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Theorem metequiv2 18072
Description: If there is a sequence of radii approaching zero for which the balls of both metrics coincide, then the generated topologies are equivalent. (Contributed by Mario Carneiro, 26-Aug-2015.)
Hypotheses
Ref Expression
metequiv.3  |-  J  =  ( MetOpen `  C )
metequiv.4  |-  K  =  ( MetOpen `  D )
Assertion
Ref Expression
metequiv2  |-  ( ( C  e.  ( * Met `  X )  /\  D  e.  ( * Met `  X
) )  ->  ( A. x  e.  X  A. r  e.  RR+  E. s  e.  RR+  ( s  <_ 
r  /\  ( x
( ball `  C )
s )  =  ( x ( ball `  D
) s ) )  ->  J  =  K ) )
Distinct variable groups:    s, r, x, C    J, r, s, x    K, r, s, x    D, r, s, x    X, r, s, x

Proof of Theorem metequiv2
StepHypRef Expression
1 simprrr 741 . . . . . . . . . . 11  |-  ( ( ( ( C  e.  ( * Met `  X
)  /\  D  e.  ( * Met `  X
) )  /\  x  e.  X )  /\  (
( r  e.  RR+  /\  s  e.  RR+ )  /\  ( s  <_  r  /\  ( x ( ball `  C ) s )  =  ( x (
ball `  D )
s ) ) ) )  ->  ( x
( ball `  C )
s )  =  ( x ( ball `  D
) s ) )
2 simplll 734 . . . . . . . . . . . 12  |-  ( ( ( ( C  e.  ( * Met `  X
)  /\  D  e.  ( * Met `  X
) )  /\  x  e.  X )  /\  (
( r  e.  RR+  /\  s  e.  RR+ )  /\  ( s  <_  r  /\  ( x ( ball `  C ) s )  =  ( x (
ball `  D )
s ) ) ) )  ->  C  e.  ( * Met `  X
) )
3 simplr 731 . . . . . . . . . . . 12  |-  ( ( ( ( C  e.  ( * Met `  X
)  /\  D  e.  ( * Met `  X
) )  /\  x  e.  X )  /\  (
( r  e.  RR+  /\  s  e.  RR+ )  /\  ( s  <_  r  /\  ( x ( ball `  C ) s )  =  ( x (
ball `  D )
s ) ) ) )  ->  x  e.  X )
4 simprlr 739 . . . . . . . . . . . . 13  |-  ( ( ( ( C  e.  ( * Met `  X
)  /\  D  e.  ( * Met `  X
) )  /\  x  e.  X )  /\  (
( r  e.  RR+  /\  s  e.  RR+ )  /\  ( s  <_  r  /\  ( x ( ball `  C ) s )  =  ( x (
ball `  D )
s ) ) ) )  ->  s  e.  RR+ )
54rpxrd 10407 . . . . . . . . . . . 12  |-  ( ( ( ( C  e.  ( * Met `  X
)  /\  D  e.  ( * Met `  X
) )  /\  x  e.  X )  /\  (
( r  e.  RR+  /\  s  e.  RR+ )  /\  ( s  <_  r  /\  ( x ( ball `  C ) s )  =  ( x (
ball `  D )
s ) ) ) )  ->  s  e.  RR* )
6 simprll 738 . . . . . . . . . . . . 13  |-  ( ( ( ( C  e.  ( * Met `  X
)  /\  D  e.  ( * Met `  X
) )  /\  x  e.  X )  /\  (
( r  e.  RR+  /\  s  e.  RR+ )  /\  ( s  <_  r  /\  ( x ( ball `  C ) s )  =  ( x (
ball `  D )
s ) ) ) )  ->  r  e.  RR+ )
76rpxrd 10407 . . . . . . . . . . . 12  |-  ( ( ( ( C  e.  ( * Met `  X
)  /\  D  e.  ( * Met `  X
