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Theorem equivcau 23830
Description: If the metric 𝐷 is "strongly finer" than 𝐶 (meaning that there is a positive real constant 𝑅 such that 𝐶(𝑥, 𝑦) ≤ 𝑅 · 𝐷(𝑥, 𝑦)), all the 𝐷-Cauchy sequences are also 𝐶-Cauchy. (Using this theorem twice in each direction states that if two metrics are strongly equivalent, then they have the same Cauchy sequences.) (Contributed by Mario Carneiro, 14-Sep-2015.)
Hypotheses
Ref Expression
equivcau.1 (𝜑𝐶 ∈ (Met‘𝑋))
equivcau.2 (𝜑𝐷 ∈ (Met‘𝑋))
equivcau.3 (𝜑𝑅 ∈ ℝ+)
equivcau.4 ((𝜑 ∧ (𝑥𝑋𝑦𝑋)) → (𝑥𝐶𝑦) ≤ (𝑅 · (𝑥𝐷𝑦)))
Assertion
Ref Expression
equivcau (𝜑 → (Cau‘𝐷) ⊆ (Cau‘𝐶))
Distinct variable groups:   𝑥,𝑦,𝐶   𝑥,𝐷,𝑦   𝜑,𝑥,𝑦   𝑥,𝑅,𝑦   𝑥,𝑋,𝑦

Proof of Theorem equivcau
Dummy variables 𝑓 𝑘 𝑟 𝑠 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simpr 485 . . . . . . 7 (((𝜑𝑓 ∈ (𝑋pm ℂ)) ∧ 𝑟 ∈ ℝ+) → 𝑟 ∈ ℝ+)
2 equivcau.3 . . . . . . . 8 (𝜑𝑅 ∈ ℝ+)
32ad2antrr 722 . . . . . . 7 (((𝜑𝑓 ∈ (𝑋pm ℂ)) ∧ 𝑟 ∈ ℝ+) → 𝑅 ∈ ℝ+)
41, 3rpdivcld 12436 . . . . . 6 (((𝜑𝑓 ∈ (𝑋pm ℂ)) ∧ 𝑟 ∈ ℝ+) → (𝑟 / 𝑅) ∈ ℝ+)
5 oveq2 7153 . . . . . . . . 9 (𝑠 = (𝑟 / 𝑅) → ((𝑓𝑘)(ball‘𝐷)𝑠) = ((𝑓𝑘)(ball‘𝐷)(𝑟 / 𝑅)))
65feq3d 6494 . . . . . . . 8 (𝑠 = (𝑟 / 𝑅) → ((𝑓 ↾ (ℤ𝑘)):(ℤ𝑘)⟶((𝑓𝑘)(ball‘𝐷)𝑠) ↔ (𝑓 ↾ (ℤ𝑘)):(ℤ𝑘)⟶((𝑓𝑘)(ball‘𝐷)(𝑟 / 𝑅))))
76rexbidv 3294 . . . . . . 7 (𝑠 = (𝑟 / 𝑅) → (∃𝑘 ∈ ℤ (𝑓 ↾ (ℤ𝑘)):(ℤ𝑘)⟶((𝑓𝑘)(ball‘𝐷)𝑠) ↔ ∃𝑘 ∈ ℤ (𝑓 ↾ (ℤ𝑘)):(ℤ𝑘)⟶((𝑓𝑘)(ball‘𝐷)(𝑟 / 𝑅))))
87rspcv 3615 . . . . . 6 ((𝑟 / 𝑅) ∈ ℝ+ → (∀𝑠 ∈ ℝ+𝑘 ∈ ℤ (𝑓 ↾ (ℤ𝑘)):(ℤ𝑘)⟶((𝑓𝑘)(ball‘𝐷)𝑠) → ∃𝑘 ∈ ℤ (𝑓 ↾ (ℤ𝑘)):(ℤ𝑘)⟶((𝑓𝑘)(ball‘𝐷)(𝑟 / 𝑅))))
