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Theorem bdbl 14006
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 13855 . . . . . 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 11280 . . . . 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 13853 . . . . . . . . 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 14003 . . . . . 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 4014 . . . 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 8022 . . . . . . . . 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 2726 . 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 14004 . . . 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 13892 . . 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 13892 . . 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 3594   class class class wbr 4004    X. cxp 4625   -->wf 5213   ` cfv 5217  (class class class)co 5875    e. cmpo 5877  infcinf 6982   0cc0 7811   RR*cxr 7991    < clt 7992    <_ cle 7993   *Metcxmet 13443   ballcbl 13445
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 4119  ax-sep 4122  ax-nul 4130  ax-pow 4175  ax-pr 4210  ax-un 4434  ax-setind 4537  ax-iinf 4588  ax-cnex 7902  ax-resscn 7903  ax-1cn 7904  ax-1re 7905  ax-icn 7906  ax-addcl 7907  ax-addrcl 7908  ax-mulcl 7909  ax-mulrcl 7910  ax-addcom 7911  ax-mulcom 7912  ax-addass 7913  ax-mulass 7914  ax-distr 7915  ax-i2m1 7916  ax-0lt1 7917  ax-1rid 7918  ax-0id 7919  ax-rnegex 7920  ax-precex 7921  ax-cnre 7922  ax-pre-ltirr 7923  ax-pre-ltwlin 7924  ax-pre-lttrn 7925  ax-pre-apti 7926  ax-pre-ltadd 7927  ax-pre-mulgt0 7928  ax-pre-mulext 7929  ax-arch 7930  ax-caucvg 7931
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 2740  df-sbc 2964  df-csb 3059  df-dif 3132  df-un 3134  df-in 3136  df-ss 3143  df-nul 3424  df-if 3536  df-pw 3578  df-sn 3599  df-pr 3600  df-op 3602  df-uni 3811  df-int 3846  df-iun 3889  df-br 4005  df-opab 4066  df-mpt 4067  df-tr 4103  df-id 4294  df-po 4297  df-iso 4298  df-iord 4367  df-on 4369  df-ilim 4370  df-suc 4372  df-iom 4591  df-xp 4633  df-rel 4634  df-cnv 4635  df-co 4636  df-dm 4637  df-rn 4638  df-res 4639  df-ima 4640  df-iota 5179  df-fun 5219  df-fn 5220  df-f 5221  df-f1 5222  df-fo 5223  df-f1o 5224  df-fv 5225  df-isom 5226  df-riota 5831  df-ov 5878  df-oprab 5879  df-mpo 5880  df-1st 6141  df-2nd 6142  df-recs 6306  df-frec 6392  df-map 6650  df-sup 6983  df-inf 6984  df-pnf 7994  df-mnf 7995  df-xr 7996  df-ltxr 7997  df-le 7998  df-sub 8130  df-neg 8131  df-reap 8532  df-ap 8539  df-div 8630  df-inn 8920  df-2 8978  df-3 8979  df-4 8980  df-n0 9177  df-z 9254  df-uz 9529  df-rp 9654  df-xneg 9772  df-xadd 9773  df-icc 9895  df-seqfrec 10446  df-exp 10520  df-cj 10851  df-re 10852  df-im 10853  df-rsqrt 11007  df-abs 11008  df-psmet 13450  df-xmet 13451  df-bl 13453
This theorem is referenced by:  bdmopn  14007
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