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

Theorem equivtotbnd 26502
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  x.  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  |-  ( ph  ->  M  e.  ( TotBnd `  X ) )
equivtotbnd.2  |-  ( ph  ->  N  e.  ( Met `  X ) )
equivtotbnd.3  |-  ( ph  ->  R  e.  RR+ )
equivtotbnd.4  |-  ( (
ph  /\  ( x  e.  X  /\  y  e.  X ) )  -> 
( x N y )  <_  ( R  x.  ( x M y ) ) )
Assertion
Ref Expression
equivtotbnd  |-  ( ph  ->  N  e.  ( TotBnd `  X ) )
Distinct variable groups:    x, y, M    x, N, y    ph, 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  |-  ( ph  ->  N  e.  ( Met `  X ) )
2 simpr 447 . . . . . 6  |-  ( (
ph  /\  r  e.  RR+ )  ->  r  e.  RR+ )
3 equivtotbnd.3 . . . . . . 7  |-  ( ph  ->  R  e.  RR+ )
43adantr 451 . . . . . 6  |-  ( (
ph  /\  r  e.  RR+ )  ->  R  e.  RR+ )
5 rpdivcl 10376 . . . . . 6  |-  ( ( r  e.  RR+  /\  R  e.  RR+ )  ->  (
r  /  R )  e.  RR+ )
62, 4, 5syl2anc 642 . . . . 5  |-  ( (
ph  /\  r  e.  RR+ )  ->  ( r  /  R )  e.  RR+ )
7 equivtotbnd.1 . . . . . . 7  |-  ( ph  ->  M  e.  ( TotBnd `  X ) )
87adantr 451 . . . . . 6  |-  ( (
ph  /\  r  e.  RR+ )  ->  M  e.  ( TotBnd `  X )
)
9 istotbnd3 26495 . . . . . . 7  |-  ( M  e.  ( TotBnd `  X
)  <->  ( M  e.  ( Met `  X
)  /\  A. s  e.  RR+  E. v  e.  ( ~P X  i^i  Fin ) U_ x  e.  v  ( x (
ball `  M )
s )  =  X ) )
109simprbi 450 . . . . . 6  |-  ( M  e.  ( TotBnd `  X
)  ->  A. s  e.  RR+  E. v  e.  ( ~P X  i^i  Fin ) U_ x  e.  v  ( x (
ball `  M )
s )  =  X )
118, 10syl 15 . . . . 5  |-  ( (
ph  /\  r  e.  RR+ )  ->  A. s  e.  RR+  E. v  e.  ( ~P X  i^i  Fin ) U_ x  e.  v  ( x (
ball `  M )
s )  =  X )
12 oveq2 5866 . . . . . . . . 9  |-  ( s  =  ( r  /  R )  ->  (
x ( ball `  M
) s )  =  ( x ( ball `  M ) ( r  /  R ) ) )
1312iuneq2d 3930 . . . . . . . 8  |-  ( s  =  ( r  /  R )  ->  U_ x  e.  v  ( x
( ball `  M )
s )  =  U_ x  e.  v  (
x ( ball `  M
) ( r  /  R ) ) )
1413eqeq1d 2291 . . . . . . 7  |-  ( s  =  ( r  /  R )  ->  ( U_ x  e.  v 
( x ( ball `  M ) s )  =  X  <->  U_ x  e.  v  ( x (
ball `  M )
( r  /  R
) )  =  X ) )
1514rexbidv 2564 . . . . . 6  |-  ( s  =  ( r  /  R )  ->  ( E. v  e.  ( ~P X  i^i  Fin ) U_ x  e.  v 
( x ( ball `  M ) s )  =  X  <->  E. v  e.  ( ~P X  i^i  Fin ) U_ x  e.  v  ( x (
ball `  M )
( r  /  R
) )  =  X ) )
1615rspcv 2880 . . . . 5  |-  ( ( r  /  R )  e.  RR+  ->  ( A. s  e.  RR+  E. v  e.  ( ~P X  i^i  Fin ) U_ x  e.  v  ( x (
ball `  M )
s )  =  X  ->  E. v  e.  ( ~P X  i^i  Fin ) U_ x  e.  v  ( x ( ball `  M ) ( r  /  R ) )  =  X ) )
176, 11, 16sylc 56 . . . 4  |-  ( (
ph  /\  r  e.  RR+ )  ->  E. v  e.  ( ~P X  i^i  Fin ) U_ x  e.  v  ( x (
ball `  M )
( r  /  R
) )  =  X )
18 elfpw 7157 . . . . . . . . . . . . . 14  |-  ( v  e.  ( ~P X  i^i  Fin )  <->  ( v  C_  X  /\  v  e. 
