Theorem lemin ≪ | index | src | ≫

theorem lemin (a b c: nat): $ a <= min b c <-> a <= b /\ a <= c $;

Step	Hyp	Ref	Expression
1		minle1	min b c <= b
2		letr	a <= min b c -> min b c <= b -> a <= b
3	1, 2	mpi	a <= min b c -> a <= b
4		minle2	min b c <= c
5		letr	a <= min b c -> min b c <= c -> a <= c
6	4, 5	mpi	a <= min b c -> a <= c
7	3, 6	iand	a <= min b c -> a <= b /\ a <= c
8		ifpos	b < c -> if (b < c) b c = b
9	8	conv min	b < c -> min b c = b
10	9	leeq2d	b < c -> (a <= min b c <-> a <= b)
11	10	anwr	a <= b /\ a <= c /\ b < c -> (a <= min b c <-> a <= b)
12		anll	a <= b /\ a <= c /\ b < c -> a <= b
13	11, 12	mpbird	a <= b /\ a <= c /\ b < c -> a <= min b c
14		ifneg	~b < c -> if (b < c) b c = c
15	14	conv min	~b < c -> min b c = c
16	15	leeq2d	~b < c -> (a <= min b c <-> a <= c)
17	16	anwr	a <= b /\ a <= c /\ ~b < c -> (a <= min b c <-> a <= c)
18		anlr	a <= b /\ a <= c /\ ~b < c -> a <= c
19	17, 18	mpbird	a <= b /\ a <= c /\ ~b < c -> a <= min b c
20	13, 19	casesda	a <= b /\ a <= c -> a <= min b c
21	7, 20	ibii	a <= min 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_set (elab, ax_8), axs_the (theid, the0), axs_peano (peano1, peano2, peano5, addeq, add0, addS)