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Theorem xmetresbl 18035
Description: An extended metric restricted to any ball (in particular the infinity ball) is a proper metric. Together with xmetec 18032, 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 18020 . . . 4  |-  ( ( D  e.  ( * Met `  X )  /\  P  e.  X  /\  R  e.  RR* )  ->  ( P ( ball `  D ) R ) 
C_  X )
42, 3syl5eqss 3256 . . 3  |-  ( ( D  e.  ( * Met `  X )  /\  P  e.  X  /\  R  e.  RR* )  ->  B  C_  X )
5 xmetres2 17977 . . 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 17946 . . . . . 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 4829 . . . . . 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 5446 . . . . 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 5427 . . . 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 6029 . . . . . 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 2316 . . . . . . . . . 10  |-  ( `' D " RR )  =  ( `' D " RR )
1918xmeter 18031 . . . . . . . . 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 18033 . . . . . . . . . . . 12  |-  ( ( D  e.  ( * Met `  X )  /\  P  e.  X  /\  R  e.  RR* )  ->  ( P ( ball `  D ) R ) 
C_  [ P ]
( `' D " RR ) )
222, 21syl5eqss 3256 . . . . . . . . . . 11  |-  ( ( D  e.  ( * Met `  X )  /\  P  e.  X  /\  R  e.  RR* )  ->  B  C_  [ P ] ( `' D " RR ) )
2322sselda 3214 . . . . . . . . . 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 6740 . . . . . . . . . 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 3214 . . . . . . . . . 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 6740 . . . . . . . . . 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 6720 . . . . . . 7  |-  ( ( ( D  e.  ( * Met `  X
)  /\  P  e.  X  /\  R  e.  RR* )  /\  ( x  e.  B  /\  y  e.  B ) )  ->  x ( `' D " RR ) y )
3518xmeterval 18030 . . . . . . . 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 2390 . . . 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 2669 . . 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 5990 . . 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 17950 . 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 1633    e. wcel 1701   A.wral 2577    C_ wss 3186   class class class wbr 4060    X. cxp 4724   `'ccnv 4725    |` cres 4728   "cima 4729    Fn wfn 5287   -->wf 5288   ` cfv 5292  (class class class)co 5900    Er wer 6699   [cec 6700   RRcr 8781   RR*cxr 8911   * Metcxmt 16418   Metcme 16419   ballcbl 16420
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-13 1703  ax-14 1705  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297  ax-sep 4178  ax-nul 4186  ax-pow 4225  ax-pr 4251  ax-un 4549  ax-cnex 8838  ax-resscn 8839  ax-1cn 8840  ax-icn 8841  ax-addcl 8842  ax-addrcl 8843  ax-mulcl 8844  ax-mulrcl 8845  ax-mulcom 8846  ax-addass 8847  ax-mulass 8848  ax-distr 8849  ax-i2m1 8850  ax-1ne0 8851  ax-1rid 8852  ax-rnegex 8853  ax-rrecex 8854  ax-cnre 8855  ax-pre-lttri 8856  ax-pre-lttrn 8857  ax-pre-ltadd 8858  ax-pre-mulgt0 8859
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 1533  df-nf 1536  df-sb 1640  df-eu 2180  df-mo 2181  df-clab 2303  df-cleq 2309  df-clel 2312  df-nfc 2441  df-ne 2481  df-nel 2482  df-ral 2582  df-rex 2583  df-reu 2584  df-rmo 2585  df-rab 2586  df-v 2824  df-sbc 3026  df-csb 3116  df-dif 3189  df-un 3191  df-in 3193  df-ss 3200  df-nul 3490  df-if 3600  df-pw 3661  df-sn 3680  df-pr 3681  df-op 3683  df-uni 3865  df-iun 3944  df-br 4061  df-opab 4115  df-mpt 4116  df-id 4346  df-po 4351  df-so 4352  df-xp 4732  df-rel 4733  df-cnv 4734  df-co 4735  df-dm 4736  df-rn 4737  df-res 4738  df-ima 4739  df-iota 5256  df-fun 5294  df-fn 5295  df-f 5296  df-f1 5297  df-fo 5298  df-f1o 5299  df-fv 5300  df-ov 5903  df-oprab 5904  df-mpt2 5905  df-1st 6164  df-2nd 6165  df-riota 6346  df-er 6702  df-ec 6704  df-map 6817  df-en 6907  df-dom 6908  df-sdom 6909  df-pnf 8914  df-mnf 8915  df-xr 8916  df-ltxr 8917  df-le 8918  df-sub 9084  df-neg 9085  df-div 9469  df-2 9849  df-rp 10402  df-xneg 10499  df-xadd 10500  df-xmul 10501  df-xmet 16425  df-met 16426  df-bl 16427
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