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

Theorem stdbdbl 18063
Description: The standard bounded metric corresponding to  C generates the same balls as  C for radii less than  R. (Contributed by Mario Carneiro, 26-Aug-2015.)
Hypothesis
Ref Expression
stdbdmet.1  |-  D  =  ( x  e.  X ,  y  e.  X  |->  if ( ( x C y )  <_  R ,  ( x C y ) ,  R ) )
Assertion
Ref Expression
stdbdbl  |-  ( ( ( C  e.  ( * Met `  X
)  /\  R  e.  RR* 
/\  0  <  R
)  /\  ( P  e.  X  /\  S  e. 
RR*  /\  S  <_  R ) )  ->  ( P ( ball `  D
) S )  =  ( P ( ball `  C ) S ) )
Distinct variable groups:    x, y, C    x, P, y    x, R, y    x, X, y
Allowed substitution hints:    D( x, y)    S( x, y)

Proof of Theorem stdbdbl
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 simpll2 995 . . . . . 6  |-  ( ( ( ( C  e.  ( * Met `  X
)  /\  R  e.  RR* 
/\  0  <  R
)  /\  ( P  e.  X  /\  S  e. 
RR*  /\  S  <_  R ) )  /\  z  e.  X )  ->  R  e.  RR* )
2 simpr1 961 . . . . . . 7  |-  ( ( ( C  e.  ( * Met `  X
)  /\  R  e.  RR* 
/\  0  <  R
)  /\  ( P  e.  X  /\  S  e. 
RR*  /\  S  <_  R ) )  ->  P  e.  X )
32adantr 451 . . . . . 6  |-  ( ( ( ( C  e.  ( * Met `  X
)  /\  R  e.  RR* 
/\  0  <  R
)  /\  ( P  e.  X  /\  S  e. 
RR*  /\  S  <_  R ) )  /\  z  e.  X )  ->  P  e.  X )
4 simpr 447 . . . . . 6  |-  ( ( ( ( C  e.  ( * Met `  X
)  /\  R  e.  RR* 
/\  0  <  R
)  /\  ( P  e.  X  /\  S  e. 
RR*  /\  S  <_  R ) )  /\  z  e.  X )  ->  z  e.  X )
5 stdbdmet.1 . . . . . . 7  |-  D  =  ( x  e.  X ,  y  e.  X  |->  if ( ( x C y )  <_  R ,  ( x C y ) ,  R ) )
65stdbdmetval 18060 . . . . . 6  |-  ( ( R  e.  RR*  /\  P  e.  X  /\  z  e.  X )  ->  ( P D z )  =  if ( ( P C z )  <_  R ,  ( P C z ) ,  R ) )
71, 3, 4, 6syl3anc 1182 . . . . 5  |-  ( ( ( ( C  e.  ( * Met `  X
)  /\  R  e.  RR* 
/\  0  <  R
)  /\  ( P  e.  X  /\  S  e. 
RR*  /\  S  <_  R ) )  /\  z  e.  X )  ->  ( P D z )  =  if ( ( P C z )  <_  R ,  ( P C z ) ,  R ) )
87breq1d 4033 . . . 4  |-  ( ( ( ( C  e.  ( * Met `  X
)  /\  R  e.  RR* 
/\  0  <  R
)  /\  ( P  e.  X  /\  S  e. 
RR*  /\  S  <_  R ) )  /\  z  e.  X )  ->  (
( P D z )  <  S  <->  if (
( P C z )  <_  R , 
( P C z ) ,  R )  <  S ) )
9 simplr3 999 . . . . . . . 8  |-  ( ( ( ( C  e.  ( * Met `  X
)  /\  R  e.  RR* 
/\  0  <  R
)  /\  ( P  e.  X  /\  S  e. 
RR*  /\  S  <_  R ) )  /\  z  e.  X )  ->  S  <_  R )
109biantrud 493 . . . . . . 7  |-  ( ( ( ( C  e.  ( * Met `  X
)  /\  R  e.  RR* 
/\  0  <  R
)  /\  ( P  e.  X  /\  S  e. 
RR*  /\  S  <_  R ) )  /\  z  e.  X )  ->  ( S  <_  ( P C z )  <->  ( S  <_  ( P C z )  /\  S  <_  R ) ) )
11 simpr2 962 . . . . . . . . 9  |-  ( ( ( C  e.  ( * Met `  X
)  /\  R  e.  RR* 
/\  0  <  R
)  /\  ( P  e.  X  /\  S  e. 
RR*  /\  S  <_  R ) )  ->  S  e.  RR* )
1211adantr 451 . . . . . . . 8  |-  ( ( ( ( C  e.  ( * Met `  X
)  /\  R  e.  RR* 
/\  0  <  R
)  /\  ( P  e.  X  /\  S  e. 
