Step | Hyp | Ref
| Expression |
1 | | nfv 1830 |
. . . . . 6
⊢
Ⅎ𝑡(𝜑 ∧ 𝑥 ∈ 𝑋) |
2 | | nfcv 2751 |
. . . . . 6
⊢
Ⅎ𝑡((𝐻‘𝑍)‘𝑥) |
3 | | dvnprodlem2.t |
. . . . . . . 8
⊢ (𝜑 → 𝑇 ∈ Fin) |
4 | | dvnprodlem2.r |
. . . . . . . 8
⊢ (𝜑 → 𝑅 ⊆ 𝑇) |
5 | | ssfi 8065 |
. . . . . . . 8
⊢ ((𝑇 ∈ Fin ∧ 𝑅 ⊆ 𝑇) → 𝑅 ∈ Fin) |
6 | 3, 4, 5 | syl2anc 691 |
. . . . . . 7
⊢ (𝜑 → 𝑅 ∈ Fin) |
7 | 6 | adantr 480 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝑅 ∈ Fin) |
8 | | dvnprodlem2.z |
. . . . . . 7
⊢ (𝜑 → 𝑍 ∈ (𝑇 ∖ 𝑅)) |
9 | 8 | adantr 480 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝑍 ∈ (𝑇 ∖ 𝑅)) |
10 | 8 | eldifbd 3553 |
. . . . . . 7
⊢ (𝜑 → ¬ 𝑍 ∈ 𝑅) |
11 | 10 | adantr 480 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ¬ 𝑍 ∈ 𝑅) |
12 | | simpl 472 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑡 ∈ 𝑅) → 𝜑) |
13 | 4 | sselda 3568 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑡 ∈ 𝑅) → 𝑡 ∈ 𝑇) |
14 | | dvnprodlem2.h |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → (𝐻‘𝑡):𝑋⟶ℂ) |
15 | 12, 13, 14 | syl2anc 691 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑡 ∈ 𝑅) → (𝐻‘𝑡):𝑋⟶ℂ) |
16 | 15 | adantlr 747 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑡 ∈ 𝑅) → (𝐻‘𝑡):𝑋⟶ℂ) |
17 | | simplr 788 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑡 ∈ 𝑅) → 𝑥 ∈ 𝑋) |
18 | 16, 17 | ffvelrnd 6268 |
. . . . . 6
⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑡 ∈ 𝑅) → ((𝐻‘𝑡)‘𝑥) ∈ ℂ) |
19 | | fveq2 6103 |
. . . . . . 7
⊢ (𝑡 = 𝑍 → (𝐻‘𝑡) = (𝐻‘𝑍)) |
20 | 19 | fveq1d 6105 |
. . . . . 6
⊢ (𝑡 = 𝑍 → ((𝐻‘𝑡)‘𝑥) = ((𝐻‘𝑍)‘𝑥)) |
21 | | id 22 |
. . . . . . . . 9
⊢ (𝜑 → 𝜑) |
22 | | eldifi 3694 |
. . . . . . . . . 10
⊢ (𝑍 ∈ (𝑇 ∖ 𝑅) → 𝑍 ∈ 𝑇) |
23 | 8, 22 | syl 17 |
. . . . . . . . 9
⊢ (𝜑 → 𝑍 ∈ 𝑇) |
24 | | simpr 476 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑍 ∈ 𝑇) → 𝑍 ∈ 𝑇) |
25 | | id 22 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑍 ∈ 𝑇) → (𝜑 ∧ 𝑍 ∈ 𝑇)) |
26 | | eleq1 2676 |
. . . . . . . . . . . . 13
⊢ (𝑡 = 𝑍 → (𝑡 ∈ 𝑇 ↔ 𝑍 ∈ 𝑇)) |
27 | 26 | anbi2d 736 |
. . . . . . . . . . . 12
⊢ (𝑡 = 𝑍 → ((𝜑 ∧ 𝑡 ∈ 𝑇) ↔ (𝜑 ∧ 𝑍 ∈ 𝑇))) |
28 | 19 | feq1d 5943 |
. . . . . . . . . . . 12
⊢ (𝑡 = 𝑍 → ((𝐻‘𝑡):𝑋⟶ℂ ↔ (𝐻‘𝑍):𝑋⟶ℂ)) |
29 | 27, 28 | imbi12d 333 |
. . . . . . . . . . 11
⊢ (𝑡 = 𝑍 → (((𝜑 ∧ 𝑡 ∈ 𝑇) → (𝐻‘𝑡):𝑋⟶ℂ) ↔ ((𝜑 ∧ 𝑍 ∈ 𝑇) → (𝐻‘𝑍):𝑋⟶ℂ))) |
30 | 29, 14 | vtoclg 3239 |
. . . . . . . . . 10
⊢ (𝑍 ∈ 𝑇 → ((𝜑 ∧ 𝑍 ∈ 𝑇) → (𝐻‘𝑍):𝑋⟶ℂ)) |
31 | 24, 25, 30 | sylc 63 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑍 ∈ 𝑇) → (𝐻‘𝑍):𝑋⟶ℂ) |
32 | 21, 23, 31 | syl2anc 691 |
. . . . . . . 8
⊢ (𝜑 → (𝐻‘𝑍):𝑋⟶ℂ) |
33 | 32 | adantr 480 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (𝐻‘𝑍):𝑋⟶ℂ) |
34 | | simpr 476 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝑥 ∈ 𝑋) |
35 | 33, 34 | ffvelrnd 6268 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ((𝐻‘𝑍)‘𝑥) ∈ ℂ) |
36 | 1, 2, 7, 9, 11, 18, 20, 35 | fprodsplitsn 14559 |
. . . . 5
⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ∏𝑡 ∈ (𝑅 ∪ {𝑍})((𝐻‘𝑡)‘𝑥) = (∏𝑡 ∈ 𝑅 ((𝐻‘𝑡)‘𝑥) · ((𝐻‘𝑍)‘𝑥))) |
37 | 36 | mpteq2dva 4672 |
. . . 4
⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ ∏𝑡 ∈ (𝑅 ∪ {𝑍})((𝐻‘𝑡)‘𝑥)) = (𝑥 ∈ 𝑋 ↦ (∏𝑡 ∈ 𝑅 ((𝐻‘𝑡)‘𝑥) · ((𝐻‘𝑍)‘𝑥)))) |
38 | 37 | oveq2d 6565 |
. . 3
⊢ (𝜑 → (𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ ∏𝑡 ∈ (𝑅 ∪ {𝑍})((𝐻‘𝑡)‘𝑥))) = (𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (∏𝑡 ∈ 𝑅 ((𝐻‘𝑡)‘𝑥) · ((𝐻‘𝑍)‘𝑥))))) |
39 | 38 | fveq1d 6105 |
. 2
⊢ (𝜑 → ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ ∏𝑡 ∈ (𝑅 ∪ {𝑍})((𝐻‘𝑡)‘𝑥)))‘𝐽) = ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (∏𝑡 ∈ 𝑅 ((𝐻‘𝑡)‘𝑥) · ((𝐻‘𝑍)‘𝑥))))‘𝐽)) |
40 | | dvnprodlem2.s |
. . 3
⊢ (𝜑 → 𝑆 ∈ {ℝ, ℂ}) |
41 | | dvnprodlem2.x |
. . 3
⊢ (𝜑 → 𝑋 ∈
((TopOpen‘ℂfld) ↾t 𝑆)) |
42 | 1, 7, 18 | fprodclf 14562 |
. . 3
⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ∏𝑡 ∈ 𝑅 ((𝐻‘𝑡)‘𝑥) ∈ ℂ) |
43 | | dvnprodlem2.j |
. . . 4
⊢ (𝜑 → 𝐽 ∈ (0...𝑁)) |
44 | | elfznn0 12302 |
. . . 4
⊢ (𝐽 ∈ (0...𝑁) → 𝐽 ∈
ℕ0) |
45 | 43, 44 | syl 17 |
. . 3
⊢ (𝜑 → 𝐽 ∈
ℕ0) |
46 | | eqid 2610 |
. . 3
⊢ (𝑥 ∈ 𝑋 ↦ ∏𝑡 ∈ 𝑅 ((𝐻‘𝑡)‘𝑥)) = (𝑥 ∈ 𝑋 ↦ ∏𝑡 ∈ 𝑅 ((𝐻‘𝑡)‘𝑥)) |
47 | | eqid 2610 |
. . 3
⊢ (𝑥 ∈ 𝑋 ↦ ((𝐻‘𝑍)‘𝑥)) = (𝑥 ∈ 𝑋 ↦ ((𝐻‘𝑍)‘𝑥)) |
48 | | dvnprodlem2.c |
. . . . . . . . . . 11
⊢ 𝐶 = (𝑠 ∈ 𝒫 𝑇 ↦ (𝑛 ∈ ℕ0 ↦ {𝑐 ∈ ((0...𝑛) ↑𝑚 𝑠) ∣ Σ𝑡 ∈ 𝑠 (𝑐‘𝑡) = 𝑛})) |
