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Theorem lkrf0 34199
Description: The value of a functional at a member of its kernel is zero. (Contributed by NM, 16-Apr-2014.)
Hypotheses
Ref Expression
lkrf0.d 𝐷 = (Scalar‘𝑊)
lkrf0.o 0 = (0g𝐷)
lkrf0.f 𝐹 = (LFnl‘𝑊)
lkrf0.k 𝐾 = (LKer‘𝑊)
Assertion
Ref Expression
lkrf0 ((𝑊𝑌𝐺𝐹𝑋 ∈ (𝐾𝐺)) → (𝐺𝑋) = 0 )

Proof of Theorem lkrf0
StepHypRef Expression
1 eqid 2620 . . . 4 (Base‘𝑊) = (Base‘𝑊)
2 lkrf0.d . . . 4 𝐷 = (Scalar‘𝑊)
3 lkrf0.o . . . 4 0 = (0g𝐷)
4 lkrf0.f . . . 4 𝐹 = (LFnl‘𝑊)
5 lkrf0.k . . . 4 𝐾 = (LKer‘𝑊)
61, 2, 3, 4, 5ellkr 34195 . . 3 ((𝑊𝑌𝐺𝐹) → (𝑋 ∈ (𝐾𝐺) ↔ (𝑋 ∈ (Base‘𝑊) ∧ (𝐺𝑋) = 0 )))
76simplbda 653 . 2 (((𝑊𝑌𝐺𝐹) ∧ 𝑋 ∈ (𝐾𝐺)) → (𝐺𝑋) = 0 )
873impa 1257 1 ((𝑊𝑌𝐺𝐹𝑋 ∈ (𝐾𝐺)) → (𝐺𝑋) = 0 )
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  w3a 1036   = wceq 1481  wcel 1988  cfv 5876  Basecbs 15838  Scalarcsca 15925  0gc0g 16081  LFnlclfn 34163  LKerclk 34191
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-8 1990  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-rep 4762  ax-sep 4772  ax-nul 4780  ax-pow 4834  ax-pr 4897  ax-un 6934
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ne 2792  df-ral 2914  df-rex 2915  df-reu 2916  df-rab 2918  df-v 3197  df-sbc 3430  df-csb 3527  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-nul 3908  df-if 4078  df-pw 4151  df-sn 4169  df-pr 4171  df-op 4175  df-uni 4428  df-iun 4513  df-br 4645  df-opab 4704  df-mpt 4721  df-id 5014  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-rn 5115  df-res 5116  df-ima 5117  df-iota 5839  df-fun 5878  df-fn 5879  df-f 5880  df-f1 5881  df-fo 5882  df-f1o 5883  df-fv 5884  df-ov 6638  df-oprab 6639  df-mpt2 6640  df-map 7844  df-lfl 34164  df-lkr 34192
This theorem is referenced by:  lkrlss  34201  lkrshp  34211  lkrin  34270  lcfrlem12N  36662
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