This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

Metamath Proof Explorer

Theorem shftval5

Description: Value of a shifted sequence. (Contributed by NM, 19-Aug-2005) (Revised by Mario Carneiro, 5-Nov-2013)

Ref Expression
Hypothesis shftfval.1 ⊢ F ∈ V
Assertion shftval5 ⊢ A ∈ ℂ ∧ B ∈ ℂ → F shift A ⁡ B + A = F ⁡ B

Proof

Step Hyp Ref Expression
1 shftfval.1 ⊢ F ∈ V
2 simpr ⊢ B ∈ ℂ ∧ A ∈ ℂ → A ∈ ℂ
3 addcl ⊢ B ∈ ℂ ∧ A ∈ ℂ → B + A ∈ ℂ
4 1 shftval ⊢ A ∈ ℂ ∧ B + A ∈ ℂ → F shift A ⁡ B + A = F ⁡ B + A - A
5 2 3 4 syl2anc ⊢ B ∈ ℂ ∧ A ∈ ℂ → F shift A ⁡ B + A = F ⁡ B + A - A
6 pncan ⊢ B ∈ ℂ ∧ A ∈ ℂ → B + A - A = B
7 6 fveq2d ⊢ B ∈ ℂ ∧ A ∈ ℂ → F ⁡ B + A - A = F ⁡ B
8 5 7 eqtrd ⊢ B ∈ ℂ ∧ A ∈ ℂ → F shift A ⁡ B + A = F ⁡ B
9 8 ancoms ⊢ A ∈ ℂ ∧ B ∈ ℂ → F shift A ⁡ B + A = F ⁡ B