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Theorem equivtotbnd 29864
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 461 . . . . . 6  |-  ( (
ph  /\  r  e.  RR+ )  ->  r  e.  RR+ )
3 equivtotbnd.3 . . . . . . 7  |-  ( ph  ->  R  e.  RR+ )
43adantr 465 . . . . . 6  |-  ( (
ph  /\  r  e.  RR+ )  ->  R  e.  RR+ )
52, 4rpdivcld 11262 . . . . 5  |-  ( (
ph  /\  r  e.  RR+ )  ->  ( r  /  R )  e.  RR+ )
6 equivtotbnd.1 . . . . . . 7  |-  ( ph  ->  M  e.  ( TotBnd `  X ) )
76adantr 465 . . . . . 6  |-  ( (
ph  /\  r  e.  RR+ )  ->  M  e.  ( TotBnd `  X )
)
8 istotbnd3 29857 . . . . . . 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 464 . . . . . 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 6283 . . . . . . . . 9  |-  ( s  =  ( r  /  R )  ->  (
x ( ball `  M
) s )  =  ( x ( ball `  M ) ( r  /  R ) ) )
1211iuneq2d 4345 . . . . . . . 8  |-  ( s  =  ( r  /  R )  ->  U_ x  e.  v  ( x
( ball `  M )
s )  =  U_ x  e.  v  (
x ( ball `  M
) ( r  /  R ) ) )
1312eqeq1d 2462 . . . . . . 7  |-  ( s  =  ( r  /  R )  ->  ( U_ x  e.  v 
( x ( ball `  M ) s )  =  X  <->  U_ x  e.  v  ( x (
ball `  M )
( r  /  R
) )  =  X ) )
1413rexbidv 2966 . . . . . 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 3203 . . . . 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 60 . . . 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 7811 . . . . . . . . . . . . . 14  |-  ( v  e.  ( ~P X  i^i  Fin )  <->  ( v  C_  X  /\  v  e. 
Fin ) )
1817simplbi 460 . . . . . . . . . . . . 13  |-  ( v  e.  ( ~P X  i^i  Fin )  ->  v  C_  X )
1918adantl 466 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  r  e.  RR+ )  /\  v  e.  ( ~P X  i^i  Fin ) )  ->  v  C_  X )
2019sselda 3497 . . . . . . . . . . 11  |-  ( ( ( ( ph  /\  r  e.  RR+ )  /\  v  e.  ( ~P X  i^i  Fin ) )  /\  x  e.  v )  ->  x  e.  X )
21 eqid 2460 . . . . . . . . . . . . . 14  |-  ( MetOpen `  N )  =  (
MetOpen `  N )
22 eqid 2460 . . . . . . . . . . . . . 14  |-  ( MetOpen `  M )  =  (
MetOpen `  M )
238simplbi 460 . . . . . . . . . . . . . . 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 20742 . . . . . . . . . . . . 13  |-  ( (
ph  /\  ( x  e.  X  /\  r  e.  RR+ ) )  -> 
( x ( ball `  M ) ( r  /  R ) ) 
C_  ( x (
ball `  N )
r ) )
2726anass1rs 805 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  r  e.  RR+ )  /\  x  e.  X )  ->  (
x ( ball `  M
) ( r  /  R ) )  C_  ( x ( ball `  N ) r ) )
2827adantlr 714 . . . . . . . . . . 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 470 . . . . . . . . . 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 2871 . . . . . . . . 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 4334 . . . . . . . . 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 3518 . . . . . . . 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 220 . . . . . . 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 729 . . . . . . . . . . 11  |-  ( ( ( ( ph  /\  r  e.  RR+ )  /\  v  e.  ( ~P X  i^i  Fin ) )  /\  x  e.  v )  ->  N  e.  ( Met `  X ) )
36 metxmet 20565 . . . . . . . . . . 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 758 . . . . . . . . . . 11  |-  ( ( ( ( ph  /\  r  e.  RR+ )  /\  v  e.  ( ~P X  i^i  Fin ) )  /\  x  e.  v )  ->  r  e.  RR+ )
3938rpxrd 11246 . . . . . . . . . 10  |-  ( ( ( ( ph  /\  r  e.  RR+ )  /\  v  e.  ( ~P X  i^i  Fin ) )  /\  x  e.  v )  ->  r  e.  RR* )
40 blssm 20649 . . . . . . . . . 10  |-  ( ( N  e.  ( *Met `  X )  /\  x  e.  X  /\  r  e.  RR* )  ->  ( x ( ball `  N ) r ) 
C_  X )
4137, 20, 39, 40syl3anc 1223 . . . . . . . . 9  |-  ( ( ( ( ph  /\  r  e.  RR+ )  /\  v  e.  ( ~P X  i^i  Fin ) )  /\  x  e.  v )  ->  ( x
( ball `  N )
r )  C_  X
)
4241ralrimiva 2871 . . . . . . . 8  |-  ( ( ( ph  /\  r  e.  RR+ )  /\  v  e.  ( ~P X  i^i  Fin ) )  ->  A. x  e.  v  ( x
( ball `  N )
r )  C_  X
)
43 iunss 4359 . . . . . . . 8  |-  ( U_ x  e.  v  (
x ( ball `  N
) r )  C_  X 
<-> 
A. x  e.  v  ( x ( ball `  N ) r ) 
C_  X )
4442, 43sylibr 212 . . . . . . 7  |-  ( ( ( ph  /\  r  e.  RR+ )  /\  v  e.  ( ~P X  i^i  Fin ) )  ->  U_ x  e.  v  ( x
( ball `  N )
r )  C_  X
)
4534, 44jctild 543 . . . . . 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 3512 . . . . . 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 227 . . . . 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 2931 . . . 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 2871 . 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 29857 . 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 664 1  |-  ( ph  ->  N  e.  ( TotBnd `  X ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1374    e. wcel 1762   A.wral 2807   E.wrex 2808    i^i cin 3468    C_ wss 3469   ~Pcpw 4003   U_ciun 4318   class class class wbr 4440   ` cfv 5579  (class class class)co 6275   Fincfn 7506    x. cmul 9486   RR*cxr 9616    <_ cle 9618    / cdiv 10195   RR+crp 11209   *Metcxmt 18167   Metcme 18168   ballcbl 18169   MetOpencmopn 18172   TotBndctotbnd 29852
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-nel 2658  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-pss 3485  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-tp 4025  df-op 4027  df-uni 4239  df-int 4276  df-iun 4320  df-br 4441  df-opab 4499  df-mpt 4500  df-tr 4534  df-eprel 4784  df-id 4788  df-po 4793  df-so 4794  df-fr 4831  df-we 4833  df-ord 4874  df-on 4875  df-lim 4876  df-suc 4877  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-riota 6236  df-ov 6278  df-oprab 6279  df-mpt2 6280  df-om 6672  df-1st 6774  df-2nd 6775  df-recs 7032  df-rdg 7066  df-1o 7120  df-oadd 7124  df-er 7301  df-map 7412  df-en 7507  df-dom 7508  df-sdom 7509  df-fin 7510  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9796  df-neg 9797  df-div 10196  df-rp 11210  df-xadd 11308  df-psmet 18175  df-xmet 18176  df-met 18177  df-bl 18178  df-totbnd 29854
This theorem is referenced by:  equivbnd2  29878
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