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

Theorem isbndx 26381
Description: A "bounded extended metric" (meaning that it satisfies the same condition as a bounded metric, but with "metric" replaced with "extended metric") is a metric and thus is bounded in the conventional sense. (Contributed by Mario Carneiro, 12-Sep-2015.)
Assertion
Ref Expression
isbndx  |-  ( M  e.  ( Bnd `  X
)  <->  ( M  e.  ( * Met `  X
)  /\  A. x  e.  X  E. r  e.  RR+  X  =  ( x ( ball `  M
) r ) ) )
Distinct variable groups:    x, r, M    X, r, x

Proof of Theorem isbndx
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 isbnd 26379 . 2  |-  ( M  e.  ( Bnd `  X
)  <->  ( M  e.  ( Met `  X
)  /\  A. x  e.  X  E. r  e.  RR+  X  =  ( x ( ball `  M
) r ) ) )
2 metxmet 18317 . . . 4  |-  ( M  e.  ( Met `  X
)  ->  M  e.  ( * Met `  X
) )
3 simpr 448 . . . . . 6  |-  ( ( A. x  e.  X  E. r  e.  RR+  X  =  ( x ( ball `  M ) r )  /\  M  e.  ( * Met `  X
) )  ->  M  e.  ( * Met `  X
) )
4 xmetf 18312 . . . . . . . 8  |-  ( M  e.  ( * Met `  X )  ->  M : ( X  X.  X ) --> RR* )
5 ffn 5550 . . . . . . . 8  |-  ( M : ( X  X.  X ) --> RR*  ->  M  Fn  ( X  X.  X ) )
63, 4, 53syl 19 . . . . . . 7  |-  ( ( A. x  e.  X  E. r  e.  RR+  X  =  ( x ( ball `  M ) r )  /\  M  e.  ( * Met `  X
) )  ->  M  Fn  ( X  X.  X
) )
7 simprr 734 . . . . . . . . . . . . . . . 16  |-  ( ( ( M  e.  ( * Met `  X
)  /\  x  e.  X )  /\  (
r  e.  RR+  /\  X  =  ( x (
ball `  M )
r ) ) )  ->  X  =  ( x ( ball `  M
) r ) )
8 rpxr 10575 . . . . . . . . . . . . . . . . . 18  |-  ( r  e.  RR+  ->  r  e. 
RR* )
9 eqid 2404 . . . . . . . . . . . . . . . . . . . 20  |-  ( `' M " RR )  =  ( `' M " RR )
109blssec 18418 . . . . . . . . . . . . . . . . . . 19  |-  ( ( M  e.  ( * Met `  X )  /\  x  e.  X  /\  r  e.  RR* )  ->  ( x ( ball `  M ) r ) 
C_  [ x ]
( `' M " RR ) )
11103expa 1153 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( M  e.  ( * Met `  X
)  /\  x  e.  X )  /\  r  e.  RR* )  ->  (
x ( ball `  M
) r )  C_  [ x ] ( `' M " RR ) )
128, 11sylan2 461 . . . . . . . . . . . . . . . . 17  |-  ( ( ( M  e.  ( * Met `  X
)  /\  x  e.  X )  /\  r  e.  RR+ )  ->  (
x ( ball `  M
) r )  C_  [ x ] ( `' M " RR ) )
1312adantrr 698 . . . . . . . . . . . . . . . 16  |-  ( ( ( M  e.  ( * Met `  X
)  /\  x  e.  X )  /\  (
r  e.  RR+  /\  X  =  ( x (
ball `  M )
r ) ) )  ->  ( x (
ball `  M )
r )  C_  [ x ] ( `' M " RR ) )
147, 13eqsstrd 3342 . . . . . . . . . . . . . . 15  |-  ( ( ( M  e.  ( * Met `  X
)  /\  x  e.  X )  /\  (
r  e.  RR+  /\  X  =  ( x (
ball `  M )
r ) ) )  ->  X  C_  [ x ] ( `' M " RR ) )
1514sselda 3308 . . . . . . . . . . . . . 14  |-  ( ( ( ( M  e.  ( * Met `  X
)  /\  x  e.  X )  /\  (
r  e.  RR+  /\  X  =  ( x (
ball `  M )
r ) ) )  /\  y  e.  X
)  ->  y  e.  [ x ] ( `' M " RR ) )
16 vex 2919 . . . . . . . . . . . . . . 15  |-  y  e. 
_V
17 vex 2919 . . . . . . . . . . . . . . 15  |-  x  e. 
