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Theorem equivtotbnd 33707
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 11927 . . . . 5 ((𝜑𝑟 ∈ ℝ+) → (𝑟 / 𝑅) ∈ ℝ+)
6 equivtotbnd.1 . . . . . . 7 (𝜑𝑀 ∈ (TotBnd‘𝑋))
76adantr 480 . . . . . 6 ((𝜑𝑟 ∈ ℝ+) → 𝑀 ∈ (TotBnd‘𝑋))
8 istotbnd3 33700 . . . . . . 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 6698 . . . . . . . . 9 (𝑠 = (𝑟 / 𝑅) → (𝑥(ball‘𝑀)𝑠) = (𝑥(ball‘𝑀)(𝑟 / 𝑅)))
1211iuneq2d 4579 . . . . . . . 8 (𝑠 = (𝑟 / 𝑅) → 𝑥𝑣 (𝑥(ball‘𝑀)𝑠) = 𝑥𝑣 (𝑥(ball‘𝑀)(𝑟 / 𝑅)))
1312eqeq1d 2653 . . . . . . 7 (𝑠 = (𝑟 / 𝑅) → ( 𝑥𝑣 (𝑥(ball‘𝑀)𝑠) = 𝑋 𝑥𝑣 (𝑥(ball‘𝑀)(𝑟 / 𝑅)) = 𝑋))
1413rexbidv 3081 . . . . . 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 3336 . . . . 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 65 . . . 4 ((𝜑𝑟 ∈ ℝ+) → ∃𝑣 ∈ (𝒫 𝑋 ∩ Fin) 𝑥𝑣 (𝑥(ball‘𝑀)(𝑟 / 𝑅)) = 𝑋)
17 elfpw 8309 . . . . . . . . . . . . . 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 3636 . . . . . . . . . . 11 ((((𝜑𝑟 ∈ ℝ+) ∧ 𝑣 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑥𝑣) → 𝑥𝑋)
21 eqid 2651 . . . . . . . . . . . . . 14 (MetOpen‘𝑁) = (MetOpen‘𝑁)
22 eqid 2651 . . . . . . . . . . . . . 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 22363 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑥𝑋𝑟 ∈ ℝ+)) → (𝑥(ball‘𝑀)(𝑟 / 𝑅)) ⊆ (𝑥(ball‘𝑁)𝑟))
2726anass1rs 866 . . . . . . . . . . . 12 (((𝜑𝑟 ∈ ℝ+) ∧ 𝑥𝑋) → (𝑥(ball‘𝑀)(𝑟 / 𝑅)) ⊆ (𝑥(ball‘𝑁)𝑟))
2827adantlr 751 . . . . . . . . . . 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 2995 . . . . . . . . 9 (((𝜑𝑟 ∈ ℝ+) ∧ 𝑣 ∈ (𝒫 𝑋 ∩ Fin)) → ∀𝑥𝑣 (𝑥(ball‘𝑀)(𝑟 / 𝑅)) ⊆ (𝑥(ball‘𝑁)𝑟))
31 ss2iun 4568 . . . . . . . . 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 3659 . . . . . . . 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 235 . . . . . . 7 (((𝜑𝑟 ∈ ℝ+) ∧ 𝑣 ∈ (𝒫 𝑋 ∩ Fin)) → ( 𝑥𝑣 (𝑥(ball‘𝑀)(𝑟 / 𝑅)) = 𝑋𝑋 𝑥𝑣 (𝑥(ball‘𝑁)𝑟)))
351ad3antrrr 766 . . . . . . . . . . 11 ((((𝜑𝑟 ∈ ℝ+) ∧ 𝑣 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑥𝑣) → 𝑁 ∈ (Met‘𝑋))
36 metxmet 22186 . . . . . . . . . . 11 (𝑁 ∈ (Met‘𝑋) → 𝑁 ∈ (∞Met‘𝑋))
3735, 36syl 17 . . . . . . . . . 10 ((((𝜑𝑟 ∈ ℝ+) ∧ 𝑣 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑥𝑣) → 𝑁 ∈ (∞Met‘𝑋))
38 simpllr 815 . . . . . . . . . . 11 ((((𝜑𝑟 ∈ ℝ+) ∧ 𝑣 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑥𝑣) → 𝑟 ∈ ℝ+)
