Theorem zfstsndeq0 ≪ | index | src | ≫

theorem zfstsndeq0 (n: nat): $ zfst n = 0 /\ zsnd n = 0 <-> n = 0 $;

Step	Hyp	Ref	Expression
1		zfstsnd	zfst n -ZN zsnd n = n
2		znsubid	0 -ZN 0 = 0
3		anl	zfst n = 0 /\ zsnd n = 0 -> zfst n = 0
4		anr	zfst n = 0 /\ zsnd n = 0 -> zsnd n = 0
5	3, 4	znsubeqd	zfst n = 0 /\ zsnd n = 0 -> zfst n -ZN zsnd n = 0 -ZN 0
6	2, 5	syl6eq	zfst n = 0 /\ zsnd n = 0 -> zfst n -ZN zsnd n = 0
7	1, 6	syl5eqr	zfst n = 0 /\ zsnd n = 0 -> n = 0
8		zfst0	zfst 0 = 0
9		zfsteq	n = 0 -> zfst n = zfst 0
10	8, 9	syl6eq	n = 0 -> zfst n = 0
11		zsnd0	zsnd 0 = 0
12		zsndeq	n = 0 -> zsnd n = zsnd 0
13	11, 12	syl6eq	n = 0 -> zsnd n = 0
14	10, 13	iand	n = 0 -> zfst n = 0 /\ zsnd n = 0
15	7, 14	ibii	zfst n = 0 /\ zsnd 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)