MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  equivcmet Structured version   Visualization version   GIF version

Theorem equivcmet 23924
Description: If two metrics are strongly equivalent, one is complete iff the other is. Unlike equivcau 23907, metss2 23122, 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 268 . . 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 23906 . . . . 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 23906 . . . . 5 (𝜑 → (CauFil‘𝐷) ⊆ (CauFil‘𝐶))
106, 9eqssd 3970 . . . 4 (𝜑 → (CauFil‘𝐶) = (CauFil‘𝐷))
11 eqid 2824 . . . . . . . 8 (MetOpen‘𝐶) = (MetOpen‘𝐶)
12 eqid 2824 . . . . . . . 8 (MetOpen‘𝐷) = (MetOpen‘𝐷)
1311, 12, 1, 2, 7, 8metss2 23122 . . . . . . 7 (𝜑 → (MetOpen‘𝐶) ⊆ (MetOpen‘𝐷))
1412, 11, 2, 1, 4, 5metss2 23122 . . . . . . 7 (𝜑 → (MetOpen‘𝐷) ⊆ (MetOpen‘𝐶))
1513, 14eqssd 3970 . . . . . 6 (𝜑 → (MetOpen‘𝐶) = (MetOpen‘𝐷))
1615oveq1d 7164 . . . . 5 (𝜑 → ((MetOpen‘𝐶) fLim 𝑓) = ((MetOpen‘𝐷) fLim 𝑓))
1716neeq1d 3073 . . . 4 (𝜑 → (((MetOpen‘𝐶) fLim 𝑓) ≠ ∅ ↔ ((MetOpen‘𝐷) fLim 𝑓) ≠ ∅))
1810, 17raleqbidv 3392 . . 3 (𝜑 → (∀𝑓 ∈ (CauFil‘𝐶)((MetOpen‘𝐶) fLim 𝑓) ≠ ∅ ↔ ∀𝑓 ∈ (CauFil‘𝐷)((MetOpen‘𝐷) fLim 𝑓) ≠ ∅))
193, 18anbi12d 633 . 2 (𝜑 → ((𝐶 ∈ (Met‘𝑋) ∧ ∀𝑓 ∈ (CauFil‘𝐶)((MetOpen‘𝐶) fLim 𝑓) ≠ ∅) ↔ (𝐷 ∈ (Met‘𝑋) ∧ ∀𝑓 ∈ (CauFil‘𝐷)((MetOpen‘𝐷) fLim 𝑓) ≠ ∅)))
2011iscmet 23891 . 2 (𝐶 ∈ (CMet‘𝑋) ↔ (𝐶 ∈ (Met‘𝑋) ∧ ∀𝑓 ∈ (CauFil‘𝐶)((MetOpen‘𝐶) fLim 𝑓) ≠ ∅))
2112iscmet 23891 . 2 (𝐷 ∈ (CMet‘𝑋) ↔ (𝐷 ∈ (Met‘𝑋) ∧ ∀𝑓 ∈ (CauFil‘𝐷)((MetOpen‘𝐷) fLim 𝑓) ≠ ∅))
2219, 20, 213bitr4g 317 1 (𝜑 → (𝐶 ∈ (CMet‘𝑋) ↔ 𝐷 ∈ (CMet‘𝑋)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399  wcel 2115  wne 3014  wral 3133  c0 4276   class class class wbr 5052  cfv 6343  (class class class)co 7149   · cmul 10540  cle 10674  +crp 12386  Metcmet 20531  MetOpencmopn 20535   fLim cflim 22542  CauFilccfil 23859  CMetccmet 23861
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5189  ax-nul 5196  ax-pow 5253  ax-pr 5317  ax-un 7455  ax-cnex 10591  ax-resscn 10592  ax-1cn 10593  ax-icn 10594  ax-addcl 10595  ax-addrcl 10596  ax-mulcl 10597  ax-mulrcl 10598  ax-mulcom 10599  ax-addass 10600  ax-mulass 10601  ax-distr 10602  ax-i2m1 10603  ax-1ne0 10604  ax-1rid 10605  ax-rnegex 10606  ax-rrecex 10607  ax-cnre 10608  ax-pre-lttri 10609  ax-pre-lttrn 10610  ax-pre-ltadd 10611  ax-pre-mulgt0 10612  ax-pre-sup 10613
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-nel 3119  df-ral 3138  df-rex 3139  df-reu 3140  df-rmo 3141  df-rab 3142  df-v 3482  df-sbc 3759  df-csb 3867  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-pss 3938  df-nul 4277  df-if 4451  df-pw 4524  df-sn 4551  df-pr 4553  df-tp 4555  df-op 4557  df-uni 4825  df-iun 4907  df-br 5053  df-opab 5115  df-mpt 5133  df-tr 5159  df-id 5447  df-eprel 5452  df-po 5461  df-so 5462  df-fr 5501  df-we 5503  df-xp 5548  df-rel 5549  df-cnv 5550  df-co 5551  df-dm 5552  df-rn 5553  df-res 5554  df-ima 5555  df-pred 6135  df-ord 6181  df-on 6182  df-lim 6183  df-suc 6184  df-iota 6302  df-fun 6345  df-fn 6346  df-f 6347  df-f1 6348  df-fo 6349  df-f1o 6350  df-fv 6351  df-riota 7107  df-ov 7152  df-oprab 7153  df-mpo 7154  df-om 7575  df-1st 7684  df-2nd 7685  df-wrecs 7943  df-recs 8004  df-rdg 8042  df-er 8285  df-map 8404  df-en 8506  df-dom 8507  df-sdom 8508  df-sup 8903  df-inf 8904  df-pnf 10675  df-mnf 10676  df-xr 10677  df-ltxr 10678  df-le 10679  df-sub 10870  df-neg 10871  df-div 11296  df-nn 11635  df-2 11697  df-n0 11895  df-z 11979  df-uz 12241  df-q 12346  df-rp 12387  df-xneg 12504  df-xadd 12505  df-xmul 12506  df-ico 12741  df-topgen 16717  df-psmet 20537  df-xmet 20538  df-met 20539  df-bl 20540  df-mopn 20541  df-fbas 20542  df-bases 21554  df-fil 22454  df-cfil 23862  df-cmet 23864
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator