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Theorem isbndx 32178
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 32176 . 2  |-  ( M  e.  ( Bnd `  X
)  <->  ( M  e.  ( Met `  X
)  /\  A. x  e.  X  E. r  e.  RR+  X  =  ( x ( ball `  M
) r ) ) )
2 metxmet 21427 . . . 4  |-  ( M  e.  ( Met `  X
)  ->  M  e.  ( *Met `  X
) )
3 simpr 468 . . . . . 6  |-  ( ( A. x  e.  X  E. r  e.  RR+  X  =  ( x ( ball `  M ) r )  /\  M  e.  ( *Met `  X
) )  ->  M  e.  ( *Met `  X ) )
4 xmetf 21422 . . . . . . . 8  |-  ( M  e.  ( *Met `  X )  ->  M : ( X  X.  X ) --> RR* )
5 ffn 5739 . . . . . . . 8  |-  ( M : ( X  X.  X ) --> RR*  ->  M  Fn  ( X  X.  X ) )
63, 4, 53syl 18 . . . . . . 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 774 . . . . . . . . . . . . . . . 16  |-  ( ( ( M  e.  ( *Met `  X
)  /\  x  e.  X )  /\  (
r  e.  RR+  /\  X  =  ( x (
ball `  M )
r ) ) )  ->  X  =  ( x ( ball `  M
) r ) )
8 rpxr 11332 . . . . . . . . . . . . . . . . . 18  |-  ( r  e.  RR+  ->  r  e. 
RR* )
9 eqid 2471 . . . . . . . . . . . . . . . . . . . 20  |-  ( `' M " RR )  =  ( `' M " RR )
109blssec 21528 . . . . . . . . . . . . . . . . . . 19  |-  ( ( M  e.  ( *Met `  X )  /\  x  e.  X  /\  r  e.  RR* )  ->  ( x ( ball `  M ) r ) 
C_  [ x ]
( `' M " RR ) )
11103expa 1231 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( M  e.  ( *Met `  X
)  /\  x  e.  X )  /\  r  e.  RR* )  ->  (
x ( ball `  M
) r )  C_  [ x ] ( `' M " RR ) )
128, 11sylan2 482 . . . . . . . . . . . . . . . . 17  |-  ( ( ( M  e.  ( *Met `  X
)  /\  x  e.  X )  /\  r  e.  RR+ )  ->  (
x ( ball `  M
) r )  C_  [ x ] ( `' M " RR ) )
1312adantrr 731 . . . . . . . . . . . . . . . 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 3452 . . . . . . . . . . . . . . 15  |-  ( ( ( M  e.  ( *Met `  X
)  /\  x  e.  X )  /\  (
r  e.  RR+  /\  X  =  ( x (
ball `  M )
r ) ) )  ->  X  C_  [ x ] ( `' M " RR ) )
1514sselda 3418 . . . . . . . . . . . . . 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 3034 . . . . . . . . . . . . . . 15  |-  y  e. 
_V
17 vex 3034 . . . . . . . . . . . . . . 15  |-  x  e. 
_V
1816, 17elec 7421 . . . . . . . . . . . . . 14  |-  ( y  e.  [ x ]
( `' M " RR )  <->  x ( `' M " RR ) y )
1915, 18sylib 201 . . . . . . . . . . . . 13  |-  ( ( ( ( M  e.  ( *Met `  X )  /\  x  e.  X )  /\  (
r  e.  RR+  /\  X  =  ( x (
ball `  M )
r ) ) )  /\  y  e.  X
)  ->  x ( `' M " RR ) y )
209xmeterval 21525 . . . . . . . . . . . . . 14  |-  ( M  e.  ( *Met `  X )  ->  (
x ( `' M " RR ) y  <->  ( x  e.  X  /\  y  e.  X  /\  (
x M y )  e.  RR ) ) )
2120ad3antrrr 744 . . . . . . . . . . . . 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 215 . . . . . . . . . . . 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 1044 . . . . . . . . . . 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 2809 . . . . . . . . . 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 2872 . . . . . . . . 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 2805 . . . . . . . 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 437 . . . . . . 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 6419 . . . . . . 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 677 . . . . . 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 21426 . . . . . 6  |-  ( M  e.  ( Met `  X
)  <->  ( M  e.  ( *Met `  X )  /\  M : ( X  X.  X ) --> RR ) )
313, 29, 30sylanbrc 677 . . . . 5  |-  ( ( A. x  e.  X  E. r  e.  RR+  X  =  ( x ( ball `  M ) r )  /\  M  e.  ( *Met `  X
) )  ->  M  e.  ( Met `  X
) )
3231ex 441 . . . 4  |-  ( A. x  e.  X  E. r  e.  RR+  X  =  ( x ( ball `  M ) r )  ->  ( M  e.  ( *Met `  X )  ->  M  e.  ( Met `  X
) ) )
332, 32impbid2 209 . . 3  |-  ( A. x  e.  X  E. r  e.  RR+  X  =  ( x ( ball `  M ) r )  ->  ( M  e.  ( Met `  X
)  <->  M  e.  ( *Met `  X ) ) )
3433pm5.32ri 650 . 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 257 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 setvar class
Syntax hints:    <-> wb 189    /\ wa 376    /\ w3a 1007    = wceq 1452    e. wcel 1904   A.wral 2756   E.wrex 2757    C_ wss 3390   class class class wbr 4395    X. cxp 4837   `'ccnv 4838   "cima 4842    Fn wfn 5584   -->wf 5585   ` cfv 5589  (class class class)co 6308   [cec 7379   RRcr 9556   RR*cxr 9692   RR+crp 11325   *Metcxmt 19032   Metcme 19033   ballcbl 19034   Bndcbnd 32163
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602  ax-cnex 9613  ax-resscn 9614  ax-1cn 9615  ax-icn 9616  ax-addcl 9617  ax-addrcl 9618  ax-mulcl 9619  ax-mulrcl 9620  ax-mulcom 9621  ax-addass 9622  ax-mulass 9623  ax-distr 9624  ax-i2m1 9625  ax-1ne0 9626  ax-1rid 9627  ax-rnegex 9628  ax-rrecex 9629  ax-cnre 9630  ax-pre-lttri 9631  ax-pre-lttrn 9632  ax-pre-ltadd 9633  ax-pre-mulgt0 9634
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-nel 2644  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-op 3966  df-uni 4191  df-iun 4271  df-br 4396  df-opab 4455  df-mpt 4456  df-id 4754  df-po 4760  df-so 4761  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-riota 6270  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-1st 6812  df-2nd 6813  df-er 7381  df-ec 7383  df-map 7492  df-en 7588  df-dom 7589  df-sdom 7590  df-pnf 9695  df-mnf 9696  df-xr 9697  df-ltxr 9698  df-le 9699  df-sub 9882  df-neg 9883  df-div 10292  df-2 10690  df-rp 11326  df-xneg 11432  df-xadd 11433  df-xmul 11434  df-psmet 19039  df-xmet 19040  df-met 19041  df-bl 19042  df-bnd 32175
This theorem is referenced by:  isbnd2  32179  blbnd  32183  ismtybndlem  32202
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