Theorem blssexps 15103

Description: Two ways to express the existence of a ball subset. (Contributed by NM, 5-May-2007.) (Revised by Mario Carneiro, 12-Nov-2013.) (Revised by Thierry Arnoux, 11-Mar-2018.)

Assertion

Ref	Expression
blssexps	|- ( ( D e. (PsMet ` X ) /\ P e. X ) -> ( E. x e. ran ( ball ` D ) ( P e. x /\ x C_ A ) <-> E. r e. RR+ ( P ( ball ` D ) r ) C_ A ) )

Distinct variable groups:   x, r, A   D, r, x   P, r, x   X, r, x

Proof of Theorem blssexps

Step	Hyp	Ref	Expression
1		blssps 15101	. . . . . . 7 |- ( ( D e. (PsMet ` X ) /\ x e. ran ( ball ` D ) /\ P e. x ) -> E. r e. RR+ ( P ( ball ` D ) r ) C_ x )
2		sstr 3232	. . . . . . . . 9 |- ( ( ( P ( ball ` D ) r ) C_ x /\ x C_ A ) -> ( P ( ball ` D ) r ) C_ A )
3	2	expcom 116	. . . . . . . 8 |- ( x C_ A -> ( ( P ( ball ` D ) r ) C_ x -> ( P ( ball ` D ) r ) C_ A ) )
4	3	reximdv 2631	. . . . . . 7 |- ( x C_ A -> ( E. r e. RR+ ( P ( ball ` D ) r ) C_ x -> E. r e. RR+ ( P ( ball ` D ) r ) C_ A ) )
5	1, 4	syl5com 29	. . . . . 6 |- ( ( D e. (PsMet ` X ) /\ x e. ran ( ball ` D ) /\ P e. x ) -> ( x C_ A -> E. r e. RR+ ( P ( ball ` D ) r ) C_ A ) )
6	5	3expa 1227	. . . . 5 |- ( ( ( D e. (PsMet ` X ) /\ x e. ran ( ball ` D ) ) /\ P e. x ) -> ( x C_ A -> E. r e. RR+ ( P ( ball ` D ) r ) C_ A ) )
7	6	expimpd 363	. . . 4 |- ( ( D e. (PsMet ` X ) /\ x e. ran ( ball ` D ) ) -> ( ( P e. x /\ x C_ A ) -> E. r e. RR+ ( P ( ball ` D ) r ) C_ A ) )
8	7	adantlr 477	. . 3 |- ( ( ( D e. (PsMet ` X ) /\ P e. X ) /\ x e. ran ( ball ` D ) ) -> ( ( P e. x /\ x C_ A ) -> E. r e. RR+ ( P ( ball ` D ) r ) C_ A ) )
9	8	rexlimdva 2648	. 2 |- ( ( D e. (PsMet ` X ) /\ P e. X ) -> ( E. x e. ran ( ball ` D ) ( P e. x /\ x C_ A ) -> E. r e. RR+ ( P ( ball ` D ) r ) C_ A ) )
10		simpll 527	. . . . 5 |- ( ( ( D e. (PsMet ` X ) /\ P e. X ) /\ ( r e. RR+ /\ ( P ( ball ` D ) r ) C_ A ) ) -> D e. (PsMet ` X ) )
11		simplr 528	. . . . 5 |- ( ( ( D e. (PsMet ` X ) /\ P e. X ) /\ ( r e. RR+ /\ ( P ( ball ` D ) r ) C_ A ) ) -> P e. X )
12		rpxr 9857	. . . . . 6 |- ( r e. RR+ -> r e. RR* )
13	12	ad2antrl 490	. . . . 5 |- ( ( ( D e. (PsMet ` X ) /\ P e. X ) /\ ( r e. RR+ /\ ( P ( ball ` D ) r ) C_ A ) ) -> r e. RR* )
14		blelrnps 15093	. . . . 5 |- ( ( D e. (PsMet ` X ) /\ P e. X /\ r e. RR* ) -> ( P ( ball ` D ) r ) e. ran ( ball ` D ) )
15	10, 11, 13, 14	syl3anc 1271	. . . 4 |- ( ( ( D e. (PsMet ` X ) /\ P e. X ) /\ ( r e. RR+ /\ ( P ( ball ` D ) r ) C_ A ) ) -> ( P ( ball ` D ) r ) e. ran ( ball ` D ) )
16		simprl 529	. . . . 5 |- ( ( ( D e. (PsMet ` X ) /\ P e. X ) /\ ( r e. RR+ /\ ( P ( ball ` D ) r ) C_ A ) ) -> r e. RR+ )
17		blcntrps 15089	. . . . 5 |- ( ( D e. (PsMet ` X ) /\ P e. X /\ r e. RR+ ) -> P e. ( P ( ball ` D ) r ) )
18	10, 11, 16, 17	syl3anc 1271	. . . 4 |- ( ( ( D e. (PsMet ` X ) /\ P e. X ) /\ ( r e. RR+ /\ ( P ( ball ` D ) r ) C_ A ) ) -> P e. ( P ( ball ` D ) r ) )
19		simprr 531	. . . 4 |- ( ( ( D e. (PsMet ` X ) /\ P e. X ) /\ ( r e. RR+ /\ ( P ( ball ` D ) r ) C_ A ) ) -> ( P ( ball ` D ) r ) C_ A )
20		eleq2 2293	. . . . . 6 |- ( x = ( P ( ball ` D ) r ) -> ( P e. x <-> P e. ( P ( ball ` D ) r ) ) )
21		sseq1 3247	. . . . . 6 |- ( x = ( P ( ball ` D ) r ) -> ( x C_ A <-> ( P ( ball ` D ) r ) C_ A ) )
22	20, 21	anbi12d 473	. . . . 5 |- ( x = ( P ( ball ` D ) r ) -> ( ( P e. x /\ x C_ A ) <-> ( P e. ( P ( ball ` D ) r ) /\ ( P ( ball ` D ) r ) C_ A ) ) )
23	22	rspcev 2907	. . . 4 |- ( ( ( P ( ball ` D ) r ) e. ran ( ball ` D ) /\ ( P e. ( P ( ball ` D ) r ) /\ ( P ( ball ` D ) r ) C_ A ) ) -> E. x e. ran ( ball ` D ) ( P e. x /\ x C_ A ) )
24	15, 18, 19, 23	syl12anc 1269	. . 3 |- ( ( ( D e. (PsMet ` X ) /\ P e. X ) /\ ( r e. RR+ /\ ( P ( ball ` D ) r ) C_ A ) ) -> E. x e. ran ( ball ` D ) ( P e. x /\ x C_ A ) )
25	24	rexlimdvaa 2649	. 2 |- ( ( D e. (PsMet ` X ) /\ P e. X ) -> ( E. r e. RR+ ( P ( ball ` D ) r ) C_ A -> E. x e. ran ( ball ` D ) ( P e. x /\ x C_ A ) ) )
26	9, 25	impbid 129	1 |- ( ( D e. (PsMet ` X ) /\ P e. X ) -> ( E. x e. ran ( ball ` D ) ( P e. x /\ x C_ A ) <-> E. r e. RR+ ( P ( ball ` D ) r ) C_ A ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 1002   = wceq 1395   e. wcel 2200   E.wrex 2509   C_ wss 3197   ran crn 4720   ` cfv 5318  (class class class)co 6001   RR*cxr 8180   RR+crp 9849  PsMetcpsmet 14499   ballcbl 14502

