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Theorem xmetres2 20592
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 elfvdm 5883 . . . 4  |-  ( D  e.  ( *Met `  X )  ->  X  e.  dom  *Met )
21adantr 465 . . 3  |-  ( ( D  e.  ( *Met `  X )  /\  R  C_  X
)  ->  X  e.  dom  *Met )
3 simpr 461 . . 3  |-  ( ( D  e.  ( *Met `  X )  /\  R  C_  X
)  ->  R  C_  X
)
42, 3ssexd 4587 . 2  |-  ( ( D  e.  ( *Met `  X )  /\  R  C_  X
)  ->  R  e.  _V )
5 xmetf 20560 . . . 4  |-  ( D  e.  ( *Met `  X )  ->  D : ( X  X.  X ) --> RR* )
65adantr 465 . . 3  |-  ( ( D  e.  ( *Met `  X )  /\  R  C_  X
)  ->  D :
( X  X.  X
) --> RR* )
7 xpss12 5099 . . . 4  |-  ( ( R  C_  X  /\  R  C_  X )  -> 
( R  X.  R
)  C_  ( X  X.  X ) )
83, 3, 7syl2anc 661 . . 3  |-  ( ( D  e.  ( *Met `  X )  /\  R  C_  X
)  ->  ( R  X.  R )  C_  ( X  X.  X ) )
9 fssres 5742 . . 3  |-  ( ( D : ( X  X.  X ) --> RR* 
/\  ( R  X.  R )  C_  ( X  X.  X ) )  ->  ( D  |`  ( R  X.  R
) ) : ( R  X.  R ) -->
RR* )
106, 8, 9syl2anc 661 . 2  |-  ( ( D  e.  ( *Met `  X )  /\  R  C_  X
)  ->  ( D  |`  ( R  X.  R
) ) : ( R  X.  R ) -->
RR* )
11 ovres 6417 . . . . 5  |-  ( ( x  e.  R  /\  y  e.  R )  ->  ( x ( D  |`  ( R  X.  R
) ) y )  =  ( x D y ) )
1211adantl 466 . . . 4  |-  ( ( ( D  e.  ( *Met `  X
)  /\  R  C_  X
)  /\  ( x  e.  R  /\  y  e.  R ) )  -> 
( x ( D  |`  ( R  X.  R
) ) y )  =  ( x D y ) )
1312eqeq1d 2462 . . 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 ) )
14 simpll 753 . . . 4  |-  ( ( ( D  e.  ( *Met `  X
)  /\  R  C_  X
)  /\  ( x  e.  R  /\  y  e.  R ) )  ->  D  e.  ( *Met `  X ) )
15 simplr 754 . . . . 5  |-  ( ( ( D  e.  ( *Met `  X
)  /\  R  C_  X
)  /\  ( x  e.  R  /\  y  e.  R ) )  ->  R  C_  X )
16 simprl 755 . . . . 5  |-  ( ( ( D  e.  ( *Met `  X
)  /\  R  C_  X
)  /\  ( x  e.  R  /\  y  e.  R ) )  ->  x  e.  R )
1715, 16sseldd 3498 . . . 4  |-  ( ( ( D  e.  ( *Met `  X
)  /\  R  C_  X
)  /\  ( x  e.  R  /\  y  e.  R ) )  ->  x  e.  X )
18 simprr 756 . . . . 5  |-  ( ( ( D  e.  ( *Met `  X
)  /\  R  C_  X
)  /\  ( x  e.  R  /\  y  e.  R ) )  -> 
y  e.  R )
1915, 18sseldd 3498 . . . 4  |-  ( ( ( D  e.  ( *Met `  X
)  /\  R  C_  X
)  /\  ( x  e.  R  /\  y  e.  R ) )  -> 
y  e.  X )
20 xmeteq0 20569 . . . 4  |-  ( ( D  e.  ( *Met `  X )  /\  x  e.  X  /\  y  e.  X
)  ->  ( (
x D y )  =  0  <->  x  =  y ) )
2114, 17, 19, 20syl3anc 1223 . . 3  |-  ( ( ( D  e.  ( *Met `  X
)  /\  R  C_  X
)  /\  ( x  e.  R  /\  y  e.  R ) )  -> 
( ( x D y )  =  0  <-> 
x  =  y ) )
2213, 21bitrd 253 . 2  |-  ( ( ( D  e.  ( *Met `  X
)  /\  R  C_  X
)  /\  ( x  e.  R  /\  y  e.  R ) )  -> 
( ( x ( D  |`  ( R  X.  R ) ) y )  =  0  <->  x  =  y ) )
23 simpll 753 . . . 4  |-  ( ( ( D  e.  ( *Met `  X
)  /\  R  C_  X
)  /\  ( x  e.  R  /\  y  e.  R  /\  z  e.  R ) )  ->  D  e.  ( *Met `  X ) )
24 simplr 754 . . . . 5  |-  ( ( ( D  e.  ( *Met `  X
)  /\  R  C_  X
)  /\  ( x  e.  R  /\  y  e.  R  /\  z  e.  R ) )  ->  R  C_  X )
