Step |
Hyp |
Ref |
Expression |
1 |
|
shftfval.1 |
⊢ 𝐹 ∈ V |
2 |
|
negcl |
⊢ ( 𝐴 ∈ ℂ → - 𝐴 ∈ ℂ ) |
3 |
1
|
shftval |
⊢ ( ( - 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → ( ( 𝐹 shift - 𝐴 ) ‘ 𝐵 ) = ( 𝐹 ‘ ( 𝐵 − - 𝐴 ) ) ) |
4 |
2 3
|
sylan |
⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → ( ( 𝐹 shift - 𝐴 ) ‘ 𝐵 ) = ( 𝐹 ‘ ( 𝐵 − - 𝐴 ) ) ) |
5 |
|
subneg |
⊢ ( ( 𝐵 ∈ ℂ ∧ 𝐴 ∈ ℂ ) → ( 𝐵 − - 𝐴 ) = ( 𝐵 + 𝐴 ) ) |
6 |
5
|
ancoms |
⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → ( 𝐵 − - 𝐴 ) = ( 𝐵 + 𝐴 ) ) |
7 |
|
addcom |
⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → ( 𝐴 + 𝐵 ) = ( 𝐵 + 𝐴 ) ) |
8 |
6 7
|
eqtr4d |
⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → ( 𝐵 − - 𝐴 ) = ( 𝐴 + 𝐵 ) ) |
9 |
8
|
fveq2d |
⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → ( 𝐹 ‘ ( 𝐵 − - 𝐴 ) ) = ( 𝐹 ‘ ( 𝐴 + 𝐵 ) ) ) |
10 |
4 9
|
eqtrd |
⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → ( ( 𝐹 shift - 𝐴 ) ‘ 𝐵 ) = ( 𝐹 ‘ ( 𝐴 + 𝐵 ) ) ) |