Step |
Hyp |
Ref |
Expression |
1 |
|
sge0pnffsumgt.k |
|- F/ k ph |
2 |
|
sge0pnffsumgt.a |
|- ( ph -> A e. V ) |
3 |
|
sge0pnffsumgt.b |
|- ( ( ph /\ k e. A ) -> B e. ( 0 [,) +oo ) ) |
4 |
|
sge0pnffsumgt.p |
|- ( ph -> ( sum^ ` ( k e. A |-> B ) ) = +oo ) |
5 |
|
sge0pnffsumgt.y |
|- ( ph -> Y e. RR ) |
6 |
|
icossicc |
|- ( 0 [,) +oo ) C_ ( 0 [,] +oo ) |
7 |
6 3
|
sseldi |
|- ( ( ph /\ k e. A ) -> B e. ( 0 [,] +oo ) ) |
8 |
1 2 7 4 5
|
sge0pnffigtmpt |
|- ( ph -> E. x e. ( ~P A i^i Fin ) Y < ( sum^ ` ( k e. x |-> B ) ) ) |
9 |
|
simpr |
|- ( ( ( ph /\ x e. ( ~P A i^i Fin ) ) /\ Y < ( sum^ ` ( k e. x |-> B ) ) ) -> Y < ( sum^ ` ( k e. x |-> B ) ) ) |
10 |
|
nfv |
|- F/ k x e. ( ~P A i^i Fin ) |
11 |
1 10
|
nfan |
|- F/ k ( ph /\ x e. ( ~P A i^i Fin ) ) |
12 |
|
elinel2 |
|- ( x e. ( ~P A i^i Fin ) -> x e. Fin ) |
13 |
12
|
adantl |
|- ( ( ph /\ x e. ( ~P A i^i Fin ) ) -> x e. Fin ) |
14 |
|
simpll |
|- ( ( ( ph /\ x e. ( ~P A i^i Fin ) ) /\ k e. x ) -> ph ) |
15 |
|
elpwinss |
|- ( x e. ( ~P A i^i Fin ) -> x C_ A ) |
16 |
15
|
sselda |
|- ( ( x e. ( ~P A i^i Fin ) /\ k e. x ) -> k e. A ) |
17 |
16
|
adantll |
|- ( ( ( ph /\ x e. ( ~P A i^i Fin ) ) /\ k e. x ) -> k e. A ) |
18 |
14 17 3
|
syl2anc |
|- ( ( ( ph /\ x e. ( ~P A i^i Fin ) ) /\ k e. x ) -> B e. ( 0 [,) +oo ) ) |
19 |
11 13 18
|
sge0fsummptf |
|- ( ( ph /\ x e. ( ~P A i^i Fin ) ) -> ( sum^ ` ( k e. x |-> B ) ) = sum_ k e. x B ) |
20 |
19
|
adantr |
|- ( ( ( ph /\ x e. ( ~P A i^i Fin ) ) /\ Y < ( sum^ ` ( k e. x |-> B ) ) ) -> ( sum^ ` ( k e. x |-> B ) ) = sum_ k e. x B ) |
21 |
9 20
|
breqtrd |
|- ( ( ( ph /\ x e. ( ~P A i^i Fin ) ) /\ Y < ( sum^ ` ( k e. x |-> B ) ) ) -> Y < sum_ k e. x B ) |
22 |
21
|
ex |
|- ( ( ph /\ x e. ( ~P A i^i Fin ) ) -> ( Y < ( sum^ ` ( k e. x |-> B ) ) -> Y < sum_ k e. x B ) ) |
23 |
22
|
reximdva |
|- ( ph -> ( E. x e. ( ~P A i^i Fin ) Y < ( sum^ ` ( k e. x |-> B ) ) -> E. x e. ( ~P A i^i Fin ) Y < sum_ k e. x B ) ) |
24 |
8 23
|
mpd |
|- ( ph -> E. x e. ( ~P A i^i Fin ) Y < sum_ k e. x B ) |