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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 23293 | . 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 486 | 1 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝐹 ∈ (Cau‘𝐷)) → 𝐹 ∈ (𝑋 ↑pm ℂ)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 382 ∈ wcel 2145 ∀wral 3061 ∃wrex 3062 ↾ cres 5251 ⟶wf 6027 ‘cfv 6031 (class class class)co 6793 ↑pm cpm 8010 ℂcc 10136 ℤcz 11579 ℤ≥cuz 11888 ℝ+crp 12035 ∞Metcxmt 19946 ballcbl 19948 Caucca 23270 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1870 ax-4 1885 ax-5 1991 ax-6 2057 ax-7 2093 ax-8 2147 ax-9 2154 ax-10 2174 ax-11 2190 ax-12 2203 ax-13 2408 ax-ext 2751 ax-sep 4915 ax-nul 4923 ax-pow 4974 ax-pr 5034 ax-un 7096 ax-cnex 10194 ax-resscn 10195 |
This theorem depends on definitions: df-bi 197 df-an 383 df-or 837 df-3an 1073 df-tru 1634 df-ex 1853 df-nf 1858 df-sb 2050 df-eu 2622 df-mo 2623 df-clab 2758 df-cleq 2764 df-clel 2767 df-nfc 2902 df-ne 2944 df-ral 3066 df-rex 3067 df-rab 3070 df-v 3353 df-sbc 3588 df-csb 3683 df-dif 3726 df-un 3728 df-in 3730 df-ss 3737 df-nul 4064 df-if 4226 df-pw 4299 df-sn 4317 df-pr 4319 df-op 4323 df-uni 4575 df-br 4787 df-opab 4847 df-mpt 4864 df-id 5157 df-xp 5255 df-rel 5256 df-cnv 5257 df-co 5258 df-dm 5259 df-rn 5260 df-res 5261 df-iota 5994 df-fun 6033 df-fn 6034 df-f 6035 df-fv 6039 df-ov 6796 df-oprab 6797 df-mpt2 6798 df-map 8011 df-xr 10280 df-xmet 19954 df-cau 23273 |
This theorem is referenced by: cmetcaulem 23305 causs 23315 |
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