3938rpxrd 11911 . . . . . . . . . 10 ((((𝜑𝑟 ∈ ℝ+) ∧ 𝑣 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑥𝑣) → 𝑟 ∈ ℝ*)
40 blssm 22270 . . . . . . . . . 10 ((𝑁 ∈ (∞Met‘𝑋) ∧ 𝑥𝑋𝑟 ∈ ℝ*) → (𝑥(ball‘𝑁)𝑟) ⊆ 𝑋)
4137, 20, 39, 40syl3anc 1366 . . . . . . . . 9 ((((𝜑𝑟 ∈ ℝ+) ∧ 𝑣 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑥𝑣) → (𝑥(ball‘𝑁)𝑟) ⊆ 𝑋)
4241ralrimiva 2995 . . . . . . . 8 (((𝜑𝑟 ∈ ℝ+) ∧ 𝑣 ∈ (𝒫 𝑋 ∩ Fin)) → ∀𝑥𝑣 (𝑥(ball‘𝑁)𝑟) ⊆ 𝑋)
43 iunss 4593 . . . . . . . 8 ( 𝑥𝑣 (𝑥(ball‘𝑁)𝑟) ⊆ 𝑋 ↔ ∀𝑥𝑣 (𝑥(ball‘𝑁)𝑟) ⊆ 𝑋)
4442, 43sylibr 224 . . . . . . 7 (((𝜑𝑟 ∈ ℝ+) ∧ 𝑣 ∈ (𝒫 𝑋 ∩ Fin)) → 𝑥𝑣 (𝑥(ball‘𝑁)𝑟) ⊆ 𝑋)
4534, 44jctild 565 . . . . . 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 3651 . . . . . 6 ( 𝑥𝑣 (𝑥(ball‘𝑁)𝑟) = 𝑋 ↔ ( 𝑥𝑣 (𝑥(ball‘𝑁)𝑟) ⊆ 𝑋𝑋 𝑥𝑣 (𝑥(ball‘𝑁)𝑟)))
4745, 46syl6ibr 242 . . . . 5 (((𝜑𝑟 ∈ ℝ+) ∧ 𝑣 ∈ (𝒫 𝑋 ∩ Fin)) → ( 𝑥𝑣 (𝑥(ball‘𝑀)(𝑟 / 𝑅)) = 𝑋 𝑥𝑣 (𝑥(ball‘𝑁)𝑟) = 𝑋))
4847reximdva 3046 . . . 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 2995 . 2 (𝜑 → ∀𝑟 ∈ ℝ+𝑣 ∈ (𝒫 𝑋 ∩ Fin) 𝑥𝑣 (𝑥(ball‘𝑁)𝑟) = 𝑋)
51 istotbnd3 33700 . 2 (𝑁 ∈ (TotBnd‘𝑋) ↔ (𝑁 ∈ (Met‘𝑋) ∧ ∀𝑟 ∈ ℝ+𝑣 ∈ (𝒫 𝑋 ∩ Fin) 𝑥𝑣 (𝑥(ball‘𝑁)𝑟) = 𝑋))
521, 50, 51sylanbrc 699 1 (𝜑𝑁 ∈ (TotBnd‘𝑋))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1523  wcel 2030  wral 2941  wrex 2942  cin 3606  wss 3607  𝒫 cpw 4191   ciun 4552   class class class wbr 4685  cfv 5926  (class class class)co 6690  Fincfn 7997   · cmul 9979  *cxr 10111  cle 10113   / cdiv 10722  +crp 11870  ∞Metcxmt 19779  Metcme 19780  ballcbl 19781  MetOpencmopn 19784  TotBndctotbnd 33695
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-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  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-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  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-om 7108  df-1st 7210  df-2nd 7211  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-1o 7605  df-oadd 7609  df-er 7787  df-map 7901  df-en 7998  df-dom 7999  df-sdom 8000  df-fin 8001  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-rp 11871  df-xadd 11985  df-psmet 19786  df-xmet 19787  df-met 19788  df-bl 19789  df-totbnd 33697
This theorem is referenced by:  equivbnd2  33721
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