Step | Hyp | Ref
| Expression |
1 | | oveq2 6658 |
. . . . . 6
⊢ (𝑥 = 𝐶 → ((𝐴 +𝑜 𝐵) +𝑜 𝑥) = ((𝐴 +𝑜 𝐵) +𝑜 𝐶)) |
2 | | oveq2 6658 |
. . . . . . 7
⊢ (𝑥 = 𝐶 → (𝐵 +𝑜 𝑥) = (𝐵 +𝑜 𝐶)) |
3 | 2 | oveq2d 6666 |
. . . . . 6
⊢ (𝑥 = 𝐶 → (𝐴 +𝑜 (𝐵 +𝑜 𝑥)) = (𝐴 +𝑜 (𝐵 +𝑜 𝐶))) |
4 | 1, 3 | eqeq12d 2637 |
. . . . 5
⊢ (𝑥 = 𝐶 → (((𝐴 +𝑜 𝐵) +𝑜 𝑥) = (𝐴 +𝑜 (𝐵 +𝑜 𝑥)) ↔ ((𝐴 +𝑜 𝐵) +𝑜 𝐶) = (𝐴 +𝑜 (𝐵 +𝑜 𝐶)))) |
5 | 4 | imbi2d 330 |
. . . 4
⊢ (𝑥 = 𝐶 → (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → ((𝐴 +𝑜 𝐵) +𝑜 𝑥) = (𝐴 +𝑜 (𝐵 +𝑜 𝑥))) ↔ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → ((𝐴 +𝑜 𝐵) +𝑜 𝐶) = (𝐴 +𝑜 (𝐵 +𝑜 𝐶))))) |
6 | | oveq2 6658 |
. . . . . 6
⊢ (𝑥 = ∅ → ((𝐴 +𝑜 𝐵) +𝑜 𝑥) = ((𝐴 +𝑜 𝐵) +𝑜
∅)) |
7 | | oveq2 6658 |
. . . . . . 7
⊢ (𝑥 = ∅ → (𝐵 +𝑜 𝑥) = (𝐵 +𝑜
∅)) |
8 | 7 | oveq2d 6666 |
. . . . . 6
⊢ (𝑥 = ∅ → (𝐴 +𝑜 (𝐵 +𝑜 𝑥)) = (𝐴 +𝑜 (𝐵 +𝑜
∅))) |
9 | 6, 8 | eqeq12d 2637 |
. . . . 5
⊢ (𝑥 = ∅ → (((𝐴 +𝑜 𝐵) +𝑜 𝑥) = (𝐴 +𝑜 (𝐵 +𝑜 𝑥)) ↔ ((𝐴 +𝑜 𝐵) +𝑜 ∅) = (𝐴 +𝑜 (𝐵 +𝑜
∅)))) |
10 | | oveq2 6658 |
. . . . . 6
⊢ (𝑥 = 𝑦 → ((𝐴 +𝑜 𝐵) +𝑜 𝑥) = ((𝐴 +𝑜 𝐵) +𝑜 𝑦)) |
11 | | oveq2 6658 |
. . . . . . 7
⊢ (𝑥 = 𝑦 → (𝐵 +𝑜 𝑥) = (𝐵 +𝑜 𝑦)) |
12 | 11 | oveq2d 6666 |
. . . . . 6
⊢ (𝑥 = 𝑦 → (𝐴 +𝑜 (𝐵 +𝑜 𝑥)) = (𝐴 +𝑜 (𝐵 +𝑜 𝑦))) |
13 | 10, 12 | eqeq12d 2637 |
. . . . 5
⊢ (𝑥 = 𝑦 → (((𝐴 +𝑜 𝐵) +𝑜 𝑥) = (𝐴 +𝑜 (𝐵 +𝑜 𝑥)) ↔ ((𝐴 +𝑜 𝐵) +𝑜 𝑦) = (𝐴 +𝑜 (𝐵 +𝑜 𝑦)))) |
14 | | oveq2 6658 |
. . . . . 6
⊢ (𝑥 = suc 𝑦 → ((𝐴 +𝑜 𝐵) +𝑜 𝑥) = ((𝐴 +𝑜 𝐵) +𝑜 suc 𝑦)) |
15 | | oveq2 6658 |
. . . . . . 7
⊢ (𝑥 = suc 𝑦 → (𝐵 +𝑜 𝑥) = (𝐵 +𝑜 suc 𝑦)) |
16 | 15 | oveq2d 6666 |
. . . . . 6
⊢ (𝑥 = suc 𝑦 → (𝐴 +𝑜 (𝐵 +𝑜 𝑥)) = (𝐴 +𝑜 (𝐵 +𝑜 suc 𝑦))) |
17 | 14, 16 | eqeq12d 2637 |
