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Theorem equivcmet 23847
Description: If two metrics are strongly equivalent, one is complete iff the other is. Unlike equivcau 23830, metss2 23049, 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 induced by the usual homeomorphism from (0, 1) against the usual metric 𝐶 on and against the discrete metric 𝐸 on . 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 (𝜑𝐶 ∈ (Met‘𝑋))
equivcmet.2 (𝜑𝐷 ∈ (Met‘𝑋))
equivcmet.3 (𝜑𝑅 ∈ ℝ+)
equivcmet.4 (𝜑𝑆 ∈ ℝ+)
equivcmet.5 ((𝜑 ∧ (𝑥𝑋𝑦𝑋)) → (𝑥𝐶𝑦) ≤ (𝑅 · (𝑥𝐷𝑦)))
equivcmet.6 ((𝜑 ∧ (𝑥𝑋𝑦𝑋)) → (𝑥𝐷𝑦) ≤ (𝑆 · (𝑥𝐶𝑦)))
Assertion
Ref Expression
equivcmet (𝜑 → (𝐶 ∈ (CMet‘𝑋) ↔ 𝐷 ∈ (CMet‘𝑋)))
Distinct variable groups:   𝑥,𝑦,𝐶   𝑥,𝐷,𝑦   𝜑,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 (𝜑𝐶 ∈ (Met‘𝑋))
2 equivcmet.2 . . . 4 (𝜑𝐷 ∈ (Met‘𝑋))
31, 22thd 266 . . 3 (𝜑 → (𝐶 ∈ (Met‘𝑋) ↔ 𝐷 ∈ (Met‘𝑋)))
4 equivcmet.4 . . . . . 6 (𝜑𝑆 ∈ ℝ+)
5 equivcmet.6 . . . . . 6 ((𝜑 ∧ (𝑥𝑋𝑦𝑋)) → (𝑥𝐷𝑦) ≤ (𝑆 · (𝑥𝐶𝑦)))
62, 1, 4, 5equivcfil 23829 . . . . 5 (𝜑 → (CauFil‘𝐶) ⊆ (CauFil‘𝐷))
7 equivcmet.3 . . . . . 6 (𝜑𝑅 ∈ ℝ+)
8 equivcmet.5 . . . . . 6 ((𝜑 ∧ (𝑥𝑋𝑦𝑋)) → (𝑥𝐶𝑦) ≤ (𝑅 · (𝑥𝐷𝑦)))
91, 2, 7, 8equivcfil 23829 . . . . 5 (𝜑 → (CauFil‘𝐷) ⊆ (CauFil‘𝐶))
106, 9eqssd 3981 . . . 4 (𝜑 → (CauFil‘𝐶) = (CauFil‘𝐷))
11 eqid 2818 . . . . . . . 8 (MetOpen‘𝐶) = (MetOpen‘𝐶)
12 eqid 2818 . . . . . . . 8 (MetOpen‘𝐷) = (MetOpen‘𝐷)
1311, 12, 1, 2, 7, 8metss2 23049 . . . . . . 7 (𝜑 → (MetOpen‘𝐶) ⊆ (MetOpen‘𝐷))
1412, 11, 2, 1, 4, 5metss2 23049 . . . . . . 7 (𝜑 → (MetOpen‘𝐷) ⊆ (MetOpen‘𝐶))
1513, 14eqssd 3981 . . . . . 6 (𝜑 → (MetOpen‘𝐶) = (MetOpen‘𝐷))
1615oveq1d 7160 . . . . 5 (𝜑 → ((MetOpen‘𝐶) fLim 𝑓) = ((MetOpen‘𝐷) fLim 𝑓))
1716neeq1d 3072 . . . 4 (𝜑 → (((MetOpen‘𝐶) fLim 𝑓) ≠ ∅ ↔ ((MetOpen‘𝐷) fLim 𝑓) ≠ ∅))
1810, 17raleqbidv 3399 . . 3 (𝜑 → (∀𝑓 ∈ (CauFil‘𝐶)((MetOpen‘𝐶) fLim 𝑓) ≠ ∅ ↔ ∀𝑓 ∈ (CauFil‘𝐷)((MetOpen‘𝐷) fLim 𝑓) ≠ ∅))
193, 18anbi12d 630 . 2 (𝜑 → ((𝐶 ∈ (Met‘𝑋) ∧ ∀𝑓 ∈ (CauFil‘𝐶)((MetOpen‘𝐶) fLim 𝑓) ≠ ∅) ↔ (𝐷 ∈ (Met‘𝑋) ∧ ∀𝑓 ∈ (CauFil‘𝐷)((MetOpen‘𝐷) fLim 𝑓) ≠ ∅)))
2011iscmet 23814 . 2 (𝐶 ∈ (CMet‘𝑋) ↔ (𝐶 ∈ (Met‘𝑋) ∧ ∀𝑓 ∈ (CauFil‘𝐶)((MetOpen‘𝐶) fLim 𝑓) ≠ ∅))
