Step |
Hyp |
Ref |
Expression |
1 |
|
fprodsub2cncf.k |
⊢ Ⅎ 𝑘 𝜑 |
2 |
|
fprodsub2cncf.a |
⊢ ( 𝜑 → 𝐴 ∈ Fin ) |
3 |
|
fprodsub2cncf.b |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝐴 ) → 𝐵 ∈ ℂ ) |
4 |
|
fprodsub2cncf.f |
⊢ 𝐹 = ( 𝑥 ∈ ℂ ↦ ∏ 𝑘 ∈ 𝐴 ( 𝐵 − 𝑥 ) ) |
5 |
4
|
a1i |
⊢ ( 𝜑 → 𝐹 = ( 𝑥 ∈ ℂ ↦ ∏ 𝑘 ∈ 𝐴 ( 𝐵 − 𝑥 ) ) ) |
6 |
|
eqid |
⊢ ( TopOpen ‘ ℂfld ) = ( TopOpen ‘ ℂfld ) |
7 |
6
|
cnfldtopon |
⊢ ( TopOpen ‘ ℂfld ) ∈ ( TopOn ‘ ℂ ) |
8 |
7
|
a1i |
⊢ ( 𝜑 → ( TopOpen ‘ ℂfld ) ∈ ( TopOn ‘ ℂ ) ) |
9 |
|
eqid |
⊢ ( 𝑥 ∈ ℂ ↦ ( 𝐵 − 𝑥 ) ) = ( 𝑥 ∈ ℂ ↦ ( 𝐵 − 𝑥 ) ) |
10 |
3 9
|
sub2cncfd |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝐴 ) → ( 𝑥 ∈ ℂ ↦ ( 𝐵 − 𝑥 ) ) ∈ ( ℂ –cn→ ℂ ) ) |
11 |
6
|
cncfcn1 |
⊢ ( ℂ –cn→ ℂ ) = ( ( TopOpen ‘ ℂfld ) Cn ( TopOpen ‘ ℂfld ) ) |
12 |
11
|
a1i |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝐴 ) → ( ℂ –cn→ ℂ ) = ( ( TopOpen ‘ ℂfld ) Cn ( TopOpen ‘ ℂfld ) ) ) |
13 |
10 12
|
eleqtrd |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝐴 ) → ( 𝑥 ∈ ℂ ↦ ( 𝐵 − 𝑥 ) ) ∈ ( ( TopOpen ‘ ℂfld ) Cn ( TopOpen ‘ ℂfld ) ) ) |
14 |
1 6 8 2 13
|
fprodcn |
⊢ ( 𝜑 → ( 𝑥 ∈ ℂ ↦ ∏ 𝑘 ∈ 𝐴 ( 𝐵 − 𝑥 ) ) ∈ ( ( TopOpen ‘ ℂfld ) Cn ( TopOpen ‘ ℂfld ) ) ) |
15 |
5 14
|
eqeltrd |
⊢ ( 𝜑 → 𝐹 ∈ ( ( TopOpen ‘ ℂfld ) Cn ( TopOpen ‘ ℂfld ) ) ) |
16 |
11
|
a1i |
⊢ ( 𝜑 → ( ℂ –cn→ ℂ ) = ( ( TopOpen ‘ ℂfld ) Cn ( TopOpen ‘ ℂfld ) ) ) |
17 |
16
|
eqcomd |
⊢ ( 𝜑 → ( ( TopOpen ‘ ℂfld ) Cn ( TopOpen ‘ ℂfld ) ) = ( ℂ –cn→ ℂ ) ) |
18 |
15 17
|
eleqtrd |
⊢ ( 𝜑 → 𝐹 ∈ ( ℂ –cn→ ℂ ) ) |