Theorem ltadd1 ≪ | index | src | ≫

theorem ltadd1 (a b c: nat): $ a < b <-> a + c < b + c $;

Step	Hyp	Ref	Expression
1		bitr	(a < b <-> suc a + c <= b + c) -> (suc a + c <= b + c <-> a + c < b + c) -> (a < b <-> a + c < b + c)
2		leadd1	suc a <= b <-> suc a + c <= b + c
3	2	conv lt	a < b <-> suc a + c <= b + c
4	1, 3	ax_mp	(suc a + c <= b + c <-> a + c < b + c) -> (a < b <-> a + c < b + c)
5		leeq1	suc a + c = suc (a + c) -> (suc a + c <= b + c <-> suc (a + c) <= b + c)
6	5	conv lt	suc a + c = suc (a + c) -> (suc a + c <= b + c <-> a + c < b + c)
7		addS1	suc a + c = suc (a + c)
8	6, 7	ax_mp	suc a + c <= b + c <-> a + c < b + c
9	4, 8	ax_mp	a < b <-> 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, add0, addS)