49 | 48 | a1i 11 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽)) → 𝐶 = (𝑠 ∈ 𝒫 𝑇 ↦ (𝑛 ∈ ℕ0 ↦ {𝑐 ∈ ((0...𝑛) ↑𝑚 𝑠) ∣ Σ𝑡 ∈ 𝑠 (𝑐‘𝑡) = 𝑛}))) |
50 | | oveq2 6557 |
. . . . . . . . . . . . . 14
⊢ (𝑠 = 𝑅 → ((0...𝑛) ↑𝑚 𝑠) = ((0...𝑛) ↑𝑚 𝑅)) |
51 | | rabeq 3166 |
. . . . . . . . . . . . . 14
⊢
(((0...𝑛)
↑𝑚 𝑠) = ((0...𝑛) ↑𝑚 𝑅) → {𝑐 ∈ ((0...𝑛) ↑𝑚 𝑠) ∣ Σ𝑡 ∈ 𝑠 (𝑐‘𝑡) = 𝑛} = {𝑐 ∈ ((0...𝑛) ↑𝑚 𝑅) ∣ Σ𝑡 ∈ 𝑠 (𝑐‘𝑡) = 𝑛}) |
52 | 50, 51 | syl 17 |
. . . . . . . . . . . . 13
⊢ (𝑠 = 𝑅 → {𝑐 ∈ ((0...𝑛) ↑𝑚 𝑠) ∣ Σ𝑡 ∈ 𝑠 (𝑐‘𝑡) = 𝑛} = {𝑐 ∈ ((0...𝑛) ↑𝑚 𝑅) ∣ Σ𝑡 ∈ 𝑠 (𝑐‘𝑡) = 𝑛}) |
53 | | sumeq1 14267 |
. . . . . . . . . . . . . . 15
⊢ (𝑠 = 𝑅 → Σ𝑡 ∈ 𝑠 (𝑐‘𝑡) = Σ𝑡 ∈ 𝑅 (𝑐‘𝑡)) |
54 | 53 | eqeq1d 2612 |
. . . . . . . . . . . . . 14
⊢ (𝑠 = 𝑅 → (Σ𝑡 ∈ 𝑠 (𝑐‘𝑡) = 𝑛 ↔ Σ𝑡 ∈ 𝑅 (𝑐‘𝑡) = 𝑛)) |
55 | 54 | rabbidv 3164 |
. . . . . . . . . . . . 13
⊢ (𝑠 = 𝑅 → {𝑐 ∈ ((0...𝑛) ↑𝑚 𝑅) ∣ Σ𝑡 ∈ 𝑠 (𝑐‘𝑡) = 𝑛} = {𝑐 ∈ ((0...𝑛) ↑𝑚 𝑅) ∣ Σ𝑡 ∈ 𝑅 (𝑐‘𝑡) = 𝑛}) |
56 | 52, 55 | eqtrd 2644 |
. . . . . . . . . . . 12
⊢ (𝑠 = 𝑅 → {𝑐 ∈ ((0...𝑛) ↑𝑚 𝑠) ∣ Σ𝑡 ∈ 𝑠 (𝑐‘𝑡) = 𝑛} = {𝑐 ∈ ((0...𝑛) ↑𝑚 𝑅) ∣ Σ𝑡 ∈ 𝑅 (𝑐‘𝑡) = 𝑛}) |
57 | 56 | mpteq2dv 4673 |
. . . . . . . . . . 11
⊢ (𝑠 = 𝑅 → (𝑛 ∈ ℕ0 ↦ {𝑐 ∈ ((0...𝑛) ↑𝑚 𝑠) ∣ Σ𝑡 ∈ 𝑠 (𝑐‘𝑡) = 𝑛}) = (𝑛 ∈ ℕ0 ↦ {𝑐 ∈ ((0...𝑛) ↑𝑚 𝑅) ∣ Σ𝑡 ∈ 𝑅 (𝑐‘𝑡) = 𝑛})) |
58 | 57 | adantl 481 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑘 ∈ (0...𝐽)) ∧ 𝑠 = 𝑅) → (𝑛 ∈ ℕ0 ↦ {𝑐 ∈ ((0...𝑛) ↑𝑚 𝑠) ∣ Σ𝑡 ∈ 𝑠 (𝑐‘𝑡) = 𝑛}) = (𝑛 ∈ ℕ0 ↦ {𝑐 ∈ ((0...𝑛) ↑𝑚 𝑅) ∣ Σ𝑡 ∈ 𝑅 (𝑐‘𝑡) = 𝑛})) |
59 | | ssexg 4732 |
. . . . . . . . . . . . . 14
⊢ ((𝑅 ⊆ 𝑇 ∧ 𝑇 ∈ Fin) → 𝑅 ∈ V) |
60 | 4, 3, 59 | syl2anc 691 |
. . . . . . . . . . . . 13
⊢ (𝜑 → 𝑅 ∈ V) |
61 | | elpwg 4116 |
. . . . . . . . . . . . 13
⊢ (𝑅 ∈ V → (𝑅 ∈ 𝒫 𝑇 ↔ 𝑅 ⊆ 𝑇)) |
62 | 60, 61 | syl 17 |
. . . . . . . . . . . 12
⊢ (𝜑 → (𝑅 ∈ 𝒫 𝑇 ↔ 𝑅 ⊆ 𝑇)) |
63 | 4, 62 | mpbird 246 |
. . . . . . . . . . 11
⊢ (𝜑 → 𝑅 ∈ 𝒫 𝑇) |
64 | 63 | adantr 480 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽)) → 𝑅 ∈ 𝒫 𝑇) |
65 | | nn0ex 11175 |
. . . . . . . . . . . 12
⊢
ℕ0 ∈ V |
66 | 65 | mptex 6390 |
. . . . . . . . . . 11
⊢ (𝑛 ∈ ℕ0
↦ {𝑐 ∈
((0...𝑛)
↑𝑚 𝑅) ∣ Σ𝑡 ∈ 𝑅 (𝑐‘𝑡) = 𝑛}) ∈ V |
67 | 66 | a1i 11 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽)) → (𝑛 ∈ ℕ0 ↦ {𝑐 ∈ ((0...𝑛) ↑𝑚 𝑅) ∣ Σ𝑡 ∈ 𝑅 (𝑐‘𝑡) = 𝑛}) ∈ V) |
68 | 49, 58, 64, 67 | fvmptd 6197 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽)) → (𝐶‘𝑅) = (𝑛 ∈ ℕ0 ↦ {𝑐 ∈ ((0...𝑛) ↑𝑚 𝑅) ∣ Σ𝑡 ∈ 𝑅 (𝑐‘𝑡) = 𝑛})) |
69 | | oveq2 6557 |
. . . . . . . . . . . . 13
⊢ (𝑛 = 𝑘 → (0...𝑛) = (0...𝑘)) |
70 | 69 | oveq1d 6564 |
. . . . . . . . . . . 12
⊢ (𝑛 = 𝑘 → ((0...𝑛) ↑𝑚 𝑅) = ((0...𝑘) ↑𝑚 𝑅)) |
71 | | rabeq 3166 |
. . . . . . . . . . . 12
⊢
(((0...𝑛)
↑𝑚 𝑅) = ((0...𝑘) ↑𝑚 𝑅) → {𝑐 ∈ ((0...𝑛) ↑𝑚 𝑅) ∣ Σ𝑡 ∈ 𝑅 (𝑐‘𝑡) = 𝑛} = {𝑐 ∈ ((0...𝑘) ↑𝑚 𝑅) ∣ Σ𝑡 ∈ 𝑅 (𝑐‘𝑡) = 𝑛}) |
72 | 70, 71 | syl 17 |
. . . . . . . . . . 11
⊢ (𝑛 = 𝑘 → {𝑐 ∈ ((0...𝑛) ↑𝑚 𝑅) ∣ Σ𝑡 ∈ 𝑅 (𝑐‘𝑡) = 𝑛} = {𝑐 ∈ ((0...𝑘) ↑𝑚 𝑅) ∣ Σ𝑡 ∈ 𝑅 (𝑐‘𝑡) = 𝑛}) |
73 | | eqeq2 2621 |
. . . . . . . . . . . 12
⊢ (𝑛 = 𝑘 → (Σ𝑡 ∈ 𝑅 (𝑐‘𝑡) = 𝑛 ↔ Σ𝑡 ∈ 𝑅 (𝑐‘𝑡) = 𝑘)) |
74 | 73 | rabbidv 3164 |
. . . . . . . . . . 11
⊢ (𝑛 = 𝑘 → {𝑐 ∈ ((0...𝑘) ↑𝑚 𝑅) ∣ Σ𝑡 ∈ 𝑅 (𝑐‘𝑡) = 𝑛} = {𝑐 ∈ ((0...𝑘) ↑𝑚 𝑅) ∣ Σ𝑡 ∈ 𝑅 (𝑐‘𝑡) = 𝑘}) |
75 | 72, 74 | eqtrd 2644 |
. . . . . . . . . 10
⊢ (𝑛 = 𝑘 → {𝑐 ∈ ((0...𝑛) ↑𝑚 𝑅) ∣ Σ𝑡 ∈ 𝑅 (𝑐‘𝑡) = 𝑛} = {𝑐 ∈ ((0...𝑘) ↑𝑚 𝑅) ∣ Σ𝑡 ∈ 𝑅 (𝑐‘𝑡) = 𝑘}) |
76 | 75 | adantl 481 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑘 ∈ (0...𝐽)) ∧ 𝑛 = 𝑘) → {𝑐 ∈ ((0...𝑛) ↑𝑚 𝑅) ∣ Σ𝑡 ∈ 𝑅 (𝑐‘𝑡) = 𝑛} = {𝑐 ∈ ((0...𝑘) ↑𝑚 𝑅) ∣ Σ𝑡 ∈ 𝑅 (𝑐‘𝑡) = 𝑘}) |
77 | | elfznn0 12302 |
. . . . . . . . . 10
⊢ (𝑘 ∈ (0...𝐽) → 𝑘 ∈ ℕ0) |
78 | 77 | adantl 481 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽)) → 𝑘 ∈ ℕ0) |
79 | | fzfid 12634 |
. . . . . . . . . . . 12
⊢ (𝜑 → (0...𝑘) ∈ Fin) |
80 | | mapfi 8145 |
. . . . . . . . . . . 12
⊢
(((0...𝑘) ∈ Fin
∧ 𝑅 ∈ Fin) →
((0...𝑘)
↑𝑚 𝑅) ∈ Fin) |
81 | 79, 6, 80 | syl2anc 691 |
. . . . . . . . . . 11
⊢ (𝜑 → ((0...𝑘) ↑𝑚 𝑅) ∈ Fin) |
82 | 81 | adantr 480 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽)) → ((0...𝑘) ↑𝑚 𝑅) ∈ Fin) |
83 | | ssrab2 3650 |
. . . . . . . . . . 11
⊢ {𝑐 ∈ ((0...𝑘) ↑𝑚 𝑅) ∣ Σ𝑡 ∈ 𝑅 (𝑐‘𝑡) = 𝑘} ⊆ ((0...𝑘) ↑𝑚 𝑅) |
84 | 83 | a1i 11 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽)) → {𝑐 ∈ ((0...𝑘) ↑𝑚 𝑅) ∣ Σ𝑡 ∈ 𝑅 (𝑐‘𝑡) = 𝑘} ⊆ ((0...𝑘) ↑𝑚 𝑅)) |
