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Theorem equivcau 23144
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 476 . . . . . . 7 (((𝜑𝑓 ∈ (𝑋pm ℂ)) ∧ 𝑟 ∈ ℝ+) → 𝑟 ∈ ℝ+)
2 equivcau.3 . . . . . . . 8 (𝜑𝑅 ∈ ℝ+)
32ad2antrr 762 . . . . . . 7 (((𝜑𝑓 ∈ (𝑋pm ℂ)) ∧ 𝑟 ∈ ℝ+) → 𝑅 ∈ ℝ+)
41, 3rpdivcld 11927 . . . . . 6 (((𝜑𝑓 ∈ (𝑋pm ℂ)) ∧ 𝑟 ∈ ℝ+) → (𝑟 / 𝑅) ∈ ℝ+)
5 oveq2 6698 . . . . . . . . 9 (𝑠 = (𝑟 / 𝑅) → ((𝑓𝑘)(ball‘𝐷)𝑠) = ((𝑓𝑘)(ball‘𝐷)(𝑟 / 𝑅)))
65feq3d 6070 . . . . . . . 8 (𝑠 = (𝑟 / 𝑅) → ((𝑓 ↾ (ℤ𝑘)):(ℤ𝑘)⟶((𝑓𝑘)(ball‘𝐷)𝑠) ↔ (𝑓 ↾ (ℤ𝑘)):(ℤ𝑘)⟶((𝑓𝑘)(ball‘𝐷)(𝑟 / 𝑅))))
76rexbidv 3081 . . . . . . 7 (𝑠 = (𝑟 / 𝑅) → (∃𝑘 ∈ ℤ (𝑓 ↾ (ℤ𝑘)):(ℤ𝑘)⟶((𝑓𝑘)(ball‘𝐷)𝑠) ↔ ∃𝑘 ∈ ℤ (𝑓 ↾ (ℤ𝑘)):(ℤ𝑘)⟶((𝑓𝑘)(ball‘𝐷)(𝑟 / 𝑅))))
87rspcv 3336 . . . . . 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 811 . . . . . . . 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 7918 . . . . . . . . . . . 12 (𝑓 ∈ (𝑋pm ℂ) → (𝑓:dom 𝑓𝑋 ∧ dom 𝑓 ⊆ ℂ))
1211simpld 474 . . . . . . . . . . 11 (𝑓 ∈ (𝑋pm ℂ) → 𝑓:dom 𝑓𝑋)
1312ad3antlr 767 . . . . . . . . . 10 ((((𝜑𝑓 ∈ (𝑋pm ℂ)) ∧ 𝑟 ∈ ℝ+) ∧ (𝑘 ∈ ℤ ∧ (𝑓 ↾ (ℤ𝑘)):(ℤ𝑘)⟶((𝑓𝑘)(ball‘𝐷)(𝑟 / 𝑅)))) → 𝑓:dom 𝑓𝑋)
14 resss 5457 . . . . . . . . . . . 12 (𝑓 ↾ (ℤ𝑘)) ⊆ 𝑓
15 dmss 5355 . . . . . . . . . . . 12 ((𝑓 ↾ (ℤ𝑘)) ⊆ 𝑓 → dom (𝑓 ↾ (ℤ𝑘)) ⊆ dom 𝑓)
1614, 15ax-mp 5 . . . . . . . . . . 11 dom (𝑓 ↾ (ℤ𝑘)) ⊆ dom 𝑓
17 uzid 11740 . . . . . . . . . . . . 13 (𝑘 ∈ ℤ → 𝑘 ∈ (ℤ𝑘))
1817ad2antrl 764 . . . . . . . . . . . 12 ((((𝜑𝑓 ∈ (𝑋pm ℂ)) ∧ 𝑟 ∈ ℝ+) ∧ (𝑘 ∈ ℤ ∧ (𝑓 ↾ (ℤ𝑘)):(ℤ𝑘)⟶((𝑓𝑘)(ball‘𝐷)(𝑟 / 𝑅)))) → 𝑘 ∈ (ℤ𝑘))
19 fdm 6089 . . . . . . . . . . . . 13 ((𝑓 ↾ (ℤ𝑘)):(ℤ𝑘)⟶((𝑓𝑘)(ball‘𝐷)(𝑟 / 𝑅)) → dom (𝑓 ↾ (ℤ𝑘)) = (ℤ𝑘))
2019ad2antll 765 . . . . . . . . . . . 12 ((((𝜑𝑓 ∈ (𝑋pm ℂ)) ∧ 𝑟 ∈ ℝ+) ∧ (𝑘 ∈ ℤ ∧ (𝑓 ↾ (ℤ𝑘)):(ℤ𝑘)⟶((𝑓𝑘)(ball‘𝐷)(𝑟 / 𝑅)))) → dom (𝑓 ↾ (ℤ𝑘)) = (ℤ𝑘))
