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Theorem equivtotbnd 33908
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 479 . . . . . 6 ((𝜑𝑟 ∈ ℝ+) → 𝑟 ∈ ℝ+)
3 equivtotbnd.3 . . . . . . 7 (𝜑𝑅 ∈ ℝ+)
43adantr 472 . . . . . 6 ((𝜑𝑟 ∈ ℝ+) → 𝑅 ∈ ℝ+)
52, 4rpdivcld 12102 . . . . 5 ((𝜑𝑟 ∈ ℝ+) → (𝑟 / 𝑅) ∈ ℝ+)
6 equivtotbnd.1 . . . . . . 7 (𝜑𝑀 ∈ (TotBnd‘𝑋))
76adantr 472 . . . . . 6 ((𝜑𝑟 ∈ ℝ+) → 𝑀 ∈ (TotBnd‘𝑋))
8 istotbnd3 33901 . . . . . . 7 (𝑀 ∈ (TotBnd‘𝑋) ↔ (𝑀 ∈ (Met‘𝑋) ∧ ∀𝑠 ∈ ℝ+𝑣 ∈ (𝒫 𝑋 ∩ Fin) 𝑥𝑣 (𝑥(ball‘𝑀)𝑠) = 𝑋))
98simprbi 483 . . . . . 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 6822 . . . . . . . . 9 (𝑠 = (𝑟 / 𝑅) → (𝑥(ball‘𝑀)𝑠) = (𝑥(ball‘𝑀)(𝑟 / 𝑅)))
1211iuneq2d 4699 . . . . . . . 8 (𝑠 = (𝑟 / 𝑅) → 𝑥𝑣 (𝑥(ball‘𝑀)𝑠) = 𝑥𝑣 (𝑥(ball‘𝑀)(𝑟 / 𝑅)))
1312eqeq1d 2762 . . . . . . 7 (𝑠 = (𝑟 / 𝑅) → ( 𝑥𝑣 (𝑥(ball‘𝑀)𝑠) = 𝑋 𝑥𝑣 (𝑥(ball‘𝑀)(𝑟 / 𝑅)) = 𝑋))
1413rexbidv 3190 . . . . . 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 3445 . . . . 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 8435 . . . . . . . . . . . . . 14 (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ↔ (𝑣𝑋𝑣 ∈ Fin))
1817simplbi 478 . . . . . . . . . . . . 13 (𝑣 ∈ (𝒫 𝑋 ∩ Fin) → 𝑣𝑋)
1918adantl 473 . . . . . . . . . . . 12 (((𝜑𝑟 ∈ ℝ+) ∧ 𝑣 ∈ (𝒫 𝑋 ∩ Fin)) → 𝑣𝑋)
2019sselda 3744 . . . . . . . . . . 11 ((((𝜑𝑟 ∈ ℝ+) ∧ 𝑣 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑥𝑣) → 𝑥𝑋)
21 eqid 2760 . . . . . . . . . . . . . 14 (MetOpen‘𝑁) = (MetOpen‘𝑁)
22 eqid 2760 . . . . . . . . . . . . . 14 (MetOpen‘𝑀) = (MetOpen‘𝑀)
238simplbi 478 . . . . . . . . . . . . . . 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 22537 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑥𝑋𝑟 ∈ ℝ+)) → (𝑥(ball‘𝑀)(𝑟 / 𝑅)) ⊆ (𝑥(ball‘𝑁)𝑟))
2726anass1rs 884 . . . . . . . . . . . 12 (((𝜑𝑟 ∈ ℝ+) ∧ 𝑥𝑋) → (𝑥(ball‘𝑀)(𝑟 / 𝑅)) ⊆ (𝑥(ball‘𝑁)𝑟))
2827adantlr 753 . . . . . . . . . . 11 ((((𝜑𝑟 ∈ ℝ+) ∧ 𝑣 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑥𝑋) → (𝑥(ball‘𝑀)(𝑟 / 𝑅)) ⊆ (𝑥(ball‘𝑁)𝑟))
2920, 28syldan 488 . . . . . . . . . 10 ((((𝜑𝑟 ∈ ℝ+) ∧ 𝑣 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑥𝑣) → (𝑥(ball‘𝑀)(𝑟 / 𝑅)) ⊆ (𝑥(ball‘𝑁)𝑟))
3029ralrimiva 3104 . . . . . . . . 9 (((𝜑𝑟 ∈ ℝ+) ∧ 𝑣 ∈ (𝒫 𝑋 ∩ Fin)) → ∀𝑥𝑣 (𝑥(ball‘𝑀)(𝑟 / 𝑅)) ⊆ (𝑥(ball‘𝑁)𝑟))
31 ss2iun 4688 . . . . . . . . 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 3767 . . . . . . . 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 768 . . . . . . . . . . 11 ((((𝜑𝑟 ∈ ℝ+) ∧ 𝑣 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑥𝑣) → 𝑁 ∈ (Met‘𝑋))
36 metxmet 22360 . . . . . . . . . . 11 (𝑁 ∈ (Met‘𝑋) → 𝑁 ∈ (∞Met‘𝑋))
3735, 36syl 17 . . . . . . . . . 10 ((((𝜑𝑟 ∈ ℝ+) ∧ 𝑣 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑥𝑣) → 𝑁 ∈ (∞Met‘𝑋))
38 simpllr 817 . . . . . . . . . . 11 ((((𝜑𝑟 ∈ ℝ+) ∧ 𝑣 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑥𝑣) → 𝑟 ∈ ℝ+)
