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Theorem elnlfn 28657
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 28610 . . . . . 6 (𝑇: ℋ⟶ℂ → (null‘𝑇) = (𝑇 “ {0}))
2 cnvimass 5449 . . . . . 6 (𝑇 “ {0}) ⊆ dom 𝑇
31, 2syl6eqss 3639 . . . . 5 (𝑇: ℋ⟶ℂ → (null‘𝑇) ⊆ dom 𝑇)
4 fdm 6013 . . . . 5 (𝑇: ℋ⟶ℂ → dom 𝑇 = ℋ)
53, 4sseqtrd 3625 . . . 4 (𝑇: ℋ⟶ℂ → (null‘𝑇) ⊆ ℋ)
65sseld 3586 . . 3 (𝑇: ℋ⟶ℂ → (𝐴 ∈ (null‘𝑇) → 𝐴 ∈ ℋ))
76pm4.71rd 666 . 2 (𝑇: ℋ⟶ℂ → (𝐴 ∈ (null‘𝑇) ↔ (𝐴 ∈ ℋ ∧ 𝐴 ∈ (null‘𝑇))))
81eleq2d 2684 . . . . 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 6007 . . . . 5 (𝑇: ℋ⟶ℂ → 𝑇 Fn ℋ)
11 eleq1 2686 . . . . . . . 8 (𝑥 = 𝐴 → (𝑥 ∈ (𝑇 “ {0}) ↔ 𝐴 ∈ (𝑇 “ {0})))
12 fveq2 6153 . . . . . . . . 9 (𝑥 = 𝐴 → (𝑇𝑥) = (𝑇𝐴))
1312eqeq1d 2623 . . . . . . . 8 (𝑥 = 𝐴 → ((𝑇𝑥) = 0 ↔ (𝑇𝐴) = 0))
1411, 13bibi12d 335 . . . . . . 7 (𝑥 = 𝐴 → ((𝑥 ∈ (𝑇 “ {0}) ↔ (𝑇𝑥) = 0) ↔ (𝐴 ∈ (𝑇 “ {0}) ↔ (𝑇𝐴) = 0)))
1514imbi2d 330 . . . . . 6 (𝑥 = 𝐴 → ((𝑇 Fn ℋ → (𝑥 ∈ (𝑇 “ {0}) ↔ (𝑇𝑥) = 0)) ↔ (𝑇 Fn ℋ → (𝐴 ∈ (𝑇 “ {0}) ↔ (𝑇𝐴) = 0))))
16 fnbrfvb 6198 . . . . . . . 8 ((𝑇 Fn ℋ ∧ 𝑥 ∈ ℋ) → ((𝑇𝑥) = 0 ↔ 𝑥𝑇0))
17 0cn 9984 . . . . . . . . 9 0 ∈ ℂ
18 vex 3192 . . . . . . . . . 10 𝑥 ∈ V
1918eliniseg 5458 . . . . . . . . 9 (0 ∈ ℂ → (𝑥 ∈ (𝑇 “ {0}) ↔ 𝑥𝑇0))
2017, 19ax-mp 5 . . . . . . . 8 (𝑥 ∈ (𝑇 “ {0}) ↔ 𝑥𝑇0)
2116, 20syl6rbbr 279 . . . . . . 7 ((𝑇 Fn ℋ ∧ 𝑥 ∈ ℋ) → (𝑥 ∈ (𝑇 “ {0}) ↔ (𝑇𝑥) = 0))
2221expcom 451 . . . . . 6 (𝑥 ∈ ℋ → (𝑇 Fn ℋ → (𝑥 ∈ (𝑇 “ {0}) ↔ (𝑇𝑥) = 0)))
2315, 22vtoclga 3261 . . . . 5 (𝐴 ∈ ℋ → (𝑇 Fn ℋ → (𝐴 ∈ (𝑇 “ {0}) ↔ (𝑇𝐴) = 0)))
2410, 23mpan9 486 . . . 4 ((𝑇: ℋ⟶ℂ ∧ 𝐴 ∈ ℋ) → (𝐴 ∈ (𝑇 “ {0}) ↔ (𝑇𝐴) = 0))
259, 24bitrd 268 . . 3 ((𝑇: ℋ⟶ℂ ∧ 𝐴 ∈ ℋ) → (𝐴 ∈ (null‘𝑇) ↔ (𝑇𝐴) = 0))
2625pm5.32da 672 . 2 (𝑇: ℋ⟶ℂ → ((𝐴 ∈ ℋ ∧ 𝐴 ∈ (null‘𝑇)) ↔ (𝐴 ∈ ℋ ∧ (𝑇𝐴) = 0)))
277, 26bitrd 268 1 (𝑇: ℋ⟶ℂ → (𝐴 ∈ (null‘𝑇) ↔ (𝐴 ∈ ℋ ∧ (𝑇𝐴) = 0)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384   = wceq 1480  wcel 1987  {csn 4153   class class class wbr 4618  ccnv 5078  dom cdm 5079  cima 5082   Fn wfn 5847  wf 5848  cfv 5852  cc 9886  0cc0 9888  chil 27646  nullcnl 27679
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6909  ax-cnex 9944  ax-1cn 9946  ax-icn 9947  ax-addcl 9948  ax-mulcl 9950  ax-i2m1 9956  ax-hilex 27726
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3191  df-sbc 3422  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-br 4619  df-opab 4679  df-mpt 4680  df-id 4994  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-iota 5815  df-fun 5854  df-fn 5855  df-f 5856  df-fv 5860  df-ov 6613  df-oprab 6614  df-mpt2 6615  df-map 7811  df-nlfn 28575
This theorem is referenced by:  elnlfn2  28658  nlelshi  28789  nlelchi  28790  riesz3i  28791
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