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Theorem xmetres2 22166
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 elfvdm 6220 . . . 4 |- ( D e. ( *Met ` X ) -> X e. dom *Met )
21adantr 481 . . 3 |- ( ( D e. ( *Met ` X ) /\ R C_ X ) -> X e. dom *Met )
3 simpr 477 . . 3 |- ( ( D e. ( *Met ` X ) /\ R C_ X ) -> R C_ X )
42, 3ssexd 4805 . 2 |- ( ( D e. ( *Met ` X ) /\ R C_ X ) -> R e. _V )
5 xmetf 22134 . . . 4 |- ( D e. ( *Met ` X ) -> D : ( X X. X ) --> RR* )
65adantr 481 . . 3 |- ( ( D e. ( *Met ` X ) /\ R C_ X ) -> D : ( X X. X ) --> RR* )
7 xpss12 5225 . . . 4 |- ( ( R C_ X /\ R C_ X ) -> ( R X. R ) C_ ( X X. X ) )
83, 7sylancom 701 . . 3 |- ( ( D e. ( *Met ` X ) /\ R C_ X ) -> ( R X. R ) C_ ( X X. X ) )
96, 8fssresd 6071 . 2 |- ( ( D e. ( *Met ` X ) /\ R C_ X ) -> ( D |` ( R X. R ) ) : ( R X. R ) --> RR* )
10 ovres 6800 . . . . 5 |- ( ( x e. R /\ y e. R ) -> ( x ( D |` ( R X. R ) ) y ) = ( x D y ) )
1110adantl 482 . . . 4 |- ( ( ( D e. ( *Met ` X ) /\ R C_ X ) /\ ( x e. R /\ y e. R ) ) -> ( x ( D |` ( R X. R ) ) y ) = ( x D y ) )
1211eqeq1d 2624 . . 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 ) )
13 simpll 790 . . . 4 |- ( ( ( D e. ( *Met ` X ) /\ R C_ X ) /\ ( x e. R /\ y e. R ) ) -> D e. ( *Met ` X ) )
14 simplr 792 . . . . 5 |- ( ( ( D e. ( *Met ` X ) /\ R C_ X ) /\ ( x e. R /\ y e. R ) ) -> R C_ X )
15 simprl 794 . . . . 5 |- ( ( ( D e. ( *Met ` X ) /\ R C_ X ) /\ ( x e. R /\ y e. R ) ) -> x e. R )
1614, 15sseldd 3604 . . . 4 |- ( ( ( D e. ( *Met ` X ) /\ R C_ X ) /\ ( x e. R /\ y e. R ) ) -> x e. X )
17 simprr 796 . . . . 5 |- ( ( ( D e. ( *Met ` X ) /\ R C_ X ) /\ ( x e. R /\ y e. R ) ) -> y e. R )
1814, 17sseldd 3604 . . . 4 |- ( ( ( D e. ( *Met ` X ) /\ R C_ X ) /\ ( x e. R /\ y e. R ) ) -> y e. X )
19 xmeteq0 22143 . . . 4 |- ( ( D e. ( *Met ` X ) /\ x e. X /\ y e. X ) -> ( ( x D y ) = 0 <-> x = y ) )
2013, 16, 18, 19syl3anc 1326 . . 3 |- ( ( ( D e. ( *Met ` X ) /\ R C_ X ) /\ ( x e. R /\ y e. R ) ) -> ( ( x D y ) = 0 <-> x = y ) )
2112, 20bitrd 268 . 2 |- ( ( ( D e. ( *Met ` X ) /\ R C_ X ) /\ ( x e. R /\ y e. R ) ) -> ( ( x ( D |` ( R X. R ) ) y ) = 0 <-> x = y ) )
22 simpll 790 . . . 4 |- ( ( ( D e. ( *Met ` X ) /\ R C_ X ) /\ ( x e. R /\ y e. R /\ z e. R ) ) -> D e. ( *Met ` X ) )
23 simplr 792 . . . . 5 |- ( ( ( D e. ( *Met ` X ) /\ R C_ X ) /\ ( x e. R /\ y e. R /\ z e. R ) ) -> R C_ X )
24 simpr3 1069 . . . . 5 |- ( ( ( D e. ( *Met ` X ) /\ R C_ X ) /\ ( x e. R /\ y e. R /\ z e. R ) ) -> z e. R )
2523, 24sseldd 3604 . . . 4 |- ( ( ( D e. ( *Met ` X ) /\ R C_ X ) /\ ( x e. R /\ y e. R /\ z e. R ) ) -> z e. X )
26163adantr3 1222 . . . 4 |- ( ( ( D e. ( *Met ` X ) /\ R C_ X ) /\ ( x e. R /\ y e. R /\ z e. R ) ) -> x e. X )
27183adantr3 1222 . . . 4 |- ( ( ( D e. ( *Met ` X ) /\ R C_ X ) /\ ( x e. R /\ y e. R /\ z e. R ) ) -> y e. X )
28 xmettri2 22145 . . . 4 |- ( ( D e. ( *Met ` X ) /\ ( z e. X /\ x e. X /\ y e. X ) ) -> ( x D y ) <_ ( ( z D x ) +e ( z D y ) ) )
2922, 25, 26, 27, 28syl13anc 1328 . . 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 ) ) )
30113adantr3 1222 . . 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 ) )
31 simpr1 1067 . . . . 5 |- ( ( ( D e. ( *Met ` X ) /\ R C_ X ) /\ ( x e. R /\ y e. R /\ z e. R ) ) -> x e. R )
3224, 31ovresd 6801 . . . 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 ) )
33 simpr2 1068 . . . . 5 |- ( ( ( D e. ( *Met ` X ) /\ R C_ X ) /\ ( x e. R /\ y e. R /\ z e. R ) ) -> y e. R )
3424, 33ovresd 6801 . . . 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 ) )
3532, 34oveq12d 6668 . . 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 ) ) )
3629, 30, 353brtr4d 4685 . 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 ) ) )
374, 9, 21, 36isxmetd 22131 1 |- ( ( D e. ( *Met ` X ) /\ R C_ X ) -> ( D |` ( R X. R ) ) e. ( *Met ` R ) )
Colors of variables: wff setvar class
Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   C_ wss 3574   class class class wbr 4653   X. cxp 5112   dom cdm 5114   |` cres 5116   -->wf 5884   ` cfv 5888  (class class class)co 6650   0cc0 9936   RR*cxr 10073   <_ cle 10075   +ecxad 11944   *Metcxmt 19731
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-map 7859  df-xr 10078  df-xmet 19739
This theorem is referenced by:  metres2  22168  xmetres  22169  xpsxmet  22185  xpsdsval  22186  xmetresbl  22242  tmsxms  22291  imasf1oxms  22294  metrest  22329  prdsxms  22335  tmsxpsval  22343  nrginvrcn  22496  divcn  22671  iitopon  22682  cncfmet  22711  cfilres  23094  dvlip2  23758  ftc1lem6  23804  ulmdvlem1  24154  ulmdvlem3  24156  abelth  24195  cxpcn3  24489  rlimcnp  24692  minvecolem4b  27734  minvecolem4  27736  ftc1cnnc  33484  blbnd  33586  ismtyres  33607  reheibor  33638
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