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Theorem blin2 18188
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 730 . . 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 732 . . 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 731 . . . . 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 3446 . . . . 5  |-  ( P  e.  ( B  i^i  C )  <->  ( P  e.  B  /\  P  e.  C ) )
53, 4sylib 188 . . . 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 445 . . 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 18185 . . 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 1183 . 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 733 . . 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 449 . . 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 18185 . . 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 1183 . 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 2792 . . 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 3484 . . . . 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 3477 . . . . . . . . . . 11  |-  ( B  i^i  C )  C_  B
16 blf 18174 . . . . . . . . . . . . . 14  |-  ( D  e.  ( * Met `  X )  ->  ( ball `  D ) : ( X  X.  RR* )
--> ~P X )
17 frn 5501 . . . . . . . . . . . . . 14  |-  ( (
ball `  D ) : ( X  X.  RR* ) --> ~P X  ->  ran  ( ball `  D
)  C_  ~P X
)
181, 16, 173syl 18 . . . . . . . . . . . . 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 3267 . . . . . . . . . . . 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
)
20 elpwi 3722 . . . . . . . . . . . 12  |-  ( B  e.  ~P X  ->  B  C_  X )
2119, 20syl 15 . . . . . . . . . . 11  |-  ( ( ( D  e.  ( * Met `  X
)  /\  P  e.  ( B  i^i  C ) )  /\  ( B  e.  ran  ( ball `  D )  /\  C  e.  ran  ( ball `  D
) ) )  ->  B  C_  X )
2215, 21syl5ss 3276 . . . . . . . . . 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 )
2322, 3sseldd 3267 . . . . . . . . 9  |-  ( ( ( D  e.  ( * Met `  X
)  /\  P  e.  ( B  i^i  C ) )  /\  ( B  e.  ran  ( ball `  D )  /\  C  e.  ran  ( ball `  D
) ) )  ->  P  e.  X )
241, 23jca 518 . . . . . . . 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 ) )
25 rpxr 10512 . . . . . . . . 9  |-  ( y  e.  RR+  ->  y  e. 
RR* )
26 rpxr 10512 . . . . . . . . 9  |-  ( z  e.  RR+  ->  z  e. 
RR* )
2725, 26anim12i 549 . . . . . . . 8  |-  ( ( y  e.  RR+  /\  z  e.  RR+ )  ->  (
y  e.  RR*  /\  z  e.  RR* ) )
28 blin 18183 . . . . . . . 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 ) ) )
2924, 27, 28syl2an 463 . . . . . . 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 ) ) )
3029sseq1d 3291 . . . . . 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 ) ) )
31 ifcl 3690 . . . . . . . 8  |-  ( ( y  e.  RR+  /\  z  e.  RR+ )  ->  if ( y  <_  z ,  y ,  z )  e.  RR+ )
32 oveq2 5989 . . . . . . . . . . 11  |-  ( x  =  if ( y  <_  z ,  y ,  z )  -> 
( P ( ball `  D ) x )  =  ( P (
ball `  D ) if ( y  <_  z ,  y ,  z ) ) )
3332sseq1d 3291 . . . . . . . . . 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 ) ) )
3433rspcev 2969 . . . . . . . . 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 ) )
3534ex 423 . . . . . . . 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 ) ) )
3631, 35syl 15 . . . . . . 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 ) ) )
3736adantl 452 . . . . . 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 ) ) )
3830, 37sylbid 206 . . . . 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 ) ) )
3914, 38syl5 28 . . . 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 ) ) )
4039rexlimdvva 2759 . . 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 ) ) )
4113, 40syl5bir 209 . 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 ) ) )
428, 12, 41mp2and 660 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 358    = wceq 1647    e. wcel 1715   E.wrex 2629    i^i cin 3237    C_ wss 3238   ifcif 3654   ~Pcpw 3714   class class class wbr 4125    X. cxp 4790   ran crn 4793   -->wf 5354   ` cfv 5358  (class class class)co 5981   RR*cxr 9013    <_ cle 9015   RR+crp 10505   * Metcxmt 16579   ballcbl 16581
This theorem is referenced by:  blbas  18189
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1551  ax-5 1562  ax-17 1621  ax-9 1659  ax-8 1680  ax-13 1717  ax-14 1719  ax-6 1734  ax-7 1739  ax-11 1751  ax-12 1937  ax-ext 2347  ax-sep 4243  ax-nul 4251  ax-pow 4290  ax-pr 4316  ax-un 4615  ax-cnex 8940  ax-resscn 8941  ax-1cn 8942  ax-icn 8943  ax-addcl 8944  ax-addrcl 8945  ax-mulcl 8946  ax-mulrcl 8947  ax-mulcom 8948  ax-addass 8949  ax-mulass 8950  ax-distr 8951  ax-i2m1 8952  ax-1ne0 8953  ax-1rid 8954  ax-rnegex 8955  ax-rrecex 8956  ax-cnre 8957  ax-pre-lttri 8958  ax-pre-lttrn 8959  ax-pre-ltadd 8960  ax-pre-mulgt0 8961  ax-pre-sup 8962
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 936  df-3an 937  df-tru 1324  df-ex 1547  df-nf 1550  df-sb 1654  df-eu 2221  df-mo 2222  df-clab 2353  df-cleq 2359  df-clel 2362  df-nfc 2491  df-ne 2531  df-nel 2532  df-ral 2633  df-rex 2634  df-reu 2635  df-rmo 2636  df-rab 2637  df-v 2875  df-sbc 3078  df-csb 3168  df-dif 3241  df-un 3243  df-in 3245  df-ss 3252  df-pss 3254  df-nul 3544  df-if 3655  df-pw 3716  df-sn 3735  df-pr 3736  df-tp 3737  df-op 3738  df-uni 3930  df-iun 4009  df-br 4126  df-opab 4180  df-mpt 4181  df-tr 4216  df-eprel 4408  df-id 4412  df-po 4417  df-so 4418  df-fr 4455  df-we 4457  df-ord 4498  df-on 4499  df-lim 4500  df-suc 4501  df-om 4760  df-xp 4798  df-rel 4799  df-cnv 4800  df-co 4801  df-dm 4802  df-rn 4803  df-res 4804  df-ima 4805  df-iota 5322  df-fun 5360  df-fn 5361  df-f 5362  df-f1 5363  df-fo 5364  df-f1o 5365  df-fv 5366  df-ov 5984  df-oprab 5985  df-mpt2 5986  df-1st 6249  df-2nd 6250  df-riota 6446  df-recs 6530  df-rdg 6565  df-er 6802  df-map 6917  df-en 7007  df-dom 7008  df-sdom 7009  df-sup 7341  df-pnf 9016  df-mnf 9017  df-xr 9018  df-ltxr 9019  df-le 9020  df-sub 9186  df-neg 9187  df-div 9571  df-nn 9894  df-2 9951  df-n0 10115  df-z 10176  df-uz 10382  df-q 10468  df-rp 10506  df-xneg 10603  df-xadd 10604  df-xmul 10605  df-xmet 16586  df-bl 16588
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