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Theorem equivcau 23006
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 477 . . . . . . 7 (((𝜑𝑓 ∈ (𝑋pm ℂ)) ∧ 𝑟 ∈ ℝ+) → 𝑟 ∈ ℝ+)
2 equivcau.3 . . . . . . . 8 (𝜑𝑅 ∈ ℝ+)
32ad2antrr 761 . . . . . . 7 (((𝜑𝑓 ∈ (𝑋pm ℂ)) ∧ 𝑟 ∈ ℝ+) → 𝑅 ∈ ℝ+)
41, 3rpdivcld 11833 . . . . . 6 (((𝜑𝑓 ∈ (𝑋pm ℂ)) ∧ 𝑟 ∈ ℝ+) → (𝑟 / 𝑅) ∈ ℝ+)
5 oveq2 6612 . . . . . . . . 9 (𝑠 = (𝑟 / 𝑅) → ((𝑓𝑘)(ball‘𝐷)𝑠) = ((𝑓𝑘)(ball‘𝐷)(𝑟 / 𝑅)))
65feq3d 5989 . . . . . . . 8 (𝑠 = (𝑟 / 𝑅) → ((𝑓 ↾ (ℤ𝑘)):(ℤ𝑘)⟶((𝑓𝑘)(ball‘𝐷)𝑠) ↔ (𝑓 ↾ (ℤ𝑘)):(ℤ𝑘)⟶((𝑓𝑘)(ball‘𝐷)(𝑟 / 𝑅))))
76rexbidv 3045 . . . . . . 7 (𝑠 = (𝑟 / 𝑅) → (∃𝑘 ∈ ℤ (𝑓 ↾ (ℤ𝑘)):(ℤ𝑘)⟶((𝑓𝑘)(ball‘𝐷)𝑠) ↔ ∃𝑘 ∈ ℤ (𝑓 ↾ (ℤ𝑘)):(ℤ𝑘)⟶((𝑓𝑘)(ball‘𝐷)(𝑟 / 𝑅))))
87rspcv 3291 . . . . . 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 795 . . . . . . . 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 7820 . . . . . . . . . . . 12 (𝑓 ∈ (𝑋pm ℂ) → (𝑓:dom 𝑓𝑋 ∧ dom 𝑓 ⊆ ℂ))
1211simpld 475 . . . . . . . . . . 11 (𝑓 ∈ (𝑋pm ℂ) → 𝑓:dom 𝑓𝑋)
1312ad3antlr 766 . . . . . . . . . 10 ((((𝜑𝑓 ∈ (𝑋pm ℂ)) ∧ 𝑟 ∈ ℝ+) ∧ (𝑘 ∈ ℤ ∧ (𝑓 ↾ (ℤ𝑘)):(ℤ𝑘)⟶((𝑓𝑘)(ball‘𝐷)(𝑟 / 𝑅)))) → 𝑓:dom 𝑓𝑋)
14 resss 5381 . . . . . . . . . . . 12 (𝑓 ↾ (ℤ𝑘)) ⊆ 𝑓
15 dmss 5283 . . . . . . . . . . . 12 ((𝑓 ↾ (ℤ𝑘)) ⊆ 𝑓 → dom (𝑓 ↾ (ℤ𝑘)) ⊆ dom 𝑓)
1614, 15ax-mp 5 . . . . . . . . . . 11 dom (𝑓 ↾ (ℤ𝑘)) ⊆ dom 𝑓
17 uzid 11646 . . . . . . . . . . . . 13 (𝑘 ∈ ℤ → 𝑘 ∈ (ℤ𝑘))
1817ad2antrl 763 . . . . . . . . . . . 12 ((((𝜑𝑓 ∈ (𝑋pm ℂ)) ∧ 𝑟 ∈ ℝ+) ∧ (𝑘 ∈ ℤ ∧ (𝑓 ↾ (ℤ𝑘)):(ℤ𝑘)⟶((𝑓𝑘)(ball‘𝐷)(𝑟 / 𝑅)))) → 𝑘 ∈ (ℤ𝑘))
19 fdm 6008 . . . . . . . . . . . . 13 ((𝑓 ↾ (ℤ𝑘)):(ℤ𝑘)⟶((𝑓𝑘)(ball‘𝐷)(𝑟 / 𝑅)) → dom (𝑓 ↾ (ℤ𝑘)) = (ℤ𝑘))
2019ad2antll 764 . . . . . . . . . . . 12 ((((𝜑𝑓 ∈ (𝑋pm ℂ)) ∧ 𝑟 ∈ ℝ+) ∧ (𝑘 ∈ ℤ ∧ (𝑓 ↾ (ℤ𝑘)):(ℤ𝑘)⟶((𝑓𝑘)(ball‘𝐷)(𝑟 / 𝑅)))) → dom (𝑓 ↾ (ℤ𝑘)) = (ℤ𝑘))
2118, 20eleqtrrd 2701 . . . . . . . . . . 11 ((((𝜑𝑓 ∈ (𝑋pm ℂ)) ∧ 𝑟 ∈ ℝ+) ∧ (𝑘 ∈ ℤ ∧ (𝑓 ↾ (ℤ𝑘)):(ℤ𝑘)⟶((𝑓𝑘)(ball‘𝐷)(𝑟 / 𝑅)))) → 𝑘 ∈ dom (𝑓 ↾ (ℤ𝑘)))
2216, 21sseldi 3581 . . . . . . . . . 10 ((((𝜑𝑓 ∈ (𝑋pm ℂ)) ∧ 𝑟 ∈ ℝ+) ∧ (𝑘 ∈ ℤ ∧ (𝑓 ↾ (ℤ𝑘)):(ℤ𝑘)⟶((𝑓𝑘)(ball‘𝐷)(𝑟 / 𝑅)))) → 𝑘 ∈ dom 𝑓)
2313, 22ffvelrnd 6316 . . . . . . . . 9 ((((𝜑𝑓 ∈ (𝑋pm ℂ)) ∧ 𝑟 ∈ ℝ+) ∧ (𝑘 ∈ ℤ ∧ (𝑓 ↾ (ℤ𝑘)):(ℤ𝑘)⟶((𝑓𝑘)(ball‘𝐷)(𝑟 / 𝑅)))) → (𝑓𝑘) ∈ 𝑋)
24 eqid 2621 . . . . . . . . . . . . 13 (MetOpen‘𝐶) = (MetOpen‘𝐶)
25 eqid 2621 . . . . . . . . . . . . 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 22226 . . . . . . . . . . . 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 2960 . . . . . . . . . 10 (𝜑 → ∀𝑥𝑋 (𝑟 ∈ ℝ+ → (𝑥(ball‘𝐷)(𝑟 / 𝑅)) ⊆ (𝑥(ball‘𝐶)𝑟)))
3231ad3antrrr 765 . . . . . . . . 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 791 . . . . . . . . 9 ((((𝜑𝑓 ∈ (𝑋pm ℂ)) ∧ 𝑟 ∈ ℝ+) ∧ (𝑘 ∈ ℤ ∧ (𝑓 ↾ (ℤ𝑘)):(ℤ𝑘)⟶((𝑓𝑘)(ball‘𝐷)(𝑟 / 𝑅)))) → 𝑟 ∈ ℝ+)
