Step |
Hyp |
Ref |
Expression |
1 |
|
oesuclem.1 |
|- Lim X |
2 |
|
oesuclem.2 |
|- ( B e. X -> ( rec ( ( x e. _V |-> ( x .o A ) ) , 1o ) ` suc B ) = ( ( x e. _V |-> ( x .o A ) ) ` ( rec ( ( x e. _V |-> ( x .o A ) ) , 1o ) ` B ) ) ) |
3 |
|
oveq1 |
|- ( A = (/) -> ( A ^o suc B ) = ( (/) ^o suc B ) ) |
4 |
|
limord |
|- ( Lim X -> Ord X ) |
5 |
1 4
|
ax-mp |
|- Ord X |
6 |
|
ordelord |
|- ( ( Ord X /\ B e. X ) -> Ord B ) |
7 |
5 6
|
mpan |
|- ( B e. X -> Ord B ) |
8 |
|
0elsuc |
|- ( Ord B -> (/) e. suc B ) |
9 |
7 8
|
syl |
|- ( B e. X -> (/) e. suc B ) |
10 |
|
limsuc |
|- ( Lim X -> ( B e. X <-> suc B e. X ) ) |
11 |
1 10
|
ax-mp |
|- ( B e. X <-> suc B e. X ) |
12 |
|
ordelon |
|- ( ( Ord X /\ suc B e. X ) -> suc B e. On ) |
13 |
5 12
|
mpan |
|- ( suc B e. X -> suc B e. On ) |
14 |
|
oe0m1 |
|- ( suc B e. On -> ( (/) e. suc B <-> ( (/) ^o suc B ) = (/) ) ) |
15 |
13 14
|
syl |
|- ( suc B e. X -> ( (/) e. suc B <-> ( (/) ^o suc B ) = (/) ) ) |
16 |
11 15
|
sylbi |
|- ( B e. X -> ( (/) e. suc B <-> ( (/) ^o suc B ) = (/) ) ) |
17 |
9 16
|
mpbid |
|- ( B e. X -> ( (/) ^o suc B ) = (/) ) |
18 |
3 17
|
sylan9eqr |
|- ( ( B e. X /\ A = (/) ) -> ( A ^o suc B ) = (/) ) |
19 |
|
oveq1 |
|- ( A = (/) -> ( A ^o B ) = ( (/) ^o B ) ) |
20 |
|
id |
|- ( A = (/) -> A = (/) ) |
21 |
19 20
|
oveq12d |
|- ( A = (/) -> ( ( A ^o B ) .o A ) = ( ( (/) ^o B ) .o (/) ) ) |
22 |
|
ordelon |
|- ( ( Ord X /\ B e. X ) -> B e. On ) |
23 |
5 22
|
mpan |
|- ( B e. X -> B e. On ) |
24 |
|
oveq2 |
|- ( B = (/) -> ( (/) ^o B ) = ( (/) ^o (/) ) ) |
25 |
|
oe0m0 |
|- ( (/) ^o (/) ) = 1o |
26 |
|
1on |
|- 1o e. On |
27 |
25 26
|
eqeltri |
|- ( (/) ^o (/) ) e. On |
28 |
24 27
|
eqeltrdi |
|- ( B = (/) -> ( (/) ^o B ) e. On ) |
29 |
28
|
adantl |
|- ( ( B e. X /\ B = (/) ) -> ( (/) ^o B ) e. On ) |
30 |
|
oe0m1 |
|- ( B e. On -> ( (/) e. B <-> ( (/) ^o B ) = (/) ) ) |
31 |
23 30
|
syl |
|- ( B e. X -> ( (/) e. B <-> ( (/) ^o B ) = (/) ) ) |
32 |
31
|
biimpa |
|- ( ( B e. X /\ (/) e. B ) -> ( (/) ^o B ) = (/) ) |
33 |
|
0elon |
|- (/) e. On |
34 |
32 33
|
eqeltrdi |
|- ( ( B e. X /\ (/) e. B ) -> ( (/) ^o B ) e. On ) |
35 |
34
|
adantll |
|- ( ( ( B e. On /\ B e. X ) /\ (/) e. B ) -> ( (/) ^o B ) e. On ) |
36 |
29 35
|
oe0lem |
