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Theorem stdbdmopn 20866
Description: The standard bounded metric corresponding to  C generates the same topology as  C. (Contributed by Mario Carneiro, 26-Aug-2015.)
Hypotheses
Ref Expression
stdbdmet.1  |-  D  =  ( x  e.  X ,  y  e.  X  |->  if ( ( x C y )  <_  R ,  ( x C y ) ,  R ) )
stdbdmopn.2  |-  J  =  ( MetOpen `  C )
Assertion
Ref Expression
stdbdmopn  |-  ( ( 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 stdbdmopn
Dummy variables  r 
s  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 rpxr 11237 . . . . . . . 8  |-  ( r  e.  RR+  ->  r  e. 
RR* )
21ad2antll 728 . . . . . . 7  |-  ( ( ( C  e.  ( *Met `  X
)  /\  R  e.  RR* 
/\  0  <  R
)  /\  ( z  e.  X  /\  r  e.  RR+ ) )  -> 
r  e.  RR* )
3 simpl2 1000 . . . . . . 7  |-  ( ( ( C  e.  ( *Met `  X
)  /\  R  e.  RR* 
/\  0  <  R
)  /\  ( z  e.  X  /\  r  e.  RR+ ) )  ->  R  e.  RR* )
4 ifcl 3986 . . . . . . 7  |-  ( ( r  e.  RR*  /\  R  e.  RR* )  ->  if ( r  <_  R ,  r ,  R
)  e.  RR* )
52, 3, 4syl2anc 661 . . . . . 6  |-  ( ( ( C  e.  ( *Met `  X
)  /\  R  e.  RR* 
/\  0  <  R
)  /\  ( z  e.  X  /\  r  e.  RR+ ) )  ->  if ( r  <_  R ,  r ,  R
)  e.  RR* )
6 rpre 11236 . . . . . . 7  |-  ( r  e.  RR+  ->  r  e.  RR )
76ad2antll 728 . . . . . 6  |-  ( ( ( C  e.  ( *Met `  X
)  /\  R  e.  RR* 
/\  0  <  R
)  /\  ( z  e.  X  /\  r  e.  RR+ ) )  -> 
r  e.  RR )
8 rpgt0 11241 . . . . . . . . 9  |-  ( r  e.  RR+  ->  0  < 
r )
98ad2antll 728 . . . . . . . 8  |-  ( ( ( C  e.  ( *Met `  X
)  /\  R  e.  RR* 
/\  0  <  R
)  /\  ( z  e.  X  /\  r  e.  RR+ ) )  -> 
0  <  r )
10 simpl3 1001 . . . . . . . 8  |-  ( ( ( C  e.  ( *Met `  X
)  /\  R  e.  RR* 
/\  0  <  R
)  /\  ( z  e.  X  /\  r  e.  RR+ ) )  -> 
0  <  R )
11 breq2 4456 . . . . . . . . 9  |-  ( r  =  if ( r  <_  R ,  r ,  R )  -> 
( 0  <  r  <->  0  <  if ( r  <_  R ,  r ,  R ) ) )
12 breq2 4456 . . . . . . . . 9  |-  ( R  =  if ( r  <_  R ,  r ,  R )  -> 
( 0  <  R  <->  0  <  if ( r  <_  R ,  r ,  R ) ) )
1311, 12ifboth 3980 . . . . . . . 8  |-  ( ( 0  <  r  /\  0  <  R )  -> 
0  <  if (
r  <_  R , 
r ,  R ) )
149, 10, 13syl2anc 661 . . . . . . 7  |-  ( ( ( C  e.  ( *Met `  X
)  /\  R  e.  RR* 
/\  0  <  R
)  /\  ( z  e.  X  /\  r  e.  RR+ ) )  -> 
0  <  if (
r  <_  R , 
r ,  R ) )
15 0xr 9650 . . . . . . . 8  |-  0  e.  RR*
16 xrltle 11365 . . . . . . . 8  |-  ( ( 0  e.  RR*  /\  if ( r  <_  R ,  r ,  R
)  e.  RR* )  ->  ( 0  <  if ( r  <_  R ,  r ,  R
)  ->  0  <_  if ( r  <_  R ,  r ,  R
) ) )
1715, 5, 16sylancr 663 . . . . . . 7  |-  ( ( ( C  e.  ( *Met `  X
)  /\  R  e.  RR* 
/\  0  <  R
)  /\  ( z  e.  X  /\  r  e.  RR+ ) )  -> 
( 0  <  if ( r  <_  R ,  r ,  R
)  ->  0  <_  if ( r  <_  R ,  r ,  R
) ) )
1814, 17mpd 15 . . . . . 6  |-  ( ( ( C  e.  ( *Met `  X
)  /\  R  e.  RR* 
/\  0  <  R
)  /\  ( z  e.  X  /\  r  e.  RR+ ) )  -> 
0  <_  if (
r  <_  R , 
r ,  R ) )
19 xrmin1 11388 . . . . . . 7  |-  ( ( r  e.  RR*  /\  R  e.  RR* )  ->  if ( r  <_  R ,  r ,  R
)  <_  r )
202, 3, 19syl2anc 661 . . . . . 6  |-  ( ( ( C  e.  ( *Met `  X
)  /\  R  e.  RR* 
/\  0  <  R
)  /\  ( z  e.  X  /\  r  e.  RR+ ) )  ->  if ( r  <_  R ,  r ,  R
)  <_  r )
21 xrrege0 11385 . . . . . 6  |-  ( ( ( if ( r  <_  R ,  r ,  R )  e. 
