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Theorem dia1dim 31873
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 443 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T
)  ->  ( K  e.  HL  /\  W  e.  H ) )
2 eqid 2296 . . . 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 30975 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T
)  ->  ( R `  F )  e.  (
Base `  K )
)
7 eqid 2296 . . . 4  |-  ( le
`  K )  =  ( le `  K
)
87, 3, 4, 5trlle 30995 . . 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 31844 . . 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 1180 . 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 31796 . 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 2331 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 358    = wceq 1632    e. wcel 1696   {cab 2282   E.wrex 2557   {crab 2560   class class class wbr 4039   ` cfv 5271   Basecbs 13164   lecple 13231   HLchlt 30162   LHypclh 30795   LTrncltrn 30912   trLctrl 30969   TEndoctendo 31563   DIsoAcdia 31840
This theorem is referenced by:  dia1dim2  31874  dib1dim  31977
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-iin 3924  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-undef 6314  df-riota 6320  df-map 6790  df-poset 14096  df-plt 14108  df-lub 14124  df-glb 14125  df-join 14126  df-meet 14127  df-p0 14161  df-p1 14162  df-lat 14168  df-clat 14230  df-oposet 29988  df-ol 29990  df-oml 29991  df-covers 30078  df-ats 30079  df-atl 30110  df-cvlat 30134  df-hlat 30163  df-llines 30309  df-lplanes 30310  df-lvols 30311  df-lines 30312  df-psubsp 30314  df-pmap 30315  df-padd 30607  df-lhyp 30799  df-laut 30800  df-ldil 30915  df-ltrn 30916  df-trl 30970  df-tendo 31566  df-disoa 31841
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