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Theorem ovresd 6428
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 6427 . 2  |-  ( ( A  e.  X  /\  B  e.  X )  ->  ( A ( D  |`  ( X  X.  X
) ) B )  =  ( A D B ) )
41, 2, 3syl2anc 661 1  |-  ( ph  ->  ( A ( D  |`  ( X  X.  X
) ) B )  =  ( A D B ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1383    e. wcel 1804    X. cxp 4987    |` cres 4991  (class class class)co 6281
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-sep 4558  ax-nul 4566  ax-pr 4676
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-ral 2798  df-rex 2799  df-rab 2802  df-v 3097  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3771  df-if 3927  df-sn 4015  df-pr 4017  df-op 4021  df-uni 4235  df-br 4438  df-opab 4496  df-xp 4995  df-res 5001  df-iota 5541  df-fv 5586  df-ov 6284
This theorem is referenced by:  sscres  15069  fullsubc  15093  fullresc  15094  funcres2c  15144  psmetres2  20691  xmetres2  20737  prdsdsf  20743  xpsdsval  20757  xmssym  20841  xmstri2  20842  mstri2  20843  xmstri  20844  mstri  20845  xmstri3  20846  mstri3  20847  msrtri  20848  tmsxpsval  20914  ngptgp  21023  nlmvscn  21069  nrginvrcn  21073  nghmcn  21125  cnmpt1ds  21220  cnmpt2ds  21221  ipcn  21559  caussi  21609  causs  21610  minveclem2  21714  minveclem3b  21716  minveclem3  21717  minveclem4  21720  minveclem6  21722  ftc1lem6  22315  ulmdvlem1  22667  abelth  22708  cxpcn3  22994  rlimcnp  23167  hhssnv  26052  qqhcn  27845  qqhucn  27846  ftc1cnnc  30064  ismtyres  30279  isdrngo2  30336  rngchom  32515  ringchom  32558  rhmsubclem4  32630
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