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Theorem dia1N 31788
Description: The value of the partial isomorphism A at the fiducial co-atom is the set of all translations i.e. the entire vector space. (Contributed by NM, 26-Nov-2013.) (New usage is discouraged.)
Hypotheses
Ref Expression
dia1.h  |-  H  =  ( LHyp `  K
)
dia1.t  |-  T  =  ( ( LTrn `  K
) `  W )
dia1.i  |-  I  =  ( ( DIsoA `  K
) `  W )
Assertion
Ref Expression
dia1N  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  ( I `  W
)  =  T )

Proof of Theorem dia1N
Dummy variable  f is distinct from all other variables.
StepHypRef Expression
1 id 20 . . 3  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  ( K  e.  HL  /\  W  e.  H ) )
2 eqid 2435 . . . . 5  |-  ( Base `  K )  =  (
Base `  K )
3 dia1.h . . . . 5  |-  H  =  ( LHyp `  K
)
42, 3lhpbase 30732 . . . 4  |-  ( W  e.  H  ->  W  e.  ( Base `  K
) )
54adantl 453 . . 3  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  W  e.  ( Base `  K ) )
6 hllat 30098 . . . 4  |-  ( K  e.  HL  ->  K  e.  Lat )
7 eqid 2435 . . . . 5  |-  ( le
`  K )  =  ( le `  K
)
82, 7latref 14474 . . . 4  |-  ( ( K  e.  Lat  /\  W  e.  ( Base `  K ) )  ->  W ( le `  K ) W )
96, 4, 8syl2an 464 . . 3  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  W ( le `  K ) W )
10 dia1.t . . . 4  |-  T  =  ( ( LTrn `  K
) `  W )
11 eqid 2435 . . . 4  |-  ( ( trL `  K ) `
 W )  =  ( ( trL `  K
) `  W )
12 dia1.i . . . 4  |-  I  =  ( ( DIsoA `  K
) `  W )
132, 7, 3, 10, 11, 12diaval 31767 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( W  e.  ( Base `  K
)  /\  W ( le `  K ) W ) )  ->  (
I `  W )  =  { f  e.  T  |  ( ( ( trL `  K ) `
 W ) `  f ) ( le
`  K ) W } )
141, 5, 9, 13syl12anc 1182 . 2  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  ( I `  W
)  =  { f  e.  T  |  ( ( ( trL `  K
) `  W ) `  f ) ( le
`  K ) W } )
157, 3, 10, 11trlle 30918 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  f  e.  T
)  ->  ( (
( trL `  K
) `  W ) `  f ) ( le
`  K ) W )
1615ralrimiva 2781 . . 3  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  A. f  e.  T  ( ( ( trL `  K ) `  W
) `  f )
( le `  K
) W )
17 rabid2 2877 . . 3  |-  ( T  =  { f  e.  T  |  ( ( ( trL `  K
) `  W ) `  f ) ( le
`  K ) W }  <->  A. f  e.  T  ( ( ( trL `  K ) `  W
) `  f )
( le `  K
) W )
1816, 17sylibr 204 . 2  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  T  =  { f  e.  T  |  ( ( ( trL `  K
) `  W ) `  f ) ( le
`  K ) W } )
1914, 18eqtr4d 2470 1  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  ( I `  W
)  =  T )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725   A.wral 2697   {crab 2701   class class class wbr 4204   ` cfv 5446   Basecbs 13461   lecple 13528   Latclat 14466   HLchlt 30085   LHypclh 30718   LTrncltrn 30835   trLctrl 30892   DIsoAcdia 31763
This theorem is referenced by:  dia1elN  31789
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-undef 6535  df-riota 6541  df-map 7012  df-poset 14395  df-plt 14407  df-lub 14423  df-glb 14424  df-meet 14426  df-p0 14460  df-p1 14461  df-lat 14467  df-oposet 29911  df-ol 29913  df-oml 29914  df-covers 30001  df-ats 30002  df-atl 30033  df-cvlat 30057  df-hlat 30086  df-lhyp 30722  df-laut 30723  df-ldil 30838  df-ltrn 30839  df-trl 30893  df-disoa 31764
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