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Theorem equivtotbnd 36646
Description: If the metric 𝑀 is "strongly finer" than 𝑁 (meaning that there is a positive real constant 𝑅 such that 𝑁(π‘₯, 𝑦) ≀ 𝑅 Β· 𝑀(π‘₯, 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 (πœ‘ β†’ 𝑀 ∈ (TotBndβ€˜π‘‹))
equivtotbnd.2 (πœ‘ β†’ 𝑁 ∈ (Metβ€˜π‘‹))
equivtotbnd.3 (πœ‘ β†’ 𝑅 ∈ ℝ+)
equivtotbnd.4 ((πœ‘ ∧ (π‘₯ ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) β†’ (π‘₯𝑁𝑦) ≀ (𝑅 Β· (π‘₯𝑀𝑦)))
Assertion
Ref Expression
equivtotbnd (πœ‘ β†’ 𝑁 ∈ (TotBndβ€˜π‘‹))
Distinct variable groups:   π‘₯,𝑦,𝑀   π‘₯,𝑁,𝑦   πœ‘,π‘₯,𝑦   π‘₯,𝑋,𝑦   π‘₯,𝑅,𝑦
Proof of Theorem equivtotbnd
Dummy variables 𝑣 𝑠 π‘Ÿ are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 equivtotbnd.2 . 2 (πœ‘ β†’ 𝑁 ∈ (Metβ€˜π‘‹))
2 simpr 486 . . . . . 6 ((πœ‘ ∧ π‘Ÿ ∈ ℝ+) β†’ π‘Ÿ ∈ ℝ+)
3 equivtotbnd.3 . . . . . . 7 (πœ‘ β†’ 𝑅 ∈ ℝ+)
43adantr 482 . . . . . 6 ((πœ‘ ∧ π‘Ÿ ∈ ℝ+) β†’ 𝑅 ∈ ℝ+)
52, 4rpdivcld 13033 . . . . 5 ((πœ‘ ∧ π‘Ÿ ∈ ℝ+) β†’ (π‘Ÿ / 𝑅) ∈ ℝ+)
6 equivtotbnd.1 . . . . . . 7 (πœ‘ β†’ 𝑀 ∈ (TotBndβ€˜π‘‹))
76adantr 482 . . . . . 6 ((πœ‘ ∧ π‘Ÿ ∈ ℝ+) β†’ 𝑀 ∈ (TotBndβ€˜π‘‹))
8 istotbnd3 36639 . . . . . . 7 (𝑀 ∈ (TotBndβ€˜π‘‹) ↔ (𝑀 ∈ (Metβ€˜π‘‹) ∧ βˆ€π‘  ∈ ℝ+ βˆƒπ‘£ ∈ (𝒫 𝑋 ∩ Fin)βˆͺ π‘₯ ∈ 𝑣 (π‘₯(ballβ€˜π‘€)𝑠) = 𝑋))
98simprbi 498 . . . . . 6 (𝑀 ∈ (TotBndβ€˜π‘‹) β†’ βˆ€π‘  ∈ ℝ+ βˆƒπ‘£ ∈ (𝒫 𝑋 ∩ Fin)βˆͺ π‘₯ ∈ 𝑣 (π‘₯(ballβ€˜π‘€)𝑠) = 𝑋)
107, 9syl 17 . . . . 5 ((πœ‘ ∧ π‘Ÿ ∈ ℝ+) β†’ βˆ€π‘  ∈ ℝ+ βˆƒπ‘£ ∈ (𝒫 𝑋 ∩ Fin)βˆͺ π‘₯ ∈ 𝑣 (π‘₯(ballβ€˜π‘€)𝑠) = 𝑋)
11 oveq2 7417 . . . . . . . . 9 (𝑠 = (π‘Ÿ / 𝑅) β†’ (π‘₯(ballβ€˜π‘€)𝑠) = (π‘₯(ballβ€˜π‘€)(π‘Ÿ / 𝑅)))
