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Theorem metdsge 18353
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 18351 . . . 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 4035 . 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 3180 . . . . . 6  |-  ( ( ( ( D  e.  ( * Met `  X
)  /\  S  C_  X  /\  A  e.  X
)  /\  R  e.  RR* )  /\  w  e.  S )  ->  w  e.  X )
10 xmetcl 17896 . . . . . 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 5866 . . . . . 6  |-  ( y  =  w  ->  ( A D y )  =  ( A D w ) )
1312cbvmptv 4111 . . . . 5  |-  ( y  e.  S  |->  ( A D y ) )  =  ( w  e.  S  |->  ( A D w ) )
1411, 13fmptd 5684 . . . 4  |-  ( ( ( D  e.  ( * Met `  X
)  /\  S  C_  X  /\  A  e.  X
)  /\  R  e.  RR* )  ->  ( y  e.  S  |->  ( A D y ) ) : S --> RR* )
15 frn 5395 . . . 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 10653 . . 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 17950 . . . . . . 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 8891 . . . . . . 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 2559 . . 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 5883 . . . . 5  |-  ( A D w )  e. 
_V
2928rgenw 2610 . . . 4  |-  A. w  e.  S  ( A D w )  e. 
_V
30 breq2 4027 . . . . 5  |-  ( z  =  ( A D w )  ->  ( R  <_  z  <->  R  <_  ( A D w ) ) )
3113, 30ralrnmpt 5669 . . . 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 3495 . . 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 1623    e. wcel 1684   A.wral 2543   _Vcvv 2788    i^i cin 3151    C_ wss 3152   (/)c0 3455   class class class wbr 4023    e. cmpt 4077   `'ccnv 4688   ran crn 4690   -->wf 5251   ` cfv 5255  (class class class)co 5858   supcsup 7193   RR*cxr 8866    < clt 8867    <_ cle 8868   * Metcxmt 16369   ballcbl 16371
This theorem is referenced by:  metds0  18354  metdstri  18355  metdseq0  18358  lebnumlem3  18461
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814  ax-pre-sup 8815
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 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-po 4314  df-so 4315  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-er 6660  df-map 6774  df-en 6864  df-dom 6865  df-sdom 6866  df-sup 7194  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-xmet 16373  df-bl 16375
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