3938rpxrd 12086 . . . . . . . . . 10 ((((𝜑𝑟 ∈ ℝ+) ∧ 𝑣 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑥𝑣) → 𝑟 ∈ ℝ*)
40 blssm 22444 . . . . . . . . . 10 ((𝑁 ∈ (∞Met‘𝑋) ∧ 𝑥𝑋𝑟 ∈ ℝ*) → (𝑥(ball‘𝑁)𝑟) ⊆ 𝑋)
4137, 20, 39, 40syl3anc 1477 . . . . . . . . 9 ((((𝜑𝑟 ∈ ℝ+) ∧ 𝑣 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑥𝑣) → (𝑥(ball‘𝑁)𝑟) ⊆ 𝑋)
4241ralrimiva 3104 . . . . . . . 8 (((𝜑𝑟 ∈ ℝ+) ∧ 𝑣 ∈ (𝒫 𝑋 ∩ Fin)) → ∀𝑥𝑣 (𝑥(ball‘𝑁)𝑟) ⊆ 𝑋)
43 iunss 4713 . . . . . . . 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 567 . . . . . 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 3759 . . . . . 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 3155 . . . 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 3104 . 2 (𝜑 → ∀𝑟 ∈ ℝ+𝑣 ∈ (𝒫 𝑋 ∩ Fin) 𝑥𝑣 (𝑥(ball‘𝑁)𝑟) = 𝑋)
51 istotbnd3 33901 . 2 (𝑁 ∈ (TotBnd‘𝑋) ↔ (𝑁 ∈ (Met‘𝑋) ∧ ∀𝑟 ∈ ℝ+𝑣 ∈ (𝒫 𝑋 ∩ Fin) 𝑥𝑣 (𝑥(ball‘𝑁)𝑟) = 𝑋))
521, 50, 51sylanbrc 701 1 (𝜑𝑁 ∈ (TotBnd‘𝑋))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1632  wcel 2139  wral 3050  wrex 3051  cin 3714  wss 3715  𝒫 cpw 4302   ciun 4672   class class class wbr 4804  cfv 6049  (class class class)co 6814  Fincfn 8123   · cmul 10153  *cxr 10285  cle 10287   / cdiv 10896  +crp 12045  ∞Metcxmt 19953  Metcme 19954  ballcbl 19955  MetOpencmopn 19958  TotBndctotbnd 33896
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7115  ax-cnex 10204  ax-resscn 10205  ax-1cn 10206  ax-icn 10207  ax-addcl 10208  ax-addrcl 10209  ax-mulcl 10210  ax-mulrcl 10211  ax-mulcom 10212  ax-addass 10213  ax-mulass 10214  ax-distr 10215  ax-i2m1 10216  ax-1ne0 10217  ax-1rid 10218  ax-rnegex 10219  ax-rrecex 10220  ax-cnre 10221  ax-pre-lttri 10222  ax-pre-lttrn 10223  ax-pre-ltadd 10224  ax-pre-mulgt0 10225
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-nel 3036  df-ral 3055  df-rex 3056  df-reu 3057  df-rmo 3058  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-pss 3731  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-tp 4326  df-op 4328  df-uni 4589  df-int 4628  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-tr 4905  df-id 5174  df-eprel 5179  df-po 5187  df-so 5188  df-fr 5225  df-we 5227  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-pred 5841  df-ord 5887  df-on 5888  df-lim 5889  df-suc 5890  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-riota 6775  df-ov 6817  df-oprab 6818  df-mpt2 6819  df-om 7232  df-1st 7334  df-2nd 7335  df-wrecs 7577  df-recs 7638  df-rdg 7676  df-1o 7730  df-oadd 7734  df-er 7913  df-map 8027  df-en 8124  df-dom 8125  df-sdom 8126  df-fin 8127  df-pnf 10288  df-mnf 10289  df-xr 10290  df-ltxr 10291  df-le 10292  df-sub 10480  df-neg 10481  df-div 10897  df-rp 12046  df-xadd 12160  df-psmet 19960  df-xmet 19961  df-met 19962  df-bl 19963  df-totbnd 33898
This theorem is referenced by:  equivbnd2  33922
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