85 | 82, 84 | ssexd 4733 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽)) → {𝑐 ∈ ((0...𝑘) ↑𝑚 𝑅) ∣ Σ𝑡 ∈ 𝑅 (𝑐‘𝑡) = 𝑘} ∈ V) |
86 | 68, 76, 78, 85 | fvmptd 6197 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽)) → ((𝐶‘𝑅)‘𝑘) = {𝑐 ∈ ((0...𝑘) ↑𝑚 𝑅) ∣ Σ𝑡 ∈ 𝑅 (𝑐‘𝑡) = 𝑘}) |
87 | | ssfi 8065 |
. . . . . . . . . 10
⊢
((((0...𝑘)
↑𝑚 𝑅) ∈ Fin ∧ {𝑐 ∈ ((0...𝑘) ↑𝑚 𝑅) ∣ Σ𝑡 ∈ 𝑅 (𝑐‘𝑡) = 𝑘} ⊆ ((0...𝑘) ↑𝑚 𝑅)) → {𝑐 ∈ ((0...𝑘) ↑𝑚 𝑅) ∣ Σ𝑡 ∈ 𝑅 (𝑐‘𝑡) = 𝑘} ∈ Fin) |
88 | 81, 83, 87 | sylancl 693 |
. . . . . . . . 9
⊢ (𝜑 → {𝑐 ∈ ((0...𝑘) ↑𝑚 𝑅) ∣ Σ𝑡 ∈ 𝑅 (𝑐‘𝑡) = 𝑘} ∈ Fin) |
89 | 88 | adantr 480 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽)) → {𝑐 ∈ ((0...𝑘) ↑𝑚 𝑅) ∣ Σ𝑡 ∈ 𝑅 (𝑐‘𝑡) = 𝑘} ∈ Fin) |
90 | 86, 89 | eqeltrd 2688 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽)) → ((𝐶‘𝑅)‘𝑘) ∈ Fin) |
91 | 90 | adantr 480 |
. . . . . 6
⊢ (((𝜑 ∧ 𝑘 ∈ (0...𝐽)) ∧ 𝑥 ∈ 𝑋) → ((𝐶‘𝑅)‘𝑘) ∈ Fin) |
92 | 77 | faccld 12933 |
. . . . . . . . . . 11
⊢ (𝑘 ∈ (0...𝐽) → (!‘𝑘) ∈ ℕ) |
93 | 92 | nncnd 10913 |
. . . . . . . . . 10
⊢ (𝑘 ∈ (0...𝐽) → (!‘𝑘) ∈ ℂ) |
94 | 93 | ad2antlr 759 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑘 ∈ (0...𝐽)) ∧ 𝑐 ∈ ((𝐶‘𝑅)‘𝑘)) → (!‘𝑘) ∈ ℂ) |
95 | 6 | adantr 480 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘𝑅)‘𝑘)) → 𝑅 ∈ Fin) |
96 | 95 | adantlr 747 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑘 ∈ (0...𝐽)) ∧ 𝑐 ∈ ((𝐶‘𝑅)‘𝑘)) → 𝑅 ∈ Fin) |
97 | | elfznn0 12302 |
. . . . . . . . . . . . . . 15
⊢ (𝑧 ∈ (0...𝑘) → 𝑧 ∈ ℕ0) |
98 | 97 | ssriv 3572 |
. . . . . . . . . . . . . 14
⊢
(0...𝑘) ⊆
ℕ0 |
99 | 98 | a1i 11 |
. . . . . . . . . . . . 13
⊢ ((((𝜑 ∧ 𝑘 ∈ (0...𝐽)) ∧ 𝑐 ∈ ((𝐶‘𝑅)‘𝑘)) ∧ 𝑡 ∈ 𝑅) → (0...𝑘) ⊆
ℕ0) |
100 | | simpr 476 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝜑 ∧ 𝑘 ∈ (0...𝐽)) ∧ 𝑐 ∈ ((𝐶‘𝑅)‘𝑘)) → 𝑐 ∈ ((𝐶‘𝑅)‘𝑘)) |
101 | 86 | eleq2d 2673 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽)) → (𝑐 ∈ ((𝐶‘𝑅)‘𝑘) ↔ 𝑐 ∈ {𝑐 ∈ ((0...𝑘) ↑𝑚 𝑅) ∣ Σ𝑡 ∈ 𝑅 (𝑐‘𝑡) = 𝑘})) |
102 | 101 | adantr 480 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝜑 ∧ 𝑘 ∈ (0...𝐽)) ∧ 𝑐 ∈ ((𝐶‘𝑅)‘𝑘)) → (𝑐 ∈ ((𝐶‘𝑅)‘𝑘) ↔ 𝑐 ∈ {𝑐 ∈ ((0...𝑘) ↑𝑚 𝑅) ∣ Σ𝑡 ∈ 𝑅 (𝑐‘𝑡) = 𝑘})) |
103 | 100, 102 | mpbid 221 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ 𝑘 ∈ (0...𝐽)) ∧ 𝑐 ∈ ((𝐶‘𝑅)‘𝑘)) → 𝑐 ∈ {𝑐 ∈ ((0...𝑘) ↑𝑚 𝑅) ∣ Σ𝑡 ∈ 𝑅 (𝑐‘𝑡) = 𝑘}) |
104 | 83 | sseli 3564 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑐 ∈ {𝑐 ∈ ((0...𝑘) ↑𝑚 𝑅) ∣ Σ𝑡 ∈ 𝑅 (𝑐‘𝑡) = 𝑘} → 𝑐 ∈ ((0...𝑘) ↑𝑚 𝑅)) |
105 | 103, 104 | syl 17 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ 𝑘 ∈ (0...𝐽)) ∧ 𝑐 ∈ ((𝐶‘𝑅)‘𝑘)) → 𝑐 ∈ ((0...𝑘) ↑𝑚 𝑅)) |
106 | | elmapi 7765 |
. . . . . . . . . . . . . . . 16
⊢ (𝑐 ∈ ((0...𝑘) ↑𝑚 𝑅) → 𝑐:𝑅⟶(0...𝑘)) |
107 | 105, 106 | syl 17 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑘 ∈ (0...𝐽)) ∧ 𝑐 ∈ ((𝐶‘𝑅)‘𝑘)) → 𝑐:𝑅⟶(0...𝑘)) |
108 | 107 | adantr 480 |
. . . . . . . . . . . . . 14
⊢ ((((𝜑 ∧ 𝑘 ∈ (0...𝐽)) ∧ 𝑐 ∈ ((𝐶‘𝑅)‘𝑘)) ∧ 𝑡 ∈ 𝑅) → 𝑐:𝑅⟶(0...𝑘)) |
109 | | simpr 476 |
. . . . . . . . . . . . . 14
⊢ ((((𝜑 ∧ 𝑘 ∈ (0...𝐽)) ∧ 𝑐 ∈ ((𝐶‘𝑅)‘𝑘)) ∧ 𝑡 ∈ 𝑅) → 𝑡 ∈ 𝑅) |
110 | 108, 109 | ffvelrnd 6268 |
. . . . . . . . . . . . 13
⊢ ((((𝜑 ∧ 𝑘 ∈ (0...𝐽)) ∧ 𝑐 ∈ ((𝐶‘𝑅)‘𝑘)) ∧ 𝑡 ∈ 𝑅) → (𝑐‘𝑡) ∈ (0...𝑘)) |
111 | 99, 110 | sseldd 3569 |
. . . . . . . . . . . 12
⊢ ((((𝜑 ∧ 𝑘 ∈ (0...𝐽)) ∧ 𝑐 ∈ ((𝐶‘𝑅)‘𝑘)) ∧ 𝑡 ∈ 𝑅) → (𝑐‘𝑡) ∈
ℕ0) |
112 | 111 | faccld 12933 |
. . . . . . . . . . 11
⊢ ((((𝜑 ∧ 𝑘 ∈ (0...𝐽)) ∧ 𝑐 ∈ ((𝐶‘𝑅)‘𝑘)) ∧ 𝑡 ∈ 𝑅) → (!‘(𝑐‘𝑡)) ∈ ℕ) |
113 | 112 | nncnd 10913 |
. . . . . . . . . 10
⊢ ((((𝜑 ∧ 𝑘 ∈ (0...𝐽)) ∧ 𝑐 ∈ ((𝐶‘𝑅)‘𝑘)) ∧ 𝑡 ∈ 𝑅) → (!‘(𝑐‘𝑡)) ∈ ℂ) |
114 | 96, 113 | fprodcl 14521 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑘 ∈ (0...𝐽)) ∧ 𝑐 ∈ ((𝐶‘𝑅)‘𝑘)) → ∏𝑡 ∈ 𝑅 (!‘(𝑐‘𝑡)) ∈ ℂ) |
115 | 112 | nnne0d 10942 |
. . . . . . . . . 10
⊢ ((((𝜑 ∧ 𝑘 ∈ (0...𝐽)) ∧ 𝑐 ∈ ((𝐶‘𝑅)‘𝑘)) ∧ 𝑡 ∈ 𝑅) → (!‘(𝑐‘𝑡)) ≠ 0) |
116 | 96, 113, 115 | fprodn0 14548 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑘 ∈ (0...𝐽)) ∧ 𝑐 ∈ ((𝐶‘𝑅)‘𝑘)) → ∏𝑡 ∈ 𝑅 (!‘(𝑐‘𝑡)) ≠ 0) |
117 | 94, 114, 116 | divcld 10680 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑘 ∈ (0...𝐽)) ∧ 𝑐 ∈ ((𝐶‘𝑅)‘𝑘)) → ((!‘𝑘) / ∏𝑡 ∈ 𝑅 (!‘(𝑐‘𝑡))) ∈ ℂ) |
118 | 117 | adantlr 747 |
. . . . . . 7
⊢ ((((𝜑 ∧ 𝑘 ∈ (0...𝐽)) ∧ 𝑥 ∈ 𝑋) ∧ 𝑐 ∈ ((𝐶‘𝑅)‘𝑘)) → ((!‘𝑘) / ∏𝑡 ∈ 𝑅 (!‘(𝑐‘𝑡))) ∈ ℂ) |
119 | 96 | adantlr 747 |
. . . . . . . 8
⊢ ((((𝜑 ∧ 𝑘 ∈ (0...𝐽)) ∧ 𝑥 ∈ 𝑋) ∧ 𝑐 ∈ ((𝐶‘𝑅)‘𝑘)) → 𝑅 ∈ Fin) |
