Theorem blin2 14752

Description: Given any two balls and a point in their intersection, there is a ball contained in the intersection with the given center point. (Contributed by Mario Carneiro, 12-Nov-2013.)

Assertion

Ref	Expression
blin2	|- ( ( ( D e. ( *Met ` X ) /\ P e. ( B i^i C ) ) /\ ( B e. ran ( ball ` D ) /\ C e. ran ( ball ` D ) ) ) -> E. x e. RR+ ( P ( ball ` D ) x ) C_ ( B i^i C ) )

Distinct variable groups:   x, B   x, C   x, D   x, P   x, X

Proof of Theorem blin2

Dummy variables y z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpll 527	. . 3 |- ( ( ( D e. ( *Met ` X ) /\ P e. ( B i^i C ) ) /\ ( B e. ran ( ball ` D ) /\ C e. ran ( ball ` D ) ) ) -> D e. ( *Met ` X ) )
2		simprl 529	. . 3 |- ( ( ( D e. ( *Met ` X ) /\ P e. ( B i^i C ) ) /\ ( B e. ran ( ball ` D ) /\ C e. ran ( ball ` D ) ) ) -> B e. ran ( ball ` D ) )
3		simplr 528	. . . 4 |- ( ( ( D e. ( *Met ` X ) /\ P e. ( B i^i C ) ) /\ ( B e. ran ( ball ` D ) /\ C e. ran ( ball ` D ) ) ) -> P e. ( B i^i C ) )
4	3	elin1d 3353	. . 3 |- ( ( ( D e. ( *Met ` X ) /\ P e. ( B i^i C ) ) /\ ( B e. ran ( ball ` D ) /\ C e. ran ( ball ` D ) ) ) -> P e. B )
5		blss 14748	. . 3 |- ( ( D e. ( *Met ` X ) /\ B e. ran ( ball ` D ) /\ P e. B ) -> E. y e. RR+ ( P ( ball ` D ) y ) C_ B )
6	1, 2, 4, 5	syl3anc 1249	. 2 |- ( ( ( D e. ( *Met ` X ) /\ P e. ( B i^i C ) ) /\ ( B e. ran ( ball ` D ) /\ C e. ran ( ball ` D ) ) ) -> E. y e. RR+ ( P ( ball ` D ) y ) C_ B )
7		simprr 531	. . 3 |- ( ( ( D e. ( *Met ` X ) /\ P e. ( B i^i C ) ) /\ ( B e. ran ( ball ` D ) /\ C e. ran ( ball ` D ) ) ) -> C e. ran ( ball ` D ) )
8	3	elin2d 3354	. . 3 |- ( ( ( D e. ( *Met ` X ) /\ P e. ( B i^i C ) ) /\ ( B e. ran ( ball ` D ) /\ C e. ran ( ball ` D ) ) ) -> P e. C )
9		blss 14748	. . 3 |- ( ( D e. ( *Met ` X ) /\ C e. ran ( ball ` D ) /\ P e. C ) -> E. z e. RR+ ( P ( ball ` D ) z ) C_ C )
10	1, 7, 8, 9	syl3anc 1249	. 2 |- ( ( ( D e. ( *Met ` X ) /\ P e. ( B i^i C ) ) /\ ( B e. ran ( ball ` D ) /\ C e. ran ( ball ` D ) ) ) -> E. z e. RR+ ( P ( ball ` D ) z ) C_ C )
11		reeanv 2667	. . 3 |- ( E. y e. RR+ E. z e. RR+ ( ( P ( ball ` D ) y ) C_ B /\ ( P ( ball ` D ) z ) C_ C ) <-> ( E. y e. RR+ ( P ( ball ` D ) y ) C_ B /\ E. z e. RR+ ( P ( ball ` D ) z ) C_ C ) )
12		ss2in 3392	. . . . 5 |- ( ( ( P ( ball ` D ) y ) C_ B /\ ( P ( ball ` D ) z ) C_ C ) -> ( ( P ( ball ` D ) y ) i^i ( P ( ball ` D ) z ) ) C_ ( B i^i C ) )
13		inss1 3384	. . . . . . . . . . 11 |- ( B i^i C ) C_ B
14		blf 14730	. . . . . . . . . . . . . 14 |- ( D e. ( *Met ` X ) -> ( ball ` D ) : ( X X. RR* ) --> ~P X )
15		frn 5419	. . . . . . . . . . . . . 14 |- ( ( ball ` D ) : ( X X. RR* ) --> ~P X -> ran ( ball ` D ) C_ ~P X )
16	1, 14, 15	3syl 17	. . . . . . . . . . . . 13 |- ( ( ( D e. ( *Met ` X ) /\ P e. ( B i^i C ) ) /\ ( B e. ran ( ball ` D ) /\ C e. ran ( ball ` D ) ) ) -> ran ( ball ` D ) C_ ~P X )
17	16, 2	sseldd 3185	. . . . . . . . . . . 12 |- ( ( ( D e. ( *Met ` X ) /\ P e. ( B i^i C ) ) /\ ( B e. ran ( ball ` D ) /\ C e. ran ( ball ` D ) ) ) -> B e. ~P X )
18	17	elpwid 3617	. . . . . . . . . . 11 |- ( ( ( D e. ( *Met ` X ) /\ P e. ( B i^i C ) ) /\ ( B e. ran ( ball ` D ) /\ C e. ran ( ball ` D ) ) ) -> B C_ X )
19	13, 18	sstrid 3195	. . . . . . . . . 10 |- ( ( ( D e. ( *Met ` X ) /\ P e. ( B i^i C ) ) /\ ( B e. ran ( ball ` D ) /\ C e. ran ( ball ` D ) ) ) -> ( B i^i C ) C_ X )
20	19, 3	sseldd 3185	. . . . . . . . 9 |- ( ( ( D e. ( *Met ` X ) /\ P e. ( B i^i C ) ) /\ ( B e. ran ( ball ` D ) /\ C e. ran ( ball ` D ) ) ) -> P e. X )
21	1, 20	jca 306	. . . . . . . 8 |- ( ( ( D e. ( *Met ` X ) /\ P e. ( B i^i C ) ) /\ ( B e. ran ( ball ` D ) /\ C e. ran ( ball ` D ) ) ) -> ( D e. ( *Met ` X ) /\ P e. X ) )
22		rpxr 9753	. . . . . . . . 9 |- ( y e. RR+ -> y e. RR* )
23		rpxr 9753	. . . . . . . . 9 |- ( z e. RR+ -> z e. RR* )
24	22, 23	anim12i 338	. . . . . . . 8 |- ( ( y e. RR+ /\ z e. RR+ ) -> ( y e. RR* /\ z e. RR* ) )
