Theorem minadd1 ≪ | index | src | ≫

theorem minadd1 (a b c: nat): $ min a b + c = min (a + c) (b + c) $;

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	2	addeq1d	a < b -> min a b + c = a + c
4		ltadd1	a < b <-> a + c < b + c
5		ifpos	a + c < b + c -> if (a + c < b + c) (a + c) (b + c) = a + c
6	5	conv min	a + c < b + c -> min (a + c) (b + c) = a + c
7	4, 6	sylbi	a < b -> min (a + c) (b + c) = a + c
8	3, 7	eqtr4d	a < b -> min a b + c = min (a + c) (b + c)
9		ifneg	~a < b -> if (a < b) a b = b
10	9	conv min	~a < b -> min a b = b
11	10	addeq1d	~a < b -> min a b + c = b + c
12		noteq	(a < b <-> a + c < b + c) -> (~a < b <-> ~a + c < b + c)
13	12, 4	ax_mp	~a < b <-> ~a + c < b + c
14		ifneg	~a + c < b + c -> if (a + c < b + c) (a + c) (b + c) = b + c
15	14	conv min	~a + c < b + c -> min (a + c) (b + c) = b + c
16	13, 15	sylbi	~a < b -> min (a + c) (b + c) = b + c
17	11, 16	eqtr4d	~a < b -> min a b + c = min (a + c) (b + c)
18	8, 17	cases	min a b + c = min (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_set (elab, ax_8), axs_the (theid), axs_peano (peano2, peano5, addeq, add0, addS)