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Theorem stdbdmopn 20098
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 11003 . . . . . . . 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 992 . . . . . . 7  |-  ( ( ( C  e.  ( *Met `  X
)  /\  R  e.  RR* 
/\  0  <  R
)  /\  ( z  e.  X  /\  r  e.  RR+ ) )  ->  R  e.  RR* )
4 ifcl 3836 . . . . . . 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 11002 . . . . . . 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 11007 . . . . . . . . 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 993 . . . . . . . 8  |-  ( ( ( C  e.  ( *Met `  X
)  /\  R  e.  RR* 
/\  0  <  R
)  /\  ( z  e.  X  /\  r  e.  RR+ ) )  -> 
0  <  R )
11 breq2 4301 . . . . . . . . 9  |-  ( r  =  if ( r  <_  R ,  r ,  R )  -> 
( 0  <  r  <->  0  <  if ( r  <_  R ,  r ,  R ) ) )
12 breq2 4301 . . . . . . . . 9  |-  ( R  =  if ( r  <_  R ,  r ,  R )  -> 
( 0  <  R  <->  0  <  if ( r  <_  R ,  r ,  R ) ) )
1311, 12ifboth 3830 . . . . . . . 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 9435 . . . . . . . 8  |-  0  e.  RR*
16 xrltle 11131 . . . . . . . 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 11154 . . . . . . 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 11151 . . . . . 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 1219 . . . . 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 11030 . . . 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 11155 . . . . . . . 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 1168 . . . . . 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 20097 . . . . . 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 2448 . . . 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 4300 . . . . . 6  |-  ( s  =  if ( r  <_  R ,  r ,  R )  -> 
( s  <_  r  <->  if ( r  <_  R ,  r ,  R
)  <_  r )
)
33 oveq2 6104 . . . . . . 7  |-  ( s  =  if ( r  <_  R ,  r ,  R )  -> 
( z ( ball `  C ) s )  =  ( z (
ball `  C ) if ( r  <_  R ,  r ,  R
) ) )
34 oveq2 6104 . . . . . . 7  |-  ( s  =  if ( r  <_  R ,  r ,  R )  -> 
( z ( ball `  D ) s )  =  ( z (
ball `  D ) if ( r  <_  R ,  r ,  R
) ) )
3533, 34eqeq12d 2457 . . . . . 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 3078 . . . 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 1216 . . 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 2813 . 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 988 . . 3  |-  ( ( C  e.  ( *Met `  X )  /\  R  e.  RR*  /\  0  <  R )  ->  C  e.  ( *Met `  X
) )
4128stdbdxmet 20095 . . 3  |-  ( ( C  e.  ( *Met `  X )  /\  R  e.  RR*  /\  0  <  R )  ->  D  e.  ( *Met `  X
) )
42 stdbdmopn.2 . . . 4  |-  J  =  ( MetOpen `  C )
43 eqid 2443 . . . 4  |-  ( MetOpen `  D )  =  (
MetOpen `  D )
4442, 43metequiv2 20090 . . 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 965    = wceq 1369    e. wcel 1756   A.wral 2720   E.wrex 2721   ifcif 3796   class class class wbr 4297   ` cfv 5423  (class class class)co 6096    e. cmpt2 6098   RRcr 9286   0cc0 9287   RR*cxr 9422    < clt 9423    <_ cle 9424   RR+crp 10996   *Metcxmt 17806   ballcbl 17808   MetOpencmopn 17811
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4418  ax-nul 4426  ax-pow 4475  ax-pr 4536  ax-un 6377  ax-cnex 9343  ax-resscn 9344  ax-1cn 9345  ax-icn 9346  ax-addcl 9347  ax-addrcl 9348  ax-mulcl 9349  ax-mulrcl 9350  ax-mulcom 9351  ax-addass 9352  ax-mulass 9353  ax-distr 9354  ax-i2m1 9355  ax-1ne0 9356  ax-1rid 9357  ax-rnegex 9358  ax-rrecex 9359  ax-cnre 9360  ax-pre-lttri 9361  ax-pre-lttrn 9362  ax-pre-ltadd 9363  ax-pre-mulgt0 9364  ax-pre-sup 9365
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-nel 2614  df-ral 2725  df-rex 2726  df-reu 2727  df-rmo 2728  df-rab 2729  df-v 2979  df-sbc 3192  df-csb 3294  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-pss 3349  df-nul 3643  df-if 3797  df-pw 3867  df-sn 3883  df-pr 3885  df-tp 3887  df-op 3889  df-uni 4097  df-iun 4178  df-br 4298  df-opab 4356  df-mpt 4357  df-tr 4391  df-eprel 4637  df-id 4641  df-po 4646  df-so 4647  df-fr 4684  df-we 4686  df-ord 4727  df-on 4728  df-lim 4729  df-suc 4730  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5386  df-fun 5425  df-fn 5426  df-f 5427  df-f1 5428  df-fo 5429  df-f1o 5430  df-fv 5431  df-riota 6057  df-ov 6099  df-oprab 6100  df-mpt2 6101  df-om 6482  df-1st 6582  df-2nd 6583  df-recs 6837  df-rdg 6871  df-er 7106  df-map 7221  df-en 7316  df-dom 7317  df-sdom 7318  df-sup 7696  df-pnf 9425  df-mnf 9426  df-xr 9427  df-ltxr 9428  df-le 9429  df-sub 9602  df-neg 9603  df-div 9999  df-nn 10328  df-2 10385  df-n0 10585  df-z 10652  df-uz 10867  df-q 10959  df-rp 10997  df-xneg 11094  df-xadd 11095  df-xmul 11096  df-icc 11312  df-topgen 14387  df-psmet 17814  df-xmet 17815  df-bl 17817  df-mopn 17818  df-bases 18510
This theorem is referenced by:  mopnex  20099  xlebnum  20542
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