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Theorem equivtotbnd 32068
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 463 . . . . . 6  |-  ( (
ph  /\  r  e.  RR+ )  ->  r  e.  RR+ )
3 equivtotbnd.3 . . . . . . 7  |-  ( ph  ->  R  e.  RR+ )
43adantr 467 . . . . . 6  |-  ( (
ph  /\  r  e.  RR+ )  ->  R  e.  RR+ )
52, 4rpdivcld 11360 . . . . 5  |-  ( (
ph  /\  r  e.  RR+ )  ->  ( r  /  R )  e.  RR+ )
6 equivtotbnd.1 . . . . . . 7  |-  ( ph  ->  M  e.  ( TotBnd `  X ) )
76adantr 467 . . . . . 6  |-  ( (
ph  /\  r  e.  RR+ )  ->  M  e.  ( TotBnd `  X )
)
8 istotbnd3 32061 . . . . . . 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 466 . . . . . 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 17 . . . . 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 6311 . . . . . . . . 9  |-  ( s  =  ( r  /  R )  ->  (
x ( ball `  M
) s )  =  ( x ( ball `  M ) ( r  /  R ) ) )
1211iuneq2d 4324 . . . . . . . 8  |-  ( s  =  ( r  /  R )  ->  U_ x  e.  v  ( x
( ball `  M )
s )  =  U_ x  e.  v  (
x ( ball `  M
) ( r  /  R ) ) )
1312eqeq1d 2425 . . . . . . 7  |-  ( s  =  ( r  /  R )  ->  ( U_ x  e.  v 
( x ( ball `  M ) s )  =  X  <->  U_ x  e.  v  ( x (
ball `  M )
( r  /  R
) )  =  X ) )
1413rexbidv 2940 . . . . . 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 3179 . . . . 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 63 . . . 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 7880 . . . . . . . . . . . . . 14  |-  ( v  e.  ( ~P X  i^i  Fin )  <->  ( v  C_  X  /\  v  e. 
Fin ) )
1817simplbi 462 . . . . . . . . . . . . 13  |-  ( v  e.  ( ~P X  i^i  Fin )  ->  v  C_  X )
1918adantl 468 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  r  e.  RR+ )  /\  v  e.  ( ~P X  i^i  Fin ) )  ->  v  C_  X )
2019sselda 3465 . . . . . . . . . . 11  |-  ( ( ( ( ph  /\  r  e.  RR+ )  /\  v  e.  ( ~P X  i^i  Fin ) )  /\  x  e.  v )  ->  x  e.  X )
21 eqid 2423 . . . . . . . . . . . . . 14  |-  ( MetOpen `  N )  =  (
MetOpen `  N )
22 eqid 2423 . . . . . . . . . . . . . 14  |-  ( MetOpen `  M )  =  (
MetOpen `  M )
238simplbi 462 . . . . . . . . . . . . . . 15  |-  ( M  e.  ( TotBnd `  X
)  ->  M  e.  ( Met `  X ) )
246, 23syl 17 . . . . . . . . . . . . . 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 21518 . . . . . . . . . . . . 13  |-  ( (
ph  /\  ( x  e.  X  /\  r  e.  RR+ ) )  -> 
( x ( ball `  M ) ( r  /  R ) ) 
C_  ( x (
ball `  N )
r ) )
2726anass1rs 815 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  r  e.  RR+ )  /\  x  e.  X )  ->  (
x ( ball `  M
) ( r  /  R ) )  C_  ( x ( ball `  N ) r ) )
2827adantlr 720 . . . . . . . . . . 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 473 . . . . . . . . . 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 2840 . . . . . . . . 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 4313 . . . . . . . . 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 17 . . . . . . . 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 3486 . . . . . . . 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 224 . . . . . . 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 735 . . . . . . . . . . 11  |-  ( ( ( ( ph  /\  r  e.  RR+ )  /\  v  e.  ( ~P X  i^i  Fin ) )  /\  x  e.  v )  ->  N  e.  ( Met `  X ) )
36 metxmet 21341 . . . . . . . . . . 11  |-  ( N  e.  ( Met `  X
)  ->  N  e.  ( *Met `  X
) )
3735, 36syl 17 . . . . . . . . . 10  |-  ( ( ( ( ph  /\  r  e.  RR+ )  /\  v  e.  ( ~P X  i^i  Fin ) )  /\  x  e.  v )  ->  N  e.  ( *Met `  X
) )
38 simpllr 768 . . . . . . . . . . 11  |-  ( ( ( ( ph  /\  r  e.  RR+ )  /\  v  e.  ( ~P X  i^i  Fin ) )  /\  x  e.  v )  ->  r  e.  RR+ )
3938rpxrd 11344 . . . . . . . . . 10  |-  ( ( ( ( ph  /\  r  e.  RR+ )  /\  v  e.  ( ~P X  i^i  Fin ) )  /\  x  e.  v )  ->  r  e.  RR* )
