Step | Hyp | Ref
| Expression |
1 | | rfcnpre3.3 |
. . . . . . . 8
⊢ 𝐾 = (topGen‘ran
(,)) |
2 | | rfcnpre3.4 |
. . . . . . . 8
⊢ 𝑇 = ∪
𝐽 |
3 | | eqid 2737 |
. . . . . . . 8
⊢ (𝐽 Cn 𝐾) = (𝐽 Cn 𝐾) |
4 | | rfcnpre3.8 |
. . . . . . . 8
⊢ (𝜑 → 𝐹 ∈ (𝐽 Cn 𝐾)) |
5 | 1, 2, 3, 4 | fcnre 42241 |
. . . . . . 7
⊢ (𝜑 → 𝐹:𝑇⟶ℝ) |
6 | | ffn 6545 |
. . . . . . 7
⊢ (𝐹:𝑇⟶ℝ → 𝐹 Fn 𝑇) |
7 | | elpreima 6878 |
. . . . . . 7
⊢ (𝐹 Fn 𝑇 → (𝑠 ∈ (◡𝐹 “ (𝐵[,)+∞)) ↔ (𝑠 ∈ 𝑇 ∧ (𝐹‘𝑠) ∈ (𝐵[,)+∞)))) |
8 | 5, 6, 7 | 3syl 18 |
. . . . . 6
⊢ (𝜑 → (𝑠 ∈ (◡𝐹 “ (𝐵[,)+∞)) ↔ (𝑠 ∈ 𝑇 ∧ (𝐹‘𝑠) ∈ (𝐵[,)+∞)))) |
9 | | rfcnpre3.6 |
. . . . . . . . . . 11
⊢ (𝜑 → 𝐵 ∈ ℝ) |
10 | 9 | rexrd 10883 |
. . . . . . . . . 10
⊢ (𝜑 → 𝐵 ∈
ℝ*) |
11 | 10 | adantr 484 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑠 ∈ 𝑇) → 𝐵 ∈
ℝ*) |
12 | | pnfxr 10887 |
. . . . . . . . 9
⊢ +∞
∈ ℝ* |
13 | | elico1 12978 |
. . . . . . . . 9
⊢ ((𝐵 ∈ ℝ*
∧ +∞ ∈ ℝ*) → ((𝐹‘𝑠) ∈ (𝐵[,)+∞) ↔ ((𝐹‘𝑠) ∈ ℝ* ∧ 𝐵 ≤ (𝐹‘𝑠) ∧ (𝐹‘𝑠) < +∞))) |
14 | 11, 12, 13 | sylancl 589 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑠 ∈ 𝑇) → ((𝐹‘𝑠) ∈ (𝐵[,)+∞) ↔ ((𝐹‘𝑠) ∈ ℝ* ∧ 𝐵 ≤ (𝐹‘𝑠) ∧ (𝐹‘𝑠) < +∞))) |
15 | | simpr2 1197 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑠 ∈ 𝑇) ∧ ((𝐹‘𝑠) ∈ ℝ* ∧ 𝐵 ≤ (𝐹‘𝑠) ∧ (𝐹‘𝑠) < +∞)) → 𝐵 ≤ (𝐹‘𝑠)) |
16 | 5 | ffvelrnda 6904 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑠 ∈ 𝑇) → (𝐹‘𝑠) ∈ ℝ) |
17 | 16 | rexrd 10883 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑠 ∈ 𝑇) → (𝐹‘𝑠) ∈
ℝ*) |
18 | 17 | adantr 484 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑠 ∈ 𝑇) ∧ 𝐵 ≤ (𝐹‘𝑠)) → (𝐹‘𝑠) ∈
ℝ*) |
19 | | simpr 488 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑠 ∈ 𝑇) ∧ 𝐵 ≤ (𝐹‘𝑠)) → 𝐵 ≤ (𝐹‘𝑠)) |
20 | 16 | adantr 484 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑠 ∈ 𝑇) ∧ 𝐵 ≤ (𝐹‘𝑠)) → (𝐹‘𝑠) ∈ ℝ) |
21 | | ltpnf 12712 |
. . . . . . . . . . 11
⊢ ((𝐹‘𝑠) ∈ ℝ → (𝐹‘𝑠) < +∞) |
22 | 20, 21 | syl 17 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑠 ∈ 𝑇) ∧ 𝐵 ≤ (𝐹‘𝑠)) → (𝐹‘𝑠) < +∞) |
