Step | Hyp | Ref
| Expression |
1 | | elq 9568 |
. 2
⊢ (𝐴 ∈ ℚ ↔
∃𝑥 ∈ ℤ
∃𝑦 ∈ ℕ
𝐴 = (𝑥 / 𝑦)) |
2 | | elq 9568 |
. 2
⊢ (𝐵 ∈ ℚ ↔
∃𝑧 ∈ ℤ
∃𝑤 ∈ ℕ
𝐵 = (𝑧 / 𝑤)) |
3 | | zmulcl 9252 |
. . . . . . . . . . 11
⊢ ((𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ) → (𝑥 · 𝑧) ∈ ℤ) |
4 | | nnmulcl 8886 |
. . . . . . . . . . 11
⊢ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) → (𝑦 · 𝑤) ∈ ℕ) |
5 | 3, 4 | anim12i 336 |
. . . . . . . . . 10
⊢ (((𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ) ∧ (𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ)) → ((𝑥 · 𝑧) ∈ ℤ ∧ (𝑦 · 𝑤) ∈ ℕ)) |
6 | 5 | an4s 583 |
. . . . . . . . 9
⊢ (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℕ) ∧ (𝑧 ∈ ℤ ∧ 𝑤 ∈ ℕ)) → ((𝑥 · 𝑧) ∈ ℤ ∧ (𝑦 · 𝑤) ∈ ℕ)) |
7 | 6 | adantr 274 |
. . . . . . . 8
⊢ ((((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℕ) ∧ (𝑧 ∈ ℤ ∧ 𝑤 ∈ ℕ)) ∧ (𝐴 = (𝑥 / 𝑦) ∧ 𝐵 = (𝑧 / 𝑤))) → ((𝑥 · 𝑧) ∈ ℤ ∧ (𝑦 · 𝑤) ∈ ℕ)) |
8 | | oveq12 5859 |
. . . . . . . . 9
⊢ ((𝐴 = (𝑥 / 𝑦) ∧ 𝐵 = (𝑧 / 𝑤)) → (𝐴 · 𝐵) = ((𝑥 / 𝑦) · (𝑧 / 𝑤))) |
9 | | zcn 9204 |
. . . . . . . . . . . 12
⊢ (𝑥 ∈ ℤ → 𝑥 ∈
ℂ) |
10 | | zcn 9204 |
. . . . . . . . . . . 12
⊢ (𝑧 ∈ ℤ → 𝑧 ∈
ℂ) |
11 | 9, 10 | anim12i 336 |
. . . . . . . . . . 11
⊢ ((𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ) → (𝑥 ∈ ℂ ∧ 𝑧 ∈
ℂ)) |
12 | 11 | ad2ant2r 506 |
. . . . . . . . . 10
⊢ (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℕ) ∧ (𝑧 ∈ ℤ ∧ 𝑤 ∈ ℕ)) → (𝑥 ∈ ℂ ∧ 𝑧 ∈
ℂ)) |
13 | | nncn 8873 |
. . . . . . . . . . . . 13
⊢ (𝑦 ∈ ℕ → 𝑦 ∈
ℂ) |
14 | | nnap0 8894 |
. . . . . . . . . . . . 13
⊢ (𝑦 ∈ ℕ → 𝑦 # 0) |
15 | 13, 14 | jca 304 |
. . . . . . . . . . . 12
⊢ (𝑦 ∈ ℕ → (𝑦 ∈ ℂ ∧ 𝑦 # 0)) |
16 | | nncn 8873 |
. . . . . . . . . . . . 13
⊢ (𝑤 ∈ ℕ → 𝑤 ∈
ℂ) |
17 | | nnap0 8894 |
. . . . . . . . . . . . 13
⊢ (𝑤 ∈ ℕ → 𝑤 # 0) |
18 | 16, 17 | jca 304 |
. . . . . . . . . . . 12
⊢ (𝑤 ∈ ℕ → (𝑤 ∈ ℂ ∧ 𝑤 # 0)) |
19 | 15, 18 | anim12i 336 |
. . . . . . . . . . 11
⊢ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) → ((𝑦 ∈ ℂ ∧ 𝑦 # 0) ∧ (𝑤 ∈ ℂ ∧ 𝑤 # 0))) |
20 | 19 | ad2ant2l 505 |
. . . . . . . . . 10
