Step | Hyp | Ref
| Expression |
1 | | nndi 6465 |
. . . . . . . 8
⊢ ((𝑥 ∈ ω ∧ 𝑦 ∈ ω ∧ 𝑧 ∈ ω) → (𝑥 ·o (𝑦 +o 𝑧)) = ((𝑥 ·o 𝑦) +o (𝑥 ·o 𝑧))) |
2 | 1 | adantl 275 |
. . . . . . 7
⊢
(((((𝐴 ∈
ω ∧ 𝐵 ∈
N) ∧ (𝐶
∈ ω ∧ 𝐷
∈ N)) ∧ ((𝐹 ∈ ω ∧ 𝐺 ∈ N) ∧ (𝑅 ∈ ω ∧ 𝑆 ∈ N))) ∧
(𝑥 ∈ ω ∧
𝑦 ∈ ω ∧
𝑧 ∈ ω)) →
(𝑥 ·o
(𝑦 +o 𝑧)) = ((𝑥 ·o 𝑦) +o (𝑥 ·o 𝑧))) |
3 | | simplll 528 |
. . . . . . . 8
⊢ ((((𝐴 ∈ ω ∧ 𝐵 ∈ N) ∧
(𝐶 ∈ ω ∧
𝐷 ∈ N))
∧ ((𝐹 ∈ ω
∧ 𝐺 ∈
N) ∧ (𝑅
∈ ω ∧ 𝑆
∈ N))) → 𝐴 ∈ ω) |
4 | | simprlr 533 |
. . . . . . . . 9
⊢ ((((𝐴 ∈ ω ∧ 𝐵 ∈ N) ∧
(𝐶 ∈ ω ∧
𝐷 ∈ N))
∧ ((𝐹 ∈ ω
∧ 𝐺 ∈
N) ∧ (𝑅
∈ ω ∧ 𝑆
∈ N))) → 𝐺 ∈ N) |
5 | | pinn 7271 |
. . . . . . . . 9
⊢ (𝐺 ∈ N →
𝐺 ∈
ω) |
6 | 4, 5 | syl 14 |
. . . . . . . 8
⊢ ((((𝐴 ∈ ω ∧ 𝐵 ∈ N) ∧
(𝐶 ∈ ω ∧
𝐷 ∈ N))
∧ ((𝐹 ∈ ω
∧ 𝐺 ∈
N) ∧ (𝑅
∈ ω ∧ 𝑆
∈ N))) → 𝐺 ∈ ω) |
7 | | nnmcl 6460 |
. . . . . . . 8
⊢ ((𝐴 ∈ ω ∧ 𝐺 ∈ ω) → (𝐴 ·o 𝐺) ∈
ω) |
8 | 3, 6, 7 | syl2anc 409 |
. . . . . . 7
⊢ ((((𝐴 ∈ ω ∧ 𝐵 ∈ N) ∧
(𝐶 ∈ ω ∧
𝐷 ∈ N))
∧ ((𝐹 ∈ ω
∧ 𝐺 ∈
N) ∧ (𝑅
∈ ω ∧ 𝑆
∈ N))) → (𝐴 ·o 𝐺) ∈ ω) |
9 | | simpllr 529 |
. . . . . . . . 9
⊢ ((((𝐴 ∈ ω ∧ 𝐵 ∈ N) ∧
(𝐶 ∈ ω ∧
𝐷 ∈ N))
∧ ((𝐹 ∈ ω
∧ 𝐺 ∈
N) ∧ (𝑅
∈ ω ∧ 𝑆
∈ N))) → 𝐵 ∈ N) |
10 | | pinn 7271 |
. . . . . . . . 9
⊢ (𝐵 ∈ N →
𝐵 ∈
ω) |
11 | 9, 10 | syl 14 |
. . . . . . . 8
⊢ ((((𝐴 ∈ ω ∧ 𝐵 ∈ N) ∧
(𝐶 ∈ ω ∧
𝐷 ∈ N))
∧ ((𝐹 ∈ ω
∧ 𝐺 ∈
N) ∧ (𝑅
∈ ω ∧ 𝑆
∈ N))) → 𝐵 ∈ ω) |
12 | | simprll 532 |
. . . . . . . 8