25		blininf 14744	. . . . . . . 8 |- ( ( ( D e. ( *Met ` X ) /\ P e. X ) /\ ( y e. RR* /\ z e. RR* ) ) -> ( ( P ( ball ` D ) y ) i^i ( P ( ball ` D ) z ) ) = ( P ( ball ` D )inf ( { y , z } , RR* , < ) ) )
26	21, 24, 25	syl2an 289	. . . . . . 7 |- ( ( ( ( D e. ( *Met ` X ) /\ P e. ( B i^i C ) ) /\ ( B e. ran ( ball ` D ) /\ C e. ran ( ball ` D ) ) ) /\ ( y e. RR+ /\ z e. RR+ ) ) -> ( ( P ( ball ` D ) y ) i^i ( P ( ball ` D ) z ) ) = ( P ( ball ` D )inf ( { y , z } , RR* , < ) ) )
27	26	sseq1d 3213	. . . . . 6 |- ( ( ( ( D e. ( *Met ` X ) /\ P e. ( B i^i C ) ) /\ ( B e. ran ( ball ` D ) /\ C e. ran ( ball ` D ) ) ) /\ ( y e. RR+ /\ z e. RR+ ) ) -> ( ( ( P ( ball ` D ) y ) i^i ( P ( ball ` D ) z ) ) C_ ( B i^i C ) <-> ( P ( ball ` D )inf ( { y , z } , RR* , < ) ) C_ ( B i^i C ) ) )
28		xrminrpcl 11456	. . . . . . . 8 |- ( ( y e. RR+ /\ z e. RR+ ) -> inf ( { y , z } , RR* , < ) e. RR+ )
29		oveq2 5933	. . . . . . . . . . 11 |- ( x = inf ( { y , z } , RR* , < ) -> ( P ( ball ` D ) x ) = ( P ( ball ` D )inf ( { y , z } , RR* , < ) ) )
30	29	sseq1d 3213	. . . . . . . . . 10 |- ( x = inf ( { y , z } , RR* , < ) -> ( ( P ( ball ` D ) x ) C_ ( B i^i C ) <-> ( P ( ball ` D )inf ( { y , z } , RR* , < ) ) C_ ( B i^i C ) ) )
31	30	rspcev 2868	. . . . . . . . 9 |- ( (inf ( { y , z } , RR* , < ) e. RR+ /\ ( P ( ball ` D )inf ( { y , z } , RR* , < ) ) C_ ( B i^i C ) ) -> E. x e. RR+ ( P ( ball ` D ) x ) C_ ( B i^i C ) )
32	31	ex 115	. . . . . . . 8 |- (inf ( { y , z } , RR* , < ) e. RR+ -> ( ( P ( ball ` D )inf ( { y , z } , RR* , < ) ) C_ ( B i^i C ) -> E. x e. RR+ ( P ( ball ` D ) x ) C_ ( B i^i C ) ) )
33	28, 32	syl 14	. . . . . . 7 |- ( ( y e. RR+ /\ z e. RR+ ) -> ( ( P ( ball ` D )inf ( { y , z } , RR* , < ) ) C_ ( B i^i C ) -> E. x e. RR+ ( P ( ball ` D ) x ) C_ ( B i^i C ) ) )
34	33	adantl 277	. . . . . 6 |- ( ( ( ( D e. ( *Met ` X ) /\ P e. ( B i^i C ) ) /\ ( B e. ran ( ball ` D ) /\ C e. ran ( ball ` D ) ) ) /\ ( y e. RR+ /\ z e. RR+ ) ) -> ( ( P ( ball ` D )inf ( { y , z } , RR* , < ) ) C_ ( B i^i C ) -> E. x e. RR+ ( P ( ball ` D ) x ) C_ ( B i^i C ) ) )
35	27, 34	sylbid 150	. . . . 5 |- ( ( ( ( D e. ( *Met ` X ) /\ P e. ( B i^i C ) ) /\ ( B e. ran ( ball ` D ) /\ C e. ran ( ball ` D ) ) ) /\ ( y e. RR+ /\ z e. RR+ ) ) -> ( ( ( P ( ball ` D ) y ) i^i ( P ( ball ` D ) z ) ) C_ ( B i^i C ) -> E. x e. RR+ ( P ( ball ` D ) x ) C_ ( B i^i C ) ) )
36	12, 35	syl5 32	. . . 4 |- ( ( ( ( D e. ( *Met ` X ) /\ P e. ( B i^i C ) ) /\ ( B e. ran ( ball ` D ) /\ C e. ran ( ball ` D ) ) ) /\ ( y e. RR+ /\ z e. RR+ ) ) -> ( ( ( P ( ball ` D ) y ) C_ B /\ ( P ( ball ` D ) z ) C_ C ) -> E. x e. RR+ ( P ( ball ` D ) x ) C_ ( B i^i C ) ) )
37	36	rexlimdvva 2622	. . 3 |- ( ( ( D e. ( *Met ` X ) /\ P e. ( B i^i C ) ) /\ ( B e. ran ( ball ` D ) /\ C e. ran ( ball ` D ) ) ) -> ( E. y e. RR+ E. z e. RR+ ( ( P ( ball ` D ) y ) C_ B /\ ( P ( ball ` D ) z ) C_ C ) -> E. x e. RR+ ( P ( ball ` D ) x ) C_ ( B i^i C ) ) )
38	11, 37	biimtrrid 153	. 2 |- ( ( ( D e. ( *Met ` X ) /\ P e. ( B i^i C ) ) /\ ( B e. ran ( ball ` D ) /\ C e. ran ( ball ` D ) ) ) -> ( ( E. y e. RR+ ( P ( ball ` D ) y ) C_ B /\ E. z e. RR+ ( P ( ball ` D ) z ) C_ C ) -> E. x e. RR+ ( P ( ball ` D ) x ) C_ ( B i^i C ) ) )
39	6, 10, 38	mp2and 433	1 |- ( ( ( D e. ( *Met ` X ) /\ P e. ( B i^i C ) ) /\ ( B e. ran ( ball ` D ) /\ C e. ran ( ball ` D ) ) ) -> E. x e. RR+ ( P ( ball ` D ) x ) C_ ( B i^i C ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1364   e. wcel 2167   E.wrex 2476   i^i cin 3156   C_ wss 3157   ~Pcpw 3606   {cpr 3624   X. cxp 4662   ran crn 4665   -->wf 5255   ` cfv 5259  (class class class)co 5925  infcinf 7058   RR*cxr 8077   < clt 8078   RR+crp 9745   *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-coll 4149  ax-sep 4152  ax-nul 4160  ax-pow 4208  ax-pr 4243  ax-un 4469  ax-setind 4574  ax-iinf 4625  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-apti 8011  ax-pre-ltadd 8012  ax-pre-mulgt0 8013  ax-pre-mulext 8014  ax-arch 8015  ax-caucvg 8016

