Step |
Hyp |
Ref |
Expression |
1 |
|
cnelprrecn |
⊢ ℂ ∈ { ℝ , ℂ } |
2 |
1
|
a1i |
⊢ ( ⊤ → ℂ ∈ { ℝ , ℂ } ) |
3 |
|
1cnd |
⊢ ( ( ⊤ ∧ 𝑥 ∈ ℂ ) → 1 ∈ ℂ ) |
4 |
|
0cnd |
⊢ ( ( ⊤ ∧ 𝑥 ∈ ℂ ) → 0 ∈ ℂ ) |
5 |
|
1cnd |
⊢ ( ⊤ → 1 ∈ ℂ ) |
6 |
2 5
|
dvmptc |
⊢ ( ⊤ → ( ℂ D ( 𝑥 ∈ ℂ ↦ 1 ) ) = ( 𝑥 ∈ ℂ ↦ 0 ) ) |
7 |
|
simpr |
⊢ ( ( ⊤ ∧ 𝑥 ∈ ℂ ) → 𝑥 ∈ ℂ ) |
8 |
2
|
dvmptid |
⊢ ( ⊤ → ( ℂ D ( 𝑥 ∈ ℂ ↦ 𝑥 ) ) = ( 𝑥 ∈ ℂ ↦ 1 ) ) |
9 |
2 3 4 6 7 3 8
|
dvmptsub |
⊢ ( ⊤ → ( ℂ D ( 𝑥 ∈ ℂ ↦ ( 1 − 𝑥 ) ) ) = ( 𝑥 ∈ ℂ ↦ ( 0 − 1 ) ) ) |
10 |
|
df-neg |
⊢ - 1 = ( 0 − 1 ) |
11 |
10
|
a1i |
⊢ ( ⊤ → - 1 = ( 0 − 1 ) ) |
12 |
11
|
eqcomd |
⊢ ( ⊤ → ( 0 − 1 ) = - 1 ) |
13 |
12
|
mpteq2dv |
⊢ ( ⊤ → ( 𝑥 ∈ ℂ ↦ ( 0 − 1 ) ) = ( 𝑥 ∈ ℂ ↦ - 1 ) ) |
14 |
9 13
|
eqtrd |
⊢ ( ⊤ → ( ℂ D ( 𝑥 ∈ ℂ ↦ ( 1 − 𝑥 ) ) ) = ( 𝑥 ∈ ℂ ↦ - 1 ) ) |
15 |
14
|
mptru |
⊢ ( ℂ D ( 𝑥 ∈ ℂ ↦ ( 1 − 𝑥 ) ) ) = ( 𝑥 ∈ ℂ ↦ - 1 ) |