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Mirrors > Home > MPE Home > Th. List > caufpm | Structured version Visualization version GIF version |
Description: Inclusion of a Cauchy sequence, under our definition. (Contributed by NM, 7-Dec-2006.) (Revised by Mario Carneiro, 24-Dec-2013.) |
Ref | Expression |
---|---|
caufpm | ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝐹 ∈ (Cau‘𝐷)) → 𝐹 ∈ (𝑋 ↑pm ℂ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | iscau 24186 | . 2 ⊢ (𝐷 ∈ (∞Met‘𝑋) → (𝐹 ∈ (Cau‘𝐷) ↔ (𝐹 ∈ (𝑋 ↑pm ℂ) ∧ ∀𝑥 ∈ ℝ+ ∃𝑦 ∈ ℤ (𝐹 ↾ (ℤ≥‘𝑦)):(ℤ≥‘𝑦)⟶((𝐹‘𝑦)(ball‘𝐷)𝑥)))) | |
2 | 1 | simprbda 502 | 1 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝐹 ∈ (Cau‘𝐷)) → 𝐹 ∈ (𝑋 ↑pm 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 ℂcc 10740 ℤcz 12189 ℤ≥cuz 12451 ℝ+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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