This theorem depends on definitions:  df-bi 117  df-stab 832  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-rmo 2483  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-nul 3452  df-if 3563  df-pw 3608  df-sn 3629  df-pr 3630  df-op 3632  df-uni 3841  df-int 3876  df-iun 3919  df-br 4035  df-opab 4096  df-mpt 4097  df-tr 4133  df-id 4329  df-po 4332  df-iso 4333  df-iord 4402  df-on 4404  df-ilim 4405  df-suc 4407  df-iom 4628  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-f1 5264  df-fo 5265  df-f1o 5266  df-fv 5267  df-isom 5268  df-riota 5880  df-ov 5928  df-oprab 5929  df-mpo 5930  df-1st 6207  df-2nd 6208  df-recs 6372  df-frec 6458  df-map 6718  df-sup 7059  df-inf 7060  df-pnf 8080  df-mnf 8081  df-xr 8082  df-ltxr 8083  df-le 8084  df-sub 8216  df-neg 8217  df-reap 8619  df-ap 8626  df-div 8717  df-inn 9008  df-2 9066  df-3 9067  df-4 9068  df-n0 9267  df-z 9344  df-uz 9619  df-q 9711  df-rp 9746  df-xneg 9864  df-xadd 9865  df-seqfrec 10557  df-exp 10648  df-cj 11024  df-re 11025  df-im 11026  df-rsqrt 11180  df-abs 11181  df-psmet 14175  df-xmet 14176  df-bl 14178

This theorem is referenced by:  blbas  14753