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Theorem xmetres2 15605
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 15569 . . . . 5 |- Rel *Met
2 relelfvdm 5730 . . . . 5 |- ( ( Rel *Met /\ D e. ( *Met ` X ) ) -> X e. dom *Met )
31, 2mpan 428 . . . 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 4275 . 2 |- ( ( D e. ( *Met ` X ) /\ R C_ X ) -> R e. _V )
7 xmetf 15576 . . . 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 4884 . . . 4 |- ( ( R C_ X /\ R C_ X ) -> ( R X. R ) C_ ( X X. X ) )
105, 9sylancom 424 . . 3 |- ( ( D e. ( *Met ` X ) /\ R C_ X ) -> ( R X. R ) C_ ( X X. X ) )
118, 10fssresd 5569 . 2 |- ( ( D e. ( *Met ` X ) /\ R C_ X ) -> ( D |` ( R X. R ) ) : ( R X. R ) --> RR* )
12 ovres 6232 . . . . 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 2247 . . 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 531 . . . 4 |- ( ( ( D e. ( *Met ` X ) /\ R C_ X ) /\ ( x e. R /\ y e. R ) ) -> D e. ( *Met ` X ) )
16 simplr 533 . . . . 5 |- ( ( ( D e. ( *Met ` X ) /\ R C_ X ) /\ ( x e. R /\ y e. R ) ) -> R C_ X )
17 simprl 535 . . . . 5 |- ( ( ( D e. ( *Met ` X ) /\ R C_ X ) /\ ( x e. R /\ y e. R ) ) -> x e. R )
1816, 17sseldd 3249 . . . 4 |- ( ( ( D e. ( *Met ` X ) /\ R C_ X ) /\ ( x e. R /\ y e. R ) ) -> x e. X )
19 simprr 537 . . . . 5 |- ( ( ( D e. ( *Met ` X ) /\ R C_ X ) /\ ( x e. R /\ y e. R ) ) -> y e. R )
2016, 19sseldd 3249 . . . 4 |- ( ( ( D e. ( *Met ` X ) /\ R C_ X ) /\ ( x e. R /\ y e. R ) ) -> y e. X )
21 xmeteq0 15585 . . . 4 |- ( ( D e. ( *Met ` X ) /\ x e. X /\ y e. X ) -> ( ( x D y ) = 0 <-> x = y ) )
2215, 18, 20, 21syl3anc 1278 . . 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 531 . . . 4 |- ( ( ( D e. ( *Met ` X ) /\ R C_ X ) /\ ( x e. R /\ y e. R /\ z e. R ) ) -> D e. ( *Met ` X ) )
25 simplr 533 . . . . 5 |- ( ( ( D e. ( *Met ` X ) /\ R C_ X ) /\ ( x e. R /\ y e. R /\ z e. R ) ) -> R C_ X )
26 simpr3 1036 . . . . 5 |- ( ( ( D e. ( *Met ` X ) /\ R C_ X ) /\ ( x e. R /\ y e. R /\ z e. R ) ) -> z e. R )
2725, 26sseldd 3249 . . . 4 |- ( ( ( D e. ( *Met ` X ) /\ R C_ X ) /\ ( x e. R /\ y e. R /\ z e. R ) ) -> z e. X )
28183adantr3 1189 . . . 4 |- ( ( ( D e. ( *Met ` X ) /\ R C_ X ) /\ ( x e. R /\ y e. R /\ z e. R ) ) -> x e. X )
29203adantr3 1189 . . . 4 |- ( ( ( D e. ( *Met ` X ) /\ R C_ X ) /\ ( x e. R /\ y e. R /\ z e. R ) ) -> y e. X )
30 xmettri2 15587 . . . 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 1280 . . 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 1189 . . 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 1034 . . . . 5 |- ( ( ( D e. ( *Met ` X ) /\ R C_ X ) /\ ( x e. R /\ y e. R /\ z e. R ) ) -> x e. R )
3426, 33ovresd 6233 . . . 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 1035 . . . . 5 |- ( ( ( D e. ( *Met ` X ) /\ R C_ X ) /\ ( x e. R /\ y e. R /\ z e. R ) ) -> y e. R )
3626, 35ovresd 6233 . . . 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 6106 . . 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 4163 . 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 15573 1 |- ( ( D e. ( *Met ` X ) /\ R C_ X ) -> ( D |` ( R X. R ) ) e. ( *Met ` R ) )
Colors of variables:    wff set class
This proof depends on syntax axioms:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 1009   = wceq 1402   e. wcel 2209   C_ wss 3220   class class class wbr 4131   X. cxp 4774   dom cdm 4776   |` cres 4778   Rel wrel 4781   -->wf 5376   ` cfv 5380  (class class class)co 6088   0cc0 8185   RR*cxr 8365   <_ cle 8367   +ecxad 10189   *Metcxmet 14982
This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4251  ax-pow 4313  ax-pr 4348  ax-un 4580  ax-setind 4686  ax-cnex 8276  ax-resscn 8277
This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-ral 2533  df-rex 2534  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-iun 4014  df-br 4132  df-opab 4194  df-mpt 4195  df-id 4440  df-xp 4782  df-rel 4783  df-cnv 4784  df-co 4785  df-dm 4786  df-rn 4787  df-res 4788  df-ima 4789  df-iota 5340  df-fun 5382  df-fn 5383  df-f 5384  df-fv 5388  df-ov 6091  df-oprab 6092  df-mpo 6093  df-1st 6378  df-2nd 6379  df-map 6928  df-pnf 8368  df-mnf 8369  df-xr 8370  df-xmet 14990
This theorem is used by:  metres2  15607  xmetres  15608  xmetresbl  15666  metrest  15732  divcnap  15791  cncfmet  15818  limcimolemlt  15890  cnplimcim  15893  cnplimclemr  15895  limccnpcntop  15901  limccnp2cntop  15903
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