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Theorem metequiv2 18540
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 742 . . . . . . . . . . 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 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 ) ) ) )  ->  C  e.  ( * Met `  X
) )
3 simplr 732 . . . . . . . . . . . 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 740 . . . . . . . . . . . . 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 10649 . . . . . . . . . . . 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 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 ) ) ) )  ->  r  e.  RR+ )
76rpxrd 10649 . . . . . . . . . . . 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 741 . . . . . . . . . . . 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 18453 . . . . . . . . . . . 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 1195 . . . . . . . . . . 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 3383 . . . . . . . . . 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 736 . . . . . . . . . . . 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 18453 . . . . . . . . . . . 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 1195 . . . . . . . . . . 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 3382 . . . . . . . . . 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 519 . . . . . . . . 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 599 . . . . . . . 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 630 . . . . . . 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 2818 . . . . . 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 2859 . . . . . 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 31 . . . . 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 2784 . . . 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 2838 . . . 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 218 . . 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 2784 . 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 18539 . 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 226 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 359    = wceq 1652    e. wcel 1725   A.wral 2705   E.wrex 2706    C_ wss 3320   class class class wbr 4212   ` cfv 5454  (class class class)co 6081   RR*cxr 9119    <_ cle 9121   RR+crp 10612   * Metcxmt 16686   ballcbl 16688   MetOpencmopn 16691
This theorem is referenced by:  stdbdmopn  18548
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-cnex 9046  ax-resscn 9047  ax-1cn 9048  ax-icn 9049  ax-addcl 9050  ax-addrcl 9051  ax-mulcl 9052  ax-mulrcl 9053  ax-mulcom 9054  ax-addass 9055  ax-mulass 9056  ax-distr 9057  ax-i2m1 9058  ax-1ne0 9059  ax-1rid 9060  ax-rnegex 9061  ax-rrecex 9062  ax-cnre 9063  ax-pre-lttri 9064  ax-pre-lttrn 9065  ax-pre-ltadd 9066  ax-pre-mulgt0 9067  ax-pre-sup 9068
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-om 4846  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-1st 6349  df-2nd 6350  df-riota 6549  df-recs 6633  df-rdg 6668  df-er 6905  df-map 7020  df-en 7110  df-dom 7111  df-sdom 7112  df-sup 7446  df-pnf 9122  df-mnf 9123  df-xr 9124  df-ltxr 9125  df-le 9126  df-sub 9293  df-neg 9294  df-div 9678  df-nn 10001  df-2 10058  df-n0 10222  df-z 10283  df-uz 10489  df-q 10575  df-rp 10613  df-xneg 10710  df-xadd 10711  df-xmul 10712  df-topgen 13667  df-psmet 16694  df-xmet 16695  df-bl 16697  df-mopn 16698  df-bases 16965
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