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| Mirrors > Home > MPE Home > Th. List > xblcntrps | 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.) (Revised by Thierry Arnoux, 11-Mar-2018.) |
| Ref | Expression |
|---|---|
| xblcntrps | β’ ((π· β (PsMetβπ) 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 (PsMetβπ) β§ π β π β§ (π β β* β§ 0 < π )) β π β π) | |
| 2 | psmet0 23713 | . . . 4 β’ ((π· β (PsMetβπ) β§ π β π) β (ππ·π) = 0) | |
| 3 | 2 | 3adant3 1132 | . . 3 β’ ((π· β (PsMetβπ) β§ π β π β§ (π β β* β§ 0 < π )) β (ππ·π) = 0) |
| 4 | simp3r 1202 | . . 3 β’ ((π· β (PsMetβπ) β§ π β π β§ (π β β* β§ 0 < π )) β 0 < π ) | |
| 5 | 3, 4 | eqbrtrd 5147 | . 2 β’ ((π· β (PsMetβπ) β§ π β π β§ (π β β* β§ 0 < π )) β (ππ·π) < π ) |
| 6 | elblps 23792 | . . 3 β’ ((π· β (PsMetβπ) 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 (PsMetβπ) 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 (PsMetβπ) 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 5125 βcfv 6516 (class class class)co 7377 0cc0 11075 β*cxr 11212 < clt 11213 PsMetcpsmet 20832 ballcbl 20835 |
| 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 2702 ax-sep 5276 ax-nul 5283 ax-pow 5340 ax-pr 5404 ax-un 7692 ax-cnex 11131 ax-resscn 11132 |
| 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 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-rab 3419 df-v 3461 df-sbc 3758 df-csb 3874 df-dif 3931 df-un 3933 df-in 3935 df-ss 3945 df-nul 4303 df-if 4507 df-pw 4582 df-sn 4607 df-pr 4609 df-op 4613 df-uni 4886 df-iun 4976 df-br 5126 df-opab 5188 df-mpt 5209 df-id 5551 df-xp 5659 df-rel 5660 df-cnv 5661 df-co 5662 df-dm 5663 df-rn 5664 df-res 5665 df-ima 5666 df-iota 6468 df-fun 6518 df-fn 6519 df-f 6520 df-fv 6524 df-ov 7380 df-oprab 7381 df-mpo 7382 df-1st 7941 df-2nd 7942 df-map 8789 df-xr 11217 df-psmet 20840 df-bl 20843 |
| This theorem is referenced by: blcntrps 23817 |
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