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Theorem xmetres2 12307
Description: Restriction of an extended metric. (Contributed by Mario Carneiro, 20-Aug-2015.)
Assertion
Ref Expression
xmetres2  |-  ( ( D  e.  ( *Met `  X )  /\  R  C_  X
)  ->  ( D  |`  ( R  X.  R
) )  e.  ( *Met `  R
) )

Proof of Theorem xmetres2
Dummy variables  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 xmetrel 12271 . . . . 5  |-  Rel  *Met
2 relelfvdm 5385 . . . . 5  |-  ( ( Rel  *Met  /\  D  e.  ( *Met `  X ) )  ->  X  e.  dom  *Met )
31, 2mpan 418 . . . 4  |-  ( D  e.  ( *Met `  X )  ->  X  e.  dom  *Met )
43adantr 272 . . 3  |-  ( ( D  e.  ( *Met `  X )  /\  R  C_  X
)  ->  X  e.  dom  *Met )
5 simpr 109 . . 3  |-  ( ( D  e.  ( *Met `  X )  /\  R  C_  X
)  ->  R  C_  X
)
64, 5ssexd 4008 . 2  |-  ( ( D  e.  ( *Met `  X )  /\  R  C_  X
)  ->  R  e.  _V )
7 xmetf 12278 . . . 4  |-  ( D  e.  ( *Met `  X )  ->  D : ( X  X.  X ) --> RR* )
87adantr 272 . . 3  |-  ( ( D  e.  ( *Met `  X )  /\  R  C_  X
)  ->  D :
( X  X.  X
) --> RR* )
9 xpss12 4584 . . . 4  |-  ( ( R  C_  X  /\  R  C_  X )  -> 
( R  X.  R
)  C_  ( X  X.  X ) )
105, 9sylancom 414 . . 3  |-  ( ( D  e.  ( *Met `  X )  /\  R  C_  X
)  ->  ( R  X.  R )  C_  ( X  X.  X ) )
118, 10fssresd 5235 . 2  |-  ( ( D  e.  ( *Met `  X )  /\  R  C_  X
)  ->  ( D  |`  ( R  X.  R
) ) : ( R  X.  R ) -->
RR* )
12 ovres 5842 . . . . 5  |-  ( ( x  e.  R  /\  y  e.  R )  ->  ( x ( D  |`  ( R  X.  R
) ) y )  =  ( x D y ) )
1312adantl 273 . . . 4  |-  ( ( ( D  e.  ( *Met `  X
)  /\  R  C_  X
)  /\  ( x  e.  R  /\  y  e.  R ) )  -> 
( x ( D  |`  ( R  X.  R
) ) y )  =  ( x D y ) )
1413eqeq1d 2108 . . 3  |-  ( ( ( D  e.  ( *Met `  X
)  /\  R  C_  X
)  /\  ( x  e.  R  /\  y  e.  R ) )  -> 
( ( x ( D  |`  ( R  X.  R ) ) y )  =  0  <->  (
x D y )  =  0 ) )
15 simpll 499 . . . 4  |-  ( ( ( D  e.  ( *Met `  X
)  /\  R  C_  X
)  /\  ( x  e.  R  /\  y  e.  R ) )  ->  D  e.  ( *Met `  X ) )
16 simplr 500 . . . . 5  |-  ( ( ( D  e.  ( *Met `  X
)  /\  R  C_  X
)  /\  ( x  e.  R  /\  y  e.  R ) )  ->  R  C_  X )
17 simprl 501 . . . . 5  |-  ( ( ( D  e.  ( *Met `  X
)  /\  R  C_  X
)  /\  ( x  e.  R  /\  y  e.  R ) )  ->  x  e.  R )
1816, 17sseldd 3048 . . . 4  |-  ( ( ( D  e.  ( *Met `  X
)  /\  R  C_  X
)  /\  ( x  e.  R  /\  y  e.  R ) )  ->  x  e.  X )
19 simprr 502 . . . . 5  |-  ( ( ( D  e.  ( *Met `  X
)  /\  R  C_  X
