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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 1138 | . 2 β’ ((π· β (βMetβπ) β§ π β π β§ (π β β* β§ 0 < π )) β π β π) | |
2 | xmet0 23711 | . . . 4 β’ ((π· β (βMetβπ) β§ π β π) β (ππ·π) = 0) | |
3 | 2 | 3adant3 1133 | . . 3 β’ ((π· β (βMetβπ) β§ π β π β§ (π β β* β§ 0 < π )) β (ππ·π) = 0) |
4 | simp3r 1203 | . . 3 β’ ((π· β (βMetβπ) β§ π β π β§ (π β β* β§ 0 < π )) β 0 < π ) | |
5 | 3, 4 | eqbrtrd 5128 | . 2 β’ ((π· β (βMetβπ) β§ π β π β§ (π β β* β§ 0 < π )) β (ππ·π) < π ) |
6 | elbl 23757 | . . 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 1182 | . 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 712 | 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 397 β§ w3a 1088 = wceq 1542 β wcel 2107 class class class wbr 5106 βcfv 6497 (class class class)co 7358 0cc0 11056 β*cxr 11193 < clt 11194 βMetcxmet 20797 ballcbl 20799 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 ax-cnex 11112 ax-resscn 11113 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3407 df-v 3446 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-iun 4957 df-br 5107 df-opab 5169 df-mpt 5190 df-id 5532 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-fv 6505 df-ov 7361 df-oprab 7362 df-mpo 7363 df-1st 7922 df-2nd 7923 df-map 8770 df-xr 11198 df-psmet 20804 df-xmet 20805 df-bl 20807 |
This theorem is referenced by: blcntr 23782 xbln0 23783 blcld 23877 metds0 24229 metdseq0 24233 heicant 36159 qndenserrnbl 44622 |
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