Mathbox for Norm Megill < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  lkrval2 Structured version   Visualization version   GIF version

Theorem lkrval2 36426
 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 3459 . 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 36425 . . . 4 ((𝑊 ∈ V ∧ 𝐺𝐹) → (𝑥 ∈ (𝐾𝐺) ↔ (𝑥𝑉 ∧ (𝐺𝑥) = 0 )))
87abbi2dv 2927 . . 3 ((𝑊 ∈ V ∧ 𝐺𝐹) → (𝐾𝐺) = {𝑥 ∣ (𝑥𝑉 ∧ (𝐺𝑥) = 0 )})
9 df-rab 3115 . . 3 {𝑥𝑉 ∣ (𝐺𝑥) = 0 } = {𝑥 ∣ (𝑥𝑉 ∧ (𝐺𝑥) = 0 )}
108, 9eqtr4di 2851 . 2 ((𝑊 ∈ V ∧ 𝐺𝐹) → (𝐾𝐺) = {𝑥𝑉 ∣ (𝐺𝑥) = 0 })
111, 10sylan 583 1 ((𝑊𝑋𝐺𝐹) → (𝐾𝐺) = {𝑥𝑉 ∣ (𝐺𝑥) = 0 })
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399   = wceq 1538   ∈ wcel 2111  {cab 2776  {crab 3110  Vcvv 3441  ‘cfv 6327  Basecbs 16482  Scalarcsca 16567  0gc0g 16712  LFnlclfn 36393  LKerclk 36421 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5155  ax-sep 5168  ax-nul 5175  ax-pow 5232  ax-pr 5296  ax-un 7448 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-reu 3113  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4802  df-iun 4884  df-br 5032  df-opab 5094  df-mpt 5112  df-id 5426  df-xp 5526  df-rel 5527  df-cnv 5528  df-co 5529  df-dm 5530  df-rn 5531  df-res 5532  df-ima 5533  df-iota 6286  df-fun 6329  df-fn 6330  df-f 6331  df-f1 6332  df-fo 6333  df-f1o 6334  df-fv 6335  df-ov 7143  df-oprab 7144  df-mpo 7145  df-map 8398  df-lfl 36394  df-lkr 36422 This theorem is referenced by:  lkrlss  36431
 Copyright terms: Public domain W3C validator