Theorem hlimi 31277

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 31061	. . . 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 5674	. 2 ⊢ (𝐹 ⇝𝑣 𝐴 → 𝐹 ∈ V)
4		nnex 12171	. . . 4 ⊢ ℕ ∈ V
5		fex 7170	. . . 4 ⊢ ((𝐹:ℕ⟶ ℋ ∧ ℕ ∈ V) → 𝐹 ∈ V)
6	4, 5	mpan2 697	. . 3 ⊢ (𝐹:ℕ⟶ ℋ → 𝐹 ∈ V)
7	6	ad2antrr 732	. 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 6633	. . . . . 6 ⊢ (𝑓 = 𝐹 → (𝑓:ℕ⟶ ℋ ↔ 𝐹:ℕ⟶ ℋ))
10		eleq1 2827	. . . . . 6 ⊢ (𝑤 = 𝐴 → (𝑤 ∈ ℋ ↔ 𝐴 ∈ ℋ))
11	9, 10	bi2anan9 644	. . . . 5 ⊢ ((𝑓 = 𝐹 ∧ 𝑤 = 𝐴) → ((𝑓:ℕ⟶ ℋ ∧ 𝑤 ∈ ℋ) ↔ (𝐹:ℕ⟶ ℋ ∧ 𝐴 ∈ ℋ)))
12		fveq1 6826	. . . . . . . . . 10 ⊢ (𝑓 = 𝐹 → (𝑓‘𝑧) = (𝐹‘𝑧))
13		oveq12 7365	. . . . . . . . . 10 ⊢ (((𝑓‘𝑧) = (𝐹‘𝑧) ∧ 𝑤 = 𝐴) → ((𝑓‘𝑧) −ℎ 𝑤) = ((𝐹‘𝑧) −ℎ 𝐴))
14	12, 13	sylan 586	. . . . . . . . 9 ⊢ ((𝑓 = 𝐹 ∧ 𝑤 = 𝐴) → ((𝑓‘𝑧) −ℎ 𝑤) = ((𝐹‘𝑧) −ℎ 𝐴))
15	14	fveq2d 6831	. . . . . . . 8 ⊢ ((𝑓 = 𝐹 ∧ 𝑤 = 𝐴) → (normℎ‘((𝑓‘𝑧) −ℎ 𝑤)) = (normℎ‘((𝐹‘𝑧) −ℎ 𝐴)))
16	15	breq1d 5082	. . . . . . 7 ⊢ ((𝑓 = 𝐹 ∧ 𝑤 = 𝐴) → ((normℎ‘((𝑓‘𝑧) −ℎ 𝑤)) < 𝑥 ↔ (normℎ‘((𝐹‘𝑧) −ℎ 𝐴)) < 𝑥))
17	16	rexralbidv 3205	. . . . . 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 638	. . . 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 5476	. . 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 697	. 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 379	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 207   ∧ wa 396   = wceq 1547   ∈ wcel 2119  ∀wral 3053  ∃wrex 3063  Vcvv 3431   class class class wbr 5072  ⟶wf 6481  ‘cfv 6485  (class class class)co 7356   < clt 11170  ℕcn 12165  ℤ_≥cuz 12779  ℝ⁺crp 12933   ℋchba 31008  norm_ℎcno 31012   −_ℎ cmv 31014   ⇝_𝑣 chli 31016

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2711  ax-rep 5199  ax-sep 5218  ax-nul 5228  ax-pr 5362  ax-un 7678  ax-cnex 11085  ax-1cn 11087  ax-addcl 11089

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3or 1093  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2718  df-cleq 2731  df-clel 2814  df-nfc 2888  df-ne 2935  df-ral 3054  df-rex 3064  df-reu 3345  df-rab 3392  df-v 3433  df-sbc 3724  df-csb 3832  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3903  df-nul 4262  df-if 4455  df-pw 4531  df-sn 4556  df-pr 4558  df-op 4562  df-uni 4839  df-iun 4923  df-br 5073  df-opab 5135  df-mpt 5154  df-tr 5180  df-id 5513  df-eprel 5518  df-po 5526  df-so 5527  df-fr 5571  df-we 5573  df-xp 5624  df-rel 5625  df-cnv 5626  df-co 5627  df-dm 5628  df-rn 5629  df-res 5630  df-ima 5631  df-pred 6252  df-ord 6313  df-on 6314  df-lim 6315  df-suc 6316  df-iota 6441  df-fun 6487  df-fn 6488  df-f 6489  df-f1 6490  df-fo 6491  df-f1o 6492  df-fv 6493  df-ov 7359  df-om 7807  df-2nd 7932  df-frecs 8221  df-wrecs 8252  df-recs 8301  df-rdg 8339  df-nn 12166  df-hlim 31061

This theorem is referenced by:  hlimseqi  31278  hlimveci  31279  hlimconvi  31280  hlim2  31281