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Theorem equivcmet 22922
 Description: If two metrics are strongly equivalent, one is complete iff the other is. Unlike equivcau 22906, metss2 22127, 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 ℝ induced by the usual homeomorphism from (0, 1) against the usual metric 𝐶 on ℝ and against the discrete metric 𝐸 on ℝ. 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 254 . . 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 22905 . . . . 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 22905 . . . . 5 (𝜑 → (CauFil‘𝐷) ⊆ (CauFil‘𝐶))
106, 9eqssd 3585 . . . 4 (𝜑 → (CauFil‘𝐶) = (CauFil‘𝐷))
11 eqid 2610 . . . . . . . 8 (MetOpen‘𝐶) = (MetOpen‘𝐶)
12 eqid 2610 . . . . . . . 8 (MetOpen‘𝐷) = (MetOpen‘𝐷)
1311, 12, 1, 2, 7, 8metss2 22127 . . . . . . 7 (𝜑 → (MetOpen‘𝐶) ⊆ (MetOpen‘𝐷))
1412, 11, 2, 1, 4, 5metss2 22127 . . . . . . 7 (𝜑 → (MetOpen‘𝐷) ⊆ (MetOpen‘𝐶))
1513, 14eqssd 3585 . . . . . 6 (𝜑 → (MetOpen‘𝐶) = (MetOpen‘𝐷))
1615oveq1d 6564 . . . . 5 (𝜑 → ((MetOpen‘𝐶) fLim 𝑓) = ((MetOpen‘𝐷) fLim 𝑓))
1716neeq1d 2841 . . . 4 (𝜑 → (((MetOpen‘𝐶) fLim 𝑓) ≠ ∅ ↔ ((MetOpen‘𝐷) fLim 𝑓) ≠ ∅))
1810, 17raleqbidv 3129 . . 3 (𝜑 → (∀𝑓 ∈ (CauFil‘𝐶)((MetOpen‘𝐶) fLim 𝑓) ≠ ∅ ↔ ∀𝑓 ∈ (CauFil‘𝐷)((MetOpen‘𝐷) fLim 𝑓) ≠ ∅))
193, 18anbi12d 743 . 2 (𝜑 → ((𝐶 ∈ (Met‘𝑋) ∧ ∀𝑓 ∈ (CauFil‘𝐶)((MetOpen‘𝐶) fLim 𝑓) ≠ ∅) ↔ (𝐷 ∈ (Met‘𝑋) ∧ ∀𝑓 ∈ (CauFil‘𝐷)((MetOpen‘𝐷) fLim 𝑓) ≠ ∅)))
2011iscmet 22890 . 2 (𝐶 ∈ (CMet‘𝑋) ↔ (𝐶 ∈ (Met‘𝑋) ∧ ∀𝑓 ∈ (CauFil‘𝐶)((MetOpen‘𝐶) fLim 𝑓) ≠ ∅))
2112iscmet 22890 . 2 (𝐷 ∈ (CMet‘𝑋) ↔ (𝐷 ∈ (Met‘𝑋) ∧ ∀𝑓 ∈ (CauFil‘𝐷)((MetOpen‘𝐷) fLim 𝑓) ≠ ∅))
2219, 20, 213bitr4g 302 1 (𝜑 → (𝐶 ∈ (CMet‘𝑋) ↔ 𝐷 ∈ (CMet‘𝑋)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   ∈ wcel 1977   ≠ wne 2780  ∀wral 2896  ∅c0 3874   class class class wbr 4583  ‘cfv 5804  (class class class)co 6549   · cmul 9820   ≤ cle 9954  ℝ+crp 11708  Metcme 19553  MetOpencmopn 19557   fLim cflim 21548  CauFilccfil 22858  CMetcms 22860 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892  ax-pre-sup 9893 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-er 7629  df-map 7746  df-en 7842  df-dom 7843  df-sdom 7844  df-sup 8231  df-inf 8232  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-div 10564  df-nn 10898  df-2 10956  df-n0 11170  df-z 11255  df-uz 11564  df-q 11665  df-rp 11709  df-xneg 11822  df-xadd 11823  df-xmul 11824  df-ico 12052  df-topgen 15927  df-psmet 19559  df-xmet 19560  df-met 19561  df-bl 19562  df-mopn 19563  df-fbas 19564  df-bases 20522  df-fil 21460  df-cfil 22861  df-cmet 22863 This theorem is referenced by: (None)
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