Step |
Hyp |
Ref |
Expression |
1 |
|
esumdivc.a |
|- ( ph -> A e. V ) |
2 |
|
esumdivc.b |
|- ( ( ph /\ k e. A ) -> B e. ( 0 [,] +oo ) ) |
3 |
|
esumdivc.c |
|- ( ph -> C e. RR+ ) |
4 |
|
1red |
|- ( ph -> 1 e. RR ) |
5 |
3
|
rpred |
|- ( ph -> C e. RR ) |
6 |
3
|
rpne0d |
|- ( ph -> C =/= 0 ) |
7 |
|
rexdiv |
|- ( ( 1 e. RR /\ C e. RR /\ C =/= 0 ) -> ( 1 /e C ) = ( 1 / C ) ) |
8 |
4 5 6 7
|
syl3anc |
|- ( ph -> ( 1 /e C ) = ( 1 / C ) ) |
9 |
|
ioorp |
|- ( 0 (,) +oo ) = RR+ |
10 |
|
ioossico |
|- ( 0 (,) +oo ) C_ ( 0 [,) +oo ) |
11 |
9 10
|
eqsstrri |
|- RR+ C_ ( 0 [,) +oo ) |
12 |
3
|
rpreccld |
|- ( ph -> ( 1 / C ) e. RR+ ) |
13 |
11 12
|
sselid |
|- ( ph -> ( 1 / C ) e. ( 0 [,) +oo ) ) |
14 |
8 13
|
eqeltrd |
|- ( ph -> ( 1 /e C ) e. ( 0 [,) +oo ) ) |
15 |
1 2 14
|
esummulc1 |
|- ( ph -> ( sum* k e. A B *e ( 1 /e C ) ) = sum* k e. A ( B *e ( 1 /e C ) ) ) |
16 |
|
iccssxr |
|- ( 0 [,] +oo ) C_ RR* |
17 |
2
|
ralrimiva |
|- ( ph -> A. k e. A B e. ( 0 [,] +oo ) ) |
18 |
|
nfcv |
|- F/_ k A |
19 |
18
|
esumcl |
|- ( ( A e. V /\ A. k e. A B e. ( 0 [,] +oo ) ) -> sum* k e. A B e. ( 0 [,] +oo ) ) |
20 |
1 17 19
|
syl2anc |
|- ( ph -> sum* k e. A B e. ( 0 [,] +oo ) ) |
21 |
16 20
|
sselid |
|- ( ph -> sum* k e. A B e. RR* ) |
22 |
|
xdivrec |
|- ( ( sum* k e. A B e. RR* /\ C e. RR /\ C =/= 0 ) -> ( sum* k e. A B /e C ) = ( sum* k e. A B *e ( 1 /e C ) ) ) |
23 |
21 5 6 22
|
syl3anc |
|- ( ph -> ( sum* k e. A B /e C ) = ( sum* k e. A B *e ( 1 /e C ) ) ) |
24 |
16 2
|
sselid |
|- ( ( ph /\ k e. A ) -> B e. RR* ) |
25 |
5
|
adantr |
|- ( ( ph /\ k e. A ) -> C e. RR ) |
26 |
6
|
adantr |
|- ( ( ph /\ k e. A ) -> C =/= 0 ) |
27 |
|
xdivrec |
|- ( ( B e. RR* /\ C e. RR /\ C =/= 0 ) -> ( B /e C ) = ( B *e ( 1 /e C ) ) ) |
28 |
24 25 26 27
|
syl3anc |
|- ( ( ph /\ k e. A ) -> ( B /e C ) = ( B *e ( 1 /e C ) ) ) |
29 |
28
|
esumeq2dv |
|- ( ph -> sum* k e. A ( B /e C ) = sum* k e. A ( B *e ( 1 /e C ) ) ) |
30 |
15 23 29
|
3eqtr4d |
|- ( ph -> ( sum* k e. A B /e C ) = sum* k e. A ( B /e C ) ) |