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Theorem xmetresbl 18420
Description: An extended metric restricted to any ball (in particular the infinity ball) is a proper metric. Together with xmetec 18417, 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 957 . . 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 18401 . . . 4  |-  ( ( D  e.  ( * Met `  X )  /\  P  e.  X  /\  R  e.  RR* )  ->  ( P ( ball `  D ) R ) 
C_  X )
42, 3syl5eqss 3352 . . 3  |-  ( ( D  e.  ( * Met `  X )  /\  P  e.  X  /\  R  e.  RR* )  ->  B  C_  X )
5 xmetres2 18344 . . 3  |-  ( ( D  e.  ( * Met `  X )  /\  B  C_  X
)  ->  ( D  |`  ( B  X.  B
) )  e.  ( * Met `  B
) )
61, 4, 5syl2anc 643 . 2  |-  ( ( D  e.  ( * Met `  X )  /\  P  e.  X  /\  R  e.  RR* )  ->  ( D  |`  ( B  X.  B ) )  e.  ( * Met `  B ) )
7 xmetf 18312 . . . . . 6  |-  ( D  e.  ( * Met `  X )  ->  D : ( X  X.  X ) --> RR* )
81, 7syl 16 . . . . 5  |-  ( ( D  e.  ( * Met `  X )  /\  P  e.  X  /\  R  e.  RR* )  ->  D : ( X  X.  X ) --> RR* )
9 xpss12 4940 . . . . . 6  |-  ( ( B  C_  X  /\  B  C_  X )  -> 
( B  X.  B
)  C_  ( X  X.  X ) )
104, 4, 9syl2anc 643 . . . . 5  |-  ( ( D  e.  ( * Met `  X )  /\  P  e.  X  /\  R  e.  RR* )  ->  ( B  X.  B
)  C_  ( X  X.  X ) )
11 fssres 5569 . . . . 5  |-  ( ( D : ( X  X.  X ) --> RR* 
/\  ( B  X.  B )  C_  ( X  X.  X ) )  ->  ( D  |`  ( B  X.  B
) ) : ( B  X.  B ) -->
RR* )
128, 10, 11syl2anc 643 . . . 4  |-  ( ( D  e.  ( * Met `  X )  /\  P  e.  X  /\  R  e.  RR* )  ->  ( D  |`  ( B  X.  B ) ) : ( B  X.  B ) --> RR* )
13 ffn 5550 . . . 4  |-  ( ( D  |`  ( B  X.  B ) ) : ( B  X.  B
) --> RR*  ->  ( D  |`  ( B  X.  B
) )  Fn  ( B  X.  B ) )
1412, 13syl 16 . . 3  |-  ( ( D  e.  ( * Met `  X )  /\  P  e.  X  /\  R  e.  RR* )  ->  ( D  |`  ( B  X.  B ) )  Fn  ( B  X.  B ) )
15 ovres 6172 . . . . . 6  |-  ( ( x  e.  B  /\  y  e.  B )  ->  ( x ( D  |`  ( B  X.  B
) ) y )  =  ( x D y ) )
1615adantl 453 . . . . 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 ) )
17 simpl1 960 . . . . . . . . 9  |-  ( ( ( D  e.  ( * Met `  X
)  /\  P  e.  X  /\  R  e.  RR* )  /\  ( x  e.  B  /\  y  e.  B ) )  ->  D  e.  ( * Met `  X ) )
18 eqid 2404 . . . . . . . . . 10  |-  ( `' D " RR )  =  ( `' D " RR )
1918xmeter 18416 . . . . . . . . 9  |-  ( D  e.  ( * Met `  X )  ->  ( `' D " RR )  Er  X )
2017, 19syl 16 . . . . . . . 8  |-  ( ( ( D  e.  ( * Met `  X
)  /\  P  e.  X  /\  R  e.  RR* )  /\  ( x  e.  B  /\  y  e.  B ) )  -> 
( `' D " RR )  Er  X
)
2118blssec 18418 . . . . . . . . . . . 12  |-  ( ( D  e.  ( * Met `  X )  /\  P  e.  X  /\  R  e.  RR* )  ->  ( P ( ball `  D ) R ) 
C_  [ P ]
( `' D " RR ) )
222, 21syl5eqss 3352 . . . . . . . . . . 11  |-  ( ( D  e.  ( * Met `  X )  /\  P  e.  X  /\  R  e.  RR* )  ->  B  C_  [ P ] ( `' D " RR ) )
2322sselda 3308 . . . . . . . . . 10  |-  ( ( ( D  e.  ( * Met `  X
)  /\  P  e.  X  /\  R  e.  RR* )  /\  x  e.  B
)  ->  x  e.  [ P ] ( `' D " RR ) )
2423adantrr 698 . . . . . . . . 9  |-  ( ( ( D  e.  ( * Met `  X
)  /\  P  e.  X  /\  R  e.  RR* )  /\  ( x  e.  B  /\  y  e.  B ) )  ->  x  e.  [ P ] ( `' D " RR ) )
25 simpl2 961 . . . . . . . . . 10  |-  ( ( ( D  e.  ( * Met `  X
)  /\  P  e.  X  /\  R  e.  RR* )  /\  ( x  e.  B  /\  y  e.  B ) )  ->  P  e.  X )
26 elecg 6902 . . . . . . . . . 10  |-  ( ( x  e.  [ P ] ( `' D " RR )  /\  P  e.  X )  ->  (
