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Theorem dia1N 31170
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 2389 . . . . 5  |-  ( Base `  K )  =  (
Base `  K )
3 dia1.h . . . . 5  |-  H  =  ( LHyp `  K
)
42, 3lhpbase 30114 . . . 4  |-  ( W  e.  H  ->  W  e.  ( Base `  K
) )
54adantl 453 . . 3  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  W  e.  ( Base `  K ) )
6 hllat 29480 . . . 4  |-  ( K  e.  HL  ->  K  e.  Lat )
7 eqid 2389 . . . . 5  |-  ( le
`  K )  =  ( le `  K
)
82, 7latref 14411 . . . 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 2389 . . . 4  |-  ( ( trL `  K ) `
 W )  =  ( ( trL `  K
) `  W )
12 dia1.i . . . 4  |-  I  =  ( ( DIsoA `  K
) `  W )
132, 7, 3, 10, 11, 12diaval 31149 . . 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 30300 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  f  e.  T
)  ->  ( (
( trL `  K
) `  W ) `  f ) ( le
`  K ) W )
1615ralrimiva 2734 . . 3  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  A. f  e.  T  ( ( ( trL `  K ) `  W
) `  f )
( le `  K
) W )
17 rabid2 2830 . . 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 2424 1  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  ( I `  W
)  =  T )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1717   A.wral 2651   {crab 2655   class class class wbr 4155   ` cfv 5396   Basecbs 13398   lecple 13465   Latclat 14403   HLchlt 29467   LHypclh 30100   LTrncltrn 30217   trLctrl 30274   DIsoAcdia 31145
This theorem is referenced by:  dia1elN  31171
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2370  ax-rep 4263  ax-sep 4273  ax-nul 4281  ax-pow 4320  ax-pr 4346  ax-un 4643
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2244  df-mo 2245  df-clab 2376  df-cleq 2382  df-clel 2385  df-nfc 2514  df-ne 2554  df-nel 2555  df-ral 2656  df-rex 2657  df-reu 2658  df-rab 2660  df-v 2903  df-sbc 3107  df-csb 3197  df-dif 3268  df-un 3270  df-in 3272  df-ss 3279  df-nul 3574  df-if 3685  df-pw 3746  df-sn 3765  df-pr 3766  df-op 3768  df-uni 3960  df-iun 4039  df-br 4156  df-opab 4210  df-mpt 4211  df-id 4441  df-xp 4826  df-rel 4827  df-cnv 4828  df-co 4829  df-dm 4830  df-rn 4831  df-res 4832  df-ima 4833  df-iota 5360  df-fun 5398  df-fn 5399  df-f 5400  df-f1 5401  df-fo 5402  df-f1o 5403  df-fv 5404  df-ov 6025  df-oprab 6026  df-mpt2 6027  df-1st 6290  df-2nd 6291  df-undef 6481  df-riota 6487  df-map 6958  df-poset 14332  df-plt 14344  df-lub 14360  df-glb 14361  df-meet 14363  df-p0 14397  df-p1 14398  df-lat 14404  df-oposet 29293  df-ol 29295  df-oml 29296  df-covers 29383  df-ats 29384  df-atl 29415  df-cvlat 29439  df-hlat 29468  df-lhyp 30104  df-laut 30105  df-ldil 30220  df-ltrn 30221  df-trl 30275  df-disoa 31146
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