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Theorem equivtotbnd 26529
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 449 . . . . . 6  |-  ( (
ph  /\  r  e.  RR+ )  ->  r  e.  RR+ )
3 equivtotbnd.3 . . . . . . 7  |-  ( ph  ->  R  e.  RR+ )
43adantr 453 . . . . . 6  |-  ( (
ph  /\  r  e.  RR+ )  ->  R  e.  RR+ )
52, 4rpdivcld 10703 . . . . 5  |-  ( (
ph  /\  r  e.  RR+ )  ->  ( r  /  R )  e.  RR+ )
6 equivtotbnd.1 . . . . . . 7  |-  ( ph  ->  M  e.  ( TotBnd `  X ) )
76adantr 453 . . . . . 6  |-  ( (
ph  /\  r  e.  RR+ )  ->  M  e.  ( TotBnd `  X )
)
8 istotbnd3 26522 . . . . . . 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 ) )
98simprbi 452 . . . . . 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 )
107, 9syl 16 . . . . 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 )
11 oveq2 6125 . . . . . . . . 9  |-  ( s  =  ( r  /  R )  ->  (
x ( ball `  M
) s )  =  ( x ( ball `  M ) ( r  /  R ) ) )
1211iuneq2d 4148 . . . . . . . 8  |-  ( s  =  ( r  /  R )  ->  U_ x  e.  v  ( x
( ball `  M )
s )  =  U_ x  e.  v  (
x ( ball `  M
) ( r  /  R ) ) )
1312eqeq1d 2451 . . . . . . 7  |-  ( s  =  ( r  /  R )  ->  ( U_ x  e.  v 
( x ( ball `  M ) s )  =  X  <->  U_ x  e.  v  ( x (
ball `  M )
( r  /  R
) )  =  X ) )
1413rexbidv 2733 . . . . . 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 ) )
1514rspcv 3057 . . . . 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 ) )
165, 10, 15sylc 59 . . . 4  |-  ( (
ph  /\  r  e.  RR+ )  ->  E. v  e.  ( ~P X  i^i  Fin ) U_ x  e.  v  ( x (
ball `  M )
( r  /  R
) )  =  X )
17 elfpw 7444 . . . . . . . . . . . . . 14  |-  ( v  e.  ( ~P X  i^i  Fin )  <->  ( v  C_  X  /\  v  e. 
Fin ) )
1817simplbi 448 . . . . . . . . . . . . 13  |-  ( v  e.  ( ~P X  i^i  Fin )  ->  v  C_  X )
1918adantl 454 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  r  e.  RR+ )  /\  v  e.  ( ~P X  i^i  Fin ) )  ->  v  C_  X )
2019sselda 3337 . . . . . . . . . . 11  |-  ( ( ( ( ph  /\  r  e.  RR+ )  /\  v  e.  ( ~P X  i^i  Fin ) )  /\  x  e.  v )  ->  x  e.  X )
21 eqid 2443 . . . . . . . . . . . . . 14  |-  ( MetOpen `  N )  =  (
MetOpen `  N )
22 eqid 2443 . . . . . . . . . . . . . 14  |-  ( MetOpen `  M )  =  (
MetOpen `  M )
238simplbi 448 . . . . . . . . . . . . . . 15  |-  ( M  e.  ( TotBnd `  X
)  ->  M  e.  ( Met `  X ) )
246, 23syl 16 . . . . . . . . . . . . . 14  |-  ( ph  ->  M  e.  ( Met `  X ) )
25 equivtotbnd.4 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  ( x  e.  X  /\  y  e.  X ) )  -> 
( x N y )  <_  ( R  x.  ( x M y ) ) )
2621, 22, 1, 24, 3, 25metss2lem 18579 . . . . . . . . . . . . 13  |-  ( (
ph  /\  ( x  e.  X  /\  r  e.  RR+ ) )  -> 
( x ( ball `  M ) ( r  /  R ) ) 
C_  ( x (
ball `  N )
r ) )
2726anass1rs 784 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  r  e.  RR+ )  /\  x  e.  X )  ->  (
x ( ball `  M
) ( r  /  R ) )  C_  ( x ( ball `  N ) r ) )
2827adantlr 697 . . . . . . . . . . 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 ) )
2920, 28syldan 458 . . . . . . . . . 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 ) )
3029ralrimiva 2796 . . . . . . . . 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 ) )
31 ss2iun 4137 . . . . . . . . 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 ) )
3230, 31syl 16 . . . . . . . 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 ) )
33 sseq1 3358 . . . . . . . 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 ) ) )
3432, 33syl5ibcom 213 . . . . . . 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 ) ) )
351ad3antrrr 712 . . . . . . . . . . 11  |-  ( ( ( ( ph  /\  r  e.  RR+ )  /\  v  e.  ( ~P X  i^i  Fin ) )  /\  x  e.  v )  ->  N  e.  ( Met `  X ) )
36 metxmet 18402 . . . . . . . . . . 11  |-  ( N  e.  ( Met `  X
)  ->  N  e.  ( * Met `  X
) )
3735, 36syl 16 . . . . . . . . . 10  |-  ( ( ( ( ph  /\  r  e.  RR+ )  /\  v  e.  ( ~P X  i^i  Fin ) )  /\  x  e.  v )  ->  N  e.  ( * Met `  X
) )
38 simpllr 737 . . . . . . . . . . 11  |-  ( ( ( ( ph  /\  r  e.  RR+ )  /\  v  e.  ( ~P X  i^i  Fin ) )  /\  x  e.  v )  ->  r  e.  RR+ )
