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Theorem equivcmet 20959
Description: If two metrics are strongly equivalent, one is complete iff the other is. Unlike equivcau 20944, metss2 20220, 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 240 . . 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 20943 . . . . 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 20943 . . . . 5  |-  ( ph  ->  (CauFil `  D )  C_  (CauFil `  C )
)
106, 9eqssd 3482 . . . 4  |-  ( ph  ->  (CauFil `  C )  =  (CauFil `  D )
)
11 eqid 2454 . . . . . . . 8  |-  ( MetOpen `  C )  =  (
MetOpen `  C )
12 eqid 2454 . . . . . . . 8  |-  ( MetOpen `  D )  =  (
MetOpen `  D )
1311, 12, 1, 2, 7, 8metss2 20220 . . . . . . 7  |-  ( ph  ->  ( MetOpen `  C )  C_  ( MetOpen `  D )
)
1412, 11, 2, 1, 4, 5metss2 20220 . . . . . . 7  |-  ( ph  ->  ( MetOpen `  D )  C_  ( MetOpen `  C )
)
1513, 14eqssd 3482 . . . . . 6  |-  ( ph  ->  ( MetOpen `  C )  =  ( MetOpen `  D
) )
1615oveq1d 6216 . . . . 5  |-  ( ph  ->  ( ( MetOpen `  C
)  fLim  f )  =  ( ( MetOpen `  D )  fLim  f
) )
1716neeq1d 2729 . . . 4  |-  ( ph  ->  ( ( ( MetOpen `  C )  fLim  f
)  =/=  (/)  <->  ( ( MetOpen
`  D )  fLim  f )  =/=  (/) ) )
1810, 17raleqbidv 3037 . . 3  |-  ( ph  ->  ( A. f  e.  (CauFil `  C )
( ( MetOpen `  C
)  fLim  f )  =/=  (/)  <->  A. f  e.  (CauFil `  D ) ( (
MetOpen `  D )  fLim  f )  =/=  (/) ) )
193, 18anbi12d 710 . 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 20928 . 2  |-  ( C  e.  ( CMet `  X
)  <->  ( C  e.  ( Met `  X
)  /\  A. f  e.  (CauFil `  C )
( ( MetOpen `  C
)  fLim  f )  =/=  (/) ) )
2112iscmet 20928 . 2  |-  ( D  e.  ( CMet `  X
)  <->  ( D  e.  ( Met `  X
)  /\  A. f  e.  (CauFil `  D )
( ( MetOpen `  D
)  fLim  f )  =/=  (/) ) )
2219, 20, 213bitr4g 288 1  |-  ( ph  ->  ( C  e.  (
CMet `  X )  <->  D  e.  ( CMet `  X
) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    e. wcel 1758    =/= wne 2648   A.wral 2799   (/)c0 3746   class class class wbr 4401   ` cfv 5527  (class class class)co 6201    x. cmul 9399    <_ cle 9531   RR+crp 11103   Metcme 17928   MetOpencmopn 17932    fLim cflim 19640  CauFilccfil 20896   CMetcms 20898
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4522  ax-nul 4530  ax-pow 4579  ax-pr 4640  ax-un 6483  ax-cnex 9450  ax-resscn 9451  ax-1cn 9452  ax-icn 9453  ax-addcl 9454  ax-addrcl 9455  ax-mulcl 9456  ax-mulrcl 9457  ax-mulcom 9458  ax-addass 9459  ax-mulass 9460  ax-distr 9461  ax-i2m1 9462  ax-1ne0 9463  ax-1rid 9464  ax-rnegex 9465  ax-rrecex 9466  ax-cnre 9467  ax-pre-lttri 9468  ax-pre-lttrn 9469  ax-pre-ltadd 9470  ax-pre-mulgt0 9471  ax-pre-sup 9472
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-nel 2651  df-ral 2804  df-rex 2805  df-reu 2806  df-rmo 2807  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3397  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-pss 3453  df-nul 3747  df-if 3901  df-pw 3971  df-sn 3987  df-pr 3989  df-tp 3991  df-op 3993  df-uni 4201  df-iun 4282  df-br 4402  df-opab 4460  df-mpt 4461  df-tr 4495  df-eprel 4741  df-id 4745  df-po 4750  df-so 4751  df-fr 4788  df-we 4790  df-ord 4831  df-on 4832  df-lim 4833  df-suc 4834  df-xp 4955  df-rel 4956  df-cnv 4957  df-co 4958  df-dm 4959  df-rn 4960  df-res 4961  df-ima 4962  df-iota 5490  df-fun 5529  df-fn 5530  df-f 5531  df-f1 5532  df-fo 5533  df-f1o 5534  df-fv 5535  df-riota 6162  df-ov 6204  df-oprab 6205  df-mpt2 6206  df-om 6588  df-1st 6688  df-2nd 6689  df-recs 6943  df-rdg 6977  df-er 7212  df-map 7327  df-en 7422  df-dom 7423  df-sdom 7424  df-sup 7803  df-pnf 9532  df-mnf 9533  df-xr 9534  df-ltxr 9535  df-le 9536  df-sub 9709  df-neg 9710  df-div 10106  df-nn 10435  df-2 10492  df-n0 10692  df-z 10759  df-uz 10974  df-q 11066  df-rp 11104  df-xneg 11201  df-xadd 11202  df-xmul 11203  df-ico 11418  df-topgen 14502  df-psmet 17935  df-xmet 17936  df-met 17937  df-bl 17938  df-mopn 17939  df-fbas 17940  df-bases 18638  df-fil 19552  df-cfil 20899  df-cmet 20901
This theorem is referenced by: (None)
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