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Theorem xmetresbl 20675
Description: An extended metric restricted to any ball (in particular the infinity ball) is a proper metric. Together with xmetec 20672, 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 996 . . 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 20656 . . . 4  |-  ( ( D  e.  ( *Met `  X )  /\  P  e.  X  /\  R  e.  RR* )  ->  ( P ( ball `  D ) R ) 
C_  X )
42, 3syl5eqss 3548 . . 3  |-  ( ( D  e.  ( *Met `  X )  /\  P  e.  X  /\  R  e.  RR* )  ->  B  C_  X )
5 xmetres2 20599 . . 3  |-  ( ( D  e.  ( *Met `  X )  /\  B  C_  X
)  ->  ( D  |`  ( B  X.  B
) )  e.  ( *Met `  B
) )
61, 4, 5syl2anc 661 . 2  |-  ( ( D  e.  ( *Met `  X )  /\  P  e.  X  /\  R  e.  RR* )  ->  ( D  |`  ( B  X.  B ) )  e.  ( *Met `  B ) )
7 xmetf 20567 . . . . . 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 5106 . . . . . 6  |-  ( ( B  C_  X  /\  B  C_  X )  -> 
( B  X.  B
)  C_  ( X  X.  X ) )
104, 4, 9syl2anc 661 . . . . 5  |-  ( ( D  e.  ( *Met `  X )  /\  P  e.  X  /\  R  e.  RR* )  ->  ( B  X.  B
)  C_  ( X  X.  X ) )
11 fssres 5749 . . . . 5  |-  ( ( D : ( X  X.  X ) --> RR* 
/\  ( B  X.  B )  C_  ( X  X.  X ) )  ->  ( D  |`  ( B  X.  B
) ) : ( B  X.  B ) -->
RR* )
128, 10, 11syl2anc 661 . . . 4  |-  ( ( D  e.  ( *Met `  X )  /\  P  e.  X  /\  R  e.  RR* )  ->  ( D  |`  ( B  X.  B ) ) : ( B  X.  B ) --> RR* )
13 ffn 5729 . . . 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 6424 . . . . . 6  |-  ( ( x  e.  B  /\  y  e.  B )  ->  ( x ( D  |`  ( B  X.  B
) ) y )  =  ( x D y ) )
1615adantl 466 . . . . 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 999 . . . . . . . . 9  |-  ( ( ( D  e.  ( *Met `  X
)  /\  P  e.  X  /\  R  e.  RR* )  /\  ( x  e.  B  /\  y  e.  B ) )  ->  D  e.  ( *Met `  X ) )
18 eqid 2467 . . . . . . . . . 10  |-  ( `' D " RR )  =  ( `' D " RR )
1918xmeter 20671 . . . . . . . . 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 20673 . . . . . . . . . . . 12  |-  ( ( D  e.  ( *Met `  X )  /\  P  e.  X  /\  R  e.  RR* )  ->  ( P ( ball `  D ) R ) 
C_  [ P ]
( `' D " RR ) )
222, 21syl5eqss 3548 . . . . . . . . . . 11  |-  ( ( D  e.  ( *Met `  X )  /\  P  e.  X  /\  R  e.  RR* )  ->  B  C_  [ P ] ( `' D " RR ) )
2322sselda 3504 . . . . . . . . . 10  |-  ( ( ( D  e.  ( *Met `  X
)  /\  P  e.  X  /\  R  e.  RR* )  /\  x  e.  B
)  ->  x  e.  [ P ] ( `' D " RR ) )
2423adantrr 716 . . . . . . . . 9  |-  ( ( ( D  e.  ( *Met `  X
)  /\  P  e.  X  /\  R  e.  RR* )  /\  ( x  e.  B  /\  y  e.  B ) )  ->  x  e.  [ P ] ( `' D " RR ) )
25 simpl2 1000 . . . . . . . . . 10  |-  ( ( ( D  e.  ( *Met `  X
)  /\  P  e.  X  /\  R  e.  RR* )  /\  ( x  e.  B  /\  y  e.  B ) )  ->  P  e.  X )
26 elecg 7347 . . . . . . . . . 10  |-  ( ( x  e.  [ P ] ( `' D " RR )  /\  P  e.  X )  ->  (
x  e.  [ P ] ( `' D " RR )  <->  P ( `' D " RR ) x ) )
2724, 25, 26syl2anc 661 . . . . . . . . 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 210 . . . . . . . 8  |-  ( ( ( D  e.  ( *Met `  X
)  /\  P  e.  X  /\  R  e.  RR* )  /\  ( x  e.  B  /\  y  e.  B ) )  ->  P ( `' D " RR ) x )
