Metamath Proof Explorer

Theorem seqabs

Description: Generalized triangle inequality: the absolute value of a finite sum is less than or equal to the sum of absolute values. (Contributed by Mario Carneiro, 26-Mar-2014) (Revised by Mario Carneiro, 27-May-2014)

Ref Expression
Hypotheses seqabs.1 ⊢ φ → N ∈ ℤ ≥ M
seqabs.2 ⊢ φ ∧ k ∈ M … N → F ⁡ k ∈ ℂ
seqabs.3 ⊢ φ ∧ k ∈ M … N → G ⁡ k = F ⁡ k
Assertion seqabs ⊢ φ → seq M + F ⁡ N ≤ seq M + G ⁡ N

Proof

Step Hyp Ref Expression
1 seqabs.1 ⊢ φ → N ∈ ℤ ≥ M
2 seqabs.2 ⊢ φ ∧ k ∈ M … N → F ⁡ k ∈ ℂ
3 seqabs.3 ⊢ φ ∧ k ∈ M … N → G ⁡ k = F ⁡ k
4 fzfid ⊢ φ → M … N ∈ Fin
5 4 2 fsumabs ⊢ φ → ∑ k = M N F ⁡ k ≤ ∑ k = M N F ⁡ k
6 eqidd ⊢ φ ∧ k ∈ M … N → F ⁡ k = F ⁡ k
7 6 1 2 fsumser ⊢ φ → ∑ k = M N F ⁡ k = seq M + F ⁡ N
8 7 fveq2d ⊢ φ → ∑ k = M N F ⁡ k = seq M + F ⁡ N
9 abscl ⊢ F ⁡ k ∈ ℂ → F ⁡ k ∈ ℝ
10 9 recnd ⊢ F ⁡ k ∈ ℂ → F ⁡ k ∈ ℂ
11 2 10 syl ⊢ φ ∧ k ∈ M … N → F ⁡ k ∈ ℂ
12 3 1 11 fsumser ⊢ φ → ∑ k = M N F ⁡ k = seq M + G ⁡ N
13 5 8 12 3brtr3d ⊢ φ → seq M + F ⁡ N ≤ seq M + G ⁡ N