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Theorem metdsge 18369
Description: The distance from the point  A to the set  S is greater than  R iff the  R-ball around  A misses  S. (Contributed by Mario Carneiro, 4-Sep-2015.)
Hypothesis
Ref Expression
metdscn.f  |-  F  =  ( x  e.  X  |->  sup ( ran  (
y  e.  S  |->  ( x D y ) ) ,  RR* ,  `'  <  ) )
Assertion
Ref Expression
metdsge  |-  ( ( ( D  e.  ( * Met `  X
)  /\  S  C_  X  /\  A  e.  X
)  /\  R  e.  RR* )  ->  ( R  <_  ( F `  A
)  <->  ( S  i^i  ( A ( ball `  D
) R ) )  =  (/) ) )
Distinct variable groups:    x, y, A    x, D, y    x, S, y    x, X, y
Allowed substitution hints:    R( x, y)    F( x, y)

Proof of Theorem metdsge
Dummy variables  w  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simpl3 960 . . . 4  |-  ( ( ( D  e.  ( * Met `  X
)  /\  S  C_  X  /\  A  e.  X
)  /\  R  e.  RR* )  ->  A  e.  X )
2 metdscn.f . . . . 5  |-  F  =  ( x  e.  X  |->  sup ( ran  (
y  e.  S  |->  ( x D y ) ) ,  RR* ,  `'  <  ) )
32metdsval 18367 . . . 4  |-  ( A  e.  X  ->  ( F `  A )  =  sup ( ran  (
y  e.  S  |->  ( A D y ) ) ,  RR* ,  `'  <  ) )
41, 3syl 15 . . 3  |-  ( ( ( D  e.  ( * Met `  X
)  /\  S  C_  X  /\  A  e.  X
)  /\  R  e.  RR* )  ->  ( F `  A )  =  sup ( ran  ( y  e.  S  |->  ( A D y ) ) , 
RR* ,  `'  <  ) )
54breq2d 4051 . 2  |-  ( ( ( D  e.  ( * Met `  X
)  /\  S  C_  X  /\  A  e.  X
)  /\  R  e.  RR* )  ->  ( R  <_  ( F `  A
)  <->  R  <_  sup ( ran  ( y  e.  S  |->  ( A D y ) ) ,  RR* ,  `'  <  ) ) )
6 simpll1 994 . . . . . 6  |-  ( ( ( ( D  e.  ( * Met `  X
)  /\  S  C_  X  /\  A  e.  X
)  /\  R  e.  RR* )  /\  w  e.  S )  ->  D  e.  ( * Met `  X
) )
71adantr 451 . . . . . 6  |-  ( ( ( ( D  e.  ( * Met `  X
)  /\  S  C_  X  /\  A  e.  X
)  /\  R  e.  RR* )  /\  w  e.  S )  ->  A  e.  X )
8 simpl2 959 . . . . . . 7  |-  ( ( ( D  e.  ( * Met `  X
)  /\  S  C_  X  /\  A  e.  X
)  /\  R  e.  RR* )  ->  S  C_  X
)
98sselda 3193 . . . . . 6  |-  ( ( ( ( D  e.  ( * Met `  X
)  /\  S  C_  X  /\  A  e.  X
)  /\  R  e.  RR* )  /\  w  e.  S )  ->  w  e.  X )
10 xmetcl 17912 . . . . . 6  |-  ( ( D  e.  ( * Met `  X )  /\  A  e.  X  /\  w  e.  X
)  ->  ( A D w )  e. 
RR* )
116, 7, 9, 10syl3anc 1182 . . . . 5  |-  ( ( ( ( D  e.  ( * Met `  X
)  /\  S  C_  X  /\  A  e.  X
)  /\  R  e.  RR* )  /\  w  e.  S )  ->  ( A D w )  e. 
RR* )
12 oveq2 5882 . . . . . 6  |-  ( y  =  w  ->  ( A D y )  =  ( A D w ) )
1312cbvmptv 4127 . . . . 5  |-  ( y  e.  S  |->  ( A D y ) )  =  ( w  e.  S  |->  ( A D w ) )
1411, 13fmptd 5700 . . . 4  |-  ( ( ( D  e.  ( * Met `  X
)  /\  S  C_  X  /\  A  e.  X
)  /\  R  e.  RR* )  ->  ( y  e.  S  |->  ( A D y ) ) : S --> RR* )
15 frn 5411 . . . 4  |-  ( ( y  e.  S  |->  ( A D y ) ) : S --> RR*  ->  ran  ( y  e.  S  |->  ( A D y ) )  C_  RR* )
1614, 15syl 15 . . 3  |-  ( ( ( D  e.  ( * Met `  X
)  /\  S  C_  X  /\  A  e.  X
)  /\  R  e.  RR* )  ->  ran  ( y  e.  S  |->  ( A D y ) ) 
C_  RR* )
17 simpr 447 . . 3  |-  ( ( ( D  e.  ( * Met `  X
)  /\  S  C_  X  /\  A  e.  X
)  /\  R  e.  RR* )  ->  R  e.  RR* )
18 infmxrgelb 10669 . . 3  |-  ( ( ran  ( y  e.  S  |->  ( A D y ) )  C_  RR* 
/\  R  e.  RR* )  ->  ( R  <_  sup ( ran  ( y  e.  S  |->  ( A D y ) ) ,  RR* ,  `'  <  )  <->  A. z  e.  ran  ( y  e.  S  |->  ( A D y ) ) R  <_ 
z ) )
1916, 17, 18syl2anc 642 . 2  |-  ( ( ( D  e.  ( * Met `  X
)  /\  S  C_  X  /\  A  e.  X
)  /\  R  e.  RR* )  ->  ( R  <_  sup ( ran  (
