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Theorem 2times 10527
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 10467 . . . 4  |-  2  =  ( 1  +  1 )
21oveq1i 6186 . . 3  |-  ( 2  x.  A )  =  ( ( 1  +  1 )  x.  A
)
3 ax-1cn 9427 . . . . 5  |-  1  e.  CC
43a1i 11 . . . 4  |-  ( A  e.  CC  ->  1  e.  CC )
5 id 22 . . . 4  |-  ( A  e.  CC  ->  A  e.  CC )
64, 4, 5adddird 9498 . . 3  |-  ( A  e.  CC  ->  (
( 1  +  1 )  x.  A )  =  ( ( 1  x.  A )  +  ( 1  x.  A
) ) )
72, 6syl5eq 2502 . 2  |-  ( A  e.  CC  ->  (
2  x.  A )  =  ( ( 1  x.  A )  +  ( 1  x.  A
) ) )
8 mulid2 9471 . . 3  |-  ( A  e.  CC  ->  (
1  x.  A )  =  A )
98, 8oveq12d 6194 . 2  |-  ( A  e.  CC  ->  (
( 1  x.  A
)  +  ( 1  x.  A ) )  =  ( A  +  A ) )
107, 9eqtrd 2490 1  |-  ( A  e.  CC  ->  (
2  x.  A )  =  ( A  +  A ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1370    e. wcel 1757  (class class class)co 6176   CCcc 9367   1c1 9370    + caddc 9372    x. cmul 9374   2c2 10458
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1709  ax-7 1729  ax-10 1776  ax-11 1781  ax-12 1793  ax-13 1944  ax-ext 2429  ax-resscn 9426  ax-1cn 9427  ax-icn 9428  ax-addcl 9429  ax-mulcl 9431  ax-mulcom 9433  ax-mulass 9435  ax-distr 9436  ax-1rid 9439  ax-cnre 9442
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1702  df-clab 2436  df-cleq 2442  df-clel 2445  df-nfc 2598  df-ral 2797  df-rex 2798  df-rab 2801  df-v 3056  df-dif 3415  df-un 3417  df-in 3419  df-ss 3426  df-nul 3722  df-if 3876  df-sn 3962  df-pr 3964  df-op 3968  df-uni 4176  df-br 4377  df-iota 5465  df-fv 5510  df-ov 6179  df-2 10467
This theorem is referenced by:  times2  10528  2timesi  10529  2halves  10640  halfaddsub  10645  avglt2  10650  2timesd  10654  expubnd  12011  subsq2  12061  absmax  12905  sinmul  13544  sin2t  13549  cos2t  13550  sadadd2lem2  13734  pythagtriplem4  13974  pythagtriplem14  13983  pythagtriplem16  13985  cncph  24340  pellexlem2  29295  acongrep  29447  2txmxeqx  30295
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