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Theorem psmetres2 12261
Description: Restriction of a pseudometric. (Contributed by Thierry Arnoux, 11-Feb-2018.)
Assertion
Ref Expression
psmetres2  |-  ( ( D  e.  (PsMet `  X )  /\  R  C_  X )  ->  ( D  |`  ( R  X.  R ) )  e.  (PsMet `  R )
)

Proof of Theorem psmetres2
Dummy variables  a  b  c  d  e are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 psmetf 12253 . . . 4  |-  ( D  e.  (PsMet `  X
)  ->  D :
( X  X.  X
) --> RR* )
21adantr 272 . . 3  |-  ( ( D  e.  (PsMet `  X )  /\  R  C_  X )  ->  D : ( X  X.  X ) --> RR* )
3 simpr 109 . . . 4  |-  ( ( D  e.  (PsMet `  X )  /\  R  C_  X )  ->  R  C_  X )
4 xpss12 4584 . . . 4  |-  ( ( R  C_  X  /\  R  C_  X )  -> 
( R  X.  R
)  C_  ( X  X.  X ) )
53, 3, 4syl2anc 406 . . 3  |-  ( ( D  e.  (PsMet `  X )  /\  R  C_  X )  ->  ( R  X.  R )  C_  ( X  X.  X
) )
62, 5fssresd 5235 . 2  |-  ( ( D  e.  (PsMet `  X )  /\  R  C_  X )  ->  ( D  |`  ( R  X.  R ) ) : ( R  X.  R
) --> RR* )
7 simpr 109 . . . . . 6  |-  ( ( ( D  e.  (PsMet `  X )  /\  R  C_  X )  /\  a  e.  R )  ->  a  e.  R )
87, 7ovresd 5843 . . . . 5  |-  ( ( ( D  e.  (PsMet `  X )  /\  R  C_  X )  /\  a  e.  R )  ->  (
a ( D  |`  ( R  X.  R
) ) a )  =  ( a D a ) )
9 simpll 499 . . . . . 6  |-  ( ( ( D  e.  (PsMet `  X )  /\  R  C_  X )  /\  a  e.  R )  ->  D  e.  (PsMet `  X )
)
103sselda 3047 . . . . . 6  |-  ( ( ( D  e.  (PsMet `  X )  /\  R  C_  X )  /\  a  e.  R )  ->  a  e.  X )
11 psmet0 12255 . . . . . 6  |-  ( ( D  e.  (PsMet `  X )  /\  a  e.  X )  ->  (
a D a )  =  0 )
129, 10, 11syl2anc 406 . . . . 5  |-  ( ( ( D  e.  (PsMet `  X )  /\  R  C_  X )  /\  a  e.  R )  ->  (
a D a )  =  0 )
138, 12eqtrd 2132 . . . 4  |-  ( ( ( D  e.  (PsMet `  X )  /\  R  C_  X )  /\  a  e.  R )  ->  (
a ( D  |`  ( R  X.  R
) ) a )  =  0 )
149ad2antrr 475 . . . . . . . 8  |-  ( ( ( ( ( D  e.  (PsMet `  X
)  /\  R  C_  X
)  /\  a  e.  R )  /\  b  e.  R )  /\  c  e.  R )  ->  D  e.  (PsMet `  X )
)
153ad2antrr 475 . . . . . . . . 9  |-  ( ( ( ( D  e.  (PsMet `  X )  /\  R  C_  X )  /\  a  e.  R
)  /\  b  e.  R )  ->  R  C_  X )
1615sselda 3047 . . . . . . . 8  |-  ( ( ( ( ( D  e.  (PsMet `  X
)  /\  R  C_  X
)  /\  a  e.  R )  /\  b  e.  R )  /\  c  e.  R )  ->  c  e.  X )
1710ad2antrr 475 . . . . . . . 8  |-  ( ( ( ( ( D  e.  (PsMet `  X
)  /\  R  C_  X
)  /\  a  e.  R )  /\  b  e.  R )  /\  c  e.  R )  ->  a  e.  X )
183adantr 272 . . . . . . . . . 10  |-  ( ( ( D  e.  (PsMet `  X )  /\  R  C_  X )  /\  a  e.  R )  ->  R  C_  X )
1918sselda 3047 . . . . . . . . 9  |-  ( ( ( ( D  e.  (PsMet `  X )  /\  R  C_  X )  /\  a  e.  R
)  /\  b  e.  R )  ->  b  e.  X )
2019adantr 272 . . . . . . . 8  |-  ( ( ( ( ( D  e.  (PsMet `  X
)  /\  R  C_  X
)  /\  a  e.  R )  /\  b  e.  R )  /\  c  e.  R )  ->  b  e.  X )
21 psmettri2 12256 . . . . . . . 8  |-  ( ( D  e.  (PsMet `  X )  /\  (
c  e.  X  /\  a  e.  X  /\  b  e.  X )
)  ->  ( a D b )  <_ 
( ( c D a ) +e
