Users' Mathboxes Mathbox for Jeff Madsen < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  equivtotbnd Structured version   Visualization version   GIF version

Theorem equivtotbnd 32747
Description: If the metric 𝑀 is "strongly finer" than 𝑁 (meaning that there is a positive real constant 𝑅 such that 𝑁(𝑥, 𝑦) ≤ 𝑅 · 𝑀(𝑥, 𝑦)), then total boundedness of 𝑀 implies total boundedness of 𝑁. (Using this theorem twice in each direction states that if two metrics are strongly equivalent, then one is totally bounded iff the other is.) (Contributed by Mario Carneiro, 14-Sep-2015.)
Hypotheses
Ref Expression
equivtotbnd.1 (𝜑𝑀 ∈ (TotBnd‘𝑋))
equivtotbnd.2 (𝜑𝑁 ∈ (Met‘𝑋))
equivtotbnd.3 (𝜑𝑅 ∈ ℝ+)
equivtotbnd.4 ((𝜑 ∧ (𝑥𝑋𝑦𝑋)) → (𝑥𝑁𝑦) ≤ (𝑅 · (𝑥𝑀𝑦)))
Assertion
Ref Expression
equivtotbnd (𝜑𝑁 ∈ (TotBnd‘𝑋))
Distinct variable groups:   𝑥,𝑦,𝑀   𝑥,𝑁,𝑦   𝜑,𝑥,𝑦   𝑥,𝑋,𝑦   𝑥,𝑅,𝑦

Proof of Theorem equivtotbnd
Dummy variables 𝑣 𝑠 𝑟 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 equivtotbnd.2 . 2 (𝜑𝑁 ∈ (Met‘𝑋))
2 simpr 476 . . . . . 6 ((𝜑𝑟 ∈ ℝ+) → 𝑟 ∈ ℝ+)
3 equivtotbnd.3 . . . . . . 7 (𝜑𝑅 ∈ ℝ+)
43adantr 480 . . . . . 6 ((𝜑𝑟 ∈ ℝ+) → 𝑅 ∈ ℝ+)
52, 4rpdivcld 11765 . . . . 5 ((𝜑𝑟 ∈ ℝ+) → (𝑟 / 𝑅) ∈ ℝ+)
6 equivtotbnd.1 . . . . . . 7 (𝜑𝑀 ∈ (TotBnd‘𝑋))
76adantr 480 . . . . . 6 ((𝜑𝑟 ∈ ℝ+) → 𝑀 ∈ (TotBnd‘𝑋))
8 istotbnd3 32740 . . . . . . 7 (𝑀 ∈ (TotBnd‘𝑋) ↔ (𝑀 ∈ (Met‘𝑋) ∧ ∀𝑠 ∈ ℝ+𝑣 ∈ (𝒫 𝑋 ∩ Fin) 𝑥𝑣 (𝑥(ball‘𝑀)𝑠) = 𝑋))
98simprbi 479 . . . . . 6 (𝑀 ∈ (TotBnd‘𝑋) → ∀𝑠 ∈ ℝ+𝑣 ∈ (𝒫 𝑋 ∩ Fin) 𝑥𝑣 (𝑥(ball‘𝑀)𝑠) = 𝑋)
107, 9syl 17 . . . . 5 ((𝜑𝑟 ∈ ℝ+) → ∀𝑠 ∈ ℝ+𝑣 ∈ (𝒫 𝑋 ∩ Fin) 𝑥𝑣 (𝑥(ball‘𝑀)𝑠) = 𝑋)
11 oveq2 6557 . . . . . . . . 9 (𝑠 = (𝑟 / 𝑅) → (𝑥(ball‘𝑀)𝑠) = (𝑥(ball‘𝑀)(𝑟 / 𝑅)))
