Theorem 2times 9238

Description: Two times a number. (Contributed by NM, 10-Oct-2004.) (Revised by Mario Carneiro, 27-May-2016.) (Proof shortened by AV, 26-Feb-2020.)

Assertion

Ref	Expression
2times	|- ( A e. CC -> ( 2 x. A ) = ( A + A ) )

Proof of Theorem 2times

Step	Hyp	Ref	Expression
1		df-2 9169	. . 3 |- 2 = ( 1 + 1 )
2	1	oveq1i 6011	. 2 |- ( 2 x. A ) = ( ( 1 + 1 ) x. A )
3		1p1times 8280	. 2 |- ( A e. CC -> ( ( 1 + 1 ) x. A ) = ( A + A ) )
4	2, 3	eqtrid 2274	1 |- ( A e. CC -> ( 2 x. A ) = ( A + A ) )

Syntax hints:   -> wi 4   = wceq 1395   e. wcel 2200  (class class class)co 6001   CCcc 7997   1c1 8000   + caddc 8002   x. cmul 8004   2c2 9161

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211  ax-resscn 8091  ax-1cn 8092  ax-icn 8094  ax-addcl 8095  ax-mulcl 8097  ax-mulcom 8100  ax-mulass 8102  ax-distr 8103  ax-1rid 8106  ax-cnre 8110

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-v 2801  df-un 3201  df-in 3203  df-ss 3210  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-br 4084  df-iota 5278  df-fv 5326  df-ov 6004  df-2 9169

This theorem is referenced by:  times2  9239  2timesi  9240  2txmxeqx  9242  2halves  9340  halfaddsub  9345  avglt2  9351  2timesd  9354  expubnd  10818  subsq2  10869  sinmul  12255  sin2t  12260  cos2t  12261  pythagtriplem4  12791  pythagtriplem14  12800  pythagtriplem16  12802