Theorem allappend ≪ | index | src | ≫ 

theorem allappend (A: set) (a b: nat):
  $ all A (a ++ b) <-> all A a /\ all A b $;
StepHypRefExpression
1 bitr4 
(all A (a ++ b) <-> A. x (x IN a ++ b -> x e. A)) -> (all A a /\ all A b <-> A. x (x IN a ++ b -> x e. A)) -> (all A (a ++ b) <-> all A a /\ all A b)
2 allal2 
all A (a ++ b) <-> A. x (x IN a ++ b -> x e. A)
3 1, 2 ax_mp 
(all A a /\ all A b <-> A. x (x IN a ++ b -> x e. A)) -> (all A (a ++ b) <-> all A a /\ all A b)
4 bitr4 
(all A a /\ all A b <-> A. x (x IN a -> x e. A) /\ A. x (x IN b -> x e. A)) ->
  (A. x (x IN a ++ b -> x e. A) <-> A. x (x IN a -> x e. A) /\ A. x (x IN b -> x e. A)) ->
  (all A a /\ all A b <-> A. x (x IN a ++ b -> x e. A))
5 aneq 
(all A a <-> A. x (x IN a -> x e. A)) -> (all A b <-> A. x (x IN b -> x e. A)) -> (all A a /\ all A b <-> A. x (x IN a -> x e. A) /\ A. x (x IN b -> x e. A))
6 allal2 
all A a <-> A. x (x IN a -> x e. A)
7 5, 6 ax_mp 
(all A b <-> A. x (x IN b -> x e. A)) -> (all A a /\ all A b <-> A. x (x IN a -> x e. A) /\ A. x (x IN b -> x e. A))
8 allal2 
all A b <-> A. x (x IN b -> x e. A)
9 7, 8 ax_mp 
all A a /\ all A b <-> A. x (x IN a -> x e. A) /\ A. x (x IN b -> x e. A)
10 4, 9 ax_mp 
(A. x (x IN a ++ b -> x e. A) <-> A. x (x IN a -> x e. A) /\ A. x (x IN b -> x e. A)) -> (all A a /\ all A b <-> A. x (x IN a ++ b -> x e. A))
11 bitr 
(A. x (x IN a ++ b -> x e. A) <-> A. x ((x IN a -> x e. A) /\ (x IN b -> x e. A))) ->
  (A. x ((x IN a -> x e. A) /\ (x IN b -> x e. A)) <-> A. x (x IN a -> x e. A) /\ A. x (x IN b -> x e. A)) ->
  (A. x (x IN a ++ b -> x e. A) <-> A. x (x IN a -> x e. A) /\ A. x (x IN b -> x e. A))
12 bitr 
(x IN a ++ b -> x e. A <-> x IN a \/ x IN b -> x e. A) ->
  (x IN a \/ x IN b -> x e. A <-> (x IN a -> x e. A) /\ (x IN b -> x e. A)) ->
  (x IN a ++ b -> x e. A <-> (x IN a -> x e. A) /\ (x IN b -> x e. A))
13 lmemappend 
x IN a ++ b <-> x IN a \/ x IN b
14 13 imeq1i 
x IN a ++ b -> x e. A <-> x IN a \/ x IN b -> x e. A
15 12, 14 ax_mp 
(x IN a \/ x IN b -> x e. A <-> (x IN a -> x e. A) /\ (x IN b -> x e. A)) -> (x IN a ++ b -> x e. A <-> (x IN a -> x e. A) /\ (x IN b -> x e. A))
16 imor 
x IN a \/ x IN b -> x e. A <-> (x IN a -> x e. A) /\ (x IN b -> x e. A)
17 15, 16 ax_mp 
x IN a ++ b -> x e. A <-> (x IN a -> x e. A) /\ (x IN b -> x e. A)
18 17 aleqi 
A. x (x IN a ++ b -> x e. A) <-> A. x ((x IN a -> x e. A) /\ (x IN b -> x e. A))
19 11, 18 ax_mp 
(A. x ((x IN a -> x e. A) /\ (x IN b -> x e. A)) <-> A. x (x IN a -> x e. A) /\ A. x (x IN b -> x e. A)) ->
  (A. x (x IN a ++ b -> x e. A) <-> A. x (x IN a -> x e. A) /\ A. x (x IN b -> x e. A))
20 alan 
A. x ((x IN a -> x e. A) /\ (x IN b -> x e. A)) <-> A. x (x IN a -> x e. A) /\ A. x (x IN b -> x e. A)
21 19, 20 ax_mp 
A. x (x IN a ++ b -> x e. A) <-> A. x (x IN a -> x e. A) /\ A. x (x IN b -> x e. A)
22 10, 21 ax_mp 
all A a /\ all A b <-> A. x (x IN a ++ b -> x e. A)
23 3, 22 ax_mp 
all A (a ++ b) <-> all A a /\ all A b

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)