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Theorem xmetresbl 17985
Description: An extended metric restricted to any ball (in particular the infinity ball) is a proper metric. Together with xmetec 17982, 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 955 . . 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 17970 . . . 4  |-  ( ( D  e.  ( * Met `  X )  /\  P  e.  X  /\  R  e.  RR* )  ->  ( P ( ball `  D ) R ) 
C_  X )
42, 3syl5eqss 3224 . . 3  |-  ( ( D  e.  ( * Met `  X )  /\  P  e.  X  /\  R  e.  RR* )  ->  B  C_  X )
5 xmetres2 17927 . . 3  |-  ( ( D  e.  ( * Met `  X )  /\  B  C_  X
)  ->  ( D  |`  ( B  X.  B
) )  e.  ( * Met `  B
) )
61, 4, 5syl2anc 642 . 2  |-  ( ( D  e.  ( * Met `  X )  /\  P  e.  X  /\  R  e.  RR* )  ->  ( D  |`  ( B  X.  B ) )  e.  ( * Met `  B ) )
7 xmetf 17896 . . . . . 6  |-  ( D  e.  ( * Met `  X )  ->  D : ( X  X.  X ) --> RR* )
81, 7syl 15 . . . . 5  |-  ( ( D  e.  ( * Met `  X )  /\  P  e.  X  /\  R  e.  RR* )  ->  D : ( X  X.  X ) --> RR* )
9 xpss12 4794 . . . . . 6  |-  ( ( B  C_  X  /\  B  C_  X )  -> 
( B  X.  B
)  C_  ( X  X.  X ) )
104, 4, 9syl2anc 642 . . . . 5  |-  ( ( D  e.  ( * Met `  X )  /\  P  e.  X  /\  R  e.  RR* )  ->  ( B  X.  B
)  C_  ( X  X.  X ) )
11 fssres 5410 . . . . 5  |-  ( ( D : ( X  X.  X ) --> RR* 
/\  ( B  X.  B )  C_  ( X  X.  X ) )  ->  ( D  |`  ( B  X.  B
) ) : ( B  X.  B ) -->
RR* )
128, 10, 11syl2anc 642 . . . 4  |-  ( ( D  e.  ( * Met `  X )  /\  P  e.  X  /\  R  e.  RR* )  ->  ( D  |`  ( B  X.  B ) ) : ( B  X.  B ) --> RR* )
13 ffn 5391 . . . 4  |-  ( ( D  |`  ( B  X.  B ) ) : ( B  X.  B
) --> RR*  ->  ( D  |`  ( B  X.  B
) )  Fn  ( B  X.  B ) )
1412, 13syl 15 . . 3  |-  ( ( D  e.  ( * Met `  X )  /\  P  e.  X  /\  R  e.  RR* )  ->  ( D  |`  ( B  X.  B ) )  Fn  ( B  X.  B ) )
15 ovres 5989 . . . . . 6  |-  ( ( x  e.  B  /\  y  e.  B )  ->  ( x ( D  |`  ( B  X.  B
) ) y )  =  ( x D y ) )
1615adantl 452 . . . . 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 958 . . . . . . . . 9  |-  ( ( ( D  e.  ( * Met `  X
)  /\  P  e.  X  /\  R  e.  RR* )  /\  ( x  e.  B  /\  y  e.  B ) )  ->  D  e.  ( * Met `  X ) )
18 eqid 2285 . . . . . . . . . 10  |-  ( `' D " RR )  =  ( `' D " RR )
1918xmeter 17981 . . . . . . . . 9  |-  ( D  e.  ( * Met `  X )  ->  ( `' D " RR )  Er  X )
2017, 19syl 15 . . . . . . . 8  |-  ( ( ( D  e.  ( * Met `  X
)  /\  P  e.  X  /\  R  e.  RR* )  /\  ( x  e.  B  /\  y  e.  B ) )  -> 
( `' D " RR )  Er  X
)
2118blssec 17983 . . . . . . . . . . . 12  |-  ( ( D  e.  ( * Met `  X )  /\  P  e.  X  /\  R  e.  RR* )  ->  ( P ( ball `  D ) R ) 
C_  [ P ]
( `' D " RR ) )
222, 21syl5eqss 3224 . . . . . . . . . . 11  |-  ( ( D  e.  ( * Met `  X )  /\  P  e.  X  /\  R  e.  RR* )  ->  B  C_  [ P ] ( `' D " RR ) )
2322sselda 3182 . . . . . . . . . 10  |-  ( ( ( D  e.  ( * Met `  X
)  /\  P  e.  X  /\  R  e.  RR* )  /\  x  e.  B
)  ->  x  e.  [ P ] ( `' D " RR ) )
2423adantrr 697 . . . . . . . . 9  |-  ( ( ( D  e.  ( * Met `  X
)  /\  P  e.  X  /\  R  e.  RR* )  /\  ( x  e.  B  /\  y  e.  B ) )  ->  x  e.  [ P ] ( `' D " RR ) )
25 simpl2 959 . . . . . . . . . 10  |-  ( ( ( D  e.  ( * Met `  X
)  /\  P  e.  X  /\  R  e.  RR* )  /\  ( x  e.  B  /\  y  e.  B ) )  ->  P  e.  X )
26 elecg 6700 . . . . . . . . . 10  |-  ( ( x  e.  [ P ] ( `' D " RR )  /\  P  e.  X )  ->  (
x  e.  [ P ] ( `' D " RR )  <->  P ( `' D " RR ) x ) )
2724, 25, 26syl2anc 642 . . . . . . . . 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 201 . . . . . . . 8  |-  ( ( ( D  e.  ( * Met `  X
)  /\  P  e.  X  /\  R  e.  RR* )  /\  ( x  e.  B  /\  y  e.  B ) )  ->  P ( `' D " RR ) x )
2922sselda 3182 . . . . . . . . . 10  |-  ( ( ( D  e.  ( * Met `  X
)  /\  P  e.  X  /\  R  e.  RR* )  /\  y  e.  B
