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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 24198 | . . . 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 5163 | . 2 β’ ((π· β (βMetβπ) β§ π β π β§ (π β β* β§ 0 < π )) β (ππ·π) < π ) |
| 6 | elbl 24244 | . . 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 710 | 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 395 β§ w3a 1084 = wceq 1533 β wcel 2098 class class class wbr 5141 βcfv 6536 (class class class)co 7404 0cc0 11109 β*cxr 11248 < clt 11249 βMetcxmet 21220 ballcbl 21222 |
| 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 2163 ax-ext 2697 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7721 ax-cnex 11165 ax-resscn 11166 |
| This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-ral 3056 df-rex 3065 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-id 5567 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-iota 6488 df-fun 6538 df-fn 6539 df-f 6540 df-fv 6544 df-ov 7407 df-oprab 7408 df-mpo 7409 df-1st 7971 df-2nd 7972 df-map 8821 df-xr 11253 df-psmet 21227 df-xmet 21228 df-bl 21230 |
| This theorem is referenced by: blcntr 24269 xbln0 24270 blcld 24364 metds0 24716 metdseq0 24720 heicant 37035 qndenserrnbl 45565 |
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