| Step |
Hyp |
Ref |
Expression |
| 1 |
|
oveq2 |
⊢ ( 𝑥 = ∅ → ( 𝐵 +o 𝑥 ) = ( 𝐵 +o ∅ ) ) |
| 2 |
1
|
oveq2d |
⊢ ( 𝑥 = ∅ → ( 𝐴 ·o ( 𝐵 +o 𝑥 ) ) = ( 𝐴 ·o ( 𝐵 +o ∅ ) ) ) |
| 3 |
|
oveq2 |
⊢ ( 𝑥 = ∅ → ( 𝐴 ·o 𝑥 ) = ( 𝐴 ·o ∅ ) ) |
| 4 |
3
|
oveq2d |
⊢ ( 𝑥 = ∅ → ( ( 𝐴 ·o 𝐵 ) +o ( 𝐴 ·o 𝑥 ) ) = ( ( 𝐴 ·o 𝐵 ) +o ( 𝐴 ·o ∅ ) ) ) |
| 5 |
2 4
|
eqeq12d |
⊢ ( 𝑥 = ∅ → ( ( 𝐴 ·o ( 𝐵 +o 𝑥 ) ) = ( ( 𝐴 ·o 𝐵 ) +o ( 𝐴 ·o 𝑥 ) ) ↔ ( 𝐴 ·o ( 𝐵 +o ∅ ) ) = ( ( 𝐴 ·o 𝐵 ) +o ( 𝐴 ·o ∅ ) ) ) ) |
| 6 |
|
oveq2 |
⊢ ( 𝑥 = 𝑦 → ( 𝐵 +o 𝑥 ) = ( 𝐵 +o 𝑦 ) ) |
| 7 |
6
|
oveq2d |
⊢ ( 𝑥 = 𝑦 → ( 𝐴 ·o ( 𝐵 +o 𝑥 ) ) = ( 𝐴 ·o ( 𝐵 +o 𝑦 ) ) ) |
| 8 |
|
oveq2 |
⊢ ( 𝑥 = 𝑦 → ( 𝐴 ·o 𝑥 ) = ( 𝐴 ·o 𝑦 ) ) |
| 9 |
8
|
oveq2d |
⊢ ( 𝑥 = 𝑦 → ( ( 𝐴 ·o 𝐵 ) +o ( 𝐴 ·o 𝑥 ) ) = ( ( 𝐴 ·o 𝐵 ) +o ( 𝐴 ·o 𝑦 ) ) ) |
| 10 |
7 9
|
eqeq12d |
⊢ ( 𝑥 = 𝑦 → ( ( 𝐴 ·o ( 𝐵 +o 𝑥 ) ) = ( ( 𝐴 ·o 𝐵 ) +o ( 𝐴 ·o 𝑥 ) ) ↔ ( 𝐴 ·o ( 𝐵 +o 𝑦 ) ) = ( ( 𝐴 ·o 𝐵 ) +o ( 𝐴 ·o 𝑦 ) ) ) ) |
| 11 |
|
oveq2 |
⊢ ( 𝑥 = suc 𝑦 → ( 𝐵 +o 𝑥 ) = ( 𝐵 +o suc 𝑦 ) ) |
| 12 |
11
|
oveq2d |
⊢ ( 𝑥 = suc 𝑦 → ( 𝐴 ·o ( 𝐵 +o 𝑥 ) ) = ( 𝐴 ·o ( 𝐵 +o suc 𝑦 ) ) ) |
| 13 |
|
oveq2 |
⊢ ( 𝑥 = suc 𝑦 → ( 𝐴 ·o 𝑥 ) = ( 𝐴 ·o suc 𝑦 ) ) |
| 14 |
13
|
oveq2d |
⊢ ( 𝑥 = suc 𝑦 → ( ( 𝐴 ·o 𝐵 ) +o ( 𝐴 ·o 𝑥 ) ) = ( ( 𝐴 ·o 𝐵 ) +o ( 𝐴 ·o suc 𝑦 ) ) ) |
| 15 |
12 14
|
eqeq12d |
⊢ ( 𝑥 = suc 𝑦 → ( ( 𝐴 ·o ( 𝐵 +o 𝑥 ) ) = ( ( 𝐴 ·o 𝐵 ) +o ( 𝐴 ·o 𝑥 ) ) ↔ ( 𝐴 ·o ( 𝐵 +o suc 𝑦 ) ) = ( ( 𝐴 ·o 𝐵 ) +o ( 𝐴 ·o suc 𝑦 ) ) ) ) |
| 16 |
|
oveq2 |
⊢ ( 𝑥 = 𝐶 → ( 𝐵 +o 𝑥 ) = ( 𝐵 +o 𝐶 ) ) |
| 17 |
16
|
oveq2d |
⊢ ( 𝑥 = 𝐶 → ( 𝐴 ·o ( 𝐵 +o 𝑥 ) ) = ( 𝐴 ·o ( 𝐵 +o 𝐶 ) ) ) |
| 18 |
|
oveq2 |
⊢ ( 𝑥 = 𝐶 → ( 𝐴 ·o 𝑥 ) = ( 𝐴 ·o 𝐶 ) ) |
| 19 |
18
|
oveq2d |
⊢ ( 𝑥 = 𝐶 → ( ( 𝐴 ·o 𝐵 ) +o ( 𝐴 ·o 𝑥 ) ) = ( ( 𝐴 ·o 𝐵 ) +o ( 𝐴 ·o 𝐶 ) ) ) |
| 20 |
17 19
|
eqeq12d |
⊢ ( 𝑥 = 𝐶 → ( ( 𝐴 ·o ( 𝐵 +o 𝑥 ) ) = ( ( 𝐴 ·o 𝐵 ) +o ( 𝐴 ·o 𝑥 ) ) ↔ ( 𝐴 ·o ( 𝐵 +o 𝐶 ) ) = ( ( 𝐴 ·o 𝐵 ) +o ( 𝐴 ·o 𝐶 ) ) ) ) |
| 21 |
|
omcl |
⊢ ( ( 𝐴 ∈ On ∧ 𝐵 ∈ On ) → ( 𝐴 ·o 𝐵 ) ∈ On ) |
| 22 |
|
oa0 |
⊢ ( ( 𝐴 ·o 𝐵 ) ∈ On → ( ( 𝐴 ·o 𝐵 ) +o ∅ ) = ( 𝐴 ·o 𝐵 ) ) |
| 23 |
21 22
|
syl |
⊢ ( ( 𝐴 ∈ On ∧ 𝐵 ∈ On ) → ( ( 𝐴 ·o 𝐵 ) +o ∅ ) = ( 𝐴 ·o 𝐵 ) ) |
| 24 |
|
om0 |
⊢ ( 𝐴 ∈ On → ( 𝐴 ·o ∅ ) = ∅ ) |
| 25 |
24
|
adantr |
⊢ ( ( 𝐴 ∈ On ∧ 𝐵 ∈ On ) → ( 𝐴 ·o ∅ ) = ∅ ) |
| 26 |
25
|
oveq2d |
⊢ ( ( 𝐴 ∈ On ∧ 𝐵 ∈ On ) → ( ( 𝐴 ·o 𝐵 ) +o ( 𝐴 ·o ∅ ) ) = ( ( 𝐴 ·o 𝐵 ) +o ∅ ) ) |
| 27 |
|
oa0 |
⊢ ( 𝐵 ∈ On → ( 𝐵 +o ∅ ) = 𝐵 ) |
| 28 |
27
|
adantl |
⊢ ( ( 𝐴 ∈ On ∧ 𝐵 ∈ On ) → ( 𝐵 +o ∅ ) = 𝐵 ) |
| 29 |
28
|
oveq2d |
⊢ ( ( 𝐴 ∈ On ∧ 𝐵 ∈ On ) → ( 𝐴 ·o ( 𝐵 +o ∅ ) ) = ( 𝐴 ·o 𝐵 ) ) |
| 30 |
23 26 29
|
3eqtr4rd |
⊢ ( ( 𝐴 ∈ On ∧ 𝐵 ∈ On ) → ( 𝐴 ·o ( 𝐵 +o ∅ ) ) = ( ( 𝐴 ·o 𝐵 ) +o ( 𝐴 ·o ∅ ) ) ) |
| 31 |
|
oveq1 |
⊢ ( ( 𝐴 ·o ( 𝐵 +o 𝑦 ) ) = ( ( 𝐴 ·o 𝐵 ) +o ( 𝐴 ·o 𝑦 ) ) → ( ( 𝐴 ·o ( 𝐵 +o 𝑦 ) ) +o 𝐴 ) = ( ( ( 𝐴 ·o 𝐵 ) +o ( 𝐴 ·o 𝑦 ) ) +o 𝐴 ) ) |
| 32 |
|
oasuc |
⊢ ( ( 𝐵 ∈ On ∧ 𝑦 ∈ On ) → ( 𝐵 +o suc 𝑦 ) = suc ( 𝐵 +o 𝑦 ) ) |
| 33 |
32
|
3adant1 |
⊢ ( ( 𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑦 ∈ On ) → ( 𝐵 +o suc 𝑦 ) = suc ( 𝐵 +o 𝑦 ) ) |
| 34 |
33
|
oveq2d |
⊢ ( ( 𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑦 ∈ On ) → ( 𝐴 ·o ( 𝐵 +o suc 𝑦 ) ) = ( 𝐴 ·o suc ( 𝐵 +o 𝑦 ) ) ) |
| 35 |
|
oacl |
⊢ ( ( 𝐵 ∈ On ∧ 𝑦 ∈ On ) → ( 𝐵 +o 𝑦 ) ∈ On ) |
| 36 |
|
omsuc |
⊢ ( ( 𝐴 ∈ On ∧ ( 𝐵 +o 𝑦 ) ∈ On ) → ( 𝐴 ·o suc ( 𝐵 +o 𝑦 ) ) = ( ( 𝐴 ·o ( 𝐵 +o 𝑦 ) ) +o 𝐴 ) ) |
| 37 |
35 36
|
sylan2 |
⊢ ( ( 𝐴 ∈ On ∧ ( 𝐵 ∈ On ∧ 𝑦 ∈ On ) ) → ( 𝐴 ·o suc ( 𝐵 +o 𝑦 ) ) = ( ( 𝐴 ·o ( 𝐵 +o 𝑦 ) ) +o 𝐴 ) ) |
| 38 |
37
|
3impb |
⊢ ( ( 𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑦 ∈ On ) → ( 𝐴 ·o suc ( 𝐵 +o 𝑦 ) ) = ( ( 𝐴 ·o ( 𝐵 +o 𝑦 ) ) +o 𝐴 ) ) |
| 39 |
34 38
|
eqtrd |
⊢ ( ( 𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑦 ∈ On ) → ( 𝐴 ·o ( 𝐵 +o suc 𝑦 ) ) = ( ( 𝐴 ·o ( 𝐵 +o 𝑦 ) ) +o 𝐴 ) ) |
| 40 |
|
omsuc |
⊢ ( ( 𝐴 ∈ On ∧ 𝑦 ∈ On ) → ( 𝐴 ·o suc 𝑦 ) = ( ( 𝐴 ·o 𝑦 ) +o 𝐴 ) ) |
| 41 |
40
|
3adant2 |
