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Theorem bdmopn 14401
Description: The standard bounded metric corresponding to  C generates the same topology as  C. (Contributed by Mario Carneiro, 26-Aug-2015.) (Revised by Jim Kingdon, 19-May-2023.)
Hypotheses
Ref Expression
stdbdmet.1  |-  D  =  ( x  e.  X ,  y  e.  X  |-> inf ( { ( x C y ) ,  R } ,  RR* ,  <  ) )
stdbdmopn.2  |-  J  =  ( MetOpen `  C )
Assertion
Ref Expression
bdmopn  |-  ( ( C  e.  ( *Met `  X )  /\  R  e.  RR*  /\  0  <  R )  ->  J  =  (
MetOpen `  D ) )
Distinct variable groups:    x, y, C   
x, R, y    x, X, y
Allowed substitution hints:    D( x, y)    J( x, y)

Proof of Theorem bdmopn
Dummy variables  r  s  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 rpxr 9679 . . . . . . . 8  |-  ( r  e.  RR+  ->  r  e. 
RR* )
21ad2antll 491 . . . . . . 7  |-  ( ( ( C  e.  ( *Met `  X
)  /\  R  e.  RR* 
/\  0  <  R
)  /\  ( z  e.  X  /\  r  e.  RR+ ) )  -> 
r  e.  RR* )
3 simpl2 1003 . . . . . . 7  |-  ( ( ( C  e.  ( *Met `  X
)  /\  R  e.  RR* 
/\  0  <  R
)  /\  ( z  e.  X  /\  r  e.  RR+ ) )  ->  R  e.  RR* )
4 xrmincl 11292 . . . . . . 7  |-  ( ( r  e.  RR*  /\  R  e.  RR* )  -> inf ( { r ,  R } ,  RR* ,  <  )  e.  RR* )
52, 3, 4syl2anc 411 . . . . . 6  |-  ( ( ( C  e.  ( *Met `  X
)  /\  R  e.  RR* 
/\  0  <  R
)  /\  ( z  e.  X  /\  r  e.  RR+ ) )  -> inf ( { r ,  R } ,  RR* ,  <  )  e.  RR* )
6 rpre 9678 . . . . . . 7  |-  ( r  e.  RR+  ->  r  e.  RR )
76ad2antll 491 . . . . . 6  |-  ( ( ( C  e.  ( *Met `  X
)  /\  R  e.  RR* 
/\  0  <  R
)  /\  ( z  e.  X  /\  r  e.  RR+ ) )  -> 
r  e.  RR )
8 0xr 8022 . . . . . . . 8  |-  0  e.  RR*
98a1i 9 . . . . . . 7  |-  ( ( ( C  e.  ( *Met `  X
)  /\  R  e.  RR* 
/\  0  <  R
)  /\  ( z  e.  X  /\  r  e.  RR+ ) )  -> 
0  e.  RR* )
10 rpgt0 9683 . . . . . . . . 9  |-  ( r  e.  RR+  ->  0  < 
r )
1110ad2antll 491 . . . . . . . 8  |-  ( ( ( C  e.  ( *Met `  X
)  /\  R  e.  RR* 
/\  0  <  R
)  /\  ( z  e.  X  /\  r  e.  RR+ ) )  -> 
0  <  r )
12 simpl3 1004 . . . . . . . 8  |-  ( ( ( C  e.  ( *Met `  X
)  /\  R  e.  RR* 
/\  0  <  R
)  /\  ( z  e.  X  /\  r  e.  RR+ ) )  -> 
0  <  R )
13 xrltmininf 11296 . . . . . . . . 9  |-  ( ( 0  e.  RR*  /\  r  e.  RR*  /\  R  e. 
