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Theorem blin2 18416
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 731 . . 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 733 . . 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 732 . . . . 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 3494 . . . . 5  |-  ( P  e.  ( B  i^i  C )  <->  ( P  e.  B  /\  P  e.  C ) )
53, 4sylib 189 . . . 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 446 . . 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 18412 . . 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 1184 . 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 734 . . 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 450 . . 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 18412 . . 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 1184 . 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 2839 . . 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 3532 . . . . 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 3525 . . . . . . . . . . 11  |-  ( B  i^i  C )  C_  B
16 blf 18394 . . . . . . . . . . . . . 14  |-  ( D  e.  ( * Met `  X )  ->  ( ball `  D ) : ( X  X.  RR* )
--> ~P X )
17 frn 5560 . . . . . . . . . . . . . 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 3313 . . . . . . . . . . . 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 3772 . . . . . . . . . . 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 3323 . . . . . . . . . 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 3313 . . . . . . . . 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 519 . . . . . . . 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 10579 . . . . . . . . 9  |-  ( y  e.  RR+  ->  y  e. 
RR* )
25 rpxr 10579 . . . . . . . . 9  |-  ( z  e.  RR+  ->  z  e. 
RR* )
2624, 25anim12i 550 . . . . . . . 8  |-  ( ( y  e.  RR+  /\  z  e.  RR+ )  ->  (
y  e.  RR*  /\  z  e.  RR* ) )
27 blin 18408 . . . . . . . 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 464 . . . . . . 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 3339 . . . . . 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 3739 . . . . . . . 8  |-  ( ( y  e.  RR+  /\  z  e.  RR+ )  ->  if ( y  <_  z ,  y ,  z )  e.  RR+ )
31 oveq2 6052 . . . . . . . . . . 11  |-  ( x  =  if ( y  <_  z ,  y ,  z )  -> 
( P ( ball `  D ) x )  =  ( P (
ball `  D ) if ( y  <_  z ,  y ,  z ) ) )
3231sseq1d 3339 . . . . . . . . . 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 3016 . . . . . . . . 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 424 . . . . . . . 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 453 . . . . . 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 207 . . . . 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 30 . . . 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 2801 . . 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 210 . 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 661 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 359    = wceq 1649    e. wcel 1721   E.wrex 2671    i^i cin 3283    C_ wss 3284   ifcif 3703   ~Pcpw 3763   class class class wbr 4176    X. cxp 4839   ran crn 4842   -->wf 5413   ` cfv 5417  (class class class)co 6044   RR*cxr 9079    <_ cle 9081   RR+crp 10572   * Metcxmt 16645   ballcbl 16647
This theorem is referenced by:  blbas  18417
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2389  ax-sep 4294  ax-nul 4302  ax-pow 4341  ax-pr 4367  ax-un 4664  ax-cnex 9006  ax-resscn 9007  ax-1cn 9008  ax-icn 9009  ax-addcl 9010  ax-addrcl 9011  ax-mulcl 9012  ax-mulrcl 9013  ax-mulcom 9014  ax-addass 9015  ax-mulass 9016  ax-distr 9017  ax-i2m1 9018  ax-1ne0 9019  ax-1rid 9020  ax-rnegex 9021  ax-rrecex 9022  ax-cnre 9023  ax-pre-lttri 9024  ax-pre-lttrn 9025  ax-pre-ltadd 9026  ax-pre-mulgt0 9027  ax-pre-sup 9028
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2262  df-mo 2263  df-clab 2395  df-cleq 2401  df-clel 2404  df-nfc 2533  df-ne 2573  df-nel 2574  df-ral 2675  df-rex 2676  df-reu 2677  df-rmo 2678  df-rab 2679  df-v 2922  df-sbc 3126  df-csb 3216  df-dif 3287  df-un 3289  df-in 3291  df-ss 3298  df-pss 3300  df-nul 3593  df-if 3704  df-pw 3765  df-sn 3784  df-pr 3785  df-tp 3786  df-op 3787  df-uni 3980  df-iun 4059  df-br 4177  df-opab 4231  df-mpt 4232  df-tr 4267  df-eprel 4458  df-id 4462  df-po 4467  df-so 4468  df-fr 4505  df-we 4507  df-ord 4548  df-on 4549  df-lim 4550  df-suc 4551  df-om 4809  df-xp 4847  df-rel 4848  df-cnv 4849  df-co 4850  df-dm 4851  df-rn 4852  df-res 4853  df-ima 4854  df-iota 5381  df-fun 5419  df-fn 5420  df-f 5421  df-f1 5422  df-fo 5423  df-f1o 5424  df-fv 5425  df-ov 6047  df-oprab 6048  df-mpt2 6049  df-1st 6312  df-2nd 6313  df-riota 6512  df-recs 6596  df-rdg 6631  df-er 6868  df-map 6983  df-en 7073  df-dom 7074  df-sdom 7075  df-sup 7408  df-pnf 9082  df-mnf 9083  df-xr 9084  df-ltxr 9085  df-le 9086  df-sub 9253  df-neg 9254  df-div 9638  df-nn 9961  df-2 10018  df-n0 10182  df-z 10243  df-uz 10449  df-q 10535  df-rp 10573  df-xneg 10670  df-xadd 10671  df-xmul 10672  df-psmet 16653  df-xmet 16654  df-bl 16656
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