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Theorem dva1dim 30441
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 30019. 
E is the division ring base by erngdv 30449, and  s `  F is the scalar product by dvavsca 30473. 
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 30223 . . . . . . . . 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 30217 . . . . . . . . 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 1154 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( s  e.  E  /\  F  e.  T ) )  -> 
( ( s `  F )  e.  T  /\  ( R `  (
s `  F )
)  .<_  ( R `  F ) ) )
109anass1rs 784 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T )  /\  s  e.  E )  ->  (
( s `  F
)  e.  T  /\  ( R `  ( s `
 F ) ) 
.<_  ( R `  F
) ) )
11 eleq1 2344 . . . . . . 7  |-  ( g  =  ( s `  F )  ->  (
g  e.  T  <->  ( s `  F )  e.  T
) )
12 fveq2 5485 . . . . . . . 8  |-  ( g  =  ( s `  F )  ->  ( R `  g )  =  ( R `  ( s `  F
) ) )
1312breq1d 4034 . . . . . . 7  |-  ( g  =  ( s `  F )  ->  (
( R `  g
)  .<_  ( R `  F )  <->  ( R `  ( s `  F
) )  .<_  ( R `
 F ) ) )
1411, 13anbi12d 693 . . . . . 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 2668 . . . 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 732 . . . . . . 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 733 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T )  /\  (
g  e.  T  /\  ( R `  g ) 
.<_  ( R `  F
) ) )  ->  F  e.  T )
19 simprl 734 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T )  /\  (
g  e.  T  /\  ( R `  g ) 
.<_  ( R `  F
) ) )  -> 
g  e.  T )
20 simprr 735 . . . . . . 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 30431 . . . . . . 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 2286 . . . . . . 7  |-  ( ( s `  F )  =  g  <->  g  =  ( s `  F
) )
2423rexbii 2569 . . . . . 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 2398 . 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 2553 . 2  |-  { g  e.  T  |  ( R `  g ) 
.<_  ( R `  F
) }  =  {
g  |  ( g  e.  T  /\  ( R `  g )  .<_  ( R `  F
) ) }
3028, 29syl6eqr 2334 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 936    = wceq 1624    e. wcel 1685   {cab 2270   E.wrex 2545   {crab 2548   class class class wbr 4024   ` cfv 5221   lecple 13209   HLchlt 28807   LHypclh 29440   LTrncltrn 29557   trLctrl 29614   TEndoctendo 30208
This theorem is referenced by:  dvhb1dimN  30442  dia1dim  30518
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1534  ax-5 1545  ax-17 1604  ax-9 1637  ax-8 1645  ax-13 1687  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1867  ax-ext 2265  ax-rep 4132  ax-sep 4142  ax-nul 4150  ax-pow 4187  ax-pr 4213  ax-un 4511
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 937  df-3an 938  df-tru 1312  df-ex 1530  df-nf 1533  df-sb 1632  df-eu 2148  df-mo 2149  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-nel 2450  df-ral 2549  df-rex 2550  df-reu 2551  df-rmo 2552  df-rab 2553  df-v 2791  df-sbc 2993  df-csb 3083  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3457  df-if 3567  df-pw 3628  df-sn 3647  df-pr 3648  df-op 3650  df-uni 3829  df-iun 3908  df-iin 3909  df-br 4025  df-opab 4079  df-mpt 4080  df-id 4308  df-xp 4694  df-rel 4695  df-cnv 4696  df-co 4697  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701  df-fun 5223  df-fn 5224  df-f 5225  df-f1 5226  df-fo 5227  df-f1o 5228  df-fv 5229  df-ov 5822  df-oprab 5823  df-mpt2 5824  df-1st 6083  df-2nd 6084  df-iota 6252  df-undef 6291  df-riota 6299  df-map 6769  df-poset 14074  df-plt 14086  df-lub 14102  df-glb 14103  df-join 14104  df-meet 14105  df-p0 14139  df-p1 14140  df-lat 14146  df-clat 14208  df-oposet 28633  df-ol 28635  df-oml 28636  df-covers 28723  df-ats 28724  df-atl 28755  df-cvlat 28779  df-hlat 28808  df-llines 28954  df-lplanes 28955  df-lvols 28956  df-lines 28957  df-psubsp 28959  df-pmap 28960  df-padd 29252  df-lhyp 29444  df-laut 29445  df-ldil 29560  df-ltrn 29561  df-trl 29615  df-tendo 30211
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