⊢ (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℕ) ∧ (𝑧 ∈ ℤ ∧ 𝑤 ∈ ℕ)) → ((𝑦 ∈ ℂ ∧ 𝑦 # 0) ∧ (𝑤 ∈ ℂ ∧ 𝑤 # 0))) |
21 | | divmuldivap 8616 |
. . . . . . . . . 10
⊢ (((𝑥 ∈ ℂ ∧ 𝑧 ∈ ℂ) ∧ ((𝑦 ∈ ℂ ∧ 𝑦 # 0) ∧ (𝑤 ∈ ℂ ∧ 𝑤 # 0))) → ((𝑥 / 𝑦) · (𝑧 / 𝑤)) = ((𝑥 · 𝑧) / (𝑦 · 𝑤))) |
22 | 12, 20, 21 | syl2anc 409 |
. . . . . . . . 9
⊢ (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℕ) ∧ (𝑧 ∈ ℤ ∧ 𝑤 ∈ ℕ)) → ((𝑥 / 𝑦) · (𝑧 / 𝑤)) = ((𝑥 · 𝑧) / (𝑦 · 𝑤))) |
23 | 8, 22 | sylan9eqr 2225 |
. . . . . . . 8
⊢ ((((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℕ) ∧ (𝑧 ∈ ℤ ∧ 𝑤 ∈ ℕ)) ∧ (𝐴 = (𝑥 / 𝑦) ∧ 𝐵 = (𝑧 / 𝑤))) → (𝐴 · 𝐵) = ((𝑥 · 𝑧) / (𝑦 · 𝑤))) |
24 | | rspceov 5892 |
. . . . . . . . . 10
⊢ (((𝑥 · 𝑧) ∈ ℤ ∧ (𝑦 · 𝑤) ∈ ℕ ∧ (𝐴 · 𝐵) = ((𝑥 · 𝑧) / (𝑦 · 𝑤))) → ∃𝑣 ∈ ℤ ∃𝑢 ∈ ℕ (𝐴 · 𝐵) = (𝑣 / 𝑢)) |
25 | 24 | 3expa 1198 |
. . . . . . . . 9
⊢ ((((𝑥 · 𝑧) ∈ ℤ ∧ (𝑦 · 𝑤) ∈ ℕ) ∧ (𝐴 · 𝐵) = ((𝑥 · 𝑧) / (𝑦 · 𝑤))) → ∃𝑣 ∈ ℤ ∃𝑢 ∈ ℕ (𝐴 · 𝐵) = (𝑣 / 𝑢)) |
26 | | elq 9568 |
. . . . . . . . 9
⊢ ((𝐴 · 𝐵) ∈ ℚ ↔ ∃𝑣 ∈ ℤ ∃𝑢 ∈ ℕ (𝐴 · 𝐵) = (𝑣 / 𝑢)) |
27 | 25, 26 | sylibr 133 |
. . . . . . . 8
⊢ ((((𝑥 · 𝑧) ∈ ℤ ∧ (𝑦 · 𝑤) ∈ ℕ) ∧ (𝐴 · 𝐵) = ((𝑥 · 𝑧) / (𝑦 · 𝑤))) → (𝐴 · 𝐵) ∈ ℚ) |
28 | 7, 23, 27 | syl2anc 409 |
. . . . . . 7
⊢ ((((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℕ) ∧ (𝑧 ∈ ℤ ∧ 𝑤 ∈ ℕ)) ∧ (𝐴 = (𝑥 / 𝑦) ∧ 𝐵 = (𝑧 / 𝑤))) → (𝐴 · 𝐵) ∈ ℚ) |
29 | 28 | an4s 583 |
. . . . . 6
⊢ ((((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℕ) ∧ 𝐴 = (𝑥 / 𝑦)) ∧ ((𝑧 ∈ ℤ ∧ 𝑤 ∈ ℕ) ∧ 𝐵 = (𝑧 / 𝑤))) → (𝐴 · 𝐵) ∈ ℚ) |
30 | 29 | exp43 370 |
. . . . 5
⊢ ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℕ) → (𝐴 = (𝑥 / 𝑦) → ((𝑧 ∈ ℤ ∧ 𝑤 ∈ ℕ) → (𝐵 = (𝑧 / 𝑤) → (𝐴 · 𝐵) ∈ ℚ)))) |
31 | 30 | rexlimivv 2593 |
. . . 4
⊢
(∃𝑥 ∈
ℤ ∃𝑦 ∈
ℕ 𝐴 = (𝑥 / 𝑦) → ((𝑧 ∈ ℤ ∧ 𝑤 ∈ ℕ) → (𝐵 = (𝑧 / 𝑤) → (𝐴 · 𝐵) ∈ ℚ))) |
32 | 31 | rexlimdvv 2594 |
. . 3
⊢
(∃𝑥 ∈
ℤ ∃𝑦 ∈
ℕ 𝐴 = (𝑥 / 𝑦) → (∃𝑧 ∈ ℤ ∃𝑤 ∈ ℕ 𝐵 = (𝑧 / 𝑤) → (𝐴 · 𝐵) ∈ ℚ)) |
33 | 32 | imp 123 |
. 2
⊢
((∃𝑥 ∈
ℤ ∃𝑦 ∈
ℕ 𝐴 = (𝑥 / 𝑦) ∧ ∃𝑧 ∈ ℤ ∃𝑤 ∈ ℕ 𝐵 = (𝑧 / 𝑤)) → (𝐴 · 𝐵) ∈ ℚ) |
34 | 1, 2, 33 | syl2anb 289 |
1
⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) → (𝐴 · 𝐵) ∈ ℚ) |