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 ) ) )