Theorem xblm 12400

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 12374	. . . 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 12348	. . . . . . . 8 |- ( ( D e. ( *Met ` X ) /\ P e. X /\ x e. X ) -> 0 <_ ( P D x ) )
3	2	3expa 1162	. . . . . . 7 |- ( ( ( D e. ( *Met ` X ) /\ P e. X ) /\ x e. X ) -> 0 <_ ( P D x ) )
4	3	3adantl3 1120	. . . . . 6 |- ( ( ( D e. ( *Met ` X ) /\ P e. X /\ R e. RR* ) /\ x e. X ) -> 0 <_ ( P D x ) )
5		0xr 7730	. . . . . . 7 |- 0 e. RR*
6		xmetcl 12335	. . . . . . . . 9 |- ( ( D e. ( *Met ` X ) /\ P e. X /\ x e. X ) -> ( P D x ) e. RR* )
7	6	3expa 1162	. . . . . . . 8 |- ( ( ( D e. ( *Met ` X ) /\ P e. X ) /\ x e. X ) -> ( P D x ) e. RR* )
8	7	3adantl3 1120	. . . . . . 7 |- ( ( ( D e. ( *Met ` X ) /\ P e. X /\ R e. RR* ) /\ x e. X ) -> ( P D x ) e. RR* )
9		simpl3 967	. . . . . . 7 |- ( ( ( D e. ( *Met ` X ) /\ P e. X /\ R e. RR* ) /\ x e. X ) -> R e. RR* )
10		xrlelttr 9476	. . . . . . 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 1301	. . . . . 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 423	. . . . 5 |- ( ( ( D e. ( *Met ` X ) /\ P e. X /\ R e. RR* ) /\ x e. X ) -> ( ( P D x ) < R -> 0 < R ) )
13	12	expimpd 358	. . . 4 |- ( ( D e. ( *Met ` X ) /\ P e. X /\ R e. RR* ) -> ( ( x e. X /\ ( P D x ) < R ) -> 0 < R ) )
14	1, 13	sylbid 149	. . 3 |- ( ( D e. ( *Met ` X ) /\ P e. X /\ R e. RR* ) -> ( x e. ( P ( ball ` D ) R ) -> 0 < R ) )
15	14	exlimdv 1771	. 2 |- ( ( D e. ( *Met ` X ) /\ P e. X /\ R e. RR* ) -> ( E. x x e. ( P ( ball ` D ) R ) -> 0 < R ) )
16		simpl2 966	. . . 4 |- ( ( ( D e. ( *Met ` X ) /\ P e. X /\ R e. RR* ) /\ 0 < R ) -> P e. X )
17		simpl1 965	. . . . 5 |- ( ( ( D e. ( *Met ` X ) /\ P e. X /\ R e. RR* ) /\ 0 < R ) -> D e. ( *Met ` X ) )
18		simpl3 967	. . . . 5 |- ( ( ( D e. ( *Met ` X ) /\ P e. X /\ R e. RR* ) /\ 0 < R ) -> R e. RR* )
19		simpr 109	. . . . 5 |- ( ( ( D e. ( *Met ` X ) /\ P e. X /\ R e. RR* ) /\ 0 < R ) -> 0 < R )
20		xblcntr 12397	. . . . 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 1201	. . . 4 |- ( ( ( D e. ( *Met ` X ) /\ P e. X /\ R e. RR* ) /\ 0 < R ) -> P e. ( P ( ball ` D ) R ) )
22		eleq1 2175	. . . . 5 |- ( x = P -> ( x e. ( P ( ball ` D ) R ) <-> P e. ( P ( ball ` D ) R ) ) )
23	22	spcegv 2743	. . . 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 114	. 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 128	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 103   <-> wb 104   /\ w3a 943   E.wex 1449   e. wcel 1461   class class class wbr 3893   ` cfv 5079  (class class class)co 5726   0cc0 7541   RR*cxr 7717   < clt 7718   <_ cle 7719   *Metcxmet 11986   ballcbl 11988

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 586  ax-in2 587  ax-io 681  ax-5 1404  ax-7 1405  ax-gen 1406  ax-ie1 1450  ax-ie2 1451  ax-8 1463  ax-10 1464  ax-11 1465  ax-i12 1466  ax-bndl 1467  ax-4 1468  ax-13 1472  ax-14 1473  ax-17 1487  ax-i9 1491  ax-ial 1495  ax-i5r 1496  ax-ext 2095  ax-sep 4004  ax-pow 4056  ax-pr 4089  ax-un 4313  ax-setind 4410  ax-cnex 7630  ax-resscn 7631  ax-1cn 7632  ax-1re 7633  ax-icn 7634  ax-addcl 7635  ax-addrcl 7636  ax-mulcl 7637  ax-mulrcl 7638  ax-addcom 7639  ax-mulcom 7640  ax-addass 7641  ax-mulass 7642  ax-distr 7643  ax-i2m1 7644  ax-0lt1 7645  ax-1rid 7646  ax-0id 7647  ax-rnegex 7648  ax-precex 7649  ax-cnre 7650  ax-pre-ltirr 7651  ax-pre-ltwlin 7652  ax-pre-lttrn 7653  ax-pre-ltadd 7655  ax-pre-mulgt0 7656

This theorem depends on definitions:  df-bi 116  df-dc 803  df-3or 944  df-3an 945  df-tru 1315  df-fal 1318  df-nf 1418  df-sb 1717  df-eu 1976  df-mo 1977  df-clab 2100  df-cleq 2106  df-clel 2109  df-nfc 2242  df-ne 2281  df-nel 2376  df-ral 2393  df-rex 2394  df-reu 2395  df-rab 2397  df-v 2657  df-sbc 2877  df-csb 2970  df-dif 3037  df-un 3039  df-in 3041  df-ss 3048  df-if 3439  df-pw 3476  df-sn 3497  df-pr 3498  df-op 3500  df-uni 3701  df-iun 3779  df-br 3894  df-opab 3948  df-mpt 3949  df-id 4173  df-po 4176  df-iso 4177  df-xp 4503  df-rel 4504  df-cnv 4505  df-co 4506  df-dm 4507  df-rn 4508  df-res 4509  df-ima 4510  df-iota 5044  df-fun 5081  df-fn 5082  df-f 5083  df-fv 5087  df-riota 5682  df-ov 5729  df-oprab 5730  df-mpo 5731  df-1st 5990  df-2nd 5991  df-map 6496  df-pnf 7720  df-mnf 7721  df-xr 7722  df-ltxr 7723  df-le 7724  df-sub 7852  df-neg 7853  df-2 8683  df-xadd 9447  df-psmet 11993  df-xmet 11994  df-bl 11996

This theorem is referenced by:  blssioo  12525