Metamath Proof Explorer


Theorem sge0fsummpt

Description: The arbitrary sum of a finite set of nonnegative extended real numbers is equal to the sum of those numbers, when none of them is +oo (Contributed by Glauco Siliprandi, 17-Aug-2020)

Ref Expression
Hypotheses sge0fsummpt.a
|- ( ph -> A e. Fin )
sge0fsummpt.b
|- ( ( ph /\ k e. A ) -> B e. ( 0 [,) +oo ) )
Assertion sge0fsummpt
|- ( ph -> ( sum^ ` ( k e. A |-> B ) ) = sum_ k e. A B )

Proof

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 )