Nearby theorems
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| Mirrors > Home > ILE Home > Th. List > blgt0 | 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 8006 | . . 3 β’ 0 β β* | |
| 2 | 1 | a1i 9 | . 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 1000 | . . 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 1001 | . . 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 13976 | . . . 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 383 | . . 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 13937 | . . 3 β’ ((π· β (βMetβπ) β§ π β π β§ π΄ β π) β (ππ·π΄) β β*) | |
| 8 | 3, 4, 6, 7 | syl3anc 1238 | . 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 1002 | . 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 13950 | . . 3 β’ ((π· β (βMetβπ) β§ π β π β§ π΄ β π) β 0 β€ (ππ·π΄)) | |
| 11 | 3, 4, 6, 10 | syl3anc 1238 | . 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 384 | . 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 9812 | 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 set class |
| Syntax hints: β wi 4 β§ wa 104 β§ w3a 978 β wcel 2148 class class class wbr 4005 βcfv 5218 (class class class)co 5877 0cc0 7813 β*cxr 7993 < clt 7994 β€ cle 7995 βMetcxmet 13525 ballcbl 13527 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-sep 4123 ax-pow 4176 ax-pr 4211 ax-un 4435 ax-setind 4538 ax-cnex 7904 ax-resscn 7905 ax-1cn 7906 ax-1re 7907 ax-icn 7908 ax-addcl 7909 ax-addrcl 7910 ax-mulcl 7911 ax-mulrcl 7912 ax-addcom 7913 ax-mulcom 7914 ax-addass 7915 ax-mulass 7916 ax-distr 7917 ax-i2m1 7918 ax-0lt1 7919 ax-1rid 7920 ax-0id 7921 ax-rnegex 7922 ax-precex 7923 ax-cnre 7924 ax-pre-ltirr 7925 ax-pre-ltwlin 7926 ax-pre-lttrn 7927 ax-pre-ltadd 7929 ax-pre-mulgt0 7930 |
| This theorem depends on definitions: df-bi 117 df-dc 835 df-3or 979 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-nel 2443 df-ral 2460 df-rex 2461 df-reu 2462 df-rab 2464 df-v 2741 df-sbc 2965 df-csb 3060 df-dif 3133 df-un 3135 df-in 3137 df-ss 3144 df-if 3537 df-pw 3579 df-sn 3600 df-pr 3601 df-op 3603 df-uni 3812 df-iun 3890 df-br 4006 df-opab 4067 df-mpt 4068 df-id 4295 df-po 4298 df-iso 4299 df-xp 4634 df-rel 4635 df-cnv 4636 df-co 4637 df-dm 4638 df-rn 4639 df-res 4640 df-ima 4641 df-iota 5180 df-fun 5220 df-fn 5221 df-f 5222 df-fv 5226 df-riota 5833 df-ov 5880 df-oprab 5881 df-mpo 5882 df-1st 6143 df-2nd 6144 df-map 6652 df-pnf 7996 df-mnf 7997 df-xr 7998 df-ltxr 7999 df-le 8000 df-sub 8132 df-neg 8133 df-2 8980 df-xadd 9775 df-psmet 13532 df-xmet 13533 df-bl 13535 |
| This theorem is referenced by: (None) |
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