Theorem hlimi 31132

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 30916	. . . 4 ⊢ ⇝𝑣 = {⟨𝑓, 𝑤⟩ ∣ ((𝑓:ℕ⟶ ℋ ∧ 𝑤 ∈ ℋ) ∧ ∀𝑥 ∈ ℝ+ ∃𝑦 ∈ ℕ ∀𝑧 ∈ (ℤ≥‘𝑦)(normℎ‘((𝑓‘𝑧) −ℎ 𝑤)) < 𝑥)}
2	1	relopabiv 5763	. . 3 ⊢ Rel ⇝𝑣
3	2	brrelex1i 5675	. 2 ⊢ (𝐹 ⇝𝑣 𝐴 → 𝐹 ∈ V)
4		nnex 12134	. . . 4 ⊢ ℕ ∈ V
5		fex 7162	. . . 4 ⊢ ((𝐹:ℕ⟶ ℋ ∧ ℕ ∈ V) → 𝐹 ∈ V)
6	4, 5	mpan2 691	. . 3 ⊢ (𝐹:ℕ⟶ ℋ → 𝐹 ∈ V)
7	6	ad2antrr 726	. 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 6630	. . . . . 6 ⊢ (𝑓 = 𝐹 → (𝑓:ℕ⟶ ℋ ↔ 𝐹:ℕ⟶ ℋ))
10		eleq1 2816	. . . . . 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 6821	. . . . . . . . . 10 ⊢ (𝑓 = 𝐹 → (𝑓‘𝑧) = (𝐹‘𝑧))
13		oveq12 7358	. . . . . . . . . 10 ⊢ (((𝑓‘𝑧) = (𝐹‘𝑧) ∧ 𝑤 = 𝐴) → ((𝑓‘𝑧) −ℎ 𝑤) = ((𝐹‘𝑧) −ℎ 𝐴))
14	12, 13	sylan 580	. . . . . . . . 9 ⊢ ((𝑓 = 𝐹 ∧ 𝑤 = 𝐴) → ((𝑓‘𝑧) −ℎ 𝑤) = ((𝐹‘𝑧) −ℎ 𝐴))
15	14	fveq2d 6826	. . . . . . . 8 ⊢ ((𝑓 = 𝐹 ∧ 𝑤 = 𝐴) → (normℎ‘((𝑓‘𝑧) −ℎ 𝑤)) = (normℎ‘((𝐹‘𝑧) −ℎ 𝐴)))
16	15	breq1d 5102	. . . . . . 7 ⊢ ((𝑓 = 𝐹 ∧ 𝑤 = 𝐴) → ((normℎ‘((𝑓‘𝑧) −ℎ 𝑤)) < 𝑥 ↔ (normℎ‘((𝐹‘𝑧) −ℎ 𝐴)) < 𝑥))
17	16	rexralbidv 3195	. . . . . 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 3152	. . . . 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 632	. . . 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 5477	. . 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 691	. 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 378	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 206   ∧ wa 395   = wceq 1540   ∈ wcel 2109  ∀wral 3044  ∃wrex 3053  Vcvv 3436   class class class wbr 5092  ⟶wf 6478  ‘cfv 6482  (class class class)co 7349   < clt 11149  ℕcn 12128  ℤ_≥cuz 12735  ℝ⁺crp 12893   ℋchba 30863  norm_ℎcno 30867   −_ℎ cmv 30869   ⇝_𝑣 chli 30871

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5218  ax-sep 5235  ax-nul 5245  ax-pr 5371  ax-un 7671  ax-cnex 11065  ax-1cn 11067  ax-addcl 11069

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-reu 3344  df-rab 3395  df-v 3438  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4285  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4859  df-iun 4943  df-br 5093  df-opab 5155  df-mpt 5174  df-tr 5200  df-id 5514  df-eprel 5519  df-po 5527  df-so 5528  df-fr 5572  df-we 5574  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-pred 6249  df-ord 6310  df-on 6311  df-lim 6312  df-suc 6313  df-iota 6438  df-fun 6484  df-fn 6485  df-f 6486  df-f1 6487  df-fo 6488  df-f1o 6489  df-fv 6490  df-ov 7352  df-om 7800  df-2nd 7925  df-frecs 8214  df-wrecs 8245  df-recs 8294  df-rdg 8332  df-nn 12129  df-hlim 30916

This theorem is referenced by:  hlimseqi  31133  hlimveci  31134  hlimconvi  31135  hlim2  31136