Fin ) )
1918simplbi 446 . . . . . . . . . . . . 13  |-  ( v  e.  ( ~P X  i^i  Fin )  ->  v  C_  X )
2019adantl 452 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  r  e.  RR+ )  /\  v  e.  ( ~P X  i^i  Fin ) )  ->  v  C_  X )
2120sselda 3180 . . . . . . . . . . 11  |-  ( ( ( ( ph  /\  r  e.  RR+ )  /\  v  e.  ( ~P X  i^i  Fin ) )  /\  x  e.  v )  ->  x  e.  X )
22 eqid 2283 . . . . . . . . . . . . . 14  |-  ( MetOpen `  N )  =  (
MetOpen `  N )
23 eqid 2283 . . . . . . . . . . . . . 14  |-  ( MetOpen `  M )  =  (
MetOpen `  M )
249simplbi 446 . . . . . . . . . . . . . . 15  |-  ( M  e.  ( TotBnd `  X
)  ->  M  e.  ( Met `  X ) )
257, 24syl 15 . . . . . . . . . . . . . 14  |-  ( ph  ->  M  e.  ( Met `  X ) )
26 equivtotbnd.4 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  ( x  e.  X  /\  y  e.  X ) )  -> 
( x N y )  <_  ( R  x.  ( x M y ) ) )
2722, 23, 1, 25, 3, 26metss2lem 18057 . . . . . . . . . . . . 13  |-  ( (
ph  /\  ( x  e.  X  /\  r  e.  RR+ ) )  -> 
( x ( ball `  M ) ( r  /  R ) ) 
C_  ( x (
ball `  N )
r ) )
2827anass1rs 782 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  r  e.  RR+ )  /\  x  e.  X )  ->  (
x ( ball `  M
) ( r  /  R ) )  C_  ( x ( ball `  N ) r ) )
2928adantlr 695 . . . . . . . . . . 11  |-  ( ( ( ( ph  /\  r  e.  RR+ )  /\  v  e.  ( ~P X  i^i  Fin ) )  /\  x  e.  X
)  ->  ( x
( ball `  M )
( r  /  R
) )  C_  (
x ( ball `  N
) r ) )
3021, 29syldan 456 . . . . . . . . . 10  |-  ( ( ( ( ph  /\  r  e.  RR+ )  /\  v  e.  ( ~P X  i^i  Fin ) )  /\  x  e.  v )  ->  ( x
( ball `  M )
( r  /  R
) )  C_  (
x ( ball `  N
) r ) )
3130ralrimiva 2626 . . . . . . . . 9  |-  ( ( ( ph  /\  r  e.  RR+ )  /\  v  e.  ( ~P X  i^i  Fin ) )  ->  A. x  e.  v  ( x
( ball `  M )
( r  /  R
) )  C_  (
x ( ball `  N
) r ) )
32 ss2iun 3920 . . . . . . . . 9  |-  ( A. x  e.  v  (
x ( ball `  M
) ( r  /  R ) )  C_  ( x ( ball `  N ) r )  ->  U_ x  e.  v  ( x ( ball `  M ) ( r  /  R ) ) 
C_  U_ x  e.  v  ( x ( ball `  N ) r ) )
3331, 32syl 15 . . . . . . . 8  |-  ( ( ( ph  /\  r  e.  RR+ )  /\  v  e.  ( ~P X  i^i  Fin ) )  ->  U_ x  e.  v  ( x
( ball `  M )
( r  /  R
) )  C_  U_ x  e.  v  ( x
( ball `  N )
r ) )
34 sseq1 3199 . . . . . . . 8  |-  ( U_ x  e.  v  (
x ( ball `  M
) ( r  /  R ) )  =  X  ->  ( U_ x  e.  v  (
x ( ball `  M
) ( r  /  R ) )  C_  U_ x  e.  v  ( x ( ball `  N
) r )  <->  X  C_  U_ x  e.  v  ( x
( ball `  N )
r ) ) )
3533, 34syl5ibcom 211 . . . . . . 7  |-  ( ( ( ph  /\  r  e.  RR+ )  /\  v  e.  ( ~P X  i^i  Fin ) )  ->  ( U_ x  e.  v 
( x ( ball `  M ) ( r  /  R ) )  =  X  ->  X  C_ 
U_ x  e.  v  ( x ( ball `  N ) r ) ) )
361ad3antrrr 710 . . . . . . . . . . 11  |-  ( ( ( ( ph  /\  r  e.  RR+ )  /\  v  e.  ( ~P X  i^i  Fin ) )  /\  x  e.  v )  ->  N  e.  ( Met `  X ) )
37 metxmet 17899 . . . . . . . . . . 11  |-  ( N  e.  ( Met `  X
)  ->  N  e.  ( * Met `  X
) )
3836, 37syl 15 . . . . . . . . . 10  |-  ( ( ( ( ph  /\  r  e.  RR+ )  /\  v  e.  ( ~P X  i^i  Fin ) )  /\  x  e.  v )  ->  N  e.  ( * Met `  X
) )
39 simpllr 735 . . . . . . . . . . 11  |-  ( ( ( ( ph  /\  r  e.  RR+ )  /\  v  e.  ( ~P X  i^i  Fin ) )  /\  x  e.  v )  ->  r  e.  RR+ )
40 rpxr 10361 . . . . . . . . . . 11  |-  ( r  e.  RR+  ->  r  e. 
