Proof of Theorem fzopth
Step | Hyp | Ref
| Expression |
1 | | eluzfz1 9939 |
. . . . . . . . 9
⊢ (𝑁 ∈
(ℤ≥‘𝑀) → 𝑀 ∈ (𝑀...𝑁)) |
2 | 1 | adantr 274 |
. . . . . . . 8
⊢ ((𝑁 ∈
(ℤ≥‘𝑀) ∧ (𝑀...𝑁) = (𝐽...𝐾)) → 𝑀 ∈ (𝑀...𝑁)) |
3 | | simpr 109 |
. . . . . . . 8
⊢ ((𝑁 ∈
(ℤ≥‘𝑀) ∧ (𝑀...𝑁) = (𝐽...𝐾)) → (𝑀...𝑁) = (𝐽...𝐾)) |
4 | 2, 3 | eleqtrd 2236 |
. . . . . . 7
⊢ ((𝑁 ∈
(ℤ≥‘𝑀) ∧ (𝑀...𝑁) = (𝐽...𝐾)) → 𝑀 ∈ (𝐽...𝐾)) |
5 | | elfzuz 9930 |
. . . . . . 7
⊢ (𝑀 ∈ (𝐽...𝐾) → 𝑀 ∈ (ℤ≥‘𝐽)) |
6 | | uzss 9464 |
. . . . . . 7
⊢ (𝑀 ∈
(ℤ≥‘𝐽) → (ℤ≥‘𝑀) ⊆
(ℤ≥‘𝐽)) |
7 | 4, 5, 6 | 3syl 17 |
. . . . . 6
⊢ ((𝑁 ∈
(ℤ≥‘𝑀) ∧ (𝑀...𝑁) = (𝐽...𝐾)) →
(ℤ≥‘𝑀) ⊆
(ℤ≥‘𝐽)) |
8 | | elfzuz2 9937 |
. . . . . . . . 9
⊢ (𝑀 ∈ (𝐽...𝐾) → 𝐾 ∈ (ℤ≥‘𝐽)) |
9 | | eluzfz1 9939 |
. . . . . . . . 9
⊢ (𝐾 ∈
(ℤ≥‘𝐽) → 𝐽 ∈ (𝐽...𝐾)) |
10 | 4, 8, 9 | 3syl 17 |
. . . . . . . 8
⊢ ((𝑁 ∈
(ℤ≥‘𝑀) ∧ (𝑀...𝑁) = (𝐽...𝐾)) → 𝐽 ∈ (𝐽...𝐾)) |
11 | 10, 3 | eleqtrrd 2237 |
. . . . . . 7
⊢ ((𝑁 ∈
(ℤ≥‘𝑀) ∧ (𝑀...𝑁) = (𝐽...𝐾)) → 𝐽 ∈ (𝑀...𝑁)) |
12 | | elfzuz 9930 |
. . . . . . 7
⊢ (𝐽 ∈ (𝑀...𝑁) → 𝐽 ∈ (ℤ≥‘𝑀)) |
13 | | uzss 9464 |
. . . . . . 7
⊢ (𝐽 ∈
(ℤ≥‘𝑀) → (ℤ≥‘𝐽) ⊆
(ℤ≥‘𝑀)) |
14 | 11, 12, 13 | 3syl 17 |
. . . . . 6
⊢ ((𝑁 ∈
(ℤ≥‘𝑀) ∧ (𝑀...𝑁) = (𝐽...𝐾)) →
(ℤ≥‘𝐽) ⊆
(ℤ≥‘𝑀)) |
15 | 7, 14 | eqssd 3145 |
. . . . 5
⊢ ((𝑁 ∈
(ℤ≥‘𝑀) ∧ (𝑀...𝑁) = (𝐽...𝐾)) →
(ℤ≥‘𝑀) = (ℤ≥‘𝐽)) |
16 | | eluzel2 9449 |
. . . . . . 7
⊢ (𝑁 ∈
(ℤ≥‘𝑀) → 𝑀 ∈ ℤ) |
17 | 16 | adantr 274 |
. . . . . 6
⊢ ((𝑁 ∈
(ℤ≥‘𝑀) ∧ (𝑀...𝑁) = (𝐽...𝐾)) → 𝑀 ∈ ℤ) |
18 | | uz11 9466 |
. . . . . 6
⊢ (𝑀 ∈ ℤ →
((ℤ≥‘𝑀) = (ℤ≥‘𝐽) ↔ 𝑀 = 𝐽)) |
19 | 17, 18 | syl 14 |
. . . . 5
⊢ ((𝑁 ∈
(ℤ≥‘𝑀) ∧ (𝑀...𝑁) = (𝐽...𝐾)) →
((ℤ≥‘𝑀) = (ℤ≥‘𝐽) ↔ 𝑀 = 𝐽)) |
20 | 15, 19 | mpbid 146 |
. . . 4
⊢ ((𝑁 ∈
(ℤ≥‘𝑀) ∧ (𝑀...𝑁) = (𝐽...𝐾)) → 𝑀 = 𝐽) |
21 | | eluzfz2 9940 |
. . . . . . . . 9
⊢ (𝐾 ∈
