Theorem zmodeqmod ≪ | index | src | ≫

theorem zmodeqmod (a b n: nat):
  $ n != 0 -> (a %Z n = b %Z n <-> modZ(n): a = b) $;

Step	Hyp	Ref	Expression
1		eqzeqm	b0 (a %Z n) = b0 (b %Z n) -> modZ(n): b0 (a %Z n) = b0 (b %Z n)
2		b0eq	a %Z n = b %Z n -> b0 (a %Z n) = b0 (b %Z n)
3	1, 2	syl	a %Z n = b %Z n -> modZ(n): b0 (a %Z n) = b0 (b %Z n)
4		zmodmodid	a %Z n % n = a %Z n
5		zmodmodid	b %Z n % n = b %Z n
6		zeqmeqm	modZ(n): b0 (a %Z n) = b0 (b %Z n) <-> mod(n): a %Z n = b %Z n
7	6	conv eqm	modZ(n): b0 (a %Z n) = b0 (b %Z n) <-> a %Z n % n = b %Z n % n
8	7	bi1i	modZ(n): b0 (a %Z n) = b0 (b %Z n) -> a %Z n % n = b %Z n % n
9	4, 5, 8	eqtr3g	modZ(n): b0 (a %Z n) = b0 (b %Z n) -> a %Z n = b %Z n
10	3, 9	ibii	a %Z n = b %Z n <-> modZ(n): b0 (a %Z n) = b0 (b %Z n)
11		zeqmmod	n != 0 -> modZ(n): b0 (a %Z n) = a
12		zeqmmod	n != 0 -> modZ(n): b0 (b %Z n) = b
13	11, 12	zeqmeqm23d	n != 0 -> (modZ(n): b0 (a %Z n) = b0 (b %Z n) <-> modZ(n): a = b)
14	10, 13	syl5bb	n != 0 -> (a %Z n = b %Z n <-> modZ(n): a = b)

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)