ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  bdmopn Unicode version

Theorem bdmopn 12432
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 9298 . . . . . . . 8  |-  ( r  e.  RR+  ->  r  e. 
RR* )
21ad2antll 478 . . . . . . 7  |-  ( ( ( C  e.  ( *Met `  X
)  /\  R  e.  RR* 
/\  0  <  R
)  /\  ( z  e.  X  /\  r  e.  RR+ ) )  -> 
r  e.  RR* )
3 simpl2 953 . . . . . . 7  |-  ( ( ( C  e.  ( *Met `  X
)  /\  R  e.  RR* 
/\  0  <  R
)  /\  ( z  e.  X  /\  r  e.  RR+ ) )  ->  R  e.  RR* )
4 xrmincl 10874 . . . . . . 7  |-  ( ( r  e.  RR*  /\  R  e.  RR* )  -> inf ( { r ,  R } ,  RR* ,  <  )  e.  RR* )
52, 3, 4syl2anc 406 . . . . . 6  |-  ( ( ( C  e.  ( *Met `  X
)  /\  R  e.  RR* 
/\  0  <  R
)  /\  ( z  e.  X  /\  r  e.  RR+ ) )  -> inf ( { r ,  R } ,  RR* ,  <  )  e.  RR* )
6 rpre 9297 . . . . . . 7  |-  ( r  e.  RR+  ->  r  e.  RR )
76ad2antll 478 . . . . . 6  |-  ( ( ( C  e.  ( *Met `  X
)  /\  R  e.  RR* 
/\  0  <  R
)  /\  ( z  e.  X  /\  r  e.  RR+ ) )  -> 
r  e.  RR )
8 0xr 7684 . . . . . . . 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 9302 . . . . . . . . 9  |-  ( r  e.  RR+  ->  0  < 
r )
1110ad2antll 478 . . . . . . . 8  |-  ( ( ( C  e.  ( *Met `  X
)  /\  R  e.  RR* 
/\  0  <  R
)  /\  ( z  e.  X  /\  r  e.  RR+ ) )  -> 
0  <  r )
12 simpl3 954 . . . . . . . 8  |-  ( ( ( C  e.  ( *Met `  X
)  /\  R  e.  RR* 
/\  0  <  R
)  /\  ( z  e.  X  /\  r  e.  RR+ ) )  -> 
0  <  R )
13 xrltmininf 10878 . . . . . . . . 9  |-  ( ( 0  e.  RR*  /\  r  e.  RR*  /\  R  e. 
RR* )  ->  (
0  < inf ( {
r ,  R } ,  RR* ,  <  )  <->  ( 0  <  r  /\  0  <  R ) ) )
148, 2, 3, 13mp3an2i 1288 . . . . . . . 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 896 . . . . . . 7  |-  ( ( ( C  e.  ( *Met `  X
)  /\  R  e.  RR* 
/\  0  <  R
)  /\  ( z  e.  X  /\  r  e.  RR+ ) )  -> 
0  < inf ( {
r ,  R } ,  RR* ,  <  )
)
169, 5, 15xrltled 9426 . . . . . 6  |-  ( ( ( C  e.  ( *Met `  X
)  /\  R  e.  RR* 
/\  0  <  R
)  /\  ( z  e.  X  /\  r  e.  RR+ ) )  -> 
0  <_ inf ( {
r ,  R } ,  RR* ,  <  )
)
17 xrmin1inf 10875 . . . . . . 7  |-  ( ( r  e.  RR*  /\  R  e.  RR* )  -> inf ( { r ,  R } ,  RR* ,  <  )  <_  r )
182, 3, 17syl2anc 406 . . . . . 6  |-  ( ( ( C  e.  ( *Met `  X
)  /\  R  e.  RR* 
/\  0  <  R
)  /\  ( z  e.  X  /\  r  e.  RR+ ) )  -> inf ( { r ,  R } ,  RR* ,  <  )  <_  r )
19 xrrege0 9449 . . . . . 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 1185 . . . . 5  |-  ( ( ( C  e.  ( *Met `  X
)  /\  R  e.  RR* 
/\  0  <  R
)  /\  ( z  e.  X  /\  r  e.  RR+ ) )  -> inf ( { r ,  R } ,  RR* ,  <  )  e.  RR )
2120, 15elrpd 9328 . . . 4  |-  ( ( ( C  e.  ( *Met `  X
)  /\  R  e.  RR* 
/\  0  <  R
)  /\  ( z  e.  X  /\  r  e.  RR+ ) )  -> inf ( { r ,  R } ,  RR* ,  <  )  e.  RR+ )
22 simprl 501 . . . . . . 7  |-  ( ( ( C  e.  ( *Met `  X
)  /\  R  e.  RR* 
/\  0  <  R
)  /\  ( z  e.  X  /\  r  e.  RR+ ) )  -> 
z  e.  X )
23 xrmin2inf 10876 . . . . . . . 8  |-  ( ( r  e.  RR*  /\  R  e.  RR* )  -> inf ( { r ,  R } ,  RR* ,  <  )  <_  R )
242, 3, 23syl2anc 406 . . . . . . 7  |-  ( ( ( C  e.  ( *Met `  X
)  /\  R  e.  RR* 
/\  0  <  R
)  /\  ( z  e.  X  /\  r  e.  RR+ ) )  -> inf ( { r ,  R } ,  RR* ,  <  )  <_  R )
2522, 5, 243jca 1129 . . . . . 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 12431 . . . . . 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 278 . . . . 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 2105 . . . 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 3878 . . . . . 6  |-  ( s  = inf ( { r ,  R } ,  RR* ,  <  )  -> 
( s  <_  r  <-> inf ( { r ,  R } ,  RR* ,  <  )  <_  r ) )
31 oveq2 5714 . . . . . . 7  |-  ( s  = inf ( { r ,  R } ,  RR* ,  <  )  -> 
( z ( ball `  C ) s )  =  ( z (
