Step |
Hyp |
Ref |
Expression |
1 |
|
fsumlt.1 |
|- ( ph -> A e. Fin ) |
2 |
|
fsumlt.2 |
|- ( ph -> A =/= (/) ) |
3 |
|
fsumlt.3 |
|- ( ( ph /\ k e. A ) -> B e. RR ) |
4 |
|
fsumlt.4 |
|- ( ( ph /\ k e. A ) -> C e. RR ) |
5 |
|
fsumlt.5 |
|- ( ( ph /\ k e. A ) -> B < C ) |
6 |
|
difrp |
|- ( ( B e. RR /\ C e. RR ) -> ( B < C <-> ( C - B ) e. RR+ ) ) |
7 |
3 4 6
|
syl2anc |
|- ( ( ph /\ k e. A ) -> ( B < C <-> ( C - B ) e. RR+ ) ) |
8 |
5 7
|
mpbid |
|- ( ( ph /\ k e. A ) -> ( C - B ) e. RR+ ) |
9 |
1 2 8
|
fsumrpcl |
|- ( ph -> sum_ k e. A ( C - B ) e. RR+ ) |
10 |
9
|
rpgt0d |
|- ( ph -> 0 < sum_ k e. A ( C - B ) ) |
11 |
4
|
recnd |
|- ( ( ph /\ k e. A ) -> C e. CC ) |
12 |
3
|
recnd |
|- ( ( ph /\ k e. A ) -> B e. CC ) |
13 |
1 11 12
|
fsumsub |
|- ( ph -> sum_ k e. A ( C - B ) = ( sum_ k e. A C - sum_ k e. A B ) ) |
14 |
10 13
|
breqtrd |
|- ( ph -> 0 < ( sum_ k e. A C - sum_ k e. A B ) ) |
15 |
1 3
|
fsumrecl |
|- ( ph -> sum_ k e. A B e. RR ) |
16 |
1 4
|
fsumrecl |
|- ( ph -> sum_ k e. A C e. RR ) |
17 |
15 16
|
posdifd |
|- ( ph -> ( sum_ k e. A B < sum_ k e. A C <-> 0 < ( sum_ k e. A C - sum_ k e. A B ) ) ) |
18 |
14 17
|
mpbird |
|- ( ph -> sum_ k e. A B < sum_ k e. A C ) |