ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  ssblex Unicode version

Theorem ssblex 13595
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 529 . . . 4  |-  ( ( ( D  e.  ( *Met `  X
)  /\  P  e.  X )  /\  ( R  e.  RR+  /\  S  e.  RR+ ) )  ->  R  e.  RR+ )
21rphalfcld 9693 . . 3  |-  ( ( ( D  e.  ( *Met `  X
)  /\  P  e.  X )  /\  ( R  e.  RR+  /\  S  e.  RR+ ) )  -> 
( R  /  2
)  e.  RR+ )
3 simprr 531 . . 3  |-  ( ( ( D  e.  ( *Met `  X
)  /\  P  e.  X )  /\  ( R  e.  RR+  /\  S  e.  RR+ ) )  ->  S  e.  RR+ )
4 rpmincl 11227 . . 3  |-  ( ( ( R  /  2
)  e.  RR+  /\  S  e.  RR+ )  -> inf ( { ( R  /  2
) ,  S } ,  RR ,  <  )  e.  RR+ )
52, 3, 4syl2anc 411 . 2  |-  ( ( ( D  e.  ( *Met `  X
)  /\  P  e.  X )  /\  ( R  e.  RR+  /\  S  e.  RR+ ) )  -> inf ( { ( R  / 
2 ) ,  S } ,  RR ,  <  )  e.  RR+ )
65rpred 9680 . . 3  |-  ( ( ( D  e.  ( *Met `  X
)  /\  P  e.  X )  /\  ( R  e.  RR+  /\  S  e.  RR+ ) )  -> inf ( { ( R  / 
2 ) ,  S } ,  RR ,  <  )  e.  RR )
72rpred 9680 . . 3  |-  ( ( ( D  e.  ( *Met `  X
)  /\  P  e.  X )  /\  ( R  e.  RR+  /\  S  e.  RR+ ) )  -> 
( R  /  2
)  e.  RR )
81rpred 9680 . . 3  |-  ( ( ( D  e.  ( *Met `  X
)  /\  P  e.  X )  /\  ( R  e.  RR+  /\  S  e.  RR+ ) )  ->  R  e.  RR )
93rpred 9680 . . . 4  |-  ( ( ( D  e.  ( *Met `  X
)  /\  P  e.  X )  /\  ( R  e.  RR+  /\  S  e.  RR+ ) )  ->  S  e.  RR )
10 min1inf 11221 . . . 4  |-  ( ( ( R  /  2
)  e.  RR  /\  S  e.  RR )  -> inf ( { ( R  /  2 ) ,  S } ,  RR ,  <  )  <_  ( R  /  2 ) )
117, 9, 10syl2anc 411 . . 3  |-  ( ( ( D  e.  ( *Met `  X
)  /\  P  e.  X )  /\  ( R  e.  RR+  /\  S  e.  RR+ ) )  -> inf ( { ( R  / 
2 ) ,  S } ,  RR ,  <  )  <_  ( R  /  2 ) )
121rpgt0d 9683 . . . 4  |-  ( ( ( D  e.  ( *Met `  X
)  /\  P  e.  X )  /\  ( R  e.  RR+  /\  S  e.  RR+ ) )  -> 
0  <  R )
13 halfpos 9136 . . . . 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 147 . . 3  |-  ( ( ( D  e.  ( *Met `  X
)  /\  P  e.  X )  /\  ( R  e.  RR+  /\  S  e.  RR+ ) )  -> 
( R  /  2
)  <  R )
166, 7, 8, 11, 15lelttrd 8069 . 2  |-  ( ( ( D  e.  ( *Met `  X
)  /\  P  e.  X )  /\  ( R  e.  RR+  /\  S  e.  RR+ ) )  -> inf ( { ( R  / 
2 ) ,  S } ,  RR ,  <  )  <  R )
17 simpl 109 . . 3  |-  ( ( ( D  e.  ( *Met `  X
)  /\  P  e.  X )  /\  ( R  e.  RR+  /\  S  e.  RR+ ) )  -> 
( D  e.  ( *Met `  X
)  /\  P  e.  X ) )
185rpxrd 9681 . . 3  |-  ( ( ( D  e.  ( *Met `  X
)  /\  P  e.  X )  /\  ( R  e.  RR+  /\  S  e.  RR+ ) )  -> inf ( { ( R  / 
2 ) ,  S } ,  RR ,  <  )  e.  RR* )
193rpxrd 9681 . . 3  |-  ( ( ( D  e.  ( *Met `  X
)  /\  P  e.  X )  /\  ( R  e.  RR+  /\  S  e.  RR+ ) )  ->  S  e.  RR* )
20 min2inf 11222 . . . 4  |-  ( ( ( R  /  2
)  e.  RR  /\  S  e.  RR )  -> inf ( { ( R  /  2 ) ,  S } ,  RR ,  <  )  <_  S
)
217, 9, 20syl2anc 411 . . 3  |-  ( ( ( D  e.  ( *Met `  X
)  /\  P  e.  X )  /\  ( R  e.  RR+  /\  S  e.  RR+ ) )  -> inf ( { ( R  / 
2 ) ,  S } ,  RR ,  <  )  <_  S )
