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Theorem ssblex 21115
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 756 . . . 4  |-  ( ( ( D  e.  ( *Met `  X
)  /\  P  e.  X )  /\  ( R  e.  RR+  /\  S  e.  RR+ ) )  ->  R  e.  RR+ )
21rphalfcld 11234 . . 3  |-  ( ( ( D  e.  ( *Met `  X
)  /\  P  e.  X )  /\  ( R  e.  RR+  /\  S  e.  RR+ ) )  -> 
( R  /  2
)  e.  RR+ )
3 simprr 758 . . 3  |-  ( ( ( D  e.  ( *Met `  X
)  /\  P  e.  X )  /\  ( R  e.  RR+  /\  S  e.  RR+ ) )  ->  S  e.  RR+ )
42, 3ifcld 3927 . 2  |-  ( ( ( D  e.  ( *Met `  X
)  /\  P  e.  X )  /\  ( R  e.  RR+  /\  S  e.  RR+ ) )  ->  if ( ( R  / 
2 )  <_  S ,  ( R  / 
2 ) ,  S
)  e.  RR+ )
54rpred 11222 . . 3  |-  ( ( ( D  e.  ( *Met `  X
)  /\  P  e.  X )  /\  ( R  e.  RR+  /\  S  e.  RR+ ) )  ->  if ( ( R  / 
2 )  <_  S ,  ( R  / 
2 ) ,  S
)  e.  RR )
62rpred 11222 . . 3  |-  ( ( ( D  e.  ( *Met `  X
)  /\  P  e.  X )  /\  ( R  e.  RR+  /\  S  e.  RR+ ) )  -> 
( R  /  2
)  e.  RR )
71rpred 11222 . . 3  |-  ( ( ( D  e.  ( *Met `  X
)  /\  P  e.  X )  /\  ( R  e.  RR+  /\  S  e.  RR+ ) )  ->  R  e.  RR )
83rpred 11222 . . . 4  |-  ( ( ( D  e.  ( *Met `  X
)  /\  P  e.  X )  /\  ( R  e.  RR+  /\  S  e.  RR+ ) )  ->  S  e.  RR )
9 min1 11360 . . . 4  |-  ( ( ( R  /  2
)  e.  RR  /\  S  e.  RR )  ->  if ( ( R  /  2 )  <_  S ,  ( R  /  2 ) ,  S )  <_  ( R  /  2 ) )
106, 8, 9syl2anc 659 . . 3  |-  ( ( ( D  e.  ( *Met `  X
)  /\  P  e.  X )  /\  ( R  e.  RR+  /\  S  e.  RR+ ) )  ->  if ( ( R  / 
2 )  <_  S ,  ( R  / 
2 ) ,  S
)  <_  ( R  /  2 ) )
111rpgt0d 11225 . . . 4  |-  ( ( ( D  e.  ( *Met `  X
)  /\  P  e.  X )  /\  ( R  e.  RR+  /\  S  e.  RR+ ) )  -> 
0  <  R )
12 halfpos 10730 . . . . 5  |-  ( R  e.  RR  ->  (
0  <  R  <->  ( R  /  2 )  < 
R ) )
137, 12syl 17 . . . 4  |-  ( ( ( D  e.  ( *Met `  X
)  /\  P  e.  X )  /\  ( R  e.  RR+  /\  S  e.  RR+ ) )  -> 
( 0  <  R  <->  ( R  /  2 )  <  R ) )
1411, 13mpbid 210 . . 3  |-  ( ( ( D  e.  ( *Met `  X
)  /\  P  e.  X )  /\  ( R  e.  RR+  /\  S  e.  RR+ ) )  -> 
( R  /  2
)  <  R )
155, 6, 7, 10, 14lelttrd 9694 . 2  |-  ( ( ( D  e.  ( *Met `  X
)  /\  P  e.  X )  /\  ( R  e.  RR+  /\  S  e.  RR+ ) )  ->  if ( ( R  / 
2 )  <_  S ,  ( R  / 
2 ) ,  S
)  <  R )
16 simpl 455 . . 3  |-  ( ( ( D  e.  ( *Met `  X
)  /\  P  e.  X )  /\  ( R  e.  RR+  /\  S  e.  RR+ ) )  -> 
( D  e.  ( *Met `  X
)  /\  P  e.  X ) )
174rpxrd 11223 . . 3  |-  ( ( ( D  e.  ( *Met `  X
)  /\  P  e.  X )  /\  ( R  e.  RR+  /\  S  e.  RR+ ) )  ->  if ( ( R  / 
2 )  <_  S ,  ( R  / 
2 ) ,  S
)  e.  RR* )
183rpxrd 11223 . . 3  |-  ( ( ( D  e.  ( *Met `  X
)  /\  P  e.  X )  /\  ( R  e.  RR+  /\  S  e.  RR+ ) )  ->  S  e.  RR* )
19 min2 11361 . . . 4  |-  ( ( ( R  /  2
)  e.  RR  /\  S  e.  RR )  ->  if ( ( R  /  2 )  <_  S ,  ( R  /  2 ) ,  S )  <_  S
)
206, 8, 19syl2anc 659 . . 3  |-  ( ( ( D  e.  ( *Met `  X
)  /\  P  e.  X )  /\  ( R  e.  RR+  /\  S  e.  RR+ ) )  ->  if ( ( R  / 
2 )  <_  S ,  ( R  / 
2 ) ,  S
)  <_  S )
21 ssbl 21110 . . 3  |-  ( ( ( D  e.  ( *Met `  X
)  /\  P  e.  X )  /\  ( if ( ( R  / 
2 )  <_  S ,  ( R  / 
2 ) ,  S
)  e.  RR*  /\  S  e.  RR* )  /\  if ( ( R  / 
2 )  <_  S ,  ( R  / 
2 ) ,  S
)  <_  S )  ->  ( P ( ball `  D ) if ( ( R  /  2
)  <_  S , 
( R  /  2
) ,  S ) )  C_  ( P
( ball `  D ) S ) )
2216, 17, 18, 20, 21syl121anc 1235 . 2  |-  ( ( ( D  e.  ( *Met `  X