3938rpxrd 10687 . . . . . . . . . 10  |-  ( ( ( ( ph  /\  r  e.  RR+ )  /\  v  e.  ( ~P X  i^i  Fin ) )  /\  x  e.  v )  ->  r  e.  RR* )
40 blssm 18486 . . . . . . . . . 10  |-  ( ( N  e.  ( * Met `  X )  /\  x  e.  X  /\  r  e.  RR* )  ->  ( x ( ball `  N ) r ) 
C_  X )
4137, 20, 39, 40syl3anc 1185 . . . . . . . . 9  |-  ( ( ( ( ph  /\  r  e.  RR+ )  /\  v  e.  ( ~P X  i^i  Fin ) )  /\  x  e.  v )  ->  ( x
( ball `  N )
r )  C_  X
)
4241ralrimiva 2796 . . . . . . . 8  |-  ( ( ( ph  /\  r  e.  RR+ )  /\  v  e.  ( ~P X  i^i  Fin ) )  ->  A. x  e.  v  ( x
( ball `  N )
r )  C_  X
)
43 iunss 4162 . . . . . . . 8  |-  ( U_ x  e.  v  (
x ( ball `  N
) r )  C_  X 
<-> 
A. x  e.  v  ( x ( ball `  N ) r ) 
C_  X )
4442, 43sylibr 205 . . . . . . 7  |-  ( ( ( ph  /\  r  e.  RR+ )  /\  v  e.  ( ~P X  i^i  Fin ) )  ->  U_ x  e.  v  ( x
( ball `  N )
r )  C_  X
)
4534, 44jctild 529 . . . . . 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 ) ) ) )
46 eqss 3352 . . . . . 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 ) ) )
4745, 46syl6ibr 220 . . . . 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 ) )
4847reximdva 2825 . . . 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 ) )
4916, 48mpd 15 . . 3  |-  ( (
ph  /\  r  e.  RR+ )  ->  E. v  e.  ( ~P X  i^i  Fin ) U_ x  e.  v  ( x (
ball `  N )
r )  =  X )
5049ralrimiva 2796 . 2  |-  ( ph  ->  A. r  e.  RR+  E. v  e.  ( ~P X  i^i  Fin ) U_ x  e.  v 
( x ( ball `  N ) r )  =  X )
51 istotbnd3 26522 . 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 ) )
521, 50, 51sylanbrc 647 1  |-  ( ph  ->  N  e.  ( TotBnd `  X ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360    = wceq 1654    e. wcel 1728   A.wral 2712   E.wrex 2713    i^i cin 3308    C_ wss 3309   ~Pcpw 3828   U_ciun 4122   class class class wbr 4243   ` cfv 5489  (class class class)co 6117   Fincfn 7145    x. cmul 9033   RR*cxr 9157    <_ cle 9159    / cdiv 9715   RR+crp 10650   * Metcxmt 16724   Metcme 16725   ballcbl 16726   MetOpencmopn 16729   TotBndctotbnd 26517
This theorem is referenced by:  equivbnd2  26543
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1628  ax-9 1669  ax-8 1690  ax-13 1730  ax-14 1732  ax-6 1747  ax-7 1752  ax-11 1764  ax-12 1954  ax-ext 2424  ax-sep 4361  ax-nul 4369  ax-pow 4412  ax-pr 4438  ax-un 4736  ax-cnex 9084  ax-resscn 9085  ax-1cn 9086  ax-icn 9087  ax-addcl 9088  ax-addrcl 9089  ax-mulcl 9090  ax-mulrcl 9091  ax-mulcom 9092  ax-addass 9093  ax-mulass 9094  ax-distr 9095  ax-i2m1 9096  ax-1ne0 9097  ax-1rid 9098  ax-rnegex 9099  ax-rrecex 9100  ax-cnre 9101  ax-pre-lttri 9102  ax-pre-lttrn 9103  ax-pre-ltadd 9104  ax-pre-mulgt0 9105
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1661  df-eu 2292  df-mo 2293  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-nel 2609  df-ral 2717  df-rex 2718  df-reu 2719  df-rmo 2720  df-rab 2721  df-v 2967  df-sbc 3171  df-csb 3271  df-dif 3312  df-un 3314  df-in 3316  df-ss 3323  df-pss 3325  df-nul 3617  df-if 3768  df-pw 3830  df-sn 3849  df-pr 3850  df-tp 3851  df-op 3852  df-uni 4045  df-int 4080  df-iun 4124  df-br 4244  df-opab 4298  df-mpt 4299  df-tr 4334  df-eprel 4529  df-id 4533  df-po 4538  df-so 4539  df-fr 4576  df-we 4578  df-ord 4619  df-on 4620  df-lim 4621  df-suc 4622  df-om 4881  df-xp 4919  df-rel 4920  df-cnv 4921  df-co 4922  df-dm 4923  df-rn 4924  df-res 4925  df-ima 4926  df-iota 5453  df-fun 5491  df-fn 5492  df-f 5493  df-f1 5494  df-fo 5495  df-f1o 5496  df-fv 5497  df-ov 6120  df-oprab 6121  df-mpt2 6122  df-1st 6385  df-2nd 6386  df-riota 6585  df-recs 6669  df-rdg 6704  df-1o 6760  df-oadd 6764  df-er 6941  df-map 7056  df-en 7146  df-dom 7147  df-sdom 7148  df-fin 7149  df-pnf 9160  df-mnf 9161  df-xr 9162  df-ltxr 9163  df-le 9164  df-sub 9331  df-neg 9332  df-div 9716  df-rp 10651  df-xadd 10749  df-psmet 16732  df-xmet 16733  df-met 16734  df-bl 16735  df-totbnd 26519
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