Step |
Hyp |
Ref |
Expression |
1 |
|
rmulccn.1 |
⊢ 𝐽 = ( topGen ‘ ran (,) ) |
2 |
|
rmulccn.2 |
⊢ ( 𝜑 → 𝐶 ∈ ℝ ) |
3 |
|
eqid |
⊢ ( TopOpen ‘ ℂfld ) = ( TopOpen ‘ ℂfld ) |
4 |
3
|
cnfldtopon |
⊢ ( TopOpen ‘ ℂfld ) ∈ ( TopOn ‘ ℂ ) |
5 |
4
|
a1i |
⊢ ( 𝜑 → ( TopOpen ‘ ℂfld ) ∈ ( TopOn ‘ ℂ ) ) |
6 |
5
|
cnmptid |
⊢ ( 𝜑 → ( 𝑥 ∈ ℂ ↦ 𝑥 ) ∈ ( ( TopOpen ‘ ℂfld ) Cn ( TopOpen ‘ ℂfld ) ) ) |
7 |
2
|
recnd |
⊢ ( 𝜑 → 𝐶 ∈ ℂ ) |
8 |
5 5 7
|
cnmptc |
⊢ ( 𝜑 → ( 𝑥 ∈ ℂ ↦ 𝐶 ) ∈ ( ( TopOpen ‘ ℂfld ) Cn ( TopOpen ‘ ℂfld ) ) ) |
9 |
3
|
mpomulcn |
⊢ ( 𝑦 ∈ ℂ , 𝑧 ∈ ℂ ↦ ( 𝑦 · 𝑧 ) ) ∈ ( ( ( TopOpen ‘ ℂfld ) ×t ( TopOpen ‘ ℂfld ) ) Cn ( TopOpen ‘ ℂfld ) ) |
10 |
9
|
a1i |
⊢ ( 𝜑 → ( 𝑦 ∈ ℂ , 𝑧 ∈ ℂ ↦ ( 𝑦 · 𝑧 ) ) ∈ ( ( ( TopOpen ‘ ℂfld ) ×t ( TopOpen ‘ ℂfld ) ) Cn ( TopOpen ‘ ℂfld ) ) ) |
11 |
|
oveq12 |
⊢ ( ( 𝑦 = 𝑥 ∧ 𝑧 = 𝐶 ) → ( 𝑦 · 𝑧 ) = ( 𝑥 · 𝐶 ) ) |
12 |
5 6 8 5 5 10 11
|
cnmpt12 |
⊢ ( 𝜑 → ( 𝑥 ∈ ℂ ↦ ( 𝑥 · 𝐶 ) ) ∈ ( ( TopOpen ‘ ℂfld ) Cn ( TopOpen ‘ ℂfld ) ) ) |
13 |
|
ax-resscn |
⊢ ℝ ⊆ ℂ |
14 |
|
unicntop |
⊢ ℂ = ∪ ( TopOpen ‘ ℂfld ) |
15 |
14
|
cnrest |
⊢ ( ( ( 𝑥 ∈ ℂ ↦ ( 𝑥 · 𝐶 ) ) ∈ ( ( TopOpen ‘ ℂfld ) Cn ( TopOpen ‘ ℂfld ) ) ∧ ℝ ⊆ ℂ ) → ( ( 𝑥 ∈ ℂ ↦ ( 𝑥 · 𝐶 ) ) ↾ ℝ ) ∈ ( ( ( TopOpen ‘ ℂfld ) ↾t ℝ ) Cn ( TopOpen ‘ ℂfld ) ) ) |
16 |
12 13 15
|
sylancl |
⊢ ( 𝜑 → ( ( 𝑥 ∈ ℂ ↦ ( 𝑥 · 𝐶 ) ) ↾ ℝ ) ∈ ( ( ( TopOpen ‘ ℂfld ) ↾t ℝ ) Cn ( TopOpen ‘ ℂfld ) ) ) |
17 |