1211iuneq2d 4483 . . . . . . . 8 (𝑠 = (𝑟 / 𝑅) → 𝑥𝑣 (𝑥(ball‘𝑀)𝑠) = 𝑥𝑣 (𝑥(ball‘𝑀)(𝑟 / 𝑅)))
1312eqeq1d 2612 . . . . . . 7 (𝑠 = (𝑟 / 𝑅) → ( 𝑥𝑣 (𝑥(ball‘𝑀)𝑠) = 𝑋 𝑥𝑣 (𝑥(ball‘𝑀)(𝑟 / 𝑅)) = 𝑋))
1413rexbidv 3034 . . . . . 6 (𝑠 = (𝑟 / 𝑅) → (∃𝑣 ∈ (𝒫 𝑋 ∩ Fin) 𝑥𝑣 (𝑥(ball‘𝑀)𝑠) = 𝑋 ↔ ∃𝑣 ∈ (𝒫 𝑋 ∩ Fin) 𝑥𝑣 (𝑥(ball‘𝑀)(𝑟 / 𝑅)) = 𝑋))
1514rspcv 3278 . . . . 5 ((𝑟 / 𝑅) ∈ ℝ+ → (∀𝑠 ∈ ℝ+𝑣 ∈ (𝒫 𝑋 ∩ Fin) 𝑥𝑣 (𝑥(ball‘𝑀)𝑠) = 𝑋 → ∃𝑣 ∈ (𝒫 𝑋 ∩ Fin) 𝑥𝑣 (𝑥(ball‘𝑀)(𝑟 / 𝑅)) = 𝑋))
165, 10, 15sylc 63 . . . 4 ((𝜑𝑟 ∈ ℝ+) → ∃𝑣 ∈ (𝒫 𝑋 ∩ Fin) 𝑥𝑣 (𝑥(ball‘𝑀)(𝑟 / 𝑅)) = 𝑋)
17 elfpw 8151 . . . . . . . . . . . . . 14 (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ↔ (𝑣𝑋𝑣 ∈ Fin))
1817simplbi 475 . . . . . . . . . . . . 13 (𝑣 ∈ (𝒫 𝑋 ∩ Fin) → 𝑣𝑋)
1918adantl 481 . . . . . . . . . . . 12 (((𝜑𝑟 ∈ ℝ+) ∧ 𝑣 ∈ (𝒫 𝑋 ∩ Fin)) → 𝑣𝑋)
2019sselda 3568 . . . . . . . . . . 11 ((((𝜑𝑟 ∈ ℝ+) ∧ 𝑣 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑥𝑣) → 𝑥𝑋)
21 eqid 2610 . . . . . . . . . . . . . 14 (MetOpen‘𝑁) = (MetOpen‘𝑁)
22 eqid 2610 . . . . . . . . . . . . . 14 (MetOpen‘𝑀) = (MetOpen‘𝑀)
238simplbi 475 . . . . . . . . . . . . . . 15 (𝑀 ∈ (TotBnd‘𝑋) → 𝑀 ∈ (Met‘𝑋))
246, 23syl 17 . . . . . . . . . . . . . 14 (𝜑𝑀 ∈ (Met‘𝑋))
25 equivtotbnd.4 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑥𝑋𝑦𝑋)) → (𝑥𝑁𝑦) ≤ (𝑅 · (𝑥𝑀𝑦)))
2621, 22, 1, 24, 3, 25metss2lem 22126 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑥𝑋𝑟 ∈ ℝ+)) → (𝑥(ball‘𝑀)(𝑟 / 𝑅)) ⊆ (𝑥(ball‘𝑁)𝑟))
2726anass1rs 845 . . . . . . . . . . . 12 (((𝜑𝑟 ∈ ℝ+) ∧ 𝑥𝑋) → (𝑥(ball‘𝑀)(𝑟 / 𝑅)) ⊆ (𝑥(ball‘𝑁)𝑟))
2827adantlr 747 . . . . . . . . . . 11 ((((𝜑𝑟 ∈ ℝ+) ∧ 𝑣 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑥𝑋) → (𝑥(ball‘𝑀)(𝑟 / 𝑅)) ⊆ (𝑥(ball‘𝑁)𝑟))