34 oveq1 6611 . . . . . . . . . . . 12 (𝑥 = (𝑓𝑘) → (𝑥(ball‘𝐷)(𝑟 / 𝑅)) = ((𝑓𝑘)(ball‘𝐷)(𝑟 / 𝑅)))
35 oveq1 6611 . . . . . . . . . . . 12 (𝑥 = (𝑓𝑘) → (𝑥(ball‘𝐶)𝑟) = ((𝑓𝑘)(ball‘𝐶)𝑟))
3634, 35sseq12d 3613 . . . . . . . . . . 11 (𝑥 = (𝑓𝑘) → ((𝑥(ball‘𝐷)(𝑟 / 𝑅)) ⊆ (𝑥(ball‘𝐶)𝑟) ↔ ((𝑓𝑘)(ball‘𝐷)(𝑟 / 𝑅)) ⊆ ((𝑓𝑘)(ball‘𝐶)𝑟)))
3736imbi2d 330 . . . . . . . . . 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 3291 . . . . . . . . 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 6014 . . . . . . 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 3011 . . . . 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 2963 . . 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 3662 . 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 22049 . . 3 (𝐷 ∈ (Met‘𝑋) → 𝐷 ∈ (∞Met‘𝑋))
47 caufval 22981 . . 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 22049 . . 3 (𝐶 ∈ (Met‘𝑋) → 𝐶 ∈ (∞Met‘𝑋))
50 caufval 22981 . . 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 3627 1 (𝜑 → (Cau‘𝐷) ⊆ (Cau‘𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1480  wcel 1987  wral 2907  wrex 2908  {crab 2911  wss 3555   class class class wbr 4613  dom cdm 5074  cres 5076  wf 5843  cfv 5847  (class class class)co 6604  pm cpm 7803  cc 9878   · cmul 9885  cle 10019   / cdiv 10628  cz 11321  cuz 11631  +crp 11776  ∞Metcxmt 19650  Metcme 19651  ballcbl 19652  MetOpencmopn 19655  Caucca 22959
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902  ax-cnex 9936  ax-resscn 9937  ax-1cn 9938  ax-icn 9939  ax-addcl 9940  ax-addrcl 9941  ax-mulcl 9942  ax-mulrcl 9943  ax-mulcom 9944  ax-addass 9945  ax-mulass 9946  ax-distr 9947  ax-i2m1 9948  ax-1ne0 9949  ax-1rid 9950  ax-rnegex 9951  ax-rrecex 9952  ax-cnre 9953  ax-pre-lttri 9954  ax-pre-lttrn 9955  ax-pre-ltadd 9956  ax-pre-mulgt0 9957
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2912  df-rex 2913  df-reu 2914  df-rmo 2915  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-id 4989  df-po 4995  df-so 4996  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-riota 6565  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-1st 7113  df-2nd 7114  df-er 7687  df-map 7804  df-pm 7805  df-en 7900  df-dom 7901  df-sdom 7902  df-pnf 10020  df-mnf 10021  df-xr 10022  df-ltxr 10023  df-le 10024  df-sub 10212  df-neg 10213  df-div 10629  df-z 11322  df-uz 11632  df-rp 11777  df-xadd 11891  df-psmet 19657  df-xmet 19658  df-met 19659  df-bl 19660  df-cau 22962
This theorem is referenced by: (None)
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