Theorem ltmul2 ≪ | index | src | ≫

theorem ltmul2 (a b c: nat): $ 0 < a -> (b < c <-> a * b < a * c) $;

Step	Hyp	Ref	Expression
1		lteq	b * a = a * b -> c * a = a * c -> (b * a < c * a <-> a * b < a * c)
2		mulcom	b * a = a * b
3	1, 2	ax_mp	c * a = a * c -> (b * a < c * a <-> a * b < a * c)
4		mulcom	c * a = a * c
5	3, 4	ax_mp	b * a < c * a <-> a * b < a * c
6		ltmul1	0 < a -> (b < c <-> b * a < c * a)
7	5, 6	syl6bb	0 < a -> (b < c <-> a * b < a * 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 (peano1, peano2, peano5, addeq, muleq, add0, addS, mul0, mulS)