Theorem shftval3

Description: Value of a sequence shifted by A - B . (Contributed by NM, 20-Jul-2005)

	Ref	Expression
Hypothesis	shftfval.1	⊢ 𝐹 ∈ V
Assertion	shftval3	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → ( ( 𝐹 shift ( 𝐴 − 𝐵 ) ) ‘ 𝐴 ) = ( 𝐹 ‘ 𝐵 ) )

Proof

Step	Hyp	Ref	Expression
1		shftfval.1	⊢ 𝐹 ∈ V
2		0cn	⊢ 0 ∈ ℂ
3	1	shftval2	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 0 ∈ ℂ ) → ( ( 𝐹 shift ( 𝐴 − 𝐵 ) ) ‘ ( 𝐴 + 0 ) ) = ( 𝐹 ‘ ( 𝐵 + 0 ) ) )
4	2 3	mp3an3	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → ( ( 𝐹 shift ( 𝐴 − 𝐵 ) ) ‘ ( 𝐴 + 0 ) ) = ( 𝐹 ‘ ( 𝐵 + 0 ) ) )
5		addrid	⊢ ( 𝐴 ∈ ℂ → ( 𝐴 + 0 ) = 𝐴 )
6	5	adantr	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → ( 𝐴 + 0 ) = 𝐴 )
7	6	fveq2d	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → ( ( 𝐹 shift ( 𝐴 − 𝐵 ) ) ‘ ( 𝐴 + 0 ) ) = ( ( 𝐹 shift ( 𝐴 − 𝐵 ) ) ‘ 𝐴 ) )
8		addrid	⊢ ( 𝐵 ∈ ℂ → ( 𝐵 + 0 ) = 𝐵 )
9	8	adantl	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → ( 𝐵 + 0 ) = 𝐵 )
10	9	fveq2d	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → ( 𝐹 ‘ ( 𝐵 + 0 ) ) = ( 𝐹 ‘ 𝐵 ) )
11	4 7 10	3eqtr3d	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → ( ( 𝐹 shift ( 𝐴 − 𝐵 ) ) ‘ 𝐴 ) = ( 𝐹 ‘ 𝐵 ) )