RR* )  ->  (
0  < inf ( {
r ,  R } ,  RR* ,  <  )  <->  ( 0  <  r  /\  0  <  R ) ) )
148, 2, 3, 13mp3an2i 1353 . . . . . . . 8  |-  ( ( ( C  e.  ( *Met `  X
)  /\  R  e.  RR* 
/\  0  <  R
)  /\  ( z  e.  X  /\  r  e.  RR+ ) )  -> 
( 0  < inf ( { r ,  R } ,  RR* ,  <  )  <-> 
( 0  <  r  /\  0  <  R ) ) )
1511, 12, 14mpbir2and 946 . . . . . . 7  |-  ( ( ( C  e.  ( *Met `  X
)  /\  R  e.  RR* 
/\  0  <  R
)  /\  ( z  e.  X  /\  r  e.  RR+ ) )  -> 
0  < inf ( {
r ,  R } ,  RR* ,  <  )
)
169, 5, 15xrltled 9817 . . . . . 6  |-  ( ( ( C  e.  ( *Met `  X
)  /\  R  e.  RR* 
/\  0  <  R
)  /\  ( z  e.  X  /\  r  e.  RR+ ) )  -> 
0  <_ inf ( {
r ,  R } ,  RR* ,  <  )
)
17 xrmin1inf 11293 . . . . . . 7  |-  ( ( r  e.  RR*  /\  R  e.  RR* )  -> inf ( { r ,  R } ,  RR* ,  <  )  <_  r )
182, 3, 17syl2anc 411 . . . . . 6  |-  ( ( ( C  e.  ( *Met `  X
)  /\  R  e.  RR* 
/\  0  <  R
)  /\  ( z  e.  X  /\  r  e.  RR+ ) )  -> inf ( { r ,  R } ,  RR* ,  <  )  <_  r )
19 xrrege0 9843 . . . . . 6  |-  ( ( (inf ( { r ,  R } ,  RR* ,  <  )  e. 
RR*  /\  r  e.  RR )  /\  (
0  <_ inf ( {
r ,  R } ,  RR* ,  <  )  /\ inf ( { r ,  R } ,  RR* ,  <  )  <_  r
) )  -> inf ( { r ,  R } ,  RR* ,  <  )  e.  RR )
205, 7, 16, 18, 19syl22anc 1250 . . . . 5  |-  ( ( ( C  e.  ( *Met `  X
)  /\  R  e.  RR* 
/\  0  <  R
)  /\  ( z  e.  X  /\  r  e.  RR+ ) )  -> inf ( { r ,  R } ,  RR* ,  <  )  e.  RR )
2120, 15elrpd 9711 . . . 4  |-  ( ( ( C  e.  ( *Met `  X
)  /\  R  e.  RR* 
/\  0  <  R
)  /\  ( z  e.  X  /\  r  e.  RR+ ) )  -> inf ( { r ,  R } ,  RR* ,  <  )  e.  RR+ )
22 simprl 529 . . . . . . 7  |-  ( ( ( C  e.  ( *Met `  X
)  /\  R  e.  RR* 
/\  0  <  R
)  /\  ( z  e.  X  /\  r  e.  RR+ ) )  -> 
z  e.  X )
23 xrmin2inf 11294 . . . . . . . 8  |-  ( ( r  e.  RR*  /\  R  e.  RR* )  -> inf ( { r ,  R } ,  RR* ,  <  )  <_  R )
242, 3, 23syl2anc 411 . . . . . . 7  |-  ( ( ( C  e.  ( *Met `  X
)  /\  R  e.  RR* 
/\  0  <  R
)  /\  ( z  e.  X  /\  r  e.  RR+ ) )  -> inf ( { r ,  R } ,  RR* ,  <  )  <_  R )
2522, 5, 243jca 1179 . . . . . 6  |-  ( ( ( C  e.  ( *Met `  X
)  /\  R  e.  RR* 
/\  0  <  R
)  /\  ( z  e.  X  /\  r  e.  RR+ ) )  -> 
( z  e.  X  /\ inf ( { r ,  R } ,  RR* ,  <  )  e.  RR*  /\ inf ( { r ,  R } ,  RR* ,  <  )  <_  R
) )
26 stdbdmet.1 . . . . . . 7  |-  D  =  ( x  e.  X ,  y  e.  X  |-> inf ( { ( x C y ) ,  R } ,  RR* ,  <  ) )
2726bdbl 14400 . . . . . 6  |-  ( ( ( C  e.  ( *Met `  X
)  /\  R  e.  RR* 
/\  0  <  R
)  /\  ( z  e.  X  /\ inf ( { r ,  R } ,  RR* ,  <  )  e.  RR*  /\ inf ( {
r ,  R } ,  RR* ,  <  )  <_  R ) )  -> 
( z ( ball `  D )inf ( { r ,  R } ,  RR* ,  <  )
)  =  ( z ( ball `  C
)inf ( { r ,  R } ,  RR* ,  <  ) ) )
2825, 27syldan 282 . . . . 5  |-  ( ( ( C  e.  ( *Met `  X
)  /\  R  e.  RR* 
/\  0  <  R
)  /\  ( z  e.  X  /\  r  e.  RR+ ) )  -> 
