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Theorem equivcmet 19132
Description: If two metrics are strongly equivalent, one is complete iff the other is. Unlike equivcau 19117, metss2 18425, 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 19116 . . . . 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 19116 . . . . 5  |-  ( ph  ->  (CauFil `  D )  C_  (CauFil `  C )
)
106, 9eqssd 3301 . . . 4  |-  ( ph  ->  (CauFil `  C )  =  (CauFil `  D )
)
11 eqid 2380 . . . . . . . 8  |-  ( MetOpen `  C )  =  (
MetOpen `  C )
12 eqid 2380 . . . . . . . 8  |-  ( MetOpen `  D )  =  (
MetOpen `  D )
1311, 12, 1, 2, 7, 8metss2 18425 . . . . . . 7  |-  ( ph  ->  ( MetOpen `  C )  C_  ( MetOpen `  D )
)
1412, 11, 2, 1, 4, 5metss2 18425 . . . . . . 7  |-  ( ph  ->  ( MetOpen `  D )  C_  ( MetOpen `  C )
)
1513, 14eqssd 3301 . . . . . 6  |-  ( ph  ->  ( MetOpen `  C )  =  ( MetOpen `  D
) )
1615oveq1d 6028 . . . . 5  |-  ( ph  ->  ( ( MetOpen `  C
)  fLim  f )  =  ( ( MetOpen `  D )  fLim  f
) )
1716neeq1d 2556 . . . 4  |-  ( ph  ->  ( ( ( MetOpen `  C )  fLim  f
)  =/=  (/)  <->  ( ( MetOpen
`  D )  fLim  f )  =/=  (/) ) )
1810, 17raleqbidv 2852 . . 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 19101 . 2  |-  ( C  e.  ( CMet `  X
)  <->  ( C  e.  ( Met `  X
)  /\  A. f  e.  (CauFil `  C )
( ( MetOpen `  C
)  fLim  f )  =/=  (/) ) )
2112iscmet 19101 . 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 1717    =/= wne 2543   A.wral 2642   (/)c0 3564   class class class wbr 4146   ` cfv 5387  (class class class)co 6013    x. cmul 8921    <_ cle 9047   RR+crp 10537   Metcme 16606   MetOpencmopn 16610    fLim cflim 17880  CauFilccfil 19069   CMetcms 19071
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361  ax-sep 4264  ax-nul 4272  ax-pow 4311  ax-pr 4337  ax-un 4634  ax-cnex 8972  ax-resscn 8973  ax-1cn 8974  ax-icn 8975  ax-addcl 8976  ax-addrcl 8977  ax-mulcl 8978  ax-mulrcl 8979  ax-mulcom 8980  ax-addass 8981  ax-mulass 8982  ax-distr 8983  ax-i2m1 8984  ax-1ne0 8985  ax-1rid 8986  ax-rnegex 8987  ax-rrecex 8988  ax-cnre 8989  ax-pre-lttri 8990  ax-pre-lttrn 8991  ax-pre-ltadd 8992  ax-pre-mulgt0 8993  ax-pre-sup 8994
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2235  df-mo 2236  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ne 2545  df-nel 2546  df-ral 2647  df-rex 2648  df-reu 2649  df-rmo 2650  df-rab 2651  df-v 2894  df-sbc 3098  df-csb 3188  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-pss 3272  df-nul 3565  df-if 3676  df-pw 3737  df-sn 3756  df-pr 3757  df-tp 3758  df-op 3759  df-uni 3951  df-iun 4030  df-br 4147  df-opab 4201  df-mpt 4202  df-tr 4237  df-eprel 4428  df-id 4432  df-po 4437  df-so 4438  df-fr 4475  df-we 4477  df-ord 4518  df-on 4519  df-lim 4520  df-suc 4521  df-om 4779  df-xp 4817  df-rel 4818  df-cnv 4819  df-co 4820  df-dm 4821  df-rn 4822  df-res 4823  df-ima 4824  df-iota 5351  df-fun 5389  df-fn 5390  df-f 5391  df-f1 5392  df-fo 5393  df-f1o 5394  df-fv 5395  df-ov 6016  df-oprab 6017  df-mpt2 6018  df-1st 6281  df-2nd 6282  df-riota 6478  df-recs 6562  df-rdg 6597  df-er 6834  df-map 6949  df-en 7039  df-dom 7040  df-sdom 7041  df-sup 7374  df-pnf 9048  df-mnf 9049  df-xr 9050  df-ltxr 9051  df-le 9052  df-sub 9218  df-neg 9219  df-div 9603  df-nn 9926  df-2 9983  df-n0 10147  df-z 10208  df-uz 10414  df-q 10500  df-rp 10538  df-xneg 10635  df-xadd 10636  df-xmul 10637  df-ico 10847  df-topgen 13587  df-xmet 16612  df-met 16613  df-bl 16614  df-mopn 16615  df-fbas 16616  df-bases 16881  df-fil 17792  df-cfil 19072  df-cmet 19074
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