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Theorem ssblex 12609
Description: A nested ball exists whose radius is less than any desired amount. (Contributed by NM, 20-Sep-2007.) (Revised by Mario Carneiro, 12-Nov-2013.)
Assertion
Ref Expression
ssblex  |-  ( ( ( D  e.  ( *Met `  X
)  /\  P  e.  X )  /\  ( R  e.  RR+  /\  S  e.  RR+ ) )  ->  E. x  e.  RR+  (
x  <  R  /\  ( P ( ball `  D
) x )  C_  ( P ( ball `  D
) S ) ) )
Distinct variable groups:    x, D    x, R    x, P    x, S    x, X

Proof of Theorem ssblex
StepHypRef Expression
1 simprl 520 . . . 4  |-  ( ( ( D  e.  ( *Met `  X
)  /\  P  e.  X )  /\  ( R  e.  RR+  /\  S  e.  RR+ ) )  ->  R  e.  RR+ )
21rphalfcld 9503 . . 3  |-  ( ( ( D  e.  ( *Met `  X
)  /\  P  e.  X )  /\  ( R  e.  RR+  /\  S  e.  RR+ ) )  -> 
( R  /  2
)  e.  RR+ )
3 simprr 521 . . 3  |-  ( ( ( D  e.  ( *Met `  X
)  /\  P  e.  X )  /\  ( R  e.  RR+  /\  S  e.  RR+ ) )  ->  S  e.  RR+ )
4 rpmincl 11016 . . 3  |-  ( ( ( R  /  2
)  e.  RR+  /\  S  e.  RR+ )  -> inf ( { ( R  /  2
) ,  S } ,  RR ,  <  )  e.  RR+ )
52, 3, 4syl2anc 408 . 2  |-  ( ( ( D  e.  ( *Met `  X
)  /\  P  e.  X )  /\  ( R  e.  RR+  /\  S  e.  RR+ ) )  -> inf ( { ( R  / 
2 ) ,  S } ,  RR ,  <  )  e.  RR+ )
65rpred 9490 . . 3  |-  ( ( ( D  e.  ( *Met `  X
)  /\  P  e.  X )  /\  ( R  e.  RR+  /\  S  e.  RR+ ) )  -> inf ( { ( R  / 
2 ) ,  S } ,  RR ,  <  )  e.  RR )
72rpred 9490 . . 3  |-  ( ( ( D  e.  ( *Met `  X
)  /\  P  e.  X )  /\  ( R  e.  RR+  /\  S  e.  RR+ ) )  -> 
( R  /  2
)  e.  RR )
81rpred 9490 . . 3  |-  ( ( ( D  e.  ( *Met `  X
)  /\  P  e.  X )  /\  ( R  e.  RR+  /\  S  e.  RR+ ) )  ->  R  e.  RR )
93rpred 9490 . . . 4  |-  ( ( ( D  e.  ( *Met `  X
)  /\  P  e.  X )  /\  ( R  e.  RR+  /\  S  e.  RR+ ) )  ->  S  e.  RR )
10 min1inf 11010 . . . 4  |-  ( ( ( R  /  2
)  e.  RR  /\  S  e.  RR )  -> inf ( { ( R  /  2 ) ,  S } ,  RR ,  <  )  <_  ( R  /  2 ) )
117, 9, 10syl2anc 408 . . 3  |-  ( ( ( D  e.  ( *Met `  X
)  /\  P  e.  X )  /\  ( R  e.  RR+  /\  S  e.  RR+ ) )  -> inf ( { ( R  / 
2 ) ,  S } ,  RR ,  <  )  <_  ( R  /  2 ) )
121rpgt0d 9493 . . . 4  |-  ( ( ( D  e.  ( *Met `  X
)  /\  P  e.  X )  /\  ( R  e.  RR+  /\  S  e.  RR+ ) )  -> 
0  <  R )
13 halfpos 8958 . . . . 5  |-  ( R  e.  RR  ->  (
0  <  R  <->  ( R  /  2 )  < 
R ) )
148, 13syl 14 . . . 4  |-  ( ( ( D  e.  ( *Met `  X
)  /\  P  e.  X )  /\  ( R  e.  RR+  /\  S  e.  RR+ ) )  -> 
( 0  <  R  <->  ( R  /  2 )  <  R ) )
1512, 14mpbid 146 . . 3  |-  ( ( ( D  e.  ( *Met `  X
)  /\  P  e.  X )  /\  ( R  e.  RR+  /\  S  e.  RR+ ) )  -> 
