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Theorem ellkr2 33396
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 = (0g𝐷)
lkrfval2.f 𝐹 = (LFnl‘𝑊)
lkrfval2.k 𝐾 = (LKer‘𝑊)
ellkr2.w (𝜑𝑊𝑌)
ellkr2.g (𝜑𝐺𝐹)
ellkr2.x (𝜑𝑋𝑉)
Assertion
Ref Expression
ellkr2 (𝜑 → (𝑋 ∈ (𝐾𝐺) ↔ (𝐺𝑋) = 0 ))

Proof of Theorem ellkr2
StepHypRef Expression
1 ellkr2.w . . 3 (𝜑𝑊𝑌)
2 ellkr2.g . . 3 (𝜑𝐺𝐹)
3 lkrfval2.v . . . 4 𝑉 = (Base‘𝑊)
4 lkrfval2.d . . . 4 𝐷 = (Scalar‘𝑊)
5 lkrfval2.o . . . 4 0 = (0g𝐷)
6 lkrfval2.f . . . 4 𝐹 = (LFnl‘𝑊)
7 lkrfval2.k . . . 4 𝐾 = (LKer‘𝑊)
83, 4, 5, 6, 7ellkr 33394 . . 3 ((𝑊𝑌𝐺𝐹) → (𝑋 ∈ (𝐾𝐺) ↔ (𝑋𝑉 ∧ (𝐺𝑋) = 0 )))
91, 2, 8syl2anc 691 . 2 (𝜑 → (𝑋 ∈ (𝐾𝐺) ↔ (𝑋𝑉 ∧ (𝐺𝑋) = 0 )))
10 ellkr2.x . . 3 (𝜑𝑋𝑉)
1110biantrurd 528 . 2 (𝜑 → ((𝐺𝑋) = 0 ↔ (𝑋𝑉 ∧ (𝐺𝑋) = 0 )))
129, 11bitr4d 270 1 (𝜑 → (𝑋 ∈ (𝐾𝐺) ↔ (𝐺𝑋) = 0 ))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 195  wa 383   = wceq 1475  wcel 1977  cfv 5804  Basecbs 15695  Scalarcsca 15771  0gc0g 15923  LFnlclfn 33362  LKerclk 33390
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-map 7746  df-lfl 33363  df-lkr 33391
This theorem is referenced by:  lclkrlem2f  35819  lclkrlem2n  35827  lcfrlem3  35851  lcfrlem25  35874  hdmapellkr  36224  hdmapip0  36225  hdmapinvlem1  36228
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