Theorem 1p1times 8177

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 7989	. . . 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 8069	. 2 |- ( A e. CC -> ( ( 1 + 1 ) x. A ) = ( ( 1 x. A ) + ( 1 x. A ) ) )
5		mullid 8041	. . 3 |- ( A e. CC -> ( 1 x. A ) = A )
6	5, 5	oveq12d 5943	. 2 |- ( A e. CC -> ( ( 1 x. A ) + ( 1 x. A ) ) = ( A + A ) )
7	4, 6	eqtrd 2229	1 |- ( A e. CC -> ( ( 1 + 1 ) x. A ) = ( A + A ) )

Syntax hints:   -> wi 4   = wceq 1364   e. wcel 2167  (class class class)co 5925   CCcc 7894   1c1 7897   + caddc 7899   x. cmul 7901

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 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178  ax-resscn 7988  ax-1cn 7989  ax-icn 7991  ax-addcl 7992  ax-mulcl 7994  ax-mulcom 7997  ax-mulass 7999  ax-distr 8000  ax-1rid 8003  ax-cnre 8007

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-rex 2481  df-v 2765  df-un 3161  df-in 3163  df-ss 3170  df-sn 3629  df-pr 3630  df-op 3632  df-uni 3841  df-br 4035  df-iota 5220  df-fv 5267  df-ov 5928

This theorem is referenced by:  eqneg  8776  2times  9135