Proof of Theorem bezoutlem2
Step | Hyp | Ref
| Expression |
1 | | bezout.2 |
. 2
⊢ 𝐺 = inf(𝑀, ℝ, < ) |
2 | | bezout.1 |
. . . . 5
⊢ 𝑀 = {𝑧 ∈ ℕ ∣ ∃𝑥 ∈ ℤ ∃𝑦 ∈ ℤ 𝑧 = ((𝐴 · 𝑥) + (𝐵 · 𝑦))} |
3 | 2 | ssrab3 4056 |
. . . 4
⊢ 𝑀 ⊆
ℕ |
4 | | nnuz 12275 |
. . . 4
⊢ ℕ =
(ℤ≥‘1) |
5 | 3, 4 | sseqtri 4002 |
. . 3
⊢ 𝑀 ⊆
(ℤ≥‘1) |
6 | | bezout.3 |
. . . . . 6
⊢ (𝜑 → 𝐴 ∈ ℤ) |
7 | | bezout.4 |
. . . . . 6
⊢ (𝜑 → 𝐵 ∈ ℤ) |
8 | 2, 6, 7 | bezoutlem1 15881 |
. . . . 5
⊢ (𝜑 → (𝐴 ≠ 0 → (abs‘𝐴) ∈ 𝑀)) |
9 | | ne0i 4299 |
. . . . 5
⊢
((abs‘𝐴)
∈ 𝑀 → 𝑀 ≠ ∅) |
10 | 8, 9 | syl6 35 |
. . . 4
⊢ (𝜑 → (𝐴 ≠ 0 → 𝑀 ≠ ∅)) |
11 | | eqid 2821 |
. . . . . . 7
⊢ {𝑧 ∈ ℕ ∣
∃𝑦 ∈ ℤ
∃𝑥 ∈ ℤ
𝑧 = ((𝐵 · 𝑦) + (𝐴 · 𝑥))} = {𝑧 ∈ ℕ ∣ ∃𝑦 ∈ ℤ ∃𝑥 ∈ ℤ 𝑧 = ((𝐵 · 𝑦) + (𝐴 · 𝑥))} |
12 | 11, 7, 6 | bezoutlem1 15881 |
. . . . . 6
⊢ (𝜑 → (𝐵 ≠ 0 → (abs‘𝐵) ∈ {𝑧 ∈ ℕ ∣ ∃𝑦 ∈ ℤ ∃𝑥 ∈ ℤ 𝑧 = ((𝐵 · 𝑦) + (𝐴 · 𝑥))})) |
13 | | rexcom 3355 |
. . . . . . . . . 10
⊢
(∃𝑥 ∈
ℤ ∃𝑦 ∈
ℤ 𝑧 = ((𝐴 · 𝑥) + (𝐵 · 𝑦)) ↔ ∃𝑦 ∈ ℤ ∃𝑥 ∈ ℤ 𝑧 = ((𝐴 · 𝑥) + (𝐵 · 𝑦))) |
14 | 6 | zcnd 12082 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → 𝐴 ∈ ℂ) |
15 | 14 | adantr 483 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ (𝑦 ∈ ℤ ∧ 𝑥 ∈ ℤ)) → 𝐴 ∈ ℂ) |
16 | | zcn 11980 |
. . . . . . . . . . . . . . 15
⊢ (𝑥 ∈ ℤ → 𝑥 ∈
ℂ) |
17 | 16 | ad2antll 727 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ (𝑦 ∈ ℤ ∧ 𝑥 ∈ ℤ)) → 𝑥 ∈ ℂ) |
18 | 15, 17 | mulcld 10655 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ (𝑦 ∈ ℤ ∧ 𝑥 ∈ ℤ)) → (𝐴 · 𝑥) ∈ ℂ) |
19 | 7 | zcnd 12082 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → 𝐵 ∈ ℂ) |
20 | 19 | adantr 483 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ (𝑦 ∈ ℤ ∧ 𝑥 ∈ ℤ)) → 𝐵 ∈ ℂ) |
21 | | zcn 11980 |
. . . . . . . . . . . . . . 15
⊢ (𝑦 ∈ ℤ → 𝑦 ∈
ℂ) |