1211iuneq2d 5027 . . . . . . . 8 (𝑠 = (π‘Ÿ / 𝑅) β†’ βˆͺ π‘₯ ∈ 𝑣 (π‘₯(ballβ€˜π‘€)𝑠) = βˆͺ π‘₯ ∈ 𝑣 (π‘₯(ballβ€˜π‘€)(π‘Ÿ / 𝑅)))
1312eqeq1d 2735 . . . . . . 7 (𝑠 = (π‘Ÿ / 𝑅) β†’ (βˆͺ π‘₯ ∈ 𝑣 (π‘₯(ballβ€˜π‘€)𝑠) = 𝑋 ↔ βˆͺ π‘₯ ∈ 𝑣 (π‘₯(ballβ€˜π‘€)(π‘Ÿ / 𝑅)) = 𝑋))
1413rexbidv 3179 . . . . . 6 (𝑠 = (π‘Ÿ / 𝑅) β†’ (βˆƒπ‘£ ∈ (𝒫 𝑋 ∩ Fin)βˆͺ π‘₯ ∈ 𝑣 (π‘₯(ballβ€˜π‘€)𝑠) = 𝑋 ↔ βˆƒπ‘£ ∈ (𝒫 𝑋 ∩ Fin)βˆͺ π‘₯ ∈ 𝑣 (π‘₯(ballβ€˜π‘€)(π‘Ÿ / 𝑅)) = 𝑋))
1514rspcv 3609 . . . . 5 ((π‘Ÿ / 𝑅) ∈ ℝ+ β†’ (βˆ€π‘  ∈ ℝ+ βˆƒπ‘£ ∈ (𝒫 𝑋 ∩ Fin)βˆͺ π‘₯ ∈ 𝑣 (π‘₯(ballβ€˜π‘€)𝑠) = 𝑋 β†’ βˆƒπ‘£ ∈ (𝒫 𝑋 ∩ Fin)βˆͺ π‘₯ ∈ 𝑣 (π‘₯(ballβ€˜π‘€)(π‘Ÿ / 𝑅)) = 𝑋))
165, 10, 15sylc 65 . . . 4 ((πœ‘ ∧ π‘Ÿ ∈ ℝ+) β†’ βˆƒπ‘£ ∈ (𝒫 𝑋 ∩ Fin)βˆͺ π‘₯ ∈ 𝑣 (π‘₯(ballβ€˜π‘€)(π‘Ÿ / 𝑅)) = 𝑋)
17 elfpw 9354 . . . . . . . . . . . . . 14 (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ↔ (𝑣 βŠ† 𝑋 ∧ 𝑣 ∈ Fin))
1817simplbi 499 . . . . . . . . . . . . 13 (𝑣 ∈ (𝒫 𝑋 ∩ Fin) β†’ 𝑣 βŠ† 𝑋)
1918adantl 483 . . . . . . . . . . . 12 (((πœ‘ ∧ π‘Ÿ ∈ ℝ+) ∧ 𝑣 ∈ (𝒫 𝑋 ∩ Fin)) β†’ 𝑣 βŠ† 𝑋)
2019sselda 3983 . . . . . . . . . . 11 ((((πœ‘ ∧ π‘Ÿ ∈ ℝ+) ∧ 𝑣 ∈ (𝒫 𝑋 ∩ Fin)) ∧ π‘₯ ∈ 𝑣) β†’ π‘₯ ∈ 𝑋)
21 eqid 2733 . . . . . . . . . . . . . 14 (MetOpenβ€˜π‘) = (MetOpenβ€˜π‘)
22 eqid 2733 . . . . . . . . . . . . . 14 (MetOpenβ€˜π‘€) = (MetOpenβ€˜π‘€)
238simplbi 499 . . . . . . . . . . . . . . 15 (𝑀 ∈ (TotBndβ€˜π‘‹) β†’ 𝑀 ∈ (Metβ€˜π‘‹))
246, 23syl 17 . . . . . . . . . . . . . 14 (πœ‘ β†’ 𝑀 ∈ (Metβ€˜π‘‹))
25 equivtotbnd.4 . . . . . . . . . . . . . 14 ((πœ‘ ∧ (π‘₯ ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) β†’ (π‘₯𝑁𝑦) ≀ (𝑅 Β· (π‘₯𝑀𝑦)))
2621, 22, 1, 24, 3, 25metss2lem 24020 . . . . . . . . . . . . 13 ((πœ‘ ∧ (π‘₯ ∈ 𝑋 ∧ π‘Ÿ ∈ ℝ+)) β†’ (π‘₯(ballβ€˜π‘€)(π‘Ÿ / 𝑅)) βŠ† (π‘₯(ballβ€˜π‘)π‘Ÿ))