)  /\  P  e.  X )  /\  ( R  e.  RR+  /\  S  e.  RR+ ) )  -> 
( P ( ball `  D ) if ( ( R  /  2
)  <_  S , 
( R  /  2
) ,  S ) )  C_  ( P
( ball `  D ) S ) )
23 breq1 4397 . . . 4  |-  ( x  =  if ( ( R  /  2 )  <_  S ,  ( R  /  2 ) ,  S )  -> 
( x  <  R  <->  if ( ( R  / 
2 )  <_  S ,  ( R  / 
2 ) ,  S
)  <  R )
)
24 oveq2 6242 . . . . 5  |-  ( x  =  if ( ( R  /  2 )  <_  S ,  ( R  /  2 ) ,  S )  -> 
( P ( ball `  D ) x )  =  ( P (
ball `  D ) if ( ( R  / 
2 )  <_  S ,  ( R  / 
2 ) ,  S
) ) )
2524sseq1d 3468 . . . 4  |-  ( x  =  if ( ( R  /  2 )  <_  S ,  ( R  /  2 ) ,  S )  -> 
( ( P (
ball `  D )
x )  C_  ( P ( ball `  D
) S )  <->  ( P
( ball `  D ) if ( ( R  / 
2 )  <_  S ,  ( R  / 
2 ) ,  S
) )  C_  ( P ( ball `  D
) S ) ) )
2623, 25anbi12d 709 . . 3  |-  ( x  =  if ( ( R  /  2 )  <_  S ,  ( R  /  2 ) ,  S )  -> 
( ( x  < 
R  /\  ( P
( ball `  D )
x )  C_  ( P ( ball `  D
) S ) )  <-> 
( if ( ( R  /  2 )  <_  S ,  ( R  /  2 ) ,  S )  < 
R  /\  ( P
( ball `  D ) if ( ( R  / 
2 )  <_  S ,  ( R  / 
2 ) ,  S
) )  C_  ( P ( ball `  D
) S ) ) ) )
2726rspcev 3159 . 2  |-  ( ( if ( ( R  /  2 )  <_  S ,  ( R  /  2 ) ,  S )  e.  RR+  /\  ( if ( ( R  /  2 )  <_  S ,  ( R  /  2 ) ,  S )  < 
R  /\  ( P
( ball `  D ) if ( ( R  / 
2 )  <_  S ,  ( R  / 
2 ) ,  S
) )  C_  ( P ( ball `  D
) S ) ) )  ->  E. x  e.  RR+  ( x  < 
R  /\  ( P
( ball `  D )
x )  C_  ( P ( ball `  D
) S ) ) )
284, 15, 22, 27syl12anc 1228 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 setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367    = wceq 1405    e. wcel 1842   E.wrex 2754    C_ wss 3413   ifcif 3884   class class class wbr 4394   ` cfv 5525  (class class class)co 6234   RRcr 9441   0cc0 9442   RR*cxr 9577    < clt 9578    <_ cle 9579    / cdiv 10167   2c2 10546   RR+crp 11183   *Metcxmt 18615   ballcbl 18617
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-sep 4516  ax-nul 4524  ax-pow 4571  ax-pr 4629  ax-un 6530  ax-cnex 9498  ax-resscn 9499  ax-1cn 9500  ax-icn 9501  ax-addcl 9502  ax-addrcl 9503  ax-mulcl 9504  ax-mulrcl 9505  ax-mulcom 9506  ax-addass 9507  ax-mulass 9508  ax-distr 9509  ax-i2m1 9510  ax-1ne0 9511  ax-1rid 9512  ax-rnegex 9513  ax-rrecex 9514  ax-cnre 9515  ax-pre-lttri 9516  ax-pre-lttrn 9517  ax-pre-ltadd 9518  ax-pre-mulgt0 9519
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 975  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-nel 2601  df-ral 2758  df-rex 2759  df-reu 2760  df-rmo 2761  df-rab 2762  df-v 3060  df-sbc 3277  df-csb 3373  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-nul 3738  df-if 3885  df-pw 3956  df-sn 3972  df-pr 3974  df-op 3978  df-uni 4191  df-iun 4272  df-br 4395  df-opab 4453  df-mpt 4454  df-id 4737  df-po 4743  df-so 4744  df-xp 4948  df-rel 4949  df-cnv 4950  df-co 4951  df-dm 4952  df-rn 4953  df-res 4954  df-ima 4955  df-iota 5489  df-fun 5527  df-fn 5528  df-f 5529  df-f1 5530  df-fo 5531  df-f1o 5532  df-fv 5533  df-riota 6196  df-ov 6237  df-oprab 6238  df-mpt2 6239  df-1st 6738  df-2nd 6739  df-er 7268  df-map 7379  df-en 7475  df-dom 7476  df-sdom 7477  df-pnf 9580  df-mnf 9581  df-xr 9582  df-ltxr 9583  df-le 9584  df-sub 9763  df-neg 9764  df-div 10168  df-2 10555  df-rp 11184  df-xneg 11289  df-xadd 11290  df-xmul 11291  df-psmet 18623  df-xmet 18624  df-bl 18626
This theorem is referenced by:  mopni3  21181
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