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Theorem blin2 18464
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.
StepHypRef Expression
1 simpll 732 . . 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 734 . . 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 733 . . . . 5  |-  ( ( ( 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 elin 3532 . . . . 5  |-  ( P  e.  ( B  i^i  C )  <->  ( P  e.  B  /\  P  e.  C ) )
53, 4sylib 190 . . . 4  |-  ( ( ( D  e.  ( * Met `  X
)  /\  P  e.  ( B  i^i  C ) )  /\  ( B  e.  ran  ( ball `  D )  /\  C  e.  ran  ( ball `  D
) ) )  -> 
( P  e.  B  /\  P  e.  C
) )
65simpld 447 . . 3  |-  ( ( ( D  e.  ( * Met `  X
)  /\  P  e.  ( B  i^i  C ) )  /\  ( B  e.  ran  ( ball `  D )  /\  C  e.  ran  ( ball `  D
) ) )  ->  P  e.  B )
7 blss 18460 . . 3  |-  ( ( D  e.  ( * Met `  X )  /\  B  e.  ran  ( ball `  D )  /\  P  e.  B
)  ->  E. y  e.  RR+  ( P (
ball `  D )
y )  C_  B
)
81, 2, 6, 7syl3anc 1185 . 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 )
9 simprr 735 . . 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 ) )
105simprd 451 . . 3  |-  ( ( ( D  e.  ( * Met `  X
)  /\  P  e.  ( B  i^i  C ) )  /\  ( B  e.  ran  ( ball `  D )  /\  C  e.  ran  ( ball `  D
) ) )  ->  P  e.  C )
11 blss 18460 . . 3  |-  ( ( D  e.  ( * Met `  X )  /\  C  e.  ran  ( ball `  D )  /\  P  e.  C
)  ->  E. z  e.  RR+  ( P (
ball `  D )
z )  C_  C
)
121, 9, 10, 11syl3anc 1185 . 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 )
13 reeanv 2877 . . 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 ) )
14 ss2in 3570 . . . . 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 ) )
15 inss1 3563 . . . . . . . . . . 11  |-  ( B  i^i  C )  C_  B
16 blf 18442 . . . . . . . . . . . . . 14  |-  ( D  e.  ( * Met `  X )  ->  ( ball `  D ) : ( X  X.  RR* )
--> ~P X )
17 frn 5600 . . . . . . . . . . . . . 14  |-  ( (
ball `  D ) : ( X  X.  RR* ) --> ~P X  ->  ran  ( ball `  D
)  C_  ~P X
)
181, 16, 173syl 19 . . . . . . . . . . . . 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
)
1918, 2sseldd 3351 . . . . . . . . . . . 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
)
2019elpwid 3810 . . . . . . . . . . 11  |-  ( ( ( D  e.  ( * Met `  X
)  /\  P  e.  ( B  i^i  C ) )  /\  ( B  e.  ran  ( ball `  D )  /\  C  e.  ran  ( ball `  D
) ) )  ->  B  C_  X )
2115, 20syl5ss 3361 . . . . . . . . . 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 )
2221, 3sseldd 3351 . . . . . . . . 9  |-  ( ( ( D  e.  ( * Met `  X
)  /\  P  e.  ( B  i^i  C ) )  /\  ( B  e.  ran  ( ball `  D )  /\  C  e.  ran  ( ball `  D
) ) )  ->  P  e.  X )
231, 22jca 520 . . . . . . . 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 ) )
24 rpxr 10624 . . . . . . . . 9  |-  ( y  e.  RR+  ->  y  e. 
RR* )
25 rpxr 10624 . . . . . . . . 9  |-  ( z  e.  RR+  ->  z  e. 
