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Theorem nlfnval 29196
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 10270 . . 3 ℂ ∈ V
2 ax-hilex 28312 . . 3 ℋ ∈ V
31, 2elmap 8089 . 2 (𝑇 ∈ (ℂ ↑𝑚 ℋ) ↔ 𝑇: ℋ⟶ℂ)
4 cnvexg 7310 . . . 4 (𝑇 ∈ (ℂ ↑𝑚 ℋ) → 𝑇 ∈ V)
5 imaexg 7301 . . . 4 (𝑇 ∈ V → (𝑇 “ {0}) ∈ V)
64, 5syl 17 . . 3 (𝑇 ∈ (ℂ ↑𝑚 ℋ) → (𝑇 “ {0}) ∈ V)
7 cnveq 5464 . . . . 5 (𝑡 = 𝑇𝑡 = 𝑇)
87imaeq1d 5647 . . . 4 (𝑡 = 𝑇 → (𝑡 “ {0}) = (𝑇 “ {0}))
9 df-nlfn 29161 . . . 4 null = (𝑡 ∈ (ℂ ↑𝑚 ℋ) ↦ (𝑡 “ {0}))
108, 9fvmptg 6469 . . 3 ((𝑇 ∈ (ℂ ↑𝑚 ℋ) ∧ (𝑇 “ {0}) ∈ V) → (null‘𝑇) = (𝑇 “ {0}))
116, 10mpdan 678 . 2 (𝑇 ∈ (ℂ ↑𝑚 ℋ) → (null‘𝑇) = (𝑇 “ {0}))
123, 11sylbir 226 1 (𝑇: ℋ⟶ℂ → (null‘𝑇) = (𝑇 “ {0}))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1652  wcel 2155  Vcvv 3350  {csn 4334  ccnv 5276  cima 5280  wf 6064  cfv 6068  (class class class)co 6842  𝑚 cmap 8060  cc 10187  0cc0 10189  chba 28232  nullcnl 28265
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-sep 4941  ax-nul 4949  ax-pow 5001  ax-pr 5062  ax-un 7147  ax-cnex 10245  ax-hilex 28312
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ral 3060  df-rex 3061  df-rab 3064  df-v 3352  df-sbc 3597  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-nul 4080  df-if 4244  df-pw 4317  df-sn 4335  df-pr 4337  df-op 4341  df-uni 4595  df-br 4810  df-opab 4872  df-mpt 4889  df-id 5185  df-xp 5283  df-rel 5284  df-cnv 5285  df-co 5286  df-dm 5287  df-rn 5288  df-res 5289  df-ima 5290  df-iota 6031  df-fun 6070  df-fn 6071  df-f 6072  df-fv 6076  df-ov 6845  df-oprab 6846  df-mpt2 6847  df-map 8062  df-nlfn 29161
This theorem is referenced by:  elnlfn  29243  nlelshi  29375
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