ball `  C )inf ( { r ,  R } ,  RR* ,  <  ) ) )
32 oveq2 5714 . . . . . . 7  |-  ( s  = inf ( { r ,  R } ,  RR* ,  <  )  -> 
( z ( ball `  D ) s )  =  ( z (
ball `  D )inf ( { r ,  R } ,  RR* ,  <  ) ) )
3331, 32eqeq12d 2114 . . . . . 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 460 . . . . 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 2744 . . . 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 1182 . . 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 2473 . 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 949 . . 3  |-  ( ( C  e.  ( *Met `  X )  /\  R  e.  RR*  /\  0  <  R )  ->  C  e.  ( *Met `  X
) )
3926bdxmet 12429 . . 3  |-  ( ( C  e.  ( *Met `  X )  /\  R  e.  RR*  /\  0  <  R )  ->  D  e.  ( *Met `  X
) )
40 stdbdmopn.2 . . . 4  |-  J  =  ( MetOpen `  C )
41 eqid 2100 . . . 4  |-  ( MetOpen `  D )  =  (
MetOpen `  D )
4240, 41metequiv2 12424 . . 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 406 . 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 103    <-> wb 104    /\ w3a 930    = wceq 1299    e. wcel 1448   A.wral 2375   E.wrex 2376   {cpr 3475   class class class wbr 3875   ` cfv 5059  (class class class)co 5706    e. cmpo 5708  infcinf 6785   RRcr 7499   0cc0 7500   RR*cxr 7671    < clt 7672    <_ cle 7673   RR+crp 9291   *Metcxmet 11931   ballcbl 11933   MetOpencmopn 11936
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 584  ax-in2 585  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-13 1459  ax-14 1460  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082  ax-coll 3983  ax-sep 3986  ax-nul 3994  ax-pow 4038  ax-pr 4069  ax-un 4293  ax-setind 4390  ax-iinf 4440  ax-cnex 7586  ax-resscn 7587  ax-1cn 7588  ax-1re 7589  ax-icn 7590  ax-addcl 7591  ax-addrcl 7592  ax-mulcl 7593  ax-mulrcl 7594  ax-addcom 7595  ax-mulcom 7596  ax-addass 7597  ax-mulass 7598  ax-distr 7599  ax-i2m1 7600  ax-0lt1 7601  ax-1rid 7602  ax-0id 7603  ax-rnegex 7604  ax-precex 7605  ax-cnre 7606  ax-pre-ltirr 7607  ax-pre-ltwlin 7608  ax-pre-lttrn 7609  ax-pre-apti 7610  ax-pre-ltadd 7611  ax-pre-mulgt0 7612  ax-pre-mulext 7613  ax-arch 7614  ax-caucvg 7615
This theorem depends on definitions:  df-bi 116  df-dc 787  df-3or 931  df-3an 932  df-tru 1302  df-fal 1305  df-nf 1405  df-sb 1704  df-eu 1963  df-mo 1964  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-ne 2268  df-nel 2363  df-ral 2380  df-rex 2381  df-reu 2382  df-rmo 2383  df-rab 2384  df-v 2643  df-sbc 2863  df-csb 2956  df-dif 3023  df-un 3025  df-in 3027  df-ss 3034  df-nul 3311  df-if 3422  df-pw 3459  df-sn 3480  df-pr 3481  df-op 3483  df-uni 3684  df-int 3719  df-iun 3762  df-br 3876  df-opab 3930  df-mpt 3931  df-tr 3967  df-id 4153  df-po 4156  df-iso 4157  df-iord 4226  df-on 4228  df-ilim 4229  df-suc 4231  df-iom 4443  df-xp 4483  df-rel 4484  df-cnv 4485  df-co 4486  df-dm 4487  df-rn 4488  df-res 4489  df-ima 4490  df-iota 5024  df-fun 5061  df-fn 5062  df-f 5063  df-f1 5064  df-fo 5065  df-f1o 5066  df-fv 5067  df-isom 5068  df-riota 5662  df-ov 5709  df-oprab 5710  df-mpo 5711  df-1st 5969  df-2nd 5970  df-recs 6132  df-frec 6218  df-map 6474  df-sup 6786  df-inf 6787  df-pnf 7674  df-mnf 7675  df-xr 7676  df-ltxr 7677  df-le 7678  df-sub 7806  df-neg 7807  df-reap 8203  df-ap 8210  df-div 8294  df-inn 8579  df-2 8637  df-3 8638  df-4 8639  df-n0 8830  df-z 8907  df-uz 9177  df-q 9262  df-rp 9292  df-xneg 9400  df-xadd 9401  df-icc 9519  df-seqfrec 10060  df-exp 10134  df-cj 10455  df-re 10456  df-im 10457  df-rsqrt 10610  df-abs 10611  df-topgen 11923  df-psmet 11938  df-xmet 11939  df-bl 11941  df-mopn 11942  df-top 11947  df-bases 11992
This theorem is referenced by:  mopnex  12433
  Copyright terms: Public domain W3C validator