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Theorem blhalf 18062
Description: A ball of radius  R  / 
2 is contained in a ball of radius  R centered at any point inside the smaller ball. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 14-Jan-2014.)
Assertion
Ref Expression
blhalf  |-  ( ( ( M  e.  ( * Met `  X
)  /\  Y  e.  X )  /\  ( R  e.  RR  /\  Z  e.  ( Y ( ball `  M ) ( R  /  2 ) ) ) )  ->  ( Y ( ball `  M
) ( R  / 
2 ) )  C_  ( Z ( ball `  M
) R ) )

Proof of Theorem blhalf
StepHypRef Expression
1 simpll 730 . 2  |-  ( ( ( M  e.  ( * Met `  X
)  /\  Y  e.  X )  /\  ( R  e.  RR  /\  Z  e.  ( Y ( ball `  M ) ( R  /  2 ) ) ) )  ->  M  e.  ( * Met `  X
) )
2 simplr 731 . 2  |-  ( ( ( M  e.  ( * Met `  X
)  /\  Y  e.  X )  /\  ( R  e.  RR  /\  Z  e.  ( Y ( ball `  M ) ( R  /  2 ) ) ) )  ->  Y  e.  X )
3 simprr 733 . . . 4  |-  ( ( ( M  e.  ( * Met `  X
)  /\  Y  e.  X )  /\  ( R  e.  RR  /\  Z  e.  ( Y ( ball `  M ) ( R  /  2 ) ) ) )  ->  Z  e.  ( Y ( ball `  M ) ( R  /  2 ) ) )
4 simprl 732 . . . . . . 7  |-  ( ( ( M  e.  ( * Met `  X
)  /\  Y  e.  X )  /\  ( R  e.  RR  /\  Z  e.  ( Y ( ball `  M ) ( R  /  2 ) ) ) )  ->  R  e.  RR )
54rehalfcld 10050 . . . . . 6  |-  ( ( ( M  e.  ( * Met `  X
)  /\  Y  e.  X )  /\  ( R  e.  RR  /\  Z  e.  ( Y ( ball `  M ) ( R  /  2 ) ) ) )  ->  ( R  /  2 )  e.  RR )
65rexrd 8971 . . . . 5  |-  ( ( ( M  e.  ( * Met `  X
)  /\  Y  e.  X )  /\  ( R  e.  RR  /\  Z  e.  ( Y ( ball `  M ) ( R  /  2 ) ) ) )  ->  ( R  /  2 )  e. 
RR* )
7 elbl 18051 . . . . 5  |-  ( ( M  e.  ( * Met `  X )  /\  Y  e.  X  /\  ( R  /  2
)  e.  RR* )  ->  ( Z  e.  ( Y ( ball `  M
) ( R  / 
2 ) )  <->  ( Z  e.  X  /\  ( Y M Z )  < 
( R  /  2
) ) ) )
81, 2, 6, 7syl3anc 1182 . . . 4  |-  ( ( ( M  e.  ( * Met `  X
)  /\  Y  e.  X )  /\  ( R  e.  RR  /\  Z  e.  ( Y ( ball `  M ) ( R  /  2 ) ) ) )  ->  ( Z  e.  ( Y
( ball `  M )
( R  /  2
) )  <->  ( Z  e.  X  /\  ( Y M Z )  < 
( R  /  2
) ) ) )
93, 8mpbid 201 . . 3  |-  ( ( ( M  e.  ( * Met `  X
)  /\  Y  e.  X )  /\  ( R  e.  RR  /\  Z  e.  ( Y ( ball `  M ) ( R  /  2 ) ) ) )  ->  ( Z  e.  X  /\  ( Y M Z )  <  ( R  / 
2 ) ) )
109simpld 445 . 2  |-  ( ( ( M  e.  ( * Met `  X
)  /\  Y  e.  X )  /\  ( R  e.  RR  /\  Z  e.  ( Y ( ball `  M ) ( R  /  2 ) ) ) )  ->  Z  e.  X )
119simprd 449 . . . 4  |-  ( ( ( M  e.  ( * Met `  X
)  /\  Y  e.  X )  /\  ( R  e.  RR  /\  Z  e.  ( Y ( ball `  M ) ( R  /  2 ) ) ) )  ->  ( Y M Z )  < 
( R  /  2
) )
12 xmetcl 17998 . . . . . 6  |-  ( ( M  e.  ( * Met `  X )  /\  Y  e.  X  /\  Z  e.  X
)  ->  ( Y M Z )  e.  RR* )
131, 2, 10, 12syl3anc 1182 . . . . 5  |-  ( ( ( M  e.  ( * Met `  X
)  /\  Y  e.  X )  /\  ( R  e.  RR  /\  Z  e.  ( Y ( ball `  M ) ( R  /  2 ) ) ) )  ->  ( Y M Z )  e. 