⊢ ((((𝐴 ∈ ω ∧ 𝐵 ∈ N) ∧
(𝐶 ∈ ω ∧
𝐷 ∈ N))
∧ ((𝐹 ∈ ω
∧ 𝐺 ∈
N) ∧ (𝑅
∈ ω ∧ 𝑆
∈ N))) → 𝐹 ∈ ω) |
13 | | nnmcl 6460 |
. . . . . . . 8
⊢ ((𝐵 ∈ ω ∧ 𝐹 ∈ ω) → (𝐵 ·o 𝐹) ∈
ω) |
14 | 11, 12, 13 | syl2anc 409 |
. . . . . . 7
⊢ ((((𝐴 ∈ ω ∧ 𝐵 ∈ N) ∧
(𝐶 ∈ ω ∧
𝐷 ∈ N))
∧ ((𝐹 ∈ ω
∧ 𝐺 ∈
N) ∧ (𝑅
∈ ω ∧ 𝑆
∈ N))) → (𝐵 ·o 𝐹) ∈ ω) |
15 | | simplrr 531 |
. . . . . . . . 9
⊢ ((((𝐴 ∈ ω ∧ 𝐵 ∈ N) ∧
(𝐶 ∈ ω ∧
𝐷 ∈ N))
∧ ((𝐹 ∈ ω
∧ 𝐺 ∈
N) ∧ (𝑅
∈ ω ∧ 𝑆
∈ N))) → 𝐷 ∈ N) |
16 | | pinn 7271 |
. . . . . . . . 9
⊢ (𝐷 ∈ N →
𝐷 ∈
ω) |
17 | 15, 16 | syl 14 |
. . . . . . . 8
⊢ ((((𝐴 ∈ ω ∧ 𝐵 ∈ N) ∧
(𝐶 ∈ ω ∧
𝐷 ∈ N))
∧ ((𝐹 ∈ ω
∧ 𝐺 ∈
N) ∧ (𝑅
∈ ω ∧ 𝑆
∈ N))) → 𝐷 ∈ ω) |
18 | | simprrr 535 |
. . . . . . . . 9
⊢ ((((𝐴 ∈ ω ∧ 𝐵 ∈ N) ∧
(𝐶 ∈ ω ∧
𝐷 ∈ N))
∧ ((𝐹 ∈ ω
∧ 𝐺 ∈
N) ∧ (𝑅
∈ ω ∧ 𝑆
∈ N))) → 𝑆 ∈ N) |
19 | | pinn 7271 |
. . . . . . . . 9
⊢ (𝑆 ∈ N →
𝑆 ∈
ω) |
20 | 18, 19 | syl 14 |
. . . . . . . 8
⊢ ((((𝐴 ∈ ω ∧ 𝐵 ∈ N) ∧
(𝐶 ∈ ω ∧
𝐷 ∈ N))
∧ ((𝐹 ∈ ω
∧ 𝐺 ∈
N) ∧ (𝑅
∈ ω ∧ 𝑆
∈ N))) → 𝑆 ∈ ω) |
21 | | nnmcl 6460 |
. . . . . . . 8
⊢ ((𝐷 ∈ ω ∧ 𝑆 ∈ ω) → (𝐷 ·o 𝑆) ∈
ω) |
22 | 17, 20, 21 | syl2anc 409 |
. . . . . . 7
⊢ ((((𝐴 ∈ ω ∧ 𝐵 ∈ N) ∧
(𝐶 ∈ ω ∧
𝐷 ∈ N))
∧ ((𝐹 ∈ ω
∧ 𝐺 ∈
N) ∧ (𝑅
∈ ω ∧ 𝑆
∈ N))) → (𝐷 ·o 𝑆) ∈ ω) |
23 | | nnacl 6459 |
. . . . . . . 8
⊢ ((𝑥 ∈ ω ∧ 𝑦 ∈ ω) → (𝑥 +o 𝑦) ∈
ω) |
24 | 23 | adantl 275 |
. . . . . . 7
⊢
(((((𝐴 ∈
ω ∧ 𝐵 ∈
N) ∧ (𝐶
∈ ω ∧ 𝐷
∈ N)) ∧ ((𝐹 ∈ ω ∧ 𝐺 ∈ N) ∧ (𝑅 ∈ ω ∧ 𝑆 ∈ N))) ∧