y  e.  S  |->  ( A D y ) ) ,  RR* ,  `'  <  )  <->  A. z  e.  ran  ( y  e.  S  |->  ( A D y ) ) R  <_ 
z ) )
2017adantr 451 . . . . . . 7  |-  ( ( ( ( D  e.  ( * Met `  X
)  /\  S  C_  X  /\  A  e.  X
)  /\  R  e.  RR* )  /\  w  e.  S )  ->  R  e.  RR* )
21 elbl2 17966 . . . . . . 7  |-  ( ( ( D  e.  ( * Met `  X
)  /\  R  e.  RR* )  /\  ( A  e.  X  /\  w  e.  X ) )  -> 
( w  e.  ( A ( ball `  D
) R )  <->  ( A D w )  < 
R ) )
226, 20, 7, 9, 21syl22anc 1183 . . . . . 6  |-  ( ( ( ( D  e.  ( * Met `  X
)  /\  S  C_  X  /\  A  e.  X
)  /\  R  e.  RR* )  /\  w  e.  S )  ->  (
w  e.  ( A ( ball `  D
) R )  <->  ( A D w )  < 
R ) )
23 xrltnle 8907 . . . . . . 7  |-  ( ( ( A D w )  e.  RR*  /\  R  e.  RR* )  ->  (
( A D w )  <  R  <->  -.  R  <_  ( A D w ) ) )
2411, 20, 23syl2anc 642 . . . . . 6  |-  ( ( ( ( D  e.  ( * Met `  X
)  /\  S  C_  X  /\  A  e.  X
)  /\  R  e.  RR* )  /\  w  e.  S )  ->  (
( A D w )  <  R  <->  -.  R  <_  ( A D w ) ) )
2522, 24bitrd 244 . . . . 5  |-  ( ( ( ( D  e.  ( * Met `  X
)  /\  S  C_  X  /\  A  e.  X
)  /\  R  e.  RR* )  /\  w  e.  S )  ->  (
w  e.  ( A ( ball `  D
) R )  <->  -.  R  <_  ( A D w ) ) )
2625con2bid 319 . . . 4  |-  ( ( ( ( D  e.  ( * Met `  X
)  /\  S  C_  X  /\  A  e.  X
)  /\  R  e.  RR* )  /\  w  e.  S )  ->  ( R  <_  ( A D w )  <->  -.  w  e.  ( A ( ball `  D ) R ) ) )
2726ralbidva 2572 . . 3  |-  ( ( ( D  e.  ( * Met `  X
)  /\  S  C_  X  /\  A  e.  X
)  /\  R  e.  RR* )  ->  ( A. w  e.  S  R  <_  ( A D w )  <->  A. w  e.  S  -.  w  e.  ( A ( ball `  D
) R ) ) )
28 ovex 5899 . . . . 5  |-  ( A D w )  e. 
_V
2928rgenw 2623 . . . 4  |-  A. w  e.  S  ( A D w )  e. 
_V
30 breq2 4043 . . . . 5  |-  ( z  =  ( A D w )  ->  ( R  <_  z  <->  R  <_  ( A D w ) ) )
3113, 30ralrnmpt 5685 . . . 4  |-  ( A. w  e.  S  ( A D w )  e. 
_V  ->  ( A. z  e.  ran  ( y  e.  S  |->  ( A D y ) ) R  <_  z  <->  A. w  e.  S  R  <_  ( A D w ) ) )
3229, 31ax-mp 8 . . 3  |-  ( A. z  e.  ran  ( y  e.  S  |->  ( A D y ) ) R  <_  z  <->  A. w  e.  S  R  <_  ( A D w ) )
33 disj 3508 . . 3  |-  ( ( S  i^i  ( A ( ball `  D
) R ) )  =  (/)  <->  A. w  e.  S  -.  w  e.  ( A ( ball `  D
) R ) )
3427, 32, 333bitr4g 279 . 2  |-  ( ( ( D  e.  ( * Met `  X
)  /\  S  C_  X  /\  A  e.  X
)  /\  R  e.  RR* )  ->  ( A. z  e.  ran  ( y  e.  S  |->  ( A D y ) ) R  <_  z  <->  ( S  i^i  ( A ( ball `  D ) R ) )  =  (/) ) )
355, 19, 343bitrd 270 1  |-  ( ( ( D  e.  ( * Met `  X
)  /\  S  C_  X  /\  A  e.  X
)  /\  R  e.  RR* )  ->  ( R  <_  ( F `  A
)  <->  ( S  i^i  ( A ( ball `  D
) R ) )  =  (/) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696   A.wral 2556   _Vcvv 2801    i^i cin 3164    C_ wss 3165   (/)c0 3468   class class class wbr 4039    e. cmpt 4093   `'ccnv 4704   ran crn 4706   -->wf 5267   ` cfv 5271  (class class class)co 5874   supcsup 7209   RR*cxr 8882    < clt 8883    <_ cle 8884   * Metcxmt 16385   ballcbl 16387
This theorem is referenced by:  metds0  18370  metdstri  18371  metdseq0  18374  lebnumlem3  18477
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830  ax-pre-sup 8831
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-po 4330  df-so 4331  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-riota 6320  df-er 6676  df-map 6790  df-en 6880  df-dom 6881  df-sdom 6882  df-sup 7210  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-xmet 16389  df-bl 16391
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