Theorem divlemul1r ≪ | index | src | ≫

theorem divlemul1r (a b c: nat): $ b != 0 -> a <= b * c -> a // b <= c $;

Step	Hyp	Ref	Expression
1		muldiv2	b != 0 -> b * c // b = c
2	1	anwl	b != 0 /\ a <= b * c -> b * c // b = c
3	2	leeq2d	b != 0 /\ a <= b * c -> (a // b <= b * c // b <-> a // b <= c)
4		lediv1	a <= b * c -> a // b <= b * c // b
5	4	anwr	b != 0 /\ a <= b * c -> a // b <= b * c // b
6	3, 5	mpbid	b != 0 /\ a <= b * c -> a // b <= c
7	6	exp	b != 0 -> a <= b * c -> a // 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 (peano1, peano2, peano5, addeq, muleq, add0, addS, mul0, mulS)