RR*  /\  S  <_  R ) )  /\  z  e.  X )  ->  S  e.  RR* )
13 simpl1 958 . . . . . . . . . 10  |-  ( ( ( C  e.  ( * Met `  X
)  /\  R  e.  RR* 
/\  0  <  R
)  /\  ( P  e.  X  /\  S  e. 
RR*  /\  S  <_  R ) )  ->  C  e.  ( * Met `  X
) )
1413adantr 451 . . . . . . . . 9  |-  ( ( ( ( C  e.  ( * Met `  X
)  /\  R  e.  RR* 
/\  0  <  R
)  /\  ( P  e.  X  /\  S  e. 
RR*  /\  S  <_  R ) )  /\  z  e.  X )  ->  C  e.  ( * Met `  X
) )
15 xmetcl 17896 . . . . . . . . 9  |-  ( ( C  e.  ( * Met `  X )  /\  P  e.  X  /\  z  e.  X
)  ->  ( P C z )  e. 
RR* )
1614, 3, 4, 15syl3anc 1182 . . . . . . . 8  |-  ( ( ( ( C  e.  ( * Met `  X
)  /\  R  e.  RR* 
/\  0  <  R
)  /\  ( P  e.  X  /\  S  e. 
RR*  /\  S  <_  R ) )  /\  z  e.  X )  ->  ( P C z )  e. 
RR* )
17 xrlemin 10513 . . . . . . . 8  |-  ( ( S  e.  RR*  /\  ( P C z )  e. 
RR*  /\  R  e.  RR* )  ->  ( S  <_  if ( ( P C z )  <_  R ,  ( P C z ) ,  R )  <->  ( S  <_  ( P C z )  /\  S  <_  R ) ) )
1812, 16, 1, 17syl3anc 1182 . . . . . . 7  |-  ( ( ( ( C  e.  ( * Met `  X
)  /\  R  e.  RR* 
/\  0  <  R
)  /\  ( P  e.  X  /\  S  e. 
RR*  /\  S  <_  R ) )  /\  z  e.  X )  ->  ( S  <_  if ( ( P C z )  <_  R ,  ( P C z ) ,  R )  <->  ( S  <_  ( P C z )  /\  S  <_  R ) ) )
1910, 18bitr4d 247 . . . . . 6  |-  ( ( ( ( C  e.  ( * Met `  X
)  /\  R  e.  RR* 
/\  0  <  R
)  /\  ( P  e.  X  /\  S  e. 
RR*  /\  S  <_  R ) )  /\  z  e.  X )  ->  ( S  <_  ( P C z )  <->  S  <_  if ( ( P C z )  <_  R ,  ( P C z ) ,  R
) ) )
2019notbid 285 . . . . 5  |-  ( ( ( ( C  e.  ( * Met `  X
)  /\  R  e.  RR* 
/\  0  <  R
)  /\  ( P  e.  X  /\  S  e. 
RR*  /\  S  <_  R ) )  /\  z  e.  X )  ->  ( -.  S  <_  ( P C z )  <->  -.  S  <_  if ( ( P C z )  <_  R ,  ( P C z ) ,  R ) ) )
21 xrltnle 8891 . . . . . 6  |-  ( ( ( P C z )  e.  RR*  /\  S  e.  RR* )  ->  (
( P C z )  <  S  <->  -.  S  <_  ( P C z ) ) )
2216, 12, 21syl2anc 642 . . . . 5  |-  ( ( ( ( C  e.  ( * Met `  X
)  /\  R  e.  RR* 
/\  0  <  R
)  /\  ( P  e.  X  /\  S  e. 
RR*  /\  S  <_  R ) )  /\  z  e.  X )  ->  (
( P C z )  <  S  <->  -.  S  <_  ( P C z ) ) )
23 ifcl 3601 . . . . . . 7  |-  ( ( ( P C z )  e.  RR*  /\  R  e.  RR* )  ->  if ( ( P C z )  <_  R ,  ( P C z ) ,  R
)  e.  RR* )
2416, 1, 23syl2anc 642 . . . . . 6  |-  ( ( ( ( C  e.  ( * Met `  X
)  /\  R  e.  RR* 
/\  0  <  R
)  /\  ( P  e.  X  /\  S  e. 