|
simpr |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ℂ ) → 𝑥 ∈ ℂ ) |
18 |
7
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ℂ ) → 𝐶 ∈ ℂ ) |
19 |
17 18
|
mulcld |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ℂ ) → ( 𝑥 · 𝐶 ) ∈ ℂ ) |
20 |
19
|
ralrimiva |
⊢ ( 𝜑 → ∀ 𝑥 ∈ ℂ ( 𝑥 · 𝐶 ) ∈ ℂ ) |
21 |
|
eqid |
⊢ ( 𝑥 ∈ ℂ ↦ ( 𝑥 · 𝐶 ) ) = ( 𝑥 ∈ ℂ ↦ ( 𝑥 · 𝐶 ) ) |
22 |
21
|
fnmpt |
⊢ ( ∀ 𝑥 ∈ ℂ ( 𝑥 · 𝐶 ) ∈ ℂ → ( 𝑥 ∈ ℂ ↦ ( 𝑥 · 𝐶 ) ) Fn ℂ ) |
23 |
20 22
|
syl |
⊢ ( 𝜑 → ( 𝑥 ∈ ℂ ↦ ( 𝑥 · 𝐶 ) ) Fn ℂ ) |
24 |
13
|
a1i |
⊢ ( 𝜑 → ℝ ⊆ ℂ ) |
25 |
23 24
|
fnssresd |
⊢ ( 𝜑 → ( ( 𝑥 ∈ ℂ ↦ ( 𝑥 · 𝐶 ) ) ↾ ℝ ) Fn ℝ ) |
26 |
|
simpr |
⊢ ( ( 𝜑 ∧ 𝑤 ∈ ℝ ) → 𝑤 ∈ ℝ ) |
27 |
|
oveq1 |
⊢ ( 𝑥 = 𝑤 → ( 𝑥 · 𝐶 ) = ( 𝑤 · 𝐶 ) ) |
28 |
|
resmpt |
⊢ ( ℝ ⊆ ℂ → ( ( 𝑥 ∈ ℂ ↦ ( 𝑥 · 𝐶 ) ) ↾ ℝ ) = ( 𝑥 ∈ ℝ ↦ ( 𝑥 · 𝐶 ) ) ) |
29 |
13 28
|
ax-mp |
⊢ ( ( 𝑥 ∈ ℂ ↦ ( 𝑥 · 𝐶 ) ) ↾ ℝ ) = ( 𝑥 ∈ ℝ ↦ ( 𝑥 · 𝐶 ) ) |
30 |
|
ovex |
⊢ ( 𝑤 · 𝐶 ) ∈ V |
31 |
27 29 30
|
fvmpt |
⊢ ( 𝑤 ∈ ℝ → ( ( ( 𝑥 ∈ ℂ ↦ ( 𝑥 · 𝐶 ) ) ↾ ℝ ) ‘ 𝑤 ) = ( 𝑤 · 𝐶 ) ) |
32 |
26 31
|
syl |
⊢ ( ( 𝜑 ∧ 𝑤 ∈ ℝ ) → ( ( ( 𝑥 ∈ ℂ ↦ ( 𝑥 · 𝐶 ) ) ↾ ℝ ) ‘ 𝑤 ) = ( 𝑤 · 𝐶 ) ) |
33 |
2
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑤 ∈ ℝ ) → 𝐶 ∈ ℝ ) |
34 |
26 33
|
remulcld |
⊢ ( ( 𝜑 ∧ 𝑤 ∈ ℝ ) → ( 𝑤 · 𝐶 ) ∈ ℝ ) |
35 |
32 34
|
eqeltrd |
⊢ ( ( 𝜑 ∧ 𝑤 ∈ ℝ ) → ( ( ( 𝑥 ∈ ℂ ↦ ( 𝑥 · 𝐶 ) ) ↾ ℝ ) ‘ 𝑤 ) ∈ ℝ ) |
36 |
35
|
ralrimiva |
