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Theorem equivtotbnd 33209
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 477 . . . . . 6 ((𝜑𝑟 ∈ ℝ+) → 𝑟 ∈ ℝ+)
3 equivtotbnd.3 . . . . . . 7 (𝜑𝑅 ∈ ℝ+)
43adantr 481 . . . . . 6 ((𝜑𝑟 ∈ ℝ+) → 𝑅 ∈ ℝ+)
52, 4rpdivcld 11833 . . . . 5 ((𝜑𝑟 ∈ ℝ+) → (𝑟 / 𝑅) ∈ ℝ+)
6 equivtotbnd.1 . . . . . . 7 (𝜑𝑀 ∈ (TotBnd‘𝑋))
76adantr 481 . . . . . 6 ((𝜑𝑟 ∈ ℝ+) → 𝑀 ∈ (TotBnd‘𝑋))
8 istotbnd3 33202 . . . . . . 7 (𝑀 ∈ (TotBnd‘𝑋) ↔ (𝑀 ∈ (Met‘𝑋) ∧ ∀𝑠 ∈ ℝ+𝑣 ∈ (𝒫 𝑋 ∩ Fin) 𝑥𝑣 (𝑥(ball‘𝑀)𝑠) = 𝑋))
98simprbi 480 . . . . . 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 6612 . . . . . . . . 9 (𝑠 = (𝑟 / 𝑅) → (𝑥(ball‘𝑀)𝑠) = (𝑥(ball‘𝑀)(𝑟 / 𝑅)))
1211iuneq2d 4513 . . . . . . . 8 (𝑠 = (𝑟 / 𝑅) → 𝑥𝑣 (𝑥(ball‘𝑀)𝑠) = 𝑥𝑣 (𝑥(ball‘𝑀)(𝑟 / 𝑅)))
1312eqeq1d 2623 . . . . . . 7 (𝑠 = (𝑟 / 𝑅) → ( 𝑥𝑣 (𝑥(ball‘𝑀)𝑠) = 𝑋 𝑥𝑣 (𝑥(ball‘𝑀)(𝑟 / 𝑅)) = 𝑋))
1413rexbidv 3045 . . . . . 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 3291 . . . . 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 8212 . . . . . . . . . . . . . 14 (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ↔ (𝑣𝑋𝑣 ∈ Fin))
1817simplbi 476 . . . . . . . . . . . . 13 (𝑣 ∈ (𝒫 𝑋 ∩ Fin) → 𝑣𝑋)
1918adantl 482 . . . . . . . . . . . 12 (((𝜑𝑟 ∈ ℝ+) ∧ 𝑣 ∈ (𝒫 𝑋 ∩ Fin)) → 𝑣𝑋)
2019sselda 3583 . . . . . . . . . . 11 ((((𝜑𝑟 ∈ ℝ+) ∧ 𝑣 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑥𝑣) → 𝑥𝑋)
21 eqid 2621 . . . . . . . . . . . . . 14 (MetOpen‘𝑁) = (MetOpen‘𝑁)
22 eqid 2621 . . . . . . . . . . . . . 14 (MetOpen‘𝑀) = (MetOpen‘𝑀)
238simplbi 476 . . . . . . . . . . . . . . 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 22226 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑥𝑋𝑟 ∈ ℝ+)) → (𝑥(ball‘𝑀)(𝑟 / 𝑅)) ⊆ (𝑥(ball‘𝑁)𝑟))
2726anass1rs 848 . . . . . . . . . . . 12 (((𝜑𝑟 ∈ ℝ+) ∧ 𝑥𝑋) → (𝑥(ball‘𝑀)(𝑟 / 𝑅)) ⊆ (𝑥(ball‘𝑁)𝑟))
2827adantlr 750 . . . . . . . . . . 11 ((((𝜑𝑟 ∈ ℝ+) ∧ 𝑣 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑥𝑋) → (𝑥(ball‘𝑀)(𝑟 / 𝑅)) ⊆ (𝑥(ball‘𝑁)𝑟))
2920, 28syldan 487 . . . . . . . . . 10 ((((𝜑𝑟 ∈ ℝ+) ∧ 𝑣 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑥𝑣) → (𝑥(ball‘𝑀)(𝑟 / 𝑅)) ⊆ (𝑥(ball‘𝑁)𝑟))
3029ralrimiva 2960 . . . . . . . . 9 (((𝜑𝑟 ∈ ℝ+) ∧ 𝑣 ∈ (𝒫 𝑋 ∩ Fin)) → ∀𝑥𝑣 (𝑥(ball‘𝑀)(𝑟 / 𝑅)) ⊆ (𝑥(ball‘𝑁)𝑟))
31 ss2iun 4502 . . . . . . . . 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 3605 . . . . . . . 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 765 . . . . . . . . . . 11 ((((𝜑𝑟 ∈ ℝ+) ∧ 𝑣 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑥𝑣) → 𝑁 ∈ (Met‘𝑋))
