Theorem shftcan1

Description: Cancellation law for the shift operation. (Contributed by NM, 4-Aug-2005) (Revised by Mario Carneiro, 5-Nov-2013)

	Ref	Expression
Hypothesis	shftfval.1	⊢ 𝐹 ∈ V
Assertion	shftcan1	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → ( ( ( 𝐹 shift 𝐴 ) shift - 𝐴 ) ‘ 𝐵 ) = ( 𝐹 ‘ 𝐵 ) )

Proof

Step	Hyp	Ref	Expression
1		shftfval.1	⊢ 𝐹 ∈ V
2		negcl	⊢ ( 𝐴 ∈ ℂ → - 𝐴 ∈ ℂ )
3	1	2shfti	⊢ ( ( 𝐴 ∈ ℂ ∧ - 𝐴 ∈ ℂ ) → ( ( 𝐹 shift 𝐴 ) shift - 𝐴 ) = ( 𝐹 shift ( 𝐴 + - 𝐴 ) ) )
4	2 3	mpdan	⊢ ( 𝐴 ∈ ℂ → ( ( 𝐹 shift 𝐴 ) shift - 𝐴 ) = ( 𝐹 shift ( 𝐴 + - 𝐴 ) ) )
5		negid	⊢ ( 𝐴 ∈ ℂ → ( 𝐴 + - 𝐴 ) = 0 )
6	5	oveq2d	⊢ ( 𝐴 ∈ ℂ → ( 𝐹 shift ( 𝐴 + - 𝐴 ) ) = ( 𝐹 shift 0 ) )
7	4 6	eqtrd	⊢ ( 𝐴 ∈ ℂ → ( ( 𝐹 shift 𝐴 ) shift - 𝐴 ) = ( 𝐹 shift 0 ) )
8	7	fveq1d	⊢ ( 𝐴 ∈ ℂ → ( ( ( 𝐹 shift 𝐴 ) shift - 𝐴 ) ‘ 𝐵 ) = ( ( 𝐹 shift 0 ) ‘ 𝐵 ) )
9	1	shftidt	⊢ ( 𝐵 ∈ ℂ → ( ( 𝐹 shift 0 ) ‘ 𝐵 ) = ( 𝐹 ‘ 𝐵 ) )
10	8 9	sylan9eq	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → ( ( ( 𝐹 shift 𝐴 ) shift - 𝐴 ) ‘ 𝐵 ) = ( 𝐹 ‘ 𝐵 ) )