Theorem subaddmin ≪ | index | src | ≫ 

theorem subaddmin (a b: nat): $ a - b + min a b = a $;
StepHypRefExpression
1 eor 
(b <= a -> a - b + min a b = a) -> (a <= b -> a - b + min a b = a) -> b <= a \/ a <= b -> a - b + min a b = a
2 eqmin2 
b <= a -> min a b = b
3 2 addeq2d 
b <= a -> a - b + min a b = a - b + b
4 npcan 
b <= a -> a - b + b = a
5 3, 4 eqtrd 
b <= a -> a - b + min a b = a
6 1, 5 ax_mp 
(a <= b -> a - b + min a b = a) -> b <= a \/ a <= b -> a - b + min a b = a
7 lesubeq0 
a <= b <-> a - b = 0
8 addeq1 
a - b = 0 -> a - b + min a b = 0 + min a b
9 7, 8 sylbi 
a <= b -> a - b + min a b = 0 + min a b
10 add01 
0 + min a b = min a b
11 eqmin1 
a <= b -> min a b = a
12 10, 11 syl5eq 
a <= b -> 0 + min a b = a
13 9, 12 eqtrd 
a <= b -> a - b + min a b = a
14 6, 13 ax_mp 
b <= a \/ a <= b -> a - b + min a b = a
15 leorle 
b <= a \/ a <= b
16 14, 15 ax_mp 
a - b + min a b = a

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, add0, addS)