23 | 18, 19, 22 | 3jca 1130 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑠 ∈ 𝑇) ∧ 𝐵 ≤ (𝐹‘𝑠)) → ((𝐹‘𝑠) ∈ ℝ* ∧ 𝐵 ≤ (𝐹‘𝑠) ∧ (𝐹‘𝑠) < +∞)) |
24 | 15, 23 | impbida 801 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑠 ∈ 𝑇) → (((𝐹‘𝑠) ∈ ℝ* ∧ 𝐵 ≤ (𝐹‘𝑠) ∧ (𝐹‘𝑠) < +∞) ↔ 𝐵 ≤ (𝐹‘𝑠))) |
25 | 14, 24 | bitrd 282 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑠 ∈ 𝑇) → ((𝐹‘𝑠) ∈ (𝐵[,)+∞) ↔ 𝐵 ≤ (𝐹‘𝑠))) |
26 | 25 | pm5.32da 582 |
. . . . . 6
⊢ (𝜑 → ((𝑠 ∈ 𝑇 ∧ (𝐹‘𝑠) ∈ (𝐵[,)+∞)) ↔ (𝑠 ∈ 𝑇 ∧ 𝐵 ≤ (𝐹‘𝑠)))) |
27 | 8, 26 | bitrd 282 |
. . . . 5
⊢ (𝜑 → (𝑠 ∈ (◡𝐹 “ (𝐵[,)+∞)) ↔ (𝑠 ∈ 𝑇 ∧ 𝐵 ≤ (𝐹‘𝑠)))) |
28 | | nfcv 2904 |
. . . . . 6
⊢
Ⅎ𝑡𝑠 |
29 | | nfcv 2904 |
. . . . . 6
⊢
Ⅎ𝑡𝑇 |
30 | | nfcv 2904 |
. . . . . . 7
⊢
Ⅎ𝑡𝐵 |
31 | | nfcv 2904 |
. . . . . . 7
⊢
Ⅎ𝑡
≤ |
32 | | rfcnpre3.2 |
. . . . . . . 8
⊢
Ⅎ𝑡𝐹 |
33 | 32, 28 | nffv 6727 |
. . . . . . 7
⊢
Ⅎ𝑡(𝐹‘𝑠) |
34 | 30, 31, 33 | nfbr 5100 |
. . . . . 6
⊢
Ⅎ𝑡 𝐵 ≤ (𝐹‘𝑠) |
35 | | fveq2 6717 |
. . . . . . 7
⊢ (𝑡 = 𝑠 → (𝐹‘𝑡) = (𝐹‘𝑠)) |
36 | 35 | breq2d 5065 |
. . . . . 6
⊢ (𝑡 = 𝑠 → (𝐵 ≤ (𝐹‘𝑡) ↔ 𝐵 ≤ (𝐹‘𝑠))) |
37 | 28, 29, 34, 36 | elrabf 3598 |
. . . . 5
⊢ (𝑠 ∈ {𝑡 ∈ 𝑇 ∣ 𝐵 ≤ (𝐹‘𝑡)} ↔ (𝑠 ∈ 𝑇 ∧ 𝐵 ≤ (𝐹‘𝑠))) |
38 | 27, 37 | bitr4di 292 |
. . . 4
⊢ (𝜑 → (𝑠 ∈ (◡𝐹 “ (𝐵[,)+∞)) ↔ 𝑠 ∈ {𝑡 ∈ 𝑇 ∣ 𝐵 ≤ (𝐹‘𝑡)})) |
39 | 38 | eqrdv 2735 |
. . 3
⊢ (𝜑 → (◡𝐹 “ (𝐵[,)+∞)) = {𝑡 ∈ 𝑇 ∣ 𝐵 ≤ (𝐹‘𝑡)}) |
40 | | rfcnpre3.5 |
. . 3
⊢ 𝐴 = {𝑡 ∈ 𝑇 ∣ 𝐵 ≤ (𝐹‘𝑡)} |
41 | 39, 40 | eqtr4di 2796 |
. 2
⊢ (𝜑 → (◡𝐹 “ (𝐵[,)+∞)) = 𝐴) |
42 | | icopnfcld 23665 |
. . . . 5
⊢ (𝐵 ∈ ℝ → (𝐵[,)+∞) ∈
(Clsd‘(topGen‘ran (,)))) |
43 | 9, 42 | syl 17 |
. . . 4
⊢ (𝜑 → (𝐵[,)+∞) ∈
(Clsd‘(topGen‘ran (,)))) |
44 | 1 | fveq2i 6720 |
. . . 4
⊢
(Clsd‘𝐾) =
(Clsd‘(topGen‘ran (,))) |
45 | 43, 44 | eleqtrrdi 2849 |
. . 3
⊢ (𝜑 → (𝐵[,)+∞) ∈ (Clsd‘𝐾)) |
46 | | cnclima 22165 |
. . 3
⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ (𝐵[,)+∞) ∈ (Clsd‘𝐾)) → (◡𝐹 “ (𝐵[,)+∞)) ∈ (Clsd‘𝐽)) |
47 | 4, 45, 46 | syl2anc 587 |
. 2
⊢ (𝜑 → (◡𝐹 “ (𝐵[,)+∞)) ∈ (Clsd‘𝐽)) |
48 | 41, 47 | eqeltrrd 2839 |
1
⊢ (𝜑 → 𝐴 ∈ (Clsd‘𝐽)) |