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Theorem bdbl 13896
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.) (Revised by Jim Kingdon, 19-May-2023.)
Hypothesis
Ref Expression
stdbdmet.1  |-  D  =  ( x  e.  X ,  y  e.  X  |-> inf ( { ( x C y ) ,  R } ,  RR* ,  <  ) )
Assertion
Ref Expression
bdbl  |-  ( ( ( 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 bdbl
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 simpr2 1004 . . . . . 6  |-  ( ( ( C  e.  ( *Met `  X
)  /\  R  e.  RR* 
/\  0  <  R
)  /\  ( P  e.  X  /\  S  e. 
RR*  /\  S  <_  R ) )  ->  S  e.  RR* )
21adantr 276 . . . . 5  |-  ( ( ( ( C  e.  ( *Met `  X )  /\  R  e.  RR*  /\  0  < 
R )  /\  ( P  e.  X  /\  S  e.  RR*  /\  S  <_  R ) )  /\  z  e.  X )  ->  S  e.  RR* )
3 simpl1 1000 . . . . . . 7  |-  ( ( ( C  e.  ( *Met `  X
)  /\  R  e.  RR* 
/\  0  <  R
)  /\  ( P  e.  X  /\  S  e. 
RR*  /\  S  <_  R ) )  ->  C  e.  ( *Met `  X ) )
43adantr 276 . . . . . 6  |-  ( ( ( ( C  e.  ( *Met `  X )  /\  R  e.  RR*  /\  0  < 
R )  /\  ( P  e.  X  /\  S  e.  RR*  /\  S  <_  R ) )  /\  z  e.  X )  ->  C  e.  ( *Met `  X ) )
5 simpr1 1003 . . . . . . 7  |-  ( ( ( C  e.  ( *Met `  X
)  /\  R  e.  RR* 
/\  0  <  R
)  /\  ( P  e.  X  /\  S  e. 
RR*  /\  S  <_  R ) )  ->  P  e.  X )
65adantr 276 . . . . . 6  |-  ( ( ( ( C  e.  ( *Met `  X )  /\  R  e.  RR*  /\  0  < 
R )  /\  ( P  e.  X  /\  S  e.  RR*  /\  S  <_  R ) )  /\  z  e.  X )  ->  P  e.  X )
7 simpr 110 . . . . . 6  |-  ( ( ( ( C  e.  ( *Met `  X )  /\  R  e.  RR*  /\  0  < 
R )  /\  ( P  e.  X  /\  S  e.  RR*  /\  S  <_  R ) )  /\  z  e.  X )  ->  z  e.  X )
8 xmetcl 13745 . . . . . 6  |-  ( ( C  e.  ( *Met `  X )  /\  P  e.  X  /\  z  e.  X
)  ->  ( P C z )  e. 
RR* )
94, 6, 7, 8syl3anc 1238 . . . . 5  |-  ( ( ( ( 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* )
10 simpll2 1037 . . . . 5  |-  ( ( ( ( C  e.  ( *Met `  X )  /\  R  e.  RR*  /\  0  < 
R )  /\  ( P  e.  X  /\  S  e.  RR*  /\  S  <_  R ) )  /\  z  e.  X )  ->  R  e.  RR* )
11 xrminltinf 11275 . . . . 5  |-  ( ( S  e.  RR*  /\  ( P C z )  e. 
RR*  /\  R  e.  RR* )  ->  (inf ( { ( P C z ) ,  R } ,  RR* ,  <  )  <  S  <->  ( ( P C z )  < 
S  \/  R  < 
S ) ) )
122, 9, 10, 11syl3anc 1238 . . . 4  |-  ( ( ( ( C  e.  ( *Met `  X )  /\  R  e.  RR*  /\  0  < 
R )  /\  ( P  e.  X  /\  S  e.  RR*  /\  S  <_  R ) )  /\  z  e.  X )  ->  (inf ( { ( P C z ) ,  R } ,  RR* ,  <  )  < 
S  <->  ( ( P C z )  < 
S  \/  R  < 
S ) ) )
13 xmetf 13743 . . . . . . . . 9  |-  ( C  e.  ( *Met `  X )  ->  C : ( X  X.  X ) --> RR* )
14133ad2ant1 1018 . . . . . . . 8  |-  ( ( C  e.  ( *Met `  X )  /\  R  e.  RR*  /\  0  <  R )  ->  C : ( X  X.  X ) -->
RR* )
1514adantr 276 . . . . . . 7  |-  ( ( ( C  e.  ( *Met `  X
)  /\  R  e.  RR* 
/\  0  <  R
)  /\  ( P  e.  X  /\  S  e. 
