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Theorem times2 10100
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 10070 . . 3  |-  2  e.  CC
2 mulcom 9076 . . 3  |-  ( ( A  e.  CC  /\  2  e.  CC )  ->  ( A  x.  2 )  =  ( 2  x.  A ) )
31, 2mpan2 653 . 2  |-  ( A  e.  CC  ->  ( A  x.  2 )  =  ( 2  x.  A ) )
4 2times 10099 . 2  |-  ( A  e.  CC  ->  (
2  x.  A )  =  ( A  +  A ) )
53, 4eqtrd 2468 1  |-  ( A  e.  CC  ->  ( A  x.  2 )  =  ( A  +  A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1652    e. wcel 1725  (class class class)co 6081   CCcc 8988    + caddc 8993    x. cmul 8995   2c2 10049
This theorem is referenced by:  times2i  10102  avglt1  10205  times2d  10211
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-resscn 9047  ax-1cn 9048  ax-icn 9049  ax-addcl 9050  ax-addrcl 9051  ax-mulcl 9052  ax-mulrcl 9053  ax-mulcom 9054  ax-mulass 9056  ax-distr 9057  ax-i2m1 9058  ax-1ne0 9059  ax-1rid 9060  ax-rrecex 9062  ax-cnre 9063
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-br 4213  df-iota 5418  df-fv 5462  df-ov 6084  df-2 10058
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