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Theorem ellkr2 29903
Description: Membership in the kernel of a functional. (Contributed by NM, 12-Jan-2015.)
Hypotheses
Ref Expression
lkrfval2.v  |-  V  =  ( Base `  W
)
lkrfval2.d  |-  D  =  (Scalar `  W )
lkrfval2.o  |-  .0.  =  ( 0g `  D )
lkrfval2.f  |-  F  =  (LFnl `  W )
lkrfval2.k  |-  K  =  (LKer `  W )
ellkr2.w  |-  ( ph  ->  W  e.  Y )
ellkr2.g  |-  ( ph  ->  G  e.  F )
ellkr2.x  |-  ( ph  ->  X  e.  V )
Assertion
Ref Expression
ellkr2  |-  ( ph  ->  ( X  e.  ( K `  G )  <-> 
( G `  X
)  =  .0.  )
)

Proof of Theorem ellkr2
StepHypRef Expression
1 ellkr2.w . . 3  |-  ( ph  ->  W  e.  Y )
2 ellkr2.g . . 3  |-  ( ph  ->  G  e.  F )
3 lkrfval2.v . . . 4  |-  V  =  ( Base `  W
)
4 lkrfval2.d . . . 4  |-  D  =  (Scalar `  W )
5 lkrfval2.o . . . 4  |-  .0.  =  ( 0g `  D )
6 lkrfval2.f . . . 4  |-  F  =  (LFnl `  W )
7 lkrfval2.k . . . 4  |-  K  =  (LKer `  W )
83, 4, 5, 6, 7ellkr 29901 . . 3  |-  ( ( W  e.  Y  /\  G  e.  F )  ->  ( X  e.  ( K `  G )  <-> 
( X  e.  V  /\  ( G `  X
)  =  .0.  )
) )
91, 2, 8syl2anc 642 . 2  |-  ( ph  ->  ( X  e.  ( K `  G )  <-> 
( X  e.  V  /\  ( G `  X
)  =  .0.  )
) )
10 ellkr2.x . . 3  |-  ( ph  ->  X  e.  V )
1110biantrurd 494 . 2  |-  ( ph  ->  ( ( G `  X )  =  .0.  <->  ( X  e.  V  /\  ( G `  X )  =  .0.  ) ) )
129, 11bitr4d 247 1  |-  ( ph  ->  ( X  e.  ( K `  G )  <-> 
( G `  X
)  =  .0.  )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1632    e. wcel 1696   ` cfv 5271   Basecbs 13164  Scalarcsca 13227   0gc0g 13416  LFnlclfn 29869  LKerclk 29897
This theorem is referenced by:  lclkrlem2f  32324  lclkrlem2n  32332  lcfrlem3  32356  lcfrlem25  32379  hdmapellkr  32729  hdmapip0  32730  hdmapinvlem1  32733
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-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-ral 2561  df-rex 2562  df-reu 2563  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-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-map 6790  df-lfl 29870  df-lkr 29898
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