Theorem equivcmet 23920

Description: If two metrics are strongly equivalent, one is complete iff the other is. Unlike equivcau 23903, 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 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	⊢ (𝜑 → 𝐶 ∈ (Met‘𝑋))
equivcmet.2	⊢ (𝜑 → 𝐷 ∈ (Met‘𝑋))
equivcmet.3	⊢ (𝜑 → 𝑅 ∈ ℝ+)
equivcmet.4	⊢ (𝜑 → 𝑆 ∈ ℝ+)
equivcmet.5	⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → (𝑥𝐶𝑦) ≤ (𝑅 · (𝑥𝐷𝑦)))
equivcmet.6	⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → (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.

Step	Hyp	Ref	Expression
1		equivcmet.1	. . . 4 ⊢ (𝜑 → 𝐶 ∈ (Met‘𝑋))
2		equivcmet.2	. . . 4 ⊢ (𝜑 → 𝐷 ∈ (Met‘𝑋))
3	1, 2	2thd 267	. . 3 ⊢ (𝜑 → (𝐶 ∈ (Met‘𝑋) ↔ 𝐷 ∈ (Met‘𝑋)))
4		equivcmet.4	. . . . . 6 ⊢ (𝜑 → 𝑆 ∈ ℝ+)
5		equivcmet.6	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → (𝑥𝐷𝑦) ≤ (𝑆 · (𝑥𝐶𝑦)))
6	2, 1, 4, 5	equivcfil 23902	. . . . 5 ⊢ (𝜑 → (CauFil‘𝐶) ⊆ (CauFil‘𝐷))
7		equivcmet.3	. . . . . 6 ⊢ (𝜑 → 𝑅 ∈ ℝ+)
8		equivcmet.5	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → (𝑥𝐶𝑦) ≤ (𝑅 · (𝑥𝐷𝑦)))
9	1, 2, 7, 8	equivcfil 23902	. . . . 5 ⊢ (𝜑 → (CauFil‘𝐷) ⊆ (CauFil‘𝐶))
10	6, 9	eqssd 3984	. . . 4 ⊢ (𝜑 → (CauFil‘𝐶) = (CauFil‘𝐷))
11		eqid 2821	. . . . . . . 8 ⊢ (MetOpen‘𝐶) = (MetOpen‘𝐶)
12		eqid 2821	. . . . . . . 8 ⊢ (MetOpen‘𝐷) = (MetOpen‘𝐷)
13	11, 12, 1, 2, 7, 8	metss2 23122	. . . . . . 7 ⊢ (𝜑 → (MetOpen‘𝐶) ⊆ (MetOpen‘𝐷))
14	12, 11, 2, 1, 4, 5	metss2 23122	. . . . . . 7 ⊢ (𝜑 → (MetOpen‘𝐷) ⊆ (MetOpen‘𝐶))
15	13, 14	eqssd 3984	. . . . . 6 ⊢ (𝜑 → (MetOpen‘𝐶) = (MetOpen‘𝐷))
16	15	oveq1d 7171	. . . . 5 ⊢ (𝜑 → ((MetOpen‘𝐶) fLim 𝑓) = ((MetOpen‘𝐷) fLim 𝑓))
17	16	neeq1d 3075	. . . 4 ⊢ (𝜑 → (((MetOpen‘𝐶) fLim 𝑓) ≠ ∅ ↔ ((MetOpen‘𝐷) fLim 𝑓) ≠ ∅))
18	10, 17	raleqbidv 3401	. . 3 ⊢ (𝜑 → (∀𝑓 ∈ (CauFil‘𝐶)((MetOpen‘𝐶) fLim 𝑓) ≠ ∅ ↔ ∀𝑓 ∈ (CauFil‘𝐷)((MetOpen‘𝐷) fLim 𝑓) ≠ ∅))
19	3, 18	anbi12d 632	. 2 ⊢ (𝜑 → ((𝐶 ∈ (Met‘𝑋) ∧ ∀𝑓 ∈ (CauFil‘𝐶)((MetOpen‘𝐶) fLim 𝑓) ≠ ∅) ↔ (𝐷 ∈ (Met‘𝑋) ∧ ∀𝑓 ∈ (CauFil‘𝐷)((MetOpen‘𝐷) fLim 𝑓) ≠ ∅)))
20	11	iscmet 23887	. 2 ⊢ (𝐶 ∈ (CMet‘𝑋) ↔ (𝐶 ∈ (Met‘𝑋) ∧ ∀𝑓 ∈ (CauFil‘𝐶)((MetOpen‘𝐶) fLim 𝑓) ≠ ∅))
21	12	iscmet 23887	. 2 ⊢ (𝐷 ∈ (CMet‘𝑋) ↔ (𝐷 ∈ (Met‘𝑋) ∧ ∀𝑓 ∈ (CauFil‘𝐷)((MetOpen‘𝐷) fLim 𝑓) ≠ ∅))
22	19, 20, 21	3bitr4g 316	1 ⊢ (𝜑 → (𝐶 ∈ (CMet‘𝑋) ↔ 𝐷 ∈ (CMet‘𝑋)))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398   ∈ wcel 2114   ≠ wne 3016  ∀wral 3138  ∅c0 4291   class class class wbr 5066  ‘cfv 6355  (class class class)co 7156   · cmul 10542   ≤ cle 10676  ℝ⁺crp 12390  Metcmet 20531  MetOpencmopn 20535   fLim cflim 22542  CauFilccfil 23855  CMetccmet 23857

