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Mirrors > Home > MPE Home > Th. List > equivcmet | Structured version Visualization version GIF version |
Description: If two metrics are strongly equivalent, one is complete iff the other is. Unlike equivcau 23830, metss2 23049, 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.) |
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))) |
Ref | Expression |
---|---|
equivcmet | ⊢ (𝜑 → (𝐶 ∈ (CMet‘𝑋) ↔ 𝐷 ∈ (CMet‘𝑋))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | equivcmet.1 | . . . 4 ⊢ (𝜑 → 𝐶 ∈ (Met‘𝑋)) | |
2 | equivcmet.2 | . . . 4 ⊢ (𝜑 → 𝐷 ∈ (Met‘𝑋)) | |
3 | 1, 2 | 2thd 266 | . . 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 23829 | . . . . 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 23829 | . . . . 5 ⊢ (𝜑 → (CauFil‘𝐷) ⊆ (CauFil‘𝐶)) |
10 | 6, 9 | eqssd 3981 | . . . 4 ⊢ (𝜑 → (CauFil‘𝐶) = (CauFil‘𝐷)) |
11 | eqid 2818 | . . . . . . . 8 ⊢ (MetOpen‘𝐶) = (MetOpen‘𝐶) | |
12 | eqid 2818 | . . . . . . . 8 ⊢ (MetOpen‘𝐷) = (MetOpen‘𝐷) | |
13 | 11, 12, 1, 2, 7, 8 | metss2 23049 | . . . . . . 7 ⊢ (𝜑 → (MetOpen‘𝐶) ⊆ (MetOpen‘𝐷)) |
14 | 12, 11, 2, 1, 4, 5 | metss2 23049 | . . . . . . 7 ⊢ (𝜑 → (MetOpen‘𝐷) ⊆ (MetOpen‘𝐶)) |
15 | 13, 14 | eqssd 3981 | . . . . . 6 ⊢ (𝜑 → (MetOpen‘𝐶) = (MetOpen‘𝐷)) |
16 | 15 | oveq1d 7160 | . . . . 5 ⊢ (𝜑 → ((MetOpen‘𝐶) fLim 𝑓) = ((MetOpen‘𝐷) fLim 𝑓)) |
17 | 16 | neeq1d 3072 | . . . 4 ⊢ (𝜑 → (((MetOpen‘𝐶) fLim 𝑓) ≠ ∅ ↔ ((MetOpen‘𝐷) fLim 𝑓) ≠ ∅)) |
18 | 10, 17 | raleqbidv 3399 | . . 3 ⊢ (𝜑 → (∀𝑓 ∈ (CauFil‘𝐶)((MetOpen‘𝐶) fLim 𝑓) ≠ ∅ ↔ ∀𝑓 ∈ (CauFil‘𝐷)((MetOpen‘𝐷) fLim 𝑓) ≠ ∅)) |
19 | 3, 18 | anbi12d 630 | . 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 23814 | . 2 ⊢ (𝐶 ∈ (CMet‘𝑋) ↔ (𝐶 ∈ (Met‘𝑋) ∧ ∀𝑓 ∈ (CauFil‘𝐶)((MetOpen‘𝐶) fLim 𝑓) ≠ ∅)) |
21 | 12 | iscmet 23814 | . 2 ⊢ (𝐷 ∈ (CMet‘𝑋) ↔ (𝐷 ∈ (Met‘𝑋) ∧ ∀𝑓 ∈ (CauFil‘𝐷)((MetOpen‘𝐷) fLim 𝑓) ≠ ∅)) |
22 | 19, 20, 21 | 3bitr4g 315 | 1 ⊢ (𝜑 → (𝐶 ∈ (CMet‘𝑋) ↔ 𝐷 ∈ (CMet‘𝑋))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 207 ∧ wa 396 ∈ wcel 2105 ≠ wne 3013 ∀wral 3135 ∅c0 4288 class class class wbr 5057 ‘cfv 6348 (class class class)co 7145 · cmul 10530 ≤ cle 10664 ℝ+crp 12377 Metcmet 20459 MetOpencmopn 20463 fLim cflim 22470 CauFilccfil 23782 CMetccmet 23784 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-cnex 10581 ax-resscn 10582 ax-1cn 10583 ax-icn 10584 ax-addcl 10585 ax-addrcl 10586 ax-mulcl 10587 ax-mulrcl 10588 ax-mulcom 10589 ax-addass 10590 ax-mulass 10591 ax-distr 10592 ax-i2m1 10593 ax-1ne0 10594 ax-1rid 10595 ax-rnegex 10596 ax-rrecex 10597 ax-cnre 10598 ax-pre-lttri 10599 ax-pre-lttrn 10600 ax-pre-ltadd 10601 ax-pre-mulgt0 10602 ax-pre-sup 10603 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-reu 3142 df-rmo 3143 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-om 7570 df-1st 7678 df-2nd 7679 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-er 8278 df-map 8397 df-en 8498 df-dom 8499 df-sdom 8500 df-sup 8894 df-inf 8895 df-pnf 10665 df-mnf 10666 df-xr 10667 df-ltxr 10668 df-le 10669 df-sub 10860 df-neg 10861 df-div 11286 df-nn 11627 df-2 11688 df-n0 11886 df-z 11970 df-uz 12232 df-q 12337 df-rp 12378 df-xneg 12495 df-xadd 12496 df-xmul 12497 df-ico 12732 df-topgen 16705 df-psmet 20465 df-xmet 20466 df-met 20467 df-bl 20468 df-mopn 20469 df-fbas 20470 df-bases 21482 df-fil 22382 df-cfil 23785 df-cmet 23787 |
This theorem is referenced by: (None) |
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