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Mirrors > Home > MPE Home > Th. List > iscau | Structured version Visualization version GIF version |
Description: Express the property "πΉ is a Cauchy sequence of metric π·". Part of Definition 1.4-3 of [Kreyszig] p. 28. The condition πΉ β (β Γ π) allows to use objects more general than sequences when convenient; see the comment in df-lm 22603. (Contributed by NM, 7-Dec-2006.) (Revised by Mario Carneiro, 14-Nov-2013.) |
Ref | Expression |
---|---|
iscau | 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₯)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | caufval 24662 | . . 3 β’ (π· β (βMetβπ) β (Cauβπ·) = {π β (π βpm β) β£ βπ₯ β β+ βπ β β€ (π βΎ (β€β₯βπ)):(β€β₯βπ)βΆ((πβπ)(ballβπ·)π₯)}) | |
2 | 1 | eleq2d 2820 | . 2 β’ (π· β (βMetβπ) β (πΉ β (Cauβπ·) β πΉ β {π β (π βpm β) β£ βπ₯ β β+ βπ β β€ (π βΎ (β€β₯βπ)):(β€β₯βπ)βΆ((πβπ)(ballβπ·)π₯)})) |
3 | reseq1 5935 | . . . . . 6 β’ (π = πΉ β (π βΎ (β€β₯βπ)) = (πΉ βΎ (β€β₯βπ))) | |
4 | eqidd 2734 | . . . . . 6 β’ (π = πΉ β (β€β₯βπ) = (β€β₯βπ)) | |
5 | fveq1 6845 | . . . . . . 7 β’ (π = πΉ β (πβπ) = (πΉβπ)) | |
6 | 5 | oveq1d 7376 | . . . . . 6 β’ (π = πΉ β ((πβπ)(ballβπ·)π₯) = ((πΉβπ)(ballβπ·)π₯)) |
7 | 3, 4, 6 | feq123d 6661 | . . . . 5 β’ (π = πΉ β ((π βΎ (β€β₯βπ)):(β€β₯βπ)βΆ((πβπ)(ballβπ·)π₯) β (πΉ βΎ (β€β₯βπ)):(β€β₯βπ)βΆ((πΉβπ)(ballβπ·)π₯))) |
8 | 7 | rexbidv 3172 | . . . 4 β’ (π = πΉ β (βπ β β€ (π βΎ (β€β₯βπ)):(β€β₯βπ)βΆ((πβπ)(ballβπ·)π₯) β βπ β β€ (πΉ βΎ (β€β₯βπ)):(β€β₯βπ)βΆ((πΉβπ)(ballβπ·)π₯))) |
9 | 8 | ralbidv 3171 | . . 3 β’ (π = πΉ β (βπ₯ β β+ βπ β β€ (π βΎ (β€β₯βπ)):(β€β₯βπ)βΆ((πβπ)(ballβπ·)π₯) β βπ₯ β β+ βπ β β€ (πΉ βΎ (β€β₯βπ)):(β€β₯βπ)βΆ((πΉβπ)(ballβπ·)π₯))) |
10 | 9 | elrab 3649 | . 2 β’ (πΉ β {π β (π βpm β) β£ βπ₯ β β+ βπ β β€ (π βΎ (β€β₯βπ)):(β€β₯βπ)βΆ((πβπ)(ballβπ·)π₯)} 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₯))) |
11 | 2, 10 | bitrdi 287 | 1 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₯)))) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β wb 205 β§ wa 397 = wceq 1542 β wcel 2107 βwral 3061 βwrex 3070 {crab 3406 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: iscau2 24664 caufpm 24669 lmcau 24700 |
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