Theorem sge0lessmpt

Description: A shorter sum of nonnegative extended reals is smaller than a longer one. (Contributed by Glauco Siliprandi, 17-Aug-2020)

	Ref	Expression
Hypotheses	sge0lessmpt.a	|- ( ph -> A e. V )
	sge0lessmpt.b	|- ( ( ph /\ x e. A ) -> B e. ( 0 [,] +oo ) )
	sge0lessmpt.c	|- ( ph -> C C_ A )
Assertion	sge0lessmpt	|- ( ph -> ( sum^ ` ( x e. C |-> B ) ) <_ ( sum^ ` ( x e. A |-> B ) ) )

Proof

Step	Hyp	Ref	Expression
1		sge0lessmpt.a	|- ( ph -> A e. V )
2		sge0lessmpt.b	|- ( ( ph /\ x e. A ) -> B e. ( 0 [,] +oo ) )
3		sge0lessmpt.c	|- ( ph -> C C_ A )
4		id	|- ( ph -> ph )
5	3	resmptd	|- ( ph -> ( ( x e. A |-> B ) |` C ) = ( x e. C |-> B ) )
6	5	eqcomd	|- ( ph -> ( x e. C |-> B ) = ( ( x e. A |-> B ) |` C ) )
7	4 6	syl	|- ( ph -> ( x e. C |-> B ) = ( ( x e. A |-> B ) |` C ) )
8	7	fveq2d	|- ( ph -> ( sum^ ` ( x e. C |-> B ) ) = ( sum^ ` ( ( x e. A |-> B ) |` C ) ) )
9		eqid	|- ( x e. A |-> B ) = ( x e. A |-> B )
10	2 9	fmptd	|- ( ph -> ( x e. A |-> B ) : A --> ( 0 [,] +oo ) )
11	1 10	sge0less	|- ( ph -> ( sum^ ` ( ( x e. A |-> B ) |` C ) ) <_ ( sum^ ` ( x e. A |-> B ) ) )
12	8 11	eqbrtrd	|- ( ph -> ( sum^ ` ( x e. C |-> B ) ) <_ ( sum^ ` ( x e. A |-> B ) ) )