| Step | Hyp | Ref | Expression | 
|---|
| 1 |  | prodeq1 15944 | . . . 4
⊢ (𝑦 = ∅ → ∏𝑘 ∈ 𝑦 𝐵 = ∏𝑘 ∈ ∅ 𝐵) | 
| 2 | 1 | mpteq2dv 5243 | . . 3
⊢ (𝑦 = ∅ → (𝑥 ∈ 𝑋 ↦ ∏𝑘 ∈ 𝑦 𝐵) = (𝑥 ∈ 𝑋 ↦ ∏𝑘 ∈ ∅ 𝐵)) | 
| 3 | 2 | eleq1d 2825 | . 2
⊢ (𝑦 = ∅ → ((𝑥 ∈ 𝑋 ↦ ∏𝑘 ∈ 𝑦 𝐵) ∈ (𝐽 Cn 𝐾) ↔ (𝑥 ∈ 𝑋 ↦ ∏𝑘 ∈ ∅ 𝐵) ∈ (𝐽 Cn 𝐾))) | 
| 4 |  | prodeq1 15944 | . . . 4
⊢ (𝑦 = 𝑧 → ∏𝑘 ∈ 𝑦 𝐵 = ∏𝑘 ∈ 𝑧 𝐵) | 
| 5 | 4 | mpteq2dv 5243 | . . 3
⊢ (𝑦 = 𝑧 → (𝑥 ∈ 𝑋 ↦ ∏𝑘 ∈ 𝑦 𝐵) = (𝑥 ∈ 𝑋 ↦ ∏𝑘 ∈ 𝑧 𝐵)) | 
| 6 | 5 | eleq1d 2825 | . 2
⊢ (𝑦 = 𝑧 → ((𝑥 ∈ 𝑋 ↦ ∏𝑘 ∈ 𝑦 𝐵) ∈ (𝐽 Cn 𝐾) ↔ (𝑥 ∈ 𝑋 ↦ ∏𝑘 ∈ 𝑧 𝐵) ∈ (𝐽 Cn 𝐾))) | 
| 7 |  | prodeq1 15944 | . . . 4
⊢ (𝑦 = (𝑧 ∪ {𝑤}) → ∏𝑘 ∈ 𝑦 𝐵 = ∏𝑘 ∈ (𝑧 ∪ {𝑤})𝐵) | 
| 8 | 7 | mpteq2dv 5243 | . . 3
⊢ (𝑦 = (𝑧 ∪ {𝑤}) → (𝑥 ∈ 𝑋 ↦ ∏𝑘 ∈ 𝑦 𝐵) = (𝑥 ∈ 𝑋 ↦ ∏𝑘 ∈ (𝑧 ∪ {𝑤})𝐵)) | 
| 9 | 8 | eleq1d 2825 | . 2
⊢ (𝑦 = (𝑧 ∪ {𝑤}) → ((𝑥 ∈ 𝑋 ↦ ∏𝑘 ∈ 𝑦 𝐵) ∈ (𝐽 Cn 𝐾) ↔ (𝑥 ∈ 𝑋 ↦ ∏𝑘 ∈ (𝑧 ∪ {𝑤})𝐵) ∈ (𝐽 Cn 𝐾))) | 
| 10 |  | prodeq1 15944 | . . . 4
⊢ (𝑦 = 𝐴 → ∏𝑘 ∈ 𝑦 𝐵 = ∏𝑘 ∈ 𝐴 𝐵) | 
| 11 | 10 | mpteq2dv 5243 | . . 3
⊢ (𝑦 = 𝐴 → (𝑥 ∈ 𝑋 ↦ ∏𝑘 ∈ 𝑦 𝐵) = (𝑥 ∈ 𝑋 ↦ ∏𝑘 ∈ 𝐴 𝐵)) | 
| 12 | 11 | eleq1d 2825 | . 2
⊢ (𝑦 = 𝐴 → ((𝑥 ∈ 𝑋 ↦ ∏𝑘 ∈ 𝑦 𝐵) ∈ (𝐽 Cn 𝐾) ↔ (𝑥 ∈ 𝑋 ↦ ∏𝑘 ∈ 𝐴 𝐵) ∈ (𝐽 Cn 𝐾))) | 
| 13 |  | prod0 15980 | . . . . . 6
⊢
∏𝑘 ∈
∅ 𝐵 =
1 | 
| 14 | 13 | mpteq2i 5246 | . . . . 5
⊢ (𝑥 ∈ 𝑋 ↦ ∏𝑘 ∈ ∅ 𝐵) = (𝑥 ∈ 𝑋 ↦ 1) | 
| 15 |  | eqidd 2737 | . . . . . 6
⊢ (𝑥 = 𝑦 → 1 = 1) | 
| 16 | 15 | cbvmptv 5254 | . . . . 5
⊢ (𝑥 ∈ 𝑋 ↦ 1) = (𝑦 ∈ 𝑋 ↦ 1) | 
| 17 | 14, 16 | eqtri 2764 | . . . 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 24804 | . . . . 5
⊢ 𝐾 ∈
(TopOn‘ℂ) | 
| 22 | 21 | a1i 11 | . . . 4
⊢ (𝜑 → 𝐾 ∈
(TopOn‘ℂ)) | 
| 23 |  | 1cnd 11257 | . . . 4
⊢ (𝜑 → 1 ∈
ℂ) | 
| 24 | 19, 22, 23 | cnmptc 23671 | . . 3
⊢ (𝜑 → (𝑦 ∈ 𝑋 ↦ 1) ∈ (𝐽 Cn 𝐾)) | 
| 25 | 18, 24 | eqeltrd 2840 | . 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 3922 | . . . . . . 7
⊢
Ⅎ𝑥⦋𝑦 / 𝑥⦌𝐵 | 
| 29 | 27, 28 | nfcprod 15946 | . . . . . 6
⊢
Ⅎ𝑥∏𝑘 ∈ (𝑧 ∪ {𝑤})⦋𝑦 / 𝑥⦌𝐵 | 
| 30 |  | csbeq1a 3912 | . . . . . . 7
⊢ (𝑥 = 𝑦 → 𝐵 = ⦋𝑦 / 𝑥⦌𝐵) | 
| 31 | 30 | prodeq2ad 45612 | . . . . . 6