_V
1816, 17elec 6903 . . . . . . . . . . . . . 14  |-  ( y  e.  [ x ]
( `' M " RR )  <->  x ( `' M " RR ) y )
1915, 18sylib 189 . . . . . . . . . . . . 13  |-  ( ( ( ( M  e.  ( * Met `  X
)  /\  x  e.  X )  /\  (
r  e.  RR+  /\  X  =  ( x (
ball `  M )
r ) ) )  /\  y  e.  X
)  ->  x ( `' M " RR ) y )
209xmeterval 18415 . . . . . . . . . . . . . 14  |-  ( M  e.  ( * Met `  X )  ->  (
x ( `' M " RR ) y  <->  ( x  e.  X  /\  y  e.  X  /\  (
x M y )  e.  RR ) ) )
2120ad3antrrr 711 . . . . . . . . . . . . 13  |-  ( ( ( ( M  e.  ( * Met `  X
)  /\  x  e.  X )  /\  (
r  e.  RR+  /\  X  =  ( x (
ball `  M )
r ) ) )  /\  y  e.  X
)  ->  ( x
( `' M " RR ) y  <->  ( x  e.  X  /\  y  e.  X  /\  (
x M y )  e.  RR ) ) )
2219, 21mpbid 202 . . . . . . . . . . . 12  |-  ( ( ( ( M  e.  ( * Met `  X
)  /\  x  e.  X )  /\  (
r  e.  RR+  /\  X  =  ( x (
ball `  M )
r ) ) )  /\  y  e.  X
)  ->  ( x  e.  X  /\  y  e.  X  /\  (
x M y )  e.  RR ) )
2322simp3d 971 . . . . . . . . . . 11  |-  ( ( ( ( M  e.  ( * Met `  X
)  /\  x  e.  X )  /\  (
r  e.  RR+  /\  X  =  ( x (
ball `  M )
r ) ) )  /\  y  e.  X
)  ->  ( x M y )  e.  RR )
2423ralrimiva 2749 . . . . . . . . . 10  |-  ( ( ( M  e.  ( * Met `  X
)  /\  x  e.  X )  /\  (
r  e.  RR+  /\  X  =  ( x (
ball `  M )
r ) ) )  ->  A. y  e.  X  ( x M y )  e.  RR )
2524rexlimdvaa 2791 . . . . . . . . 9  |-  ( ( M  e.  ( * Met `  X )  /\  x  e.  X
)  ->  ( E. r  e.  RR+  X  =  ( x ( ball `  M ) r )  ->  A. y  e.  X  ( x M y )  e.  RR ) )
2625ralimdva 2744 . . . . . . . 8  |-  ( M  e.  ( * Met `  X )  ->  ( A. x  e.  X  E. r  e.  RR+  X  =  ( x ( ball `  M ) r )  ->  A. x  e.  X  A. y  e.  X  ( x M y )  e.  RR ) )
2726impcom 420 . . . . . . 7  |-  ( ( A. x  e.  X  E. r  e.  RR+  X  =  ( x ( ball `  M ) r )  /\  M  e.  ( * Met `  X
) )  ->  A. x  e.  X  A. y  e.  X  ( x M y )  e.  RR )
28 ffnov 6133 . . . . . . 7  |-  ( M : ( X  X.  X ) --> RR  <->  ( M  Fn  ( X  X.  X
)  /\  A. x  e.  X  A. y  e.  X  ( x M y )  e.  RR ) )
296, 27, 28sylanbrc 646 . . . . . 6  |-  ( ( A. x  e.  X  E. r  e.  RR+  X  =  ( x ( ball `  M ) r )  /\  M  e.  ( * Met `  X
) )  ->  M : ( X  X.  X ) --> RR )
30 ismet2 18316 . . . . . 6  |-  ( M  e.  ( Met `  X
)  <->  ( M  e.  ( * Met `  X
)  /\  M :
( X  X.  X
) --> RR ) )
313, 29, 30sylanbrc 646 . . . . 5  |-  ( ( A. x  e.  X  E. r  e.  RR+  X  =  ( x ( ball `  M ) r )  /\  M  e.  ( * Met `  X
) )  ->  M  e.  ( Met `  X
) )
3231ex 424 . . . 4  |-  ( A. x  e.  X  E. r  e.  RR+  X  =  ( x ( ball `  M ) r )  ->  ( M  e.  ( * Met `  X
)  ->  M  e.  ( Met `  X ) ) )
332, 32impbid2 196 . . 3  |-  ( A. x  e.  X  E. r  e.  RR+  X  =  ( x ( ball `  M ) r )  ->  ( M  e.  ( Met `  X
)  <->  M  e.  ( * Met `  X ) ) )
3433pm5.32ri 620 . 2  |-  ( ( M  e.  ( Met `  X )  /\  A. x  e.  X  E. r  e.  RR+  X  =  ( x ( ball `  M ) r ) )  <->  ( M  e.  ( * Met `  X
)  /\  A. x  e.  X  E. r  e.  RR+  X  =  ( x ( ball `  M
) r ) ) )
351, 34bitri 241 1  |-  ( M  e.  ( Bnd `  X
)  <->  ( M  e.  ( * Met `  X
)  /\  A. x  e.  X  E. r  e.  RR+  X  =  ( x ( ball `  M
) r ) ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1721   A.wral 2666   E.wrex 2667    C_ wss 3280   class class class wbr 4172    X. cxp 4835   `'ccnv 4836   "cima 4840    Fn wfn 5408   -->wf 5409   ` cfv 5413  (class class class)co 6040   [cec 6862   RRcr 8945   RR*cxr 9075   RR+crp 10568   * Metcxmt 16641   Metcme 16642   ballcbl 16643   Bndcbnd 26366
This theorem is referenced by:  isbnd2  26382  blbnd  26386  ismtybndlem  26405
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-id 4458  df-po 4463  df-so 4464  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-riota 6508  df-er 6864  df-ec 6866  df-map 6979  df-en 7069  df-dom 7070  df-sdom 7071  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-div 9634  df-2 10014  df-rp 10569  df-xneg 10666  df-xadd 10667  df-xmul 10668  df-psmet 16649  df-xmet 16650  df-met 16651  df-bl 16652  df-bnd 26378
  Copyright terms: Public domain W3C validator