Theorem znegeqcom ≪ | index | src | ≫

theorem znegeqcom (a b: nat): $ -uZ a = b <-> -uZ b = a $;

Step	Hyp	Ref	Expression
1		bitr4	(-uZ a = b <-> a +Z b = 0) -> (-uZ b = a <-> a +Z b = 0) -> (-uZ a = b <-> -uZ b = a)
2		eqzneg	-uZ a = b <-> a +Z b = 0
3	2	bitr4*	(-uZ b = a <-> a +Z b = 0) -> (-uZ a = b <-> -uZ b = a)
4		bitr4	(-uZ b = a <-> b +Z a = 0) -> (a +Z b = 0 <-> b +Z a = 0) -> (-uZ b = a <-> a +Z b = 0)
5		eqzneg	-uZ b = a <-> b +Z a = 0
6	5	bitr4*	(a +Z b = 0 <-> b +Z a = 0) -> (-uZ b = a <-> a +Z b = 0)
7		eqeq1	a +Z b = b +Z a -> (a +Z b = 0 <-> b +Z a = 0)
8		zaddcom	a +Z b = b +Z a
9	8	eqeq1*	a +Z b = 0 <-> b +Z a = 0
10	5, 9	bitr4*	-uZ b = a <-> a +Z b = 0
11	2, 10	bitr4*	-uZ a = b <-> -uZ b = 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)