Theorem mulmoddir ≪ | index | src | ≫ 

theorem mulmoddir (a b c: nat): $ a % b * c = a * c % (b * c) $;
StepHypRefExpression
1 eqtr 
a % b * c = c * (a % b) -> c * (a % b) = a * c % (b * c) -> a % b * c = a * c % (b * c)
2 mulcom 
a % b * c = c * (a % b)
3 1, 2 ax_mp 
c * (a % b) = a * c % (b * c) -> a % b * c = a * c % (b * c)
4 eqtr 
c * (a % b) = c * a % (c * b) -> c * a % (c * b) = a * c % (b * c) -> c * (a % b) = a * c % (b * c)
5 mulmoddi 
c * (a % b) = c * a % (c * b)
6 4, 5 ax_mp 
c * a % (c * b) = a * c % (b * c) -> c * (a % b) = a * c % (b * c)
7 modeq 
c * a = a * c -> c * b = b * c -> c * a % (c * b) = a * c % (b * c)
8 mulcom 
c * a = a * c
9 7, 8 ax_mp 
c * b = b * c -> c * a % (c * b) = a * c % (b * c)
10 mulcom 
c * b = b * c
11 9, 10 ax_mp 
c * a % (c * b) = a * c % (b * c)
12 6, 11 ax_mp 
c * (a % b) = a * c % (b * c)
13 3, 12 ax_mp 
a % b * c = a * c % (b * 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)