| Step | Hyp | Ref | Expression | 
|---|
| 1 |  | oveq2 7439 | . . . . . . 7
⊢ (𝑥 = 𝐶 → (𝐵 +o 𝑥) = (𝐵 +o 𝐶)) | 
| 2 | 1 | oveq2d 7447 | . . . . . 6
⊢ (𝑥 = 𝐶 → (𝐴 ·o (𝐵 +o 𝑥)) = (𝐴 ·o (𝐵 +o 𝐶))) | 
| 3 |  | oveq2 7439 | . . . . . . 7
⊢ (𝑥 = 𝐶 → (𝐴 ·o 𝑥) = (𝐴 ·o 𝐶)) | 
| 4 | 3 | oveq2d 7447 | . . . . . 6
⊢ (𝑥 = 𝐶 → ((𝐴 ·o 𝐵) +o (𝐴 ·o 𝑥)) = ((𝐴 ·o 𝐵) +o (𝐴 ·o 𝐶))) | 
| 5 | 2, 4 | eqeq12d 2753 | . . . . 5
⊢ (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)))) | 
| 6 | 5 | imbi2d 340 | . . . 4
⊢ (𝑥 = 𝐶 → (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 ·o (𝐵 +o 𝑥)) = ((𝐴 ·o 𝐵) +o (𝐴 ·o 𝑥))) ↔ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 ·o (𝐵 +o 𝐶)) = ((𝐴 ·o 𝐵) +o (𝐴 ·o 𝐶))))) | 
| 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 |  | oveq2 7439 | . . . . . . 7
⊢ (𝑥 = ∅ → (𝐴 ·o 𝑥) = (𝐴 ·o
∅)) | 
| 10 | 9 | oveq2d 7447 | . . . . . 6
⊢ (𝑥 = ∅ → ((𝐴 ·o 𝐵) +o (𝐴 ·o 𝑥)) = ((𝐴 ·o 𝐵) +o (𝐴 ·o
∅))) | 
| 11 | 8, 10 | eqeq12d 2753 | . . . . 5
⊢ (𝑥 = ∅ → ((𝐴 ·o (𝐵 +o 𝑥)) = ((𝐴 ·o 𝐵) +o (𝐴 ·o 𝑥)) ↔ (𝐴 ·o (𝐵 +o ∅)) = ((𝐴 ·o 𝐵) +o (𝐴 ·o
∅)))) | 
| 12 |  | oveq2 7439 | . . . . . . 7
⊢ (𝑥 = 𝑦 → (𝐵 +o 𝑥) = (𝐵 +o 𝑦)) | 
| 13 | 12 | oveq2d 7447 | . . . . . 6
⊢ (𝑥 = 𝑦 → (𝐴 ·o (𝐵 +o 𝑥)) = (𝐴 ·o (𝐵 +o 𝑦))) | 
| 14 |  | oveq2 7439 | . . . . . . 7
⊢ (𝑥 = 𝑦 → (𝐴 ·o 𝑥) = (𝐴 ·o 𝑦)) | 
| 15 | 14 | oveq2d 7447 | . . . . . 6
⊢ (𝑥 = 𝑦 → ((𝐴 ·o 𝐵) +o (𝐴 ·o 𝑥)) = ((𝐴 ·o 𝐵) +o (𝐴 ·o 𝑦))) | 
| 16 | 13, 15 | eqeq12d 2753 | . . . . 5
⊢ (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)))) | 
| 17 |  | oveq2 7439 | . . . . . . 7
⊢ (𝑥 = suc 𝑦 → (𝐵 +o 𝑥) = (𝐵 +o suc 𝑦)) | 
| 18 | 17 | oveq2d 7447 | . . . . . 6
⊢ (𝑥 = suc 𝑦 → (𝐴 ·o (𝐵 +o 𝑥)) = (𝐴 ·o (𝐵 +o suc 𝑦))) | 
| 19 |  | oveq2 7439 | . . . . . . 7
⊢ (𝑥 = suc 𝑦 → (𝐴 ·o 𝑥) = (𝐴 ·o suc 𝑦)) | 
| 20 | 19 | oveq2d 7447 | . . . . . 6
⊢ (𝑥 = suc 𝑦 → ((𝐴 ·o 𝐵) +o (𝐴 ·o 𝑥)) = ((𝐴 ·o 𝐵) +o (𝐴 ·o suc 𝑦))) | 
| 21 | 18, 20 | eqeq12d 2753 | . . . . 5
⊢ (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)))) | 
| 22 |  | nna0 8642 | . . . . . . . . 9
