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Theorem elnlfn 31957
Description: Membership in the null space of a Hilbert space functional. (Contributed by NM, 11-Feb-2006.) (Revised by Mario Carneiro, 17-Nov-2013.) (New usage is discouraged.)
Assertion
Ref Expression
elnlfn (𝑇: ℋ⟶ℂ → (𝐴 ∈ (null‘𝑇) ↔ (𝐴 ∈ ℋ ∧ (𝑇𝐴) = 0)))

Proof of Theorem elnlfn
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 nlfnval 31910 . . . . . 6 (𝑇: ℋ⟶ℂ → (null‘𝑇) = (𝑇 “ {0}))
2 cnvimass 6102 . . . . . 6 (𝑇 “ {0}) ⊆ dom 𝑇
31, 2eqsstrdi 4050 . . . . 5 (𝑇: ℋ⟶ℂ → (null‘𝑇) ⊆ dom 𝑇)
4 fdm 6746 . . . . 5 (𝑇: ℋ⟶ℂ → dom 𝑇 = ℋ)
53, 4sseqtrd 4036 . . . 4 (𝑇: ℋ⟶ℂ → (null‘𝑇) ⊆ ℋ)
65sseld 3994 . . 3 (𝑇: ℋ⟶ℂ → (𝐴 ∈ (null‘𝑇) → 𝐴 ∈ ℋ))
76pm4.71rd 562 . 2 (𝑇: ℋ⟶ℂ → (𝐴 ∈ (null‘𝑇) ↔ (𝐴 ∈ ℋ ∧ 𝐴 ∈ (null‘𝑇))))
81eleq2d 2825 . . . . 5 (𝑇: ℋ⟶ℂ → (𝐴 ∈ (null‘𝑇) ↔ 𝐴 ∈ (𝑇 “ {0})))
98adantr 480 . . . 4 ((𝑇: ℋ⟶ℂ ∧ 𝐴 ∈ ℋ) → (𝐴 ∈ (null‘𝑇) ↔ 𝐴 ∈ (𝑇 “ {0})))
10 ffn 6737 . . . . 5 (𝑇: ℋ⟶ℂ → 𝑇 Fn ℋ)
11 eleq1 2827 . . . . . . . 8 (𝑥 = 𝐴 → (𝑥 ∈ (𝑇 “ {0}) ↔ 𝐴 ∈ (𝑇 “ {0})))
12 fveqeq2 6916 . . . . . . . 8 (𝑥 = 𝐴 → ((𝑇𝑥) = 0 ↔ (𝑇𝐴) = 0))
1311, 12bibi12d 345 . . . . . . 7 (𝑥 = 𝐴 → ((𝑥 ∈ (𝑇 “ {0}) ↔ (𝑇𝑥) = 0) ↔ (𝐴 ∈ (𝑇 “ {0}) ↔ (𝑇𝐴) = 0)))
1413imbi2d 340 . . . . . 6 (𝑥 = 𝐴 → ((𝑇 Fn ℋ → (𝑥 ∈ (𝑇 “ {0}) ↔ (𝑇𝑥) = 0)) ↔ (𝑇 Fn ℋ → (𝐴 ∈ (𝑇 “ {0}) ↔ (𝑇𝐴) = 0))))
15 0cn 11251 . . . . . . . . 9 0 ∈ ℂ
16 vex 3482 . . . . . . . . . 10 𝑥 ∈ V
1716eliniseg 6115 . . . . . . . . 9 (0 ∈ ℂ → (𝑥 ∈ (𝑇 “ {0}) ↔ 𝑥𝑇0))
1815, 17ax-mp 5 . . . . . . . 8 (𝑥 ∈ (𝑇 “ {0}) ↔ 𝑥𝑇0)
19 fnbrfvb 6960 . . . . . . . 8 ((𝑇 Fn ℋ ∧ 𝑥 ∈ ℋ) → ((𝑇𝑥) = 0 ↔ 𝑥𝑇0))
2018, 19bitr4id 290 . . . . . . 7 ((𝑇 Fn ℋ ∧ 𝑥 ∈ ℋ) → (𝑥 ∈ (𝑇 “ {0}) ↔ (𝑇𝑥) = 0))
2120expcom 413 . . . . . 6 (𝑥 ∈ ℋ → (𝑇 Fn ℋ → (𝑥 ∈ (𝑇 “ {0}) ↔ (𝑇𝑥) = 0)))
2214, 21vtoclga 3577 . . . . 5 (𝐴 ∈ ℋ → (𝑇 Fn ℋ → (𝐴 ∈ (𝑇 “ {0}) ↔ (𝑇𝐴) = 0)))
2310, 22mpan9 506 . . . 4 ((𝑇: ℋ⟶ℂ ∧ 𝐴 ∈ ℋ) → (𝐴 ∈ (𝑇 “ {0}) ↔ (𝑇𝐴) = 0))
249, 23bitrd 279 . . 3 ((𝑇: ℋ⟶ℂ ∧ 𝐴 ∈ ℋ) → (𝐴 ∈ (null‘𝑇) ↔ (𝑇𝐴) = 0))
2524pm5.32da 579 . 2 (𝑇: ℋ⟶ℂ → ((𝐴 ∈ ℋ ∧ 𝐴 ∈ (null‘𝑇)) ↔ (𝐴 ∈ ℋ ∧ (𝑇𝐴) = 0)))
267, 25bitrd 279 1 (𝑇: ℋ⟶ℂ → (𝐴 ∈ (null‘𝑇) ↔ (𝐴 ∈ ℋ ∧ (𝑇𝐴) = 0)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1537  wcel 2106  {csn 4631   class class class wbr 5148  ccnv 5688  dom cdm 5689  cima 5692   Fn wfn 6558  wf 6559  cfv 6563  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-1cn 11211  ax-icn 11212  ax-addcl 11213  ax-mulcl 11215  ax-i2m1 11221  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-ne 2939  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:  elnlfn2  31958  nlelshi  32089  nlelchi  32090  riesz3i  32091
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