Theorem sge0ssrempt

Description: If a sum of nonnegative extended reals is real, than any subsum is real. (Contributed by Glauco Siliprandi, 17-Aug-2020)

	Ref	Expression
Hypotheses	sge0ssrempt.xph	⊢ Ⅎ 𝑥 𝜑
	sge0ssrempt.a	⊢ ( 𝜑 → 𝐴 ∈ 𝑉 )
	sge0ssrempt.b	⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → 𝐵 ∈ ( 0 [,] +∞ ) )
	sge0ssrempt.re	⊢ ( 𝜑 → ( Σ^ ‘ ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ) ∈ ℝ )
	sge0ssrempt.c	⊢ ( 𝜑 → 𝐶 ⊆ 𝐴 )
Assertion	sge0ssrempt	⊢ ( 𝜑 → ( Σ^ ‘ ( 𝑥 ∈ 𝐶 ↦ 𝐵 ) ) ∈ ℝ )

Proof

Step	Hyp	Ref	Expression
1		sge0ssrempt.xph	⊢ Ⅎ 𝑥 𝜑
2		sge0ssrempt.a	⊢ ( 𝜑 → 𝐴 ∈ 𝑉 )
3		sge0ssrempt.b	⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → 𝐵 ∈ ( 0 [,] +∞ ) )
4		sge0ssrempt.re	⊢ ( 𝜑 → ( Σ^ ‘ ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ) ∈ ℝ )
5		sge0ssrempt.c	⊢ ( 𝜑 → 𝐶 ⊆ 𝐴 )
6	5	resmptd	⊢ ( 𝜑 → ( ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ↾ 𝐶 ) = ( 𝑥 ∈ 𝐶 ↦ 𝐵 ) )
7	6	fveq2d	⊢ ( 𝜑 → ( Σ^ ‘ ( ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ↾ 𝐶 ) ) = ( Σ^ ‘ ( 𝑥 ∈ 𝐶 ↦ 𝐵 ) ) )
8	7	eqcomd	⊢ ( 𝜑 → ( Σ^ ‘ ( 𝑥 ∈ 𝐶 ↦ 𝐵 ) ) = ( Σ^ ‘ ( ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ↾ 𝐶 ) ) )
9		eqid	⊢ ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) = ( 𝑥 ∈ 𝐴 ↦ 𝐵 )
10	1 3 9	fmptdf	⊢ ( 𝜑 → ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) : 𝐴 ⟶ ( 0 [,] +∞ ) )
11	2 10 4	sge0ssre	⊢ ( 𝜑 → ( Σ^ ‘ ( ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ↾ 𝐶 ) ) ∈ ℝ )
12	8 11	eqeltrd	⊢ ( 𝜑 → ( Σ^ ‘ ( 𝑥 ∈ 𝐶 ↦ 𝐵 ) ) ∈ ℝ )