Step |
Hyp |
Ref |
Expression |
1 |
|
sge0fsummpt.a |
|- ( ph -> A e. Fin ) |
2 |
|
sge0fsummpt.b |
|- ( ( ph /\ k e. A ) -> B e. ( 0 [,) +oo ) ) |
3 |
|
eqid |
|- ( k e. A |-> B ) = ( k e. A |-> B ) |
4 |
2 3
|
fmptd |
|- ( ph -> ( k e. A |-> B ) : A --> ( 0 [,) +oo ) ) |
5 |
1 4
|
sge0fsum |
|- ( ph -> ( sum^ ` ( k e. A |-> B ) ) = sum_ j e. A ( ( k e. A |-> B ) ` j ) ) |
6 |
|
fveq2 |
|- ( j = k -> ( ( k e. A |-> B ) ` j ) = ( ( k e. A |-> B ) ` k ) ) |
7 |
|
nfcv |
|- F/_ k A |
8 |
|
nfcv |
|- F/_ j A |
9 |
|
nfmpt1 |
|- F/_ k ( k e. A |-> B ) |
10 |
|
nfcv |
|- F/_ k j |
11 |
9 10
|
nffv |
|- F/_ k ( ( k e. A |-> B ) ` j ) |
12 |
|
nfcv |
|- F/_ j ( ( k e. A |-> B ) ` k ) |
13 |
6 7 8 11 12
|
cbvsum |
|- sum_ j e. A ( ( k e. A |-> B ) ` j ) = sum_ k e. A ( ( k e. A |-> B ) ` k ) |
14 |
13
|
a1i |
|- ( ph -> sum_ j e. A ( ( k e. A |-> B ) ` j ) = sum_ k e. A ( ( k e. A |-> B ) ` k ) ) |
15 |
|
simpr |
|- ( ( ph /\ k e. A ) -> k e. A ) |
16 |
3
|
fvmpt2 |
|- ( ( k e. A /\ B e. ( 0 [,) +oo ) ) -> ( ( k e. A |-> B ) ` k ) = B ) |
17 |
15 2 16
|
syl2anc |
|- ( ( ph /\ k e. A ) -> ( ( k e. A |-> B ) ` k ) = B ) |
18 |
17
|
sumeq2dv |
|- ( ph -> sum_ k e. A ( ( k e. A |-> B ) ` k ) = sum_ k e. A B ) |
19 |
5 14 18
|
3eqtrd |
|- ( ph -> ( sum^ ` ( k e. A |-> B ) ) = sum_ k e. A B ) |