Theorem shl12 ≪ | index | src | ≫

theorem shl12 (a: nat): $ shl a 1 = b0 a $;

Step	Hyp	Ref	Expression
1		eqtr	shl a 1 = a * 2 -> a * 2 = b0 a -> shl a 1 = b0 a
2		muleq2	2 ^ 1 = 2 -> a * 2 ^ 1 = a * 2
3	2	conv shl	2 ^ 1 = 2 -> shl a 1 = a * 2
4		pow12	2 ^ 1 = 2
5	3, 4	ax_mp	shl a 1 = a * 2
6	1, 5	ax_mp	a * 2 = b0 a -> shl a 1 = b0 a
7		b0mul22	a * 2 = b0 a
8	6, 7	ax_mp	shl a 1 = b0 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)