22 ssbl 13590 . . 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 1243 . 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 4003 . . . 4  |-  ( x  = inf ( { ( R  /  2 ) ,  S } ,  RR ,  <  )  -> 
( x  <  R  <-> inf ( { ( R  / 
2 ) ,  S } ,  RR ,  <  )  <  R ) )
25 oveq2 5877 . . . . 5  |-  ( x  = inf ( { ( R  /  2 ) ,  S } ,  RR ,  <  )  -> 
( P ( ball `  D ) x )  =  ( P (
ball `  D )inf ( { ( R  / 
2 ) ,  S } ,  RR ,  <  ) ) )
2625sseq1d 3184 . . . 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 473 . . 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 2841 . 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 1236 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 104    <-> wb 105    = wceq 1353    e. wcel 2148   E.wrex 2456    C_ wss 3129   {cpr 3592   class class class wbr 4000   ` cfv 5212  (class class class)co 5869  infcinf 6976   RRcr 7798   0cc0 7799   RR*cxr 7978    < clt 7979    <_ cle 7980    / cdiv 8615   2c2 8956   RR+crp 9637   *Metcxmet 13144   ballcbl 13146
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4115  ax-sep 4118  ax-nul 4126  ax-pow 4171  ax-pr 4206  ax-un 4430  ax-setind 4533  ax-iinf 4584  ax-cnex 7890  ax-resscn 7891  ax-1cn 7892  ax-1re 7893  ax-icn 7894  ax-addcl 7895  ax-addrcl 7896  ax-mulcl 7897  ax-mulrcl 7898  ax-addcom 7899  ax-mulcom 7900  ax-addass 7901  ax-mulass 7902  ax-distr 7903  ax-i2m1 7904  ax-0lt1 7905  ax-1rid 7906  ax-0id 7907  ax-rnegex 7908  ax-precex 7909  ax-cnre 7910  ax-pre-ltirr 7911  ax-pre-ltwlin 7912  ax-pre-lttrn 7913  ax-pre-apti 7914  ax-pre-ltadd 7915  ax-pre-mulgt0 7916  ax-pre-mulext 7917  ax-arch 7918  ax-caucvg 7919
This theorem depends on definitions:  df-bi 117  df-stab 831  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rmo 2463  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-if 3535  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-int 3843  df-iun 3886  df-br 4001  df-opab 4062  df-mpt 4063  df-tr 4099  df-id 4290  df-po 4293  df-iso 4294  df-iord 4363  df-on 4365  df-ilim 4366  df-suc 4368  df-iom 4587  df-xp 4629  df-rel 4630  df-cnv 4631  df-co 4632  df-dm 4633  df-rn 4634  df-res 4635  df-ima 4636  df-iota 5174  df-fun 5214  df-fn 5215  df-f 5216  df-f1 5217  df-fo 5218  df-f1o 5219  df-fv 5220  df-isom 5221  df-riota 5825  df-ov 5872  df-oprab 5873  df-mpo 5874  df-1st 6135  df-2nd 6136  df-recs 6300  df-frec 6386  df-map 6644  df-sup 6977  df-inf 6978  df-pnf 7981  df-mnf 7982  df-xr 7983  df-ltxr 7984  df-le 7985  df-sub 8117  df-neg 8118  df-reap 8519  df-ap 8526  df-div 8616  df-inn 8906  df-2 8964  df-3 8965  df-4 8966  df-n0 9163  df-z 9240  df-uz 9515  df-rp 9638  df-xneg 9756  df-xadd 9757  df-seqfrec 10429  df-exp 10503  df-cj 10832  df-re 10833  df-im 10834  df-rsqrt 10988  df-abs 10989  df-psmet 13151  df-xmet 13152  df-bl 13154
This theorem is referenced by:  mopni3  13648
  Copyright terms: Public domain W3C validator