MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ovresd Structured version   Unicode version

Theorem ovresd 6230
Description: Lemma for converting metric theorems to metric space theorems. (Contributed by Mario Carneiro, 2-Oct-2015.)
Hypotheses
Ref Expression
ovresd.1  |-  ( ph  ->  A  e.  X )
ovresd.2  |-  ( ph  ->  B  e.  X )
Assertion
Ref Expression
ovresd  |-  ( ph  ->  ( A ( D  |`  ( X  X.  X
) ) B )  =  ( A D B ) )

Proof of Theorem ovresd
StepHypRef Expression
1 ovresd.1 . 2  |-  ( ph  ->  A  e.  X )
2 ovresd.2 . 2  |-  ( ph  ->  B  e.  X )
3 ovres 6229 . 2  |-  ( ( A  e.  X  /\  B  e.  X )  ->  ( A ( D  |`  ( X  X.  X
) ) B )  =  ( A D B ) )
41, 2, 3syl2anc 656 1  |-  ( ph  ->  ( A ( D  |`  ( X  X.  X
) ) B )  =  ( A D B ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1364    e. wcel 1761    X. cxp 4834    |` cres 4838  (class class class)co 6090
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-sep 4410  ax-nul 4418  ax-pr 4528
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-ral 2718  df-rex 2719  df-rab 2722  df-v 2972  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-nul 3635  df-if 3789  df-sn 3875  df-pr 3877  df-op 3881  df-uni 4089  df-br 4290  df-opab 4348  df-xp 4842  df-res 4848  df-iota 5378  df-fv 5423  df-ov 6093
This theorem is referenced by:  sscres  14732  fullsubc  14756  fullresc  14757  funcres2c  14807  psmetres2  19849  xmetres2  19895  prdsdsf  19901  xpsdsval  19915  xmssym  19999  xmstri2  20000  mstri2  20001  xmstri  20002  mstri  20003  xmstri3  20004  mstri3  20005  msrtri  20006  tmsxpsval  20072  ngptgp  20181  nlmvscn  20227  nrginvrcn  20231  nghmcn  20283  cnmpt1ds  20378  cnmpt2ds  20379  ipcn  20717  caussi  20767  causs  20768  minveclem2  20872  minveclem3b  20874  minveclem3  20875  minveclem4  20878  minveclem6  20880  ftc1lem6  21472  ulmdvlem1  21824  abelth  21865  cxpcn3  22145  rlimcnp  22318  hhssnv  24600  qqhcn  26356  qqhucn  26357  ftc1cnnc  28391  ismtyres  28632  isdrngo2  28689
  Copyright terms: Public domain W3C validator