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Theorem elnlfn 31168
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 31121 . . . . . 6 (𝑇: ℋ⟶ℂ → (null‘𝑇) = (𝑇 “ {0}))
2 cnvimass 6077 . . . . . 6 (𝑇 “ {0}) ⊆ dom 𝑇
31, 2eqsstrdi 4035 . . . . 5 (𝑇: ℋ⟶ℂ → (null‘𝑇) ⊆ dom 𝑇)
4 fdm 6723 . . . . 5 (𝑇: ℋ⟶ℂ → dom 𝑇 = ℋ)
53, 4sseqtrd 4021 . . . 4 (𝑇: ℋ⟶ℂ → (null‘𝑇) ⊆ ℋ)
65sseld 3980 . . 3 (𝑇: ℋ⟶ℂ → (𝐴 ∈ (null‘𝑇) → 𝐴 ∈ ℋ))
76pm4.71rd 563 . 2 (𝑇: ℋ⟶ℂ → (𝐴 ∈ (null‘𝑇) ↔ (𝐴 ∈ ℋ ∧ 𝐴 ∈ (null‘𝑇))))
81eleq2d 2819 . . . . 5 (𝑇: ℋ⟶ℂ → (𝐴 ∈ (null‘𝑇) ↔ 𝐴 ∈ (𝑇 “ {0})))
98adantr 481 . . . 4 ((𝑇: ℋ⟶ℂ ∧ 𝐴 ∈ ℋ) → (𝐴 ∈ (null‘𝑇) ↔ 𝐴 ∈ (𝑇 “ {0})))
10 ffn 6714 . . . . 5 (𝑇: ℋ⟶ℂ → 𝑇 Fn ℋ)
11 eleq1 2821 . . . . . . . 8 (𝑥 = 𝐴 → (𝑥 ∈ (𝑇 “ {0}) ↔ 𝐴 ∈ (𝑇 “ {0})))
12 fveqeq2 6897 . . . . . . . 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 11202 . . . . . . . . 9 0 ∈ ℂ
16 vex 3478 . . . . . . . . . 10 𝑥 ∈ V
1716eliniseg 6090 . . . . . . . . 9 (0 ∈ ℂ → (𝑥 ∈ (𝑇 “ {0}) ↔ 𝑥𝑇0))
1815, 17ax-mp 5 . . . . . . . 8 (𝑥 ∈ (𝑇 “ {0}) ↔ 𝑥𝑇0)
19 fnbrfvb 6941 . . . . . . . 8 ((𝑇 Fn ℋ ∧ 𝑥 ∈ ℋ) → ((𝑇𝑥) = 0 ↔ 𝑥𝑇0))
2018, 19bitr4id 289 . . . . . . 7 ((𝑇 Fn ℋ ∧ 𝑥 ∈ ℋ) → (𝑥 ∈ (𝑇 “ {0}) ↔ (𝑇𝑥) = 0))
2120expcom 414 . . . . . 6 (𝑥 ∈ ℋ → (𝑇 Fn ℋ → (𝑥 ∈ (𝑇 “ {0}) ↔ (𝑇𝑥) = 0)))
2214, 21vtoclga 3565 . . . . 5 (𝐴 ∈ ℋ → (𝑇 Fn ℋ → (𝐴 ∈ (𝑇 “ {0}) ↔ (𝑇𝐴) = 0)))
2310, 22mpan9 507 . . . 4 ((𝑇: ℋ⟶ℂ ∧ 𝐴 ∈ ℋ) → (𝐴 ∈ (𝑇 “ {0}) ↔ (𝑇𝐴) = 0))
249, 23bitrd 278 . . 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 278 1 (𝑇: ℋ⟶ℂ → (𝐴 ∈ (null‘𝑇) ↔ (𝐴 ∈ ℋ ∧ (𝑇𝐴) = 0)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396   = wceq 1541  wcel 2106  {csn 4627   class class class wbr 5147  ccnv 5674  dom cdm 5675  cima 5678   Fn wfn 6535  wf 6536  cfv 6540  cc 11104  0cc0 11106  chba 30159  nullcnl 30192
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7721  ax-cnex 11162  ax-1cn 11164  ax-icn 11165  ax-addcl 11166  ax-mulcl 11168  ax-i2m1 11174  ax-hilex 30239
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-sbc 3777  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-fv 6548  df-ov 7408  df-oprab 7409  df-mpo 7410  df-map 8818  df-nlfn 31086
This theorem is referenced by:  elnlfn2  31169  nlelshi  31300  nlelchi  31301  riesz3i  31302
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