)  /\  ( x  e.  R  /\  y  e.  R ) )  -> 
y  e.  R )
2016, 19sseldd 3048 . . . 4  |-  ( ( ( D  e.  ( *Met `  X
)  /\  R  C_  X
)  /\  ( x  e.  R  /\  y  e.  R ) )  -> 
y  e.  X )
21 xmeteq0 12287 . . . 4  |-  ( ( D  e.  ( *Met `  X )  /\  x  e.  X  /\  y  e.  X
)  ->  ( (
x D y )  =  0  <->  x  =  y ) )
2215, 18, 20, 21syl3anc 1184 . . 3  |-  ( ( ( D  e.  ( *Met `  X
)  /\  R  C_  X
)  /\  ( x  e.  R  /\  y  e.  R ) )  -> 
( ( x D y )  =  0  <-> 
x  =  y ) )
2314, 22bitrd 187 . 2  |-  ( ( ( D  e.  ( *Met `  X
)  /\  R  C_  X
)  /\  ( x  e.  R  /\  y  e.  R ) )  -> 
( ( x ( D  |`  ( R  X.  R ) ) y )  =  0  <->  x  =  y ) )
24 simpll 499 . . . 4  |-  ( ( ( D  e.  ( *Met `  X
)  /\  R  C_  X
)  /\  ( x  e.  R  /\  y  e.  R  /\  z  e.  R ) )  ->  D  e.  ( *Met `  X ) )
25 simplr 500 . . . . 5  |-  ( ( ( D  e.  ( *Met `  X
)  /\  R  C_  X
)  /\  ( x  e.  R  /\  y  e.  R  /\  z  e.  R ) )  ->  R  C_  X )
26 simpr3 957 . . . . 5  |-  ( ( ( D  e.  ( *Met `  X
)  /\  R  C_  X
)  /\  ( x  e.  R  /\  y  e.  R  /\  z  e.  R ) )  -> 
z  e.  R )
2725, 26sseldd 3048 . . . 4  |-  ( ( ( D  e.  ( *Met `  X
)  /\  R  C_  X
)  /\  ( x  e.  R  /\  y  e.  R  /\  z  e.  R ) )  -> 
z  e.  X )
28183adantr3 1110 . . . 4  |-  ( ( ( D  e.  ( *Met `  X
)  /\  R  C_  X
)  /\  ( x  e.  R  /\  y  e.  R  /\  z  e.  R ) )  ->  x  e.  X )
29203adantr3 1110 . . . 4  |-  ( ( ( D  e.  ( *Met `  X
)  /\  R  C_  X
)  /\  ( x  e.  R  /\  y  e.  R  /\  z  e.  R ) )  -> 
y  e.  X )
30 xmettri2 12289 . . . 4  |-  ( ( D  e.  ( *Met `  X )  /\  ( z  e.  X  /\  x  e.  X  /\  y  e.  X ) )  -> 
( x D y )  <_  ( (
z D x ) +e ( z D y ) ) )
3124, 27, 28, 29, 30syl13anc 1186 . . 3  |-  ( ( ( D  e.  ( *Met `  X
)  /\  R  C_  X
)  /\  ( x  e.  R  /\  y  e.  R  /\  z  e.  R ) )  -> 
( x D y )  <_  ( (
z D x ) +e ( z D y ) ) )
32133adantr3 1110 . . 3  |-  ( ( ( D  e.  ( *Met `  X
)  /\  R  C_  X
)  /\  ( x  e.  R  /\  y  e.  R  /\  z  e.  R ) )  -> 
( x ( D  |`  ( R  X.  R
) ) y )  =  ( x D y ) )
33 simpr1 955 . . . . 5  |-  ( ( ( D  e.  ( *Met `  X
)  /\  R  C_  X
)  /\  ( x  e.  R  /\  y  e.  R  /\  z  e.  R ) )  ->  x  e.  R )
3426, 33ovresd 5843 . . . 4  |-  ( ( ( D  e.  ( *Met `  X
)  /\  R  C_  X
)  /\  ( x  e.  R  /\  y  e.  R  /\  z  e.  R ) )  -> 
( z ( D  |`  ( R  X.  R
) ) x )  =  ( z D x ) )
35 simpr2 956 . . . . 5  |-  ( ( ( D  e.  ( *Met `  X
)  /\  R  C_  X
)  /\  ( x  e.  R  /\  y  e.  R  /\  z  e.  R ) )  -> 
