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Theorem times2 9079
Description: A number times 2. (Contributed by NM, 16-Oct-2007.)
Assertion
Ref Expression
times2  |-  ( A  e.  CC  ->  ( A  x.  2 )  =  ( A  +  A ) )

Proof of Theorem times2
StepHypRef Expression
1 2cn 9021 . . 3  |-  2  e.  CC
2 mulcom 7971 . . 3  |-  ( ( A  e.  CC  /\  2  e.  CC )  ->  ( A  x.  2 )  =  ( 2  x.  A ) )
31, 2mpan2 425 . 2  |-  ( A  e.  CC  ->  ( A  x.  2 )  =  ( 2  x.  A ) )
4 2times 9078 . 2  |-  ( A  e.  CC  ->  (
2  x.  A )  =  ( A  +  A ) )
53, 4eqtrd 2222 1  |-  ( A  e.  CC  ->  ( A  x.  2 )  =  ( A  +  A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1364    e. wcel 2160  (class class class)co 5897   CCcc 7840    + caddc 7845    x. cmul 7847   2c2 9001
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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171  ax-resscn 7934  ax-1cn 7935  ax-1re 7936  ax-icn 7937  ax-addcl 7938  ax-addrcl 7939  ax-mulcl 7940  ax-mulcom 7943  ax-mulass 7945  ax-distr 7946  ax-1rid 7949  ax-cnre 7953
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-v 2754  df-un 3148  df-in 3150  df-ss 3157  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-br 4019  df-iota 5196  df-fv 5243  df-ov 5900  df-2 9009
This theorem is referenced by:  times2i  9081  avglt1  9188  times2d  9193
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