)  ->  y  e.  [ P ] ( `' D " RR ) )
3029adantrl 696 . . . . . . . . 9  |-  ( ( ( D  e.  ( * Met `  X
)  /\  P  e.  X  /\  R  e.  RR* )  /\  ( x  e.  B  /\  y  e.  B ) )  -> 
y  e.  [ P ] ( `' D " RR ) )
31 elecg 6700 . . . . . . . . . 10  |-  ( ( y  e.  [ P ] ( `' D " RR )  /\  P  e.  X )  ->  (
y  e.  [ P ] ( `' D " RR )  <->  P ( `' D " RR ) y ) )
3230, 25, 31syl2anc 642 . . . . . . . . 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 201 . . . . . . . 8  |-  ( ( ( D  e.  ( * Met `  X
)  /\  P  e.  X  /\  R  e.  RR* )  /\  ( x  e.  B  /\  y  e.  B ) )  ->  P ( `' D " RR ) y )
3420, 28, 33ertr3d 6680 . . . . . . 7  |-  ( ( ( D  e.  ( * Met `  X
)  /\  P  e.  X  /\  R  e.  RR* )  /\  ( x  e.  B  /\  y  e.  B ) )  ->  x ( `' D " RR ) y )
3518xmeterval 17980 . . . . . . . 8  |-  ( D  e.  ( * Met `  X )  ->  (
x ( `' D " RR ) y  <->  ( x  e.  X  /\  y  e.  X  /\  (
x D y )  e.  RR ) ) )
3617, 35syl 15 . . . . . . 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 201 . . . . . 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 969 . . . . 5  |-  ( ( ( D  e.  ( * Met `  X
)  /\  P  e.  X  /\  R  e.  RR* )  /\  ( x  e.  B  /\  y  e.  B ) )  -> 
( x D y )  e.  RR )
3916, 38eqeltrd 2359 . . . 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 2637 . . 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 5950 . . 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 645 . 2  |-  ( ( D  e.  ( * Met `  X )  /\  P  e.  X  /\  R  e.  RR* )  ->  ( D  |`  ( B  X.  B ) ) : ( B  X.  B ) --> RR )
43 ismet2 17900 . 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 645 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 176    /\ wa 358    /\ w3a 934    = wceq 1625    e. wcel 1686   A.wral 2545    C_ wss 3154   class class class wbr 4025    X. cxp 4689   `'ccnv 4690    |` cres 4693   "cima 4694    Fn wfn 5252   -->wf 5253   ` cfv 5257  (class class class)co 5860    Er wer 6659   [cec 6660   RRcr 8738   RR*cxr 8868   * Metcxmt 16371   Metcme 16372   ballcbl 16373
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1535  ax-5 1546  ax-17 1605  ax-9 1637  ax-8 1645  ax-13 1688  ax-14 1690  ax-6 1705  ax-7 1710  ax-11 1717  ax-12 1868  ax-ext 2266  ax-sep 4143  ax-nul 4151  ax-pow 4190  ax-pr 4216  ax-un 4514  ax-cnex 8795  ax-resscn 8796  ax-1cn 8797  ax-icn 8798  ax-addcl 8799  ax-addrcl 8800  ax-mulcl 8801  ax-mulrcl 8802  ax-mulcom 8803  ax-addass 8804  ax-mulass 8805  ax-distr 8806  ax-i2m1 8807  ax-1ne0 8808  ax-1rid 8809  ax-rnegex 8810  ax-rrecex 8811  ax-cnre 8812  ax-pre-lttri 8813  ax-pre-lttrn 8814  ax-pre-ltadd 8815  ax-pre-mulgt0 8816
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 1531  df-nf 1534  df-sb 1632  df-eu 2149  df-mo 2150  df-clab 2272  df-cleq 2278  df-clel 2281  df-nfc 2410  df-ne 2450  df-nel 2451  df-ral 2550  df-rex 2551  df-reu 2552  df-rmo 2553  df-rab 2554  df-v 2792  df-sbc 2994  df-csb 3084  df-dif 3157  df-un 3159  df-in 3161  df-ss 3168  df-nul 3458  df-if 3568  df-pw 3629  df-sn 3648  df-pr 3649  df-op 3651  df-uni 3830  df-iun 3909  df-br 4026  df-opab 4080  df-mpt 4081  df-id 4311  df-po 4316  df-so 4317  df-xp 4697  df-rel 4698  df-cnv 4699  df-co 4700  df-dm 4701  df-rn 4702  df-res 4703  df-ima 4704  df-iota 5221  df-fun 5259  df-fn 5260  df-f 5261  df-f1 5262  df-fo 5263  df-f1o 5264  df-fv 5265  df-ov 5863  df-oprab 5864  df-mpt2 5865  df-1st 6124  df-2nd 6125  df-riota 6306  df-er 6662  df-ec 6664  df-map 6776  df-en 6866  df-dom 6867  df-sdom 6868  df-pnf 8871  df-mnf 8872  df-xr 8873  df-ltxr 8874  df-le 8875  df-sub 9041  df-neg 9042  df-div 9426  df-2 9806  df-rp 10357  df-xneg 10454  df-xadd 10455  df-xmul 10456  df-xmet 16375  df-met 16376  df-bl 16377
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