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Theorem equivtotbnd 26489
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 10667 . . . . 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 26482 . . . . . . 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 6091 . . . . . . . . 9  |-  ( s  =  ( r  /  R )  ->  (
x ( ball `  M
) s )  =  ( x ( ball `  M ) ( r  /  R ) ) )
1211iuneq2d 4120 . . . . . . . 8  |-  ( s  =  ( r  /  R )  ->  U_ x  e.  v  ( x
( ball `  M )
s )  =  U_ x  e.  v  (
x ( ball `  M
) ( r  /  R ) ) )
1312eqeq1d 2446 . . . . . . 7  |-  ( s  =  ( r  /  R )  ->  ( U_ x  e.  v 
( x ( ball `  M ) s )  =  X  <->  U_ x  e.  v  ( x (
ball `  M )
( r  /  R
) )  =  X ) )
1413rexbidv 2728 . . . . . 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 3050 . . . . 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 7410 . . . . . . . . . . . . . 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 3350 . . . . . . . . . . 11  |-  ( ( ( ( ph  /\  r  e.  RR+ )  /\  v  e.  ( ~P X  i^i  Fin ) )  /\  x  e.  v )  ->  x  e.  X )
21 eqid 2438 . . . . . . . . . . . . . 14  |-  ( MetOpen `  N )  =  (
MetOpen `  N )
22 eqid 2438 . . . . . . . . . . . . . 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 18543 . . . . . . . . . . . . 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 2791 . . . . . . . . 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 4110 . . . . . . . . 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 3371 . . . . . . . 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 18366 . . . . . . . . . . 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 10651 . . . . . . . . . 10  |-  ( ( ( ( ph  /\  r  e.  RR+ )  /\  v  e.  ( ~P X  i^i  Fin ) )  /\  x  e.  v )  ->  r  e.  RR* )
40 blssm 18450 . . . . . . . . . 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 2791 . . . . . . . 8  |-  ( ( ( ph  /\  r  e.  RR+ )  /\  v  e.  ( ~P X  i^i  Fin ) )  ->  A. x  e.  v  ( x
( ball `  N )
r )  C_  X
)
43 iunss 4134 . . . . . . . 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 3365 . . . . . 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 2820 . . . 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 2791 . 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 26482 . 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 1653    e. wcel 1726   A.wral 2707   E.wrex 2708    i^i cin 3321    C_ wss 3322   ~Pcpw 3801   U_ciun 4095   class class class wbr 4214   ` cfv 5456  (class class class)co 6083   Fincfn 7111    x. cmul 8997   RR*cxr 9121    <_ cle 9123    / cdiv 9679   RR+crp 10614   * Metcxmt 16688   Metcme 16689   ballcbl 16690   MetOpencmopn 16693   TotBndctotbnd 26477
This theorem is referenced by:  equivbnd2  26503
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703  ax-cnex 9048  ax-resscn 9049  ax-1cn 9050  ax-icn 9051  ax-addcl 9052  ax-addrcl 9053  ax-mulcl 9054  ax-mulrcl 9055  ax-mulcom 9056  ax-addass 9057  ax-mulass 9058  ax-distr 9059  ax-i2m1 9060  ax-1ne0 9061  ax-1rid 9062  ax-rnegex 9063  ax-rrecex 9064  ax-cnre 9065  ax-pre-lttri 9066  ax-pre-lttrn 9067  ax-pre-ltadd 9068  ax-pre-mulgt0 9069
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 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rmo 2715  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-int 4053  df-iun 4097  df-br 4215  df-opab 4269  df-mpt 4270  df-tr 4305  df-eprel 4496  df-id 4500  df-po 4505  df-so 4506  df-fr 4543  df-we 4545  df-ord 4586  df-on 4587  df-lim 4588  df-suc 4589  df-om 4848  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-ov 6086  df-oprab 6087  df-mpt2 6088  df-1st 6351  df-2nd 6352  df-riota 6551  df-recs 6635  df-rdg 6670  df-1o 6726  df-oadd 6730  df-er 6907  df-map 7022  df-en 7112  df-dom 7113  df-sdom 7114  df-fin 7115  df-pnf 9124  df-mnf 9125  df-xr 9126  df-ltxr 9127  df-le 9128  df-sub 9295  df-neg 9296  df-div 9680  df-rp 10615  df-xadd 10713  df-psmet 16696  df-xmet 16697  df-met 16698  df-bl 16699  df-totbnd 26479
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