Theorem 2times 9414

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 9345	. . 3 |- 2 = ( 1 + 1 )
2	1	oveq1i 6088	. 2 |- ( 2 x. A ) = ( ( 1 + 1 ) x. A )
3		1p1times 8453	. 2 |- ( A e. CC -> ( ( 1 + 1 ) x. A ) = ( A + A ) )
4	2, 3	eqtrid 2283	1 |- ( A e. CC -> ( 2 x. A ) = ( A + A ) )

Syntax hints:   -> wi 4   = wceq 1402   e. wcel 2209  (class class class)co 6078   CCcc 8170   1c1 8173   + caddc 8175   x. cmul 8177   2c2 9337

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220  ax-resscn 8264  ax-1cn 8265  ax-icn 8267  ax-addcl 8268  ax-mulcl 8270  ax-mulcom 8273  ax-mulass 8275  ax-distr 8276  ax-1rid 8279  ax-cnre 8283

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-v 2823  df-un 3224  df-in 3226  df-ss 3233  df-sn 3714  df-pr 3715  df-op 3717  df-uni 3934  df-br 4129  df-iota 5335  df-fv 5383  df-ov 6081  df-2 9345

This theorem is referenced by:  times2  9415  2timesi  9416  2txmxeqx  9418  2halves  9516  halfaddsub  9521  avglt2  9527  2timesd  9530  expubnd  11014  subsq2  11065  sinmul  12492  sin2t  12497  cos2t  12498  pythagtriplem4  13028  pythagtriplem14  13037  pythagtriplem16  13039  pellexlem2  16009