Theorem blssexps 15016

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 15014	. . . . . . 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 3209	. . . . . . . . 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 2609	. . . . . . 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 1206	. . . . 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 2625	. 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 9818	. . . . . 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 15006	. . . . 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 1250	. . . 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 15002	. . . . 5 |- ( ( D e. (PsMet ` X ) /\ P e. X /\ r e. RR+ ) -> P e. ( P ( ball ` D ) r ) )
18	10, 11, 16, 17	syl3anc 1250	. . . 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 2271	. . . . . 6 |- ( x = ( P ( ball ` D ) r ) -> ( P e. x <-> P e. ( P ( ball ` D ) r ) ) )
21		sseq1 3224	. . . . . 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 2884	. . . 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 1248	. . 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 2626	. 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 981   = wceq 1373   e. wcel 2178   E.wrex 2487   C_ wss 3174   ran crn 4694   ` cfv 5290  (class class class)co 5967   RR*cxr 8141   RR+crp 9810  PsMetcpsmet 14412   ballcbl 14415

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2180  ax-14 2181  ax-ext 2189  ax-sep 4178  ax-pow 4234  ax-pr 4269  ax-un 4498  ax-setind 4603  ax-cnex 8051  ax-resscn 8052  ax-1cn 8053  ax-1re 8054  ax-icn 8055  ax-addcl 8056  ax-addrcl 8057  ax-mulcl 8058  ax-mulrcl 8059  ax-addcom 8060  ax-mulcom 8061  ax-addass 8062  ax-mulass 8063  ax-distr 8064  ax-i2m1 8065  ax-0lt1 8066  ax-1rid 8067  ax-0id 8068  ax-rnegex 8069  ax-precex 8070  ax-cnre 8071  ax-pre-ltirr 8072  ax-pre-ltwlin 8073  ax-pre-lttrn 8074  ax-pre-apti 8075  ax-pre-ltadd 8076  ax-pre-mulgt0 8077  ax-pre-mulext 8078  ax-arch 8079

This theorem depends on definitions:  df-bi 117  df-stab 833  df-dc 837  df-3or 982  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ne 2379  df-nel 2474  df-ral 2491  df-rex 2492  df-reu 2493  df-rmo 2494  df-rab 2495  df-v 2778  df-sbc 3006  df-csb 3102  df-dif 3176  df-un 3178  df-in 3180  df-ss 3187  df-if 3580  df-pw 3628  df-sn 3649  df-pr 3650  df-op 3652  df-uni 3865  df-int 3900  df-iun 3943  df-br 4060  df-opab 4122  df-mpt 4123  df-id 4358  df-po 4361  df-iso 4362  df-xp 4699  df-rel 4700  df-cnv 4701  df-co 4702  df-dm 4703  df-rn 4704  df-res 4705  df-ima 4706  df-iota 5251  df-fun 5292  df-fn 5293  df-f 5294  df-fv 5298  df-riota 5922  df-ov 5970  df-oprab 5971  df-mpo 5972  df-1st 6249  df-2nd 6250  df-map 6760  df-pnf 8144  df-mnf 8145  df-xr 8146  df-ltxr 8147  df-le 8148  df-sub 8280  df-neg 8281  df-reap 8683  df-ap 8690  df-div 8781  df-inn 9072  df-2 9130  df-n0 9331  df-z 9408  df-uz 9684  df-q 9776  df-rp 9811  df-xneg 9929  df-xadd 9930  df-psmet 14420  df-bl 14423

This theorem is referenced by: (None)