Theorem blssexps 13188

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 13186	. . . . . . 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 3155	. . . . . . . . 9 |- ( ( ( P ( ball ` D ) r ) C_ x /\ x C_ A ) -> ( P ( ball ` D ) r ) C_ A )
3	2	expcom 115	. . . . . . . 8 |- ( x C_ A -> ( ( P ( ball ` D ) r ) C_ x -> ( P ( ball ` D ) r ) C_ A ) )
4	3	reximdv 2571	. . . . . . 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 1198	. . . . 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 361	. . . 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 474	. . 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 2587	. 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 524	. . . . 5 |- ( ( ( D e. (PsMet ` X ) /\ P e. X ) /\ ( r e. RR+ /\ ( P ( ball ` D ) r ) C_ A ) ) -> D e. (PsMet ` X ) )
11		simplr 525	. . . . 5 |- ( ( ( D e. (PsMet ` X ) /\ P e. X ) /\ ( r e. RR+ /\ ( P ( ball ` D ) r ) C_ A ) ) -> P e. X )
12		rpxr 9611	. . . . . 6 |- ( r e. RR+ -> r e. RR* )
13	12	ad2antrl 487	. . . . 5 |- ( ( ( D e. (PsMet ` X ) /\ P e. X ) /\ ( r e. RR+ /\ ( P ( ball ` D ) r ) C_ A ) ) -> r e. RR* )
14		blelrnps 13178	. . . . 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 1233	. . . 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 526	. . . . 5 |- ( ( ( D e. (PsMet ` X ) /\ P e. X ) /\ ( r e. RR+ /\ ( P ( ball ` D ) r ) C_ A ) ) -> r e. RR+ )
17		blcntrps 13174	. . . . 5 |- ( ( D e. (PsMet ` X ) /\ P e. X /\ r e. RR+ ) -> P e. ( P ( ball ` D ) r ) )
18	10, 11, 16, 17	syl3anc 1233	. . . 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 527	. . . 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 2234	. . . . . 6 |- ( x = ( P ( ball ` D ) r ) -> ( P e. x <-> P e. ( P ( ball ` D ) r ) ) )
21		sseq1 3170	. . . . . 6 |- ( x = ( P ( ball ` D ) r ) -> ( x C_ A <-> ( P ( ball ` D ) r ) C_ A ) )
22	20, 21	anbi12d 470	. . . . 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 2834	. . . 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 1231	. . 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 2588	. 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 128	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 103   <-> wb 104   /\ w3a 973   = wceq 1348   e. wcel 2141   E.wrex 2449   C_ wss 3121   ran crn 4610   ` cfv 5196  (class class class)co 5851   RR*cxr 7946   RR+crp 9603  PsMetcpsmet 12738   ballcbl 12741

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-sep 4105  ax-pow 4158  ax-pr 4192  ax-un 4416  ax-setind 4519  ax-cnex 7858  ax-resscn 7859  ax-1cn 7860  ax-1re 7861  ax-icn 7862  ax-addcl 7863  ax-addrcl 7864  ax-mulcl 7865  ax-mulrcl 7866  ax-addcom 7867  ax-mulcom 7868  ax-addass 7869  ax-mulass 7870  ax-distr 7871  ax-i2m1 7872  ax-0lt1 7873  ax-1rid 7874  ax-0id 7875  ax-rnegex 7876  ax-precex 7877  ax-cnre 7878  ax-pre-ltirr 7879  ax-pre-ltwlin 7880  ax-pre-lttrn 7881  ax-pre-apti 7882  ax-pre-ltadd 7883  ax-pre-mulgt0 7884  ax-pre-mulext 7885  ax-arch 7886

This theorem depends on definitions:  df-bi 116  df-stab 826  df-dc 830  df-3or 974  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-nel 2436  df-ral 2453  df-rex 2454  df-reu 2455  df-rmo 2456  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-if 3526  df-pw 3566  df-sn 3587  df-pr 3588  df-op 3590  df-uni 3795  df-int 3830  df-iun 3873  df-br 3988  df-opab 4049  df-mpt 4050  df-id 4276  df-po 4279  df-iso 4280  df-xp 4615  df-rel 4616  df-cnv 4617  df-co 4618  df-dm 4619  df-rn 4620  df-res 4621  df-ima 4622  df-iota 5158  df-fun 5198  df-fn 5199  df-f 5200  df-fv 5204  df-riota 5807  df-ov 5854  df-oprab 5855  df-mpo 5856  df-1st 6117  df-2nd 6118  df-map 6626  df-pnf 7949  df-mnf 7950  df-xr 7951  df-ltxr 7952  df-le 7953  df-sub 8085  df-neg 8086  df-reap 8487  df-ap 8494  df-div 8583  df-inn 8872  df-2 8930  df-n0 9129  df-z 9206  df-uz 9481  df-q 9572  df-rp 9604  df-xneg 9722  df-xadd 9723  df-psmet 12746  df-bl 12749

This theorem is referenced by: (None)