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Theorem equivcfil 21473
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 461 . . . . . . . 8  |-  ( ( ( ph  /\  f  e.  ( Fil `  X
) )  /\  r  e.  RR+ )  ->  r  e.  RR+ )
2 equivcau.3 . . . . . . . . 9  |-  ( ph  ->  R  e.  RR+ )
32ad2antrr 725 . . . . . . . 8  |-  ( ( ( ph  /\  f  e.  ( Fil `  X
) )  /\  r  e.  RR+ )  ->  R  e.  RR+ )
41, 3rpdivcld 11269 . . . . . . 7  |-  ( ( ( ph  /\  f  e.  ( Fil `  X
) )  /\  r  e.  RR+ )  ->  (
r  /  R )  e.  RR+ )
5 oveq2 6290 . . . . . . . . . 10  |-  ( s  =  ( r  /  R )  ->  (
x ( ball `  D
) s )  =  ( x ( ball `  D ) ( r  /  R ) ) )
65eleq1d 2536 . . . . . . . . 9  |-  ( s  =  ( r  /  R )  ->  (
( x ( ball `  D ) s )  e.  f  <->  ( x
( ball `  D )
( r  /  R
) )  e.  f ) )
76rexbidv 2973 . . . . . . . 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 3210 . . . . . . 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 758 . . . . . . . 8  |-  ( ( ( ( ph  /\  f  e.  ( Fil `  X ) )  /\  r  e.  RR+ )  /\  x  e.  X )  ->  f  e.  ( Fil `  X ) )
11 eqid 2467 . . . . . . . . . . . 12  |-  ( MetOpen `  C )  =  (
MetOpen `  C )
12 eqid 2467 . . . . . . . . . . . 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 20749 . . . . . . . . . . 11  |-  ( (
ph  /\  ( x  e.  X  /\  r  e.  RR+ ) )  -> 
( x ( ball `  D ) ( r  /  R ) ) 
C_  ( x (
ball `  C )
r ) )
1716ancom2s 800 . . . . . . . . . 10  |-  ( (
ph  /\  ( r  e.  RR+  /\  x  e.  X ) )  -> 
( x ( ball `  D ) ( r  /  R ) ) 
C_  ( x (
ball `  C )
r ) )
1817adantlr 714 . . . . . . . . 9  |-  ( ( ( ph  /\  f  e.  ( Fil `  X
) )  /\  (
r  e.  RR+  /\  x  e.  X ) )  -> 
( x ( ball `  D ) ( r  /  R ) ) 
C_  ( x (
ball `  C )
r ) )
1918anassrs 648 . . . . . . . 8  |-  ( ( ( ( ph  /\  f  e.  ( Fil `  X ) )  /\  r  e.  RR+ )  /\  x  e.  X )  ->  ( x ( ball `  D ) ( r  /  R ) ) 
C_  ( x (
ball `  C )
r ) )
2013ad3antrrr 729 . . . . . . . . . 10  |-  ( ( ( ( ph  /\  f  e.  ( Fil `  X ) )  /\  r  e.  RR+ )  /\  x  e.  X )  ->  C  e.  ( Met `  X ) )
21 metxmet 20572 . . . . . . . . . 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 461 . . . . . . . . 9  |-  ( ( ( ( ph  /\  f  e.  ( Fil `  X ) )  /\  r  e.  RR+ )  /\  x  e.  X )  ->  x  e.  X )
24 rpxr 11223 . . . . . . . . . 10  |-  ( r  e.  RR+  ->  r  e. 
RR* )
2524ad2antlr 726 . . . . . . . . 9  |-  ( ( ( ( ph  /\  f  e.  ( Fil `  X ) )  /\  r  e.  RR+ )  /\  x  e.  X )  ->  r  e.  RR* )
26 blssm 20656 . . . . . . . . 9  |-  ( ( C  e.  ( *Met `  X )  /\  x  e.  X  /\  r  e.  RR* )  ->  ( x ( ball `  C ) r ) 
C_  X )
2722, 23, 25, 26syl3anc 1228 . . . . . . . 8  |-  ( ( ( ( ph  /\  f  e.  ( Fil `  X ) )  /\  r  e.  RR+ )  /\  x  e.  X )  ->  ( x ( ball `  C ) r ) 
C_  X )
28 filss 20089 . . . . . . . . . 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 1214 . . . . . . . . 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 61 . . . . . . 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 2938 . . . . . 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 44 . . . . 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 2882 . . . 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 693 . . 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 20572 . . . 4  |-  ( D  e.  ( Met `  X
)  ->  D  e.  ( *Met `  X
) )
37 iscfil3 21447 . . . 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 21447 . . . 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 3510 1  |-  ( ph  ->  (CauFil `  D )  C_  (CauFil `  C )
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1379    e. wcel 1767   A.wral 2814   E.wrex 2815    C_ wss 3476   class class class wbr 4447   ` cfv 5586  (class class class)co 6282    x. cmul 9493   RR*cxr 9623    <_ cle 9625    / cdiv 10202   RR+crp 11216   *Metcxmt 18174   Metcme 18175   ballcbl 18176   MetOpencmopn 18179   Filcfil 20081  CauFilccfil 21426
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574  ax-cnex 9544  ax-resscn 9545  ax-1cn 9546  ax-icn 9547  ax-addcl 9548  ax-addrcl 9549  ax-mulcl 9550  ax-mulrcl 9551  ax-mulcom 9552  ax-addass 9553  ax-mulass 9554  ax-distr 9555  ax-i2m1 9556  ax-1ne0 9557  ax-1rid 9558  ax-rnegex 9559  ax-rrecex 9560  ax-cnre 9561  ax-pre-lttri 9562  ax-pre-lttrn 9563  ax-pre-ltadd 9564  ax-pre-mulgt0 9565
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-po 4800  df-so 4801  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-riota 6243  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-1st 6781  df-2nd 6782  df-er 7308  df-map 7419  df-en 7514  df-dom 7515  df-sdom 7516  df-pnf 9626  df-mnf 9627  df-xr 9628  df-ltxr 9629  df-le 9630  df-sub 9803  df-neg 9804  df-div 10203  df-2 10590  df-rp 11217  df-xneg 11314  df-xadd 11315  df-xmul 11316  df-ico 11531  df-psmet 18182  df-xmet 18183  df-met 18184  df-bl 18185  df-fbas 18187  df-fil 20082  df-cfil 21429
This theorem is referenced by:  equivcmet  21489
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