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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 1137 | . 2 β’ ((π· β (βMetβπ) β§ π β π β§ (π β β* β§ 0 < π )) β π β π) | |
| 2 | xmet0 23847 | . . . 4 β’ ((π· β (βMetβπ) β§ π β π) β (ππ·π) = 0) | |
| 3 | 2 | 3adant3 1132 | . . 3 β’ ((π· β (βMetβπ) β§ π β π β§ (π β β* β§ 0 < π )) β (ππ·π) = 0) |
| 4 | simp3r 1202 | . . 3 β’ ((π· β (βMetβπ) β§ π β π β§ (π β β* β§ 0 < π )) β 0 < π ) | |
| 5 | 3, 4 | eqbrtrd 5170 | . 2 β’ ((π· β (βMetβπ) β§ π β π β§ (π β β* β§ 0 < π )) β (ππ·π) < π ) |
| 6 | elbl 23893 | . . 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 1181 | . 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 396 β§ w3a 1087 = wceq 1541 β wcel 2106 class class class wbr 5148 βcfv 6543 (class class class)co 7408 0cc0 11109 β*cxr 11246 < clt 11247 βMetcxmet 20928 ballcbl 20930 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7724 ax-cnex 11165 ax-resscn 11166 |
| This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3433 df-v 3476 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 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-id 5574 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-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-fv 6551 df-ov 7411 df-oprab 7412 df-mpo 7413 df-1st 7974 df-2nd 7975 df-map 8821 df-xr 11251 df-psmet 20935 df-xmet 20936 df-bl 20938 |
| This theorem is referenced by: blcntr 23918 xbln0 23919 blcld 24013 metds0 24365 metdseq0 24369 heicant 36518 qndenserrnbl 45001 |
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