2726anass1rs 654 . . . . . . . . . . . 12 (((πœ‘ ∧ π‘Ÿ ∈ ℝ+) ∧ π‘₯ ∈ 𝑋) β†’ (π‘₯(ballβ€˜π‘€)(π‘Ÿ / 𝑅)) βŠ† (π‘₯(ballβ€˜π‘)π‘Ÿ))
2827adantlr 714 . . . . . . . . . . 11 ((((πœ‘ ∧ π‘Ÿ ∈ ℝ+) ∧ 𝑣 ∈ (𝒫 𝑋 ∩ Fin)) ∧ π‘₯ ∈ 𝑋) β†’ (π‘₯(ballβ€˜π‘€)(π‘Ÿ / 𝑅)) βŠ† (π‘₯(ballβ€˜π‘)π‘Ÿ))
2920, 28syldan 592 . . . . . . . . . 10 ((((πœ‘ ∧ π‘Ÿ ∈ ℝ+) ∧ 𝑣 ∈ (𝒫 𝑋 ∩ Fin)) ∧ π‘₯ ∈ 𝑣) β†’ (π‘₯(ballβ€˜π‘€)(π‘Ÿ / 𝑅)) βŠ† (π‘₯(ballβ€˜π‘)π‘Ÿ))
3029ralrimiva 3147 . . . . . . . . 9 (((πœ‘ ∧ π‘Ÿ ∈ ℝ+) ∧ 𝑣 ∈ (𝒫 𝑋 ∩ Fin)) β†’ βˆ€π‘₯ ∈ 𝑣 (π‘₯(ballβ€˜π‘€)(π‘Ÿ / 𝑅)) βŠ† (π‘₯(ballβ€˜π‘)π‘Ÿ))
31 ss2iun 5016 . . . . . . . . 9 (βˆ€π‘₯ ∈ 𝑣 (π‘₯(ballβ€˜π‘€)(π‘Ÿ / 𝑅)) βŠ† (π‘₯(ballβ€˜π‘)π‘Ÿ) β†’ βˆͺ π‘₯ ∈ 𝑣 (π‘₯(ballβ€˜π‘€)(π‘Ÿ / 𝑅)) βŠ† βˆͺ π‘₯ ∈ 𝑣 (π‘₯(ballβ€˜π‘)π‘Ÿ))
3230, 31syl 17 . . . . . . . 8 (((πœ‘ ∧ π‘Ÿ ∈ ℝ+) ∧ 𝑣 ∈ (𝒫 𝑋 ∩ Fin)) β†’ βˆͺ π‘₯ ∈ 𝑣 (π‘₯(ballβ€˜π‘€)(π‘Ÿ / 𝑅)) βŠ† βˆͺ π‘₯ ∈ 𝑣 (π‘₯(ballβ€˜π‘)π‘Ÿ))
33 sseq1 4008 . . . . . . . 8 (βˆͺ π‘₯ ∈ 𝑣 (π‘₯(ballβ€˜π‘€)(π‘Ÿ / 𝑅)) = 𝑋 β†’ (βˆͺ π‘₯ ∈ 𝑣 (π‘₯(ballβ€˜π‘€)(π‘Ÿ / 𝑅)) βŠ† βˆͺ π‘₯ ∈ 𝑣 (π‘₯(ballβ€˜π‘)π‘Ÿ) ↔ 𝑋 βŠ† βˆͺ π‘₯ ∈ 𝑣 (π‘₯(ballβ€˜π‘)π‘Ÿ)))
3432, 33syl5ibcom 244 . . . . . . 7 (((πœ‘ ∧ π‘Ÿ ∈ ℝ+) ∧ 𝑣 ∈ (𝒫 𝑋 ∩ Fin)) β†’ (βˆͺ π‘₯ ∈ 𝑣 (π‘₯(ballβ€˜π‘€)(π‘Ÿ / 𝑅)) = 𝑋 β†’ 𝑋 βŠ† βˆͺ π‘₯ ∈ 𝑣 (π‘₯(ballβ€˜π‘)π‘Ÿ)))
351ad3antrrr 729 . . . . . . . . . . 11 ((((πœ‘ ∧ π‘Ÿ ∈ ℝ+) ∧ 𝑣 ∈ (𝒫 𝑋 ∩ Fin)) ∧ π‘₯ ∈ 𝑣) β†’ 𝑁 ∈ (Metβ€˜π‘‹))
36 metxmet 23840 . . . . . . . . . . 11 (𝑁 ∈ (Metβ€˜π‘‹) β†’ 𝑁 ∈ (∞Metβ€˜π‘‹))
3735, 36syl 17 . . . . . . . . . 10 ((((πœ‘ ∧ π‘Ÿ ∈ ℝ+) ∧ 𝑣 ∈ (𝒫 𝑋 ∩ Fin)) ∧ π‘₯ ∈ 𝑣) β†’ 𝑁 ∈ (∞Metβ€˜π‘‹))
38 simpllr 775 . . . . . . . . . . 11 ((((πœ‘ ∧ π‘Ÿ ∈ ℝ+) ∧ 𝑣 ∈ (𝒫 𝑋 ∩ Fin)) ∧ π‘₯ ∈ 𝑣) β†’ π‘Ÿ ∈ ℝ+)
