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Theorem ovresd 6014
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 6013 . 2  |-  ( ( A  e.  X  /\  B  e.  X )  ->  ( A ( D  |`  ( X  X.  X
) ) B )  =  ( A D B ) )
41, 2, 3syl2anc 411 1  |-  ( ph  ->  ( A ( D  |`  ( X  X.  X
) ) B )  =  ( A D B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1353    e. wcel 2148    X. cxp 4624    |` cres 4628  (class class class)co 5874
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4121  ax-pow 4174  ax-pr 4209
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2739  df-un 3133  df-in 3135  df-ss 3142  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-br 4004  df-opab 4065  df-xp 4632  df-res 4638  df-iota 5178  df-fv 5224  df-ov 5877
This theorem is referenced by:  psmetres2  13803  xmetres2  13849  xmssym  13939  xmstri2  13940  mstri2  13941  xmstri  13942  mstri  13943  xmstri3  13944  mstri3  13945  msrtri  13946  limcimolemlt  14103  cnplimcim  14106  cnplimclemr  14108  limccnpcntop  14114  limccnp2lem  14115
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