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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 ) ) )