Theorem xblm 15140

Description: A ball is inhabited iff the radius is positive. (Contributed by Mario Carneiro, 23-Aug-2015.)

Assertion

Ref	Expression
xblm	|- ( ( D e. ( *Met ` X ) /\ P e. X /\ R e. RR* ) -> ( E. x x e. ( P ( ball ` D ) R ) <-> 0 < R ) )

Distinct variable groups:   x, D   x, R   x, P   x, X

Proof of Theorem xblm

Step	Hyp	Ref	Expression
1		elbl 15114	. . . 4 |- ( ( D e. ( *Met ` X ) /\ P e. X /\ R e. RR* ) -> ( x e. ( P ( ball ` D ) R ) <-> ( x e. X /\ ( P D x ) < R ) ) )
2		xmetge0 15088	. . . . . . . 8 |- ( ( D e. ( *Met ` X ) /\ P e. X /\ x e. X ) -> 0 <_ ( P D x ) )
3	2	3expa 1229	. . . . . . 7 |- ( ( ( D e. ( *Met ` X ) /\ P e. X ) /\ x e. X ) -> 0 <_ ( P D x ) )
4	3	3adantl3 1181	. . . . . 6 |- ( ( ( D e. ( *Met ` X ) /\ P e. X /\ R e. RR* ) /\ x e. X ) -> 0 <_ ( P D x ) )
5		0xr 8225	. . . . . . 7 |- 0 e. RR*
6		xmetcl 15075	. . . . . . . . 9 |- ( ( D e. ( *Met ` X ) /\ P e. X /\ x e. X ) -> ( P D x ) e. RR* )
7	6	3expa 1229	. . . . . . . 8 |- ( ( ( D e. ( *Met ` X ) /\ P e. X ) /\ x e. X ) -> ( P D x ) e. RR* )
8	7	3adantl3 1181	. . . . . . 7 |- ( ( ( D e. ( *Met ` X ) /\ P e. X /\ R e. RR* ) /\ x e. X ) -> ( P D x ) e. RR* )
9		simpl3 1028	. . . . . . 7 |- ( ( ( D e. ( *Met ` X ) /\ P e. X /\ R e. RR* ) /\ x e. X ) -> R e. RR* )
10		xrlelttr 10040	. . . . . . 7 |- ( ( 0 e. RR* /\ ( P D x ) e. RR* /\ R e. RR* ) -> ( ( 0 <_ ( P D x ) /\ ( P D x ) < R ) -> 0 < R ) )
11	5, 8, 9, 10	mp3an2i 1378	. . . . . 6 |- ( ( ( D e. ( *Met ` X ) /\ P e. X /\ R e. RR* ) /\ x e. X ) -> ( ( 0 <_ ( P D x ) /\ ( P D x ) < R ) -> 0 < R ) )
12	4, 11	mpand 429	. . . . 5 |- ( ( ( D e. ( *Met ` X ) /\ P e. X /\ R e. RR* ) /\ x e. X ) -> ( ( P D x ) < R -> 0 < R ) )
13	12	expimpd 363	. . . 4 |- ( ( D e. ( *Met ` X ) /\ P e. X /\ R e. RR* ) -> ( ( x e. X /\ ( P D x ) < R ) -> 0 < R ) )
14	1, 13	sylbid 150	. . 3 |- ( ( D e. ( *Met ` X ) /\ P e. X /\ R e. RR* ) -> ( x e. ( P ( ball ` D ) R ) -> 0 < R ) )
15	14	exlimdv 1867	. 2 |- ( ( D e. ( *Met ` X ) /\ P e. X /\ R e. RR* ) -> ( E. x x e. ( P ( ball ` D ) R ) -> 0 < R ) )
16		simpl2 1027	. . . 4 |- ( ( ( D e. ( *Met ` X ) /\ P e. X /\ R e. RR* ) /\ 0 < R ) -> P e. X )
17		simpl1 1026	. . . . 5 |- ( ( ( D e. ( *Met ` X ) /\ P e. X /\ R e. RR* ) /\ 0 < R ) -> D e. ( *Met ` X ) )
18		simpl3 1028	. . . . 5 |- ( ( ( D e. ( *Met ` X ) /\ P e. X /\ R e. RR* ) /\ 0 < R ) -> R e. RR* )
19		simpr 110	. . . . 5 |- ( ( ( D e. ( *Met ` X ) /\ P e. X /\ R e. RR* ) /\ 0 < R ) -> 0 < R )
20		xblcntr 15137	. . . . 5 |- ( ( D e. ( *Met ` X ) /\ P e. X /\ ( R e. RR* /\ 0 < R ) ) -> P e. ( P ( ball ` D ) R ) )
21	17, 16, 18, 19, 20	syl112anc 1277	. . . 4 |- ( ( ( D e. ( *Met ` X ) /\ P e. X /\ R e. RR* ) /\ 0 < R ) -> P e. ( P ( ball ` D ) R ) )
22		eleq1 2294	. . . . 5 |- ( x = P -> ( x e. ( P ( ball ` D ) R ) <-> P e. ( P ( ball ` D ) R ) ) )
23	22	spcegv 2894	. . . 4 |- ( P e. X -> ( P e. ( P ( ball ` D ) R ) -> E. x x e. ( P ( ball ` D ) R ) ) )
24	16, 21, 23	sylc 62	. . 3 |- ( ( ( D e. ( *Met ` X ) /\ P e. X /\ R e. RR* ) /\ 0 < R ) -> E. x x e. ( P ( ball ` D ) R ) )
25	24	ex 115	. 2 |- ( ( D e. ( *Met ` X ) /\ P e. X /\ R e. RR* ) -> ( 0 < R -> E. x x e. ( P ( ball ` D ) R ) ) )
26	15, 25	impbid 129	1 |- ( ( D e. ( *Met ` X ) /\ P e. X /\ R e. RR* ) -> ( E. x x e. ( P ( ball ` D ) R ) <-> 0 < R ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 1004   E.wex 1540   e. wcel 2202   class class class wbr 4088   ` cfv 5326  (class class class)co 6017   0cc0 8031   RR*cxr 8212   < clt 8213   <_ cle 8214   *Metcxmet 14549   ballcbl 14551

