| Step | Hyp | Ref
| Expression |
| 1 | | eqid 2737 |
. . . 4
⊢
(TopOpen‘ℂfld) =
(TopOpen‘ℂfld) |
| 2 | | refsumcn.3 |
. . . 4
⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋)) |
| 3 | | refsumcn.4 |
. . . 4
⊢ (𝜑 → 𝐴 ∈ Fin) |
| 4 | | refsumcn.5 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → (𝑥 ∈ 𝑋 ↦ 𝐵) ∈ (𝐽 Cn 𝐾)) |
| 5 | | refsumcn.2 |
. . . . . . . 8
⊢ 𝐾 = (topGen‘ran
(,)) |
| 6 | | tgioo4 24826 |
. . . . . . . 8
⊢
(topGen‘ran (,)) = ((TopOpen‘ℂfld)
↾t ℝ) |
| 7 | 5, 6 | eqtri 2765 |
. . . . . . 7
⊢ 𝐾 =
((TopOpen‘ℂfld) ↾t
ℝ) |
| 8 | 7 | oveq2i 7442 |
. . . . . 6
⊢ (𝐽 Cn 𝐾) = (𝐽 Cn ((TopOpen‘ℂfld)
↾t ℝ)) |
| 9 | 4, 8 | eleqtrdi 2851 |
. . . . 5
⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → (𝑥 ∈ 𝑋 ↦ 𝐵) ∈ (𝐽 Cn ((TopOpen‘ℂfld)
↾t ℝ))) |
| 10 | 1 | cnfldtopon 24803 |
. . . . . . 7
⊢
(TopOpen‘ℂfld) ∈
(TopOn‘ℂ) |
| 11 | 10 | a1i 11 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) →
(TopOpen‘ℂfld) ∈
(TopOn‘ℂ)) |
| 12 | 2 | adantr 480 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐽 ∈ (TopOn‘𝑋)) |
| 13 | | retopon 24784 |
. . . . . . . . . 10
⊢
(topGen‘ran (,)) ∈ (TopOn‘ℝ) |
| 14 | 5, 13 | eqeltri 2837 |
. . . . . . . . 9
⊢ 𝐾 ∈
(TopOn‘ℝ) |
| 15 | 14 | a1i 11 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐾 ∈
(TopOn‘ℝ)) |
| 16 | | cnf2 23257 |
. . . . . . . 8
⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘ℝ) ∧ (𝑥 ∈ 𝑋 ↦ 𝐵) ∈ (𝐽 Cn 𝐾)) → (𝑥 ∈ 𝑋 ↦ 𝐵):𝑋⟶ℝ) |
| 17 | 12, 15, 4, 16 | syl3anc 1373 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → (𝑥 ∈ 𝑋 ↦ 𝐵):𝑋⟶ℝ) |
| 18 | 17 | frnd 6744 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → ran (𝑥 ∈ 𝑋 ↦ 𝐵) ⊆ ℝ) |
| 19 | | ax-resscn 11212 |
. . . . . . 7
⊢ ℝ
⊆ ℂ |
| 20 | 19 | a1i 11 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → ℝ ⊆
ℂ) |
| 21 | | cnrest2 23294 |
. . . . . 6
⊢
(((TopOpen‘ℂfld) ∈ (TopOn‘ℂ)
∧ ran (𝑥 ∈ 𝑋 ↦ 𝐵) ⊆ ℝ ∧ ℝ ⊆
ℂ) → ((𝑥 ∈
𝑋 ↦ 𝐵) ∈ (𝐽 Cn (TopOpen‘ℂfld))
↔ (𝑥 ∈ 𝑋 ↦ 𝐵) ∈ (𝐽 Cn ((TopOpen‘ℂfld)
↾t ℝ)))) |
| 22 | 11, 18, 20, 21 | syl3anc 1373 |
. . . . 5
⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → ((𝑥 ∈ 𝑋 ↦ 𝐵) ∈ (𝐽 Cn (TopOpen‘ℂfld))
↔ (𝑥 ∈ 𝑋 ↦ 𝐵) ∈ (𝐽 Cn ((TopOpen‘ℂfld)
↾t ℝ)))) |
| 23 | 9, 22 | mpbird 257 |
. . . 4
⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → (𝑥 ∈ 𝑋 ↦ 𝐵) ∈ (𝐽 Cn
(TopOpen‘ℂfld))) |
| 24 | 1, 2, 3, 23 | fsumcnf 45026 |
. . 3
⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ 𝐴 𝐵) ∈ (𝐽 Cn
(TopOpen‘ℂfld))) |
| 25 | 10 | a1i 11 |
. . . 4
⊢ (𝜑 →
(TopOpen‘ℂfld) ∈
(TopOn‘ℂ)) |
| 26 | | refsumcn.1 |
. . . . . . . . . . 11
⊢
Ⅎ𝑥𝜑 |
| 27 | 3 | adantr 480 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐴 ∈ Fin) |
| 28 | | simpll 767 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ 𝐴) → 𝜑) |
| 29 | | simpr 484 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ 𝐴) → 𝑘 ∈ 𝐴) |
| 30 | 28, 29 | jca 511 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ 𝐴) → (𝜑 ∧ 𝑘 ∈ 𝐴)) |
| 31 | | simplr 769 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ 𝐴) → 𝑥 ∈ 𝑋) |
