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Theorem iscau 24663
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 βŠ† (β„‚ Γ— 𝑋) allows to use objects more general than sequences when convenient; see the comment in df-lm 22603. (Contributed by NM, 7-Dec-2006.) (Revised by Mario Carneiro, 14-Nov-2013.)
Assertion
Ref Expression
iscau (𝐷 ∈ (∞Metβ€˜π‘‹) β†’ (𝐹 ∈ (Cauβ€˜π·) ↔ (𝐹 ∈ (𝑋 ↑pm β„‚) ∧ βˆ€π‘₯ ∈ ℝ+ βˆƒπ‘˜ ∈ β„€ (𝐹 β†Ύ (β„€β‰₯β€˜π‘˜)):(β„€β‰₯β€˜π‘˜)⟢((πΉβ€˜π‘˜)(ballβ€˜π·)π‘₯))))
Distinct variable groups:   π‘₯,π‘˜,𝐷   π‘˜,𝐹,π‘₯   π‘˜,𝑋,π‘₯

Proof of Theorem iscau
Dummy variable 𝑓 is distinct from all other variables.
StepHypRef Expression
1 caufval 24662 . . 3 (𝐷 ∈ (∞Metβ€˜π‘‹) β†’ (Cauβ€˜π·) = {𝑓 ∈ (𝑋 ↑pm β„‚) ∣ βˆ€π‘₯ ∈ ℝ+ βˆƒπ‘˜ ∈ β„€ (𝑓 β†Ύ (β„€β‰₯β€˜π‘˜)):(β„€β‰₯β€˜π‘˜)⟢((π‘“β€˜π‘˜)(ballβ€˜π·)π‘₯)})
21eleq2d 2820 . 2 (𝐷 ∈ (∞Metβ€˜π‘‹) β†’ (𝐹 ∈ (Cauβ€˜π·) ↔ 𝐹 ∈ {𝑓 ∈ (𝑋 ↑pm β„‚) ∣ βˆ€π‘₯ ∈ ℝ+ βˆƒπ‘˜ ∈ β„€ (𝑓 β†Ύ (β„€β‰₯β€˜π‘˜)):(β„€β‰₯β€˜π‘˜)⟢((π‘“β€˜π‘˜)(ballβ€˜π·)π‘₯)}))
3 reseq1 5935 . . . . . 6 (𝑓 = 𝐹 β†’ (𝑓 β†Ύ (β„€β‰₯β€˜π‘˜)) = (𝐹 β†Ύ (β„€β‰₯β€˜π‘˜)))
4 eqidd 2734 . . . . . 6 (𝑓 = 𝐹 β†’ (β„€β‰₯β€˜π‘˜) = (β„€β‰₯β€˜π‘˜))
5 fveq1 6845 . . . . . . 7 (𝑓 = 𝐹 β†’ (π‘“β€˜π‘˜) = (πΉβ€˜π‘˜))
65oveq1d 7376 . . . . . 6 (𝑓 = 𝐹 β†’ ((π‘“β€˜π‘˜)(ballβ€˜π·)π‘₯) = ((πΉβ€˜π‘˜)(ballβ€˜π·)π‘₯))
73, 4, 6feq123d 6661 . . . . 5 (𝑓 = 𝐹 β†’ ((𝑓 β†Ύ (β„€β‰₯β€˜π‘˜)):(β„€β‰₯β€˜π‘˜)⟢((π‘“β€˜π‘˜)(ballβ€˜π·)π‘₯) ↔ (𝐹 β†Ύ (β„€β‰₯β€˜π‘˜)):(β„€β‰₯β€˜π‘˜)⟢((πΉβ€˜π‘˜)(ballβ€˜π·)π‘₯)))
87rexbidv 3172 . . . 4 (𝑓 = 𝐹 β†’ (βˆƒπ‘˜ ∈ β„€ (𝑓 β†Ύ (β„€β‰₯β€˜π‘˜)):(β„€β‰₯β€˜π‘˜)⟢((π‘“β€˜π‘˜)(ballβ€˜π·)π‘₯) ↔ βˆƒπ‘˜ ∈ β„€ (𝐹 β†Ύ (β„€β‰₯β€˜π‘˜)):(β„€β‰₯β€˜π‘˜)⟢((πΉβ€˜π‘˜)(ballβ€˜π·)π‘₯)))