3938rpxrd 13017 . . . . . . . . . 10 ((((πœ‘ ∧ π‘Ÿ ∈ ℝ+) ∧ 𝑣 ∈ (𝒫 𝑋 ∩ Fin)) ∧ π‘₯ ∈ 𝑣) β†’ π‘Ÿ ∈ ℝ*)
40 blssm 23924 . . . . . . . . . 10 ((𝑁 ∈ (∞Metβ€˜π‘‹) ∧ π‘₯ ∈ 𝑋 ∧ π‘Ÿ ∈ ℝ*) β†’ (π‘₯(ballβ€˜π‘)π‘Ÿ) βŠ† 𝑋)
4137, 20, 39, 40syl3anc 1372 . . . . . . . . 9 ((((πœ‘ ∧ π‘Ÿ ∈ ℝ+) ∧ 𝑣 ∈ (𝒫 𝑋 ∩ Fin)) ∧ π‘₯ ∈ 𝑣) β†’ (π‘₯(ballβ€˜π‘)π‘Ÿ) βŠ† 𝑋)
4241ralrimiva 3147 . . . . . . . 8 (((πœ‘ ∧ π‘Ÿ ∈ ℝ+) ∧ 𝑣 ∈ (𝒫 𝑋 ∩ Fin)) β†’ βˆ€π‘₯ ∈ 𝑣 (π‘₯(ballβ€˜π‘)π‘Ÿ) βŠ† 𝑋)
43 iunss 5049 . . . . . . . 8 (βˆͺ π‘₯ ∈ 𝑣 (π‘₯(ballβ€˜π‘)π‘Ÿ) βŠ† 𝑋 ↔ βˆ€π‘₯ ∈ 𝑣 (π‘₯(ballβ€˜π‘)π‘Ÿ) βŠ† 𝑋)
4442, 43sylibr 233 . . . . . . 7 (((πœ‘ ∧ π‘Ÿ ∈ ℝ+) ∧ 𝑣 ∈ (𝒫 𝑋 ∩ Fin)) β†’ βˆͺ π‘₯ ∈ 𝑣 (π‘₯(ballβ€˜π‘)π‘Ÿ) βŠ† 𝑋)
4534, 44jctild 527 . . . . . 6 (((πœ‘ ∧ π‘Ÿ ∈ ℝ+) ∧ 𝑣 ∈ (𝒫 𝑋 ∩ Fin)) β†’ (βˆͺ π‘₯ ∈ 𝑣 (π‘₯(ballβ€˜π‘€)(π‘Ÿ / 𝑅)) = 𝑋 β†’ (βˆͺ π‘₯ ∈ 𝑣 (π‘₯(ballβ€˜π‘)π‘Ÿ) βŠ† 𝑋 ∧ 𝑋 βŠ† βˆͺ π‘₯ ∈ 𝑣 (π‘₯(ballβ€˜π‘)π‘Ÿ))))
46 eqss 3998 . . . . . 6 (βˆͺ π‘₯ ∈ 𝑣 (π‘₯(ballβ€˜π‘)π‘Ÿ) = 𝑋 ↔ (βˆͺ π‘₯ ∈ 𝑣 (π‘₯(ballβ€˜π‘)π‘Ÿ) βŠ† 𝑋 ∧ 𝑋 βŠ† βˆͺ π‘₯ ∈ 𝑣 (π‘₯(ballβ€˜π‘)π‘Ÿ)))
4745, 46syl6ibr 252 . . . . 5 (((πœ‘ ∧ π‘Ÿ ∈ ℝ+) ∧ 𝑣 ∈ (𝒫 𝑋 ∩ Fin)) β†’ (βˆͺ π‘₯ ∈ 𝑣 (π‘₯(ballβ€˜π‘€)(π‘Ÿ / 𝑅)) = 𝑋 β†’ βˆͺ π‘₯ ∈ 𝑣 (π‘₯(ballβ€˜π‘)π‘Ÿ) = 𝑋))
4847reximdva 3169 . . . 4 ((πœ‘ ∧ π‘Ÿ ∈ ℝ+) β†’ (βˆƒπ‘£ ∈ (𝒫 𝑋 ∩ Fin)βˆͺ π‘₯ ∈ 𝑣 (π‘₯(ballβ€˜π‘€)(π‘Ÿ / 𝑅)) = 𝑋 β†’ βˆƒπ‘£ ∈ (𝒫 𝑋 ∩ Fin)βˆͺ π‘₯ ∈ 𝑣 (π‘₯(ballβ€˜π‘)π‘Ÿ) = 𝑋))
4916, 48mpd 15 . . 3 ((πœ‘ ∧ π‘Ÿ ∈ ℝ+) β†’ βˆƒπ‘£ ∈ (𝒫 𝑋 ∩ Fin)βˆͺ π‘₯ ∈ 𝑣 (π‘₯(ballβ€˜π‘)π‘Ÿ) = 𝑋)
5049ralrimiva 3147 . 2 (πœ‘ β†’ βˆ€π‘Ÿ ∈ ℝ+ βˆƒπ‘£ ∈ (𝒫 𝑋 ∩ Fin)βˆͺ π‘₯ ∈ 𝑣 (π‘₯(ballβ€˜π‘)π‘Ÿ) = 𝑋)
51 istotbnd3 36639 . 2 (𝑁 ∈ (TotBndβ€˜π‘‹) ↔ (𝑁 ∈ (Metβ€˜π‘‹) ∧ βˆ€π‘Ÿ ∈ ℝ+ βˆƒπ‘£ ∈ (𝒫 𝑋 ∩ Fin)βˆͺ π‘₯ ∈ 𝑣 (π‘₯(ballβ€˜π‘)π‘Ÿ) = 𝑋))