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 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-cnex 8122  ax-resscn 8123  ax-1cn 8124  ax-1re 8125  ax-icn 8126  ax-addcl 8127  ax-addrcl 8128  ax-mulcl 8129  ax-mulrcl 8130  ax-addcom 8131  ax-mulcom 8132  ax-addass 8133  ax-mulass 8134  ax-distr 8135  ax-i2m1 8136  ax-0lt1 8137  ax-1rid 8138  ax-0id 8139  ax-rnegex 8140  ax-precex 8141  ax-cnre 8142  ax-pre-ltirr 8143  ax-pre-ltwlin 8144  ax-pre-lttrn 8145  ax-pre-ltadd 8147  ax-pre-mulgt0 8148

This theorem depends on definitions:  df-bi 117  df-dc 842  df-3or 1005  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-nel 2498  df-ral 2515  df-rex 2516  df-reu 2517  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-if 3606  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-iun 3972  df-br 4089  df-opab 4151  df-mpt 4152  df-id 4390  df-po 4393  df-iso 4394  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-fv 5334  df-riota 5970  df-ov 6020  df-oprab 6021  df-mpo 6022  df-1st 6302  df-2nd 6303  df-map 6818  df-pnf 8215  df-mnf 8216  df-xr 8217  df-ltxr 8218  df-le 8219  df-sub 8351  df-neg 8352  df-2 9201  df-xadd 10007  df-psmet 14556  df-xmet 14557  df-bl 14559

This theorem is referenced by:  blssioo  15276