Theorem hlimi 28971

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	⊢ (𝐹 ⇝𝑣 𝐴 ↔ ((𝐹:ℕ⟶ ℋ ∧ 𝐴 ∈ ℋ) ∧ ∀𝑥 ∈ ℝ+ ∃𝑦 ∈ ℕ ∀𝑧 ∈ (ℤ≥‘𝑦)(normℎ‘((𝐹‘𝑧) −ℎ 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.

Step	Hyp	Ref	Expression
1		df-hlim 28755	. . . 4 ⊢ ⇝𝑣 = {⟨𝑓, 𝑤⟩ ∣ ((𝑓:ℕ⟶ ℋ ∧ 𝑤 ∈ ℋ) ∧ ∀𝑥 ∈ ℝ+ ∃𝑦 ∈ ℕ ∀𝑧 ∈ (ℤ≥‘𝑦)(normℎ‘((𝑓‘𝑧) −ℎ 𝑤)) < 𝑥)}
2	1	relopabi 5658	. . 3 ⊢ Rel ⇝𝑣
3	2	brrelex1i 5572	. 2 ⊢ (𝐹 ⇝𝑣 𝐴 → 𝐹 ∈ V)
4		nnex 11631	. . . 4 ⊢ ℕ ∈ V
5		fex 6966	. . . 4 ⊢ ((𝐹:ℕ⟶ ℋ ∧ ℕ ∈ V) → 𝐹 ∈ V)
6	4, 5	mpan2 690	. . 3 ⊢ (𝐹:ℕ⟶ ℋ → 𝐹 ∈ V)
7	6	ad2antrr 725	. 2 ⊢ (((𝐹:ℕ⟶ ℋ ∧ 𝐴 ∈ ℋ) ∧ ∀𝑥 ∈ ℝ+ ∃𝑦 ∈ ℕ ∀𝑧 ∈ (ℤ≥‘𝑦)(normℎ‘((𝐹‘𝑧) −ℎ 𝐴)) < 𝑥) → 𝐹 ∈ V)
8		hlim.1	. . 3 ⊢ 𝐴 ∈ V
9		feq1 6468	. . . . . 6 ⊢ (𝑓 = 𝐹 → (𝑓:ℕ⟶ ℋ ↔ 𝐹:ℕ⟶ ℋ))
10		eleq1 2877	. . . . . 6 ⊢ (𝑤 = 𝐴 → (𝑤 ∈ ℋ ↔ 𝐴 ∈ ℋ))
11	9, 10	bi2anan9 638	. . . . 5 ⊢ ((𝑓 = 𝐹 ∧ 𝑤 = 𝐴) → ((𝑓:ℕ⟶ ℋ ∧ 𝑤 ∈ ℋ) ↔ (𝐹:ℕ⟶ ℋ ∧ 𝐴 ∈ ℋ)))
12		fveq1 6644	. . . . . . . . . 10 ⊢ (𝑓 = 𝐹 → (𝑓‘𝑧) = (𝐹‘𝑧))
13		oveq12 7144	. . . . . . . . . 10 ⊢ (((𝑓‘𝑧) = (𝐹‘𝑧) ∧ 𝑤 = 𝐴) → ((𝑓‘𝑧) −ℎ 𝑤) = ((𝐹‘𝑧) −ℎ 𝐴))
14	12, 13	sylan 583	. . . . . . . . 9 ⊢ ((𝑓 = 𝐹 ∧ 𝑤 = 𝐴) → ((𝑓‘𝑧) −ℎ 𝑤) = ((𝐹‘𝑧) −ℎ 𝐴))
15	14	fveq2d 6649	. . . . . . . 8 ⊢ ((𝑓 = 𝐹 ∧ 𝑤 = 𝐴) → (normℎ‘((𝑓‘𝑧) −ℎ 𝑤)) = (normℎ‘((𝐹‘𝑧) −ℎ 𝐴)))
16	15	breq1d 5040	. . . . . . 7 ⊢ ((𝑓 = 𝐹 ∧ 𝑤 = 𝐴) → ((normℎ‘((𝑓‘𝑧) −ℎ 𝑤)) < 𝑥 ↔ (normℎ‘((𝐹‘𝑧) −ℎ 𝐴)) < 𝑥))
17	16	rexralbidv 3260	. . . . . 6 ⊢ ((𝑓 = 𝐹 ∧ 𝑤 = 𝐴) → (∃𝑦 ∈ ℕ ∀𝑧 ∈ (ℤ≥‘𝑦)(normℎ‘((𝑓‘𝑧) −ℎ 𝑤)) < 𝑥 ↔ ∃𝑦 ∈ ℕ ∀𝑧 ∈ (ℤ≥‘𝑦)(normℎ‘((𝐹‘𝑧) −ℎ 𝐴)) < 𝑥))
18	17	ralbidv 3162	. . . . 5 ⊢ ((𝑓 = 𝐹 ∧ 𝑤 = 𝐴) → (∀𝑥 ∈ ℝ+ ∃𝑦 ∈ ℕ ∀𝑧 ∈ (ℤ≥‘𝑦)(normℎ‘((𝑓‘𝑧) −ℎ 𝑤)) < 𝑥 ↔ ∀𝑥 ∈ ℝ+ ∃𝑦 ∈ ℕ ∀𝑧 ∈ (ℤ≥‘𝑦)(normℎ‘((𝐹‘𝑧) −ℎ 𝐴)) < 𝑥))
