Theorem zmul01 | index | src |

theorem zmul01 (a: nat): $ 0 *Z a = 0 $;
StepHypRefExpression
1 eqtr3
(0 -Z 0) *Z a = 0 *Z a -> (0 -Z 0) *Z a = 0 -> 0 *Z a = 0
2 zmuleq1
0 -Z 0 = 0 -> (0 -Z 0) *Z a = 0 *Z a
3 zsubid
0 -Z 0 = 0
4 2, 3 ax_mp
(0 -Z 0) *Z a = 0 *Z a
5 1, 4 ax_mp
(0 -Z 0) *Z a = 0 -> 0 *Z a = 0
6 eqtr
(0 -Z 0) *Z a = 0 *Z a -Z 0 *Z a -> 0 *Z a -Z 0 *Z a = 0 -> (0 -Z 0) *Z a = 0
7 zsubmul
(0 -Z 0) *Z a = 0 *Z a -Z 0 *Z a
8 6, 7 ax_mp
0 *Z a -Z 0 *Z a = 0 -> (0 -Z 0) *Z a = 0
9 zsubid
0 *Z a -Z 0 *Z a = 0
10 8, 9 ax_mp
(0 -Z 0) *Z a = 0
11 5, 10 ax_mp
0 *Z a = 0

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)