Theorem iscau 25598

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 to use objects more general than sequences when convenient; see the comment in df-lm 23548. (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 25597	. . 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 2847	. 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 5964	. . . . . 6 ⊢ (𝑓 = 𝐹 → (𝑓 ↾ (ℤ≥‘𝑘)) = (𝐹 ↾ (ℤ≥‘𝑘)))
4		eqidd 2762	. . . . . 6 ⊢ (𝑓 = 𝐹 → (ℤ≥‘𝑘) = (ℤ≥‘𝑘))
5		fveq1 6885	. . . . . . 7 ⊢ (𝑓 = 𝐹 → (𝑓‘𝑘) = (𝐹‘𝑘))
6	5	oveq1d 7436	. . . . . 6 ⊢ (𝑓 = 𝐹 → ((𝑓‘𝑘)(ball‘𝐷)𝑥) = ((𝐹‘𝑘)(ball‘𝐷)𝑥))
7	3, 4, 6	feq123d 6699	. . . . 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 3187	. . . 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 3186	. . 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 3645	. 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	bitrdi 290	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))))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2145  ∀wral 3077  ∃wrex 3087  {crab 3413   ↾ cres 5653  ⟶wf 6535  ‘cfv 6539  (class class class)co 7421   ↑_pm cpm 8849  ℂcc 11199  ℤcz 12694  ℤ_≥cuz 12966  ℝ⁺crp 13121  ∞Metcxmet 21664  ballcbl 21666  Cauccau 25575

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391  ax-un 7752  ax-cnex 11257  ax-resscn 11258

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3900  df-un 3902  df-in 3904  df-ss 3915  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5104  df-opab 5168  df-mpt 5187  df-id 5546  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-iota 6495  df-fun 6541  df-fn 6542  df-f 6543  df-fv 6547  df-ov 7424  df-oprab 7425  df-mpo 7426  df-map 8850  df-xr 11348  df-xmet 21672  df-cau 25578

This theorem is used by:  iscau2  25599  caufpm  25604  lmcau  25635