. . . . 5
⊢ (𝑥 = suc 𝑦 → (((𝐴 +𝑜 𝐵) +𝑜 𝑥) = (𝐴 +𝑜 (𝐵 +𝑜 𝑥)) ↔ ((𝐴 +𝑜 𝐵) +𝑜 suc 𝑦) = (𝐴 +𝑜 (𝐵 +𝑜 suc 𝑦)))) |
18 | | nnacl 7691 |
. . . . . . 7
⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 +𝑜 𝐵) ∈
ω) |
19 | | nna0 7684 |
. . . . . . 7
⊢ ((𝐴 +𝑜 𝐵) ∈ ω → ((𝐴 +𝑜 𝐵) +𝑜 ∅)
= (𝐴 +𝑜
𝐵)) |
20 | 18, 19 | syl 17 |
. . . . . 6
⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → ((𝐴 +𝑜 𝐵) +𝑜 ∅)
= (𝐴 +𝑜
𝐵)) |
21 | | nna0 7684 |
. . . . . . . 8
⊢ (𝐵 ∈ ω → (𝐵 +𝑜 ∅)
= 𝐵) |
22 | 21 | oveq2d 6666 |
. . . . . . 7
⊢ (𝐵 ∈ ω → (𝐴 +𝑜 (𝐵 +𝑜 ∅))
= (𝐴 +𝑜
𝐵)) |
23 | 22 | adantl 482 |
. . . . . 6
⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 +𝑜 (𝐵 +𝑜 ∅))
= (𝐴 +𝑜
𝐵)) |
24 | 20, 23 | eqtr4d 2659 |
. . . . 5
⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → ((𝐴 +𝑜 𝐵) +𝑜 ∅)
= (𝐴 +𝑜
(𝐵 +𝑜
∅))) |
25 | | suceq 5790 |
. . . . . . 7
⊢ (((𝐴 +𝑜 𝐵) +𝑜 𝑦) = (𝐴 +𝑜 (𝐵 +𝑜 𝑦)) → suc ((𝐴 +𝑜 𝐵) +𝑜 𝑦) = suc (𝐴 +𝑜 (𝐵 +𝑜 𝑦))) |
26 | | nnasuc 7686 |
. . . . . . . . 9
⊢ (((𝐴 +𝑜 𝐵) ∈ ω ∧ 𝑦 ∈ ω) → ((𝐴 +𝑜 𝐵) +𝑜 suc
𝑦) = suc ((𝐴 +𝑜 𝐵) +𝑜 𝑦)) |
27 | 18, 26 | sylan 488 |
. . . . . . . 8
⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝑦 ∈ ω) → ((𝐴 +𝑜 𝐵) +𝑜 suc
𝑦) = suc ((𝐴 +𝑜 𝐵) +𝑜 𝑦)) |
28 | | nnasuc 7686 |
. . . . . . . . . . . 12
⊢ ((𝐵 ∈ ω ∧ 𝑦 ∈ ω) → (𝐵 +𝑜 suc 𝑦) = suc (𝐵 +𝑜 𝑦)) |
29 | 28 | oveq2d 6666 |
. . . . . . . . . . 11
⊢ ((𝐵 ∈ ω ∧ 𝑦 ∈ ω) → (𝐴 +𝑜 (𝐵 +𝑜 suc 𝑦)) = (𝐴 +𝑜 suc (𝐵 +𝑜 𝑦))) |
30 | 29 | adantl 482 |
. . . . . . . . . 10
⊢ ((𝐴 ∈ ω ∧ (𝐵 ∈ ω ∧ 𝑦 ∈ ω)) → (𝐴 +𝑜 (𝐵 +𝑜 suc 𝑦)) = (𝐴 +𝑜 suc (𝐵 +𝑜 𝑦))) |
31 | | nnacl 7691 |
. . . . . . . . . . 11
⊢ ((𝐵 ∈ ω ∧ 𝑦 ∈ ω) → (𝐵 +𝑜 𝑦) ∈
ω) |
32 | | nnasuc 7686 |
