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Theorem dvhb1dimN 31783
Description: Two expressions for the 1-dimensional subspaces of vector space H, in the isomorphism B case where the 2nd vector component is zero. (Contributed by NM, 23-Feb-2014.) (New usage is discouraged.)
Hypotheses
Ref Expression
dvhb1dim.l  |-  .<_  =  ( le `  K )
dvhb1dim.h  |-  H  =  ( LHyp `  K
)
dvhb1dim.t  |-  T  =  ( ( LTrn `  K
) `  W )
dvhb1dim.r  |-  R  =  ( ( trL `  K
) `  W )
dvhb1dim.e  |-  E  =  ( ( TEndo `  K
) `  W )
dvhb1dim.o  |-  .0.  =  ( h  e.  T  |->  (  _I  |`  B ) )
Assertion
Ref Expression
dvhb1dimN  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T
)  ->  { g  e.  ( T  X.  E
)  |  E. s  e.  E  g  =  <. ( s `  F
) ,  .0.  >. }  =  { g  e.  ( T  X.  E
)  |  ( ( R `  ( 1st `  g ) )  .<_  ( R `  F )  /\  ( 2nd `  g
)  =  .0.  ) } )
Distinct variable groups:    .<_ , s    E, s    g, s, F    g, H, s    g, K, s    .0. , s    R, s    g, h, T, s    g, W, s
Allowed substitution hints:    B( g, h, s)    R( g, h)    E( g, h)    F( h)    H( h)    K( h)    .<_ ( g, h)    W( h)    .0. ( g, h)

Proof of Theorem dvhb1dimN
Dummy variable  f is distinct from all other variables.
StepHypRef Expression
1 eqop 6389 . . . . 5  |-  ( g  e.  ( T  X.  E )  ->  (
g  =  <. (
s `  F ) ,  .0.  >. 
<->  ( ( 1st `  g
)  =  ( s `
 F )  /\  ( 2nd `  g )  =  .0.  ) ) )
21adantl 453 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T )  /\  g  e.  ( T  X.  E
) )  ->  (
g  =  <. (
s `  F ) ,  .0.  >. 
<->  ( ( 1st `  g
)  =  ( s `
 F )  /\  ( 2nd `  g )  =  .0.  ) ) )
32rexbidv 2726 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T )  /\  g  e.  ( T  X.  E
) )  ->  ( E. s  e.  E  g  =  <. ( s `
 F ) ,  .0.  >. 
<->  E. s  e.  E  ( ( 1st `  g
)  =  ( s `
 F )  /\  ( 2nd `  g )  =  .0.  ) ) )
4 r19.41v 2861 . . . 4  |-  ( E. s  e.  E  ( ( 1st `  g
)  =  ( s `
 F )  /\  ( 2nd `  g )  =  .0.  )  <->  ( E. s  e.  E  ( 1st `  g )  =  ( s `  F
)  /\  ( 2nd `  g )  =  .0.  ) )
5 fvex 5742 . . . . . . . 8  |-  ( 1st `  g )  e.  _V
6 eqeq1 2442 . . . . . . . . 9  |-  ( f  =  ( 1st `  g
)  ->  ( f  =  ( s `  F )  <->  ( 1st `  g )  =  ( s `  F ) ) )
76rexbidv 2726 . . . . . . . 8  |-  ( f  =  ( 1st `  g
)  ->  ( E. s  e.  E  f  =  ( s `  F )  <->  E. s  e.  E  ( 1st `  g )  =  ( s `  F ) ) )
85, 7elab 3082 . . . . . . 7  |-  ( ( 1st `  g )  e.  { f  |  E. s  e.  E  f  =  ( s `  F ) }  <->  E. s  e.  E  ( 1st `  g )  =  ( s `  F ) )
9 dvhb1dim.l . . . . . . . . . 10  |-  .<_  =  ( le `  K )
10 dvhb1dim.h . . . . . . . . . 10  |-  H  =  ( LHyp `  K
)
11 dvhb1dim.t . . . . . . . . . 10  |-  T  =  ( ( LTrn `  K
) `  W )
12 dvhb1dim.r . . . . . . . . . 10  |-  R  =  ( ( trL `  K
) `  W )
13 dvhb1dim.e . . . . . . . . . 10  |-  E  =  ( ( TEndo `  K
) `  W )
149, 10, 11, 12, 13dva1dim 31782 . . . . . . . . 9  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T
)  ->  { f  |  E. s  e.  E  f  =  ( s `  F ) }  =  { f  e.  T  |  ( R `  f )  .<_  ( R `
 F ) } )
1514adantr 452 . . . . . . . 8  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T )  /\  g  e.  ( T  X.  E
) )  ->  { f  |  E. s  e.  E  f  =  ( s `  F ) }  =  { f  e.  T  |  ( R `  f ) 
.<_  ( R `  F
) } )
1615eleq2d 2503 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T )  /\  g  e.  ( T  X.  E
) )  ->  (
( 1st `  g
)  e.  { f  |  E. s  e.  E  f  =  ( s `  F ) }  <->  ( 1st `  g
)  e.  { f  e.  T  |  ( R `  f ) 
.<_  ( R `  F
) } ) )
