Step |
Hyp |
Ref |
Expression |
1 |
|
eluzfz1 |
⊢ ( 𝑁 ∈ ( ℤ≥ ‘ 𝑀 ) → 𝑀 ∈ ( 𝑀 ... 𝑁 ) ) |
2 |
1
|
adantr |
⊢ ( ( 𝑁 ∈ ( ℤ≥ ‘ 𝑀 ) ∧ ( 𝑀 ... 𝑁 ) = ( 𝐽 ... 𝐾 ) ) → 𝑀 ∈ ( 𝑀 ... 𝑁 ) ) |
3 |
|
simpr |
⊢ ( ( 𝑁 ∈ ( ℤ≥ ‘ 𝑀 ) ∧ ( 𝑀 ... 𝑁 ) = ( 𝐽 ... 𝐾 ) ) → ( 𝑀 ... 𝑁 ) = ( 𝐽 ... 𝐾 ) ) |
4 |
2 3
|
eleqtrd |
⊢ ( ( 𝑁 ∈ ( ℤ≥ ‘ 𝑀 ) ∧ ( 𝑀 ... 𝑁 ) = ( 𝐽 ... 𝐾 ) ) → 𝑀 ∈ ( 𝐽 ... 𝐾 ) ) |
5 |
|
elfzuz |
⊢ ( 𝑀 ∈ ( 𝐽 ... 𝐾 ) → 𝑀 ∈ ( ℤ≥ ‘ 𝐽 ) ) |
6 |
|
uzss |
⊢ ( 𝑀 ∈ ( ℤ≥ ‘ 𝐽 ) → ( ℤ≥ ‘ 𝑀 ) ⊆ ( ℤ≥ ‘ 𝐽 ) ) |
7 |
4 5 6
|
3syl |
⊢ ( ( 𝑁 ∈ ( ℤ≥ ‘ 𝑀 ) ∧ ( 𝑀 ... 𝑁 ) = ( 𝐽 ... 𝐾 ) ) → ( ℤ≥ ‘ 𝑀 ) ⊆ ( ℤ≥ ‘ 𝐽 ) ) |
8 |
|
elfzuz2 |
⊢ ( 𝑀 ∈ ( 𝐽 ... 𝐾 ) → 𝐾 ∈ ( ℤ≥ ‘ 𝐽 ) ) |
9 |
|
eluzfz1 |
⊢ ( 𝐾 ∈ ( ℤ≥ ‘ 𝐽 ) → 𝐽 ∈ ( 𝐽 ... 𝐾 ) ) |
10 |
4 8 9
|
3syl |
⊢ ( ( 𝑁 ∈ ( ℤ≥ ‘ 𝑀 ) ∧ ( 𝑀 ... 𝑁 ) = ( 𝐽 ... 𝐾 ) ) → 𝐽 ∈ ( 𝐽 ... 𝐾 ) ) |
11 |
10 3
|
eleqtrrd |
⊢ ( ( 𝑁 ∈ ( ℤ≥ ‘ 𝑀 ) ∧ ( 𝑀 ... 𝑁 ) = ( 𝐽 ... 𝐾 ) ) → 𝐽 ∈ ( 𝑀 ... 𝑁 ) ) |
12 |
|
elfzuz |
⊢ ( 𝐽 ∈ ( 𝑀 ... 𝑁 ) → 𝐽 ∈ ( ℤ≥ ‘ 𝑀 ) ) |
13 |
|
uzss |
⊢ ( 𝐽 ∈ ( ℤ≥ ‘ 𝑀 ) → ( ℤ≥ ‘ 𝐽 ) ⊆ ( ℤ≥ ‘ 𝑀 ) ) |
14 |
11 12 13
|
3syl |
⊢ ( ( 𝑁 ∈ ( ℤ≥ ‘ 𝑀 ) ∧ ( 𝑀 ... 𝑁 ) = ( 𝐽 ... 𝐾 ) ) → ( ℤ≥ ‘ 𝐽 ) ⊆ ( ℤ≥ ‘ 𝑀 ) ) |
15 |
7 14
|
eqssd |
