Step | Hyp | Ref
| Expression |
1 | | prodeq1 14478 |
. . . . . . . . 9
⊢ (𝑠 = ∅ → ∏𝑡 ∈ 𝑠 ((𝐻‘𝑡)‘𝑥) = ∏𝑡 ∈ ∅ ((𝐻‘𝑡)‘𝑥)) |
2 | 1 | mpteq2dv 4673 |
. . . . . . . 8
⊢ (𝑠 = ∅ → (𝑥 ∈ 𝑋 ↦ ∏𝑡 ∈ 𝑠 ((𝐻‘𝑡)‘𝑥)) = (𝑥 ∈ 𝑋 ↦ ∏𝑡 ∈ ∅ ((𝐻‘𝑡)‘𝑥))) |
3 | 2 | oveq2d 6565 |
. . . . . . 7
⊢ (𝑠 = ∅ → (𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ ∏𝑡 ∈ 𝑠 ((𝐻‘𝑡)‘𝑥))) = (𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ ∏𝑡 ∈ ∅ ((𝐻‘𝑡)‘𝑥)))) |
4 | 3 | fveq1d 6105 |
. . . . . 6
⊢ (𝑠 = ∅ → ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ ∏𝑡 ∈ 𝑠 ((𝐻‘𝑡)‘𝑥)))‘𝑘) = ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ ∏𝑡 ∈ ∅ ((𝐻‘𝑡)‘𝑥)))‘𝑘)) |
5 | | fveq2 6103 |
. . . . . . . . . 10
⊢ (𝑠 = ∅ → (𝐷‘𝑠) = (𝐷‘∅)) |
6 | 5 | fveq1d 6105 |
. . . . . . . . 9
⊢ (𝑠 = ∅ → ((𝐷‘𝑠)‘𝑘) = ((𝐷‘∅)‘𝑘)) |
7 | 6 | sumeq1d 14279 |
. . . . . . . 8
⊢ (𝑠 = ∅ → Σ𝑐 ∈ ((𝐷‘𝑠)‘𝑘)(((!‘𝑘) / ∏𝑡 ∈ 𝑠 (!‘(𝑐‘𝑡))) · ∏𝑡 ∈ 𝑠 (((𝑆 D𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥)) = Σ𝑐 ∈ ((𝐷‘∅)‘𝑘)(((!‘𝑘) / ∏𝑡 ∈ 𝑠 (!‘(𝑐‘𝑡))) · ∏𝑡 ∈ 𝑠 (((𝑆 D𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥))) |
8 | | prodeq1 14478 |
. . . . . . . . . . 11
⊢ (𝑠 = ∅ → ∏𝑡 ∈ 𝑠 (!‘(𝑐‘𝑡)) = ∏𝑡 ∈ ∅ (!‘(𝑐‘𝑡))) |
9 | 8 | oveq2d 6565 |
. . . . . . . . . 10
⊢ (𝑠 = ∅ →
((!‘𝑘) / ∏𝑡 ∈ 𝑠 (!‘(𝑐‘𝑡))) = ((!‘𝑘) / ∏𝑡 ∈ ∅ (!‘(𝑐‘𝑡)))) |
10 | | prodeq1 14478 |
. . . . . . . . . 10
⊢ (𝑠 = ∅ → ∏𝑡 ∈ 𝑠 (((𝑆 D𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥) = ∏𝑡 ∈ ∅ (((𝑆 D𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥)) |
11 | 9, 10 | oveq12d 6567 |
. . . . . . . . 9
⊢ (𝑠 = ∅ →
(((!‘𝑘) /
∏𝑡 ∈ 𝑠 (!‘(𝑐‘𝑡))) · ∏𝑡 ∈ 𝑠 (((𝑆 D𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥)) = (((!‘𝑘) / ∏𝑡 ∈ ∅ (!‘(𝑐‘𝑡))) · ∏𝑡 ∈ ∅ (((𝑆 D𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥))) |
12 | 11 | sumeq2ad 38632 |
. . . . . . . 8
⊢ (𝑠 = ∅ → Σ𝑐 ∈ ((𝐷‘∅)‘𝑘)(((!‘𝑘) / ∏𝑡 ∈ 𝑠 (!‘(𝑐‘𝑡))) · ∏𝑡 ∈ 𝑠 (((𝑆 D𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥)) = Σ𝑐 ∈ ((𝐷‘∅)‘𝑘)(((!‘𝑘) / ∏𝑡 ∈ ∅ (!‘(𝑐‘𝑡))) · ∏𝑡 ∈ ∅ (((𝑆 D𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥))) |
13 | 7, 12 | eqtrd 2644 |
. . . . . . 7
⊢ (𝑠 = ∅ → Σ𝑐 ∈ ((𝐷‘𝑠)‘𝑘)(((!‘𝑘) / ∏𝑡 ∈ 𝑠 (!‘(𝑐‘𝑡))) · ∏𝑡 ∈ 𝑠 (((𝑆 D𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥)) = Σ𝑐 ∈ ((𝐷‘∅)‘𝑘)(((!‘𝑘) / ∏𝑡 ∈ ∅ (!‘(𝑐‘𝑡))) · ∏𝑡 ∈ ∅ (((𝑆 D𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥))) |
14 | 13 | mpteq2dv 4673 |
. . . . . 6
⊢ (𝑠 = ∅ → (𝑥 ∈ 𝑋 ↦ Σ𝑐 ∈ ((𝐷‘𝑠)‘𝑘)(((!‘𝑘) / ∏𝑡 ∈ 𝑠 (!‘(𝑐‘𝑡))) · ∏𝑡 ∈ 𝑠 (((𝑆 D𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥))) = (𝑥 ∈ 𝑋 ↦ Σ𝑐 ∈ ((𝐷‘∅)‘𝑘)(((!‘𝑘) / ∏𝑡 ∈ ∅ (!‘(𝑐‘𝑡))) · ∏𝑡 ∈ ∅ (((𝑆 D𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥)))) |
15 | 4, 14 | eqeq12d 2625 |
. . . . 5
⊢ (𝑠 = ∅ → (((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ ∏𝑡 ∈ 𝑠 ((𝐻‘𝑡)‘𝑥)))‘𝑘) = (𝑥 ∈ 𝑋 ↦ Σ𝑐 ∈ ((𝐷‘𝑠)‘𝑘)(((!‘𝑘) / ∏𝑡 ∈ 𝑠 (!‘(𝑐‘𝑡))) · ∏𝑡 ∈ 𝑠 (((𝑆 D𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥))) ↔ ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ ∏𝑡 ∈ ∅ ((𝐻‘𝑡)‘𝑥)))‘𝑘) = (𝑥 ∈ 𝑋 ↦ Σ𝑐 ∈ ((𝐷‘∅)‘𝑘)(((!‘𝑘) / ∏𝑡 ∈ ∅ (!‘(𝑐‘𝑡))) · ∏𝑡 ∈ ∅ (((𝑆 D𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥))))) |
16 | 15 | ralbidv 2969 |
. . . 4
⊢ (𝑠 = ∅ → (∀𝑘 ∈ (0...𝑁)((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ ∏𝑡 ∈ 𝑠 ((𝐻‘𝑡)‘𝑥)))‘𝑘) = (𝑥 ∈ 𝑋 ↦ Σ𝑐 ∈ ((𝐷‘𝑠)‘𝑘)(((!‘𝑘) / ∏𝑡 ∈ 𝑠 (!‘(𝑐‘𝑡))) · ∏𝑡 ∈ 𝑠 (((𝑆 D𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥))) ↔ ∀𝑘 ∈ (0...𝑁)((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ ∏𝑡 ∈ ∅ ((𝐻‘𝑡)‘𝑥)))‘𝑘) = (𝑥 ∈ 𝑋 ↦ Σ𝑐 ∈ ((𝐷‘∅)‘𝑘)(((!‘𝑘) / ∏𝑡 ∈ ∅ (!‘(𝑐‘𝑡))) · ∏𝑡 ∈ ∅ (((𝑆 D𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥))))) |
17 | | prodeq1 14478 |
. . . . . . . . 9
⊢ (𝑠 = 𝑟 → ∏𝑡 ∈ 𝑠 ((𝐻‘𝑡)‘𝑥) = ∏𝑡 ∈ 𝑟 ((𝐻‘𝑡)‘𝑥)) |
18 | 17 | mpteq2dv 4673 |
. . . . . . . 8
⊢ (𝑠 = 𝑟 → (𝑥 ∈ 𝑋 ↦ ∏𝑡 ∈ 𝑠 ((𝐻‘𝑡)‘𝑥)) = (𝑥 ∈ 𝑋 ↦ ∏𝑡 ∈ 𝑟 ((𝐻‘𝑡)‘𝑥))) |
19 | 18 | oveq2d 6565 |
. . . . . . 7
⊢ (𝑠 = 𝑟 → (𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ ∏𝑡 ∈ 𝑠 ((𝐻‘𝑡)‘𝑥))) = (𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ ∏𝑡 ∈ 𝑟 ((𝐻‘𝑡)‘𝑥)))) |
20 | 19 | fveq1d 6105 |
. . . . . 6
⊢ (𝑠 = 𝑟 → ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ ∏𝑡 ∈ 𝑠 ((𝐻‘𝑡)‘𝑥)))‘𝑘) = ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ ∏𝑡 ∈ 𝑟 ((𝐻‘𝑡)‘𝑥)))‘𝑘)) |
21 | | fveq2 6103 |
. . . . . . . . . 10
⊢ (𝑠 = 𝑟 → (𝐷‘𝑠) = (𝐷‘𝑟)) |
22 | 21 | fveq1d 6105 |
. . . . . . . . 9
⊢ (𝑠 = 𝑟 → ((𝐷‘𝑠)‘𝑘) = ((𝐷‘𝑟)‘𝑘)) |
23 | 22 | sumeq1d 14279 |
. . . . . . . 8
⊢ (𝑠 = 𝑟 → Σ𝑐 ∈ ((𝐷‘𝑠)‘𝑘)(((!‘𝑘) / ∏𝑡 ∈ 𝑠 (!‘(𝑐‘𝑡))) · ∏𝑡 ∈ 𝑠 (((𝑆 D𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥)) = Σ𝑐 ∈ ((𝐷‘𝑟)‘𝑘)(((!‘𝑘) / ∏𝑡 ∈ 𝑠 (!‘(𝑐‘𝑡))) · ∏𝑡 ∈ 𝑠 (((𝑆 D𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥))) |
24 | | prodeq1 14478 |
. . . . . . . . . . 11
⊢ (𝑠 = 𝑟 → ∏𝑡 ∈ 𝑠 (!‘(𝑐‘𝑡)) = ∏𝑡 ∈ 𝑟 (!‘(𝑐‘𝑡))) |
25 | 24 | oveq2d 6565 |
. . . . . . . . . 10
⊢ (𝑠 = 𝑟 → ((!‘𝑘) / ∏𝑡 ∈ 𝑠 (!‘(𝑐‘𝑡))) = ((!‘𝑘) / ∏𝑡 ∈ 𝑟 (!‘(𝑐‘𝑡)))) |
26 | | prodeq1 14478 |
. . . . . . . . . 10
⊢ (𝑠 = 𝑟 → ∏𝑡 ∈ 𝑠 (((𝑆 D𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥) = ∏𝑡 ∈ 𝑟 (((𝑆 D𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥)) |
27 | 25, 26 | oveq12d 6567 |
. . . . . . . . 9
⊢ (𝑠 = 𝑟 → (((!‘𝑘) / ∏𝑡 ∈ 𝑠 (!‘(𝑐‘𝑡))) · ∏𝑡 ∈ 𝑠 (((𝑆 D𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥)) = (((!‘𝑘) / ∏𝑡 ∈ 𝑟 (!‘(𝑐‘𝑡))) · ∏𝑡 ∈ 𝑟 (((𝑆 D𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥))) |
28 | 27 | sumeq2ad 38632 |
. . . . . . . 8
⊢ (𝑠 = 𝑟 → Σ𝑐 ∈ ((𝐷‘𝑟)‘𝑘)(((!‘𝑘) / ∏𝑡 ∈ 𝑠 (!‘(𝑐‘𝑡))) · ∏𝑡 ∈ 𝑠 (((𝑆 D𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥)) = Σ𝑐 ∈ ((𝐷‘𝑟)‘𝑘)(((!‘𝑘) / ∏𝑡 ∈ 𝑟 (!‘(𝑐‘𝑡))) · ∏𝑡 ∈ 𝑟 (((𝑆 D𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥))) |
29 | 23, 28 | eqtrd 2644 |
. . . . . . 7
⊢ (𝑠 = 𝑟 → Σ𝑐 ∈ ((𝐷‘𝑠)‘𝑘)(((!‘𝑘) / ∏𝑡 ∈ 𝑠 (!‘(𝑐‘𝑡))) · ∏𝑡 ∈ 𝑠 (((𝑆 D𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥)) = Σ𝑐 ∈ ((𝐷‘𝑟)‘𝑘)(((!‘𝑘) / ∏𝑡 ∈ 𝑟 (!‘(𝑐‘𝑡))) · ∏𝑡 ∈ 𝑟 (((𝑆 D𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥))) |
30 | 29 | mpteq2dv 4673 |
. . . . . 6
⊢ (𝑠 = 𝑟 → (𝑥 ∈ 𝑋 ↦ Σ𝑐 ∈ ((𝐷‘𝑠)‘𝑘)(((!‘𝑘) / ∏𝑡 ∈ 𝑠 (!‘(𝑐‘𝑡))) · ∏𝑡 ∈ 𝑠 (((𝑆 D𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥))) = (𝑥 ∈ 𝑋 ↦ Σ𝑐 ∈ ((𝐷‘𝑟)‘𝑘)(((!‘𝑘) / ∏𝑡 ∈ 𝑟 (!‘(𝑐‘𝑡))) · ∏𝑡 ∈ 𝑟 (((𝑆 D𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥)))) |
31 | 20, 30 | eqeq12d 2625 |
. . . . 5
⊢ (𝑠 = 𝑟 → (((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ ∏𝑡 ∈ 𝑠 ((𝐻‘𝑡)‘𝑥)))‘𝑘) = (𝑥 ∈ 𝑋 ↦ Σ𝑐 ∈ ((𝐷‘𝑠)‘𝑘)(((!‘𝑘) / ∏𝑡 ∈ 𝑠 (!‘(𝑐‘𝑡))) · ∏𝑡 ∈ 𝑠 (((𝑆 D𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥))) ↔ ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ ∏𝑡 ∈ 𝑟 ((𝐻‘𝑡)‘𝑥)))‘𝑘) = (𝑥 ∈ 𝑋 ↦ Σ𝑐 ∈ ((𝐷‘𝑟)‘𝑘)(((!‘𝑘) / ∏𝑡 ∈ 𝑟 (!‘(𝑐‘𝑡))) · ∏𝑡 ∈ 𝑟 (((𝑆 D𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥))))) |
32 | 31 | ralbidv 2969 |
. . . 4
⊢ (𝑠 = 𝑟 → (∀𝑘 ∈ (0...𝑁)((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ ∏𝑡 ∈ 𝑠 ((𝐻‘𝑡)‘𝑥)))‘𝑘) = (𝑥 ∈ 𝑋 ↦ Σ𝑐 ∈ ((𝐷‘𝑠)‘𝑘)(((!‘𝑘) / ∏𝑡 ∈ 𝑠 (!‘(𝑐‘𝑡))) · ∏𝑡 ∈ 𝑠 (((𝑆 D𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥))) ↔ ∀𝑘 ∈ (0...𝑁)((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ ∏𝑡 ∈ 𝑟 ((𝐻‘𝑡)‘𝑥)))‘𝑘) = (𝑥 ∈ 𝑋 ↦ Σ𝑐 ∈ ((𝐷‘𝑟)‘𝑘)(((!‘𝑘) / ∏𝑡 ∈ 𝑟 (!‘(𝑐‘𝑡))) · ∏𝑡 ∈ 𝑟 (((𝑆 D𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥))))) |
33 | | prodeq1 14478 |
. . . . . . . . 9
⊢ (𝑠 = (𝑟 ∪ {𝑧}) → ∏𝑡 ∈ 𝑠 ((𝐻‘𝑡)‘𝑥) = ∏𝑡 ∈ (𝑟 ∪ {𝑧})((𝐻‘𝑡)‘𝑥)) |
34 | 33 | mpteq2dv 4673 |
. . . . . . . 8
⊢ (𝑠 = (𝑟 ∪ {𝑧}) → (𝑥 ∈ 𝑋 ↦ ∏𝑡 ∈ 𝑠 ((𝐻‘𝑡)‘𝑥)) = (𝑥 ∈ 𝑋 ↦ ∏𝑡 ∈ (𝑟 ∪ {𝑧})((𝐻‘𝑡)‘𝑥))) |
35 | 34 | oveq2d 6565 |
. . . . . . 7
⊢ (𝑠 = (𝑟 ∪ {𝑧}) → (𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ ∏𝑡 ∈ 𝑠 ((𝐻‘𝑡)‘𝑥))) = (𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ ∏𝑡 ∈ (𝑟 ∪ {𝑧})((𝐻‘𝑡)‘𝑥)))) |
36 | 35 | fveq1d 6105 |
. . . . . 6
⊢ (𝑠 = (𝑟 ∪ {𝑧}) → ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ ∏𝑡 ∈ 𝑠 ((𝐻‘𝑡)‘𝑥)))‘𝑘) = ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ ∏𝑡 ∈ (𝑟 ∪ {𝑧})((𝐻‘𝑡)‘𝑥)))‘𝑘)) |
37 | | fveq2 6103 |
. . . . . . . . . 10
⊢ (𝑠 = (𝑟 ∪ {𝑧}) → (𝐷‘𝑠) = (𝐷‘(𝑟 ∪ {𝑧}))) |
38 | 37 | fveq1d 6105 |
. . . . . . . . 9
⊢ (𝑠 = (𝑟 ∪ {𝑧}) → ((𝐷‘𝑠)‘𝑘) = ((𝐷‘(𝑟 ∪ {𝑧}))‘𝑘)) |
39 | 38 | sumeq1d 14279 |
. . . . . . . 8
⊢ (𝑠 = (𝑟 ∪ {𝑧}) → Σ𝑐 ∈ ((𝐷‘𝑠)‘𝑘)(((!‘𝑘) / ∏𝑡 ∈ 𝑠 (!‘(𝑐‘𝑡))) · ∏𝑡 ∈ 𝑠 (((𝑆 D𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥)) = Σ𝑐 ∈ ((𝐷‘(𝑟 ∪ {𝑧}))‘𝑘)(((!‘𝑘) / ∏𝑡 ∈ 𝑠 (!‘(𝑐‘𝑡))) · ∏𝑡 ∈ 𝑠 (((𝑆 D𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥))) |
40 | | prodeq1 14478 |
. . . . . . . . . . 11
⊢ (𝑠 = (𝑟 ∪ {𝑧}) → ∏𝑡 ∈ 𝑠 (!‘(𝑐‘𝑡)) = ∏𝑡 ∈ (𝑟 ∪ {𝑧})(!‘(𝑐‘𝑡))) |
41 | 40 | oveq2d 6565 |
. . . . . . . . . 10
⊢ (𝑠 = (𝑟 ∪ {𝑧}) → ((!‘𝑘) / ∏𝑡 ∈ 𝑠 (!‘(𝑐‘𝑡))) = ((!‘𝑘) / ∏𝑡 ∈ (𝑟 ∪ {𝑧})(!‘(𝑐‘𝑡)))) |
