Hilbert Space Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  HSE Home  >  Th. List  >  hlimi Structured version   Visualization version   GIF version

Theorem hlimi 28601
 Description: Express the predicate: The limit of vector sequence 𝐹 in a Hilbert space is 𝐴, i.e. 𝐹 converges to 𝐴. This means that for any real 𝑥, no matter how small, there always exists an integer 𝑦 such that the norm of any later vector in the sequence minus the limit is less than 𝑥. Definition of converge in [Beran] p. 96. (Contributed by NM, 16-Aug-1999.) (Revised by Mario Carneiro, 14-May-2014.) (New usage is discouraged.)
Hypothesis
Ref Expression
hlim.1 𝐴 ∈ V
Assertion
Ref Expression
hlimi (𝐹𝑣 𝐴 ↔ ((𝐹:ℕ⟶ ℋ ∧ 𝐴 ∈ ℋ) ∧ ∀𝑥 ∈ ℝ+𝑦 ∈ ℕ ∀𝑧 ∈ (ℤ𝑦)(norm‘((𝐹𝑧) − 𝐴)) < 𝑥))
Distinct variable groups:   𝑥,𝑦,𝑧,𝐹   𝑥,𝐴,𝑦,𝑧

Proof of Theorem hlimi
Dummy variables 𝑤 𝑓 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-hlim 28385 . . . 4 𝑣 = {⟨𝑓, 𝑤⟩ ∣ ((𝑓:ℕ⟶ ℋ ∧ 𝑤 ∈ ℋ) ∧ ∀𝑥 ∈ ℝ+𝑦 ∈ ℕ ∀𝑧 ∈ (ℤ𝑦)(norm‘((𝑓𝑧) − 𝑤)) < 𝑥)}
21relopabi 5479 . . 3 Rel ⇝𝑣
32brrelex1i 5394 . 2 (𝐹𝑣 𝐴𝐹 ∈ V)
4 nnex 11358 . . . 4 ℕ ∈ V
5 fex 6746 . . . 4 ((𝐹:ℕ⟶ ℋ ∧ ℕ ∈ V) → 𝐹 ∈ V)
64, 5mpan2 684 . . 3 (𝐹:ℕ⟶ ℋ → 𝐹 ∈ V)
76ad2antrr 719 . 2 (((𝐹:ℕ⟶ ℋ ∧ 𝐴 ∈ ℋ) ∧ ∀𝑥 ∈ ℝ+𝑦 ∈ ℕ ∀𝑧 ∈ (ℤ𝑦)(norm‘((𝐹𝑧) − 𝐴)) < 𝑥) → 𝐹 ∈ V)
8 hlim.1 . . 3 𝐴 ∈ V
9 feq1 6260 . . . . . 6 (𝑓 = 𝐹 → (𝑓:ℕ⟶ ℋ ↔ 𝐹:ℕ⟶ ℋ))
10 eleq1 2895 . . . . . 6 (𝑤 = 𝐴 → (𝑤 ∈ ℋ ↔ 𝐴 ∈ ℋ))
119, 10bi2anan9 631 . . . . 5 ((𝑓 = 𝐹𝑤 = 𝐴) → ((𝑓:ℕ⟶ ℋ ∧ 𝑤 ∈ ℋ) ↔ (𝐹:ℕ⟶ ℋ ∧ 𝐴 ∈ ℋ)))
12 fveq1 6433 . . . . . . . . . 10 (𝑓 = 𝐹 → (𝑓𝑧) = (𝐹𝑧))
13 oveq12 6915 . . . . . . . . . 10 (((𝑓𝑧) = (𝐹𝑧) ∧ 𝑤 = 𝐴) → ((𝑓𝑧) − 𝑤) = ((𝐹𝑧) − 𝐴))
1412, 13sylan 577 . . . . . . . . 9 ((𝑓 = 𝐹𝑤 = 𝐴) → ((𝑓𝑧) − 𝑤) = ((𝐹𝑧) − 𝐴))
1514fveq2d 6438 . . . . . . . 8 ((𝑓 = 𝐹𝑤 = 𝐴) → (norm‘((𝑓𝑧) − 𝑤)) = (norm‘((𝐹𝑧) − 𝐴)))
1615breq1d 4884 . . . . . . 7 ((𝑓 = 𝐹𝑤 = 𝐴) → ((norm‘((𝑓𝑧) − 𝑤)) < 𝑥 ↔ (norm‘((𝐹𝑧) − 𝐴)) < 𝑥))
1716rexralbidv 3269 . . . . . 6 ((𝑓 = 𝐹𝑤 = 𝐴) → (∃𝑦 ∈ ℕ ∀𝑧 ∈ (ℤ𝑦)(norm‘((𝑓𝑧) − 𝑤)) < 𝑥 ↔ ∃𝑦 ∈ ℕ ∀𝑧 ∈ (ℤ𝑦)(norm‘((𝐹𝑧) − 𝐴)) < 𝑥))
1817ralbidv 3196 . . . . 5 ((𝑓 = 𝐹𝑤 = 𝐴) → (∀𝑥 ∈ ℝ+𝑦 ∈ ℕ ∀𝑧 ∈ (ℤ𝑦)(norm‘((𝑓𝑧) − 𝑤)) < 𝑥 ↔ ∀𝑥 ∈ ℝ+𝑦 ∈ ℕ ∀𝑧 ∈ (ℤ𝑦)(norm‘((𝐹𝑧) − 𝐴)) < 𝑥))
