Theorem submul ≪ | index | src | ≫

theorem submul (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		mulsub	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		subeq	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_set (elab, ax_8), axs_the (theid, the0), axs_peano (peano1, peano2, peano5, addeq, muleq, add0, addS, mul0, mulS)