Theorem xblm 14270

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 14244	. . . 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 14218	. . . . . . . 8 |- ( ( D e. ( *Met ` X ) /\ P e. X /\ x e. X ) -> 0 <_ ( P D x ) )
3	2	3expa 1204	. . . . . . 7 |- ( ( ( D e. ( *Met ` X ) /\ P e. X ) /\ x e. X ) -> 0 <_ ( P D x ) )
4	3	3adantl3 1156	. . . . . 6 |- ( ( ( D e. ( *Met ` X ) /\ P e. X /\ R e. RR* ) /\ x e. X ) -> 0 <_ ( P D x ) )
5		0xr 8018	. . . . . . 7 |- 0 e. RR*
6		xmetcl 14205	. . . . . . . . 9 |- ( ( D e. ( *Met ` X ) /\ P e. X /\ x e. X ) -> ( P D x ) e. RR* )
7	6	3expa 1204	. . . . . . . 8 |- ( ( ( D e. ( *Met ` X ) /\ P e. X ) /\ x e. X ) -> ( P D x ) e. RR* )
8	7	3adantl3 1156	. . . . . . 7 |- ( ( ( D e. ( *Met ` X ) /\ P e. X /\ R e. RR* ) /\ x e. X ) -> ( P D x ) e. RR* )
9		simpl3 1003	. . . . . . 7 |- ( ( ( D e. ( *Met ` X ) /\ P e. X /\ R e. RR* ) /\ x e. X ) -> R e. RR* )
10		xrlelttr 9820	. . . . . . 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 1352	. . . . . 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 1829	. 2 |- ( ( D e. ( *Met ` X ) /\ P e. X /\ R e. RR* ) -> ( E. x x e. ( P ( ball ` D ) R ) -> 0 < R ) )
16		simpl2 1002	. . . 4 |- ( ( ( D e. ( *Met ` X ) /\ P e. X /\ R e. RR* ) /\ 0 < R ) -> P e. X )
17		simpl1 1001	. . . . 5 |- ( ( ( D e. ( *Met ` X ) /\ P e. X /\ R e. RR* ) /\ 0 < R ) -> D e. ( *Met ` X ) )
18		simpl3 1003	. . . . 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 14267	. . . . 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 1252	. . . 4 |- ( ( ( D e. ( *Met ` X ) /\ P e. X /\ R e. RR* ) /\ 0 < R ) -> P e. ( P ( ball ` D ) R ) )
22		eleq1 2250	. . . . 5 |- ( x = P -> ( x e. ( P ( ball ` D ) R ) <-> P e. ( P ( ball ` D ) R ) ) )
23	22	spcegv 2837	. . . 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 979   E.wex 1502   e. wcel 2158   class class class wbr 4015   ` cfv 5228  (class class class)co 5888   0cc0 7825   RR*cxr 8005   < clt 8006   <_ cle 8007   *Metcxmet 13779   ballcbl 13781

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 615  ax-in2 616  ax-io 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-13 2160  ax-14 2161  ax-ext 2169  ax-sep 4133  ax-pow 4186  ax-pr 4221  ax-un 4445  ax-setind 4548  ax-cnex 7916  ax-resscn 7917  ax-1cn 7918  ax-1re 7919  ax-icn 7920  ax-addcl 7921  ax-addrcl 7922  ax-mulcl 7923  ax-mulrcl 7924  ax-addcom 7925  ax-mulcom 7926  ax-addass 7927  ax-mulass 7928  ax-distr 7929  ax-i2m1 7930  ax-0lt1 7931  ax-1rid 7932  ax-0id 7933  ax-rnegex 7934  ax-precex 7935  ax-cnre 7936  ax-pre-ltirr 7937  ax-pre-ltwlin 7938  ax-pre-lttrn 7939  ax-pre-ltadd 7941  ax-pre-mulgt0 7942

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 980  df-3an 981  df-tru 1366  df-fal 1369  df-nf 1471  df-sb 1773  df-eu 2039  df-mo 2040  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ne 2358  df-nel 2453  df-ral 2470  df-rex 2471  df-reu 2472  df-rab 2474  df-v 2751  df-sbc 2975  df-csb 3070  df-dif 3143  df-un 3145  df-in 3147  df-ss 3154  df-if 3547  df-pw 3589  df-sn 3610  df-pr 3611  df-op 3613  df-uni 3822  df-iun 3900  df-br 4016  df-opab 4077  df-mpt 4078  df-id 4305  df-po 4308  df-iso 4309  df-xp 4644  df-rel 4645  df-cnv 4646  df-co 4647  df-dm 4648  df-rn 4649  df-res 4650  df-ima 4651  df-iota 5190  df-fun 5230  df-fn 5231  df-f 5232  df-fv 5236  df-riota 5844  df-ov 5891  df-oprab 5892  df-mpo 5893  df-1st 6155  df-2nd 6156  df-map 6664  df-pnf 8008  df-mnf 8009  df-xr 8010  df-ltxr 8011  df-le 8012  df-sub 8144  df-neg 8145  df-2 8992  df-xadd 9787  df-psmet 13786  df-xmet 13787  df-bl 13789

This theorem is referenced by:  blssioo  14398