Theorem iscau 22882

Description: Express the property "𝐹 is a Cauchy sequence of metric 𝐷." Part of Definition 1.4-3 of [Kreyszig] p. 28. The condition 𝐹 ⊆ (ℂ × 𝑋) allows us to use objects more general than sequences when convenient; see the comment in df-lm 20843. (Contributed by NM, 7-Dec-2006.) (Revised by Mario Carneiro, 14-Nov-2013.)

Assertion

Ref	Expression
iscau	⊢ (𝐷 ∈ (∞Met‘𝑋) → (𝐹 ∈ (Cau‘𝐷) ↔ (𝐹 ∈ (𝑋 ↑pm ℂ) ∧ ∀𝑥 ∈ ℝ+ ∃𝑘 ∈ ℤ (𝐹 ↾ (ℤ≥‘𝑘)):(ℤ≥‘𝑘)⟶((𝐹‘𝑘)(ball‘𝐷)𝑥))))

Distinct variable groups:   𝑥,𝑘,𝐷   𝑘,𝐹,𝑥   𝑘,𝑋,𝑥

Proof of Theorem iscau

Dummy variable 𝑓 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		caufval 22881	. . 3 ⊢ (𝐷 ∈ (∞Met‘𝑋) → (Cau‘𝐷) = {𝑓 ∈ (𝑋 ↑pm ℂ) ∣ ∀𝑥 ∈ ℝ+ ∃𝑘 ∈ ℤ (𝑓 ↾ (ℤ≥‘𝑘)):(ℤ≥‘𝑘)⟶((𝑓‘𝑘)(ball‘𝐷)𝑥)})
2	1	eleq2d 2673	. 2 ⊢ (𝐷 ∈ (∞Met‘𝑋) → (𝐹 ∈ (Cau‘𝐷) ↔ 𝐹 ∈ {𝑓 ∈ (𝑋 ↑pm ℂ) ∣ ∀𝑥 ∈ ℝ+ ∃𝑘 ∈ ℤ (𝑓 ↾ (ℤ≥‘𝑘)):(ℤ≥‘𝑘)⟶((𝑓‘𝑘)(ball‘𝐷)𝑥)}))
3		reseq1 5311	. . . . . 6 ⊢ (𝑓 = 𝐹 → (𝑓 ↾ (ℤ≥‘𝑘)) = (𝐹 ↾ (ℤ≥‘𝑘)))
4		eqidd 2611	. . . . . 6 ⊢ (𝑓 = 𝐹 → (ℤ≥‘𝑘) = (ℤ≥‘𝑘))
5		fveq1 6102	. . . . . . 7 ⊢ (𝑓 = 𝐹 → (𝑓‘𝑘) = (𝐹‘𝑘))
6	5	oveq1d 6564	. . . . . 6 ⊢ (𝑓 = 𝐹 → ((𝑓‘𝑘)(ball‘𝐷)𝑥) = ((𝐹‘𝑘)(ball‘𝐷)𝑥))
7	3, 4, 6	feq123d 5947	. . . . 5 ⊢ (𝑓 = 𝐹 → ((𝑓 ↾ (ℤ≥‘𝑘)):(ℤ≥‘𝑘)⟶((𝑓‘𝑘)(ball‘𝐷)𝑥) ↔ (𝐹 ↾ (ℤ≥‘𝑘)):(ℤ≥‘𝑘)⟶((𝐹‘𝑘)(ball‘𝐷)𝑥)))
8	7	rexbidv 3034	. . . 4 ⊢ (𝑓 = 𝐹 → (∃𝑘 ∈ ℤ (𝑓 ↾ (ℤ≥‘𝑘)):(ℤ≥‘𝑘)⟶((𝑓‘𝑘)(ball‘𝐷)𝑥) ↔ ∃𝑘 ∈ ℤ (𝐹 ↾ (ℤ≥‘𝑘)):(ℤ≥‘𝑘)⟶((𝐹‘𝑘)(ball‘𝐷)𝑥)))
9	8	ralbidv 2969	. . 3 ⊢ (𝑓 = 𝐹 → (∀𝑥 ∈ ℝ+ ∃𝑘 ∈ ℤ (𝑓 ↾ (ℤ≥‘𝑘)):(ℤ≥‘𝑘)⟶((𝑓‘𝑘)(ball‘𝐷)𝑥) ↔ ∀𝑥 ∈ ℝ+ ∃𝑘 ∈ ℤ (𝐹 ↾ (ℤ≥‘𝑘)):(ℤ≥‘𝑘)⟶((𝐹‘𝑘)(ball‘𝐷)𝑥)))
10	9	elrab 3331	. 2 ⊢ (𝐹 ∈ {𝑓 ∈ (𝑋 ↑pm ℂ) ∣ ∀𝑥 ∈ ℝ+ ∃𝑘 ∈ ℤ (𝑓 ↾ (ℤ≥‘𝑘)):(ℤ≥‘𝑘)⟶((𝑓‘𝑘)(ball‘𝐷)𝑥)} ↔ (𝐹 ∈ (𝑋 ↑pm ℂ) ∧ ∀𝑥 ∈ ℝ+ ∃𝑘 ∈ ℤ (𝐹 ↾ (ℤ≥‘𝑘)):(ℤ≥‘𝑘)⟶((𝐹‘𝑘)(ball‘𝐷)𝑥)))
11	2, 10	syl6bb 275	1 ⊢ (𝐷 ∈ (∞Met‘𝑋) → (𝐹 ∈ (Cau‘𝐷) ↔ (𝐹 ∈ (𝑋 ↑pm ℂ) ∧ ∀𝑥 ∈ ℝ+ ∃𝑘 ∈ ℤ (𝐹 ↾ (ℤ≥‘𝑘)):(ℤ≥‘𝑘)⟶((𝐹‘𝑘)(ball‘𝐷)𝑥))))

Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ∀wral 2896  ∃wrex 2897  {crab 2900   ↾ cres 5040  ⟶wf 5800  ‘cfv 5804  (class class class)co 6549   ↑_pm cpm 7745  ℂcc 9813  ℤcz 11254  ℤ_≥cuz 11563  ℝ⁺crp 11708  ∞Metcxmt 19552  ballcbl 19554  Caucca 22859

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-map 7746  df-xr 9957  df-xmet 19560  df-cau 22862

This theorem is referenced by:  iscau2  22883  caufpm  22888  lmcau  22919