Theorem equivcmet 25306

Description: If two metrics are strongly equivalent, one is complete iff the other is. Unlike equivcau 25289, metss2 24488, 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 265	. . 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 25288	. . . . 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 25288	. . . . 5 ⊢ (𝜑 → (CauFil‘𝐷) ⊆ (CauFil‘𝐶))
10	6, 9	eqssd 3983	. . . 4 ⊢ (𝜑 → (CauFil‘𝐶) = (CauFil‘𝐷))
11		eqid 2734	. . . . . . . 8 ⊢ (MetOpen‘𝐶) = (MetOpen‘𝐶)
12		eqid 2734	. . . . . . . 8 ⊢ (MetOpen‘𝐷) = (MetOpen‘𝐷)
13	11, 12, 1, 2, 7, 8	metss2 24488	. . . . . . 7 ⊢ (𝜑 → (MetOpen‘𝐶) ⊆ (MetOpen‘𝐷))
14	12, 11, 2, 1, 4, 5	metss2 24488	. . . . . . 7 ⊢ (𝜑 → (MetOpen‘𝐷) ⊆ (MetOpen‘𝐶))
15	13, 14	eqssd 3983	. . . . . 6 ⊢ (𝜑 → (MetOpen‘𝐶) = (MetOpen‘𝐷))
16	15	oveq1d 7429	. . . . 5 ⊢ (𝜑 → ((MetOpen‘𝐶) fLim 𝑓) = ((MetOpen‘𝐷) fLim 𝑓))
17	16	neeq1d 2990	. . . 4 ⊢ (𝜑 → (((MetOpen‘𝐶) fLim 𝑓) ≠ ∅ ↔ ((MetOpen‘𝐷) fLim 𝑓) ≠ ∅))
18	10, 17	raleqbidv 3330	. . 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 25273	. 2 ⊢ (𝐶 ∈ (CMet‘𝑋) ↔ (𝐶 ∈ (Met‘𝑋) ∧ ∀𝑓 ∈ (CauFil‘𝐶)((MetOpen‘𝐶) fLim 𝑓) ≠ ∅))
21	12	iscmet 25273	. 2 ⊢ (𝐷 ∈ (CMet‘𝑋) ↔ (𝐷 ∈ (Met‘𝑋) ∧ ∀𝑓 ∈ (CauFil‘𝐷)((MetOpen‘𝐷) fLim 𝑓) ≠ ∅))
22	19, 20, 21	3bitr4g 314	1 ⊢ (𝜑 → (𝐶 ∈ (CMet‘𝑋) ↔ 𝐷 ∈ (CMet‘𝑋)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∈ wcel 2107   ≠ wne 2931  ∀wral 3050  ∅c0 4315   class class class wbr 5125  ‘cfv 6542  (class class class)co 7414   · cmul 11143   ≤ cle 11279  ℝ⁺crp 13017  Metcmet 21317  MetOpencmopn 21321   fLim cflim 23907  CauFilccfil 25241  CMetccmet 25243

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2706  ax-sep 5278  ax-nul 5288  ax-pow 5347  ax-pr 5414  ax-un 7738  ax-cnex 11194  ax-resscn 11195  ax-1cn 11196  ax-icn 11197  ax-addcl 11198  ax-addrcl 11199  ax-mulcl 11200  ax-mulrcl 11201  ax-mulcom 11202  ax-addass 11203  ax-mulass 11204  ax-distr 11205  ax-i2m1 11206  ax-1ne0 11207  ax-1rid 11208  ax-rnegex 11209  ax-rrecex 11210  ax-cnre 11211  ax-pre-lttri 11212  ax-pre-lttrn 11213  ax-pre-ltadd 11214  ax-pre-mulgt0 11215  ax-pre-sup 11216

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2808  df-nfc 2884  df-ne 2932  df-nel 3036  df-ral 3051  df-rex 3060  df-rmo 3364  df-reu 3365  df-rab 3421  df-v 3466  df-sbc 3773  df-csb 3882  df-dif 3936  df-un 3938  df-in 3940  df-ss 3950  df-pss 3953  df-nul 4316  df-if 4508  df-pw 4584  df-sn 4609  df-pr 4611  df-op 4615  df-uni 4890  df-iun 4975  df-br 5126  df-opab 5188  df-mpt 5208  df-tr 5242  df-id 5560  df-eprel 5566  df-po 5574  df-so 5575  df-fr 5619  df-we 5621  df-xp 5673  df-rel 5674  df-cnv 5675  df-co 5676  df-dm 5677  df-rn 5678  df-res 5679  df-ima 5680  df-pred 6303  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6495  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-riota 7371  df-ov 7417  df-oprab 7418  df-mpo 7419  df-om 7871  df-1st 7997  df-2nd 7998  df-frecs 8289  df-wrecs 8320  df-recs 8394  df-rdg 8433  df-er 8728  df-map 8851  df-en 8969  df-dom 8970  df-sdom 8971  df-sup 9465  df-inf 9466  df-pnf 11280  df-mnf 11281  df-xr 11282  df-ltxr 11283  df-le 11284  df-sub 11477  df-neg 11478  df-div 11904  df-nn 12250  df-2 12312  df-n0 12511  df-z 12598  df-uz 12862  df-q 12974  df-rp 13018  df-xneg 13137  df-xadd 13138  df-xmul 13139  df-ico 13376  df-topgen 17464  df-psmet 21323  df-xmet 21324  df-met 21325  df-bl 21326  df-mopn 21327  df-fbas 21328  df-bases 22919  df-fil 23819  df-cfil 25244  df-cmet 25246

This theorem is referenced by: (None)