Theorem zaddmul ≪ | index | src | ≫ 

theorem zaddmul (a b c: nat): $ (a +Z b) *Z c = a *Z c +Z b *Z c $;
StepHypRefExpression
1 eqtr 
(a +Z b) *Z c = c *Z (a +Z b) -> c *Z (a +Z b) = a *Z c +Z b *Z c -> (a +Z b) *Z c = a *Z c +Z b *Z c
2 zmulcom 
(a +Z b) *Z c = c *Z (a +Z b)
3 1, 2 ax_mp 
c *Z (a +Z b) = a *Z c +Z b *Z c -> (a +Z b) *Z c = a *Z c +Z b *Z c
4 eqtr 
c *Z (a +Z b) = c *Z a +Z c *Z b -> c *Z a +Z c *Z b = a *Z c +Z b *Z c -> c *Z (a +Z b) = a *Z c +Z b *Z c
5 zmuladd 
c *Z (a +Z b) = c *Z a +Z c *Z b
6 4, 5 ax_mp 
c *Z a +Z c *Z b = a *Z c +Z b *Z c -> c *Z (a +Z b) = a *Z c +Z b *Z c
7 zaddeq 
c *Z a = a *Z c -> c *Z b = b *Z c -> c *Z a +Z c *Z b = a *Z c +Z b *Z c
8 zmulcom 
c *Z a = a *Z c
9 7, 8 ax_mp 
c *Z b = b *Z c -> c *Z a +Z c *Z b = a *Z c +Z b *Z c
10 zmulcom 
c *Z b = b *Z c
11 9, 10 ax_mp 
c *Z a +Z c *Z b = a *Z c +Z b *Z c
12 6, 11 ax_mp 
c *Z (a +Z b) = a *Z c +Z b *Z c
13 3, 12 ax_mp 
(a +Z b) *Z c = a *Z c +Z b *Z c

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)