Theorem znegsub ≪ | index | src | ≫

theorem znegsub (a b: nat): $ -uZ (a -Z b) = b -Z a $;

Step	Hyp	Ref	Expression
1		eqtr	-uZ (a -Z b) = -uZ a -Z -uZ b -> -uZ a -Z -uZ b = b -Z a -> -uZ (a -Z b) = b -Z a
2		znegadd2	-uZ (a +Z -uZ b) = -uZ a -Z -uZ b
3	2	conv zsub	-uZ (a -Z b) = -uZ a -Z -uZ b
4	1, 3	ax_mp	-uZ a -Z -uZ b = b -Z a -> -uZ (a -Z b) = b -Z a
5		eqtr	-uZ a -Z -uZ b = -uZ -uZ b +Z -uZ a -> -uZ -uZ b +Z -uZ a = b -Z a -> -uZ a -Z -uZ b = b -Z a
6		zaddcom	-uZ a +Z -uZ -uZ b = -uZ -uZ b +Z -uZ a
7	6	conv zsub	-uZ a -Z -uZ b = -uZ -uZ b +Z -uZ a
8	5, 7	ax_mp	-uZ -uZ b +Z -uZ a = b -Z a -> -uZ a -Z -uZ b = b -Z a
9		zaddeq1	-uZ -uZ b = b -> -uZ -uZ b +Z -uZ a = b +Z -uZ a
10	9	conv zsub	-uZ -uZ b = b -> -uZ -uZ b +Z -uZ a = b -Z a
11		znegneg	-uZ -uZ b = b
12	10, 11	ax_mp	-uZ -uZ b +Z -uZ a = b -Z a
13	8, 12	ax_mp	-uZ a -Z -uZ b = b -Z a
14	4, 13	ax_mp	-uZ (a -Z b) = b -Z 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)