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Theorem equivcfil 19252
Description: If the metric  D is "strongly finer" than  C (meaning that there is a positive real constant 
R such that  C ( x ,  y )  <_  R  x.  D (
x ,  y )), all the  D-Cauchy filters are also  C-Cauchy. (Using this theorem twice in each direction states that if two metrics are strongly equivalent, then they have the same Cauchy sequences.) (Contributed by Mario Carneiro, 14-Sep-2015.)
Hypotheses
Ref Expression
equivcau.1  |-  ( ph  ->  C  e.  ( Met `  X ) )
equivcau.2  |-  ( ph  ->  D  e.  ( Met `  X ) )
equivcau.3  |-  ( ph  ->  R  e.  RR+ )
equivcau.4  |-  ( (
ph  /\  ( x  e.  X  /\  y  e.  X ) )  -> 
( x C y )  <_  ( R  x.  ( x D y ) ) )
Assertion
Ref Expression
equivcfil  |-  ( ph  ->  (CauFil `  D )  C_  (CauFil `  C )
)
Distinct variable groups:    x, y, C    x, D, y    ph, x, y    x, R, y    x, X, y

Proof of Theorem equivcfil
Dummy variables  f 
r  s are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simpr 448 . . . . . . . 8  |-  ( ( ( ph  /\  f  e.  ( Fil `  X
) )  /\  r  e.  RR+ )  ->  r  e.  RR+ )
2 equivcau.3 . . . . . . . . 9  |-  ( ph  ->  R  e.  RR+ )
32ad2antrr 707 . . . . . . . 8  |-  ( ( ( ph  /\  f  e.  ( Fil `  X
) )  /\  r  e.  RR+ )  ->  R  e.  RR+ )
41, 3rpdivcld 10665 . . . . . . 7  |-  ( ( ( ph  /\  f  e.  ( Fil `  X
) )  /\  r  e.  RR+ )  ->  (
r  /  R )  e.  RR+ )
5 oveq2 6089 . . . . . . . . . 10  |-  ( s  =  ( r  /  R )  ->  (
x ( ball `  D
) s )  =  ( x ( ball `  D ) ( r  /  R ) ) )
65eleq1d 2502 . . . . . . . . 9  |-  ( s  =  ( r  /  R )  ->  (
( x ( ball `  D ) s )  e.  f  <->  ( x
( ball `  D )
( r  /  R
) )  e.  f ) )
76rexbidv 2726 . . . . . . . 8  |-  ( s  =  ( r  /  R )  ->  ( E. x  e.  X  ( x ( ball `  D ) s )  e.  f  <->  E. x  e.  X  ( x
( ball `  D )
( r  /  R
) )  e.  f ) )
87rspcv 3048 . . . . . . 7  |-  ( ( r  /  R )  e.  RR+  ->  ( A. s  e.  RR+  E. x  e.  X  ( x
( ball `  D )
s )  e.  f  ->  E. x  e.  X  ( x ( ball `  D ) ( r  /  R ) )  e.  f ) )
94, 8syl 16 . . . . . 6  |-  ( ( ( ph  /\  f  e.  ( Fil `  X
) )  /\  r  e.  RR+ )  ->  ( A. s  e.  RR+  E. x  e.  X  ( x
( ball `  D )
s )  e.  f  ->  E. x  e.  X  ( x ( ball `  D ) ( r  /  R ) )  e.  f ) )
10 simpllr 736 . . . . . . . 8  |-  ( ( ( ( ph  /\  f  e.  ( Fil `  X ) )  /\  r  e.  RR+ )  /\  x  e.  X )  ->  f  e.  ( Fil `  X ) )
11 eqid 2436 . . . . . . . . . . . 12  |-  ( MetOpen `  C )  =  (
MetOpen `  C )
12 eqid 2436 . . . . . . . . . . . 12  |-  ( MetOpen `  D )  =  (
MetOpen `  D )
13 equivcau.1 . . . . . . . . . . . 12  |-  ( ph  ->  C  e.  ( Met `  X ) )
14 equivcau.2 . . . . . . . . . . . 12  |-  ( ph  ->  D  e.  ( Met `  X ) )
