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Theorem dva1dim 31796
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 31374. 
E is the division ring base by erngdv 31804, and  s `  F is the scalar product by dvavsca 31828. 
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 31578 . . . . . . . . 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 31572 . . . . . . . . 9  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  s  e.  E  /\  F  e.  T
)  ->  ( R `  ( s `  F
) )  .<_  ( R `
 F ) )
84, 7jca 518 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  s  e.  E  /\  F  e.  T
)  ->  ( (
s `  F )  e.  T  /\  ( R `  ( s `  F ) )  .<_  ( R `  F ) ) )
983expb 1152 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( s  e.  E  /\  F  e.  T ) )  -> 
( ( s `  F )  e.  T  /\  ( R `  (
s `  F )
)  .<_  ( R `  F ) ) )
109anass1rs 782 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T )  /\  s  e.  E )  ->  (
( s `  F
)  e.  T  /\  ( R `  ( s `
 F ) ) 
.<_  ( R `  F
) ) )
11 eleq1 2356 . . . . . . 7  |-  ( g  =  ( s `  F )  ->  (
g  e.  T  <->  ( s `  F )  e.  T
) )
12 fveq2 5541 . . . . . . . 8  |-  ( g  =  ( s `  F )  ->  ( R `  g )  =  ( R `  ( s `  F
) ) )
1312breq1d 4049 . . . . . . 7  |-  ( g  =  ( s `  F )  ->  (
( R `  g
)  .<_  ( R `  F )  <->  ( R `  ( s `  F
) )  .<_  ( R `
 F ) ) )
1411, 13anbi12d 691 . . . . . 6  |-  ( g  =  ( s `  F )  ->  (
( g  e.  T  /\  ( R `  g
)  .<_  ( R `  F ) )  <->  ( (
s `  F )  e.  T  /\  ( R `  ( s `  F ) )  .<_  ( R `  F ) ) ) )
1510, 14syl5ibrcom 213 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T )  /\  s  e.  E )  ->  (
g  =  ( s `
 F )  -> 
( g  e.  T  /\  ( R `  g
)  .<_  ( R `  F ) ) ) )
1615rexlimdva 2680 . . . 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 730 . . . . . . 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 731 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T )  /\  (
g  e.  T  /\  ( R `  g ) 
.<_  ( R `  F
) ) )  ->  F  e.  T )
19 simprl 732 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T )  /\  (
g  e.  T  /\  ( R `  g ) 
.<_  ( R `  F
) ) )  -> 
g  e.  T )
20 simprr 733 . . . . . . 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 31786 . . . . . . 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 1187 . . . . . 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 2298 . . . . . . 7  |-  ( ( s `  F )  =  g  <->  g  =  ( s `  F
) )
2423rexbii 2581 . . . . . 6  |-  ( E. s  e.  E  ( s `  F )  =  g  <->  E. s  e.  E  g  =  ( s `  F
) )
2522, 24sylib 188 . . . . 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 423 . . . 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 183 . . 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 2410 . 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 2565 . 2  |-  { g  e.  T  |  ( R `  g ) 
.<_  ( R `  F
) }  =  {
g  |  ( g  e.  T  /\  ( R `  g )  .<_  ( R `  F
) ) }
3028, 29syl6eqr 2346 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 358    /\ w3a 934    = wceq 1632    e. wcel 1696   {cab 2282   E.wrex 2557   {crab 2560   class class class wbr 4039   ` cfv 5271   lecple 13231   HLchlt 30162   LHypclh 30795   LTrncltrn 30912   trLctrl 30969   TEndoctendo 31563
This theorem is referenced by:  dvhb1dimN  31797  dia1dim  31873
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-iin 3924  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-undef 6314  df-riota 6320  df-map 6790  df-poset 14096  df-plt 14108  df-lub 14124  df-glb 14125  df-join 14126  df-meet 14127  df-p0 14161  df-p1 14162  df-lat 14168  df-clat 14230  df-oposet 29988  df-ol 29990  df-oml 29991  df-covers 30078  df-ats 30079  df-atl 30110  df-cvlat 30134  df-hlat 30163  df-llines 30309  df-lplanes 30310  df-lvols 30311  df-lines 30312  df-psubsp 30314  df-pmap 30315  df-padd 30607  df-lhyp 30799  df-laut 30800  df-ldil 30915  df-ltrn 30916  df-trl 30970  df-tendo 31566
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