Step | Hyp | Ref
| Expression |
1 | | prodeq1 15471 |
. . . 4
⊢ (𝑦 = ∅ → ∏𝑘 ∈ 𝑦 𝐵 = ∏𝑘 ∈ ∅ 𝐵) |
2 | 1 | mpteq2dv 5151 |
. . 3
⊢ (𝑦 = ∅ → (𝑥 ∈ 𝑋 ↦ ∏𝑘 ∈ 𝑦 𝐵) = (𝑥 ∈ 𝑋 ↦ ∏𝑘 ∈ ∅ 𝐵)) |
3 | 2 | eleq1d 2822 |
. 2
⊢ (𝑦 = ∅ → ((𝑥 ∈ 𝑋 ↦ ∏𝑘 ∈ 𝑦 𝐵) ∈ (𝐽 Cn 𝐾) ↔ (𝑥 ∈ 𝑋 ↦ ∏𝑘 ∈ ∅ 𝐵) ∈ (𝐽 Cn 𝐾))) |
4 | | prodeq1 15471 |
. . . 4
⊢ (𝑦 = 𝑧 → ∏𝑘 ∈ 𝑦 𝐵 = ∏𝑘 ∈ 𝑧 𝐵) |
5 | 4 | mpteq2dv 5151 |
. . 3
⊢ (𝑦 = 𝑧 → (𝑥 ∈ 𝑋 ↦ ∏𝑘 ∈ 𝑦 𝐵) = (𝑥 ∈ 𝑋 ↦ ∏𝑘 ∈ 𝑧 𝐵)) |
6 | 5 | eleq1d 2822 |
. 2
⊢ (𝑦 = 𝑧 → ((𝑥 ∈ 𝑋 ↦ ∏𝑘 ∈ 𝑦 𝐵) ∈ (𝐽 Cn 𝐾) ↔ (𝑥 ∈ 𝑋 ↦ ∏𝑘 ∈ 𝑧 𝐵) ∈ (𝐽 Cn 𝐾))) |
7 | | prodeq1 15471 |
. . . 4
⊢ (𝑦 = (𝑧 ∪ {𝑤}) → ∏𝑘 ∈ 𝑦 𝐵 = ∏𝑘 ∈ (𝑧 ∪ {𝑤})𝐵) |
8 | 7 | mpteq2dv 5151 |
. . 3
⊢ (𝑦 = (𝑧 ∪ {𝑤}) → (𝑥 ∈ 𝑋 ↦ ∏𝑘 ∈ 𝑦 𝐵) = (𝑥 ∈ 𝑋 ↦ ∏𝑘 ∈ (𝑧 ∪ {𝑤})𝐵)) |
9 | 8 | eleq1d 2822 |
. 2
⊢ (𝑦 = (𝑧 ∪ {𝑤}) → ((𝑥 ∈ 𝑋 ↦ ∏𝑘 ∈ 𝑦 𝐵) ∈ (𝐽 Cn 𝐾) ↔ (𝑥 ∈ 𝑋 ↦ ∏𝑘 ∈ (𝑧 ∪ {𝑤})𝐵) ∈ (𝐽 Cn 𝐾))) |
10 | | prodeq1 15471 |
. . . 4
⊢ (𝑦 = 𝐴 → ∏𝑘 ∈ 𝑦 𝐵 = ∏𝑘 ∈ 𝐴 𝐵) |
11 | 10 | mpteq2dv 5151 |
. . 3
⊢ (𝑦 = 𝐴 → (𝑥 ∈ 𝑋 ↦ ∏𝑘 ∈ 𝑦 𝐵) = (𝑥 ∈ 𝑋 ↦ ∏𝑘 ∈ 𝐴 𝐵)) |
12 | 11 | eleq1d 2822 |
. 2
⊢ (𝑦 = 𝐴 → ((𝑥 ∈ 𝑋 ↦ ∏𝑘 ∈ 𝑦 𝐵) ∈ (𝐽 Cn 𝐾) ↔ (𝑥 ∈ 𝑋 ↦ ∏𝑘 ∈ 𝐴 𝐵) ∈ (𝐽 Cn 𝐾))) |
13 | | prod0 15505 |
. . . . . 6
⊢
∏𝑘 ∈
∅ 𝐵 =
1 |
14 | 13 | mpteq2i 5147 |
. . . . 5
⊢ (𝑥 ∈ 𝑋 ↦ ∏𝑘 ∈ ∅ 𝐵) = (𝑥 ∈ 𝑋 ↦ 1) |
15 | | eqidd 2738 |
. . . . . 6
⊢ (𝑥 = 𝑦 → 1 = 1) |
16 | 15 | cbvmptv 5158 |
. . . . 5
⊢ (𝑥 ∈ 𝑋 ↦ 1) = (𝑦 ∈ 𝑋 ↦ 1) |
17 | 14, 16 | eqtri 2765 |
. . . 4
⊢ (𝑥 ∈ 𝑋 ↦ ∏𝑘 ∈ ∅ 𝐵) = (𝑦 ∈ 𝑋 ↦ 1) |
18 | 17 | a1i 11 |
. . 3
⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ ∏𝑘 ∈ ∅ 𝐵) = (𝑦 ∈ 𝑋 ↦ 1)) |
19 | | fprodcn.j |
. . . 4
⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋)) |
20 | | fprodcn.k |
. . . . . 6
⊢ 𝐾 =
(TopOpen‘ℂfld) |
21 | 20 | cnfldtopon 23680 |
. . . . 5
⊢ 𝐾 ∈
(TopOn‘ℂ) |
22 | 21 | a1i 11 |
. . . 4
⊢ (𝜑 → 𝐾 ∈
(TopOn‘ℂ)) |
23 | | 1cnd 10828 |
. . . 4
⊢ (𝜑 → 1 ∈
ℂ) |
24 | 19, 22, 23 | cnmptc 22559 |
. . 3
⊢ (𝜑 → (𝑦 ∈ 𝑋 ↦ 1) ∈ (𝐽 Cn 𝐾)) |
25 | 18, 24 | eqeltrd 2838 |
. 2
⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ ∏𝑘 ∈ ∅ 𝐵) ∈ (𝐽 Cn 𝐾)) |
26 | | nfcv 2904 |
. . . . . 6
⊢
Ⅎ𝑦∏𝑘 ∈ (𝑧 ∪ {𝑤})𝐵 |
27 | | nfcv 2904 |
. . . . . . 7
⊢
Ⅎ𝑥(𝑧 ∪ {𝑤}) |
28 | | nfcsb1v 3836 |
. . . . . . 7
⊢
Ⅎ𝑥⦋𝑦 / 𝑥⦌𝐵 |
29 | 27, 28 | nfcprod 15473 |
. . . . . 6
⊢
Ⅎ𝑥∏𝑘 ∈ (𝑧 ∪ {𝑤})⦋𝑦 / 𝑥⦌𝐵 |
30 | | csbeq1a 3825 |
. . . . . . 7
⊢ (𝑥 = 𝑦 → 𝐵 = ⦋𝑦 / 𝑥⦌𝐵) |
31 | 30 | prodeq2ad 42808 |
. . . . . 6
⊢ (𝑥 = 𝑦 → ∏𝑘 ∈ (𝑧 ∪ {𝑤})𝐵 = ∏𝑘 ∈ (𝑧 ∪ {𝑤})⦋𝑦 / 𝑥⦌𝐵) |
32 | 26, 29, 31 | cbvmpt 5156 |
. . . . 5
⊢ (𝑥 ∈ 𝑋 ↦ ∏𝑘 ∈ (𝑧 ∪ {𝑤})𝐵) = (𝑦 ∈ 𝑋 ↦ ∏𝑘 ∈ (𝑧 ∪ {𝑤})⦋𝑦 / 𝑥⦌𝐵) |
33 | 32 | a1i 11 |
. . . 4
⊢ (((𝜑 ∧ (𝑧 ⊆ 𝐴 ∧ 𝑤 ∈ (𝐴 ∖ 𝑧))) ∧ (𝑥 ∈ 𝑋 ↦ ∏𝑘 ∈ 𝑧 𝐵) ∈ (𝐽 Cn 𝐾)) → (𝑥 ∈ 𝑋 ↦ ∏𝑘 ∈ (𝑧 ∪ {𝑤})𝐵) = (𝑦 ∈ 𝑋 ↦ ∏𝑘 ∈ (𝑧 ∪ {𝑤})⦋𝑦 / 𝑥⦌𝐵)) |
34 | | fprodcn.d |
. . . . . . 7
⊢
Ⅎ𝑘𝜑 |
35 | | nfv 1922 |
. . . . . . 7
⊢
Ⅎ𝑘(𝑧 ⊆ 𝐴 ∧ 𝑤 ∈ (𝐴 ∖ 𝑧)) |
36 | 34, 35 | nfan 1907 |
. . . . . 6
⊢
Ⅎ𝑘(𝜑 ∧ (𝑧 ⊆ 𝐴 ∧ 𝑤 ∈ (𝐴 ∖ 𝑧))) |
37 | | nfcv 2904 |
. . . . . . . 8
⊢
Ⅎ𝑘𝑋 |
38 | | nfcv 2904 |