40 blssm 21425 . . . . . . . . . 10  |-  ( ( N  e.  ( *Met `  X )  /\  x  e.  X  /\  r  e.  RR* )  ->  ( x ( ball `  N ) r ) 
C_  X )
4137, 20, 39, 40syl3anc 1265 . . . . . . . . 9  |-  ( ( ( ( ph  /\  r  e.  RR+ )  /\  v  e.  ( ~P X  i^i  Fin ) )  /\  x  e.  v )  ->  ( x
( ball `  N )
r )  C_  X
)
4241ralrimiva 2840 . . . . . . . 8  |-  ( ( ( ph  /\  r  e.  RR+ )  /\  v  e.  ( ~P X  i^i  Fin ) )  ->  A. x  e.  v  ( x
( ball `  N )
r )  C_  X
)
43 iunss 4338 . . . . . . . 8  |-  ( U_ x  e.  v  (
x ( ball `  N
) r )  C_  X 
<-> 
A. x  e.  v  ( x ( ball `  N ) r ) 
C_  X )
4442, 43sylibr 216 . . . . . . 7  |-  ( ( ( ph  /\  r  e.  RR+ )  /\  v  e.  ( ~P X  i^i  Fin ) )  ->  U_ x  e.  v  ( x
( ball `  N )
r )  C_  X
)
4534, 44jctild 546 . . . . . 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 3480 . . . . . 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 231 . . . . 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 2901 . . . 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 2840 . 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 32061 . 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 669 1  |-  ( ph  ->  N  e.  ( TotBnd `  X ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 371    = wceq 1438    e. wcel 1869   A.wral 2776   E.wrex 2777    i^i cin 3436    C_ wss 3437   ~Pcpw 3980   U_ciun 4297   class class class wbr 4421   ` cfv 5599  (class class class)co 6303   Fincfn 7575    x. cmul 9546   RR*cxr 9676    <_ cle 9678    / cdiv 10271   RR+crp 11304   *Metcxmt 18948   Metcme 18949   ballcbl 18950   MetOpencmopn 18953   TotBndctotbnd 32056
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1666  ax-4 1679  ax-5 1749  ax-6 1795  ax-7 1840  ax-8 1871  ax-9 1873  ax-10 1888  ax-11 1893  ax-12 1906  ax-13 2054  ax-ext 2401  ax-sep 4544  ax-nul 4553  ax-pow 4600  ax-pr 4658  ax-un 6595  ax-cnex 9597  ax-resscn 9598  ax-1cn 9599  ax-icn 9600  ax-addcl 9601  ax-addrcl 9602  ax-mulcl 9603  ax-mulrcl 9604  ax-mulcom 9605  ax-addass 9606  ax-mulass 9607  ax-distr 9608  ax-i2m1 9609  ax-1ne0 9610  ax-1rid 9611  ax-rnegex 9612  ax-rrecex 9613  ax-cnre 9614  ax-pre-lttri 9615  ax-pre-lttrn 9616  ax-pre-ltadd 9617  ax-pre-mulgt0 9618
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 984  df-3an 985  df-tru 1441  df-ex 1661  df-nf 1665  df-sb 1788  df-eu 2270  df-mo 2271  df-clab 2409  df-cleq 2415  df-clel 2418  df-nfc 2573  df-ne 2621  df-nel 2622  df-ral 2781  df-rex 2782  df-reu 2783  df-rmo 2784  df-rab 2785  df-v 3084  df-sbc 3301  df-csb 3397  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-pss 3453  df-nul 3763  df-if 3911  df-pw 3982  df-sn 3998  df-pr 4000  df-tp 4002  df-op 4004  df-uni 4218  df-int 4254  df-iun 4299  df-br 4422  df-opab 4481  df-mpt 4482  df-tr 4517  df-eprel 4762  df-id 4766  df-po 4772  df-so 4773  df-fr 4810  df-we 4812  df-xp 4857  df-rel 4858  df-cnv 4859  df-co 4860  df-dm 4861  df-rn 4862  df-res 4863  df-ima 4864  df-pred 5397  df-ord 5443  df-on 5444  df-lim 5445  df-suc 5446  df-iota 5563  df-fun 5601  df-fn 5602  df-f 5603  df-f1 5604  df-fo 5605  df-f1o 5606  df-fv 5607  df-riota 6265  df-ov 6306  df-oprab 6307  df-mpt2 6308  df-om 6705  df-1st 6805  df-2nd 6806  df-wrecs 7034  df-recs 7096  df-rdg 7134  df-1o 7188  df-oadd 7192  df-er 7369  df-map 7480  df-en 7576  df-dom 7577  df-sdom 7578  df-fin 7579  df-pnf 9679  df-mnf 9680  df-xr 9681  df-ltxr 9682  df-le 9683  df-sub 9864  df-neg 9865  df-div 10272  df-rp 11305  df-xadd 11412  df-psmet 18955  df-xmet 18956  df-met 18957  df-bl 18958  df-totbnd 32058
This theorem is referenced by:  equivbnd2  32082
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