Theorem leadd2 ≪ | index | src | ≫

theorem leadd2 (a b c: nat): $ b <= c <-> a + b <= a + c $;

Step	Hyp	Ref	Expression
1		bitr	(b <= c <-> b + a <= c + a) -> (b + a <= c + a <-> a + b <= a + c) -> (b <= c <-> a + b <= a + c)
2		leadd1	b <= c <-> b + a <= c + a
3	1, 2	ax_mp	(b + a <= c + a <-> a + b <= a + c) -> (b <= c <-> a + b <= a + c)
4		leeq	b + a = a + b -> c + a = a + c -> (b + a <= c + a <-> a + b <= a + c)
5		addcom	b + a = a + b
6	4, 5	ax_mp	c + a = a + c -> (b + a <= c + a <-> a + b <= a + c)
7		addcom	c + a = a + c
8	6, 7	ax_mp	b + a <= c + a <-> a + b <= a + c
9	3, 8	ax_mp	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 (peano2, peano5, addeq, add0, addS)