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Theorem lkrval 38786
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 38785 . . 3 (𝑊𝑋𝐾 = (𝑓𝐹 ↦ (𝑓 “ { 0 })))
65fveq1d 6903 . 2 (𝑊𝑋 → (𝐾𝐺) = ((𝑓𝐹 ↦ (𝑓 “ { 0 }))‘𝐺))
7 cnvexg 7937 . . . 4 (𝐺𝐹𝐺 ∈ V)
8 imaexg 7926 . . . 4 (𝐺 ∈ V → (𝐺 “ { 0 }) ∈ V)
97, 8syl 17 . . 3 (𝐺𝐹 → (𝐺 “ { 0 }) ∈ V)
10 cnveq 5880 . . . . 5 (𝑓 = 𝐺𝑓 = 𝐺)
1110imaeq1d 6068 . . . 4 (𝑓 = 𝐺 → (𝑓 “ { 0 }) = (𝐺 “ { 0 }))
12 eqid 2726 . . . 4 (𝑓𝐹 ↦ (𝑓 “ { 0 })) = (𝑓𝐹 ↦ (𝑓 “ { 0 }))
1311, 12fvmptg 7007 . . 3 ((𝐺𝐹 ∧ (𝐺 “ { 0 }) ∈ V) → ((𝑓𝐹 ↦ (𝑓 “ { 0 }))‘𝐺) = (𝐺 “ { 0 }))
149, 13mpdan 685 . 2 (𝐺𝐹 → ((𝑓𝐹 ↦ (𝑓 “ { 0 }))‘𝐺) = (𝐺 “ { 0 }))
156, 14sylan9eq 2786 1 ((𝑊𝑋𝐺𝐹) → (𝐾𝐺) = (𝐺 “ { 0 }))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 394   = wceq 1534  wcel 2099  Vcvv 3462  {csn 4633  cmpt 5236  ccnv 5681  cima 5685  cfv 6554  Scalarcsca 17269  0gc0g 17454  LFnlclfn 38755  LKerclk 38783
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-rep 5290  ax-sep 5304  ax-nul 5311  ax-pow 5369  ax-pr 5433  ax-un 7746
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ne 2931  df-ral 3052  df-rex 3061  df-reu 3365  df-rab 3420  df-v 3464  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4326  df-if 4534  df-pw 4609  df-sn 4634  df-pr 4636  df-op 4640  df-uni 4914  df-iun 5003  df-br 5154  df-opab 5216  df-mpt 5237  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-iota 6506  df-fun 6556  df-fn 6557  df-f 6558  df-f1 6559  df-fo 6560  df-f1o 6561  df-fv 6562  df-lkr 38784
This theorem is referenced by:  ellkr  38787  lkr0f  38792
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