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Metamath Proof Explorer

Theorem shftval4

Description: Value of a sequence shifted by -u A . (Contributed by NM, 18-Aug-2005) (Revised by Mario Carneiro, 5-Nov-2013)

Ref Expression
Hypothesis shftfval.1 ⊢ F ∈ V
Assertion shftval4 ⊢ A ∈ ℂ ∧ B ∈ ℂ → F shift − A ⁡ B = F ⁡ A + B

Proof

Step Hyp Ref Expression
1 shftfval.1 ⊢ F ∈ V
2 negcl ⊢ A ∈ ℂ → − A ∈ ℂ
3 1 shftval ⊢ − A ∈ ℂ ∧ B ∈ ℂ → F shift − A ⁡ B = F ⁡ B − − A
4 2 3 sylan ⊢ A ∈ ℂ ∧ B ∈ ℂ → F shift − A ⁡ B = F ⁡ B − − A
5 subneg ⊢ B ∈ ℂ ∧ A ∈ ℂ → B − − A = B + A
6 5 ancoms ⊢ A ∈ ℂ ∧ B ∈ ℂ → B − − A = B + A
7 addcom ⊢ A ∈ ℂ ∧ B ∈ ℂ → A + B = B + A
8 6 7 eqtr4d ⊢ A ∈ ℂ ∧ B ∈ ℂ → B − − A = A + B
9 8 fveq2d ⊢ A ∈ ℂ ∧ B ∈ ℂ → F ⁡ B − − A = F ⁡ A + B
10 4 9 eqtrd ⊢ A ∈ ℂ ∧ B ∈ ℂ → F shift − A ⁡ B = F ⁡ A + B