⊢ ( ( 𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑦 ∈ On ) → ( 𝐴 ·o suc 𝑦 ) = ( ( 𝐴 ·o 𝑦 ) +o 𝐴 ) ) |
| 42 |
41
|
oveq2d |
⊢ ( ( 𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑦 ∈ On ) → ( ( 𝐴 ·o 𝐵 ) +o ( 𝐴 ·o suc 𝑦 ) ) = ( ( 𝐴 ·o 𝐵 ) +o ( ( 𝐴 ·o 𝑦 ) +o 𝐴 ) ) ) |
| 43 |
|
omcl |
⊢ ( ( 𝐴 ∈ On ∧ 𝑦 ∈ On ) → ( 𝐴 ·o 𝑦 ) ∈ On ) |
| 44 |
|
oaass |
⊢ ( ( ( 𝐴 ·o 𝐵 ) ∈ On ∧ ( 𝐴 ·o 𝑦 ) ∈ On ∧ 𝐴 ∈ On ) → ( ( ( 𝐴 ·o 𝐵 ) +o ( 𝐴 ·o 𝑦 ) ) +o 𝐴 ) = ( ( 𝐴 ·o 𝐵 ) +o ( ( 𝐴 ·o 𝑦 ) +o 𝐴 ) ) ) |
| 45 |
21 44
|
syl3an1 |
⊢ ( ( ( 𝐴 ∈ On ∧ 𝐵 ∈ On ) ∧ ( 𝐴 ·o 𝑦 ) ∈ On ∧ 𝐴 ∈ On ) → ( ( ( 𝐴 ·o 𝐵 ) +o ( 𝐴 ·o 𝑦 ) ) +o 𝐴 ) = ( ( 𝐴 ·o 𝐵 ) +o ( ( 𝐴 ·o 𝑦 ) +o 𝐴 ) ) ) |
| 46 |
43 45
|
syl3an2 |
⊢ ( ( ( 𝐴 ∈ On ∧ 𝐵 ∈ On ) ∧ ( 𝐴 ∈ On ∧ 𝑦 ∈ On ) ∧ 𝐴 ∈ On ) → ( ( ( 𝐴 ·o 𝐵 ) +o ( 𝐴 ·o 𝑦 ) ) +o 𝐴 ) = ( ( 𝐴 ·o 𝐵 ) +o ( ( 𝐴 ·o 𝑦 ) +o 𝐴 ) ) ) |
| 47 |
46
|
3exp |
⊢ ( ( 𝐴 ∈ On ∧ 𝐵 ∈ On ) → ( ( 𝐴 ∈ On ∧ 𝑦 ∈ On ) → ( 𝐴 ∈ On → ( ( ( 𝐴 ·o 𝐵 ) +o ( 𝐴 ·o 𝑦 ) ) +o 𝐴 ) = ( ( 𝐴 ·o 𝐵 ) +o ( ( 𝐴 ·o 𝑦 ) +o 𝐴 ) ) ) ) ) |
| 48 |
47
|
exp4b |
⊢ ( 𝐴 ∈ On → ( 𝐵 ∈ On → ( 𝐴 ∈ On → ( 𝑦 ∈ On → ( 𝐴 ∈ On → ( ( ( 𝐴 ·o 𝐵 ) +o ( 𝐴 ·o 𝑦 ) ) +o 𝐴 ) = ( ( 𝐴 ·o 𝐵 ) +o ( ( 𝐴 ·o 𝑦 ) +o 𝐴 ) ) ) ) ) ) ) |
| 49 |
48
|
pm2.43a |
⊢ ( 𝐴 ∈ On → ( 𝐵 ∈ On → ( 𝑦 ∈ On → ( 𝐴 ∈ On → ( ( ( 𝐴 ·o 𝐵 ) +o ( 𝐴 ·o 𝑦 ) ) +o 𝐴 ) = ( ( 𝐴 ·o 𝐵 ) +o ( ( 𝐴 ·o 𝑦 ) +o 𝐴 ) ) ) ) ) ) |
| 50 |
49
|
com4r |
⊢ ( 𝐴 ∈ On → ( 𝐴 ∈ On → ( 𝐵 ∈ On → ( 𝑦 ∈ On → ( ( ( 𝐴 ·o 𝐵 ) +o ( 𝐴 ·o 𝑦 ) ) +o 𝐴 ) = ( ( 𝐴 ·o 𝐵 ) +o ( ( 𝐴 ·o 𝑦 ) +o 𝐴 ) ) ) ) ) ) |
| 51 |
50
|
pm2.43i |
⊢ ( 𝐴 ∈ On → ( 𝐵 ∈ On → ( 𝑦 ∈ On → ( ( ( 𝐴 ·o 𝐵 ) +o ( 𝐴 ·o 𝑦 ) ) +o 𝐴 ) = ( ( 𝐴 ·o 𝐵 ) +o ( ( 𝐴 ·o 𝑦 ) +o 𝐴 ) ) ) ) ) |
| 52 |
51
|
3imp |
⊢ ( ( 𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑦 ∈ On ) → ( ( ( 𝐴 ·o 𝐵 ) +o ( 𝐴 ·o 𝑦 ) ) +o 𝐴 ) = ( ( 𝐴 ·o 𝐵 ) +o ( ( 𝐴 ·o 𝑦 ) +o 𝐴 ) ) ) |
| 53 |
42 52
|
eqtr4d |
⊢ ( ( 𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑦 ∈ On ) → ( ( 𝐴 ·o 𝐵 ) +o ( 𝐴 ·o suc 𝑦 ) ) = ( ( ( 𝐴 ·o 𝐵 ) +o ( 𝐴 ·o 𝑦 ) ) +o 𝐴 ) ) |
| 54 |
39 53
|
eqeq12d |
⊢ ( ( 𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑦 ∈ On ) → ( ( 𝐴 ·o ( 𝐵 +o suc 𝑦 ) ) = ( ( 𝐴 ·o 𝐵 ) +o ( 𝐴 ·o suc 𝑦 ) ) ↔ ( ( 𝐴 ·o ( 𝐵 +o 𝑦 ) ) +o 𝐴 ) = ( ( ( 𝐴 ·o 𝐵 ) +o ( 𝐴 ·o 𝑦 ) ) +o 𝐴 ) ) ) |
| 55 |
31 54
|
imbitrrid |
⊢ ( ( 𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑦 ∈ On ) → ( ( 𝐴 ·o ( 𝐵 +o 𝑦 ) ) = ( ( 𝐴 ·o 𝐵 ) +o ( 𝐴 ·o 𝑦 ) ) → ( 𝐴 ·o ( 𝐵 +o suc 𝑦 ) ) = ( ( 𝐴 ·o 𝐵 ) +o ( 𝐴 ·o suc 𝑦 ) ) ) ) |
| 56 |
55
|
3exp |
⊢ ( 𝐴 ∈ On → ( 𝐵 ∈ On → ( 𝑦 ∈ On → ( ( 𝐴 ·o ( 𝐵 +o 𝑦 ) ) = ( ( 𝐴 ·o 𝐵 ) +o ( 𝐴 ·o 𝑦 ) ) → ( 𝐴 ·o ( 𝐵 +o suc 𝑦 ) ) = ( ( 𝐴 ·o 𝐵 ) +o ( 𝐴 ·o suc 𝑦 ) ) ) ) ) ) |
| 57 |
56
|
com3r |
⊢ ( 𝑦 ∈ On → ( 𝐴 ∈ On → ( 𝐵 ∈ On → ( ( 𝐴 ·o ( 𝐵 +o 𝑦 ) ) = ( ( 𝐴 ·o 𝐵 ) +o ( 𝐴 ·o 𝑦 ) ) → ( 𝐴 ·o ( 𝐵 +o suc 𝑦 ) ) = ( ( 𝐴 ·o 𝐵 ) +o ( 𝐴 ·o suc 𝑦 ) ) ) ) ) ) |
| 58 |
57
|
impd |
⊢ ( 𝑦 ∈ On → ( ( 𝐴 ∈ On ∧ 𝐵 ∈ On ) → ( ( 𝐴 ·o ( 𝐵 +o 𝑦 ) ) = ( ( 𝐴 ·o 𝐵 ) +o ( 𝐴 ·o 𝑦 ) ) → ( 𝐴 ·o ( 𝐵 +o suc 𝑦 ) ) = ( ( 𝐴 ·o 𝐵 ) +o ( 𝐴 ·o suc 𝑦 ) ) ) ) ) |
| 59 |
|
vex |
⊢ 𝑥 ∈ V |
| 60 |
|
limelon |
⊢ ( ( 𝑥 ∈ V ∧ Lim 𝑥 ) → 𝑥 ∈ On ) |
| 61 |
59 60
|
mpan |
⊢ ( Lim 𝑥 → 𝑥 ∈ On ) |
| 62 |
|
oacl |
⊢ ( ( 𝐵 ∈ On ∧ 𝑥 ∈ On ) → ( 𝐵 +o 𝑥 ) ∈ On ) |
| 63 |
|
om0r |
⊢ ( ( 𝐵 +o 𝑥 ) ∈ On → ( ∅ ·o ( 𝐵 +o 𝑥 ) ) = ∅ ) |
| 64 |
62 63
|
syl |
⊢ ( ( 𝐵 ∈ On ∧ 𝑥 ∈ On ) → ( ∅ ·o ( 𝐵 +o 𝑥 ) ) = ∅ ) |
| 65 |
|
om0r |
⊢ ( 𝐵 ∈ On → ( ∅ ·o 𝐵 ) = ∅ ) |
| 66 |
|
om0r |
⊢ ( 𝑥 ∈ On → ( ∅ ·o 𝑥 ) = ∅ ) |
| 67 |
65 66
|
oveqan12d |
⊢ ( ( 𝐵 ∈ On ∧ 𝑥 ∈ On ) → ( ( ∅ ·o 𝐵 ) +o ( ∅ ·o 𝑥 ) ) = ( ∅ +o ∅ ) ) |
| 68 |
|
0elon |
⊢ ∅ ∈ On |
| 69 |
|
oa0 |
⊢ ( ∅ ∈ On → ( ∅ +o ∅ ) = ∅ ) |
| 70 |
68 69
|
ax-mp |
⊢ ( ∅ +o ∅ ) = ∅ |
| 71 |
67 70
|
eqtr2di |
⊢ ( ( 𝐵 ∈ On ∧ 𝑥 ∈ On ) → ∅ = ( ( ∅ ·o 𝐵 ) +o ( ∅ ·o 𝑥 ) ) ) |
| 72 |
64 71
|
eqtrd |
⊢ ( ( 𝐵 ∈ On ∧ 𝑥 ∈ On ) → ( ∅ ·o ( 𝐵 +o 𝑥 ) ) = ( ( ∅ ·o 𝐵 ) +o ( ∅ ·o 𝑥 ) ) ) |
| 73 |
61 72
|
sylan2 |
⊢ ( ( 𝐵 ∈ On ∧ Lim 𝑥 ) → ( ∅ ·o ( 𝐵 +o 𝑥 ) ) = ( ( ∅ ·o 𝐵 ) +o ( ∅ ·o 𝑥 ) ) ) |