(𝑥 ∈ ω ∧
𝑦 ∈ ω)) →
(𝑥 +o 𝑦) ∈
ω) |
25 | | nnmcom 6468 |
. . . . . . . 8
⊢ ((𝑥 ∈ ω ∧ 𝑦 ∈ ω) → (𝑥 ·o 𝑦) = (𝑦 ·o 𝑥)) |
26 | 25 | adantl 275 |
. . . . . . 7
⊢
(((((𝐴 ∈
ω ∧ 𝐵 ∈
N) ∧ (𝐶
∈ ω ∧ 𝐷
∈ N)) ∧ ((𝐹 ∈ ω ∧ 𝐺 ∈ N) ∧ (𝑅 ∈ ω ∧ 𝑆 ∈ N))) ∧
(𝑥 ∈ ω ∧
𝑦 ∈ ω)) →
(𝑥 ·o
𝑦) = (𝑦 ·o 𝑥)) |
27 | 2, 8, 14, 22, 24, 26 | caovdir2d 6029 |
. . . . . 6
⊢ ((((𝐴 ∈ ω ∧ 𝐵 ∈ N) ∧
(𝐶 ∈ ω ∧
𝐷 ∈ N))
∧ ((𝐹 ∈ ω
∧ 𝐺 ∈
N) ∧ (𝑅
∈ ω ∧ 𝑆
∈ N))) → (((𝐴 ·o 𝐺) +o (𝐵 ·o 𝐹)) ·o (𝐷 ·o 𝑆)) = (((𝐴 ·o 𝐺) ·o (𝐷 ·o 𝑆)) +o ((𝐵 ·o 𝐹) ·o (𝐷 ·o 𝑆)))) |
28 | | nnmass 6466 |
. . . . . . . . 9
⊢ ((𝑥 ∈ ω ∧ 𝑦 ∈ ω ∧ 𝑧 ∈ ω) → ((𝑥 ·o 𝑦) ·o 𝑧) = (𝑥 ·o (𝑦 ·o 𝑧))) |
29 | 28 | adantl 275 |
. . . . . . . 8
⊢
(((((𝐴 ∈
ω ∧ 𝐵 ∈
N) ∧ (𝐶
∈ ω ∧ 𝐷
∈ N)) ∧ ((𝐹 ∈ ω ∧ 𝐺 ∈ N) ∧ (𝑅 ∈ ω ∧ 𝑆 ∈ N))) ∧
(𝑥 ∈ ω ∧
𝑦 ∈ ω ∧
𝑧 ∈ ω)) →
((𝑥 ·o
𝑦) ·o
𝑧) = (𝑥 ·o (𝑦 ·o 𝑧))) |
30 | | nnmcl 6460 |
. . . . . . . . 9
⊢ ((𝑥 ∈ ω ∧ 𝑦 ∈ ω) → (𝑥 ·o 𝑦) ∈
ω) |
31 | 30 | adantl 275 |
. . . . . . . 8
⊢
(((((𝐴 ∈
ω ∧ 𝐵 ∈
N) ∧ (𝐶
∈ ω ∧ 𝐷
∈ N)) ∧ ((𝐹 ∈ ω ∧ 𝐺 ∈ N) ∧ (𝑅 ∈ ω ∧ 𝑆 ∈ N))) ∧
(𝑥 ∈ ω ∧
𝑦 ∈ ω)) →
(𝑥 ·o
𝑦) ∈
ω) |
32 | 3, 6, 17, 26, 29, 20, 31 | caov4d 6037 |
. . . . . . 7
⊢ ((((𝐴 ∈ ω ∧ 𝐵 ∈ N) ∧
(𝐶 ∈ ω ∧
𝐷 ∈ N))
∧ ((𝐹 ∈ ω
∧ 𝐺 ∈
N) ∧ (𝑅
∈ ω ∧ 𝑆
∈ N))) → ((𝐴 ·o 𝐺) ·o (𝐷 ·o 𝑆)) = ((𝐴 ·o 𝐷) ·o (𝐺 ·o 𝑆))) |
33 | 11, 12, 17, 26, 29, 20, 31 | caov4d 6037 |
. . . . . . 7
⊢ ((((𝐴 ∈ ω ∧ 𝐵 ∈ N) ∧
(𝐶 ∈ ω ∧
𝐷 ∈ N))