42 | | prodeq1 14478 |
. . . . . . . . . 10
⊢ (𝑠 = (𝑟 ∪ {𝑧}) → ∏𝑡 ∈ 𝑠 (((𝑆 D𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥) = ∏𝑡 ∈ (𝑟 ∪ {𝑧})(((𝑆 D𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥)) |
43 | 41, 42 | oveq12d 6567 |
. . . . . . . . 9
⊢ (𝑠 = (𝑟 ∪ {𝑧}) → (((!‘𝑘) / ∏𝑡 ∈ 𝑠 (!‘(𝑐‘𝑡))) · ∏𝑡 ∈ 𝑠 (((𝑆 D𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥)) = (((!‘𝑘) / ∏𝑡 ∈ (𝑟 ∪ {𝑧})(!‘(𝑐‘𝑡))) · ∏𝑡 ∈ (𝑟 ∪ {𝑧})(((𝑆 D𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥))) |
44 | 43 | sumeq2ad 38632 |
. . . . . . . 8
⊢ (𝑠 = (𝑟 ∪ {𝑧}) → Σ𝑐 ∈ ((𝐷‘(𝑟 ∪ {𝑧}))‘𝑘)(((!‘𝑘) / ∏𝑡 ∈ 𝑠 (!‘(𝑐‘𝑡))) · ∏𝑡 ∈ 𝑠 (((𝑆 D𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥)) = Σ𝑐 ∈ ((𝐷‘(𝑟 ∪ {𝑧}))‘𝑘)(((!‘𝑘) / ∏𝑡 ∈ (𝑟 ∪ {𝑧})(!‘(𝑐‘𝑡))) · ∏𝑡 ∈ (𝑟 ∪ {𝑧})(((𝑆 D𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥))) |
45 | 39, 44 | eqtrd 2644 |
. . . . . . 7
⊢ (𝑠 = (𝑟 ∪ {𝑧}) → Σ𝑐 ∈ ((𝐷‘𝑠)‘𝑘)(((!‘𝑘) / ∏𝑡 ∈ 𝑠 (!‘(𝑐‘𝑡))) · ∏𝑡 ∈ 𝑠 (((𝑆 D𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥)) = Σ𝑐 ∈ ((𝐷‘(𝑟 ∪ {𝑧}))‘𝑘)(((!‘𝑘) / ∏𝑡 ∈ (𝑟 ∪ {𝑧})(!‘(𝑐‘𝑡))) · ∏𝑡 ∈ (𝑟 ∪ {𝑧})(((𝑆 D𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥))) |
46 | 45 | mpteq2dv 4673 |
. . . . . 6
⊢ (𝑠 = (𝑟 ∪ {𝑧}) → (𝑥 ∈ 𝑋 ↦ Σ𝑐 ∈ ((𝐷‘𝑠)‘𝑘)(((!‘𝑘) / ∏𝑡 ∈ 𝑠 (!‘(𝑐‘𝑡))) · ∏𝑡 ∈ 𝑠 (((𝑆 D𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥))) = (𝑥 ∈ 𝑋 ↦ Σ𝑐 ∈ ((𝐷‘(𝑟 ∪ {𝑧}))‘𝑘)(((!‘𝑘) / ∏𝑡 ∈ (𝑟 ∪ {𝑧})(!‘(𝑐‘𝑡))) · ∏𝑡 ∈ (𝑟 ∪ {𝑧})(((𝑆 D𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥)))) |
47 | 36, 46 | eqeq12d 2625 |
. . . . 5
⊢ (𝑠 = (𝑟 ∪ {𝑧}) → (((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ ∏𝑡 ∈ 𝑠 ((𝐻‘𝑡)‘𝑥)))‘𝑘) = (𝑥 ∈ 𝑋 ↦ Σ𝑐 ∈ ((𝐷‘𝑠)‘𝑘)(((!‘𝑘) / ∏𝑡 ∈ 𝑠 (!‘(𝑐‘𝑡))) · ∏𝑡 ∈ 𝑠 (((𝑆 D𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥))) ↔ ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ ∏𝑡 ∈ (𝑟 ∪ {𝑧})((𝐻‘𝑡)‘𝑥)))‘𝑘) = (𝑥 ∈ 𝑋 ↦ Σ𝑐 ∈ ((𝐷‘(𝑟 ∪ {𝑧}))‘𝑘)(((!‘𝑘) / ∏𝑡 ∈ (𝑟 ∪ {𝑧})(!‘(𝑐‘𝑡))) · ∏𝑡 ∈ (𝑟 ∪ {𝑧})(((𝑆 D𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥))))) |
48 | 47 | ralbidv 2969 |
. . . 4
⊢ (𝑠 = (𝑟 ∪ {𝑧}) → (∀𝑘 ∈ (0...𝑁)((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ ∏𝑡 ∈ 𝑠 ((𝐻‘𝑡)‘𝑥)))‘𝑘) = (𝑥 ∈ 𝑋 ↦ Σ𝑐 ∈ ((𝐷‘𝑠)‘𝑘)(((!‘𝑘) / ∏𝑡 ∈ 𝑠 (!‘(𝑐‘𝑡))) · ∏𝑡 ∈ 𝑠 (((𝑆 D𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥))) ↔ ∀𝑘 ∈ (0...𝑁)((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ ∏𝑡 ∈ (𝑟 ∪ {𝑧})((𝐻‘𝑡)‘𝑥)))‘𝑘) = (𝑥 ∈ 𝑋 ↦ Σ𝑐 ∈ ((𝐷‘(𝑟 ∪ {𝑧}))‘𝑘)(((!‘𝑘) / ∏𝑡 ∈ (𝑟 ∪ {𝑧})(!‘(𝑐‘𝑡))) · ∏𝑡 ∈ (𝑟 ∪ {𝑧})(((𝑆 D𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥))))) |
49 | | prodeq1 14478 |
. . . . . . . . . 10
⊢ (𝑠 = 𝑇 → ∏𝑡 ∈ 𝑠 ((𝐻‘𝑡)‘𝑥) = ∏𝑡 ∈ 𝑇 ((𝐻‘𝑡)‘𝑥)) |
50 | 49 | mpteq2dv 4673 |
. . . . . . . . 9
⊢ (𝑠 = 𝑇 → (𝑥 ∈ 𝑋 ↦ ∏𝑡 ∈ 𝑠 ((𝐻‘𝑡)‘𝑥)) = (𝑥 ∈ 𝑋 ↦ ∏𝑡 ∈ 𝑇 ((𝐻‘𝑡)‘𝑥))) |
51 | | dvnprodlem3.f |
. . . . . . . . . . 11
⊢ 𝐹 = (𝑥 ∈ 𝑋 ↦ ∏𝑡 ∈ 𝑇 ((𝐻‘𝑡)‘𝑥)) |
52 | 51 | a1i 11 |
. . . . . . . . . 10
⊢ (𝑠 = 𝑇 → 𝐹 = (𝑥 ∈ 𝑋 ↦ ∏𝑡 ∈ 𝑇 ((𝐻‘𝑡)‘𝑥))) |
53 | 52 | eqcomd 2616 |
. . . . . . . . 9
⊢ (𝑠 = 𝑇 → (𝑥 ∈ 𝑋 ↦ ∏𝑡 ∈ 𝑇 ((𝐻‘𝑡)‘𝑥)) = 𝐹) |
54 | 50, 53 | eqtrd 2644 |
. . . . . . . 8
⊢ (𝑠 = 𝑇 → (𝑥 ∈ 𝑋 ↦ ∏𝑡 ∈ 𝑠 ((𝐻‘𝑡)‘𝑥)) = 𝐹) |
55 | 54 | oveq2d 6565 |
. . . . . . 7
⊢ (𝑠 = 𝑇 → (𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ ∏𝑡 ∈ 𝑠 ((𝐻‘𝑡)‘𝑥))) = (𝑆 D𝑛 𝐹)) |
56 | 55 | fveq1d 6105 |
. . . . . 6
⊢ (𝑠 = 𝑇 → ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ ∏𝑡 ∈ 𝑠 ((𝐻‘𝑡)‘𝑥)))‘𝑘) = ((𝑆 D𝑛 𝐹)‘𝑘)) |
57 | | fveq2 6103 |
. . . . . . . . . 10
⊢ (𝑠 = 𝑇 → (𝐷‘𝑠) = (𝐷‘𝑇)) |
58 | 57 | fveq1d 6105 |
. . . . . . . . 9
⊢ (𝑠 = 𝑇 → ((𝐷‘𝑠)‘𝑘) = ((𝐷‘𝑇)‘𝑘)) |
59 | 58 | sumeq1d 14279 |
. . . . . . . 8
⊢ (𝑠 = 𝑇 → Σ𝑐 ∈ ((𝐷‘𝑠)‘𝑘)(((!‘𝑘) / ∏𝑡 ∈ 𝑠 (!‘(𝑐‘𝑡))) · ∏𝑡 ∈ 𝑠 (((𝑆 D𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥)) = Σ𝑐 ∈ ((𝐷‘𝑇)‘𝑘)(((!‘𝑘) / ∏𝑡 ∈ 𝑠 (!‘(𝑐‘𝑡))) · ∏𝑡 ∈ 𝑠 (((𝑆 D𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥))) |
60 | | prodeq1 14478 |
. . . . . . . . . . 11
⊢ (𝑠 = 𝑇 → ∏𝑡 ∈ 𝑠 (!‘(𝑐‘𝑡)) = ∏𝑡 ∈ 𝑇 (!‘(𝑐‘𝑡))) |
61 | 60 | oveq2d 6565 |
. . . . . . . . . 10
⊢ (𝑠 = 𝑇 → ((!‘𝑘) / ∏𝑡 ∈ 𝑠 (!‘(𝑐‘𝑡))) = ((!‘𝑘) / ∏𝑡 ∈ 𝑇 (!‘(𝑐‘𝑡)))) |
62 | | prodeq1 14478 |
. . . . . . . . . 10
⊢ (𝑠 = 𝑇 → ∏𝑡 ∈ 𝑠 (((𝑆 D𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥) = ∏𝑡 ∈ 𝑇 (((𝑆 D𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥)) |
63 | 61, 62 | oveq12d 6567 |
. . . . . . . . 9
⊢ (𝑠 = 𝑇 → (((!‘𝑘) / ∏𝑡 ∈ 𝑠 (!‘(𝑐‘𝑡))) · ∏𝑡 ∈ 𝑠 (((𝑆 D𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥)) = (((!‘𝑘) / ∏𝑡 ∈ 𝑇 (!‘(𝑐‘𝑡))) · ∏𝑡 ∈ 𝑇 (((𝑆 D𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥))) |
64 | 63 | sumeq2ad 38632 |
. . . . . . . 8
⊢ (𝑠 = 𝑇 → Σ𝑐 ∈ ((𝐷‘𝑇)‘𝑘)(((!‘𝑘) / ∏𝑡 ∈ 𝑠 (!‘(𝑐‘𝑡))) · ∏𝑡 ∈ 𝑠 (((𝑆 D𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥)) = Σ𝑐 ∈ ((𝐷‘𝑇)‘𝑘)(((!‘𝑘) / ∏𝑡 ∈ 𝑇 (!‘(𝑐‘𝑡))) · ∏𝑡 ∈ 𝑇 (((𝑆 D𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥))) |
65 | 59, 64 | eqtrd 2644 |
. . . . . . 7
⊢ (𝑠 = 𝑇 → Σ𝑐 ∈ ((𝐷‘𝑠)‘𝑘)(((!‘𝑘) / ∏𝑡 ∈ 𝑠 (!‘(𝑐‘𝑡))) · ∏𝑡 ∈ 𝑠 (((𝑆 D𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥)) = Σ𝑐 ∈ ((𝐷‘𝑇)‘𝑘)(((!‘𝑘) / ∏𝑡 ∈ 𝑇 (!‘(𝑐‘𝑡))) · ∏𝑡 ∈ 𝑇 (((𝑆 D𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥))) |
66 | 65 | mpteq2dv 4673 |
. . . . . 6
⊢ (𝑠 = 𝑇 → (𝑥 ∈ 𝑋 ↦ Σ𝑐 ∈ ((𝐷‘𝑠)‘𝑘)(((!‘𝑘) / ∏𝑡 ∈ 𝑠 (!‘(𝑐‘𝑡))) · ∏𝑡 ∈ 𝑠 (((𝑆 D𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥))) = (𝑥 ∈ 𝑋 ↦ Σ𝑐 ∈ ((𝐷‘𝑇)‘𝑘)(((!‘𝑘) / ∏𝑡 ∈ 𝑇 (!‘(𝑐‘𝑡))) · ∏𝑡 ∈ 𝑇 (((𝑆 D𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥)))) |
67 | 56, 66 | eqeq12d 2625 |
. . . . 5
⊢ (𝑠 = 𝑇 → (((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ ∏𝑡 ∈ 𝑠 ((𝐻‘𝑡)‘𝑥)))‘𝑘) = (𝑥 ∈ 𝑋 ↦ Σ𝑐 ∈ ((𝐷‘𝑠)‘𝑘)(((!‘𝑘) / ∏𝑡 ∈ 𝑠 (!‘(𝑐‘𝑡))) · ∏𝑡 ∈ 𝑠 (((𝑆 D𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥))) ↔ ((𝑆 D𝑛 𝐹)‘𝑘) = (𝑥 ∈ 𝑋 ↦ Σ𝑐 ∈ ((𝐷‘𝑇)‘𝑘)(((!‘𝑘) / ∏𝑡 ∈ 𝑇 (!‘(𝑐‘𝑡))) · ∏𝑡 ∈ 𝑇 (((𝑆 D𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥))))) |
68 | 67 | ralbidv 2969 |
. . . 4
⊢ (𝑠 = 𝑇 → (∀𝑘 ∈ (0...𝑁)((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ ∏𝑡 ∈ 𝑠 ((𝐻‘𝑡)‘𝑥)))‘𝑘) = (𝑥 ∈ 𝑋 ↦ Σ𝑐 ∈ ((𝐷‘𝑠)‘𝑘)(((!‘𝑘) / ∏𝑡 ∈ 𝑠 (!‘(𝑐‘𝑡))) · ∏𝑡 ∈ 𝑠 (((𝑆 D𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥))) ↔ ∀𝑘 ∈ (0...𝑁)((𝑆 D𝑛 𝐹)‘𝑘) = (𝑥 ∈ 𝑋 ↦ Σ𝑐 ∈ ((𝐷‘𝑇)‘𝑘)(((!‘𝑘) / ∏𝑡 ∈ 𝑇 (!‘(𝑐‘𝑡))) · ∏𝑡 ∈ 𝑇 (((𝑆 D𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥))))) |
69 | | prod0 14512 |
. . . . . . . . . . . . 13
⊢
∏𝑡 ∈
∅ ((𝐻‘𝑡)‘𝑥) = 1 |
70 | 69 | mpteq2i 4669 |
. . . . . . . . . . . 12
⊢ (𝑥 ∈ 𝑋 ↦ ∏𝑡 ∈ ∅ ((𝐻‘𝑡)‘𝑥)) = (𝑥 ∈ 𝑋 ↦ 1) |
71 | 70 | oveq2i 6560 |
. . . . . . . . . . 11
⊢ (𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ ∏𝑡 ∈ ∅ ((𝐻‘𝑡)‘𝑥))) = (𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ 1)) |
72 | 71 | a1i 11 |
. . . . . . . . . 10
⊢ (𝑘 = 0 → (𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ ∏𝑡 ∈ ∅ ((𝐻‘𝑡)‘𝑥))) = (𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ 1))) |
73 | | id 22 |
. . . . . . . . . 10
⊢ (𝑘 = 0 → 𝑘 = 0) |
74 | 72, 73 | fveq12d 6109 |
. . . . . . . . 9
⊢ (𝑘 = 0 → ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ ∏𝑡 ∈ ∅ ((𝐻‘𝑡)‘𝑥)))‘𝑘) = ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ 1))‘0)) |
75 | 74 | adantl 481 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑘 = 0) → ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ ∏𝑡 ∈ ∅ ((𝐻‘𝑡)‘𝑥)))‘𝑘) = ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ 1))‘0)) |
76 | | dvnprodlem3.s |
. . . . . . . . . . 11
⊢ (𝜑 → 𝑆 ∈ {ℝ, ℂ}) |
77 | | recnprss 23474 |
. . . . . . . . . . 11
⊢ (𝑆 ∈ {ℝ, ℂ}
→ 𝑆 ⊆
ℂ) |
78 | 76, 77 | syl 17 |
. . . . . . . . . 10
⊢ (𝜑 → 𝑆 ⊆ ℂ) |
79 | | 1cnd 9935 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 1 ∈ ℂ) |
80 | | eqid 2610 |
. . . . . . . . . . . . . 14
⊢ (𝑥 ∈ 𝑋 ↦ 1) = (𝑥 ∈ 𝑋 ↦ 1) |
81 | 79, 80 | fmptd 6292 |
. . . . . . . . . . . . 13
⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 1):𝑋⟶ℂ) |
82 | | 1re 9918 |
. . . . . . . . . . . . . . . . 17
⊢ 1 ∈
ℝ |
83 | 82 | rgenw 2908 |
. . . . . . . . . . . . . . . 16
⊢
∀𝑥 ∈
𝑋 1 ∈
ℝ |
84 | | dmmptg 5549 |
. . . . . . . . . . . . . . . 16
⊢
(∀𝑥 ∈
𝑋 1 ∈ ℝ →
dom (𝑥 ∈ 𝑋 ↦ 1) = 𝑋) |
85 | 83, 84 | ax-mp 5 |
. . . . . . . . . . . . . . 15
⊢ dom
(𝑥 ∈ 𝑋 ↦ 1) = 𝑋 |
86 | 85 | a1i 11 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → dom (𝑥 ∈ 𝑋 ↦ 1) = 𝑋) |
87 | 86 | feq2d 5944 |
. . . . . . . . . . . . 13
⊢ (𝜑 → ((𝑥 ∈ 𝑋 ↦ 1):dom (𝑥 ∈ 𝑋 ↦ 1)⟶ℂ ↔ (𝑥 ∈ 𝑋 ↦ 1):𝑋⟶ℂ)) |
88 | 81, 87 | mpbird 246 |
. . . . . . . . . . . 12
⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 1):dom (𝑥 ∈ 𝑋 ↦ 1)⟶ℂ) |
89 | | restsspw 15915 |
. . . . . . . . . . . . . . 15
⊢
((TopOpen‘ℂfld) ↾t 𝑆) ⊆ 𝒫 𝑆 |
90 | | dvnprodlem3.x |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → 𝑋 ∈
((TopOpen‘ℂfld) ↾t 𝑆)) |
