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Theorem xmetres2 19936
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 5716 . . . 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 4439 . 2  |-  ( ( D  e.  ( *Met `  X )  /\  R  C_  X
)  ->  R  e.  _V )
5 xmetf 19904 . . . 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 4945 . . . 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 5578 . . 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 6230 . . . . 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 2451 . . 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 3357 . . . 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 3357 . . . 4  |-  ( ( ( D  e.  ( *Met `  X
)  /\  R  C_  X
)  /\  ( x  e.  R  /\  y  e.  R ) )  -> 
y  e.  X )
20 xmeteq0 19913 . . . 4  |-  ( ( D  e.  ( *Met `  X )  /\  x  e.  X  /\  y  e.  X
)  ->  ( (
x D y )  =  0  <->  x  =  y ) )
2114, 17, 19, 20syl3anc 1218 . . 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 996 . . . . 5  |-  ( ( ( D  e.  ( *Met `  X
)  /\  R  C_  X
)  /\  ( x  e.  R  /\  y  e.  R  /\  z  e.  R ) )  -> 
z  e.  R )
2624, 25sseldd 3357 . . . 4  |-  ( ( ( D  e.  ( *Met `  X
)  /\  R  C_  X
)  /\  ( x  e.  R  /\  y  e.  R  /\  z  e.  R ) )  -> 
z  e.  X )
27173adantr3 1149 . . . 4  |-  ( ( ( D  e.  ( *Met `  X
)  /\  R  C_  X
)  /\  ( x  e.  R  /\  y  e.  R  /\  z  e.  R ) )  ->  x  e.  X )
28193adantr3 1149 . . . 4  |-  ( ( ( D  e.  ( *Met `  X
)  /\  R  C_  X
)  /\  ( x  e.  R  /\  y  e.  R  /\  z  e.  R ) )  -> 
y  e.  X )
29 xmettri2 19915 . . . 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 1220 . . 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 1149 . . 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 994 . . . . 5  |-  ( ( ( D  e.  ( *Met `  X
)  /\  R  C_  X
)  /\  ( x  e.  R  /\  y  e.  R  /\  z  e.  R ) )  ->  x  e.  R )
3325, 32ovresd 6231 . . . 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 995 . . . . 5  |-  ( ( ( D  e.  ( *Met `  X
)  /\  R  C_  X
)  /\  ( x  e.  R  /\  y  e.  R  /\  z  e.  R ) )  -> 
y  e.  R )
3525, 34ovresd 6231 . . . 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 6109 . . 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 4322 . 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 19901 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 965    = wceq 1369    e. wcel 1756    C_ wss 3328   class class class wbr 4292    X. cxp 4838   dom cdm 4840    |` cres 4842   -->wf 5414   ` cfv 5418  (class class class)co 6091   0cc0 9282   RR*cxr 9417    <_ cle 9419   +ecxad 11087   *Metcxmt 17801
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 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531  ax-un 6372  ax-cnex 9338  ax-resscn 9339
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 2568  df-ne 2608  df-ral 2720  df-rex 2721  df-rab 2724  df-v 2974  df-sbc 3187  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-nul 3638  df-if 3792  df-pw 3862  df-sn 3878  df-pr 3880  df-op 3884  df-uni 4092  df-br 4293  df-opab 4351  df-mpt 4352  df-id 4636  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-iota 5381  df-fun 5420  df-fn 5421  df-f 5422  df-fv 5426  df-ov 6094  df-oprab 6095  df-mpt2 6096  df-map 7216  df-xr 9422  df-xmet 17810
This theorem is referenced by:  metres2  19938  xmetres  19939  xpsxmet  19955  xpsdsval  19956  xmetresbl  20012  tmsxms  20061  imasf1oxms  20064  metrest  20099  prdsxms  20105  tmsxpsval  20113  nrginvrcn  20272  divcn  20444  iitopon  20455  cncfmet  20484  cfilres  20807  dvlip2  21467  ftc1lem6  21513  ulmdvlem1  21865  ulmdvlem3  21867  abelth  21906  cxpcn3  22186  rlimcnp  22359  minvecolem4b  24279  minvecolem4  24281  ftc1cnnc  28466  blbnd  28686  ismtyres  28707  reheibor  28738
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