Step |
Hyp |
Ref |
Expression |
1 |
|
uzinico2.1 |
⊢ ( 𝜑 → 𝑁 ∈ ( ℤ≥ ‘ 𝑀 ) ) |
2 |
|
inass |
⊢ ( ( ( ℤ≥ ‘ 𝑀 ) ∩ ℤ ) ∩ ( 𝑁 [,) +∞ ) ) = ( ( ℤ≥ ‘ 𝑀 ) ∩ ( ℤ ∩ ( 𝑁 [,) +∞ ) ) ) |
3 |
2
|
a1i |
⊢ ( 𝜑 → ( ( ( ℤ≥ ‘ 𝑀 ) ∩ ℤ ) ∩ ( 𝑁 [,) +∞ ) ) = ( ( ℤ≥ ‘ 𝑀 ) ∩ ( ℤ ∩ ( 𝑁 [,) +∞ ) ) ) ) |
4 |
|
incom |
⊢ ( ( ℤ≥ ‘ 𝑀 ) ∩ ( ℤ ∩ ( 𝑁 [,) +∞ ) ) ) = ( ( ℤ ∩ ( 𝑁 [,) +∞ ) ) ∩ ( ℤ≥ ‘ 𝑀 ) ) |
5 |
4
|
a1i |
⊢ ( 𝜑 → ( ( ℤ≥ ‘ 𝑀 ) ∩ ( ℤ ∩ ( 𝑁 [,) +∞ ) ) ) = ( ( ℤ ∩ ( 𝑁 [,) +∞ ) ) ∩ ( ℤ≥ ‘ 𝑀 ) ) ) |
6 |
|
uzssz |
⊢ ( ℤ≥ ‘ 𝑀 ) ⊆ ℤ |
7 |
6
|
a1i |
⊢ ( 𝜑 → ( ℤ≥ ‘ 𝑀 ) ⊆ ℤ ) |
8 |
7 1
|
sseldd |
⊢ ( 𝜑 → 𝑁 ∈ ℤ ) |
9 |
|
eqid |
⊢ ( ℤ≥ ‘ 𝑁 ) = ( ℤ≥ ‘ 𝑁 ) |
10 |
8 9
|
uzinico |
⊢ ( 𝜑 → ( ℤ≥ ‘ 𝑁 ) = ( ℤ ∩ ( 𝑁 [,) +∞ ) ) ) |
11 |
10
|
eqcomd |
⊢ ( 𝜑 → ( ℤ ∩ ( 𝑁 [,) +∞ ) ) = ( ℤ≥ ‘ 𝑁 ) ) |
12 |
11
|
ineq1d |
⊢ ( 𝜑 → ( ( ℤ ∩ ( 𝑁 [,) +∞ ) ) ∩ ( ℤ≥ ‘ 𝑀 ) ) = ( ( ℤ≥ ‘ 𝑁 ) ∩ ( ℤ≥ ‘ 𝑀 ) ) ) |
13 |
1
|
uzssd |
⊢ ( 𝜑 → ( ℤ≥ ‘ 𝑁 ) ⊆ ( ℤ≥ ‘ 𝑀 ) ) |
14 |
|
df-ss |
⊢ ( ( ℤ≥ ‘ 𝑁 ) ⊆ ( ℤ≥ ‘ 𝑀 ) ↔ ( ( ℤ≥ ‘ 𝑁 ) ∩ ( ℤ≥ ‘ 𝑀 ) ) = ( ℤ≥ ‘ 𝑁 ) ) |
15 |
13 14
|
sylib |
⊢ ( 𝜑 → ( ( ℤ≥ ‘ 𝑁 ) ∩ ( ℤ≥ ‘ 𝑀 ) ) = ( ℤ≥ ‘ 𝑁 ) ) |
16 |
5 12 15
|
3eqtrd |
⊢ ( 𝜑 → ( ( ℤ≥ ‘ 𝑀 ) ∩ ( ℤ ∩ ( 𝑁 [,) +∞ ) ) ) = ( ℤ≥ ‘ 𝑁 ) ) |
17 |
|
uzssz |
⊢ ( ℤ≥ ‘ 𝑁 ) ⊆ ℤ |
18 |
|
df-ss |
⊢ ( ( ℤ≥ ‘ 𝑁 ) ⊆ ℤ ↔ ( ( ℤ≥ ‘ 𝑁 ) ∩ ℤ ) = ( ℤ≥ ‘ 𝑁 ) ) |
19 |
17 18
|
mpbi |
⊢ ( ( ℤ≥ ‘ 𝑁 ) ∩ ℤ ) = ( ℤ≥ ‘ 𝑁 ) |
20 |
19
|
a1i |
⊢ ( 𝜑 → ( ( ℤ≥ ‘ 𝑁 ) ∩ ℤ ) = ( ℤ≥ ‘ 𝑁 ) ) |
21 |
20
|
eqcomd |
⊢ ( 𝜑 → ( ℤ≥ ‘ 𝑁 ) = ( ( ℤ≥ ‘ 𝑁 ) ∩ ℤ ) ) |
22 |
3 16 21
|
3eqtrrd |
⊢ ( 𝜑 → ( ( ℤ≥ ‘ 𝑁 ) ∩ ℤ ) = ( ( ( ℤ≥ ‘ 𝑀 ) ∩ ℤ ) ∩ ( 𝑁 [,) +∞ ) ) ) |
23 |
|
df-ss |
⊢ ( ( ℤ≥ ‘ 𝑀 ) ⊆ ℤ ↔ ( ( ℤ≥ ‘ 𝑀 ) ∩ ℤ ) = ( ℤ≥ ‘ 𝑀 ) ) |
24 |
6 23
|
mpbi |
⊢ ( ( ℤ≥ ‘ 𝑀 ) ∩ ℤ ) = ( ℤ≥ ‘ 𝑀 ) |
25 |
24
|
ineq1i |
⊢ ( ( ( ℤ≥ ‘ 𝑀 ) ∩ ℤ ) ∩ ( 𝑁 [,) +∞ ) ) = ( ( ℤ≥ ‘ 𝑀 ) ∩ ( 𝑁 [,) +∞ ) ) |
26 |
25
|
a1i |
⊢ ( 𝜑 → ( ( ( ℤ≥ ‘ 𝑀 ) ∩ ℤ ) ∩ ( 𝑁 [,) +∞ ) ) = ( ( ℤ≥ ‘ 𝑀 ) ∩ ( 𝑁 [,) +∞ ) ) ) |
27 |
22 20 26
|
3eqtr3d |
⊢ ( 𝜑 → ( ℤ≥ ‘ 𝑁 ) = ( ( ℤ≥ ‘ 𝑀 ) ∩ ( 𝑁 [,) +∞ ) ) ) |