Theorem znpncan ≪ | index | src | ≫

theorem znpncan (a b c: nat): $ a -Z b +Z (b -Z c) = a -Z c $;

Step	Hyp	Ref	Expression
1		eqtr3	a -Z b +Z b -Z c = a -Z b +Z (b -Z c) -> a -Z b +Z b -Z c = a -Z c -> a -Z b +Z (b -Z c) = a -Z c
2		zaddsubass	a -Z b +Z b -Z c = a -Z b +Z (b -Z c)
3	1, 2	ax_mp	a -Z b +Z b -Z c = a -Z c -> a -Z b +Z (b -Z c) = a -Z c
4		zsubeq1	a -Z b +Z b = a -> a -Z b +Z b -Z c = a -Z c
5		znpcan	a -Z b +Z b = a
6	4, 5	ax_mp	a -Z b +Z b -Z c = a -Z c
7	3, 6	ax_mp	a -Z b +Z (b -Z c) = a -Z 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)