) )  /\  x  e.  X )  /\  (
( r  e.  RR+  /\  s  e.  RR+ )  /\  ( s  <_  r  /\  ( x ( ball `  C ) s )  =  ( x (
ball `  D )
s ) ) ) )  ->  r  e.  RR* )
8 simprrl 740 . . . . . . . . . . . 12  |-  ( ( ( ( C  e.  ( * Met `  X
)  /\  D  e.  ( * Met `  X
) )  /\  x  e.  X )  /\  (
( r  e.  RR+  /\  s  e.  RR+ )  /\  ( s  <_  r  /\  ( x ( ball `  C ) s )  =  ( x (
ball `  D )
s ) ) ) )  ->  s  <_  r )
9 ssbl 17987 . . . . . . . . . . . 12  |-  ( ( ( C  e.  ( * Met `  X
)  /\  x  e.  X )  /\  (
s  e.  RR*  /\  r  e.  RR* )  /\  s  <_  r )  ->  (
x ( ball `  C
) s )  C_  ( x ( ball `  C ) r ) )
102, 3, 5, 7, 8, 9syl221anc 1193 . . . . . . . . . . 11  |-  ( ( ( ( C  e.  ( * Met `  X
)  /\  D  e.  ( * Met `  X
) )  /\  x  e.  X )  /\  (
( r  e.  RR+  /\  s  e.  RR+ )  /\  ( s  <_  r  /\  ( x ( ball `  C ) s )  =  ( x (
ball `  D )
s ) ) ) )  ->  ( x
( ball `  C )
s )  C_  (
x ( ball `  C
) r ) )
111, 10eqsstr3d 3226 . . . . . . . . . 10  |-  ( ( ( ( C  e.  ( * Met `  X
)  /\  D  e.  ( * Met `  X
) )  /\  x  e.  X )  /\  (
( r  e.  RR+  /\  s  e.  RR+ )  /\  ( s  <_  r  /\  ( x ( ball `  C ) s )  =  ( x (
ball `  D )
s ) ) ) )  ->  ( x
( ball `  D )
s )  C_  (
x ( ball `  C
) r ) )
12 simpllr 735 . . . . . . . . . . . 12  |-  ( ( ( ( C  e.  ( * Met `  X
)  /\  D  e.  ( * Met `  X
) )  /\  x  e.  X )  /\  (
( r  e.  RR+  /\  s  e.  RR+ )  /\  ( s  <_  r  /\  ( x ( ball `  C ) s )  =  ( x (
ball `  D )
s ) ) ) )  ->  D  e.  ( * Met `  X
) )
13 ssbl 17987 . . . . . . . . . . . 12  |-  ( ( ( D  e.  ( * Met `  X
)  /\  x  e.  X )  /\  (
s  e.  RR*  /\  r  e.  RR* )  /\  s  <_  r )  ->  (
x ( ball `  D
) s )  C_  ( x ( ball `  D ) r ) )
1412, 3, 5, 7, 8, 13syl221anc 1193 . . . . . . . . . . 11  |-  ( ( ( ( C  e.  ( * Met `  X
)  /\  D  e.  ( * Met `  X
) )  /\  x  e.  X )  /\  (
( r  e.  RR+  /\  s  e.  RR+ )  /\  ( s  <_  r  /\  ( x ( ball `  C ) s )  =  ( x (
ball `  D )
s ) ) ) )  ->  ( x
( ball `  D )
s )  C_  (
x ( ball `  D
) r ) )
151, 14eqsstrd 3225 . . . . . . . . . 10  |-  ( ( ( ( C  e.  ( * Met `  X
)  /\  D  e.  ( * Met `  X
) )  /\  x  e.  X )  /\  (
( r  e.  RR+  /\  s  e.  RR+ )  /\  ( s  <_  r  /\  ( x ( ball `  C ) s )  =  ( x (
ball `  D )
s ) ) ) )  ->  ( x
( ball `  C )
s )  C_  (
x ( ball `  D
) r ) )
1611, 15jca 518 . . . . . . . . 9  |-  ( ( ( ( C  e.  ( * Met `  X
)  /\  D  e.  ( * Met `  X
) )  /\  x  e.  X )  /\  (