| 32 | | eqid 2737 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑥 ∈ 𝑋 ↦ 𝐵) = (𝑥 ∈ 𝑋 ↦ 𝐵) |
| 33 | 32 | fmpt 7130 |
. . . . . . . . . . . . . . . 16
⊢
(∀𝑥 ∈
𝑋 𝐵 ∈ ℝ ↔ (𝑥 ∈ 𝑋 ↦ 𝐵):𝑋⟶ℝ) |
| 34 | 17, 33 | sylibr 234 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → ∀𝑥 ∈ 𝑋 𝐵 ∈ ℝ) |
| 35 | | rsp 3247 |
. . . . . . . . . . . . . . 15
⊢
(∀𝑥 ∈
𝑋 𝐵 ∈ ℝ → (𝑥 ∈ 𝑋 → 𝐵 ∈ ℝ)) |
| 36 | 34, 35 | syl 17 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → (𝑥 ∈ 𝑋 → 𝐵 ∈ ℝ)) |
| 37 | 30, 31, 36 | sylc 65 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℝ) |
| 38 | 27, 37 | fsumrecl 15770 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → Σ𝑘 ∈ 𝐴 𝐵 ∈ ℝ) |
| 39 | 38 | ex 412 |
. . . . . . . . . . 11
⊢ (𝜑 → (𝑥 ∈ 𝑋 → Σ𝑘 ∈ 𝐴 𝐵 ∈ ℝ)) |
| 40 | 26, 39 | ralrimi 3257 |
. . . . . . . . . 10
⊢ (𝜑 → ∀𝑥 ∈ 𝑋 Σ𝑘 ∈ 𝐴 𝐵 ∈ ℝ) |
| 41 | | eqid 2737 |
. . . . . . . . . . 11
⊢ (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ 𝐴 𝐵) = (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ 𝐴 𝐵) |
| 42 | 41 | fnmpt 6708 |
. . . . . . . . . 10
⊢
(∀𝑥 ∈
𝑋 Σ𝑘 ∈ 𝐴 𝐵 ∈ ℝ → (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ 𝐴 𝐵) Fn 𝑋) |
| 43 | 40, 42 | syl 17 |
. . . . . . . . 9
⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ 𝐴 𝐵) Fn 𝑋) |
| 44 | | nfcv 2905 |
. . . . . . . . . 10
⊢
Ⅎ𝑥𝑋 |
| 45 | | nfcv 2905 |
. . . . . . . . . 10
⊢
Ⅎ𝑥𝑦 |
| 46 | | nfmpt1 5250 |
. . . . . . . . . 10
⊢
Ⅎ𝑥(𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ 𝐴 𝐵) |
| 47 | 44, 45, 46 | fvelrnbf 45023 |
. . . . . . . . 9
⊢ ((𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ 𝐴 𝐵) Fn 𝑋 → (𝑦 ∈ ran (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ 𝐴 𝐵) ↔ ∃𝑥 ∈ 𝑋 ((𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ 𝐴 𝐵)‘𝑥) = 𝑦)) |
| 48 | 43, 47 | syl 17 |
. . . . . . . 8
⊢ (𝜑 → (𝑦 ∈ ran (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ 𝐴 𝐵) ↔ ∃𝑥 ∈ 𝑋 ((𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ 𝐴 𝐵)‘𝑥) = 𝑦)) |
| 49 | 48 | biimpa 476 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑦 ∈ ran (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ 𝐴 𝐵)) → ∃𝑥 ∈ 𝑋 ((𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ 𝐴 𝐵)‘𝑥) = 𝑦) |
| 50 | 46 | nfrn 5963 |
. . . . . . . . . 10
⊢
Ⅎ𝑥ran
(𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ 𝐴 𝐵) |
| 51 | 50 | nfcri 2897 |
. . . . . . . . 9
⊢
Ⅎ𝑥 𝑦 ∈ ran (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ 𝐴 𝐵) |
| 52 | 26, 51 | nfan 1899 |
. . . . . . . 8
⊢
Ⅎ𝑥(𝜑 ∧ 𝑦 ∈ ran (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ 𝐴 𝐵)) |
| 53 | | nfcv 2905 |
. . . . . . . . 9
⊢
Ⅎ𝑥ℝ |
| 54 | 53 | nfcri 2897 |
. . . . . . . 8
⊢
Ⅎ𝑥 𝑦 ∈ ℝ |
| 55 | | simpr 484 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝑥 ∈ 𝑋) |
| 56 | 55, 38 | jca 511 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (𝑥 ∈ 𝑋 ∧ Σ𝑘 ∈ 𝐴 𝐵 ∈ ℝ)) |
| 57 | 41 | fvmpt2 7027 |
. . . . . . . . . . . . . 14
⊢ ((𝑥 ∈ 𝑋 ∧ Σ𝑘 ∈ 𝐴 𝐵 ∈ ℝ) → ((𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ 𝐴 𝐵)‘𝑥) = Σ𝑘 ∈ 𝐴 𝐵) |
