Theorem shftidt

Description: Identity law for the shift operation. (Contributed by NM, 19-Aug-2005) (Revised by Mario Carneiro, 5-Nov-2013)

	Ref	Expression
Hypothesis	shftfval.1	⊢ 𝐹 ∈ V
Assertion	shftidt	⊢ ( 𝐴 ∈ ℂ → ( ( 𝐹 shift 0 ) ‘ 𝐴 ) = ( 𝐹 ‘ 𝐴 ) )

Proof

Step	Hyp	Ref	Expression
1		shftfval.1	⊢ 𝐹 ∈ V
2	1	shftidt2	⊢ ( 𝐹 shift 0 ) = ( 𝐹 ↾ ℂ )
3	2	fveq1i	⊢ ( ( 𝐹 shift 0 ) ‘ 𝐴 ) = ( ( 𝐹 ↾ ℂ ) ‘ 𝐴 )
4		fvres	⊢ ( 𝐴 ∈ ℂ → ( ( 𝐹 ↾ ℂ ) ‘ 𝐴 ) = ( 𝐹 ‘ 𝐴 ) )
5	3 4	syl5eq	⊢ ( 𝐴 ∈ ℂ → ( ( 𝐹 shift 0 ) ‘ 𝐴 ) = ( 𝐹 ‘ 𝐴 ) )