91 | 89, 90 | sseldi 3566 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → 𝑋 ∈ 𝒫 𝑆) |
92 | | elpwi 4117 |
. . . . . . . . . . . . . 14
⊢ (𝑋 ∈ 𝒫 𝑆 → 𝑋 ⊆ 𝑆) |
93 | 91, 92 | syl 17 |
. . . . . . . . . . . . 13
⊢ (𝜑 → 𝑋 ⊆ 𝑆) |
94 | 86, 93 | eqsstrd 3602 |
. . . . . . . . . . . 12
⊢ (𝜑 → dom (𝑥 ∈ 𝑋 ↦ 1) ⊆ 𝑆) |
95 | 88, 94 | jca 553 |
. . . . . . . . . . 11
⊢ (𝜑 → ((𝑥 ∈ 𝑋 ↦ 1):dom (𝑥 ∈ 𝑋 ↦ 1)⟶ℂ ∧ dom (𝑥 ∈ 𝑋 ↦ 1) ⊆ 𝑆)) |
96 | | cnex 9896 |
. . . . . . . . . . . . 13
⊢ ℂ
∈ V |
97 | 96 | a1i 11 |
. . . . . . . . . . . 12
⊢ (𝜑 → ℂ ∈
V) |
98 | | elpm2g 7760 |
. . . . . . . . . . . 12
⊢ ((ℂ
∈ V ∧ 𝑆 ∈
{ℝ, ℂ}) → ((𝑥 ∈ 𝑋 ↦ 1) ∈ (ℂ
↑pm 𝑆) ↔ ((𝑥 ∈ 𝑋 ↦ 1):dom (𝑥 ∈ 𝑋 ↦ 1)⟶ℂ ∧ dom (𝑥 ∈ 𝑋 ↦ 1) ⊆ 𝑆))) |
99 | 97, 76, 98 | syl2anc 691 |
. . . . . . . . . . 11
⊢ (𝜑 → ((𝑥 ∈ 𝑋 ↦ 1) ∈ (ℂ
↑pm 𝑆) ↔ ((𝑥 ∈ 𝑋 ↦ 1):dom (𝑥 ∈ 𝑋 ↦ 1)⟶ℂ ∧ dom (𝑥 ∈ 𝑋 ↦ 1) ⊆ 𝑆))) |
100 | 95, 99 | mpbird 246 |
. . . . . . . . . 10
⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 1) ∈ (ℂ
↑pm 𝑆)) |
101 | | dvn0 23493 |
. . . . . . . . . 10
⊢ ((𝑆 ⊆ ℂ ∧ (𝑥 ∈ 𝑋 ↦ 1) ∈ (ℂ
↑pm 𝑆)) → ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ 1))‘0) = (𝑥 ∈ 𝑋 ↦ 1)) |
102 | 78, 100, 101 | syl2anc 691 |
. . . . . . . . 9
⊢ (𝜑 → ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ 1))‘0) = (𝑥 ∈ 𝑋 ↦ 1)) |
103 | 102 | adantr 480 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑘 = 0) → ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ 1))‘0) = (𝑥 ∈ 𝑋 ↦ 1)) |
104 | | fveq2 6103 |
. . . . . . . . . . . . . . 15
⊢ (𝑘 = 0 → ((𝐷‘∅)‘𝑘) = ((𝐷‘∅)‘0)) |
105 | 104 | adantl 481 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑘 = 0) → ((𝐷‘∅)‘𝑘) = ((𝐷‘∅)‘0)) |
106 | | dvnprodlem3.d |
. . . . . . . . . . . . . . . . . 18
⊢ 𝐷 = (𝑠 ∈ 𝒫 𝑇 ↦ (𝑛 ∈ ℕ0 ↦ {𝑐 ∈ ((0...𝑛) ↑𝑚 𝑠) ∣ Σ𝑡 ∈ 𝑠 (𝑐‘𝑡) = 𝑛})) |
107 | 106 | a1i 11 |
. . . . . . . . . . . . . . . . 17
⊢ (𝜑 → 𝐷 = (𝑠 ∈ 𝒫 𝑇 ↦ (𝑛 ∈ ℕ0 ↦ {𝑐 ∈ ((0...𝑛) ↑𝑚 𝑠) ∣ Σ𝑡 ∈ 𝑠 (𝑐‘𝑡) = 𝑛}))) |
108 | | oveq2 6557 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑠 = ∅ → ((0...𝑛) ↑𝑚
𝑠) = ((0...𝑛) ↑𝑚
∅)) |
109 | | elmapfn 7766 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ (𝑥 ∈ ((0...𝑛) ↑𝑚 ∅) →
𝑥 Fn
∅) |
110 | | fn0 5924 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ (𝑥 Fn ∅ ↔ 𝑥 = ∅) |
111 | 109, 110 | sylib 207 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ (𝑥 ∈ ((0...𝑛) ↑𝑚 ∅) →
𝑥 =
∅) |
112 | | velsn 4141 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ (𝑥 ∈ {∅} ↔ 𝑥 = ∅) |
113 | 111, 112 | sylibr 223 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (𝑥 ∈ ((0...𝑛) ↑𝑚 ∅) →
𝑥 ∈
{∅}) |
114 | 112 | biimpi 205 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ (𝑥 ∈ {∅} → 𝑥 = ∅) |
115 | | id 22 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ (𝑥 = ∅ → 𝑥 = ∅) |
116 | | f0 5999 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
⊢
∅:∅⟶(0...𝑛) |
117 | | ovex 6577 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
31
⊢
(0...𝑛) ∈
V |
118 | | 0ex 4718 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
31
⊢ ∅
∈ V |
119 | 117, 118 | elmap 7772 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
⊢ (∅
∈ ((0...𝑛)
↑𝑚 ∅) ↔ ∅:∅⟶(0...𝑛)) |
120 | 116, 119 | mpbir 220 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
⊢ ∅
∈ ((0...𝑛)
↑𝑚 ∅) |
121 | 120 | a1i 11 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ (𝑥 = ∅ → ∅ ∈
((0...𝑛)
↑𝑚 ∅)) |
122 | 115, 121 | eqeltrd 2688 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ (𝑥 = ∅ → 𝑥 ∈ ((0...𝑛) ↑𝑚
∅)) |
123 | 114, 122 | syl 17 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (𝑥 ∈ {∅} → 𝑥 ∈ ((0...𝑛) ↑𝑚
∅)) |
124 | 113, 123 | impbii 198 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (𝑥 ∈ ((0...𝑛) ↑𝑚 ∅) ↔
𝑥 ∈
{∅}) |
125 | 124 | ax-gen 1713 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢
∀𝑥(𝑥 ∈ ((0...𝑛) ↑𝑚 ∅) ↔
𝑥 ∈
{∅}) |
126 | | dfcleq 2604 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢
(((0...𝑛)
↑𝑚 ∅) = {∅} ↔ ∀𝑥(𝑥 ∈ ((0...𝑛) ↑𝑚 ∅) ↔
𝑥 ∈
{∅})) |
127 | 125, 126 | mpbir 220 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢
((0...𝑛)
↑𝑚 ∅) = {∅} |
128 | 127 | a1i 11 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑠 = ∅ → ((0...𝑛) ↑𝑚
∅) = {∅}) |
129 | 108, 128 | eqtrd 2644 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑠 = ∅ → ((0...𝑛) ↑𝑚
𝑠) =
{∅}) |
130 | | rabeq 3166 |
. . . . . . . . . . . . . . . . . . . . 21
⊢
(((0...𝑛)
↑𝑚 𝑠) = {∅} → {𝑐 ∈ ((0...𝑛) ↑𝑚 𝑠) ∣ Σ𝑡 ∈ 𝑠 (𝑐‘𝑡) = 𝑛} = {𝑐 ∈ {∅} ∣ Σ𝑡 ∈ 𝑠 (𝑐‘𝑡) = 𝑛}) |
131 | 129, 130 | syl 17 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑠 = ∅ → {𝑐 ∈ ((0...𝑛) ↑𝑚 𝑠) ∣ Σ𝑡 ∈ 𝑠 (𝑐‘𝑡) = 𝑛} = {𝑐 ∈ {∅} ∣ Σ𝑡 ∈ 𝑠 (𝑐‘𝑡) = 𝑛}) |
132 | | sumeq1 14267 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑠 = ∅ → Σ𝑡 ∈ 𝑠 (𝑐‘𝑡) = Σ𝑡 ∈ ∅ (𝑐‘𝑡)) |
133 | 132 | eqeq1d 2612 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑠 = ∅ → (Σ𝑡 ∈ 𝑠 (𝑐‘𝑡) = 𝑛 ↔ Σ𝑡 ∈ ∅ (𝑐‘𝑡) = 𝑛)) |
134 | 133 | rabbidv 3164 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑠 = ∅ → {𝑐 ∈ {∅} ∣
Σ𝑡 ∈ 𝑠 (𝑐‘𝑡) = 𝑛} = {𝑐 ∈ {∅} ∣ Σ𝑡 ∈ ∅ (𝑐‘𝑡) = 𝑛}) |
135 | 131, 134 | eqtrd 2644 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑠 = ∅ → {𝑐 ∈ ((0...𝑛) ↑𝑚 𝑠) ∣ Σ𝑡 ∈ 𝑠 (𝑐‘𝑡) = 𝑛} = {𝑐 ∈ {∅} ∣ Σ𝑡 ∈ ∅ (𝑐‘𝑡) = 𝑛}) |
136 | 135 | mpteq2dv 4673 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑠 = ∅ → (𝑛 ∈ ℕ0
↦ {𝑐 ∈
((0...𝑛)
↑𝑚 𝑠) ∣ Σ𝑡 ∈ 𝑠 (𝑐‘𝑡) = 𝑛}) = (𝑛 ∈ ℕ0 ↦ {𝑐 ∈ {∅} ∣
Σ𝑡 ∈ ∅
(𝑐‘𝑡) = 𝑛})) |
137 | 136 | adantl 481 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ 𝑠 = ∅) → (𝑛 ∈ ℕ0 ↦ {𝑐 ∈ ((0...𝑛) ↑𝑚 𝑠) ∣ Σ𝑡 ∈ 𝑠 (𝑐‘𝑡) = 𝑛}) = (𝑛 ∈ ℕ0 ↦ {𝑐 ∈ {∅} ∣
Σ𝑡 ∈ ∅
(𝑐‘𝑡) = 𝑛})) |
138 | | 0elpw 4760 |
. . . . . . . . . . . . . . . . . 18
⊢ ∅
∈ 𝒫 𝑇 |
139 | 138 | a1i 11 |
. . . . . . . . . . . . . . . . 17
⊢ (𝜑 → ∅ ∈ 𝒫
𝑇) |
140 | | nn0ex 11175 |
. . . . . . . . . . . . . . . . . . 19
⊢
ℕ0 ∈ V |
141 | 140 | mptex 6390 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑛 ∈ ℕ0
↦ {𝑐 ∈ {∅}
∣ Σ𝑡 ∈
∅ (𝑐‘𝑡) = 𝑛}) ∈ V |
142 | 141 | a1i 11 |
. . . . . . . . . . . . . . . . 17
⊢ (𝜑 → (𝑛 ∈ ℕ0 ↦ {𝑐 ∈ {∅} ∣
Σ𝑡 ∈ ∅
(𝑐‘𝑡) = 𝑛}) ∈ V) |
143 | 107, 137,
139, 142 | fvmptd 6197 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → (𝐷‘∅) = (𝑛 ∈ ℕ0 ↦ {𝑐 ∈ {∅} ∣
Σ𝑡 ∈ ∅
(𝑐‘𝑡) = 𝑛})) |
144 | | eqeq2 2621 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑛 = 0 → (Σ𝑡 ∈ ∅ (𝑐‘𝑡) = 𝑛 ↔ Σ𝑡 ∈ ∅ (𝑐‘𝑡) = 0)) |
145 | 144 | rabbidv 3164 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑛 = 0 → {𝑐 ∈ {∅} ∣ Σ𝑡 ∈ ∅ (𝑐‘𝑡) = 𝑛} = {𝑐 ∈ {∅} ∣ Σ𝑡 ∈ ∅ (𝑐‘𝑡) = 0}) |
146 | 145 | adantl 481 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑛 = 0) → {𝑐 ∈ {∅} ∣ Σ𝑡 ∈ ∅ (𝑐‘𝑡) = 𝑛} = {𝑐 ∈ {∅} ∣ Σ𝑡 ∈ ∅ (𝑐‘𝑡) = 0}) |
147 | | 0nn0 11184 |
. . . . . . . . . . . . . . . . 17
⊢ 0 ∈
ℕ0 |
148 | 147 | a1i 11 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → 0 ∈
ℕ0) |
149 | | p0ex 4779 |
. . . . . . . . . . . . . . . . . 18
⊢ {∅}
∈ V |
150 | 149 | rabex 4740 |
. . . . . . . . . . . . . . . . 17
⊢ {𝑐 ∈ {∅} ∣
Σ𝑡 ∈ ∅
(𝑐‘𝑡) = 0} ∈ V |
151 | 150 | a1i 11 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → {𝑐 ∈ {∅} ∣ Σ𝑡 ∈ ∅ (𝑐‘𝑡) = 0} ∈ V) |
152 | 143, 146,
148, 151 | fvmptd 6197 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → ((𝐷‘∅)‘0) = {𝑐 ∈ {∅} ∣
Σ𝑡 ∈ ∅
(𝑐‘𝑡) = 0}) |
153 | 152 | adantr 480 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑘 = 0) → ((𝐷‘∅)‘0) = {𝑐 ∈ {∅} ∣
Σ𝑡 ∈ ∅
(𝑐‘𝑡) = 0}) |
154 | | snidg 4153 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (∅
∈ V → ∅ ∈ {∅}) |
155 | 118, 154 | ax-mp 5 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ∅
∈ {∅} |
156 | | eqid 2610 |
. . . . . . . . . . . . . . . . . . . 20
⊢ 0 =
0 |
157 | 155, 156 | pm3.2i 470 |
. . . . . . . . . . . . . . . . . . 19
⊢ (∅
∈ {∅} ∧ 0 = 0) |
158 | | sum0 14299 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢
Σ𝑡 ∈
∅ (𝑐‘𝑡) = 0 |
159 | 158 | a1i 11 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑐 = ∅ → Σ𝑡 ∈ ∅ (𝑐‘𝑡) = 0) |
160 | 159 | eqeq1d 2612 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑐 = ∅ → (Σ𝑡 ∈ ∅ (𝑐‘𝑡) = 0 ↔ 0 = 0)) |
161 | 160 | elrab 3331 |
. . . . . . . . . . . . . . . . . . 19
⊢ (∅
∈ {𝑐 ∈ {∅}
∣ Σ𝑡 ∈
∅ (𝑐‘𝑡) = 0} ↔ (∅ ∈
{∅} ∧ 0 = 0)) |
162 | 157, 161 | mpbir 220 |
. . . . . . . . . . . . . . . . . 18
⊢ ∅
∈ {𝑐 ∈ {∅}
∣ Σ𝑡 ∈
∅ (𝑐‘𝑡) = 0} |
163 | 162 | n0ii 3881 |
. . . . . . . . . . . . . . . . 17
⊢ ¬
{𝑐 ∈ {∅} ∣
Σ𝑡 ∈ ∅
(𝑐‘𝑡) = 0} = ∅ |
164 | | eqid 2610 |
. . . . . . . . . . . . . . . . . 18
⊢ {𝑐 ∈ {∅} ∣
Σ𝑡 ∈ ∅
(𝑐‘𝑡) = 0} = {𝑐 ∈ {∅} ∣ Σ𝑡 ∈ ∅ (𝑐‘𝑡) = 0} |
165 | | rabrsn 4203 |
. . . . . . . . . . . . . . . . . 18
⊢ ({𝑐 ∈ {∅} ∣
Σ𝑡 ∈ ∅
(𝑐‘𝑡) = 0} = {𝑐 ∈ {∅} ∣ Σ𝑡 ∈ ∅ (𝑐‘𝑡) = 0} → ({𝑐 ∈ {∅} ∣ Σ𝑡 ∈ ∅ (𝑐‘𝑡) = 0} = ∅ ∨ {𝑐 ∈ {∅} ∣ Σ𝑡 ∈ ∅ (𝑐‘𝑡) = 0} = {∅})) |
166 | 164, 165 | ax-mp 5 |
. . . . . . . . . . . . . . . . 17
⊢ ({𝑐 ∈ {∅} ∣