RR*  /\  r  e.  RR )  /\  (
0  <_  if (
r  <_  R , 
r ,  R )  /\  if ( r  <_  R ,  r ,  R )  <_ 
r ) )  ->  if ( r  <_  R ,  r ,  R
)  e.  RR )
225, 7, 18, 20, 21syl22anc 1229 . . . . 5  |-  ( ( ( C  e.  ( *Met `  X
)  /\  R  e.  RR* 
/\  0  <  R
)  /\  ( z  e.  X  /\  r  e.  RR+ ) )  ->  if ( r  <_  R ,  r ,  R
)  e.  RR )
2322, 14elrpd 11264 . . . 4  |-  ( ( ( C  e.  ( *Met `  X
)  /\  R  e.  RR* 
/\  0  <  R
)  /\  ( z  e.  X  /\  r  e.  RR+ ) )  ->  if ( r  <_  R ,  r ,  R
)  e.  RR+ )
24 simprl 755 . . . . . . 7  |-  ( ( ( C  e.  ( *Met `  X
)  /\  R  e.  RR* 
/\  0  <  R
)  /\  ( z  e.  X  /\  r  e.  RR+ ) )  -> 
z  e.  X )
25 xrmin2 11389 . . . . . . . 8  |-  ( ( r  e.  RR*  /\  R  e.  RR* )  ->  if ( r  <_  R ,  r ,  R
)  <_  R )
262, 3, 25syl2anc 661 . . . . . . 7  |-  ( ( ( C  e.  ( *Met `  X
)  /\  R  e.  RR* 
/\  0  <  R
)  /\  ( z  e.  X  /\  r  e.  RR+ ) )  ->  if ( r  <_  R ,  r ,  R
)  <_  R )
2724, 5, 263jca 1176 . . . . . 6  |-  ( ( ( C  e.  ( *Met `  X
)  /\  R  e.  RR* 
/\  0  <  R
)  /\  ( z  e.  X  /\  r  e.  RR+ ) )  -> 
( z  e.  X  /\  if ( r  <_  R ,  r ,  R )  e.  RR*  /\  if ( r  <_  R ,  r ,  R )  <_  R
) )
28 stdbdmet.1 . . . . . . 7  |-  D  =  ( x  e.  X ,  y  e.  X  |->  if ( ( x C y )  <_  R ,  ( x C y ) ,  R ) )
2928stdbdbl 20865 . . . . . 6  |-  ( ( ( C  e.  ( *Met `  X
)  /\  R  e.  RR* 
/\  0  <  R
)  /\  ( z  e.  X  /\  if ( r  <_  R , 
r ,  R )  e.  RR*  /\  if ( r  <_  R , 
r ,  R )  <_  R ) )  ->  ( z (
ball `  D ) if ( r  <_  R ,  r ,  R
) )  =  ( z ( ball `  C
) if ( r  <_  R ,  r ,  R ) ) )
3027, 29syldan 470 . . . . 5  |-  ( ( ( C  e.  ( *Met `  X
)  /\  R  e.  RR* 
/\  0  <  R
)  /\  ( z  e.  X  /\  r  e.  RR+ ) )  -> 
( z ( ball `  D ) if ( r  <_  R , 
r ,  R ) )  =  ( z ( ball `  C
) if ( r  <_  R ,  r ,  R ) ) )
3130eqcomd 2475 . . . 4  |-  ( ( ( C  e.  ( *Met `  X
)  /\  R  e.  RR* 
/\  0  <  R
)  /\  ( z  e.  X  /\  r  e.  RR+ ) )  -> 
( z ( ball `  C ) if ( r  <_  R , 
r ,  R ) )  =  ( z ( ball `  D
) if ( r  <_  R ,  r ,  R ) ) )
32 breq1 4455 . . . . . 6  |-  ( s  =  if ( r  <_  R ,  r ,  R )  -> 
( s  <_  r  <->  if ( r  <_  R ,  r ,  R
)  <_  r )
)
33 oveq2 6302 . . . . . . 7  |-  ( s  =  if ( r  <_  R ,  r ,  R )  -> 
( z ( ball `  C ) s )  =  ( z (
ball `  C ) if ( r  <_  R ,  r ,  R
) ) )
34 oveq2 6302 . . . . . . 7  |-  ( s  =  if ( r  <_  R ,  r ,  R )  -> 
( z ( ball `  D ) s )  =  ( z (
ball `  D ) if ( r  <_  R ,  r ,  R
) ) )
3533, 34eqeq12d 2489 . . . . . 6  |-  ( s  =  if ( r  <_  R ,  r ,  R )  -> 
( ( z (
ball `  C )
s )  =  ( z ( ball `  D
) s )  <->  ( z
( ball `  C ) if ( r  <_  R ,  r ,  R
) )  =  ( z ( ball `  D
) if ( r  <_  R ,  r ,  R ) ) ) )
3632, 35anbi12d 710 . . . . 5  |-  ( s  =  if ( r  <_  R ,  r ,  R )  -> 
( ( s  <_ 
r  /\  ( z
( ball `  C )
s )  =  ( z ( ball `  D
) s ) )  <-> 
( if ( r  <_  R ,  r ,  R )  <_ 
r  /\  ( z
( ball `  C ) if ( r  <_  R ,  r ,  R
) )  =  ( z ( ball `  D
) if ( r  <_  R ,  r ,  R ) ) ) ) )
3736rspcev 3219 . . . 4  |-  ( ( if ( r  <_  R ,  r ,  R )  e.  RR+  /\  ( if ( r  <_  R ,  r ,  R )  <_ 
