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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 11123 | . . 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 1190 | . . 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 1191 | . . 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 23647 | . . . 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 23590 | . . 3 β’ ((π· β (βMetβπ) β§ π β π β§ π΄ β π) β (ππ·π΄) β β*) | |
8 | 3, 4, 6, 7 | syl3anc 1370 | . 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 1192 | . 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 23603 | . . 3 β’ ((π· β (βMetβπ) β§ π β π β§ π΄ β π) β 0 β€ (ππ·π΄)) | |
11 | 3, 4, 6, 10 | syl3anc 1370 | . 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 12995 | 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 1086 β wcel 2105 class class class wbr 5092 βcfv 6479 (class class class)co 7337 0cc0 10972 β*cxr 11109 < clt 11110 β€ cle 11111 βMetcxmet 20688 ballcbl 20690 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2707 ax-sep 5243 ax-nul 5250 ax-pow 5308 ax-pr 5372 ax-un 7650 ax-cnex 11028 ax-resscn 11029 ax-1cn 11030 ax-icn 11031 ax-addcl 11032 ax-addrcl 11033 ax-mulcl 11034 ax-mulrcl 11035 ax-mulcom 11036 ax-addass 11037 ax-mulass 11038 ax-distr 11039 ax-i2m1 11040 ax-1ne0 11041 ax-1rid 11042 ax-rnegex 11043 ax-rrecex 11044 ax-cnre 11045 ax-pre-lttri 11046 ax-pre-lttrn 11047 ax-pre-ltadd 11048 ax-pre-mulgt0 11049 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3349 df-reu 3350 df-rab 3404 df-v 3443 df-sbc 3728 df-csb 3844 df-dif 3901 df-un 3903 df-in 3905 df-ss 3915 df-nul 4270 df-if 4474 df-pw 4549 df-sn 4574 df-pr 4576 df-op 4580 df-uni 4853 df-iun 4943 df-br 5093 df-opab 5155 df-mpt 5176 df-id 5518 df-po 5532 df-so 5533 df-xp 5626 df-rel 5627 df-cnv 5628 df-co 5629 df-dm 5630 df-rn 5631 df-res 5632 df-ima 5633 df-iota 6431 df-fun 6481 df-fn 6482 df-f 6483 df-f1 6484 df-fo 6485 df-f1o 6486 df-fv 6487 df-riota 7293 df-ov 7340 df-oprab 7341 df-mpo 7342 df-1st 7899 df-2nd 7900 df-er 8569 df-map 8688 df-en 8805 df-dom 8806 df-sdom 8807 df-pnf 11112 df-mnf 11113 df-xr 11114 df-ltxr 11115 df-le 11116 df-sub 11308 df-neg 11309 df-div 11734 df-2 12137 df-rp 12832 df-xneg 12949 df-xadd 12950 df-xmul 12951 df-psmet 20695 df-xmet 20696 df-bl 20698 |
This theorem is referenced by: (None) |
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