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Theorem dva1dim 30325
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 29903. 
E is the division ring base by erngdv 30333, and  s `  F is the scalar product by dvavsca 30357. 
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 30107 . . . . . . . . 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 30101 . . . . . . . . 9  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  s  e.  E  /\  F  e.  T
)  ->  ( R `  ( s `  F
) )  .<_  ( R `
 F ) )
84, 7jca 520 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  s  e.  E  /\  F  e.  T
)  ->  ( (
s `  F )  e.  T  /\  ( R `  ( s `  F ) )  .<_  ( R `  F ) ) )
983expb 1157 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( s  e.  E  /\  F  e.  T ) )  -> 
( ( s `  F )  e.  T  /\  ( R `  (
s `  F )
)  .<_  ( R `  F ) ) )
109anass1rs 785 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T )  /\  s  e.  E )  ->  (
( s `  F
)  e.  T  /\  ( R `  ( s `
 F ) ) 
.<_  ( R `  F
) ) )
11 eleq1 2316 . . . . . . 7  |-  ( g  =  ( s `  F )  ->  (
g  e.  T  <->  ( s `  F )  e.  T
) )
12 fveq2 5444 . . . . . . . 8  |-  ( g  =  ( s `  F )  ->  ( R `  g )  =  ( R `  ( s `  F
) ) )
1312breq1d 3993 . . . . . . 7  |-  ( g  =  ( s `  F )  ->  (
( R `  g
)  .<_  ( R `  F )  <->  ( R `  ( s `  F
) )  .<_  ( R `
 F ) ) )
1411, 13anbi12d 694 . . . . . 6  |-  ( g  =  ( s `  F )  ->  (
( g  e.  T  /\  ( R `  g
)  .<_  ( R `  F ) )  <->  ( (
s `  F )  e.  T  /\  ( R `  ( s `  F ) )  .<_  ( R `  F ) ) ) )
1510, 14syl5ibrcom 215 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T )  /\  s  e.  E )  ->  (
g  =  ( s `
 F )  -> 
( g  e.  T  /\  ( R `  g
)  .<_  ( R `  F ) ) ) )
1615rexlimdva 2640 . . . 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 733 . . . . . . 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 734 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T )  /\  (
g  e.  T  /\  ( R `  g ) 
.<_  ( R `  F
) ) )  ->  F  e.  T )
19 simprl 735 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T )  /\  (
g  e.  T  /\  ( R `  g ) 
.<_  ( R `  F
) ) )  -> 
g  e.  T )
20 simprr 736 . . . . . . 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 30315 . . . . . . 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 1192 . . . . . 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 2258 . . . . . . 7  |-  ( ( s `  F )  =  g  <->  g  =  ( s `  F
) )
2423rexbii 2541 . . . . . 6  |-  ( E. s  e.  E  ( s `  F )  =  g  <->  E. s  e.  E  g  =  ( s `  F
) )
2522, 24sylib 190 . . . . 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 425 . . . 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 185 . . 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 2370 . 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 2525 . 2  |-  { g  e.  T  |  ( R `  g ) 
.<_  ( R `  F
) }  =  {
g  |  ( g  e.  T  /\  ( R `  g )  .<_  ( R `  F
) ) }
3028, 29syl6eqr 2306 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 6    /\ wa 360    /\ w3a 939    = wceq 1619    e. wcel 1621   {cab 2242   E.wrex 2517   {crab 2520   class class class wbr 3983   ` cfv 4659   lecple 13163   HLchlt 28691   LHypclh 29324   LTrncltrn 29441   trLctrl 29498   TEndoctendo 30092
This theorem is referenced by:  dvhb1dimN  30326  dia1dim  30402
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 4091  ax-sep 4101  ax-nul 4109  ax-pow 4146  ax-pr 4172  ax-un 4470
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 2521  df-rex 2522  df-reu 2523  df-rmo 2524  df-rab 2525  df-v 2759  df-sbc 2953  df-csb 3043  df-dif 3116  df-un 3118  df-in 3120  df-ss 3127  df-nul 3417  df-if 3526  df-pw 3587  df-sn 3606  df-pr 3607  df-op 3609  df-uni 3788  df-iun 3867  df-iin 3868  df-br 3984  df-opab 4038  df-mpt 4039  df-id 4267  df-xp 4661  df-rel 4662  df-cnv 4663  df-co 4664  df-dm 4665  df-rn 4666  df-res 4667  df-ima 4668  df-fun 4669  df-fn 4670  df-f 4671  df-f1 4672  df-fo 4673  df-f1o 4674  df-fv 4675  df-ov 5781  df-oprab 5782  df-mpt2 5783  df-1st 6042  df-2nd 6043  df-iota 6211  df-undef 6250  df-riota 6258  df-map 6728  df-poset 14028  df-plt 14040  df-lub 14056  df-glb 14057  df-join 14058  df-meet 14059  df-p0 14093  df-p1 14094  df-lat 14100  df-clat 14162  df-oposet 28517  df-ol 28519  df-oml 28520  df-covers 28607  df-ats 28608  df-atl 28639  df-cvlat 28663  df-hlat 28692  df-llines 28838  df-lplanes 28839  df-lvols 28840  df-lines 28841  df-psubsp 28843  df-pmap 28844  df-padd 29136  df-lhyp 29328  df-laut 29329  df-ldil 29444  df-ltrn 29445  df-trl 29499  df-tendo 30095
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