RR* )
14 xrltle 10575 . . . . 5  |-  ( ( ( Y M Z )  e.  RR*  /\  ( R  /  2 )  e. 
RR* )  ->  (
( Y M Z )  <  ( R  /  2 )  -> 
( Y M Z )  <_  ( R  /  2 ) ) )
1513, 6, 14syl2anc 642 . . . 4  |-  ( ( ( M  e.  ( * Met `  X
)  /\  Y  e.  X )  /\  ( R  e.  RR  /\  Z  e.  ( Y ( ball `  M ) ( R  /  2 ) ) ) )  ->  (
( Y M Z )  <  ( R  /  2 )  -> 
( Y M Z )  <_  ( R  /  2 ) ) )
1611, 15mpd 14 . . 3  |-  ( ( ( M  e.  ( * Met `  X
)  /\  Y  e.  X )  /\  ( R  e.  RR  /\  Z  e.  ( Y ( ball `  M ) ( R  /  2 ) ) ) )  ->  ( Y M Z )  <_ 
( R  /  2
) )
175recnd 8951 . . . . 5  |-  ( ( ( M  e.  ( * Met `  X
)  /\  Y  e.  X )  /\  ( R  e.  RR  /\  Z  e.  ( Y ( ball `  M ) ( R  /  2 ) ) ) )  ->  ( R  /  2 )  e.  CC )
1817, 17pncand 9248 . . . 4  |-  ( ( ( M  e.  ( * Met `  X
)  /\  Y  e.  X )  /\  ( R  e.  RR  /\  Z  e.  ( Y ( ball `  M ) ( R  /  2 ) ) ) )  ->  (
( ( R  / 
2 )  +  ( R  /  2 ) )  -  ( R  /  2 ) )  =  ( R  / 
2 ) )
194recnd 8951 . . . . . 6  |-  ( ( ( M  e.  ( * Met `  X
)  /\  Y  e.  X )  /\  ( R  e.  RR  /\  Z  e.  ( Y ( ball `  M ) ( R  /  2 ) ) ) )  ->  R  e.  CC )
20192halvesd 10049 . . . . 5  |-  ( ( ( M  e.  ( * Met `  X
)  /\  Y  e.  X )  /\  ( R  e.  RR  /\  Z  e.  ( Y ( ball `  M ) ( R  /  2 ) ) ) )  ->  (
( R  /  2
)  +  ( R  /  2 ) )  =  R )
2120oveq1d 5960 . . . 4  |-  ( ( ( M  e.  ( * Met `  X
)  /\  Y  e.  X )  /\  ( R  e.  RR  /\  Z  e.  ( Y ( ball `  M ) ( R  /  2 ) ) ) )  ->  (
( ( R  / 
2 )  +  ( R  /  2 ) )  -  ( R  /  2 ) )  =  ( R  -  ( R  /  2
) ) )
2218, 21eqtr3d 2392 . . 3  |-  ( ( ( M  e.  ( * Met `  X
)  /\  Y  e.  X )  /\  ( R  e.  RR  /\  Z  e.  ( Y ( ball `  M ) ( R  /  2 ) ) ) )  ->  ( R  /  2 )  =  ( R  -  ( R  /  2 ) ) )
2316, 22breqtrd 4128 . 2  |-  ( ( ( M  e.  ( * Met `  X
)  /\  Y  e.  X )  /\  ( R  e.  RR  /\  Z  e.  ( Y ( ball `  M ) ( R  /  2 ) ) ) )  ->  ( Y M Z )  <_ 
( R  -  ( R  /  2 ) ) )
24 blss2 18061 . 2  |-  ( ( ( M  e.  ( * Met `  X
)  /\  Y  e.  X  /\  Z  e.  X
)  /\  ( ( R  /  2 )  e.  RR  /\  R  e.  RR  /\  ( Y M Z )  <_ 
( R  -  ( R  /  2 ) ) ) )  ->  ( Y ( ball `  M
) ( R  / 
2 ) )  C_  ( Z ( ball `  M
) R ) )
251, 2, 10, 5, 4, 23, 24syl33anc 1197 1  |-  ( ( ( M  e.  ( * Met `  X