. . . . . . . . . . 11
⊢ ((𝐴 ∈ ω ∧ (𝐵 +𝑜 𝑦) ∈ ω) → (𝐴 +𝑜 suc
(𝐵 +𝑜
𝑦)) = suc (𝐴 +𝑜 (𝐵 +𝑜 𝑦))) |
33 | 31, 32 | sylan2 491 |
. . . . . . . . . 10
⊢ ((𝐴 ∈ ω ∧ (𝐵 ∈ ω ∧ 𝑦 ∈ ω)) → (𝐴 +𝑜 suc
(𝐵 +𝑜
𝑦)) = suc (𝐴 +𝑜 (𝐵 +𝑜 𝑦))) |
34 | 30, 33 | eqtrd 2656 |
. . . . . . . . 9
⊢ ((𝐴 ∈ ω ∧ (𝐵 ∈ ω ∧ 𝑦 ∈ ω)) → (𝐴 +𝑜 (𝐵 +𝑜 suc 𝑦)) = suc (𝐴 +𝑜 (𝐵 +𝑜 𝑦))) |
35 | 34 | anassrs 680 |
. . . . . . . 8
⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝑦 ∈ ω) → (𝐴 +𝑜 (𝐵 +𝑜 suc 𝑦)) = suc (𝐴 +𝑜 (𝐵 +𝑜 𝑦))) |
36 | 27, 35 | eqeq12d 2637 |
. . . . . . 7
⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝑦 ∈ ω) → (((𝐴 +𝑜 𝐵) +𝑜 suc
𝑦) = (𝐴 +𝑜 (𝐵 +𝑜 suc 𝑦)) ↔ suc ((𝐴 +𝑜 𝐵) +𝑜 𝑦) = suc (𝐴 +𝑜 (𝐵 +𝑜 𝑦)))) |
37 | 25, 36 | syl5ibr 236 |
. . . . . 6
⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝑦 ∈ ω) → (((𝐴 +𝑜 𝐵) +𝑜 𝑦) = (𝐴 +𝑜 (𝐵 +𝑜 𝑦)) → ((𝐴 +𝑜 𝐵) +𝑜 suc 𝑦) = (𝐴 +𝑜 (𝐵 +𝑜 suc 𝑦)))) |
38 | 37 | expcom 451 |
. . . . 5
⊢ (𝑦 ∈ ω → ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (((𝐴 +𝑜 𝐵) +𝑜 𝑦) = (𝐴 +𝑜 (𝐵 +𝑜 𝑦)) → ((𝐴 +𝑜 𝐵) +𝑜 suc 𝑦) = (𝐴 +𝑜 (𝐵 +𝑜 suc 𝑦))))) |
39 | 9, 13, 17, 24, 38 | finds2 7094 |
. . . 4
⊢ (𝑥 ∈ ω → ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → ((𝐴 +𝑜 𝐵) +𝑜 𝑥) = (𝐴 +𝑜 (𝐵 +𝑜 𝑥)))) |
40 | 5, 39 | vtoclga 3272 |
. . 3
⊢ (𝐶 ∈ ω → ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → ((𝐴 +𝑜 𝐵) +𝑜 𝐶) = (𝐴 +𝑜 (𝐵 +𝑜 𝐶)))) |
41 | 40 | com12 32 |
. 2
⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐶 ∈ ω → ((𝐴 +𝑜 𝐵) +𝑜 𝐶) = (𝐴 +𝑜 (𝐵 +𝑜 𝐶)))) |
42 | 41 | 3impia 1261 |
1
⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) → ((𝐴 +𝑜 𝐵) +𝑜 𝐶) = (𝐴 +𝑜 (𝐵 +𝑜 𝐶))) |