Theorem lmemssn ≪ | index | src | ≫

theorem lmemssn (a: nat): $ lmems (a : 0) = sn a $;

Step	Hyp	Ref	Expression
1		eqtr	lmems (a : 0) = a ; lmems 0 -> a ; lmems 0 = sn a -> lmems (a : 0) = sn a
2		lmemsS	lmems (a : 0) = a ; lmems 0
3	1, 2	ax_mp	a ; lmems 0 = sn a -> lmems (a : 0) = sn a
4		eqtr	a ; lmems 0 = a ; 0 -> a ; 0 = sn a -> a ; lmems 0 = sn a
5		inseq2	lmems 0 = 0 -> a ; lmems 0 = a ; 0
6		lmems0	lmems 0 = 0
7	5, 6	ax_mp	a ; lmems 0 = a ; 0
8	4, 7	ax_mp	a ; 0 = sn a -> a ; lmems 0 = sn a
9		ins02	a ; 0 = sn a
10	8, 9	ax_mp	a ; lmems 0 = sn a
11	3, 10	ax_mp	lmems (a : 0) = sn 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)