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Theorem dva1dim 31621
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 31199. 
E is the division ring base by erngdv 31629, and  s `  F is the scalar product by dvavsca 31653. 
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 31403 . . . . . . . . 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 31397 . . . . . . . . 9  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  s  e.  E  /\  F  e.  T
)  ->  ( R `  ( s `  F
) )  .<_  ( R `
 F ) )
84, 7jca 519 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  s  e.  E  /\  F  e.  T
)  ->  ( (
s `  F )  e.  T  /\  ( R `  ( s `  F ) )  .<_  ( R `  F ) ) )
983expb 1154 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( s  e.  E  /\  F  e.  T ) )  -> 
( ( s `  F )  e.  T  /\  ( R `  (
s `  F )
)  .<_  ( R `  F ) ) )
109anass1rs 783 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T )  /\  s  e.  E )  ->  (
( s `  F
)  e.  T  /\  ( R `  ( s `
 F ) ) 
.<_  ( R `  F
) ) )
11 eleq1 2495 . . . . . . 7  |-  ( g  =  ( s `  F )  ->  (
g  e.  T  <->  ( s `  F )  e.  T
) )
12 fveq2 5719 . . . . . . . 8  |-  ( g  =  ( s `  F )  ->  ( R `  g )  =  ( R `  ( s `  F
) ) )
1312breq1d 4214 . . . . . . 7  |-  ( g  =  ( s `  F )  ->  (
( R `  g
)  .<_  ( R `  F )  <->  ( R `  ( s `  F
) )  .<_  ( R `
 F ) ) )
1411, 13anbi12d 692 . . . . . 6  |-  ( g  =  ( s `  F )  ->  (
( g  e.  T  /\  ( R `  g
)  .<_  ( R `  F ) )  <->  ( (
s `  F )  e.  T  /\  ( R `  ( s `  F ) )  .<_  ( R `  F ) ) ) )
1510, 14syl5ibrcom 214 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T )  /\  s  e.  E )  ->  (
g  =  ( s `
 F )  -> 
( g  e.  T  /\  ( R `  g
)  .<_  ( R `  F ) ) ) )
1615rexlimdva 2822 . . . 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 731 . . . . . . 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 732 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T )  /\  (
g  e.  T  /\  ( R `  g ) 
.<_  ( R `  F
) ) )  ->  F  e.  T )
19 simprl 733 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T )  /\  (
g  e.  T  /\  ( R `  g ) 
.<_  ( R `  F
) ) )  -> 
g  e.  T )
20 simprr 734 . . . . . . 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 31611 . . . . . . 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 1189 . . . . . 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 2437 . . . . . . 7  |-  ( ( s `  F )  =  g  <->  g  =  ( s `  F
) )
2423rexbii 2722 . . . . . 6  |-  ( E. s  e.  E  ( s `  F )  =  g  <->  E. s  e.  E  g  =  ( s `  F
) )
2522, 24sylib 189 . . . . 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 424 . . . 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 184 . . 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 2549 . 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 2706 . 2  |-  { g  e.  T  |  ( R `  g ) 
.<_  ( R `  F
) }  =  {
g  |  ( g  e.  T  /\  ( R `  g )  .<_  ( R `  F
) ) }
3028, 29syl6eqr 2485 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 set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725   {cab 2421   E.wrex 2698   {crab 2701   class class class wbr 4204   ` cfv 5445   lecple 13524   HLchlt 29987   LHypclh 30620   LTrncltrn 30737   trLctrl 30794   TEndoctendo 31388
This theorem is referenced by:  dvhb1dimN  31622  dia1dim  31698
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4692
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-iun 4087  df-iin 4088  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4875  df-rel 4876  df-cnv 4877  df-co 4878  df-dm 4879  df-rn 4880  df-res 4881  df-ima 4882  df-iota 5409  df-fun 5447  df-fn 5448  df-f 5449  df-f1 5450  df-fo 5451  df-f1o 5452  df-fv 5453  df-ov 6075  df-oprab 6076  df-mpt2 6077  df-1st 6340  df-2nd 6341  df-undef 6534  df-riota 6540  df-map 7011  df-poset 14391  df-plt 14403  df-lub 14419  df-glb 14420  df-join 14421  df-meet 14422  df-p0 14456  df-p1 14457  df-lat 14463  df-clat 14525  df-oposet 29813  df-ol 29815  df-oml 29816  df-covers 29903  df-ats 29904  df-atl 29935  df-cvlat 29959  df-hlat 29988  df-llines 30134  df-lplanes 30135  df-lvols 30136  df-lines 30137  df-psubsp 30139  df-pmap 30140  df-padd 30432  df-lhyp 30624  df-laut 30625  df-ldil 30740  df-ltrn 30741  df-trl 30795  df-tendo 31391
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