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Theorem lkrfval 34877
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 3352 . 2 (𝑊𝑋𝑊 ∈ V)
2 lkrfval.k . . 3 𝐾 = (LKer‘𝑊)
3 fveq2 6352 . . . . . 6 (𝑤 = 𝑊 → (LFnl‘𝑤) = (LFnl‘𝑊))
4 lkrfval.f . . . . . 6 𝐹 = (LFnl‘𝑊)
53, 4syl6eqr 2812 . . . . 5 (𝑤 = 𝑊 → (LFnl‘𝑤) = 𝐹)
6 fveq2 6352 . . . . . . . . . 10 (𝑤 = 𝑊 → (Scalar‘𝑤) = (Scalar‘𝑊))
7 lkrfval.d . . . . . . . . . 10 𝐷 = (Scalar‘𝑊)
86, 7syl6eqr 2812 . . . . . . . . 9 (𝑤 = 𝑊 → (Scalar‘𝑤) = 𝐷)
98fveq2d 6356 . . . . . . . 8 (𝑤 = 𝑊 → (0g‘(Scalar‘𝑤)) = (0g𝐷))
10 lkrfval.o . . . . . . . 8 0 = (0g𝐷)
119, 10syl6eqr 2812 . . . . . . 7 (𝑤 = 𝑊 → (0g‘(Scalar‘𝑤)) = 0 )
1211sneqd 4333 . . . . . 6 (𝑤 = 𝑊 → {(0g‘(Scalar‘𝑤))} = { 0 })
1312imaeq2d 5624 . . . . 5 (𝑤 = 𝑊 → (𝑓 “ {(0g‘(Scalar‘𝑤))}) = (𝑓 “ { 0 }))
145, 13mpteq12dv 4885 . . . 4 (𝑤 = 𝑊 → (𝑓 ∈ (LFnl‘𝑤) ↦ (𝑓 “ {(0g‘(Scalar‘𝑤))})) = (𝑓𝐹 ↦ (𝑓 “ { 0 })))
15 df-lkr 34876 . . . 4 LKer = (𝑤 ∈ V ↦ (𝑓 ∈ (LFnl‘𝑤) ↦ (𝑓 “ {(0g‘(Scalar‘𝑤))})))
16 fvex 6362 . . . . . 6 (LFnl‘𝑊) ∈ V
174, 16eqeltri 2835 . . . . 5 𝐹 ∈ V
1817mptex 6650 . . . 4 (𝑓𝐹 ↦ (𝑓 “ { 0 })) ∈ V
1914, 15, 18fvmpt 6444 . . 3 (𝑊 ∈ V → (LKer‘𝑊) = (𝑓𝐹 ↦ (𝑓 “ { 0 })))
202, 19syl5eq 2806 . 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 1632  wcel 2139  Vcvv 3340  {csn 4321  cmpt 4881  ccnv 5265  cima 5269  cfv 6049  Scalarcsca 16146  0gc0g 16302  LFnlclfn 34847  LKerclk 34875
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pr 5055
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-reu 3057  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-lkr 34876
This theorem is referenced by:  lkrval  34878
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