Theorem equivcmet 25370

Description: If two metrics are strongly equivalent, one is complete iff the other is. Unlike equivcau 25353, metss2 24546, 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 25352	. . . . 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 25352	. . . . 5 ⊢ (𝜑 → (CauFil‘𝐷) ⊆ (CauFil‘𝐶))
10	6, 9	eqssd 4026	. . . 4 ⊢ (𝜑 → (CauFil‘𝐶) = (CauFil‘𝐷))
11		eqid 2740	. . . . . . . 8 ⊢ (MetOpen‘𝐶) = (MetOpen‘𝐶)
12		eqid 2740	. . . . . . . 8 ⊢ (MetOpen‘𝐷) = (MetOpen‘𝐷)
13	11, 12, 1, 2, 7, 8	metss2 24546	. . . . . . 7 ⊢ (𝜑 → (MetOpen‘𝐶) ⊆ (MetOpen‘𝐷))
14	12, 11, 2, 1, 4, 5	metss2 24546	. . . . . . 7 ⊢ (𝜑 → (MetOpen‘𝐷) ⊆ (MetOpen‘𝐶))
15	13, 14	eqssd 4026	. . . . . 6 ⊢ (𝜑 → (MetOpen‘𝐶) = (MetOpen‘𝐷))
16	15	oveq1d 7463	. . . . 5 ⊢ (𝜑 → ((MetOpen‘𝐶) fLim 𝑓) = ((MetOpen‘𝐷) fLim 𝑓))
17	16	neeq1d 3006	. . . 4 ⊢ (𝜑 → (((MetOpen‘𝐶) fLim 𝑓) ≠ ∅ ↔ ((MetOpen‘𝐷) fLim 𝑓) ≠ ∅))
18	10, 17	raleqbidv 3354	. . 3 ⊢ (𝜑 → (∀𝑓 ∈ (CauFil‘𝐶)((MetOpen‘𝐶) fLim 𝑓) ≠ ∅ ↔ ∀𝑓 ∈ (CauFil‘𝐷)((MetOpen‘𝐷) fLim 𝑓) ≠ ∅))
19	3, 18	anbi12d 631	. 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 25337	. 2 ⊢ (𝐶 ∈ (CMet‘𝑋) ↔ (𝐶 ∈ (Met‘𝑋) ∧ ∀𝑓 ∈ (CauFil‘𝐶)((MetOpen‘𝐶) fLim 𝑓) ≠ ∅))
21	12	iscmet 25337	. 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 2108   ≠ wne 2946  ∀wral 3067  ∅c0 4352   class class class wbr 5166  ‘cfv 6573  (class class class)co 7448   · cmul 11189   ≤ cle 11325  ℝ⁺crp 13057  Metcmet 21373  MetOpencmopn 21377   fLim cflim 23963  CauFilccfil 25305  CMetccmet 25307

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770  ax-cnex 11240  ax-resscn 11241  ax-1cn 11242  ax-icn 11243  ax-addcl 11244  ax-addrcl 11245  ax-mulcl 11246  ax-mulrcl 11247  ax-mulcom 11248  ax-addass 11249  ax-mulass 11250  ax-distr 11251  ax-i2m1 11252  ax-1ne0 11253  ax-1rid 11254  ax-rnegex 11255  ax-rrecex 11256  ax-cnre 11257  ax-pre-lttri 11258  ax-pre-lttrn 11259  ax-pre-ltadd 11260  ax-pre-mulgt0 11261  ax-pre-sup 11262

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-nel 3053  df-ral 3068  df-rex 3077  df-rmo 3388  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-pss 3996  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-iun 5017  df-br 5167  df-opab 5229  df-mpt 5250  df-tr 5284  df-id 5593  df-eprel 5599  df-po 5607  df-so 5608  df-fr 5652  df-we 5654  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-pred 6332  df-ord 6398  df-on 6399  df-lim 6400  df-suc 6401  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-riota 7404  df-ov 7451  df-oprab 7452  df-mpo 7453  df-om 7904  df-1st 8030  df-2nd 8031  df-frecs 8322  df-wrecs 8353  df-recs 8427  df-rdg 8466  df-er 8763  df-map 8886  df-en 9004  df-dom 9005  df-sdom 9006  df-sup 9511  df-inf 9512  df-pnf 11326  df-mnf 11327  df-xr 11328  df-ltxr 11329  df-le 11330  df-sub 11522  df-neg 11523  df-div 11948  df-nn 12294  df-2 12356  df-n0 12554  df-z 12640  df-uz 12904  df-q 13014  df-rp 13058  df-xneg 13175  df-xadd 13176  df-xmul 13177  df-ico 13413  df-topgen 17503  df-psmet 21379  df-xmet 21380  df-met 21381  df-bl 21382  df-mopn 21383  df-fbas 21384  df-bases 22974  df-fil 23875  df-cfil 25308  df-cmet 25310

This theorem is referenced by: (None)