25 simpr3 999 . . . . 5  |-  ( ( ( D  e.  ( *Met `  X
)  /\  R  C_  X
)  /\  ( x  e.  R  /\  y  e.  R  /\  z  e.  R ) )  -> 
z  e.  R )
2624, 25sseldd 3498 . . . 4  |-  ( ( ( D  e.  ( *Met `  X
)  /\  R  C_  X
)  /\  ( x  e.  R  /\  y  e.  R  /\  z  e.  R ) )  -> 
z  e.  X )
27173adantr3 1152 . . . 4  |-  ( ( ( D  e.  ( *Met `  X
)  /\  R  C_  X
)  /\  ( x  e.  R  /\  y  e.  R  /\  z  e.  R ) )  ->  x  e.  X )
28193adantr3 1152 . . . 4  |-  ( ( ( D  e.  ( *Met `  X
)  /\  R  C_  X
)  /\  ( x  e.  R  /\  y  e.  R  /\  z  e.  R ) )  -> 
y  e.  X )
29 xmettri2 20571 . . . 4  |-  ( ( D  e.  ( *Met `  X )  /\  ( z  e.  X  /\  x  e.  X  /\  y  e.  X ) )  -> 
( x D y )  <_  ( (
z D x ) +e ( z D y ) ) )
3023, 26, 27, 28, 29syl13anc 1225 . . 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 ) ) )
31123adantr3 1152 . . 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 ) )
32 simpr1 997 . . . . 5  |-  ( ( ( D  e.  ( *Met `  X
)  /\  R  C_  X
)  /\  ( x  e.  R  /\  y  e.  R  /\  z  e.  R ) )  ->  x  e.  R )
3325, 32ovresd 6418 . . . 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 ) )
34 simpr2 998 . . . . 5  |-  ( ( ( D  e.  ( *Met `  X
)  /\  R  C_  X
)  /\  ( x  e.  R  /\  y  e.  R  /\  z  e.  R ) )  -> 
y  e.  R )
3525, 34ovresd 6418 . . . 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 ) )
3633, 35oveq12d 6293 . . 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 ) ) )
3730, 31, 363brtr4d 4470 . 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 ) ) )
384, 10, 22, 37isxmetd 20557 1  |-  ( ( D  e.  ( *Met `  X )  /\  R  C_  X
)  ->  ( D  |`  ( R  X.  R
) )  e.  ( *Met `  R
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 968    = wceq 1374    e. wcel 1762    C_ wss 3469   class class class wbr 4440    X. cxp 4990   dom cdm 4992    |` cres 4994   -->wf 5575   ` cfv 5579  (class class class)co 6275   0cc0 9481   RR*cxr 9616    <_ cle 9618   +ecxad 11305   *Metcxmt 18167
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567  ax-cnex 9537  ax-resscn 9538
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3108  df-sbc 3325  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-op 4027  df-uni 4239  df-br 4441  df-opab 4499  df-mpt 4500  df-id 4788  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-fv 5587  df-ov 6278  df-oprab 6279  df-mpt2 6280  df-map 7412  df-xr 9621  df-xmet 18176
This theorem is referenced by:  metres2  20594  xmetres  20595  xpsxmet  20611  xpsdsval  20612  xmetresbl  20668  tmsxms  20717  imasf1oxms  20720  metrest  20755  prdsxms  20761  tmsxpsval  20769  nrginvrcn  20928  divcn  21100  iitopon  21111  cncfmet  21140  cfilres  21463  dvlip2  22124  ftc1lem6  22170  ulmdvlem1  22522  ulmdvlem3  22524  abelth  22563  cxpcn3  22843  rlimcnp  23016  minvecolem4b  25456  minvecolem4  25458  ftc1cnnc  29653  blbnd  29873  ismtyres  29894  reheibor  29925
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