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Theorem dva1dim 36812
Description: Two expressions for the 1-dimensional subspaces of partial vector space A. Remark in [Crawley] p. 120 line 21, but using a non-identity translation (nonzero vector) 
F whose trace is  P rather than  P itself;  F exists by cdlemf 36390. 
E is the division ring base by erngdv 36820, and  s `  F is the scalar product by dvavsca 36844. 
F must be a non-identity translation for the expression to be a 1-dimensional subspace, although the theorem doesn't require it. (Contributed by NM, 14-Oct-2013.)
Hypotheses
Ref Expression
dva1dim.l  |-  .<_  =  ( le `  K )
dva1dim.h  |-  H  =  ( LHyp `  K
)
dva1dim.t  |-  T  =  ( ( LTrn `  K
) `  W )
dva1dim.r  |-  R  =  ( ( trL `  K
) `  W )
dva1dim.e  |-  E  =  ( ( TEndo `  K
) `  W )
Assertion
Ref Expression
dva1dim  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T
)  ->  { g  |  E. s  e.  E  g  =  ( s `  F ) }  =  { g  e.  T  |  ( R `  g )  .<_  ( R `
 F ) } )
Distinct variable groups:    .<_ , s    E, s    g, s, F    g, H, s    g, K, s    R, s    T, g, s   
g, W, s
Allowed substitution hints:    R( g)    E( g)   
.<_ ( g)

Proof of Theorem dva1dim
StepHypRef Expression
1 dva1dim.h . . . . . . . . . 10  |-  H  =  ( LHyp `  K
)
2 dva1dim.t . . . . . . . . . 10  |-  T  =  ( ( LTrn `  K
) `  W )
3 dva1dim.e . . . . . . . . . 10  |-  E  =  ( ( TEndo `  K
) `  W )
41, 2, 3tendocl 36594 . . . . . . . . 9  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  s  e.  E  /\  F  e.  T
)  ->  ( s `  F )  e.  T
)
5 dva1dim.l . . . . . . . . . 10  |-  .<_  =  ( le `  K )
6 dva1dim.r . . . . . . . . . 10  |-  R  =  ( ( trL `  K
) `  W )
75, 1, 2, 6, 3tendotp 36588 . . . . . . . . 9  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  s  e.  E  /\  F  e.  T
)  ->  ( R `  ( s `  F
) )  .<_  ( R `
 F ) )
84, 7jca 532 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  s  e.  E  /\  F  e.  T
)  ->  ( (
s `  F )  e.  T  /\  ( R `  ( s `  F ) )  .<_  ( R `  F ) ) )
983expb 1197 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( s  e.  E  /\  F  e.  T ) )  -> 
( ( s `  F )  e.  T  /\  ( R `  (
s `  F )
)  .<_  ( R `  F ) ) )
109anass1rs 807 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T )  /\  s  e.  E )  ->  (
( s `  F
)  e.  T  /\  ( R `  ( s `
 F ) ) 
.<_  ( R `  F
) ) )
11 eleq1 2529 . . . . . . 7  |-  ( g  =  ( s `  F )  ->  (
g  e.  T  <->  ( s `  F )  e.  T
) )
12 fveq2 5872 . . . . . . . 8  |-  ( g  =  ( s `  F )  ->  ( R `  g )  =  ( R `  ( s `  F
) ) )
1312breq1d 4466 . . . . . . 7  |-  ( g  =  ( s `  F )  ->  (
( R `  g
)  .<_  ( R `  F )  <->  ( R `  ( s `  F
) )  .<_  ( R `
 F ) ) )
1411, 13anbi12d 710 . . . . . 6  |-  ( g  =  ( s `  F )  ->  (
( g  e.  T  /\  ( R `  g
)  .<_  ( R `  F ) )  <->  ( (
s `  F )  e.  T  /\  ( R `  ( s `  F ) )  .<_  ( R `  F ) ) ) )
1510, 14syl5ibrcom 222 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T )  /\  s  e.  E )  ->  (
g  =  ( s `
 F )  -> 
( g  e.  T  /\  ( R `  g
)  .<_  ( R `  F ) ) ) )
1615rexlimdva 2949 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T
)  ->  ( E. s  e.  E  g  =  ( s `  F )  ->  (
g  e.  T  /\  ( R `  g ) 
.<_  ( R `  F
) ) ) )
17 simpll 753 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T )  /\  (
