Theorem equivcmet 25215

Description: If two metrics are strongly equivalent, one is complete iff the other is. Unlike equivcau 25198, metss2 24398, 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 25197	. . . . 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 25197	. . . . 5 ⊢ (𝜑 → (CauFil‘𝐷) ⊆ (CauFil‘𝐶))
10	6, 9	eqssd 3953	. . . 4 ⊢ (𝜑 → (CauFil‘𝐶) = (CauFil‘𝐷))
11		eqid 2729	. . . . . . . 8 ⊢ (MetOpen‘𝐶) = (MetOpen‘𝐶)
12		eqid 2729	. . . . . . . 8 ⊢ (MetOpen‘𝐷) = (MetOpen‘𝐷)
13	11, 12, 1, 2, 7, 8	metss2 24398	. . . . . . 7 ⊢ (𝜑 → (MetOpen‘𝐶) ⊆ (MetOpen‘𝐷))
14	12, 11, 2, 1, 4, 5	metss2 24398	. . . . . . 7 ⊢ (𝜑 → (MetOpen‘𝐷) ⊆ (MetOpen‘𝐶))
15	13, 14	eqssd 3953	. . . . . 6 ⊢ (𝜑 → (MetOpen‘𝐶) = (MetOpen‘𝐷))
16	15	oveq1d 7364	. . . . 5 ⊢ (𝜑 → ((MetOpen‘𝐶) fLim 𝑓) = ((MetOpen‘𝐷) fLim 𝑓))
17	16	neeq1d 2984	. . . 4 ⊢ (𝜑 → (((MetOpen‘𝐶) fLim 𝑓) ≠ ∅ ↔ ((MetOpen‘𝐷) fLim 𝑓) ≠ ∅))
18	10, 17	raleqbidv 3309	. . 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 25182	. 2 ⊢ (𝐶 ∈ (CMet‘𝑋) ↔ (𝐶 ∈ (Met‘𝑋) ∧ ∀𝑓 ∈ (CauFil‘𝐶)((MetOpen‘𝐶) fLim 𝑓) ≠ ∅))
21	12	iscmet 25182	. 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 2109   ≠ wne 2925  ∀wral 3044  ∅c0 4284   class class class wbr 5092  ‘cfv 6482  (class class class)co 7349   · cmul 11014   ≤ cle 11150  ℝ⁺crp 12893  Metcmet 21247  MetOpencmopn 21251   fLim cflim 23819  CauFilccfil 25150  CMetccmet 25152

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5235  ax-nul 5245  ax-pow 5304  ax-pr 5371  ax-un 7671  ax-cnex 11065  ax-resscn 11066  ax-1cn 11067  ax-icn 11068  ax-addcl 11069  ax-addrcl 11070  ax-mulcl 11071  ax-mulrcl 11072  ax-mulcom 11073  ax-addass 11074  ax-mulass 11075  ax-distr 11076  ax-i2m1 11077  ax-1ne0 11078  ax-1rid 11079  ax-rnegex 11080  ax-rrecex 11081  ax-cnre 11082  ax-pre-lttri 11083  ax-pre-lttrn 11084  ax-pre-ltadd 11085  ax-pre-mulgt0 11086  ax-pre-sup 11087

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-nel 3030  df-ral 3045  df-rex 3054  df-rmo 3343  df-reu 3344  df-rab 3395  df-v 3438  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4285  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4859  df-iun 4943  df-br 5093  df-opab 5155  df-mpt 5174  df-tr 5200  df-id 5514  df-eprel 5519  df-po 5527  df-so 5528  df-fr 5572  df-we 5574  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-pred 6249  df-ord 6310  df-on 6311  df-lim 6312  df-suc 6313  df-iota 6438  df-fun 6484  df-fn 6485  df-f 6486  df-f1 6487  df-fo 6488  df-f1o 6489  df-fv 6490  df-riota 7306  df-ov 7352  df-oprab 7353  df-mpo 7354  df-om 7800  df-1st 7924  df-2nd 7925  df-frecs 8214  df-wrecs 8245  df-recs 8294  df-rdg 8332  df-er 8625  df-map 8755  df-en 8873  df-dom 8874  df-sdom 8875  df-sup 9332  df-inf 9333  df-pnf 11151  df-mnf 11152  df-xr 11153  df-ltxr 11154  df-le 11155  df-sub 11349  df-neg 11350  df-div 11778  df-nn 12129  df-2 12191  df-n0 12385  df-z 12472  df-uz 12736  df-q 12850  df-rp 12894  df-xneg 13014  df-xadd 13015  df-xmul 13016  df-ico 13254  df-topgen 17347  df-psmet 21253  df-xmet 21254  df-met 21255  df-bl 21256  df-mopn 21257  df-fbas 21258  df-bases 22831  df-fil 23731  df-cfil 25153  df-cmet 25155

This theorem is referenced by: (None)