| 74 |
73
|
ancoms |
⊢ ( ( Lim 𝑥 ∧ 𝐵 ∈ On ) → ( ∅ ·o ( 𝐵 +o 𝑥 ) ) = ( ( ∅ ·o 𝐵 ) +o ( ∅ ·o 𝑥 ) ) ) |
| 75 |
|
oveq1 |
⊢ ( 𝐴 = ∅ → ( 𝐴 ·o ( 𝐵 +o 𝑥 ) ) = ( ∅ ·o ( 𝐵 +o 𝑥 ) ) ) |
| 76 |
|
oveq1 |
⊢ ( 𝐴 = ∅ → ( 𝐴 ·o 𝐵 ) = ( ∅ ·o 𝐵 ) ) |
| 77 |
|
oveq1 |
⊢ ( 𝐴 = ∅ → ( 𝐴 ·o 𝑥 ) = ( ∅ ·o 𝑥 ) ) |
| 78 |
76 77
|
oveq12d |
⊢ ( 𝐴 = ∅ → ( ( 𝐴 ·o 𝐵 ) +o ( 𝐴 ·o 𝑥 ) ) = ( ( ∅ ·o 𝐵 ) +o ( ∅ ·o 𝑥 ) ) ) |
| 79 |
75 78
|
eqeq12d |
⊢ ( 𝐴 = ∅ → ( ( 𝐴 ·o ( 𝐵 +o 𝑥 ) ) = ( ( 𝐴 ·o 𝐵 ) +o ( 𝐴 ·o 𝑥 ) ) ↔ ( ∅ ·o ( 𝐵 +o 𝑥 ) ) = ( ( ∅ ·o 𝐵 ) +o ( ∅ ·o 𝑥 ) ) ) ) |
| 80 |
74 79
|
imbitrrid |
⊢ ( 𝐴 = ∅ → ( ( Lim 𝑥 ∧ 𝐵 ∈ On ) → ( 𝐴 ·o ( 𝐵 +o 𝑥 ) ) = ( ( 𝐴 ·o 𝐵 ) +o ( 𝐴 ·o 𝑥 ) ) ) ) |
| 81 |
80
|
expd |
⊢ ( 𝐴 = ∅ → ( Lim 𝑥 → ( 𝐵 ∈ On → ( 𝐴 ·o ( 𝐵 +o 𝑥 ) ) = ( ( 𝐴 ·o 𝐵 ) +o ( 𝐴 ·o 𝑥 ) ) ) ) ) |
| 82 |
81
|
com3r |
⊢ ( 𝐵 ∈ On → ( 𝐴 = ∅ → ( Lim 𝑥 → ( 𝐴 ·o ( 𝐵 +o 𝑥 ) ) = ( ( 𝐴 ·o 𝐵 ) +o ( 𝐴 ·o 𝑥 ) ) ) ) ) |
| 83 |
82
|
imp |
⊢ ( ( 𝐵 ∈ On ∧ 𝐴 = ∅ ) → ( Lim 𝑥 → ( 𝐴 ·o ( 𝐵 +o 𝑥 ) ) = ( ( 𝐴 ·o 𝐵 ) +o ( 𝐴 ·o 𝑥 ) ) ) ) |
| 84 |
83
|
a1dd |
⊢ ( ( 𝐵 ∈ On ∧ 𝐴 = ∅ ) → ( Lim 𝑥 → ( ∀ 𝑦 ∈ 𝑥 ( 𝐴 ·o ( 𝐵 +o 𝑦 ) ) = ( ( 𝐴 ·o 𝐵 ) +o ( 𝐴 ·o 𝑦 ) ) → ( 𝐴 ·o ( 𝐵 +o 𝑥 ) ) = ( ( 𝐴 ·o 𝐵 ) +o ( 𝐴 ·o 𝑥 ) ) ) ) ) |
| 85 |
|
simplr |
⊢ ( ( ( 𝑥 ∈ On ∧ 𝐵 ∈ On ) ∧ 𝑧 ∈ ( 𝐵 +o 𝑥 ) ) → 𝐵 ∈ On ) |
| 86 |
62
|
ancoms |
⊢ ( ( 𝑥 ∈ On ∧ 𝐵 ∈ On ) → ( 𝐵 +o 𝑥 ) ∈ On ) |
| 87 |
|
onelon |
⊢ ( ( ( 𝐵 +o 𝑥 ) ∈ On ∧ 𝑧 ∈ ( 𝐵 +o 𝑥 ) ) → 𝑧 ∈ On ) |
| 88 |
86 87
|
sylan |
⊢ ( ( ( 𝑥 ∈ On ∧ 𝐵 ∈ On ) ∧ 𝑧 ∈ ( 𝐵 +o 𝑥 ) ) → 𝑧 ∈ On ) |
| 89 |
|
ontri1 |
⊢ ( ( 𝐵 ∈ On ∧ 𝑧 ∈ On ) → ( 𝐵 ⊆ 𝑧 ↔ ¬ 𝑧 ∈ 𝐵 ) ) |
| 90 |
|
oawordex |
⊢ ( ( 𝐵 ∈ On ∧ 𝑧 ∈ On ) → ( 𝐵 ⊆ 𝑧 ↔ ∃ 𝑣 ∈ On ( 𝐵 +o 𝑣 ) = 𝑧 ) ) |
| 91 |
89 90
|
bitr3d |
⊢ ( ( 𝐵 ∈ On ∧ 𝑧 ∈ On ) → ( ¬ 𝑧 ∈ 𝐵 ↔ ∃ 𝑣 ∈ On ( 𝐵 +o 𝑣 ) = 𝑧 ) ) |
| 92 |
85 88 91
|
syl2anc |
⊢ ( ( ( 𝑥 ∈ On ∧ 𝐵 ∈ On ) ∧ 𝑧 ∈ ( 𝐵 +o 𝑥 ) ) → ( ¬ 𝑧 ∈ 𝐵 ↔ ∃ 𝑣 ∈ On ( 𝐵 +o 𝑣 ) = 𝑧 ) ) |
| 93 |
|
oaord |
⊢ ( ( 𝑣 ∈ On ∧ 𝑥 ∈ On ∧ 𝐵 ∈ On ) → ( 𝑣 ∈ 𝑥 ↔ ( 𝐵 +o 𝑣 ) ∈ ( 𝐵 +o 𝑥 ) ) ) |
| 94 |
93
|
3expb |
⊢ ( ( 𝑣 ∈ On ∧ ( 𝑥 ∈ On ∧ 𝐵 ∈ On ) ) → ( 𝑣 ∈ 𝑥 ↔ ( 𝐵 +o 𝑣 ) ∈ ( 𝐵 +o 𝑥 ) ) ) |
| 95 |
|
eleq1 |
⊢ ( ( 𝐵 +o 𝑣 ) = 𝑧 → ( ( 𝐵 +o 𝑣 ) ∈ ( 𝐵 +o 𝑥 ) ↔ 𝑧 ∈ ( 𝐵 +o 𝑥 ) ) ) |
| 96 |
94 95
|
sylan9bb |
⊢ ( ( ( 𝑣 ∈ On ∧ ( 𝑥 ∈ On ∧ 𝐵 ∈ On ) ) ∧ ( 𝐵 +o 𝑣 ) = 𝑧 ) → ( 𝑣 ∈ 𝑥 ↔ 𝑧 ∈ ( 𝐵 +o 𝑥 ) ) ) |
| 97 |
|
iba |
⊢ ( ( 𝐵 +o 𝑣 ) = 𝑧 → ( 𝑣 ∈ 𝑥 ↔ ( 𝑣 ∈ 𝑥 ∧ ( 𝐵 +o 𝑣 ) = 𝑧 ) ) ) |
| 98 |
97
|
adantl |
⊢ ( ( ( 𝑣 ∈ On ∧ ( 𝑥 ∈ On ∧ 𝐵 ∈ On ) ) ∧ ( 𝐵 +o 𝑣 ) = 𝑧 ) → ( 𝑣 ∈ 𝑥 ↔ ( 𝑣 ∈ 𝑥 ∧ ( 𝐵 +o 𝑣 ) = 𝑧 ) ) ) |
| 99 |
96 98
|
bitr3d |
⊢ ( ( ( 𝑣 ∈ On ∧ ( 𝑥 ∈ On ∧ 𝐵 ∈ On ) ) ∧ ( 𝐵 +o 𝑣 ) = 𝑧 ) → ( 𝑧 ∈ ( 𝐵 +o 𝑥 ) ↔ ( 𝑣 ∈ 𝑥 ∧ ( 𝐵 +o 𝑣 ) = 𝑧 ) ) ) |
| 100 |
99
|
an32s |
⊢ ( ( ( 𝑣 ∈ On ∧ ( 𝐵 +o 𝑣 ) = 𝑧 ) ∧ ( 𝑥 ∈ On ∧ 𝐵 ∈ On ) ) → ( 𝑧 ∈ ( 𝐵 +o 𝑥 ) ↔ ( 𝑣 ∈ 𝑥 ∧ ( 𝐵 +o 𝑣 ) = 𝑧 ) ) ) |
| 101 |
100
|
biimpcd |
⊢ ( 𝑧 ∈ ( 𝐵 +o 𝑥 ) → ( ( ( 𝑣 ∈ On ∧ ( 𝐵 +o 𝑣 ) = 𝑧 ) ∧ ( 𝑥 ∈ On ∧ 𝐵 ∈ On ) ) → ( 𝑣 ∈ 𝑥 ∧ ( 𝐵 +o 𝑣 ) = 𝑧 ) ) ) |
| 102 |
101
|
exp4c |
⊢ ( 𝑧 ∈ ( 𝐵 +o 𝑥 ) → ( 𝑣 ∈ On → ( ( 𝐵 +o 𝑣 ) = 𝑧 → ( ( 𝑥 ∈ On ∧ 𝐵 ∈ On ) → ( 𝑣 ∈ 𝑥 ∧ ( 𝐵 +o 𝑣 ) = 𝑧 ) ) ) ) ) |
| 103 |
102
|
com4r |
⊢ ( ( 𝑥 ∈ On ∧ 𝐵 ∈ On ) → ( 𝑧 ∈ ( 𝐵 +o 𝑥 ) → ( 𝑣 ∈ On → ( ( 𝐵 +o 𝑣 ) = 𝑧 → ( 𝑣 ∈ 𝑥 ∧ ( 𝐵 +o 𝑣 ) = 𝑧 ) ) ) ) ) |
| 104 |
103
|
imp |
⊢ ( ( ( 𝑥 ∈ On ∧ 𝐵 ∈ On ) ∧ 𝑧 ∈ ( 𝐵 +o 𝑥 ) ) → ( 𝑣 ∈ On → ( ( 𝐵 +o 𝑣 ) = 𝑧 → ( 𝑣 ∈ 𝑥 ∧ ( 𝐵 +o 𝑣 ) = 𝑧 ) ) ) ) |
| 105 |
104
|
reximdvai |
⊢ ( ( ( 𝑥 ∈ On ∧ 𝐵 ∈ On ) ∧ 𝑧 ∈ ( 𝐵 +o 𝑥 ) ) → ( ∃ 𝑣 ∈ On ( 𝐵 +o 𝑣 ) = 𝑧 → ∃ 𝑣 ∈ On ( 𝑣 ∈ 𝑥 ∧ ( 𝐵 +o 𝑣 ) = 𝑧 ) ) ) |
| 106 |
92 105
|
sylbid |
⊢ ( ( ( 𝑥 ∈ On ∧ 𝐵 ∈ On ) ∧ 𝑧 ∈ ( 𝐵 +o 𝑥 ) ) → ( ¬ 𝑧 ∈ 𝐵 → ∃ 𝑣 ∈ On ( 𝑣 ∈ 𝑥 ∧ ( 𝐵 +o 𝑣 ) = 𝑧 ) ) ) |
| 107 |
106
|
orrd |
⊢ ( ( ( 𝑥 ∈ On ∧ 𝐵 ∈ On ) ∧ 𝑧 ∈ ( 𝐵 +o 𝑥 ) ) → ( 𝑧 ∈ 𝐵 ∨ ∃ 𝑣 ∈ On ( 𝑣 ∈ 𝑥 ∧ ( 𝐵 +o 𝑣 ) = 𝑧 ) ) ) |
| 108 |
61 107
|
sylanl1 |
⊢ ( ( ( Lim 𝑥 ∧ 𝐵 ∈ On ) ∧ 𝑧 ∈ ( 𝐵 +o 𝑥 ) ) → ( 𝑧 ∈ 𝐵 ∨ ∃ 𝑣 ∈ On ( 𝑣 ∈ 𝑥 ∧ ( 𝐵 +o 𝑣 ) = 𝑧 ) ) ) |
| 109 |
108
|
adantlrl |
