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Theorem equivcfil 22028
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 459 . . . . . . . 8  |-  ( ( ( ph  /\  f  e.  ( Fil `  X
) )  /\  r  e.  RR+ )  ->  r  e.  RR+ )
2 equivcau.3 . . . . . . . . 9  |-  ( ph  ->  R  e.  RR+ )
32ad2antrr 724 . . . . . . . 8  |-  ( ( ( ph  /\  f  e.  ( Fil `  X
) )  /\  r  e.  RR+ )  ->  R  e.  RR+ )
41, 3rpdivcld 11320 . . . . . . 7  |-  ( ( ( ph  /\  f  e.  ( Fil `  X
) )  /\  r  e.  RR+ )  ->  (
r  /  R )  e.  RR+ )
5 oveq2 6285 . . . . . . . . . 10  |-  ( s  =  ( r  /  R )  ->  (
x ( ball `  D
) s )  =  ( x ( ball `  D ) ( r  /  R ) ) )
65eleq1d 2471 . . . . . . . . 9  |-  ( s  =  ( r  /  R )  ->  (
( x ( ball `  D ) s )  e.  f  <->  ( x
( ball `  D )
( r  /  R
) )  e.  f ) )
76rexbidv 2917 . . . . . . . 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 3155 . . . . . . 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 17 . . . . . 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 761 . . . . . . . 8  |-  ( ( ( ( ph  /\  f  e.  ( Fil `  X ) )  /\  r  e.  RR+ )  /\  x  e.  X )  ->  f  e.  ( Fil `  X ) )
11 eqid 2402 . . . . . . . . . . . 12  |-  ( MetOpen `  C )  =  (
MetOpen `  C )
12 eqid 2402 . . . . . . . . . . . 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 21304 . . . . . . . . . . 11  |-  ( (
ph  /\  ( x  e.  X  /\  r  e.  RR+ ) )  -> 
( x ( ball `  D ) ( r  /  R ) ) 
C_  ( x (
ball `  C )
r ) )
1716ancom2s 803 . . . . . . . . . 10  |-  ( (
ph  /\  ( r  e.  RR+  /\  x  e.  X ) )  -> 
( x ( ball `  D ) ( r  /  R ) ) 
C_  ( x (
ball `  C )
r ) )
1817adantlr 713 . . . . . . . . 9  |-  ( ( ( ph  /\  f  e.  ( Fil `  X
) )  /\  (
r  e.  RR+  /\  x  e.  X ) )  -> 
( x ( ball `  D ) ( r  /  R ) ) 
C_  ( x (
ball `  C )
r ) )
1918anassrs 646 . . . . . . . 8  |-  ( ( ( ( ph  /\  f  e.  ( Fil `  X ) )  /\  r  e.  RR+ )  /\  x  e.  X )  ->  ( x ( ball `  D ) ( r  /  R ) ) 
C_  ( x (
ball `  C )
r ) )
2013ad3antrrr 728 . . . . . . . . . 10  |-  ( ( ( ( ph  /\  f  e.  ( Fil `  X ) )  /\  r  e.  RR+ )  /\  x  e.  X )  ->  C  e.  ( Met `  X ) )
21 metxmet 21127 . . . . . . . . . 10  |-  ( C  e.  ( Met `  X
)  ->  C  e.  ( *Met `  X
) )
2220, 21syl 17 . . . . . . . . 9  |-  ( ( ( ( ph  /\  f  e.  ( Fil `  X ) )  /\  r  e.  RR+ )  /\  x  e.  X )  ->  C  e.  ( *Met `  X ) )
23 simpr 459 . . . . . . . . 9  |-  ( ( ( ( ph  /\  f  e.  ( Fil `  X ) )  /\  r  e.  RR+ )  /\  x  e.  X )  ->  x  e.  X )
24 rpxr 11271 . . . . . . . . . 10  |-  ( r  e.  RR+  ->  r  e. 
