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Theorem 2times 9076
Description: Two times a number. (Contributed by NM, 10-Oct-2004.) (Revised by Mario Carneiro, 27-May-2016.) (Proof shortened by AV, 26-Feb-2020.)
Assertion
Ref Expression
2times  |-  ( A  e.  CC  ->  (
2  x.  A )  =  ( A  +  A ) )

Proof of Theorem 2times
StepHypRef Expression
1 df-2 9007 . . 3  |-  2  =  ( 1  +  1 )
21oveq1i 5905 . 2  |-  ( 2  x.  A )  =  ( ( 1  +  1 )  x.  A
)
3 1p1times 8120 . 2  |-  ( A  e.  CC  ->  (
( 1  +  1 )  x.  A )  =  ( A  +  A ) )
42, 3eqtrid 2234 1  |-  ( A  e.  CC  ->  (
2  x.  A )  =  ( A  +  A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1364    e. wcel 2160  (class class class)co 5895   CCcc 7838   1c1 7841    + caddc 7843    x. cmul 7845   2c2 8999
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 7932  ax-1cn 7933  ax-icn 7935  ax-addcl 7936  ax-mulcl 7938  ax-mulcom 7941  ax-mulass 7943  ax-distr 7944  ax-1rid 7947  ax-cnre 7951
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 5898  df-2 9007
This theorem is referenced by:  times2  9077  2timesi  9078  2halves  9177  halfaddsub  9182  avglt2  9187  2timesd  9190  expubnd  10607  subsq2  10658  sinmul  11783  sin2t  11788  cos2t  11789  pythagtriplem4  12299  pythagtriplem14  12308  pythagtriplem16  12310
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