Σ𝑡 ∈ ∅
(𝑐‘𝑡) = 0} = ∅ ∨ {𝑐 ∈ {∅} ∣ Σ𝑡 ∈ ∅ (𝑐‘𝑡) = 0} = {∅}) |
167 | 163, 166 | mtpor 1686 |
. . . . . . . . . . . . . . . 16
⊢ {𝑐 ∈ {∅} ∣
Σ𝑡 ∈ ∅
(𝑐‘𝑡) = 0} = {∅} |
168 | 167 | a1i 11 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑘 = 0) → {𝑐 ∈ {∅} ∣ Σ𝑡 ∈ ∅ (𝑐‘𝑡) = 0} = {∅}) |
169 | | iftrue 4042 |
. . . . . . . . . . . . . . . 16
⊢ (𝑘 = 0 → if(𝑘 = 0, {∅}, ∅) =
{∅}) |
170 | 169 | adantl 481 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑘 = 0) → if(𝑘 = 0, {∅}, ∅) =
{∅}) |
171 | 168, 170 | eqtr4d 2647 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑘 = 0) → {𝑐 ∈ {∅} ∣ Σ𝑡 ∈ ∅ (𝑐‘𝑡) = 0} = if(𝑘 = 0, {∅}, ∅)) |
172 | 105, 153,
171 | 3eqtrd 2648 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑘 = 0) → ((𝐷‘∅)‘𝑘) = if(𝑘 = 0, {∅}, ∅)) |
173 | 172, 170 | eqtrd 2644 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑘 = 0) → ((𝐷‘∅)‘𝑘) = {∅}) |
174 | 173 | sumeq1d 14279 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑘 = 0) → Σ𝑐 ∈ ((𝐷‘∅)‘𝑘)(((!‘𝑘) / ∏𝑡 ∈ ∅ (!‘(𝑐‘𝑡))) · ∏𝑡 ∈ ∅ (((𝑆 D𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥)) = Σ𝑐 ∈ {∅} (((!‘𝑘) / ∏𝑡 ∈ ∅ (!‘(𝑐‘𝑡))) · ∏𝑡 ∈ ∅ (((𝑆 D𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥))) |
175 | | fveq2 6103 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑘 = 0 → (!‘𝑘) =
(!‘0)) |
176 | | fac0 12925 |
. . . . . . . . . . . . . . . . . . 19
⊢
(!‘0) = 1 |
177 | 176 | a1i 11 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑘 = 0 → (!‘0) =
1) |
178 | 175, 177 | eqtrd 2644 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑘 = 0 → (!‘𝑘) = 1) |
179 | 178 | oveq1d 6564 |
. . . . . . . . . . . . . . . 16
⊢ (𝑘 = 0 → ((!‘𝑘) / ∏𝑡 ∈ ∅ (!‘(𝑐‘𝑡))) = (1 / ∏𝑡 ∈ ∅ (!‘(𝑐‘𝑡)))) |
180 | | prod0 14512 |
. . . . . . . . . . . . . . . . . 18
⊢
∏𝑡 ∈
∅ (!‘(𝑐‘𝑡)) = 1 |
181 | 180 | oveq2i 6560 |
. . . . . . . . . . . . . . . . 17
⊢ (1 /
∏𝑡 ∈ ∅
(!‘(𝑐‘𝑡))) = (1 / 1) |
182 | | 1div1e1 10596 |
. . . . . . . . . . . . . . . . 17
⊢ (1 / 1) =
1 |
183 | 181, 182 | eqtri 2632 |
. . . . . . . . . . . . . . . 16
⊢ (1 /
∏𝑡 ∈ ∅
(!‘(𝑐‘𝑡))) = 1 |
184 | 179, 183 | syl6eq 2660 |
. . . . . . . . . . . . . . 15
⊢ (𝑘 = 0 → ((!‘𝑘) / ∏𝑡 ∈ ∅ (!‘(𝑐‘𝑡))) = 1) |
185 | | prod0 14512 |
. . . . . . . . . . . . . . . 16
⊢
∏𝑡 ∈
∅ (((𝑆
D𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥) = 1 |
186 | 185 | a1i 11 |
. . . . . . . . . . . . . . 15
⊢ (𝑘 = 0 → ∏𝑡 ∈ ∅ (((𝑆 D𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥) = 1) |
187 | 184, 186 | oveq12d 6567 |
. . . . . . . . . . . . . 14
⊢ (𝑘 = 0 → (((!‘𝑘) / ∏𝑡 ∈ ∅ (!‘(𝑐‘𝑡))) · ∏𝑡 ∈ ∅ (((𝑆 D𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥)) = (1 · 1)) |
188 | 187 | ad2antlr 759 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑘 = 0) ∧ 𝑐 ∈ {∅}) → (((!‘𝑘) / ∏𝑡 ∈ ∅ (!‘(𝑐‘𝑡))) · ∏𝑡 ∈ ∅ (((𝑆 D𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥)) = (1 · 1)) |
189 | | 1t1e1 11052 |
. . . . . . . . . . . . . 14
⊢ (1
· 1) = 1 |
190 | 189 | a1i 11 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑘 = 0) ∧ 𝑐 ∈ {∅}) → (1 · 1) =
1) |
191 | 188, 190 | eqtrd 2644 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑘 = 0) ∧ 𝑐 ∈ {∅}) → (((!‘𝑘) / ∏𝑡 ∈ ∅ (!‘(𝑐‘𝑡))) · ∏𝑡 ∈ ∅ (((𝑆 D𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥)) = 1) |
192 | 191 | sumeq2dv 14281 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑘 = 0) → Σ𝑐 ∈ {∅} (((!‘𝑘) / ∏𝑡 ∈ ∅ (!‘(𝑐‘𝑡))) · ∏𝑡 ∈ ∅ (((𝑆 D𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥)) = Σ𝑐 ∈ {∅}1) |
193 | | ax-1cn 9873 |
. . . . . . . . . . . . 13
⊢ 1 ∈
ℂ |
194 | | eqidd 2611 |
. . . . . . . . . . . . . 14
⊢ (𝑐 = ∅ → 1 =
1) |
195 | 194 | sumsn 14319 |
. . . . . . . . . . . . 13
⊢ ((∅
∈ V ∧ 1 ∈ ℂ) → Σ𝑐 ∈ {∅}1 = 1) |
196 | 118, 193,
195 | mp2an 704 |
. . . . . . . . . . . 12
⊢
Σ𝑐 ∈
{∅}1 = 1 |
197 | 196 | a1i 11 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑘 = 0) → Σ𝑐 ∈ {∅}1 = 1) |
198 | 174, 192,
197 | 3eqtrd 2648 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑘 = 0) → Σ𝑐 ∈ ((𝐷‘∅)‘𝑘)(((!‘𝑘) / ∏𝑡 ∈ ∅ (!‘(𝑐‘𝑡))) · ∏𝑡 ∈ ∅ (((𝑆 D𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥)) = 1) |
199 | 198 | mpteq2dv 4673 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑘 = 0) → (𝑥 ∈ 𝑋 ↦ Σ𝑐 ∈ ((𝐷‘∅)‘𝑘)(((!‘𝑘) / ∏𝑡 ∈ ∅ (!‘(𝑐‘𝑡))) · ∏𝑡 ∈ ∅ (((𝑆 D𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥))) = (𝑥 ∈ 𝑋 ↦ 1)) |
200 | 199 | eqcomd 2616 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑘 = 0) → (𝑥 ∈ 𝑋 ↦ 1) = (𝑥 ∈ 𝑋 ↦ Σ𝑐 ∈ ((𝐷‘∅)‘𝑘)(((!‘𝑘) / ∏𝑡 ∈ ∅ (!‘(𝑐‘𝑡))) · ∏𝑡 ∈ ∅ (((𝑆 D𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥)))) |
201 | 75, 103, 200 | 3eqtrd 2648 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑘 = 0) → ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ ∏𝑡 ∈ ∅ ((𝐻‘𝑡)‘𝑥)))‘𝑘) = (𝑥 ∈ 𝑋 ↦ Σ𝑐 ∈ ((𝐷‘∅)‘𝑘)(((!‘𝑘) / ∏𝑡 ∈ ∅ (!‘(𝑐‘𝑡))) · ∏𝑡 ∈ ∅ (((𝑆 D𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥)))) |
202 | 201 | a1d 25 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑘 = 0) → (𝑘 ∈ (0...𝑁) → ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ ∏𝑡 ∈ ∅ ((𝐻‘𝑡)‘𝑥)))‘𝑘) = (𝑥 ∈ 𝑋 ↦ Σ𝑐 ∈ ((𝐷‘∅)‘𝑘)(((!‘𝑘) / ∏𝑡 ∈ ∅ (!‘(𝑐‘𝑡))) · ∏𝑡 ∈ ∅ (((𝑆 D𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥))))) |
203 | 71 | fveq1i 6104 |
. . . . . . . . 9
⊢ ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ ∏𝑡 ∈ ∅ ((𝐻‘𝑡)‘𝑥)))‘𝑘) = ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ 1))‘𝑘) |
204 | 203 | a1i 11 |
. . . . . . . 8
⊢ (((𝜑 ∧ ¬ 𝑘 = 0) ∧ 𝑘 ∈ (0...𝑁)) → ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ ∏𝑡 ∈ ∅ ((𝐻‘𝑡)‘𝑥)))‘𝑘) = ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ 1))‘𝑘)) |
205 | 76 | adantr 480 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ ¬ 𝑘 = 0) → 𝑆 ∈ {ℝ, ℂ}) |
206 | 205 | adantr 480 |
. . . . . . . . 9
⊢ (((𝜑 ∧ ¬ 𝑘 = 0) ∧ 𝑘 ∈ (0...𝑁)) → 𝑆 ∈ {ℝ, ℂ}) |
207 | 90 | adantr 480 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ ¬ 𝑘 = 0) → 𝑋 ∈
((TopOpen‘ℂfld) ↾t 𝑆)) |
208 | 207 | adantr 480 |
. . . . . . . . 9
⊢ (((𝜑 ∧ ¬ 𝑘 = 0) ∧ 𝑘 ∈ (0...𝑁)) → 𝑋 ∈
((TopOpen‘ℂfld) ↾t 𝑆)) |
209 | 193 | a1i 11 |
. . . . . . . . 9
⊢ (((𝜑 ∧ ¬ 𝑘 = 0) ∧ 𝑘 ∈ (0...𝑁)) → 1 ∈ ℂ) |
210 | | elfznn0 12302 |
. . . . . . . . . . . . 13
⊢ (𝑘 ∈ (0...𝑁) → 𝑘 ∈ ℕ0) |
211 | 210 | adantl 481 |
. . . . . . . . . . . 12
⊢ ((¬
𝑘 = 0 ∧ 𝑘 ∈ (0...𝑁)) → 𝑘 ∈ ℕ0) |
212 | | neqne 2790 |
. . . . . . . . . . . . 13
⊢ (¬
𝑘 = 0 → 𝑘 ≠ 0) |
213 | 212 | adantr 480 |
. . . . . . . . . . . 12
⊢ ((¬
𝑘 = 0 ∧ 𝑘 ∈ (0...𝑁)) → 𝑘 ≠ 0) |
214 | 211, 213 | jca 553 |
. . . . . . . . . . 11
⊢ ((¬
𝑘 = 0 ∧ 𝑘 ∈ (0...𝑁)) → (𝑘 ∈ ℕ0 ∧ 𝑘 ≠ 0)) |
215 | | elnnne0 11183 |
. . . . . . . . . . 11
⊢ (𝑘 ∈ ℕ ↔ (𝑘 ∈ ℕ0
∧ 𝑘 ≠
0)) |
216 | 214, 215 | sylibr 223 |
. . . . . . . . . 10
⊢ ((¬
𝑘 = 0 ∧ 𝑘 ∈ (0...𝑁)) → 𝑘 ∈ ℕ) |
217 | 216 | adantll 746 |
. . . . . . . . 9
⊢ (((𝜑 ∧ ¬ 𝑘 = 0) ∧ 𝑘 ∈ (0...𝑁)) → 𝑘 ∈ ℕ) |
218 | 206, 208,
209, 217 | dvnmptconst 38831 |
. . . . . . . 8
⊢ (((𝜑 ∧ ¬ 𝑘 = 0) ∧ 𝑘 ∈ (0...𝑁)) → ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ 1))‘𝑘) = (𝑥 ∈ 𝑋 ↦ 0)) |
219 | 143 | ad2antrr 758 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ ¬ 𝑘 = 0) ∧ 𝑘 ∈ (0...𝑁)) → (𝐷‘∅) = (𝑛 ∈ ℕ0 ↦ {𝑐 ∈ {∅} ∣
Σ𝑡 ∈ ∅
(𝑐‘𝑡) = 𝑛})) |
220 | | eqeq2 2621 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑛 = 𝑘 → (Σ𝑡 ∈ ∅ (𝑐‘𝑡) = 𝑛 ↔ Σ𝑡 ∈ ∅ (𝑐‘𝑡) = 𝑘)) |
221 | 220 | rabbidv 3164 |
. . . . . . . . . . . . . . . 16
⊢ (𝑛 = 𝑘 → {𝑐 ∈ {∅} ∣ Σ𝑡 ∈ ∅ (𝑐‘𝑡) = 𝑛} = {𝑐 ∈ {∅} ∣ Σ𝑡 ∈ ∅ (𝑐‘𝑡) = 𝑘}) |
222 | 221 | adantl 481 |
. . . . . . . . . . . . . . 15
⊢ ((¬
𝑘 = 0 ∧ 𝑛 = 𝑘) → {𝑐 ∈ {∅} ∣ Σ𝑡 ∈ ∅ (𝑐‘𝑡) = 𝑛} = {𝑐 ∈ {∅} ∣ Σ𝑡 ∈ ∅ (𝑐‘𝑡) = 𝑘}) |
223 | | eqidd 2611 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢
(Σ𝑡 ∈
∅ (𝑐‘𝑡) = 𝑘 → 𝑘 = 𝑘) |
224 | | id 22 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢
(Σ𝑡 ∈
∅ (𝑐‘𝑡) = 𝑘 → Σ𝑡 ∈ ∅ (𝑐‘𝑡) = 𝑘) |
225 | 224 | eqcomd 2616 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢
(Σ𝑡 ∈
∅ (𝑐‘𝑡) = 𝑘 → 𝑘 = Σ𝑡 ∈ ∅ (𝑐‘𝑡)) |
226 | 158 | a1i 11 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢
(Σ𝑡 ∈
∅ (𝑐‘𝑡) = 𝑘 → Σ𝑡 ∈ ∅ (𝑐‘𝑡) = 0) |
227 | 223, 225,
226 | 3eqtrd 2648 |
. . . . . . . . . . . . . . . . . . . . 21
⊢
(Σ𝑡 ∈
∅ (𝑐‘𝑡) = 𝑘 → 𝑘 = 0) |
228 | 227 | adantl 481 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝑐 ∈ {∅} ∧
Σ𝑡 ∈ ∅
(𝑐‘𝑡) = 𝑘) → 𝑘 = 0) |
229 | 228 | adantll 746 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((¬
𝑘 = 0 ∧ 𝑐 ∈ {∅}) ∧
Σ𝑡 ∈ ∅
(𝑐‘𝑡) = 𝑘) → 𝑘 = 0) |
230 | | simpll 786 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((¬
𝑘 = 0 ∧ 𝑐 ∈ {∅}) ∧
Σ𝑡 ∈ ∅
(𝑐‘𝑡) = 𝑘) → ¬ 𝑘 = 0) |
231 | 229, 230 | pm2.65da 598 |
. . . . . . . . . . . . . . . . . 18
⊢ ((¬
𝑘 = 0 ∧ 𝑐 ∈ {∅}) → ¬
Σ𝑡 ∈ ∅
(𝑐‘𝑡) = 𝑘) |
232 | 231 | ralrimiva 2949 |
. . . . . . . . . . . . . . . . 17