∧ ((𝐹 ∈ ω
∧ 𝐺 ∈
N) ∧ (𝑅
∈ ω ∧ 𝑆
∈ N))) → ((𝐵 ·o 𝐹) ·o (𝐷 ·o 𝑆)) = ((𝐵 ·o 𝐷) ·o (𝐹 ·o 𝑆))) |
34 | 32, 33 | oveq12d 5871 |
. . . . . 6
⊢ ((((𝐴 ∈ ω ∧ 𝐵 ∈ N) ∧
(𝐶 ∈ ω ∧
𝐷 ∈ N))
∧ ((𝐹 ∈ ω
∧ 𝐺 ∈
N) ∧ (𝑅
∈ ω ∧ 𝑆
∈ N))) → (((𝐴 ·o 𝐺) ·o (𝐷 ·o 𝑆)) +o ((𝐵 ·o 𝐹) ·o (𝐷 ·o 𝑆))) = (((𝐴 ·o 𝐷) ·o (𝐺 ·o 𝑆)) +o ((𝐵 ·o 𝐷) ·o (𝐹 ·o 𝑆)))) |
35 | 27, 34 | eqtrd 2203 |
. . . . 5
⊢ ((((𝐴 ∈ ω ∧ 𝐵 ∈ N) ∧
(𝐶 ∈ ω ∧
𝐷 ∈ N))
∧ ((𝐹 ∈ ω
∧ 𝐺 ∈
N) ∧ (𝑅
∈ ω ∧ 𝑆
∈ N))) → (((𝐴 ·o 𝐺) +o (𝐵 ·o 𝐹)) ·o (𝐷 ·o 𝑆)) = (((𝐴 ·o 𝐷) ·o (𝐺 ·o 𝑆)) +o ((𝐵 ·o 𝐷) ·o (𝐹 ·o 𝑆)))) |
36 | | oveq1 5860 |
. . . . . 6
⊢ ((𝐴 ·o 𝐷) = (𝐵 ·o 𝐶) → ((𝐴 ·o 𝐷) ·o (𝐺 ·o 𝑆)) = ((𝐵 ·o 𝐶) ·o (𝐺 ·o 𝑆))) |
37 | | oveq2 5861 |
. . . . . 6
⊢ ((𝐹 ·o 𝑆) = (𝐺 ·o 𝑅) → ((𝐵 ·o 𝐷) ·o (𝐹 ·o 𝑆)) = ((𝐵 ·o 𝐷) ·o (𝐺 ·o 𝑅))) |
38 | 36, 37 | oveqan12d 5872 |
. . . . 5
⊢ (((𝐴 ·o 𝐷) = (𝐵 ·o 𝐶) ∧ (𝐹 ·o 𝑆) = (𝐺 ·o 𝑅)) → (((𝐴 ·o 𝐷) ·o (𝐺 ·o 𝑆)) +o ((𝐵 ·o 𝐷) ·o (𝐹 ·o 𝑆))) = (((𝐵 ·o 𝐶) ·o (𝐺 ·o 𝑆)) +o ((𝐵 ·o 𝐷) ·o (𝐺 ·o 𝑅)))) |
39 | 35, 38 | sylan9eq 2223 |
. . . 4
⊢
(((((𝐴 ∈
ω ∧ 𝐵 ∈
N) ∧ (𝐶
∈ ω ∧ 𝐷
∈ N)) ∧ ((𝐹 ∈ ω ∧ 𝐺 ∈ N) ∧ (𝑅 ∈ ω ∧ 𝑆 ∈ N))) ∧
((𝐴 ·o
𝐷) = (𝐵 ·o 𝐶) ∧ (𝐹 ·o 𝑆) = (𝐺 ·o 𝑅))) → (((𝐴 ·o 𝐺) +o (𝐵 ·o 𝐹)) ·o (𝐷 ·o 𝑆)) = (((𝐵 ·o 𝐶) ·o (𝐺 ·o 𝑆)) +o ((𝐵 ·o 𝐷) ·o (𝐺 ·o 𝑅)))) |
40 | | nnmcl 6460 |
. . . . . . . 8
⊢ ((𝐵 ∈ ω ∧ 𝐺 ∈ ω) → (𝐵 ·o 𝐺) ∈
ω) |
41 | 11, 6, 40 | syl2anc 409 |
. . . . . . 7
