Theorem xblm 14737

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 14711	. . . 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 14685	. . . . . . . 8 |- ( ( D e. ( *Met ` X ) /\ P e. X /\ x e. X ) -> 0 <_ ( P D x ) )
3	2	3expa 1205	. . . . . . 7 |- ( ( ( D e. ( *Met ` X ) /\ P e. X ) /\ x e. X ) -> 0 <_ ( P D x ) )
4	3	3adantl3 1157	. . . . . 6 |- ( ( ( D e. ( *Met ` X ) /\ P e. X /\ R e. RR* ) /\ x e. X ) -> 0 <_ ( P D x ) )
5		0xr 8090	. . . . . . 7 |- 0 e. RR*
6		xmetcl 14672	. . . . . . . . 9 |- ( ( D e. ( *Met ` X ) /\ P e. X /\ x e. X ) -> ( P D x ) e. RR* )
7	6	3expa 1205	. . . . . . . 8 |- ( ( ( D e. ( *Met ` X ) /\ P e. X ) /\ x e. X ) -> ( P D x ) e. RR* )
8	7	3adantl3 1157	. . . . . . 7 |- ( ( ( D e. ( *Met ` X ) /\ P e. X /\ R e. RR* ) /\ x e. X ) -> ( P D x ) e. RR* )
9		simpl3 1004	. . . . . . 7 |- ( ( ( D e. ( *Met ` X ) /\ P e. X /\ R e. RR* ) /\ x e. X ) -> R e. RR* )
10		xrlelttr 9898	. . . . . . 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 1353	. . . . . 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 1833	. 2 |- ( ( D e. ( *Met ` X ) /\ P e. X /\ R e. RR* ) -> ( E. x x e. ( P ( ball ` D ) R ) -> 0 < R ) )
16		simpl2 1003	. . . 4 |- ( ( ( D e. ( *Met ` X ) /\ P e. X /\ R e. RR* ) /\ 0 < R ) -> P e. X )
17		simpl1 1002	. . . . 5 |- ( ( ( D e. ( *Met ` X ) /\ P e. X /\ R e. RR* ) /\ 0 < R ) -> D e. ( *Met ` X ) )
18		simpl3 1004	. . . . 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 14734	. . . . 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 1253	. . . 4 |- ( ( ( D e. ( *Met ` X ) /\ P e. X /\ R e. RR* ) /\ 0 < R ) -> P e. ( P ( ball ` D ) R ) )
22		eleq1 2259	. . . . 5 |- ( x = P -> ( x e. ( P ( ball ` D ) R ) <-> P e. ( P ( ball ` D ) R ) ) )
23	22	spcegv 2852	. . . 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 980   E.wex 1506   e. wcel 2167   class class class wbr 4034   ` cfv 5259  (class class class)co 5925   0cc0 7896   RR*cxr 8077   < clt 8078   <_ cle 8079   *Metcxmet 14168   ballcbl 14170

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-sep 4152  ax-pow 4208  ax-pr 4243  ax-un 4469  ax-setind 4574  ax-cnex 7987  ax-resscn 7988  ax-1cn 7989  ax-1re 7990  ax-icn 7991  ax-addcl 7992  ax-addrcl 7993  ax-mulcl 7994  ax-mulrcl 7995  ax-addcom 7996  ax-mulcom 7997  ax-addass 7998  ax-mulass 7999  ax-distr 8000  ax-i2m1 8001  ax-0lt1 8002  ax-1rid 8003  ax-0id 8004  ax-rnegex 8005  ax-precex 8006  ax-cnre 8007  ax-pre-ltirr 8008  ax-pre-ltwlin 8009  ax-pre-lttrn 8010  ax-pre-ltadd 8012  ax-pre-mulgt0 8013

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-nel 2463  df-ral 2480  df-rex 2481  df-reu 2482  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-if 3563  df-pw 3608  df-sn 3629  df-pr 3630  df-op 3632  df-uni 3841  df-iun 3919  df-br 4035  df-opab 4096  df-mpt 4097  df-id 4329  df-po 4332  df-iso 4333  df-xp 4670  df-rel 4671  df-cnv 4672  df-co 4673  df-dm 4674  df-rn 4675  df-res 4676  df-ima 4677  df-iota 5220  df-fun 5261  df-fn 5262  df-f 5263  df-fv 5267  df-riota 5880  df-ov 5928  df-oprab 5929  df-mpo 5930  df-1st 6207  df-2nd 6208  df-map 6718  df-pnf 8080  df-mnf 8081  df-xr 8082  df-ltxr 8083  df-le 8084  df-sub 8216  df-neg 8217  df-2 9066  df-xadd 9865  df-psmet 14175  df-xmet 14176  df-bl 14178

This theorem is referenced by:  blssioo  14873