|- ( ( B e. On /\ B e. X ) -> ( (/) ^o B ) e. On ) |
37 |
23 36
|
mpancom |
|- ( B e. X -> ( (/) ^o B ) e. On ) |
38 |
|
om0 |
|- ( ( (/) ^o B ) e. On -> ( ( (/) ^o B ) .o (/) ) = (/) ) |
39 |
37 38
|
syl |
|- ( B e. X -> ( ( (/) ^o B ) .o (/) ) = (/) ) |
40 |
21 39
|
sylan9eqr |
|- ( ( B e. X /\ A = (/) ) -> ( ( A ^o B ) .o A ) = (/) ) |
41 |
18 40
|
eqtr4d |
|- ( ( B e. X /\ A = (/) ) -> ( A ^o suc B ) = ( ( A ^o B ) .o A ) ) |
42 |
2
|
ad2antlr |
|- ( ( ( A e. On /\ B e. X ) /\ (/) e. A ) -> ( rec ( ( x e. _V |-> ( x .o A ) ) , 1o ) ` suc B ) = ( ( x e. _V |-> ( x .o A ) ) ` ( rec ( ( x e. _V |-> ( x .o A ) ) , 1o ) ` B ) ) ) |
43 |
11 13
|
sylbi |
|- ( B e. X -> suc B e. On ) |
44 |
|
oevn0 |
|- ( ( ( A e. On /\ suc B e. On ) /\ (/) e. A ) -> ( A ^o suc B ) = ( rec ( ( x e. _V |-> ( x .o A ) ) , 1o ) ` suc B ) ) |
45 |
43 44
|
sylanl2 |
|- ( ( ( A e. On /\ B e. X ) /\ (/) e. A ) -> ( A ^o suc B ) = ( rec ( ( x e. _V |-> ( x .o A ) ) , 1o ) ` suc B ) ) |
46 |
|
ovex |
|- ( A ^o B ) e. _V |
47 |
|
oveq1 |
|- ( x = ( A ^o B ) -> ( x .o A ) = ( ( A ^o B ) .o A ) ) |
48 |
|
eqid |
|- ( x e. _V |-> ( x .o A ) ) = ( x e. _V |-> ( x .o A ) ) |
49 |
|
ovex |
|- ( ( A ^o B ) .o A ) e. _V |
50 |
47 48 49
|
fvmpt |
|- ( ( A ^o B ) e. _V -> ( ( x e. _V |-> ( x .o A ) ) ` ( A ^o B ) ) = ( ( A ^o B ) .o A ) ) |
51 |
46 50
|
ax-mp |
|- ( ( x e. _V |-> ( x .o A ) ) ` ( A ^o B ) ) = ( ( A ^o B ) .o A ) |
52 |
|
oevn0 |
|- ( ( ( A e. On /\ B e. On ) /\ (/) e. A ) -> ( A ^o B ) = ( rec ( ( x e. _V |-> ( x .o A ) ) , 1o ) ` B ) ) |
53 |
23 52
|
sylanl2 |
|- ( ( ( A e. On /\ B e. X ) /\ (/) e. A ) -> ( A ^o B ) = ( rec ( ( x e. _V |-> ( x .o A ) ) , 1o ) ` B ) ) |
54 |
53
|
fveq2d |
|- ( ( ( A e. On /\ B e. X ) /\ (/) e. A ) -> ( ( x e. _V |-> ( x .o A ) ) ` ( A ^o B ) ) = ( ( x e. _V |-> ( x .o A ) ) ` ( rec ( ( x e. _V |-> ( x .o A ) ) , 1o ) ` B ) ) ) |
55 |
51 54
|
eqtr3id |
|- ( ( ( A e. On /\ B e. X ) /\ (/) e. A ) -> ( ( A ^o B ) .o A ) = ( ( x e. _V |-> ( x .o A ) ) ` ( rec ( ( x e. _V |-> ( x .o A ) ) , 1o ) ` B ) ) ) |
56 |
42 45 55
|
3eqtr4d |
|- ( ( ( A e. On /\ B e. X ) /\ (/) e. A ) -> ( A ^o suc B ) = ( ( A ^o B ) .o A ) ) |
57 |
41 56
|
oe0lem |
|- ( ( A e. On /\ B e. X ) -> ( A ^o suc B ) = ( ( A ^o B ) .o A ) ) |