19	11, 18	anbi12d 633	. . . 4 ⊢ ((𝑓 = 𝐹 ∧ 𝑤 = 𝐴) → (((𝑓:ℕ⟶ ℋ ∧ 𝑤 ∈ ℋ) ∧ ∀𝑥 ∈ ℝ+ ∃𝑦 ∈ ℕ ∀𝑧 ∈ (ℤ≥‘𝑦)(normℎ‘((𝑓‘𝑧) −ℎ 𝑤)) < 𝑥) ↔ ((𝐹:ℕ⟶ ℋ ∧ 𝐴 ∈ ℋ) ∧ ∀𝑥 ∈ ℝ+ ∃𝑦 ∈ ℕ ∀𝑧 ∈ (ℤ≥‘𝑦)(normℎ‘((𝐹‘𝑧) −ℎ 𝐴)) < 𝑥)))
20	19, 1	brabga 5386	. . 3 ⊢ ((𝐹 ∈ V ∧ 𝐴 ∈ V) → (𝐹 ⇝𝑣 𝐴 ↔ ((𝐹:ℕ⟶ ℋ ∧ 𝐴 ∈ ℋ) ∧ ∀𝑥 ∈ ℝ+ ∃𝑦 ∈ ℕ ∀𝑧 ∈ (ℤ≥‘𝑦)(normℎ‘((𝐹‘𝑧) −ℎ 𝐴)) < 𝑥)))
21	8, 20	mpan2 690	. 2 ⊢ (𝐹 ∈ V → (𝐹 ⇝𝑣 𝐴 ↔ ((𝐹:ℕ⟶ ℋ ∧ 𝐴 ∈ ℋ) ∧ ∀𝑥 ∈ ℝ+ ∃𝑦 ∈ ℕ ∀𝑧 ∈ (ℤ≥‘𝑦)(normℎ‘((𝐹‘𝑧) −ℎ 𝐴)) < 𝑥)))
22	3, 7, 21	pm5.21nii 383	1 ⊢ (𝐹 ⇝𝑣 𝐴 ↔ ((𝐹:ℕ⟶ ℋ ∧ 𝐴 ∈ ℋ) ∧ ∀𝑥 ∈ ℝ+ ∃𝑦 ∈ ℕ ∀𝑧 ∈ (ℤ≥‘𝑦)(normℎ‘((𝐹‘𝑧) −ℎ 𝐴)) < 𝑥))

Syntax hints:   ↔ wb 209   ∧ wa 399   = wceq 1538   ∈ wcel 2111  ∀wral 3106  ∃wrex 3107  Vcvv 3441   class class class wbr 5030  ⟶wf 6320  ‘cfv 6324  (class class class)co 7135   < clt 10664  ℕcn 11625  ℤ_≥cuz 12231  ℝ⁺crp 12377   ℋchba 28702  norm_ℎcno 28706   −_ℎ cmv 28708   ⇝_𝑣 chli 28710

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 2770  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441  ax-cnex 10582  ax-1cn 10584  ax-addcl 10586

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-reu 3113  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-pss 3900  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4801  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-tr 5137  df-id 5425  df-eprel 5430  df-po 5438  df-so 5439  df-fr 5478  df-we 5480  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-pred 6116  df-ord 6162  df-on 6163  df-lim 6164  df-suc 6165  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332  df-ov 7138  df-om 7561  df-wrecs 7930  df-recs 7991  df-rdg 8029  df-nn 11626  df-hlim 28755

This theorem is referenced by:  hlimseqi  28972  hlimveci  28973  hlimconvi  28974  hlim2  28975