Metamath Proof Explorer


Theorem lcmineqlem7

Description: Derivative of 1-x for chain rule application. (Contributed by metakunt, 12-May-2024)

Ref Expression
Assertion lcmineqlem7 ( ℂ D ( 𝑥 ∈ ℂ ↦ ( 1 − 𝑥 ) ) ) = ( 𝑥 ∈ ℂ ↦ - 1 )

Proof

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 )