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Theorem isbndx 26506
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 26504 . 2  |-  ( M  e.  ( Bnd `  X
)  <->  ( M  e.  ( Met `  X
)  /\  A. x  e.  X  E. r  e.  RR+  X  =  ( x ( ball `  M
) r ) ) )
2 metxmet 17899 . . . 4  |-  ( M  e.  ( Met `  X
)  ->  M  e.  ( * Met `  X
) )
3 simpr 447 . . . . . 6  |-  ( ( A. x  e.  X  E. r  e.  RR+  X  =  ( x ( ball `  M ) r )  /\  M  e.  ( * Met `  X
) )  ->  M  e.  ( * Met `  X
) )
4 xmetf 17894 . . . . . . . 8  |-  ( M  e.  ( * Met `  X )  ->  M : ( X  X.  X ) --> RR* )
5 ffn 5389 . . . . . . . 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 733 . . . . . . . . . . . . . . . . 17  |-  ( ( ( M  e.  ( * Met `  X
)  /\  x  e.  X )  /\  (
r  e.  RR+  /\  X  =  ( x (
ball `  M )
r ) ) )  ->  X  =  ( x ( ball `  M
) r ) )
8 rpxr 10361 . . . . . . . . . . . . . . . . . . 19  |-  ( r  e.  RR+  ->  r  e. 
RR* )
9 eqid 2283 . . . . . . . . . . . . . . . . . . . . 21  |-  ( `' M " RR )  =  ( `' M " RR )
109blssec 17981 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( M  e.  ( * Met `  X )  /\  x  e.  X  /\  r  e.  RR* )  ->  ( x ( ball `  M ) r ) 
C_  [ x ]
( `' M " RR ) )
11103expa 1151 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( M  e.  ( * Met `  X
)  /\  x  e.  X )  /\  r  e.  RR* )  ->  (
x ( ball `  M
) r )  C_  [ x ] ( `' M " RR ) )
128, 11sylan2 460 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( M  e.  ( * Met `  X
)  /\  x  e.  X )  /\  r  e.  RR+ )  ->  (
x ( ball `  M
) r )  C_  [ x ] ( `' M " RR ) )
1312adantrr 697 . . . . . . . . . . . . . . . . 17  |-  ( ( ( 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 3212 . . . . . . . . . . . . . . . 16  |-  ( ( ( M  e.  ( * Met `  X
)  /\  x  e.  X )  /\  (
r  e.  RR+  /\  X  =  ( x (
ball `  M )
r ) ) )  ->  X  C_  [ x ] ( `' M " RR ) )
1514sselda 3180 . . . . . . . . . . . . . . 15  |-  ( ( ( ( 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 2791 . . . . . . . . . . . . . . . 16  |-  y  e. 
_V
17 vex 2791 . . . . . . . . . . . . . . . 16  |-  x  e. 
_V
1816, 17elec 6699 . . . . . . . . . . . . . . 15  |-  ( y  e.  [ x ]
( `' M " RR )  <->  x ( `' M " RR ) y )
1915, 18sylib 188 . . . . . . . . . . . . . 14  |-  ( ( ( ( M  e.  ( * Met `  X
)  /\  x  e.  X )  /\  (
r  e.  RR+  /\  X  =  ( x (
ball `  M )
r ) ) )  /\  y  e.  X
)  ->  x ( `' M " RR ) y )
209xmeterval 17978 . . . . . . . . . . . . . . 15  |-  ( M  e.  ( * Met `  X )  ->  (
x ( `' M " RR ) y  <->  ( x  e.  X  /\  y  e.  X  /\  (
x M y )  e.  RR ) ) )
2120ad3antrrr 710 . . . . . . . . . . . . . 14  |-  ( ( ( ( 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 201 . . . . . . . . . . . . 13  |-  ( ( ( ( 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 969 . . . . . . . . . . . 12  |-  ( ( ( ( M  e.  ( * Met `  X
)  /\  x  e.  X )  /\  (
r  e.  RR+  /\  X  =  ( x (
ball `  M )
r ) ) )  /\  y  e.  X
)  ->  ( x M y )  e.  RR )
2423ralrimiva 2626 . . . . . . . . . . 11  |-  ( ( ( M  e.  ( * Met `  X
)  /\  x  e.  X )  /\  (
r  e.  RR+  /\  X  =  ( x (
ball `  M )
r ) ) )  ->  A. y  e.  X  ( x M y )  e.  RR )
2524expr 598 . . . . . . . . . 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 ) )
2625rexlimdva 2667 . . . . . . . . 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 ) )
2726ralimdva 2621 . . . . . . . 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 ) )
2827impcom 419 . . . . . . 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 )
29 ffnov 5948 . . . . . . 7  |-  ( M : ( X  X.  X ) --> RR  <->  ( M  Fn  ( X  X.  X
)  /\  A. x  e.  X  A. y  e.  X  ( x M y )  e.  RR ) )
306, 28, 29sylanbrc 645 . . . . . 6  |-  ( ( A. x  e.  X  E. r  e.  RR+  X  =  ( x ( ball `  M ) r )  /\  M  e.  ( * Met `  X
) )  ->  M : ( X  X.  X ) --> RR )
31 ismet2 17898 . . . . . 6  |-  ( M  e.  ( Met `  X
)  <->  ( M  e.  ( * Met `  X
)  /\  M :
( X  X.  X
) --> RR ) )
323, 30, 31sylanbrc 645 . . . . 5  |-  ( ( A. x  e.  X  E. r  e.  RR+  X  =  ( x ( ball `  M ) r )  /\  M  e.  ( * Met `  X
) )  ->  M  e.  ( Met `  X
) )
3332ex 423 . . . 4  |-  ( A. x  e.  X  E. r  e.  RR+  X  =  ( x ( ball `  M ) r )  ->  ( M  e.  ( * Met `  X
)  ->  M  e.  ( Met `  X ) ) )
342, 33impbid2 195 . . 3  |-  ( A. x  e.  X  E. r  e.  RR+  X  =  ( x ( ball `  M ) r )  ->  ( M  e.  ( Met `  X
)  <->  M  e.  ( * Met `  X ) ) )
3534pm5.32ri 619 . 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 ) ) )
361, 35bitri 240 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 176    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   A.wral 2543   E.wrex 2544    C_ wss 3152   class class class wbr 4023    X. cxp 4687   `'ccnv 4688   "cima 4692    Fn wfn 5250   -->wf 5251   ` cfv 5255  (class class class)co 5858   [cec 6658   RRcr 8736   RR*cxr 8866   RR+crp 10354   * Metcxmt 16369   Metcme 16370   ballcbl 16371   Bndcbnd 26491
This theorem is referenced by:  isbnd2  26507  blbnd  26511  ismtybndlem  26530
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-po 4314  df-so 4315  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-er 6660  df-ec 6662  df-map 6774  df-en 6864  df-dom 6865  df-sdom 6866  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-2 9804  df-rp 10355  df-xneg 10452  df-xadd 10453  df-xmul 10454  df-xmet 16373  df-met 16374  df-bl 16375  df-bnd 26503
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