120 | 21 | ad4antr 764 |
. . . . . . . . . 10
⊢
(((((𝜑 ∧ 𝑘 ∈ (0...𝐽)) ∧ 𝑥 ∈ 𝑋) ∧ 𝑐 ∈ ((𝐶‘𝑅)‘𝑘)) ∧ 𝑡 ∈ 𝑅) → 𝜑) |
121 | 120, 13 | sylancom 698 |
. . . . . . . . . 10
⊢
(((((𝜑 ∧ 𝑘 ∈ (0...𝐽)) ∧ 𝑥 ∈ 𝑋) ∧ 𝑐 ∈ ((𝐶‘𝑅)‘𝑘)) ∧ 𝑡 ∈ 𝑅) → 𝑡 ∈ 𝑇) |
122 | | elfzuz3 12210 |
. . . . . . . . . . . . . . . 16
⊢ (𝑘 ∈ (0...𝐽) → 𝐽 ∈ (ℤ≥‘𝑘)) |
123 | | fzss2 12252 |
. . . . . . . . . . . . . . . 16
⊢ (𝐽 ∈
(ℤ≥‘𝑘) → (0...𝑘) ⊆ (0...𝐽)) |
124 | 122, 123 | syl 17 |
. . . . . . . . . . . . . . 15
⊢ (𝑘 ∈ (0...𝐽) → (0...𝑘) ⊆ (0...𝐽)) |
125 | 124 | adantl 481 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽)) → (0...𝑘) ⊆ (0...𝐽)) |
126 | 45 | nn0zd 11356 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝜑 → 𝐽 ∈ ℤ) |
127 | | dvnprodlem2.n |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝜑 → 𝑁 ∈
ℕ0) |
128 | 127 | nn0zd 11356 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝜑 → 𝑁 ∈ ℤ) |
129 | | elfzle2 12216 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝐽 ∈ (0...𝑁) → 𝐽 ≤ 𝑁) |
130 | 43, 129 | syl 17 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝜑 → 𝐽 ≤ 𝑁) |
131 | 126, 128,
130 | 3jca 1235 |
. . . . . . . . . . . . . . . . 17
⊢ (𝜑 → (𝐽 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝐽 ≤ 𝑁)) |
132 | | eluz2 11569 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑁 ∈
(ℤ≥‘𝐽) ↔ (𝐽 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝐽 ≤ 𝑁)) |
133 | 131, 132 | sylibr 223 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝐽)) |
134 | | fzss2 12252 |
. . . . . . . . . . . . . . . 16
⊢ (𝑁 ∈
(ℤ≥‘𝐽) → (0...𝐽) ⊆ (0...𝑁)) |
135 | 133, 134 | syl 17 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → (0...𝐽) ⊆ (0...𝑁)) |
136 | 135 | adantr 480 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽)) → (0...𝐽) ⊆ (0...𝑁)) |
137 | 125, 136 | sstrd 3578 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽)) → (0...𝑘) ⊆ (0...𝑁)) |
138 | 137 | ad2antrr 758 |
. . . . . . . . . . . 12
⊢ ((((𝜑 ∧ 𝑘 ∈ (0...𝐽)) ∧ 𝑐 ∈ ((𝐶‘𝑅)‘𝑘)) ∧ 𝑡 ∈ 𝑅) → (0...𝑘) ⊆ (0...𝑁)) |
139 | 138, 110 | sseldd 3569 |
. . . . . . . . . . 11
⊢ ((((𝜑 ∧ 𝑘 ∈ (0...𝐽)) ∧ 𝑐 ∈ ((𝐶‘𝑅)‘𝑘)) ∧ 𝑡 ∈ 𝑅) → (𝑐‘𝑡) ∈ (0...𝑁)) |
140 | 139 | adantllr 751 |
. . . . . . . . . 10
⊢
(((((𝜑 ∧ 𝑘 ∈ (0...𝐽)) ∧ 𝑥 ∈ 𝑋) ∧ 𝑐 ∈ ((𝐶‘𝑅)‘𝑘)) ∧ 𝑡 ∈ 𝑅) → (𝑐‘𝑡) ∈ (0...𝑁)) |
141 | | fvex 6113 |
. . . . . . . . . . 11
⊢ (𝑐‘𝑡) ∈ V |
142 | | eleq1 2676 |
. . . . . . . . . . . . 13
⊢ (𝑗 = (𝑐‘𝑡) → (𝑗 ∈ (0...𝑁) ↔ (𝑐‘𝑡) ∈ (0...𝑁))) |
143 | 142 | 3anbi3d 1397 |
. . . . . . . . . . . 12
⊢ (𝑗 = (𝑐‘𝑡) → ((𝜑 ∧ 𝑡 ∈ 𝑇 ∧ 𝑗 ∈ (0...𝑁)) ↔ (𝜑 ∧ 𝑡 ∈ 𝑇 ∧ (𝑐‘𝑡) ∈ (0...𝑁)))) |
144 | | fveq2 6103 |
. . . . . . . . . . . . 13
⊢ (𝑗 = (𝑐‘𝑡) → ((𝑆 D𝑛 (𝐻‘𝑡))‘𝑗) = ((𝑆 D𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))) |
145 | 144 | feq1d 5943 |
. . . . . . . . . . . 12
⊢ (𝑗 = (𝑐‘𝑡) → (((𝑆 D𝑛 (𝐻‘𝑡))‘𝑗):𝑋⟶ℂ ↔ ((𝑆 D𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡)):𝑋⟶ℂ)) |
146 | 143, 145 | imbi12d 333 |
. . . . . . . . . . 11
⊢ (𝑗 = (𝑐‘𝑡) → (((𝜑 ∧ 𝑡 ∈ 𝑇 ∧ 𝑗 ∈ (0...𝑁)) → ((𝑆 D𝑛 (𝐻‘𝑡))‘𝑗):𝑋⟶ℂ) ↔ ((𝜑 ∧ 𝑡 ∈ 𝑇 ∧ (𝑐‘𝑡) ∈ (0...𝑁)) → ((𝑆 D𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡)):𝑋⟶ℂ))) |
147 | | dvnprodlem2.dvnh |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇 ∧ 𝑗 ∈ (0...𝑁)) → ((𝑆 D𝑛 (𝐻‘𝑡))‘𝑗):𝑋⟶ℂ) |
148 | 141, 146,
147 | vtocl 3232 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇 ∧ (𝑐‘𝑡) ∈ (0...𝑁)) → ((𝑆 D𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡)):𝑋⟶ℂ) |
149 | 120, 121,
140, 148 | syl3anc 1318 |
. . . . . . . . 9
⊢
(((((𝜑 ∧ 𝑘 ∈ (0...𝐽)) ∧ 𝑥 ∈ 𝑋) ∧ 𝑐 ∈ ((𝐶‘𝑅)‘𝑘)) ∧ 𝑡 ∈ 𝑅) → ((𝑆 D𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡)):𝑋⟶ℂ) |
150 | | simpllr 795 |
. . . . . . . . 9
⊢
(((((𝜑 ∧ 𝑘 ∈ (0...𝐽)) ∧ 𝑥 ∈ 𝑋) ∧ 𝑐 ∈ ((𝐶‘𝑅)‘𝑘)) ∧ 𝑡 ∈ 𝑅) → 𝑥 ∈ 𝑋) |
151 | 149, 150 | ffvelrnd 6268 |
. . . . . . . 8
⊢
(((((𝜑 ∧ 𝑘 ∈ (0...𝐽)) ∧ 𝑥 ∈ 𝑋) ∧ 𝑐 ∈ ((𝐶‘𝑅)‘𝑘)) ∧ 𝑡 ∈ 𝑅) → (((𝑆 D𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥) ∈ ℂ) |
152 | 119, 151 | fprodcl 14521 |
. . . . . . 7
⊢ ((((𝜑 ∧ 𝑘 ∈ (0...𝐽)) ∧ 𝑥 ∈ 𝑋) ∧ 𝑐 ∈ ((𝐶‘𝑅)‘𝑘)) → ∏𝑡 ∈ 𝑅 (((𝑆 D𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥) ∈ ℂ) |
153 | 118, 152 | mulcld 9939 |
. . . . . 6
⊢ ((((𝜑 ∧ 𝑘 ∈ (0...𝐽)) ∧ 𝑥 ∈ 𝑋) ∧ 𝑐 ∈ ((𝐶‘𝑅)‘𝑘)) → (((!‘𝑘) / ∏𝑡 ∈ 𝑅 (!‘(𝑐‘𝑡))) · ∏𝑡 ∈ 𝑅 (((𝑆 D𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥)) ∈ ℂ) |
