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Theorem 2times 8752
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 8689 . . 3  |-  2  =  ( 1  +  1 )
21oveq1i 5738 . 2  |-  ( 2  x.  A )  =  ( ( 1  +  1 )  x.  A
)
3 1p1times 7819 . 2  |-  ( A  e.  CC  ->  (
( 1  +  1 )  x.  A )  =  ( A  +  A ) )
42, 3syl5eq 2159 1  |-  ( A  e.  CC  ->  (
2  x.  A )  =  ( A  +  A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1314    e. wcel 1463  (class class class)co 5728   CCcc 7545   1c1 7548    + caddc 7550    x. cmul 7552   2c2 8681
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-resscn 7637  ax-1cn 7638  ax-icn 7640  ax-addcl 7641  ax-mulcl 7643  ax-mulcom 7646  ax-mulass 7648  ax-distr 7649  ax-1rid 7652  ax-cnre 7656
This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2244  df-ral 2395  df-rex 2396  df-v 2659  df-un 3041  df-in 3043  df-ss 3050  df-sn 3499  df-pr 3500  df-op 3502  df-uni 3703  df-br 3896  df-iota 5046  df-fv 5089  df-ov 5731  df-2 8689
This theorem is referenced by:  times2  8753  2timesi  8754  2halves  8853  halfaddsub  8858  avglt2  8863  2timesd  8866  expubnd  10243  subsq2  10293  sinmul  11302  sin2t  11307  cos2t  11308
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