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Theorem dvhb1dimN 37125
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 6739 . . . . 5  |-  ( g  e.  ( T  X.  E )  ->  (
g  =  <. (
s `  F ) ,  .0.  >. 
<->  ( ( 1st `  g
)  =  ( s `
 F )  /\  ( 2nd `  g )  =  .0.  ) ) )
21adantl 464 . . . 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 2893 . . 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 2934 . . . 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 5784 . . . . . . . 8  |-  ( 1st `  g )  e.  _V
6 eqeq1 2386 . . . . . . . . 9  |-  ( f  =  ( 1st `  g
)  ->  ( f  =  ( s `  F )  <->  ( 1st `  g )  =  ( s `  F ) ) )
76rexbidv 2893 . . . . . . . 8  |-  ( f  =  ( 1st `  g
)  ->  ( E. s  e.  E  f  =  ( s `  F )  <->  E. s  e.  E  ( 1st `  g )  =  ( s `  F ) ) )
85, 7elab 3171 . . . . . . 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 37124 . . . . . . . . 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 463 . . . . . . . 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 2452 . . . . . . 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 259 . . . . . 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 6729 . . . . . . . 8  |-  ( g  e.  ( T  X.  E )  ->  ( 1st `  g )  e.  T )
1918adantl 464 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T )  /\  g  e.  ( T  X.  E
) )  ->  ( 1st `  g )  e.  T )
20 fveq2 5774 . . . . . . . . 9  |-  ( f  =  ( 1st `  g
)  ->  ( R `  f )  =  ( R `  ( 1st `  g ) ) )
2120breq1d 4377 . . . . . . . 8  |-  ( f  =  ( 1st `  g
)  ->  ( ( R `  f )  .<_  ( R `  F
)  <->  ( R `  ( 1st `  g ) )  .<_  ( R `  F ) ) )
2221elrab3 3183 . . . . . . 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 253 . . . . 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 702 . . . 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 257 . . 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 253 . 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 3025 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 setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367    = wceq 1399    e. wcel 1826   {cab 2367   E.wrex 2733   {crab 2736   <.cop 3950   class class class wbr 4367    |-> cmpt 4425    _I cid 4704    X. cxp 4911    |` cres 4915   ` cfv 5496   1stc1st 6697   2ndc2nd 6698   lecple 14709   HLchlt 35488   LHypclh 36121   LTrncltrn 36238   trLctrl 36296   TEndoctendo 36891
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-8 1828  ax-9 1830  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006  ax-ext 2360  ax-rep 4478  ax-sep 4488  ax-nul 4496  ax-pow 4543  ax-pr 4601  ax-un 6491  ax-riotaBAD 35097
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1402  df-fal 1405  df-ex 1621  df-nf 1625  df-sb 1748  df-eu 2222  df-mo 2223  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2532  df-ne 2579  df-nel 2580  df-ral 2737  df-rex 2738  df-reu 2739  df-rmo 2740  df-rab 2741  df-v 3036  df-sbc 3253  df-csb 3349  df-dif 3392  df-un 3394  df-in 3396  df-ss 3403  df-nul 3712  df-if 3858  df-pw 3929  df-sn 3945  df-pr 3947  df-op 3951  df-uni 4164  df-iun 4245  df-iin 4246  df-br 4368  df-opab 4426  df-mpt 4427  df-id 4709  df-xp 4919  df-rel 4920  df-cnv 4921  df-co 4922  df-dm 4923  df-rn 4924  df-res 4925  df-ima 4926  df-iota 5460  df-fun 5498  df-fn 5499  df-f 5500  df-f1 5501  df-fo 5502  df-f1o 5503  df-fv 5504  df-riota 6158  df-ov 6199  df-oprab 6200  df-mpt2 6201  df-1st 6699  df-2nd 6700  df-undef 6920  df-map 7340  df-preset 15674  df-poset 15692  df-plt 15705  df-lub 15721  df-glb 15722  df-join 15723  df-meet 15724  df-p0 15786  df-p1 15787  df-lat 15793  df-clat 15855  df-oposet 35314  df-ol 35316  df-oml 35317  df-covers 35404  df-ats 35405  df-atl 35436  df-cvlat 35460  df-hlat 35489  df-llines 35635  df-lplanes 35636  df-lvols 35637  df-lines 35638  df-psubsp 35640  df-pmap 35641  df-padd 35933  df-lhyp 36125  df-laut 36126  df-ldil 36241  df-ltrn 36242  df-trl 36297  df-tendo 36894
This theorem is referenced by: (None)
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