =  ( ( DIsoB `  K ) `  W )
163, 10, 4, 14, 15dihvalb 37107 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( W  e.  B  /\  W ( le `  K ) W ) )  -> 
( I `  W
)  =  ( ( ( DIsoB `  K ) `  W ) `  W
) )
171, 13, 16syl2anc 661 . 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 2457 . . . 4  |-  ( (
DIsoA `  K ) `  W )  =  ( ( DIsoA `  K ) `  W )
213, 10, 4, 18, 19, 20, 15dibval2 37014 . . 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 661 . 2  |-  ( ph  ->  ( ( ( DIsoB `  K ) `  W
) `  W )  =  ( ( ( ( DIsoA `  K ) `  W ) `  W
)  X.  {  .0.  } ) )
23 eqid 2457 . . . . . 6  |-  ( ( trL `  K ) `
 W )  =  ( ( trL `  K
) `  W )
243, 10, 4, 18, 23, 20diaval 36902 . . . . 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 661 . . . 4  |-  ( ph  ->  ( ( ( DIsoA `  K ) `  W
) `  W )  =  { g  e.  T  |  ( ( ( trL `  K ) `
 W ) `  g ) ( le
`  K ) W } )
2610, 4, 18, 23trlle 36052 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  g  e.  T
)  ->  ( (
( trL `  K
) `  W ) `  g ) ( le
`  K ) W )
271, 26sylan 471 . . . . . 6  |-  ( (
ph  /\  g  e.  T )  ->  (
( ( trL `  K
) `  W ) `  g ) ( le
`  K ) W )
2827ralrimiva 2871 . . . . 5  |-  ( ph  ->  A. g  e.  T  ( ( ( trL `  K ) `  W
) `  g )
( le `  K
) W )
29 rabid2 3035 . . . . 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 212 . . . 4  |-  ( ph  ->  T  =  { g  e.  T  |  ( ( ( trL `  K
) `  W ) `  g ) ( le
`  K ) W } )
3125, 30eqtr4d 2501 . . 3  |-  ( ph  ->  ( ( ( DIsoA `  K ) `  W
) `  W )  =  T )
3231xpeq1d 5031 . 2  |-  ( ph  ->  ( ( ( (
DIsoA `  K ) `  W ) `  W
)  X.  {  .0.  } )  =  ( T  X.  {  .0.  }
) )
3317, 22, 323eqtrd 2502 1  |-  ( ph  ->  ( I `  W
)  =  ( T  X.  {  .0.  }
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1395    e. wcel 1819   A.wral 2807   {crab 2811   {csn 4032   class class class wbr 4456    |-> cmpt 4515    _I cid 4799    X. cxp 5006    |` cres 5010   ` cfv 5594   Basecbs 14644   lecple 14719   Latclat 15802   HLchlt 35218   LHypclh 35851   LTrncltrn 35968   trLctrl 36026   DIsoAcdia 36898   DIsoBcdib 37008   DIsoHcdih 37098
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-id 4804  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-map 7440  df-preset 15684  df-poset 15702  df-plt 15715  df-lub 15731  df-glb 15732  df-join 15733  df-meet 15734  df-p0 15796  df-p1 15797  df-lat 15803  df-oposet 35044  df-ol 35046  df-oml 35047  df-covers 35134  df-ats 35135  df-atl 35166  df-cvlat 35190  df-hlat 35219  df-lhyp 35855  df-laut 35856  df-ldil 35971  df-ltrn 35972  df-trl 36027  df-disoa 36899  df-dib 37009  df-dih 37099
This theorem is referenced by: (None)
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