Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  lkrval Structured version   Visualization version   GIF version

Theorem lkrval 33841
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 33840 . . 3 (𝑊𝑋𝐾 = (𝑓𝐹 ↦ (𝑓 “ { 0 })))
65fveq1d 6152 . 2 (𝑊𝑋 → (𝐾𝐺) = ((𝑓𝐹 ↦ (𝑓 “ { 0 }))‘𝐺))
7 cnvexg 7062 . . . 4 (𝐺𝐹𝐺 ∈ V)
8 imaexg 7051 . . . 4 (𝐺 ∈ V → (𝐺 “ { 0 }) ∈ V)
97, 8syl 17 . . 3 (𝐺𝐹 → (𝐺 “ { 0 }) ∈ V)
10 cnveq 5261 . . . . 5 (𝑓 = 𝐺𝑓 = 𝐺)
1110imaeq1d 5428 . . . 4 (𝑓 = 𝐺 → (𝑓 “ { 0 }) = (𝐺 “ { 0 }))
12 eqid 2626 . . . 4 (𝑓𝐹 ↦ (𝑓 “ { 0 })) = (𝑓𝐹 ↦ (𝑓 “ { 0 }))
1311, 12fvmptg 6238 . . 3 ((𝐺𝐹 ∧ (𝐺 “ { 0 }) ∈ V) → ((𝑓𝐹 ↦ (𝑓 “ { 0 }))‘𝐺) = (𝐺 “ { 0 }))
149, 13mpdan 701 . 2 (𝐺𝐹 → ((𝑓𝐹 ↦ (𝑓 “ { 0 }))‘𝐺) = (𝐺 “ { 0 }))
156, 14sylan9eq 2680 1 ((𝑊𝑋𝐺𝐹) → (𝐾𝐺) = (𝐺 “ { 0 }))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1480  wcel 1992  Vcvv 3191  {csn 4153  cmpt 4678  ccnv 5078  cima 5082  cfv 5850  Scalarcsca 15860  0gc0g 16016  LFnlclfn 33810  LKerclk 33838
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-8 1994  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606  ax-rep 4736  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6903
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1883  df-eu 2478  df-mo 2479  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ne 2797  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3193  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-id 4994  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-iota 5813  df-fun 5852  df-fn 5853  df-f 5854  df-f1 5855  df-fo 5856  df-f1o 5857  df-fv 5858  df-lkr 33839
This theorem is referenced by:  ellkr  33842  lkr0f  33847
  Copyright terms: Public domain W3C validator