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Theorem isbndx 29909
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 29907 . 2  |-  ( M  e.  ( Bnd `  X
)  <->  ( M  e.  ( Met `  X
)  /\  A. x  e.  X  E. r  e.  RR+  X  =  ( x ( ball `  M
) r ) ) )
2 metxmet 20600 . . . 4  |-  ( M  e.  ( Met `  X
)  ->  M  e.  ( *Met `  X
) )
3 simpr 461 . . . . . 6  |-  ( ( A. x  e.  X  E. r  e.  RR+  X  =  ( x ( ball `  M ) r )  /\  M  e.  ( *Met `  X
) )  ->  M  e.  ( *Met `  X ) )
4 xmetf 20595 . . . . . . . 8  |-  ( M  e.  ( *Met `  X )  ->  M : ( X  X.  X ) --> RR* )
5 ffn 5731 . . . . . . . 8  |-  ( M : ( X  X.  X ) --> RR*  ->  M  Fn  ( X  X.  X ) )
63, 4, 53syl 20 . . . . . . 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 756 . . . . . . . . . . . . . . . 16  |-  ( ( ( M  e.  ( *Met `  X
)  /\  x  e.  X )  /\  (
r  e.  RR+  /\  X  =  ( x (
ball `  M )
r ) ) )  ->  X  =  ( x ( ball `  M
) r ) )
8 rpxr 11227 . . . . . . . . . . . . . . . . . 18  |-  ( r  e.  RR+  ->  r  e. 
RR* )
9 eqid 2467 . . . . . . . . . . . . . . . . . . . 20  |-  ( `' M " RR )  =  ( `' M " RR )
109blssec 20701 . . . . . . . . . . . . . . . . . . 19  |-  ( ( M  e.  ( *Met `  X )  /\  x  e.  X  /\  r  e.  RR* )  ->  ( x ( ball `  M ) r ) 
C_  [ x ]
( `' M " RR ) )
11103expa 1196 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( M  e.  ( *Met `  X
)  /\  x  e.  X )  /\  r  e.  RR* )  ->  (
x ( ball `  M
) r )  C_  [ x ] ( `' M " RR ) )
128, 11sylan2 474 . . . . . . . . . . . . . . . . 17  |-  ( ( ( M  e.  ( *Met `  X
)  /\  x  e.  X )  /\  r  e.  RR+ )  ->  (
x ( ball `  M
) r )  C_  [ x ] ( `' M " RR ) )
1312adantrr 716 . . . . . . . . . . . . . . . 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 3538 . . . . . . . . . . . . . . 15  |-  ( ( ( M  e.  ( *Met `  X
)  /\  x  e.  X )  /\  (
r  e.  RR+  /\  X  =  ( x (
ball `  M )
r ) ) )  ->  X  C_  [ x ] ( `' M " RR ) )
1514sselda 3504 . . . . . . . . . . . . . 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 3116 . . . . . . . . . . . . . . 15  |-  y  e. 
_V
17 vex 3116 . . . . . . . . . . . . . . 15  |-  x  e. 
_V
1816, 17elec 7351 . . . . . . . . . . . . . 14  |-  ( y  e.  [ x ]
( `' M " RR )  <->  x ( `' M " RR ) y )
1915, 18sylib 196 . . . . . . . . . . . . 13  |-  ( ( ( ( M  e.  ( *Met `  X )  /\  x  e.  X )  /\  (
r  e.  RR+  /\  X  =  ( x (
ball `  M )
r ) ) )  /\  y  e.  X
)  ->  x ( `' M " RR ) y )
209xmeterval 20698 . . . . . . . . . . . . . 14  |-  ( M  e.  ( *Met `  X )  ->  (
x ( `' M " RR ) y  <->  ( x  e.  X  /\  y  e.  X  /\  (
x M y )  e.  RR ) ) )
2120ad3antrrr 729 . . . . . . . . . . . . 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 210 . . . . . . . . . . . 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 1010 . . . . . . . . . . 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 2878 . . . . . . . . . 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 2956 . . . . . . . . 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 2872 . . . . . . . 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 430 . . . . . . 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 6390 . . . . . . 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 664 . . . . . 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 20599 . . . . . 6  |-  ( M  e.  ( Met `  X
)  <->  ( M  e.  ( *Met `  X )  /\  M : ( X  X.  X ) --> RR ) )
313, 29, 30sylanbrc 664 . . . . 5  |-  ( ( A. x  e.  X  E. r  e.  RR+  X  =  ( x ( ball `  M ) r )  /\  M  e.  ( *Met `  X
) )  ->  M  e.  ( Met `  X
) )
3231ex 434 . . . 4  |-  ( A. x  e.  X  E. r  e.  RR+  X  =  ( x ( ball `  M ) r )  ->  ( M  e.  ( *Met `  X )  ->  M  e.  ( Met `  X
) ) )
332, 32impbid2 204 . . 3  |-  ( A. x  e.  X  E. r  e.  RR+  X  =  ( x ( ball `  M ) r )  ->  ( M  e.  ( Met `  X
)  <->  M  e.  ( *Met `  X ) ) )
3433pm5.32ri 638 . 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 249 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 184    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767   A.wral 2814   E.wrex 2815    C_ wss 3476   class class class wbr 4447    X. cxp 4997   `'ccnv 4998   "cima 5002    Fn wfn 5583   -->wf 5584   ` cfv 5588  (class class class)co 6284   [cec 7309   RRcr 9491   RR*cxr 9627   RR+crp 11220   *Metcxmt 18202   Metcme 18203   ballcbl 18204   Bndcbnd 29894
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6576  ax-cnex 9548  ax-resscn 9549  ax-1cn 9550  ax-icn 9551  ax-addcl 9552  ax-addrcl 9553  ax-mulcl 9554  ax-mulrcl 9555  ax-mulcom 9556  ax-addass 9557  ax-mulass 9558  ax-distr 9559  ax-i2m1 9560  ax-1ne0 9561  ax-1rid 9562  ax-rnegex 9563  ax-rrecex 9564  ax-cnre 9565  ax-pre-lttri 9566  ax-pre-lttrn 9567  ax-pre-ltadd 9568  ax-pre-mulgt0 9569
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-po 4800  df-so 4801  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-riota 6245  df-ov 6287  df-oprab 6288  df-mpt2 6289  df-1st 6784  df-2nd 6785  df-er 7311  df-ec 7313  df-map 7422  df-en 7517  df-dom 7518  df-sdom 7519  df-pnf 9630  df-mnf 9631  df-xr 9632  df-ltxr 9633  df-le 9634  df-sub 9807  df-neg 9808  df-div 10207  df-2 10594  df-rp 11221  df-xneg 11318  df-xadd 11319  df-xmul 11320  df-psmet 18210  df-xmet 18211  df-met 18212  df-bl 18213  df-bnd 29906
This theorem is referenced by:  isbnd2  29910  blbnd  29914  ismtybndlem  29933
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