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Theorem 2times 9843
Description: Two times a number. (Contributed by NM, 10-Oct-2004.) (Revised by Mario Carneiro, 27-May-2016.)
Assertion
Ref Expression
2times  |-  ( A  e.  CC  ->  (
2  x.  A )  =  ( A  +  A ) )

Proof of Theorem 2times
StepHypRef Expression
1 df-2 9804 . . . 4  |-  2  =  ( 1  +  1 )
21oveq1i 5868 . . 3  |-  ( 2  x.  A )  =  ( ( 1  +  1 )  x.  A
)
3 ax-1cn 8795 . . . . 5  |-  1  e.  CC
43a1i 10 . . . 4  |-  ( A  e.  CC  ->  1  e.  CC )
5 id 19 . . . 4  |-  ( A  e.  CC  ->  A  e.  CC )
64, 4, 5adddird 8860 . . 3  |-  ( A  e.  CC  ->  (
( 1  +  1 )  x.  A )  =  ( ( 1  x.  A )  +  ( 1  x.  A
) ) )
72, 6syl5eq 2327 . 2  |-  ( A  e.  CC  ->  (
2  x.  A )  =  ( ( 1  x.  A )  +  ( 1  x.  A
) ) )
8 mulid2 8836 . . 3  |-  ( A  e.  CC  ->  (
1  x.  A )  =  A )
98, 8oveq12d 5876 . 2  |-  ( A  e.  CC  ->  (
( 1  x.  A
)  +  ( 1  x.  A ) )  =  ( A  +  A ) )
107, 9eqtrd 2315 1  |-  ( A  e.  CC  ->  (
2  x.  A )  =  ( A  +  A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1623    e. wcel 1684  (class class class)co 5858   CCcc 8735   1c1 8738    + caddc 8740    x. cmul 8742   2c2 9795
This theorem is referenced by:  times2  9844  2timesi  9845  2halves  9940  halfaddsub  9945  avglt2  9950  2timesd  9954  expubnd  11162  subsq2  11211  absmax  11813  sinmul  12452  sin2t  12457  cos2t  12458  sadadd2lem2  12641  pythagtriplem4  12872  pythagtriplem14  12881  pythagtriplem16  12883  cncph  21397  sqsscirc1  23292  pellexlem2  26915  acongrep  27067
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-mulcl 8799  ax-mulcom 8801  ax-mulass 8803  ax-distr 8804  ax-1rid 8807  ax-cnre 8810
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-iota 5219  df-fv 5263  df-ov 5861  df-2 9804
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