. . . . . . . . 9
⊢
Ⅎ𝑘𝑧 |
39 | 38 | nfcprod1 15472 |
. . . . . . . 8
⊢
Ⅎ𝑘∏𝑘 ∈ 𝑧 𝐵 |
40 | 37, 39 | nfmpt 5152 |
. . . . . . 7
⊢
Ⅎ𝑘(𝑥 ∈ 𝑋 ↦ ∏𝑘 ∈ 𝑧 𝐵) |
41 | | nfcv 2904 |
. . . . . . 7
⊢
Ⅎ𝑘(𝐽 Cn 𝐾) |
42 | 40, 41 | nfel 2918 |
. . . . . 6
⊢
Ⅎ𝑘(𝑥 ∈ 𝑋 ↦ ∏𝑘 ∈ 𝑧 𝐵) ∈ (𝐽 Cn 𝐾) |
43 | 36, 42 | nfan 1907 |
. . . . 5
⊢
Ⅎ𝑘((𝜑 ∧ (𝑧 ⊆ 𝐴 ∧ 𝑤 ∈ (𝐴 ∖ 𝑧))) ∧ (𝑥 ∈ 𝑋 ↦ ∏𝑘 ∈ 𝑧 𝐵) ∈ (𝐽 Cn 𝐾)) |
44 | 19 | ad2antrr 726 |
. . . . 5
⊢ (((𝜑 ∧ (𝑧 ⊆ 𝐴 ∧ 𝑤 ∈ (𝐴 ∖ 𝑧))) ∧ (𝑥 ∈ 𝑋 ↦ ∏𝑘 ∈ 𝑧 𝐵) ∈ (𝐽 Cn 𝐾)) → 𝐽 ∈ (TopOn‘𝑋)) |
45 | | fprodcn.a |
. . . . . 6
⊢ (𝜑 → 𝐴 ∈ Fin) |
46 | 45 | ad2antrr 726 |
. . . . 5
⊢ (((𝜑 ∧ (𝑧 ⊆ 𝐴 ∧ 𝑤 ∈ (𝐴 ∖ 𝑧))) ∧ (𝑥 ∈ 𝑋 ↦ ∏𝑘 ∈ 𝑧 𝐵) ∈ (𝐽 Cn 𝐾)) → 𝐴 ∈ Fin) |
47 | | nfcv 2904 |
. . . . . . . . . 10
⊢
Ⅎ𝑦𝐵 |
48 | 47, 28, 30 | cbvmpt 5156 |
. . . . . . . . 9
⊢ (𝑥 ∈ 𝑋 ↦ 𝐵) = (𝑦 ∈ 𝑋 ↦ ⦋𝑦 / 𝑥⦌𝐵) |
49 | 48 | eqcomi 2746 |
. . . . . . . 8
⊢ (𝑦 ∈ 𝑋 ↦ ⦋𝑦 / 𝑥⦌𝐵) = (𝑥 ∈ 𝑋 ↦ 𝐵) |
50 | 49 | a1i 11 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → (𝑦 ∈ 𝑋 ↦ ⦋𝑦 / 𝑥⦌𝐵) = (𝑥 ∈ 𝑋 ↦ 𝐵)) |
51 | | fprodcn.b |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → (𝑥 ∈ 𝑋 ↦ 𝐵) ∈ (𝐽 Cn 𝐾)) |
52 | 50, 51 | eqeltrd 2838 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → (𝑦 ∈ 𝑋 ↦ ⦋𝑦 / 𝑥⦌𝐵) ∈ (𝐽 Cn 𝐾)) |
53 | 52 | ad4ant14 752 |
. . . . 5
⊢ ((((𝜑 ∧ (𝑧 ⊆ 𝐴 ∧ 𝑤 ∈ (𝐴 ∖ 𝑧))) ∧ (𝑥 ∈ 𝑋 ↦ ∏𝑘 ∈ 𝑧 𝐵) ∈ (𝐽 Cn 𝐾)) ∧ 𝑘 ∈ 𝐴) → (𝑦 ∈ 𝑋 ↦ ⦋𝑦 / 𝑥⦌𝐵) ∈ (𝐽 Cn 𝐾)) |
54 | | simplrl 777 |
. . . . 5
⊢ (((𝜑 ∧ (𝑧 ⊆ 𝐴 ∧ 𝑤 ∈ (𝐴 ∖ 𝑧))) ∧ (𝑥 ∈ 𝑋 ↦ ∏𝑘 ∈ 𝑧 𝐵) ∈ (𝐽 Cn 𝐾)) → 𝑧 ⊆ 𝐴) |
