Theorem eqmin1 ≪ | index | src | ≫ 

theorem eqmin1 (a b: nat): $ a <= b -> min a b = a $;
StepHypRefExpression
1 leloe 
a <= b <-> a < b \/ a = b
2 eor 
(a < b -> min a b = a) -> (a = b -> min a b = a) -> a < b \/ a = b -> min a b = a
3 ifpos 
a < b -> if (a < b) a b = a
4 3 conv min 
a < b -> min a b = a
5 2, 4 ax_mp 
(a = b -> min a b = a) -> a < b \/ a = b -> min a b = a
6 ifid 
if (a < b) a a = a
7 eqcom 
a = b -> b = a
8 7 ifeq3d 
a = b -> if (a < b) a b = if (a < b) a a
9 8 conv min 
a = b -> min a b = if (a < b) a a
10 6, 9 syl6eq 
a = b -> min a b = a
11 5, 10 ax_mp 
a < b \/ a = b -> min a b = a
12 1, 11 sylbi 
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)