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Theorem hlimi 31018
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 30802 . . . 4 𝑣 = {⟨𝑓, 𝑤⟩ ∣ ((𝑓:ℕ⟶ ℋ ∧ 𝑤 ∈ ℋ) ∧ ∀𝑥 ∈ ℝ+𝑦 ∈ ℕ ∀𝑧 ∈ (ℤ𝑦)(norm‘((𝑓𝑧) − 𝑤)) < 𝑥)}
21relopabiv 5826 . . 3 Rel ⇝𝑣
32brrelex1i 5738 . 2 (𝐹𝑣 𝐴𝐹 ∈ V)
4 nnex 12256 . . . 4 ℕ ∈ V
5 fex 7244 . . . 4 ((𝐹:ℕ⟶ ℋ ∧ ℕ ∈ V) → 𝐹 ∈ V)
64, 5mpan2 689 . . 3 (𝐹:ℕ⟶ ℋ → 𝐹 ∈ V)
76ad2antrr 724 . 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 6708 . . . . . 6 (𝑓 = 𝐹 → (𝑓:ℕ⟶ ℋ ↔ 𝐹:ℕ⟶ ℋ))
10 eleq1 2817 . . . . . 6 (𝑤 = 𝐴 → (𝑤 ∈ ℋ ↔ 𝐴 ∈ ℋ))
119, 10bi2anan9 636 . . . . 5 ((𝑓 = 𝐹𝑤 = 𝐴) → ((𝑓:ℕ⟶ ℋ ∧ 𝑤 ∈ ℋ) ↔ (𝐹:ℕ⟶ ℋ ∧ 𝐴 ∈ ℋ)))
12 fveq1 6901 . . . . . . . . . 10 (𝑓 = 𝐹 → (𝑓𝑧) = (𝐹𝑧))
13 oveq12 7435 . . . . . . . . . 10 (((𝑓𝑧) = (𝐹𝑧) ∧ 𝑤 = 𝐴) → ((𝑓𝑧) − 𝑤) = ((𝐹𝑧) − 𝐴))
1412, 13sylan 578 . . . . . . . . 9 ((𝑓 = 𝐹𝑤 = 𝐴) → ((𝑓𝑧) − 𝑤) = ((𝐹𝑧) − 𝐴))
1514fveq2d 6906 . . . . . . . 8 ((𝑓 = 𝐹𝑤 = 𝐴) → (norm‘((𝑓𝑧) − 𝑤)) = (norm‘((𝐹𝑧) − 𝐴)))
1615breq1d 5162 . . . . . . 7 ((𝑓 = 𝐹𝑤 = 𝐴) → ((norm‘((𝑓𝑧) − 𝑤)) < 𝑥 ↔ (norm‘((𝐹𝑧) − 𝐴)) < 𝑥))
1716rexralbidv 3218 . . . . . 6 ((𝑓 = 𝐹𝑤 = 𝐴) → (∃𝑦 ∈ ℕ ∀𝑧 ∈ (ℤ𝑦)(norm‘((𝑓𝑧) − 𝑤)) < 𝑥 ↔ ∃𝑦 ∈ ℕ ∀𝑧 ∈ (ℤ𝑦)(norm‘((𝐹𝑧) − 𝐴)) < 𝑥))
1817ralbidv 3175 . . . . 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 630 . . . 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 5540 . . 3 ((𝐹 ∈ V ∧ 𝐴 ∈ V) → (𝐹𝑣 𝐴 ↔ ((𝐹:ℕ⟶ ℋ ∧ 𝐴 ∈ ℋ) ∧ ∀𝑥 ∈ ℝ+𝑦 ∈ ℕ ∀𝑧 ∈ (ℤ𝑦)(norm‘((𝐹𝑧) − 𝐴)) < 𝑥)))
218, 20mpan2 689 . 2 (𝐹 ∈ V → (𝐹𝑣 𝐴 ↔ ((𝐹:ℕ⟶ ℋ ∧ 𝐴 ∈ ℋ) ∧ ∀𝑥 ∈ ℝ+𝑦 ∈ ℕ ∀𝑧 ∈ (ℤ𝑦)(norm‘((𝐹𝑧) − 𝐴)) < 𝑥)))
223, 7, 21pm5.21nii 377 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 205  wa 394   = wceq 1533  wcel 2098  wral 3058  wrex 3067  Vcvv 3473   class class class wbr 5152  wf 6549  cfv 6553  (class class class)co 7426   < clt 11286  cn 12250  cuz 12860  +crp 13014  chba 30749  normcno 30753   cmv 30755  𝑣 chli 30757
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2699  ax-rep 5289  ax-sep 5303  ax-nul 5310  ax-pr 5433  ax-un 7746  ax-cnex 11202  ax-1cn 11204  ax-addcl 11206
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-reu 3375  df-rab 3431  df-v 3475  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4327  df-if 4533  df-pw 4608  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-iun 5002  df-br 5153  df-opab 5215  df-mpt 5236  df-tr 5270  df-id 5580  df-eprel 5586  df-po 5594  df-so 5595  df-fr 5637  df-we 5639  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-pred 6310  df-ord 6377  df-on 6378  df-lim 6379  df-suc 6380  df-iota 6505  df-fun 6555  df-fn 6556  df-f 6557  df-f1 6558  df-fo 6559  df-f1o 6560  df-fv 6561  df-ov 7429  df-om 7877  df-2nd 8000  df-frecs 8293  df-wrecs 8324  df-recs 8398  df-rdg 8437  df-nn 12251  df-hlim 30802
This theorem is referenced by:  hlimseqi  31019  hlimveci  31020  hlimconvi  31021  hlim2  31022
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