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4202  ax-pow 4258  ax-pr 4293  ax-un 4524  ax-setind 4629  ax-cnex 8090  ax-resscn 8091  ax-1cn 8092  ax-1re 8093  ax-icn 8094  ax-addcl 8095  ax-addrcl 8096  ax-mulcl 8097  ax-mulrcl 8098  ax-addcom 8099  ax-mulcom 8100  ax-addass 8101  ax-mulass 8102  ax-distr 8103  ax-i2m1 8104  ax-0lt1 8105  ax-1rid 8106  ax-0id 8107  ax-rnegex 8108  ax-precex 8109  ax-cnre 8110  ax-pre-ltirr 8111  ax-pre-ltwlin 8112  ax-pre-lttrn 8113  ax-pre-apti 8114  ax-pre-ltadd 8115  ax-pre-mulgt0 8116  ax-pre-mulext 8117  ax-arch 8118

This theorem depends on definitions:  df-bi 117  df-stab 836  df-dc 840  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rmo 2516  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-if 3603  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-int 3924  df-iun 3967  df-br 4084  df-opab 4146  df-mpt 4147  df-id 4384  df-po 4387  df-iso 4388  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-res 4731  df-ima 4732  df-iota 5278  df-fun 5320  df-fn 5321  df-f 5322  df-fv 5326  df-riota 5954  df-ov 6004  df-oprab 6005  df-mpo 6006  df-1st 6286  df-2nd 6287  df-map 6797  df-pnf 8183  df-mnf 8184  df-xr 8185  df-ltxr 8186  df-le 8187  df-sub 8319  df-neg 8320  df-reap 8722  df-ap 8729  df-div 8820  df-inn 9111  df-2 9169  df-n0 9370  df-z 9447  df-uz 9723  df-q 9815  df-rp 9850  df-xneg 9968  df-xadd 9969  df-psmet 14507  df-bl 14510

This theorem is referenced by: (None)