Theorem shlshl | index | src |

theorem shlshl (a b c: nat): $ shl (shl a b) c = shl a (b + c) $;
StepHypRefExpression
2
a * 2 ^ b * 2 ^ c = a * (2 ^ b * 2 ^ c)
3
conv shl
shl (shl a b) c = a * (2 ^ b * 2 ^ c)
5
2 ^ (b + c) = 2 ^ b * 2 ^ c -> a * 2 ^ (b + c) = a * (2 ^ b * 2 ^ c)
6
conv shl
2 ^ (b + c) = 2 ^ b * 2 ^ c -> shl a (b + c) = a * (2 ^ b * 2 ^ c)
7
2 ^ (b + c) = 2 ^ b * 2 ^ c
8
6, 7
shl a (b + c) = a * (2 ^ b * 2 ^ c)
9
3, 8
shl (shl a b) c = shl 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)