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Theorem metequiv2 18056
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 10391 . . . . . . . . . . . 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 10391 . . . . . . . . . . . 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 17971 . . . . . . . . . . . 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 3213 . . . . . . . . . 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 17971 . . . . . . . . . . . 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 3212 . . . . . . . . . 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 2655 . . . . . 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 2691 . . . . . 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 2621 . . . 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 2675 . . . 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 2621 . 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 18055 . 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 1623    e. wcel 1684   A.wral 2543   E.wrex 2544    C_ wss 3152   class class class wbr 4023   ` cfv 5255  (class class class)co 5858   RR*cxr 8866    <_ cle 8868   RR+crp 10354   * Metcxmt 16369   ballcbl 16371   MetOpencmopn 16372
This theorem is referenced by:  stdbdmopn  18064
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814  ax-pre-sup 8815
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 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-er 6660  df-map 6774  df-en 6864  df-dom 6865  df-sdom 6866  df-sup 7194  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-nn 9747  df-2 9804  df-n0 9966  df-z 10025  df-uz 10231  df-q 10317  df-rp 10355  df-xneg 10452  df-xadd 10453  df-xmul 10454  df-topgen 13344  df-xmet 16373  df-bl 16375  df-mopn 16376  df-bases 16638
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