Theorem 2times 10059

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

Step	Hyp	Ref	Expression
1		df-2 10018	. . . 4 |- 2 = ( 1 + 1 )
2	1	oveq1i 6054	. . 3 |- ( 2 x. A ) = ( ( 1 + 1 ) x. A )
3		ax-1cn 9008	. . . . 5 |- 1 e. CC
4	3	a1i 11	. . . 4 |- ( A e. CC -> 1 e. CC )
5		id 20	. . . 4 |- ( A e. CC -> A e. CC )
6	4, 4, 5	adddird 9073	. . 3 |- ( A e. CC -> ( ( 1 + 1 ) x. A ) = ( ( 1 x. A ) + ( 1 x. A ) ) )
7	2, 6	syl5eq 2452	. 2 |- ( A e. CC -> ( 2 x. A ) = ( ( 1 x. A ) + ( 1 x. A ) ) )
8		mulid2 9049	. . 3 |- ( A e. CC -> ( 1 x. A ) = A )
9	8, 8	oveq12d 6062	. 2 |- ( A e. CC -> ( ( 1 x. A ) + ( 1 x. A ) ) = ( A + A ) )
10	7, 9	eqtrd 2440	1 |- ( A e. CC -> ( 2 x. A ) = ( A + A ) )

Syntax hints:   -> wi 4   = wceq 1649   e. wcel 1721  (class class class)co 6044   CCcc 8948   1c1 8951   + caddc 8953   x. cmul 8955   2c2 10009

This theorem is referenced by:  times2  10060  2timesi  10061  2halves  10156  halfaddsub  10161  avglt2  10166  2timesd  10170  expubnd  11399  subsq2  11448  absmax  12092  sinmul  12732  sin2t  12737  cos2t  12738  sadadd2lem2  12921  pythagtriplem4  13152  pythagtriplem14  13161  pythagtriplem16  13163  cncph  22277  pellexlem2  26787  acongrep  26939

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2389  ax-resscn 9007  ax-1cn 9008  ax-icn 9009  ax-addcl 9010  ax-mulcl 9012  ax-mulcom 9014  ax-mulass 9016  ax-distr 9017  ax-1rid 9020  ax-cnre 9023

This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2395  df-cleq 2401  df-clel 2404  df-nfc 2533  df-ral 2675  df-rex 2676  df-rab 2679  df-v 2922  df-dif 3287  df-un 3289  df-in 3291  df-ss 3298  df-nul 3593  df-if 3704  df-sn 3784  df-pr 3785  df-op 3787  df-uni 3980  df-br 4177  df-iota 5381  df-fv 5425  df-ov 6047  df-2 10018