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 ) ) |