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Theorem lkrval 33393
 Description: Value of the kernel of a functional. (Contributed by NM, 15-Apr-2014.)
Hypotheses
Ref Expression
lkrfval.d 𝐷 = (Scalar‘𝑊)
lkrfval.o 0 = (0g𝐷)
lkrfval.f 𝐹 = (LFnl‘𝑊)
lkrfval.k 𝐾 = (LKer‘𝑊)
Assertion
Ref Expression
lkrval ((𝑊𝑋𝐺𝐹) → (𝐾𝐺) = (𝐺 “ { 0 }))

Proof of Theorem lkrval
Dummy variable 𝑓 is distinct from all other variables.
StepHypRef Expression
1 lkrfval.d . . . 4 𝐷 = (Scalar‘𝑊)
2 lkrfval.o . . . 4 0 = (0g𝐷)
3 lkrfval.f . . . 4 𝐹 = (LFnl‘𝑊)
4 lkrfval.k . . . 4 𝐾 = (LKer‘𝑊)
51, 2, 3, 4lkrfval 33392 . . 3 (𝑊𝑋𝐾 = (𝑓𝐹 ↦ (𝑓 “ { 0 })))
65fveq1d 6105 . 2 (𝑊𝑋 → (𝐾𝐺) = ((𝑓𝐹 ↦ (𝑓 “ { 0 }))‘𝐺))
7 cnvexg 7005 . . . 4 (𝐺𝐹𝐺 ∈ V)
8 imaexg 6995 . . . 4 (𝐺 ∈ V → (𝐺 “ { 0 }) ∈ V)
97, 8syl 17 . . 3 (𝐺𝐹 → (𝐺 “ { 0 }) ∈ V)
10 cnveq 5218 . . . . 5 (𝑓 = 𝐺𝑓 = 𝐺)
1110imaeq1d 5384 . . . 4 (𝑓 = 𝐺 → (𝑓 “ { 0 }) = (𝐺 “ { 0 }))
12 eqid 2610 . . . 4 (𝑓𝐹 ↦ (𝑓 “ { 0 })) = (𝑓𝐹 ↦ (𝑓 “ { 0 }))
1311, 12fvmptg 6189 . . 3 ((𝐺𝐹 ∧ (𝐺 “ { 0 }) ∈ V) → ((𝑓𝐹 ↦ (𝑓 “ { 0 }))‘𝐺) = (𝐺 “ { 0 }))
149, 13mpdan 699 . 2 (𝐺𝐹 → ((𝑓𝐹 ↦ (𝑓 “ { 0 }))‘𝐺) = (𝐺 “ { 0 }))
156, 14sylan9eq 2664 1 ((𝑊𝑋𝐺𝐹) → (𝐾𝐺) = (𝐺 “ { 0 }))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977  Vcvv 3173  {csn 4125   ↦ cmpt 4643  ◡ccnv 5037   “ cima 5041  ‘cfv 5804  Scalarcsca 15771  0gc0g 15923  LFnlclfn 33362  LKerclk 33390 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-lkr 33391 This theorem is referenced by:  ellkr  33394  lkr0f  33399
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