Theorem ltappend2 | index | src |

theorem ltappend2 (a b c: nat): $ b < c <-> a ++ b < a ++ c $;
StepHypRefExpression
1 bitr4
(b < c <-> ~c <= b) -> (a ++ b < a ++ c <-> ~c <= b) -> (b < c <-> a ++ b < a ++ c)
2 ltnle
b < c <-> ~c <= b
3 1, 2 ax_mp
(a ++ b < a ++ c <-> ~c <= b) -> (b < c <-> a ++ b < a ++ c)
4 bitr4
(a ++ b < a ++ c <-> ~a ++ c <= a ++ b) -> (~c <= b <-> ~a ++ c <= a ++ b) -> (a ++ b < a ++ c <-> ~c <= b)
5 ltnle
a ++ b < a ++ c <-> ~a ++ c <= a ++ b
6 4, 5 ax_mp
(~c <= b <-> ~a ++ c <= a ++ b) -> (a ++ b < a ++ c <-> ~c <= b)
7 noteq
(c <= b <-> a ++ c <= a ++ b) -> (~c <= b <-> ~a ++ c <= a ++ b)
8 leappend2
c <= b <-> a ++ c <= a ++ b
9 7, 8 ax_mp
~c <= b <-> ~a ++ c <= a ++ b
10 6, 9 ax_mp
a ++ b < a ++ c <-> ~c <= b
11 3, 10 ax_mp
b < c <-> a ++ b < a ++ 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)