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Theorem xmetres 19961
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 19926 . . 3  |-  ( D  e.  ( *Met `  X )  ->  D : ( X  X.  X ) --> RR* )
2 fdm 5584 . . 3  |-  ( D : ( X  X.  X ) --> RR*  ->  dom 
D  =  ( X  X.  X ) )
3 metreslem 19959 . . 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 3591 . . 3  |-  ( X  i^i  R )  C_  X
6 xmetres2 19958 . . 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 2517 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 1369    e. wcel 1756    i^i cin 3348    C_ wss 3349    X. cxp 4859   dom cdm 4861    |` cres 4863   -->wf 5435   ` cfv 5439   RR*cxr 9438   *Metcxmt 17823
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4434  ax-nul 4442  ax-pow 4491  ax-pr 4552  ax-un 6393  ax-cnex 9359  ax-resscn 9360
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2622  df-ral 2741  df-rex 2742  df-rab 2745  df-v 2995  df-sbc 3208  df-dif 3352  df-un 3354  df-in 3356  df-ss 3363  df-nul 3659  df-if 3813  df-pw 3883  df-sn 3899  df-pr 3901  df-op 3905  df-uni 4113  df-br 4314  df-opab 4372  df-mpt 4373  df-id 4657  df-xp 4867  df-rel 4868  df-cnv 4869  df-co 4870  df-dm 4871  df-rn 4872  df-res 4873  df-iota 5402  df-fun 5441  df-fn 5442  df-f 5443  df-fv 5447  df-ov 6115  df-oprab 6116  df-mpt2 6117  df-map 7237  df-xr 9443  df-xmet 17832
This theorem is referenced by:  blres  20028  ressxms  20122  cfilresi  20828  caussi  20830  causs  20831  minvecolem4a  24300
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