( R  /  2
)  <  R )
166, 7, 8, 11, 15lelttrd 7894 . 2  |-  ( ( ( D  e.  ( *Met `  X
)  /\  P  e.  X )  /\  ( R  e.  RR+  /\  S  e.  RR+ ) )  -> inf ( { ( R  / 
2 ) ,  S } ,  RR ,  <  )  <  R )
17 simpl 108 . . 3  |-  ( ( ( D  e.  ( *Met `  X
)  /\  P  e.  X )  /\  ( R  e.  RR+  /\  S  e.  RR+ ) )  -> 
( D  e.  ( *Met `  X
)  /\  P  e.  X ) )
185rpxrd 9491 . . 3  |-  ( ( ( D  e.  ( *Met `  X
)  /\  P  e.  X )  /\  ( R  e.  RR+  /\  S  e.  RR+ ) )  -> inf ( { ( R  / 
2 ) ,  S } ,  RR ,  <  )  e.  RR* )
193rpxrd 9491 . . 3  |-  ( ( ( D  e.  ( *Met `  X
)  /\  P  e.  X )  /\  ( R  e.  RR+  /\  S  e.  RR+ ) )  ->  S  e.  RR* )
20 min2inf 11011 . . . 4  |-  ( ( ( R  /  2
)  e.  RR  /\  S  e.  RR )  -> inf ( { ( R  /  2 ) ,  S } ,  RR ,  <  )  <_  S
)
217, 9, 20syl2anc 408 . . 3  |-  ( ( ( D  e.  ( *Met `  X
)  /\  P  e.  X )  /\  ( R  e.  RR+  /\  S  e.  RR+ ) )  -> inf ( { ( R  / 
2 ) ,  S } ,  RR ,  <  )  <_  S )
22 ssbl 12604 . . 3  |-  ( ( ( D  e.  ( *Met `  X
)  /\  P  e.  X )  /\  (inf ( { ( R  / 
2 ) ,  S } ,  RR ,  <  )  e.  RR*  /\  S  e.  RR* )  /\ inf ( { ( R  / 
2 ) ,  S } ,  RR ,  <  )  <_  S )  ->  ( P ( ball `  D )inf ( { ( R  /  2
) ,  S } ,  RR ,  <  )
)  C_  ( P
( ball `  D ) S ) )
2317, 18, 19, 21, 22syl121anc 1221 . 2  |-  ( ( ( D  e.  ( *Met `  X
)  /\  P  e.  X )  /\  ( R  e.  RR+  /\  S  e.  RR+ ) )  -> 
( P ( ball `  D )inf ( { ( R  /  2
) ,  S } ,  RR ,  <  )
)  C_  ( P
( ball `  D ) S ) )
24 breq1 3932 . . . 4  |-  ( x  = inf ( { ( R  /  2 ) ,  S } ,  RR ,  <  )  -> 
( x  <  R  <-> inf ( { ( R  / 
2 ) ,  S } ,  RR ,  <  )  <  R ) )
25 oveq2 5782 . . . . 5  |-  ( x  = inf ( { ( R  /  2 ) ,  S } ,  RR ,  <  )  -> 
( P ( ball `  D ) x )  =  ( P (
ball `  D )inf ( { ( R  / 
2 ) ,  S } ,  RR ,  <  ) ) )
2625sseq1d 3126 . . . 4  |-  ( x  = inf ( { ( R  /  2 ) ,  S } ,  RR ,  <  )  -> 
( ( P (
ball `  D )
x )  C_  ( P ( ball `  D
) S )  <->  ( P
( ball `  D )inf ( { ( R  / 
2 ) ,  S } ,  RR ,  <  ) )  C_  ( P ( ball `  D
) S ) ) )
2724, 26anbi12d 464 . . 3  |-  ( x  = inf ( { ( R  /  2 ) ,  S } ,  RR ,  <  )  -> 
( ( x  < 
R  /\  ( P
( ball `  D )
x )  C_  ( P ( ball `  D
) S ) )  <-> 
(inf ( { ( R  /  2 ) ,  S } ,  RR ,  <  )  < 
R  /\  ( P
( ball `  D )inf ( { ( R  / 
2 ) ,  S } ,  RR ,  <  ) )  C_  ( P ( ball `  D
) S ) ) ) )
2827rspcev 2789 . 2  |-  ( (inf ( { ( R  /  2 ) ,  S } ,  RR ,  <  )  e.  RR+  /\  (inf ( { ( R  /  2 ) ,  S } ,  RR ,  <  )  < 
R  /\  ( P
( ball `  D )inf ( { ( R  / 
2 ) ,  S } ,  RR ,  <  ) )  C_  ( P ( ball `  D
) S ) ) )  ->  E. x  e.  RR+  ( x  < 
