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Theorem equivcmet 23022
Description: If two metrics are strongly equivalent, one is complete iff the other is. Unlike equivcau 23006, metss2 22227, 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 255 . . 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 23005 . . . . 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 23005 . . . . 5 (𝜑 → (CauFil‘𝐷) ⊆ (CauFil‘𝐶))
106, 9eqssd 3600 . . . 4 (𝜑 → (CauFil‘𝐶) = (CauFil‘𝐷))
11 eqid 2621 . . . . . . . 8 (MetOpen‘𝐶) = (MetOpen‘𝐶)
12 eqid 2621 . . . . . . . 8 (MetOpen‘𝐷) = (MetOpen‘𝐷)
1311, 12, 1, 2, 7, 8metss2 22227 . . . . . . 7 (𝜑 → (MetOpen‘𝐶) ⊆ (MetOpen‘𝐷))
1412, 11, 2, 1, 4, 5metss2 22227 . . . . . . 7 (𝜑 → (MetOpen‘𝐷) ⊆ (MetOpen‘𝐶))
1513, 14eqssd 3600 . . . . . 6 (𝜑 → (MetOpen‘𝐶) = (MetOpen‘𝐷))
1615oveq1d 6619 . . . . 5 (𝜑 → ((MetOpen‘𝐶) fLim 𝑓) = ((MetOpen‘𝐷) fLim 𝑓))
1716neeq1d 2849 . . . 4 (𝜑 → (((MetOpen‘𝐶) fLim 𝑓) ≠ ∅ ↔ ((MetOpen‘𝐷) fLim 𝑓) ≠ ∅))
1810, 17raleqbidv 3141 . . 3 (𝜑 → (∀𝑓 ∈ (CauFil‘𝐶)((MetOpen‘𝐶) fLim 𝑓) ≠ ∅ ↔ ∀𝑓 ∈ (CauFil‘𝐷)((MetOpen‘𝐷) fLim 𝑓) ≠ ∅))
193, 18anbi12d 746 . 2 (𝜑 → ((𝐶 ∈ (Met‘𝑋) ∧ ∀𝑓 ∈ (CauFil‘𝐶)((MetOpen‘𝐶) fLim 𝑓) ≠ ∅) ↔ (𝐷 ∈ (Met‘𝑋) ∧ ∀𝑓 ∈ (CauFil‘𝐷)((MetOpen‘𝐷) fLim 𝑓) ≠ ∅)))
2011iscmet 22990 . 2 (𝐶 ∈ (CMet‘𝑋) ↔ (𝐶 ∈ (Met‘𝑋) ∧ ∀𝑓 ∈ (CauFil‘𝐶)((MetOpen‘𝐶) fLim 𝑓) ≠ ∅))
2112iscmet 22990 . 2 (𝐷 ∈ (CMet‘𝑋) ↔ (𝐷 ∈ (Met‘𝑋) ∧ ∀𝑓 ∈ (CauFil‘𝐷)((MetOpen‘𝐷) fLim 𝑓) ≠ ∅))
2219, 20, 213bitr4g 303 1 (𝜑 → (𝐶 ∈ (CMet‘𝑋) ↔ 𝐷 ∈ (CMet‘𝑋)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384  wcel 1987  wne 2790  wral 2907  c0 3891   class class class wbr 4613  cfv 5847  (class class class)co 6604   · cmul 9885  cle 10019  +crp 11776  Metcme 19651  MetOpencmopn 19655   fLim cflim 21648  CauFilccfil 22958  CMetcms 22960
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902  ax-cnex 9936  ax-resscn 9937  ax-1cn 9938  ax-icn 9939  ax-addcl 9940  ax-addrcl 9941  ax-mulcl 9942  ax-mulrcl 9943  ax-mulcom 9944  ax-addass 9945  ax-mulass 9946  ax-distr 9947  ax-i2m1 9948  ax-1ne0 9949  ax-1rid 9950  ax-rnegex 9951  ax-rrecex 9952  ax-cnre 9953  ax-pre-lttri 9954  ax-pre-lttrn 9955  ax-pre-ltadd 9956  ax-pre-mulgt0 9957  ax-pre-sup 9958
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2912  df-rex 2913  df-reu 2914  df-rmo 2915  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-pss 3571  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-tp 4153  df-op 4155  df-uni 4403  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-tr 4713  df-eprel 4985  df-id 4989  df-po 4995  df-so 4996  df-fr 5033  df-we 5035  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-pred 5639  df-ord 5685  df-on 5686  df-lim 5687  df-suc 5688  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-riota 6565  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-om 7013  df-1st 7113  df-2nd 7114  df-wrecs 7352  df-recs 7413  df-rdg 7451  df-er 7687  df-map 7804  df-en 7900  df-dom 7901  df-sdom 7902  df-sup 8292  df-inf 8293  df-pnf 10020  df-mnf 10021  df-xr 10022  df-ltxr 10023  df-le 10024  df-sub 10212  df-neg 10213  df-div 10629  df-nn 10965  df-2 11023  df-n0 11237  df-z 11322  df-uz 11632  df-q 11733  df-rp 11777  df-xneg 11890  df-xadd 11891  df-xmul 11892  df-ico 12123  df-topgen 16025  df-psmet 19657  df-xmet 19658  df-met 19659  df-bl 19660  df-mopn 19661  df-fbas 19662  df-bases 20622  df-fil 21560  df-cfil 22961  df-cmet 22963
This theorem is referenced by: (None)
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