Database
SUPPLEMENTARY MATERIAL (USERS' MATHBOXES)
Mathbox for Stefan O'Rear
Algebraic integers I
itgoval
Next ⟩
aaitgo
Metamath Proof Explorer
Ascii
Unicode
Theorem
itgoval
Description:
Value of the integral-over function.
(Contributed by
Stefan O'Rear
, 27-Nov-2014)
Ref
Expression
Assertion
itgoval
⊢
S
⊆
ℂ
→
IntgOver
⁡
S
=
x
∈
ℂ
|
∃
p
∈
Poly
⁡
S
p
⁡
x
=
0
∧
coeff
⁡
p
⁡
deg
⁡
p
=
1
Proof
Step
Hyp
Ref
Expression
1
cnex
⊢
ℂ
∈
V
2
1
elpw2
⊢
S
∈
𝒫
ℂ
↔
S
⊆
ℂ
3
fveq2
⊢
s
=
S
→
Poly
⁡
s
=
Poly
⁡
S
4
3
rexeqdv
⊢
s
=
S
→
∃
p
∈
Poly
⁡
s
p
⁡
x
=
0
∧
coeff
⁡
p
⁡
deg
⁡
p
=
1
↔
∃
p
∈
Poly
⁡
S
p
⁡
x
=
0
∧
coeff
⁡
p
⁡
deg
⁡
p
=
1
5
4
rabbidv
⊢
s
=
S
→
x
∈
ℂ
|
∃
p
∈
Poly
⁡
s
p
⁡
x
=
0
∧
coeff
⁡
p
⁡
deg
⁡
p
=
1
=
x
∈
ℂ
|
∃
p
∈
Poly
⁡
S
p
⁡
x
=
0
∧
coeff
⁡
p
⁡
deg
⁡
p
=
1
6
df-itgo
⊢
IntgOver
=
s
∈
𝒫
ℂ
⟼
x
∈
ℂ
|
∃
p
∈
Poly
⁡
s
p
⁡
x
=
0
∧
coeff
⁡
p
⁡
deg
⁡
p
=
1
7
1
rabex
⊢
x
∈
ℂ
|
∃
p
∈
Poly
⁡
S
p
⁡
x
=
0
∧
coeff
⁡
p
⁡
deg
⁡
p
=
1
∈
V
8
5
6
7
fvmpt
⊢
S
∈
𝒫
ℂ
→
IntgOver
⁡
S
=
x
∈
ℂ
|
∃
p
∈
Poly
⁡
S
p
⁡
x
=
0
∧
coeff
⁡
p
⁡
deg
⁡
p
=
1
9
2
8
sylbir
⊢
S
⊆
ℂ
→
IntgOver
⁡
S
=
x
∈
ℂ
|
∃
p
∈
Poly
⁡
S
p
⁡
x
=
0
∧
coeff
⁡
p
⁡
deg
⁡
p
=
1