Theorem zsndznsub ≪ | index | src | ≫

theorem zsndznsub (m n: nat): $ zsnd (m -ZN n) = n - m $;

Step	Hyp	Ref	Expression
1		eqtr3	zfst (-uZ (m -ZN n)) = zsnd (m -ZN n) -> zfst (-uZ (m -ZN n)) = n - m -> zsnd (m -ZN n) = n - m
2		zfstneg	zfst (-uZ (m -ZN n)) = zsnd (m -ZN n)
3	2	eqtr3*	zfst (-uZ (m -ZN n)) = n - m -> zsnd (m -ZN n) = n - m
4		eqtr	zfst (-uZ (m -ZN n)) = zfst (n -ZN m) -> zfst (n -ZN m) = n - m -> zfst (-uZ (m -ZN n)) = n - m
5		zfsteq	-uZ (m -ZN n) = n -ZN m -> zfst (-uZ (m -ZN n)) = zfst (n -ZN m)
6		znegzn	-uZ (m -ZN n) = n -ZN m
7	6	zfsteq*	zfst (-uZ (m -ZN n)) = zfst (n -ZN m)
8	7	eqtr*	zfst (n -ZN m) = n - m -> zfst (-uZ (m -ZN n)) = n - m
9		zfstznsub	zfst (n -ZN m) = n - m
10	7, 9	eqtr*	zfst (-uZ (m -ZN n)) = n - m
11	2, 10	eqtr3*	zsnd (m -ZN n) = n - m

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)