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Theorem ellkr2 33075
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 33073 . . 3  |-  ( ( W  e.  Y  /\  G  e.  F )  ->  ( X  e.  ( K `  G )  <-> 
( X  e.  V  /\  ( G `  X
)  =  .0.  )
) )
91, 2, 8syl2anc 661 . 2  |-  ( ph  ->  ( X  e.  ( K `  G )  <-> 
( X  e.  V  /\  ( G `  X
)  =  .0.  )
) )
10 ellkr2.x . . 3  |-  ( ph  ->  X  e.  V )
1110biantrurd 508 . 2  |-  ( ph  ->  ( ( G `  X )  =  .0.  <->  ( X  e.  V  /\  ( G `  X )  =  .0.  ) ) )
129, 11bitr4d 256 1  |-  ( ph  ->  ( X  e.  ( K `  G )  <-> 
( G `  X
)  =  .0.  )
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1370    e. wcel 1758   ` cfv 5527   Basecbs 14293  Scalarcsca 14361   0gc0g 14498  LFnlclfn 33041  LKerclk 33069
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4512  ax-sep 4522  ax-nul 4530  ax-pow 4579  ax-pr 4640  ax-un 6483
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-reu 2806  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3397  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-nul 3747  df-if 3901  df-pw 3971  df-sn 3987  df-pr 3989  df-op 3993  df-uni 4201  df-iun 4282  df-br 4402  df-opab 4460  df-mpt 4461  df-id 4745  df-xp 4955  df-rel 4956  df-cnv 4957  df-co 4958  df-dm 4959  df-rn 4960  df-res 4961  df-ima 4962  df-iota 5490  df-fun 5529  df-fn 5530  df-f 5531  df-f1 5532  df-fo 5533  df-f1o 5534  df-fv 5535  df-ov 6204  df-oprab 6205  df-mpt2 6206  df-map 7327  df-lfl 33042  df-lkr 33070
This theorem is referenced by:  lclkrlem2f  35496  lclkrlem2n  35504  lcfrlem3  35528  lcfrlem25  35551  hdmapellkr  35901  hdmapip0  35902  hdmapinvlem1  35905
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