( z ( ball `  D )inf ( { r ,  R } ,  RR* ,  <  )
)  =  ( z ( ball `  C
)inf ( { r ,  R } ,  RR* ,  <  ) ) )
2928eqcomd 2195 . . . 4  |-  ( ( ( C  e.  ( *Met `  X
)  /\  R  e.  RR* 
/\  0  <  R
)  /\  ( z  e.  X  /\  r  e.  RR+ ) )  -> 
( z ( ball `  C )inf ( { r ,  R } ,  RR* ,  <  )
)  =  ( z ( ball `  D
)inf ( { r ,  R } ,  RR* ,  <  ) ) )
30 breq1 4021 . . . . . 6  |-  ( s  = inf ( { r ,  R } ,  RR* ,  <  )  -> 
( s  <_  r  <-> inf ( { r ,  R } ,  RR* ,  <  )  <_  r ) )
31 oveq2 5899 . . . . . . 7  |-  ( s  = inf ( { r ,  R } ,  RR* ,  <  )  -> 
( z ( ball `  C ) s )  =  ( z (
ball `  C )inf ( { r ,  R } ,  RR* ,  <  ) ) )
32 oveq2 5899 . . . . . . 7  |-  ( s  = inf ( { r ,  R } ,  RR* ,  <  )  -> 
( z ( ball `  D ) s )  =  ( z (
ball `  D )inf ( { r ,  R } ,  RR* ,  <  ) ) )
3331, 32eqeq12d 2204 . . . . . 6  |-  ( s  = inf ( { r ,  R } ,  RR* ,  <  )  -> 
( ( z (
ball `  C )
s )  =  ( z ( ball `  D
) s )  <->  ( z
( ball `  C )inf ( { r ,  R } ,  RR* ,  <  ) )  =  ( z ( ball `  D
)inf ( { r ,  R } ,  RR* ,  <  ) ) ) )
3430, 33anbi12d 473 . . . . 5  |-  ( s  = inf ( { r ,  R } ,  RR* ,  <  )  -> 
( ( s  <_ 
r  /\  ( z
( ball `  C )
s )  =  ( z ( ball `  D
) s ) )  <-> 
(inf ( { r ,  R } ,  RR* ,  <  )  <_ 
r  /\  ( z
( ball `  C )inf ( { r ,  R } ,  RR* ,  <  ) )  =  ( z ( ball `  D
)inf ( { r ,  R } ,  RR* ,  <  ) ) ) ) )
3534rspcev 2856 . . . 4  |-  ( (inf ( { r ,  R } ,  RR* ,  <  )  e.  RR+  /\  (inf ( { r ,  R } ,  RR* ,  <  )  <_ 
r  /\  ( z
( ball `  C )inf ( { r ,  R } ,  RR* ,  <  ) )  =  ( z ( ball `  D
)inf ( { r ,  R } ,  RR* ,  <  ) ) ) )  ->  E. s  e.  RR+  ( s  <_ 
r  /\  ( z
( ball `  C )
s )  =  ( z ( ball `  D
) s ) ) )
3621, 18, 29, 35syl12anc 1247 . . 3  |-  ( ( ( C  e.  ( *Met `  X
)  /\  R  e.  RR* 
/\  0  <  R
)  /\  ( z  e.  X  /\  r  e.  RR+ ) )  ->  E. s  e.  RR+  (
s  <_  r  /\  ( z ( ball `  C ) s )  =  ( z (
ball `  D )
s ) ) )
3736ralrimivva 2572 . 2  |-  ( ( C  e.  ( *Met `  X )  /\  R  e.  RR*  /\  0  <  R )  ->  A. z  e.  X  A. r  e.  RR+  E. s  e.  RR+  ( s  <_ 
r  /\  ( z
( ball `  C )
s )  =  ( z ( ball `  D
) s ) ) )
38 simp1 999 . . 3  |-  ( ( C  e.  ( *Met `  X )  /\  R  e.  RR*  /\  0  <  R )  ->  C  e.  ( *Met `  X
) )
3926bdxmet 14398 . . 3  |-  ( ( C  e.  ( *Met `  X )  /\  R  e.  RR*  /\  0  <  R )  ->  D  e.  ( *Met `  X
) )
40 stdbdmopn.2 . . . 4  |-  J  =  ( MetOpen `  C )
41 eqid 2189 . . . 4  |-  ( MetOpen `  D )  =  (
MetOpen `  D )
4240, 41metequiv2 14393 . . 3  |-  ( ( C  e.  ( *Met `  X )  /\  D  e.  ( *Met `  X
) )  ->  ( A. z  e.  X  A. r  e.  RR+  E. s  e.  RR+  ( s  <_ 
r  /\  ( z
( ball `  C )
s )  =  ( z ( ball `  D
) s ) )  ->  J  =  (
MetOpen `  D ) ) )
4338, 39, 42syl2anc 411 . 2  |-  ( ( C  e.  ( *Met `  X )  /\  R  e.  RR*  /\  0  <  R )  ->  ( A. z  e.  X  A. r  e.  RR+  E. s  e.  RR+  ( s  <_  r  /\  ( z ( ball `  C ) s )  =  ( z (
