Theorem zleasymb ≪ | index | src | ≫ 

theorem zleasymb (a b: nat): $ a = b <-> a <=Z b /\ b <=Z a $;
StepHypRefExpression
1 bitr3 
(a -Z b = 0 <-> a = b) -> (a -Z b = 0 <-> a <=Z b /\ b <=Z a) -> (a = b <-> a <=Z b /\ b <=Z a)
2 zsubeq0 
a -Z b = 0 <-> a = b
3 1, 2 ax_mp 
(a -Z b = 0 <-> a <=Z b /\ b <=Z a) -> (a = b <-> a <=Z b /\ b <=Z a)
4 bitr3 
(zfst (a -Z b) = 0 /\ zsnd (a -Z b) = 0 <-> a -Z b = 0) ->
  (zfst (a -Z b) = 0 /\ zsnd (a -Z b) = 0 <-> a <=Z b /\ b <=Z a) ->
  (a -Z b = 0 <-> a <=Z b /\ b <=Z a)
5 zfstsndeq0 
zfst (a -Z b) = 0 /\ zsnd (a -Z b) = 0 <-> a -Z b = 0
6 4, 5 ax_mp 
(zfst (a -Z b) = 0 /\ zsnd (a -Z b) = 0 <-> a <=Z b /\ b <=Z a) -> (a -Z b = 0 <-> a <=Z b /\ b <=Z a)
7 bitr 
(zfst (a -Z b) = 0 /\ zsnd (a -Z b) = 0 <-> a -Z b <=Z 0 /\ 0 <=Z a -Z b) ->
  (a -Z b <=Z 0 /\ 0 <=Z a -Z b <-> a <=Z b /\ b <=Z a) ->
  (zfst (a -Z b) = 0 /\ zsnd (a -Z b) = 0 <-> a <=Z b /\ b <=Z a)
8 aneq 
(zfst (a -Z b) = 0 <-> a -Z b <=Z 0) -> (zsnd (a -Z b) = 0 <-> 0 <=Z a -Z b) -> (zfst (a -Z b) = 0 /\ zsnd (a -Z b) = 0 <-> a -Z b <=Z 0 /\ 0 <=Z a -Z b)
9 zfsteq0 
zfst (a -Z b) = 0 <-> a -Z b <=Z 0
10 8, 9 ax_mp 
(zsnd (a -Z b) = 0 <-> 0 <=Z a -Z b) -> (zfst (a -Z b) = 0 /\ zsnd (a -Z b) = 0 <-> a -Z b <=Z 0 /\ 0 <=Z a -Z b)
11 zsndeq0 
zsnd (a -Z b) = 0 <-> 0 <=Z a -Z b
12 10, 11 ax_mp 
zfst (a -Z b) = 0 /\ zsnd (a -Z b) = 0 <-> a -Z b <=Z 0 /\ 0 <=Z a -Z b
13 7, 12 ax_mp 
(a -Z b <=Z 0 /\ 0 <=Z a -Z b <-> a <=Z b /\ b <=Z a) -> (zfst (a -Z b) = 0 /\ zsnd (a -Z b) = 0 <-> a <=Z b /\ b <=Z a)
14 aneq 
(a -Z b <=Z 0 <-> a <=Z b) -> (0 <=Z a -Z b <-> b <=Z a) -> (a -Z b <=Z 0 /\ 0 <=Z a -Z b <-> a <=Z b /\ b <=Z a)
15 zlesub0 
a -Z b <=Z 0 <-> a <=Z b
16 14, 15 ax_mp 
(0 <=Z a -Z b <-> b <=Z a) -> (a -Z b <=Z 0 /\ 0 <=Z a -Z b <-> a <=Z b /\ b <=Z a)
17 zle0sub 
0 <=Z a -Z b <-> b <=Z a
18 16, 17 ax_mp 
a -Z b <=Z 0 /\ 0 <=Z a -Z b <-> a <=Z b /\ b <=Z a
19 13, 18 ax_mp 
zfst (a -Z b) = 0 /\ zsnd (a -Z b) = 0 <-> a <=Z b /\ b <=Z a
20 6, 19 ax_mp 
a -Z b = 0 <-> a <=Z b /\ b <=Z a
21 3, 20 ax_mp 
a = b <-> a <=Z b /\ b <=Z 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, muleq, add0, addS, mul0, mulS)