g  e.  T  /\  ( R `  g ) 
.<_  ( R `  F
) ) )  -> 
( K  e.  HL  /\  W  e.  H ) )
18 simplr 755 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T )  /\  (
g  e.  T  /\  ( R `  g ) 
.<_  ( R `  F
) ) )  ->  F  e.  T )
19 simprl 756 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T )  /\  (
g  e.  T  /\  ( R `  g ) 
.<_  ( R `  F
) ) )  -> 
g  e.  T )
20 simprr 757 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T )  /\  (
g  e.  T  /\  ( R `  g ) 
.<_  ( R `  F
) ) )  -> 
( R `  g
)  .<_  ( R `  F ) )
215, 1, 2, 6, 3tendoex 36802 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  g  e.  T )  /\  ( R `  g )  .<_  ( R `  F
) )  ->  E. s  e.  E  ( s `  F )  =  g )
2217, 18, 19, 20, 21syl121anc 1233 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T )  /\  (
g  e.  T  /\  ( R `  g ) 
.<_  ( R `  F
) ) )  ->  E. s  e.  E  ( s `  F
)  =  g )
23 eqcom 2466 . . . . . . 7  |-  ( ( s `  F )  =  g  <->  g  =  ( s `  F
) )
2423rexbii 2959 . . . . . 6  |-  ( E. s  e.  E  ( s `  F )  =  g  <->  E. s  e.  E  g  =  ( s `  F
) )
2522, 24sylib 196 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T )  /\  (
g  e.  T  /\  ( R `  g ) 
.<_  ( R `  F
) ) )  ->  E. s  e.  E  g  =  ( s `  F ) )
2625ex 434 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T
)  ->  ( (
g  e.  T  /\  ( R `  g ) 
.<_  ( R `  F
) )  ->  E. s  e.  E  g  =  ( s `  F
) ) )
2716, 26impbid 191 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T
)  ->  ( E. s  e.  E  g  =  ( s `  F )  <->  ( g  e.  T  /\  ( R `  g )  .<_  ( R `  F
) ) ) )
2827abbidv 2593 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T
)  ->  { g  |  E. s  e.  E  g  =  ( s `  F ) }  =  { g  |  ( g  e.  T  /\  ( R `  g ) 
.<_  ( R `  F
) ) } )
29 df-rab 2816 . 2  |-  { g  e.  T  |  ( R `  g ) 
.<_  ( R `  F
) }  =  {
g  |  ( g  e.  T  /\  ( R `  g )  .<_  ( R `  F
) ) }
3028, 29syl6eqr 2516 1  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T
)  ->  { g  |  E. s  e.  E  g  =  ( s `  F ) }  =  { g  e.  T  |  ( R `  g )  .<_  ( R `
 F ) } )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 973    = wceq 1395    e. wcel 1819   {cab 2442   E.wrex 2808   {crab 2811   class class class wbr 4456   ` cfv 5594   lecple 14718   HLchlt 35176   LHypclh 35809   LTrncltrn 35926   trLctrl 35984   TEndoctendo 36579
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-riotaBAD 34785
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-fal 1401  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-iun 4334  df-iin 4335  df-br 4457  df-opab 4516  df-mpt 4517  df-id 4804  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-1st 6799  df-2nd 6800  df-undef 7020  df-map 7440  df-preset 15683  df-poset 15701  df-plt 15714  df-lub 15730  df-glb 15731  df-join 15732  df-meet 15733  df-p0 15795  df-p1 15796  df-lat 15802  df-clat 15864  df-oposet 35002  df-ol 35004  df-oml 35005  df-covers 35092  df-ats 35093  df-atl 35124  df-cvlat 35148  df-hlat 35177  df-llines 35323  df-lplanes 35324  df-lvols 35325  df-lines 35326  df-psubsp 35328  df-pmap 35329  df-padd 35621  df-lhyp 35813  df-laut 35814  df-ldil 35929  df-ltrn 35930  df-trl 35985  df-tendo 36582
This theorem is referenced by:  dvhb1dimN  36813  dia1dim  36889
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