RR* )
4139, 40syl 15 . . . . . . . . . 10  |-  ( ( ( ( ph  /\  r  e.  RR+ )  /\  v  e.  ( ~P X  i^i  Fin ) )  /\  x  e.  v )  ->  r  e.  RR* )
42 blssm 17968 . . . . . . . . . 10  |-  ( ( N  e.  ( * Met `  X )  /\  x  e.  X  /\  r  e.  RR* )  ->  ( x ( ball `  N ) r ) 
C_  X )
4338, 21, 41, 42syl3anc 1182 . . . . . . . . 9  |-  ( ( ( ( ph  /\  r  e.  RR+ )  /\  v  e.  ( ~P X  i^i  Fin ) )  /\  x  e.  v )  ->  ( x
( ball `  N )
r )  C_  X
)
4443ralrimiva 2626 . . . . . . . 8  |-  ( ( ( ph  /\  r  e.  RR+ )  /\  v  e.  ( ~P X  i^i  Fin ) )  ->  A. x  e.  v  ( x
( ball `  N )
r )  C_  X
)
45 iunss 3943 . . . . . . . 8  |-  ( U_ x  e.  v  (
x ( ball `  N
) r )  C_  X 
<-> 
A. x  e.  v  ( x ( ball `  N ) r ) 
C_  X )
4644, 45sylibr 203 . . . . . . 7  |-  ( ( ( ph  /\  r  e.  RR+ )  /\  v  e.  ( ~P X  i^i  Fin ) )  ->  U_ x  e.  v  ( x
( ball `  N )
r )  C_  X
)
4735, 46jctild 527 . . . . . 6  |-  ( ( ( ph  /\  r  e.  RR+ )  /\  v  e.  ( ~P X  i^i  Fin ) )  ->  ( U_ x  e.  v 
( x ( ball `  M ) ( r  /  R ) )  =  X  ->  ( U_ x  e.  v 
( x ( ball `  N ) r ) 
C_  X  /\  X  C_ 
U_ x  e.  v  ( x ( ball `  N ) r ) ) ) )
48 eqss 3194 . . . . . 6  |-  ( U_ x  e.  v  (
x ( ball `  N
) r )  =  X  <->  ( U_ x  e.  v  ( x
( ball `  N )
r )  C_  X  /\  X  C_  U_ x  e.  v  ( x
( ball `  N )
r ) ) )
4947, 48syl6ibr 218 . . . . 5  |-  ( ( ( ph  /\  r  e.  RR+ )  /\  v  e.  ( ~P X  i^i  Fin ) )  ->  ( U_ x  e.  v 
( x ( ball `  M ) ( r  /  R ) )  =  X  ->  U_ x  e.  v  ( x
( ball `  N )
r )  =  X ) )
5049reximdva 2655 . . . 4  |-  ( (
ph  /\  r  e.  RR+ )  ->  ( E. v  e.  ( ~P X  i^i  Fin ) U_ x  e.  v  (
x ( ball `  M
) ( r  /  R ) )  =  X  ->  E. v  e.  ( ~P X  i^i  Fin ) U_ x  e.  v  ( x (
ball `  N )
r )  =  X ) )
5117, 50mpd 14 . . 3  |-  ( (
ph  /\  r  e.  RR+ )  ->  E. v  e.  ( ~P X  i^i  Fin ) U_ x  e.  v  ( x (
ball `  N )
r )  =  X )
5251ralrimiva 2626 . 2  |-  ( ph  ->  A. r  e.  RR+  E. v  e.  ( ~P X  i^i  Fin ) U_ x  e.  v 
( x ( ball `  N ) r )  =  X )
53 istotbnd3 26495 . 2  |-  ( N  e.  ( TotBnd `  X
)  <->  ( N  e.  ( Met `  X
)  /\  A. r  e.  RR+  E. v  e.  ( ~P X  i^i  Fin ) U_ x  e.  v  ( x (
ball `  N )
r )  =  X ) )
541, 52, 53sylanbrc 645 1  |-  ( ph  ->  N  e.  ( TotBnd `  X ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684   A.wral 2543   E.wrex 2544    i^i cin 3151    C_ wss 3152   ~Pcpw 3625   U_ciun 3905   class class class wbr 4023   ` cfv 5255  (class class class)co 5858   Fincfn 6863    x. cmul 8742   RR*cxr 8866    <_ cle 8868    / cdiv 9423   RR+crp 10354   * Metcxmt 16369   Metcme 16370   ballcbl 16371   MetOpencmopn 16372   TotBndctotbnd 26490
This theorem is referenced by:  equivbnd2  26516
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-oadd 6483  df-er 6660  df-map 6774  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-rp 10355  df-xadd 10453  df-xmet 16373  df-met 16374  df-bl 16375  df-totbnd 26492
  Copyright terms: Public domain W3C validator