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 ) ) )