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Theorem xmetres2 14364
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 14328 . . . . 5  |-  Rel  *Met
2 relelfvdm 5569 . . . . 5  |-  ( ( Rel  *Met  /\  D  e.  ( *Met `  X ) )  ->  X  e.  dom  *Met )
31, 2mpan 424 . . . 4  |-  ( D  e.  ( *Met `  X )  ->  X  e.  dom  *Met )
43adantr 276 . . 3  |-  ( ( D  e.  ( *Met `  X )  /\  R  C_  X
)  ->  X  e.  dom  *Met )
5 simpr 110 . . 3  |-  ( ( D  e.  ( *Met `  X )  /\  R  C_  X
)  ->  R  C_  X
)
64, 5ssexd 4161 . 2  |-  ( ( D  e.  ( *Met `  X )  /\  R  C_  X
)  ->  R  e.  _V )
7 xmetf 14335 . . . 4  |-  ( D  e.  ( *Met `  X )  ->  D : ( X  X.  X ) --> RR* )
87adantr 276 . . 3  |-  ( ( D  e.  ( *Met `  X )  /\  R  C_  X
)  ->  D :
( X  X.  X
) --> RR* )
9 xpss12 4754 . . . 4  |-  ( ( R  C_  X  /\  R  C_  X )  -> 
( R  X.  R
)  C_  ( X  X.  X ) )
105, 9sylancom 420 . . 3  |-  ( ( D  e.  ( *Met `  X )  /\  R  C_  X
)  ->  ( R  X.  R )  C_  ( X  X.  X ) )
118, 10fssresd 5414 . 2  |-  ( ( D  e.  ( *Met `  X )  /\  R  C_  X
)  ->  ( D  |`  ( R  X.  R
) ) : ( R  X.  R ) -->
RR* )
12 ovres 6040 . . . . 5  |-  ( ( x  e.  R  /\  y  e.  R )  ->  ( x ( D  |`  ( R  X.  R
) ) y )  =  ( x D y ) )
1312adantl 277 . . . 4  |-  ( ( ( D  e.  ( *Met `  X
)  /\  R  C_  X
)  /\  ( x  e.  R  /\  y  e.  R ) )  -> 
( x ( D  |`  ( R  X.  R
) ) y )  =  ( x D y ) )
1413eqeq1d 2198 . . 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 527 . . . 4  |-  ( ( ( D  e.  ( *Met `  X
)  /\  R  C_  X
)  /\  ( x  e.  R  /\  y  e.  R ) )  ->  D  e.  ( *Met `  X ) )
16 simplr 528 . . . . 5  |-  ( ( ( D  e.  ( *Met `  X
)  /\  R  C_  X
)  /\  ( x  e.  R  /\  y  e.  R ) )  ->  R  C_  X )
17 simprl 529 . . . . 5  |-  ( ( ( D  e.  ( *Met `  X
)  /\  R  C_  X
)  /\  ( x  e.  R  /\  y  e.  R ) )  ->  x  e.  R )
1816, 17sseldd 3171 . . . 4  |-  ( ( ( D  e.  ( *Met `  X
)  /\  R  C_  X
)  /\  ( x  e.  R  /\  y  e.  R ) )  ->  x  e.  X )
19 simprr 531 . . . . 5  |-  ( ( ( D  e.  ( *Met `  X
)  /\  R  C_  X
)  /\  ( x  e.  R  /\  y  e.  R ) )  -> 
y  e.  R )
2016, 19sseldd 3171 . . . 4  |-  ( ( ( D  e.  ( *Met `  X
)  /\  R  C_  X
)  /\  ( x  e.  R  /\  y  e.  R ) )  -> 
y  e.  X )
21 xmeteq0 14344 . . . 4  |-  ( ( D  e.  ( *Met `  X )  /\  x  e.  X  /\  y  e.  X
)  ->  ( (
x D y )  =  0  <->  x  =  y ) )
2215, 18, 20, 21syl3anc 1249 . . 3  |-  ( ( ( D  e.  ( *Met `  X
)  /\  R  C_  X
)  /\  ( x  e.  R  /\  y  e.  R ) )  -> 
( ( x D y )  =  0  <-> 
x  =  y ) )
2314, 22bitrd 188 . 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 527 . . . 4  |-  ( ( ( D  e.  ( *Met `  X
)  /\  R  C_  X
)  /\  ( x  e.  R  /\  y  e.  R  /\  z  e.  R ) )  ->  D  e.  ( *Met `  X ) )
25 simplr 528 . . . . 5  |-  ( ( ( D  e.  ( *Met `  X
)  /\  R  C_  X
