Theorem blin2 12604

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 518	. . 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 520	. . 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 519	. . . 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 3265	. . 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 12600	. . 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 1216	. 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 521	. . 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 3266	. . 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 12600	. . 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 1216	. 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 2600	. . 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 3304	. . . . 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 3296	. . . . . . . . . . 11 |- ( B i^i C ) C_ B
14		blf 12582	. . . . . . . . . . . . . 14 |- ( D e. ( *Met ` X ) -> ( ball ` D ) : ( X X. RR* ) --> ~P X )
15		frn 5281	. . . . . . . . . . . . . 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 3098	. . . . . . . . . . . 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 3521	. . . . . . . . . . 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 3108	. . . . . . . . . 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 3098	. . . . . . . . 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 304	. . . . . . . 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 9452	. . . . . . . . 9 |- ( y e. RR+ -> y e. RR* )
23		rpxr 9452	. . . . . . . . 9 |- ( z e. RR+ -> z e. RR* )
24	22, 23	anim12i 336	. . . . . . . 8 |- ( ( y e. RR+ /\ z e. RR+ ) -> ( y e. RR* /\ z e. RR* ) )
25		blininf 12596	. . . . . . . 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 287	. . . . . . 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 3126	. . . . . 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 11046	. . . . . . . 8 |- ( ( y e. RR+ /\ z e. RR+ ) -> inf ( { y , z } , RR* , < ) e. RR+ )
29		oveq2 5782	. . . . . . . . . . 11 |- ( x = inf ( { y , z } , RR* , < ) -> ( P ( ball ` D ) x ) = ( P ( ball ` D )inf ( { y , z } , RR* , < ) ) )
30	29	sseq1d 3126	. . . . . . . . . 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 2789	. . . . . . . . 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 114	. . . . . . . 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 275	. . . . . 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 149	. . . . 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 2557	. . 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	syl5bir 152	. 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 429	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 103   = wceq 1331   e. wcel 1480   E.wrex 2417   i^i cin 3070   C_ wss 3071   ~Pcpw 3510   {cpr 3528   X. cxp 4537   ran crn 4540   -->wf 5119   ` cfv 5123  (class class class)co 5774  infcinf 6870   RR*cxr 7802   < clt 7803   RR+crp 9444   *Metcxmet 12152   ballcbl 12154

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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-coll 4043  ax-sep 4046  ax-nul 4054  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452  ax-iinf 4502  ax-cnex 7714  ax-resscn 7715  ax-1cn 7716  ax-1re 7717  ax-icn 7718  ax-addcl 7719  ax-addrcl 7720  ax-mulcl 7721  ax-mulrcl 7722  ax-addcom 7723  ax-mulcom 7724  ax-addass 7725  ax-mulass 7726  ax-distr 7727  ax-i2m1 7728  ax-0lt1 7729  ax-1rid 7730  ax-0id 7731  ax-rnegex 7732  ax-precex 7733  ax-cnre 7734  ax-pre-ltirr 7735  ax-pre-ltwlin 7736  ax-pre-lttrn 7737  ax-pre-apti 7738  ax-pre-ltadd 7739  ax-pre-mulgt0 7740  ax-pre-mulext 7741  ax-arch 7742  ax-caucvg 7743

This theorem depends on definitions:  df-bi 116  df-stab 816  df-dc 820  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-nel 2404  df-ral 2421  df-rex 2422  df-reu 2423  df-rmo 2424  df-rab 2425  df-v 2688  df-sbc 2910  df-csb 3004  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-nul 3364  df-if 3475  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-int 3772  df-iun 3815  df-br 3930  df-opab 3990  df-mpt 3991  df-tr 4027  df-id 4215  df-po 4218  df-iso 4219  df-iord 4288  df-on 4290  df-ilim 4291  df-suc 4293  df-iom 4505  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-f1 5128  df-fo 5129  df-f1o 5130  df-fv 5131  df-isom 5132  df-riota 5730  df-ov 5777  df-oprab 5778  df-mpo 5779  df-1st 6038  df-2nd 6039  df-recs 6202  df-frec 6288  df-map 6544  df-sup 6871  df-inf 6872  df-pnf 7805  df-mnf 7806  df-xr 7807  df-ltxr 7808  df-le 7809  df-sub 7938  df-neg 7939  df-reap 8340  df-ap 8347  df-div 8436  df-inn 8724  df-2 8782  df-3 8783  df-4 8784  df-n0 8981  df-z 9058  df-uz 9330  df-q 9415  df-rp 9445  df-xneg 9562  df-xadd 9563  df-seqfrec 10222  df-exp 10296  df-cj 10617  df-re 10618  df-im 10619  df-rsqrt 10773  df-abs 10774  df-psmet 12159  df-xmet 12160  df-bl 12162

This theorem is referenced by:  blbas  12605