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Theorem nlfnval 29071
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 10230 . . 3 ℂ ∈ V
2 ax-hilex 28187 . . 3 ℋ ∈ V
31, 2elmap 8055 . 2 (𝑇 ∈ (ℂ ↑𝑚 ℋ) ↔ 𝑇: ℋ⟶ℂ)
4 cnvexg 7279 . . . 4 (𝑇 ∈ (ℂ ↑𝑚 ℋ) → 𝑇 ∈ V)
5 imaexg 7270 . . . 4 (𝑇 ∈ V → (𝑇 “ {0}) ∈ V)
64, 5syl 17 . . 3 (𝑇 ∈ (ℂ ↑𝑚 ℋ) → (𝑇 “ {0}) ∈ V)
7 cnveq 5452 . . . . 5 (𝑡 = 𝑇𝑡 = 𝑇)
87imaeq1d 5624 . . . 4 (𝑡 = 𝑇 → (𝑡 “ {0}) = (𝑇 “ {0}))
9 df-nlfn 29036 . . . 4 null = (𝑡 ∈ (ℂ ↑𝑚 ℋ) ↦ (𝑡 “ {0}))
108, 9fvmptg 6444 . . 3 ((𝑇 ∈ (ℂ ↑𝑚 ℋ) ∧ (𝑇 “ {0}) ∈ V) → (null‘𝑇) = (𝑇 “ {0}))
116, 10mpdan 705 . 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 1632  wcel 2140  Vcvv 3341  {csn 4322  ccnv 5266  cima 5270  wf 6046  cfv 6050  (class class class)co 6815  𝑚 cmap 8026  cc 10147  0cc0 10149  chil 28107  nullcnl 28140
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1989  ax-6 2055  ax-7 2091  ax-8 2142  ax-9 2149  ax-10 2169  ax-11 2184  ax-12 2197  ax-13 2392  ax-ext 2741  ax-sep 4934  ax-nul 4942  ax-pow 4993  ax-pr 5056  ax-un 7116  ax-cnex 10205  ax-hilex 28187
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2048  df-eu 2612  df-mo 2613  df-clab 2748  df-cleq 2754  df-clel 2757  df-nfc 2892  df-ral 3056  df-rex 3057  df-rab 3060  df-v 3343  df-sbc 3578  df-dif 3719  df-un 3721  df-in 3723  df-ss 3730  df-nul 4060  df-if 4232  df-pw 4305  df-sn 4323  df-pr 4325  df-op 4329  df-uni 4590  df-br 4806  df-opab 4866  df-mpt 4883  df-id 5175  df-xp 5273  df-rel 5274  df-cnv 5275  df-co 5276  df-dm 5277  df-rn 5278  df-res 5279  df-ima 5280  df-iota 6013  df-fun 6052  df-fn 6053  df-f 6054  df-fv 6058  df-ov 6818  df-oprab 6819  df-mpt2 6820  df-map 8028  df-nlfn 29036
This theorem is referenced by:  elnlfn  29118  nlelshi  29250
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