Theorem znpnpcan1 ≪ | index | src | ≫

theorem znpnpcan1 (a b c: nat): $ a + c -ZN (b + c) = a -ZN b $;

Step	Hyp	Ref	Expression
1		zneqb	a + c -ZN (b + c) = a -ZN b <-> a + c + b = a + (b + c)
2		eqtr	a + c + b = a + b + c -> a + b + c = a + (b + c) -> a + c + b = a + (b + c)
3		addrass	a + c + b = a + b + c
4	2, 3	ax_mp	a + b + c = a + (b + c) -> a + c + b = a + (b + c)
5		addass	a + b + c = a + (b + c)
6	4, 5	ax_mp	a + c + b = a + (b + c)
7	1, 6	mpbir	a + c -ZN (b + c) = a -ZN 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)