55 | | simplrr 778 |
. . . . 5
⊢ (((𝜑 ∧ (𝑧 ⊆ 𝐴 ∧ 𝑤 ∈ (𝐴 ∖ 𝑧))) ∧ (𝑥 ∈ 𝑋 ↦ ∏𝑘 ∈ 𝑧 𝐵) ∈ (𝐽 Cn 𝐾)) → 𝑤 ∈ (𝐴 ∖ 𝑧)) |
56 | | nfcv 2904 |
. . . . . . . . 9
⊢
Ⅎ𝑦∏𝑘 ∈ 𝑧 𝐵 |
57 | | nfcv 2904 |
. . . . . . . . . 10
⊢
Ⅎ𝑥𝑧 |
58 | 57, 28 | nfcprod 15473 |
. . . . . . . . 9
⊢
Ⅎ𝑥∏𝑘 ∈ 𝑧 ⦋𝑦 / 𝑥⦌𝐵 |
59 | 30 | prodeq2sdv 15486 |
. . . . . . . . 9
⊢ (𝑥 = 𝑦 → ∏𝑘 ∈ 𝑧 𝐵 = ∏𝑘 ∈ 𝑧 ⦋𝑦 / 𝑥⦌𝐵) |
60 | 56, 58, 59 | cbvmpt 5156 |
. . . . . . . 8
⊢ (𝑥 ∈ 𝑋 ↦ ∏𝑘 ∈ 𝑧 𝐵) = (𝑦 ∈ 𝑋 ↦ ∏𝑘 ∈ 𝑧 ⦋𝑦 / 𝑥⦌𝐵) |
61 | 60 | eleq1i 2828 |
. . . . . . 7
⊢ ((𝑥 ∈ 𝑋 ↦ ∏𝑘 ∈ 𝑧 𝐵) ∈ (𝐽 Cn 𝐾) ↔ (𝑦 ∈ 𝑋 ↦ ∏𝑘 ∈ 𝑧 ⦋𝑦 / 𝑥⦌𝐵) ∈ (𝐽 Cn 𝐾)) |
62 | 61 | biimpi 219 |
. . . . . 6
⊢ ((𝑥 ∈ 𝑋 ↦ ∏𝑘 ∈ 𝑧 𝐵) ∈ (𝐽 Cn 𝐾) → (𝑦 ∈ 𝑋 ↦ ∏𝑘 ∈ 𝑧 ⦋𝑦 / 𝑥⦌𝐵) ∈ (𝐽 Cn 𝐾)) |
63 | 62 | adantl 485 |
. . . . 5
⊢ (((𝜑 ∧ (𝑧 ⊆ 𝐴 ∧ 𝑤 ∈ (𝐴 ∖ 𝑧))) ∧ (𝑥 ∈ 𝑋 ↦ ∏𝑘 ∈ 𝑧 𝐵) ∈ (𝐽 Cn 𝐾)) → (𝑦 ∈ 𝑋 ↦ ∏𝑘 ∈ 𝑧 ⦋𝑦 / 𝑥⦌𝐵) ∈ (𝐽 Cn 𝐾)) |
64 | 43, 20, 44, 46, 53, 54, 55, 63 | fprodcnlem 42815 |
. . . 4
⊢ (((𝜑 ∧ (𝑧 ⊆ 𝐴 ∧ 𝑤 ∈ (𝐴 ∖ 𝑧))) ∧ (𝑥 ∈ 𝑋 ↦ ∏𝑘 ∈ 𝑧 𝐵) ∈ (𝐽 Cn 𝐾)) → (𝑦 ∈ 𝑋 ↦ ∏𝑘 ∈ (𝑧 ∪ {𝑤})⦋𝑦 / 𝑥⦌𝐵) ∈ (𝐽 Cn 𝐾)) |
65 | 33, 64 | eqeltrd 2838 |
. . 3
⊢ (((𝜑 ∧ (𝑧 ⊆ 𝐴 ∧ 𝑤 ∈ (𝐴 ∖ 𝑧))) ∧ (𝑥 ∈ 𝑋 ↦ ∏𝑘 ∈ 𝑧 𝐵) ∈ (𝐽 Cn 𝐾)) → (𝑥 ∈ 𝑋 ↦ ∏𝑘 ∈ (𝑧 ∪ {𝑤})𝐵) ∈ (𝐽 Cn 𝐾)) |
66 | 65 | ex 416 |
. 2
⊢ ((𝜑 ∧ (𝑧 ⊆ 𝐴 ∧ 𝑤 ∈ (𝐴 ∖ 𝑧))) → ((𝑥 ∈ 𝑋 ↦ ∏𝑘 ∈ 𝑧 𝐵) ∈ (𝐽 Cn 𝐾) → (𝑥 ∈ 𝑋 ↦ ∏𝑘 ∈ (𝑧 ∪ {𝑤})𝐵) ∈ (𝐽 Cn 𝐾))) |
67 | 3, 6, 9, 12, 25, 66, 45 | findcard2d 8844 |
1
⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ ∏𝑘 ∈ 𝐴 𝐵) ∈ (𝐽 Cn 𝐾)) |