Theorem shftvalg

Description: Value of a sequence shifted by A . (Contributed by Scott Fenton, 16-Dec-2017)

	Ref	Expression
Assertion	shftvalg	⊢ ( ( 𝐹 ∈ 𝑉 ∧ 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → ( ( 𝐹 shift 𝐴 ) ‘ 𝐵 ) = ( 𝐹 ‘ ( 𝐵 − 𝐴 ) ) )

Proof

Step	Hyp	Ref	Expression
1		oveq1	⊢ ( 𝑓 = 𝐹 → ( 𝑓 shift 𝐴 ) = ( 𝐹 shift 𝐴 ) )
2	1	fveq1d	⊢ ( 𝑓 = 𝐹 → ( ( 𝑓 shift 𝐴 ) ‘ 𝐵 ) = ( ( 𝐹 shift 𝐴 ) ‘ 𝐵 ) )
3		fveq1	⊢ ( 𝑓 = 𝐹 → ( 𝑓 ‘ ( 𝐵 − 𝐴 ) ) = ( 𝐹 ‘ ( 𝐵 − 𝐴 ) ) )
4	2 3	eqeq12d	⊢ ( 𝑓 = 𝐹 → ( ( ( 𝑓 shift 𝐴 ) ‘ 𝐵 ) = ( 𝑓 ‘ ( 𝐵 − 𝐴 ) ) ↔ ( ( 𝐹 shift 𝐴 ) ‘ 𝐵 ) = ( 𝐹 ‘ ( 𝐵 − 𝐴 ) ) ) )
5	4	imbi2d	⊢ ( 𝑓 = 𝐹 → ( ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → ( ( 𝑓 shift 𝐴 ) ‘ 𝐵 ) = ( 𝑓 ‘ ( 𝐵 − 𝐴 ) ) ) ↔ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → ( ( 𝐹 shift 𝐴 ) ‘ 𝐵 ) = ( 𝐹 ‘ ( 𝐵 − 𝐴 ) ) ) ) )
6		vex	⊢ 𝑓 ∈ V
7	6	shftval	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → ( ( 𝑓 shift 𝐴 ) ‘ 𝐵 ) = ( 𝑓 ‘ ( 𝐵 − 𝐴 ) ) )
8	5 7	vtoclg	⊢ ( 𝐹 ∈ 𝑉 → ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → ( ( 𝐹 shift 𝐴 ) ‘ 𝐵 ) = ( 𝐹 ‘ ( 𝐵 − 𝐴 ) ) ) )
9	8	3impib	⊢ ( ( 𝐹 ∈ 𝑉 ∧ 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → ( ( 𝐹 shift 𝐴 ) ‘ 𝐵 ) = ( 𝐹 ‘ ( 𝐵 − 𝐴 ) ) )