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Theorem 2times 8871
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 8802 . . 3  |-  2  =  ( 1  +  1 )
21oveq1i 5791 . 2  |-  ( 2  x.  A )  =  ( ( 1  +  1 )  x.  A
)
3 1p1times 7919 . 2  |-  ( A  e.  CC  ->  (
( 1  +  1 )  x.  A )  =  ( A  +  A ) )
42, 3syl5eq 2185 1  |-  ( A  e.  CC  ->  (
2  x.  A )  =  ( A  +  A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1332    e. wcel 1481  (class class class)co 5781   CCcc 7641   1c1 7644    + caddc 7646    x. cmul 7648   2c2 8794
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-resscn 7735  ax-1cn 7736  ax-icn 7738  ax-addcl 7739  ax-mulcl 7741  ax-mulcom 7744  ax-mulass 7746  ax-distr 7747  ax-1rid 7750  ax-cnre 7754
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rex 2423  df-v 2691  df-un 3079  df-in 3081  df-ss 3088  df-sn 3537  df-pr 3538  df-op 3540  df-uni 3744  df-br 3937  df-iota 5095  df-fv 5138  df-ov 5784  df-2 8802
This theorem is referenced by:  times2  8872  2timesi  8873  2halves  8972  halfaddsub  8977  avglt2  8982  2timesd  8985  expubnd  10380  subsq2  10430  sinmul  11485  sin2t  11490  cos2t  11491
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