( c D b ) ) )
2214, 16, 17, 20, 21syl13anc 1186 . . . . . . 7  |-  ( ( ( ( ( D  e.  (PsMet `  X
)  /\  R  C_  X
)  /\  a  e.  R )  /\  b  e.  R )  /\  c  e.  R )  ->  (
a D b )  <_  ( ( c D a ) +e ( c D b ) ) )
237adantr 272 . . . . . . . . 9  |-  ( ( ( ( D  e.  (PsMet `  X )  /\  R  C_  X )  /\  a  e.  R
)  /\  b  e.  R )  ->  a  e.  R )
24 simpr 109 . . . . . . . . 9  |-  ( ( ( ( D  e.  (PsMet `  X )  /\  R  C_  X )  /\  a  e.  R
)  /\  b  e.  R )  ->  b  e.  R )
2523, 24ovresd 5843 . . . . . . . 8  |-  ( ( ( ( D  e.  (PsMet `  X )  /\  R  C_  X )  /\  a  e.  R
)  /\  b  e.  R )  ->  (
a ( D  |`  ( R  X.  R
) ) b )  =  ( a D b ) )
2625adantr 272 . . . . . . 7  |-  ( ( ( ( ( D  e.  (PsMet `  X
)  /\  R  C_  X
)  /\  a  e.  R )  /\  b  e.  R )  /\  c  e.  R )  ->  (
a ( D  |`  ( R  X.  R
) ) b )  =  ( a D b ) )
27 simpr 109 . . . . . . . . 9  |-  ( ( ( ( ( D  e.  (PsMet `  X
)  /\  R  C_  X
)  /\  a  e.  R )  /\  b  e.  R )  /\  c  e.  R )  ->  c  e.  R )
287ad2antrr 475 . . . . . . . . 9  |-  ( ( ( ( ( D  e.  (PsMet `  X
)  /\  R  C_  X
)  /\  a  e.  R )  /\  b  e.  R )  /\  c  e.  R )  ->  a  e.  R )
2927, 28ovresd 5843 . . . . . . . 8  |-  ( ( ( ( ( D  e.  (PsMet `  X
)  /\  R  C_  X
)  /\  a  e.  R )  /\  b  e.  R )  /\  c  e.  R )  ->  (
c ( D  |`  ( R  X.  R
) ) a )  =  ( c D a ) )
3024adantr 272 . . . . . . . . 9  |-  ( ( ( ( ( D  e.  (PsMet `  X
)  /\  R  C_  X
)  /\  a  e.  R )  /\  b  e.  R )  /\  c  e.  R )  ->  b  e.  R )
3127, 30ovresd 5843 . . . . . . . 8  |-  ( ( ( ( ( D  e.  (PsMet `  X
)  /\  R  C_  X
)  /\  a  e.  R )  /\  b  e.  R )  /\  c  e.  R )  ->  (
c ( D  |`  ( R  X.  R
) ) b )  =  ( c D b ) )
3229, 31oveq12d 5724 . . . . . . 7  |-  ( ( ( ( ( D  e.  (PsMet `  X
)  /\  R  C_  X
)  /\  a  e.  R )  /\  b  e.  R )  /\  c  e.  R )  ->  (
( c ( D  |`  ( R  X.  R
) ) a ) +e ( c ( D  |`  ( R  X.  R ) ) b ) )  =  ( ( c D a ) +e
( c D b ) ) )
3322, 26, 323brtr4d 3905 . . . . . 6  |-  ( ( ( ( ( D  e.  (PsMet `  X
)  /\  R  C_  X
)  /\  a  e.  R )  /\  b  e.  R )  /\  c  e.  R )  ->  (
a ( D  |`  ( R  X.  R
) ) b )  <_  ( ( c ( D  |`  ( R  X.  R ) ) a ) +e
( c ( D  |`  ( R  X.  R
) ) b ) ) )
3433ralrimiva 2464 . . . . 5  |-  ( ( ( ( D  e.  (PsMet `  X )  /\  R  C_  X )  /\  a  e.  R
)  /\  b  e.  R )  ->  A. c  e.  R  ( a
( D  |`  ( R  X.  R ) ) b )  <_  (
( c ( D  |`  ( R  X.  R
) ) a ) +e ( c ( D  |`  ( R  X.  R ) ) b ) ) )
3534ralrimiva 2464 . . . 4  |-  ( ( ( D  e.  (PsMet `  X )  /\  R  C_  X )  /\  a  e.  R )  ->  A. b  e.  R  A. c  e.  R  ( a
( D  |`  ( R  X.  R ) ) b )  <_  (
( c ( D  |`  ( R  X.  R
) ) a ) +e ( c ( D  |`  ( R  X.  R ) ) b ) ) )
3613, 35jca 302 . . 3  |-  ( ( ( D  e.  (PsMet `  X )  /\  R  C_  X )  /\  a  e.  R )  ->  (
( a ( D  |`  ( R  X.  R
) ) a )  =  0  /\  A. b  e.  R  A. c  e.  R  (
a ( D  |`  ( R  X.  R
) ) b )  <_  ( ( c ( D  |`  ( R  X.  R ) ) a ) +e
( c ( D  |`  ( R  X.  R
) ) b ) ) ) )
3736ralrimiva 2464 . 2  |-  ( ( D  e.  (PsMet `  X )  /\  R  C_  X )  ->  A. a  e.  R  ( (
