Theorem append02 ≪ | index | src | ≫

theorem append02 (a: nat): $ a ++ 0 = a $;

Step	Hyp	Ref	Expression
1		id	_1 = a -> _1 = a
2	1	appendeq1d	_1 = a -> _1 ++ 0 = a ++ 0
3	2, 1	eqeqd	_1 = a -> (_1 ++ 0 = _1 <-> a ++ 0 = a)
4		id	_1 = 0 -> _1 = 0
5	4	appendeq1d	_1 = 0 -> _1 ++ 0 = 0 ++ 0
6	5, 4	eqeqd	_1 = 0 -> (_1 ++ 0 = _1 <-> 0 ++ 0 = 0)
7		id	_1 = a2 -> _1 = a2
8	7	appendeq1d	_1 = a2 -> _1 ++ 0 = a2 ++ 0
9	8, 7	eqeqd	_1 = a2 -> (_1 ++ 0 = _1 <-> a2 ++ 0 = a2)
10		id	_1 = a1 : a2 -> _1 = a1 : a2
11	10	appendeq1d	_1 = a1 : a2 -> _1 ++ 0 = a1 : a2 ++ 0
12	11, 10	eqeqd	_1 = a1 : a2 -> (_1 ++ 0 = _1 <-> a1 : a2 ++ 0 = a1 : a2)
13		append0	0 ++ 0 = 0
14		appendS	a1 : a2 ++ 0 = a1 : (a2 ++ 0)
15		conseq2	a2 ++ 0 = a2 -> a1 : (a2 ++ 0) = a1 : a2
16	14, 15	syl5eq	a2 ++ 0 = a2 -> a1 : a2 ++ 0 = a1 : a2
17	3, 6, 9, 12, 13, 16	listind	a ++ 0 = a

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)