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Theorem nlfnval 28124
 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 9896 . . 3 ℂ ∈ V
2 ax-hilex 27240 . . 3 ℋ ∈ V
31, 2elmap 7772 . 2 (𝑇 ∈ (ℂ ↑𝑚 ℋ) ↔ 𝑇: ℋ⟶ℂ)
4 cnvexg 7005 . . . 4 (𝑇 ∈ (ℂ ↑𝑚 ℋ) → 𝑇 ∈ V)
5 imaexg 6995 . . . 4 (𝑇 ∈ V → (𝑇 “ {0}) ∈ V)
64, 5syl 17 . . 3 (𝑇 ∈ (ℂ ↑𝑚 ℋ) → (𝑇 “ {0}) ∈ V)
7 cnveq 5218 . . . . 5 (𝑡 = 𝑇𝑡 = 𝑇)
87imaeq1d 5384 . . . 4 (𝑡 = 𝑇 → (𝑡 “ {0}) = (𝑇 “ {0}))
9 df-nlfn 28089 . . . 4 null = (𝑡 ∈ (ℂ ↑𝑚 ℋ) ↦ (𝑡 “ {0}))
108, 9fvmptg 6189 . . 3 ((𝑇 ∈ (ℂ ↑𝑚 ℋ) ∧ (𝑇 “ {0}) ∈ V) → (null‘𝑇) = (𝑇 “ {0}))
116, 10mpdan 699 . 2 (𝑇 ∈ (ℂ ↑𝑚 ℋ) → (null‘𝑇) = (𝑇 “ {0}))
123, 11sylbir 224 1 (𝑇: ℋ⟶ℂ → (null‘𝑇) = (𝑇 “ {0}))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1475   ∈ wcel 1977  Vcvv 3173  {csn 4125  ◡ccnv 5037   “ cima 5041  ⟶wf 5800  ‘cfv 5804  (class class class)co 6549   ↑𝑚 cmap 7744  ℂcc 9813  0cc0 9815   ℋchil 27160  nullcnl 27193 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-hilex 27240 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-map 7746  df-nlfn 28089 This theorem is referenced by:  elnlfn  28171  nlelshi  28303
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