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Theorem xmetresbl 14392
Description: An extended metric restricted to any ball (in particular the infinity ball) is a proper metric. Together with xmetec 14389, this shows that any extended metric space can be "factored" into the disjoint union of proper metric spaces, with points in the same region measured by that region's metric, and points in different regions being distance +oo from each other. (Contributed by Mario Carneiro, 23-Aug-2015.)
Hypothesis
Ref Expression
xmetresbl.1  |-  B  =  ( P ( ball `  D ) R )
Assertion
Ref Expression
xmetresbl  |-  ( ( D  e.  ( *Met `  X )  /\  P  e.  X  /\  R  e.  RR* )  ->  ( D  |`  ( B  X.  B ) )  e.  ( Met `  B
) )

Proof of Theorem xmetresbl
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simp1 999 . . 3  |-  ( ( D  e.  ( *Met `  X )  /\  P  e.  X  /\  R  e.  RR* )  ->  D  e.  ( *Met `  X ) )
2 xmetresbl.1 . . . 4  |-  B  =  ( P ( ball `  D ) R )
3 blssm 14373 . . . 4  |-  ( ( D  e.  ( *Met `  X )  /\  P  e.  X  /\  R  e.  RR* )  ->  ( P ( ball `  D ) R ) 
C_  X )
42, 3eqsstrid 3216 . . 3  |-  ( ( D  e.  ( *Met `  X )  /\  P  e.  X  /\  R  e.  RR* )  ->  B  C_  X )
5 xmetres2 14331 . . 3  |-  ( ( D  e.  ( *Met `  X )  /\  B  C_  X
)  ->  ( D  |`  ( B  X.  B
) )  e.  ( *Met `  B
) )
61, 4, 5syl2anc 411 . 2  |-  ( ( D  e.  ( *Met `  X )  /\  P  e.  X  /\  R  e.  RR* )  ->  ( D  |`  ( B  X.  B ) )  e.  ( *Met `  B ) )
7 xmetf 14302 . . . . . 6  |-  ( D  e.  ( *Met `  X )  ->  D : ( X  X.  X ) --> RR* )
81, 7syl 14 . . . . 5  |-  ( ( D  e.  ( *Met `  X )  /\  P  e.  X  /\  R  e.  RR* )  ->  D : ( X  X.  X ) --> RR* )
9 xpss12 4751 . . . . . 6  |-  ( ( B  C_  X  /\  B  C_  X )  -> 
( B  X.  B
)  C_  ( X  X.  X ) )
104, 4, 9syl2anc 411 . . . . 5  |-  ( ( D  e.  ( *Met `  X )  /\  P  e.  X  /\  R  e.  RR* )  ->  ( B  X.  B
)  C_  ( X  X.  X ) )
118, 10fssresd 5411 . . . 4  |-  ( ( D  e.  ( *Met `  X )  /\  P  e.  X  /\  R  e.  RR* )  ->  ( D  |`  ( B  X.  B ) ) : ( B  X.  B ) --> RR* )
1211ffnd 5385 . . 3  |-  ( ( D  e.  ( *Met `  X )  /\  P  e.  X  /\  R  e.  RR* )  ->  ( D  |`  ( B  X.  B ) )  Fn  ( B  X.  B ) )
13 ovres 6035 . . . . . 6  |-  ( ( x  e.  B  /\  y  e.  B )  ->  ( x ( D  |`  ( B  X.  B
) ) y )  =  ( x D y ) )
1413adantl 277 . . . . 5  |-  ( ( ( D  e.  ( *Met `  X
)  /\  P  e.  X  /\  R  e.  RR* )  /\  ( x  e.  B  /\  y  e.  B ) )  -> 
( x ( D  |`  ( B  X.  B
) ) y )  =  ( x D y ) )
15 simpl1 1002 . . . . . . . . 9  |-  ( ( ( D  e.  ( *Met `  X
)  /\  P  e.  X  /\  R  e.  RR* )  /\  ( x  e.  B  /\  y  e.  B ) )  ->  D  e.  ( *Met `  X ) )
16 eqid 2189 . . . . . . . . . 10  |-  ( `' D " RR )  =  ( `' D " RR )
1716xmeter 14388 . . . . . . . . 9  |-  ( D  e.  ( *Met `  X )  ->  ( `' D " RR )  Er  X )
1815, 17syl 14 . . . . . . . 8  |-  ( ( ( D  e.  ( *Met `  X
)  /\  P  e.  X  /\  R  e.  RR* )  /\  ( x  e.  B  /\  y  e.  B ) )  -> 
( `' D " RR )  Er  X
)