521, 50, 51sylanbrc 584 1 (πœ‘ β†’ 𝑁 ∈ (TotBndβ€˜π‘‹))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 397   = wceq 1542   ∈ wcel 2107  βˆ€wral 3062  βˆƒwrex 3071   ∩ cin 3948   βŠ† wss 3949  π’« cpw 4603  βˆͺ ciun 4998   class class class wbr 5149  β€˜cfv 6544  (class class class)co 7409  Fincfn 8939   Β· cmul 11115  β„*cxr 11247   ≀ cle 11249   / cdiv 11871  β„+crp 12974  βˆžMetcxmet 20929  Metcmet 20930  ballcbl 20931  MetOpencmopn 20934  TotBndctotbnd 36634
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725  ax-cnex 11166  ax-resscn 11167  ax-1cn 11168  ax-icn 11169  ax-addcl 11170  ax-addrcl 11171  ax-mulcl 11172  ax-mulrcl 11173  ax-mulcom 11174  ax-addass 11175  ax-mulass 11176  ax-distr 11177  ax-i2m1 11178  ax-1ne0 11179  ax-1rid 11180  ax-rnegex 11181  ax-rrecex 11182  ax-cnre 11183  ax-pre-lttri 11184  ax-pre-lttrn 11185  ax-pre-ltadd 11186  ax-pre-mulgt0 11187
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-nel 3048  df-ral 3063  df-rex 3072  df-rmo 3377  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5575  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-we 5634  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-riota 7365  df-ov 7412  df-oprab 7413  df-mpo 7414  df-om 7856  df-1st 7975  df-2nd 7976  df-1o 8466  df-er 8703  df-map 8822  df-en 8940  df-dom 8941  df-sdom 8942  df-fin 8943  df-pnf 11250  df-mnf 11251  df-xr 11252  df-ltxr 11253  df-le 11254  df-sub 11446  df-neg 11447  df-div 11872  df-rp 12975  df-xadd 13093  df-psmet 20936  df-xmet 20937  df-met 20938  df-bl 20939  df-totbnd 36636
This theorem is referenced by:  equivbnd2  36660
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