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Theorem dvhb1dimN 30993
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 6204 . . . . 5  |-  ( g  e.  ( T  X.  E )  ->  (
g  =  <. (
s `  F ) ,  .0.  >. 
<->  ( ( 1st `  g
)  =  ( s `
 F )  /\  ( 2nd `  g )  =  .0.  ) ) )
21adantl 452 . . . 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 2598 . . 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 2727 . . . 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 5577 . . . . . . . 8  |-  ( 1st `  g )  e.  _V
6 eqeq1 2322 . . . . . . . . 9  |-  ( f  =  ( 1st `  g
)  ->  ( f  =  ( s `  F )  <->  ( 1st `  g )  =  ( s `  F ) ) )
76rexbidv 2598 . . . . . . . 8  |-  ( f  =  ( 1st `  g
)  ->  ( E. s  e.  E  f  =  ( s `  F )  <->  E. s  e.  E  ( 1st `  g )  =  ( s `  F ) ) )
85, 7elab 2948 . . . . . . 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 30992 . . . . . . . . 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 451 . . . . . . . 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 2383 . . . . . . 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 250 . . . . . 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 6191 . . . . . . . 8  |-  ( g  e.  ( T  X.  E )  ->  ( 1st `  g )  e.  T )
1918adantl 452 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T )  /\  g  e.  ( T  X.  E
) )  ->  ( 1st `  g )  e.  T )
20 fveq2 5563 . . . . . . . . 9  |-  ( f  =  ( 1st `  g
)  ->  ( R `  f )  =  ( R `  ( 1st `  g ) ) )
2120breq1d 4070 . . . . . . . 8  |-  ( f  =  ( 1st `  g
)  ->  ( ( R `  f )  .<_  ( R `  F
)  <->  ( R `  ( 1st `  g ) )  .<_  ( R `  F ) ) )
2221elrab3 2958 . . . . . . 7  |-  ( ( 1st `  g )  e.  T  ->  (
( 1st `  g
)  e.  { f  e.  T  |  ( R `  f ) 
.<_  ( R `  F
) }  <->  ( R `  ( 1st `  g
) )  .<_  ( R `
 F ) ) )
2319, 22syl 15 . . . . . 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 244 . . . . 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 685 . . . 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 248 . . 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 244 . 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 2813 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 176    /\ wa 358    = wceq 1633    e. wcel 1701   {cab 2302   E.wrex 2578   {crab 2581   <.cop 3677   class class class wbr 4060    e. cmpt 4114    _I cid 4341    X. cxp 4724    |` cres 4728   ` cfv 5292   1stc1st 6162   2ndc2nd 6163   lecple 13262   HLchlt 29358   LHypclh 29991   LTrncltrn 30108   trLctrl 30165   TEndoctendo 30759
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-13 1703  ax-14 1705  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297  ax-rep 4168  ax-sep 4178  ax-nul 4186  ax-pow 4225  ax-pr 4251  ax-un 4549
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 1533  df-nf 1536  df-sb 1640  df-eu 2180  df-mo 2181  df-clab 2303  df-cleq 2309  df-clel 2312  df-nfc 2441  df-ne 2481  df-nel 2482  df-ral 2582  df-rex 2583  df-reu 2584  df-rmo 2585  df-rab 2586  df-v 2824  df-sbc 3026  df-csb 3116  df-dif 3189  df-un 3191  df-in 3193  df-ss 3200  df-nul 3490  df-if 3600  df-pw 3661  df-sn 3680  df-pr 3681  df-op 3683  df-uni 3865  df-iun 3944  df-iin 3945  df-br 4061  df-opab 4115  df-mpt 4116  df-id 4346  df-xp 4732  df-rel 4733  df-cnv 4734  df-co 4735  df-dm 4736  df-rn 4737  df-res 4738  df-ima 4739  df-iota 5256  df-fun 5294  df-fn 5295  df-f 5296  df-f1 5297  df-fo 5298  df-f1o 5299  df-fv 5300  df-ov 5903  df-oprab 5904  df-mpt2 5905  df-1st 6164  df-2nd 6165  df-undef 6340  df-riota 6346  df-map 6817  df-poset 14129  df-plt 14141  df-lub 14157  df-glb 14158  df-join 14159  df-meet 14160  df-p0 14194  df-p1 14195  df-lat 14201  df-clat 14263  df-oposet 29184  df-ol 29186  df-oml 29187  df-covers 29274  df-ats 29275  df-atl 29306  df-cvlat 29330  df-hlat 29359  df-llines 29505  df-lplanes 29506  df-lvols 29507  df-lines 29508  df-psubsp 29510  df-pmap 29511  df-padd 29803  df-lhyp 29995  df-laut 29996  df-ldil 30111  df-ltrn 30112  df-trl 30166  df-tendo 30762
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