⊢ ((((𝐴 ∈ ω ∧ 𝐵 ∈ N) ∧
(𝐶 ∈ ω ∧
𝐷 ∈ N))
∧ ((𝐹 ∈ ω
∧ 𝐺 ∈
N) ∧ (𝑅
∈ ω ∧ 𝑆
∈ N))) → (𝐵 ·o 𝐺) ∈ ω) |
42 | | simplrl 530 |
. . . . . . . 8
⊢ ((((𝐴 ∈ ω ∧ 𝐵 ∈ N) ∧
(𝐶 ∈ ω ∧
𝐷 ∈ N))
∧ ((𝐹 ∈ ω
∧ 𝐺 ∈
N) ∧ (𝑅
∈ ω ∧ 𝑆
∈ N))) → 𝐶 ∈ ω) |
43 | | nnmcl 6460 |
. . . . . . . 8
⊢ ((𝐶 ∈ ω ∧ 𝑆 ∈ ω) → (𝐶 ·o 𝑆) ∈
ω) |
44 | 42, 20, 43 | syl2anc 409 |
. . . . . . 7
⊢ ((((𝐴 ∈ ω ∧ 𝐵 ∈ N) ∧
(𝐶 ∈ ω ∧
𝐷 ∈ N))
∧ ((𝐹 ∈ ω
∧ 𝐺 ∈
N) ∧ (𝑅
∈ ω ∧ 𝑆
∈ N))) → (𝐶 ·o 𝑆) ∈ ω) |
45 | | simprrl 534 |
. . . . . . . 8
⊢ ((((𝐴 ∈ ω ∧ 𝐵 ∈ N) ∧
(𝐶 ∈ ω ∧
𝐷 ∈ N))
∧ ((𝐹 ∈ ω
∧ 𝐺 ∈
N) ∧ (𝑅
∈ ω ∧ 𝑆
∈ N))) → 𝑅 ∈ ω) |
46 | | nnmcl 6460 |
. . . . . . . 8
⊢ ((𝐷 ∈ ω ∧ 𝑅 ∈ ω) → (𝐷 ·o 𝑅) ∈
ω) |
47 | 17, 45, 46 | syl2anc 409 |
. . . . . . 7
⊢ ((((𝐴 ∈ ω ∧ 𝐵 ∈ N) ∧
(𝐶 ∈ ω ∧
𝐷 ∈ N))
∧ ((𝐹 ∈ ω
∧ 𝐺 ∈
N) ∧ (𝑅
∈ ω ∧ 𝑆
∈ N))) → (𝐷 ·o 𝑅) ∈ ω) |
48 | | nndi 6465 |
. . . . . . 7
⊢ (((𝐵 ·o 𝐺) ∈ ω ∧ (𝐶 ·o 𝑆) ∈ ω ∧ (𝐷 ·o 𝑅) ∈ ω) → ((𝐵 ·o 𝐺) ·o ((𝐶 ·o 𝑆) +o (𝐷 ·o 𝑅))) = (((𝐵 ·o 𝐺) ·o (𝐶 ·o 𝑆)) +o ((𝐵 ·o 𝐺) ·o (𝐷 ·o 𝑅)))) |
49 | 41, 44, 47, 48 | syl3anc 1233 |
. . . . . 6
⊢ ((((𝐴 ∈ ω ∧ 𝐵 ∈ N) ∧
(𝐶 ∈ ω ∧
𝐷 ∈ N))
∧ ((𝐹 ∈ ω
∧ 𝐺 ∈
N) ∧ (𝑅
∈ ω ∧ 𝑆
∈ N))) → ((𝐵 ·o 𝐺) ·o ((𝐶 ·o 𝑆) +o (𝐷 ·o 𝑅))) = (((𝐵 ·o 𝐺) ·o (𝐶 ·o 𝑆)) +o ((𝐵 ·o 𝐺) ·o (𝐷 ·o 𝑅)))) |
50 | 11, 6, 42, 26, 29, 20, 31 | caov4d 6037 |
. . . . . . 7
⊢ ((((𝐴 ∈ ω ∧ 𝐵 ∈ N) ∧
(𝐶 ∈ ω ∧
𝐷 ∈ N))
∧ ((𝐹 ∈ ω
∧ 𝐺 ∈
N) ∧ (𝑅
∈ ω ∧ 𝑆
