Theorem blin2 15516

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 531	. . 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 535	. . 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 533	. . . 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 3418	. . 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 15512	. . 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 1278	. 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 537	. . 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 3419	. . 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 15512	. . 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 1278	. 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 2721	. . 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 3459	. . . . 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 3451	. . . . . . . . . . 11 |- ( B i^i C ) C_ B
14		blf 15494	. . . . . . . . . . . . . 14 |- ( D e. ( *Met ` X ) -> ( ball ` D ) : ( X X. RR* ) --> ~P X )
15		frn 5540	. . . . . . . . . . . . . 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 3249	. . . . . . . . . . . 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 3699	. . . . . . . . . . 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 3259	. . . . . . . . . 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 3249	. . . . . . . . 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 10045	. . . . . . . . 9 |- ( y e. RR+ -> y e. RR* )
23		rpxr 10045	. . . . . . . . 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 15508	. . . . . . . 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 3277	. . . . . 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 12023	. . . . . . . 8 |- ( ( y e. RR+ /\ z e. RR+ ) -> inf ( { y , z } , RR* , < ) e. RR+ )
29		oveq2 6087	. . . . . . . . . . 11 |- ( x = inf ( { y , z } , RR* , < ) -> ( P ( ball ` D ) x ) = ( P ( ball ` D )inf ( { y , z } , RR* , < ) ) )
30	29	sseq1d 3277	. . . . . . . . . 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 2929	. . . . . . . . 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 2676	. . 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 437	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 1402   e. wcel 2209   E.wrex 2529   i^i cin 3219   C_ wss 3220   ~Pcpw 3688   {cpr 3709   X. cxp 4770   ran crn 4773   -->wf 5371   ` cfv 5375  (class class class)co 6079  infcinf 7317   RR*cxr 8353   < clt 8354   RR+crp 10037   *Metcxmet 14856   ballcbl 14858

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4244  ax-sep 4247  ax-nul 4257  ax-pow 4309  ax-pr 4344  ax-un 4576  ax-setind 4682  ax-iinf 4733  ax-cnex 8264  ax-resscn 8265  ax-1cn 8266  ax-1re 8267  ax-icn 8268  ax-addcl 8269  ax-addrcl 8270  ax-mulcl 8271  ax-mulrcl 8272  ax-addcom 8273  ax-mulcom 8274  ax-addass 8275  ax-mulass 8276  ax-distr 8277  ax-i2m1 8278  ax-0lt1 8279  ax-1rid 8280  ax-0id 8281  ax-rnegex 8282  ax-precex 8283  ax-cnre 8284  ax-pre-ltirr 8285  ax-pre-ltwlin 8286  ax-pre-lttrn 8287  ax-pre-apti 8288  ax-pre-ltadd 8289  ax-pre-mulgt0 8290  ax-pre-mulext 8291  ax-arch 8292  ax-caucvg 8293

This theorem depends on definitions:  df-bi 117  df-stab 843  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-if 3639  df-pw 3690  df-sn 3714  df-pr 3715  df-op 3717  df-uni 3934  df-int 3969  df-iun 4012  df-br 4129  df-opab 4191  df-mpt 4192  df-tr 4228  df-id 4436  df-po 4439  df-iso 4440  df-iord 4509  df-on 4511  df-ilim 4512  df-suc 4514  df-iom 4736  df-xp 4778  df-rel 4779  df-cnv 4780  df-co 4781  df-dm 4782  df-rn 4783  df-res 4784  df-ima 4785  df-iota 5335  df-fun 5377  df-fn 5378  df-f 5379  df-f1 5380  df-fo 5381  df-f1o 5382  df-fv 5383  df-isom 5384  df-riota 6032  df-ov 6082  df-oprab 6083  df-mpo 6084  df-1st 6368  df-2nd 6369  df-recs 6570  df-frec 6656  df-map 6918  df-sup 7318  df-inf 7319  df-pnf 8356  df-mnf 8357  df-xr 8358  df-ltxr 8359  df-le 8360  df-sub 8493  df-neg 8494  df-reap 8897  df-ap 8904  df-div 8997  df-inn 9288  df-2 9346  df-3 9347  df-4 9348  df-n0 9547  df-z 9628  df-uz 9905  df-q 10003  df-rp 10038  df-xneg 10157  df-xadd 10158  df-seqfrec 10868  df-exp 10959  df-cj 11590  df-re 11591  df-im 11592  df-rsqrt 11747  df-abs 11748  df-psmet 14863  df-xmet 14864  df-bl 14866

This theorem is referenced by:  blbas  15517