⊢ (𝐵 ∈ ω → (𝐵 +o ∅) = 𝐵) | 
| 23 | 22 | adantl 481 | . . . . . . . 8
⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐵 +o ∅) = 𝐵) | 
| 24 | 23 | oveq2d 7447 | . . . . . . 7
⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 ·o (𝐵 +o ∅)) =
(𝐴 ·o
𝐵)) | 
| 25 |  | nnmcl 8650 | . . . . . . . 8
⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 ·o 𝐵) ∈
ω) | 
| 26 |  | nna0 8642 | . . . . . . . 8
⊢ ((𝐴 ·o 𝐵) ∈ ω → ((𝐴 ·o 𝐵) +o ∅) =
(𝐴 ·o
𝐵)) | 
| 27 | 25, 26 | syl 17 | . . . . . . 7
⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → ((𝐴 ·o 𝐵) +o ∅) =
(𝐴 ·o
𝐵)) | 
| 28 | 24, 27 | eqtr4d 2780 | . . . . . 6
⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 ·o (𝐵 +o ∅)) =
((𝐴 ·o
𝐵) +o
∅)) | 
| 29 |  | nnm0 8643 | . . . . . . . 8
⊢ (𝐴 ∈ ω → (𝐴 ·o ∅) =
∅) | 
| 30 | 29 | adantr 480 | . . . . . . 7
⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 ·o ∅) =
∅) | 
| 31 | 30 | oveq2d 7447 | . . . . . 6
⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → ((𝐴 ·o 𝐵) +o (𝐴 ·o ∅))
= ((𝐴 ·o
𝐵) +o
∅)) | 
| 32 | 28, 31 | eqtr4d 2780 | . . . . 5
⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 ·o (𝐵 +o ∅)) =
((𝐴 ·o
𝐵) +o (𝐴 ·o
∅))) | 
| 33 |  | oveq1 7438 | . . . . . . . . 9
⊢ ((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)) | 
| 34 |  | nnasuc 8644 | . . . . . . . . . . . . 13
⊢ ((𝐵 ∈ ω ∧ 𝑦 ∈ ω) → (𝐵 +o suc 𝑦) = suc (𝐵 +o 𝑦)) | 
| 35 | 34 | 3adant1 1131 | . . . . . . . . . . . 12
⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝑦 ∈ ω) → (𝐵 +o suc 𝑦) = suc (𝐵 +o 𝑦)) | 
| 36 | 35 | oveq2d 7447 | . . . . . . . . . . 11
⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝑦 ∈ ω) → (𝐴 ·o (𝐵 +o suc 𝑦)) = (𝐴 ·o suc (𝐵 +o 𝑦))) | 
| 37 |  | nnacl 8649 | . . . . . . . . . . . . 13
⊢ ((𝐵 ∈ ω ∧ 𝑦 ∈ ω) → (𝐵 +o 𝑦) ∈
ω) | 
| 38 |  | nnmsuc 8645 | . . . . . . . . . . . . 13
⊢ ((𝐴 ∈ ω ∧ (𝐵 +o 𝑦) ∈ ω) → (𝐴 ·o suc (𝐵 +o 𝑦)) = ((𝐴 ·o (𝐵 +o 𝑦)) +o 𝐴)) | 
| 39 | 37, 38 | sylan2 593 | . . . . . . . . . . . 12
