Theorem 1p1times 8091

Description: Two times a number. (Contributed by NM, 18-May-1999.) (Revised by Mario Carneiro, 27-May-2016.)

Assertion

Ref	Expression
1p1times	|- ( A e. CC -> ( ( 1 + 1 ) x. A ) = ( A + A ) )

Proof of Theorem 1p1times

Step	Hyp	Ref	Expression
1		ax-1cn 7904	. . . 4 |- 1 e. CC
2	1	a1i 9	. . 3 |- ( A e. CC -> 1 e. CC )
3		id 19	. . 3 |- ( A e. CC -> A e. CC )
4	2, 2, 3	adddird 7983	. 2 |- ( A e. CC -> ( ( 1 + 1 ) x. A ) = ( ( 1 x. A ) + ( 1 x. A ) ) )
5		mullid 7955	. . 3 |- ( A e. CC -> ( 1 x. A ) = A )
6	5, 5	oveq12d 5893	. 2 |- ( A e. CC -> ( ( 1 x. A ) + ( 1 x. A ) ) = ( A + A ) )
7	4, 6	eqtrd 2210	1 |- ( A e. CC -> ( ( 1 + 1 ) x. A ) = ( A + A ) )

Syntax hints:   -> wi 4   = wceq 1353   e. wcel 2148  (class class class)co 5875   CCcc 7809   1c1 7812   + caddc 7814   x. cmul 7816

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159  ax-resscn 7903  ax-1cn 7904  ax-icn 7906  ax-addcl 7907  ax-mulcl 7909  ax-mulcom 7912  ax-mulass 7914  ax-distr 7915  ax-1rid 7918  ax-cnre 7922

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2740  df-un 3134  df-in 3136  df-ss 3143  df-sn 3599  df-pr 3600  df-op 3602  df-uni 3811  df-br 4005  df-iota 5179  df-fv 5225  df-ov 5878

This theorem is referenced by:  eqneg  8689  2times  9047