Proof of Theorem fzopth
| Step | Hyp | Ref | Expression | 
|---|
| 1 |  | eluzfz1 13571 | . . . . . . . . 9
⊢ (𝑁 ∈
(ℤ≥‘𝑀) → 𝑀 ∈ (𝑀...𝑁)) | 
| 2 | 1 | adantr 480 | . . . . . . . 8
⊢ ((𝑁 ∈
(ℤ≥‘𝑀) ∧ (𝑀...𝑁) = (𝐽...𝐾)) → 𝑀 ∈ (𝑀...𝑁)) | 
| 3 |  | simpr 484 | . . . . . . . 8
⊢ ((𝑁 ∈
(ℤ≥‘𝑀) ∧ (𝑀...𝑁) = (𝐽...𝐾)) → (𝑀...𝑁) = (𝐽...𝐾)) | 
| 4 | 2, 3 | eleqtrd 2843 | . . . . . . 7
⊢ ((𝑁 ∈
(ℤ≥‘𝑀) ∧ (𝑀...𝑁) = (𝐽...𝐾)) → 𝑀 ∈ (𝐽...𝐾)) | 
| 5 |  | elfzuz 13560 | . . . . . . 7
⊢ (𝑀 ∈ (𝐽...𝐾) → 𝑀 ∈ (ℤ≥‘𝐽)) | 
| 6 |  | uzss 12901 | . . . . . . 7
⊢ (𝑀 ∈
(ℤ≥‘𝐽) → (ℤ≥‘𝑀) ⊆
(ℤ≥‘𝐽)) | 
| 7 | 4, 5, 6 | 3syl 18 | . . . . . 6
⊢ ((𝑁 ∈
(ℤ≥‘𝑀) ∧ (𝑀...𝑁) = (𝐽...𝐾)) →
(ℤ≥‘𝑀) ⊆
(ℤ≥‘𝐽)) | 
| 8 |  | elfzuz2 13569 | . . . . . . . . 9
⊢ (𝑀 ∈ (𝐽...𝐾) → 𝐾 ∈ (ℤ≥‘𝐽)) | 
| 9 |  | eluzfz1 13571 | . . . . . . . . 9
⊢ (𝐾 ∈
(ℤ≥‘𝐽) → 𝐽 ∈ (𝐽...𝐾)) | 
| 10 | 4, 8, 9 | 3syl 18 | . . . . . . . 8
⊢ ((𝑁 ∈
(ℤ≥‘𝑀) ∧ (𝑀...𝑁) = (𝐽...𝐾)) → 𝐽 ∈ (𝐽...𝐾)) | 
| 11 | 10, 3 | eleqtrrd 2844 | . . . . . . 7
⊢ ((𝑁 ∈
(ℤ≥‘𝑀) ∧ (𝑀...𝑁) = (𝐽...𝐾)) → 𝐽 ∈ (𝑀...𝑁)) | 
| 12 |  | elfzuz 13560 | . . . . . . 7
⊢ (𝐽 ∈ (𝑀...𝑁) → 𝐽 ∈ (ℤ≥‘𝑀)) | 
| 13 |  | uzss 12901 | . . . . . . 7
⊢ (𝐽 ∈
(ℤ≥‘𝑀) → (ℤ≥‘𝐽) ⊆
(ℤ≥‘𝑀)) | 
| 14 | 11, 12, 13 | 3syl 18 | . . . . . 6
⊢ ((𝑁 ∈
(ℤ≥‘𝑀) ∧ (𝑀...𝑁) = (𝐽...𝐾)) →
(ℤ≥‘𝐽) ⊆
(ℤ≥‘𝑀)) | 
| 15 | 7, 14 | eqssd 4001 | . . . . 5
⊢ ((𝑁 ∈
(ℤ≥‘𝑀) ∧ (𝑀...𝑁) = (𝐽...𝐾)) →
(ℤ≥‘𝑀) = (ℤ≥‘𝐽)) | 
| 16 |  | eluzel2 12883 | . . . . . . 7
⊢ (𝑁 ∈
(ℤ≥‘𝑀) → 𝑀 ∈ ℤ) | 
| 17 | 16 | adantr 480 | . . . . . 6
⊢ ((𝑁 ∈
(ℤ≥‘𝑀) ∧ (𝑀...𝑁) = (𝐽...𝐾)) → 𝑀 ∈ ℤ) | 
| 18 |  | uz11 12903 | . . . . . 6
⊢ (𝑀 ∈ ℤ →
((ℤ≥‘𝑀) = (ℤ≥‘𝐽) ↔ 𝑀 = 𝐽)) | 
| 19 | 17, 18 | syl 17 | . . . . 5
⊢ ((𝑁 ∈
(ℤ≥‘𝑀) ∧ (𝑀...𝑁) = (𝐽...𝐾)) →
((ℤ≥‘𝑀) = (ℤ≥‘𝐽) ↔ 𝑀 = 𝐽)) | 
| 20 | 15, 19 | mpbid 232 | . . . 4
⊢ ((𝑁 ∈
(ℤ≥‘𝑀) ∧ (𝑀...𝑁) = (𝐽...𝐾)) → 𝑀 = 𝐽) | 
| 21 |  | eluzfz2 13572 | . . . . . . . . 9
⊢ (𝐾 ∈