154 | 91, 153 | fsumcl 14311 |
. . . . 5
⊢ (((𝜑 ∧ 𝑘 ∈ (0...𝐽)) ∧ 𝑥 ∈ 𝑋) → Σ𝑐 ∈ ((𝐶‘𝑅)‘𝑘)(((!‘𝑘) / ∏𝑡 ∈ 𝑅 (!‘(𝑐‘𝑡))) · ∏𝑡 ∈ 𝑅 (((𝑆 D𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥)) ∈ ℂ) |
155 | | eqid 2610 |
. . . . 5
⊢ (𝑥 ∈ 𝑋 ↦ Σ𝑐 ∈ ((𝐶‘𝑅)‘𝑘)(((!‘𝑘) / ∏𝑡 ∈ 𝑅 (!‘(𝑐‘𝑡))) · ∏𝑡 ∈ 𝑅 (((𝑆 D𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥))) = (𝑥 ∈ 𝑋 ↦ Σ𝑐 ∈ ((𝐶‘𝑅)‘𝑘)(((!‘𝑘) / ∏𝑡 ∈ 𝑅 (!‘(𝑐‘𝑡))) · ∏𝑡 ∈ 𝑅 (((𝑆 D𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥))) |
156 | 154, 155 | fmptd 6292 |
. . . 4
⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽)) → (𝑥 ∈ 𝑋 ↦ Σ𝑐 ∈ ((𝐶‘𝑅)‘𝑘)(((!‘𝑘) / ∏𝑡 ∈ 𝑅 (!‘(𝑐‘𝑡))) · ∏𝑡 ∈ 𝑅 (((𝑆 D𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥))):𝑋⟶ℂ) |
157 | | dvnprodlem2.ind |
. . . . . . 7
⊢ (𝜑 → ∀𝑘 ∈ (0...𝑁)((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ ∏𝑡 ∈ 𝑅 ((𝐻‘𝑡)‘𝑥)))‘𝑘) = (𝑥 ∈ 𝑋 ↦ Σ𝑐 ∈ ((𝐶‘𝑅)‘𝑘)(((!‘𝑘) / ∏𝑡 ∈ 𝑅 (!‘(𝑐‘𝑡))) · ∏𝑡 ∈ 𝑅 (((𝑆 D𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥)))) |
158 | 157 | adantr 480 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽)) → ∀𝑘 ∈ (0...𝑁)((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ ∏𝑡 ∈ 𝑅 ((𝐻‘𝑡)‘𝑥)))‘𝑘) = (𝑥 ∈ 𝑋 ↦ Σ𝑐 ∈ ((𝐶‘𝑅)‘𝑘)(((!‘𝑘) / ∏𝑡 ∈ 𝑅 (!‘(𝑐‘𝑡))) · ∏𝑡 ∈ 𝑅 (((𝑆 D𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥)))) |
159 | | 0zd 11266 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽)) → 0 ∈ ℤ) |
160 | 128 | adantr 480 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽)) → 𝑁 ∈ ℤ) |
161 | | elfzelz 12213 |
. . . . . . . . . 10
⊢ (𝑘 ∈ (0...𝐽) → 𝑘 ∈ ℤ) |
162 | 161 | adantl 481 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽)) → 𝑘 ∈ ℤ) |
163 | 159, 160,
162 | 3jca 1235 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽)) → (0 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑘 ∈
ℤ)) |
164 | | elfzle1 12215 |
. . . . . . . . 9
⊢ (𝑘 ∈ (0...𝐽) → 0 ≤ 𝑘) |
165 | 164 | adantl 481 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽)) → 0 ≤ 𝑘) |
166 | 162 | zred 11358 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽)) → 𝑘 ∈ ℝ) |
167 | 45 | nn0red 11229 |
. . . . . . . . . 10
⊢ (𝜑 → 𝐽 ∈ ℝ) |
168 | 167 | adantr 480 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽)) → 𝐽 ∈ ℝ) |
169 | 160 | zred 11358 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽)) → 𝑁 ∈ ℝ) |
170 | | elfzle2 12216 |
. . . . . . . . . 10
⊢ (𝑘 ∈ (0...𝐽) → 𝑘 ≤ 𝐽) |
171 | 170 | adantl 481 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽)) → 𝑘 ≤ 𝐽) |
172 | 130 | adantr 480 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽)) → 𝐽 ≤ 𝑁) |
173 | 166, 168,
169, 171, 172 | letrd 10073 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽)) → 𝑘 ≤ 𝑁) |
174 | 163, 165,
173 | jca32 556 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽)) → ((0 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑘 ∈ ℤ) ∧ (0 ≤
𝑘 ∧ 𝑘 ≤ 𝑁))) |
175 | | elfz2 12204 |
. . . . . . 7
⊢ (𝑘 ∈ (0...𝑁) ↔ ((0 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑘 ∈ ℤ) ∧ (0 ≤
𝑘 ∧ 𝑘 ≤ 𝑁))) |
176 | 174, 175 | sylibr 223 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽)) → 𝑘 ∈ (0...𝑁)) |
177 | | rspa 2914 |
. . . . . 6
⊢
((∀𝑘 ∈
(0...𝑁)((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ ∏𝑡 ∈ 𝑅 ((𝐻‘𝑡)‘𝑥)))‘𝑘) = (𝑥 ∈ 𝑋 ↦ Σ𝑐 ∈ ((𝐶‘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 ((𝐻‘𝑡)‘𝑥)))‘𝑘) = (𝑥 ∈ 𝑋 ↦ Σ𝑐 ∈ ((𝐶‘𝑅)‘𝑘)(((!‘𝑘) / ∏𝑡 ∈ 𝑅 (!‘(𝑐‘𝑡))) · ∏𝑡 ∈ 𝑅 (((𝑆 D𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥)))) |
178 | 158, 176,
177 | syl2anc 691 |
. . . . 5
⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽)) → ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ ∏𝑡 ∈ 𝑅 ((𝐻‘𝑡)‘𝑥)))‘𝑘) = (𝑥 ∈ 𝑋 ↦ Σ𝑐 ∈ ((𝐶‘𝑅)‘𝑘)(((!‘𝑘) / ∏𝑡 ∈ 𝑅 (!‘(𝑐‘𝑡))) · ∏𝑡 ∈ 𝑅 (((𝑆 D𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥)))) |
179 | 178 | feq1d 5943 |
. . . 4
⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽)) → (((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ ∏𝑡 ∈ 𝑅 ((𝐻‘𝑡)‘𝑥)))‘𝑘):𝑋⟶ℂ ↔ (𝑥 ∈ 𝑋 ↦ Σ𝑐 ∈ ((𝐶‘𝑅)‘𝑘)(((!‘𝑘) / ∏𝑡 ∈ 𝑅 (!‘(𝑐‘𝑡))) · ∏𝑡 ∈ 𝑅 (((𝑆 D𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥))):𝑋⟶ℂ)) |
180 | 156, 179 | mpbird 246 |
. . 3
⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽)) → ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ ∏𝑡 ∈ 𝑅 ((𝐻‘𝑡)‘𝑥)))‘𝑘):𝑋⟶ℂ) |
