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Theorem dvhb1dimN 31175
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 6162 . . . . 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 2564 . . 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 2693 . . . 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 5539 . . . . . . . 8  |-  ( 1st `  g )  e.  _V
6 eqeq1 2289 . . . . . . . . 9  |-  ( f  =  ( 1st `  g
)  ->  ( f  =  ( s `  F )  <->  ( 1st `  g )  =  ( s `  F ) ) )
76rexbidv 2564 . . . . . . . 8  |-  ( f  =  ( 1st `  g
)  ->  ( E. s  e.  E  f  =  ( s `  F )  <->  E. s  e.  E  ( 1st `  g )  =  ( s `  F ) ) )
85, 7elab 2914 . . . . . . 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 31174 . . . . . . . . 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 2350 . . . . . . 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 6149 . . . . . . . 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 5525 . . . . . . . . 9  |-  ( f  =  ( 1st `  g
)  ->  ( R `  f )  =  ( R `  ( 1st `  g ) ) )
2120breq1d 4033 . . . . . . . 8  |-  ( f  =  ( 1st `  g
)  ->  ( ( R `  f )  .<_  ( R `  F
)  <->  ( R `  ( 1st `  g ) )  .<_  ( R `  F ) ) )
2221elrab3 2924 . . . . . . 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 2779 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 1623    e. wcel 1684   {cab 2269   E.wrex 2544   {crab 2547   <.cop 3643   class class class wbr 4023    e. cmpt 4077    _I cid 4304    X. cxp 4687    |` cres 4691   ` cfv 5255   1stc1st 6120   2ndc2nd 6121   lecple 13215   HLchlt 29540   LHypclh 30173   LTrncltrn 30290   trLctrl 30347   TEndoctendo 30941
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
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 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-iin 3908  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-undef 6298  df-riota 6304  df-map 6774  df-poset 14080  df-plt 14092  df-lub 14108  df-glb 14109  df-join 14110  df-meet 14111  df-p0 14145  df-p1 14146  df-lat 14152  df-clat 14214  df-oposet 29366  df-ol 29368  df-oml 29369  df-covers 29456  df-ats 29457  df-atl 29488  df-cvlat 29512  df-hlat 29541  df-llines 29687  df-lplanes 29688  df-lvols 29689  df-lines 29690  df-psubsp 29692  df-pmap 29693  df-padd 29985  df-lhyp 30177  df-laut 30178  df-ldil 30293  df-ltrn 30294  df-trl 30348  df-tendo 30944
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