a ( D  |`  ( R  X.  R
) ) a )  =  0  /\  A. b  e.  R  A. c  e.  R  (
a ( D  |`  ( R  X.  R
) ) b )  <_  ( ( c ( D  |`  ( R  X.  R ) ) a ) +e
( c ( D  |`  ( R  X.  R
) ) b ) ) ) )
38 df-psmet 11938 . . . . . 6  |- PsMet  =  ( a  e.  _V  |->  { b  e.  ( RR*  ^m  ( a  X.  a
) )  |  A. c  e.  a  (
( c b c )  =  0  /\ 
A. d  e.  a 
A. e  e.  a  ( c b d )  <_  ( (
e b c ) +e ( e b d ) ) ) } )
3938mptrcl 5435 . . . . 5  |-  ( D  e.  (PsMet `  X
)  ->  X  e.  _V )
4039adantr 272 . . . 4  |-  ( ( D  e.  (PsMet `  X )  /\  R  C_  X )  ->  X  e.  _V )
4140, 3ssexd 4008 . . 3  |-  ( ( D  e.  (PsMet `  X )  /\  R  C_  X )  ->  R  e.  _V )
42 ispsmet 12251 . . 3  |-  ( R  e.  _V  ->  (
( D  |`  ( R  X.  R ) )  e.  (PsMet `  R
)  <->  ( ( D  |`  ( R  X.  R
) ) : ( R  X.  R ) -->
RR*  /\  A. a  e.  R  ( (
a ( D  |`  ( R  X.  R
) ) a )  =  0  /\  A. b  e.  R  A. c  e.  R  (
a ( D  |`  ( R  X.  R
) ) b )  <_  ( ( c ( D  |`  ( R  X.  R ) ) a ) +e
( c ( D  |`  ( R  X.  R
) ) b ) ) ) ) ) )
4341, 42syl 14 . 2  |-  ( ( D  e.  (PsMet `  X )  /\  R  C_  X )  ->  (
( D  |`  ( R  X.  R ) )  e.  (PsMet `  R
)  <->  ( ( D  |`  ( R  X.  R
) ) : ( R  X.  R ) -->
RR*  /\  A. a  e.  R  ( (
a ( D  |`  ( R  X.  R
) ) a )  =  0  /\  A. b  e.  R  A. c  e.  R  (
a ( D  |`  ( R  X.  R
) ) b )  <_  ( ( c ( D  |`  ( R  X.  R ) ) a ) +e
( c ( D  |`  ( R  X.  R
) ) b ) ) ) ) ) )
446, 37, 43mpbir2and 896 1  |-  ( ( D  e.  (PsMet `  X )  /\  R  C_  X )  ->  ( D  |`  ( R  X.  R ) )  e.  (PsMet `  R )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104    = wceq 1299    e. wcel 1448   A.wral 2375   {crab 2379   _Vcvv 2641    C_ wss 3021   class class class wbr 3875    X. cxp 4475    |` cres 4479   -->wf 5055   ` cfv 5059  (class class class)co 5706    ^m cmap 6472   0cc0 7500   RR*cxr 7671    <_ cle 7673   +ecxad 9398  PsMetcpsmet 11930
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 584  ax-in2 585  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-13 1459  ax-14 1460  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082  ax-sep 3986  ax-pow 4038  ax-pr 4069  ax-un 4293  ax-setind 4390  ax-cnex 7586  ax-resscn 7587
This theorem depends on definitions:  df-bi 116  df-3an 932  df-tru 1302  df-fal 1305  df-nf 1405  df-sb 1704  df-eu 1963  df-mo 1964  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-ne 2268  df-ral 2380  df-rex 2381  df-rab 2384  df-v 2643  df-sbc 2863  df-csb 2956  df-dif 3023  df-un 3025  df-in 3027  df-ss 3034  df-pw 3459  df-sn 3480  df-pr 3481  df-op 3483  df-uni 3684  df-br 3876  df-opab 3930  df-mpt 3931  df-id 4153  df-xp 4483  df-rel 4484  df-cnv 4485  df-co 4486  df-dm 4487  df-rn 4488  df-res 4489  df-ima 4490  df-iota 5024  df-fun 5061  df-fn 5062  df-f 5063  df-fv 5067  df-ov 5709  df-oprab 5710  df-mpo 5711  df-map 6474  df-pnf 7674  df-mnf 7675  df-xr 7676  df-psmet 11938
This theorem is referenced by: (None)
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