2112iscmet 23814 . 2 (𝐷 ∈ (CMet‘𝑋) ↔ (𝐷 ∈ (Met‘𝑋) ∧ ∀𝑓 ∈ (CauFil‘𝐷)((MetOpen‘𝐷) fLim 𝑓) ≠ ∅))
2219, 20, 213bitr4g 315 1 (𝜑 → (𝐶 ∈ (CMet‘𝑋) ↔ 𝐷 ∈ (CMet‘𝑋)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396  wcel 2105  wne 3013  wral 3135  c0 4288   class class class wbr 5057  cfv 6348  (class class class)co 7145   · cmul 10530  cle 10664  +crp 12377  Metcmet 20459  MetOpencmopn 20463   fLim cflim 22470  CauFilccfil 23782  CMetccmet 23784
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450  ax-cnex 10581  ax-resscn 10582  ax-1cn 10583  ax-icn 10584  ax-addcl 10585  ax-addrcl 10586  ax-mulcl 10587  ax-mulrcl 10588  ax-mulcom 10589  ax-addass 10590  ax-mulass 10591  ax-distr 10592  ax-i2m1 10593  ax-1ne0 10594  ax-1rid 10595  ax-rnegex 10596  ax-rrecex 10597  ax-cnre 10598  ax-pre-lttri 10599  ax-pre-lttrn 10600  ax-pre-ltadd 10601  ax-pre-mulgt0 10602  ax-pre-sup 10603
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3or 1080  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-nel 3121  df-ral 3140  df-rex 3141  df-reu 3142  df-rmo 3143  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-pss 3951  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-tp 4562  df-op 4564  df-uni 4831  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-tr 5164  df-id 5453  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-pred 6141  df-ord 6187  df-on 6188  df-lim 6189  df-suc 6190  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-riota 7103  df-ov 7148  df-oprab 7149  df-mpo 7150  df-om 7570  df-1st 7678  df-2nd 7679  df-wrecs 7936  df-recs 7997  df-rdg 8035  df-er 8278  df-map 8397  df-en 8498  df-dom 8499  df-sdom 8500  df-sup 8894  df-inf 8895  df-pnf 10665  df-mnf 10666  df-xr 10667  df-ltxr 10668  df-le 10669  df-sub 10860  df-neg 10861  df-div 11286  df-nn 11627  df-2 11688  df-n0 11886  df-z 11970  df-uz 12232  df-q 12337  df-rp 12378  df-xneg 12495  df-xadd 12496  df-xmul 12497  df-ico 12732  df-topgen 16705  df-psmet 20465  df-xmet 20466  df-met 20467  df-bl 20468  df-mopn 20469  df-fbas 20470  df-bases 21482  df-fil 22382  df-cfil 23785  df-cmet 23787
This theorem is referenced by: (None)
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