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Theorem ellkr2 39073
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 39071 . . 3 ((𝑊𝑌𝐺𝐹) → (𝑋 ∈ (𝐾𝐺) ↔ (𝑋𝑉 ∧ (𝐺𝑋) = 0 )))
91, 2, 8syl2anc 584 . 2 (𝜑 → (𝑋 ∈ (𝐾𝐺) ↔ (𝑋𝑉 ∧ (𝐺𝑋) = 0 )))
10 ellkr2.x . . 3 (𝜑𝑋𝑉)
1110biantrurd 532 . 2 (𝜑 → ((𝐺𝑋) = 0 ↔ (𝑋𝑉 ∧ (𝐺𝑋) = 0 )))
129, 11bitr4d 282 1 (𝜑 → (𝑋 ∈ (𝐾𝐺) ↔ (𝐺𝑋) = 0 ))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1537  wcel 2106  cfv 6563  Basecbs 17245  Scalarcsca 17301  0gc0g 17486  LFnlclfn 39039  LKerclk 39067
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-map 8867  df-lfl 39040  df-lkr 39068
This theorem is referenced by:  lclkrlem2f  41495  lclkrlem2n  41503  lcfrlem3  41527  lcfrlem25  41550  hdmapellkr  41897  hdmapip0  41898  hdmapinvlem1  41901
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