⊢ ( ( ( Lim 𝑥 ∧ ( 𝐴 ∈ On ∧ 𝐵 ∈ On ) ) ∧ 𝑧 ∈ ( 𝐵 +o 𝑥 ) ) → ( 𝑧 ∈ 𝐵 ∨ ∃ 𝑣 ∈ On ( 𝑣 ∈ 𝑥 ∧ ( 𝐵 +o 𝑣 ) = 𝑧 ) ) ) |
| 110 |
109
|
adantlr |
⊢ ( ( ( ( Lim 𝑥 ∧ ( 𝐴 ∈ On ∧ 𝐵 ∈ On ) ) ∧ ( ∅ ∈ 𝐴 ∧ ∀ 𝑦 ∈ 𝑥 ( 𝐴 ·o ( 𝐵 +o 𝑦 ) ) = ( ( 𝐴 ·o 𝐵 ) +o ( 𝐴 ·o 𝑦 ) ) ) ) ∧ 𝑧 ∈ ( 𝐵 +o 𝑥 ) ) → ( 𝑧 ∈ 𝐵 ∨ ∃ 𝑣 ∈ On ( 𝑣 ∈ 𝑥 ∧ ( 𝐵 +o 𝑣 ) = 𝑧 ) ) ) |
| 111 |
|
0ellim |
⊢ ( Lim 𝑥 → ∅ ∈ 𝑥 ) |
| 112 |
|
om00el |
⊢ ( ( 𝐴 ∈ On ∧ 𝑥 ∈ On ) → ( ∅ ∈ ( 𝐴 ·o 𝑥 ) ↔ ( ∅ ∈ 𝐴 ∧ ∅ ∈ 𝑥 ) ) ) |
| 113 |
112
|
biimprd |
⊢ ( ( 𝐴 ∈ On ∧ 𝑥 ∈ On ) → ( ( ∅ ∈ 𝐴 ∧ ∅ ∈ 𝑥 ) → ∅ ∈ ( 𝐴 ·o 𝑥 ) ) ) |
| 114 |
111 113
|
sylan2i |
⊢ ( ( 𝐴 ∈ On ∧ 𝑥 ∈ On ) → ( ( ∅ ∈ 𝐴 ∧ Lim 𝑥 ) → ∅ ∈ ( 𝐴 ·o 𝑥 ) ) ) |
| 115 |
61 114
|
sylan2 |
⊢ ( ( 𝐴 ∈ On ∧ Lim 𝑥 ) → ( ( ∅ ∈ 𝐴 ∧ Lim 𝑥 ) → ∅ ∈ ( 𝐴 ·o 𝑥 ) ) ) |
| 116 |
115
|
exp4b |
⊢ ( 𝐴 ∈ On → ( Lim 𝑥 → ( ∅ ∈ 𝐴 → ( Lim 𝑥 → ∅ ∈ ( 𝐴 ·o 𝑥 ) ) ) ) ) |
| 117 |
116
|
com4r |
⊢ ( Lim 𝑥 → ( 𝐴 ∈ On → ( Lim 𝑥 → ( ∅ ∈ 𝐴 → ∅ ∈ ( 𝐴 ·o 𝑥 ) ) ) ) ) |
| 118 |
117
|
pm2.43a |
⊢ ( Lim 𝑥 → ( 𝐴 ∈ On → ( ∅ ∈ 𝐴 → ∅ ∈ ( 𝐴 ·o 𝑥 ) ) ) ) |
| 119 |
118
|
imp31 |
⊢ ( ( ( Lim 𝑥 ∧ 𝐴 ∈ On ) ∧ ∅ ∈ 𝐴 ) → ∅ ∈ ( 𝐴 ·o 𝑥 ) ) |
| 120 |
119
|
a1d |
⊢ ( ( ( Lim 𝑥 ∧ 𝐴 ∈ On ) ∧ ∅ ∈ 𝐴 ) → ( 𝑧 ∈ 𝐵 → ∅ ∈ ( 𝐴 ·o 𝑥 ) ) ) |
| 121 |
120
|
adantlrr |
⊢ ( ( ( Lim 𝑥 ∧ ( 𝐴 ∈ On ∧ 𝐵 ∈ On ) ) ∧ ∅ ∈ 𝐴 ) → ( 𝑧 ∈ 𝐵 → ∅ ∈ ( 𝐴 ·o 𝑥 ) ) ) |
| 122 |
|
omordi |
⊢ ( ( ( 𝐵 ∈ On ∧ 𝐴 ∈ On ) ∧ ∅ ∈ 𝐴 ) → ( 𝑧 ∈ 𝐵 → ( 𝐴 ·o 𝑧 ) ∈ ( 𝐴 ·o 𝐵 ) ) ) |
| 123 |
122
|
ancom1s |
⊢ ( ( ( 𝐴 ∈ On ∧ 𝐵 ∈ On ) ∧ ∅ ∈ 𝐴 ) → ( 𝑧 ∈ 𝐵 → ( 𝐴 ·o 𝑧 ) ∈ ( 𝐴 ·o 𝐵 ) ) ) |
| 124 |
|
onelss |
⊢ ( ( 𝐴 ·o 𝐵 ) ∈ On → ( ( 𝐴 ·o 𝑧 ) ∈ ( 𝐴 ·o 𝐵 ) → ( 𝐴 ·o 𝑧 ) ⊆ ( 𝐴 ·o 𝐵 ) ) ) |
| 125 |
22
|
sseq2d |
⊢ ( ( 𝐴 ·o 𝐵 ) ∈ On → ( ( 𝐴 ·o 𝑧 ) ⊆ ( ( 𝐴 ·o 𝐵 ) +o ∅ ) ↔ ( 𝐴 ·o 𝑧 ) ⊆ ( 𝐴 ·o 𝐵 ) ) ) |
| 126 |
124 125
|
sylibrd |
⊢ ( ( 𝐴 ·o 𝐵 ) ∈ On → ( ( 𝐴 ·o 𝑧 ) ∈ ( 𝐴 ·o 𝐵 ) → ( 𝐴 ·o 𝑧 ) ⊆ ( ( 𝐴 ·o 𝐵 ) +o ∅ ) ) ) |
| 127 |
21 126
|
syl |
⊢ ( ( 𝐴 ∈ On ∧ 𝐵 ∈ On ) → ( ( 𝐴 ·o 𝑧 ) ∈ ( 𝐴 ·o 𝐵 ) → ( 𝐴 ·o 𝑧 ) ⊆ ( ( 𝐴 ·o 𝐵 ) +o ∅ ) ) ) |
| 128 |
127
|
adantr |
⊢ ( ( ( 𝐴 ∈ On ∧ 𝐵 ∈ On ) ∧ ∅ ∈ 𝐴 ) → ( ( 𝐴 ·o 𝑧 ) ∈ ( 𝐴 ·o 𝐵 ) → ( 𝐴 ·o 𝑧 ) ⊆ ( ( 𝐴 ·o 𝐵 ) +o ∅ ) ) ) |
| 129 |
123 128
|
syld |
⊢ ( ( ( 𝐴 ∈ On ∧ 𝐵 ∈ On ) ∧ ∅ ∈ 𝐴 ) → ( 𝑧 ∈ 𝐵 → ( 𝐴 ·o 𝑧 ) ⊆ ( ( 𝐴 ·o 𝐵 ) +o ∅ ) ) ) |
| 130 |
129
|
adantll |
⊢ ( ( ( Lim 𝑥 ∧ ( 𝐴 ∈ On ∧ 𝐵 ∈ On ) ) ∧ ∅ ∈ 𝐴 ) → ( 𝑧 ∈ 𝐵 → ( 𝐴 ·o 𝑧 ) ⊆ ( ( 𝐴 ·o 𝐵 ) +o ∅ ) ) ) |
| 131 |
121 130
|
jcad |
⊢ ( ( ( Lim 𝑥 ∧ ( 𝐴 ∈ On ∧ 𝐵 ∈ On ) ) ∧ ∅ ∈ 𝐴 ) → ( 𝑧 ∈ 𝐵 → ( ∅ ∈ ( 𝐴 ·o 𝑥 ) ∧ ( 𝐴 ·o 𝑧 ) ⊆ ( ( 𝐴 ·o 𝐵 ) +o ∅ ) ) ) ) |
| 132 |
|
oveq2 |
⊢ ( 𝑤 = ∅ → ( ( 𝐴 ·o 𝐵 ) +o 𝑤 ) = ( ( 𝐴 ·o 𝐵 ) +o ∅ ) ) |
| 133 |
132
|
sseq2d |
⊢ ( 𝑤 = ∅ → ( ( 𝐴 ·o 𝑧 ) ⊆ ( ( 𝐴 ·o 𝐵 ) +o 𝑤 ) ↔ ( 𝐴 ·o 𝑧 ) ⊆ ( ( 𝐴 ·o 𝐵 ) +o ∅ ) ) ) |
| 134 |
133
|
rspcev |
⊢ ( ( ∅ ∈ ( 𝐴 ·o 𝑥 ) ∧ ( 𝐴 ·o 𝑧 ) ⊆ ( ( 𝐴 ·o 𝐵 ) +o ∅ ) ) → ∃ 𝑤 ∈ ( 𝐴 ·o 𝑥 ) ( 𝐴 ·o 𝑧 ) ⊆ ( ( 𝐴 ·o 𝐵 ) +o 𝑤 ) ) |
| 135 |
131 134
|
syl6 |
⊢ ( ( ( Lim 𝑥 ∧ ( 𝐴 ∈ On ∧ 𝐵 ∈ On ) ) ∧ ∅ ∈ 𝐴 ) → ( 𝑧 ∈ 𝐵 → ∃ 𝑤 ∈ ( 𝐴 ·o 𝑥 ) ( 𝐴 ·o 𝑧 ) ⊆ ( ( 𝐴 ·o 𝐵 ) +o 𝑤 ) ) ) |
| 136 |
135
|
adantrr |
⊢ ( ( ( Lim 𝑥 ∧ ( 𝐴 ∈ On ∧ 𝐵 ∈ On ) ) ∧ ( ∅ ∈ 𝐴 ∧ ∀ 𝑦 ∈ 𝑥 ( 𝐴 ·o ( 𝐵 +o 𝑦 ) ) = ( ( 𝐴 ·o 𝐵 ) +o ( 𝐴 ·o 𝑦 ) ) ) ) → ( 𝑧 ∈ 𝐵 → ∃ 𝑤 ∈ ( 𝐴 ·o 𝑥 ) ( 𝐴 ·o 𝑧 ) ⊆ ( ( 𝐴 ·o 𝐵 ) +o 𝑤 ) ) ) |
| 137 |
|
omordi |
⊢ ( ( ( 𝑥 ∈ On ∧ 𝐴 ∈ On ) ∧ ∅ ∈ 𝐴 ) → ( 𝑣 ∈ 𝑥 → ( 𝐴 ·o 𝑣 ) ∈ ( 𝐴 ·o 𝑥 ) ) ) |
| 138 |
61 137
|
sylanl1 |
⊢ ( ( ( Lim 𝑥 ∧ 𝐴 ∈ On ) ∧ ∅ ∈ 𝐴 ) → ( 𝑣 ∈ 𝑥 → ( 𝐴 ·o 𝑣 ) ∈ ( 𝐴 ·o 𝑥 ) ) ) |
| 139 |
138
|
adantrd |
⊢ ( ( ( Lim 𝑥 ∧ 𝐴 ∈ On ) ∧ ∅ ∈ 𝐴 ) → ( ( 𝑣 ∈ 𝑥 ∧ ( 𝐵 +o 𝑣 ) = 𝑧 ) → ( 𝐴 ·o 𝑣 ) ∈ ( 𝐴 ·o 𝑥 ) ) ) |
| 140 |
139
|
adantrr |
⊢ ( ( ( Lim 𝑥 ∧ 𝐴 ∈ On ) ∧ ( ∅ ∈ 𝐴 ∧ ∀ 𝑦 ∈ 𝑥 ( 𝐴 ·o ( 𝐵 +o 𝑦 ) ) = ( ( 𝐴 ·o 𝐵 ) +o ( 𝐴 ·o 𝑦 ) ) ) ) → ( ( 𝑣 ∈ 𝑥 ∧ ( 𝐵 +o 𝑣 ) = 𝑧 ) → ( 𝐴 ·o 𝑣 ) ∈ ( 𝐴 ·o 𝑥 ) ) ) |
| 141 |
|
oveq2 |
⊢ ( 𝑦 = 𝑣 → ( 𝐵 +o 𝑦 ) = ( 𝐵 +o 𝑣 ) ) |
| 142 |
141
|
oveq2d |
⊢ ( 𝑦 = 𝑣 → ( 𝐴 ·o ( 𝐵 +o 𝑦 ) ) = ( 𝐴 ·o ( 𝐵 +o 𝑣 ) ) ) |
| 143 |
|
oveq2 |
⊢ ( 𝑦 = 𝑣 → ( 𝐴 ·o 𝑦 ) = ( 𝐴 ·o 𝑣 ) ) |
| 144 |
143
|
oveq2d |
⊢ ( 𝑦 = 𝑣 → ( ( 𝐴 ·o 𝐵 ) +o ( 𝐴 ·o 𝑦 ) ) = ( ( 𝐴 ·o 𝐵 ) +o ( 𝐴 ·o 𝑣 ) ) ) |