181 | 23 | adantr 480 |
. . . . 5
⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽)) → 𝑍 ∈ 𝑇) |
182 | | simpl 472 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽)) → 𝜑) |
183 | 182, 181,
176 | 3jca 1235 |
. . . . 5
⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽)) → (𝜑 ∧ 𝑍 ∈ 𝑇 ∧ 𝑘 ∈ (0...𝑁))) |
184 | 26 | 3anbi2d 1396 |
. . . . . . 7
⊢ (𝑡 = 𝑍 → ((𝜑 ∧ 𝑡 ∈ 𝑇 ∧ 𝑘 ∈ (0...𝑁)) ↔ (𝜑 ∧ 𝑍 ∈ 𝑇 ∧ 𝑘 ∈ (0...𝑁)))) |
185 | 19 | oveq2d 6565 |
. . . . . . . . 9
⊢ (𝑡 = 𝑍 → (𝑆 D𝑛 (𝐻‘𝑡)) = (𝑆 D𝑛 (𝐻‘𝑍))) |
186 | 185 | fveq1d 6105 |
. . . . . . . 8
⊢ (𝑡 = 𝑍 → ((𝑆 D𝑛 (𝐻‘𝑡))‘𝑘) = ((𝑆 D𝑛 (𝐻‘𝑍))‘𝑘)) |
187 | 186 | feq1d 5943 |
. . . . . . 7
⊢ (𝑡 = 𝑍 → (((𝑆 D𝑛 (𝐻‘𝑡))‘𝑘):𝑋⟶ℂ ↔ ((𝑆 D𝑛 (𝐻‘𝑍))‘𝑘):𝑋⟶ℂ)) |
188 | 184, 187 | imbi12d 333 |
. . . . . 6
⊢ (𝑡 = 𝑍 → (((𝜑 ∧ 𝑡 ∈ 𝑇 ∧ 𝑘 ∈ (0...𝑁)) → ((𝑆 D𝑛 (𝐻‘𝑡))‘𝑘):𝑋⟶ℂ) ↔ ((𝜑 ∧ 𝑍 ∈ 𝑇 ∧ 𝑘 ∈ (0...𝑁)) → ((𝑆 D𝑛 (𝐻‘𝑍))‘𝑘):𝑋⟶ℂ))) |
189 | | eleq1 2676 |
. . . . . . . . 9
⊢ (𝑗 = 𝑘 → (𝑗 ∈ (0...𝑁) ↔ 𝑘 ∈ (0...𝑁))) |
190 | 189 | 3anbi3d 1397 |
. . . . . . . 8
⊢ (𝑗 = 𝑘 → ((𝜑 ∧ 𝑡 ∈ 𝑇 ∧ 𝑗 ∈ (0...𝑁)) ↔ (𝜑 ∧ 𝑡 ∈ 𝑇 ∧ 𝑘 ∈ (0...𝑁)))) |
191 | | fveq2 6103 |
. . . . . . . . 9
⊢ (𝑗 = 𝑘 → ((𝑆 D𝑛 (𝐻‘𝑡))‘𝑗) = ((𝑆 D𝑛 (𝐻‘𝑡))‘𝑘)) |
192 | 191 | feq1d 5943 |
. . . . . . . 8
⊢ (𝑗 = 𝑘 → (((𝑆 D𝑛 (𝐻‘𝑡))‘𝑗):𝑋⟶ℂ ↔ ((𝑆 D𝑛 (𝐻‘𝑡))‘𝑘):𝑋⟶ℂ)) |
193 | 190, 192 | imbi12d 333 |
. . . . . . 7
⊢ (𝑗 = 𝑘 → (((𝜑 ∧ 𝑡 ∈ 𝑇 ∧ 𝑗 ∈ (0...𝑁)) → ((𝑆 D𝑛 (𝐻‘𝑡))‘𝑗):𝑋⟶ℂ) ↔ ((𝜑 ∧ 𝑡 ∈ 𝑇 ∧ 𝑘 ∈ (0...𝑁)) → ((𝑆 D𝑛 (𝐻‘𝑡))‘𝑘):𝑋⟶ℂ))) |
194 | 193, 147 | chvarv 2251 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇 ∧ 𝑘 ∈ (0...𝑁)) → ((𝑆 D𝑛 (𝐻‘𝑡))‘𝑘):𝑋⟶ℂ) |
195 | 188, 194 | vtoclg 3239 |
. . . . 5
⊢ (𝑍 ∈ 𝑇 → ((𝜑 ∧ 𝑍 ∈ 𝑇 ∧ 𝑘 ∈ (0...𝑁)) → ((𝑆 D𝑛 (𝐻‘𝑍))‘𝑘):𝑋⟶ℂ)) |
196 | 181, 183,
195 | sylc 63 |
. . . 4
⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽)) → ((𝑆 D𝑛 (𝐻‘𝑍))‘𝑘):𝑋⟶ℂ) |
197 | 32 | feqmptd 6159 |
. . . . . . . . 9
⊢ (𝜑 → (𝐻‘𝑍) = (𝑥 ∈ 𝑋 ↦ ((𝐻‘𝑍)‘𝑥))) |
198 | 197 | eqcomd 2616 |
. . . . . . . 8
⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ ((𝐻‘𝑍)‘𝑥)) = (𝐻‘𝑍)) |
199 | 198 | oveq2d 6565 |
. . . . . . 7
⊢ (𝜑 → (𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ ((𝐻‘𝑍)‘𝑥))) = (𝑆 D𝑛 (𝐻‘𝑍))) |
200 | 199 | fveq1d 6105 |
. . . . . 6
⊢ (𝜑 → ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ ((𝐻‘𝑍)‘𝑥)))‘𝑘) = ((𝑆 D𝑛 (𝐻‘𝑍))‘𝑘)) |
201 | 200 | adantr 480 |
. . . . 5
⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽)) → ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ ((𝐻‘𝑍)‘𝑥)))‘𝑘) = ((𝑆 D𝑛 (𝐻‘𝑍))‘𝑘)) |
202 | 201 | feq1d 5943 |
. . . 4
⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽)) → (((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ ((𝐻‘𝑍)‘𝑥)))‘𝑘):𝑋⟶ℂ ↔ ((𝑆 D𝑛 (𝐻‘𝑍))‘𝑘):𝑋⟶ℂ)) |
203 | 196, 202 | mpbird 246 |
. . 3
⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽)) → ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ ((𝐻‘𝑍)‘𝑥)))‘𝑘):𝑋⟶ℂ) |
204 | | fveq2 6103 |
. . . . . . . 8
⊢ (𝑦 = 𝑥 → ((𝐻‘𝑡)‘𝑦) = ((𝐻‘𝑡)‘𝑥)) |
205 | 204 | prodeq2ad 38659 |
. . . . . . 7
⊢ (𝑦 = 𝑥 → ∏𝑡 ∈ 𝑅 ((𝐻‘𝑡)‘𝑦) = ∏𝑡 ∈ 𝑅 ((𝐻‘𝑡)‘𝑥)) |
206 | 205 | cbvmptv 4678 |
. . . . . 6
⊢ (𝑦 ∈ 𝑋 ↦ ∏𝑡 ∈ 𝑅 ((𝐻‘𝑡)‘𝑦)) = (𝑥 ∈ 𝑋 ↦ ∏𝑡 ∈ 𝑅 ((𝐻‘𝑡)‘𝑥)) |
207 | 206 | oveq2i 6560 |
. . . . 5
⊢ (𝑆 D𝑛 (𝑦 ∈ 𝑋 ↦ ∏𝑡 ∈ 𝑅 ((𝐻‘𝑡)‘𝑦))) = (𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ ∏𝑡 ∈ 𝑅 ((𝐻‘𝑡)‘𝑥))) |
208 | 207 | fveq1i 6104 |
. . . 4
⊢ ((𝑆 D𝑛 (𝑦 ∈ 𝑋 ↦ ∏𝑡 ∈ 𝑅 ((𝐻‘𝑡)‘𝑦)))‘𝑘) = ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ ∏𝑡 ∈ 𝑅 ((𝐻‘𝑡)‘𝑥)))‘𝑘) |
209 | 208 | mpteq2i 4669 |
. . 3
⊢ (𝑘 ∈ (0...𝐽) ↦ ((𝑆 D𝑛 (𝑦 ∈ 𝑋 ↦ ∏𝑡 ∈ 𝑅 ((𝐻‘𝑡)‘𝑦)))‘𝑘)) = (𝑘 ∈ (0...𝐽) ↦ ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ ∏𝑡 ∈ 𝑅 ((𝐻‘𝑡)‘𝑥)))‘𝑘)) |
210 | | fveq2 6103 |
. . . . . . 7
⊢ (𝑦 = 𝑥 → ((𝐻‘𝑍)‘𝑦) = ((𝐻‘𝑍)‘𝑥)) |
211 | 210 | cbvmptv 4678 |
. . . . . 6
⊢ (𝑦 ∈ 𝑋 ↦ ((𝐻‘𝑍)‘𝑦)) = (𝑥 ∈ 𝑋 ↦ ((𝐻‘𝑍)‘𝑥)) |
212 | 211 | oveq2i 6560 |
. . . . 5
⊢ (𝑆 D𝑛 (𝑦 ∈ 𝑋 ↦ ((𝐻‘𝑍)‘𝑦))) = (𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ ((𝐻‘𝑍)‘𝑥))) |
213 | 212 | fveq1i 6104 |
. . . 4
⊢ ((𝑆 D𝑛 (𝑦 ∈ 𝑋 ↦ ((𝐻‘𝑍)‘𝑦)))‘𝑘) = ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ ((𝐻‘𝑍)‘𝑥)))‘𝑘) |
214 | 213 | mpteq2i 4669 |
. . 3
⊢ (𝑘 ∈ (0...𝐽) ↦ ((𝑆 D𝑛 (𝑦 ∈ 𝑋 ↦ ((𝐻‘𝑍)‘𝑦)))‘𝑘)) = (𝑘 ∈ (0...𝐽) ↦ ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ ((𝐻‘𝑍)‘𝑥)))‘𝑘)) |