R  /\  ( P
( ball `  D )
x )  C_  ( P ( ball `  D
) S ) ) )
295, 16, 23, 28syl12anc 1214 1  |-  ( ( ( D  e.  ( *Met `  X
)  /\  P  e.  X )  /\  ( R  e.  RR+  /\  S  e.  RR+ ) )  ->  E. x  e.  RR+  (
x  <  R  /\  ( P ( ball `  D
) x )  C_  ( P ( ball `  D
) S ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104    = wceq 1331    e. wcel 1480   E.wrex 2417    C_ wss 3071   {cpr 3528   class class class wbr 3929   ` cfv 5123  (class class class)co 5774  infcinf 6870   RRcr 7626   0cc0 7627   RR*cxr 7806    < clt 7807    <_ cle 7808    / cdiv 8439   2c2 8778   RR+crp 9448   *Metcxmet 12158   ballcbl 12160
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-coll 4043  ax-sep 4046  ax-nul 4054  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452  ax-iinf 4502  ax-cnex 7718  ax-resscn 7719  ax-1cn 7720  ax-1re 7721  ax-icn 7722  ax-addcl 7723  ax-addrcl 7724  ax-mulcl 7725  ax-mulrcl 7726  ax-addcom 7727  ax-mulcom 7728  ax-addass 7729  ax-mulass 7730  ax-distr 7731  ax-i2m1 7732  ax-0lt1 7733  ax-1rid 7734  ax-0id 7735  ax-rnegex 7736  ax-precex 7737  ax-cnre 7738  ax-pre-ltirr 7739  ax-pre-ltwlin 7740  ax-pre-lttrn 7741  ax-pre-apti 7742  ax-pre-ltadd 7743  ax-pre-mulgt0 7744  ax-pre-mulext 7745  ax-arch 7746  ax-caucvg 7747
This theorem depends on definitions:  df-bi 116  df-stab 816  df-dc 820  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-nel 2404  df-ral 2421  df-rex 2422  df-reu 2423  df-rmo 2424  df-rab 2425  df-v 2688  df-sbc 2910  df-csb 3004  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-nul 3364  df-if 3475  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-int 3772  df-iun 3815  df-br 3930  df-opab 3990  df-mpt 3991  df-tr 4027  df-id 4215  df-po 4218  df-iso 4219  df-iord 4288  df-on 4290  df-ilim 4291  df-suc 4293  df-iom 4505  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-f1 5128  df-fo 5129  df-f1o 5130  df-fv 5131  df-isom 5132  df-riota 5730  df-ov 5777  df-oprab 5778  df-mpo 5779  df-1st 6038  df-2nd 6039  df-recs 6202  df-frec 6288  df-map 6544  df-sup 6871  df-inf 6872  df-pnf 7809  df-mnf 7810  df-xr 7811  df-ltxr 7812  df-le 7813  df-sub 7942  df-neg 7943  df-reap 8344  df-ap 8351  df-div 8440  df-inn 8728  df-2 8786  df-3 8787  df-4 8788  df-n0 8985  df-z 9062  df-uz 9334  df-rp 9449  df-xneg 9566  df-xadd 9567  df-seqfrec 10226  df-exp 10300  df-cj 10621  df-re 10622  df-im 10623  df-rsqrt 10777  df-abs 10778  df-psmet 12165  df-xmet 12166  df-bl 12168
This theorem is referenced by:  mopni3  12662
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