94, 8syl 17 . . . . 5 (((𝜑𝑓 ∈ (𝑋pm ℂ)) ∧ 𝑟 ∈ ℝ+) → (∀𝑠 ∈ ℝ+𝑘 ∈ ℤ (𝑓 ↾ (ℤ𝑘)):(ℤ𝑘)⟶((𝑓𝑘)(ball‘𝐷)𝑠) → ∃𝑘 ∈ ℤ (𝑓 ↾ (ℤ𝑘)):(ℤ𝑘)⟶((𝑓𝑘)(ball‘𝐷)(𝑟 / 𝑅))))
10 simprr 769 . . . . . . . 8 ((((𝜑𝑓 ∈ (𝑋pm ℂ)) ∧ 𝑟 ∈ ℝ+) ∧ (𝑘 ∈ ℤ ∧ (𝑓 ↾ (ℤ𝑘)):(ℤ𝑘)⟶((𝑓𝑘)(ball‘𝐷)(𝑟 / 𝑅)))) → (𝑓 ↾ (ℤ𝑘)):(ℤ𝑘)⟶((𝑓𝑘)(ball‘𝐷)(𝑟 / 𝑅)))
11 elpmi 8414 . . . . . . . . . . . 12 (𝑓 ∈ (𝑋pm ℂ) → (𝑓:dom 𝑓𝑋 ∧ dom 𝑓 ⊆ ℂ))
1211simpld 495 . . . . . . . . . . 11 (𝑓 ∈ (𝑋pm ℂ) → 𝑓:dom 𝑓𝑋)
1312ad3antlr 727 . . . . . . . . . 10 ((((𝜑𝑓 ∈ (𝑋pm ℂ)) ∧ 𝑟 ∈ ℝ+) ∧ (𝑘 ∈ ℤ ∧ (𝑓 ↾ (ℤ𝑘)):(ℤ𝑘)⟶((𝑓𝑘)(ball‘𝐷)(𝑟 / 𝑅)))) → 𝑓:dom 𝑓𝑋)
14 resss 5871 . . . . . . . . . . . 12 (𝑓 ↾ (ℤ𝑘)) ⊆ 𝑓
15 dmss 5764 . . . . . . . . . . . 12 ((𝑓 ↾ (ℤ𝑘)) ⊆ 𝑓 → dom (𝑓 ↾ (ℤ𝑘)) ⊆ dom 𝑓)
1614, 15ax-mp 5 . . . . . . . . . . 11 dom (𝑓 ↾ (ℤ𝑘)) ⊆ dom 𝑓
17 uzid 12246 . . . . . . . . . . . . 13 (𝑘 ∈ ℤ → 𝑘 ∈ (ℤ𝑘))
1817ad2antrl 724 . . . . . . . . . . . 12 ((((𝜑𝑓 ∈ (𝑋pm ℂ)) ∧ 𝑟 ∈ ℝ+) ∧ (𝑘 ∈ ℤ ∧ (𝑓 ↾ (ℤ𝑘)):(ℤ𝑘)⟶((𝑓𝑘)(ball‘𝐷)(𝑟 / 𝑅)))) → 𝑘 ∈ (ℤ𝑘))
19 fdm 6515 . . . . . . . . . . . . 13 ((𝑓 ↾ (ℤ𝑘)):(ℤ𝑘)⟶((𝑓𝑘)(ball‘𝐷)(𝑟 / 𝑅)) → dom (𝑓 ↾ (ℤ𝑘)) = (ℤ𝑘))
2019ad2antll 725 . . . . . . . . . . . 12 ((((𝜑𝑓 ∈ (𝑋pm ℂ)) ∧ 𝑟 ∈ ℝ+) ∧ (𝑘 ∈ ℤ ∧ (𝑓 ↾ (ℤ𝑘)):(ℤ𝑘)⟶((𝑓𝑘)(ball‘𝐷)(𝑟 / 𝑅)))) → dom (𝑓 ↾ (ℤ𝑘)) = (ℤ𝑘))
2118, 20eleqtrrd 2913 . . . . . . . . . . 11 ((((𝜑𝑓 ∈ (𝑋pm ℂ)) ∧ 𝑟 ∈ ℝ+) ∧ (𝑘 ∈ ℤ ∧ (𝑓 ↾ (ℤ𝑘)):(ℤ𝑘)⟶((𝑓𝑘)(ball‘𝐷)(𝑟 / 𝑅)))) → 𝑘 ∈ dom (𝑓 ↾ (ℤ𝑘)))