1911, 18anbi12d 626 . . . 4 ((𝑓 = 𝐹𝑤 = 𝐴) → (((𝑓:ℕ⟶ ℋ ∧ 𝑤 ∈ ℋ) ∧ ∀𝑥 ∈ ℝ+𝑦 ∈ ℕ ∀𝑧 ∈ (ℤ𝑦)(norm‘((𝑓𝑧) − 𝑤)) < 𝑥) ↔ ((𝐹:ℕ⟶ ℋ ∧ 𝐴 ∈ ℋ) ∧ ∀𝑥 ∈ ℝ+𝑦 ∈ ℕ ∀𝑧 ∈ (ℤ𝑦)(norm‘((𝐹𝑧) − 𝐴)) < 𝑥)))
2019, 1brabga 5216 . . 3 ((𝐹 ∈ V ∧ 𝐴 ∈ V) → (𝐹𝑣 𝐴 ↔ ((𝐹:ℕ⟶ ℋ ∧ 𝐴 ∈ ℋ) ∧ ∀𝑥 ∈ ℝ+𝑦 ∈ ℕ ∀𝑧 ∈ (ℤ𝑦)(norm‘((𝐹𝑧) − 𝐴)) < 𝑥)))
218, 20mpan2 684 . 2 (𝐹 ∈ V → (𝐹𝑣 𝐴 ↔ ((𝐹:ℕ⟶ ℋ ∧ 𝐴 ∈ ℋ) ∧ ∀𝑥 ∈ ℝ+𝑦 ∈ ℕ ∀𝑧 ∈ (ℤ𝑦)(norm‘((𝐹𝑧) − 𝐴)) < 𝑥)))
223, 7, 21pm5.21nii 370 1 (𝐹𝑣 𝐴 ↔ ((𝐹:ℕ⟶ ℋ ∧ 𝐴 ∈ ℋ) ∧ ∀𝑥 ∈ ℝ+𝑦 ∈ ℕ ∀𝑧 ∈ (ℤ𝑦)(norm‘((𝐹𝑧) − 𝐴)) < 𝑥))
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 198   ∧ wa 386   = wceq 1658   ∈ wcel 2166  ∀wral 3118  ∃wrex 3119  Vcvv 3415   class class class wbr 4874  ⟶wf 6120  ‘cfv 6124  (class class class)co 6906   < clt 10392  ℕcn 11351  ℤ≥cuz 11969  ℝ+crp 12113   ℋchba 28332  normℎcno 28336   −ℎ cmv 28338   ⇝𝑣 chli 28340 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-8 2168  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2391  ax-ext 2804  ax-rep 4995  ax-sep 5006  ax-nul 5014  ax-pow 5066  ax-pr 5128  ax-un 7210  ax-cnex 10309  ax-1cn 10311  ax-addcl 10313 This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3or 1114  df-3an 1115  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-mo 2606  df-eu 2641  df-clab 2813  df-cleq 2819  df-clel 2822  df-nfc 2959  df-ne 3001  df-ral 3123  df-rex 3124  df-reu 3125  df-rab 3127  df-v 3417  df-sbc 3664  df-csb 3759  df-dif 3802  df-un 3804  df-in 3806  df-ss 3813  df-pss 3815  df-nul 4146  df-if 4308  df-pw 4381  df-sn 4399  df-pr 4401  df-tp 4403  df-op 4405  df-uni 4660  df-iun 4743  df-br 4875  df-opab 4937  df-mpt 4954  df-tr 4977  df-id 5251  df-eprel 5256  df-po 5264  df-so 5265  df-fr 5302  df-we 5304  df-xp 5349  df-rel 5350  df-cnv 5351  df-co 5352  df-dm 5353  df-rn 5354  df-res 5355  df-ima 5356  df-pred 5921  df-ord 5967  df-on 5968  df-lim 5969  df-suc 5970  df-iota 6087  df-fun 6126  df-fn 6127  df-f 6128  df-f1 6129  df-fo 6130  df-f1o 6131  df-fv 6132  df-ov 6909  df-om 7328  df-wrecs 7673  df-recs 7735  df-rdg 7773  df-nn 11352  df-hlim 28385 This theorem is referenced by:  hlimseqi  28602  hlimveci  28603  hlimconvi  28604  hlim2  28605
 Copyright terms: Public domain W3C validator