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Theorem equivcmet 19260
Description: If two metrics are strongly equivalent, one is complete iff the other is. Unlike equivcau 19245, metss2 18534, this theorem does not have a one-directional form - it is possible for a metric  C that is strongly finer than the complete metric  D to be incomplete and vice versa. Consider  D  = the metric on  RR induced by the usual homeomorphism from  ( 0 ,  1 ) against the usual metric 
C on  RR and against the discrete metric  E on  RR. Then both  C and  E are complete but  D is not, and  C is strongly finer than  D, which is strongly finer than  E. (Contributed by Mario Carneiro, 15-Sep-2015.)
Hypotheses
Ref Expression
equivcmet.1  |-  ( ph  ->  C  e.  ( Met `  X ) )
equivcmet.2  |-  ( ph  ->  D  e.  ( Met `  X ) )
equivcmet.3  |-  ( ph  ->  R  e.  RR+ )
equivcmet.4  |-  ( ph  ->  S  e.  RR+ )
equivcmet.5  |-  ( (
ph  /\  ( x  e.  X  /\  y  e.  X ) )  -> 
( x C y )  <_  ( R  x.  ( x D y ) ) )
equivcmet.6  |-  ( (
ph  /\  ( x  e.  X  /\  y  e.  X ) )  -> 
( x D y )  <_  ( S  x.  ( x C y ) ) )
Assertion
Ref Expression
equivcmet  |-  ( ph  ->  ( C  e.  (
CMet `  X )  <->  D  e.  ( CMet `  X
) ) )
Distinct variable groups:    x, y, C    x, D, y    ph, x, y    x, R, y    x, X, y    x, S, y

Proof of Theorem equivcmet
Dummy variable  f is distinct from all other variables.
StepHypRef Expression
1 equivcmet.1 . . . 4  |-  ( ph  ->  C  e.  ( Met `  X ) )
2 equivcmet.2 . . . 4  |-  ( ph  ->  D  e.  ( Met `  X ) )
31, 22thd 232 . . 3  |-  ( ph  ->  ( C  e.  ( Met `  X )  <-> 
D  e.  ( Met `  X ) ) )
4 equivcmet.4 . . . . . 6  |-  ( ph  ->  S  e.  RR+ )
5 equivcmet.6 . . . . . 6  |-  ( (
ph  /\  ( x  e.  X  /\  y  e.  X ) )  -> 
( x D y )  <_  ( S  x.  ( x C y ) ) )
62, 1, 4, 5equivcfil 19244 . . . . 5  |-  ( ph  ->  (CauFil `  C )  C_  (CauFil `  D )
)
7 equivcmet.3 . . . . . 6  |-  ( ph  ->  R  e.  RR+ )
8 equivcmet.5 . . . . . 6  |-  ( (
ph  /\  ( x  e.  X  /\  y  e.  X ) )  -> 
( x C y )  <_  ( R  x.  ( x D y ) ) )
91, 2, 7, 8equivcfil 19244 . . . . 5  |-  ( ph  ->  (CauFil `  D )  C_  (CauFil `  C )
)
106, 9eqssd 3357 . . . 4  |-  ( ph  ->  (CauFil `  C )  =  (CauFil `  D )
)
11 eqid 2435 . . . . . . . 8  |-  ( MetOpen `  C )  =  (
MetOpen `  C )
12 eqid 2435 . . . . . . . 8  |-  ( MetOpen `  D )  =  (
MetOpen `  D )
1311, 12, 1, 2, 7, 8metss2 18534 . . . . . . 7  |-  ( ph  ->  ( MetOpen `  C )  C_  ( MetOpen `  D )
)
1412, 11, 2, 1, 4, 5metss2 18534 . . . . . . 7  |-  ( ph  ->  ( MetOpen `  D )  C_  ( MetOpen `  C )
)
1513, 14eqssd 3357 . . . . . 6  |-  ( ph  ->  ( MetOpen `  C )  =  ( MetOpen `  D
) )
1615oveq1d 6088 . . . . 5  |-  ( ph  ->  ( ( MetOpen `  C
)  fLim  f )  =  ( ( MetOpen `  D )  fLim  f
) )
1716neeq1d 2611 . . . 4  |-  ( ph  ->  ( ( ( MetOpen `  C )  fLim  f
)  =/=  (/)  <->  ( ( MetOpen
`  D )  fLim  f )  =/=  (/) ) )
1810, 17raleqbidv 2908 . . 3  |-  ( ph  ->  ( A. f  e.  (CauFil `  C )
( ( MetOpen `  C
)  fLim  f )  =/=  (/)  <->  A. f  e.  (CauFil `  D ) ( (
MetOpen `  D )  fLim  f )  =/=  (/) ) )
193, 18anbi12d 692 . 2  |-  ( ph  ->  ( ( C  e.  ( Met `  X
)  /\  A. f  e.  (CauFil `  C )
( ( MetOpen `  C
)  fLim  f )  =/=  (/) )  <->  ( D  e.  ( Met `  X
)  /\  A. f  e.  (CauFil `  D )
( ( MetOpen `  D
)  fLim  f )  =/=  (/) ) ) )
2011iscmet 19229 . 2  |-  ( C  e.  ( CMet `  X
)  <->  ( C  e.  ( Met `  X
)  /\  A. f  e.  (CauFil `  C )
( ( MetOpen `  C
)  fLim  f )  =/=  (/) ) )
2112iscmet 19229 . 2  |-  ( D  e.  ( CMet `  X
)  <->  ( D  e.  ( Met `  X
)  /\  A. f  e.  (CauFil `  D )
( ( MetOpen `  D
)  fLim  f )  =/=  (/) ) )
2219, 20, 213bitr4g 280 1  |-  ( ph  ->  ( C  e.  (
CMet `  X )  <->  D  e.  ( CMet `  X
) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    e. wcel 1725    =/= wne 2598   A.wral 2697   (/)c0 3620   class class class wbr 4204   ` cfv 5446  (class class class)co 6073    x. cmul 8987    <_ cle 9113   RR+crp 10604   Metcme 16679   MetOpencmopn 16683    fLim cflim 17958  CauFilccfil 19197   CMetcms 19199
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 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-cnex 9038  ax-resscn 9039  ax-1cn 9040  ax-icn 9041  ax-addcl 9042  ax-addrcl 9043  ax-mulcl 9044  ax-mulrcl 9045  ax-mulcom 9046  ax-addass 9047  ax-mulass 9048  ax-distr 9049  ax-i2m1 9050  ax-1ne0 9051  ax-1rid 9052  ax-rnegex 9053  ax-rrecex 9054  ax-cnre 9055  ax-pre-lttri 9056  ax-pre-lttrn 9057  ax-pre-ltadd 9058  ax-pre-mulgt0 9059  ax-pre-sup 9060
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 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-riota 6541  df-recs 6625  df-rdg 6660  df-er 6897  df-map 7012  df-en 7102  df-dom 7103  df-sdom 7104  df-sup 7438  df-pnf 9114  df-mnf 9115  df-xr 9116  df-ltxr 9117  df-le 9118  df-sub 9285  df-neg 9286  df-div 9670  df-nn 9993  df-2 10050  df-n0 10214  df-z 10275  df-uz 10481  df-q 10567  df-rp 10605  df-xneg 10702  df-xadd 10703  df-xmul 10704  df-ico 10914  df-topgen 13659  df-psmet 16686  df-xmet 16687  df-met 16688  df-bl 16689  df-mopn 16690  df-fbas 16691  df-bases 16957  df-fil 17870  df-cfil 19200  df-cmet 19202
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