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Theorem dia1dim 31251
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 2283 . . . 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 30353 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T
)  ->  ( R `  F )  e.  (
Base `  K )
)
7 eqid 2283 . . . 4  |-  ( le
`  K )  =  ( le `  K
)
87, 3, 4, 5trlle 30373 . . 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 31222 . . 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 31174 . 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 2318 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 1623    e. wcel 1684   {cab 2269   E.wrex 2544   {crab 2547   class class class wbr 4023   ` cfv 5255   Basecbs 13148   lecple 13215   HLchlt 29540   LHypclh 30173   LTrncltrn 30290   trLctrl 30347   TEndoctendo 30941   DIsoAcdia 31218
This theorem is referenced by:  dia1dim2  31252  dib1dim  31355
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
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 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-iin 3908  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-undef 6298  df-riota 6304  df-map 6774  df-poset 14080  df-plt 14092  df-lub 14108  df-glb 14109  df-join 14110  df-meet 14111  df-p0 14145  df-p1 14146  df-lat 14152  df-clat 14214  df-oposet 29366  df-ol 29368  df-oml 29369  df-covers 29456  df-ats 29457  df-atl 29488  df-cvlat 29512  df-hlat 29541  df-llines 29687  df-lplanes 29688  df-lvols 29689  df-lines 29690  df-psubsp 29692  df-pmap 29693  df-padd 29985  df-lhyp 30177  df-laut 30178  df-ldil 30293  df-ltrn 30294  df-trl 30348  df-tendo 30944  df-disoa 31219
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