⊢ ((𝐴 ∈ ω ∧ (𝐵 ∈ ω ∧ 𝑦 ∈ ω)) → (𝐴 ·o suc (𝐵 +o 𝑦)) = ((𝐴 ·o (𝐵 +o 𝑦)) +o 𝐴)) | 
| 40 | 39 | 3impb 1115 | . . . . . . . . . . 11
⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝑦 ∈ ω) → (𝐴 ·o suc (𝐵 +o 𝑦)) = ((𝐴 ·o (𝐵 +o 𝑦)) +o 𝐴)) | 
| 41 | 36, 40 | eqtrd 2777 | . . . . . . . . . 10
⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝑦 ∈ ω) → (𝐴 ·o (𝐵 +o suc 𝑦)) = ((𝐴 ·o (𝐵 +o 𝑦)) +o 𝐴)) | 
| 42 |  | nnmsuc 8645 | . . . . . . . . . . . . 13
⊢ ((𝐴 ∈ ω ∧ 𝑦 ∈ ω) → (𝐴 ·o suc 𝑦) = ((𝐴 ·o 𝑦) +o 𝐴)) | 
| 43 | 42 | 3adant2 1132 | . . . . . . . . . . . 12
⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝑦 ∈ ω) → (𝐴 ·o suc 𝑦) = ((𝐴 ·o 𝑦) +o 𝐴)) | 
| 44 | 43 | oveq2d 7447 | . . . . . . . . . . 11
⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝑦 ∈ ω) → ((𝐴 ·o 𝐵) +o (𝐴 ·o suc 𝑦)) = ((𝐴 ·o 𝐵) +o ((𝐴 ·o 𝑦) +o 𝐴))) | 
| 45 |  | nnmcl 8650 | . . . . . . . . . . . . . . . . . 18
⊢ ((𝐴 ∈ ω ∧ 𝑦 ∈ ω) → (𝐴 ·o 𝑦) ∈
ω) | 
| 46 |  | nnaass 8660 | . . . . . . . . . . . . . . . . . . 19
⊢ (((𝐴 ·o 𝐵) ∈ ω ∧ (𝐴 ·o 𝑦) ∈ ω ∧ 𝐴 ∈ ω) → (((𝐴 ·o 𝐵) +o (𝐴 ·o 𝑦)) +o 𝐴) = ((𝐴 ·o 𝐵) +o ((𝐴 ·o 𝑦) +o 𝐴))) | 
| 47 | 25, 46 | syl3an1 1164 | . . . . . . . . . . . . . . . . . 18
⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ (𝐴 ·o 𝑦) ∈ ω ∧ 𝐴 ∈ ω) → (((𝐴 ·o 𝐵) +o (𝐴 ·o 𝑦)) +o 𝐴) = ((𝐴 ·o 𝐵) +o ((𝐴 ·o 𝑦) +o 𝐴))) | 
| 48 | 45, 47 | syl3an2 1165 | . . . . . . . . . . . . . . . . 17
⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ (𝐴 ∈ ω ∧ 𝑦 ∈ ω) ∧ 𝐴 ∈ ω) → (((𝐴 ·o 𝐵) +o (𝐴 ·o 𝑦)) +o 𝐴) = ((𝐴 ·o 𝐵) +o ((𝐴 ·o 𝑦) +o 𝐴))) | 
| 49 | 48 | 3exp 1120 | . . . . . . . . . . . . . . . 16
⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → ((𝐴 ∈ ω ∧ 𝑦 ∈ ω) → (𝐴 ∈ ω → (((𝐴 ·o 𝐵) +o (𝐴 ·o 𝑦)) +o 𝐴) = ((𝐴 ·o 𝐵) +o ((𝐴 ·o 𝑦) +o 𝐴))))) | 
| 50 | 49 | exp4b 430 | . . . . . . . . . . . . . . 15
⊢ (𝐴 ∈ ω → (𝐵 ∈ ω → (𝐴 ∈ ω → (𝑦 ∈ ω → (𝐴 ∈ ω → (((𝐴 ·o 𝐵) +o (𝐴 ·o 𝑦)) +o 𝐴) = ((𝐴 ·o 𝐵) +o ((𝐴 ·o 𝑦) +o 𝐴))))))) | 
| 51 | 50 | pm2.43a 54 | . . . . . . . . . . . . . 14
