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Theorem 2times 10104
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 10063 . . . 4  |-  2  =  ( 1  +  1 )
21oveq1i 6094 . . 3  |-  ( 2  x.  A )  =  ( ( 1  +  1 )  x.  A
)
3 ax-1cn 9053 . . . . 5  |-  1  e.  CC
43a1i 11 . . . 4  |-  ( A  e.  CC  ->  1  e.  CC )
5 id 21 . . . 4  |-  ( A  e.  CC  ->  A  e.  CC )
64, 4, 5adddird 9118 . . 3  |-  ( A  e.  CC  ->  (
( 1  +  1 )  x.  A )  =  ( ( 1  x.  A )  +  ( 1  x.  A
) ) )
72, 6syl5eq 2482 . 2  |-  ( A  e.  CC  ->  (
2  x.  A )  =  ( ( 1  x.  A )  +  ( 1  x.  A
) ) )
8 mulid2 9094 . . 3  |-  ( A  e.  CC  ->  (
1  x.  A )  =  A )
98, 8oveq12d 6102 . 2  |-  ( A  e.  CC  ->  (
( 1  x.  A
)  +  ( 1  x.  A ) )  =  ( A  +  A ) )
107, 9eqtrd 2470 1  |-  ( A  e.  CC  ->  (
2  x.  A )  =  ( A  +  A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1653    e. wcel 1726  (class class class)co 6084   CCcc 8993   1c1 8996    + caddc 8998    x. cmul 9000   2c2 10054
This theorem is referenced by:  times2  10105  2timesi  10106  2halves  10201  halfaddsub  10206  avglt2  10211  2timesd  10215  expubnd  11445  subsq2  11494  absmax  12138  sinmul  12778  sin2t  12783  cos2t  12784  sadadd2lem2  12967  pythagtriplem4  13198  pythagtriplem14  13207  pythagtriplem16  13209  cncph  22325  pellexlem2  26907  acongrep  27059  2txmxeqx  28113
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-resscn 9052  ax-1cn 9053  ax-icn 9054  ax-addcl 9055  ax-mulcl 9057  ax-mulcom 9059  ax-mulass 9061  ax-distr 9062  ax-1rid 9065  ax-cnre 9068
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4216  df-iota 5421  df-fv 5465  df-ov 6087  df-2 10063
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