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Theorem xmetres2 14699
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 14663 . . . . 5 |- Rel *Met
2 relelfvdm 5593 . . . . 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 4174 . 2 |- ( ( D e. ( *Met ` X ) /\ R C_ X ) -> R e. _V )
7 xmetf 14670 . . . 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 4771 . . . 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 5437 . 2 |- ( ( D e. ( *Met ` X ) /\ R C_ X ) -> ( D |` ( R X. R ) ) : ( R X. R ) --> RR* )
12 ovres 6067 . . . . 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 2205 . . 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 3185 . . . 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 3185 . . . 4 |- ( ( ( D e. ( *Met ` X ) /\ R C_ X ) /\ ( x e. R /\ y e. R ) ) -> y e. X )
21 xmeteq0 14679 . . . 4 |- ( ( D e. ( *Met ` X ) /\ x e. X /\ y e. X ) -> ( ( x D y ) = 0 <-> x = y ) )
2215, 18, 20, 21syl3anc 1249 . . 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 1007 . . . . 5 |- ( ( ( D e. ( *Met ` X ) /\ R C_ X ) /\ ( x e. R /\ y e. R /\ z e. R ) ) -> z e. R )
2725, 26sseldd 3185 . . . 4 |- ( ( ( D e. ( *Met ` X ) /\ R C_ X ) /\ ( x e. R /\ y e. R /\ z e. R ) ) -> z e. X )
28183adantr3 1160 . . . 4 |- ( ( ( D e. ( *Met ` X ) /\ R C_ X ) /\ ( x e. R /\ y e. R /\ z e. R ) ) -> x e. X )
29203adantr3 1160 . . . 4 |- ( ( ( D e. ( *Met ` X ) /\ R C_ X ) /\ ( x e. R /\ y e. R /\ z e. R ) ) -> y e. X )
30 xmettri2 14681 . . . 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 1251 . . 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 1160 . . 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 1005 . . . . 5 |- ( ( ( D e. ( *Met ` X ) /\ R C_ X ) /\ ( x e. R /\ y e. R /\ z e. R ) ) -> x e. R )
3426, 33ovresd 6068 . . . 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 1006 . . . . 5 |- ( ( ( D e. ( *Met ` X ) /\ R C_ X ) /\ ( x e. R /\ y e. R /\ z e. R ) ) -> y e. R )
3626, 35ovresd 6068 . . . 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 5943 . . 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 4066 . 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 14667 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 980   = wceq 1364   e. wcel 2167   C_ wss 3157   class class class wbr 4034   X. cxp 4662   dom cdm 4664   |` cres 4666   Rel wrel 4669   -->wf 5255   ` cfv 5259  (class class class)co 5925   0cc0 7896   RR*cxr 8077   <_ cle 8079   +ecxad 9862   *Metcxmet 14168
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 615  ax-in2 616  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-sep 4152  ax-pow 4208  ax-pr 4243  ax-un 4469  ax-setind 4574  ax-cnex 7987  ax-resscn 7988
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-ral 2480  df-rex 2481  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-pw 3608  df-sn 3629  df-pr 3630  df-op 3632  df-uni 3841  df-iun 3919  df-br 4035  df-opab 4096  df-mpt 4097  df-id 4329  df-xp 4670  df-rel 4671  df-cnv 4672  df-co 4673  df-dm 4674  df-rn 4675  df-res 4676  df-ima 4677  df-iota 5220  df-fun 5261  df-fn 5262  df-f 5263  df-fv 5267  df-ov 5928  df-oprab 5929  df-mpo 5930  df-1st 6207  df-2nd 6208  df-map 6718  df-pnf 8080  df-mnf 8081  df-xr 8082  df-xmet 14176
This theorem is referenced by:  metres2  14701  xmetres  14702  xmetresbl  14760  metrest  14826  divcnap  14885  cncfmet  14912  limcimolemlt  14984  cnplimcim  14987  cnplimclemr  14989  limccnpcntop  14995  limccnp2cntop  14997
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