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Theorem blin2 21114
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 752 . . 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 755 . . 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 754 . . . . 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 3623 . . . . 5  |-  ( P  e.  ( B  i^i  C )  <->  ( P  e.  B  /\  P  e.  C ) )
53, 4sylib 196 . . . 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 457 . . 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 21110 . . 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 1228 . 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 756 . . 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 461 . . 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 21110 . . 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 1228 . 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 2972 . . 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 3663 . . . . 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 3656 . . . . . . . . . . 11  |-  ( B  i^i  C )  C_  B
16 blf 21092 . . . . . . . . . . . . . 14  |-  ( D  e.  ( *Met `  X )  ->  ( ball `  D ) : ( X  X.  RR* )
--> ~P X )
17 frn 5674 . . . . . . . . . . . . . 14  |-  ( (
ball `  D ) : ( X  X.  RR* ) --> ~P X  ->  ran  ( ball `  D
)  C_  ~P X
)
181, 16, 173syl 20 . . . . . . . . . . . . 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 3440 . . . . . . . . . . . 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 3962 . . . . . . . . . . 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 3450 . . . . . . . . . 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 3440 . . . . . . . . 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 530 . . . . . . . 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 11188 . . . . . . . . 9  |-  ( y  e.  RR+  ->  y  e. 
RR* )
25 rpxr 11188 . . . . . . . . 9  |-  ( z  e.  RR+  ->  z  e. 
RR* )
2624, 25anim12i 564 . . . . . . . 8  |-  ( ( y  e.  RR+  /\  z  e.  RR+ )  ->  (
y  e.  RR*  /\  z  e.  RR* ) )
27 blin 21106 . . . . . . . 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 475 . . . . . . 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 3466 . . . . . 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 3924 . . . . . . . 8  |-  ( ( y  e.  RR+  /\  z  e.  RR+ )  ->  if ( y  <_  z ,  y ,  z )  e.  RR+ )
31 oveq2 6240 . . . . . . . . . . 11  |-  ( x  =  if ( y  <_  z ,  y ,  z )  -> 
( P ( ball `  D ) x )  =  ( P (
ball `  D ) if ( y  <_  z ,  y ,  z ) ) )
3231sseq1d 3466 . . . . . . . . . 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 3157 . . . . . . . . 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 432 . . . . . . . 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 17 . . . . . . 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 464 . . . . . 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 215 . . . . 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 2900 . . 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 218 . 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 677 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 setvar class
Syntax hints:    -> wi 4    /\ wa 367    = wceq 1403    e. wcel 1840   E.wrex 2752    i^i cin 3410    C_ wss 3411   ifcif 3882   ~Pcpw 3952   class class class wbr 4392    X. cxp 4938   ran crn 4941   -->wf 5519   ` cfv 5523  (class class class)co 6232   RR*cxr 9575    <_ cle 9577   RR+crp 11181   *Metcxmt 18613   ballcbl 18615
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1637  ax-4 1650  ax-5 1723  ax-6 1769  ax-7 1812  ax-8 1842  ax-9 1844  ax-10 1859  ax-11 1864  ax-12 1876  ax-13 2024  ax-ext 2378  ax-sep 4514  ax-nul 4522  ax-pow 4569  ax-pr 4627  ax-un 6528  ax-cnex 9496  ax-resscn 9497  ax-1cn 9498  ax-icn 9499  ax-addcl 9500  ax-addrcl 9501  ax-mulcl 9502  ax-mulrcl 9503  ax-mulcom 9504  ax-addass 9505  ax-mulass 9506  ax-distr 9507  ax-i2m1 9508  ax-1ne0 9509  ax-1rid 9510  ax-rnegex 9511  ax-rrecex 9512  ax-cnre 9513  ax-pre-lttri 9514  ax-pre-lttrn 9515  ax-pre-ltadd 9516  ax-pre-mulgt0 9517  ax-pre-sup 9518
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 973  df-3an 974  df-tru 1406  df-ex 1632  df-nf 1636  df-sb 1762  df-eu 2240  df-mo 2241  df-clab 2386  df-cleq 2392  df-clel 2395  df-nfc 2550  df-ne 2598  df-nel 2599  df-ral 2756  df-rex 2757  df-reu 2758  df-rmo 2759  df-rab 2760  df-v 3058  df-sbc 3275  df-csb 3371  df-dif 3414  df-un 3416  df-in 3418  df-ss 3425  df-pss 3427  df-nul 3736  df-if 3883  df-pw 3954  df-sn 3970  df-pr 3972  df-tp 3974  df-op 3976  df-uni 4189  df-iun 4270  df-br 4393  df-opab 4451  df-mpt 4452  df-tr 4487  df-eprel 4731  df-id 4735  df-po 4741  df-so 4742  df-fr 4779  df-we 4781  df-ord 4822  df-on 4823  df-lim 4824  df-suc 4825  df-xp 4946  df-rel 4947  df-cnv 4948  df-co 4949  df-dm 4950  df-rn 4951  df-res 4952  df-ima 4953  df-iota 5487  df-fun 5525  df-fn 5526  df-f 5527  df-f1 5528  df-fo 5529  df-f1o 5530  df-fv 5531  df-riota 6194  df-ov 6235  df-oprab 6236  df-mpt2 6237  df-om 6637  df-1st 6736  df-2nd 6737  df-recs 6997  df-rdg 7031  df-er 7266  df-map 7377  df-en 7473  df-dom 7474  df-sdom 7475  df-sup 7853  df-pnf 9578  df-mnf 9579  df-xr 9580  df-ltxr 9581  df-le 9582  df-sub 9761  df-neg 9762  df-div 10166  df-nn 10495  df-2 10553  df-n0 10755  df-z 10824  df-uz 11044  df-q 11144  df-rp 11182  df-xneg 11287  df-xadd 11288  df-xmul 11289  df-psmet 18621  df-xmet 18622  df-bl 18624
This theorem is referenced by:  blbas  21115
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