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Theorem dia1dim 31859
Description: Two expressions for the 1-dimensional subspaces of partial vector space A (when  F is a nonzero vector i.e. non-identity translation). Remark after Lemma L in [Crawley] p. 120 line 21. (Contributed by NM, 15-Oct-2013.) (Revised by Mario Carneiro, 22-Jun-2014.)
Hypotheses
Ref Expression
dia1dim.h  |-  H  =  ( LHyp `  K
)
dia1dim.t  |-  T  =  ( ( LTrn `  K
) `  W )
dia1dim.r  |-  R  =  ( ( trL `  K
) `  W )
dia1dim.e  |-  E  =  ( ( TEndo `  K
) `  W )
dia1dim.i  |-  I  =  ( ( DIsoA `  K
) `  W )
Assertion
Ref Expression
dia1dim  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T
)  ->  ( I `  ( R `  F
) )  =  {
g  |  E. s  e.  E  g  =  ( s `  F
) } )
Distinct variable groups:    E, s    g, s, F    g, H, s    g, K, s    R, g, s    T, g, s   
g, W, s
Allowed substitution hints:    E( g)    I(
g, s)

Proof of Theorem dia1dim
StepHypRef Expression
1 simpl 444 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T
)  ->  ( K  e.  HL  /\  W  e.  H ) )
2 eqid 2436 . . . 4  |-  ( Base `  K )  =  (
Base `  K )
3 dia1dim.h . . . 4  |-  H  =  ( LHyp `  K
)
4 dia1dim.t . . . 4  |-  T  =  ( ( LTrn `  K
) `  W )
5 dia1dim.r . . . 4  |-  R  =  ( ( trL `  K
) `  W )
62, 3, 4, 5trlcl 30961 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T
)  ->  ( R `  F )  e.  (
Base `  K )
)
7 eqid 2436 . . . 4  |-  ( le
`  K )  =  ( le `  K
)
87, 3, 4, 5trlle 30981 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T
)  ->  ( R `  F ) ( le
`  K ) W )
9 dia1dim.i . . . 4  |-  I  =  ( ( DIsoA `  K
) `  W )
102, 7, 3, 4, 5, 9diaval 31830 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( R `
 F )  e.  ( Base `  K
)  /\  ( R `  F ) ( le
`  K ) W ) )  ->  (
I `  ( R `  F ) )  =  { g  e.  T  |  ( R `  g ) ( le
`  K ) ( R `  F ) } )
111, 6, 8, 10syl12anc 1182 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T
)  ->  ( I `  ( R `  F
) )  =  {
g  e.  T  | 
( R `  g
) ( le `  K ) ( R `
 F ) } )
12 dia1dim.e . . 3  |-  E  =  ( ( TEndo `  K
) `  W )
137, 3, 4, 5, 12dva1dim 31782 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T
)  ->  { g  |  E. s  e.  E  g  =  ( s `  F ) }  =  { g  e.  T  |  ( R `  g ) ( le
`  K ) ( R `  F ) } )
1411, 13eqtr4d 2471 1  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T
)  ->  ( I `  ( R `  F
) )  =  {
g  |  E. s  e.  E  g  =  ( s `  F
) } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725   {cab 2422   E.wrex 2706   {crab 2709   class class class wbr 4212   ` cfv 5454   Basecbs 13469   lecple 13536   HLchlt 30148   LHypclh 30781   LTrncltrn 30898   trLctrl 30955   TEndoctendo 31549   DIsoAcdia 31826
This theorem is referenced by:  dia1dim2  31860  dib1dim  31963
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 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-iun 4095  df-iin 4096  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-1st 6349  df-2nd 6350  df-undef 6543  df-riota 6549  df-map 7020  df-poset 14403  df-plt 14415  df-lub 14431  df-glb 14432  df-join 14433  df-meet 14434  df-p0 14468  df-p1 14469  df-lat 14475  df-clat 14537  df-oposet 29974  df-ol 29976  df-oml 29977  df-covers 30064  df-ats 30065  df-atl 30096  df-cvlat 30120  df-hlat 30149  df-llines 30295  df-lplanes 30296  df-lvols 30297  df-lines 30298  df-psubsp 30300  df-pmap 30301  df-padd 30593  df-lhyp 30785  df-laut 30786  df-ldil 30901  df-ltrn 30902  df-trl 30956  df-tendo 31552  df-disoa 31827
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