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Theorem dia1dim 29940
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 445 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T
)  ->  ( K  e.  HL  /\  W  e.  H ) )
2 eqid 2253 . . . 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 29042 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T
)  ->  ( R `  F )  e.  (
Base `  K )
)
7 eqid 2253 . . . 4  |-  ( le
`  K )  =  ( le `  K
)
87, 3, 4, 5trlle 29062 . . 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 29911 . . 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 1185 . 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 29863 . 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 2288 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 6    /\ wa 360    = wceq 1619    e. wcel 1621   {cab 2239   E.wrex 2510   {crab 2512   class class class wbr 3920   ` cfv 4592   Basecbs 13022   lecple 13089   HLchlt 28229   LHypclh 28862   LTrncltrn 28979   trLctrl 29036   TEndoctendo 29630   DIsoAcdia 29907
This theorem is referenced by:  dia1dim2  29941  dib1dim  30044
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1926  ax-ext 2234  ax-rep 4028  ax-sep 4038  ax-nul 4046  ax-pow 4082  ax-pr 4108  ax-un 4403
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 940  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1883  df-eu 2118  df-mo 2119  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-ne 2414  df-nel 2415  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2516  df-v 2729  df-sbc 2922  df-csb 3010  df-dif 3081  df-un 3083  df-in 3085  df-ss 3089  df-nul 3363  df-if 3471  df-pw 3532  df-sn 3550  df-pr 3551  df-op 3553  df-uni 3728  df-iun 3805  df-iin 3806  df-br 3921  df-opab 3975  df-mpt 3976  df-id 4202  df-xp 4594  df-rel 4595  df-cnv 4596  df-co 4597  df-dm 4598  df-rn 4599  df-res 4600  df-ima 4601  df-fun 4602  df-fn 4603  df-f 4604  df-f1 4605  df-fo 4606  df-f1o 4607  df-fv 4608  df-ov 5713  df-oprab 5714  df-mpt2 5715  df-1st 5974  df-2nd 5975  df-iota 6143  df-undef 6182  df-riota 6190  df-map 6660  df-poset 13924  df-plt 13936  df-lub 13952  df-glb 13953  df-join 13954  df-meet 13955  df-p0 13989  df-p1 13990  df-lat 13996  df-clat 14058  df-oposet 28055  df-ol 28057  df-oml 28058  df-covers 28145  df-ats 28146  df-atl 28177  df-cvlat 28201  df-hlat 28230  df-llines 28376  df-lplanes 28377  df-lvols 28378  df-lines 28379  df-psubsp 28381  df-pmap 28382  df-padd 28674  df-lhyp 28866  df-laut 28867  df-ldil 28982  df-ltrn 28983  df-trl 29037  df-tendo 29633  df-disoa 29908
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