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Theorem 2times 8038
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 7973 . . 3  |-  2  =  ( 1  +  1 )
21oveq1i 5522 . 2  |-  ( 2  x.  A )  =  ( ( 1  +  1 )  x.  A
)
3 1p1times 7147 . 2  |-  ( A  e.  CC  ->  (
( 1  +  1 )  x.  A )  =  ( A  +  A ) )
42, 3syl5eq 2084 1  |-  ( A  e.  CC  ->  (
2  x.  A )  =  ( A  +  A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1243    e. wcel 1393  (class class class)co 5512   CCcc 6887   1c1 6890    + caddc 6892    x. cmul 6894   2c2 7964
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-resscn 6976  ax-1cn 6977  ax-icn 6979  ax-addcl 6980  ax-mulcl 6982  ax-mulcom 6985  ax-mulass 6987  ax-distr 6988  ax-1rid 6991  ax-cnre 6995
This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-un 2922  df-in 2924  df-ss 2931  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-br 3765  df-iota 4867  df-fv 4910  df-ov 5515  df-2 7973
This theorem is referenced by:  times2  8039  2timesi  8040  2halves  8154  halfaddsub  8159  avglt2  8164  2timesd  8167  expubnd  9311  subsq2  9359
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