∈ N))) → ((𝐵 ·o 𝐺) ·o (𝐶 ·o 𝑆)) = ((𝐵 ·o 𝐶) ·o (𝐺 ·o 𝑆))) |
51 | 11, 6, 17, 26, 29, 45, 31 | caov4d 6037 |
. . . . . . 7
⊢ ((((𝐴 ∈ ω ∧ 𝐵 ∈ N) ∧
(𝐶 ∈ ω ∧
𝐷 ∈ N))
∧ ((𝐹 ∈ ω
∧ 𝐺 ∈
N) ∧ (𝑅
∈ ω ∧ 𝑆
∈ N))) → ((𝐵 ·o 𝐺) ·o (𝐷 ·o 𝑅)) = ((𝐵 ·o 𝐷) ·o (𝐺 ·o 𝑅))) |
52 | 50, 51 | oveq12d 5871 |
. . . . . 6
⊢ ((((𝐴 ∈ ω ∧ 𝐵 ∈ N) ∧
(𝐶 ∈ ω ∧
𝐷 ∈ N))
∧ ((𝐹 ∈ ω
∧ 𝐺 ∈
N) ∧ (𝑅
∈ ω ∧ 𝑆
∈ N))) → (((𝐵 ·o 𝐺) ·o (𝐶 ·o 𝑆)) +o ((𝐵 ·o 𝐺) ·o (𝐷 ·o 𝑅))) = (((𝐵 ·o 𝐶) ·o (𝐺 ·o 𝑆)) +o ((𝐵 ·o 𝐷) ·o (𝐺 ·o 𝑅)))) |
53 | 49, 52 | eqtrd 2203 |
. . . . 5
⊢ ((((𝐴 ∈ ω ∧ 𝐵 ∈ N) ∧
(𝐶 ∈ ω ∧
𝐷 ∈ N))
∧ ((𝐹 ∈ ω
∧ 𝐺 ∈
N) ∧ (𝑅
∈ ω ∧ 𝑆
∈ N))) → ((𝐵 ·o 𝐺) ·o ((𝐶 ·o 𝑆) +o (𝐷 ·o 𝑅))) = (((𝐵 ·o 𝐶) ·o (𝐺 ·o 𝑆)) +o ((𝐵 ·o 𝐷) ·o (𝐺 ·o 𝑅)))) |
54 | 53 | adantr 274 |
. . . 4
⊢
(((((𝐴 ∈
ω ∧ 𝐵 ∈
N) ∧ (𝐶
∈ ω ∧ 𝐷
∈ N)) ∧ ((𝐹 ∈ ω ∧ 𝐺 ∈ N) ∧ (𝑅 ∈ ω ∧ 𝑆 ∈ N))) ∧
((𝐴 ·o
𝐷) = (𝐵 ·o 𝐶) ∧ (𝐹 ·o 𝑆) = (𝐺 ·o 𝑅))) → ((𝐵 ·o 𝐺) ·o ((𝐶 ·o 𝑆) +o (𝐷 ·o 𝑅))) = (((𝐵 ·o 𝐶) ·o (𝐺 ·o 𝑆)) +o ((𝐵 ·o 𝐷) ·o (𝐺 ·o 𝑅)))) |
55 | 39, 54 | eqtr4d 2206 |
. . 3
⊢
(((((𝐴 ∈
ω ∧ 𝐵 ∈
N) ∧ (𝐶
∈ ω ∧ 𝐷
∈ N)) ∧ ((𝐹 ∈ ω ∧ 𝐺 ∈ N) ∧ (𝑅 ∈ ω ∧ 𝑆 ∈ N))) ∧
((𝐴 ·o
𝐷) = (𝐵 ·o 𝐶) ∧ (𝐹 ·o 𝑆) = (𝐺 ·o 𝑅))) → (((𝐴 ·o 𝐺) +o (𝐵 ·o 𝐹)) ·o (𝐷 ·o 𝑆)) = ((𝐵 ·o 𝐺) ·o ((𝐶 ·o 𝑆) +o (𝐷 ·o 𝑅)))) |
56 | | nnacl 6459 |
. . . . . 6
⊢ (((𝐴 ·o 𝐺) ∈ ω ∧ (𝐵 ·o 𝐹) ∈ ω) → ((𝐴 ·o 𝐺) +o (𝐵 ·o 𝐹)) ∈
ω) |
57 | 8, 14, 56 | syl2anc 409 |
. . . . 5
⊢ ((((𝐴 ∈ ω ∧ 𝐵 ∈ N) ∧
