Step | Hyp | Ref
| Expression |
1 | | oveq2 6557 |
. . . . . . 7
⊢ (𝑥 = 𝐶 → (𝐵 +𝑜 𝑥) = (𝐵 +𝑜 𝐶)) |
2 | 1 | oveq2d 6565 |
. . . . . 6
⊢ (𝑥 = 𝐶 → (𝐴 ·𝑜 (𝐵 +𝑜 𝑥)) = (𝐴 ·𝑜 (𝐵 +𝑜 𝐶))) |
3 | | oveq2 6557 |
. . . . . . 7
⊢ (𝑥 = 𝐶 → (𝐴 ·𝑜 𝑥) = (𝐴 ·𝑜 𝐶)) |
4 | 3 | oveq2d 6565 |
. . . . . 6
⊢ (𝑥 = 𝐶 → ((𝐴 ·𝑜 𝐵) +𝑜 (𝐴 ·𝑜
𝑥)) = ((𝐴 ·𝑜 𝐵) +𝑜 (𝐴 ·𝑜
𝐶))) |
5 | 2, 4 | eqeq12d 2625 |
. . . . 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 329 |
. . . 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 6557 |
. . . . . . 7
⊢ (𝑥 = ∅ → (𝐵 +𝑜 𝑥) = (𝐵 +𝑜
∅)) |
8 | 7 | oveq2d 6565 |
. . . . . 6
⊢ (𝑥 = ∅ → (𝐴 ·𝑜
(𝐵 +𝑜
𝑥)) = (𝐴 ·𝑜 (𝐵 +𝑜
∅))) |
9 | | oveq2 6557 |
. . . . . . 7
⊢ (𝑥 = ∅ → (𝐴 ·𝑜
𝑥) = (𝐴 ·𝑜
∅)) |
10 | 9 | oveq2d 6565 |
. . . . . 6
⊢ (𝑥 = ∅ → ((𝐴 ·𝑜
𝐵) +𝑜
(𝐴
·𝑜 𝑥)) = ((𝐴 ·𝑜 𝐵) +𝑜 (𝐴 ·𝑜
∅))) |
11 | 8, 10 | eqeq12d 2625 |
. . . . 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 6557 |
. . . . . . 7
⊢ (𝑥 = 𝑦 → (𝐵 +𝑜 𝑥) = (𝐵 +𝑜 𝑦)) |
13 | 12 | oveq2d 6565 |
. . . . . 6
⊢ (𝑥 = 𝑦 → (𝐴 ·𝑜 (𝐵 +𝑜 𝑥)) = (𝐴 ·𝑜 (𝐵 +𝑜 𝑦))) |
14 | | oveq2 6557 |
. . . . . . 7
⊢ (𝑥 = 𝑦 → (𝐴 ·𝑜 𝑥) = (𝐴 ·𝑜 𝑦)) |
15 | 14 | oveq2d 6565 |
. . . . . 6
⊢ (𝑥 = 𝑦 → ((𝐴 ·𝑜 𝐵) +𝑜 (𝐴 ·𝑜
𝑥)) = ((𝐴 ·𝑜 𝐵) +𝑜 (𝐴 ·𝑜
𝑦))) |
16 | 13, 15 | eqeq12d 2625 |
. . . . 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 6557 |
. . . . . . 7
⊢ (𝑥 = suc 𝑦 → (𝐵 +𝑜 𝑥) = (𝐵 +𝑜 suc 𝑦)) |
18 | 17 | oveq2d 6565 |
. . . . . 6
⊢ (𝑥 = suc 𝑦 → (𝐴 ·𝑜 (𝐵 +𝑜 𝑥)) = (𝐴 ·𝑜 (𝐵 +𝑜 suc 𝑦))) |
19 | | oveq2 6557 |
. . . . . . 7
⊢ (𝑥 = suc 𝑦 → (𝐴 ·𝑜 𝑥) = (𝐴 ·𝑜 suc 𝑦)) |
20 | 19 | oveq2d 6565 |
. . . . . 6
⊢ (𝑥 = suc 𝑦 → ((𝐴 ·𝑜 𝐵) +𝑜 (𝐴 ·𝑜
𝑥)) = ((𝐴 ·𝑜 𝐵) +𝑜 (𝐴 ·𝑜
suc 𝑦))) |
21 | 18, 20 | eqeq12d 2625 |
. . . . 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 7571 |
. . . . . . . . 9
⊢ (𝐵 ∈ ω → (𝐵 +𝑜 ∅)
= 𝐵) |
23 | 22 | adantl 481 |
. . . . . . . 8
⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐵 +𝑜 ∅)
= 𝐵) |
24 | 23 | oveq2d 6565 |
. . . . . . 7
⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 ·𝑜
(𝐵 +𝑜
∅)) = (𝐴
·𝑜 𝐵)) |
25 | | nnmcl 7579 |
. . . . . . . 8
⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 ·𝑜
𝐵) ∈
ω) |
26 | | nna0 7571 |
. . . . . . . 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 2647 |
. . . . . 6
⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 ·𝑜
(𝐵 +𝑜
∅)) = ((𝐴
·𝑜 𝐵) +𝑜
∅)) |
29 | | nnm0 7572 |
. . . . . . . 8
⊢ (𝐴 ∈ ω → (𝐴 ·𝑜
∅) = ∅) |
30 | 29 | adantr 480 |
. . . . . . 7
⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 ·𝑜
∅) = ∅) |
31 | 30 | oveq2d 6565 |
. . . . . 6
⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → ((𝐴 ·𝑜
𝐵) +𝑜
(𝐴
·𝑜 ∅)) = ((𝐴 ·𝑜 𝐵) +𝑜
∅)) |
32 | 28, 31 | eqtr4d 2647 |
. . . . 5
⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 ·𝑜
(𝐵 +𝑜
∅)) = ((𝐴
·𝑜 𝐵) +𝑜 (𝐴 ·𝑜
∅))) |
33 | | oveq1 6556 |
. . . . . . . . 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 7573 |
. . . . . . . . . . . . 13
⊢ ((𝐵 ∈ ω ∧ 𝑦 ∈ ω) → (𝐵 +𝑜 suc 𝑦) = suc (𝐵 +𝑜 𝑦)) |
35 | 34 | 3adant1 1072 |
. . . . . . . . . . . 12
⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝑦 ∈ ω) → (𝐵 +𝑜 suc 𝑦) = suc (𝐵 +𝑜 𝑦)) |
