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Theorem blin2 17975
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 3358 . . . . 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 17972 . . 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 1182 . 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 17972 . . 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 1182 . 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 2707 . . 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 3396 . . . . 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 3389 . . . . . . . . . . 11  |-  ( B  i^i  C )  C_  B
16 blf 17961 . . . . . . . . . . . . . 14  |-  ( D  e.  ( * Met `  X )  ->  ( ball `  D ) : ( X  X.  RR* )
--> ~P X )
17 frn 5395 . . . . . . . . . . . . . 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 3181 . . . . . . . . . . . 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 3633 . . . . . . . . . . . 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 3190 . . . . . . . . . 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 3181 . . . . . . . . 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 10361 . . . . . . . . 9  |-  ( y  e.  RR+  ->  y  e. 
RR* )
26 rpxr 10361 . . . . . . . . 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 17970 . . . . . . . 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 3205 . . . . . 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 3601 . . . . . . . 8  |-  ( ( y  e.  RR+  /\  z  e.  RR+ )  ->  if ( y  <_  z ,  y ,  z )  e.  RR+ )
32 oveq2 5866 . . . . . . . . . . 11  |-  ( x  =  if ( y  <_  z ,  y ,  z )  -> 
( P ( ball `  D ) x )  =  ( P (
ball `  D ) if ( y  <_  z ,  y ,  z ) ) )
3332sseq1d 3205 . . . . . . . . . 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 2884 . . . . . . . . 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 2674 . . 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 1623    e. wcel 1684   E.wrex 2544    i^i cin 3151    C_ wss 3152   ifcif 3565   ~Pcpw 3625   class class class wbr 4023    X. cxp 4687   ran crn 4690   -->wf 5251   ` cfv 5255  (class class class)co 5858   RR*cxr 8866    <_ cle 8868   RR+crp 10354   * Metcxmt 16369   ballcbl 16371
This theorem is referenced by:  blbas  17976
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814  ax-pre-sup 8815
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-er 6660  df-map 6774  df-en 6864  df-dom 6865  df-sdom 6866  df-sup 7194  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-nn 9747  df-2 9804  df-n0 9966  df-z 10025  df-uz 10231  df-q 10317  df-rp 10355  df-xneg 10452  df-xadd 10453  df-xmul 10454  df-xmet 16373  df-bl 16375
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