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Theorem xmetres2 12587
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 12551 . . . . 5  |-  Rel  *Met
2 relelfvdm 5461 . . . . 5  |-  ( ( Rel  *Met  /\  D  e.  ( *Met `  X ) )  ->  X  e.  dom  *Met )
31, 2mpan 421 . . . 4  |-  ( D  e.  ( *Met `  X )  ->  X  e.  dom  *Met )
43adantr 274 . . 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 4076 . 2  |-  ( ( D  e.  ( *Met `  X )  /\  R  C_  X
)  ->  R  e.  _V )
7 xmetf 12558 . . . 4  |-  ( D  e.  ( *Met `  X )  ->  D : ( X  X.  X ) --> RR* )
87adantr 274 . . 3  |-  ( ( D  e.  ( *Met `  X )  /\  R  C_  X
)  ->  D :
( X  X.  X
) --> RR* )
9 xpss12 4654 . . . 4  |-  ( ( R  C_  X  /\  R  C_  X )  -> 
( R  X.  R
)  C_  ( X  X.  X ) )
105, 9sylancom 417 . . 3  |-  ( ( D  e.  ( *Met `  X )  /\  R  C_  X
)  ->  ( R  X.  R )  C_  ( X  X.  X ) )
118, 10fssresd 5307 . 2  |-  ( ( D  e.  ( *Met `  X )  /\  R  C_  X
)  ->  ( D  |`  ( R  X.  R
) ) : ( R  X.  R ) -->
RR* )
12 ovres 5918 . . . . 5  |-  ( ( x  e.  R  /\  y  e.  R )  ->  ( x ( D  |`  ( R  X.  R
) ) y )  =  ( x D y ) )
1312adantl 275 . . . 4  |-  ( ( ( D  e.  ( *Met `  X
)  /\  R  C_  X
)  /\  ( x  e.  R  /\  y  e.  R ) )  -> 
( x ( D  |`  ( R  X.  R
) ) y )  =  ( x D y ) )
1413eqeq1d 2149 . . 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 519 . . . 4  |-  ( ( ( D  e.  ( *Met `  X
)  /\  R  C_  X
)  /\  ( x  e.  R  /\  y  e.  R ) )  ->  D  e.  ( *Met `  X ) )
16 simplr 520 . . . . 5  |-  ( ( ( D  e.  ( *Met `  X
)  /\  R  C_  X
)  /\  ( x  e.  R  /\  y  e.  R ) )  ->  R  C_  X )
17 simprl 521 . . . . 5  |-  ( ( ( D  e.  ( *Met `  X
)  /\  R  C_  X
)  /\  ( x  e.  R  /\  y  e.  R ) )  ->  x  e.  R )
1816, 17sseldd 3103 . . . 4  |-  ( ( ( D  e.  ( *Met `  X
)  /\  R  C_  X
)  /\  ( x  e.  R  /\  y  e.  R ) )  ->  x  e.  X )
19 simprr 522 . . . . 5  |-  ( ( ( D  e.  ( *Met `  X
)  /\  R  C_  X
)  /\  ( x  e.  R  /\  y  e.  R ) )  -> 
y  e.  R )
2016, 19sseldd 3103 . . . 4  |-  ( ( ( D  e.  ( *Met `  X
)  /\  R  C_  X
)  /\  ( x  e.  R  /\  y  e.  R ) )  -> 
y  e.  X )
21 xmeteq0 12567 . . . 4  |-  ( ( D  e.  ( *Met `  X )  /\  x  e.  X  /\  y  e.  X
)  ->  ( (
x D y )  =  0  <->  x  =  y ) )
2215, 18, 20, 21syl3anc 1217 . . 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 519 . . . 4  |-  ( ( ( D  e.  ( *Met `  X
)  /\  R  C_  X
)  /\  ( x  e.  R  /\  y  e.  R  /\  z  e.  R ) )  ->  D  e.  ( *Met `  X ) )
25 simplr 520 . . . . 5  |-  ( ( ( D  e.  ( *Met `  X
)  /\  R  C_  X
)  /\  ( x  e.  R  /\  y  e.  R  /\  z  e.  R ) )  ->  R  C_  X )
26 simpr3 990 . . . . 5  |-  ( ( ( D  e.  ( *Met `  X
)  /\  R  C_  X
)  /\  ( x  e.  R  /\  y  e.  R  /\  z  e.  R ) )  -> 
z  e.  R )
2725, 26sseldd 3103 . . . 4  |-  ( ( ( D  e.  ( *Met `  X
)  /\  R  C_  X
)  /\  ( x  e.  R  /\  y  e.  R  /\  z  e.  R ) )  -> 
z  e.  X )
28183adantr3 1143 . . . 4  |-  ( ( ( D  e.  ( *Met `  X
)  /\  R  C_  X
)  /\  ( x  e.  R  /\  y  e.  R  /\  z  e.  R ) )  ->  x  e.  X )
29203adantr3 1143 . . . 4  |-  ( ( ( D  e.  ( *Met `  X
)  /\  R  C_  X
)  /\  ( x  e.  R  /\  y  e.  R  /\  z  e.  R ) )  -> 
y  e.  X )
30 xmettri2 12569 . . . 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 1219 . . 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 1143 . . 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 988 . . . . 5  |-  ( ( ( D  e.  ( *Met `  X
)  /\  R  C_  X
)  /\  ( x  e.  R  /\  y  e.  R  /\  z  e.  R ) )  ->  x  e.  R )
3426, 33ovresd 5919 . . . 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 989 . . . . 5  |-  ( ( ( D  e.  ( *Met `  X
)  /\  R  C_  X
)  /\  ( x  e.  R  /\  y  e.  R  /\  z  e.  R ) )  -> 
y  e.  R )
3626, 35ovresd 5919 . . . 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 5800 . . 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 3968 . 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 12555 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 963    = wceq 1332    e. wcel 1481    C_ wss 3076   class class class wbr 3937    X. cxp 4545   dom cdm 4547    |` cres 4549   Rel wrel 4552   -->wf 5127   ` cfv 5131  (class class class)co 5782   0cc0 7644   RR*cxr 7823    <_ cle 7825   +ecxad 9587   *Metcxmet 12188
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4054  ax-pow 4106  ax-pr 4139  ax-un 4363  ax-setind 4460  ax-cnex 7735  ax-resscn 7736
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ne 2310  df-ral 2422  df-rex 2423  df-rab 2426  df-v 2691  df-sbc 2914  df-csb 3008  df-dif 3078  df-un 3080  df-in 3082  df-ss 3089  df-pw 3517  df-sn 3538  df-pr 3539  df-op 3541  df-uni 3745  df-iun 3823  df-br 3938  df-opab 3998  df-mpt 3999  df-id 4223  df-xp 4553  df-rel 4554  df-cnv 4555  df-co 4556  df-dm 4557  df-rn 4558  df-res 4559  df-ima 4560  df-iota 5096  df-fun 5133  df-fn 5134  df-f 5135  df-fv 5139  df-ov 5785  df-oprab 5786  df-mpo 5787  df-1st 6046  df-2nd 6047  df-map 6552  df-pnf 7826  df-mnf 7827  df-xr 7828  df-xmet 12196
This theorem is referenced by:  metres2  12589  xmetres  12590  xmetresbl  12648  metrest  12714  divcnap  12763  cncfmet  12787  limcimolemlt  12841  cnplimcim  12844  cnplimclemr  12846  limccnpcntop  12852  limccnp2cntop  12854
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