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Theorem lkrval2 39091
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 3501 . 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 39090 . . . 4 ((𝑊 ∈ V ∧ 𝐺𝐹) → (𝑥 ∈ (𝐾𝐺) ↔ (𝑥𝑉 ∧ (𝐺𝑥) = 0 )))
87eqabdv 2875 . . 3 ((𝑊 ∈ V ∧ 𝐺𝐹) → (𝐾𝐺) = {𝑥 ∣ (𝑥𝑉 ∧ (𝐺𝑥) = 0 )})
9 df-rab 3437 . . 3 {𝑥𝑉 ∣ (𝐺𝑥) = 0 } = {𝑥 ∣ (𝑥𝑉 ∧ (𝐺𝑥) = 0 )}
108, 9eqtr4di 2795 . 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 1540  wcel 2108  {cab 2714  {crab 3436  Vcvv 3480  cfv 6561  Basecbs 17247  Scalarcsca 17300  0gc0g 17484  LFnlclfn 39058  LKerclk 39086
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-map 8868  df-lfl 39059  df-lkr 39087
This theorem is referenced by:  lkrlss  39096
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