⊢ (¬
𝑘 = 0 → ∀𝑐 ∈ {∅} ¬
Σ𝑡 ∈ ∅
(𝑐‘𝑡) = 𝑘) |
233 | | rabeq0 3911 |
. . . . . . . . . . . . . . . . 17
⊢ ({𝑐 ∈ {∅} ∣
Σ𝑡 ∈ ∅
(𝑐‘𝑡) = 𝑘} = ∅ ↔ ∀𝑐 ∈ {∅} ¬ Σ𝑡 ∈ ∅ (𝑐‘𝑡) = 𝑘) |
234 | 232, 233 | sylibr 223 |
. . . . . . . . . . . . . . . 16
⊢ (¬
𝑘 = 0 → {𝑐 ∈ {∅} ∣
Σ𝑡 ∈ ∅
(𝑐‘𝑡) = 𝑘} = ∅) |
235 | 234 | adantr 480 |
. . . . . . . . . . . . . . 15
⊢ ((¬
𝑘 = 0 ∧ 𝑛 = 𝑘) → {𝑐 ∈ {∅} ∣ Σ𝑡 ∈ ∅ (𝑐‘𝑡) = 𝑘} = ∅) |
236 | 222, 235 | eqtrd 2644 |
. . . . . . . . . . . . . 14
⊢ ((¬
𝑘 = 0 ∧ 𝑛 = 𝑘) → {𝑐 ∈ {∅} ∣ Σ𝑡 ∈ ∅ (𝑐‘𝑡) = 𝑛} = ∅) |
237 | 236 | adantll 746 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ ¬ 𝑘 = 0) ∧ 𝑛 = 𝑘) → {𝑐 ∈ {∅} ∣ Σ𝑡 ∈ ∅ (𝑐‘𝑡) = 𝑛} = ∅) |
238 | 237 | adantlr 747 |
. . . . . . . . . . . 12
⊢ ((((𝜑 ∧ ¬ 𝑘 = 0) ∧ 𝑘 ∈ (0...𝑁)) ∧ 𝑛 = 𝑘) → {𝑐 ∈ {∅} ∣ Σ𝑡 ∈ ∅ (𝑐‘𝑡) = 𝑛} = ∅) |
239 | 210 | adantl 481 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ ¬ 𝑘 = 0) ∧ 𝑘 ∈ (0...𝑁)) → 𝑘 ∈ ℕ0) |
240 | 118 | a1i 11 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ ¬ 𝑘 = 0) ∧ 𝑘 ∈ (0...𝑁)) → ∅ ∈ V) |
241 | 219, 238,
239, 240 | fvmptd 6197 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ ¬ 𝑘 = 0) ∧ 𝑘 ∈ (0...𝑁)) → ((𝐷‘∅)‘𝑘) = ∅) |
242 | 241 | sumeq1d 14279 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ ¬ 𝑘 = 0) ∧ 𝑘 ∈ (0...𝑁)) → Σ𝑐 ∈ ((𝐷‘∅)‘𝑘)(((!‘𝑘) / ∏𝑡 ∈ ∅ (!‘(𝑐‘𝑡))) · ∏𝑡 ∈ ∅ (((𝑆 D𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥)) = Σ𝑐 ∈ ∅ (((!‘𝑘) / ∏𝑡 ∈ ∅ (!‘(𝑐‘𝑡))) · ∏𝑡 ∈ ∅ (((𝑆 D𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥))) |
243 | | sum0 14299 |
. . . . . . . . . . 11
⊢
Σ𝑐 ∈
∅ (((!‘𝑘) /
∏𝑡 ∈ ∅
(!‘(𝑐‘𝑡))) · ∏𝑡 ∈ ∅ (((𝑆 D𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥)) = 0 |
244 | 243 | a1i 11 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ ¬ 𝑘 = 0) ∧ 𝑘 ∈ (0...𝑁)) → Σ𝑐 ∈ ∅ (((!‘𝑘) / ∏𝑡 ∈ ∅ (!‘(𝑐‘𝑡))) · ∏𝑡 ∈ ∅ (((𝑆 D𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥)) = 0) |
245 | 242, 244 | eqtr2d 2645 |
. . . . . . . . 9
⊢ (((𝜑 ∧ ¬ 𝑘 = 0) ∧ 𝑘 ∈ (0...𝑁)) → 0 = Σ𝑐 ∈ ((𝐷‘∅)‘𝑘)(((!‘𝑘) / ∏𝑡 ∈ ∅ (!‘(𝑐‘𝑡))) · ∏𝑡 ∈ ∅ (((𝑆 D𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥))) |
246 | 245 | mpteq2dv 4673 |
. . . . . . . 8
⊢ (((𝜑 ∧ ¬ 𝑘 = 0) ∧ 𝑘 ∈ (0...𝑁)) → (𝑥 ∈ 𝑋 ↦ 0) = (𝑥 ∈ 𝑋 ↦ Σ𝑐 ∈ ((𝐷‘∅)‘𝑘)(((!‘𝑘) / ∏𝑡 ∈ ∅ (!‘(𝑐‘𝑡))) · ∏𝑡 ∈ ∅ (((𝑆 D𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥)))) |
247 | 204, 218,
246 | 3eqtrd 2648 |
. . . . . . 7
⊢ (((𝜑 ∧ ¬ 𝑘 = 0) ∧ 𝑘 ∈ (0...𝑁)) → ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ ∏𝑡 ∈ ∅ ((𝐻‘𝑡)‘𝑥)))‘𝑘) = (𝑥 ∈ 𝑋 ↦ Σ𝑐 ∈ ((𝐷‘∅)‘𝑘)(((!‘𝑘) / ∏𝑡 ∈ ∅ (!‘(𝑐‘𝑡))) · ∏𝑡 ∈ ∅ (((𝑆 D𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥)))) |
248 | 247 | ex 449 |
. . . . . 6
⊢ ((𝜑 ∧ ¬ 𝑘 = 0) → (𝑘 ∈ (0...𝑁) → ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ ∏𝑡 ∈ ∅ ((𝐻‘𝑡)‘𝑥)))‘𝑘) = (𝑥 ∈ 𝑋 ↦ Σ𝑐 ∈ ((𝐷‘∅)‘𝑘)(((!‘𝑘) / ∏𝑡 ∈ ∅ (!‘(𝑐‘𝑡))) · ∏𝑡 ∈ ∅ (((𝑆 D𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥))))) |
249 | 202, 248 | pm2.61dan 828 |
. . . . 5
⊢ (𝜑 → (𝑘 ∈ (0...𝑁) → ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ ∏𝑡 ∈ ∅ ((𝐻‘𝑡)‘𝑥)))‘𝑘) = (𝑥 ∈ 𝑋 ↦ Σ𝑐 ∈ ((𝐷‘∅)‘𝑘)(((!‘𝑘) / ∏𝑡 ∈ ∅ (!‘(𝑐‘𝑡))) · ∏𝑡 ∈ ∅ (((𝑆 D𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥))))) |
250 | 249 | ralrimiv 2948 |
. . . 4
⊢ (𝜑 → ∀𝑘 ∈ (0...𝑁)((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ ∏𝑡 ∈ ∅ ((𝐻‘𝑡)‘𝑥)))‘𝑘) = (𝑥 ∈ 𝑋 ↦ Σ𝑐 ∈ ((𝐷‘∅)‘𝑘)(((!‘𝑘) / ∏𝑡 ∈ ∅ (!‘(𝑐‘𝑡))) · ∏𝑡 ∈ ∅ (((𝑆 D𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥)))) |
251 | | simpll 786 |
. . . . . . . 8
⊢ ((((𝜑 ∧ (𝑟 ⊆ 𝑇 ∧ 𝑧 ∈ (𝑇 ∖ 𝑟))) ∧ ∀𝑘 ∈ (0...𝑁)((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ ∏𝑡 ∈ 𝑟 ((𝐻‘𝑡)‘𝑥)))‘𝑘) = (𝑥 ∈ 𝑋 ↦ Σ𝑐 ∈ ((𝐷‘𝑟)‘𝑘)(((!‘𝑘) / ∏𝑡 ∈ 𝑟 (!‘(𝑐‘𝑡))) · ∏𝑡 ∈ 𝑟 (((𝑆 D𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥)))) ∧ 𝑗 ∈ (0...𝑁)) → (𝜑 ∧ (𝑟 ⊆ 𝑇 ∧ 𝑧 ∈ (𝑇 ∖ 𝑟)))) |
252 | | fveq2 6103 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑥 = 𝑦 → ((𝐻‘𝑡)‘𝑥) = ((𝐻‘𝑡)‘𝑦)) |
253 | 252 | prodeq2ad 38659 |
. . . . . . . . . . . . . . . 16
⊢ (𝑥 = 𝑦 → ∏𝑡 ∈ 𝑟 ((𝐻‘𝑡)‘𝑥) = ∏𝑡 ∈ 𝑟 ((𝐻‘𝑡)‘𝑦)) |
254 | | fveq2 6103 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑡 = 𝑢 → (𝐻‘𝑡) = (𝐻‘𝑢)) |
255 | 254 | fveq1d 6105 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑡 = 𝑢 → ((𝐻‘𝑡)‘𝑦) = ((𝐻‘𝑢)‘𝑦)) |
256 | 255 | cbvprodv 14485 |
. . . . . . . . . . . . . . . . 17
⊢
∏𝑡 ∈
𝑟 ((𝐻‘𝑡)‘𝑦) = ∏𝑢 ∈ 𝑟 ((𝐻‘𝑢)‘𝑦) |
257 | 256 | a1i 11 |
. . . . . . . . . . . . . . . 16
⊢ (𝑥 = 𝑦 → ∏𝑡 ∈ 𝑟 ((𝐻‘𝑡)‘𝑦) = ∏𝑢 ∈ 𝑟 ((𝐻‘𝑢)‘𝑦)) |
258 | 253, 257 | eqtrd 2644 |
. . . . . . . . . . . . . . 15
⊢ (𝑥 = 𝑦 → ∏𝑡 ∈ 𝑟 ((𝐻‘𝑡)‘𝑥) = ∏𝑢 ∈ 𝑟 ((𝐻‘𝑢)‘𝑦)) |
259 | 258 | cbvmptv 4678 |
. . . . . . . . . . . . . 14
⊢ (𝑥 ∈ 𝑋 ↦ ∏𝑡 ∈ 𝑟 ((𝐻‘𝑡)‘𝑥)) = (𝑦 ∈ 𝑋 ↦ ∏𝑢 ∈ 𝑟 ((𝐻‘𝑢)‘𝑦)) |
260 | 259 | oveq2i 6560 |
. . . . . . . . . . . . 13
⊢ (𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ ∏𝑡 ∈ 𝑟 ((𝐻‘𝑡)‘𝑥))) = (𝑆 D𝑛 (𝑦 ∈ 𝑋 ↦ ∏𝑢 ∈ 𝑟 ((𝐻‘𝑢)‘𝑦))) |
261 | 260 | fveq1i 6104 |
. . . . . . . . . . . 12
⊢ ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ ∏𝑡 ∈ 𝑟 ((𝐻‘𝑡)‘𝑥)))‘𝑘) = ((𝑆 D𝑛 (𝑦 ∈ 𝑋 ↦ ∏𝑢 ∈ 𝑟 ((𝐻‘𝑢)‘𝑦)))‘𝑘) |
262 | | fveq2 6103 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑡 = 𝑢 → (𝑐‘𝑡) = (𝑐‘𝑢)) |
263 | 262 | fveq2d 6107 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑡 = 𝑢 → (!‘(𝑐‘𝑡)) = (!‘(𝑐‘𝑢))) |
264 | 263 | cbvprodv 14485 |
. . . . . . . . . . . . . . . . . 18
⊢
∏𝑡 ∈
𝑟 (!‘(𝑐‘𝑡)) = ∏𝑢 ∈ 𝑟 (!‘(𝑐‘𝑢)) |
265 | 264 | oveq2i 6560 |
. . . . . . . . . . . . . . . . 17
⊢
((!‘𝑘) /
∏𝑡 ∈ 𝑟 (!‘(𝑐‘𝑡))) = ((!‘𝑘) / ∏𝑢 ∈ 𝑟 (!‘(𝑐‘𝑢))) |
266 | 265 | a1i 11 |
. . . . . . . . . . . . . . . 16
⊢ (𝑥 = 𝑦 → ((!‘𝑘) / ∏𝑡 ∈ 𝑟 (!‘(𝑐‘𝑡))) = ((!‘𝑘) / ∏𝑢 ∈ 𝑟 (!‘(𝑐‘𝑢)))) |
267 | | fveq2 6103 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑥 = 𝑦 → (((𝑆 D𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥) = (((𝑆 D𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑦)) |
268 | 267 | prodeq2ad 38659 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑥 = 𝑦 → ∏𝑡 ∈ 𝑟 (((𝑆 D𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥) = ∏𝑡 ∈ 𝑟 (((𝑆 D𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑦)) |
269 | 254 | oveq2d 6565 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑡 = 𝑢 → (𝑆 D𝑛 (𝐻‘𝑡)) = (𝑆 D𝑛 (𝐻‘𝑢))) |
270 | 269, 262 | fveq12d 6109 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑡 = 𝑢 → ((𝑆 D𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡)) = ((𝑆 D𝑛 (𝐻‘𝑢))‘(𝑐‘𝑢))) |
271 | 270 | fveq1d 6105 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑡 = 𝑢 → (((𝑆 D𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑦) = (((𝑆 D𝑛 (𝐻‘𝑢))‘(𝑐‘𝑢))‘𝑦)) |
272 | 271 | cbvprodv 14485 |
. . . . . . . . . . . . . . . . . 18
⊢
∏𝑡 ∈
𝑟 (((𝑆 D𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑦) = ∏𝑢 ∈ 𝑟 (((𝑆 D𝑛 (𝐻‘𝑢))‘(𝑐‘𝑢))‘𝑦) |
273 | 272 | a1i 11 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑥 = 𝑦 → ∏𝑡 ∈ 𝑟 (((𝑆 D𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑦) = ∏𝑢 ∈ 𝑟 (((𝑆 D𝑛 (𝐻‘𝑢))‘(𝑐‘𝑢))‘𝑦)) |
274 | 268, 273 | eqtrd 2644 |
. . . . . . . . . . . . . . . 16
⊢ (𝑥 = 𝑦 → ∏𝑡 ∈ 𝑟 (((𝑆 D𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥) = ∏𝑢 ∈ 𝑟 (((𝑆 D𝑛 (𝐻‘𝑢))‘(𝑐‘𝑢))‘𝑦)) |
275 | 266, 274 | oveq12d 6567 |
. . . . . . . . . . . . . . 15
⊢ (𝑥 = 𝑦 → (((!‘𝑘) / ∏𝑡 ∈ 𝑟 (!‘(𝑐‘𝑡))) · ∏𝑡 ∈ 𝑟 (((𝑆 D𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥)) = (((!‘𝑘) / ∏𝑢 ∈ 𝑟 (!‘(𝑐‘𝑢))) · ∏𝑢 ∈ 𝑟 (((𝑆 D𝑛 (𝐻‘𝑢))‘(𝑐‘𝑢))‘𝑦))) |
276 | 275 | sumeq2ad 38632 |
. . . . . . . . . . . . . 14
⊢ (𝑥 = 𝑦 → Σ𝑐 ∈ ((𝐷‘𝑟)‘𝑘)(((!‘𝑘) / ∏𝑡 ∈ 𝑟 (!‘(𝑐‘𝑡))) · ∏𝑡 ∈ 𝑟 (((𝑆 D𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥)) = Σ𝑐 ∈ ((𝐷‘𝑟)‘𝑘)(((!‘𝑘) / ∏𝑢 ∈ 𝑟 (!‘(𝑐‘𝑢))) · ∏𝑢 ∈ 𝑟 (((𝑆 D𝑛 (𝐻‘𝑢))‘(𝑐‘𝑢))‘𝑦))) |
277 | | fveq1 6102 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑐 = 𝑑 → (𝑐‘𝑢) = (𝑑‘𝑢)) |
278 | 277 | fveq2d 6107 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑐 = 𝑑 → (!‘(𝑐‘𝑢)) = (!‘(𝑑‘𝑢))) |
279 | 278 | prodeq2ad 38659 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑐 = 𝑑 → ∏𝑢 ∈ 𝑟 (!‘(𝑐‘𝑢)) = ∏𝑢 ∈ 𝑟 (!‘(𝑑‘𝑢))) |
280 | 279 | oveq2d 6565 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑐 = 𝑑 → ((!‘𝑘) / ∏𝑢 ∈ 𝑟 (!‘(𝑐‘𝑢))) = ((!‘𝑘) / ∏𝑢 ∈ 𝑟 (!‘(𝑑‘𝑢)))) |
281 | 277 | fveq2d 6107 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑐 = 𝑑 → ((𝑆 D𝑛 (𝐻‘𝑢))‘(𝑐‘𝑢)) = ((𝑆 D𝑛 (𝐻‘𝑢))‘(𝑑‘𝑢))) |
282 | 281 | fveq1d 6105 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑐 = 𝑑 → (((𝑆 D𝑛 (𝐻‘𝑢))‘(𝑐‘𝑢))‘𝑦) = (((𝑆 D𝑛 (𝐻‘𝑢))‘(𝑑‘𝑢))‘𝑦)) |
283 | 282 | prodeq2ad 38659 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑐 = 𝑑 → ∏𝑢 ∈ 𝑟 (((𝑆 D𝑛 (𝐻‘𝑢))‘(𝑐‘𝑢))‘𝑦) = ∏𝑢 ∈ 𝑟 (((𝑆 D𝑛 (𝐻‘𝑢))‘(𝑑‘𝑢))‘𝑦)) |
284 | 280, 283 | oveq12d 6567 |
. . . . . . . . . . . . . . . 16
⊢ (𝑐 = 𝑑 → (((!‘𝑘) / ∏𝑢 ∈ 𝑟 (!‘(𝑐‘𝑢))) · ∏𝑢 ∈ 𝑟 (((𝑆 D𝑛 (𝐻‘𝑢))‘(𝑐‘𝑢))‘𝑦)) = (((!‘𝑘) / ∏𝑢 ∈ 𝑟 (!‘(𝑑‘𝑢))) · ∏𝑢 ∈ 𝑟 (((𝑆 D𝑛 (𝐻‘𝑢))‘(𝑑‘𝑢))‘𝑦))) |
