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 } ) ) |