)  /\  Y  e.  X )  /\  ( R  e.  RR  /\  Z  e.  ( Y ( ball `  M ) ( R  /  2 ) ) ) )  ->  ( Y ( ball `  M
) ( R  / 
2 ) )  C_  ( Z ( ball `  M
) R ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    e. wcel 1710    C_ wss 3228   class class class wbr 4104   ` cfv 5337  (class class class)co 5945   RRcr 8826    + caddc 8830   RR*cxr 8956    < clt 8957    <_ cle 8958    - cmin 9127    / cdiv 9513   2c2 9885   * Metcxmt 16468   ballcbl 16470
This theorem is referenced by:  met2ndci  18170  iscfil3  18803  cfilfcls  18804  iscmet3lem2  18822  lmcau  18842  lgamucov  24071  blhalfOLD  25796  sstotbnd2  25821  isbnd2  25830  heiborlem8  25865
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-sep 4222  ax-nul 4230  ax-pow 4269  ax-pr 4295  ax-un 4594  ax-cnex 8883  ax-resscn 8884  ax-1cn 8885  ax-icn 8886  ax-addcl 8887  ax-addrcl 8888  ax-mulcl 8889  ax-mulrcl 8890  ax-mulcom 8891  ax-addass 8892  ax-mulass 8893  ax-distr 8894  ax-i2m1 8895  ax-1ne0 8896  ax-1rid 8897  ax-rnegex 8898  ax-rrecex 8899  ax-cnre 8900  ax-pre-lttri 8901  ax-pre-lttrn 8902  ax-pre-ltadd 8903  ax-pre-mulgt0 8904
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-nel 2524  df-ral 2624  df-rex 2625  df-reu 2626  df-rmo 2627  df-rab 2628  df-v 2866  df-sbc 3068  df-csb 3158  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-nul 3532  df-if 3642  df-pw 3703  df-sn 3722  df-pr 3723  df-op 3725  df-uni 3909  df-iun 3988  df-br 4105  df-opab 4159  df-mpt 4160  df-id 4391  df-po 4396  df-so 4397  df-xp 4777  df-rel 4778  df-cnv 4779  df-co 4780  df-dm 4781  df-rn 4782  df-res 4783  df-ima 4784  df-iota 5301  df-fun 5339  df-fn 5340  df-f 5341  df-f1 5342  df-fo 5343  df-f1o 5344  df-fv 5345  df-ov 5948  df-oprab 5949  df-mpt2 5950  df-1st 6209  df-2nd 6210  df-riota 6391  df-er 6747  df-map 6862  df-en 6952  df-dom 6953  df-sdom 6954  df-pnf 8959  df-mnf 8960  df-xr 8961  df-ltxr 8962  df-le 8963  df-sub 9129  df-neg 9130  df-div 9514  df-2 9894  df-rp 10447  df-xneg 10544  df-xadd 10545  df-xmul 10546  df-xmet 16475  df-bl 16477
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