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Theorem xmetres2 12307
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 xmetrel 12271 . . . . 5 |- Rel *Met
2 relelfvdm 5385 . . . . 5 |- ( ( Rel *Met /\ D e. ( *Met ` X ) ) -> X e. dom *Met )
31, 2mpan 418 . . . 4 |- ( D e. ( *Met ` X ) -> X e. dom *Met )
43adantr 272 . . 3 |- ( ( D e. ( *Met ` X ) /\ R C_ X ) -> X e. dom *Met )
5 simpr 109 . . 3 |- ( ( D e. ( *Met ` X ) /\ R C_ X ) -> R C_ X )
64, 5ssexd 4008 . 2 |- ( ( D e. ( *Met ` X ) /\ R C_ X ) -> R e. _V )
7 xmetf 12278 . . . 4 |- ( D e. ( *Met ` X ) -> D : ( X X. X ) --> RR* )
87adantr 272 . . 3 |- ( ( D e. ( *Met ` X ) /\ R C_ X ) -> D : ( X X. X ) --> RR* )
9 xpss12 4584 . . . 4 |- ( ( R C_ X /\ R C_ X ) -> ( R X. R ) C_ ( X X. X ) )
105, 9sylancom 414 . . 3 |- ( ( D e. ( *Met ` X ) /\ R C_ X ) -> ( R X. R ) C_ ( X X. X ) )
118, 10fssresd 5235 . 2 |- ( ( D e. ( *Met ` X ) /\ R C_ X ) -> ( D |` ( R X. R ) ) : ( R X. R ) --> RR* )
12 ovres 5842 . . . . 5 |- ( ( x e. R /\ y e. R ) -> ( x ( D |` ( R X. R ) ) y ) = ( x D y ) )
1312adantl 273 . . . 4 |- ( ( ( D e. ( *Met ` X ) /\ R C_ X ) /\ ( x e. R /\ y e. R ) ) -> ( x ( D |` ( R X. R ) ) y ) = ( x D y ) )
1413eqeq1d 2108 . . 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 ) )
15 simpll 499 . . . 4 |- ( ( ( D e. ( *Met ` X ) /\ R C_ X ) /\ ( x e. R /\ y e. R ) ) -> D e. ( *Met ` X ) )
16 simplr 500 . . . . 5 |- ( ( ( D e. ( *Met ` X ) /\ R C_ X ) /\ ( x e. R /\ y e. R ) ) -> R C_ X )
17 simprl 501 . . . . 5 |- ( ( ( D e. ( *Met ` X ) /\ R C_ X ) /\ ( x e. R /\ y e. R ) ) -> x e. R )
1816, 17sseldd 3048 . . . 4 |- ( ( ( D e. ( *Met ` X ) /\ R C_ X ) /\ ( x e. R /\ y e. R ) ) -> x e. X )
19 simprr 502 . . . . 5 |- ( ( ( D e. ( *Met ` X ) /\ R C_ X ) /\ ( x e. R /\ y e. R ) ) -> y e. R )
2016, 19sseldd 3048 . . . 4 |- ( ( ( D e. ( *Met ` X ) /\ R C_ X ) /\ ( x e. R /\ y e. R ) ) -> y e. X )
21 xmeteq0 12287 . . . 4 |- ( ( D e. ( *Met ` X ) /\ x e. X /\ y e. X ) -> ( ( x D y ) = 0 <-> x = y ) )
2215, 18, 20, 21syl3anc 1184 . . 3 |- ( ( ( D e. ( *Met ` X ) /\ R C_ X ) /\ ( x e. R /\ y e. R ) ) -> ( ( x D y ) = 0 <-> x = y ) )
2314, 22bitrd 187 . 2 |- ( ( ( D e. ( *Met ` X ) /\ R C_ X ) /\ ( x e. R /\ y e. R ) ) -> ( ( x ( D |` ( R X. R ) ) y ) = 0 <-> x = y ) )
24 simpll 499 . . . 4 |- ( ( ( D e. ( *Met ` X ) /\ R C_ X ) /\ ( x e. R /\ y e. R /\ z e. R ) ) -> D e. ( *Met ` X ) )
25 simplr 500 . . . . 5 |- ( ( ( D e. ( *Met ` X ) /\ R C_ X ) /\ ( x e. R /\ y e. R /\ z e. R ) ) -> R C_ X )
26 simpr3 957 . . . . 5 |- ( ( ( D e. ( *Met ` X ) /\ R C_ X ) /\ ( x e. R /\ y e. R /\ z e. R ) ) -> z e. R )
2725, 26sseldd 3048 . . . 4 |- ( ( ( D e. ( *Met ` X ) /\ R C_ X ) /\ ( x e. R /\ y e. R /\ z e. R ) ) -> z e. X )
28183adantr3 1110 . . . 4 |- ( ( ( D e. ( *Met ` X ) /\ R C_ X ) /\ ( x e. R /\ y e. R /\ z e. R ) ) -> x e. X )
29203adantr3 1110 . . . 4 |- ( ( ( D e. ( *Met ` X ) /\ R C_ X ) /\ ( x e. R /\ y e. R /\ z e. R ) ) -> y e. X )
30 xmettri2 12289 . . . 4 |- ( ( D e. ( *Met ` X ) /\ ( z e. X /\ x e. X /\ y e. X ) ) -> ( x D y ) <_ ( ( z D x ) +e ( z D y ) ) )
3124, 27, 28, 29, 30syl13anc 1186 . . 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 ) ) )
32133adantr3 1110 . . 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 ) )
33 simpr1 955 . . . . 5 |- ( ( ( D e. ( *Met ` X ) /\ R C_ X ) /\ ( x e. R /\ y e. R /\ z e. R ) ) -> x e. R )
3426, 33ovresd 5843 . . . 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 ) )
35 simpr2 956 . . . . 5 |- ( ( ( D e. ( *Met ` X ) /\ R C_ X ) /\ ( x e. R /\ y e. R /\ z e. R ) ) -> y e. R )
3626, 35ovresd 5843 . . . 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 ) )
3734, 36oveq12d 5724 . . 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 ) ) )
3831, 32, 373brtr4d 3905 . 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 ) ) )
396, 11, 23, 38isxmetd 12275 1 |- ( ( D e. ( *Met ` X ) /\ R C_ X ) -> ( D |` ( R X. R ) ) e. ( *Met ` R ) )
Colors of variables: wff set class
Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   /\ w3a 930   = wceq 1299   e. wcel 1448   C_ wss 3021   class class class wbr 3875   X. cxp 4475   dom cdm 4477   |` cres 4479   Rel wrel 4482   -->wf 5055   ` cfv 5059  (class class class)co 5706   0cc0 7500   RR*cxr 7671   <_ cle 7673   +ecxad 9398   *Metcxmet 11931
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-iun 3762  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-1st 5969  df-2nd 5970  df-map 6474  df-pnf 7674  df-mnf 7675  df-xr 7676  df-xmet 11939
This theorem is referenced by:  metres2  12309  xmetres  12310  xmetresbl  12368  metrest  12434  cncfmet  12492  limcimolemlt  12513  cnplimcim  12516  limccnpcntop  12520
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