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Theorem dihwN 32149
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 451 . . . . 5  |-  ( ph  ->  W  e.  H )
3 dihw.b . . . . . 6  |-  B  =  ( Base `  K
)
4 dihw.h . . . . . 6  |-  H  =  ( LHyp `  K
)
53, 4lhpbase 30857 . . . . 5  |-  ( W  e.  H  ->  W  e.  B )
62, 5syl 16 . . . 4  |-  ( ph  ->  W  e.  B )
71simpld 447 . . . . . 6  |-  ( ph  ->  K  e.  HL )
8 hllat 30223 . . . . . 6  |-  ( K  e.  HL  ->  K  e.  Lat )
97, 8syl 16 . . . . 5  |-  ( ph  ->  K  e.  Lat )
10 eqid 2438 . . . . . 6  |-  ( le
`  K )  =  ( le `  K
)
113, 10latref 14484 . . . . 5  |-  ( ( K  e.  Lat  /\  W  e.  B )  ->  W ( le `  K ) W )
129, 6, 11syl2anc 644 . . . 4  |-  ( ph  ->  W ( le `  K ) W )
136, 12jca 520 . . 3  |-  ( ph  ->  ( W  e.  B  /\  W ( le `  K ) W ) )
14 dihw.i . . . 4  |-  I  =  ( ( DIsoH `  K
) `  W )
15 eqid 2438 . . . 4  |-  ( (
DIsoB `  K ) `  W )  =  ( ( DIsoB `  K ) `  W )
163, 10, 4, 14, 15dihvalb 32097 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( W  e.  B  /\  W ( le `  K ) W ) )  -> 
( I `  W
)  =  ( ( ( DIsoB `  K ) `  W ) `  W
) )
171, 13, 16syl2anc 644 . 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 2438 . . . 4  |-  ( (
DIsoA `  K ) `  W )  =  ( ( DIsoA `  K ) `  W )
213, 10, 4, 18, 19, 20, 15dibval2 32004 . . 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 644 . 2  |-  ( ph  ->  ( ( ( DIsoB `  K ) `  W
) `  W )  =  ( ( ( ( DIsoA `  K ) `  W ) `  W
)  X.  {  .0.  } ) )
23 eqid 2438 . . . . . 6  |-  ( ( trL `  K ) `
 W )  =  ( ( trL `  K
) `  W )
243, 10, 4, 18, 23, 20diaval 31892 . . . . 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 644 . . . 4  |-  ( ph  ->  ( ( ( DIsoA `  K ) `  W
) `  W )  =  { g  e.  T  |  ( ( ( trL `  K ) `
 W ) `  g ) ( le
`  K ) W } )
2610, 4, 18, 23trlle 31043 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  g  e.  T
)  ->  ( (
( trL `  K
) `  W ) `  g ) ( le
`  K ) W )
271, 26sylan 459 . . . . . 6  |-  ( (
ph  /\  g  e.  T )  ->  (
( ( trL `  K
) `  W ) `  g ) ( le
`  K ) W )
2827ralrimiva 2791 . . . . 5  |-  ( ph  ->  A. g  e.  T  ( ( ( trL `  K ) `  W
) `  g )
( le `  K
) W )
29 rabid2 2887 . . . . 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 205 . . . 4  |-  ( ph  ->  T  =  { g  e.  T  |  ( ( ( trL `  K
) `  W ) `  g ) ( le
`  K ) W } )
3125, 30eqtr4d 2473 . . 3  |-  ( ph  ->  ( ( ( DIsoA `  K ) `  W
) `  W )  =  T )
3231xpeq1d 4903 . 2  |-  ( ph  ->  ( ( ( (
DIsoA `  K ) `  W ) `  W
)  X.  {  .0.  } )  =  ( T  X.  {  .0.  }
) )
3317, 22, 323eqtrd 2474 1  |-  ( ph  ->  ( I `  W
)  =  ( T  X.  {  .0.  }
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360    = wceq 1653    e. wcel 1726   A.wral 2707   {crab 2711   {csn 3816   class class class wbr 4214    e. cmpt 4268    _I cid 4495    X. cxp 4878    |` cres 4882   ` cfv 5456   Basecbs 13471   lecple 13538   Latclat 14476   HLchlt 30210   LHypclh 30843   LTrncltrn 30960   trLctrl 31017   DIsoAcdia 31888   DIsoBcdib 31998   DIsoHcdih 32088
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-13 1728  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-pow 4379  ax-pr 4405  ax-un 4703
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-nel 2604  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-pw 3803  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-oprab 6087  df-mpt2 6088  df-1st 6351  df-2nd 6352  df-undef 6545  df-riota 6551  df-map 7022  df-poset 14405  df-plt 14417  df-lub 14433  df-glb 14434  df-meet 14436  df-p0 14470  df-p1 14471  df-lat 14477  df-oposet 30036  df-ol 30038  df-oml 30039  df-covers 30126  df-ats 30127  df-atl 30158  df-cvlat 30182  df-hlat 30211  df-lhyp 30847  df-laut 30848  df-ldil 30963  df-ltrn 30964  df-trl 31018  df-disoa 31889  df-dib 31999  df-dih 32089
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