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Theorem lkrval 35781
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 35780 . . 3 (𝑊𝑋𝐾 = (𝑓𝐹 ↦ (𝑓 “ { 0 })))
65fveq1d 6545 . 2 (𝑊𝑋 → (𝐾𝐺) = ((𝑓𝐹 ↦ (𝑓 “ { 0 }))‘𝐺))
7 cnvexg 7490 . . . 4 (𝐺𝐹𝐺 ∈ V)
8 imaexg 7481 . . . 4 (𝐺 ∈ V → (𝐺 “ { 0 }) ∈ V)
97, 8syl 17 . . 3 (𝐺𝐹 → (𝐺 “ { 0 }) ∈ V)
10 cnveq 5635 . . . . 5 (𝑓 = 𝐺𝑓 = 𝐺)
1110imaeq1d 5810 . . . 4 (𝑓 = 𝐺 → (𝑓 “ { 0 }) = (𝐺 “ { 0 }))
12 eqid 2795 . . . 4 (𝑓𝐹 ↦ (𝑓 “ { 0 })) = (𝑓𝐹 ↦ (𝑓 “ { 0 }))
1311, 12fvmptg 6638 . . 3 ((𝐺𝐹 ∧ (𝐺 “ { 0 }) ∈ V) → ((𝑓𝐹 ↦ (𝑓 “ { 0 }))‘𝐺) = (𝐺 “ { 0 }))
149, 13mpdan 683 . 2 (𝐺𝐹 → ((𝑓𝐹 ↦ (𝑓 “ { 0 }))‘𝐺) = (𝐺 “ { 0 }))
156, 14sylan9eq 2851 1 ((𝑊𝑋𝐺𝐹) → (𝐾𝐺) = (𝐺 “ { 0 }))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1522  wcel 2081  Vcvv 3437  {csn 4476  cmpt 5045  ccnv 5447  cima 5451  cfv 6230  Scalarcsca 16402  0gc0g 16547  LFnlclfn 35750  LKerclk 35778
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1777  ax-4 1791  ax-5 1888  ax-6 1947  ax-7 1992  ax-8 2083  ax-9 2091  ax-10 2112  ax-11 2126  ax-12 2141  ax-13 2344  ax-ext 2769  ax-rep 5086  ax-sep 5099  ax-nul 5106  ax-pow 5162  ax-pr 5226  ax-un 7324
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3an 1082  df-tru 1525  df-ex 1762  df-nf 1766  df-sb 2043  df-mo 2576  df-eu 2612  df-clab 2776  df-cleq 2788  df-clel 2863  df-nfc 2935  df-ne 2985  df-ral 3110  df-rex 3111  df-reu 3112  df-rab 3114  df-v 3439  df-sbc 3710  df-csb 3816  df-dif 3866  df-un 3868  df-in 3870  df-ss 3878  df-nul 4216  df-if 4386  df-pw 4459  df-sn 4477  df-pr 4479  df-op 4483  df-uni 4750  df-iun 4831  df-br 4967  df-opab 5029  df-mpt 5046  df-id 5353  df-xp 5454  df-rel 5455  df-cnv 5456  df-co 5457  df-dm 5458  df-rn 5459  df-res 5460  df-ima 5461  df-iota 6194  df-fun 6232  df-fn 6233  df-f 6234  df-f1 6235  df-fo 6236  df-f1o 6237  df-fv 6238  df-lkr 35779
This theorem is referenced by:  ellkr  35782  lkr0f  35787
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