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Theorem ovresd 5962
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 5961 . 2  |-  ( ( A  e.  X  /\  B  e.  X )  ->  ( A ( D  |`  ( X  X.  X
) ) B )  =  ( A D B ) )
41, 2, 3syl2anc 409 1  |-  ( ph  ->  ( A ( D  |`  ( X  X.  X
) ) B )  =  ( A D B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1335    e. wcel 2128    X. cxp 4585    |` cres 4589  (class class class)co 5825
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-14 2131  ax-ext 2139  ax-sep 4083  ax-pow 4136  ax-pr 4170
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ral 2440  df-rex 2441  df-v 2714  df-un 3106  df-in 3108  df-ss 3115  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-uni 3774  df-br 3967  df-opab 4027  df-xp 4593  df-res 4599  df-iota 5136  df-fv 5179  df-ov 5828
This theorem is referenced by:  psmetres2  12775  xmetres2  12821  xmssym  12911  xmstri2  12912  mstri2  12913  xmstri  12914  mstri  12915  xmstri3  12916  mstri3  12917  msrtri  12918  limcimolemlt  13075  cnplimcim  13078  cnplimclemr  13080  limccnpcntop  13086  limccnp2lem  13087
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