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Theorem lkrval2 33857
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 3198 . 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 33856 . . . 4 ((𝑊 ∈ V ∧ 𝐺𝐹) → (𝑥 ∈ (𝐾𝐺) ↔ (𝑥𝑉 ∧ (𝐺𝑥) = 0 )))
87abbi2dv 2739 . . 3 ((𝑊 ∈ V ∧ 𝐺𝐹) → (𝐾𝐺) = {𝑥 ∣ (𝑥𝑉 ∧ (𝐺𝑥) = 0 )})
9 df-rab 2916 . . 3 {𝑥𝑉 ∣ (𝐺𝑥) = 0 } = {𝑥 ∣ (𝑥𝑉 ∧ (𝐺𝑥) = 0 )}
108, 9syl6eqr 2673 . 2 ((𝑊 ∈ V ∧ 𝐺𝐹) → (𝐾𝐺) = {𝑥𝑉 ∣ (𝐺𝑥) = 0 })
111, 10sylan 488 1 ((𝑊𝑋𝐺𝐹) → (𝐾𝐺) = {𝑥𝑉 ∣ (𝐺𝑥) = 0 })
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1480  wcel 1987  {cab 2607  {crab 2911  Vcvv 3186  cfv 5847  Basecbs 15781  Scalarcsca 15865  0gc0g 16021  LFnlclfn 33824  LKerclk 33852
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4731  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-map 7804  df-lfl 33825  df-lkr 33853
This theorem is referenced by:  lkrlss  33862
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