| 145 |
142 144
|
eqeq12d |
⊢ ( 𝑦 = 𝑣 → ( ( 𝐴 ·o ( 𝐵 +o 𝑦 ) ) = ( ( 𝐴 ·o 𝐵 ) +o ( 𝐴 ·o 𝑦 ) ) ↔ ( 𝐴 ·o ( 𝐵 +o 𝑣 ) ) = ( ( 𝐴 ·o 𝐵 ) +o ( 𝐴 ·o 𝑣 ) ) ) ) |
| 146 |
145
|
rspccv |
⊢ ( ∀ 𝑦 ∈ 𝑥 ( 𝐴 ·o ( 𝐵 +o 𝑦 ) ) = ( ( 𝐴 ·o 𝐵 ) +o ( 𝐴 ·o 𝑦 ) ) → ( 𝑣 ∈ 𝑥 → ( 𝐴 ·o ( 𝐵 +o 𝑣 ) ) = ( ( 𝐴 ·o 𝐵 ) +o ( 𝐴 ·o 𝑣 ) ) ) ) |
| 147 |
|
oveq2 |
⊢ ( ( 𝐵 +o 𝑣 ) = 𝑧 → ( 𝐴 ·o ( 𝐵 +o 𝑣 ) ) = ( 𝐴 ·o 𝑧 ) ) |
| 148 |
|
eqeq1 |
⊢ ( ( 𝐴 ·o ( 𝐵 +o 𝑣 ) ) = ( ( 𝐴 ·o 𝐵 ) +o ( 𝐴 ·o 𝑣 ) ) → ( ( 𝐴 ·o ( 𝐵 +o 𝑣 ) ) = ( 𝐴 ·o 𝑧 ) ↔ ( ( 𝐴 ·o 𝐵 ) +o ( 𝐴 ·o 𝑣 ) ) = ( 𝐴 ·o 𝑧 ) ) ) |
| 149 |
147 148
|
imbitrid |
⊢ ( ( 𝐴 ·o ( 𝐵 +o 𝑣 ) ) = ( ( 𝐴 ·o 𝐵 ) +o ( 𝐴 ·o 𝑣 ) ) → ( ( 𝐵 +o 𝑣 ) = 𝑧 → ( ( 𝐴 ·o 𝐵 ) +o ( 𝐴 ·o 𝑣 ) ) = ( 𝐴 ·o 𝑧 ) ) ) |
| 150 |
|
eqimss2 |
⊢ ( ( ( 𝐴 ·o 𝐵 ) +o ( 𝐴 ·o 𝑣 ) ) = ( 𝐴 ·o 𝑧 ) → ( 𝐴 ·o 𝑧 ) ⊆ ( ( 𝐴 ·o 𝐵 ) +o ( 𝐴 ·o 𝑣 ) ) ) |
| 151 |
149 150
|
syl6 |
⊢ ( ( 𝐴 ·o ( 𝐵 +o 𝑣 ) ) = ( ( 𝐴 ·o 𝐵 ) +o ( 𝐴 ·o 𝑣 ) ) → ( ( 𝐵 +o 𝑣 ) = 𝑧 → ( 𝐴 ·o 𝑧 ) ⊆ ( ( 𝐴 ·o 𝐵 ) +o ( 𝐴 ·o 𝑣 ) ) ) ) |
| 152 |
151
|
imim2i |
⊢ ( ( 𝑣 ∈ 𝑥 → ( 𝐴 ·o ( 𝐵 +o 𝑣 ) ) = ( ( 𝐴 ·o 𝐵 ) +o ( 𝐴 ·o 𝑣 ) ) ) → ( 𝑣 ∈ 𝑥 → ( ( 𝐵 +o 𝑣 ) = 𝑧 → ( 𝐴 ·o 𝑧 ) ⊆ ( ( 𝐴 ·o 𝐵 ) +o ( 𝐴 ·o 𝑣 ) ) ) ) ) |
| 153 |
152
|
impd |
⊢ ( ( 𝑣 ∈ 𝑥 → ( 𝐴 ·o ( 𝐵 +o 𝑣 ) ) = ( ( 𝐴 ·o 𝐵 ) +o ( 𝐴 ·o 𝑣 ) ) ) → ( ( 𝑣 ∈ 𝑥 ∧ ( 𝐵 +o 𝑣 ) = 𝑧 ) → ( 𝐴 ·o 𝑧 ) ⊆ ( ( 𝐴 ·o 𝐵 ) +o ( 𝐴 ·o 𝑣 ) ) ) ) |
| 154 |
146 153
|
syl |
⊢ ( ∀ 𝑦 ∈ 𝑥 ( 𝐴 ·o ( 𝐵 +o 𝑦 ) ) = ( ( 𝐴 ·o 𝐵 ) +o ( 𝐴 ·o 𝑦 ) ) → ( ( 𝑣 ∈ 𝑥 ∧ ( 𝐵 +o 𝑣 ) = 𝑧 ) → ( 𝐴 ·o 𝑧 ) ⊆ ( ( 𝐴 ·o 𝐵 ) +o ( 𝐴 ·o 𝑣 ) ) ) ) |
| 155 |
154
|
ad2antll |
⊢ ( ( ( Lim 𝑥 ∧ 𝐴 ∈ On ) ∧ ( ∅ ∈ 𝐴 ∧ ∀ 𝑦 ∈ 𝑥 ( 𝐴 ·o ( 𝐵 +o 𝑦 ) ) = ( ( 𝐴 ·o 𝐵 ) +o ( 𝐴 ·o 𝑦 ) ) ) ) → ( ( 𝑣 ∈ 𝑥 ∧ ( 𝐵 +o 𝑣 ) = 𝑧 ) → ( 𝐴 ·o 𝑧 ) ⊆ ( ( 𝐴 ·o 𝐵 ) +o ( 𝐴 ·o 𝑣 ) ) ) ) |
| 156 |
140 155
|
jcad |
⊢ ( ( ( Lim 𝑥 ∧ 𝐴 ∈ On ) ∧ ( ∅ ∈ 𝐴 ∧ ∀ 𝑦 ∈ 𝑥 ( 𝐴 ·o ( 𝐵 +o 𝑦 ) ) = ( ( 𝐴 ·o 𝐵 ) +o ( 𝐴 ·o 𝑦 ) ) ) ) → ( ( 𝑣 ∈ 𝑥 ∧ ( 𝐵 +o 𝑣 ) = 𝑧 ) → ( ( 𝐴 ·o 𝑣 ) ∈ ( 𝐴 ·o 𝑥 ) ∧ ( 𝐴 ·o 𝑧 ) ⊆ ( ( 𝐴 ·o 𝐵 ) +o ( 𝐴 ·o 𝑣 ) ) ) ) ) |
| 157 |
|
oveq2 |
⊢ ( 𝑤 = ( 𝐴 ·o 𝑣 ) → ( ( 𝐴 ·o 𝐵 ) +o 𝑤 ) = ( ( 𝐴 ·o 𝐵 ) +o ( 𝐴 ·o 𝑣 ) ) ) |
| 158 |
157
|
sseq2d |
⊢ ( 𝑤 = ( 𝐴 ·o 𝑣 ) → ( ( 𝐴 ·o 𝑧 ) ⊆ ( ( 𝐴 ·o 𝐵 ) +o 𝑤 ) ↔ ( 𝐴 ·o 𝑧 ) ⊆ ( ( 𝐴 ·o 𝐵 ) +o ( 𝐴 ·o 𝑣 ) ) ) ) |
| 159 |
158
|
rspcev |
⊢ ( ( ( 𝐴 ·o 𝑣 ) ∈ ( 𝐴 ·o 𝑥 ) ∧ ( 𝐴 ·o 𝑧 ) ⊆ ( ( 𝐴 ·o 𝐵 ) +o ( 𝐴 ·o 𝑣 ) ) ) → ∃ 𝑤 ∈ ( 𝐴 ·o 𝑥 ) ( 𝐴 ·o 𝑧 ) ⊆ ( ( 𝐴 ·o 𝐵 ) +o 𝑤 ) ) |
| 160 |
156 159
|
syl6 |
⊢ ( ( ( Lim 𝑥 ∧ 𝐴 ∈ On ) ∧ ( ∅ ∈ 𝐴 ∧ ∀ 𝑦 ∈ 𝑥 ( 𝐴 ·o ( 𝐵 +o 𝑦 ) ) = ( ( 𝐴 ·o 𝐵 ) +o ( 𝐴 ·o 𝑦 ) ) ) ) → ( ( 𝑣 ∈ 𝑥 ∧ ( 𝐵 +o 𝑣 ) = 𝑧 ) → ∃ 𝑤 ∈ ( 𝐴 ·o 𝑥 ) ( 𝐴 ·o 𝑧 ) ⊆ ( ( 𝐴 ·o 𝐵 ) +o 𝑤 ) ) ) |
| 161 |
160
|
rexlimdvw |
⊢ ( ( ( Lim 𝑥 ∧ 𝐴 ∈ On ) ∧ ( ∅ ∈ 𝐴 ∧ ∀ 𝑦 ∈ 𝑥 ( 𝐴 ·o ( 𝐵 +o 𝑦 ) ) = ( ( 𝐴 ·o 𝐵 ) +o ( 𝐴 ·o 𝑦 ) ) ) ) → ( ∃ 𝑣 ∈ On ( 𝑣 ∈ 𝑥 ∧ ( 𝐵 +o 𝑣 ) = 𝑧 ) → ∃ 𝑤 ∈ ( 𝐴 ·o 𝑥 ) ( 𝐴 ·o 𝑧 ) ⊆ ( ( 𝐴 ·o 𝐵 ) +o 𝑤 ) ) ) |
| 162 |
161
|
adantlrr |
⊢ ( ( ( Lim 𝑥 ∧ ( 𝐴 ∈ On ∧ 𝐵 ∈ On ) ) ∧ ( ∅ ∈ 𝐴 ∧ ∀ 𝑦 ∈ 𝑥 ( 𝐴 ·o ( 𝐵 +o 𝑦 ) ) = ( ( 𝐴 ·o 𝐵 ) +o ( 𝐴 ·o 𝑦 ) ) ) ) → ( ∃ 𝑣 ∈ On ( 𝑣 ∈ 𝑥 ∧ ( 𝐵 +o 𝑣 ) = 𝑧 ) → ∃ 𝑤 ∈ ( 𝐴 ·o 𝑥 ) ( 𝐴 ·o 𝑧 ) ⊆ ( ( 𝐴 ·o 𝐵 ) +o 𝑤 ) ) ) |
| 163 |
136 162
|
jaod |
⊢ ( ( ( Lim 𝑥 ∧ ( 𝐴 ∈ On ∧ 𝐵 ∈ On ) ) ∧ ( ∅ ∈ 𝐴 ∧ ∀ 𝑦 ∈ 𝑥 ( 𝐴 ·o ( 𝐵 +o 𝑦 ) ) = ( ( 𝐴 ·o 𝐵 ) +o ( 𝐴 ·o 𝑦 ) ) ) ) → ( ( 𝑧 ∈ 𝐵 ∨ ∃ 𝑣 ∈ On ( 𝑣 ∈ 𝑥 ∧ ( 𝐵 +o 𝑣 ) = 𝑧 ) ) → ∃ 𝑤 ∈ ( 𝐴 ·o 𝑥 ) ( 𝐴 ·o 𝑧 ) ⊆ ( ( 𝐴 ·o 𝐵 ) +o 𝑤 ) ) ) |
| 164 |
163
|
adantr |
⊢ ( ( ( ( Lim 𝑥 ∧ ( 𝐴 ∈ On ∧ 𝐵 ∈ On ) ) ∧ ( ∅ ∈ 𝐴 ∧ ∀ 𝑦 ∈ 𝑥 ( 𝐴 ·o ( 𝐵 +o 𝑦 ) ) = ( ( 𝐴 ·o 𝐵 ) +o ( 𝐴 ·o 𝑦 ) ) ) ) ∧ 𝑧 ∈ ( 𝐵 +o 𝑥 ) ) → ( ( 𝑧 ∈ 𝐵 ∨ ∃ 𝑣 ∈ On ( 𝑣 ∈ 𝑥 ∧ ( 𝐵 +o 𝑣 ) = 𝑧 ) ) → ∃ 𝑤 ∈ ( 𝐴 ·o 𝑥 ) ( 𝐴 ·o 𝑧 ) ⊆ ( ( 𝐴 ·o 𝐵 ) +o 𝑤 ) ) ) |
| 165 |
110 164
|
mpd |
⊢ ( ( ( ( Lim 𝑥 ∧ ( 𝐴 ∈ On ∧ 𝐵 ∈ On ) ) ∧ ( ∅ ∈ 𝐴 ∧ ∀ 𝑦 ∈ 𝑥 ( 𝐴 ·o ( 𝐵 +o 𝑦 ) ) = ( ( 𝐴 ·o 𝐵 ) +o ( 𝐴 ·o 𝑦 ) ) ) ) ∧ 𝑧 ∈ ( 𝐵 +o 𝑥 ) ) → ∃ 𝑤 ∈ ( 𝐴 ·o 𝑥 ) ( 𝐴 ·o 𝑧 ) ⊆ ( ( 𝐴 ·o 𝐵 ) +o 𝑤 ) ) |
| 166 |
165
|
ralrimiva |