(ℤ≥‘𝐽) → 𝐾 ∈ (𝐽...𝐾)) |
22 | 4, 8, 21 | 3syl 17 |
. . . . . . . 8
⊢ ((𝑁 ∈
(ℤ≥‘𝑀) ∧ (𝑀...𝑁) = (𝐽...𝐾)) → 𝐾 ∈ (𝐽...𝐾)) |
23 | 22, 3 | eleqtrrd 2237 |
. . . . . . 7
⊢ ((𝑁 ∈
(ℤ≥‘𝑀) ∧ (𝑀...𝑁) = (𝐽...𝐾)) → 𝐾 ∈ (𝑀...𝑁)) |
24 | | elfzuz3 9931 |
. . . . . . 7
⊢ (𝐾 ∈ (𝑀...𝑁) → 𝑁 ∈ (ℤ≥‘𝐾)) |
25 | | uzss 9464 |
. . . . . . 7
⊢ (𝑁 ∈
(ℤ≥‘𝐾) → (ℤ≥‘𝑁) ⊆
(ℤ≥‘𝐾)) |
26 | 23, 24, 25 | 3syl 17 |
. . . . . 6
⊢ ((𝑁 ∈
(ℤ≥‘𝑀) ∧ (𝑀...𝑁) = (𝐽...𝐾)) →
(ℤ≥‘𝑁) ⊆
(ℤ≥‘𝐾)) |
27 | | eluzfz2 9940 |
. . . . . . . . 9
⊢ (𝑁 ∈
(ℤ≥‘𝑀) → 𝑁 ∈ (𝑀...𝑁)) |
28 | 27 | adantr 274 |
. . . . . . . 8
⊢ ((𝑁 ∈
(ℤ≥‘𝑀) ∧ (𝑀...𝑁) = (𝐽...𝐾)) → 𝑁 ∈ (𝑀...𝑁)) |
29 | 28, 3 | eleqtrd 2236 |
. . . . . . 7
⊢ ((𝑁 ∈
(ℤ≥‘𝑀) ∧ (𝑀...𝑁) = (𝐽...𝐾)) → 𝑁 ∈ (𝐽...𝐾)) |
30 | | elfzuz3 9931 |
. . . . . . 7
⊢ (𝑁 ∈ (𝐽...𝐾) → 𝐾 ∈ (ℤ≥‘𝑁)) |
31 | | uzss 9464 |
. . . . . . 7
⊢ (𝐾 ∈
(ℤ≥‘𝑁) → (ℤ≥‘𝐾) ⊆
(ℤ≥‘𝑁)) |
32 | 29, 30, 31 | 3syl 17 |
. . . . . 6
⊢ ((𝑁 ∈
(ℤ≥‘𝑀) ∧ (𝑀...𝑁) = (𝐽...𝐾)) →
(ℤ≥‘𝐾) ⊆
(ℤ≥‘𝑁)) |
33 | 26, 32 | eqssd 3145 |
. . . . 5
⊢ ((𝑁 ∈
(ℤ≥‘𝑀) ∧ (𝑀...𝑁) = (𝐽...𝐾)) →
(ℤ≥‘𝑁) = (ℤ≥‘𝐾)) |
34 | | eluzelz 9453 |
. . . . . . 7
⊢ (𝑁 ∈
(ℤ≥‘𝑀) → 𝑁 ∈ ℤ) |
35 | 34 | adantr 274 |
. . . . . 6
⊢ ((𝑁 ∈
(ℤ≥‘𝑀) ∧ (𝑀...𝑁) = (𝐽...𝐾)) → 𝑁 ∈ ℤ) |
36 | | uz11 9466 |
. . . . . 6
⊢ (𝑁 ∈ ℤ →
((ℤ≥‘𝑁) = (ℤ≥‘𝐾) ↔ 𝑁 = 𝐾)) |
37 | 35, 36 | syl 14 |
. . . . 5
⊢ ((𝑁 ∈
(ℤ≥‘𝑀) ∧ (𝑀...𝑁) = (𝐽...𝐾)) →
((ℤ≥‘𝑁) = (ℤ≥‘𝐾) ↔ 𝑁 = 𝐾)) |
38 | 33, 37 | mpbid 146 |
. . . 4
⊢ ((𝑁 ∈
(ℤ≥‘𝑀) ∧ (𝑀...𝑁) = (𝐽...𝐾)) → 𝑁 = 𝐾) |
39 | 20, 38 | jca 304 |
. . 3
⊢ ((𝑁 ∈
(ℤ≥‘𝑀) ∧ (𝑀...𝑁) = (𝐽...𝐾)) → (𝑀 = 𝐽 ∧ 𝑁 = 𝐾)) |
40 | 39 | ex 114 |
. 2
⊢ (𝑁 ∈
(ℤ≥‘𝑀) → ((𝑀...𝑁) = (𝐽...𝐾) → (𝑀 = 𝐽 ∧ 𝑁 = 𝐾))) |
41 | | oveq12 5835 |
. 2
⊢ ((𝑀 = 𝐽 ∧ 𝑁 = 𝐾) → (𝑀...𝑁) = (𝐽...𝐾)) |
42 | 40, 41 | impbid1 141 |
1
⊢ (𝑁 ∈
(ℤ≥‘𝑀) → ((𝑀...𝑁) = (𝐽...𝐾) ↔ (𝑀 = 𝐽 ∧ 𝑁 = 𝐾))) |