22 | 21 | ad2antrl 726 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ (𝑦 ∈ ℤ ∧ 𝑥 ∈ ℤ)) → 𝑦 ∈ ℂ) |
23 | 20, 22 | mulcld 10655 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ (𝑦 ∈ ℤ ∧ 𝑥 ∈ ℤ)) → (𝐵 · 𝑦) ∈ ℂ) |
24 | 18, 23 | addcomd 10836 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ (𝑦 ∈ ℤ ∧ 𝑥 ∈ ℤ)) → ((𝐴 · 𝑥) + (𝐵 · 𝑦)) = ((𝐵 · 𝑦) + (𝐴 · 𝑥))) |
25 | 24 | eqeq2d 2832 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ (𝑦 ∈ ℤ ∧ 𝑥 ∈ ℤ)) → (𝑧 = ((𝐴 · 𝑥) + (𝐵 · 𝑦)) ↔ 𝑧 = ((𝐵 · 𝑦) + (𝐴 · 𝑥)))) |
26 | 25 | 2rexbidva 3299 |
. . . . . . . . . 10
⊢ (𝜑 → (∃𝑦 ∈ ℤ ∃𝑥 ∈ ℤ 𝑧 = ((𝐴 · 𝑥) + (𝐵 · 𝑦)) ↔ ∃𝑦 ∈ ℤ ∃𝑥 ∈ ℤ 𝑧 = ((𝐵 · 𝑦) + (𝐴 · 𝑥)))) |
27 | 13, 26 | syl5bb 285 |
. . . . . . . . 9
⊢ (𝜑 → (∃𝑥 ∈ ℤ ∃𝑦 ∈ ℤ 𝑧 = ((𝐴 · 𝑥) + (𝐵 · 𝑦)) ↔ ∃𝑦 ∈ ℤ ∃𝑥 ∈ ℤ 𝑧 = ((𝐵 · 𝑦) + (𝐴 · 𝑥)))) |
28 | 27 | rabbidv 3480 |
. . . . . . . 8
⊢ (𝜑 → {𝑧 ∈ ℕ ∣ ∃𝑥 ∈ ℤ ∃𝑦 ∈ ℤ 𝑧 = ((𝐴 · 𝑥) + (𝐵 · 𝑦))} = {𝑧 ∈ ℕ ∣ ∃𝑦 ∈ ℤ ∃𝑥 ∈ ℤ 𝑧 = ((𝐵 · 𝑦) + (𝐴 · 𝑥))}) |
29 | 2, 28 | syl5eq 2868 |
. . . . . . 7
⊢ (𝜑 → 𝑀 = {𝑧 ∈ ℕ ∣ ∃𝑦 ∈ ℤ ∃𝑥 ∈ ℤ 𝑧 = ((𝐵 · 𝑦) + (𝐴 · 𝑥))}) |
30 | 29 | eleq2d 2898 |
. . . . . 6
⊢ (𝜑 → ((abs‘𝐵) ∈ 𝑀 ↔ (abs‘𝐵) ∈ {𝑧 ∈ ℕ ∣ ∃𝑦 ∈ ℤ ∃𝑥 ∈ ℤ 𝑧 = ((𝐵 · 𝑦) + (𝐴 · 𝑥))})) |
31 | 12, 30 | sylibrd 261 |
. . . . 5
⊢ (𝜑 → (𝐵 ≠ 0 → (abs‘𝐵) ∈ 𝑀)) |
32 | | ne0i 4299 |
. . . . 5
⊢
((abs‘𝐵)
∈ 𝑀 → 𝑀 ≠ ∅) |
33 | 31, 32 | syl6 35 |
. . . 4
⊢ (𝜑 → (𝐵 ≠ 0 → 𝑀 ≠ ∅)) |
34 | | bezout.5 |
. . . . 5
⊢ (𝜑 → ¬ (𝐴 = 0 ∧ 𝐵 = 0)) |
35 | | neorian 3111 |
. . . . 5
⊢ ((𝐴 ≠ 0 ∨ 𝐵 ≠ 0) ↔ ¬ (𝐴 = 0 ∧ 𝐵 = 0)) |
36 | 34, 35 | sylibr 236 |
. . . 4
⊢ (𝜑 → (𝐴 ≠ 0 ∨ 𝐵 ≠ 0)) |
37 | 10, 33, 36 | mpjaod 856 |
. . 3
⊢ (𝜑 → 𝑀 ≠ ∅) |
38 | | infssuzcl 12326 |
. . 3
⊢ ((𝑀 ⊆
(ℤ≥‘1) ∧ 𝑀 ≠ ∅) → inf(𝑀, ℝ, < ) ∈ 𝑀) |
39 | 5, 37, 38 | sylancr 589 |
. 2
⊢ (𝜑 → inf(𝑀, ℝ, < ) ∈ 𝑀) |
40 | 1, 39 | eqeltrid 2917 |
1
⊢ (𝜑 → 𝐺 ∈ 𝑀) |