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Theorem lkrval2 39071
Description: Value of the kernel of a functional. (Contributed by NM, 15-Apr-2014.)
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)
Assertion
Ref Expression
lkrval2 ((𝑊𝑋𝐺𝐹) → (𝐾𝐺) = {𝑥𝑉 ∣ (𝐺𝑥) = 0 })
Distinct variable groups:   𝑥,𝐹   𝑥,𝐺   𝑥,𝐾   𝑥,𝑊
Allowed substitution hints:   𝐷(𝑥)   𝑉(𝑥)   𝑋(𝑥)   0 (𝑥)

Proof of Theorem lkrval2
StepHypRef Expression
1 elex 3498 . 2 (𝑊𝑋𝑊 ∈ V)
2 lkrfval2.v . . . . 5 𝑉 = (Base‘𝑊)
3 lkrfval2.d . . . . 5 𝐷 = (Scalar‘𝑊)
4 lkrfval2.o . . . . 5 0 = (0g𝐷)
5 lkrfval2.f . . . . 5 𝐹 = (LFnl‘𝑊)
6 lkrfval2.k . . . . 5 𝐾 = (LKer‘𝑊)
72, 3, 4, 5, 6ellkr 39070 . . . 4 ((𝑊 ∈ V ∧ 𝐺𝐹) → (𝑥 ∈ (𝐾𝐺) ↔ (𝑥𝑉 ∧ (𝐺𝑥) = 0 )))
87eqabdv 2872 . . 3 ((𝑊 ∈ V ∧ 𝐺𝐹) → (𝐾𝐺) = {𝑥 ∣ (𝑥𝑉 ∧ (𝐺𝑥) = 0 )})
9 df-rab 3433 . . 3 {𝑥𝑉 ∣ (𝐺𝑥) = 0 } = {𝑥 ∣ (𝑥𝑉 ∧ (𝐺𝑥) = 0 )}
108, 9eqtr4di 2792 . 2 ((𝑊 ∈ V ∧ 𝐺𝐹) → (𝐾𝐺) = {𝑥𝑉 ∣ (𝐺𝑥) = 0 })
111, 10sylan 580 1 ((𝑊𝑋𝐺𝐹) → (𝐾𝐺) = {𝑥𝑉 ∣ (𝐺𝑥) = 0 })
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1536  wcel 2105  {cab 2711  {crab 3432  Vcvv 3477  cfv 6562  Basecbs 17244  Scalarcsca 17300  0gc0g 17485  LFnlclfn 39038  LKerclk 39066
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-rep 5284  ax-sep 5301  ax-nul 5311  ax-pow 5370  ax-pr 5437  ax-un 7753
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-ral 3059  df-rex 3068  df-reu 3378  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-iun 4997  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5582  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-f1 6567  df-fo 6568  df-f1o 6569  df-fv 6570  df-ov 7433  df-oprab 7434  df-mpo 7435  df-map 8866  df-lfl 39039  df-lkr 39067
This theorem is referenced by:  lkrlss  39076
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