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Theorem equivcmet 21920
Description: If two metrics are strongly equivalent, one is complete iff the other is. Unlike equivcau 21905, metss2 21181, 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 21904 . . . . 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 21904 . . . . 5  |-  ( ph  ->  (CauFil `  D )  C_  (CauFil `  C )
)
106, 9eqssd 3506 . . . 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 21181 . . . . . . 7  |-  ( ph  ->  ( MetOpen `  C )  C_  ( MetOpen `  D )
)
1412, 11, 2, 1, 4, 5metss2 21181 . . . . . . 7  |-  ( ph  ->  ( MetOpen `  D )  C_  ( MetOpen `  C )
)
1513, 14eqssd 3506 . . . . . 6  |-  ( ph  ->  ( MetOpen `  C )  =  ( MetOpen `  D
) )
1615oveq1d 6285 . . . . 5  |-  ( ph  ->  ( ( MetOpen `  C
)  fLim  f )  =  ( ( MetOpen `  D )  fLim  f
) )
1716neeq1d 2731 . . . 4  |-  ( ph  ->  ( ( ( MetOpen `  C )  fLim  f
)  =/=  (/)  <->  ( ( MetOpen
`  D )  fLim  f )  =/=  (/) ) )
1810, 17raleqbidv 3065 . . 3  |-  ( ph  ->  ( A. f  e.  (CauFil `  C )
( ( MetOpen `  C
)  fLim  f )  =/=  (/)  <->  A. f  e.  (CauFil `  D ) ( (
MetOpen `  D )  fLim  f )  =/=  (/) ) )
193, 18anbi12d 708 . 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 21889 . 2  |-  ( C  e.  ( CMet `  X
)  <->  ( C  e.  ( Met `  X
)  /\  A. f  e.  (CauFil `  C )
( ( MetOpen `  C
)  fLim  f )  =/=  (/) ) )
2112iscmet 21889 . 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 367    e. wcel 1823    =/= wne 2649   A.wral 2804   (/)c0 3783   class class class wbr 4439   ` cfv 5570  (class class class)co 6270    x. cmul 9486    <_ cle 9618   RR+crp 11221   Metcme 18599   MetOpencmopn 18603    fLim cflim 20601  CauFilccfil 21857   CMetcms 21859
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558  ax-pre-sup 9559
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-nel 2652  df-ral 2809  df-rex 2810  df-reu 2811  df-rmo 2812  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-riota 6232  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-om 6674  df-1st 6773  df-2nd 6774  df-recs 7034  df-rdg 7068  df-er 7303  df-map 7414  df-en 7510  df-dom 7511  df-sdom 7512  df-sup 7893  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9798  df-neg 9799  df-div 10203  df-nn 10532  df-2 10590  df-n0 10792  df-z 10861  df-uz 11083  df-q 11184  df-rp 11222  df-xneg 11321  df-xadd 11322  df-xmul 11323  df-ico 11538  df-topgen 14933  df-psmet 18606  df-xmet 18607  df-met 18608  df-bl 18609  df-mopn 18610  df-fbas 18611  df-bases 19568  df-fil 20513  df-cfil 21860  df-cmet 21862
This theorem is referenced by: (None)
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