⊢ ( ( ( Lim 𝑥 ∧ ( 𝐴 ∈ On ∧ 𝐵 ∈ On ) ) ∧ ( ∅ ∈ 𝐴 ∧ ∀ 𝑦 ∈ 𝑥 ( 𝐴 ·o ( 𝐵 +o 𝑦 ) ) = ( ( 𝐴 ·o 𝐵 ) +o ( 𝐴 ·o 𝑦 ) ) ) ) → ∀ 𝑧 ∈ ( 𝐵 +o 𝑥 ) ∃ 𝑤 ∈ ( 𝐴 ·o 𝑥 ) ( 𝐴 ·o 𝑧 ) ⊆ ( ( 𝐴 ·o 𝐵 ) +o 𝑤 ) ) |
| 167 |
|
iunss2 |
⊢ ( ∀ 𝑧 ∈ ( 𝐵 +o 𝑥 ) ∃ 𝑤 ∈ ( 𝐴 ·o 𝑥 ) ( 𝐴 ·o 𝑧 ) ⊆ ( ( 𝐴 ·o 𝐵 ) +o 𝑤 ) → ∪ 𝑧 ∈ ( 𝐵 +o 𝑥 ) ( 𝐴 ·o 𝑧 ) ⊆ ∪ 𝑤 ∈ ( 𝐴 ·o 𝑥 ) ( ( 𝐴 ·o 𝐵 ) +o 𝑤 ) ) |
| 168 |
166 167
|
syl |
⊢ ( ( ( Lim 𝑥 ∧ ( 𝐴 ∈ On ∧ 𝐵 ∈ On ) ) ∧ ( ∅ ∈ 𝐴 ∧ ∀ 𝑦 ∈ 𝑥 ( 𝐴 ·o ( 𝐵 +o 𝑦 ) ) = ( ( 𝐴 ·o 𝐵 ) +o ( 𝐴 ·o 𝑦 ) ) ) ) → ∪ 𝑧 ∈ ( 𝐵 +o 𝑥 ) ( 𝐴 ·o 𝑧 ) ⊆ ∪ 𝑤 ∈ ( 𝐴 ·o 𝑥 ) ( ( 𝐴 ·o 𝐵 ) +o 𝑤 ) ) |
| 169 |
|
omordlim |
⊢ ( ( ( 𝐴 ∈ On ∧ ( 𝑥 ∈ V ∧ Lim 𝑥 ) ) ∧ 𝑤 ∈ ( 𝐴 ·o 𝑥 ) ) → ∃ 𝑣 ∈ 𝑥 𝑤 ∈ ( 𝐴 ·o 𝑣 ) ) |
| 170 |
169
|
ex |
⊢ ( ( 𝐴 ∈ On ∧ ( 𝑥 ∈ V ∧ Lim 𝑥 ) ) → ( 𝑤 ∈ ( 𝐴 ·o 𝑥 ) → ∃ 𝑣 ∈ 𝑥 𝑤 ∈ ( 𝐴 ·o 𝑣 ) ) ) |
| 171 |
59 170
|
mpanr1 |
⊢ ( ( 𝐴 ∈ On ∧ Lim 𝑥 ) → ( 𝑤 ∈ ( 𝐴 ·o 𝑥 ) → ∃ 𝑣 ∈ 𝑥 𝑤 ∈ ( 𝐴 ·o 𝑣 ) ) ) |
| 172 |
171
|
ancoms |
⊢ ( ( Lim 𝑥 ∧ 𝐴 ∈ On ) → ( 𝑤 ∈ ( 𝐴 ·o 𝑥 ) → ∃ 𝑣 ∈ 𝑥 𝑤 ∈ ( 𝐴 ·o 𝑣 ) ) ) |
| 173 |
172
|
imp |
⊢ ( ( ( Lim 𝑥 ∧ 𝐴 ∈ On ) ∧ 𝑤 ∈ ( 𝐴 ·o 𝑥 ) ) → ∃ 𝑣 ∈ 𝑥 𝑤 ∈ ( 𝐴 ·o 𝑣 ) ) |
| 174 |
173
|
adantlrr |
⊢ ( ( ( Lim 𝑥 ∧ ( 𝐴 ∈ On ∧ 𝐵 ∈ On ) ) ∧ 𝑤 ∈ ( 𝐴 ·o 𝑥 ) ) → ∃ 𝑣 ∈ 𝑥 𝑤 ∈ ( 𝐴 ·o 𝑣 ) ) |
| 175 |
174
|
adantlr |
⊢ ( ( ( ( Lim 𝑥 ∧ ( 𝐴 ∈ On ∧ 𝐵 ∈ On ) ) ∧ ∀ 𝑦 ∈ 𝑥 ( 𝐴 ·o ( 𝐵 +o 𝑦 ) ) = ( ( 𝐴 ·o 𝐵 ) +o ( 𝐴 ·o 𝑦 ) ) ) ∧ 𝑤 ∈ ( 𝐴 ·o 𝑥 ) ) → ∃ 𝑣 ∈ 𝑥 𝑤 ∈ ( 𝐴 ·o 𝑣 ) ) |
| 176 |
|
oaordi |
⊢ ( ( 𝑥 ∈ On ∧ 𝐵 ∈ On ) → ( 𝑣 ∈ 𝑥 → ( 𝐵 +o 𝑣 ) ∈ ( 𝐵 +o 𝑥 ) ) ) |
| 177 |
61 176
|
sylan |
⊢ ( ( Lim 𝑥 ∧ 𝐵 ∈ On ) → ( 𝑣 ∈ 𝑥 → ( 𝐵 +o 𝑣 ) ∈ ( 𝐵 +o 𝑥 ) ) ) |
| 178 |
177
|
imp |
⊢ ( ( ( Lim 𝑥 ∧ 𝐵 ∈ On ) ∧ 𝑣 ∈ 𝑥 ) → ( 𝐵 +o 𝑣 ) ∈ ( 𝐵 +o 𝑥 ) ) |
| 179 |
178
|
adantlrl |
⊢ ( ( ( Lim 𝑥 ∧ ( 𝐴 ∈ On ∧ 𝐵 ∈ On ) ) ∧ 𝑣 ∈ 𝑥 ) → ( 𝐵 +o 𝑣 ) ∈ ( 𝐵 +o 𝑥 ) ) |
| 180 |
179
|
a1d |
⊢ ( ( ( Lim 𝑥 ∧ ( 𝐴 ∈ On ∧ 𝐵 ∈ On ) ) ∧ 𝑣 ∈ 𝑥 ) → ( 𝑤 ∈ ( 𝐴 ·o 𝑣 ) → ( 𝐵 +o 𝑣 ) ∈ ( 𝐵 +o 𝑥 ) ) ) |
| 181 |
180
|
adantlr |
⊢ ( ( ( ( Lim 𝑥 ∧ ( 𝐴 ∈ On ∧ 𝐵 ∈ On ) ) ∧ ∀ 𝑦 ∈ 𝑥 ( 𝐴 ·o ( 𝐵 +o 𝑦 ) ) = ( ( 𝐴 ·o 𝐵 ) +o ( 𝐴 ·o 𝑦 ) ) ) ∧ 𝑣 ∈ 𝑥 ) → ( 𝑤 ∈ ( 𝐴 ·o 𝑣 ) → ( 𝐵 +o 𝑣 ) ∈ ( 𝐵 +o 𝑥 ) ) ) |
| 182 |
|
limord |
⊢ ( Lim 𝑥 → Ord 𝑥 ) |
| 183 |
|
ordelon |
⊢ ( ( Ord 𝑥 ∧ 𝑣 ∈ 𝑥 ) → 𝑣 ∈ On ) |
| 184 |
182 183
|
sylan |
⊢ ( ( Lim 𝑥 ∧ 𝑣 ∈ 𝑥 ) → 𝑣 ∈ On ) |
| 185 |
|
omcl |
⊢ ( ( 𝐴 ∈ On ∧ 𝑣 ∈ On ) → ( 𝐴 ·o 𝑣 ) ∈ On ) |
| 186 |
185
|
ancoms |
⊢ ( ( 𝑣 ∈ On ∧ 𝐴 ∈ On ) → ( 𝐴 ·o 𝑣 ) ∈ On ) |
| 187 |
186
|
adantrr |
⊢ ( ( 𝑣 ∈ On ∧ ( 𝐴 ∈ On ∧ 𝐵 ∈ On ) ) → ( 𝐴 ·o 𝑣 ) ∈ On ) |
| 188 |
21
|
adantl |
⊢ ( ( 𝑣 ∈ On ∧ ( 𝐴 ∈ On ∧ 𝐵 ∈ On ) ) → ( 𝐴 ·o 𝐵 ) ∈ On ) |
| 189 |
|
oaordi |
⊢ ( ( ( 𝐴 ·o 𝑣 ) ∈ On ∧ ( 𝐴 ·o 𝐵 ) ∈ On ) → ( 𝑤 ∈ ( 𝐴 ·o 𝑣 ) → ( ( 𝐴 ·o 𝐵 ) +o 𝑤 ) ∈ ( ( 𝐴 ·o 𝐵 ) +o ( 𝐴 ·o 𝑣 ) ) ) ) |
| 190 |
187 188 189
|
syl2anc |
⊢ ( ( 𝑣 ∈ On ∧ ( 𝐴 ∈ On ∧ 𝐵 ∈ On ) ) → ( 𝑤 ∈ ( 𝐴 ·o 𝑣 ) → ( ( 𝐴 ·o 𝐵 ) +o 𝑤 ) ∈ ( ( 𝐴 ·o 𝐵 ) +o ( 𝐴 ·o 𝑣 ) ) ) ) |
| 191 |
184 190
|
sylan |
⊢ ( ( ( Lim 𝑥 ∧ 𝑣 ∈ 𝑥 ) ∧ ( 𝐴 ∈ On ∧ 𝐵 ∈ On ) ) → ( 𝑤 ∈ ( 𝐴 ·o 𝑣 ) → ( ( 𝐴 ·o 𝐵 ) +o 𝑤 ) ∈ ( ( 𝐴 ·o 𝐵 ) +o ( 𝐴 ·o 𝑣 ) ) ) ) |
| 192 |
191
|
an32s |
⊢ ( ( ( Lim 𝑥 ∧ ( 𝐴 ∈ On ∧ 𝐵 ∈ On ) ) ∧ 𝑣 ∈ 𝑥 ) → ( 𝑤 ∈ ( 𝐴 ·o 𝑣 ) → ( ( 𝐴 ·o 𝐵 ) +o 𝑤 ) ∈ ( ( 𝐴 ·o 𝐵 ) +o ( 𝐴 ·o 𝑣 ) ) ) ) |
| 193 |
192
|
adantlr |
⊢ ( ( ( ( Lim 𝑥 ∧ ( 𝐴 ∈ On ∧ 𝐵 ∈ On ) ) ∧ ∀ 𝑦 ∈ 𝑥 ( 𝐴 ·o ( 𝐵 +o 𝑦 ) ) = ( ( 𝐴 ·o 𝐵 ) +o ( 𝐴 ·o 𝑦 ) ) ) ∧ 𝑣 ∈ 𝑥 ) → ( 𝑤 ∈ ( 𝐴 ·o 𝑣 ) → ( ( 𝐴 ·o 𝐵 ) +o 𝑤 ) ∈ ( ( 𝐴 ·o 𝐵 ) +o ( 𝐴 ·o 𝑣 ) ) ) ) |
| 194 |
145
|
rspccva |
⊢ ( ( ∀ 𝑦 ∈ 𝑥 ( 𝐴 ·o ( 𝐵 +o 𝑦 ) ) = ( ( 𝐴 ·o 𝐵 ) +o ( 𝐴 ·o 𝑦 ) ) ∧ 𝑣 ∈ 𝑥 ) → ( 𝐴 ·o ( 𝐵 +o 𝑣 ) ) = ( ( 𝐴 ·o 𝐵 ) +o ( 𝐴 ·o 𝑣 ) ) ) |
| 195 |
194
|
eleq2d |
⊢ ( ( ∀ 𝑦 ∈ 𝑥 ( 𝐴 ·o ( 𝐵 +o 𝑦 ) ) = ( ( 𝐴 ·o 𝐵 ) +o ( 𝐴 ·o 𝑦 ) ) ∧ 𝑣 ∈ 𝑥 ) → ( ( ( 𝐴 ·o 𝐵 ) +o 𝑤 ) ∈ ( 𝐴 ·o ( 𝐵 +o 𝑣 ) ) ↔ ( ( 𝐴 ·o 𝐵 ) +o 𝑤 ) ∈ ( ( 𝐴 ·o 𝐵 ) +o ( 𝐴 ·o 𝑣 ) ) ) ) |
| 196 |
195
|
adantll |
⊢ ( ( ( ( Lim 𝑥 ∧ ( 𝐴 ∈ On ∧ 𝐵 ∈ On ) ) ∧ ∀ 𝑦 ∈ 𝑥 ( 𝐴 ·o ( 𝐵 +o 𝑦 ) ) = ( ( 𝐴 ·o 𝐵 ) +o ( 𝐴 ·o 𝑦 ) ) ) ∧ 𝑣 ∈ 𝑥 ) → ( ( ( 𝐴 ·o 𝐵 ) +o 𝑤 ) ∈ ( 𝐴 ·o ( 𝐵 +o 𝑣 ) ) ↔ ( ( 𝐴 ·o 𝐵 ) +o 𝑤 ) ∈ ( ( 𝐴 ·o 𝐵 ) +o ( 𝐴 ·o 𝑣 ) ) ) ) |
| 197 |
193 196
|
sylibrd |