r  /\  ( z
( ball `  C ) if ( r  <_  R ,  r ,  R
) )  =  ( z ( ball `  D
) if ( r  <_  R ,  r ,  R ) ) ) )  ->  E. s  e.  RR+  ( s  <_ 
r  /\  ( z
( ball `  C )
s )  =  ( z ( ball `  D
) s ) ) )
3823, 20, 31, 37syl12anc 1226 . . 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 ) ) )
3938ralrimivva 2888 . 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 ) ) )
40 simp1 996 . . 3  |-  ( ( C  e.  ( *Met `  X )  /\  R  e.  RR*  /\  0  <  R )  ->  C  e.  ( *Met `  X
) )
4128stdbdxmet 20863 . . 3  |-  ( ( C  e.  ( *Met `  X )  /\  R  e.  RR*  /\  0  <  R )  ->  D  e.  ( *Met `  X
) )
42 stdbdmopn.2 . . . 4  |-  J  =  ( MetOpen `  C )
43 eqid 2467 . . . 4  |-  ( MetOpen `  D )  =  (
MetOpen `  D )
4442, 43metequiv2 20858 . . 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 ) ) )
4540, 41, 44syl2anc 661 . 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 ) ) )
4639, 45mpd 15 1  |-  ( ( C  e.  ( *Met `  X )  /\  R  e.  RR*  /\  0  <  R )  ->  J  =  (
MetOpen `  D ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767   A.wral 2817   E.wrex 2818   ifcif 3944   class class class wbr 4452   ` cfv 5593  (class class class)co 6294    |-> cmpt2 6296   RRcr 9501   0cc0 9502   RR*cxr 9637    < clt 9638    <_ cle 9639   RR+crp 11230   *Metcxmt 18250   ballcbl 18252   MetOpencmopn 18255
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4573  ax-nul 4581  ax-pow 4630  ax-pr 4691  ax-un 6586  ax-cnex 9558  ax-resscn 9559  ax-1cn 9560  ax-icn 9561  ax-addcl 9562  ax-addrcl 9563  ax-mulcl 9564  ax-mulrcl 9565  ax-mulcom 9566  ax-addass 9567  ax-mulass 9568  ax-distr 9569  ax-i2m1 9570  ax-1ne0 9571  ax-1rid 9572  ax-rnegex 9573  ax-rrecex 9574  ax-cnre 9575  ax-pre-lttri 9576  ax-pre-lttrn 9577  ax-pre-ltadd 9578  ax-pre-mulgt0 9579  ax-pre-sup 9580
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2822  df-rex 2823  df-reu 2824  df-rmo 2825  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4251  df-iun 4332  df-br 4453  df-opab 4511  df-mpt 4512  df-tr 4546  df-eprel 4796  df-id 4800  df-po 4805  df-so 4806  df-fr 4843  df-we 4845  df-ord 4886  df-on 4887  df-lim 4888  df-suc 4889  df-xp 5010  df-rel 5011  df-cnv 5012  df-co 5013  df-dm 5014  df-rn 5015  df-res 5016  df-ima 5017  df-iota 5556  df-fun 5595  df-fn 5596  df-f 5597  df-f1 5598  df-fo 5599  df-f1o 5600  df-fv 5601  df-riota 6255  df-ov 6297  df-oprab 6298  df-mpt2 6299  df-om 6695  df-1st 6794  df-2nd 6795  df-recs 7052  df-rdg 7086  df-er 7321  df-map 7432  df-en 7527  df-dom 7528  df-sdom 7529  df-sup 7911  df-pnf 9640  df-mnf 9641  df-xr 9642  df-ltxr 9643  df-le 9644  df-sub 9817  df-neg 9818  df-div 10217  df-nn 10547  df-2 10604  df-n0 10806  df-z 10875  df-uz 11093  df-q 11193  df-rp 11231  df-xneg 11328  df-xadd 11329  df-xmul 11330  df-icc 11546  df-topgen 14711  df-psmet 18258  df-xmet 18259  df-bl 18261  df-mopn 18262  df-bases 19247
This theorem is referenced by:  mopnex  20867  xlebnum  21310
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