1916blssec 14390 . . . . . . . . . . . 12  |-  ( ( D  e.  ( *Met `  X )  /\  P  e.  X  /\  R  e.  RR* )  ->  ( P ( ball `  D ) R ) 
C_  [ P ]
( `' D " RR ) )
202, 19eqsstrid 3216 . . . . . . . . . . 11  |-  ( ( D  e.  ( *Met `  X )  /\  P  e.  X  /\  R  e.  RR* )  ->  B  C_  [ P ] ( `' D " RR ) )
2120sselda 3170 . . . . . . . . . 10  |-  ( ( ( D  e.  ( *Met `  X
)  /\  P  e.  X  /\  R  e.  RR* )  /\  x  e.  B
)  ->  x  e.  [ P ] ( `' D " RR ) )
2221adantrr 479 . . . . . . . . 9  |-  ( ( ( D  e.  ( *Met `  X
)  /\  P  e.  X  /\  R  e.  RR* )  /\  ( x  e.  B  /\  y  e.  B ) )  ->  x  e.  [ P ] ( `' D " RR ) )
23 simpl2 1003 . . . . . . . . . 10  |-  ( ( ( D  e.  ( *Met `  X
)  /\  P  e.  X  /\  R  e.  RR* )  /\  ( x  e.  B  /\  y  e.  B ) )  ->  P  e.  X )
24 elecg 6598 . . . . . . . . . 10  |-  ( ( x  e.  [ P ] ( `' D " RR )  /\  P  e.  X )  ->  (
x  e.  [ P ] ( `' D " RR )  <->  P ( `' D " RR ) x ) )
2522, 23, 24syl2anc 411 . . . . . . . . 9  |-  ( ( ( D  e.  ( *Met `  X
)  /\  P  e.  X  /\  R  e.  RR* )  /\  ( x  e.  B  /\  y  e.  B ) )  -> 
( x  e.  [ P ] ( `' D " RR )  <->  P ( `' D " RR ) x ) )
2622, 25mpbid 147 . . . . . . . 8  |-  ( ( ( D  e.  ( *Met `  X
)  /\  P  e.  X  /\  R  e.  RR* )  /\  ( x  e.  B  /\  y  e.  B ) )  ->  P ( `' D " RR ) x )
2720sselda 3170 . . . . . . . . . 10  |-  ( ( ( D  e.  ( *Met `  X
)  /\  P  e.  X  /\  R  e.  RR* )  /\  y  e.  B
)  ->  y  e.  [ P ] ( `' D " RR ) )
2827adantrl 478 . . . . . . . . 9  |-  ( ( ( D  e.  ( *Met `  X
)  /\  P  e.  X  /\  R  e.  RR* )  /\  ( x  e.  B  /\  y  e.  B ) )  -> 
y  e.  [ P ] ( `' D " RR ) )
29 elecg 6598 . . . . . . . . . 10  |-  ( ( y  e.  [ P ] ( `' D " RR )  /\  P  e.  X )  ->  (
y  e.  [ P ] ( `' D " RR )  <->  P ( `' D " RR ) y ) )
3028, 23, 29syl2anc 411 . . . . . . . . 9  |-  ( ( ( D  e.  ( *Met `  X
)  /\  P  e.  X  /\  R  e.  RR* )  /\  ( x  e.  B  /\  y  e.  B ) )  -> 
( y  e.  [ P ] ( `' D " RR )  <->  P ( `' D " RR ) y ) )
3128, 30mpbid 147 . . . . . . . 8  |-  ( ( ( D  e.  ( *Met `  X
)  /\  P  e.  X  /\  R  e.  RR* )  /\  ( x  e.  B  /\  y  e.  B ) )  ->  P ( `' D " RR ) y )
3218, 26, 31ertr3d 6576 . . . . . . 7  |-  ( ( ( D  e.  ( *Met `  X
)  /\  P  e.  X  /\  R  e.  RR* )  /\  ( x  e.  B  /\  y  e.  B ) )  ->  x ( `' D " RR ) y )
3316xmeterval 14387 . . . . . . . 8  |-  ( D  e.  ( *Met `  X )  ->  (
x ( `' D " RR ) y  <->  ( x  e.  X  /\  y  e.  X  /\  (
x D y )  e.  RR ) ) )
3415, 33syl 14 . . . . . . 7  |-  ( ( ( D  e.  ( *Met `  X
)  /\  P  e.  X  /\  R  e.  RR* )  /\  ( x  e.  B  /\  y  e.  B ) )  -> 
( x ( `' D " RR ) y  <->  ( x  e.  X  /\  y  e.  X  /\  ( x D y )  e.  RR ) ) )
3532, 34mpbid 147 . . . . . 6  |-  ( ( ( D  e.  ( *Met `  X
)  /\  P  e.  X  /\  R  e.  RR* )  /\  ( x  e.  B  /\  y  e.  B ) )  -> 
( x  e.  X  /\  y  e.  X  /\  ( x D y )  e.  RR ) )
3635simp3d 1013 . . . . 5  |-  ( ( ( D  e.  ( *Met `  X
)  /\  P  e.  X  /\  R  e.  RR* )  /\  ( x  e.  B  /\  y  e.  B ) )  -> 