RR* )
2524ad2antlr 725 . . . . . . . . 9  |-  ( ( ( ( ph  /\  f  e.  ( Fil `  X ) )  /\  r  e.  RR+ )  /\  x  e.  X )  ->  r  e.  RR* )
26 blssm 21211 . . . . . . . . 9  |-  ( ( C  e.  ( *Met `  X )  /\  x  e.  X  /\  r  e.  RR* )  ->  ( x ( ball `  C ) r ) 
C_  X )
2722, 23, 25, 26syl3anc 1230 . . . . . . . 8  |-  ( ( ( ( ph  /\  f  e.  ( Fil `  X ) )  /\  r  e.  RR+ )  /\  x  e.  X )  ->  ( x ( ball `  C ) r ) 
C_  X )
28 filss 20644 . . . . . . . . . 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 1215 . . . . . . . . 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 87 . . . . . . . 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 60 . . . . . . 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 2878 . . . . . 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 2821 . . . 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 691 . . 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 21127 . . . 4  |-  ( D  e.  ( Met `  X
)  ->  D  e.  ( *Met `  X
) )
37 iscfil3 22002 . . . 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 20 . . 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 22002 . . . 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 20 . . 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 268 . 2  |-  ( ph  ->  ( f  e.  (CauFil `  D )  ->  f  e.  (CauFil `  C )
) )
4241ssrdv 3447 1  |-  ( ph  ->  (CauFil `  D )  C_  (CauFil `  C )
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367    = wceq 1405    e. wcel 1842   A.wral 2753   E.wrex 2754    C_ wss 3413   class class class wbr 4394   ` cfv 5568  (class class class)co 6277    x. cmul 9526   RR*cxr 9656    <_ cle 9658    / cdiv 10246   RR+crp 11264   *Metcxmt 18721   Metcme 18722   ballcbl 18723   MetOpencmopn 18726   Filcfil 20636  CauFilccfil 21981
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-sep 4516  ax-nul 4524  ax-pow 4571  ax-pr 4629  ax-un 6573  ax-cnex 9577  ax-resscn 9578  ax-1cn 9579  ax-icn 9580  ax-addcl 9581  ax-addrcl 9582  ax-mulcl 9583  ax-mulrcl 9584  ax-mulcom 9585  ax-addass 9586  ax-mulass 9587  ax-distr 9588  ax-i2m1 9589  ax-1ne0 9590  ax-1rid 9591  ax-rnegex 9592  ax-rrecex 9593  ax-cnre 9594  ax-pre-lttri 9595  ax-pre-lttrn 9596  ax-pre-ltadd 9597  ax-pre-mulgt0 9598
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 975  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-nel 2601  df-ral 2758  df-rex 2759  df-reu 2760  df-rmo 2761  df-rab 2762  df-v 3060  df-sbc 3277  df-csb 3373  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-nul 3738  df-if 3885  df-pw 3956  df-sn 3972  df-pr 3974  df-op 3978  df-uni 4191  df-iun 4272  df-br 4395  df-opab 4453  df-mpt 4454  df-id 4737  df-po 4743  df-so 4744  df-xp 4828  df-rel 4829  df-cnv 4830  df-co 4831  df-dm 4832  df-rn 4833  df-res 4834  df-ima 4835  df-iota 5532  df-fun 5570  df-fn 5571  df-f 5572  df-f1 5573  df-fo 5574  df-f1o 5575  df-fv 5576  df-riota 6239  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-1st 6783  df-2nd 6784  df-er 7347  df-map 7458  df-en 7554  df-dom 7555  df-sdom 7556  df-pnf 9659  df-mnf 9660  df-xr 9661  df-ltxr 9662  df-le 9663  df-sub 9842  df-neg 9843  df-div 10247  df-2 10634  df-rp 11265  df-xneg 11370  df-xadd 11371  df-xmul 11372  df-ico 11587  df-psmet 18729  df-xmet 18730  df-met 18731  df-bl 18732  df-fbas 18734  df-fil 20637  df-cfil 21984
This theorem is referenced by:  equivcmet  22044
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