215 | 40, 41, 42, 35, 45, 46, 47, 180, 203, 209, 214 | dvnmul 38833 |
. 2
⊢ (𝜑 → ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (∏𝑡 ∈ 𝑅 ((𝐻‘𝑡)‘𝑥) · ((𝐻‘𝑍)‘𝑥))))‘𝐽) = (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ (0...𝐽)((𝐽C𝑘) · ((((𝑘 ∈ (0...𝐽) ↦ ((𝑆 D𝑛 (𝑦 ∈ 𝑋 ↦ ∏𝑡 ∈ 𝑅 ((𝐻‘𝑡)‘𝑦)))‘𝑘))‘𝑘)‘𝑥) · (((𝑘 ∈ (0...𝐽) ↦ ((𝑆 D𝑛 (𝑦 ∈ 𝑋 ↦ ((𝐻‘𝑍)‘𝑦)))‘𝑘))‘(𝐽 − 𝑘))‘𝑥))))) |
216 | 208 | a1i 11 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽)) → ((𝑆 D𝑛 (𝑦 ∈ 𝑋 ↦ ∏𝑡 ∈ 𝑅 ((𝐻‘𝑡)‘𝑦)))‘𝑘) = ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ ∏𝑡 ∈ 𝑅 ((𝐻‘𝑡)‘𝑥)))‘𝑘)) |
217 | 157 | r19.21bi 2916 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑁)) → ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ ∏𝑡 ∈ 𝑅 ((𝐻‘𝑡)‘𝑥)))‘𝑘) = (𝑥 ∈ 𝑋 ↦ Σ𝑐 ∈ ((𝐶‘𝑅)‘𝑘)(((!‘𝑘) / ∏𝑡 ∈ 𝑅 (!‘(𝑐‘𝑡))) · ∏𝑡 ∈ 𝑅 (((𝑆 D𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥)))) |
218 | 182, 176,
217 | syl2anc 691 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽)) → ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ ∏𝑡 ∈ 𝑅 ((𝐻‘𝑡)‘𝑥)))‘𝑘) = (𝑥 ∈ 𝑋 ↦ Σ𝑐 ∈ ((𝐶‘𝑅)‘𝑘)(((!‘𝑘) / ∏𝑡 ∈ 𝑅 (!‘(𝑐‘𝑡))) · ∏𝑡 ∈ 𝑅 (((𝑆 D𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥)))) |
219 | 216, 218 | eqtrd 2644 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽)) → ((𝑆 D𝑛 (𝑦 ∈ 𝑋 ↦ ∏𝑡 ∈ 𝑅 ((𝐻‘𝑡)‘𝑦)))‘𝑘) = (𝑥 ∈ 𝑋 ↦ Σ𝑐 ∈ ((𝐶‘𝑅)‘𝑘)(((!‘𝑘) / ∏𝑡 ∈ 𝑅 (!‘(𝑐‘𝑡))) · ∏𝑡 ∈ 𝑅 (((𝑆 D𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥)))) |
220 | 219 | mpteq2dva 4672 |
. . . . . . . . . . . 12
⊢ (𝜑 → (𝑘 ∈ (0...𝐽) ↦ ((𝑆 D𝑛 (𝑦 ∈ 𝑋 ↦ ∏𝑡 ∈ 𝑅 ((𝐻‘𝑡)‘𝑦)))‘𝑘)) = (𝑘 ∈ (0...𝐽) ↦ (𝑥 ∈ 𝑋 ↦ Σ𝑐 ∈ ((𝐶‘𝑅)‘𝑘)(((!‘𝑘) / ∏𝑡 ∈ 𝑅 (!‘(𝑐‘𝑡))) · ∏𝑡 ∈ 𝑅 (((𝑆 D𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥))))) |
221 | | mptexg 6389 |
. . . . . . . . . . . . . 14
⊢ (𝑋 ∈
((TopOpen‘ℂfld) ↾t 𝑆) → (𝑥 ∈ 𝑋 ↦ Σ𝑐 ∈ ((𝐶‘𝑅)‘𝑘)(((!‘𝑘) / ∏𝑡 ∈ 𝑅 (!‘(𝑐‘𝑡))) · ∏𝑡 ∈ 𝑅 (((𝑆 D𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥))) ∈ V) |
222 | 41, 221 | syl 17 |
. . . . . . . . . . . . 13
⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ Σ𝑐 ∈ ((𝐶‘𝑅)‘𝑘)(((!‘𝑘) / ∏𝑡 ∈ 𝑅 (!‘(𝑐‘𝑡))) · ∏𝑡 ∈ 𝑅 (((𝑆 D𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥))) ∈ V) |
223 | 222 | adantr 480 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽)) → (𝑥 ∈ 𝑋 ↦ Σ𝑐 ∈ ((𝐶‘𝑅)‘𝑘)(((!‘𝑘) / ∏𝑡 ∈ 𝑅 (!‘(𝑐‘𝑡))) · ∏𝑡 ∈ 𝑅 (((𝑆 D𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥))) ∈ V) |
224 | 220, 223 | fvmpt2d 6202 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽)) → ((𝑘 ∈ (0...𝐽) ↦ ((𝑆 D𝑛 (𝑦 ∈ 𝑋 ↦ ∏𝑡 ∈ 𝑅 ((𝐻‘𝑡)‘𝑦)))‘𝑘))‘𝑘) = (𝑥 ∈ 𝑋 ↦ Σ𝑐 ∈ ((𝐶‘𝑅)‘𝑘)(((!‘𝑘) / ∏𝑡 ∈ 𝑅 (!‘(𝑐‘𝑡))) · ∏𝑡 ∈ 𝑅 (((𝑆 D𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥)))) |
225 | 224 | adantlr 747 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ (0...𝐽)) → ((𝑘 ∈ (0...𝐽) ↦ ((𝑆 D𝑛 (𝑦 ∈ 𝑋 ↦ ∏𝑡 ∈ 𝑅 ((𝐻‘𝑡)‘𝑦)))‘𝑘))‘𝑘) = (𝑥 ∈ 𝑋 ↦ Σ𝑐 ∈ ((𝐶‘𝑅)‘𝑘)(((!‘𝑘) / ∏𝑡 ∈ 𝑅 (!‘(𝑐‘𝑡))) · ∏𝑡 ∈ 𝑅 (((𝑆 D𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥)))) |
226 | 225 | fveq1d 6105 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ (0...𝐽)) → (((𝑘 ∈ (0...𝐽) ↦ ((𝑆 D𝑛 (𝑦 ∈ 𝑋 ↦ ∏𝑡 ∈ 𝑅 ((𝐻‘𝑡)‘𝑦)))‘𝑘))‘𝑘)‘𝑥) = ((𝑥 ∈ 𝑋 ↦ Σ𝑐 ∈ ((𝐶‘𝑅)‘𝑘)(((!‘𝑘) / ∏𝑡 ∈ 𝑅 (!‘(𝑐‘𝑡))) · ∏𝑡 ∈ 𝑅 (((𝑆 D𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥)))‘𝑥)) |
227 | 34 | adantr 480 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ (0...𝐽)) → 𝑥 ∈ 𝑋) |
228 | 154 | an32s 842 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ (0...𝐽)) → Σ𝑐 ∈ ((𝐶‘𝑅)‘𝑘)(((!‘𝑘) / ∏𝑡 ∈ 𝑅 (!‘(𝑐‘𝑡))) · ∏𝑡 ∈ 𝑅 (((𝑆 D𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥)) ∈ ℂ) |
229 | 155 | fvmpt2 6200 |
. . . . . . . . . 10
⊢ ((𝑥 ∈ 𝑋 ∧ Σ𝑐 ∈ ((𝐶‘𝑅)‘𝑘)(((!‘𝑘) / ∏𝑡 ∈ 𝑅 (!‘(𝑐‘𝑡))) · ∏𝑡 ∈ 𝑅 (((𝑆 D𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥)) ∈ ℂ) → ((𝑥 ∈ 𝑋 ↦ Σ𝑐 ∈ ((𝐶‘𝑅)‘𝑘)(((!‘𝑘) / ∏𝑡 ∈ 𝑅 (!‘(𝑐‘𝑡))) · ∏𝑡 ∈ 𝑅 (((𝑆 D𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥)))‘𝑥) = Σ𝑐 ∈ ((𝐶‘𝑅)‘𝑘)(((!‘𝑘) / ∏𝑡 ∈ 𝑅 (!‘(𝑐‘𝑡))) · ∏𝑡 ∈ 𝑅 (((𝑆 D𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥))) |
230 | 227, 228,
229 | syl2anc 691 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ (0...𝐽)) → ((𝑥 ∈ 𝑋 ↦ Σ𝑐 ∈ ((𝐶‘𝑅)‘𝑘)(((!‘𝑘) / ∏𝑡 ∈ 𝑅 (!‘(𝑐‘𝑡))) · ∏𝑡 ∈ 𝑅 (((𝑆 D𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥)))‘𝑥) = Σ𝑐 ∈ ((𝐶‘𝑅)‘𝑘)(((!‘𝑘) / ∏𝑡 ∈ 𝑅 (!‘(𝑐‘𝑡))) · ∏𝑡 ∈ 𝑅 (((𝑆 D𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥))) |
