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Theorem caufpm 24192
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 ∈ (∞Met‘𝑋) ∧ 𝐹 ∈ (Cau‘𝐷)) → 𝐹 ∈ (𝑋pm ℂ))

Proof of Theorem caufpm
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 iscau 24186 . 2 (𝐷 ∈ (∞Met‘𝑋) → (𝐹 ∈ (Cau‘𝐷) ↔ (𝐹 ∈ (𝑋pm ℂ) ∧ ∀𝑥 ∈ ℝ+𝑦 ∈ ℤ (𝐹 ↾ (ℤ𝑦)):(ℤ𝑦)⟶((𝐹𝑦)(ball‘𝐷)𝑥))))
21simprbda 502 1 ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝐹 ∈ (Cau‘𝐷)) → 𝐹 ∈ (𝑋pm ℂ))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399  wcel 2111  wral 3062  wrex 3063  cres 5562  wf 6385  cfv 6389  (class class class)co 7222  pm cpm 8518  cc 10740  cz 12189  cuz 12451  +crp 12599  ∞Metcxmet 20361  ballcbl 20363  Cauccau 24163
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2159  ax-12 2176  ax-ext 2709  ax-sep 5201  ax-nul 5208  ax-pow 5267  ax-pr 5331  ax-un 7532  ax-cnex 10798  ax-resscn 10799
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2072  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2887  df-ne 2942  df-ral 3067  df-rex 3068  df-rab 3071  df-v 3417  df-sbc 3704  df-csb 3821  df-dif 3878  df-un 3880  df-in 3882  df-ss 3892  df-nul 4247  df-if 4449  df-pw 4524  df-sn 4551  df-pr 4553  df-op 4557  df-uni 4829  df-br 5063  df-opab 5125  df-mpt 5145  df-id 5464  df-xp 5566  df-rel 5567  df-cnv 5568  df-co 5569  df-dm 5570  df-rn 5571  df-res 5572  df-iota 6347  df-fun 6391  df-fn 6392  df-f 6393  df-fv 6397  df-ov 7225  df-oprab 7226  df-mpo 7227  df-map 8519  df-xr 10884  df-xmet 20369  df-cau 24166
This theorem is referenced by:  cmetcaulem  24198  causs  24208
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