Theorem fsumle

Description: If all of the terms of finite sums compare, so do the sums. (Contributed by NM, 11-Dec-2005) (Proof shortened by Mario Carneiro, 24-Apr-2014)

	Ref	Expression
Hypotheses	fsumle.1	⊢ ( 𝜑 → 𝐴 ∈ Fin )
	fsumle.2	⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝐴 ) → 𝐵 ∈ ℝ )
	fsumle.3	⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝐴 ) → 𝐶 ∈ ℝ )
	fsumle.4	⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝐴 ) → 𝐵 ≤ 𝐶 )
Assertion	fsumle	⊢ ( 𝜑 → Σ 𝑘 ∈ 𝐴 𝐵 ≤ Σ 𝑘 ∈ 𝐴 𝐶 )

Proof

Step	Hyp	Ref	Expression
1		fsumle.1	⊢ ( 𝜑 → 𝐴 ∈ Fin )
2		fsumle.2	⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝐴 ) → 𝐵 ∈ ℝ )
3		fsumle.3	⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝐴 ) → 𝐶 ∈ ℝ )
4		fsumle.4	⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝐴 ) → 𝐵 ≤ 𝐶 )
5	3 2	resubcld	⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝐴 ) → ( 𝐶 − 𝐵 ) ∈ ℝ )
6	3 2	subge0d	⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝐴 ) → ( 0 ≤ ( 𝐶 − 𝐵 ) ↔ 𝐵 ≤ 𝐶 ) )
7	4 6	mpbird	⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝐴 ) → 0 ≤ ( 𝐶 − 𝐵 ) )
8	1 5 7	fsumge0	⊢ ( 𝜑 → 0 ≤ Σ 𝑘 ∈ 𝐴 ( 𝐶 − 𝐵 ) )
9	3	recnd	⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝐴 ) → 𝐶 ∈ ℂ )
10	2	recnd	⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝐴 ) → 𝐵 ∈ ℂ )
11	1 9 10	fsumsub	⊢ ( 𝜑 → Σ 𝑘 ∈ 𝐴 ( 𝐶 − 𝐵 ) = ( Σ 𝑘 ∈ 𝐴 𝐶 − Σ 𝑘 ∈ 𝐴 𝐵 ) )
12	8 11	breqtrd	⊢ ( 𝜑 → 0 ≤ ( Σ 𝑘 ∈ 𝐴 𝐶 − Σ 𝑘 ∈ 𝐴 𝐵 ) )
13	1 3	fsumrecl	⊢ ( 𝜑 → Σ 𝑘 ∈ 𝐴 𝐶 ∈ ℝ )
14	1 2	fsumrecl	⊢ ( 𝜑 → Σ 𝑘 ∈ 𝐴 𝐵 ∈ ℝ )
15	13 14	subge0d	⊢ ( 𝜑 → ( 0 ≤ ( Σ 𝑘 ∈ 𝐴 𝐶 − Σ 𝑘 ∈ 𝐴 𝐵 ) ↔ Σ 𝑘 ∈ 𝐴 𝐵 ≤ Σ 𝑘 ∈ 𝐴 𝐶 ) )
16	12 15	mpbid	⊢ ( 𝜑 → Σ 𝑘 ∈ 𝐴 𝐵 ≤ Σ 𝑘 ∈ 𝐴 𝐶 )