Proof of Theorem oaltom
Step | Hyp | Ref
| Expression |
1 | | om2 42757 |
. . . . 5
⊢ (𝐵 ∈ On → (𝐵 +o 𝐵) = (𝐵 ·o
2o)) |
2 | 1 | ad2antlr 726 |
. . . 4
⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (1o
∈ 𝐴 ∧ 𝐴 ∈ 𝐵)) → (𝐵 +o 𝐵) = (𝐵 ·o
2o)) |
3 | | 2on 8494 |
. . . . . . . 8
⊢
2o ∈ On |
4 | 3 | a1i 11 |
. . . . . . 7
⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → 2o
∈ On) |
5 | | simpl 482 |
. . . . . . 7
⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → 𝐴 ∈ On) |
6 | | simpr 484 |
. . . . . . 7
⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → 𝐵 ∈ On) |
7 | 4, 5, 6 | 3jca 1126 |
. . . . . 6
⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) →
(2o ∈ On ∧ 𝐴 ∈ On ∧ 𝐵 ∈ On)) |
8 | 7 | adantr 480 |
. . . . 5
⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (1o
∈ 𝐴 ∧ 𝐴 ∈ 𝐵)) → (2o ∈ On ∧
𝐴 ∈ On ∧ 𝐵 ∈ On)) |
9 | | df-2o 8481 |
. . . . . . 7
⊢
2o = suc 1o |
10 | 9 | a1i 11 |
. . . . . 6
⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (1o
∈ 𝐴 ∧ 𝐴 ∈ 𝐵)) → 2o = suc
1o) |
11 | | simprl 770 |
. . . . . . . 8
⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (1o
∈ 𝐴 ∧ 𝐴 ∈ 𝐵)) → 1o ∈ 𝐴) |
12 | | eloni 6373 |
. . . . . . . . . 10
⊢ (𝐴 ∈ On → Ord 𝐴) |
13 | 12 | adantr 480 |
. . . . . . . . 9
⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → Ord 𝐴) |
14 | 13 | adantr 480 |
. . . . . . . 8
⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (1o
∈ 𝐴 ∧ 𝐴 ∈ 𝐵)) → Ord 𝐴) |
15 | 11, 14 | jca 511 |
. . . . . . 7
⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (1o
∈ 𝐴 ∧ 𝐴 ∈ 𝐵)) → (1o ∈ 𝐴 ∧ Ord 𝐴)) |
16 | | ordelsuc 7817 |
. . . . . . . 8
⊢
((1o ∈ 𝐴 ∧ Ord 𝐴) → (1o ∈ 𝐴 ↔ suc 1o
⊆ 𝐴)) |
17 | 16 | biimpd 228 |
. . . . . . 7
⊢
((1o ∈ 𝐴 ∧ Ord 𝐴) → (1o ∈ 𝐴 → suc 1o
⊆ 𝐴)) |
18 | 15, 11, 17 | sylc 65 |
. . . . . 6
⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (1o
∈ 𝐴 ∧ 𝐴 ∈ 𝐵)) → suc 1o ⊆ 𝐴) |
19 | 10, 18 | eqsstrd 4016 |
. . . . 5
⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (1o
∈ 𝐴 ∧ 𝐴 ∈ 𝐵)) → 2o ⊆ 𝐴) |
20 | | omwordi 8585 |
. . . . 5
⊢
((2o ∈ On ∧ 𝐴 ∈ On ∧ 𝐵 ∈ On) → (2o ⊆
𝐴 → (𝐵 ·o 2o) ⊆
(𝐵 ·o
𝐴))) |
21 | 8, 19, 20 | sylc 65 |
. . . 4
⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (1o
∈ 𝐴 ∧ 𝐴 ∈ 𝐵)) → (𝐵 ·o 2o) ⊆
(𝐵 ·o
𝐴)) |
22 | 2, 21 | eqsstrd 4016 |
. . 3
⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (1o
∈ 𝐴 ∧ 𝐴 ∈ 𝐵)) → (𝐵 +o 𝐵) ⊆ (𝐵 ·o 𝐴)) |
23 | 6, 6 | jca 511 |
. . . 4
⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐵 ∈ On ∧ 𝐵 ∈ On)) |
24 | | simpr 484 |
. . . 4
⊢
((1o ∈ 𝐴 ∧ 𝐴 ∈ 𝐵) → 𝐴 ∈ 𝐵) |
25 | | oaordi 8560 |
. . . . 5
⊢ ((𝐵 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ∈ 𝐵 → (𝐵 +o 𝐴) ∈ (𝐵 +o 𝐵))) |
26 | 25 | imp 406 |
. . . 4
⊢ (((𝐵 ∈ On ∧ 𝐵 ∈ On) ∧ 𝐴 ∈ 𝐵) → (𝐵 +o 𝐴) ∈ (𝐵 +o 𝐵)) |
27 | 23, 24, 26 | syl2an 595 |
. . 3
⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (1o
∈ 𝐴 ∧ 𝐴 ∈ 𝐵)) → (𝐵 +o 𝐴) ∈ (𝐵 +o 𝐵)) |
28 | 22, 27 | sseldd 3979 |
. 2
⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (1o
∈ 𝐴 ∧ 𝐴 ∈ 𝐵)) → (𝐵 +o 𝐴) ∈ (𝐵 ·o 𝐴)) |
29 | 28 | ex 412 |
1
⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) →
((1o ∈ 𝐴
∧ 𝐴 ∈ 𝐵) → (𝐵 +o 𝐴) ∈ (𝐵 ·o 𝐴))) |