( x D y )  e.  RR )
3714, 36eqeltrd 2266 . . . 4  |-  ( ( ( D  e.  ( *Met `  X
)  /\  P  e.  X  /\  R  e.  RR* )  /\  ( x  e.  B  /\  y  e.  B ) )  -> 
( x ( D  |`  ( B  X.  B
) ) y )  e.  RR )
3837ralrimivva 2572 . . 3  |-  ( ( D  e.  ( *Met `  X )  /\  P  e.  X  /\  R  e.  RR* )  ->  A. x  e.  B  A. y  e.  B  ( x ( D  |`  ( B  X.  B
) ) y )  e.  RR )
39 ffnov 5999 . . 3  |-  ( ( D  |`  ( B  X.  B ) ) : ( B  X.  B
) --> RR  <->  ( ( D  |`  ( B  X.  B ) )  Fn  ( B  X.  B
)  /\  A. x  e.  B  A. y  e.  B  ( x
( D  |`  ( B  X.  B ) ) y )  e.  RR ) )
4012, 38, 39sylanbrc 417 . 2  |-  ( ( D  e.  ( *Met `  X )  /\  P  e.  X  /\  R  e.  RR* )  ->  ( D  |`  ( B  X.  B ) ) : ( B  X.  B ) --> RR )
41 ismet2 14306 . 2  |-  ( ( D  |`  ( B  X.  B ) )  e.  ( Met `  B
)  <->  ( ( D  |`  ( B  X.  B
) )  e.  ( *Met `  B
)  /\  ( D  |`  ( B  X.  B
) ) : ( B  X.  B ) --> RR ) )
426, 40, 41sylanbrc 417 1  |-  ( ( D  e.  ( *Met `  X )  /\  P  e.  X  /\  R  e.  RR* )  ->  ( D  |`  ( B  X.  B ) )  e.  ( Met `  B
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    <-> wb 105    /\ w3a 980    = wceq 1364    e. wcel 2160   A.wral 2468    C_ wss 3144   class class class wbr 4018    X. cxp 4642   `'ccnv 4643    |` cres 4646   "cima 4647    Fn wfn 5230   -->wf 5231   ` cfv 5235  (class class class)co 5895    Er wer 6555   [cec 6556   RRcr 7839   RR*cxr 8020   *Metcxmet 13846   Metcmet 13847   ballcbl 13848
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4192  ax-pr 4227  ax-un 4451  ax-setind 4554  ax-cnex 7931  ax-resscn 7932  ax-1cn 7933  ax-1re 7934  ax-icn 7935  ax-addcl 7936  ax-addrcl 7937  ax-mulcl 7938  ax-mulrcl 7939  ax-addcom 7940  ax-mulcom 7941  ax-addass 7942  ax-mulass 7943  ax-distr 7944  ax-i2m1 7945  ax-0lt1 7946  ax-1rid 7947  ax-0id 7948  ax-rnegex 7949  ax-precex 7950  ax-cnre 7951  ax-pre-ltirr 7952  ax-pre-ltwlin 7953  ax-pre-lttrn 7954  ax-pre-apti 7955  ax-pre-ltadd 7956  ax-pre-mulgt0 7957
This theorem depends on definitions:  df-bi 117  df-stab 832  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-nel 2456  df-ral 2473  df-rex 2474  df-reu 2475  df-rab 2477  df-v 2754  df-sbc 2978  df-csb 3073  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-if 3550  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-iun 3903  df-br 4019  df-opab 4080  df-mpt 4081  df-id 4311  df-po 4314  df-iso 4315  df-xp 4650  df-rel 4651  df-cnv 4652  df-co 4653  df-dm 4654  df-rn 4655  df-res 4656  df-ima 4657  df-iota 5196  df-fun 5237  df-fn 5238  df-f 5239  df-fv 5243  df-riota 5851  df-ov 5898  df-oprab 5899  df-mpo 5900  df-1st 6164  df-2nd 6165  df-er 6558  df-ec 6560  df-map 6675  df-pnf 8023  df-mnf 8024  df-xr 8025  df-ltxr 8026  df-le 8027  df-sub 8159  df-neg 8160  df-2 9007  df-xneg 9801  df-xadd 9802  df-psmet 13853  df-xmet 13854  df-met 13855  df-bl 13856
This theorem is referenced by: (None)
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