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 39089
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 39088 . . 3 (𝑊𝑋𝐾 = (𝑓𝐹 ↦ (𝑓 “ { 0 })))
65fveq1d 6908 . 2 (𝑊𝑋 → (𝐾𝐺) = ((𝑓𝐹 ↦ (𝑓 “ { 0 }))‘𝐺))
7 cnvexg 7946 . . . 4 (𝐺𝐹𝐺 ∈ V)
8 imaexg 7935 . . . 4 (𝐺 ∈ V → (𝐺 “ { 0 }) ∈ V)
97, 8syl 17 . . 3 (𝐺𝐹 → (𝐺 “ { 0 }) ∈ V)
10 cnveq 5884 . . . . 5 (𝑓 = 𝐺𝑓 = 𝐺)
1110imaeq1d 6077 . . . 4 (𝑓 = 𝐺 → (𝑓 “ { 0 }) = (𝐺 “ { 0 }))
12 eqid 2737 . . . 4 (𝑓𝐹 ↦ (𝑓 “ { 0 })) = (𝑓𝐹 ↦ (𝑓 “ { 0 }))
1311, 12fvmptg 7014 . . 3 ((𝐺𝐹 ∧ (𝐺 “ { 0 }) ∈ V) → ((𝑓𝐹 ↦ (𝑓 “ { 0 }))‘𝐺) = (𝐺 “ { 0 }))
149, 13mpdan 687 . 2 (𝐺𝐹 → ((𝑓𝐹 ↦ (𝑓 “ { 0 }))‘𝐺) = (𝐺 “ { 0 }))
156, 14sylan9eq 2797 1 ((𝑊𝑋𝐺𝐹) → (𝐾𝐺) = (𝐺 “ { 0 }))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = 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-pow 5365  ax-pr 5432  ax-un 7755
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-pw 4602  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:  ellkr  39090  lkr0f  39095
  Copyright terms: Public domain W3C validator