| Step | Hyp | Ref | Expression | 
|---|
| 1 |  | oveq2 7439 | . . . . . 6
⊢ (𝑥 = 𝐶 → ((𝐴 +o 𝐵) +o 𝑥) = ((𝐴 +o 𝐵) +o 𝐶)) | 
| 2 |  | oveq2 7439 | . . . . . . 7
⊢ (𝑥 = 𝐶 → (𝐵 +o 𝑥) = (𝐵 +o 𝐶)) | 
| 3 | 2 | oveq2d 7447 | . . . . . 6
⊢ (𝑥 = 𝐶 → (𝐴 +o (𝐵 +o 𝑥)) = (𝐴 +o (𝐵 +o 𝐶))) | 
| 4 | 1, 3 | eqeq12d 2753 | . . . . 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 340 | . . . 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 7439 | . . . . . 6
⊢ (𝑥 = ∅ → ((𝐴 +o 𝐵) +o 𝑥) = ((𝐴 +o 𝐵) +o ∅)) | 
| 7 |  | oveq2 7439 | . . . . . . 7
⊢ (𝑥 = ∅ → (𝐵 +o 𝑥) = (𝐵 +o ∅)) | 
| 8 | 7 | oveq2d 7447 | . . . . . 6
⊢ (𝑥 = ∅ → (𝐴 +o (𝐵 +o 𝑥)) = (𝐴 +o (𝐵 +o ∅))) | 
| 9 | 6, 8 | eqeq12d 2753 | . . . . 5
⊢ (𝑥 = ∅ → (((𝐴 +o 𝐵) +o 𝑥) = (𝐴 +o (𝐵 +o 𝑥)) ↔ ((𝐴 +o 𝐵) +o ∅) = (𝐴 +o (𝐵 +o
∅)))) | 
| 10 |  | oveq2 7439 | . . . . . 6
⊢ (𝑥 = 𝑦 → ((𝐴 +o 𝐵) +o 𝑥) = ((𝐴 +o 𝐵) +o 𝑦)) | 
| 11 |  | oveq2 7439 | . . . . . . 7
⊢ (𝑥 = 𝑦 → (𝐵 +o 𝑥) = (𝐵 +o 𝑦)) | 
| 12 | 11 | oveq2d 7447 | . . . . . 6
⊢ (𝑥 = 𝑦 → (𝐴 +o (𝐵 +o 𝑥)) = (𝐴 +o (𝐵 +o 𝑦))) | 
| 13 | 10, 12 | eqeq12d 2753 | . . . . 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 7439 | . . . . . 6
⊢ (𝑥 = suc 𝑦 → ((𝐴 +o 𝐵) +o 𝑥) = ((𝐴 +o 𝐵) +o suc 𝑦)) | 
| 15 |  | oveq2 7439 | . . . . . . 7
⊢ (𝑥 = suc 𝑦 → (𝐵 +o 𝑥) = (𝐵 +o suc 𝑦)) | 
| 16 | 15 | oveq2d 7447 | . . . . . 6
⊢ (𝑥 = suc 𝑦 → (𝐴 +o (𝐵 +o 𝑥)) = (𝐴 +o (𝐵 +o suc 𝑦))) | 
| 17 | 14, 16 | eqeq12d 2753 | . . . . 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 8649 | . . . . . . 7
⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 +o 𝐵) ∈ ω) | 
| 19 |  | nna0 8642 | . . . . . . 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 8642 | . . . . . . . 8
⊢ (𝐵 ∈ ω → (𝐵 +o ∅) = 𝐵) | 
| 22 | 21 | oveq2d 7447 | . . . . . . 7
⊢ (𝐵 ∈ ω → (𝐴 +o (𝐵 +o ∅)) =
(𝐴 +o 𝐵)) | 
| 23 | 22 | adantl 481 | . . . . . 6
⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 +o (𝐵 +o ∅)) =
(𝐴 +o 𝐵)) | 
| 24 | 20, 23 | eqtr4d 2780 | . . . . 5
⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → ((𝐴 +o 𝐵) +o ∅) = (𝐴 +o (𝐵 +o
∅))) | 
| 25 |  | suceq 6450 | . . . . . . 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 8644 | . . . . . . . . 9
⊢ (((𝐴 +o 𝐵) ∈ ω ∧ 𝑦 ∈ ω) → ((𝐴 +o 𝐵) +o suc 𝑦) = suc ((𝐴 +o 𝐵) +o 𝑦)) | 
| 27 | 18, 26 | sylan 580 | . . . . . . . 8
⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝑦 ∈ ω) → ((𝐴 +o 𝐵) +o suc 𝑦) = suc ((𝐴 +o 𝐵) +o 𝑦)) | 
| 28 |  | nnasuc 8644 | . . . . . . . . . . . 12
⊢ ((𝐵 ∈ ω ∧ 𝑦 ∈ ω) → (𝐵 +o suc 𝑦) = suc (𝐵 +o 𝑦)) | 
| 29 | 28 | oveq2d 7447 | . . . . . . . . . . 11
⊢ ((𝐵 ∈ ω ∧ 𝑦 ∈ ω) → (𝐴 +o (𝐵 +o suc 𝑦)) = (𝐴 +o suc (𝐵 +o 𝑦))) | 
| 30 | 29 | adantl 481 | . . . . . . . . . 10
⊢ ((𝐴 ∈ ω ∧ (𝐵 ∈ ω ∧ 𝑦 ∈ ω)) → (𝐴 +o (𝐵 +o suc 𝑦)) = (𝐴 +o suc (𝐵 +o 𝑦))) | 
| 31 |  | nnacl 8649 | . . . . . . . . . . 11
⊢ ((𝐵 ∈ ω ∧ 𝑦 ∈ ω) → (𝐵 +o 𝑦) ∈
ω) | 
| 32 |  | nnasuc 8644 | . . . . . . . . . . 11
⊢ ((𝐴 ∈ ω ∧ (𝐵 +o 𝑦) ∈ ω) → (𝐴 +o suc (𝐵 +o 𝑦)) = suc (𝐴 +o (𝐵 +o 𝑦))) | 
| 33 | 31, 32 | sylan2 593 | . . . . . . . . . 10
⊢ ((𝐴 ∈ ω ∧ (𝐵 ∈ ω ∧ 𝑦 ∈ ω)) → (𝐴 +o suc (𝐵 +o 𝑦)) = suc (𝐴 +o (𝐵 +o 𝑦))) | 
| 34 | 30, 33 | eqtrd 2777 | . . . . . . . . 9
⊢ ((𝐴 ∈ ω ∧ (𝐵 ∈ ω ∧ 𝑦 ∈ ω)) → (𝐴 +o (𝐵 +o suc 𝑦)) = suc (𝐴 +o (𝐵 +o 𝑦))) | 
| 35 | 34 | anassrs 467 | . . . . . . . 8
⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝑦 ∈ ω) → (𝐴 +o (𝐵 +o suc 𝑦)) = suc (𝐴 +o (𝐵 +o 𝑦))) | 
| 36 | 27, 35 | eqeq12d 2753 | . . . . . . 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 | imbitrrid 246 | . . . . . 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 413 | . . . . 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 7920 | . . . 4
⊢ (𝑥 ∈ ω → ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → ((𝐴 +o 𝐵) +o 𝑥) = (𝐴 +o (𝐵 +o 𝑥)))) | 
| 40 | 5, 39 | vtoclga 3577 | . . 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 1118 | 1
⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) → ((𝐴 +o 𝐵) +o 𝐶) = (𝐴 +o (𝐵 +o 𝐶))) |