2216, 21sseldi 3962 . . . . . . . . . 10 ((((𝜑𝑓 ∈ (𝑋pm ℂ)) ∧ 𝑟 ∈ ℝ+) ∧ (𝑘 ∈ ℤ ∧ (𝑓 ↾ (ℤ𝑘)):(ℤ𝑘)⟶((𝑓𝑘)(ball‘𝐷)(𝑟 / 𝑅)))) → 𝑘 ∈ dom 𝑓)
2313, 22ffvelrnd 6844 . . . . . . . . 9 ((((𝜑𝑓 ∈ (𝑋pm ℂ)) ∧ 𝑟 ∈ ℝ+) ∧ (𝑘 ∈ ℤ ∧ (𝑓 ↾ (ℤ𝑘)):(ℤ𝑘)⟶((𝑓𝑘)(ball‘𝐷)(𝑟 / 𝑅)))) → (𝑓𝑘) ∈ 𝑋)
24 eqid 2818 . . . . . . . . . . . . 13 (MetOpen‘𝐶) = (MetOpen‘𝐶)
25 eqid 2818 . . . . . . . . . . . . 13 (MetOpen‘𝐷) = (MetOpen‘𝐷)
26 equivcau.1 . . . . . . . . . . . . 13 (𝜑𝐶 ∈ (Met‘𝑋))
27 equivcau.2 . . . . . . . . . . . . 13 (𝜑𝐷 ∈ (Met‘𝑋))
28 equivcau.4 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑥𝑋𝑦𝑋)) → (𝑥𝐶𝑦) ≤ (𝑅 · (𝑥𝐷𝑦)))
2924, 25, 26, 27, 2, 28metss2lem 23048 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑥𝑋𝑟 ∈ ℝ+)) → (𝑥(ball‘𝐷)(𝑟 / 𝑅)) ⊆ (𝑥(ball‘𝐶)𝑟))
3029expr 457 . . . . . . . . . . 11 ((𝜑𝑥𝑋) → (𝑟 ∈ ℝ+ → (𝑥(ball‘𝐷)(𝑟 / 𝑅)) ⊆ (𝑥(ball‘𝐶)𝑟)))
3130ralrimiva 3179 . . . . . . . . . 10 (𝜑 → ∀𝑥𝑋 (𝑟 ∈ ℝ+ → (𝑥(ball‘𝐷)(𝑟 / 𝑅)) ⊆ (𝑥(ball‘𝐶)𝑟)))
3231ad3antrrr 726 . . . . . . . . 9 ((((𝜑𝑓 ∈ (𝑋pm ℂ)) ∧ 𝑟 ∈ ℝ+) ∧ (𝑘 ∈ ℤ ∧ (𝑓 ↾ (ℤ𝑘)):(ℤ𝑘)⟶((𝑓𝑘)(ball‘𝐷)(𝑟 / 𝑅)))) → ∀𝑥𝑋 (𝑟 ∈ ℝ+ → (𝑥(ball‘𝐷)(𝑟 / 𝑅)) ⊆ (𝑥(ball‘𝐶)𝑟)))
33 simplr 765 . . . . . . . . 9 ((((𝜑𝑓 ∈ (𝑋pm ℂ)) ∧ 𝑟 ∈ ℝ+) ∧ (𝑘 ∈ ℤ ∧ (𝑓 ↾ (ℤ𝑘)):(ℤ𝑘)⟶((𝑓𝑘)(ball‘𝐷)(𝑟 / 𝑅)))) → 𝑟 ∈ ℝ+)
34 oveq1 7152 . . . . . . . . . . . 12 (𝑥 = (𝑓𝑘) → (𝑥(ball‘𝐷)(𝑟 / 𝑅)) = ((𝑓𝑘)(ball‘𝐷)(𝑟 / 𝑅)))
35 oveq1 7152 . . . . . . . . . . . 12 (𝑥 = (𝑓𝑘) → (𝑥(ball‘𝐶)𝑟) = ((𝑓𝑘)(ball‘𝐶)𝑟))
3634, 35sseq12d 3997 . . . . . . . . . . 11 (𝑥 = (𝑓𝑘) → ((𝑥(ball‘𝐷)(𝑟 / 𝑅)) ⊆ (𝑥(ball‘𝐶)𝑟) ↔ ((𝑓𝑘)(ball‘𝐷)(𝑟 / 𝑅)) ⊆ ((𝑓𝑘)(ball‘𝐶)𝑟)))
