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Theorem elnlfn 29810
 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 29763 . . . . . 6 (𝑇: ℋ⟶ℂ → (null‘𝑇) = (𝑇 “ {0}))
2 cnvimass 5921 . . . . . 6 (𝑇 “ {0}) ⊆ dom 𝑇
31, 2eqsstrdi 3946 . . . . 5 (𝑇: ℋ⟶ℂ → (null‘𝑇) ⊆ dom 𝑇)
4 fdm 6506 . . . . 5 (𝑇: ℋ⟶ℂ → dom 𝑇 = ℋ)
53, 4sseqtrd 3932 . . . 4 (𝑇: ℋ⟶ℂ → (null‘𝑇) ⊆ ℋ)
65sseld 3891 . . 3 (𝑇: ℋ⟶ℂ → (𝐴 ∈ (null‘𝑇) → 𝐴 ∈ ℋ))
76pm4.71rd 566 . 2 (𝑇: ℋ⟶ℂ → (𝐴 ∈ (null‘𝑇) ↔ (𝐴 ∈ ℋ ∧ 𝐴 ∈ (null‘𝑇))))
81eleq2d 2837 . . . . 5 (𝑇: ℋ⟶ℂ → (𝐴 ∈ (null‘𝑇) ↔ 𝐴 ∈ (𝑇 “ {0})))
98adantr 484 . . . 4 ((𝑇: ℋ⟶ℂ ∧ 𝐴 ∈ ℋ) → (𝐴 ∈ (null‘𝑇) ↔ 𝐴 ∈ (𝑇 “ {0})))
10 ffn 6498 . . . . 5 (𝑇: ℋ⟶ℂ → 𝑇 Fn ℋ)
11 eleq1 2839 . . . . . . . 8 (𝑥 = 𝐴 → (𝑥 ∈ (𝑇 “ {0}) ↔ 𝐴 ∈ (𝑇 “ {0})))
12 fveqeq2 6667 . . . . . . . 8 (𝑥 = 𝐴 → ((𝑇𝑥) = 0 ↔ (𝑇𝐴) = 0))
1311, 12bibi12d 349 . . . . . . 7 (𝑥 = 𝐴 → ((𝑥 ∈ (𝑇 “ {0}) ↔ (𝑇𝑥) = 0) ↔ (𝐴 ∈ (𝑇 “ {0}) ↔ (𝑇𝐴) = 0)))
1413imbi2d 344 . . . . . 6 (𝑥 = 𝐴 → ((𝑇 Fn ℋ → (𝑥 ∈ (𝑇 “ {0}) ↔ (𝑇𝑥) = 0)) ↔ (𝑇 Fn ℋ → (𝐴 ∈ (𝑇 “ {0}) ↔ (𝑇𝐴) = 0))))
15 0cn 10671 . . . . . . . . 9 0 ∈ ℂ
16 vex 3413 . . . . . . . . . 10 𝑥 ∈ V
1716eliniseg 5930 . . . . . . . . 9 (0 ∈ ℂ → (𝑥 ∈ (𝑇 “ {0}) ↔ 𝑥𝑇0))
1815, 17ax-mp 5 . . . . . . . 8 (𝑥 ∈ (𝑇 “ {0}) ↔ 𝑥𝑇0)
19 fnbrfvb 6706 . . . . . . . 8 ((𝑇 Fn ℋ ∧ 𝑥 ∈ ℋ) → ((𝑇𝑥) = 0 ↔ 𝑥𝑇0))
2018, 19bitr4id 293 . . . . . . 7 ((𝑇 Fn ℋ ∧ 𝑥 ∈ ℋ) → (𝑥 ∈ (𝑇 “ {0}) ↔ (𝑇𝑥) = 0))
2120expcom 417 . . . . . 6 (𝑥 ∈ ℋ → (𝑇 Fn ℋ → (𝑥 ∈ (𝑇 “ {0}) ↔ (𝑇𝑥) = 0)))
2214, 21vtoclga 3492 . . . . 5 (𝐴 ∈ ℋ → (𝑇 Fn ℋ → (𝐴 ∈ (𝑇 “ {0}) ↔ (𝑇𝐴) = 0)))
2310, 22mpan9 510 . . . 4 ((𝑇: ℋ⟶ℂ ∧ 𝐴 ∈ ℋ) → (𝐴 ∈ (𝑇 “ {0}) ↔ (𝑇𝐴) = 0))
249, 23bitrd 282 . . 3 ((𝑇: ℋ⟶ℂ ∧ 𝐴 ∈ ℋ) → (𝐴 ∈ (null‘𝑇) ↔ (𝑇𝐴) = 0))
2524pm5.32da 582 . 2 (𝑇: ℋ⟶ℂ → ((𝐴 ∈ ℋ ∧ 𝐴 ∈ (null‘𝑇)) ↔ (𝐴 ∈ ℋ ∧ (𝑇𝐴) = 0)))
267, 25bitrd 282 1 (𝑇: ℋ⟶ℂ → (𝐴 ∈ (null‘𝑇) ↔ (𝐴 ∈ ℋ ∧ (𝑇𝐴) = 0)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   = wceq 1538   ∈ wcel 2111  {csn 4522   class class class wbr 5032  ◡ccnv 5523  dom cdm 5524   “ cima 5527   Fn wfn 6330  ⟶wf 6331  ‘cfv 6335  ℂcc 10573  0cc0 10575   ℋchba 28801  nullcnl 28834 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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-sep 5169  ax-nul 5176  ax-pow 5234  ax-pr 5298  ax-un 7459  ax-cnex 10631  ax-1cn 10633  ax-icn 10634  ax-addcl 10635  ax-mulcl 10637  ax-i2m1 10643  ax-hilex 28881 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ral 3075  df-rex 3076  df-rab 3079  df-v 3411  df-sbc 3697  df-dif 3861  df-un 3863  df-in 3865  df-ss 3875  df-nul 4226  df-if 4421  df-pw 4496  df-sn 4523  df-pr 4525  df-op 4529  df-uni 4799  df-br 5033  df-opab 5095  df-mpt 5113  df-id 5430  df-xp 5530  df-rel 5531  df-cnv 5532  df-co 5533  df-dm 5534  df-rn 5535  df-res 5536  df-ima 5537  df-iota 6294  df-fun 6337  df-fn 6338  df-f 6339  df-fv 6343  df-ov 7153  df-oprab 7154  df-mpo 7155  df-map 8418  df-nlfn 29728 This theorem is referenced by:  elnlfn2  29811  nlelshi  29942  nlelchi  29943  riesz3i  29944
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