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Theorem lkrfval 39088
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 3501 . 2 (𝑊𝑋𝑊 ∈ V)
2 lkrfval.k . . 3 𝐾 = (LKer‘𝑊)
3 fveq2 6906 . . . . . 6 (𝑤 = 𝑊 → (LFnl‘𝑤) = (LFnl‘𝑊))
4 lkrfval.f . . . . . 6 𝐹 = (LFnl‘𝑊)
53, 4eqtr4di 2795 . . . . 5 (𝑤 = 𝑊 → (LFnl‘𝑤) = 𝐹)
6 fveq2 6906 . . . . . . . . . 10 (𝑤 = 𝑊 → (Scalar‘𝑤) = (Scalar‘𝑊))
7 lkrfval.d . . . . . . . . . 10 𝐷 = (Scalar‘𝑊)
86, 7eqtr4di 2795 . . . . . . . . 9 (𝑤 = 𝑊 → (Scalar‘𝑤) = 𝐷)
98fveq2d 6910 . . . . . . . 8 (𝑤 = 𝑊 → (0g‘(Scalar‘𝑤)) = (0g𝐷))
10 lkrfval.o . . . . . . . 8 0 = (0g𝐷)
119, 10eqtr4di 2795 . . . . . . 7 (𝑤 = 𝑊 → (0g‘(Scalar‘𝑤)) = 0 )
1211sneqd 4638 . . . . . 6 (𝑤 = 𝑊 → {(0g‘(Scalar‘𝑤))} = { 0 })
1312imaeq2d 6078 . . . . 5 (𝑤 = 𝑊 → (𝑓 “ {(0g‘(Scalar‘𝑤))}) = (𝑓 “ { 0 }))
145, 13mpteq12dv 5233 . . . 4 (𝑤 = 𝑊 → (𝑓 ∈ (LFnl‘𝑤) ↦ (𝑓 “ {(0g‘(Scalar‘𝑤))})) = (𝑓𝐹 ↦ (𝑓 “ { 0 })))
15 df-lkr 39087 . . . 4 LKer = (𝑤 ∈ V ↦ (𝑓 ∈ (LFnl‘𝑤) ↦ (𝑓 “ {(0g‘(Scalar‘𝑤))})))
1614, 15, 4mptfvmpt 7248 . . 3 (𝑊 ∈ V → (LKer‘𝑊) = (𝑓𝐹 ↦ (𝑓 “ { 0 })))
172, 16eqtrid 2789 . 2 (𝑊 ∈ V → 𝐾 = (𝑓𝐹 ↦ (𝑓 “ { 0 })))
181, 17syl 17 1 (𝑊𝑋𝐾 = (𝑓𝐹 ↦ (𝑓 “ { 0 })))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1540  wcel 2108  Vcvv 3480  {csn 4626  cmpt 5225  ccnv 5684  cima 5688  cfv 6561  Scalarcsca 17300  0gc0g 17484  LFnlclfn 39058  LKerclk 39086
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-lkr 39087
This theorem is referenced by:  lkrval  39089
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