⊢ ( ( ( ( Lim 𝑥 ∧ ( 𝐴 ∈ On ∧ 𝐵 ∈ On ) ) ∧ ∀ 𝑦 ∈ 𝑥 ( 𝐴 ·o ( 𝐵 +o 𝑦 ) ) = ( ( 𝐴 ·o 𝐵 ) +o ( 𝐴 ·o 𝑦 ) ) ) ∧ 𝑣 ∈ 𝑥 ) → ( 𝑤 ∈ ( 𝐴 ·o 𝑣 ) → ( ( 𝐴 ·o 𝐵 ) +o 𝑤 ) ∈ ( 𝐴 ·o ( 𝐵 +o 𝑣 ) ) ) ) |
| 198 |
|
oacl |
⊢ ( ( 𝐵 ∈ On ∧ 𝑣 ∈ On ) → ( 𝐵 +o 𝑣 ) ∈ On ) |
| 199 |
198
|
ancoms |
⊢ ( ( 𝑣 ∈ On ∧ 𝐵 ∈ On ) → ( 𝐵 +o 𝑣 ) ∈ On ) |
| 200 |
|
omcl |
⊢ ( ( 𝐴 ∈ On ∧ ( 𝐵 +o 𝑣 ) ∈ On ) → ( 𝐴 ·o ( 𝐵 +o 𝑣 ) ) ∈ On ) |
| 201 |
199 200
|
sylan2 |
⊢ ( ( 𝐴 ∈ On ∧ ( 𝑣 ∈ On ∧ 𝐵 ∈ On ) ) → ( 𝐴 ·o ( 𝐵 +o 𝑣 ) ) ∈ On ) |
| 202 |
201
|
an12s |
⊢ ( ( 𝑣 ∈ On ∧ ( 𝐴 ∈ On ∧ 𝐵 ∈ On ) ) → ( 𝐴 ·o ( 𝐵 +o 𝑣 ) ) ∈ On ) |
| 203 |
184 202
|
sylan |
⊢ ( ( ( Lim 𝑥 ∧ 𝑣 ∈ 𝑥 ) ∧ ( 𝐴 ∈ On ∧ 𝐵 ∈ On ) ) → ( 𝐴 ·o ( 𝐵 +o 𝑣 ) ) ∈ On ) |
| 204 |
203
|
an32s |
⊢ ( ( ( Lim 𝑥 ∧ ( 𝐴 ∈ On ∧ 𝐵 ∈ On ) ) ∧ 𝑣 ∈ 𝑥 ) → ( 𝐴 ·o ( 𝐵 +o 𝑣 ) ) ∈ On ) |
| 205 |
|
onelss |
⊢ ( ( 𝐴 ·o ( 𝐵 +o 𝑣 ) ) ∈ On → ( ( ( 𝐴 ·o 𝐵 ) +o 𝑤 ) ∈ ( 𝐴 ·o ( 𝐵 +o 𝑣 ) ) → ( ( 𝐴 ·o 𝐵 ) +o 𝑤 ) ⊆ ( 𝐴 ·o ( 𝐵 +o 𝑣 ) ) ) ) |
| 206 |
204 205
|
syl |
⊢ ( ( ( Lim 𝑥 ∧ ( 𝐴 ∈ On ∧ 𝐵 ∈ On ) ) ∧ 𝑣 ∈ 𝑥 ) → ( ( ( 𝐴 ·o 𝐵 ) +o 𝑤 ) ∈ ( 𝐴 ·o ( 𝐵 +o 𝑣 ) ) → ( ( 𝐴 ·o 𝐵 ) +o 𝑤 ) ⊆ ( 𝐴 ·o ( 𝐵 +o 𝑣 ) ) ) ) |
| 207 |
206
|
adantlr |
⊢ ( ( ( ( Lim 𝑥 ∧ ( 𝐴 ∈ On ∧ 𝐵 ∈ On ) ) ∧ ∀ 𝑦 ∈ 𝑥 ( 𝐴 ·o ( 𝐵 +o 𝑦 ) ) = ( ( 𝐴 ·o 𝐵 ) +o ( 𝐴 ·o 𝑦 ) ) ) ∧ 𝑣 ∈ 𝑥 ) → ( ( ( 𝐴 ·o 𝐵 ) +o 𝑤 ) ∈ ( 𝐴 ·o ( 𝐵 +o 𝑣 ) ) → ( ( 𝐴 ·o 𝐵 ) +o 𝑤 ) ⊆ ( 𝐴 ·o ( 𝐵 +o 𝑣 ) ) ) ) |
| 208 |
197 207
|
syld |
⊢ ( ( ( ( Lim 𝑥 ∧ ( 𝐴 ∈ On ∧ 𝐵 ∈ On ) ) ∧ ∀ 𝑦 ∈ 𝑥 ( 𝐴 ·o ( 𝐵 +o 𝑦 ) ) = ( ( 𝐴 ·o 𝐵 ) +o ( 𝐴 ·o 𝑦 ) ) ) ∧ 𝑣 ∈ 𝑥 ) → ( 𝑤 ∈ ( 𝐴 ·o 𝑣 ) → ( ( 𝐴 ·o 𝐵 ) +o 𝑤 ) ⊆ ( 𝐴 ·o ( 𝐵 +o 𝑣 ) ) ) ) |
| 209 |
181 208
|
jcad |
⊢ ( ( ( ( Lim 𝑥 ∧ ( 𝐴 ∈ On ∧ 𝐵 ∈ On ) ) ∧ ∀ 𝑦 ∈ 𝑥 ( 𝐴 ·o ( 𝐵 +o 𝑦 ) ) = ( ( 𝐴 ·o 𝐵 ) +o ( 𝐴 ·o 𝑦 ) ) ) ∧ 𝑣 ∈ 𝑥 ) → ( 𝑤 ∈ ( 𝐴 ·o 𝑣 ) → ( ( 𝐵 +o 𝑣 ) ∈ ( 𝐵 +o 𝑥 ) ∧ ( ( 𝐴 ·o 𝐵 ) +o 𝑤 ) ⊆ ( 𝐴 ·o ( 𝐵 +o 𝑣 ) ) ) ) ) |
| 210 |
|
oveq2 |
⊢ ( 𝑧 = ( 𝐵 +o 𝑣 ) → ( 𝐴 ·o 𝑧 ) = ( 𝐴 ·o ( 𝐵 +o 𝑣 ) ) ) |
| 211 |
210
|
sseq2d |
⊢ ( 𝑧 = ( 𝐵 +o 𝑣 ) → ( ( ( 𝐴 ·o 𝐵 ) +o 𝑤 ) ⊆ ( 𝐴 ·o 𝑧 ) ↔ ( ( 𝐴 ·o 𝐵 ) +o 𝑤 ) ⊆ ( 𝐴 ·o ( 𝐵 +o 𝑣 ) ) ) ) |
| 212 |
211
|
rspcev |
⊢ ( ( ( 𝐵 +o 𝑣 ) ∈ ( 𝐵 +o 𝑥 ) ∧ ( ( 𝐴 ·o 𝐵 ) +o 𝑤 ) ⊆ ( 𝐴 ·o ( 𝐵 +o 𝑣 ) ) ) → ∃ 𝑧 ∈ ( 𝐵 +o 𝑥 ) ( ( 𝐴 ·o 𝐵 ) +o 𝑤 ) ⊆ ( 𝐴 ·o 𝑧 ) ) |
| 213 |
209 212
|
syl6 |
⊢ ( ( ( ( Lim 𝑥 ∧ ( 𝐴 ∈ On ∧ 𝐵 ∈ On ) ) ∧ ∀ 𝑦 ∈ 𝑥 ( 𝐴 ·o ( 𝐵 +o 𝑦 ) ) = ( ( 𝐴 ·o 𝐵 ) +o ( 𝐴 ·o 𝑦 ) ) ) ∧ 𝑣 ∈ 𝑥 ) → ( 𝑤 ∈ ( 𝐴 ·o 𝑣 ) → ∃ 𝑧 ∈ ( 𝐵 +o 𝑥 ) ( ( 𝐴 ·o 𝐵 ) +o 𝑤 ) ⊆ ( 𝐴 ·o 𝑧 ) ) ) |
| 214 |
213
|
rexlimdva |
⊢ ( ( ( Lim 𝑥 ∧ ( 𝐴 ∈ On ∧ 𝐵 ∈ On ) ) ∧ ∀ 𝑦 ∈ 𝑥 ( 𝐴 ·o ( 𝐵 +o 𝑦 ) ) = ( ( 𝐴 ·o 𝐵 ) +o ( 𝐴 ·o 𝑦 ) ) ) → ( ∃ 𝑣 ∈ 𝑥 𝑤 ∈ ( 𝐴 ·o 𝑣 ) → ∃ 𝑧 ∈ ( 𝐵 +o 𝑥 ) ( ( 𝐴 ·o 𝐵 ) +o 𝑤 ) ⊆ ( 𝐴 ·o 𝑧 ) ) ) |
| 215 |
214
|
adantr |
⊢ ( ( ( ( Lim 𝑥 ∧ ( 𝐴 ∈ On ∧ 𝐵 ∈ On ) ) ∧ ∀ 𝑦 ∈ 𝑥 ( 𝐴 ·o ( 𝐵 +o 𝑦 ) ) = ( ( 𝐴 ·o 𝐵 ) +o ( 𝐴 ·o 𝑦 ) ) ) ∧ 𝑤 ∈ ( 𝐴 ·o 𝑥 ) ) → ( ∃ 𝑣 ∈ 𝑥 𝑤 ∈ ( 𝐴 ·o 𝑣 ) → ∃ 𝑧 ∈ ( 𝐵 +o 𝑥 ) ( ( 𝐴 ·o 𝐵 ) +o 𝑤 ) ⊆ ( 𝐴 ·o 𝑧 ) ) ) |
| 216 |
175 215
|
mpd |
⊢ ( ( ( ( Lim 𝑥 ∧ ( 𝐴 ∈ On ∧ 𝐵 ∈ On ) ) ∧ ∀ 𝑦 ∈ 𝑥 ( 𝐴 ·o ( 𝐵 +o 𝑦 ) ) = ( ( 𝐴 ·o 𝐵 ) +o ( 𝐴 ·o 𝑦 ) ) ) ∧ 𝑤 ∈ ( 𝐴 ·o 𝑥 ) ) → ∃ 𝑧 ∈ ( 𝐵 +o 𝑥 ) ( ( 𝐴 ·o 𝐵 ) +o 𝑤 ) ⊆ ( 𝐴 ·o 𝑧 ) ) |
| 217 |
216
|
ralrimiva |
⊢ ( ( ( Lim 𝑥 ∧ ( 𝐴 ∈ On ∧ 𝐵 ∈ On ) ) ∧ ∀ 𝑦 ∈ 𝑥 ( 𝐴 ·o ( 𝐵 +o 𝑦 ) ) = ( ( 𝐴 ·o 𝐵 ) +o ( 𝐴 ·o 𝑦 ) ) ) → ∀ 𝑤 ∈ ( 𝐴 ·o 𝑥 ) ∃ 𝑧 ∈ ( 𝐵 +o 𝑥 ) ( ( 𝐴 ·o 𝐵 ) +o 𝑤 ) ⊆ ( 𝐴 ·o 𝑧 ) ) |
| 218 |
|
iunss2 |
⊢ ( ∀ 𝑤 ∈ ( 𝐴 ·o 𝑥 ) ∃ 𝑧 ∈ ( 𝐵 +o 𝑥 ) ( ( 𝐴 ·o 𝐵 ) +o 𝑤 ) ⊆ ( 𝐴 ·o 𝑧 ) → ∪ 𝑤 ∈ ( 𝐴 ·o 𝑥 ) ( ( 𝐴 ·o 𝐵 ) +o 𝑤 ) ⊆ ∪ 𝑧 ∈ ( 𝐵 +o 𝑥 ) ( 𝐴 ·o 𝑧 ) ) |
| 219 |
217 218
|
syl |
⊢ ( ( ( Lim 𝑥 ∧ ( 𝐴 ∈ On ∧ 𝐵 ∈ On ) ) ∧ ∀ 𝑦 ∈ 𝑥 ( 𝐴 ·o ( 𝐵 +o 𝑦 ) ) = ( ( 𝐴 ·o 𝐵 ) +o ( 𝐴 ·o 𝑦 ) ) ) → ∪ 𝑤 ∈ ( 𝐴 ·o 𝑥 ) ( ( 𝐴 ·o 𝐵 ) +o 𝑤 ) ⊆ ∪ 𝑧 ∈ ( 𝐵 +o 𝑥 ) ( 𝐴 ·o 𝑧 ) ) |
| 220 |
219
|
adantrl |
⊢ ( ( ( Lim 𝑥 ∧ ( 𝐴 ∈ On ∧ 𝐵 ∈ On ) ) ∧ ( ∅ ∈ 𝐴 ∧ ∀ 𝑦 ∈ 𝑥 ( 𝐴 ·o ( 𝐵 +o 𝑦 ) ) = ( ( 𝐴 ·o 𝐵 ) +o ( 𝐴 ·o 𝑦 ) ) ) ) → ∪ 𝑤 ∈ ( 𝐴 ·o 𝑥 ) ( ( 𝐴 ·o 𝐵 ) +o 𝑤 ) ⊆ ∪ 𝑧 ∈ ( 𝐵 +o 𝑥 ) ( 𝐴 ·o 𝑧 ) ) |
| 221 |
168 220
|
eqssd |
