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Theorem nlfnval 31900
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 11236 . . 3 ℂ ∈ V
2 ax-hilex 31018 . . 3 ℋ ∈ V
31, 2elmap 8911 . 2 (𝑇 ∈ (ℂ ↑m ℋ) ↔ 𝑇: ℋ⟶ℂ)
4 cnvexg 7946 . . . 4 (𝑇 ∈ (ℂ ↑m ℋ) → 𝑇 ∈ V)
5 imaexg 7935 . . . 4 (𝑇 ∈ V → (𝑇 “ {0}) ∈ V)
64, 5syl 17 . . 3 (𝑇 ∈ (ℂ ↑m ℋ) → (𝑇 “ {0}) ∈ V)
7 cnveq 5884 . . . . 5 (𝑡 = 𝑇𝑡 = 𝑇)
87imaeq1d 6077 . . . 4 (𝑡 = 𝑇 → (𝑡 “ {0}) = (𝑇 “ {0}))
9 df-nlfn 31865 . . . 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 1540  wcel 2108  Vcvv 3480  {csn 4626  ccnv 5684  cima 5688  wf 6557  cfv 6561  (class class class)co 7431  m cmap 8866  cc 11153  0cc0 11155  chba 30938  nullcnl 30971
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-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-cnex 11211  ax-hilex 31018
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-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-sbc 3789  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-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-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-map 8868  df-nlfn 31865
This theorem is referenced by:  elnlfn  31947  nlelshi  32079
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