15 equivcau.4 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( x  e.  X  /\  y  e.  X ) )  -> 
( x C y )  <_  ( R  x.  ( x D y ) ) )
1611, 12, 13, 14, 2, 15metss2lem 18541 . . . . . . . . . . 11  |-  ( (
ph  /\  ( x  e.  X  /\  r  e.  RR+ ) )  -> 
( x ( ball `  D ) ( r  /  R ) ) 
C_  ( x (
ball `  C )
r ) )
1716ancom2s 778 . . . . . . . . . 10  |-  ( (
ph  /\  ( r  e.  RR+  /\  x  e.  X ) )  -> 
( x ( ball `  D ) ( r  /  R ) ) 
C_  ( x (
ball `  C )
r ) )
1817adantlr 696 . . . . . . . . 9  |-  ( ( ( ph  /\  f  e.  ( Fil `  X
) )  /\  (
r  e.  RR+  /\  x  e.  X ) )  -> 
( x ( ball `  D ) ( r  /  R ) ) 
C_  ( x (
ball `  C )
r ) )
1918anassrs 630 . . . . . . . 8  |-  ( ( ( ( ph  /\  f  e.  ( Fil `  X ) )  /\  r  e.  RR+ )  /\  x  e.  X )  ->  ( x ( ball `  D ) ( r  /  R ) ) 
C_  ( x (
ball `  C )
r ) )
2013ad3antrrr 711 . . . . . . . . . 10  |-  ( ( ( ( ph  /\  f  e.  ( Fil `  X ) )  /\  r  e.  RR+ )  /\  x  e.  X )  ->  C  e.  ( Met `  X ) )
21 metxmet 18364 . . . . . . . . . 10  |-  ( C  e.  ( Met `  X
)  ->  C  e.  ( * Met `  X
) )
2220, 21syl 16 . . . . . . . . 9  |-  ( ( ( ( ph  /\  f  e.  ( Fil `  X ) )  /\  r  e.  RR+ )  /\  x  e.  X )  ->  C  e.  ( * Met `  X ) )
23 simpr 448 . . . . . . . . 9  |-  ( ( ( ( ph  /\  f  e.  ( Fil `  X ) )  /\  r  e.  RR+ )  /\  x  e.  X )  ->  x  e.  X )
24 rpxr 10619 . . . . . . . . . 10  |-  ( r  e.  RR+  ->  r  e. 
RR* )
2524ad2antlr 708 . . . . . . . . 9  |-  ( ( ( ( ph  /\  f  e.  ( Fil `  X ) )  /\  r  e.  RR+ )  /\  x  e.  X )  ->  r  e.  RR* )
26 blssm 18448 . . . . . . . . 9  |-  ( ( C  e.  ( * Met `  X )  /\  x  e.  X  /\  r  e.  RR* )  ->  ( x ( ball `  C ) r ) 
C_  X )
2722, 23, 25, 26syl3anc 1184 . . . . . . . 8  |-  ( ( ( ( ph  /\  f  e.  ( Fil `  X ) )  /\  r  e.  RR+ )  /\  x  e.  X )  ->  ( x ( ball `  C ) r ) 
C_  X )
28 filss 17885 . . . . . . . . . 10  |-  ( ( f  e.  ( Fil `  X )  /\  (
( x ( ball `  D ) ( r  /  R ) )  e.  f  /\  (
x ( ball `  C
) r )  C_  X  /\  ( x (
ball `  D )
( r  /  R
) )  C_  (
x ( ball `  C
) r ) ) )  ->  ( x
( ball `  C )
r )  e.  f )
29283exp2 1171 . . . . . . . . 9  |-  ( f  e.  ( Fil `  X
)  ->  ( (
x ( ball `  D
) ( r  /  R ) )  e.  f  ->  ( (
x ( ball `  C
) r )  C_  X  ->  ( ( x ( ball `  D
) ( r  /  R ) )  C_  ( x ( ball `  C ) r )  ->  ( x (
ball `  C )
r )  e.  f ) ) ) )
3029com24 83 . . . . . . . 8  |-  ( f  e.  ( Fil `  X
)  ->  ( (
x ( ball `  D
) ( r  /  R ) )  C_  ( x ( ball `  C ) r )  ->  ( ( x ( ball `  C
) r )  C_  X  ->  ( ( x ( ball `  D
) ( r  /  R ) )  e.  f  ->  ( x
( ball `  C )
r )  e.  f ) ) ) )
3110, 19, 27, 30syl3c 59 . . . . . . 7  |-  ( ( ( ( ph  /\  f  e.  ( Fil `  X ) )  /\  r  e.  RR+ )  /\  x  e.  X )  ->  ( ( x (
ball `  D )
( r  /  R
) )  e.  f  ->  ( x (