2920, 28syldan 486 . . . . . . . . . 10 ((((𝜑𝑟 ∈ ℝ+) ∧ 𝑣 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑥𝑣) → (𝑥(ball‘𝑀)(𝑟 / 𝑅)) ⊆ (𝑥(ball‘𝑁)𝑟))
3029ralrimiva 2949 . . . . . . . . 9 (((𝜑𝑟 ∈ ℝ+) ∧ 𝑣 ∈ (𝒫 𝑋 ∩ Fin)) → ∀𝑥𝑣 (𝑥(ball‘𝑀)(𝑟 / 𝑅)) ⊆ (𝑥(ball‘𝑁)𝑟))
31 ss2iun 4472 . . . . . . . . 9 (∀𝑥𝑣 (𝑥(ball‘𝑀)(𝑟 / 𝑅)) ⊆ (𝑥(ball‘𝑁)𝑟) → 𝑥𝑣 (𝑥(ball‘𝑀)(𝑟 / 𝑅)) ⊆ 𝑥𝑣 (𝑥(ball‘𝑁)𝑟))
3230, 31syl 17 . . . . . . . 8 (((𝜑𝑟 ∈ ℝ+) ∧ 𝑣 ∈ (𝒫 𝑋 ∩ Fin)) → 𝑥𝑣 (𝑥(ball‘𝑀)(𝑟 / 𝑅)) ⊆ 𝑥𝑣 (𝑥(ball‘𝑁)𝑟))
33 sseq1 3589 . . . . . . . 8 ( 𝑥𝑣 (𝑥(ball‘𝑀)(𝑟 / 𝑅)) = 𝑋 → ( 𝑥𝑣 (𝑥(ball‘𝑀)(𝑟 / 𝑅)) ⊆ 𝑥𝑣 (𝑥(ball‘𝑁)𝑟) ↔ 𝑋 𝑥𝑣 (𝑥(ball‘𝑁)𝑟)))
3432, 33syl5ibcom 234 . . . . . . 7 (((𝜑𝑟 ∈ ℝ+) ∧ 𝑣 ∈ (𝒫 𝑋 ∩ Fin)) → ( 𝑥𝑣 (𝑥(ball‘𝑀)(𝑟 / 𝑅)) = 𝑋𝑋 𝑥𝑣 (𝑥(ball‘𝑁)𝑟)))
351ad3antrrr 762 . . . . . . . . . . 11 ((((𝜑𝑟 ∈ ℝ+) ∧ 𝑣 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑥𝑣) → 𝑁 ∈ (Met‘𝑋))
36 metxmet 21949 . . . . . . . . . . 11 (𝑁 ∈ (Met‘𝑋) → 𝑁 ∈ (∞Met‘𝑋))
3735, 36syl 17 . . . . . . . . . 10 ((((𝜑𝑟 ∈ ℝ+) ∧ 𝑣 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑥𝑣) → 𝑁 ∈ (∞Met‘𝑋))
38 simpllr 795 . . . . . . . . . . 11 ((((𝜑𝑟 ∈ ℝ+) ∧ 𝑣 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑥𝑣) → 𝑟 ∈ ℝ+)
3938rpxrd 11749 . . . . . . . . . 10 ((((𝜑𝑟 ∈ ℝ+) ∧ 𝑣 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑥𝑣) → 𝑟 ∈ ℝ*)
40 blssm 22033 . . . . . . . . . 10 ((𝑁 ∈ (∞Met‘𝑋) ∧ 𝑥𝑋𝑟 ∈ ℝ*) → (𝑥(ball‘𝑁)𝑟) ⊆ 𝑋)
4137, 20, 39, 40syl3anc 1318 . . . . . . . . 9 ((((𝜑𝑟 ∈ ℝ+) ∧ 𝑣 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑥𝑣) → (𝑥(ball‘𝑁)𝑟) ⊆ 𝑋)
4241ralrimiva 2949 . . . . . . . 8 (((𝜑𝑟 ∈ ℝ+) ∧ 𝑣 ∈ (𝒫 𝑋 ∩ Fin)) → ∀𝑥𝑣 (𝑥(ball‘𝑁)𝑟) ⊆ 𝑋)