y  e.  R )
3626, 35ovresd 5843 . . . 4  |-  ( ( ( D  e.  ( *Met `  X
)  /\  R  C_  X
)  /\  ( x  e.  R  /\  y  e.  R  /\  z  e.  R ) )  -> 
( z ( D  |`  ( R  X.  R
) ) y )  =  ( z D y ) )
3734, 36oveq12d 5724 . . 3  |-  ( ( ( D  e.  ( *Met `  X
)  /\  R  C_  X
)  /\  ( x  e.  R  /\  y  e.  R  /\  z  e.  R ) )  -> 
( ( z ( D  |`  ( R  X.  R ) ) x ) +e ( z ( D  |`  ( R  X.  R
) ) y ) )  =  ( ( z D x ) +e ( z D y ) ) )
3831, 32, 373brtr4d 3905 . 2  |-  ( ( ( D  e.  ( *Met `  X
)  /\  R  C_  X
)  /\  ( x  e.  R  /\  y  e.  R  /\  z  e.  R ) )  -> 
( x ( D  |`  ( R  X.  R
) ) y )  <_  ( ( z ( D  |`  ( R  X.  R ) ) x ) +e
( z ( D  |`  ( R  X.  R
) ) y ) ) )
396, 11, 23, 38isxmetd 12275 1  |-  ( ( D  e.  ( *Met `  X )  /\  R  C_  X
)  ->  ( D  |`  ( R  X.  R
) )  e.  ( *Met `  R
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104    /\ w3a 930    = wceq 1299    e. wcel 1448    C_ wss 3021   class class class wbr 3875    X. cxp 4475   dom cdm 4477    |` cres 4479   Rel wrel 4482   -->wf 5055   ` cfv 5059  (class class class)co 5706   0cc0 7500   RR*cxr 7671    <_ cle 7673   +ecxad 9398   *Metcxmet 11931
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 584  ax-in2 585  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-13 1459  ax-14 1460  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082  ax-sep 3986  ax-pow 4038  ax-pr 4069  ax-un 4293  ax-setind 4390  ax-cnex 7586  ax-resscn 7587
This theorem depends on definitions:  df-bi 116  df-3an 932  df-tru 1302  df-fal 1305  df-nf 1405  df-sb 1704  df-eu 1963  df-mo 1964  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-ne 2268  df-ral 2380  df-rex 2381  df-rab 2384  df-v 2643  df-sbc 2863  df-csb 2956  df-dif 3023  df-un 3025  df-in 3027  df-ss 3034  df-pw 3459  df-sn 3480  df-pr 3481  df-op 3483  df-uni 3684  df-iun 3762  df-br 3876  df-opab 3930  df-mpt 3931  df-id 4153  df-xp 4483  df-rel 4484  df-cnv 4485  df-co 4486  df-dm 4487  df-rn 4488  df-res 4489  df-ima 4490  df-iota 5024  df-fun 5061  df-fn 5062  df-f 5063  df-fv 5067  df-ov 5709  df-oprab 5710  df-mpo 5711  df-1st 5969  df-2nd 5970  df-map 6474  df-pnf 7674  df-mnf 7675  df-xr 7676  df-xmet 11939
This theorem is referenced by:  metres2  12309  xmetres  12310  xmetresbl  12368  metrest  12434  cncfmet  12492  limcimolemlt  12513  cnplimcim  12516  limccnpcntop  12520
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