Theorem xblm 15282

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 15256	. . . 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 15230	. . . . . . . 8 |- ( ( D e. ( *Met ` X ) /\ P e. X /\ x e. X ) -> 0 <_ ( P D x ) )
3	2	3expa 1230	. . . . . . 7 |- ( ( ( D e. ( *Met ` X ) /\ P e. X ) /\ x e. X ) -> 0 <_ ( P D x ) )
4	3	3adantl3 1182	. . . . . 6 |- ( ( ( D e. ( *Met ` X ) /\ P e. X /\ R e. RR* ) /\ x e. X ) -> 0 <_ ( P D x ) )
5		0xr 8320	. . . . . . 7 |- 0 e. RR*
6		xmetcl 15217	. . . . . . . . 9 |- ( ( D e. ( *Met ` X ) /\ P e. X /\ x e. X ) -> ( P D x ) e. RR* )
7	6	3expa 1230	. . . . . . . 8 |- ( ( ( D e. ( *Met ` X ) /\ P e. X ) /\ x e. X ) -> ( P D x ) e. RR* )
8	7	3adantl3 1182	. . . . . . 7 |- ( ( ( D e. ( *Met ` X ) /\ P e. X /\ R e. RR* ) /\ x e. X ) -> ( P D x ) e. RR* )
9		simpl3 1029	. . . . . . 7 |- ( ( ( D e. ( *Met ` X ) /\ P e. X /\ R e. RR* ) /\ x e. X ) -> R e. RR* )
10		xrlelttr 10139	. . . . . . 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 1379	. . . . . 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 1868	. 2 |- ( ( D e. ( *Met ` X ) /\ P e. X /\ R e. RR* ) -> ( E. x x e. ( P ( ball ` D ) R ) -> 0 < R ) )
16		simpl2 1028	. . . 4 |- ( ( ( D e. ( *Met ` X ) /\ P e. X /\ R e. RR* ) /\ 0 < R ) -> P e. X )
17		simpl1 1027	. . . . 5 |- ( ( ( D e. ( *Met ` X ) /\ P e. X /\ R e. RR* ) /\ 0 < R ) -> D e. ( *Met ` X ) )
18		simpl3 1029	. . . . 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 15279	. . . . 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 1278	. . . 4 |- ( ( ( D e. ( *Met ` X ) /\ P e. X /\ R e. RR* ) /\ 0 < R ) -> P e. ( P ( ball ` D ) R ) )
22		eleq1 2295	. . . . 5 |- ( x = P -> ( x e. ( P ( ball ` D ) R ) <-> P e. ( P ( ball ` D ) R ) ) )
23	22	spcegv 2905	. . . 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 1005   E.wex 1541   e. wcel 2203   class class class wbr 4109   ` cfv 5352  (class class class)co 6050   0cc0 8127   RR*cxr 8307   < clt 8308   <_ cle 8309   *Metcxmet 14684   ballcbl 14686

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2205  ax-14 2206  ax-ext 2214  ax-sep 4228  ax-pow 4287  ax-pr 4322  ax-un 4554  ax-setind 4659  ax-cnex 8218  ax-resscn 8219  ax-1cn 8220  ax-1re 8221  ax-icn 8222  ax-addcl 8223  ax-addrcl 8224  ax-mulcl 8225  ax-mulrcl 8226  ax-addcom 8227  ax-mulcom 8228  ax-addass 8229  ax-mulass 8230  ax-distr 8231  ax-i2m1 8232  ax-0lt1 8233  ax-1rid 8234  ax-0id 8235  ax-rnegex 8236  ax-precex 8237  ax-cnre 8238  ax-pre-ltirr 8239  ax-pre-ltwlin 8240  ax-pre-lttrn 8241  ax-pre-ltadd 8243  ax-pre-mulgt0 8244

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-nel 2508  df-ral 2525  df-rex 2526  df-reu 2527  df-rab 2529  df-v 2815  df-sbc 3043  df-csb 3139  df-dif 3213  df-un 3215  df-in 3217  df-ss 3224  df-if 3621  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-iun 3993  df-br 4110  df-opab 4172  df-mpt 4173  df-id 4414  df-po 4417  df-iso 4418  df-xp 4755  df-rel 4756  df-cnv 4757  df-co 4758  df-dm 4759  df-rn 4760  df-res 4761  df-ima 4762  df-iota 5312  df-fun 5354  df-fn 5355  df-f 5356  df-fv 5360  df-riota 6003  df-ov 6053  df-oprab 6054  df-mpo 6055  df-1st 6334  df-2nd 6335  df-map 6884  df-pnf 8310  df-mnf 8311  df-xr 8312  df-ltxr 8313  df-le 8314  df-sub 8446  df-neg 8447  df-2 9296  df-xadd 10106  df-psmet 14691  df-xmet 14692  df-bl 14694

This theorem is referenced by:  blssioo  15418