43 iunss 4497 . . . . . . . 8 ( 𝑥𝑣 (𝑥(ball‘𝑁)𝑟) ⊆ 𝑋 ↔ ∀𝑥𝑣 (𝑥(ball‘𝑁)𝑟) ⊆ 𝑋)
4442, 43sylibr 223 . . . . . . 7 (((𝜑𝑟 ∈ ℝ+) ∧ 𝑣 ∈ (𝒫 𝑋 ∩ Fin)) → 𝑥𝑣 (𝑥(ball‘𝑁)𝑟) ⊆ 𝑋)
4534, 44jctild 564 . . . . . 6 (((𝜑𝑟 ∈ ℝ+) ∧ 𝑣 ∈ (𝒫 𝑋 ∩ Fin)) → ( 𝑥𝑣 (𝑥(ball‘𝑀)(𝑟 / 𝑅)) = 𝑋 → ( 𝑥𝑣 (𝑥(ball‘𝑁)𝑟) ⊆ 𝑋𝑋 𝑥𝑣 (𝑥(ball‘𝑁)𝑟))))
46 eqss 3583 . . . . . 6 ( 𝑥𝑣 (𝑥(ball‘𝑁)𝑟) = 𝑋 ↔ ( 𝑥𝑣 (𝑥(ball‘𝑁)𝑟) ⊆ 𝑋𝑋 𝑥𝑣 (𝑥(ball‘𝑁)𝑟)))
4745, 46syl6ibr 241 . . . . 5 (((𝜑𝑟 ∈ ℝ+) ∧ 𝑣 ∈ (𝒫 𝑋 ∩ Fin)) → ( 𝑥𝑣 (𝑥(ball‘𝑀)(𝑟 / 𝑅)) = 𝑋 𝑥𝑣 (𝑥(ball‘𝑁)𝑟) = 𝑋))
4847reximdva 3000 . . . 4 ((𝜑𝑟 ∈ ℝ+) → (∃𝑣 ∈ (𝒫 𝑋 ∩ Fin) 𝑥𝑣 (𝑥(ball‘𝑀)(𝑟 / 𝑅)) = 𝑋 → ∃𝑣 ∈ (𝒫 𝑋 ∩ Fin) 𝑥𝑣 (𝑥(ball‘𝑁)𝑟) = 𝑋))
4916, 48mpd 15 . . 3 ((𝜑𝑟 ∈ ℝ+) → ∃𝑣 ∈ (𝒫 𝑋 ∩ Fin) 𝑥𝑣 (𝑥(ball‘𝑁)𝑟) = 𝑋)
5049ralrimiva 2949 . 2 (𝜑 → ∀𝑟 ∈ ℝ+𝑣 ∈ (𝒫 𝑋 ∩ Fin) 𝑥𝑣 (𝑥(ball‘𝑁)𝑟) = 𝑋)
51 istotbnd3 32740 . 2 (𝑁 ∈ (TotBnd‘𝑋) ↔ (𝑁 ∈ (Met‘𝑋) ∧ ∀𝑟 ∈ ℝ+𝑣 ∈ (𝒫 𝑋 ∩ Fin) 𝑥𝑣 (𝑥(ball‘𝑁)𝑟) = 𝑋))
521, 50, 51sylanbrc 695 1 (𝜑𝑁 ∈ (TotBnd‘𝑋))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1475  wcel 1977  wral 2896  wrex 2897  cin 3539  wss 3540  𝒫 cpw 4108   ciun 4455   class class class wbr 4583  cfv 5804  (class class class)co 6549  Fincfn 7841   · cmul 9820  *cxr 9952  cle 9954   / cdiv 10563  +crp 11708  ∞Metcxmt 19552  Metcme 19553  ballcbl 19554  MetOpencmopn 19557  TotBndctotbnd 32735
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-oadd 7451  df-er 7629  df-map 7746  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-div 10564  df-rp 11709  df-xadd 11823  df-psmet 19559  df-xmet 19560  df-met 19561  df-bl 19562  df-totbnd 32737
This theorem is referenced by:  equivbnd2  32761
  Copyright terms: Public domain W3C validator