Theorem 2times 9367

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 9298	. . 3 |- 2 = ( 1 + 1 )
2	1	oveq1i 6062	. 2 |- ( 2 x. A ) = ( ( 1 + 1 ) x. A )
3		1p1times 8409	. 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 6052   CCcc 8127   1c1 8130   + caddc 8132   x. cmul 8134   2c2 9290

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 8221  ax-1cn 8222  ax-icn 8224  ax-addcl 8225  ax-mulcl 8227  ax-mulcom 8230  ax-mulass 8232  ax-distr 8233  ax-1rid 8236  ax-cnre 8240

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 3217  df-in 3219  df-ss 3226  df-sn 3697  df-pr 3698  df-op 3700  df-uni 3917  df-br 4112  df-iota 5314  df-fv 5362  df-ov 6055  df-2 9298

This theorem is referenced by:  times2  9368  2timesi  9369  2txmxeqx  9371  2halves  9469  halfaddsub  9474  avglt2  9480  2timesd  9483  expubnd  10962  subsq2  11013  sinmul  12434  sin2t  12439  cos2t  12440  pythagtriplem4  12970  pythagtriplem14  12979  pythagtriplem16  12981  pellexlem2  15863