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Theorem nlfnval 28710
Description: Value of the null space of a Hilbert space functional. (Contributed by NM, 11-Feb-2006.) (New usage is discouraged.)
Assertion
Ref Expression
nlfnval (𝑇: ℋ⟶ℂ → (null‘𝑇) = (𝑇 “ {0}))

Proof of Theorem nlfnval
Dummy variable 𝑡 is distinct from all other variables.
StepHypRef Expression
1 cnex 10002 . . 3 ℂ ∈ V
2 ax-hilex 27826 . . 3 ℋ ∈ V
31, 2elmap 7871 . 2 (𝑇 ∈ (ℂ ↑𝑚 ℋ) ↔ 𝑇: ℋ⟶ℂ)
4 cnvexg 7097 . . . 4 (𝑇 ∈ (ℂ ↑𝑚 ℋ) → 𝑇 ∈ V)
5 imaexg 7088 . . . 4 (𝑇 ∈ V → (𝑇 “ {0}) ∈ V)
64, 5syl 17 . . 3 (𝑇 ∈ (ℂ ↑𝑚 ℋ) → (𝑇 “ {0}) ∈ V)
7 cnveq 5285 . . . . 5 (𝑡 = 𝑇𝑡 = 𝑇)
87imaeq1d 5453 . . . 4 (𝑡 = 𝑇 → (𝑡 “ {0}) = (𝑇 “ {0}))
9 df-nlfn 28675 . . . 4 null = (𝑡 ∈ (ℂ ↑𝑚 ℋ) ↦ (𝑡 “ {0}))
108, 9fvmptg 6267 . . 3 ((𝑇 ∈ (ℂ ↑𝑚 ℋ) ∧ (𝑇 “ {0}) ∈ V) → (null‘𝑇) = (𝑇 “ {0}))
116, 10mpdan 701 . 2 (𝑇 ∈ (ℂ ↑𝑚 ℋ) → (null‘𝑇) = (𝑇 “ {0}))
123, 11sylbir 225 1 (𝑇: ℋ⟶ℂ → (null‘𝑇) = (𝑇 “ {0}))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1481  wcel 1988  Vcvv 3195  {csn 4168  ccnv 5103  cima 5107  wf 5872  cfv 5876  (class class class)co 6635  𝑚 cmap 7842  cc 9919  0cc0 9921  chil 27746  nullcnl 27779
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-8 1990  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-sep 4772  ax-nul 4780  ax-pow 4834  ax-pr 4897  ax-un 6934  ax-cnex 9977  ax-hilex 27826
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ral 2914  df-rex 2915  df-rab 2918  df-v 3197  df-sbc 3430  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-nul 3908  df-if 4078  df-pw 4151  df-sn 4169  df-pr 4171  df-op 4175  df-uni 4428  df-br 4645  df-opab 4704  df-mpt 4721  df-id 5014  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-rn 5115  df-res 5116  df-ima 5117  df-iota 5839  df-fun 5878  df-fn 5879  df-f 5880  df-fv 5884  df-ov 6638  df-oprab 6639  df-mpt2 6640  df-map 7844  df-nlfn 28675
This theorem is referenced by:  elnlfn  28757  nlelshi  28889
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