Theorem lepow2 ≪ | index | src | ≫

theorem lepow2 (a b c: nat): $ 1 < a -> (b <= c <-> a ^ b <= a ^ c) $;

Step	Hyp	Ref	Expression
1		lenlt	b <= c <-> ~c < b
2		lenlt	a ^ b <= a ^ c <-> ~a ^ c < a ^ b
3		ltpow2	1 < a -> (c < b <-> a ^ c < a ^ b)
4	3	noteqd	1 < a -> (~c < b <-> ~a ^ c < a ^ b)
5	2, 4	syl6bbr	1 < a -> (~c < b <-> a ^ b <= a ^ c)
6	1, 5	syl5bb	1 < a -> (b <= c <-> a ^ b <= a ^ 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)