⊢ (𝑥 = 𝑦 → ∏𝑘 ∈ (𝑧 ∪ {𝑤})𝐵 = ∏𝑘 ∈ (𝑧 ∪ {𝑤})⦋𝑦 / 𝑥⦌𝐵) | 
| 32 | 26, 29, 31 | cbvmpt 5252 | . . . . 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 1913 | . . . . . . 7
⊢
Ⅎ𝑘(𝑧 ⊆ 𝐴 ∧ 𝑤 ∈ (𝐴 ∖ 𝑧)) | 
| 36 | 34, 35 | nfan 1898 | . . . . . 6
⊢
Ⅎ𝑘(𝜑 ∧ (𝑧 ⊆ 𝐴 ∧ 𝑤 ∈ (𝐴 ∖ 𝑧))) | 
| 37 |  | nfcv 2904 | . . . . . . . 8
⊢
Ⅎ𝑘𝑋 | 
| 38 |  | nfcv 2904 | . . . . . . . . 9
⊢
Ⅎ𝑘𝑧 | 
| 39 | 38 | nfcprod1 15945 | . . . . . . . 8
⊢
Ⅎ𝑘∏𝑘 ∈ 𝑧 𝐵 | 
| 40 | 37, 39 | nfmpt 5248 | . . . . . . 7
⊢
Ⅎ𝑘(𝑥 ∈ 𝑋 ↦ ∏𝑘 ∈ 𝑧 𝐵) | 
| 41 |  | nfcv 2904 | . . . . . . 7
⊢
Ⅎ𝑘(𝐽 Cn 𝐾) | 
| 42 | 40, 41 | nfel 2919 | . . . . . 6
⊢
Ⅎ𝑘(𝑥 ∈ 𝑋 ↦ ∏𝑘 ∈ 𝑧 𝐵) ∈ (𝐽 Cn 𝐾) | 
| 43 | 36, 42 | nfan 1898 | . . . . 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 5252 | . . . . . . . . 9
⊢ (𝑥 ∈ 𝑋 ↦ 𝐵) = (𝑦 ∈ 𝑋 ↦ ⦋𝑦 / 𝑥⦌𝐵) | 
| 49 | 48 | eqcomi 2745 | . . . . . . . 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 2840 | . . . . . 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 776 | . . . . 5
⊢ (((𝜑 ∧ (𝑧 ⊆ 𝐴 ∧ 𝑤 ∈ (𝐴 ∖ 𝑧))) ∧ (𝑥 ∈ 𝑋 ↦ ∏𝑘 ∈ 𝑧 𝐵) ∈ (𝐽 Cn 𝐾)) → 𝑧 ⊆ 𝐴) | 
| 55 |  | simplrr 777 | . . . . 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 15946 | . . . . . . . . 9
⊢
Ⅎ𝑥∏𝑘 ∈ 𝑧 ⦋𝑦 / 𝑥⦌𝐵 | 
| 59 | 30 | prodeq2sdv 15960 | . . . . . . . . 9
⊢ (𝑥 = 𝑦 → ∏𝑘 ∈ 𝑧 𝐵 = ∏𝑘 ∈ 𝑧 ⦋𝑦 / 𝑥⦌𝐵) | 
| 60 | 56, 58, 59 | cbvmpt 5252 | . . . . . . . 8
⊢ (𝑥 ∈ 𝑋 ↦ ∏𝑘 ∈ 𝑧 𝐵) = (𝑦 ∈ 𝑋 ↦ ∏𝑘 ∈ 𝑧 ⦋𝑦 / 𝑥⦌𝐵) | 
| 61 | 60 | eleq1i 2831 | . . . . . . 7
⊢ ((𝑥 ∈ 𝑋 ↦ ∏𝑘 ∈ 𝑧 𝐵) ∈ (𝐽 Cn 𝐾) ↔ (𝑦 ∈ 𝑋 ↦ ∏𝑘 ∈ 𝑧 ⦋𝑦 / 𝑥⦌𝐵) ∈ (𝐽 Cn 𝐾)) | 
| 62 | 61 | biimpi 216 | . . . . . 6
⊢ ((𝑥 ∈ 𝑋 ↦ ∏𝑘 ∈ 𝑧 𝐵) ∈ (𝐽 Cn 𝐾) → (𝑦 ∈ 𝑋 ↦ ∏𝑘 ∈ 𝑧 ⦋𝑦 / 𝑥⦌𝐵) ∈ (𝐽 Cn 𝐾)) | 
| 63 | 62 | adantl 481 | . . . . 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 45619 | . . . 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 2840 | . . 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 412 | . 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 9207 | 1
⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ ∏𝑘 ∈ 𝐴 𝐵) ∈ (𝐽 Cn 𝐾)) |