x  e.  [ P ] ( `' D " RR )  <->  P ( `' D " RR ) x ) )
2724, 25, 26syl2anc 643 . . . . . . . . 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 ) )
2824, 27mpbid 202 . . . . . . . 8  |-  ( ( ( D  e.  ( * Met `  X
)  /\  P  e.  X  /\  R  e.  RR* )  /\  ( x  e.  B  /\  y  e.  B ) )  ->  P ( `' D " RR ) x )
2922sselda 3308 . . . . . . . . . 10  |-  ( ( ( D  e.  ( * Met `  X
)  /\  P  e.  X  /\  R  e.  RR* )  /\  y  e.  B
)  ->  y  e.  [ P ] ( `' D " RR ) )
3029adantrl 697 . . . . . . . . 9  |-  ( ( ( D  e.  ( * Met `  X
)  /\  P  e.  X  /\  R  e.  RR* )  /\  ( x  e.  B  /\  y  e.  B ) )  -> 
y  e.  [ P ] ( `' D " RR ) )
31 elecg 6902 . . . . . . . . . 10  |-  ( ( y  e.  [ P ] ( `' D " RR )  /\  P  e.  X )  ->  (
y  e.  [ P ] ( `' D " RR )  <->  P ( `' D " RR ) y ) )
3230, 25, 31syl2anc 643 . . . . . . . . 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 ) )
3330, 32mpbid 202 . . . . . . . 8  |-  ( ( ( D  e.  ( * Met `  X
)  /\  P  e.  X  /\  R  e.  RR* )  /\  ( x  e.  B  /\  y  e.  B ) )  ->  P ( `' D " RR ) y )
3420, 28, 33ertr3d 6882 . . . . . . 7  |-  ( ( ( D  e.  ( * Met `  X
)  /\  P  e.  X  /\  R  e.  RR* )  /\  ( x  e.  B  /\  y  e.  B ) )  ->  x ( `' D " RR ) y )
3518xmeterval 18415 . . . . . . . 8  |-  ( D  e.  ( * Met `  X )  ->  (
x ( `' D " RR ) y  <->  ( x  e.  X  /\  y  e.  X  /\  (
x D y )  e.  RR ) ) )
3617, 35syl 16 . . . . . . 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 ) ) )
3734, 36mpbid 202 . . . . . 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 ) )
3837simp3d 971 . . . . 5  |-  ( ( ( D  e.  ( * Met `  X
)  /\  P  e.  X  /\  R  e.  RR* )  /\  ( x  e.  B  /\  y  e.  B ) )  -> 
( x D y )  e.  RR )
3916, 38eqeltrd 2478 . . . 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 )
4039ralrimivva 2758 . . 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 )
41 ffnov 6133 . . 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 ) )
4214, 40, 41sylanbrc 646 . 2  |-  ( ( D  e.  ( * Met `  X )  /\  P  e.  X  /\  R  e.  RR* )  ->  ( D  |`  ( B  X.  B ) ) : ( B  X.  B ) --> RR )
43 ismet2 18316 . 2  |-  ( ( D  |`  ( B  X.  B ) )  e.  ( Met `  B
)  <->  ( ( D  |`  ( B  X.  B
) )  e.  ( * Met `  B
)  /\  ( D  |`  ( B  X.  B
) ) : ( B  X.  B ) --> RR ) )
446, 42, 43sylanbrc 646 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    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1721   A.wral 2666    C_ wss 3280   class class class wbr 4172    X. cxp 4835   `'ccnv 4836    |` cres 4839   "cima 4840    Fn wfn 5408   -->wf 5409   ` cfv 5413  (class class class)co 6040    Er wer 6861   [cec 6862   RRcr 8945   RR*cxr 9075   * Metcxmt 16641   Metcme 16642   ballcbl 16643
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 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023
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 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-id 4458  df-po 4463  df-so 4464  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-riota 6508  df-er 6864  df-ec 6866  df-map 6979  df-en 7069  df-dom 7070  df-sdom 7071  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-div 9634  df-2 10014  df-rp 10569  df-xneg 10666  df-xadd 10667  df-xmul 10668  df-psmet 16649  df-xmet 16650  df-met 16651  df-bl 16652
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