Theorem zabseq0 ≪ | index | src | ≫

theorem zabseq0 (n: nat): $ zabs n = 0 <-> n = 0 $;

Step	Hyp	Ref	Expression
1		bitr	(zabs n = 0 <-> zfst n = 0 /\ zsnd n = 0) -> (zfst n = 0 /\ zsnd n = 0 <-> n = 0) -> (zabs n = 0 <-> n = 0)
2		addeq0	zfst n + zsnd n = 0 <-> zfst n = 0 /\ zsnd n = 0
3	2	conv zabs	zabs n = 0 <-> zfst n = 0 /\ zsnd n = 0
4	1, 3	ax_mp	(zfst n = 0 /\ zsnd n = 0 <-> n = 0) -> (zabs n = 0 <-> n = 0)
5		zfstsndeq0	zfst n = 0 /\ zsnd n = 0 <-> n = 0
6	4, 5	ax_mp	zabs n = 0 <-> n = 0

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)