Theorem lesub1i ≪ | index | src | ≫

theorem lesub1i (a b c: nat): $ a <= b -> a - c <= b - c $;

Step	Hyp	Ref	Expression
1		leadd1	a - c <= b - c <-> a - c + c <= b - c + c
2		npcan	c <= a -> a - c + c = a
3	2	anwl	c <= a /\ a <= b -> a - c + c = a
4		npcan	c <= b -> b - c + c = b
5		letr	c <= a -> a <= b -> c <= b
6	5	imp	c <= a /\ a <= b -> c <= b
7	4, 6	syl	c <= a /\ a <= b -> b - c + c = b
8	3, 7	leeqd	c <= a /\ a <= b -> (a - c + c <= b - c + c <-> a <= b)
9		anr	c <= a /\ a <= b -> a <= b
10	8, 9	mpbird	c <= a /\ a <= b -> a - c + c <= b - c + c
11	1, 10	sylibr	c <= a /\ a <= b -> a - c <= b - c
12	11	exp	c <= a -> a <= b -> a - c <= b - c
13		le01	0 <= b - c
14		nlesubeq0	~c <= a -> a - c = 0
15	14	leeq1d	~c <= a -> (a - c <= b - c <-> 0 <= b - c)
16	13, 15	mpbiri	~c <= a -> a - c <= b - c
17	16	a1d	~c <= a -> a <= b -> a - c <= b - c
18	12, 17	cases	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_set (elab, ax_8), axs_the (theid, the0), axs_peano (peano2, peano5, addeq, add0, addS)