Theorem blin2 13226

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 524	. . 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 526	. . 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 525	. . . 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 3316	. . 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 13222	. . 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 1233	. 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 527	. . 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 3317	. . 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 13222	. . 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 1233	. 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 2639	. . 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 3355	. . . . 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 3347	. . . . . . . . . . 11 |- ( B i^i C ) C_ B
14		blf 13204	. . . . . . . . . . . . . 14 |- ( D e. ( *Met ` X ) -> ( ball ` D ) : ( X X. RR* ) --> ~P X )
15		frn 5356	. . . . . . . . . . . . . 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 3148	. . . . . . . . . . . 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 3577	. . . . . . . . . . 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 3158	. . . . . . . . . 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 3148	. . . . . . . . 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 9618	. . . . . . . . 9 |- ( y e. RR+ -> y e. RR* )
23		rpxr 9618	. . . . . . . . 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 13218	. . . . . . . 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 3176	. . . . . 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 11237	. . . . . . . 8 |- ( ( y e. RR+ /\ z e. RR+ ) -> inf ( { y , z } , RR* , < ) e. RR+ )
29		oveq2 5861	. . . . . . . . . . 11 |- ( x = inf ( { y , z } , RR* , < ) -> ( P ( ball ` D ) x ) = ( P ( ball ` D )inf ( { y , z } , RR* , < ) ) )
30	29	sseq1d 3176	. . . . . . . . . 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 2834	. . . . . . . . 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 2595	. . 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 431	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 1348   e. wcel 2141   E.wrex 2449   i^i cin 3120   C_ wss 3121   ~Pcpw 3566   {cpr 3584   X. cxp 4609   ran crn 4612   -->wf 5194   ` cfv 5198  (class class class)co 5853  infcinf 6960   RR*cxr 7953   < clt 7954   RR+crp 9610   *Metcxmet 12774   ballcbl 12776

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-coll 4104  ax-sep 4107  ax-nul 4115  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521  ax-iinf 4572  ax-cnex 7865  ax-resscn 7866  ax-1cn 7867  ax-1re 7868  ax-icn 7869  ax-addcl 7870  ax-addrcl 7871  ax-mulcl 7872  ax-mulrcl 7873  ax-addcom 7874  ax-mulcom 7875  ax-addass 7876  ax-mulass 7877  ax-distr 7878  ax-i2m1 7879  ax-0lt1 7880  ax-1rid 7881  ax-0id 7882  ax-rnegex 7883  ax-precex 7884  ax-cnre 7885  ax-pre-ltirr 7886  ax-pre-ltwlin 7887  ax-pre-lttrn 7888  ax-pre-apti 7889  ax-pre-ltadd 7890  ax-pre-mulgt0 7891  ax-pre-mulext 7892  ax-arch 7893  ax-caucvg 7894

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-nul 3415  df-if 3527  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-int 3832  df-iun 3875  df-br 3990  df-opab 4051  df-mpt 4052  df-tr 4088  df-id 4278  df-po 4281  df-iso 4282  df-iord 4351  df-on 4353  df-ilim 4354  df-suc 4356  df-iom 4575  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-f1 5203  df-fo 5204  df-f1o 5205  df-fv 5206  df-isom 5207  df-riota 5809  df-ov 5856  df-oprab 5857  df-mpo 5858  df-1st 6119  df-2nd 6120  df-recs 6284  df-frec 6370  df-map 6628  df-sup 6961  df-inf 6962  df-pnf 7956  df-mnf 7957  df-xr 7958  df-ltxr 7959  df-le 7960  df-sub 8092  df-neg 8093  df-reap 8494  df-ap 8501  df-div 8590  df-inn 8879  df-2 8937  df-3 8938  df-4 8939  df-n0 9136  df-z 9213  df-uz 9488  df-q 9579  df-rp 9611  df-xneg 9729  df-xadd 9730  df-seqfrec 10402  df-exp 10476  df-cj 10806  df-re 10807  df-im 10808  df-rsqrt 10962  df-abs 10963  df-psmet 12781  df-xmet 12782  df-bl 12784

This theorem is referenced by:  blbas  13227