Theorem zletr ≪ | index | src | ≫ 

theorem zletr (a b c: nat): $ a <=Z b -> b <=Z c -> a <=Z c $;
StepHypRefExpression
1 zle0sub 
0 <=Z b -Z a <-> a <=Z b
2 zle0sub 
0 <=Z c -Z b <-> b <=Z c
3 bitr 
(0 <=Z b -Z a +Z (c -Z b) <-> 0 <=Z c -Z a) -> (0 <=Z c -Z a <-> a <=Z c) -> (0 <=Z b -Z a +Z (c -Z b) <-> a <=Z c)
4 zleeq2 
b -Z a +Z (c -Z b) = c -Z a -> (0 <=Z b -Z a +Z (c -Z b) <-> 0 <=Z c -Z a)
5 znpncan2 
b -Z a +Z (c -Z b) = c -Z a
6 4, 5 ax_mp 
0 <=Z b -Z a +Z (c -Z b) <-> 0 <=Z c -Z a
7 3, 6 ax_mp 
(0 <=Z c -Z a <-> a <=Z c) -> (0 <=Z b -Z a +Z (c -Z b) <-> a <=Z c)
8 zle0sub 
0 <=Z c -Z a <-> a <=Z c
9 7, 8 ax_mp 
0 <=Z b -Z a +Z (c -Z b) <-> a <=Z c
10 zaddpos 
0 <=Z b -Z a -> 0 <=Z c -Z b -> 0 <=Z b -Z a +Z (c -Z b)
11 9, 10 syl6ib 
0 <=Z b -Z a -> 0 <=Z c -Z b -> a <=Z c
12 2, 11 syl5bir 
0 <=Z b -Z a -> b <=Z c -> a <=Z c
13 1, 12 sylbir 
a <=Z b -> b <=Z c -> a <=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)