Theorem mincom ≪ | index | src | ≫

theorem mincom (a b: nat): $ min a b = min b a $;

Step	Hyp	Ref	Expression
1		ifpos	a < b -> if (a < b) a b = a
2	1	conv min	a < b -> min a b = a
3		ifneg	~b < a -> if (b < a) b a = a
4	3	conv min	~b < a -> min b a = a
5		ltnlt	a < b -> ~b < a
6	4, 5	syl	a < b -> min b a = a
7	2, 6	eqtr4d	a < b -> min a b = min b a
8		ifneg	~a < b -> if (a < b) a b = b
9	8	conv min	~a < b -> min a b = b
10		lenlt	b <= a <-> ~a < b
11		eqmin1	b <= a -> min b a = b
12	10, 11	sylbir	~a < b -> min b a = b
13	9, 12	eqtr4d	~a < b -> min a b = min b a
14	7, 13	cases	min a b = min b a

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)