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Theorem xmetres 20733
Description: A restriction of an extended metric is an extended metric. (Contributed by Mario Carneiro, 24-Aug-2015.)
Assertion
Ref Expression
xmetres  |-  ( D  e.  ( *Met `  X )  ->  ( D  |`  ( R  X.  R ) )  e.  ( *Met `  ( X  i^i  R ) ) )

Proof of Theorem xmetres
StepHypRef Expression
1 xmetf 20698 . . 3  |-  ( D  e.  ( *Met `  X )  ->  D : ( X  X.  X ) --> RR* )
2 fdm 5741 . . 3  |-  ( D : ( X  X.  X ) --> RR*  ->  dom 
D  =  ( X  X.  X ) )
3 metreslem 20731 . . 3  |-  ( dom 
D  =  ( X  X.  X )  -> 
( D  |`  ( R  X.  R ) )  =  ( D  |`  ( ( X  i^i  R )  X.  ( X  i^i  R ) ) ) )
41, 2, 33syl 20 . 2  |-  ( D  e.  ( *Met `  X )  ->  ( D  |`  ( R  X.  R ) )  =  ( D  |`  (
( X  i^i  R
)  X.  ( X  i^i  R ) ) ) )
5 inss1 3723 . . 3  |-  ( X  i^i  R )  C_  X
6 xmetres2 20730 . . 3  |-  ( ( D  e.  ( *Met `  X )  /\  ( X  i^i  R )  C_  X )  ->  ( D  |`  (
( X  i^i  R
)  X.  ( X  i^i  R ) ) )  e.  ( *Met `  ( X  i^i  R ) ) )
75, 6mpan2 671 . 2  |-  ( D  e.  ( *Met `  X )  ->  ( D  |`  ( ( X  i^i  R )  X.  ( X  i^i  R
) ) )  e.  ( *Met `  ( X  i^i  R ) ) )
84, 7eqeltrd 2555 1  |-  ( D  e.  ( *Met `  X )  ->  ( D  |`  ( R  X.  R ) )  e.  ( *Met `  ( X  i^i  R ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1379    e. wcel 1767    i^i cin 3480    C_ wss 3481    X. cxp 5003   dom cdm 5005    |` cres 5007   -->wf 5590   ` cfv 5594   RR*cxr 9639   *Metcxmt 18271
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587  ax-cnex 9560  ax-resscn 9561
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-rab 2826  df-v 3120  df-sbc 3337  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-op 4040  df-uni 4252  df-br 4454  df-opab 4512  df-mpt 4513  df-id 4801  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-fv 5602  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-map 7434  df-xr 9644  df-xmet 18280
This theorem is referenced by:  blres  20800  ressxms  20894  cfilresi  21600  caussi  21602  causs  21603  minvecolem4a  25624
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