Metamath Proof Explorer

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→Fshift−AshiftA⁡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→Fshift−Ashift−−A=Fshift−AshiftA
5 4 fveq1d ⊢A∈ℂ∧B∈ℂ→Fshift−Ashift−−A⁡B=Fshift−AshiftA⁡B
6 negcl ⊢A∈ℂ→−A∈ℂ
7 1 shftcan1 ⊢−A∈ℂ∧B∈ℂ→Fshift−Ashift−−A⁡B=F⁡B
8 6 7 sylan ⊢A∈ℂ∧B∈ℂ→Fshift−Ashift−−A⁡B=F⁡B
9 5 8 eqtr3d ⊢A∈ℂ∧B∈ℂ→Fshift−AshiftA⁡B=F⁡B