Theorem addmul ≪ | index | src | ≫

theorem addmul (a b c: nat): $ (a + b) * c = a * c + b * c $;

Step	Hyp	Ref	Expression
1		eqtr	(a + b) * c = c * (a + b) -> c * (a + b) = a * c + b * c -> (a + b) * c = a * c + b * c
2		mulcom	(a + b) * c = c * (a + b)
3	1, 2	ax_mp	c * (a + b) = a * c + b * c -> (a + b) * c = a * c + b * c
4		eqtr	c * (a + b) = c * a + c * b -> c * a + c * b = a * c + b * c -> c * (a + b) = a * c + b * c
5		muladd	c * (a + b) = c * a + c * b
6	4, 5	ax_mp	c * a + c * b = a * c + b * c -> c * (a + b) = a * c + b * c
7		addeq	c * a = a * c -> c * b = b * c -> c * a + c * b = a * c + b * c
8		mulcom	c * a = a * c
9	7, 8	ax_mp	c * b = b * c -> c * a + c * b = a * c + b * c
10		mulcom	c * b = b * c
11	9, 10	ax_mp	c * a + c * b = a * c + b * c
12	6, 11	ax_mp	c * (a + b) = a * c + b * c
13	3, 12	ax_mp	(a + b) * c = a * c + b * c

Axiom use

axs_prop_calc (ax_1, ax_2, ax_3, ax_mp, itru), axs_pred_calc (ax_gen, ax_4, ax_5, ax_6, ax_7, ax_10, ax_11, ax_12), axs_peano (peano2, peano5, addeq, muleq, add0, addS, mul0, mulS)