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Theorem lkrfval 34692
Description: The kernel of a functional. (Contributed by NM, 15-Apr-2014.) (Revised by Mario Carneiro, 24-Jun-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
lkrfval (𝑊𝑋𝐾 = (𝑓𝐹 ↦ (𝑓 “ { 0 })))
Distinct variable groups:   𝑓,𝐹   𝑓,𝑊
Allowed substitution hints:   𝐷(𝑓)   𝐾(𝑓)   𝑋(𝑓)   0 (𝑓)

Proof of Theorem lkrfval
Dummy variable 𝑤 is distinct from all other variables.
StepHypRef Expression
1 elex 3243 . 2 (𝑊𝑋𝑊 ∈ V)
2 lkrfval.k . . 3 𝐾 = (LKer‘𝑊)
3 fveq2 6229 . . . . . 6 (𝑤 = 𝑊 → (LFnl‘𝑤) = (LFnl‘𝑊))
4 lkrfval.f . . . . . 6 𝐹 = (LFnl‘𝑊)
53, 4syl6eqr 2703 . . . . 5 (𝑤 = 𝑊 → (LFnl‘𝑤) = 𝐹)
6 fveq2 6229 . . . . . . . . . 10 (𝑤 = 𝑊 → (Scalar‘𝑤) = (Scalar‘𝑊))
7 lkrfval.d . . . . . . . . . 10 𝐷 = (Scalar‘𝑊)
86, 7syl6eqr 2703 . . . . . . . . 9 (𝑤 = 𝑊 → (Scalar‘𝑤) = 𝐷)
98fveq2d 6233 . . . . . . . 8 (𝑤 = 𝑊 → (0g‘(Scalar‘𝑤)) = (0g𝐷))
10 lkrfval.o . . . . . . . 8 0 = (0g𝐷)
119, 10syl6eqr 2703 . . . . . . 7 (𝑤 = 𝑊 → (0g‘(Scalar‘𝑤)) = 0 )
1211sneqd 4222 . . . . . 6 (𝑤 = 𝑊 → {(0g‘(Scalar‘𝑤))} = { 0 })
1312imaeq2d 5501 . . . . 5 (𝑤 = 𝑊 → (𝑓 “ {(0g‘(Scalar‘𝑤))}) = (𝑓 “ { 0 }))
145, 13mpteq12dv 4766 . . . 4 (𝑤 = 𝑊 → (𝑓 ∈ (LFnl‘𝑤) ↦ (𝑓 “ {(0g‘(Scalar‘𝑤))})) = (𝑓𝐹 ↦ (𝑓 “ { 0 })))
15 df-lkr 34691 . . . 4 LKer = (𝑤 ∈ V ↦ (𝑓 ∈ (LFnl‘𝑤) ↦ (𝑓 “ {(0g‘(Scalar‘𝑤))})))
16 fvex 6239 . . . . . 6 (LFnl‘𝑊) ∈ V
174, 16eqeltri 2726 . . . . 5 𝐹 ∈ V
1817mptex 6527 . . . 4 (𝑓𝐹 ↦ (𝑓 “ { 0 })) ∈ V
1914, 15, 18fvmpt 6321 . . 3 (𝑊 ∈ V → (LKer‘𝑊) = (𝑓𝐹 ↦ (𝑓 “ { 0 })))
202, 19syl5eq 2697 . 2 (𝑊 ∈ V → 𝐾 = (𝑓𝐹 ↦ (𝑓 “ { 0 })))
211, 20syl 17 1 (𝑊𝑋𝐾 = (𝑓𝐹 ↦ (𝑓 “ { 0 })))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1523  wcel 2030  Vcvv 3231  {csn 4210  cmpt 4762  ccnv 5142  cima 5146  cfv 5926  Scalarcsca 15991  0gc0g 16147  LFnlclfn 34662  LKerclk 34690
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pr 4936
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-lkr 34691
This theorem is referenced by:  lkrval  34693
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