| Step |
Hyp |
Ref |
Expression |
| 1 |
|
imo72b2lem2.1 |
|- ( ph -> F : RR --> RR ) |
| 2 |
|
imo72b2lem2.2 |
|- ( ph -> C e. RR ) |
| 3 |
|
imo72b2lem2.3 |
|- ( ph -> A. z e. RR ( abs ` ( F ` z ) ) <_ C ) |
| 4 |
|
imaco |
|- ( ( abs o. F ) " RR ) = ( abs " ( F " RR ) ) |
| 5 |
4
|
eqcomi |
|- ( abs " ( F " RR ) ) = ( ( abs o. F ) " RR ) |
| 6 |
|
imassrn |
|- ( ( abs o. F ) " RR ) C_ ran ( abs o. F ) |
| 7 |
6
|
a1i |
|- ( ph -> ( ( abs o. F ) " RR ) C_ ran ( abs o. F ) ) |
| 8 |
|
absf |
|- abs : CC --> RR |
| 9 |
8
|
a1i |
|- ( ph -> abs : CC --> RR ) |
| 10 |
|
ax-resscn |
|- RR C_ CC |
| 11 |
10
|
a1i |
|- ( ph -> RR C_ CC ) |
| 12 |
9 11
|
fssresd |
|- ( ph -> ( abs |` RR ) : RR --> RR ) |
| 13 |
1 12
|
fco2d |
|- ( ph -> ( abs o. F ) : RR --> RR ) |
| 14 |
13
|
frnd |
|- ( ph -> ran ( abs o. F ) C_ RR ) |
| 15 |
7 14
|
sstrd |
|- ( ph -> ( ( abs o. F ) " RR ) C_ RR ) |
| 16 |
5 15
|
eqsstrid |
|- ( ph -> ( abs " ( F " RR ) ) C_ RR ) |
| 17 |
|
0re |
|- 0 e. RR |
| 18 |
17
|
ne0ii |
|- RR =/= (/) |
| 19 |
18
|
a1i |
|- ( ph -> RR =/= (/) ) |
| 20 |
19 13
|
wnefimgd |
|- ( ph -> ( ( abs o. F ) " RR ) =/= (/) ) |
| 21 |
20
|
necomd |
|- ( ph -> (/) =/= ( ( abs o. F ) " RR ) ) |
| 22 |
5
|
a1i |
|- ( ph -> ( abs " ( F " RR ) ) = ( ( abs o. F ) " RR ) ) |
| 23 |
21 22
|
neeqtrrd |
|- ( ph -> (/) =/= ( abs " ( F " RR ) ) ) |
| 24 |
23
|
necomd |
|- ( ph -> ( abs " ( F " RR ) ) =/= (/) ) |
| 25 |
|
simpr |
|- ( ( ph /\ c = C ) -> c = C ) |
| 26 |
25
|
breq2d |
|- ( ( ph /\ c = C ) -> ( v <_ c <-> v <_ C ) ) |
| 27 |
26
|
ralbidv |
|- ( ( ph /\ c = C ) -> ( A. v e. ( abs " ( F " RR ) ) v <_ c <-> A. v e. ( abs " ( F " RR ) ) v <_ C ) ) |
| 28 |
1 3
|
extoimad |
|- ( ph -> A. v e. ( abs " ( F " RR ) ) v <_ C ) |
| 29 |
2 27 28
|
rspcedvd |
|- ( ph -> E. c e. RR A. v e. ( abs " ( F " RR ) ) v <_ c ) |
| 30 |
1 3
|
extoimad |
|- ( ph -> A. t e. ( abs " ( F " RR ) ) t <_ C ) |
| 31 |
16 24 29 2 30
|
suprleubrd |
|- ( ph -> sup ( ( abs " ( F " RR ) ) , RR , < ) <_ C ) |