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Theorem nlfnval 31910
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 11234 . . 3 ℂ ∈ V
2 ax-hilex 31028 . . 3 ℋ ∈ V
31, 2elmap 8910 . 2 (𝑇 ∈ (ℂ ↑m ℋ) ↔ 𝑇: ℋ⟶ℂ)
4 cnvexg 7947 . . . 4 (𝑇 ∈ (ℂ ↑m ℋ) → 𝑇 ∈ V)
5 imaexg 7936 . . . 4 (𝑇 ∈ V → (𝑇 “ {0}) ∈ V)
64, 5syl 17 . . 3 (𝑇 ∈ (ℂ ↑m ℋ) → (𝑇 “ {0}) ∈ V)
7 cnveq 5887 . . . . 5 (𝑡 = 𝑇𝑡 = 𝑇)
87imaeq1d 6079 . . . 4 (𝑡 = 𝑇 → (𝑡 “ {0}) = (𝑇 “ {0}))
9 df-nlfn 31875 . . . 4 null = (𝑡 ∈ (ℂ ↑m ℋ) ↦ (𝑡 “ {0}))
108, 9fvmptg 7014 . . 3 ((𝑇 ∈ (ℂ ↑m ℋ) ∧ (𝑇 “ {0}) ∈ V) → (null‘𝑇) = (𝑇 “ {0}))
116, 10mpdan 687 . 2 (𝑇 ∈ (ℂ ↑m ℋ) → (null‘𝑇) = (𝑇 “ {0}))
123, 11sylbir 235 1 (𝑇: ℋ⟶ℂ → (null‘𝑇) = (𝑇 “ {0}))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1537  wcel 2106  Vcvv 3478  {csn 4631  ccnv 5688  cima 5692  wf 6559  cfv 6563  (class class class)co 7431  m cmap 8865  cc 11151  0cc0 11153  chba 30948  nullcnl 30981
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-cnex 11209  ax-hilex 31028
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-map 8867  df-nlfn 31875
This theorem is referenced by:  elnlfn  31957  nlelshi  32089
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