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Theorem stdbdmopn 20887
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 11231 . . . . . . . 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 999 . . . . . . 7  |-  ( ( ( C  e.  ( *Met `  X
)  /\  R  e.  RR* 
/\  0  <  R
)  /\  ( z  e.  X  /\  r  e.  RR+ ) )  ->  R  e.  RR* )
42, 3ifcld 3965 . . . . . 6  |-  ( ( ( C  e.  ( *Met `  X
)  /\  R  e.  RR* 
/\  0  <  R
)  /\  ( z  e.  X  /\  r  e.  RR+ ) )  ->  if ( r  <_  R ,  r ,  R
)  e.  RR* )
5 rpre 11230 . . . . . . 7  |-  ( r  e.  RR+  ->  r  e.  RR )
65ad2antll 728 . . . . . 6  |-  ( ( ( C  e.  ( *Met `  X
)  /\  R  e.  RR* 
/\  0  <  R
)  /\  ( z  e.  X  /\  r  e.  RR+ ) )  -> 
r  e.  RR )
7 rpgt0 11235 . . . . . . . . 9  |-  ( r  e.  RR+  ->  0  < 
r )
87ad2antll 728 . . . . . . . 8  |-  ( ( ( C  e.  ( *Met `  X
)  /\  R  e.  RR* 
/\  0  <  R
)  /\  ( z  e.  X  /\  r  e.  RR+ ) )  -> 
0  <  r )
9 simpl3 1000 . . . . . . . 8  |-  ( ( ( C  e.  ( *Met `  X
)  /\  R  e.  RR* 
/\  0  <  R
)  /\  ( z  e.  X  /\  r  e.  RR+ ) )  -> 
0  <  R )
10 breq2 4437 . . . . . . . . 9  |-  ( r  =  if ( r  <_  R ,  r ,  R )  -> 
( 0  <  r  <->  0  <  if ( r  <_  R ,  r ,  R ) ) )
11 breq2 4437 . . . . . . . . 9  |-  ( R  =  if ( r  <_  R ,  r ,  R )  -> 
( 0  <  R  <->  0  <  if ( r  <_  R ,  r ,  R ) ) )
1210, 11ifboth 3958 . . . . . . . 8  |-  ( ( 0  <  r  /\  0  <  R )  -> 
0  <  if (
r  <_  R , 
r ,  R ) )
138, 9, 12syl2anc 661 . . . . . . 7  |-  ( ( ( C  e.  ( *Met `  X
)  /\  R  e.  RR* 
/\  0  <  R
)  /\  ( z  e.  X  /\  r  e.  RR+ ) )  -> 
0  <  if (
r  <_  R , 
r ,  R ) )
14 0xr 9638 . . . . . . . 8  |-  0  e.  RR*
15 xrltle 11359 . . . . . . . 8  |-  ( ( 0  e.  RR*  /\  if ( r  <_  R ,  r ,  R
)  e.  RR* )  ->  ( 0  <  if ( r  <_  R ,  r ,  R
)  ->  0  <_  if ( r  <_  R ,  r ,  R
) ) )
1614, 4, 15sylancr 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
) ) )
1713, 16mpd 15 . . . . . 6  |-  ( ( ( C  e.  ( *Met `  X
)  /\  R  e.  RR* 
/\  0  <  R
)  /\  ( z  e.  X  /\  r  e.  RR+ ) )  -> 
0  <_  if (
r  <_  R , 
r ,  R ) )
18 xrmin1 11382 . . . . . . 7  |-  ( ( r  e.  RR*  /\  R  e.  RR* )  ->  if ( r  <_  R ,  r ,  R
)  <_  r )
192, 3, 18syl2anc 661 . . . . . 6  |-  ( ( ( C  e.  ( *Met `  X
)  /\  R  e.  RR* 
/\  0  <  R
)  /\  ( z  e.  X  /\  r  e.  RR+ ) )  ->  if ( r  <_  R ,  r ,  R
)  <_  r )
20 xrrege0 11379 . . . . . 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 )
214, 6, 17, 19, 20syl22anc 1228 . . . . 5  |-  ( ( ( C  e.  ( *Met `  X
)  /\  R  e.  RR* 
/\  0  <  R
)  /\  ( z  e.  X  /\  r  e.  RR+ ) )  ->  if ( r  <_  R ,  r ,  R
)  e.  RR )
2221, 13elrpd 11258 . . . 4  |-  ( ( ( C  e.  ( *Met `  X
)  /\  R  e.  RR* 
/\  0  <  R
)  /\  ( z  e.  X  /\  r  e.  RR+ ) )  ->  if ( r  <_  R ,  r ,  R
)  e.  RR+ )
23 simprl 755 . . . . . . 7  |-  ( ( ( C  e.  ( *Met `  X
)  /\  R  e.  RR* 
/\  0  <  R
)  /\  ( z  e.  X  /\  r  e.  RR+ ) )  -> 
z  e.  X )
24 xrmin2 11383 . . . . . . . 8  |-  ( ( r  e.  RR*  /\  R  e.  RR* )  ->  if ( r  <_  R ,  r ,  R
)  <_  R )
252, 3, 24syl2anc 661 . . . . . . 7  |-  ( ( ( C  e.  ( *Met `  X
)  /\  R  e.  RR* 
/\  0  <  R
)  /\  ( z  e.  X  /\  r  e.  RR+ ) )  ->  if ( r  <_  R ,  r ,  R
)  <_  R )
2623, 4, 253jca 1175 . . . . . 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
) )
27 stdbdmet.1 . . . . . . 7  |-  D  =  ( x  e.  X ,  y  e.  X  |->  if ( ( x C y )  <_  R ,  ( x C y ) ,  R ) )
2827stdbdbl 20886 . . . . . 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 ) ) )
2926, 28syldan 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 ) ) )
3029eqcomd 2449 . . . 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 ) ) )
31 breq1 4436 . . . . . 6  |-  ( s  =  if ( r  <_  R ,  r ,  R )  -> 
( s  <_  r  <->  if ( r  <_  R ,  r ,  R