285 | 284 | cbvsumv 14274 |
. . . . . . . . . . . . . . 15
⊢
Σ𝑐 ∈
((𝐷‘𝑟)‘𝑘)(((!‘𝑘) / ∏𝑢 ∈ 𝑟 (!‘(𝑐‘𝑢))) · ∏𝑢 ∈ 𝑟 (((𝑆 D𝑛 (𝐻‘𝑢))‘(𝑐‘𝑢))‘𝑦)) = Σ𝑑 ∈ ((𝐷‘𝑟)‘𝑘)(((!‘𝑘) / ∏𝑢 ∈ 𝑟 (!‘(𝑑‘𝑢))) · ∏𝑢 ∈ 𝑟 (((𝑆 D𝑛 (𝐻‘𝑢))‘(𝑑‘𝑢))‘𝑦)) |
286 | 285 | a1i 11 |
. . . . . . . . . . . . . 14
⊢ (𝑥 = 𝑦 → Σ𝑐 ∈ ((𝐷‘𝑟)‘𝑘)(((!‘𝑘) / ∏𝑢 ∈ 𝑟 (!‘(𝑐‘𝑢))) · ∏𝑢 ∈ 𝑟 (((𝑆 D𝑛 (𝐻‘𝑢))‘(𝑐‘𝑢))‘𝑦)) = Σ𝑑 ∈ ((𝐷‘𝑟)‘𝑘)(((!‘𝑘) / ∏𝑢 ∈ 𝑟 (!‘(𝑑‘𝑢))) · ∏𝑢 ∈ 𝑟 (((𝑆 D𝑛 (𝐻‘𝑢))‘(𝑑‘𝑢))‘𝑦))) |
287 | 276, 286 | eqtrd 2644 |
. . . . . . . . . . . . 13
⊢ (𝑥 = 𝑦 → Σ𝑐 ∈ ((𝐷‘𝑟)‘𝑘)(((!‘𝑘) / ∏𝑡 ∈ 𝑟 (!‘(𝑐‘𝑡))) · ∏𝑡 ∈ 𝑟 (((𝑆 D𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥)) = Σ𝑑 ∈ ((𝐷‘𝑟)‘𝑘)(((!‘𝑘) / ∏𝑢 ∈ 𝑟 (!‘(𝑑‘𝑢))) · ∏𝑢 ∈ 𝑟 (((𝑆 D𝑛 (𝐻‘𝑢))‘(𝑑‘𝑢))‘𝑦))) |
288 | 287 | cbvmptv 4678 |
. . . . . . . . . . . 12
⊢ (𝑥 ∈ 𝑋 ↦ Σ𝑐 ∈ ((𝐷‘𝑟)‘𝑘)(((!‘𝑘) / ∏𝑡 ∈ 𝑟 (!‘(𝑐‘𝑡))) · ∏𝑡 ∈ 𝑟 (((𝑆 D𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥))) = (𝑦 ∈ 𝑋 ↦ Σ𝑑 ∈ ((𝐷‘𝑟)‘𝑘)(((!‘𝑘) / ∏𝑢 ∈ 𝑟 (!‘(𝑑‘𝑢))) · ∏𝑢 ∈ 𝑟 (((𝑆 D𝑛 (𝐻‘𝑢))‘(𝑑‘𝑢))‘𝑦))) |
289 | 261, 288 | eqeq12i 2624 |
. . . . . . . . . . 11
⊢ (((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ ∏𝑡 ∈ 𝑟 ((𝐻‘𝑡)‘𝑥)))‘𝑘) = (𝑥 ∈ 𝑋 ↦ Σ𝑐 ∈ ((𝐷‘𝑟)‘𝑘)(((!‘𝑘) / ∏𝑡 ∈ 𝑟 (!‘(𝑐‘𝑡))) · ∏𝑡 ∈ 𝑟 (((𝑆 D𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥))) ↔ ((𝑆 D𝑛 (𝑦 ∈ 𝑋 ↦ ∏𝑢 ∈ 𝑟 ((𝐻‘𝑢)‘𝑦)))‘𝑘) = (𝑦 ∈ 𝑋 ↦ Σ𝑑 ∈ ((𝐷‘𝑟)‘𝑘)(((!‘𝑘) / ∏𝑢 ∈ 𝑟 (!‘(𝑑‘𝑢))) · ∏𝑢 ∈ 𝑟 (((𝑆 D𝑛 (𝐻‘𝑢))‘(𝑑‘𝑢))‘𝑦)))) |
290 | 289 | ralbii 2963 |
. . . . . . . . . 10
⊢
(∀𝑘 ∈
(0...𝑁)((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ ∏𝑡 ∈ 𝑟 ((𝐻‘𝑡)‘𝑥)))‘𝑘) = (𝑥 ∈ 𝑋 ↦ Σ𝑐 ∈ ((𝐷‘𝑟)‘𝑘)(((!‘𝑘) / ∏𝑡 ∈ 𝑟 (!‘(𝑐‘𝑡))) · ∏𝑡 ∈ 𝑟 (((𝑆 D𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥))) ↔ ∀𝑘 ∈ (0...𝑁)((𝑆 D𝑛 (𝑦 ∈ 𝑋 ↦ ∏𝑢 ∈ 𝑟 ((𝐻‘𝑢)‘𝑦)))‘𝑘) = (𝑦 ∈ 𝑋 ↦ Σ𝑑 ∈ ((𝐷‘𝑟)‘𝑘)(((!‘𝑘) / ∏𝑢 ∈ 𝑟 (!‘(𝑑‘𝑢))) · ∏𝑢 ∈ 𝑟 (((𝑆 D𝑛 (𝐻‘𝑢))‘(𝑑‘𝑢))‘𝑦)))) |
291 | 290 | biimpi 205 |
. . . . . . . . 9
⊢
(∀𝑘 ∈
(0...𝑁)((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ ∏𝑡 ∈ 𝑟 ((𝐻‘𝑡)‘𝑥)))‘𝑘) = (𝑥 ∈ 𝑋 ↦ Σ𝑐 ∈ ((𝐷‘𝑟)‘𝑘)(((!‘𝑘) / ∏𝑡 ∈ 𝑟 (!‘(𝑐‘𝑡))) · ∏𝑡 ∈ 𝑟 (((𝑆 D𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥))) → ∀𝑘 ∈ (0...𝑁)((𝑆 D𝑛 (𝑦 ∈ 𝑋 ↦ ∏𝑢 ∈ 𝑟 ((𝐻‘𝑢)‘𝑦)))‘𝑘) = (𝑦 ∈ 𝑋 ↦ Σ𝑑 ∈ ((𝐷‘𝑟)‘𝑘)(((!‘𝑘) / ∏𝑢 ∈ 𝑟 (!‘(𝑑‘𝑢))) · ∏𝑢 ∈ 𝑟 (((𝑆 D𝑛 (𝐻‘𝑢))‘(𝑑‘𝑢))‘𝑦)))) |
292 | 291 | ad2antlr 759 |
. . . . . . . 8
⊢ ((((𝜑 ∧ (𝑟 ⊆ 𝑇 ∧ 𝑧 ∈ (𝑇 ∖ 𝑟))) ∧ ∀𝑘 ∈ (0...𝑁)((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ ∏𝑡 ∈ 𝑟 ((𝐻‘𝑡)‘𝑥)))‘𝑘) = (𝑥 ∈ 𝑋 ↦ Σ𝑐 ∈ ((𝐷‘𝑟)‘𝑘)(((!‘𝑘) / ∏𝑡 ∈ 𝑟 (!‘(𝑐‘𝑡))) · ∏𝑡 ∈ 𝑟 (((𝑆 D𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥)))) ∧ 𝑗 ∈ (0...𝑁)) → ∀𝑘 ∈ (0...𝑁)((𝑆 D𝑛 (𝑦 ∈ 𝑋 ↦ ∏𝑢 ∈ 𝑟 ((𝐻‘𝑢)‘𝑦)))‘𝑘) = (𝑦 ∈ 𝑋 ↦ Σ𝑑 ∈ ((𝐷‘𝑟)‘𝑘)(((!‘𝑘) / ∏𝑢 ∈ 𝑟 (!‘(𝑑‘𝑢))) · ∏𝑢 ∈ 𝑟 (((𝑆 D𝑛 (𝐻‘𝑢))‘(𝑑‘𝑢))‘𝑦)))) |
293 | | simpr 476 |
. . . . . . . 8
⊢ ((((𝜑 ∧ (𝑟 ⊆ 𝑇 ∧ 𝑧 ∈ (𝑇 ∖ 𝑟))) ∧ ∀𝑘 ∈ (0...𝑁)((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ ∏𝑡 ∈ 𝑟 ((𝐻‘𝑡)‘𝑥)))‘𝑘) = (𝑥 ∈ 𝑋 ↦ Σ𝑐 ∈ ((𝐷‘𝑟)‘𝑘)(((!‘𝑘) / ∏𝑡 ∈ 𝑟 (!‘(𝑐‘𝑡))) · ∏𝑡 ∈ 𝑟 (((𝑆 D𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥)))) ∧ 𝑗 ∈ (0...𝑁)) → 𝑗 ∈ (0...𝑁)) |
294 | 76 | ad3antrrr 762 |
. . . . . . . . 9
⊢ ((((𝜑 ∧ (𝑟 ⊆ 𝑇 ∧ 𝑧 ∈ (𝑇 ∖ 𝑟))) ∧ ∀𝑘 ∈ (0...𝑁)((𝑆 D𝑛 (𝑦 ∈ 𝑋 ↦ ∏𝑢 ∈ 𝑟 ((𝐻‘𝑢)‘𝑦)))‘𝑘) = (𝑦 ∈ 𝑋 ↦ Σ𝑑 ∈ ((𝐷‘𝑟)‘𝑘)(((!‘𝑘) / ∏𝑢 ∈ 𝑟 (!‘(𝑑‘𝑢))) · ∏𝑢 ∈ 𝑟 (((𝑆 D𝑛 (𝐻‘𝑢))‘(𝑑‘𝑢))‘𝑦)))) ∧ 𝑗 ∈ (0...𝑁)) → 𝑆 ∈ {ℝ, ℂ}) |
295 | 90 | ad3antrrr 762 |
. . . . . . . . 9
⊢ ((((𝜑 ∧ (𝑟 ⊆ 𝑇 ∧ 𝑧 ∈ (𝑇 ∖ 𝑟))) ∧ ∀𝑘 ∈ (0...𝑁)((𝑆 D𝑛 (𝑦 ∈ 𝑋 ↦ ∏𝑢 ∈ 𝑟 ((𝐻‘𝑢)‘𝑦)))‘𝑘) = (𝑦 ∈ 𝑋 ↦ Σ𝑑 ∈ ((𝐷‘𝑟)‘𝑘)(((!‘𝑘) / ∏𝑢 ∈ 𝑟 (!‘(𝑑‘𝑢))) · ∏𝑢 ∈ 𝑟 (((𝑆 D𝑛 (𝐻‘𝑢))‘(𝑑‘𝑢))‘𝑦)))) ∧ 𝑗 ∈ (0...𝑁)) → 𝑋 ∈
((TopOpen‘ℂfld) ↾t 𝑆)) |
296 | | dvnprodlem3.t |
. . . . . . . . . 10
⊢ (𝜑 → 𝑇 ∈ Fin) |
297 | 296 | ad3antrrr 762 |
. . . . . . . . 9
⊢ ((((𝜑 ∧ (𝑟 ⊆ 𝑇 ∧ 𝑧 ∈ (𝑇 ∖ 𝑟))) ∧ ∀𝑘 ∈ (0...𝑁)((𝑆 D𝑛 (𝑦 ∈ 𝑋 ↦ ∏𝑢 ∈ 𝑟 ((𝐻‘𝑢)‘𝑦)))‘𝑘) = (𝑦 ∈ 𝑋 ↦ Σ𝑑 ∈ ((𝐷‘𝑟)‘𝑘)(((!‘𝑘) / ∏𝑢 ∈ 𝑟 (!‘(𝑑‘𝑢))) · ∏𝑢 ∈ 𝑟 (((𝑆 D𝑛 (𝐻‘𝑢))‘(𝑑‘𝑢))‘𝑦)))) ∧ 𝑗 ∈ (0...𝑁)) → 𝑇 ∈ Fin) |
298 | | simp-4l 802 |
. . . . . . . . . 10
⊢
(((((𝜑 ∧ (𝑟 ⊆ 𝑇 ∧ 𝑧 ∈ (𝑇 ∖ 𝑟))) ∧ ∀𝑘 ∈ (0...𝑁)((𝑆 D𝑛 (𝑦 ∈ 𝑋 ↦ ∏𝑢 ∈ 𝑟 ((𝐻‘𝑢)‘𝑦)))‘𝑘) = (𝑦 ∈ 𝑋 ↦ Σ𝑑 ∈ ((𝐷‘𝑟)‘𝑘)(((!‘𝑘) / ∏𝑢 ∈ 𝑟 (!‘(𝑑‘𝑢))) · ∏𝑢 ∈ 𝑟 (((𝑆 D𝑛 (𝐻‘𝑢))‘(𝑑‘𝑢))‘𝑦)))) ∧ 𝑗 ∈ (0...𝑁)) ∧ 𝑡 ∈ 𝑇) → 𝜑) |
299 | | simpr 476 |
. . . . . . . . . 10
⊢
(((((𝜑 ∧ (𝑟 ⊆ 𝑇 ∧ 𝑧 ∈ (𝑇 ∖ 𝑟))) ∧ ∀𝑘 ∈ (0...𝑁)((𝑆 D𝑛 (𝑦 ∈ 𝑋 ↦ ∏𝑢 ∈ 𝑟 ((𝐻‘𝑢)‘𝑦)))‘𝑘) = (𝑦 ∈ 𝑋 ↦ Σ𝑑 ∈ ((𝐷‘𝑟)‘𝑘)(((!‘𝑘) / ∏𝑢 ∈ 𝑟 (!‘(𝑑‘𝑢))) · ∏𝑢 ∈ 𝑟 (((𝑆 D𝑛 (𝐻‘𝑢))‘(𝑑‘𝑢))‘𝑦)))) ∧ 𝑗 ∈ (0...𝑁)) ∧ 𝑡 ∈ 𝑇) → 𝑡 ∈ 𝑇) |
300 | | dvnprodlem3.h |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → (𝐻‘𝑡):𝑋⟶ℂ) |
301 | 298, 299,
300 | syl2anc 691 |
. . . . . . . . 9
⊢
(((((𝜑 ∧ (𝑟 ⊆ 𝑇 ∧ 𝑧 ∈ (𝑇 ∖ 𝑟))) ∧ ∀𝑘 ∈ (0...𝑁)((𝑆 D𝑛 (𝑦 ∈ 𝑋 ↦ ∏𝑢 ∈ 𝑟 ((𝐻‘𝑢)‘𝑦)))‘𝑘) = (𝑦 ∈ 𝑋 ↦ Σ𝑑 ∈ ((𝐷‘𝑟)‘𝑘)(((!‘𝑘) / ∏𝑢 ∈ 𝑟 (!‘(𝑑‘𝑢))) · ∏𝑢 ∈ 𝑟 (((𝑆 D𝑛 (𝐻‘𝑢))‘(𝑑‘𝑢))‘𝑦)))) ∧ 𝑗 ∈ (0...𝑁)) ∧ 𝑡 ∈ 𝑇) → (𝐻‘𝑡):𝑋⟶ℂ) |
302 | | dvnprodlem3.n |
. . . . . . . . . 10
⊢ (𝜑 → 𝑁 ∈
ℕ0) |
303 | 302 | ad3antrrr 762 |
. . . . . . . . 9
⊢ ((((𝜑 ∧ (𝑟 ⊆ 𝑇 ∧ 𝑧 ∈ (𝑇 ∖ 𝑟))) ∧ ∀𝑘 ∈ (0...𝑁)((𝑆 D𝑛 (𝑦 ∈ 𝑋 ↦ ∏𝑢 ∈ 𝑟 ((𝐻‘𝑢)‘𝑦)))‘𝑘) = (𝑦 ∈ 𝑋 ↦ Σ𝑑 ∈ ((𝐷‘𝑟)‘𝑘)(((!‘𝑘) / ∏𝑢 ∈ 𝑟 (!‘(𝑑‘𝑢))) · ∏𝑢 ∈ 𝑟 (((𝑆 D𝑛 (𝐻‘𝑢))‘(𝑑‘𝑢))‘𝑦)))) ∧ 𝑗 ∈ (0...𝑁)) → 𝑁 ∈
ℕ0) |
304 | | simplll 794 |
. . . . . . . . . . 11
⊢ ((((𝜑 ∧ (𝑟 ⊆ 𝑇 ∧ 𝑧 ∈ (𝑇 ∖ 𝑟))) ∧ ∀𝑘 ∈ (0...𝑁)((𝑆 D𝑛 (𝑦 ∈ 𝑋 ↦ ∏𝑢 ∈ 𝑟 ((𝐻‘𝑢)‘𝑦)))‘𝑘) = (𝑦 ∈ 𝑋 ↦ Σ𝑑 ∈ ((𝐷‘𝑟)‘𝑘)(((!‘𝑘) / ∏𝑢 ∈ 𝑟 (!‘(𝑑‘𝑢))) · ∏𝑢 ∈ 𝑟 (((𝑆 D𝑛 (𝐻‘𝑢))‘(𝑑‘𝑢))‘𝑦)))) ∧ 𝑗 ∈ (0...𝑁)) → 𝜑) |
305 | 304 | 3ad2ant1 1075 |
. . . . . . . . . 10
⊢
(((((𝜑 ∧ (𝑟 ⊆ 𝑇 ∧ 𝑧 ∈ (𝑇 ∖ 𝑟))) ∧ ∀𝑘 ∈ (0...𝑁)((𝑆 D𝑛 (𝑦 ∈ 𝑋 ↦ ∏𝑢 ∈ 𝑟 ((𝐻‘𝑢)‘𝑦)))‘𝑘) = (𝑦 ∈ 𝑋 ↦ Σ𝑑 ∈ ((𝐷‘𝑟)‘𝑘)(((!‘𝑘) / ∏𝑢 ∈ 𝑟 (!‘(𝑑‘𝑢))) · ∏𝑢 ∈ 𝑟 (((𝑆 D𝑛 (𝐻‘𝑢))‘(𝑑‘𝑢))‘𝑦)))) ∧ 𝑗 ∈ (0...𝑁)) ∧ 𝑡 ∈ 𝑇 ∧ ℎ ∈ (0...𝑁)) → 𝜑) |
306 | | simp2 1055 |
. . . . . . . . . 10
⊢
(((((𝜑 ∧ (𝑟 ⊆ 𝑇 ∧ 𝑧 ∈ (𝑇 ∖ 𝑟))) ∧ ∀𝑘 ∈ (0...𝑁)((𝑆 D𝑛 (𝑦 ∈ 𝑋 ↦ ∏𝑢 ∈ 𝑟 ((𝐻‘𝑢)‘𝑦)))‘𝑘) = (𝑦 ∈ 𝑋 ↦ Σ𝑑 ∈ ((𝐷‘𝑟)‘𝑘)(((!‘𝑘) / ∏𝑢 ∈ 𝑟 (!‘(𝑑‘𝑢))) · ∏𝑢 ∈ 𝑟 (((𝑆 D𝑛 (𝐻‘𝑢))‘(𝑑‘𝑢))‘𝑦)))) ∧ 𝑗 ∈ (0...𝑁)) ∧ 𝑡 ∈ 𝑇 ∧ ℎ ∈ (0...𝑁)) → 𝑡 ∈ 𝑇) |
307 | | simp3 1056 |
. . . . . . . . . 10
⊢
(((((𝜑 ∧ (𝑟 ⊆ 𝑇 ∧ 𝑧 ∈ (𝑇 ∖ 𝑟))) ∧ ∀𝑘 ∈ (0...𝑁)((𝑆 D𝑛 (𝑦 ∈ 𝑋 ↦ ∏𝑢 ∈ 𝑟 ((𝐻‘𝑢)‘𝑦)))‘𝑘) = (𝑦 ∈ 𝑋 ↦ Σ𝑑 ∈ ((𝐷‘𝑟)‘𝑘)(((!‘𝑘) / ∏𝑢 ∈ 𝑟 (!‘(𝑑‘𝑢))) · ∏𝑢 ∈ 𝑟 (((𝑆 D𝑛 (𝐻‘𝑢))‘(𝑑‘𝑢))‘𝑦)))) ∧ 𝑗 ∈ (0...𝑁)) ∧ 𝑡 ∈ 𝑇 ∧ ℎ ∈ (0...𝑁)) → ℎ ∈ (0...𝑁)) |
308 | | eleq1 2676 |
. . . . . . . . . . . . 13
⊢ (𝑗 = ℎ → (𝑗 ∈ (0...𝑁) ↔ ℎ ∈ (0...𝑁))) |
309 | 308 | 3anbi3d 1397 |
. . . . . . . . . . . 12
⊢ (𝑗 = ℎ → ((𝜑 ∧ 𝑡 ∈ 𝑇 ∧ 𝑗 ∈ (0...𝑁)) ↔ (𝜑 ∧ 𝑡 ∈ 𝑇 ∧ ℎ ∈ (0...𝑁)))) |
310 | | fveq2 6103 |
. . . . . . . . . . . . 13
⊢ (𝑗 = ℎ → ((𝑆 D𝑛 (𝐻‘𝑡))‘𝑗) = ((𝑆 D𝑛 (𝐻‘𝑡))‘ℎ)) |
311 | 310 | feq1d 5943 |
. . . . . . . . . . . 12
⊢ (𝑗 = ℎ → (((𝑆 D𝑛 (𝐻‘𝑡))‘𝑗):𝑋⟶ℂ ↔ ((𝑆 D𝑛 (𝐻‘𝑡))‘ℎ):𝑋⟶ℂ)) |
312 | 309, 311 | imbi12d 333 |
. . . . . . . . . . 11
⊢ (𝑗 = ℎ → (((𝜑 ∧ 𝑡 ∈ 𝑇 ∧ 𝑗 ∈ (0...𝑁)) → ((𝑆 D𝑛 (𝐻‘𝑡))‘𝑗):𝑋⟶ℂ) ↔ ((𝜑 ∧ 𝑡 ∈ 𝑇 ∧ ℎ ∈ (0...𝑁)) → ((𝑆 D𝑛 (𝐻‘𝑡))‘ℎ):𝑋⟶ℂ))) |
313 | | dvnprodlem3.dvnh |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇 ∧ 𝑗 ∈ (0...𝑁)) → ((𝑆 D𝑛 (𝐻‘𝑡))‘𝑗):𝑋⟶ℂ) |
314 | 312, 313 | chvarv 2251 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇 ∧ ℎ ∈ (0...𝑁)) → ((𝑆 D𝑛 (𝐻‘𝑡))‘ℎ):𝑋⟶ℂ) |
315 | 305, 306,
307, 314 | syl3anc 1318 |
. . . . . . . . 9
⊢
(((((𝜑 ∧ (𝑟 ⊆ 𝑇 ∧ 𝑧 ∈ (𝑇 ∖ 𝑟))) ∧ ∀𝑘 ∈ (0...𝑁)((𝑆 D𝑛 (𝑦 ∈ 𝑋 ↦ ∏𝑢 ∈ 𝑟 ((𝐻‘𝑢)‘𝑦)))‘𝑘) = (𝑦 ∈ 𝑋 ↦ Σ𝑑 ∈ ((𝐷‘𝑟)‘𝑘)(((!‘𝑘) / ∏𝑢 ∈ 𝑟 (!‘(𝑑‘𝑢))) · ∏𝑢 ∈ 𝑟 (((𝑆 D𝑛 (𝐻‘𝑢))‘(𝑑‘𝑢))‘𝑦)))) ∧ 𝑗 ∈ (0...𝑁)) ∧ 𝑡 ∈ 𝑇 ∧ ℎ ∈ (0...𝑁)) → ((𝑆 D𝑛 (𝐻‘𝑡))‘ℎ):𝑋⟶ℂ) |
316 | | simprl 790 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ (𝑟 ⊆ 𝑇 ∧ 𝑧 ∈ (𝑇 ∖ 𝑟))) → 𝑟 ⊆ 𝑇) |
317 | 316 | ad2antrr 758 |
. . . . . . . . 9
⊢ ((((𝜑 ∧ (𝑟 ⊆ 𝑇 ∧ 𝑧 ∈ (𝑇 ∖ 𝑟))) ∧ ∀𝑘 ∈ (0...𝑁)((𝑆 D𝑛 (𝑦 ∈ 𝑋 ↦ ∏𝑢 ∈ 𝑟 ((𝐻‘𝑢)‘𝑦)))‘𝑘) = (𝑦 ∈ 𝑋 ↦ Σ𝑑 ∈ ((𝐷‘𝑟)‘𝑘)(((!‘𝑘) / ∏𝑢 ∈ 𝑟 (!‘(𝑑‘𝑢))) · ∏𝑢 ∈ 𝑟 (((𝑆 D𝑛 (𝐻‘𝑢))‘(𝑑‘𝑢))‘𝑦)))) ∧ 𝑗 ∈ (0...𝑁)) → 𝑟 ⊆ 𝑇) |
318 | | simprr 792 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ (𝑟 ⊆ 𝑇 ∧ 𝑧 ∈ (𝑇 ∖ 𝑟))) → 𝑧 ∈ (𝑇 ∖ 𝑟)) |
319 | 318 | ad2antrr 758 |
. . . . . . . . 9
⊢ ((((𝜑 ∧ (𝑟 ⊆ 𝑇 ∧ 𝑧 ∈ (𝑇 ∖ 𝑟))) ∧ ∀𝑘 ∈ (0...𝑁)((𝑆 D𝑛 (𝑦 ∈ 𝑋 ↦ ∏𝑢 ∈ 𝑟 ((𝐻‘𝑢)‘𝑦)))‘𝑘) = (𝑦 ∈ 𝑋 ↦ Σ𝑑 ∈ ((𝐷‘𝑟)‘𝑘)(((!‘𝑘) / ∏𝑢 ∈ 𝑟 (!‘(𝑑‘𝑢))) · ∏𝑢 ∈ 𝑟 (((𝑆 D𝑛 (𝐻‘𝑢))‘(𝑑‘𝑢))‘𝑦)))) ∧ 𝑗 ∈ (0...𝑁)) → 𝑧 ∈ (𝑇 ∖ 𝑟)) |
320 | 260 | eqcomi 2619 |
. . . . . . . . . . . . . . 15
⊢ (𝑆 D𝑛 (𝑦 ∈ 𝑋 ↦ ∏𝑢 ∈ 𝑟 ((𝐻‘𝑢)‘𝑦))) = (𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ ∏𝑡 ∈ 𝑟 ((𝐻‘𝑡)‘𝑥))) |
321 | 320 | a1i 11 |
. . . . . . . . . . . . . 14
⊢ (𝑘 = 𝑙 → (𝑆 D𝑛 (𝑦 ∈ 𝑋 ↦ ∏𝑢 ∈ 𝑟 ((𝐻‘𝑢)‘𝑦))) = (𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ ∏𝑡 ∈ 𝑟 ((𝐻‘𝑡)‘𝑥)))) |
322 | | id 22 |
. . . . . . . . . . . . . 14
⊢ (𝑘 = 𝑙 → 𝑘 = 𝑙) |
323 | 321, 322 | fveq12d 6109 |
. . . . . . . . . . . . 13
⊢ (𝑘 = 𝑙 → ((𝑆 D𝑛 (𝑦 ∈ 𝑋 ↦ ∏𝑢 ∈ 𝑟 ((𝐻‘𝑢)‘𝑦)))‘𝑘) = ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ ∏𝑡 ∈ 𝑟 ((𝐻‘𝑡)‘𝑥)))‘𝑙)) |
