Theorem addb01 | index | src |

theorem addb01 (a b: nat): $ b0 a + b1 b = b1 (a + b) $;
StepHypRefExpression
1 eqtr
b0 a + b1 b = suc (b0 a + b0 b) -> suc (b0 a + b0 b) = b1 (a + b) -> b0 a + b1 b = b1 (a + b)
2 addS
b0 a + suc (b0 b) = suc (b0 a + b0 b)
3 2 conv b1
b0 a + b1 b = suc (b0 a + b0 b)
4 1, 3 ax_mp
suc (b0 a + b0 b) = b1 (a + b) -> b0 a + b1 b = b1 (a + b)
5 suceq
b0 a + b0 b = b0 (a + b) -> suc (b0 a + b0 b) = suc (b0 (a + b))
6 5 conv b1
b0 a + b0 b = b0 (a + b) -> suc (b0 a + b0 b) = b1 (a + b)
7 addb00
b0 a + b0 b = b0 (a + b)
8 6, 7 ax_mp
suc (b0 a + b0 b) = b1 (a + b)
9 4, 8 ax_mp
b0 a + b1 b = b1 (a + b)

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_peano (peano2, peano5, addeq, add0, addS)