RR*  /\  S  <_  R ) )  ->  C : ( X  X.  X ) --> RR* )
1615adantr 276 . . . . . 6  |-  ( ( ( ( C  e.  ( *Met `  X )  /\  R  e.  RR*  /\  0  < 
R )  /\  ( P  e.  X  /\  S  e.  RR*  /\  S  <_  R ) )  /\  z  e.  X )  ->  C : ( X  X.  X ) --> RR* )
17 stdbdmet.1 . . . . . . 7  |-  D  =  ( x  e.  X ,  y  e.  X  |-> inf ( { ( x C y ) ,  R } ,  RR* ,  <  ) )
1817bdmetval 13893 . . . . . 6  |-  ( ( ( C : ( X  X.  X ) -->
RR*  /\  R  e.  RR* )  /\  ( P  e.  X  /\  z  e.  X ) )  -> 
( P D z )  = inf ( { ( P C z ) ,  R } ,  RR* ,  <  )
)
1916, 10, 6, 7, 18syl22anc 1239 . . . . 5  |-  ( ( ( ( C  e.  ( *Met `  X )  /\  R  e.  RR*  /\  0  < 
R )  /\  ( P  e.  X  /\  S  e.  RR*  /\  S  <_  R ) )  /\  z  e.  X )  ->  ( P D z )  = inf ( { ( P C z ) ,  R } ,  RR* ,  <  )
)
2019breq1d 4013 . . . 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  <-> inf ( { ( P C z ) ,  R } ,  RR* ,  <  )  <  S ) )
21 simpr3 1005 . . . . . . . 8  |-  ( ( ( C  e.  ( *Met `  X
)  /\  R  e.  RR* 
/\  0  <  R
)  /\  ( P  e.  X  /\  S  e. 
RR*  /\  S  <_  R ) )  ->  S  <_  R )
22 simpl2 1001 . . . . . . . . 9  |-  ( ( ( C  e.  ( *Met `  X
)  /\  R  e.  RR* 
/\  0  <  R
)  /\  ( P  e.  X  /\  S  e. 
RR*  /\  S  <_  R ) )  ->  R  e.  RR* )
23 xrlenlt 8020 . . . . . . . . 9  |-  ( ( S  e.  RR*  /\  R  e.  RR* )  ->  ( S  <_  R  <->  -.  R  <  S ) )
241, 22, 23syl2anc 411 . . . . . . . 8  |-  ( ( ( C  e.  ( *Met `  X
)  /\  R  e.  RR* 
/\  0  <  R
)  /\  ( P  e.  X  /\  S  e. 
RR*  /\  S  <_  R ) )  ->  ( S  <_  R  <->  -.  R  <  S ) )
2521, 24mpbid 147 . . . . . . 7  |-  ( ( ( C  e.  ( *Met `  X
)  /\  R  e.  RR* 
/\  0  <  R
)  /\  ( P  e.  X  /\  S  e. 
RR*  /\  S  <_  R ) )  ->  -.  R  <  S )
26 biorf 744 . . . . . . 7  |-  ( -.  R  <  S  -> 
( ( P C z )  <  S  <->  ( R  <  S  \/  ( P C z )  <  S ) ) )
2725, 26syl 14 . . . . . 6  |-  ( ( ( C  e.  ( *Met `  X
)  /\  R  e.  RR* 
/\  0  <  R
)  /\  ( P  e.  X  /\  S  e. 
RR*  /\  S  <_  R ) )  ->  (
( P C z )  <  S  <->  ( R  <  S  \/  ( P C z )  < 
S ) ) )
28 orcom 728 . . . . . 6  |-  ( ( R  <  S  \/  ( P C z )  <  S )  <->  ( ( P C z )  < 
S  \/  R  < 
S ) )
2927, 28bitrdi 196 . . . . 5  |-  ( ( ( C  e.  ( *Met `  X
)  /\  R  e.  RR* 
/\  0  <  R
)  /\  ( P  e.  X  /\  S  e. 
