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Theorem ovresd 6425
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 6424 . 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 1379    e. wcel 1767    X. cxp 4997    |` cres 5001  (class class class)co 6282
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pr 4686
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-xp 5005  df-res 5011  df-iota 5549  df-fv 5594  df-ov 6285
This theorem is referenced by:  sscres  15049  fullsubc  15073  fullresc  15074  funcres2c  15124  psmetres2  20553  xmetres2  20599  prdsdsf  20605  xpsdsval  20619  xmssym  20703  xmstri2  20704  mstri2  20705  xmstri  20706  mstri  20707  xmstri3  20708  mstri3  20709  msrtri  20710  tmsxpsval  20776  ngptgp  20885  nlmvscn  20931  nrginvrcn  20935  nghmcn  20987  cnmpt1ds  21082  cnmpt2ds  21083  ipcn  21421  caussi  21471  causs  21472  minveclem2  21576  minveclem3b  21578  minveclem3  21579  minveclem4  21582  minveclem6  21584  ftc1lem6  22177  ulmdvlem1  22529  abelth  22570  cxpcn3  22850  rlimcnp  23023  hhssnv  25856  qqhcn  27608  qqhucn  27609  ftc1cnnc  29666  ismtyres  29907  isdrngo2  29964
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