| 58 | 56, 57 | syl 17 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ((𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ 𝐴 𝐵)‘𝑥) = Σ𝑘 ∈ 𝐴 𝐵) |
| 59 | 58 | 3adant3 1133 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋 ∧ ((𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ 𝐴 𝐵)‘𝑥) = 𝑦) → ((𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ 𝐴 𝐵)‘𝑥) = Σ𝑘 ∈ 𝐴 𝐵) |
| 60 | | simp3 1139 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋 ∧ ((𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ 𝐴 𝐵)‘𝑥) = 𝑦) → ((𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ 𝐴 𝐵)‘𝑥) = 𝑦) |
| 61 | 59, 60 | eqtr3d 2779 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋 ∧ ((𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ 𝐴 𝐵)‘𝑥) = 𝑦) → Σ𝑘 ∈ 𝐴 𝐵 = 𝑦) |
| 62 | 38 | 3adant3 1133 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋 ∧ ((𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ 𝐴 𝐵)‘𝑥) = 𝑦) → Σ𝑘 ∈ 𝐴 𝐵 ∈ ℝ) |
| 63 | 61, 62 | eqeltrrd 2842 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋 ∧ ((𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ 𝐴 𝐵)‘𝑥) = 𝑦) → 𝑦 ∈ ℝ) |
| 64 | 63 | 3adant1r 1178 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑦 ∈ ran (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ 𝐴 𝐵)) ∧ 𝑥 ∈ 𝑋 ∧ ((𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ 𝐴 𝐵)‘𝑥) = 𝑦) → 𝑦 ∈ ℝ) |
| 65 | 64 | 3exp 1120 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑦 ∈ ran (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ 𝐴 𝐵)) → (𝑥 ∈ 𝑋 → (((𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ 𝐴 𝐵)‘𝑥) = 𝑦 → 𝑦 ∈ ℝ))) |
| 66 | 52, 54, 65 | rexlimd 3266 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑦 ∈ ran (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ 𝐴 𝐵)) → (∃𝑥 ∈ 𝑋 ((𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ 𝐴 𝐵)‘𝑥) = 𝑦 → 𝑦 ∈ ℝ)) |
| 67 | 49, 66 | mpd 15 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑦 ∈ ran (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ 𝐴 𝐵)) → 𝑦 ∈ ℝ) |
| 68 | 67 | ex 412 |
. . . . 5
⊢ (𝜑 → (𝑦 ∈ ran (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ 𝐴 𝐵) → 𝑦 ∈ ℝ)) |
| 69 | 68 | ssrdv 3989 |
. . . 4
⊢ (𝜑 → ran (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ 𝐴 𝐵) ⊆ ℝ) |
| 70 | 19 | a1i 11 |
. . . 4
⊢ (𝜑 → ℝ ⊆
ℂ) |
| 71 | | cnrest2 23294 |
. . . 4
⊢
(((TopOpen‘ℂfld) ∈ (TopOn‘ℂ)
∧ ran (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ 𝐴 𝐵) ⊆ ℝ ∧ ℝ ⊆
ℂ) → ((𝑥 ∈
𝑋 ↦ Σ𝑘 ∈ 𝐴 𝐵) ∈ (𝐽 Cn (TopOpen‘ℂfld))
↔ (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ 𝐴 𝐵) ∈ (𝐽 Cn ((TopOpen‘ℂfld)
↾t ℝ)))) |
| 72 | 25, 69, 70, 71 | syl3anc 1373 |
. . 3
⊢ (𝜑 → ((𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ 𝐴 𝐵) ∈ (𝐽 Cn (TopOpen‘ℂfld))
↔ (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ 𝐴 𝐵) ∈ (𝐽 Cn ((TopOpen‘ℂfld)
↾t ℝ)))) |
| 73 | 24, 72 | mpbid 232 |
. 2
⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ 𝐴 𝐵) ∈ (𝐽 Cn ((TopOpen‘ℂfld)
↾t ℝ))) |
| 74 | 73, 8 | eleqtrrdi 2852 |
1
⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ 𝐴 𝐵) ∈ (𝐽 Cn 𝐾)) |