98ralbidv 3171 . . 3 (𝑓 = 𝐹 β†’ (βˆ€π‘₯ ∈ ℝ+ βˆƒπ‘˜ ∈ β„€ (𝑓 β†Ύ (β„€β‰₯β€˜π‘˜)):(β„€β‰₯β€˜π‘˜)⟢((π‘“β€˜π‘˜)(ballβ€˜π·)π‘₯) ↔ βˆ€π‘₯ ∈ ℝ+ βˆƒπ‘˜ ∈ β„€ (𝐹 β†Ύ (β„€β‰₯β€˜π‘˜)):(β„€β‰₯β€˜π‘˜)⟢((πΉβ€˜π‘˜)(ballβ€˜π·)π‘₯)))
109elrab 3649 . 2 (𝐹 ∈ {𝑓 ∈ (𝑋 ↑pm β„‚) ∣ βˆ€π‘₯ ∈ ℝ+ βˆƒπ‘˜ ∈ β„€ (𝑓 β†Ύ (β„€β‰₯β€˜π‘˜)):(β„€β‰₯β€˜π‘˜)⟢((π‘“β€˜π‘˜)(ballβ€˜π·)π‘₯)} ↔ (𝐹 ∈ (𝑋 ↑pm β„‚) ∧ βˆ€π‘₯ ∈ ℝ+ βˆƒπ‘˜ ∈ β„€ (𝐹 β†Ύ (β„€β‰₯β€˜π‘˜)):(β„€β‰₯β€˜π‘˜)⟢((πΉβ€˜π‘˜)(ballβ€˜π·)π‘₯)))
112, 10bitrdi 287 1 (𝐷 ∈ (∞Metβ€˜π‘‹) β†’ (𝐹 ∈ (Cauβ€˜π·) ↔ (𝐹 ∈ (𝑋 ↑pm β„‚) ∧ βˆ€π‘₯ ∈ ℝ+ βˆƒπ‘˜ ∈ β„€ (𝐹 β†Ύ (β„€β‰₯β€˜π‘˜)):(β„€β‰₯β€˜π‘˜)⟢((πΉβ€˜π‘˜)(ballβ€˜π·)π‘₯))))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 397   = wceq 1542   ∈ wcel 2107  βˆ€wral 3061  βˆƒwrex 3070  {crab 3406   β†Ύ cres 5639  βŸΆwf 6496  β€˜cfv 6500  (class class class)co 7361   ↑pm cpm 8772  β„‚cc 11057  β„€cz 12507  β„€β‰₯cuz 12771  β„+crp 12923  βˆžMetcxmet 20804  ballcbl 20806  Cauccau 24640
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5260  ax-nul 5267  ax-pow 5324  ax-pr 5388  ax-un 7676  ax-cnex 11115  ax-resscn 11116
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3449  df-sbc 3744  df-csb 3860  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4287  df-if 4491  df-pw 4566  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-br 5110  df-opab 5172  df-mpt 5193  df-id 5535  df-xp 5643  df-rel 5644  df-cnv 5645  df-co 5646  df-dm 5647  df-rn 5648  df-res 5649  df-iota 6452  df-fun 6502  df-fn 6503  df-f 6504  df-fv 6508  df-ov 7364  df-oprab 7365  df-mpo 7366  df-map 8773  df-xr 11201  df-xmet 20812  df-cau 24643
This theorem is referenced by:  iscau2  24664  caufpm  24669  lmcau  24700
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