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Theorem dia1dim 30381
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 2256 . . . 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 29483 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T
)  ->  ( R `  F )  e.  (
Base `  K )
)
7 eqid 2256 . . . 4  |-  ( le
`  K )  =  ( le `  K
)
87, 3, 4, 5trlle 29503 . . 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 30352 . . 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 30304 . 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 2291 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 2242   E.wrex 2517   {crab 2519   class class class wbr 3963   ` cfv 4638   Basecbs 13075   lecple 13142   HLchlt 28670   LHypclh 29303   LTrncltrn 29420   trLctrl 29477   TEndoctendo 30071   DIsoAcdia 30348
This theorem is referenced by:  dia1dim2  30382  dib1dim  30485
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 1927  ax-ext 2237  ax-rep 4071  ax-sep 4081  ax-nul 4089  ax-pow 4126  ax-pr 4152  ax-un 4449
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 1884  df-eu 2121  df-mo 2122  df-clab 2243  df-cleq 2249  df-clel 2252  df-nfc 2381  df-ne 2421  df-nel 2422  df-ral 2520  df-rex 2521  df-reu 2522  df-rab 2523  df-v 2742  df-sbc 2936  df-csb 3024  df-dif 3097  df-un 3099  df-in 3101  df-ss 3108  df-nul 3398  df-if 3507  df-pw 3568  df-sn 3587  df-pr 3588  df-op 3590  df-uni 3769  df-iun 3848  df-iin 3849  df-br 3964  df-opab 4018  df-mpt 4019  df-id 4246  df-xp 4640  df-rel 4641  df-cnv 4642  df-co 4643  df-dm 4644  df-rn 4645  df-res 4646  df-ima 4647  df-fun 4648  df-fn 4649  df-f 4650  df-f1 4651  df-fo 4652  df-f1o 4653  df-fv 4654  df-ov 5760  df-oprab 5761  df-mpt2 5762  df-1st 6021  df-2nd 6022  df-iota 6190  df-undef 6229  df-riota 6237  df-map 6707  df-poset 14007  df-plt 14019  df-lub 14035  df-glb 14036  df-join 14037  df-meet 14038  df-p0 14072  df-p1 14073  df-lat 14079  df-clat 14141  df-oposet 28496  df-ol 28498  df-oml 28499  df-covers 28586  df-ats 28587  df-atl 28618  df-cvlat 28642  df-hlat 28671  df-llines 28817  df-lplanes 28818  df-lvols 28819  df-lines 28820  df-psubsp 28822  df-pmap 28823  df-padd 29115  df-lhyp 29307  df-laut 29308  df-ldil 29423  df-ltrn 29424  df-trl 29478  df-tendo 30074  df-disoa 30349
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