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Theorem equivcfil 18725
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 447 . . . . . . . 8  |-  ( ( ( ph  /\  f  e.  ( Fil `  X
) )  /\  r  e.  RR+ )  ->  r  e.  RR+ )
2 equivcau.3 . . . . . . . . 9  |-  ( ph  ->  R  e.  RR+ )
32ad2antrr 706 . . . . . . . 8  |-  ( ( ( ph  /\  f  e.  ( Fil `  X
) )  /\  r  e.  RR+ )  ->  R  e.  RR+ )
41, 3rpdivcld 10407 . . . . . . 7  |-  ( ( ( ph  /\  f  e.  ( Fil `  X
) )  /\  r  e.  RR+ )  ->  (
r  /  R )  e.  RR+ )
5 oveq2 5866 . . . . . . . . . 10  |-  ( s  =  ( r  /  R )  ->  (
x ( ball `  D
) s )  =  ( x ( ball `  D ) ( r  /  R ) ) )
65eleq1d 2349 . . . . . . . . 9  |-  ( s  =  ( r  /  R )  ->  (
( x ( ball `  D ) s )  e.  f  <->  ( x
( ball `  D )
( r  /  R
) )  e.  f ) )
76rexbidv 2564 . . . . . . . 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 2880 . . . . . . 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 15 . . . . . 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 735 . . . . . . . 8  |-  ( ( ( ( ph  /\  f  e.  ( Fil `  X ) )  /\  r  e.  RR+ )  /\  x  e.  X )  ->  f  e.  ( Fil `  X ) )
11 eqid 2283 . . . . . . . . . . . 12  |-  ( MetOpen `  C )  =  (
MetOpen `  C )
12 eqid 2283 . . . . . . . . . . . 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 18057 . . . . . . . . . . 11  |-  ( (
ph  /\  ( x  e.  X  /\  r  e.  RR+ ) )  -> 
( x ( ball `  D ) ( r  /  R ) ) 
C_  ( x (
ball `  C )
r ) )
1716ancom2s 777 . . . . . . . . . 10  |-  ( (
ph  /\  ( r  e.  RR+  /\  x  e.  X ) )  -> 
( x ( ball `  D ) ( r  /  R ) ) 
C_  ( x (
ball `  C )
r ) )
1817adantlr 695 . . . . . . . . 9  |-  ( ( ( ph  /\  f  e.  ( Fil `  X
) )  /\  (
r  e.  RR+  /\  x  e.  X ) )  -> 
( x ( ball `  D ) ( r  /  R ) ) 
C_  ( x (
ball `  C )
r ) )
1918anassrs 629 . . . . . . . 8  |-  ( ( ( ( ph  /\  f  e.  ( Fil `  X ) )  /\  r  e.  RR+ )  /\  x  e.  X )  ->  ( x ( ball `  D ) ( r  /  R ) ) 
C_  ( x (
ball `  C )
r ) )
2013ad3antrrr 710 . . . . . . . . . 10  |-  ( ( ( ( ph  /\  f  e.  ( Fil `  X ) )  /\  r  e.  RR+ )  /\  x  e.  X )  ->  C  e.  ( Met `  X ) )
21 metxmet 17899 . . . . . . . . . 10  |-  ( C  e.  ( Met `  X
)  ->  C  e.  ( * Met `  X
) )
2220, 21syl 15 . . . . . . . . 9  |-  ( ( ( ( ph  /\  f  e.  ( Fil `  X ) )  /\  r  e.  RR+ )  /\  x  e.  X )  ->  C  e.  ( * Met `  X ) )
23 simpr 447 . . . . . . . . 9  |-  ( ( ( ( ph  /\  f  e.  ( Fil `  X ) )  /\  r  e.  RR+ )  /\  x  e.  X )  ->  x  e.  X )
24 rpxr 10361 . . . . . . . . . 10  |-  ( r  e.  RR+  ->  r  e. 
RR* )
2524ad2antlr 707 . . . . . . . . 9  |-  ( ( ( ( ph  /\  f  e.  ( Fil `  X ) )  /\  r  e.  RR+ )  /\  x  e.  X )  ->  r  e.  RR* )
26 blssm 17968 . . . . . . . . 9  |-  ( ( C  e.  ( * Met `  X )  /\  x  e.  X  /\  r  e.  RR* )  ->  ( x ( ball `  C ) r ) 
C_  X )
2722, 23, 25, 26syl3anc 1182 . . . . . . . 8  |-  ( ( ( ( ph  /\  f  e.  ( Fil `  X ) )  /\  r  e.  RR+ )  /\  x  e.  X )  ->  ( x ( ball `  C ) r ) 
C_  X )
28 filss 17548 . . . . . . . . . 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 1169 . . . . . . . . 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 81 . . . . . . . 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 57 . . . . . . 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 2655 . . . . . 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 40 . . . . 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 2633 . . . 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 674 . . 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 17899 . . . 4  |-  ( D  e.  ( Met `  X
)  ->  D  e.  ( * Met `  X
) )
37 iscfil3 18699 . . . 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 18 . . 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 18699 . . . 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 18 . . 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 259 . 2  |-  ( ph  ->  ( f  e.  (CauFil `  D )  ->  f  e.  (CauFil `  C )
) )
4241ssrdv 3185 1  |-  ( ph  ->  (CauFil `  D )  C_  (CauFil `  C )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ 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    x. cmul 8742   RR*cxr 8866    <_ cle 8868    / cdiv 9423   RR+crp 10354   * Metcxmt 16369   Metcme 16370   ballcbl 16371   MetOpencmopn 16372   Filcfil 17540  CauFilccfil 18678
This theorem is referenced by:  equivcmet  18741
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
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-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-po 4314  df-so 4315  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-er 6660  df-map 6774  df-en 6864  df-dom 6865  df-sdom 6866  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-2 9804  df-rp 10355  df-xneg 10452  df-xadd 10453  df-xmul 10454  df-ico 10662  df-xmet 16373  df-met 16374  df-bl 16375  df-fbas 17520  df-fil 17541  df-cfil 18681
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