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Theorem lkrfval 39068
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 3498 . 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 2792 . . . . 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 2792 . . . . . . . . 9 (𝑤 = 𝑊 → (Scalar‘𝑤) = 𝐷)
98fveq2d 6910 . . . . . . . 8 (𝑤 = 𝑊 → (0g‘(Scalar‘𝑤)) = (0g𝐷))
10 lkrfval.o . . . . . . . 8 0 = (0g𝐷)
119, 10eqtr4di 2792 . . . . . . 7 (𝑤 = 𝑊 → (0g‘(Scalar‘𝑤)) = 0 )
1211sneqd 4642 . . . . . 6 (𝑤 = 𝑊 → {(0g‘(Scalar‘𝑤))} = { 0 })
1312imaeq2d 6079 . . . . 5 (𝑤 = 𝑊 → (𝑓 “ {(0g‘(Scalar‘𝑤))}) = (𝑓 “ { 0 }))
145, 13mpteq12dv 5238 . . . 4 (𝑤 = 𝑊 → (𝑓 ∈ (LFnl‘𝑤) ↦ (𝑓 “ {(0g‘(Scalar‘𝑤))})) = (𝑓𝐹 ↦ (𝑓 “ { 0 })))
15 df-lkr 39067 . . . 4 LKer = (𝑤 ∈ V ↦ (𝑓 ∈ (LFnl‘𝑤) ↦ (𝑓 “ {(0g‘(Scalar‘𝑤))})))
1614, 15, 4mptfvmpt 7247 . . 3 (𝑊 ∈ V → (LKer‘𝑊) = (𝑓𝐹 ↦ (𝑓 “ { 0 })))
172, 16eqtrid 2786 . 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 1536  wcel 2105  Vcvv 3477  {csn 4630  cmpt 5230  ccnv 5687  cima 5691  cfv 6562  Scalarcsca 17300  0gc0g 17485  LFnlclfn 39038  LKerclk 39066
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-rep 5284  ax-sep 5301  ax-nul 5311  ax-pr 5437
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-ral 3059  df-rex 3068  df-reu 3378  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-nul 4339  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-iun 4997  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5582  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-f1 6567  df-fo 6568  df-f1o 6569  df-fv 6570  df-lkr 39067
This theorem is referenced by:  lkrval  39069
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