Step |
Hyp |
Ref |
Expression |
0 |
|
co1 |
|- O(1) |
1 |
|
vf |
|- f |
2 |
|
cc |
|- CC |
3 |
|
cpm |
|- ^pm |
4 |
|
cr |
|- RR |
5 |
2 4 3
|
co |
|- ( CC ^pm RR ) |
6 |
|
vx |
|- x |
7 |
|
vm |
|- m |
8 |
|
vy |
|- y |
9 |
1
|
cv |
|- f |
10 |
9
|
cdm |
|- dom f |
11 |
6
|
cv |
|- x |
12 |
|
cico |
|- [,) |
13 |
|
cpnf |
|- +oo |
14 |
11 13 12
|
co |
|- ( x [,) +oo ) |
15 |
10 14
|
cin |
|- ( dom f i^i ( x [,) +oo ) ) |
16 |
|
cabs |
|- abs |
17 |
8
|
cv |
|- y |
18 |
17 9
|
cfv |
|- ( f ` y ) |
19 |
18 16
|
cfv |
|- ( abs ` ( f ` y ) ) |
20 |
|
cle |
|- <_ |
21 |
7
|
cv |
|- m |
22 |
19 21 20
|
wbr |
|- ( abs ` ( f ` y ) ) <_ m |
23 |
22 8 15
|
wral |
|- A. y e. ( dom f i^i ( x [,) +oo ) ) ( abs ` ( f ` y ) ) <_ m |
24 |
23 7 4
|
wrex |
|- E. m e. RR A. y e. ( dom f i^i ( x [,) +oo ) ) ( abs ` ( f ` y ) ) <_ m |
25 |
24 6 4
|
wrex |
|- E. x e. RR E. m e. RR A. y e. ( dom f i^i ( x [,) +oo ) ) ( abs ` ( f ` y ) ) <_ m |
26 |
25 1 5
|
crab |
|- { f e. ( CC ^pm RR ) | E. x e. RR E. m e. RR A. y e. ( dom f i^i ( x [,) +oo ) ) ( abs ` ( f ` y ) ) <_ m } |
27 |
0 26
|
wceq |
|- O(1) = { f e. ( CC ^pm RR ) | E. x e. RR E. m e. RR A. y e. ( dom f i^i ( x [,) +oo ) ) ( abs ` ( f ` y ) ) <_ m } |