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Theorem equivcmet 18741
Description: If two metrics are strongly equivalent, one is complete iff the other is. Unlike equivcau 18726, metss2 18058, 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 231 . . 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 18725 . . . . 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 18725 . . . . 5  |-  ( ph  ->  (CauFil `  D )  C_  (CauFil `  C )
)
106, 9eqssd 3196 . . . 4  |-  ( ph  ->  (CauFil `  C )  =  (CauFil `  D )
)
11 eqid 2283 . . . . . . . 8  |-  ( MetOpen `  C )  =  (
MetOpen `  C )
12 eqid 2283 . . . . . . . 8  |-  ( MetOpen `  D )  =  (
MetOpen `  D )
1311, 12, 1, 2, 7, 8metss2 18058 . . . . . . 7  |-  ( ph  ->  ( MetOpen `  C )  C_  ( MetOpen `  D )
)
1412, 11, 2, 1, 4, 5metss2 18058 . . . . . . 7  |-  ( ph  ->  ( MetOpen `  D )  C_  ( MetOpen `  C )
)
1513, 14eqssd 3196 . . . . . 6  |-  ( ph  ->  ( MetOpen `  C )  =  ( MetOpen `  D
) )
1615oveq1d 5873 . . . . 5  |-  ( ph  ->  ( ( MetOpen `  C
)  fLim  f )  =  ( ( MetOpen `  D )  fLim  f
) )
1716neeq1d 2459 . . . 4  |-  ( ph  ->  ( ( ( MetOpen `  C )  fLim  f
)  =/=  (/)  <->  ( ( MetOpen
`  D )  fLim  f )  =/=  (/) ) )
1810, 17raleqbidv 2748 . . 3  |-  ( ph  ->  ( A. f  e.  (CauFil `  C )
( ( MetOpen `  C
)  fLim  f )  =/=  (/)  <->  A. f  e.  (CauFil `  D ) ( (
MetOpen `  D )  fLim  f )  =/=  (/) ) )
193, 18anbi12d 691 . 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 18710 . 2  |-  ( C  e.  ( CMet `  X
)  <->  ( C  e.  ( Met `  X
)  /\  A. f  e.  (CauFil `  C )
( ( MetOpen `  C
)  fLim  f )  =/=  (/) ) )
2112iscmet 18710 . 2  |-  ( D  e.  ( CMet `  X
)  <->  ( D  e.  ( Met `  X
)  /\  A. f  e.  (CauFil `  D )
( ( MetOpen `  D
)  fLim  f )  =/=  (/) ) )
2219, 20, 213bitr4g 279 1  |-  ( ph  ->  ( C  e.  (
CMet `  X )  <->  D  e.  ( CMet `  X
) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    e. wcel 1684    =/= wne 2446   A.wral 2543   (/)c0 3455   class class class wbr 4023   ` cfv 5255  (class class class)co 5858    x. cmul 8742    <_ cle 8868   RR+crp 10354   Metcme 16370   MetOpencmopn 16372    fLim cflim 17629  CauFilccfil 18678   CMetcms 18680
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814  ax-pre-sup 8815
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 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-er 6660  df-map 6774  df-en 6864  df-dom 6865  df-sdom 6866  df-sup 7194  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-nn 9747  df-2 9804  df-n0 9966  df-z 10025  df-uz 10231  df-q 10317  df-rp 10355  df-xneg 10452  df-xadd 10453  df-xmul 10454  df-ico 10662  df-topgen 13344  df-xmet 16373  df-met 16374  df-bl 16375  df-mopn 16376  df-bases 16638  df-fbas 17520  df-fil 17541  df-cfil 18681  df-cmet 18683
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