324 | 288 | eqcomi 2619 |
. . . . . . . . . . . . . . 15
⊢ (𝑦 ∈ 𝑋 ↦ Σ𝑑 ∈ ((𝐷‘𝑟)‘𝑘)(((!‘𝑘) / ∏𝑢 ∈ 𝑟 (!‘(𝑑‘𝑢))) · ∏𝑢 ∈ 𝑟 (((𝑆 D𝑛 (𝐻‘𝑢))‘(𝑑‘𝑢))‘𝑦))) = (𝑥 ∈ 𝑋 ↦ Σ𝑐 ∈ ((𝐷‘𝑟)‘𝑘)(((!‘𝑘) / ∏𝑡 ∈ 𝑟 (!‘(𝑐‘𝑡))) · ∏𝑡 ∈ 𝑟 (((𝑆 D𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥))) |
325 | 324 | a1i 11 |
. . . . . . . . . . . . . 14
⊢ (𝑘 = 𝑙 → (𝑦 ∈ 𝑋 ↦ Σ𝑑 ∈ ((𝐷‘𝑟)‘𝑘)(((!‘𝑘) / ∏𝑢 ∈ 𝑟 (!‘(𝑑‘𝑢))) · ∏𝑢 ∈ 𝑟 (((𝑆 D𝑛 (𝐻‘𝑢))‘(𝑑‘𝑢))‘𝑦))) = (𝑥 ∈ 𝑋 ↦ Σ𝑐 ∈ ((𝐷‘𝑟)‘𝑘)(((!‘𝑘) / ∏𝑡 ∈ 𝑟 (!‘(𝑐‘𝑡))) · ∏𝑡 ∈ 𝑟 (((𝑆 D𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥)))) |
326 | | fveq2 6103 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑘 = 𝑙 → (!‘𝑘) = (!‘𝑙)) |
327 | 326 | oveq1d 6564 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑘 = 𝑙 → ((!‘𝑘) / ∏𝑡 ∈ 𝑟 (!‘(𝑐‘𝑡))) = ((!‘𝑙) / ∏𝑡 ∈ 𝑟 (!‘(𝑐‘𝑡)))) |
328 | 327 | oveq1d 6564 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑘 = 𝑙 → (((!‘𝑘) / ∏𝑡 ∈ 𝑟 (!‘(𝑐‘𝑡))) · ∏𝑡 ∈ 𝑟 (((𝑆 D𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥)) = (((!‘𝑙) / ∏𝑡 ∈ 𝑟 (!‘(𝑐‘𝑡))) · ∏𝑡 ∈ 𝑟 (((𝑆 D𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥))) |
329 | 328 | sumeq2ad 38632 |
. . . . . . . . . . . . . . . 16
⊢ (𝑘 = 𝑙 → Σ𝑐 ∈ ((𝐷‘𝑟)‘𝑘)(((!‘𝑘) / ∏𝑡 ∈ 𝑟 (!‘(𝑐‘𝑡))) · ∏𝑡 ∈ 𝑟 (((𝑆 D𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥)) = Σ𝑐 ∈ ((𝐷‘𝑟)‘𝑘)(((!‘𝑙) / ∏𝑡 ∈ 𝑟 (!‘(𝑐‘𝑡))) · ∏𝑡 ∈ 𝑟 (((𝑆 D𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥))) |
330 | | fveq2 6103 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑘 = 𝑙 → ((𝐷‘𝑟)‘𝑘) = ((𝐷‘𝑟)‘𝑙)) |
331 | 330 | sumeq1d 14279 |
. . . . . . . . . . . . . . . 16
⊢ (𝑘 = 𝑙 → Σ𝑐 ∈ ((𝐷‘𝑟)‘𝑘)(((!‘𝑙) / ∏𝑡 ∈ 𝑟 (!‘(𝑐‘𝑡))) · ∏𝑡 ∈ 𝑟 (((𝑆 D𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥)) = Σ𝑐 ∈ ((𝐷‘𝑟)‘𝑙)(((!‘𝑙) / ∏𝑡 ∈ 𝑟 (!‘(𝑐‘𝑡))) · ∏𝑡 ∈ 𝑟 (((𝑆 D𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥))) |
332 | 329, 331 | eqtrd 2644 |
. . . . . . . . . . . . . . 15
⊢ (𝑘 = 𝑙 → Σ𝑐 ∈ ((𝐷‘𝑟)‘𝑘)(((!‘𝑘) / ∏𝑡 ∈ 𝑟 (!‘(𝑐‘𝑡))) · ∏𝑡 ∈ 𝑟 (((𝑆 D𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥)) = Σ𝑐 ∈ ((𝐷‘𝑟)‘𝑙)(((!‘𝑙) / ∏𝑡 ∈ 𝑟 (!‘(𝑐‘𝑡))) · ∏𝑡 ∈ 𝑟 (((𝑆 D𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥))) |
333 | 332 | mpteq2dv 4673 |
. . . . . . . . . . . . . 14
⊢ (𝑘 = 𝑙 → (𝑥 ∈ 𝑋 ↦ Σ𝑐 ∈ ((𝐷‘𝑟)‘𝑘)(((!‘𝑘) / ∏𝑡 ∈ 𝑟 (!‘(𝑐‘𝑡))) · ∏𝑡 ∈ 𝑟 (((𝑆 D𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥))) = (𝑥 ∈ 𝑋 ↦ Σ𝑐 ∈ ((𝐷‘𝑟)‘𝑙)(((!‘𝑙) / ∏𝑡 ∈ 𝑟 (!‘(𝑐‘𝑡))) · ∏𝑡 ∈ 𝑟 (((𝑆 D𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥)))) |
334 | 325, 333 | eqtrd 2644 |
. . . . . . . . . . . . 13
⊢ (𝑘 = 𝑙 → (𝑦 ∈ 𝑋 ↦ Σ𝑑 ∈ ((𝐷‘𝑟)‘𝑘)(((!‘𝑘) / ∏𝑢 ∈ 𝑟 (!‘(𝑑‘𝑢))) · ∏𝑢 ∈ 𝑟 (((𝑆 D𝑛 (𝐻‘𝑢))‘(𝑑‘𝑢))‘𝑦))) = (𝑥 ∈ 𝑋 ↦ Σ𝑐 ∈ ((𝐷‘𝑟)‘𝑙)(((!‘𝑙) / ∏𝑡 ∈ 𝑟 (!‘(𝑐‘𝑡))) · ∏𝑡 ∈ 𝑟 (((𝑆 D𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥)))) |
335 | 323, 334 | eqeq12d 2625 |
. . . . . . . . . . . 12
⊢ (𝑘 = 𝑙 → (((𝑆 D𝑛 (𝑦 ∈ 𝑋 ↦ ∏𝑢 ∈ 𝑟 ((𝐻‘𝑢)‘𝑦)))‘𝑘) = (𝑦 ∈ 𝑋 ↦ Σ𝑑 ∈ ((𝐷‘𝑟)‘𝑘)(((!‘𝑘) / ∏𝑢 ∈ 𝑟 (!‘(𝑑‘𝑢))) · ∏𝑢 ∈ 𝑟 (((𝑆 D𝑛 (𝐻‘𝑢))‘(𝑑‘𝑢))‘𝑦))) ↔ ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ ∏𝑡 ∈ 𝑟 ((𝐻‘𝑡)‘𝑥)))‘𝑙) = (𝑥 ∈ 𝑋 ↦ Σ𝑐 ∈ ((𝐷‘𝑟)‘𝑙)(((!‘𝑙) / ∏𝑡 ∈ 𝑟 (!‘(𝑐‘𝑡))) · ∏𝑡 ∈ 𝑟 (((𝑆 D𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥))))) |
336 | 335 | cbvralv 3147 |
. . . . . . . . . . 11
⊢
(∀𝑘 ∈
(0...𝑁)((𝑆 D𝑛 (𝑦 ∈ 𝑋 ↦ ∏𝑢 ∈ 𝑟 ((𝐻‘𝑢)‘𝑦)))‘𝑘) = (𝑦 ∈ 𝑋 ↦ Σ𝑑 ∈ ((𝐷‘𝑟)‘𝑘)(((!‘𝑘) / ∏𝑢 ∈ 𝑟 (!‘(𝑑‘𝑢))) · ∏𝑢 ∈ 𝑟 (((𝑆 D𝑛 (𝐻‘𝑢))‘(𝑑‘𝑢))‘𝑦))) ↔ ∀𝑙 ∈ (0...𝑁)((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ ∏𝑡 ∈ 𝑟 ((𝐻‘𝑡)‘𝑥)))‘𝑙) = (𝑥 ∈ 𝑋 ↦ Σ𝑐 ∈ ((𝐷‘𝑟)‘𝑙)(((!‘𝑙) / ∏𝑡 ∈ 𝑟 (!‘(𝑐‘𝑡))) · ∏𝑡 ∈ 𝑟 (((𝑆 D𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥)))) |
337 | 336 | biimpi 205 |
. . . . . . . . . 10
⊢
(∀𝑘 ∈
(0...𝑁)((𝑆 D𝑛 (𝑦 ∈ 𝑋 ↦ ∏𝑢 ∈ 𝑟 ((𝐻‘𝑢)‘𝑦)))‘𝑘) = (𝑦 ∈ 𝑋 ↦ Σ𝑑 ∈ ((𝐷‘𝑟)‘𝑘)(((!‘𝑘) / ∏𝑢 ∈ 𝑟 (!‘(𝑑‘𝑢))) · ∏𝑢 ∈ 𝑟 (((𝑆 D𝑛 (𝐻‘𝑢))‘(𝑑‘𝑢))‘𝑦))) → ∀𝑙 ∈ (0...𝑁)((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ ∏𝑡 ∈ 𝑟 ((𝐻‘𝑡)‘𝑥)))‘𝑙) = (𝑥 ∈ 𝑋 ↦ Σ𝑐 ∈ ((𝐷‘𝑟)‘𝑙)(((!‘𝑙) / ∏𝑡 ∈ 𝑟 (!‘(𝑐‘𝑡))) · ∏𝑡 ∈ 𝑟 (((𝑆 D𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥)))) |
338 | 337 | ad2antlr 759 |
. . . . . . . . 9
⊢ ((((𝜑 ∧ (𝑟 ⊆ 𝑇 ∧ 𝑧 ∈ (𝑇 ∖ 𝑟))) ∧ ∀𝑘 ∈ (0...𝑁)((𝑆 D𝑛 (𝑦 ∈ 𝑋 ↦ ∏𝑢 ∈ 𝑟 ((𝐻‘𝑢)‘𝑦)))‘𝑘) = (𝑦 ∈ 𝑋 ↦ Σ𝑑 ∈ ((𝐷‘𝑟)‘𝑘)(((!‘𝑘) / ∏𝑢 ∈ 𝑟 (!‘(𝑑‘𝑢))) · ∏𝑢 ∈ 𝑟 (((𝑆 D𝑛 (𝐻‘𝑢))‘(𝑑‘𝑢))‘𝑦)))) ∧ 𝑗 ∈ (0...𝑁)) → ∀𝑙 ∈ (0...𝑁)((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ ∏𝑡 ∈ 𝑟 ((𝐻‘𝑡)‘𝑥)))‘𝑙) = (𝑥 ∈ 𝑋 ↦ Σ𝑐 ∈ ((𝐷‘𝑟)‘𝑙)(((!‘𝑙) / ∏𝑡 ∈ 𝑟 (!‘(𝑐‘𝑡))) · ∏𝑡 ∈ 𝑟 (((𝑆 D𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥)))) |
339 | | simpr 476 |
. . . . . . . . 9
⊢ ((((𝜑 ∧ (𝑟 ⊆ 𝑇 ∧ 𝑧 ∈ (𝑇 ∖ 𝑟))) ∧ ∀𝑘 ∈ (0...𝑁)((𝑆 D𝑛 (𝑦 ∈ 𝑋 ↦ ∏𝑢 ∈ 𝑟 ((𝐻‘𝑢)‘𝑦)))‘𝑘) = (𝑦 ∈ 𝑋 ↦ Σ𝑑 ∈ ((𝐷‘𝑟)‘𝑘)(((!‘𝑘) / ∏𝑢 ∈ 𝑟 (!‘(𝑑‘𝑢))) · ∏𝑢 ∈ 𝑟 (((𝑆 D𝑛 (𝐻‘𝑢))‘(𝑑‘𝑢))‘𝑦)))) ∧ 𝑗 ∈ (0...𝑁)) → 𝑗 ∈ (0...𝑁)) |
340 | | fveq1 6102 |
. . . . . . . . . . . 12
⊢ (𝑑 = 𝑐 → (𝑑‘𝑧) = (𝑐‘𝑧)) |
341 | 340 | oveq2d 6565 |
. . . . . . . . . . 11
⊢ (𝑑 = 𝑐 → (𝑗 − (𝑑‘𝑧)) = (𝑗 − (𝑐‘𝑧))) |
342 | | reseq1 5311 |
. . . . . . . . . . 11
⊢ (𝑑 = 𝑐 → (𝑑 ↾ 𝑟) = (𝑐 ↾ 𝑟)) |
343 | 341, 342 | opeq12d 4348 |
. . . . . . . . . 10
⊢ (𝑑 = 𝑐 → 〈(𝑗 − (𝑑‘𝑧)), (𝑑 ↾ 𝑟)〉 = 〈(𝑗 − (𝑐‘𝑧)), (𝑐 ↾ 𝑟)〉) |
344 | 343 | cbvmptv 4678 |
. . . . . . . . 9
⊢ (𝑑 ∈ ((𝐷‘(𝑟 ∪ {𝑧}))‘𝑗) ↦ 〈(𝑗 − (𝑑‘𝑧)), (𝑑 ↾ 𝑟)〉) = (𝑐 ∈ ((𝐷‘(𝑟 ∪ {𝑧}))‘𝑗) ↦ 〈(𝑗 − (𝑐‘𝑧)), (𝑐 ↾ 𝑟)〉) |
345 | 294, 295,
297, 301, 303, 315, 106, 317, 319, 338, 339, 344 | dvnprodlem2 38837 |
. . . . . . . 8
⊢ ((((𝜑 ∧ (𝑟 ⊆ 𝑇 ∧ 𝑧 ∈ (𝑇 ∖ 𝑟))) ∧ ∀𝑘 ∈ (0...𝑁)((𝑆 D𝑛 (𝑦 ∈ 𝑋 ↦ ∏𝑢 ∈ 𝑟 ((𝐻‘𝑢)‘𝑦)))‘𝑘) = (𝑦 ∈ 𝑋 ↦ Σ𝑑 ∈ ((𝐷‘𝑟)‘𝑘)(((!‘𝑘) / ∏𝑢 ∈ 𝑟 (!‘(𝑑‘𝑢))) · ∏𝑢 ∈ 𝑟 (((𝑆 D𝑛 (𝐻‘𝑢))‘(𝑑‘𝑢))‘𝑦)))) ∧ 𝑗 ∈ (0...𝑁)) → ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ ∏𝑡 ∈ (𝑟 ∪ {𝑧})((𝐻‘𝑡)‘𝑥)))‘𝑗) = (𝑥 ∈ 𝑋 ↦ Σ𝑐 ∈ ((𝐷‘(𝑟 ∪ {𝑧}))‘𝑗)(((!‘𝑗) / ∏𝑡 ∈ (𝑟 ∪ {𝑧})(!‘(𝑐‘𝑡))) · ∏𝑡 ∈ (𝑟 ∪ {𝑧})(((𝑆 D𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥)))) |
346 | 251, 292,
293, 345 | syl21anc 1317 |
. . . . . . 7
⊢ ((((𝜑 ∧ (𝑟 ⊆ 𝑇 ∧ 𝑧 ∈ (𝑇 ∖ 𝑟))) ∧ ∀𝑘 ∈ (0...𝑁)((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ ∏𝑡 ∈ 𝑟 ((𝐻‘𝑡)‘𝑥)))‘𝑘) = (𝑥 ∈ 𝑋 ↦ Σ𝑐 ∈ ((𝐷‘𝑟)‘𝑘)(((!‘𝑘) / ∏𝑡 ∈ 𝑟 (!‘(𝑐‘𝑡))) · ∏𝑡 ∈ 𝑟 (((𝑆 D𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥)))) ∧ 𝑗 ∈ (0...𝑁)) → ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ ∏𝑡 ∈ (𝑟 ∪ {𝑧})((𝐻‘𝑡)‘𝑥)))‘𝑗) = (𝑥 ∈ 𝑋 ↦ Σ𝑐 ∈ ((𝐷‘(𝑟 ∪ {𝑧}))‘𝑗)(((!‘𝑗) / ∏𝑡 ∈ (𝑟 ∪ {𝑧})(!‘(𝑐‘𝑡))) · ∏𝑡 ∈ (𝑟 ∪ {𝑧})(((𝑆 D𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥)))) |
347 | 346 | ralrimiva 2949 |
. . . . . 6
⊢ (((𝜑 ∧ (𝑟 ⊆ 𝑇 ∧ 𝑧 ∈ (𝑇 ∖ 𝑟))) ∧ ∀𝑘 ∈ (0...𝑁)((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ ∏𝑡 ∈ 𝑟 ((𝐻‘𝑡)‘𝑥)))‘𝑘) = (𝑥 ∈ 𝑋 ↦ Σ𝑐 ∈ ((𝐷‘𝑟)‘𝑘)(((!‘𝑘) / ∏𝑡 ∈ 𝑟 (!‘(𝑐‘𝑡))) · ∏𝑡 ∈ 𝑟 (((𝑆 D𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥)))) → ∀𝑗 ∈ (0...𝑁)((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ ∏𝑡 ∈ (𝑟 ∪ {𝑧})((𝐻‘𝑡)‘𝑥)))‘𝑗) = (𝑥 ∈ 𝑋 ↦ Σ𝑐 ∈ ((𝐷‘(𝑟 ∪ {𝑧}))‘𝑗)(((!‘𝑗) / ∏𝑡 ∈ (𝑟 ∪ {𝑧})(!‘(𝑐‘𝑡))) · ∏𝑡 ∈ (𝑟 ∪ {𝑧})(((𝑆 D𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥)))) |
348 | | fveq2 6103 |
. . . . . . . 8
⊢ (𝑗 = 𝑘 → ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ ∏𝑡 ∈ (𝑟 ∪ {𝑧})((𝐻‘𝑡)‘𝑥)))‘𝑗) = ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ ∏𝑡 ∈ (𝑟 ∪ {𝑧})((𝐻‘𝑡)‘𝑥)))‘𝑘)) |
349 | | fveq2 6103 |
. . . . . . . . . . . . 13
⊢ (𝑗 = 𝑘 → (!‘𝑗) = (!‘𝑘)) |
350 | 349 | oveq1d 6564 |
. . . . . . . . . . . 12
⊢ (𝑗 = 𝑘 → ((!‘𝑗) / ∏𝑡 ∈ (𝑟 ∪ {𝑧})(!‘(𝑐‘𝑡))) = ((!‘𝑘) / ∏𝑡 ∈ (𝑟 ∪ {𝑧})(!‘(𝑐‘𝑡)))) |
351 | 350 | oveq1d 6564 |
. . . . . . . . . . 11
⊢ (𝑗 = 𝑘 → (((!‘𝑗) / ∏𝑡 ∈ (𝑟 ∪ {𝑧})(!‘(𝑐‘𝑡))) · ∏𝑡 ∈ (𝑟 ∪ {𝑧})(((𝑆 D𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥)) = (((!‘𝑘) / ∏𝑡 ∈ (𝑟 ∪ {𝑧})(!‘(𝑐‘𝑡))) · ∏𝑡 ∈ (𝑟 ∪ {𝑧})(((𝑆 D𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥))) |
352 | 351 | sumeq2ad 38632 |
. . . . . . . . . 10
⊢ (𝑗 = 𝑘 → Σ𝑐 ∈ ((𝐷‘(𝑟 ∪ {𝑧}))‘𝑗)(((!‘𝑗) / ∏𝑡 ∈ (𝑟 ∪ {𝑧})(!‘(𝑐‘𝑡))) · ∏𝑡 ∈ (𝑟 ∪ {𝑧})(((𝑆 D𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥)) = Σ𝑐 ∈ ((𝐷‘(𝑟 ∪ {𝑧}))‘𝑗)(((!‘𝑘) / ∏𝑡 ∈ (𝑟 ∪ {𝑧})(!‘(𝑐‘𝑡))) · ∏𝑡 ∈ (𝑟 ∪ {𝑧})(((𝑆 D𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥))) |
353 | | fveq2 6103 |
. . . . . . . . . . 11
⊢ (𝑗 = 𝑘 → ((𝐷‘(𝑟 ∪ {𝑧}))‘𝑗) = ((𝐷‘(𝑟 ∪ {𝑧}))‘𝑘)) |
354 | 353 | sumeq1d 14279 |
. . . . . . . . . 10
⊢ (𝑗 = 𝑘 → Σ𝑐 ∈ ((𝐷‘(𝑟 ∪ {𝑧}))‘𝑗)(((!‘𝑘) / ∏𝑡 ∈ (𝑟 ∪ {𝑧})(!‘(𝑐‘𝑡))) · ∏𝑡 ∈ (𝑟 ∪ {𝑧})(((𝑆 D𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥)) = Σ𝑐 ∈ ((𝐷‘(𝑟 ∪ {𝑧}))‘𝑘)(((!‘𝑘) / ∏𝑡 ∈ (𝑟 ∪ {𝑧})(!‘(𝑐‘𝑡))) · ∏𝑡 ∈ (𝑟 ∪ {𝑧})(((𝑆 D𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥))) |
355 | 352, 354 | eqtrd 2644 |
. . . . . . . . 9