(𝐶 ∈ ω ∧
𝐷 ∈ N))
∧ ((𝐹 ∈ ω
∧ 𝐺 ∈
N) ∧ (𝑅
∈ ω ∧ 𝑆
∈ N))) → ((𝐴 ·o 𝐺) +o (𝐵 ·o 𝐹)) ∈ ω) |
58 | | mulpiord 7279 |
. . . . . . . 8
⊢ ((𝐵 ∈ N ∧
𝐺 ∈ N)
→ (𝐵
·N 𝐺) = (𝐵 ·o 𝐺)) |
59 | | mulclpi 7290 |
. . . . . . . 8
⊢ ((𝐵 ∈ N ∧
𝐺 ∈ N)
→ (𝐵
·N 𝐺) ∈ N) |
60 | 58, 59 | eqeltrrd 2248 |
. . . . . . 7
⊢ ((𝐵 ∈ N ∧
𝐺 ∈ N)
→ (𝐵
·o 𝐺)
∈ N) |
61 | 60 | ad2ant2l 505 |
. . . . . 6
⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ N) ∧
(𝐹 ∈ ω ∧
𝐺 ∈ N))
→ (𝐵
·o 𝐺)
∈ N) |
62 | 61 | ad2ant2r 506 |
. . . . 5
⊢ ((((𝐴 ∈ ω ∧ 𝐵 ∈ N) ∧
(𝐶 ∈ ω ∧
𝐷 ∈ N))
∧ ((𝐹 ∈ ω
∧ 𝐺 ∈
N) ∧ (𝑅
∈ ω ∧ 𝑆
∈ N))) → (𝐵 ·o 𝐺) ∈ N) |
63 | | nnacl 6459 |
. . . . . 6
⊢ (((𝐶 ·o 𝑆) ∈ ω ∧ (𝐷 ·o 𝑅) ∈ ω) → ((𝐶 ·o 𝑆) +o (𝐷 ·o 𝑅)) ∈
ω) |
64 | 44, 47, 63 | syl2anc 409 |
. . . . 5
⊢ ((((𝐴 ∈ ω ∧ 𝐵 ∈ N) ∧
(𝐶 ∈ ω ∧
𝐷 ∈ N))
∧ ((𝐹 ∈ ω
∧ 𝐺 ∈
N) ∧ (𝑅
∈ ω ∧ 𝑆
∈ N))) → ((𝐶 ·o 𝑆) +o (𝐷 ·o 𝑅)) ∈ ω) |
65 | | mulpiord 7279 |
. . . . . . . 8
⊢ ((𝐷 ∈ N ∧
𝑆 ∈ N)
→ (𝐷
·N 𝑆) = (𝐷 ·o 𝑆)) |
66 | | mulclpi 7290 |
. . . . . . . 8
⊢ ((𝐷 ∈ N ∧
𝑆 ∈ N)
→ (𝐷
·N 𝑆) ∈ N) |
67 | 65, 66 | eqeltrrd 2248 |
. . . . . . 7
⊢ ((𝐷 ∈ N ∧
𝑆 ∈ N)
→ (𝐷
·o 𝑆)
∈ N) |
68 | 67 | ad2ant2l 505 |
. . . . . 6
⊢ (((𝐶 ∈ ω ∧ 𝐷 ∈ N) ∧
(𝑅 ∈ ω ∧
𝑆 ∈ N))
→ (𝐷
·o 𝑆)
∈ N) |
69 | 68 | ad2ant2l 505 |
. . . . 5
⊢ ((((𝐴 ∈ ω ∧ 𝐵 ∈ N) ∧
(𝐶 ∈ ω ∧
𝐷 ∈ N))
∧ ((𝐹 ∈ ω
∧ 𝐺 ∈
N) ∧ (𝑅
∈ ω ∧ 𝑆
∈ N))) → (𝐷 ·o 𝑆) ∈ N) |
70 | | enq0breq 7398 |
