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| Mirrors > Home > MPE Home > Th. List > xblcntr | Structured version Visualization version GIF version | ||
| Description: A ball contains its center. (Contributed by NM, 2-Sep-2006.) (Revised by Mario Carneiro, 12-Nov-2013.) |
| Ref | Expression |
|---|---|
| xblcntr | β’ ((π· β (βMetβπ) β§ π β π β§ (π β β* β§ 0 < π )) β π β (π(ballβπ·)π )) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | simp2 1134 | . 2 β’ ((π· β (βMetβπ) β§ π β π β§ (π β β* β§ 0 < π )) β π β π) | |
| 2 | xmet0 24268 | . . . 4 β’ ((π· β (βMetβπ) β§ π β π) β (ππ·π) = 0) | |
| 3 | 2 | 3adant3 1129 | . . 3 β’ ((π· β (βMetβπ) β§ π β π β§ (π β β* β§ 0 < π )) β (ππ·π) = 0) |
| 4 | simp3r 1199 | . . 3 β’ ((π· β (βMetβπ) β§ π β π β§ (π β β* β§ 0 < π )) β 0 < π ) | |
| 5 | 3, 4 | eqbrtrd 5174 | . 2 β’ ((π· β (βMetβπ) β§ π β π β§ (π β β* β§ 0 < π )) β (ππ·π) < π ) |
| 6 | elbl 24314 | . . 3 β’ ((π· β (βMetβπ) β§ π β π β§ π β β*) β (π β (π(ballβπ·)π ) β (π β π β§ (ππ·π) < π ))) | |
| 7 | 6 | 3adant3r 1178 | . 2 β’ ((π· β (βMetβπ) β§ π β π β§ (π β β* β§ 0 < π )) β (π β (π(ballβπ·)π ) β (π β π β§ (ππ·π) < π ))) |
| 8 | 1, 5, 7 | mpbir2and 711 | 1 β’ ((π· β (βMetβπ) β§ π β π β§ (π β β* β§ 0 < π )) β π β (π(ballβπ·)π )) |
| Colors of variables: wff setvar class |
| Syntax hints: β wi 4 β wb 205 β§ wa 394 β§ w3a 1084 = wceq 1533 β wcel 2098 class class class wbr 5152 βcfv 6553 (class class class)co 7426 0cc0 11146 β*cxr 11285 < clt 11286 βMetcxmet 21271 ballcbl 21273 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2699 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-cnex 11202 ax-resscn 11203 |
| This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-ral 3059 df-rex 3068 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-iun 5002 df-br 5153 df-opab 5215 df-mpt 5236 df-id 5580 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-fv 6561 df-ov 7429 df-oprab 7430 df-mpo 7431 df-1st 7999 df-2nd 8000 df-map 8853 df-xr 11290 df-psmet 21278 df-xmet 21279 df-bl 21281 |
| This theorem is referenced by: blcntr 24339 xbln0 24340 blcld 24434 metds0 24786 metdseq0 24790 heicant 37161 qndenserrnbl 45712 |
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