178, 16syl5bbr 251 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T )  /\  g  e.  ( T  X.  E
) )  ->  ( E. s  e.  E  ( 1st `  g )  =  ( s `  F )  <->  ( 1st `  g )  e.  {
f  e.  T  | 
( R `  f
)  .<_  ( R `  F ) } ) )
18 xp1st 6376 . . . . . . . 8  |-  ( g  e.  ( T  X.  E )  ->  ( 1st `  g )  e.  T )
1918adantl 453 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T )  /\  g  e.  ( T  X.  E
) )  ->  ( 1st `  g )  e.  T )
20 fveq2 5728 . . . . . . . . 9  |-  ( f  =  ( 1st `  g
)  ->  ( R `  f )  =  ( R `  ( 1st `  g ) ) )
2120breq1d 4222 . . . . . . . 8  |-  ( f  =  ( 1st `  g
)  ->  ( ( R `  f )  .<_  ( R `  F
)  <->  ( R `  ( 1st `  g ) )  .<_  ( R `  F ) ) )
2221elrab3 3093 . . . . . . 7  |-  ( ( 1st `  g )  e.  T  ->  (
( 1st `  g
)  e.  { f  e.  T  |  ( R `  f ) 
.<_  ( R `  F
) }  <->  ( R `  ( 1st `  g
) )  .<_  ( R `
 F ) ) )
2319, 22syl 16 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T )  /\  g  e.  ( T  X.  E
) )  ->  (
( 1st `  g
)  e.  { f  e.  T  |  ( R `  f ) 
.<_  ( R `  F
) }  <->  ( R `  ( 1st `  g
) )  .<_  ( R `
 F ) ) )
2417, 23bitrd 245 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T )  /\  g  e.  ( T  X.  E
) )  ->  ( E. s  e.  E  ( 1st `  g )  =  ( s `  F )  <->  ( R `  ( 1st `  g
) )  .<_  ( R `
 F ) ) )
2524anbi1d 686 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T )  /\  g  e.  ( T  X.  E
) )  ->  (
( E. s  e.  E  ( 1st `  g
)  =  ( s `
 F )  /\  ( 2nd `  g )  =  .0.  )  <->  ( ( R `  ( 1st `  g ) )  .<_  ( R `  F )  /\  ( 2nd `  g
)  =  .0.  )
) )
264, 25syl5bb 249 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T )  /\  g  e.  ( T  X.  E
) )  ->  ( E. s  e.  E  ( ( 1st `  g
)  =  ( s `
 F )  /\  ( 2nd `  g )  =  .0.  )  <->  ( ( R `  ( 1st `  g ) )  .<_  ( R `  F )  /\  ( 2nd `  g
)  =  .0.  )
) )
273, 26bitrd 245 . 2  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T )  /\  g  e.  ( T  X.  E
) )  ->  ( E. s  e.  E  g  =  <. ( s `
 F ) ,  .0.  >. 
<->  ( ( R `  ( 1st `  g ) )  .<_  ( R `  F )  /\  ( 2nd `  g )  =  .0.  ) ) )
2827rabbidva 2947 1  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T
)  ->  { g  e.  ( T  X.  E
)  |  E. s  e.  E  g  =  <. ( s `  F
) ,  .0.  >. }  =  { g  e.  ( T  X.  E
)  |  ( ( R `  ( 1st `  g ) )  .<_  ( R `  F )  /\  ( 2nd `  g
)  =  .0.  ) } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1652    e. wcel 1725   {cab 2422   E.wrex 2706   {crab 2709   <.cop 3817   class class class wbr 4212    e. cmpt 4266    _I cid 4493    X. cxp 4876    |` cres 4880   ` cfv 5454   1stc1st 6347   2ndc2nd 6348   lecple 13536   HLchlt 30148   LHypclh 30781   LTrncltrn 30898   trLctrl 30955   TEndoctendo 31549
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 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701
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 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-iun 4095  df-iin 4096  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-1st 6349  df-2nd 6350  df-undef 6543  df-riota 6549  df-map 7020  df-poset 14403  df-plt 14415  df-lub 14431  df-glb 14432  df-join 14433  df-meet 14434  df-p0 14468  df-p1 14469  df-lat 14475  df-clat 14537  df-oposet 29974  df-ol 29976  df-oml 29977  df-covers 30064  df-ats 30065  df-atl 30096  df-cvlat 30120  df-hlat 30149  df-llines 30295  df-lplanes 30296  df-lvols 30297  df-lines 30298  df-psubsp 30300  df-pmap 30301  df-padd 30593  df-lhyp 30785  df-laut 30786  df-ldil 30901  df-ltrn 30902  df-trl 30956  df-tendo 31552
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