)  /\  ( x  e.  R  /\  y  e.  R  /\  z  e.  R ) )  ->  R  C_  X )
26 simpr3 1007 . . . . 5  |-  ( ( ( D  e.  ( *Met `  X
)  /\  R  C_  X
)  /\  ( x  e.  R  /\  y  e.  R  /\  z  e.  R ) )  -> 
z  e.  R )
2725, 26sseldd 3171 . . . 4  |-  ( ( ( D  e.  ( *Met `  X
)  /\  R  C_  X
)  /\  ( x  e.  R  /\  y  e.  R  /\  z  e.  R ) )  -> 
z  e.  X )
28183adantr3 1160 . . . 4  |-  ( ( ( D  e.  ( *Met `  X
)  /\  R  C_  X
)  /\  ( x  e.  R  /\  y  e.  R  /\  z  e.  R ) )  ->  x  e.  X )
29203adantr3 1160 . . . 4  |-  ( ( ( D  e.  ( *Met `  X
)  /\  R  C_  X
)  /\  ( x  e.  R  /\  y  e.  R  /\  z  e.  R ) )  -> 
y  e.  X )
30 xmettri2 14346 . . . 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 1251 . . 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 1160 . . 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 1005 . . . . 5  |-  ( ( ( D  e.  ( *Met `  X
)  /\  R  C_  X
)  /\  ( x  e.  R  /\  y  e.  R  /\  z  e.  R ) )  ->  x  e.  R )
3426, 33ovresd 6041 . . . 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 1006 . . . . 5  |-  ( ( ( D  e.  ( *Met `  X
)  /\  R  C_  X
)  /\  ( x  e.  R  /\  y  e.  R  /\  z  e.  R ) )  -> 
y  e.  R )
3626, 35ovresd 6041 . . . 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 5918 . . 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 4053 . 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 14332 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 104    <-> wb 105    /\ w3a 980    = wceq 1364    e. wcel 2160    C_ wss 3144   class class class wbr 4021    X. cxp 4645   dom cdm 4647    |` cres 4649   Rel wrel 4652   -->wf 5234   ` cfv 5238  (class class class)co 5900   0cc0 7846   RR*cxr 8026    <_ cle 8028   +ecxad 9806   *Metcxmet 13874
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-sep 4139  ax-pow 4195  ax-pr 4230  ax-un 4454  ax-setind 4557  ax-cnex 7937  ax-resscn 7938
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-ral 2473  df-rex 2474  df-rab 2477  df-v 2754  df-sbc 2978  df-csb 3073  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-pw 3595  df-sn 3616  df-pr 3617  df-op 3619  df-uni 3828  df-iun 3906  df-br 4022  df-opab 4083  df-mpt 4084  df-id 4314  df-xp 4653  df-rel 4654  df-cnv 4655  df-co 4656  df-dm 4657  df-rn 4658  df-res 4659  df-ima 4660  df-iota 5199  df-fun 5240  df-fn 5241  df-f 5242  df-fv 5246  df-ov 5903  df-oprab 5904  df-mpo 5905  df-1st 6169  df-2nd 6170  df-map 6680  df-pnf 8029  df-mnf 8030  df-xr 8031  df-xmet 13882
This theorem is referenced by:  metres2  14366  xmetres  14367  xmetresbl  14425  metrest  14491  divcnap  14540  cncfmet  14564  limcimolemlt  14618  cnplimcim  14621  cnplimclemr  14623  limccnpcntop  14629  limccnp2cntop  14631
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