⊢ ( ( ( Lim 𝑥 ∧ ( 𝐴 ∈ On ∧ 𝐵 ∈ On ) ) ∧ ( ∅ ∈ 𝐴 ∧ ∀ 𝑦 ∈ 𝑥 ( 𝐴 ·o ( 𝐵 +o 𝑦 ) ) = ( ( 𝐴 ·o 𝐵 ) +o ( 𝐴 ·o 𝑦 ) ) ) ) → ∪ 𝑧 ∈ ( 𝐵 +o 𝑥 ) ( 𝐴 ·o 𝑧 ) = ∪ 𝑤 ∈ ( 𝐴 ·o 𝑥 ) ( ( 𝐴 ·o 𝐵 ) +o 𝑤 ) ) |
| 222 |
|
oalimcl |
⊢ ( ( 𝐵 ∈ On ∧ ( 𝑥 ∈ V ∧ Lim 𝑥 ) ) → Lim ( 𝐵 +o 𝑥 ) ) |
| 223 |
59 222
|
mpanr1 |
⊢ ( ( 𝐵 ∈ On ∧ Lim 𝑥 ) → Lim ( 𝐵 +o 𝑥 ) ) |
| 224 |
223
|
ancoms |
⊢ ( ( Lim 𝑥 ∧ 𝐵 ∈ On ) → Lim ( 𝐵 +o 𝑥 ) ) |
| 225 |
224
|
anim2i |
⊢ ( ( 𝐴 ∈ On ∧ ( Lim 𝑥 ∧ 𝐵 ∈ On ) ) → ( 𝐴 ∈ On ∧ Lim ( 𝐵 +o 𝑥 ) ) ) |
| 226 |
225
|
an12s |
⊢ ( ( Lim 𝑥 ∧ ( 𝐴 ∈ On ∧ 𝐵 ∈ On ) ) → ( 𝐴 ∈ On ∧ Lim ( 𝐵 +o 𝑥 ) ) ) |
| 227 |
|
ovex |
⊢ ( 𝐵 +o 𝑥 ) ∈ V |
| 228 |
|
omlim |
⊢ ( ( 𝐴 ∈ On ∧ ( ( 𝐵 +o 𝑥 ) ∈ V ∧ Lim ( 𝐵 +o 𝑥 ) ) ) → ( 𝐴 ·o ( 𝐵 +o 𝑥 ) ) = ∪ 𝑧 ∈ ( 𝐵 +o 𝑥 ) ( 𝐴 ·o 𝑧 ) ) |
| 229 |
227 228
|
mpanr1 |
⊢ ( ( 𝐴 ∈ On ∧ Lim ( 𝐵 +o 𝑥 ) ) → ( 𝐴 ·o ( 𝐵 +o 𝑥 ) ) = ∪ 𝑧 ∈ ( 𝐵 +o 𝑥 ) ( 𝐴 ·o 𝑧 ) ) |
| 230 |
226 229
|
syl |
⊢ ( ( Lim 𝑥 ∧ ( 𝐴 ∈ On ∧ 𝐵 ∈ On ) ) → ( 𝐴 ·o ( 𝐵 +o 𝑥 ) ) = ∪ 𝑧 ∈ ( 𝐵 +o 𝑥 ) ( 𝐴 ·o 𝑧 ) ) |
| 231 |
230
|
adantr |
⊢ ( ( ( Lim 𝑥 ∧ ( 𝐴 ∈ On ∧ 𝐵 ∈ On ) ) ∧ ( ∅ ∈ 𝐴 ∧ ∀ 𝑦 ∈ 𝑥 ( 𝐴 ·o ( 𝐵 +o 𝑦 ) ) = ( ( 𝐴 ·o 𝐵 ) +o ( 𝐴 ·o 𝑦 ) ) ) ) → ( 𝐴 ·o ( 𝐵 +o 𝑥 ) ) = ∪ 𝑧 ∈ ( 𝐵 +o 𝑥 ) ( 𝐴 ·o 𝑧 ) ) |
| 232 |
21
|
ad2antlr |
⊢ ( ( ( Lim 𝑥 ∧ ( 𝐴 ∈ On ∧ 𝐵 ∈ On ) ) ∧ ∅ ∈ 𝐴 ) → ( 𝐴 ·o 𝐵 ) ∈ On ) |
| 233 |
59
|
jctl |
⊢ ( Lim 𝑥 → ( 𝑥 ∈ V ∧ Lim 𝑥 ) ) |
| 234 |
233
|
anim1ci |
⊢ ( ( Lim 𝑥 ∧ 𝐴 ∈ On ) → ( 𝐴 ∈ On ∧ ( 𝑥 ∈ V ∧ Lim 𝑥 ) ) ) |
| 235 |
|
omlimcl |
⊢ ( ( ( 𝐴 ∈ On ∧ ( 𝑥 ∈ V ∧ Lim 𝑥 ) ) ∧ ∅ ∈ 𝐴 ) → Lim ( 𝐴 ·o 𝑥 ) ) |
| 236 |
234 235
|
sylan |
⊢ ( ( ( Lim 𝑥 ∧ 𝐴 ∈ On ) ∧ ∅ ∈ 𝐴 ) → Lim ( 𝐴 ·o 𝑥 ) ) |
| 237 |
236
|
adantlrr |
⊢ ( ( ( Lim 𝑥 ∧ ( 𝐴 ∈ On ∧ 𝐵 ∈ On ) ) ∧ ∅ ∈ 𝐴 ) → Lim ( 𝐴 ·o 𝑥 ) ) |
| 238 |
|
ovex |
⊢ ( 𝐴 ·o 𝑥 ) ∈ V |
| 239 |
237 238
|
jctil |
⊢ ( ( ( Lim 𝑥 ∧ ( 𝐴 ∈ On ∧ 𝐵 ∈ On ) ) ∧ ∅ ∈ 𝐴 ) → ( ( 𝐴 ·o 𝑥 ) ∈ V ∧ Lim ( 𝐴 ·o 𝑥 ) ) ) |
| 240 |
|
oalim |
⊢ ( ( ( 𝐴 ·o 𝐵 ) ∈ On ∧ ( ( 𝐴 ·o 𝑥 ) ∈ V ∧ Lim ( 𝐴 ·o 𝑥 ) ) ) → ( ( 𝐴 ·o 𝐵 ) +o ( 𝐴 ·o 𝑥 ) ) = ∪ 𝑤 ∈ ( 𝐴 ·o 𝑥 ) ( ( 𝐴 ·o 𝐵 ) +o 𝑤 ) ) |
| 241 |
232 239 240
|
syl2anc |
⊢ ( ( ( Lim 𝑥 ∧ ( 𝐴 ∈ On ∧ 𝐵 ∈ On ) ) ∧ ∅ ∈ 𝐴 ) → ( ( 𝐴 ·o 𝐵 ) +o ( 𝐴 ·o 𝑥 ) ) = ∪ 𝑤 ∈ ( 𝐴 ·o 𝑥 ) ( ( 𝐴 ·o 𝐵 ) +o 𝑤 ) ) |
| 242 |
241
|
adantrr |
⊢ ( ( ( Lim 𝑥 ∧ ( 𝐴 ∈ On ∧ 𝐵 ∈ On ) ) ∧ ( ∅ ∈ 𝐴 ∧ ∀ 𝑦 ∈ 𝑥 ( 𝐴 ·o ( 𝐵 +o 𝑦 ) ) = ( ( 𝐴 ·o 𝐵 ) +o ( 𝐴 ·o 𝑦 ) ) ) ) → ( ( 𝐴 ·o 𝐵 ) +o ( 𝐴 ·o 𝑥 ) ) = ∪ 𝑤 ∈ ( 𝐴 ·o 𝑥 ) ( ( 𝐴 ·o 𝐵 ) +o 𝑤 ) ) |
| 243 |
221 231 242
|
3eqtr4d |
⊢ ( ( ( Lim 𝑥 ∧ ( 𝐴 ∈ On ∧ 𝐵 ∈ On ) ) ∧ ( ∅ ∈ 𝐴 ∧ ∀ 𝑦 ∈ 𝑥 ( 𝐴 ·o ( 𝐵 +o 𝑦 ) ) = ( ( 𝐴 ·o 𝐵 ) +o ( 𝐴 ·o 𝑦 ) ) ) ) → ( 𝐴 ·o ( 𝐵 +o 𝑥 ) ) = ( ( 𝐴 ·o 𝐵 ) +o ( 𝐴 ·o 𝑥 ) ) ) |
| 244 |
243
|
exp43 |
⊢ ( Lim 𝑥 → ( ( 𝐴 ∈ On ∧ 𝐵 ∈ On ) → ( ∅ ∈ 𝐴 → ( ∀ 𝑦 ∈ 𝑥 ( 𝐴 ·o ( 𝐵 +o 𝑦 ) ) = ( ( 𝐴 ·o 𝐵 ) +o ( 𝐴 ·o 𝑦 ) ) → ( 𝐴 ·o ( 𝐵 +o 𝑥 ) ) = ( ( 𝐴 ·o 𝐵 ) +o ( 𝐴 ·o 𝑥 ) ) ) ) ) ) |
| 245 |
244
|
com3l |
⊢ ( ( 𝐴 ∈ On ∧ 𝐵 ∈ On ) → ( ∅ ∈ 𝐴 → ( Lim 𝑥 → ( ∀ 𝑦 ∈ 𝑥 ( 𝐴 ·o ( 𝐵 +o 𝑦 ) ) = ( ( 𝐴 ·o 𝐵 ) +o ( 𝐴 ·o 𝑦 ) ) → ( 𝐴 ·o ( 𝐵 +o 𝑥 ) ) = ( ( 𝐴 ·o 𝐵 ) +o ( 𝐴 ·o 𝑥 ) ) ) ) ) ) |
| 246 |
245
|
imp |
⊢ ( ( ( 𝐴 ∈ On ∧ 𝐵 ∈ On ) ∧ ∅ ∈ 𝐴 ) → ( Lim 𝑥 → ( ∀ 𝑦 ∈ 𝑥 ( 𝐴 ·o ( 𝐵 +o 𝑦 ) ) = ( ( 𝐴 ·o 𝐵 ) +o ( 𝐴 ·o 𝑦 ) ) → ( 𝐴 ·o ( 𝐵 +o 𝑥 ) ) = ( ( 𝐴 ·o 𝐵 ) +o ( 𝐴 ·o 𝑥 ) ) ) ) ) |
| 247 |
84 246
|
oe0lem |
⊢ ( ( 𝐴 ∈ On ∧ 𝐵 ∈ On ) → ( Lim 𝑥 → ( ∀ 𝑦 ∈ 𝑥 ( 𝐴 ·o ( 𝐵 +o 𝑦 ) ) = ( ( 𝐴 ·o 𝐵 ) +o ( 𝐴 ·o 𝑦 ) ) → ( 𝐴 ·o ( 𝐵 +o 𝑥 ) ) = ( ( 𝐴 ·o 𝐵 ) +o ( 𝐴 ·o 𝑥 ) ) ) ) ) |
| 248 |
247
|
com12 |
⊢ ( Lim 𝑥 → ( ( 𝐴 ∈ On ∧ 𝐵 ∈ On ) → ( ∀ 𝑦 ∈ 𝑥 ( 𝐴 ·o ( 𝐵 +o 𝑦 ) ) = ( ( 𝐴 ·o 𝐵 ) +o ( 𝐴 ·o 𝑦 ) ) → ( 𝐴 ·o ( 𝐵 +o 𝑥 ) ) = ( ( 𝐴 ·o 𝐵 ) +o ( 𝐴 ·o 𝑥 ) ) ) ) ) |
| 249 |
5 10 15 20 30 58 248
|
tfinds3 |
⊢ ( 𝐶 ∈ On → ( ( 𝐴 ∈ On ∧ 𝐵 ∈ On ) → ( 𝐴 ·o ( 𝐵 +o 𝐶 ) ) = ( ( 𝐴 ·o 𝐵 ) +o ( 𝐴 ·o 𝐶 ) ) ) ) |
| 250 |
249
|
expdcom |
⊢ ( 𝐴 ∈ On → ( 𝐵 ∈ On → ( 𝐶 ∈ On → ( 𝐴 ·o ( 𝐵 +o 𝐶 ) ) = ( ( 𝐴 ·o 𝐵 ) +o ( 𝐴 ·o 𝐶 ) ) ) ) ) |
| 251 |
250
|
3imp |
⊢ ( ( 𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On ) → ( 𝐴 ·o ( 𝐵 +o 𝐶 ) ) = ( ( 𝐴 ·o 𝐵 ) +o ( 𝐴 ·o 𝐶 ) ) ) |