( r  e.  RR+  /\  s  e.  RR+ )  /\  ( s  <_  r  /\  ( x ( ball `  C ) s )  =  ( x (
ball `  D )
s ) ) ) )  ->  ( (
x ( ball `  D
) s )  C_  ( x ( ball `  C ) r )  /\  ( x (
ball `  C )
s )  C_  (
x ( ball `  D
) r ) ) )
1716expr 598 . . . . . . . 8  |-  ( ( ( ( C  e.  ( * Met `  X
)  /\  D  e.  ( * Met `  X
) )  /\  x  e.  X )  /\  (
r  e.  RR+  /\  s  e.  RR+ ) )  -> 
( ( s  <_ 
r  /\  ( x
( ball `  C )
s )  =  ( x ( ball `  D
) s ) )  ->  ( ( x ( ball `  D
) s )  C_  ( x ( ball `  C ) r )  /\  ( x (
ball `  C )
s )  C_  (
x ( ball `  D
) r ) ) ) )
1817anassrs 629 . . . . . . 7  |-  ( ( ( ( ( C  e.  ( * Met `  X )  /\  D  e.  ( * Met `  X
) )  /\  x  e.  X )  /\  r  e.  RR+ )  /\  s  e.  RR+ )  ->  (
( s  <_  r  /\  ( x ( ball `  C ) s )  =  ( x (
ball `  D )
s ) )  -> 
( ( x (
ball `  D )
s )  C_  (
x ( ball `  C
) r )  /\  ( x ( ball `  C ) s ) 
C_  ( x (
ball `  D )
r ) ) ) )
1918reximdva 2668 . . . . . 6  |-  ( ( ( ( C  e.  ( * Met `  X
)  /\  D  e.  ( * Met `  X
) )  /\  x  e.  X )  /\  r  e.  RR+ )  ->  ( E. s  e.  RR+  (
s  <_  r  /\  ( x ( ball `  C ) s )  =  ( x (
ball `  D )
s ) )  ->  E. s  e.  RR+  (
( x ( ball `  D ) s ) 
C_  ( x (
ball `  C )
r )  /\  (
x ( ball `  C
) s )  C_  ( x ( ball `  D ) r ) ) ) )
20 r19.40 2704 . . . . . 6  |-  ( E. s  e.  RR+  (
( x ( ball `  D ) s ) 
C_  ( x (
ball `  C )
r )  /\  (
x ( ball `  C
) s )  C_  ( x ( ball `  D ) r ) )  ->  ( E. s  e.  RR+  ( x ( ball `  D
) s )  C_  ( x ( ball `  C ) r )  /\  E. s  e.  RR+  ( x ( ball `  C ) s ) 
C_  ( x (
ball `  D )
r ) ) )
2119, 20syl6 29 . . . . 5  |-  ( ( ( ( C  e.  ( * Met `  X
)  /\  D  e.  ( * Met `  X
) )  /\  x  e.  X )  /\  r  e.  RR+ )  ->  ( E. s  e.  RR+  (
s  <_  r  /\  ( x ( ball `  C ) s )  =  ( x (
ball `  D )
s ) )  -> 
( E. s  e.  RR+  ( x ( ball `  D ) s ) 
C_  ( x (
ball `  C )
r )  /\  E. s  e.  RR+  ( x ( ball `  C
) s )  C_  ( x ( ball `  D ) r ) ) ) )
2221ralimdva 2634 . . . 4  |-  ( ( ( C  e.  ( * Met `  X
)  /\  D  e.  ( * Met `  X
) )  /\  x  e.  X )  ->  ( A. r  e.  RR+  E. s  e.  RR+  ( s  <_ 
r  /\  ( x
( ball `  C )
s )  =  ( x ( ball `  D
) s ) )  ->  A. r  e.  RR+  ( E. s  e.  RR+  ( x ( ball `  D ) s ) 
C_  ( x (
ball `  C )
r )  /\  E. s  e.  RR+  ( x ( ball `  C
) s )  C_  ( x ( ball `  D ) r ) ) ) )
23 r19.26 2688 . . . 4  |-  ( A. r  e.  RR+  ( E. s  e.  RR+  (
x ( ball `  D
) s )  C_  ( x ( ball `  C ) r )  /\  E. s  e.  RR+  ( x ( ball `  C ) s ) 
C_  ( x (
ball `  D )