36 metxmet 22049 . . . . . . . . . . 11 (𝑁 ∈ (Met‘𝑋) → 𝑁 ∈ (∞Met‘𝑋))
3735, 36syl 17 . . . . . . . . . 10 ((((𝜑𝑟 ∈ ℝ+) ∧ 𝑣 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑥𝑣) → 𝑁 ∈ (∞Met‘𝑋))
38 simpllr 798 . . . . . . . . . . 11 ((((𝜑𝑟 ∈ ℝ+) ∧ 𝑣 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑥𝑣) → 𝑟 ∈ ℝ+)
3938rpxrd 11817 . . . . . . . . . 10 ((((𝜑𝑟 ∈ ℝ+) ∧ 𝑣 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑥𝑣) → 𝑟 ∈ ℝ*)
40 blssm 22133 . . . . . . . . . 10 ((𝑁 ∈ (∞Met‘𝑋) ∧ 𝑥𝑋𝑟 ∈ ℝ*) → (𝑥(ball‘𝑁)𝑟) ⊆ 𝑋)
4137, 20, 39, 40syl3anc 1323 . . . . . . . . 9 ((((𝜑𝑟 ∈ ℝ+) ∧ 𝑣 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑥𝑣) → (𝑥(ball‘𝑁)𝑟) ⊆ 𝑋)
4241ralrimiva 2960 . . . . . . . 8 (((𝜑𝑟 ∈ ℝ+) ∧ 𝑣 ∈ (𝒫 𝑋 ∩ Fin)) → ∀𝑥𝑣 (𝑥(ball‘𝑁)𝑟) ⊆ 𝑋)
43 iunss 4527 . . . . . . . 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 3598 . . . . . 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 3011 . . . 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 2960 . 2 (𝜑 → ∀𝑟 ∈ ℝ+𝑣 ∈ (𝒫 𝑋 ∩ Fin) 𝑥𝑣 (𝑥(ball‘𝑁)𝑟) = 𝑋)
51 istotbnd3 33202 . 2 (𝑁 ∈ (TotBnd‘𝑋) ↔ (𝑁 ∈ (Met‘𝑋) ∧ ∀𝑟 ∈ ℝ+𝑣 ∈ (𝒫 𝑋 ∩ Fin) 𝑥𝑣 (𝑥(ball‘𝑁)𝑟) = 𝑋))
521, 50, 51sylanbrc 697 1 (𝜑𝑁 ∈ (TotBnd‘𝑋))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1480  wcel 1987  wral 2907  wrex 2908  cin 3554  wss 3555  𝒫 cpw 4130   ciun 4485   class class class wbr 4613  cfv 5847  (class class class)co 6604  Fincfn 7899   · cmul 9885  *cxr 10017  cle 10019   / cdiv 10628  +crp 11776  ∞Metcxmt 19650  Metcme 19651  ballcbl 19652  MetOpencmopn 19655  TotBndctotbnd 33197
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-pss 3571  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-tp 4153  df-op 4155  df-uni 4403  df-int 4441  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-tr 4713  df-eprel 4985  df-id 4989  df-po 4995  df-so 4996  df-fr 5033  df-we 5035  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-pred 5639  df-ord 5685  df-on 5686  df-lim 5687  df-suc 5688  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-om 7013  df-1st 7113  df-2nd 7114  df-wrecs 7352  df-recs 7413  df-rdg 7451  df-1o 7505  df-oadd 7509  df-er 7687  df-map 7804  df-en 7900  df-dom 7901  df-sdom 7902  df-fin 7903  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-rp 11777  df-xadd 11891  df-psmet 19657  df-xmet 19658  df-met 19659  df-bl 19660  df-totbnd 33199
This theorem is referenced by:  equivbnd2  33223
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