Metamath Proof Explorer


Theorem dvsid

Description: Derivative of the identity function on the real or complex numbers. (Contributed by Steve Rodriguez, 11-Nov-2015)

Ref Expression
Assertion dvsid ( 𝑆 ∈ { ℝ , ℂ } → ( 𝑆 D ( I ↾ 𝑆 ) ) = ( 𝑆 × { 1 } ) )

Proof

Step Hyp Ref Expression
1 fnresi ( I ↾ ℂ ) Fn ℂ
2 rnresi ran ( I ↾ ℂ ) = ℂ
3 2 eqimssi ran ( I ↾ ℂ ) ⊆ ℂ
4 df-f ( ( I ↾ ℂ ) : ℂ ⟶ ℂ ↔ ( ( I ↾ ℂ ) Fn ℂ ∧ ran ( I ↾ ℂ ) ⊆ ℂ ) )
5 1 3 4 mpbir2an ( I ↾ ℂ ) : ℂ ⟶ ℂ
6 5 jctr ( 𝑆 ∈ { ℝ , ℂ } → ( 𝑆 ∈ { ℝ , ℂ } ∧ ( I ↾ ℂ ) : ℂ ⟶ ℂ ) )
7 recnprss ( 𝑆 ∈ { ℝ , ℂ } → 𝑆 ⊆ ℂ )
8 dvid ( ℂ D ( I ↾ ℂ ) ) = ( ℂ × { 1 } )
9 8 dmeqi dom ( ℂ D ( I ↾ ℂ ) ) = dom ( ℂ × { 1 } )
10 1ex 1 ∈ V
11 10 fconst ( ℂ × { 1 } ) : ℂ ⟶ { 1 }
12 11 fdmi dom ( ℂ × { 1 } ) = ℂ
13 9 12 eqtri dom ( ℂ D ( I ↾ ℂ ) ) = ℂ
14 7 13 sseqtrrdi ( 𝑆 ∈ { ℝ , ℂ } → 𝑆 ⊆ dom ( ℂ D ( I ↾ ℂ ) ) )
15 ssid ℂ ⊆ ℂ
16 14 15 jctil ( 𝑆 ∈ { ℝ , ℂ } → ( ℂ ⊆ ℂ ∧ 𝑆 ⊆ dom ( ℂ D ( I ↾ ℂ ) ) ) )
17 dvres3 ( ( ( 𝑆 ∈ { ℝ , ℂ } ∧ ( I ↾ ℂ ) : ℂ ⟶ ℂ ) ∧ ( ℂ ⊆ ℂ ∧ 𝑆 ⊆ dom ( ℂ D ( I ↾ ℂ ) ) ) ) → ( 𝑆 D ( ( I ↾ ℂ ) ↾ 𝑆 ) ) = ( ( ℂ D ( I ↾ ℂ ) ) ↾ 𝑆 ) )
18 6 16 17 syl2anc ( 𝑆 ∈ { ℝ , ℂ } → ( 𝑆 D ( ( I ↾ ℂ ) ↾ 𝑆 ) ) = ( ( ℂ D ( I ↾ ℂ ) ) ↾ 𝑆 ) )
19 7 resabs1d ( 𝑆 ∈ { ℝ , ℂ } → ( ( I ↾ ℂ ) ↾ 𝑆 ) = ( I ↾ 𝑆 ) )
20 19 oveq2d ( 𝑆 ∈ { ℝ , ℂ } → ( 𝑆 D ( ( I ↾ ℂ ) ↾ 𝑆 ) ) = ( 𝑆 D ( I ↾ 𝑆 ) ) )
21 8 reseq1i ( ( ℂ D ( I ↾ ℂ ) ) ↾ 𝑆 ) = ( ( ℂ × { 1 } ) ↾ 𝑆 )
22 xpssres ( 𝑆 ⊆ ℂ → ( ( ℂ × { 1 } ) ↾ 𝑆 ) = ( 𝑆 × { 1 } ) )
23 21 22 syl5eq ( 𝑆 ⊆ ℂ → ( ( ℂ D ( I ↾ ℂ ) ) ↾ 𝑆 ) = ( 𝑆 × { 1 } ) )
24 7 23 syl ( 𝑆 ∈ { ℝ , ℂ } → ( ( ℂ D ( I ↾ ℂ ) ) ↾ 𝑆 ) = ( 𝑆 × { 1 } ) )
25 18 20 24 3eqtr3d ( 𝑆 ∈ { ℝ , ℂ } → ( 𝑆 D ( I ↾ 𝑆 ) ) = ( 𝑆 × { 1 } ) )