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Theorem equivcfil 18741
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 10423 . . . . . . 7  |-  ( ( ( ph  /\  f  e.  ( Fil `  X
) )  /\  r  e.  RR+ )  ->  (
r  /  R )  e.  RR+ )
5 oveq2 5882 . . . . . . . . . 10  |-  ( s  =  ( r  /  R )  ->  (
x ( ball `  D
) s )  =  ( x ( ball `  D ) ( r  /  R ) ) )
65eleq1d 2362 . . . . . . . . 9  |-  ( s  =  ( r  /  R )  ->  (
( x ( ball `  D ) s )  e.  f  <->  ( x
( ball `  D )
( r  /  R
) )  e.  f ) )
76rexbidv 2577 . . . . . . . 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 2893 . . . . . . 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 2296 . . . . . . . . . . . 12  |-  ( MetOpen `  C )  =  (
MetOpen `  C )
12 eqid 2296 . . . . . . . . . . . 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 18073 . . . . . . . . . . 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 17915 . . . . . . . . . 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 10377 . . . . . . . . . 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 17984 . . . . . . . . 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 17564 . . . . . . . . . 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 2668 . . . . . 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 2646 . . . 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 17915 . . . 4  |-  ( D  e.  ( Met `  X
)  ->  D  e.  ( * Met `  X
) )
37 iscfil3 18715 . . . 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 18715 . . . 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 3198 1  |-  ( ph  ->  (CauFil `  D )  C_  (CauFil `  C )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ 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    x. cmul 8758   RR*cxr 8882    <_ cle 8884    / cdiv 9439   RR+crp 10370   * Metcxmt 16385   Metcme 16386   ballcbl 16387   MetOpencmopn 16388   Filcfil 17556  CauFilccfil 18694
This theorem is referenced by:  equivcmet  18757
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
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-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-po 4330  df-so 4331  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-er 6676  df-map 6790  df-en 6880  df-dom 6881  df-sdom 6882  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-2 9820  df-rp 10371  df-xneg 10468  df-xadd 10469  df-xmul 10470  df-ico 10678  df-xmet 16389  df-met 16390  df-bl 16391  df-fbas 17536  df-fil 17557  df-cfil 18697
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