⊢ (𝑗 = 𝑘 → Σ𝑐 ∈ ((𝐷‘(𝑟 ∪ {𝑧}))‘𝑗)(((!‘𝑗) / ∏𝑡 ∈ (𝑟 ∪ {𝑧})(!‘(𝑐‘𝑡))) · ∏𝑡 ∈ (𝑟 ∪ {𝑧})(((𝑆 D𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥)) = Σ𝑐 ∈ ((𝐷‘(𝑟 ∪ {𝑧}))‘𝑘)(((!‘𝑘) / ∏𝑡 ∈ (𝑟 ∪ {𝑧})(!‘(𝑐‘𝑡))) · ∏𝑡 ∈ (𝑟 ∪ {𝑧})(((𝑆 D𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥))) |
356 | 355 | mpteq2dv 4673 |
. . . . . . . 8
⊢ (𝑗 = 𝑘 → (𝑥 ∈ 𝑋 ↦ Σ𝑐 ∈ ((𝐷‘(𝑟 ∪ {𝑧}))‘𝑗)(((!‘𝑗) / ∏𝑡 ∈ (𝑟 ∪ {𝑧})(!‘(𝑐‘𝑡))) · ∏𝑡 ∈ (𝑟 ∪ {𝑧})(((𝑆 D𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥))) = (𝑥 ∈ 𝑋 ↦ Σ𝑐 ∈ ((𝐷‘(𝑟 ∪ {𝑧}))‘𝑘)(((!‘𝑘) / ∏𝑡 ∈ (𝑟 ∪ {𝑧})(!‘(𝑐‘𝑡))) · ∏𝑡 ∈ (𝑟 ∪ {𝑧})(((𝑆 D𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥)))) |
357 | 348, 356 | eqeq12d 2625 |
. . . . . . 7
⊢ (𝑗 = 𝑘 → (((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ ∏𝑡 ∈ (𝑟 ∪ {𝑧})((𝐻‘𝑡)‘𝑥)))‘𝑗) = (𝑥 ∈ 𝑋 ↦ Σ𝑐 ∈ ((𝐷‘(𝑟 ∪ {𝑧}))‘𝑗)(((!‘𝑗) / ∏𝑡 ∈ (𝑟 ∪ {𝑧})(!‘(𝑐‘𝑡))) · ∏𝑡 ∈ (𝑟 ∪ {𝑧})(((𝑆 D𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥))) ↔ ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ ∏𝑡 ∈ (𝑟 ∪ {𝑧})((𝐻‘𝑡)‘𝑥)))‘𝑘) = (𝑥 ∈ 𝑋 ↦ Σ𝑐 ∈ ((𝐷‘(𝑟 ∪ {𝑧}))‘𝑘)(((!‘𝑘) / ∏𝑡 ∈ (𝑟 ∪ {𝑧})(!‘(𝑐‘𝑡))) · ∏𝑡 ∈ (𝑟 ∪ {𝑧})(((𝑆 D𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥))))) |
358 | 357 | cbvralv 3147 |
. . . . . 6
⊢
(∀𝑗 ∈
(0...𝑁)((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ ∏𝑡 ∈ (𝑟 ∪ {𝑧})((𝐻‘𝑡)‘𝑥)))‘𝑗) = (𝑥 ∈ 𝑋 ↦ Σ𝑐 ∈ ((𝐷‘(𝑟 ∪ {𝑧}))‘𝑗)(((!‘𝑗) / ∏𝑡 ∈ (𝑟 ∪ {𝑧})(!‘(𝑐‘𝑡))) · ∏𝑡 ∈ (𝑟 ∪ {𝑧})(((𝑆 D𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥))) ↔ ∀𝑘 ∈ (0...𝑁)((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ ∏𝑡 ∈ (𝑟 ∪ {𝑧})((𝐻‘𝑡)‘𝑥)))‘𝑘) = (𝑥 ∈ 𝑋 ↦ Σ𝑐 ∈ ((𝐷‘(𝑟 ∪ {𝑧}))‘𝑘)(((!‘𝑘) / ∏𝑡 ∈ (𝑟 ∪ {𝑧})(!‘(𝑐‘𝑡))) · ∏𝑡 ∈ (𝑟 ∪ {𝑧})(((𝑆 D𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥)))) |
359 | 347, 358 | sylib 207 |
. . . . 5
⊢ (((𝜑 ∧ (𝑟 ⊆ 𝑇 ∧ 𝑧 ∈ (𝑇 ∖ 𝑟))) ∧ ∀𝑘 ∈ (0...𝑁)((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ ∏𝑡 ∈ 𝑟 ((𝐻‘𝑡)‘𝑥)))‘𝑘) = (𝑥 ∈ 𝑋 ↦ Σ𝑐 ∈ ((𝐷‘𝑟)‘𝑘)(((!‘𝑘) / ∏𝑡 ∈ 𝑟 (!‘(𝑐‘𝑡))) · ∏𝑡 ∈ 𝑟 (((𝑆 D𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥)))) → ∀𝑘 ∈ (0...𝑁)((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ ∏𝑡 ∈ (𝑟 ∪ {𝑧})((𝐻‘𝑡)‘𝑥)))‘𝑘) = (𝑥 ∈ 𝑋 ↦ Σ𝑐 ∈ ((𝐷‘(𝑟 ∪ {𝑧}))‘𝑘)(((!‘𝑘) / ∏𝑡 ∈ (𝑟 ∪ {𝑧})(!‘(𝑐‘𝑡))) · ∏𝑡 ∈ (𝑟 ∪ {𝑧})(((𝑆 D𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥)))) |
360 | 359 | ex 449 |
. . . 4
⊢ ((𝜑 ∧ (𝑟 ⊆ 𝑇 ∧ 𝑧 ∈ (𝑇 ∖ 𝑟))) → (∀𝑘 ∈ (0...𝑁)((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ ∏𝑡 ∈ 𝑟 ((𝐻‘𝑡)‘𝑥)))‘𝑘) = (𝑥 ∈ 𝑋 ↦ Σ𝑐 ∈ ((𝐷‘𝑟)‘𝑘)(((!‘𝑘) / ∏𝑡 ∈ 𝑟 (!‘(𝑐‘𝑡))) · ∏𝑡 ∈ 𝑟 (((𝑆 D𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥))) → ∀𝑘 ∈ (0...𝑁)((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ ∏𝑡 ∈ (𝑟 ∪ {𝑧})((𝐻‘𝑡)‘𝑥)))‘𝑘) = (𝑥 ∈ 𝑋 ↦ Σ𝑐 ∈ ((𝐷‘(𝑟 ∪ {𝑧}))‘𝑘)(((!‘𝑘) / ∏𝑡 ∈ (𝑟 ∪ {𝑧})(!‘(𝑐‘𝑡))) · ∏𝑡 ∈ (𝑟 ∪ {𝑧})(((𝑆 D𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥))))) |
361 | 16, 32, 48, 68, 250, 360, 296 | findcard2d 8087 |
. . 3
⊢ (𝜑 → ∀𝑘 ∈ (0...𝑁)((𝑆 D𝑛 𝐹)‘𝑘) = (𝑥 ∈ 𝑋 ↦ Σ𝑐 ∈ ((𝐷‘𝑇)‘𝑘)(((!‘𝑘) / ∏𝑡 ∈ 𝑇 (!‘(𝑐‘𝑡))) · ∏𝑡 ∈ 𝑇 (((𝑆 D𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥)))) |
362 | | nn0uz 11598 |
. . . . 5
⊢
ℕ0 = (ℤ≥‘0) |
363 | 302, 362 | syl6eleq 2698 |
. . . 4
⊢ (𝜑 → 𝑁 ∈
(ℤ≥‘0)) |
364 | | eluzfz2 12220 |
. . . 4
⊢ (𝑁 ∈
(ℤ≥‘0) → 𝑁 ∈ (0...𝑁)) |
365 | 363, 364 | syl 17 |
. . 3
⊢ (𝜑 → 𝑁 ∈ (0...𝑁)) |
366 | | fveq2 6103 |
. . . . 5
⊢ (𝑘 = 𝑁 → ((𝑆 D𝑛 𝐹)‘𝑘) = ((𝑆 D𝑛 𝐹)‘𝑁)) |
367 | | fveq2 6103 |
. . . . . . . 8
⊢ (𝑘 = 𝑁 → ((𝐷‘𝑇)‘𝑘) = ((𝐷‘𝑇)‘𝑁)) |
368 | 367 | sumeq1d 14279 |
. . . . . . 7
⊢ (𝑘 = 𝑁 → Σ𝑐 ∈ ((𝐷‘𝑇)‘𝑘)(((!‘𝑘) / ∏𝑡 ∈ 𝑇 (!‘(𝑐‘𝑡))) · ∏𝑡 ∈ 𝑇 (((𝑆 D𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥)) = Σ𝑐 ∈ ((𝐷‘𝑇)‘𝑁)(((!‘𝑘) / ∏𝑡 ∈ 𝑇 (!‘(𝑐‘𝑡))) · ∏𝑡 ∈ 𝑇 (((𝑆 D𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥))) |
369 | | fveq2 6103 |
. . . . . . . . . 10
⊢ (𝑘 = 𝑁 → (!‘𝑘) = (!‘𝑁)) |
370 | 369 | oveq1d 6564 |
. . . . . . . . 9
⊢ (𝑘 = 𝑁 → ((!‘𝑘) / ∏𝑡 ∈ 𝑇 (!‘(𝑐‘𝑡))) = ((!‘𝑁) / ∏𝑡 ∈ 𝑇 (!‘(𝑐‘𝑡)))) |
371 | 370 | oveq1d 6564 |
. . . . . . . 8
⊢ (𝑘 = 𝑁 → (((!‘𝑘) / ∏𝑡 ∈ 𝑇 (!‘(𝑐‘𝑡))) · ∏𝑡 ∈ 𝑇 (((𝑆 D𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥)) = (((!‘𝑁) / ∏𝑡 ∈ 𝑇 (!‘(𝑐‘𝑡))) · ∏𝑡 ∈ 𝑇 (((𝑆 D𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥))) |
372 | 371 | sumeq2ad 38632 |
. . . . . . 7
⊢ (𝑘 = 𝑁 → Σ𝑐 ∈ ((𝐷‘𝑇)‘𝑁)(((!‘𝑘) / ∏𝑡 ∈ 𝑇 (!‘(𝑐‘𝑡))) · ∏𝑡 ∈ 𝑇 (((𝑆 D𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥)) = Σ𝑐 ∈ ((𝐷‘𝑇)‘𝑁)(((!‘𝑁) / ∏𝑡 ∈ 𝑇 (!‘(𝑐‘𝑡))) · ∏𝑡 ∈ 𝑇 (((𝑆 D𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥))) |
373 | 368, 372 | eqtrd 2644 |
. . . . . 6
⊢ (𝑘 = 𝑁 → Σ𝑐 ∈ ((𝐷‘𝑇)‘𝑘)(((!‘𝑘) / ∏𝑡 ∈ 𝑇 (!‘(𝑐‘𝑡))) · ∏𝑡 ∈ 𝑇 (((𝑆 D𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥)) = Σ𝑐 ∈ ((𝐷‘𝑇)‘𝑁)(((!‘𝑁) / ∏𝑡 ∈ 𝑇 (!‘(𝑐‘𝑡))) · ∏𝑡 ∈ 𝑇 (((𝑆 D𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥))) |
374 | 373 | mpteq2dv 4673 |
. . . . 5
⊢ (𝑘 = 𝑁 → (𝑥 ∈ 𝑋 ↦ Σ𝑐 ∈ ((𝐷‘𝑇)‘𝑘)(((!‘𝑘) / ∏𝑡 ∈ 𝑇 (!‘(𝑐‘𝑡))) · ∏𝑡 ∈ 𝑇 (((𝑆 D𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥))) = (𝑥 ∈ 𝑋 ↦ Σ𝑐 ∈ ((𝐷‘𝑇)‘𝑁)(((!‘𝑁) / ∏𝑡 ∈ 𝑇 (!‘(𝑐‘𝑡))) · ∏𝑡 ∈ 𝑇 (((𝑆 D𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥)))) |
375 | 366, 374 | eqeq12d 2625 |
. . . 4
⊢ (𝑘 = 𝑁 → (((𝑆 D𝑛 𝐹)‘𝑘) = (𝑥 ∈ 𝑋 ↦ Σ𝑐 ∈ ((𝐷‘𝑇)‘𝑘)(((!‘𝑘) / ∏𝑡 ∈ 𝑇 (!‘(𝑐‘𝑡))) · ∏𝑡 ∈ 𝑇 (((𝑆 D𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥))) ↔ ((𝑆 D𝑛 𝐹)‘𝑁) = (𝑥 ∈ 𝑋 ↦ Σ𝑐 ∈ ((𝐷‘𝑇)‘𝑁)(((!‘𝑁) / ∏𝑡 ∈ 𝑇 (!‘(𝑐‘𝑡))) · ∏𝑡 ∈ 𝑇 (((𝑆 D𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥))))) |
376 | 375 | rspccva 3281 |
. . 3
⊢
((∀𝑘 ∈
(0...𝑁)((𝑆 D𝑛 𝐹)‘𝑘) = (𝑥 ∈ 𝑋 ↦ Σ𝑐 ∈ ((𝐷‘𝑇)‘𝑘)(((!‘𝑘) / ∏𝑡 ∈ 𝑇 (!‘(𝑐‘𝑡))) · ∏𝑡 ∈ 𝑇 (((𝑆 D𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥))) ∧ 𝑁 ∈ (0...𝑁)) → ((𝑆 D𝑛 𝐹)‘𝑁) = (𝑥 ∈ 𝑋 ↦ Σ𝑐 ∈ ((𝐷‘𝑇)‘𝑁)(((!‘𝑁) / ∏𝑡 ∈ 𝑇 (!‘(𝑐‘𝑡))) · ∏𝑡 ∈ 𝑇 (((𝑆 D𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥)))) |
377 | 361, 365,
376 | syl2anc 691 |
. 2
⊢ (𝜑 → ((𝑆 D𝑛 𝐹)‘𝑁) = (𝑥 ∈ 𝑋 ↦ Σ𝑐 ∈ ((𝐷‘𝑇)‘𝑁)(((!‘𝑁) / ∏𝑡 ∈ 𝑇 (!‘(𝑐‘𝑡))) · ∏𝑡 ∈ 𝑇 (((𝑆 D𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥)))) |
378 | | oveq2 6557 |
. . . . . . . . . . 11
⊢ (𝑠 = 𝑇 → ((0...𝑛) ↑𝑚 𝑠) = ((0...𝑛) ↑𝑚 𝑇)) |
379 | | rabeq 3166 |
. . . . . . . . . . 11
⊢
(((0...𝑛)
↑𝑚 𝑠) = ((0...𝑛) ↑𝑚 𝑇) → {𝑐 ∈ ((0...𝑛) ↑𝑚 𝑠) ∣ Σ𝑡 ∈ 𝑠 (𝑐‘𝑡) = 𝑛} = {𝑐 ∈ ((0...𝑛) ↑𝑚 𝑇) ∣ Σ𝑡 ∈ 𝑠 (𝑐‘𝑡) = 𝑛}) |
380 | 378, 379 | syl 17 |
. . . . . . . . . 10
⊢ (𝑠 = 𝑇 → {𝑐 ∈ ((0...𝑛) ↑𝑚 𝑠) ∣ Σ𝑡 ∈ 𝑠 (𝑐‘𝑡) = 𝑛} = {𝑐 ∈ ((0...𝑛) ↑𝑚 𝑇) ∣ Σ𝑡 ∈ 𝑠 (𝑐‘𝑡) = 𝑛}) |
381 | | sumeq1 14267 |
. . . . . . . . . . . 12
⊢ (𝑠 = 𝑇 → Σ𝑡 ∈ 𝑠 (𝑐‘𝑡) = Σ𝑡 ∈ 𝑇 (𝑐‘𝑡)) |
382 | 381 | eqeq1d 2612 |
. . . . . . . . . . 11
⊢ (𝑠 = 𝑇 → (Σ𝑡 ∈ 𝑠 (𝑐‘𝑡) = 𝑛 ↔ Σ𝑡 ∈ 𝑇 (𝑐‘𝑡) = 𝑛)) |
383 | 382 | rabbidv 3164 |
. . . . . . . . . 10
⊢ (𝑠 = 𝑇 → {𝑐 ∈ ((0...𝑛) ↑𝑚 𝑇) ∣ Σ𝑡 ∈ 𝑠 (𝑐‘𝑡) = 𝑛} = {𝑐 ∈ ((0...𝑛) ↑𝑚 𝑇) ∣ Σ𝑡 ∈ 𝑇 (𝑐‘𝑡) = 𝑛}) |
384 | 380, 383 | eqtrd 2644 |
. . . . . . . . 9
⊢ (𝑠 = 𝑇 → {𝑐 ∈ ((0...𝑛) ↑𝑚 𝑠) ∣ Σ𝑡 ∈ 𝑠 (𝑐‘𝑡) = 𝑛} = {𝑐 ∈ ((0...𝑛) ↑𝑚 𝑇) ∣ Σ𝑡 ∈ 𝑇 (𝑐‘𝑡) = 𝑛}) |
385 | 384 | mpteq2dv 4673 |
. . . . . . . 8
⊢ (𝑠 = 𝑇 → (𝑛 ∈ ℕ0 ↦ {𝑐 ∈ ((0...𝑛) ↑𝑚 𝑠) ∣ Σ𝑡 ∈ 𝑠 (𝑐‘𝑡) = 𝑛}) = (𝑛 ∈ ℕ0 ↦ {𝑐 ∈ ((0...𝑛) ↑𝑚 𝑇) ∣ Σ𝑡 ∈ 𝑇 (𝑐‘𝑡) = 𝑛})) |
386 | 385 | adantl 481 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑠 = 𝑇) → (𝑛 ∈ ℕ0 ↦ {𝑐 ∈ ((0...𝑛) ↑𝑚 𝑠) ∣ Σ𝑡 ∈ 𝑠 (𝑐‘𝑡) = 𝑛}) = (𝑛 ∈ ℕ0 ↦ {𝑐 ∈ ((0...𝑛) ↑𝑚 𝑇) ∣ Σ𝑡 ∈ 𝑇 (𝑐‘𝑡) = 𝑛})) |
387 | | pwidg 4121 |
. . . . . . . 8
⊢ (𝑇 ∈ Fin → 𝑇 ∈ 𝒫 𝑇) |
388 | 296, 387 | syl 17 |
. . . . . . 7
⊢ (𝜑 → 𝑇 ∈ 𝒫 𝑇) |
389 | 140 | mptex 6390 |
. . . . . . . 8
⊢ (𝑛 ∈ ℕ0
↦ {𝑐 ∈
((0...𝑛)
↑𝑚 𝑇) ∣ Σ𝑡 ∈ 𝑇 (𝑐‘𝑡) = 𝑛}) ∈ V |
390 | 389 | a1i 11 |
. . . . . . 7
⊢ (𝜑 → (𝑛 ∈ ℕ0 ↦ {𝑐 ∈ ((0...𝑛) ↑𝑚 𝑇) ∣ Σ𝑡 ∈ 𝑇 (𝑐‘𝑡) = 𝑛}) ∈ V) |
391 | 107, 386,
388, 390 | fvmptd 6197 |
. . . . . 6
⊢ (𝜑 → (𝐷‘𝑇) = (𝑛 ∈ ℕ0 ↦ {𝑐 ∈ ((0...𝑛) ↑𝑚 𝑇) ∣ Σ𝑡 ∈ 𝑇 (𝑐‘𝑡) = 𝑛})) |
392 | | dvnprodlem3.c |
. . . . . . 7
⊢ 𝐶 = (𝑛 ∈ ℕ0 ↦ {𝑐 ∈ ((0...𝑛) ↑𝑚 𝑇) ∣ Σ𝑡 ∈ 𝑇 (𝑐‘𝑡) = 𝑛}) |
393 | 392 | a1i 11 |
. . . . . 6
⊢ (𝜑 → 𝐶 = (𝑛 ∈ ℕ0 ↦ {𝑐 ∈ ((0...𝑛) ↑𝑚 𝑇) ∣ Σ𝑡 ∈ 𝑇 (𝑐‘𝑡) = 𝑛})) |
394 | 391, 393 | eqtr4d 2647 |
. . . . 5
⊢ (𝜑 → (𝐷‘𝑇) = 𝐶) |
395 | 394 | fveq1d 6105 |
. . . 4
⊢ (𝜑 → ((𝐷‘𝑇)‘𝑁) = (𝐶‘𝑁)) |
396 | 395 | sumeq1d 14279 |
. . 3
⊢ (𝜑 → Σ𝑐 ∈ ((𝐷‘𝑇)‘𝑁)(((!‘𝑁) / ∏𝑡 ∈ 𝑇 (!‘(𝑐‘𝑡))) · ∏𝑡 ∈ 𝑇 (((𝑆 D𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥)) = Σ𝑐 ∈ (𝐶‘𝑁)(((!‘𝑁) / ∏𝑡 ∈ 𝑇 (!‘(𝑐‘𝑡))) · ∏𝑡 ∈ 𝑇 (((𝑆 D𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥))) |
397 | 396 | mpteq2dv 4673 |
. 2
⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ Σ𝑐 ∈ ((𝐷‘𝑇)‘𝑁)(((!‘𝑁) / ∏𝑡 ∈ 𝑇 (!‘(𝑐‘𝑡))) · ∏𝑡 ∈ 𝑇 (((𝑆 D𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥))) = (𝑥 ∈ 𝑋 ↦ Σ𝑐 ∈ (𝐶‘𝑁)(((!‘𝑁) / ∏𝑡 ∈ 𝑇 (!‘(𝑐‘𝑡))) · ∏𝑡 ∈ 𝑇 (((𝑆 D𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥)))) |
398 | 377, 397 | eqtrd 2644 |
1
⊢ (𝜑 → ((𝑆 D𝑛 𝐹)‘𝑁) = (𝑥 ∈ 𝑋 ↦ Σ𝑐 ∈ (𝐶‘𝑁)(((!‘𝑁) / ∏𝑡 ∈ 𝑇 (!‘(𝑐‘𝑡))) · ∏𝑡 ∈ 𝑇 (((𝑆 D𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥)))) |