Theorem shftval5

Description: Value of a shifted sequence. (Contributed by NM, 19-Aug-2005) (Revised by Mario Carneiro, 5-Nov-2013)

	Ref	Expression
Hypothesis	shftfval.1	⊢ 𝐹 ∈ V
Assertion	shftval5	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → ( ( 𝐹 shift 𝐴 ) ‘ ( 𝐵 + 𝐴 ) ) = ( 𝐹 ‘ 𝐵 ) )

Proof

Step	Hyp	Ref	Expression
1		shftfval.1	⊢ 𝐹 ∈ V
2		simpr	⊢ ( ( 𝐵 ∈ ℂ ∧ 𝐴 ∈ ℂ ) → 𝐴 ∈ ℂ )
3		addcl	⊢ ( ( 𝐵 ∈ ℂ ∧ 𝐴 ∈ ℂ ) → ( 𝐵 + 𝐴 ) ∈ ℂ )
4	1	shftval	⊢ ( ( 𝐴 ∈ ℂ ∧ ( 𝐵 + 𝐴 ) ∈ ℂ ) → ( ( 𝐹 shift 𝐴 ) ‘ ( 𝐵 + 𝐴 ) ) = ( 𝐹 ‘ ( ( 𝐵 + 𝐴 ) − 𝐴 ) ) )
5	2 3 4	syl2anc	⊢ ( ( 𝐵 ∈ ℂ ∧ 𝐴 ∈ ℂ ) → ( ( 𝐹 shift 𝐴 ) ‘ ( 𝐵 + 𝐴 ) ) = ( 𝐹 ‘ ( ( 𝐵 + 𝐴 ) − 𝐴 ) ) )
6		pncan	⊢ ( ( 𝐵 ∈ ℂ ∧ 𝐴 ∈ ℂ ) → ( ( 𝐵 + 𝐴 ) − 𝐴 ) = 𝐵 )
7	6	fveq2d	⊢ ( ( 𝐵 ∈ ℂ ∧ 𝐴 ∈ ℂ ) → ( 𝐹 ‘ ( ( 𝐵 + 𝐴 ) − 𝐴 ) ) = ( 𝐹 ‘ 𝐵 ) )
8	5 7	eqtrd	⊢ ( ( 𝐵 ∈ ℂ ∧ 𝐴 ∈ ℂ ) → ( ( 𝐹 shift 𝐴 ) ‘ ( 𝐵 + 𝐴 ) ) = ( 𝐹 ‘ 𝐵 ) )
9	8	ancoms	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → ( ( 𝐹 shift 𝐴 ) ‘ ( 𝐵 + 𝐴 ) ) = ( 𝐹 ‘ 𝐵 ) )