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| Mirrors > Home > MPE Home > Th. List > blgt0 | Structured version Visualization version GIF version | ||
| Description: A nonempty ball implies that the radius is positive. (Contributed by NM, 11-Mar-2007.) (Revised by Mario Carneiro, 23-Aug-2015.) |
| Ref | Expression |
|---|---|
| blgt0 | β’ (((π· β (βMetβπ) β§ π β π β§ π β β*) β§ π΄ β (π(ballβπ·)π )) β 0 < π ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | 0xr 11265 | . . 3 β’ 0 β β* | |
| 2 | 1 | a1i 11 | . 2 β’ (((π· β (βMetβπ) β§ π β π β§ π β β*) β§ π΄ β (π(ballβπ·)π )) β 0 β β*) |
| 3 | simpl1 1191 | . . 3 β’ (((π· β (βMetβπ) β§ π β π β§ π β β*) β§ π΄ β (π(ballβπ·)π )) β π· β (βMetβπ)) | |
| 4 | simpl2 1192 | . . 3 β’ (((π· β (βMetβπ) β§ π β π β§ π β β*) β§ π΄ β (π(ballβπ·)π )) β π β π) | |
| 5 | elbl 24114 | . . . 4 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 ))) | |
| 6 | 5 | simprbda 499 | . . 3 β’ (((π· β (βMetβπ) β§ π β π β§ π β β*) β§ π΄ β (π(ballβπ·)π )) β π΄ β π) |
| 7 | xmetcl 24057 | . . 3 β’ ((π· β (βMetβπ) β§ π β π β§ π΄ β π) β (ππ·π΄) β β*) | |
| 8 | 3, 4, 6, 7 | syl3anc 1371 | . 2 β’ (((π· β (βMetβπ) β§ π β π β§ π β β*) β§ π΄ β (π(ballβπ·)π )) β (ππ·π΄) β β*) |
| 9 | simpl3 1193 | . 2 β’ (((π· β (βMetβπ) β§ π β π β§ π β β*) β§ π΄ β (π(ballβπ·)π )) β π β β*) | |
| 10 | xmetge0 24070 | . . 3 β’ ((π· β (βMetβπ) β§ π β π β§ π΄ β π) β 0 β€ (ππ·π΄)) | |
| 11 | 3, 4, 6, 10 | syl3anc 1371 | . 2 β’ (((π· β (βMetβπ) β§ π β π β§ π β β*) β§ π΄ β (π(ballβπ·)π )) β 0 β€ (ππ·π΄)) |
| 12 | 5 | simplbda 500 | . 2 β’ (((π· β (βMetβπ) β§ π β π β§ π β β*) β§ π΄ β (π(ballβπ·)π )) β (ππ·π΄) < π ) |
| 13 | 2, 8, 9, 11, 12 | xrlelttrd 13143 | 1 β’ (((π· β (βMetβπ) β§ π β π β§ π β β*) β§ π΄ β (π(ballβπ·)π )) β 0 < π ) |
| Colors of variables: wff setvar class |
| Syntax hints: β wi 4 β§ wa 396 β§ w3a 1087 β wcel 2106 class class class wbr 5148 βcfv 6543 (class class class)co 7411 0cc0 11112 β*cxr 11251 < clt 11252 β€ cle 11253 βMetcxmet 21129 ballcbl 21131 |
| 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 7727 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189 |
| This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 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-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 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-po 5588 df-so 5589 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-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7367 df-ov 7414 df-oprab 7415 df-mpo 7416 df-1st 7977 df-2nd 7978 df-er 8705 df-map 8824 df-en 8942 df-dom 8943 df-sdom 8944 df-pnf 11254 df-mnf 11255 df-xr 11256 df-ltxr 11257 df-le 11258 df-sub 11450 df-neg 11451 df-div 11876 df-2 12279 df-rp 12979 df-xneg 13096 df-xadd 13097 df-xmul 13098 df-psmet 21136 df-xmet 21137 df-bl 21139 |
| This theorem is referenced by: (None) |
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