)  <_  r )
)
32 oveq2 6285 . . . . . . 7  |-  ( s  =  if ( r  <_  R ,  r ,  R )  -> 
( z ( ball `  C ) s )  =  ( z (
ball `  C ) if ( r  <_  R ,  r ,  R
) ) )
33 oveq2 6285 . . . . . . 7  |-  ( s  =  if ( r  <_  R ,  r ,  R )  -> 
( z ( ball `  D ) s )  =  ( z (
ball `  D ) if ( r  <_  R ,  r ,  R
) ) )
3432, 33eqeq12d 2463 . . . . . 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 ) ) ) )
3531, 34anbi12d 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 ) ) ) ) )
3635rspcev 3194 . . . 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 ) ) )
3722, 19, 30, 36syl12anc 1225 . . 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 ) ) )
3837ralrimivva 2862 . 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 ) ) )
39 simp1 995 . . 3  |-  ( ( C  e.  ( *Met `  X )  /\  R  e.  RR*  /\  0  <  R )  ->  C  e.  ( *Met `  X
) )
4027stdbdxmet 20884 . . 3  |-  ( ( C  e.  ( *Met `  X )  /\  R  e.  RR*  /\  0  <  R )  ->  D  e.  ( *Met `  X
) )
41 stdbdmopn.2 . . . 4  |-  J  =  ( MetOpen `  C )
42 eqid 2441 . . . 4  |-  ( MetOpen `  D )  =  (
MetOpen `  D )
4341, 42metequiv2 20879 . . 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 ) ) )
4439, 40, 43syl2anc 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 ) ) )
4538, 44mpd 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 972    = wceq 1381    e. wcel 1802   A.wral 2791   E.wrex 2792   ifcif 3922   class class class wbr 4433   ` cfv 5574  (class class class)co 6277    |-> cmpt2 6279   RRcr 9489   0cc0 9490   RR*cxr 9625    < clt 9626    <_ cle 9627   RR+crp 11224   *Metcxmt 18271   ballcbl 18273   MetOpencmopn 18276
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1603  ax-4 1616  ax-5 1689  ax-6 1732  ax-7 1774  ax-8 1804  ax-9 1806  ax-10 1821  ax-11 1826  ax-12 1838  ax-13 1983  ax-ext 2419  ax-sep 4554  ax-nul 4562  ax-pow 4611  ax-pr 4672  ax-un 6573  ax-cnex 9546  ax-resscn 9547  ax-1cn 9548  ax-icn 9549  ax-addcl 9550  ax-addrcl 9551  ax-mulcl 9552  ax-mulrcl 9553  ax-mulcom 9554  ax-addass 9555  ax-mulass 9556  ax-distr 9557  ax-i2m1 9558  ax-1ne0 9559  ax-1rid 9560  ax-rnegex 9561  ax-rrecex 9562  ax-cnre 9563  ax-pre-lttri 9564  ax-pre-lttrn 9565  ax-pre-ltadd 9566  ax-pre-mulgt0 9567  ax-pre-sup 9568
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 973  df-3an 974  df-tru 1384  df-ex 1598  df-nf 1602  df-sb 1725  df-eu 2270  df-mo 2271  df-clab 2427  df-cleq 2433  df-clel 2436  df-nfc 2591  df-ne 2638  df-nel 2639  df-ral 2796  df-rex 2797  df-reu 2798  df-rmo 2799  df-rab 2800  df-v 3095  df-sbc 3312  df-csb 3418  df-dif 3461  df-un 3463  df-in 3465  df-ss 3472  df-pss 3474  df-nul 3768  df-if 3923  df-pw 3995  df-sn 4011  df-pr 4013  df-tp 4015  df-op 4017  df-uni 4231  df-iun 4313  df-br 4434  df-opab 4492  df-mpt 4493  df-tr 4527  df-eprel 4777  df-id 4781  df-po 4786  df-so 4787  df-fr 4824  df-we 4826  df-ord 4867  df-on 4868  df-lim 4869  df-suc 4870  df-xp 4991  df-rel 4992  df-cnv 4993  df-co 4994  df-dm 4995  df-rn 4996  df-res 4997  df-ima 4998  df-iota 5537  df-fun 5576  df-fn 5577  df-f 5578  df-f1 5579  df-fo 5580  df-f1o 5581  df-fv 5582  df-riota 6238  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-om 6682  df-1st 6781  df-2nd 6782  df-recs 7040  df-rdg 7074  df-er 7309  df-map 7420  df-en 7515  df-dom 7516  df-sdom 7517  df-sup 7899  df-pnf 9628  df-mnf 9629  df-xr 9630  df-ltxr 9631  df-le 9632  df-sub 9807  df-neg 9808  df-div 10208  df-nn 10538  df-2 10595  df-n0 10797  df-z 10866  df-uz 11086  df-q 11187  df-rp 11225  df-xneg 11322  df-xadd 11323  df-xmul 11324  df-icc 11540  df-topgen 14713  df-psmet 18279  df-xmet 18280  df-bl 18282  df-mopn 18283  df-bases 19268
This theorem is referenced by:  mopnex  20888  xlebnum  21331
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