Theorem eqmaddlem ≪ | index | src | ≫

theorem eqmaddlem (a b c n: nat):
  $ a <= b -> (mod(n): a + c = b + c <-> mod(n): a = b) $;

Step	Hyp	Ref	Expression
1		leadd1	a <= b <-> a + c <= b + c
2		eqmdvdsub	a + c <= b + c -> (mod(n): a + c = b + c <-> n \|\| b + c - (a + c))
3	1, 2	sylbi	a <= b -> (mod(n): a + c = b + c <-> n \|\| b + c - (a + c))
4		dvdeq2	b + c - (a + c) = b - a -> (n \|\| b + c - (a + c) <-> n \|\| b - a)
5		pnpcan2	b + c - (a + c) = b - a
6	4, 5	ax_mp	n \|\| b + c - (a + c) <-> n \|\| b - a
7		eqmdvdsub	a <= b -> (mod(n): a = b <-> n \|\| b - a)
8	6, 7	syl6bbr	a <= b -> (mod(n): a = b <-> n \|\| b + c - (a + c))
9	3, 8	bitr4d	a <= b -> (mod(n): a + c = b + c <-> mod(n): 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_set (elab, ax_8), axs_the (theid, the0), axs_peano (peano1, peano2, peano5, addeq, muleq, add0, addS, mul0, mulS)