. . . . 5
⊢
(((((𝐴
·o 𝐺)
+o (𝐵
·o 𝐹))
∈ ω ∧ (𝐵
·o 𝐺)
∈ N) ∧ (((𝐶 ·o 𝑆) +o (𝐷 ·o 𝑅)) ∈ ω ∧ (𝐷 ·o 𝑆) ∈ N)) →
(〈((𝐴
·o 𝐺)
+o (𝐵
·o 𝐹)),
(𝐵 ·o
𝐺)〉
~Q0 〈((𝐶 ·o 𝑆) +o (𝐷 ·o 𝑅)), (𝐷 ·o 𝑆)〉 ↔ (((𝐴 ·o 𝐺) +o (𝐵 ·o 𝐹)) ·o (𝐷 ·o 𝑆)) = ((𝐵 ·o 𝐺) ·o ((𝐶 ·o 𝑆) +o (𝐷 ·o 𝑅))))) |
71 | 57, 62, 64, 69, 70 | syl22anc 1234 |
. . . 4
⊢ ((((𝐴 ∈ ω ∧ 𝐵 ∈ N) ∧
(𝐶 ∈ ω ∧
𝐷 ∈ N))
∧ ((𝐹 ∈ ω
∧ 𝐺 ∈
N) ∧ (𝑅
∈ ω ∧ 𝑆
∈ N))) → (〈((𝐴 ·o 𝐺) +o (𝐵 ·o 𝐹)), (𝐵 ·o 𝐺)〉 ~Q0
〈((𝐶
·o 𝑆)
+o (𝐷
·o 𝑅)),
(𝐷 ·o
𝑆)〉 ↔ (((𝐴 ·o 𝐺) +o (𝐵 ·o 𝐹)) ·o (𝐷 ·o 𝑆)) = ((𝐵 ·o 𝐺) ·o ((𝐶 ·o 𝑆) +o (𝐷 ·o 𝑅))))) |
72 | 71 | adantr 274 |
. . 3
⊢
(((((𝐴 ∈
ω ∧ 𝐵 ∈
N) ∧ (𝐶
∈ ω ∧ 𝐷
∈ N)) ∧ ((𝐹 ∈ ω ∧ 𝐺 ∈ N) ∧ (𝑅 ∈ ω ∧ 𝑆 ∈ N))) ∧
((𝐴 ·o
𝐷) = (𝐵 ·o 𝐶) ∧ (𝐹 ·o 𝑆) = (𝐺 ·o 𝑅))) → (〈((𝐴 ·o 𝐺) +o (𝐵 ·o 𝐹)), (𝐵 ·o 𝐺)〉 ~Q0
〈((𝐶
·o 𝑆)
+o (𝐷
·o 𝑅)),
(𝐷 ·o
𝑆)〉 ↔ (((𝐴 ·o 𝐺) +o (𝐵 ·o 𝐹)) ·o (𝐷 ·o 𝑆)) = ((𝐵 ·o 𝐺) ·o ((𝐶 ·o 𝑆) +o (𝐷 ·o 𝑅))))) |
73 | 55, 72 | mpbird 166 |
. 2
⊢
(((((𝐴 ∈
ω ∧ 𝐵 ∈
N) ∧ (𝐶
∈ ω ∧ 𝐷
∈ N)) ∧ ((𝐹 ∈ ω ∧ 𝐺 ∈ N) ∧ (𝑅 ∈ ω ∧ 𝑆 ∈ N))) ∧
((𝐴 ·o
𝐷) = (𝐵 ·o 𝐶) ∧ (𝐹 ·o 𝑆) = (𝐺 ·o 𝑅))) → 〈((𝐴 ·o 𝐺) +o (𝐵 ·o 𝐹)), (𝐵 ·o 𝐺)〉 ~Q0
〈((𝐶
·o 𝑆)
+o (𝐷
·o 𝑅)),
(𝐷 ·o
𝑆)〉) |
74 | 73 | ex 114 |
1
⊢ ((((𝐴 ∈ ω ∧ 𝐵 ∈ N) ∧
(𝐶 ∈ ω ∧
𝐷 ∈ N))
∧ ((𝐹 ∈ ω
∧ 𝐺 ∈
N) ∧ (𝑅
∈ ω ∧ 𝑆
∈ N))) → (((𝐴 ·o 𝐷) = (𝐵 ·o 𝐶) ∧ (𝐹 ·o 𝑆) = (𝐺 ·o 𝑅)) → 〈((𝐴 ·o 𝐺) +o (𝐵 ·o 𝐹)), (𝐵 ·o 𝐺)〉 ~Q0
〈((𝐶
·o 𝑆)
+o (𝐷
·o 𝑅)),
(𝐷 ·o
𝑆)〉)) |