Theorem zaddcom ≪ | index | src | ≫

theorem zaddcom (a b: nat): $ a +Z b = b +Z a $;


zfst a + zfst b = zfst b + zfst a -> zsnd a + zsnd b = zsnd b + zsnd a -> zfst a + zfst b -ZN (zsnd a + zsnd b) = zfst b + zfst a -ZN (zsnd b + zsnd a)
zfst a + zfst b = zfst b + zfst a -> zsnd a + zsnd b = zsnd b + zsnd a -> a +Z b = b +Z a
zfst a + zfst b = zfst b + zfst a
zsnd a + zsnd b = zsnd b + zsnd a -> a +Z b = b +Z a
zsnd a + zsnd b = zsnd b + zsnd a
a +Z b = b +Z a

Step	Hyp	Ref	Expression
1		znsubeq	zfst a + zfst b = zfst b + zfst a -> zsnd a + zsnd b = zsnd b + zsnd a -> zfst a + zfst b -ZN (zsnd a + zsnd b) = zfst b + zfst a -ZN (zsnd b + zsnd a)
2	1	conv zadd	zfst a + zfst b = zfst b + zfst a -> zsnd a + zsnd b = zsnd b + zsnd a -> a +Z b = b +Z a
3		addcom	zfst a + zfst b = zfst b + zfst a
4	2, 3	ax_mp	zsnd a + zsnd b = zsnd b + zsnd a -> a +Z b = b +Z a
5		addcom	zsnd a + zsnd b = zsnd b + zsnd a
6	4, 5	ax_mp	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 (peano2, peano5, addeq, add0, addS)