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Mirrors > Home > MPE Home > Th. List > Mathboxes > 0totbnd | Structured version Visualization version GIF version |
Description: The metric (there is only one) on the empty set is totally bounded. (Contributed by Mario Carneiro, 16-Sep-2015.) |
Ref | Expression |
---|---|
0totbnd | β’ (π = β β (π β (TotBndβπ) β π β (Metβπ))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fveq2 6890 | . . 3 β’ (π = β β (TotBndβπ) = (TotBndββ )) | |
2 | 1 | eleq2d 2817 | . 2 β’ (π = β β (π β (TotBndβπ) β π β (TotBndββ ))) |
3 | 0elpw 5353 | . . . . . . 7 β’ β β π« β | |
4 | 0fin 9173 | . . . . . . 7 β’ β β Fin | |
5 | elin 3963 | . . . . . . 7 β’ (β β (π« β β© Fin) β (β β π« β β§ β β Fin)) | |
6 | 3, 4, 5 | mpbir2an 707 | . . . . . 6 β’ β β (π« β β© Fin) |
7 | 0iun 5065 | . . . . . 6 β’ βͺ π₯ β β (π₯(ballβπ)π) = β | |
8 | iuneq1 5012 | . . . . . . . 8 β’ (π£ = β β βͺ π₯ β π£ (π₯(ballβπ)π) = βͺ π₯ β β (π₯(ballβπ)π)) | |
9 | 8 | eqeq1d 2732 | . . . . . . 7 β’ (π£ = β β (βͺ π₯ β π£ (π₯(ballβπ)π) = β β βͺ π₯ β β (π₯(ballβπ)π) = β )) |
10 | 9 | rspcev 3611 | . . . . . 6 β’ ((β β (π« β β© Fin) β§ βͺ π₯ β β (π₯(ballβπ)π) = β ) β βπ£ β (π« β β© Fin)βͺ π₯ β π£ (π₯(ballβπ)π) = β ) |
11 | 6, 7, 10 | mp2an 688 | . . . . 5 β’ βπ£ β (π« β β© Fin)βͺ π₯ β π£ (π₯(ballβπ)π) = β |
12 | 11 | rgenw 3063 | . . . 4 β’ βπ β β+ βπ£ β (π« β β© Fin)βͺ π₯ β π£ (π₯(ballβπ)π) = β |
13 | istotbnd3 36942 | . . . 4 β’ (π β (TotBndββ ) β (π β (Metββ ) β§ βπ β β+ βπ£ β (π« β β© Fin)βͺ π₯ β π£ (π₯(ballβπ)π) = β )) | |
14 | 12, 13 | mpbiran2 706 | . . 3 β’ (π β (TotBndββ ) β π β (Metββ )) |
15 | fveq2 6890 | . . . 4 β’ (π = β β (Metβπ) = (Metββ )) | |
16 | 15 | eleq2d 2817 | . . 3 β’ (π = β β (π β (Metβπ) β π β (Metββ ))) |
17 | 14, 16 | bitr4id 289 | . 2 β’ (π = β β (π β (TotBndββ ) β π β (Metβπ))) |
18 | 2, 17 | bitrd 278 | 1 β’ (π = β β (π β (TotBndβπ) β π β (Metβπ))) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β wb 205 = wceq 1539 β wcel 2104 βwral 3059 βwrex 3068 β© cin 3946 β c0 4321 π« cpw 4601 βͺ ciun 4996 βcfv 6542 (class class class)co 7411 Fincfn 8941 β+crp 12978 Metcmet 21130 ballcbl 21131 TotBndctotbnd 36937 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-10 2135 ax-11 2152 ax-12 2169 ax-ext 2701 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7727 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2532 df-eu 2561 df-clab 2708 df-cleq 2722 df-clel 2808 df-nfc 2883 df-ne 2939 df-ral 3060 df-rex 3069 df-reu 3375 df-rab 3431 df-v 3474 df-sbc 3777 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-ov 7414 df-om 7858 df-1st 7977 df-2nd 7978 df-1o 8468 df-er 8705 df-en 8942 df-dom 8943 df-fin 8945 df-totbnd 36939 |
This theorem is referenced by: prdsbnd2 36966 |
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