ball `  C )
r )  e.  f ) )
3231reximdva 2818 . . . . . 6  |-  ( ( ( ph  /\  f  e.  ( Fil `  X
) )  /\  r  e.  RR+ )  ->  ( E. x  e.  X  ( x ( ball `  D ) ( r  /  R ) )  e.  f  ->  E. x  e.  X  ( x
( ball `  C )
r )  e.  f ) )
339, 32syld 42 . . . . 5  |-  ( ( ( ph  /\  f  e.  ( Fil `  X
) )  /\  r  e.  RR+ )  ->  ( A. s  e.  RR+  E. x  e.  X  ( x
( ball `  D )
s )  e.  f  ->  E. x  e.  X  ( x ( ball `  C ) r )  e.  f ) )
3433ralrimdva 2796 . . . 4  |-  ( (
ph  /\  f  e.  ( Fil `  X ) )  ->  ( A. s  e.  RR+  E. x  e.  X  ( x
( ball `  D )
s )  e.  f  ->  A. r  e.  RR+  E. x  e.  X  ( x ( ball `  C
) r )  e.  f ) )
3534imdistanda 675 . . 3  |-  ( ph  ->  ( ( f  e.  ( Fil `  X
)  /\  A. s  e.  RR+  E. x  e.  X  ( x (
ball `  D )
s )  e.  f )  ->  ( f  e.  ( Fil `  X
)  /\  A. r  e.  RR+  E. x  e.  X  ( x (
ball `  C )
r )  e.  f ) ) )
36 metxmet 18364 . . . 4  |-  ( D  e.  ( Met `  X
)  ->  D  e.  ( * Met `  X
) )
37 iscfil3 19226 . . . 4  |-  ( D  e.  ( * Met `  X )  ->  (
f  e.  (CauFil `  D )  <->  ( f  e.  ( Fil `  X
)  /\  A. s  e.  RR+  E. x  e.  X  ( x (
ball `  D )
s )  e.  f ) ) )
3814, 36, 373syl 19 . . 3  |-  ( ph  ->  ( f  e.  (CauFil `  D )  <->  ( f  e.  ( Fil `  X
)  /\  A. s  e.  RR+  E. x  e.  X  ( x (
ball `  D )
s )  e.  f ) ) )
39 iscfil3 19226 . . . 4  |-  ( C  e.  ( * Met `  X )  ->  (
f  e.  (CauFil `  C )  <->  ( f  e.  ( Fil `  X
)  /\  A. r  e.  RR+  E. x  e.  X  ( x (
ball `  C )
r )  e.  f ) ) )
4013, 21, 393syl 19 . . 3  |-  ( ph  ->  ( f  e.  (CauFil `  C )  <->  ( f  e.  ( Fil `  X
)  /\  A. r  e.  RR+  E. x  e.  X  ( x (
ball `  C )
r )  e.  f ) ) )
4135, 38, 403imtr4d 260 . 2  |-  ( ph  ->  ( f  e.  (CauFil `  D )  ->  f  e.  (CauFil `  C )
) )
4241ssrdv 3354 1  |-  ( ph  ->  (CauFil `  D )  C_  (CauFil `  C )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ 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    x. cmul 8995   RR*cxr 9119    <_ cle 9121    / cdiv 9677   RR+crp 10612   * Metcxmt 16686   Metcme 16687   ballcbl 16688   MetOpencmopn 16691   Filcfil 17877  CauFilccfil 19205
This theorem is referenced by:  equivcmet  19268
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
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-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-po 4503  df-so 4504  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-er 6905  df-map 7020  df-en 7110  df-dom 7111  df-sdom 7112  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-2 10058  df-rp 10613  df-xneg 10710  df-xadd 10711  df-xmul 10712  df-ico 10922  df-psmet 16694  df-xmet 16695  df-met 16696  df-bl 16697  df-fbas 16699  df-fil 17878  df-cfil 19208
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