2922sselda 3504 . . . . . . . . . 10  |-  ( ( ( D  e.  ( *Met `  X
)  /\  P  e.  X  /\  R  e.  RR* )  /\  y  e.  B
)  ->  y  e.  [ P ] ( `' D " RR ) )
3029adantrl 715 . . . . . . . . 9  |-  ( ( ( D  e.  ( *Met `  X
)  /\  P  e.  X  /\  R  e.  RR* )  /\  ( x  e.  B  /\  y  e.  B ) )  -> 
y  e.  [ P ] ( `' D " RR ) )
31 elecg 7347 . . . . . . . . . 10  |-  ( ( y  e.  [ P ] ( `' D " RR )  /\  P  e.  X )  ->  (
y  e.  [ P ] ( `' D " RR )  <->  P ( `' D " RR ) y ) )
3230, 25, 31syl2anc 661 . . . . . . . . 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 210 . . . . . . . 8  |-  ( ( ( D  e.  ( *Met `  X
)  /\  P  e.  X  /\  R  e.  RR* )  /\  ( x  e.  B  /\  y  e.  B ) )  ->  P ( `' D " RR ) y )
3420, 28, 33ertr3d 7326 . . . . . . 7  |-  ( ( ( D  e.  ( *Met `  X
)  /\  P  e.  X  /\  R  e.  RR* )  /\  ( x  e.  B  /\  y  e.  B ) )  ->  x ( `' D " RR ) y )
3518xmeterval 20670 . . . . . . . 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 210 . . . . . 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 1010 . . . . 5  |-  ( ( ( D  e.  ( *Met `  X
)  /\  P  e.  X  /\  R  e.  RR* )  /\  ( x  e.  B  /\  y  e.  B ) )  -> 
( x D y )  e.  RR )
3916, 38eqeltrd 2555 . . . 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 2885 . . 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 6388 . . 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 664 . 2  |-  ( ( D  e.  ( *Met `  X )  /\  P  e.  X  /\  R  e.  RR* )  ->  ( D  |`  ( B  X.  B ) ) : ( B  X.  B ) --> RR )
43 ismet2 20571 . 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 664 1  |-  ( ( D  e.  ( *Met `  X )  /\  P  e.  X  /\  R  e.  RR* )  ->  ( D  |`  ( B  X.  B ) )  e.  ( Met `  B
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767   A.wral 2814    C_ wss 3476   class class class wbr 4447    X. cxp 4997   `'ccnv 4998    |` cres 5001   "cima 5002    Fn wfn 5581   -->wf 5582   ` cfv 5586  (class class class)co 6282    Er wer 7305   [cec 7306   RRcr 9487   RR*cxr 9623   *Metcxmt 18174   Metcme 18175   ballcbl 18176
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 6574  ax-cnex 9544  ax-resscn 9545  ax-1cn 9546  ax-icn 9547  ax-addcl 9548  ax-addrcl 9549  ax-mulcl 9550  ax-mulrcl 9551  ax-mulcom 9552  ax-addass 9553  ax-mulass 9554  ax-distr 9555  ax-i2m1 9556  ax-1ne0 9557  ax-1rid 9558  ax-rnegex 9559  ax-rrecex 9560  ax-cnre 9561  ax-pre-lttri 9562  ax-pre-lttrn 9563  ax-pre-ltadd 9564  ax-pre-mulgt0 9565
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 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-riota 6243  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-1st 6781  df-2nd 6782  df-er 7308  df-ec 7310  df-map 7419  df-en 7514  df-dom 7515  df-sdom 7516  df-pnf 9626  df-mnf 9627  df-xr 9628  df-ltxr 9629  df-le 9630  df-sub 9803  df-neg 9804  df-div 10203  df-2 10590  df-rp 11217  df-xneg 11314  df-xadd 11315  df-xmul 11316  df-psmet 18182  df-xmet 18183  df-met 18184  df-bl 18185
This theorem is referenced by: (None)
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