3736imbi2d 342 . . . . . . . . . 10 (𝑥 = (𝑓𝑘) → ((𝑟 ∈ ℝ+ → (𝑥(ball‘𝐷)(𝑟 / 𝑅)) ⊆ (𝑥(ball‘𝐶)𝑟)) ↔ (𝑟 ∈ ℝ+ → ((𝑓𝑘)(ball‘𝐷)(𝑟 / 𝑅)) ⊆ ((𝑓𝑘)(ball‘𝐶)𝑟))))
3837rspcv 3615 . . . . . . . . 9 ((𝑓𝑘) ∈ 𝑋 → (∀𝑥𝑋 (𝑟 ∈ ℝ+ → (𝑥(ball‘𝐷)(𝑟 / 𝑅)) ⊆ (𝑥(ball‘𝐶)𝑟)) → (𝑟 ∈ ℝ+ → ((𝑓𝑘)(ball‘𝐷)(𝑟 / 𝑅)) ⊆ ((𝑓𝑘)(ball‘𝐶)𝑟))))
3923, 32, 33, 38syl3c 66 . . . . . . . 8 ((((𝜑𝑓 ∈ (𝑋pm ℂ)) ∧ 𝑟 ∈ ℝ+) ∧ (𝑘 ∈ ℤ ∧ (𝑓 ↾ (ℤ𝑘)):(ℤ𝑘)⟶((𝑓𝑘)(ball‘𝐷)(𝑟 / 𝑅)))) → ((𝑓𝑘)(ball‘𝐷)(𝑟 / 𝑅)) ⊆ ((𝑓𝑘)(ball‘𝐶)𝑟))
4010, 39fssd 6521 . . . . . . 7 ((((𝜑𝑓 ∈ (𝑋pm ℂ)) ∧ 𝑟 ∈ ℝ+) ∧ (𝑘 ∈ ℤ ∧ (𝑓 ↾ (ℤ𝑘)):(ℤ𝑘)⟶((𝑓𝑘)(ball‘𝐷)(𝑟 / 𝑅)))) → (𝑓 ↾ (ℤ𝑘)):(ℤ𝑘)⟶((𝑓𝑘)(ball‘𝐶)𝑟))
4140expr 457 . . . . . 6 ((((𝜑𝑓 ∈ (𝑋pm ℂ)) ∧ 𝑟 ∈ ℝ+) ∧ 𝑘 ∈ ℤ) → ((𝑓 ↾ (ℤ𝑘)):(ℤ𝑘)⟶((𝑓𝑘)(ball‘𝐷)(𝑟 / 𝑅)) → (𝑓 ↾ (ℤ𝑘)):(ℤ𝑘)⟶((𝑓𝑘)(ball‘𝐶)𝑟)))
4241reximdva 3271 . . . . 5 (((𝜑𝑓 ∈ (𝑋pm ℂ)) ∧ 𝑟 ∈ ℝ+) → (∃𝑘 ∈ ℤ (𝑓 ↾ (ℤ𝑘)):(ℤ𝑘)⟶((𝑓𝑘)(ball‘𝐷)(𝑟 / 𝑅)) → ∃𝑘 ∈ ℤ (𝑓 ↾ (ℤ𝑘)):(ℤ𝑘)⟶((𝑓𝑘)(ball‘𝐶)𝑟)))
439, 42syld 47 . . . 4 (((𝜑𝑓 ∈ (𝑋pm ℂ)) ∧ 𝑟 ∈ ℝ+) → (∀𝑠 ∈ ℝ+𝑘 ∈ ℤ (𝑓 ↾ (ℤ𝑘)):(ℤ𝑘)⟶((𝑓𝑘)(ball‘𝐷)𝑠) → ∃𝑘 ∈ ℤ (𝑓 ↾ (ℤ𝑘)):(ℤ𝑘)⟶((𝑓𝑘)(ball‘𝐶)𝑟)))
4443ralrimdva 3186 . . 3 ((𝜑𝑓 ∈ (𝑋pm ℂ)) → (∀𝑠 ∈ ℝ+𝑘 ∈ ℤ (𝑓 ↾ (ℤ𝑘)):(ℤ𝑘)⟶((𝑓𝑘)(ball‘𝐷)𝑠) → ∀𝑟 ∈ ℝ+𝑘 ∈ ℤ (𝑓 ↾ (ℤ𝑘)):(ℤ𝑘)⟶((𝑓𝑘)(ball‘𝐶)𝑟)))
4544ss2rabdv 4049 . 2 (𝜑 → {𝑓 ∈ (𝑋pm ℂ) ∣ ∀𝑠 ∈ ℝ+𝑘 ∈ ℤ (𝑓 ↾ (ℤ𝑘)):(ℤ𝑘)⟶((𝑓𝑘)(ball‘𝐷)𝑠)} ⊆ {𝑓 ∈ (𝑋pm ℂ) ∣ ∀𝑟 ∈ ℝ+𝑘 ∈ ℤ (𝑓 ↾ (ℤ𝑘)):(ℤ𝑘)⟶((𝑓𝑘)(ball‘𝐶)𝑟)})
46 metxmet 22871 . . 3 (𝐷 ∈ (Met‘𝑋) → 𝐷 ∈ (∞Met‘𝑋))
47 caufval 23805 . . 3 (𝐷 ∈ (∞Met‘𝑋) → (Cau‘𝐷) = {𝑓 ∈ (𝑋pm ℂ) ∣ ∀𝑠 ∈ ℝ+𝑘 ∈ ℤ (𝑓 ↾ (ℤ𝑘)):(ℤ𝑘)⟶((𝑓𝑘)(ball‘𝐷)𝑠)})
