Theorem 2times 9249

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 9180	. . 3 |- 2 = ( 1 + 1 )
2	1	oveq1i 6017	. 2 |- ( 2 x. A ) = ( ( 1 + 1 ) x. A )
3		1p1times 8291	. 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 6007   CCcc 8008   1c1 8011   + caddc 8013   x. cmul 8015   2c2 9172

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 8102  ax-1cn 8103  ax-icn 8105  ax-addcl 8106  ax-mulcl 8108  ax-mulcom 8111  ax-mulass 8113  ax-distr 8114  ax-1rid 8117  ax-cnre 8121

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 6010  df-2 9180

This theorem is referenced by:  times2  9250  2timesi  9251  2txmxeqx  9253  2halves  9351  halfaddsub  9356  avglt2  9362  2timesd  9365  expubnd  10830  subsq2  10881  sinmul  12271  sin2t  12276  cos2t  12277  pythagtriplem4  12807  pythagtriplem14  12816  pythagtriplem16  12818