Step |
Hyp |
Ref |
Expression |
1 |
|
esumval.p |
|- F/ k ph |
2 |
|
esumval.0 |
|- F/_ k A |
3 |
|
esumval.1 |
|- ( ph -> A e. V ) |
4 |
|
esumval.2 |
|- ( ( ph /\ k e. A ) -> B e. ( 0 [,] +oo ) ) |
5 |
|
esumval.3 |
|- ( ( ph /\ x e. ( ~P A i^i Fin ) ) -> ( ( RR*s |`s ( 0 [,] +oo ) ) gsum ( k e. x |-> B ) ) = C ) |
6 |
|
df-esum |
|- sum* k e. A B = U. ( ( RR*s |`s ( 0 [,] +oo ) ) tsums ( k e. A |-> B ) ) |
7 |
|
eqid |
|- ( RR*s |`s ( 0 [,] +oo ) ) = ( RR*s |`s ( 0 [,] +oo ) ) |
8 |
|
nfcv |
|- F/_ k ( 0 [,] +oo ) |
9 |
|
eqid |
|- ( k e. A |-> B ) = ( k e. A |-> B ) |
10 |
1 2 8 4 9
|
fmptdF |
|- ( ph -> ( k e. A |-> B ) : A --> ( 0 [,] +oo ) ) |
11 |
|
inss1 |
|- ( ~P A i^i Fin ) C_ ~P A |
12 |
11
|
sseli |
|- ( x e. ( ~P A i^i Fin ) -> x e. ~P A ) |
13 |
12
|
elpwid |
|- ( x e. ( ~P A i^i Fin ) -> x C_ A ) |
14 |
13
|
adantl |
|- ( ( ph /\ x e. ( ~P A i^i Fin ) ) -> x C_ A ) |
15 |
|
nfcv |
|- F/_ k x |
16 |
2 15
|
resmptf |
|- ( x C_ A -> ( ( k e. A |-> B ) |` x ) = ( k e. x |-> B ) ) |
17 |
14 16
|
syl |
|- ( ( ph /\ x e. ( ~P A i^i Fin ) ) -> ( ( k e. A |-> B ) |` x ) = ( k e. x |-> B ) ) |
18 |
17
|
oveq2d |
|- ( ( ph /\ x e. ( ~P A i^i Fin ) ) -> ( ( RR*s |`s ( 0 [,] +oo ) ) gsum ( ( k e. A |-> B ) |` x ) ) = ( ( RR*s |`s ( 0 [,] +oo ) ) gsum ( k e. x |-> B ) ) ) |
19 |
18 5
|
eqtr2d |
|- ( ( ph /\ x e. ( ~P A i^i Fin ) ) -> C = ( ( RR*s |`s ( 0 [,] +oo ) ) gsum ( ( k e. A |-> B ) |` x ) ) ) |
20 |
19
|
mpteq2dva |
|- ( ph -> ( x e. ( ~P A i^i Fin ) |-> C ) = ( x e. ( ~P A i^i Fin ) |-> ( ( RR*s |`s ( 0 [,] +oo ) ) gsum ( ( k e. A |-> B ) |` x ) ) ) ) |
21 |
20
|
rneqd |
|- ( ph -> ran ( x e. ( ~P A i^i Fin ) |-> C ) = ran ( x e. ( ~P A i^i Fin ) |-> ( ( RR*s |`s ( 0 [,] +oo ) ) gsum ( ( k e. A |-> B ) |` x ) ) ) ) |
22 |
21
|
supeq1d |
|- ( ph -> sup ( ran ( x e. ( ~P A i^i Fin ) |-> C ) , RR* , < ) = sup ( ran ( x e. ( ~P A i^i Fin ) |-> ( ( RR*s |`s ( 0 [,] +oo ) ) gsum ( ( k e. A |-> B ) |` x ) ) ) , RR* , < ) ) |
23 |
7 3 10 22
|
xrge0tsmsd |
|- ( ph -> ( ( RR*s |`s ( 0 [,] +oo ) ) tsums ( k e. A |-> B ) ) = { sup ( ran ( x e. ( ~P A i^i Fin ) |-> C ) , RR* , < ) } ) |
24 |
23
|
unieqd |
|- ( ph -> U. ( ( RR*s |`s ( 0 [,] +oo ) ) tsums ( k e. A |-> B ) ) = U. { sup ( ran ( x e. ( ~P A i^i Fin ) |-> C ) , RR* , < ) } ) |
25 |
6 24
|
syl5eq |
|- ( ph -> sum* k e. A B = U. { sup ( ran ( x e. ( ~P A i^i Fin ) |-> C ) , RR* , < ) } ) |
26 |
|
xrltso |
|- < Or RR* |
27 |
26
|
supex |
|- sup ( ran ( x e. ( ~P A i^i Fin ) |-> C ) , RR* , < ) e. _V |
28 |
27
|
unisn |
|- U. { sup ( ran ( x e. ( ~P A i^i Fin ) |-> C ) , RR* , < ) } = sup ( ran ( x e. ( ~P A i^i Fin ) |-> C ) , RR* , < ) |
29 |
25 28
|
eqtrdi |
|- ( ph -> sum* k e. A B = sup ( ran ( x e. ( ~P A i^i Fin ) |-> C ) , RR* , < ) ) |