4827, 46, 473syl 18 . 2 (𝜑 → (Cau‘𝐷) = {𝑓 ∈ (𝑋pm ℂ) ∣ ∀𝑠 ∈ ℝ+𝑘 ∈ ℤ (𝑓 ↾ (ℤ𝑘)):(ℤ𝑘)⟶((𝑓𝑘)(ball‘𝐷)𝑠)})
49 metxmet 22871 . . 3 (𝐶 ∈ (Met‘𝑋) → 𝐶 ∈ (∞Met‘𝑋))
50 caufval 23805 . . 3 (𝐶 ∈ (∞Met‘𝑋) → (Cau‘𝐶) = {𝑓 ∈ (𝑋pm ℂ) ∣ ∀𝑟 ∈ ℝ+𝑘 ∈ ℤ (𝑓 ↾ (ℤ𝑘)):(ℤ𝑘)⟶((𝑓𝑘)(ball‘𝐶)𝑟)})
5126, 49, 503syl 18 . 2 (𝜑 → (Cau‘𝐶) = {𝑓 ∈ (𝑋pm ℂ) ∣ ∀𝑟 ∈ ℝ+𝑘 ∈ ℤ (𝑓 ↾ (ℤ𝑘)):(ℤ𝑘)⟶((𝑓𝑘)(ball‘𝐶)𝑟)})
5245, 48, 513sstr4d 4011 1 (𝜑 → (Cau‘𝐷) ⊆ (Cau‘𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1528  wcel 2105  wral 3135  wrex 3136  {crab 3139  wss 3933   class class class wbr 5057  dom cdm 5548  cres 5550  wf 6344  cfv 6348  (class class class)co 7145  pm cpm 8396  cc 10523   · cmul 10530  cle 10664   / cdiv 11285  cz 11969  cuz 12231  +crp 12377  ∞Metcxmet 20458  Metcmet 20459  ballcbl 20460  MetOpencmopn 20463  Cauccau 23783
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
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-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-po 5467  df-so 5468  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-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-1st 7678  df-2nd 7679  df-er 8278  df-map 8397  df-pm 8398  df-en 8498  df-dom 8499  df-sdom 8500  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-z 11970  df-uz 12232  df-rp 12378  df-xadd 12496  df-psmet 20465  df-xmet 20466  df-met 20467  df-bl 20468  df-cau 23786
This theorem is referenced by: (None)
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