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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 6891 | . . 3 β’ (π = β β (TotBndβπ) = (TotBndββ )) | |
2 | 1 | eleq2d 2818 | . 2 β’ (π = β β (π β (TotBndβπ) β π β (TotBndββ ))) |
3 | 0elpw 5354 | . . . . . . 7 β’ β β π« β | |
4 | 0fin 9174 | . . . . . . 7 β’ β β Fin | |
5 | elin 3964 | . . . . . . 7 β’ (β β (π« β β© Fin) β (β β π« β β§ β β Fin)) | |
6 | 3, 4, 5 | mpbir2an 708 | . . . . . 6 β’ β β (π« β β© Fin) |
7 | 0iun 5066 | . . . . . 6 β’ βͺ π₯ β β (π₯(ballβπ)π) = β | |
8 | iuneq1 5013 | . . . . . . . 8 β’ (π£ = β β βͺ π₯ β π£ (π₯(ballβπ)π) = βͺ π₯ β β (π₯(ballβπ)π)) | |
9 | 8 | eqeq1d 2733 | . . . . . . 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 3612 | . . . . . 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 689 | . . . . 5 β’ βπ£ β (π« β β© Fin)βͺ π₯ β π£ (π₯(ballβπ)π) = β |
12 | 11 | rgenw 3064 | . . . 4 β’ βπ β β+ βπ£ β (π« β β© Fin)βͺ π₯ β π£ (π₯(ballβπ)π) = β |
13 | istotbnd3 36943 | . . . 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 707 | . . 3 β’ (π β (TotBndββ ) β π β (Metββ )) |
15 | fveq2 6891 | . . . 4 β’ (π = β β (Metβπ) = (Metββ )) | |
16 | 15 | eleq2d 2818 | . . 3 β’ (π = β β (π β (Metβπ) β π β (Metββ ))) |
17 | 14, 16 | bitr4id 290 | . 2 β’ (π = β β (π β (TotBndββ ) β π β (Metβπ))) |
18 | 2, 17 | bitrd 279 | 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 1540 β wcel 2105 βwral 3060 βwrex 3069 β© cin 3947 β c0 4322 π« cpw 4602 βͺ ciun 4997 βcfv 6543 (class class class)co 7412 Fincfn 8942 β+crp 12979 Metcmet 21131 ballcbl 21132 TotBndctotbnd 36938 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7728 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-ral 3061 df-rex 3070 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3778 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-ov 7415 df-om 7859 df-1st 7978 df-2nd 7979 df-1o 8469 df-er 8706 df-en 8943 df-dom 8944 df-fin 8946 df-totbnd 36940 |
This theorem is referenced by: prdsbnd2 36967 |
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