Theorem 1p1times 8312

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 8124	. . . 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 8204	. 2 |- ( A e. CC -> ( ( 1 + 1 ) x. A ) = ( ( 1 x. A ) + ( 1 x. A ) ) )
5		mullid 8176	. . 3 |- ( A e. CC -> ( 1 x. A ) = A )
6	5, 5	oveq12d 6035	. 2 |- ( A e. CC -> ( ( 1 x. A ) + ( 1 x. A ) ) = ( A + A ) )
7	4, 6	eqtrd 2264	1 |- ( A e. CC -> ( ( 1 + 1 ) x. A ) = ( A + A ) )

Syntax hints:   -> wi 4   = wceq 1397   e. wcel 2202  (class class class)co 6017   CCcc 8029   1c1 8032   + caddc 8034   x. cmul 8036

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213  ax-resscn 8123  ax-1cn 8124  ax-icn 8126  ax-addcl 8127  ax-mulcl 8129  ax-mulcom 8132  ax-mulass 8134  ax-distr 8135  ax-1rid 8138  ax-cnre 8142

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-rex 2516  df-v 2804  df-un 3204  df-in 3206  df-ss 3213  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-br 4089  df-iota 5286  df-fv 5334  df-ov 6020

This theorem is referenced by:  eqneg  8911  2times  9270