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Theorem isbndx 26184
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 26182 . 2  |-  ( M  e.  ( Bnd `  X
)  <->  ( M  e.  ( Met `  X
)  /\  A. x  e.  X  E. r  e.  RR+  X  =  ( x ( ball `  M
) r ) ) )
2 metxmet 18275 . . . 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 18270 . . . . . . . 8  |-  ( M  e.  ( * Met `  X )  ->  M : ( X  X.  X ) --> RR* )
5 ffn 5533 . . . . . . . 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 10553 . . . . . . . . . . . . . . . . . 18  |-  ( r  e.  RR+  ->  r  e. 
RR* )
9 eqid 2389 . . . . . . . . . . . . . . . . . . . 20  |-  ( `' M " RR )  =  ( `' M " RR )
109blssec 18357 . . . . . . . . . . . . . . . . . . 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 3327 . . . . . . . . . . . . . . 15  |-  ( ( ( M  e.  ( * Met `  X
)  /\  x  e.  X )  /\  (
r  e.  RR+  /\  X  =  ( x (
ball `  M )
r ) ) )  ->  X  C_  [ x ] ( `' M " RR ) )
1514sselda 3293 . . . . . . . . . . . . . 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 2904 . . . . . . . . . . . . . . 15  |-  y  e. 
_V
17 vex 2904 . . . . . . . . . . . . . . 15  |-  x  e. 
_V
1816, 17elec 6882 . . . . . . . . . . . . . 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 18354 . . . . . . . . . . . . . 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 2734 . . . . . . . . . 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 2776 . . . . . . . . 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 2729 . . . . . . . 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 6115 . . . . . . 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 18274 . . . . . 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 1717   A.wral 2651   E.wrex 2652    C_ wss 3265   class class class wbr 4155    X. cxp 4818   `'ccnv 4819   "cima 4823    Fn wfn 5391   -->wf 5392   ` cfv 5396  (class class class)co 6022   [cec 6841   RRcr 8924   RR*cxr 9054   RR+crp 10546   * Metcxmt 16614   Metcme 16615   ballcbl 16616   Bndcbnd 26169
This theorem is referenced by:  isbnd2  26185  blbnd  26189  ismtybndlem  26208
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 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2370  ax-sep 4273  ax-nul 4281  ax-pow 4320  ax-pr 4346  ax-un 4643  ax-cnex 8981  ax-resscn 8982  ax-1cn 8983  ax-icn 8984  ax-addcl 8985  ax-addrcl 8986  ax-mulcl 8987  ax-mulrcl 8988  ax-mulcom 8989  ax-addass 8990  ax-mulass 8991  ax-distr 8992  ax-i2m1 8993  ax-1ne0 8994  ax-1rid 8995  ax-rnegex 8996  ax-rrecex 8997  ax-cnre 8998  ax-pre-lttri 8999  ax-pre-lttrn 9000  ax-pre-ltadd 9001  ax-pre-mulgt0 9002
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 2244  df-mo 2245  df-clab 2376  df-cleq 2382  df-clel 2385  df-nfc 2514  df-ne 2554  df-nel 2555  df-ral 2656  df-rex 2657  df-reu 2658  df-rmo 2659  df-rab 2660  df-v 2903  df-sbc 3107  df-csb 3197  df-dif 3268  df-un 3270  df-in 3272  df-ss 3279  df-nul 3574  df-if 3685  df-pw 3746  df-sn 3765  df-pr 3766  df-op 3768  df-uni 3960  df-iun 4039  df-br 4156  df-opab 4210  df-mpt 4211  df-id 4441  df-po 4446  df-so 4447  df-xp 4826  df-rel 4827  df-cnv 4828  df-co 4829  df-dm 4830  df-rn 4831  df-res 4832  df-ima 4833  df-iota 5360  df-fun 5398  df-fn 5399  df-f 5400  df-f1 5401  df-fo 5402  df-f1o 5403  df-fv 5404  df-ov 6025  df-oprab 6026  df-mpt2 6027  df-1st 6290  df-2nd 6291  df-riota 6487  df-er 6843  df-ec 6845  df-map 6958  df-en 7048  df-dom 7049  df-sdom 7050  df-pnf 9057  df-mnf 9058  df-xr 9059  df-ltxr 9060  df-le 9061  df-sub 9227  df-neg 9228  df-div 9612  df-2 9992  df-rp 10547  df-xneg 10644  df-xadd 10645  df-xmul 10646  df-xmet 16621  df-met 16622  df-bl 16623  df-bnd 26181
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