Theorem maxcom ≪ | index | src | ≫

theorem maxcom (a b: nat): $ max a b = max b a $;

Step	Hyp	Ref	Expression
1		ifpos	a < b -> if (a < b) b a = b
2	1	conv max	a < b -> max a b = b
3		ifneg	~b < a -> if (b < a) a b = b
4	3	conv max	~b < a -> max b a = b
5		ltnlt	a < b -> ~b < a
6	4, 5	syl	a < b -> max b a = b
7	2, 6	eqtr4d	a < b -> max a b = max b a
8		ifneg	~a < b -> if (a < b) b a = a
9	8	conv max	~a < b -> max a b = a
10		lenlt	b <= a <-> ~a < b
11		eqmax2	b <= a -> max b a = a
12	10, 11	sylbir	~a < b -> max b a = a
13	9, 12	eqtr4d	~a < b -> max a b = max b a
14	7, 13	cases	max a b = max 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)