Theorem pow12 ≪ | index | src | ≫

theorem pow12 (a: nat): $ a ^ 1 = a $;

Step	Hyp	Ref	Expression
1		eqtr	a ^ 1 = a * a ^ 0 -> a * a ^ 0 = a -> a ^ 1 = a
2		powS	a ^ suc 0 = a * a ^ 0
3	2	conv d1	a ^ 1 = a * a ^ 0
4	1, 3	ax_mp	a * a ^ 0 = a -> a ^ 1 = a
5		eqtr	a * a ^ 0 = a * 1 -> a * 1 = a -> a * a ^ 0 = a
6		muleq2	a ^ 0 = 1 -> a * a ^ 0 = a * 1
7		pow0	a ^ 0 = 1
8	6, 7	ax_mp	a * a ^ 0 = a * 1
9	5, 8	ax_mp	a * 1 = a -> a * a ^ 0 = a
10		mul12	a * 1 = a
11	9, 10	ax_mp	a * a ^ 0 = a
12	4, 11	ax_mp	a ^ 1 = 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, muleq, add0, addS, mul0, mulS)