ball `  D )
s ) )  ->  J  =  ( MetOpen `  D ) ) )
4437, 43mpd 13 1  |-  ( ( C  e.  ( *Met `  X )  /\  R  e.  RR*  /\  0  <  R )  ->  J  =  (
MetOpen `  D ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    <-> wb 105    /\ w3a 980    = wceq 1364    e. wcel 2160   A.wral 2468   E.wrex 2469   {cpr 3608   class class class wbr 4018   ` cfv 5231  (class class class)co 5891    e. cmpo 5893  infcinf 7000   RRcr 7828   0cc0 7829   RR*cxr 8009    < clt 8010    <_ cle 8011   RR+crp 9671   *Metcxmet 13810   ballcbl 13812   MetOpencmopn 13815
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 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-coll 4133  ax-sep 4136  ax-nul 4144  ax-pow 4189  ax-pr 4224  ax-un 4448  ax-setind 4551  ax-iinf 4602  ax-cnex 7920  ax-resscn 7921  ax-1cn 7922  ax-1re 7923  ax-icn 7924  ax-addcl 7925  ax-addrcl 7926  ax-mulcl 7927  ax-mulrcl 7928  ax-addcom 7929  ax-mulcom 7930  ax-addass 7931  ax-mulass 7932  ax-distr 7933  ax-i2m1 7934  ax-0lt1 7935  ax-1rid 7936  ax-0id 7937  ax-rnegex 7938  ax-precex 7939  ax-cnre 7940  ax-pre-ltirr 7941  ax-pre-ltwlin 7942  ax-pre-lttrn 7943  ax-pre-apti 7944  ax-pre-ltadd 7945  ax-pre-mulgt0 7946  ax-pre-mulext 7947  ax-arch 7948  ax-caucvg 7949
This theorem depends on definitions:  df-bi 117  df-stab 832  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-nel 2456  df-ral 2473  df-rex 2474  df-reu 2475  df-rmo 2476  df-rab 2477  df-v 2754  df-sbc 2978  df-csb 3073  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-nul 3438  df-if 3550  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-int 3860  df-iun 3903  df-br 4019  df-opab 4080  df-mpt 4081  df-tr 4117  df-id 4308  df-po 4311  df-iso 4312  df-iord 4381  df-on 4383  df-ilim 4384  df-suc 4386  df-iom 4605  df-xp 4647  df-rel 4648  df-cnv 4649  df-co 4650  df-dm 4651  df-rn 4652  df-res 4653  df-ima 4654  df-iota 5193  df-fun 5233  df-fn 5234  df-f 5235  df-f1 5236  df-fo 5237  df-f1o 5238  df-fv 5239  df-isom 5240  df-riota 5847  df-ov 5894  df-oprab 5895  df-mpo 5896  df-1st 6159  df-2nd 6160  df-recs 6324  df-frec 6410  df-map 6668  df-sup 7001  df-inf 7002  df-pnf 8012  df-mnf 8013  df-xr 8014  df-ltxr 8015  df-le 8016  df-sub 8148  df-neg 8149  df-reap 8550  df-ap 8557  df-div 8648  df-inn 8938  df-2 8996  df-3 8997  df-4 8998  df-n0 9195  df-z 9272  df-uz 9547  df-q 9638  df-rp 9672  df-xneg 9790  df-xadd 9791  df-icc 9913  df-seqfrec 10464  df-exp 10538  df-cj 10869  df-re 10870  df-im 10871  df-rsqrt 11025  df-abs 11026  df-topgen 12731  df-psmet 13817  df-xmet 13818  df-bl 13820  df-mopn 13821  df-top 13895  df-bases 13940
This theorem is referenced by:  mopnex  14402
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