⊢ ( ( 𝑁 ∈ ( ℤ≥ ‘ 𝑀 ) ∧ ( 𝑀 ... 𝑁 ) = ( 𝐽 ... 𝐾 ) ) → ( ℤ≥ ‘ 𝑀 ) = ( ℤ≥ ‘ 𝐽 ) ) |
16 |
|
eluzel2 |
⊢ ( 𝑁 ∈ ( ℤ≥ ‘ 𝑀 ) → 𝑀 ∈ ℤ ) |
17 |
16
|
adantr |
⊢ ( ( 𝑁 ∈ ( ℤ≥ ‘ 𝑀 ) ∧ ( 𝑀 ... 𝑁 ) = ( 𝐽 ... 𝐾 ) ) → 𝑀 ∈ ℤ ) |
18 |
|
uz11 |
⊢ ( 𝑀 ∈ ℤ → ( ( ℤ≥ ‘ 𝑀 ) = ( ℤ≥ ‘ 𝐽 ) ↔ 𝑀 = 𝐽 ) ) |
19 |
17 18
|
syl |
⊢ ( ( 𝑁 ∈ ( ℤ≥ ‘ 𝑀 ) ∧ ( 𝑀 ... 𝑁 ) = ( 𝐽 ... 𝐾 ) ) → ( ( ℤ≥ ‘ 𝑀 ) = ( ℤ≥ ‘ 𝐽 ) ↔ 𝑀 = 𝐽 ) ) |
20 |
15 19
|
mpbid |
⊢ ( ( 𝑁 ∈ ( ℤ≥ ‘ 𝑀 ) ∧ ( 𝑀 ... 𝑁 ) = ( 𝐽 ... 𝐾 ) ) → 𝑀 = 𝐽 ) |
21 |
|
eluzfz2 |
⊢ ( 𝐾 ∈ ( ℤ≥ ‘ 𝐽 ) → 𝐾 ∈ ( 𝐽 ... 𝐾 ) ) |
22 |
4 8 21
|
3syl |
⊢ ( ( 𝑁 ∈ ( ℤ≥ ‘ 𝑀 ) ∧ ( 𝑀 ... 𝑁 ) = ( 𝐽 ... 𝐾 ) ) → 𝐾 ∈ ( 𝐽 ... 𝐾 ) ) |
23 |
22 3
|
eleqtrrd |
⊢ ( ( 𝑁 ∈ ( ℤ≥ ‘ 𝑀 ) ∧ ( 𝑀 ... 𝑁 ) = ( 𝐽 ... 𝐾 ) ) → 𝐾 ∈ ( 𝑀 ... 𝑁 ) ) |
24 |
|
elfzuz3 |
⊢ ( 𝐾 ∈ ( 𝑀 ... 𝑁 ) → 𝑁 ∈ ( ℤ≥ ‘ 𝐾 ) ) |
25 |
|
uzss |
⊢ ( 𝑁 ∈ ( ℤ≥ ‘ 𝐾 ) → ( ℤ≥ ‘ 𝑁 ) ⊆ ( ℤ≥ ‘ 𝐾 ) ) |
26 |
23 24 25
|
3syl |
⊢ ( ( 𝑁 ∈ ( ℤ≥ ‘ 𝑀 ) ∧ ( 𝑀 ... 𝑁 ) = ( 𝐽 ... 𝐾 ) ) → ( ℤ≥ ‘ 𝑁 ) ⊆ ( ℤ≥ ‘ 𝐾 ) ) |
27 |
|
eluzfz2 |
⊢ ( 𝑁 ∈ ( ℤ≥ ‘ 𝑀 ) → 𝑁 ∈ ( 𝑀 ... 𝑁 ) ) |
28 |
27
|
adantr |
⊢ ( ( 𝑁 ∈ ( ℤ≥ ‘ 𝑀 ) ∧ ( 𝑀 ... 𝑁 ) = ( 𝐽 ... 𝐾 ) ) → 𝑁 ∈ ( 𝑀 ... 𝑁 ) ) |
29 |
28 3
|
eleqtrd |