⊢ (𝐴 ∈ ω → (𝐵 ∈ ω → (𝑦 ∈ ω → (𝐴 ∈ ω → (((𝐴 ·o 𝐵) +o (𝐴 ·o 𝑦)) +o 𝐴) = ((𝐴 ·o 𝐵) +o ((𝐴 ·o 𝑦) +o 𝐴)))))) | 
| 52 | 51 | com4r 94 | . . . . . . . . . . . . 13
⊢ (𝐴 ∈ ω → (𝐴 ∈ ω → (𝐵 ∈ ω → (𝑦 ∈ ω → (((𝐴 ·o 𝐵) +o (𝐴 ·o 𝑦)) +o 𝐴) = ((𝐴 ·o 𝐵) +o ((𝐴 ·o 𝑦) +o 𝐴)))))) | 
| 53 | 52 | pm2.43i 52 | . . . . . . . . . . . 12
⊢ (𝐴 ∈ ω → (𝐵 ∈ ω → (𝑦 ∈ ω → (((𝐴 ·o 𝐵) +o (𝐴 ·o 𝑦)) +o 𝐴) = ((𝐴 ·o 𝐵) +o ((𝐴 ·o 𝑦) +o 𝐴))))) | 
| 54 | 53 | 3imp 1111 | . . . . . . . . . . 11
⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝑦 ∈ ω) → (((𝐴 ·o 𝐵) +o (𝐴 ·o 𝑦)) +o 𝐴) = ((𝐴 ·o 𝐵) +o ((𝐴 ·o 𝑦) +o 𝐴))) | 
| 55 | 44, 54 | eqtr4d 2780 | . . . . . . . . . 10
⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝑦 ∈ ω) → ((𝐴 ·o 𝐵) +o (𝐴 ·o suc 𝑦)) = (((𝐴 ·o 𝐵) +o (𝐴 ·o 𝑦)) +o 𝐴)) | 
| 56 | 41, 55 | eqeq12d 2753 | . . . . . . . . 9
⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝑦 ∈ ω) → ((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))) | 
| 57 | 33, 56 | imbitrrid 246 | . . . . . . . 8
⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝑦 ∈ ω) → ((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 | 3exp 1120 | . . . . . . 7
⊢ (𝐴 ∈ ω → (𝐵 ∈ ω → (𝑦 ∈ ω → ((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 | 58 | com3r 87 | . . . . . 6
⊢ (𝑦 ∈ ω → (𝐴 ∈ ω → (𝐵 ∈ ω → ((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)))))) | 
| 60 | 59 | impd 410 | . . . . 5
⊢ (𝑦 ∈ ω → ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → ((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))))) | 
| 61 | 11, 16, 21, 32, 60 | finds2 7920 | . . . 4
⊢ (𝑥 ∈ ω → ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 ·o (𝐵 +o 𝑥)) = ((𝐴 ·o 𝐵) +o (𝐴 ·o 𝑥)))) | 
| 62 | 6, 61 | vtoclga 3577 | . . 3
⊢ (𝐶 ∈ ω → ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 ·o (𝐵 +o 𝐶)) = ((𝐴 ·o 𝐵) +o (𝐴 ·o 𝐶)))) | 
| 63 | 62 | expdcom 414 | . 2
⊢ (𝐴 ∈ ω → (𝐵 ∈ ω → (𝐶 ∈ ω → (𝐴 ·o (𝐵 +o 𝐶)) = ((𝐴 ·o 𝐵) +o (𝐴 ·o 𝐶))))) | 
| 64 | 63 | 3imp 1111 | 1
⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) → (𝐴 ·o (𝐵 +o 𝐶)) = ((𝐴 ·o 𝐵) +o (𝐴 ·o 𝐶))) |