Theorem 2times 9382

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 9313	. . 3 |- 2 = ( 1 + 1 )
2	1	oveq1i 6068	. 2 |- ( 2 x. A ) = ( ( 1 + 1 ) x. A )
3		1p1times 8423	. 2 |- ( A e. CC -> ( ( 1 + 1 ) x. A ) = ( A + A ) )
4	2, 3	eqtrid 2279	1 |- ( A e. CC -> ( 2 x. A ) = ( A + A ) )

Syntax hints:   -> wi 4   = wceq 1398   e. wcel 2205  (class class class)co 6058   CCcc 8141   1c1 8144   + caddc 8146   x. cmul 8148   2c2 9305

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2216  ax-resscn 8235  ax-1cn 8236  ax-icn 8238  ax-addcl 8239  ax-mulcl 8241  ax-mulcom 8244  ax-mulass 8246  ax-distr 8247  ax-1rid 8250  ax-cnre 8254

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ral 2527  df-rex 2528  df-v 2817  df-un 3218  df-in 3220  df-ss 3227  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-br 4115  df-iota 5317  df-fv 5365  df-ov 6061  df-2 9313

This theorem is referenced by:  times2  9383  2timesi  9384  2txmxeqx  9386  2halves  9484  halfaddsub  9489  avglt2  9495  2timesd  9498  expubnd  10982  subsq2  11033  sinmul  12455  sin2t  12460  cos2t  12461  pythagtriplem4  12991  pythagtriplem14  13000  pythagtriplem16  13002  pellexlem2  15972