⊢ ( 𝜑 → ∀ 𝑤 ∈ ℝ ( ( ( 𝑥 ∈ ℂ ↦ ( 𝑥 · 𝐶 ) ) ↾ ℝ ) ‘ 𝑤 ) ∈ ℝ ) |
37 |
|
fnfvrnss |
⊢ ( ( ( ( 𝑥 ∈ ℂ ↦ ( 𝑥 · 𝐶 ) ) ↾ ℝ ) Fn ℝ ∧ ∀ 𝑤 ∈ ℝ ( ( ( 𝑥 ∈ ℂ ↦ ( 𝑥 · 𝐶 ) ) ↾ ℝ ) ‘ 𝑤 ) ∈ ℝ ) → ran ( ( 𝑥 ∈ ℂ ↦ ( 𝑥 · 𝐶 ) ) ↾ ℝ ) ⊆ ℝ ) |
38 |
25 36 37
|
syl2anc |
⊢ ( 𝜑 → ran ( ( 𝑥 ∈ ℂ ↦ ( 𝑥 · 𝐶 ) ) ↾ ℝ ) ⊆ ℝ ) |
39 |
|
cnrest2 |
⊢ ( ( ( TopOpen ‘ ℂfld ) ∈ ( TopOn ‘ ℂ ) ∧ ran ( ( 𝑥 ∈ ℂ ↦ ( 𝑥 · 𝐶 ) ) ↾ ℝ ) ⊆ ℝ ∧ ℝ ⊆ ℂ ) → ( ( ( 𝑥 ∈ ℂ ↦ ( 𝑥 · 𝐶 ) ) ↾ ℝ ) ∈ ( ( ( TopOpen ‘ ℂfld ) ↾t ℝ ) Cn ( TopOpen ‘ ℂfld ) ) ↔ ( ( 𝑥 ∈ ℂ ↦ ( 𝑥 · 𝐶 ) ) ↾ ℝ ) ∈ ( ( ( TopOpen ‘ ℂfld ) ↾t ℝ ) Cn ( ( TopOpen ‘ ℂfld ) ↾t ℝ ) ) ) ) |
40 |
4 38 24 39
|
mp3an2i |
⊢ ( 𝜑 → ( ( ( 𝑥 ∈ ℂ ↦ ( 𝑥 · 𝐶 ) ) ↾ ℝ ) ∈ ( ( ( TopOpen ‘ ℂfld ) ↾t ℝ ) Cn ( TopOpen ‘ ℂfld ) ) ↔ ( ( 𝑥 ∈ ℂ ↦ ( 𝑥 · 𝐶 ) ) ↾ ℝ ) ∈ ( ( ( TopOpen ‘ ℂfld ) ↾t ℝ ) Cn ( ( TopOpen ‘ ℂfld ) ↾t ℝ ) ) ) ) |
41 |
16 40
|
mpbid |
⊢ ( 𝜑 → ( ( 𝑥 ∈ ℂ ↦ ( 𝑥 · 𝐶 ) ) ↾ ℝ ) ∈ ( ( ( TopOpen ‘ ℂfld ) ↾t ℝ ) Cn ( ( TopOpen ‘ ℂfld ) ↾t ℝ ) ) ) |
42 |
3
|
tgioo2 |
⊢ ( topGen ‘ ran (,) ) = ( ( TopOpen ‘ ℂfld ) ↾t ℝ ) |
43 |
1 42
|
eqtri |
⊢ 𝐽 = ( ( TopOpen ‘ ℂfld ) ↾t ℝ ) |
44 |
43 43
|
oveq12i |
⊢ ( 𝐽 Cn 𝐽 ) = ( ( ( TopOpen ‘ ℂfld ) ↾t ℝ ) Cn ( ( TopOpen ‘ ℂfld ) ↾t ℝ ) ) |
45 |
44
|
eqcomi |
⊢ ( ( ( TopOpen ‘ ℂfld ) ↾t ℝ ) Cn ( ( TopOpen ‘ ℂfld ) ↾t ℝ ) ) = ( 𝐽 Cn 𝐽 ) |
46 |
41 29 45
|
3eltr3g |
⊢ ( 𝜑 → ( 𝑥 ∈ ℝ ↦ ( 𝑥 · 𝐶 ) ) ∈ ( 𝐽 Cn 𝐽 ) ) |