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793  ax-sep 5203  ax-nul 5210  ax-pow 5266  ax-pr 5330  ax-un 7461  ax-cnex 10593  ax-resscn 10594  ax-1cn 10595  ax-icn 10596  ax-addcl 10597  ax-addrcl 10598  ax-mulcl 10599  ax-mulrcl 10600  ax-mulcom 10601  ax-addass 10602  ax-mulass 10603  ax-distr 10604  ax-i2m1 10605  ax-1ne0 10606  ax-1rid 10607  ax-rnegex 10608  ax-rrecex 10609  ax-cnre 10610  ax-pre-lttri 10611  ax-pre-lttrn 10612  ax-pre-ltadd 10613  ax-pre-mulgt0 10614  ax-pre-sup 10615

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-nel 3124  df-ral 3143  df-rex 3144  df-reu 3145  df-rmo 3146  df-rab 3147  df-v 3496  df-sbc 3773  df-csb 3884  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-pss 3954  df-nul 4292  df-if 4468  df-pw 4541  df-sn 4568  df-pr 4570  df-tp 4572  df-op 4574  df-uni 4839  df-iun 4921  df-br 5067  df-opab 5129  df-mpt 5147  df-tr 5173  df-id 5460  df-eprel 5465  df-po 5474  df-so 5475  df-fr 5514  df-we 5516  df-xp 5561  df-rel 5562  df-cnv 5563  df-co 5564  df-dm 5565  df-rn 5566  df-res 5567  df-ima 5568  df-pred 6148  df-ord 6194  df-on 6195  df-lim 6196  df-suc 6197  df-iota 6314  df-fun 6357  df-fn 6358  df-f 6359  df-f1 6360  df-fo 6361  df-f1o 6362  df-fv 6363  df-riota 7114  df-ov 7159  df-oprab 7160  df-mpo 7161  df-om 7581  df-1st 7689  df-2nd 7690  df-wrecs 7947  df-recs 8008  df-rdg 8046  df-er 8289  df-map 8408  df-en 8510  df-dom 8511  df-sdom 8512  df-sup 8906  df-inf 8907  df-pnf 10677  df-mnf 10678  df-xr 10679  df-ltxr 10680  df-le 10681  df-sub 10872  df-neg 10873  df-div 11298  df-nn 11639  df-2 11701  df-n0 11899  df-z 11983  df-uz 12245  df-q 12350  df-rp 12391  df-xneg 12508  df-xadd 12509  df-xmul 12510  df-ico 12745  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 23858  df-cmet 23860

This theorem is referenced by: (None)