(ℤ≥‘𝐽) → 𝐾 ∈ (𝐽...𝐾)) | 
| 22 | 4, 8, 21 | 3syl 18 | . . . . . . . 8
⊢ ((𝑁 ∈
(ℤ≥‘𝑀) ∧ (𝑀...𝑁) = (𝐽...𝐾)) → 𝐾 ∈ (𝐽...𝐾)) | 
| 23 | 22, 3 | eleqtrrd 2844 | . . . . . . 7
⊢ ((𝑁 ∈
(ℤ≥‘𝑀) ∧ (𝑀...𝑁) = (𝐽...𝐾)) → 𝐾 ∈ (𝑀...𝑁)) | 
| 24 |  | elfzuz3 13561 | . . . . . . 7
⊢ (𝐾 ∈ (𝑀...𝑁) → 𝑁 ∈ (ℤ≥‘𝐾)) | 
| 25 |  | uzss 12901 | . . . . . . 7
⊢ (𝑁 ∈
(ℤ≥‘𝐾) → (ℤ≥‘𝑁) ⊆
(ℤ≥‘𝐾)) | 
| 26 | 23, 24, 25 | 3syl 18 | . . . . . 6
⊢ ((𝑁 ∈
(ℤ≥‘𝑀) ∧ (𝑀...𝑁) = (𝐽...𝐾)) →
(ℤ≥‘𝑁) ⊆
(ℤ≥‘𝐾)) | 
| 27 |  | eluzfz2 13572 | . . . . . . . . 9
⊢ (𝑁 ∈
(ℤ≥‘𝑀) → 𝑁 ∈ (𝑀...𝑁)) | 
| 28 | 27 | adantr 480 | . . . . . . . 8
⊢ ((𝑁 ∈
(ℤ≥‘𝑀) ∧ (𝑀...𝑁) = (𝐽...𝐾)) → 𝑁 ∈ (𝑀...𝑁)) | 
| 29 | 28, 3 | eleqtrd 2843 | . . . . . . 7
⊢ ((𝑁 ∈
(ℤ≥‘𝑀) ∧ (𝑀...𝑁) = (𝐽...𝐾)) → 𝑁 ∈ (𝐽...𝐾)) | 
| 30 |  | elfzuz3 13561 | . . . . . . 7
⊢ (𝑁 ∈ (𝐽...𝐾) → 𝐾 ∈ (ℤ≥‘𝑁)) | 
| 31 |  | uzss 12901 | . . . . . . 7
⊢ (𝐾 ∈
(ℤ≥‘𝑁) → (ℤ≥‘𝐾) ⊆
(ℤ≥‘𝑁)) | 
| 32 | 29, 30, 31 | 3syl 18 | . . . . . 6
⊢ ((𝑁 ∈
(ℤ≥‘𝑀) ∧ (𝑀...𝑁) = (𝐽...𝐾)) →
(ℤ≥‘𝐾) ⊆
(ℤ≥‘𝑁)) | 
| 33 | 26, 32 | eqssd 4001 | . . . . 5
⊢ ((𝑁 ∈
(ℤ≥‘𝑀) ∧ (𝑀...𝑁) = (𝐽...𝐾)) →
(ℤ≥‘𝑁) = (ℤ≥‘𝐾)) | 
| 34 |  | eluzelz 12888 | . . . . . . 7
⊢ (𝑁 ∈
(ℤ≥‘𝑀) → 𝑁 ∈ ℤ) | 
| 35 | 34 | adantr 480 | . . . . . 6
⊢ ((𝑁 ∈
(ℤ≥‘𝑀) ∧ (𝑀...𝑁) = (𝐽...𝐾)) → 𝑁 ∈ ℤ) | 
| 36 |  | uz11 12903 | . . . . . 6
⊢ (𝑁 ∈ ℤ →
((ℤ≥‘𝑁) = (ℤ≥‘𝐾) ↔ 𝑁 = 𝐾)) | 
| 37 | 35, 36 | syl 17 | . . . . 5
⊢ ((𝑁 ∈
(ℤ≥‘𝑀) ∧ (𝑀...𝑁) = (𝐽...𝐾)) →
((ℤ≥‘𝑁) = (ℤ≥‘𝐾) ↔ 𝑁 = 𝐾)) | 
| 38 | 33, 37 | mpbid 232 | . . . 4
⊢ ((𝑁 ∈
(ℤ≥‘𝑀) ∧ (𝑀...𝑁) = (𝐽...𝐾)) → 𝑁 = 𝐾) | 
| 39 | 20, 38 | jca 511 | . . 3
⊢ ((𝑁 ∈
(ℤ≥‘𝑀) ∧ (𝑀...𝑁) = (𝐽...𝐾)) → (𝑀 = 𝐽 ∧ 𝑁 = 𝐾)) | 
| 40 | 39 | ex 412 | . 2
⊢ (𝑁 ∈
(ℤ≥‘𝑀) → ((𝑀...𝑁) = (𝐽...𝐾) → (𝑀 = 𝐽 ∧ 𝑁 = 𝐾))) | 
| 41 |  | oveq12 7440 | . 2
⊢ ((𝑀 = 𝐽 ∧ 𝑁 = 𝐾) → (𝑀...𝑁) = (𝐽...𝐾)) | 
| 42 | 40, 41 | impbid1 225 | 1
⊢ (𝑁 ∈
(ℤ≥‘𝑀) → ((𝑀...𝑁) = (𝐽...𝐾) ↔ (𝑀 = 𝐽 ∧ 𝑁 = 𝐾))) |