2118, 20eleqtrrd 2733 . . . . . . . . . . 11 ((((𝜑𝑓 ∈ (𝑋pm ℂ)) ∧ 𝑟 ∈ ℝ+) ∧ (𝑘 ∈ ℤ ∧ (𝑓 ↾ (ℤ𝑘)):(ℤ𝑘)⟶((𝑓𝑘)(ball‘𝐷)(𝑟 / 𝑅)))) → 𝑘 ∈ dom (𝑓 ↾ (ℤ𝑘)))
2216, 21sseldi 3634 . . . . . . . . . 10 ((((𝜑𝑓 ∈ (𝑋pm ℂ)) ∧ 𝑟 ∈ ℝ+) ∧ (𝑘 ∈ ℤ ∧ (𝑓 ↾ (ℤ𝑘)):(ℤ𝑘)⟶((𝑓𝑘)(ball‘𝐷)(𝑟 / 𝑅)))) → 𝑘 ∈ dom 𝑓)
2313, 22ffvelrnd 6400 . . . . . . . . 9 ((((𝜑𝑓 ∈ (𝑋pm ℂ)) ∧ 𝑟 ∈ ℝ+) ∧ (𝑘 ∈ ℤ ∧ (𝑓 ↾ (ℤ𝑘)):(ℤ𝑘)⟶((𝑓𝑘)(ball‘𝐷)(𝑟 / 𝑅)))) → (𝑓𝑘) ∈ 𝑋)
24 eqid 2651 . . . . . . . . . . . . 13 (MetOpen‘𝐶) = (MetOpen‘𝐶)
25 eqid 2651 . . . . . . . . . . . . 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 22363 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑥𝑋𝑟 ∈ ℝ+)) → (𝑥(ball‘𝐷)(𝑟 / 𝑅)) ⊆ (𝑥(ball‘𝐶)𝑟))
3029expr 642 . . . . . . . . . . 11 ((𝜑𝑥𝑋) → (𝑟 ∈ ℝ+ → (𝑥(ball‘𝐷)(𝑟 / 𝑅)) ⊆ (𝑥(ball‘𝐶)𝑟)))
3130ralrimiva 2995 . . . . . . . . . 10 (𝜑 → ∀𝑥𝑋 (𝑟 ∈ ℝ+ → (𝑥(ball‘𝐷)(𝑟 / 𝑅)) ⊆ (𝑥(ball‘𝐶)𝑟)))
3231ad3antrrr 766 . . . . . . . . 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 807 . . . . . . . . 9 ((((𝜑𝑓 ∈ (𝑋pm ℂ)) ∧ 𝑟 ∈ ℝ+) ∧ (𝑘 ∈ ℤ ∧ (𝑓 ↾ (ℤ𝑘)):(ℤ𝑘)⟶((𝑓𝑘)(ball‘𝐷)(𝑟 / 𝑅)))) → 𝑟 ∈ ℝ+)
34 oveq1 6697 . . . . . . . . . . . 12 (𝑥 = (𝑓𝑘) → (𝑥(ball‘𝐷)(𝑟 / 𝑅)) = ((𝑓𝑘)(ball‘𝐷)(𝑟 / 𝑅)))
35 oveq1 6697 . . . . . . . . . . . 12 (𝑥 = (𝑓𝑘) → (𝑥(ball‘𝐶)𝑟) = ((𝑓𝑘)(ball‘𝐶)𝑟))
3634, 35sseq12d 3667 . . . . . . . . . . 11 (𝑥 = (𝑓𝑘) → ((𝑥(ball‘𝐷)(𝑟 / 𝑅)) ⊆ (𝑥(ball‘𝐶)𝑟) ↔ ((𝑓𝑘)(ball‘𝐷)(𝑟 / 𝑅)) ⊆ ((𝑓𝑘)(ball‘𝐶)𝑟)))
3736imbi2d 329 . . . . . . . . . 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 3336 . . . . . . . . 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 6095 . . . . . . 7 ((((𝜑𝑓 ∈ (𝑋pm ℂ)) ∧ 𝑟 ∈ ℝ+) ∧ (𝑘 ∈ ℤ ∧ (𝑓 ↾ (ℤ𝑘)):(ℤ𝑘)⟶((𝑓𝑘)(ball‘𝐷)(𝑟 / 𝑅)))) → (𝑓 ↾ (ℤ𝑘)):(ℤ𝑘)⟶((𝑓𝑘)(ball‘𝐶)𝑟))
4140expr 642 . . . . . 6 ((((𝜑𝑓 ∈ (𝑋pm ℂ)) ∧ 𝑟 ∈ ℝ+) ∧ 𝑘 ∈ ℤ) → ((𝑓 ↾ (ℤ𝑘)):(ℤ𝑘)⟶((𝑓𝑘)(ball‘𝐷)(𝑟 / 𝑅)) → (𝑓 ↾ (ℤ𝑘)):(ℤ𝑘)⟶((𝑓𝑘)(ball‘𝐶)𝑟)))
