Step |
Hyp |
Ref |
Expression |
0 |
|
citg2 |
|- S.2 |
1 |
|
vf |
|- f |
2 |
|
cc0 |
|- 0 |
3 |
|
cicc |
|- [,] |
4 |
|
cpnf |
|- +oo |
5 |
2 4 3
|
co |
|- ( 0 [,] +oo ) |
6 |
|
cmap |
|- ^m |
7 |
|
cr |
|- RR |
8 |
5 7 6
|
co |
|- ( ( 0 [,] +oo ) ^m RR ) |
9 |
|
vx |
|- x |
10 |
|
vg |
|- g |
11 |
|
citg1 |
|- S.1 |
12 |
11
|
cdm |
|- dom S.1 |
13 |
10
|
cv |
|- g |
14 |
|
cle |
|- <_ |
15 |
14
|
cofr |
|- oR <_ |
16 |
1
|
cv |
|- f |
17 |
13 16 15
|
wbr |
|- g oR <_ f |
18 |
9
|
cv |
|- x |
19 |
13 11
|
cfv |
|- ( S.1 ` g ) |
20 |
18 19
|
wceq |
|- x = ( S.1 ` g ) |
21 |
17 20
|
wa |
|- ( g oR <_ f /\ x = ( S.1 ` g ) ) |
22 |
21 10 12
|
wrex |
|- E. g e. dom S.1 ( g oR <_ f /\ x = ( S.1 ` g ) ) |
23 |
22 9
|
cab |
|- { x | E. g e. dom S.1 ( g oR <_ f /\ x = ( S.1 ` g ) ) } |
24 |
|
cxr |
|- RR* |
25 |
|
clt |
|- < |
26 |
23 24 25
|
csup |
|- sup ( { x | E. g e. dom S.1 ( g oR <_ f /\ x = ( S.1 ` g ) ) } , RR* , < ) |
27 |
1 8 26
|
cmpt |
|- ( f e. ( ( 0 [,] +oo ) ^m RR ) |-> sup ( { x | E. g e. dom S.1 ( g oR <_ f /\ x = ( S.1 ` g ) ) } , RR* , < ) ) |
28 |
0 27
|
wceq |
|- S.2 = ( f e. ( ( 0 [,] +oo ) ^m RR ) |-> sup ( { x | E. g e. dom S.1 ( g oR <_ f /\ x = ( S.1 ` g ) ) } , RR* , < ) ) |