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Theorem xmetres2 13546
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 13510 . . . . 5 |- Rel *Met
2 relelfvdm 5543 . . . . 5 |- ( ( Rel *Met /\ D e. ( *Met ` X ) ) -> X e. dom *Met )
31, 2mpan 424 . . . 4 |- ( D e. ( *Met ` X ) -> X e. dom *Met )
43adantr 276 . . 3 |- ( ( D e. ( *Met ` X ) /\ R C_ X ) -> X e. dom *Met )
5 simpr 110 . . 3 |- ( ( D e. ( *Met ` X ) /\ R C_ X ) -> R C_ X )
64, 5ssexd 4140 . 2 |- ( ( D e. ( *Met ` X ) /\ R C_ X ) -> R e. _V )
7 xmetf 13517 . . . 4 |- ( D e. ( *Met ` X ) -> D : ( X X. X ) --> RR* )
87adantr 276 . . 3 |- ( ( D e. ( *Met ` X ) /\ R C_ X ) -> D : ( X X. X ) --> RR* )
9 xpss12 4730 . . . 4 |- ( ( R C_ X /\ R C_ X ) -> ( R X. R ) C_ ( X X. X ) )
105, 9sylancom 420 . . 3 |- ( ( D e. ( *Met ` X ) /\ R C_ X ) -> ( R X. R ) C_ ( X X. X ) )
118, 10fssresd 5388 . 2 |- ( ( D e. ( *Met ` X ) /\ R C_ X ) -> ( D |` ( R X. R ) ) : ( R X. R ) --> RR* )
12 ovres 6008 . . . . 5 |- ( ( x e. R /\ y e. R ) -> ( x ( D |` ( R X. R ) ) y ) = ( x D y ) )
1312adantl 277 . . . 4 |- ( ( ( D e. ( *Met ` X ) /\ R C_ X ) /\ ( x e. R /\ y e. R ) ) -> ( x ( D |` ( R X. R ) ) y ) = ( x D y ) )
1413eqeq1d 2186 . . 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 527 . . . 4 |- ( ( ( D e. ( *Met ` X ) /\ R C_ X ) /\ ( x e. R /\ y e. R ) ) -> D e. ( *Met ` X ) )
16 simplr 528 . . . . 5 |- ( ( ( D e. ( *Met ` X ) /\ R C_ X ) /\ ( x e. R /\ y e. R ) ) -> R C_ X )
17 simprl 529 . . . . 5 |- ( ( ( D e. ( *Met ` X ) /\ R C_ X ) /\ ( x e. R /\ y e. R ) ) -> x e. R )
1816, 17sseldd 3156 . . . 4 |- ( ( ( D e. ( *Met ` X ) /\ R C_ X ) /\ ( x e. R /\ y e. R ) ) -> x e. X )
19 simprr 531 . . . . 5 |- ( ( ( D e. ( *Met ` X ) /\ R C_ X ) /\ ( x e. R /\ y e. R ) ) -> y e. R )
2016, 19sseldd 3156 . . . 4 |- ( ( ( D e. ( *Met ` X ) /\ R C_ X ) /\ ( x e. R /\ y e. R ) ) -> y e. X )
21 xmeteq0 13526 . . . 4 |- ( ( D e. ( *Met ` X ) /\ x e. X /\ y e. X ) -> ( ( x D y ) = 0 <-> x = y ) )
2215, 18, 20, 21syl3anc 1238 . . 3 |- ( ( ( D e. ( *Met ` X ) /\ R C_ X ) /\ ( x e. R /\ y e. R ) ) -> ( ( x D y ) = 0 <-> x = y ) )
2314, 22bitrd 188 . 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 527 . . . 4 |- ( ( ( D e. ( *Met ` X ) /\ R C_ X ) /\ ( x e. R /\ y e. R /\ z e. R ) ) -> D e. ( *Met ` X ) )
25 simplr 528 . . . . 5 |- ( ( ( D e. ( *Met ` X ) /\ R C_ X ) /\ ( x e. R /\ y e. R /\ z e. R ) ) -> R C_ X )
26 simpr3 1005 . . . . 5 |- ( ( ( D e. ( *Met ` X ) /\ R C_ X ) /\ ( x e. R /\ y e. R /\ z e. R ) ) -> z e. R )
2725, 26sseldd 3156 . . . 4 |- ( ( ( D e. ( *Met ` X ) /\ R C_ X ) /\ ( x e. R /\ y e. R /\ z e. R ) ) -> z e. X )
28183adantr3 1158 . . . 4 |- ( ( ( D e. ( *Met ` X ) /\ R C_ X ) /\ ( x e. R /\ y e. R /\ z e. R ) ) -> x e. X )
29203adantr3 1158 . . . 4 |- ( ( ( D e. ( *Met ` X ) /\ R C_ X ) /\ ( x e. R /\ y e. R /\ z e. R ) ) -> y e. X )
30 xmettri2 13528 . . . 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 1240 . . 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 1158 . . 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 1003 . . . . 5 |- ( ( ( D e. ( *Met ` X ) /\ R C_ X ) /\ ( x e. R /\ y e. R /\ z e. R ) ) -> x e. R )
3426, 33ovresd 6009 . . . 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 1004 . . . . 5 |- ( ( ( D e. ( *Met ` X ) /\ R C_ X ) /\ ( x e. R /\ y e. R /\ z e. R ) ) -> y e. R )
3626, 35ovresd 6009 . . . 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 5887 . . 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 4032 . 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 13514 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 104   <-> wb 105   /\ w3a 978   = wceq 1353   e. wcel 2148   C_ wss 3129   class class class wbr 4000   X. cxp 4621   dom cdm 4623   |` cres 4625   Rel wrel 4628   -->wf 5208   ` cfv 5212  (class class class)co 5869   0cc0 7802   RR*cxr 7981   <_ cle 7983   +ecxad 9757   *Metcxmet 13147
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4118  ax-pow 4171  ax-pr 4206  ax-un 4430  ax-setind 4533  ax-cnex 7893  ax-resscn 7894
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-iun 3886  df-br 4001  df-opab 4062  df-mpt 4063  df-id 4290  df-xp 4629  df-rel 4630  df-cnv 4631  df-co 4632  df-dm 4633  df-rn 4634  df-res 4635  df-ima 4636  df-iota 5174  df-fun 5214  df-fn 5215  df-f 5216  df-fv 5220  df-ov 5872  df-oprab 5873  df-mpo 5874  df-1st 6135  df-2nd 6136  df-map 6644  df-pnf 7984  df-mnf 7985  df-xr 7986  df-xmet 13155
This theorem is referenced by:  metres2  13548  xmetres  13549  xmetresbl  13607  metrest  13673  divcnap  13722  cncfmet  13746  limcimolemlt  13800  cnplimcim  13803  cnplimclemr  13805  limccnpcntop  13811  limccnp2cntop  13813
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