![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 | β’ ((π· β (βMetβπ) β§ πΉ β (Cauβπ·)) β πΉ β (π βpm β)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | iscau 24663 | . 2 β’ (π· β (βMetβπ) β (πΉ β (Cauβπ·) β (πΉ β (π βpm β) β§ βπ₯ β β+ βπ¦ β β€ (πΉ βΎ (β€β₯βπ¦)):(β€β₯βπ¦)βΆ((πΉβπ¦)(ballβπ·)π₯)))) | |
2 | 1 | simprbda 500 | 1 β’ ((π· β (βMetβπ) β§ πΉ β (Cauβπ·)) β πΉ β (π βpm β)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 397 β wcel 2107 βwral 3061 βwrex 3070 βΎ cres 5639 βΆwf 6496 βcfv 6500 (class class class)co 7361 βpm cpm 8772 βcc 11057 β€cz 12507 β€β₯cuz 12771 β+crp 12923 βMetcxmet 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 |
Copyright terms: Public domain | W3C validator |