MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  metdsge Unicode version

Theorem metdsge 18751
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 962 . . . 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 18749 . . . 4  |-  ( A  e.  X  ->  ( F `  A )  =  sup ( ran  (
y  e.  S  |->  ( A D y ) ) ,  RR* ,  `'  <  ) )
41, 3syl 16 . . 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 4166 . 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 996 . . . . . 6  |-  ( ( ( ( D  e.  ( * Met `  X
)  /\  S  C_  X  /\  A  e.  X
)  /\  R  e.  RR* )  /\  w  e.  S )  ->  D  e.  ( * Met `  X
) )
71adantr 452 . . . . . 6  |-  ( ( ( ( D  e.  ( * Met `  X
)  /\  S  C_  X  /\  A  e.  X
)  /\  R  e.  RR* )  /\  w  e.  S )  ->  A  e.  X )
8 simpl2 961 . . . . . . 7  |-  ( ( ( D  e.  ( * Met `  X
)  /\  S  C_  X  /\  A  e.  X
)  /\  R  e.  RR* )  ->  S  C_  X
)
98sselda 3292 . . . . . 6  |-  ( ( ( ( D  e.  ( * Met `  X
)  /\  S  C_  X  /\  A  e.  X
)  /\  R  e.  RR* )  /\  w  e.  S )  ->  w  e.  X )
10 xmetcl 18271 . . . . . 6  |-  ( ( D  e.  ( * Met `  X )  /\  A  e.  X  /\  w  e.  X
)  ->  ( A D w )  e. 
RR* )
116, 7, 9, 10syl3anc 1184 . . . . 5  |-  ( ( ( ( D  e.  ( * Met `  X
)  /\  S  C_  X  /\  A  e.  X
)  /\  R  e.  RR* )  /\  w  e.  S )  ->  ( A D w )  e. 
RR* )
12 oveq2 6029 . . . . . 6  |-  ( y  =  w  ->  ( A D y )  =  ( A D w ) )
1312cbvmptv 4242 . . . . 5  |-  ( y  e.  S  |->  ( A D y ) )  =  ( w  e.  S  |->  ( A D w ) )
1411, 13fmptd 5833 . . . 4  |-  ( ( ( D  e.  ( * Met `  X
)  /\  S  C_  X  /\  A  e.  X
)  /\  R  e.  RR* )  ->  ( y  e.  S  |->  ( A D y ) ) : S --> RR* )
15 frn 5538 . . . 4  |-  ( ( y  e.  S  |->  ( A D y ) ) : S --> RR*  ->  ran  ( y  e.  S  |->  ( A D y ) )  C_  RR* )
1614, 15syl 16 . . 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 448 . . 3  |-  ( ( ( D  e.  ( * Met `  X
)  /\  S  C_  X  /\  A  e.  X
)  /\  R  e.  RR* )  ->  R  e.  RR* )
18 infmxrgelb 10846 . . 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 643 . 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 452 . . . . . . 7  |-  ( ( ( ( D  e.  ( * Met `  X
)  /\  S  C_  X  /\  A  e.  X
)  /\  R  e.  RR* )  /\  w  e.  S )  ->  R  e.  RR* )
21 elbl2 18325 . . . . . . 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 1185 . . . . . 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 9078 . . . . . . 7  |-  ( ( ( A D w )  e.  RR*  /\  R  e.  RR* )  ->  (
( A D w )  <  R  <->  -.  R  <_  ( A D w ) ) )
2411, 20, 23syl2anc 643 . . . . . 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 245 . . . . 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 320 . . . 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 2666 . . 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 6046 . . . . 5  |-  ( A D w )  e. 
_V
2928rgenw 2717 . . . 4  |-  A. w  e.  S  ( A D w )  e. 
_V
30 breq2 4158 . . . . 5  |-  ( z  =  ( A D w )  ->  ( R  <_  z  <->  R  <_  ( A D w ) ) )
3113, 30ralrnmpt 5818 . . . 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 3612 . . 3  |-  ( ( S  i^i  ( A ( ball `  D
) R ) )  =  (/)  <->  A. w  e.  S  -.  w  e.  ( A ( ball `  D
) R ) )
3427, 32, 333bitr4g 280 . 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 271 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 177    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1717   A.wral 2650   _Vcvv 2900    i^i cin 3263    C_ wss 3264   (/)c0 3572   class class class wbr 4154    e. cmpt 4208   `'ccnv 4818   ran crn 4820   -->wf 5391   ` cfv 5395  (class class class)co 6021   supcsup 7381   RR*cxr 9053    < clt 9054    <_ cle 9055   * Metcxmt 16613   ballcbl 16615
This theorem is referenced by:  metds0  18752  metdstri  18753  metdseq0  18756  lebnumlem3  18860
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369  ax-sep 4272  ax-nul 4280  ax-pow 4319  ax-pr 4345  ax-un 4642  ax-cnex 8980  ax-resscn 8981  ax-1cn 8982  ax-icn 8983  ax-addcl 8984  ax-addrcl 8985  ax-mulcl 8986  ax-mulrcl 8987  ax-mulcom 8988  ax-addass 8989  ax-mulass 8990  ax-distr 8991  ax-i2m1 8992  ax-1ne0 8993  ax-1rid 8994  ax-rnegex 8995  ax-rrecex 8996  ax-cnre 8997  ax-pre-lttri 8998  ax-pre-lttrn 8999  ax-pre-ltadd 9000  ax-pre-mulgt0 9001  ax-pre-sup 9002
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2243  df-mo 2244  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ne 2553  df-nel 2554  df-ral 2655  df-rex 2656  df-reu 2657  df-rmo 2658  df-rab 2659  df-v 2902  df-sbc 3106  df-csb 3196  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-nul 3573  df-if 3684  df-pw 3745  df-sn 3764  df-pr 3765  df-op 3767  df-uni 3959  df-iun 4038  df-br 4155  df-opab 4209  df-mpt 4210  df-id 4440  df-po 4445  df-so 4446  df-xp 4825  df-rel 4826  df-cnv 4827  df-co 4828  df-dm 4829  df-rn 4830  df-res 4831  df-ima 4832  df-iota 5359  df-fun 5397  df-fn 5398  df-f 5399  df-f1 5400  df-fo 5401  df-f1o 5402  df-fv 5403  df-ov 6024  df-oprab 6025  df-mpt2 6026  df-1st 6289  df-2nd 6290  df-riota 6486  df-er 6842  df-map 6957  df-en 7047  df-dom 7048  df-sdom 7049  df-sup 7382  df-pnf 9056  df-mnf 9057  df-xr 9058  df-ltxr 9059  df-le 9060  df-sub 9226  df-neg 9227  df-xmet 16620  df-bl 16622
  Copyright terms: Public domain W3C validator