Step |
Hyp |
Ref |
Expression |
1 |
|
uzinico2.1 |
|- ( ph -> N e. ( ZZ>= ` M ) ) |
2 |
|
inass |
|- ( ( ( ZZ>= ` M ) i^i ZZ ) i^i ( N [,) +oo ) ) = ( ( ZZ>= ` M ) i^i ( ZZ i^i ( N [,) +oo ) ) ) |
3 |
2
|
a1i |
|- ( ph -> ( ( ( ZZ>= ` M ) i^i ZZ ) i^i ( N [,) +oo ) ) = ( ( ZZ>= ` M ) i^i ( ZZ i^i ( N [,) +oo ) ) ) ) |
4 |
|
incom |
|- ( ( ZZ>= ` M ) i^i ( ZZ i^i ( N [,) +oo ) ) ) = ( ( ZZ i^i ( N [,) +oo ) ) i^i ( ZZ>= ` M ) ) |
5 |
4
|
a1i |
|- ( ph -> ( ( ZZ>= ` M ) i^i ( ZZ i^i ( N [,) +oo ) ) ) = ( ( ZZ i^i ( N [,) +oo ) ) i^i ( ZZ>= ` M ) ) ) |
6 |
|
uzssz |
|- ( ZZ>= ` M ) C_ ZZ |
7 |
6
|
a1i |
|- ( ph -> ( ZZ>= ` M ) C_ ZZ ) |
8 |
7 1
|
sseldd |
|- ( ph -> N e. ZZ ) |
9 |
|
eqid |
|- ( ZZ>= ` N ) = ( ZZ>= ` N ) |
10 |
8 9
|
uzinico |
|- ( ph -> ( ZZ>= ` N ) = ( ZZ i^i ( N [,) +oo ) ) ) |
11 |
10
|
eqcomd |
|- ( ph -> ( ZZ i^i ( N [,) +oo ) ) = ( ZZ>= ` N ) ) |
12 |
11
|
ineq1d |
|- ( ph -> ( ( ZZ i^i ( N [,) +oo ) ) i^i ( ZZ>= ` M ) ) = ( ( ZZ>= ` N ) i^i ( ZZ>= ` M ) ) ) |
13 |
1
|
uzssd |
|- ( ph -> ( ZZ>= ` N ) C_ ( ZZ>= ` M ) ) |
14 |
|
df-ss |
|- ( ( ZZ>= ` N ) C_ ( ZZ>= ` M ) <-> ( ( ZZ>= ` N ) i^i ( ZZ>= ` M ) ) = ( ZZ>= ` N ) ) |
15 |
13 14
|
sylib |
|- ( ph -> ( ( ZZ>= ` N ) i^i ( ZZ>= ` M ) ) = ( ZZ>= ` N ) ) |
16 |
5 12 15
|
3eqtrd |
|- ( ph -> ( ( ZZ>= ` M ) i^i ( ZZ i^i ( N [,) +oo ) ) ) = ( ZZ>= ` N ) ) |
17 |
|
uzssz |
|- ( ZZ>= ` N ) C_ ZZ |
18 |
|
df-ss |
|- ( ( ZZ>= ` N ) C_ ZZ <-> ( ( ZZ>= ` N ) i^i ZZ ) = ( ZZ>= ` N ) ) |
19 |
17 18
|
mpbi |
|- ( ( ZZ>= ` N ) i^i ZZ ) = ( ZZ>= ` N ) |
20 |
19
|
a1i |
|- ( ph -> ( ( ZZ>= ` N ) i^i ZZ ) = ( ZZ>= ` N ) ) |
21 |
20
|
eqcomd |
|- ( ph -> ( ZZ>= ` N ) = ( ( ZZ>= ` N ) i^i ZZ ) ) |
22 |
3 16 21
|
3eqtrrd |
|- ( ph -> ( ( ZZ>= ` N ) i^i ZZ ) = ( ( ( ZZ>= ` M ) i^i ZZ ) i^i ( N [,) +oo ) ) ) |
23 |
|
df-ss |
|- ( ( ZZ>= ` M ) C_ ZZ <-> ( ( ZZ>= ` M ) i^i ZZ ) = ( ZZ>= ` M ) ) |
24 |
6 23
|
mpbi |
|- ( ( ZZ>= ` M ) i^i ZZ ) = ( ZZ>= ` M ) |
25 |
24
|
ineq1i |
|- ( ( ( ZZ>= ` M ) i^i ZZ ) i^i ( N [,) +oo ) ) = ( ( ZZ>= ` M ) i^i ( N [,) +oo ) ) |
26 |
25
|
a1i |
|- ( ph -> ( ( ( ZZ>= ` M ) i^i ZZ ) i^i ( N [,) +oo ) ) = ( ( ZZ>= ` M ) i^i ( N [,) +oo ) ) ) |
27 |
22 20 26
|
3eqtr3d |
|- ( ph -> ( ZZ>= ` N ) = ( ( ZZ>= ` M ) i^i ( N [,) +oo ) ) ) |