Theorem zaddrass | index | src |

theorem zaddrass (a b c: nat): $ a +Z b +Z c = a +Z c +Z b $;
StepHypRefExpression
1 eqtr4
a +Z b +Z c = a +Z (b +Z c) -> a +Z c +Z b = a +Z (b +Z c) -> a +Z b +Z c = a +Z c +Z b
2 zaddass
a +Z b +Z c = a +Z (b +Z c)
3 1, 2 ax_mp
a +Z c +Z b = a +Z (b +Z c) -> a +Z b +Z c = a +Z c +Z b
4 eqtr4
a +Z c +Z b = a +Z (c +Z b) -> a +Z (b +Z c) = a +Z (c +Z b) -> a +Z c +Z b = a +Z (b +Z c)
5 zaddass
a +Z c +Z b = a +Z (c +Z b)
6 4, 5 ax_mp
a +Z (b +Z c) = a +Z (c +Z b) -> a +Z c +Z b = a +Z (b +Z c)
7 zaddeq2
b +Z c = c +Z b -> a +Z (b +Z c) = a +Z (c +Z b)
8 zaddcom
b +Z c = c +Z b
9 7, 8 ax_mp
a +Z (b +Z c) = a +Z (c +Z b)
10 6, 9 ax_mp
a +Z c +Z b = a +Z (b +Z c)
11 3, 10 ax_mp
a +Z b +Z c = a +Z c +Z 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_set (elab, ax_8), axs_the (theid, the0), axs_peano (peano1, peano2, peano5, addeq, muleq, add0, addS, mul0, mulS)