RR*  /\  S  <_  R ) )  ->  (
( P C z )  <  S  <->  ( ( P C z )  < 
S  \/  R  < 
S ) ) )
3029adantr 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  <->  ( ( P C z )  <  S  \/  R  <  S ) ) )
3112, 20, 303bitr4d 220 . . 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 ) )
3231rabbidva 2725 . 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 } )
3317bdxmet 13894 . . . 4  |-  ( ( C  e.  ( *Met `  X )  /\  R  e.  RR*  /\  0  <  R )  ->  D  e.  ( *Met `  X
) )
3433adantr 276 . . 3  |-  ( ( ( C  e.  ( *Met `  X
)  /\  R  e.  RR* 
/\  0  <  R
)  /\  ( P  e.  X  /\  S  e. 
RR*  /\  S  <_  R ) )  ->  D  e.  ( *Met `  X ) )
35 blval 13782 . . 3  |-  ( ( D  e.  ( *Met `  X )  /\  P  e.  X  /\  S  e.  RR* )  ->  ( P ( ball `  D ) S )  =  { z  e.  X  |  ( P D z )  < 
S } )
3634, 5, 1, 35syl3anc 1238 . 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 } )
37 blval 13782 . . 3  |-  ( ( C  e.  ( *Met `  X )  /\  P  e.  X  /\  S  e.  RR* )  ->  ( P ( ball `  C ) S )  =  { z  e.  X  |  ( P C z )  < 
S } )
383, 5, 1, 37syl3anc 1238 . 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 } )
3932, 36, 383eqtr4d 2220 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    /\ wa 104    <-> wb 105    \/ wo 708    /\ w3a 978    = wceq 1353    e. wcel 2148   {crab 2459   {cpr 3593   class class class wbr 4003    X. cxp 4624   -->wf 5212   ` cfv 5216  (class class class)co 5874    e. cmpo 5876  infcinf 6981   0cc0 7810   RR*cxr 7989    < clt 7990    <_ cle 7991   *Metcxmet 13331   ballcbl 13333
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4118  ax-sep 4121  ax-nul 4129  ax-pow 4174  ax-pr 4209  ax-un 4433  ax-setind 4536  ax-iinf 4587  ax-cnex 7901  ax-resscn 7902  ax-1cn 7903  ax-1re 7904  ax-icn 7905  ax-addcl 7906  ax-addrcl 7907  ax-mulcl 7908  ax-mulrcl 7909  ax-addcom 7910  ax-mulcom 7911  ax-addass 7912  ax-mulass 7913  ax-distr 7914  ax-i2m1 7915  ax-0lt1 7916  ax-1rid 7917  ax-0id 7918  ax-rnegex 7919  ax-precex 7920  ax-cnre 7921  ax-pre-ltirr 7922  ax-pre-ltwlin 7923  ax-pre-lttrn 7924  ax-pre-apti 7925  ax-pre-ltadd 7926  ax-pre-mulgt0 7927  ax-pre-mulext 7928  ax-arch 7929  ax-caucvg 7930
This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rmo 2463  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-if 3535  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-int 3845  df-iun 3888  df-br 4004  df-opab 4065  df-mpt 4066  df-tr 4102  df-id 4293  df-po 4296  df-iso 4297  df-iord 4366  df-on 4368  df-ilim 4369  df-suc 4371  df-iom 4590  df-xp 4632  df-rel 4633  df-cnv 4634  df-co 4635  df-dm 4636  df-rn 4637  df-res 4638  df-ima 4639  df-iota 5178  df-fun 5218  df-fn 5219  df-f 5220  df-f1 5221  df-fo 5222  df-f1o 5223  df-fv 5224  df-isom 5225  df-riota 5830  df-ov 5877  df-oprab 5878  df-mpo 5879  df-1st 6140  df-2nd 6141  df-recs 6305  df-frec 6391  df-map 6649  df-sup 6982  df-inf 6983  df-pnf 7992  df-mnf 7993  df-xr 7994  df-ltxr 7995  df-le 7996  df-sub 8128  df-neg 8129  df-reap 8530  df-ap 8537  df-div 8628  df-inn 8918  df-2 8976  df-3 8977  df-4 8978  df-n0 9175  df-z 9252  df-uz 9527  df-rp 9652  df-xneg 9770  df-xadd 9771  df-icc 9893  df-seqfrec 10443  df-exp 10517  df-cj 10846  df-re 10847  df-im 10848  df-rsqrt 11002  df-abs 11003  df-psmet 13338  df-xmet 13339  df-bl 13341
This theorem is referenced by:  bdmopn  13897
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