Theorem shftval

Description: Value of a sequence shifted by A . (Contributed by NM, 20-Jul-2005) (Revised by Mario Carneiro, 4-Nov-2013)

	Ref	Expression
Hypothesis	shftfval.1	⊢ 𝐹 ∈ V
Assertion	shftval	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → ( ( 𝐹 shift 𝐴 ) ‘ 𝐵 ) = ( 𝐹 ‘ ( 𝐵 − 𝐴 ) ) )

Proof

Step	Hyp	Ref	Expression
1		shftfval.1	⊢ 𝐹 ∈ V
2	1	shftfib	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → ( ( 𝐹 shift 𝐴 ) “ { 𝐵 } ) = ( 𝐹 “ { ( 𝐵 − 𝐴 ) } ) )
3	2	eleq2d	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → ( 𝑥 ∈ ( ( 𝐹 shift 𝐴 ) “ { 𝐵 } ) ↔ 𝑥 ∈ ( 𝐹 “ { ( 𝐵 − 𝐴 ) } ) ) )
4	3	iotabidv	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → ( ℩ 𝑥 𝑥 ∈ ( ( 𝐹 shift 𝐴 ) “ { 𝐵 } ) ) = ( ℩ 𝑥 𝑥 ∈ ( 𝐹 “ { ( 𝐵 − 𝐴 ) } ) ) )
5		dffv3	⊢ ( ( 𝐹 shift 𝐴 ) ‘ 𝐵 ) = ( ℩ 𝑥 𝑥 ∈ ( ( 𝐹 shift 𝐴 ) “ { 𝐵 } ) )
6		dffv3	⊢ ( 𝐹 ‘ ( 𝐵 − 𝐴 ) ) = ( ℩ 𝑥 𝑥 ∈ ( 𝐹 “ { ( 𝐵 − 𝐴 ) } ) )
7	4 5 6	3eqtr4g	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → ( ( 𝐹 shift 𝐴 ) ‘ 𝐵 ) = ( 𝐹 ‘ ( 𝐵 − 𝐴 ) ) )