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

Theorem lkrfval 36341
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 3487 . 2 (𝑊𝑋𝑊 ∈ V)
2 lkrfval.k . . 3 𝐾 = (LKer‘𝑊)
3 fveq2 6652 . . . . . 6 (𝑤 = 𝑊 → (LFnl‘𝑤) = (LFnl‘𝑊))
4 lkrfval.f . . . . . 6 𝐹 = (LFnl‘𝑊)
53, 4eqtr4di 2875 . . . . 5 (𝑤 = 𝑊 → (LFnl‘𝑤) = 𝐹)
6 fveq2 6652 . . . . . . . . . 10 (𝑤 = 𝑊 → (Scalar‘𝑤) = (Scalar‘𝑊))
7 lkrfval.d . . . . . . . . . 10 𝐷 = (Scalar‘𝑊)
86, 7eqtr4di 2875 . . . . . . . . 9 (𝑤 = 𝑊 → (Scalar‘𝑤) = 𝐷)
98fveq2d 6656 . . . . . . . 8 (𝑤 = 𝑊 → (0g‘(Scalar‘𝑤)) = (0g𝐷))
10 lkrfval.o . . . . . . . 8 0 = (0g𝐷)
119, 10eqtr4di 2875 . . . . . . 7 (𝑤 = 𝑊 → (0g‘(Scalar‘𝑤)) = 0 )
1211sneqd 4551 . . . . . 6 (𝑤 = 𝑊 → {(0g‘(Scalar‘𝑤))} = { 0 })
1312imaeq2d 5907 . . . . 5 (𝑤 = 𝑊 → (𝑓 “ {(0g‘(Scalar‘𝑤))}) = (𝑓 “ { 0 }))
145, 13mpteq12dv 5127 . . . 4 (𝑤 = 𝑊 → (𝑓 ∈ (LFnl‘𝑤) ↦ (𝑓 “ {(0g‘(Scalar‘𝑤))})) = (𝑓𝐹 ↦ (𝑓 “ { 0 })))
15 df-lkr 36340 . . . 4 LKer = (𝑤 ∈ V ↦ (𝑓 ∈ (LFnl‘𝑤) ↦ (𝑓 “ {(0g‘(Scalar‘𝑤))})))
1614, 15, 4mptfvmpt 6973 . . 3 (𝑊 ∈ V → (LKer‘𝑊) = (𝑓𝐹 ↦ (𝑓 “ { 0 })))
172, 16syl5eq 2869 . 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 1538  wcel 2114  Vcvv 3469  {csn 4539  cmpt 5122  ccnv 5531  cima 5535  cfv 6334  Scalarcsca 16559  0gc0g 16704  LFnlclfn 36311  LKerclk 36339
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794  ax-rep 5166  ax-sep 5179  ax-nul 5186  ax-pr 5307
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-ne 3012  df-ral 3135  df-rex 3136  df-reu 3137  df-rab 3139  df-v 3471  df-sbc 3748  df-csb 3856  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4266  df-if 4440  df-sn 4540  df-pr 4542  df-op 4546  df-uni 4814  df-iun 4896  df-br 5043  df-opab 5105  df-mpt 5123  df-id 5437  df-xp 5538  df-rel 5539  df-cnv 5540  df-co 5541  df-dm 5542  df-rn 5543  df-res 5544  df-ima 5545  df-iota 6293  df-fun 6336  df-fn 6337  df-f 6338  df-f1 6339  df-fo 6340  df-f1o 6341  df-fv 6342  df-lkr 36340
This theorem is referenced by:  lkrval  36342
  Copyright terms: Public domain W3C validator