r ) )  <->  ( A. r  e.  RR+  E. s  e.  RR+  ( x (
ball `  D )
s )  C_  (
x ( ball `  C
) r )  /\  A. r  e.  RR+  E. s  e.  RR+  ( x (
ball `  C )
s )  C_  (
x ( ball `  D
) r ) ) )
2422, 23syl6ib 217 . . 3  |-  ( ( ( C  e.  ( * Met `  X
)  /\  D  e.  ( * Met `  X
) )  /\  x  e.  X )  ->  ( A. r  e.  RR+  E. s  e.  RR+  ( s  <_ 
r  /\  ( x
( ball `  C )
s )  =  ( x ( ball `  D
) s ) )  ->  ( A. r  e.  RR+  E. s  e.  RR+  ( x ( ball `  D ) s ) 
C_  ( x (
ball `  C )
r )  /\  A. r  e.  RR+  E. s  e.  RR+  ( x (
ball `  C )
s )  C_  (
x ( ball `  D
) r ) ) ) )
2524ralimdva 2634 . 2  |-  ( ( C  e.  ( * Met `  X )  /\  D  e.  ( * Met `  X
) )  ->  ( A. x  e.  X  A. r  e.  RR+  E. s  e.  RR+  ( s  <_ 
r  /\  ( x
( ball `  C )
s )  =  ( x ( ball `  D
) s ) )  ->  A. x  e.  X  ( A. r  e.  RR+  E. s  e.  RR+  (
x ( ball `  D
) s )  C_  ( x ( ball `  C ) r )  /\  A. r  e.  RR+  E. s  e.  RR+  ( x ( ball `  C ) s ) 
C_  ( x (
ball `  D )
r ) ) ) )
26 metequiv.3 . . 3  |-  J  =  ( MetOpen `  C )
27 metequiv.4 . . 3  |-  K  =  ( MetOpen `  D )
2826, 27metequiv 18071 . 2  |-  ( ( C  e.  ( * Met `  X )  /\  D  e.  ( * Met `  X
) )  ->  ( J  =  K  <->  A. x  e.  X  ( A. r  e.  RR+  E. s  e.  RR+  ( x (
ball `  D )
s )  C_  (
x ( ball `  C
) r )  /\  A. r  e.  RR+  E. s  e.  RR+  ( x (
ball `  C )
s )  C_  (
x ( ball `  D
) r ) ) ) )
2925, 28sylibrd 225 1  |-  ( ( C  e.  ( * Met `  X )  /\  D  e.  ( * Met `  X
) )  ->  ( A. x  e.  X  A. r  e.  RR+  E. s  e.  RR+  ( s  <_ 
r  /\  ( x
( ball `  C )
s )  =  ( x ( ball `  D
) s ) )  ->  J  =  K ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1632    e. wcel 1696   A.wral 2556   E.wrex 2557    C_ wss 3165   class class class wbr 4039   ` cfv 5271  (class class class)co 5874   RR*cxr 8882    <_ cle 8884   RR+crp 10370   * Metcxmt 16385   ballcbl 16387   MetOpencmopn 16388
This theorem is referenced by:  stdbdmopn  18080
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830  ax-pre-sup 8831
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-er 6676  df-map 6790  df-en 6880  df-dom 6881  df-sdom 6882  df-sup 7210  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-div 9440  df-nn 9763  df-2 9820  df-n0 9982  df-z 10041  df-uz 10247  df-q 10333  df-rp 10371  df-xneg 10468  df-xadd 10469  df-xmul 10470  df-topgen 13360  df-xmet 16389  df-bl 16391  df-mopn 16392  df-bases 16654
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