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Theorem caufpm 24669
Description: Inclusion of a Cauchy sequence, under our definition. (Contributed by NM, 7-Dec-2006.) (Revised by Mario Carneiro, 24-Dec-2013.)
Assertion
Ref Expression
caufpm ((𝐷 ∈ (∞Metβ€˜π‘‹) ∧ 𝐹 ∈ (Cauβ€˜π·)) β†’ 𝐹 ∈ (𝑋 ↑pm β„‚))

Proof of Theorem caufpm
Dummy variables π‘₯ 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 iscau 24663 . 2 (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‘₯))))
21simprbda 500 1 ((𝐷 ∈ (∞Metβ€˜π‘‹) ∧ 𝐹 ∈ (Cauβ€˜π·)) β†’ 𝐹 ∈ (𝑋 ↑pm β„‚))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 397   ∈ wcel 2107  βˆ€wral 3061  βˆƒwrex 3070   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:  cmetcaulem  24675  causs  24685
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