36 | 35 | oveq2d 6565 |
. . . . . . . . . . 11
⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝑦 ∈ ω) → (𝐴 ·𝑜
(𝐵 +𝑜
suc 𝑦)) = (𝐴 ·𝑜
suc (𝐵
+𝑜 𝑦))) |
37 | | nnacl 7578 |
. . . . . . . . . . . . 13
⊢ ((𝐵 ∈ ω ∧ 𝑦 ∈ ω) → (𝐵 +𝑜 𝑦) ∈
ω) |
38 | | nnmsuc 7574 |
. . . . . . . . . . . . 13
⊢ ((𝐴 ∈ ω ∧ (𝐵 +𝑜 𝑦) ∈ ω) → (𝐴 ·𝑜
suc (𝐵
+𝑜 𝑦)) =
((𝐴
·𝑜 (𝐵 +𝑜 𝑦)) +𝑜 𝐴)) |
39 | 37, 38 | sylan2 490 |
. . . . . . . . . . . 12
⊢ ((𝐴 ∈ ω ∧ (𝐵 ∈ ω ∧ 𝑦 ∈ ω)) → (𝐴 ·𝑜
suc (𝐵
+𝑜 𝑦)) =
((𝐴
·𝑜 (𝐵 +𝑜 𝑦)) +𝑜 𝐴)) |
40 | 39 | 3impb 1252 |
. . . . . . . . . . 11
⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝑦 ∈ ω) → (𝐴 ·𝑜
suc (𝐵
+𝑜 𝑦)) =
((𝐴
·𝑜 (𝐵 +𝑜 𝑦)) +𝑜 𝐴)) |
41 | 36, 40 | eqtrd 2644 |
. . . . . . . . . 10
⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝑦 ∈ ω) → (𝐴 ·𝑜
(𝐵 +𝑜
suc 𝑦)) = ((𝐴 ·𝑜
(𝐵 +𝑜
𝑦)) +𝑜
𝐴)) |
42 | | nnmsuc 7574 |
. . . . . . . . . . . . 13
⊢ ((𝐴 ∈ ω ∧ 𝑦 ∈ ω) → (𝐴 ·𝑜
suc 𝑦) = ((𝐴 ·𝑜
𝑦) +𝑜
𝐴)) |
43 | 42 | 3adant2 1073 |
. . . . . . . . . . . 12
⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝑦 ∈ ω) → (𝐴 ·𝑜
suc 𝑦) = ((𝐴 ·𝑜
𝑦) +𝑜
𝐴)) |
44 | 43 | oveq2d 6565 |
. . . . . . . . . . 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 7579 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝐴 ∈ ω ∧ 𝑦 ∈ ω) → (𝐴 ·𝑜
𝑦) ∈
ω) |
46 | | nnaass 7589 |
. . . . . . . . . . . . . . . . . . 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 1351 |
. . . . . . . . . . . . . . . . . 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 1352 |
. . . . . . . . . . . . . . . . 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 1256 |
. . . . . . . . . . . . . . . 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 630 |
. . . . . . . . . . . . . . 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 52 |
. . . . . . . . . . . . . 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 92 |
. . . . . . . . . . . . 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 50 |
. . . . . . . . . . . 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 1249 |
. . . . . . . . . . 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 2647 |
. . . . . . . . . 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 2625 |
. . . . . . . . 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 | syl5ibr 235 |
. . . . . . . 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 1256 |
. . . . . . 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 85 |
. . . . . 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 446 |
. . . . 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 6986 |
. . . 4
⊢ (𝑥 ∈ ω → ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 ·𝑜
(𝐵 +𝑜
𝑥)) = ((𝐴 ·𝑜 𝐵) +𝑜 (𝐴 ·𝑜
𝑥)))) |
62 | 6, 61 | vtoclga 3245 |
. . 3
⊢ (𝐶 ∈ ω → ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 ·𝑜
(𝐵 +𝑜
𝐶)) = ((𝐴 ·𝑜 𝐵) +𝑜 (𝐴 ·𝑜
𝐶)))) |
63 | 62 | expdcom 454 |
. 2
⊢ (𝐴 ∈ ω → (𝐵 ∈ ω → (𝐶 ∈ ω → (𝐴 ·𝑜
(𝐵 +𝑜
𝐶)) = ((𝐴 ·𝑜 𝐵) +𝑜 (𝐴 ·𝑜
𝐶))))) |
64 | 63 | 3imp 1249 |
1
⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) → (𝐴 ·𝑜
(𝐵 +𝑜
𝐶)) = ((𝐴 ·𝑜 𝐵) +𝑜 (𝐴 ·𝑜
𝐶))) |