RR*  /\  S  <_  R ) )  /\  z  e.  X )  ->  if ( ( P C z )  <_  R ,  ( P C z ) ,  R
)  e.  RR* )
25 xrltnle 8891 . . . . . 6  |-  ( ( if ( ( P C z )  <_  R ,  ( P C z ) ,  R )  e.  RR*  /\  S  e.  RR* )  ->  ( if ( ( P C z )  <_  R ,  ( P C z ) ,  R )  < 
S  <->  -.  S  <_  if ( ( P C z )  <_  R ,  ( P C z ) ,  R
) ) )
2624, 12, 25syl2anc 642 . . . . 5  |-  ( ( ( ( C  e.  ( * Met `  X
)  /\  R  e.  RR* 
/\  0  <  R
)  /\  ( P  e.  X  /\  S  e. 
RR*  /\  S  <_  R ) )  /\  z  e.  X )  ->  ( if ( ( P C z )  <_  R ,  ( P C z ) ,  R
)  <  S  <->  -.  S  <_  if ( ( P C z )  <_  R ,  ( P C z ) ,  R ) ) )
2720, 22, 263bitr4d 276 . . . 4  |-  ( ( ( ( C  e.  ( * Met `  X
)  /\  R  e.  RR* 
/\  0  <  R
)  /\  ( P  e.  X  /\  S  e. 
RR*  /\  S  <_  R ) )  /\  z  e.  X )  ->  (
( P C z )  <  S  <->  if (
( P C z )  <_  R , 
( P C z ) ,  R )  <  S ) )
288, 27bitr4d 247 . . 3  |-  ( ( ( ( C  e.  ( * Met `  X
)  /\  R  e.  RR* 
/\  0  <  R
)  /\  ( P  e.  X  /\  S  e. 
RR*  /\  S  <_  R ) )  /\  z  e.  X )  ->  (
( P D z )  <  S  <->  ( P C z )  < 
S ) )
2928rabbidva 2779 . 2  |-  ( ( ( C  e.  ( * Met `  X
)  /\  R  e.  RR* 
/\  0  <  R
)  /\  ( P  e.  X  /\  S  e. 
RR*  /\  S  <_  R ) )  ->  { z  e.  X  |  ( P D z )  <  S }  =  { z  e.  X  |  ( P C z )  <  S } )
305stdbdxmet 18061 . . . 4  |-  ( ( C  e.  ( * Met `  X )  /\  R  e.  RR*  /\  0  <  R )  ->  D  e.  ( * Met `  X
) )
3130adantr 451 . . 3  |-  ( ( ( C  e.  ( * Met `  X
)  /\  R  e.  RR* 
/\  0  <  R
)  /\  ( P  e.  X  /\  S  e. 
RR*  /\  S  <_  R ) )  ->  D  e.  ( * Met `  X
) )
32 blval 17948 . . 3  |-  ( ( D  e.  ( * Met `  X )  /\  P  e.  X  /\  S  e.  RR* )  ->  ( P ( ball `  D ) S )  =  { z  e.  X  |  ( P D z )  < 
S } )
3331, 2, 11, 32syl3anc 1182 . 2  |-  ( ( ( C  e.  ( * Met `  X
)  /\  R  e.  RR* 
/\  0  <  R
)  /\  ( P  e.  X  /\  S  e. 
RR*  /\  S  <_  R ) )  ->  ( P ( ball `  D
) S )  =  { z  e.  X  |  ( P D z )  <  S } )
34 blval 17948 . . 3  |-  ( ( C  e.  ( * Met `  X )  /\  P  e.  X  /\  S  e.  RR* )  ->  ( P ( ball `  C ) S )  =  { z  e.  X  |  ( P C z )  < 
S } )
3513, 2, 11, 34syl3anc 1182 . 2  |-  ( ( ( C  e.  ( * Met `  X
)  /\  R  e.  RR* 
/\  0  <  R
)  /\  ( P  e.  X  /\  S  e. 
RR*  /\  S  <_  R ) )  ->  ( P ( ball `  C
) S )  =  { z  e.  X  |  ( P C z )  <  S } )
3629, 33, 353eqtr4d 2325 1  |-  ( ( ( C  e.  ( * Met `  X
)  /\  R  e.  RR* 
/\  0  <  R
)  /\  ( P  e.  X  /\  S  e. 
RR*  /\  S  <_  R ) )  ->  ( P ( ball `  D
) S )  =  ( P ( ball `  C ) S ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   {crab 2547   ifcif 3565   class class class wbr 4023   ` cfv 5255  (class class class)co 5858    e. cmpt2 5860   0cc0 8737   RR*cxr 8866    < clt 8867    <_ cle 8868   * Metcxmt 16369   ballcbl 16371
This theorem is referenced by:  stdbdmopn  18064  xlebnum  18463
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
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-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-2 9804  df-rp 10355  df-xneg 10452  df-xadd 10453  df-xmul 10454  df-icc 10663  df-xmet 16373  df-bl 16375
  Copyright terms: Public domain W3C validator