⊢ ( ( 𝑁 ∈ ( ℤ≥ ‘ 𝑀 ) ∧ ( 𝑀 ... 𝑁 ) = ( 𝐽 ... 𝐾 ) ) → 𝑁 ∈ ( 𝐽 ... 𝐾 ) ) |
30 |
|
elfzuz3 |
⊢ ( 𝑁 ∈ ( 𝐽 ... 𝐾 ) → 𝐾 ∈ ( ℤ≥ ‘ 𝑁 ) ) |
31 |
|
uzss |
⊢ ( 𝐾 ∈ ( ℤ≥ ‘ 𝑁 ) → ( ℤ≥ ‘ 𝐾 ) ⊆ ( ℤ≥ ‘ 𝑁 ) ) |
32 |
29 30 31
|
3syl |
⊢ ( ( 𝑁 ∈ ( ℤ≥ ‘ 𝑀 ) ∧ ( 𝑀 ... 𝑁 ) = ( 𝐽 ... 𝐾 ) ) → ( ℤ≥ ‘ 𝐾 ) ⊆ ( ℤ≥ ‘ 𝑁 ) ) |
33 |
26 32
|
eqssd |
⊢ ( ( 𝑁 ∈ ( ℤ≥ ‘ 𝑀 ) ∧ ( 𝑀 ... 𝑁 ) = ( 𝐽 ... 𝐾 ) ) → ( ℤ≥ ‘ 𝑁 ) = ( ℤ≥ ‘ 𝐾 ) ) |
34 |
|
eluzelz |
⊢ ( 𝑁 ∈ ( ℤ≥ ‘ 𝑀 ) → 𝑁 ∈ ℤ ) |
35 |
34
|
adantr |
⊢ ( ( 𝑁 ∈ ( ℤ≥ ‘ 𝑀 ) ∧ ( 𝑀 ... 𝑁 ) = ( 𝐽 ... 𝐾 ) ) → 𝑁 ∈ ℤ ) |
36 |
|
uz11 |
⊢ ( 𝑁 ∈ ℤ → ( ( ℤ≥ ‘ 𝑁 ) = ( ℤ≥ ‘ 𝐾 ) ↔ 𝑁 = 𝐾 ) ) |
37 |
35 36
|
syl |
⊢ ( ( 𝑁 ∈ ( ℤ≥ ‘ 𝑀 ) ∧ ( 𝑀 ... 𝑁 ) = ( 𝐽 ... 𝐾 ) ) → ( ( ℤ≥ ‘ 𝑁 ) = ( ℤ≥ ‘ 𝐾 ) ↔ 𝑁 = 𝐾 ) ) |
38 |
33 37
|
mpbid |
⊢ ( ( 𝑁 ∈ ( ℤ≥ ‘ 𝑀 ) ∧ ( 𝑀 ... 𝑁 ) = ( 𝐽 ... 𝐾 ) ) → 𝑁 = 𝐾 ) |
39 |
20 38
|
jca |
⊢ ( ( 𝑁 ∈ ( ℤ≥ ‘ 𝑀 ) ∧ ( 𝑀 ... 𝑁 ) = ( 𝐽 ... 𝐾 ) ) → ( 𝑀 = 𝐽 ∧ 𝑁 = 𝐾 ) ) |
40 |
39
|
ex |
⊢ ( 𝑁 ∈ ( ℤ≥ ‘ 𝑀 ) → ( ( 𝑀 ... 𝑁 ) = ( 𝐽 ... 𝐾 ) → ( 𝑀 = 𝐽 ∧ 𝑁 = 𝐾 ) ) ) |
41 |
|
oveq12 |
⊢ ( ( 𝑀 = 𝐽 ∧ 𝑁 = 𝐾 ) → ( 𝑀 ... 𝑁 ) = ( 𝐽 ... 𝐾 ) ) |
42 |
40 41
|
impbid1 |
⊢ ( 𝑁 ∈ ( ℤ≥ ‘ 𝑀 ) → ( ( 𝑀 ... 𝑁 ) = ( 𝐽 ... 𝐾 ) ↔ ( 𝑀 = 𝐽 ∧ 𝑁 = 𝐾 ) ) ) |