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Metamath Proof Explorer

Theorem climaddc2

Description: Limit of a constant C added to each term of a sequence. (Contributed by NM, 24-Sep-2005) (Revised by Mario Carneiro, 3-Feb-2014)

Ref Expression
Hypotheses climadd.1 ⊢ Z = ℤ ≥ M
climadd.2 ⊢ φ → M ∈ ℤ
climadd.4 ⊢ φ → F ⇝ A
climaddc1.5 ⊢ φ → C ∈ ℂ
climaddc1.6 ⊢ φ → G ∈ W
climaddc1.7 ⊢ φ ∧ k ∈ Z → F ⁡ k ∈ ℂ
climaddc2.h ⊢ φ ∧ k ∈ Z → G ⁡ k = C + F ⁡ k
Assertion climaddc2 ⊢ φ → G ⇝ C + A

Proof

Step Hyp Ref Expression
1 climadd.1 ⊢ Z = ℤ ≥ M
2 climadd.2 ⊢ φ → M ∈ ℤ
3 climadd.4 ⊢ φ → F ⇝ A
4 climaddc1.5 ⊢ φ → C ∈ ℂ
5 climaddc1.6 ⊢ φ → G ∈ W
6 climaddc1.7 ⊢ φ ∧ k ∈ Z → F ⁡ k ∈ ℂ
7 climaddc2.h ⊢ φ ∧ k ∈ Z → G ⁡ k = C + F ⁡ k
8 4 adantr ⊢ φ ∧ k ∈ Z → C ∈ ℂ
9 8 6 7 comraddd ⊢ φ ∧ k ∈ Z → G ⁡ k = F ⁡ k + C
10 1 2 3 4 5 6 9 climaddc1 ⊢ φ → G ⇝ A + C
11 climcl ⊢ F ⇝ A → A ∈ ℂ
12 3 11 syl ⊢ φ → A ∈ ℂ
13 12 4 addcomd ⊢ φ → A + C = C + A
14 10 13 breqtrd ⊢ φ → G ⇝ C + A