RR* )
2624, 25anim12i 551 . . . . . . . 8  |-  ( ( y  e.  RR+  /\  z  e.  RR+ )  ->  (
y  e.  RR*  /\  z  e.  RR* ) )
27 blin 18456 . . . . . . . 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 ) if ( y  <_  z ,  y ,  z ) ) )
2823, 26, 27syl2an 465 . . . . . . 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
) if ( y  <_  z ,  y ,  z ) ) )
2928sseq1d 3377 . . . . . 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 ) if ( y  <_  z , 
y ,  z ) )  C_  ( B  i^i  C ) ) )
30 ifcl 3777 . . . . . . . 8  |-  ( ( y  e.  RR+  /\  z  e.  RR+ )  ->  if ( y  <_  z ,  y ,  z )  e.  RR+ )
31 oveq2 6092 . . . . . . . . . . 11  |-  ( x  =  if ( y  <_  z ,  y ,  z )  -> 
( P ( ball `  D ) x )  =  ( P (
ball `  D ) if ( y  <_  z ,  y ,  z ) ) )
3231sseq1d 3377 . . . . . . . . . 10  |-  ( x  =  if ( y  <_  z ,  y ,  z )  -> 
( ( P (
ball `  D )
x )  C_  ( B  i^i  C )  <->  ( P
( ball `  D ) if ( y  <_  z ,  y ,  z ) )  C_  ( B  i^i  C ) ) )
3332rspcev 3054 . . . . . . . . 9  |-  ( ( if ( y  <_ 
z ,  y ,  z )  e.  RR+  /\  ( P ( ball `  D ) if ( y  <_  z , 
y ,  z ) )  C_  ( B  i^i  C ) )  ->  E. x  e.  RR+  ( P ( ball `  D
) x )  C_  ( B  i^i  C ) )
3433ex 425 . . . . . . . 8  |-  ( if ( y  <_  z ,  y ,  z )  e.  RR+  ->  ( ( P ( ball `  D ) if ( y  <_  z , 
y ,  z ) )  C_  ( B  i^i  C )  ->  E. x  e.  RR+  ( P (
ball `  D )
x )  C_  ( B  i^i  C ) ) )
3530, 34syl 16 . . . . . . 7  |-  ( ( y  e.  RR+  /\  z  e.  RR+ )  ->  (
( P ( ball `  D ) if ( y  <_  z , 
y ,  z ) )  C_  ( B  i^i  C )  ->  E. x  e.  RR+  ( P (
ball `  D )
x )  C_  ( B  i^i  C ) ) )
3635adantl 454 . . . . . 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
) if ( y  <_  z ,  y ,  z ) ) 
C_  ( B  i^i  C )  ->  E. x  e.  RR+  ( P (
ball `  D )
x )  C_  ( B  i^i  C ) ) )
3729, 36sylbid 208 . . . . 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 ) ) )
3814, 37syl5 31 . . . 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 ) ) )
3938rexlimdvva 2839 . . 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 ) ) )
4013, 39syl5bir 211 . 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 ) ) )
418, 12, 40mp2and 662 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 ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360    = wceq 1653    e. wcel 1726   E.wrex 2708    i^i cin 3321    C_ wss 3322   ifcif 3741   ~Pcpw 3801   class class class wbr 4215    X. cxp 4879   ran crn 4882   -->wf 5453   ` cfv 5457  (class class class)co 6084   RR*cxr 9124    <_ cle 9126   RR+crp 10617   * Metcxmt 16691   ballcbl 16693
This theorem is referenced by:  blbas  18465
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704  ax-cnex 9051  ax-resscn 9052  ax-1cn 9053  ax-icn 9054  ax-addcl 9055  ax-addrcl 9056  ax-mulcl 9057  ax-mulrcl 9058  ax-mulcom 9059  ax-addass 9060  ax-mulass 9061  ax-distr 9062  ax-i2m1 9063  ax-1ne0 9064  ax-1rid 9065  ax-rnegex 9066  ax-rrecex 9067  ax-cnre 9068  ax-pre-lttri 9069  ax-pre-lttrn 9070  ax-pre-ltadd 9071  ax-pre-mulgt0 9072  ax-pre-sup 9073
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rmo 2715  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-iun 4097  df-br 4216  df-opab 4270  df-mpt 4271  df-tr 4306  df-eprel 4497  df-id 4501  df-po 4506  df-so 4507  df-fr 4544  df-we 4546  df-ord 4587  df-on 4588  df-lim 4589  df-suc 4590  df-om 4849  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-f1 5462  df-fo 5463  df-f1o 5464  df-fv 5465  df-ov 6087  df-oprab 6088  df-mpt2 6089  df-1st 6352  df-2nd 6353  df-riota 6552  df-recs 6636  df-rdg 6671  df-er 6908  df-map 7023  df-en 7113  df-dom 7114  df-sdom 7115  df-sup 7449  df-pnf 9127  df-mnf 9128  df-xr 9129  df-ltxr 9130  df-le 9131  df-sub 9298  df-neg 9299  df-div 9683  df-nn 10006  df-2 10063  df-n0 10227  df-z 10288  df-uz 10494  df-q 10580  df-rp 10618  df-xneg 10715  df-xadd 10716  df-xmul 10717  df-psmet 16699  df-xmet 16700  df-bl 16702
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