4241reximdva 3046 . . . . 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 2998 . . 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 3716 . 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 22186 . . 3 (𝐷 ∈ (Met‘𝑋) → 𝐷 ∈ (∞Met‘𝑋))
47 caufval 23119 . . 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 22186 . . 3 (𝐶 ∈ (Met‘𝑋) → 𝐶 ∈ (∞Met‘𝑋))
50 caufval 23119 . . 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 3681 1 (𝜑 → (Cau‘𝐷) ⊆ (Cau‘𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1523  wcel 2030  wral 2941  wrex 2942  {crab 2945  wss 3607   class class class wbr 4685  dom cdm 5143  cres 5145  wf 5922  cfv 5926  (class class class)co 6690  pm cpm 7900  cc 9972   · cmul 9979  cle 10113   / cdiv 10722  cz 11415  cuz 11725  +crp 11870  ∞Metcxmt 19779  Metcme 19780  ballcbl 19781  MetOpencmopn 19784  Caucca 23097
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-cnex 10030  ax-resscn 10031  ax-1cn 10032  ax-icn 10033  ax-addcl 10034  ax-addrcl 10035  ax-mulcl 10036  ax-mulrcl 10037  ax-mulcom 10038  ax-addass 10039  ax-mulass 10040  ax-distr 10041  ax-i2m1 10042  ax-1ne0 10043  ax-1rid 10044  ax-rnegex 10045  ax-rrecex 10046  ax-cnre 10047  ax-pre-lttri 10048  ax-pre-lttrn 10049  ax-pre-ltadd 10050  ax-pre-mulgt0 10051
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-nel 2927  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-po 5064  df-so 5065  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-1st 7210  df-2nd 7211  df-er 7787  df-map 7901  df-pm 7902  df-en 7998  df-dom 7999  df-sdom 8000  df-pnf 10114  df-mnf 10115  df-xr 10116  df-ltxr 10117  df-le 10118  df-sub 10306  df-neg 10307  df-div 10723  df-z 11416  df-uz 11726  df-rp 11871  df-xadd 11985  df-psmet 19786  df-xmet 19787  df-met 19788  df-bl 19789  df-cau 23100
This theorem is referenced by: (None)
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