Theorem shftcan2

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	shftcan2	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → ( ( ( 𝐹 shift - 𝐴 ) shift 𝐴 ) ‘ 𝐵 ) = ( 𝐹 ‘ 𝐵 ) )

Proof

Step	Hyp	Ref	Expression
1		shftfval.1	⊢ 𝐹 ∈ V
2		negneg	⊢ ( 𝐴 ∈ ℂ → - - 𝐴 = 𝐴 )
3	2	adantr	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → - - 𝐴 = 𝐴 )
4	3	oveq2d	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → ( ( 𝐹 shift - 𝐴 ) shift - - 𝐴 ) = ( ( 𝐹 shift - 𝐴 ) shift 𝐴 ) )
5	4	fveq1d	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → ( ( ( 𝐹 shift - 𝐴 ) shift - - 𝐴 ) ‘ 𝐵 ) = ( ( ( 𝐹 shift - 𝐴 ) shift 𝐴 ) ‘ 𝐵 ) )
6		negcl	⊢ ( 𝐴 ∈ ℂ → - 𝐴 ∈ ℂ )
7	1	shftcan1	⊢ ( ( - 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → ( ( ( 𝐹 shift - 𝐴 ) shift - - 𝐴 ) ‘ 𝐵 ) = ( 𝐹 ‘ 𝐵 ) )
8	6 7	sylan	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → ( ( ( 𝐹 shift - 𝐴 ) shift - - 𝐴 ) ‘ 𝐵 ) = ( 𝐹 ‘ 𝐵 ) )
9	5 8	eqtr3d	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → ( ( ( 𝐹 shift - 𝐴 ) shift 𝐴 ) ‘ 𝐵 ) = ( 𝐹 ‘ 𝐵 ) )