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Theorem 2times 9992
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 9951 . . . 4  |-  2  =  ( 1  +  1 )
21oveq1i 5991 . . 3  |-  ( 2  x.  A )  =  ( ( 1  +  1 )  x.  A
)
3 ax-1cn 8942 . . . . 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 9007 . . 3  |-  ( A  e.  CC  ->  (
( 1  +  1 )  x.  A )  =  ( ( 1  x.  A )  +  ( 1  x.  A
) ) )
72, 6syl5eq 2410 . 2  |-  ( A  e.  CC  ->  (
2  x.  A )  =  ( ( 1  x.  A )  +  ( 1  x.  A
) ) )
8 mulid2 8983 . . 3  |-  ( A  e.  CC  ->  (
1  x.  A )  =  A )
98, 8oveq12d 5999 . 2  |-  ( A  e.  CC  ->  (
( 1  x.  A
)  +  ( 1  x.  A ) )  =  ( A  +  A ) )
107, 9eqtrd 2398 1  |-  ( A  e.  CC  ->  (
2  x.  A )  =  ( A  +  A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1647    e. wcel 1715  (class class class)co 5981   CCcc 8882   1c1 8885    + caddc 8887    x. cmul 8889   2c2 9942
This theorem is referenced by:  times2  9993  2timesi  9994  2halves  10089  halfaddsub  10094  avglt2  10099  2timesd  10103  expubnd  11327  subsq2  11376  absmax  12020  sinmul  12660  sin2t  12665  cos2t  12666  sadadd2lem2  12849  pythagtriplem4  13080  pythagtriplem14  13089  pythagtriplem16  13091  cncph  21831  sqsscirc1  23782  pellexlem2  26506  acongrep  26658
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1551  ax-5 1562  ax-17 1621  ax-9 1659  ax-8 1680  ax-6 1734  ax-7 1739  ax-11 1751  ax-12 1937  ax-ext 2347  ax-resscn 8941  ax-1cn 8942  ax-icn 8943  ax-addcl 8944  ax-mulcl 8946  ax-mulcom 8948  ax-mulass 8950  ax-distr 8951  ax-1rid 8954  ax-cnre 8957
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 937  df-tru 1324  df-ex 1547  df-nf 1550  df-sb 1654  df-clab 2353  df-cleq 2359  df-clel 2362  df-nfc 2491  df-ral 2633  df-rex 2634  df-rab 2637  df-v 2875  df-dif 3241  df-un 3243  df-in 3245  df-ss 3252  df-nul 3544  df-if 3655  df-sn 3735  df-pr 3736  df-op 3738  df-uni 3930  df-br 4126  df-iota 5322  df-fv 5366  df-ov 5984  df-2 9951
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