231 | 226, 230 | eqtrd 2644 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ (0...𝐽)) → (((𝑘 ∈ (0...𝐽) ↦ ((𝑆 D𝑛 (𝑦 ∈ 𝑋 ↦ ∏𝑡 ∈ 𝑅 ((𝐻‘𝑡)‘𝑦)))‘𝑘))‘𝑘)‘𝑥) = Σ𝑐 ∈ ((𝐶‘𝑅)‘𝑘)(((!‘𝑘) / ∏𝑡 ∈ 𝑅 (!‘(𝑐‘𝑡))) · ∏𝑡 ∈ 𝑅 (((𝑆 D𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥))) |
232 | | fveq2 6103 |
. . . . . . . . . . . . . . 15
⊢ (𝑘 = 𝑗 → ((𝑆 D𝑛 (𝑦 ∈ 𝑋 ↦ ((𝐻‘𝑍)‘𝑦)))‘𝑘) = ((𝑆 D𝑛 (𝑦 ∈ 𝑋 ↦ ((𝐻‘𝑍)‘𝑦)))‘𝑗)) |
233 | 232 | cbvmptv 4678 |
. . . . . . . . . . . . . 14
⊢ (𝑘 ∈ (0...𝐽) ↦ ((𝑆 D𝑛 (𝑦 ∈ 𝑋 ↦ ((𝐻‘𝑍)‘𝑦)))‘𝑘)) = (𝑗 ∈ (0...𝐽) ↦ ((𝑆 D𝑛 (𝑦 ∈ 𝑋 ↦ ((𝐻‘𝑍)‘𝑦)))‘𝑗)) |
234 | 233 | a1i 11 |
. . . . . . . . . . . . 13
⊢ (𝜑 → (𝑘 ∈ (0...𝐽) ↦ ((𝑆 D𝑛 (𝑦 ∈ 𝑋 ↦ ((𝐻‘𝑍)‘𝑦)))‘𝑘)) = (𝑗 ∈ (0...𝐽) ↦ ((𝑆 D𝑛 (𝑦 ∈ 𝑋 ↦ ((𝐻‘𝑍)‘𝑦)))‘𝑗))) |
235 | 212, 199 | syl5eq 2656 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → (𝑆 D𝑛 (𝑦 ∈ 𝑋 ↦ ((𝐻‘𝑍)‘𝑦))) = (𝑆 D𝑛 (𝐻‘𝑍))) |
236 | 235 | fveq1d 6105 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → ((𝑆 D𝑛 (𝑦 ∈ 𝑋 ↦ ((𝐻‘𝑍)‘𝑦)))‘𝑗) = ((𝑆 D𝑛 (𝐻‘𝑍))‘𝑗)) |
237 | 236 | mpteq2dv 4673 |
. . . . . . . . . . . . 13
⊢ (𝜑 → (𝑗 ∈ (0...𝐽) ↦ ((𝑆 D𝑛 (𝑦 ∈ 𝑋 ↦ ((𝐻‘𝑍)‘𝑦)))‘𝑗)) = (𝑗 ∈ (0...𝐽) ↦ ((𝑆 D𝑛 (𝐻‘𝑍))‘𝑗))) |
238 | 234, 237 | eqtrd 2644 |
. . . . . . . . . . . 12
⊢ (𝜑 → (𝑘 ∈ (0...𝐽) ↦ ((𝑆 D𝑛 (𝑦 ∈ 𝑋 ↦ ((𝐻‘𝑍)‘𝑦)))‘𝑘)) = (𝑗 ∈ (0...𝐽) ↦ ((𝑆 D𝑛 (𝐻‘𝑍))‘𝑗))) |
239 | 238 | adantr 480 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽)) → (𝑘 ∈ (0...𝐽) ↦ ((𝑆 D𝑛 (𝑦 ∈ 𝑋 ↦ ((𝐻‘𝑍)‘𝑦)))‘𝑘)) = (𝑗 ∈ (0...𝐽) ↦ ((𝑆 D𝑛 (𝐻‘𝑍))‘𝑗))) |
240 | | fveq2 6103 |
. . . . . . . . . . . 12
⊢ (𝑗 = (𝐽 − 𝑘) → ((𝑆 D𝑛 (𝐻‘𝑍))‘𝑗) = ((𝑆 D𝑛 (𝐻‘𝑍))‘(𝐽 − 𝑘))) |
241 | 240 | adantl 481 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑘 ∈ (0...𝐽)) ∧ 𝑗 = (𝐽 − 𝑘)) → ((𝑆 D𝑛 (𝐻‘𝑍))‘𝑗) = ((𝑆 D𝑛 (𝐻‘𝑍))‘(𝐽 − 𝑘))) |
242 | | 0zd 11266 |
. . . . . . . . . . . . . . 15
⊢ (𝑘 ∈ (0...𝐽) → 0 ∈ ℤ) |
243 | | elfzel2 12211 |
. . . . . . . . . . . . . . 15
⊢ (𝑘 ∈ (0...𝐽) → 𝐽 ∈ ℤ) |
244 | 243, 161 | zsubcld 11363 |
. . . . . . . . . . . . . . 15
⊢ (𝑘 ∈ (0...𝐽) → (𝐽 − 𝑘) ∈ ℤ) |
245 | 242, 243,
244 | 3jca 1235 |
. . . . . . . . . . . . . 14
⊢ (𝑘 ∈ (0...𝐽) → (0 ∈ ℤ ∧ 𝐽 ∈ ℤ ∧ (𝐽 − 𝑘) ∈ ℤ)) |
246 | 243 | zred 11358 |
. . . . . . . . . . . . . . . 16
⊢ (𝑘 ∈ (0...𝐽) → 𝐽 ∈ ℝ) |
247 | 77 | nn0red 11229 |
. . . . . . . . . . . . . . . 16
⊢ (𝑘 ∈ (0...𝐽) → 𝑘 ∈ ℝ) |
248 | 246, 247 | subge0d 10496 |
. . . . . . . . . . . . . . 15
⊢ (𝑘 ∈ (0...𝐽) → (0 ≤ (𝐽 − 𝑘) ↔ 𝑘 ≤ 𝐽)) |
249 | 170, 248 | mpbird 246 |
. . . . . . . . . . . . . 14
⊢ (𝑘 ∈ (0...𝐽) → 0 ≤ (𝐽 − 𝑘)) |
250 | 246, 247 | subge02d 10498 |
. . . . . . . . . . . . . . 15
⊢ (𝑘 ∈ (0...𝐽) → (0 ≤ 𝑘 ↔ (𝐽 − 𝑘) ≤ 𝐽)) |
251 | 164, 250 | mpbid 221 |
. . . . . . . . . . . . . 14
⊢ (𝑘 ∈ (0...𝐽) → (𝐽 − 𝑘) ≤ 𝐽) |
252 | 245, 249,
251 | jca32 556 |
. . . . . . . . . . . . 13
⊢ (𝑘 ∈ (0...𝐽) → ((0 ∈ ℤ ∧ 𝐽 ∈ ℤ ∧ (𝐽 − 𝑘) ∈ ℤ) ∧ (0 ≤ (𝐽 − 𝑘) ∧ (𝐽 − 𝑘) ≤ 𝐽))) |
253 | 252 | adantl 481 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽)) → ((0 ∈ ℤ ∧ 𝐽 ∈ ℤ ∧ (𝐽 − 𝑘) ∈ ℤ) ∧ (0 ≤ (𝐽 − 𝑘) ∧ (𝐽 − 𝑘) ≤ 𝐽))) |
254 | | elfz2 12204 |
. . . . . . . . . . . 12
⊢ ((𝐽 − 𝑘) ∈ (0...𝐽) ↔ ((0 ∈ ℤ ∧ 𝐽 ∈ ℤ ∧ (𝐽 − 𝑘) ∈ ℤ) ∧ (0 ≤ (𝐽 − 𝑘) ∧ (𝐽 − 𝑘) ≤ 𝐽))) |
255 | 253, 254 | sylibr 223 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽)) → (𝐽 − 𝑘) ∈ (0...𝐽)) |
256 | | fvex 6113 |
. . . . . . . . . . . 12
⊢ ((𝑆 D𝑛 (𝐻‘𝑍))‘(𝐽 − 𝑘)) ∈ V |
257 | 256 | a1i 11 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽)) → ((𝑆 D𝑛 (𝐻‘𝑍))‘(𝐽 − 𝑘)) ∈ V) |
258 | 239, 241,
255, 257 | fvmptd 6197 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽)) → ((𝑘 ∈ (0...𝐽) ↦ ((𝑆 D𝑛 (𝑦 ∈ 𝑋 ↦ ((𝐻‘𝑍)‘𝑦)))‘𝑘))‘(𝐽 − 𝑘)) = ((𝑆 D𝑛 (𝐻‘𝑍))‘(𝐽 − 𝑘))) |
259 | 258 | adantlr 747 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ (0...𝐽)) → ((𝑘 ∈ (0...𝐽) ↦ ((𝑆 D𝑛 (𝑦 ∈ 𝑋 ↦ ((𝐻‘𝑍)‘𝑦)))‘𝑘))‘(𝐽 − 𝑘)) = ((𝑆 D𝑛 (𝐻‘𝑍))‘(𝐽 − 𝑘))) |
260 | 259 | fveq1d 6105 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ (0...𝐽)) → (((𝑘 ∈ (0...𝐽) ↦ ((𝑆 D𝑛 (𝑦 ∈ 𝑋 ↦ ((𝐻‘𝑍)‘𝑦)))‘𝑘))‘(𝐽 − 𝑘))‘𝑥) = (((𝑆 D𝑛 (𝐻‘𝑍))‘(𝐽 − 𝑘))‘𝑥)) |
261 | 231, 260 | oveq12d 6567 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ (0...𝐽)) → ((((𝑘 ∈ (0...𝐽) ↦ ((𝑆 D𝑛 (𝑦 ∈ 𝑋 ↦ ∏𝑡 ∈ 𝑅 ((𝐻‘𝑡)‘𝑦)))‘𝑘))‘𝑘)‘𝑥) · (((𝑘 ∈ (0...𝐽) ↦ ((𝑆 D𝑛 (𝑦 ∈ 𝑋 ↦ ((𝐻‘𝑍)‘𝑦)))‘𝑘))‘(𝐽 − 𝑘))‘𝑥)) = (Σ𝑐 ∈ ((𝐶‘𝑅)‘𝑘)(((!‘𝑘) / ∏𝑡 ∈ 𝑅 (!‘(𝑐‘𝑡))) · ∏𝑡 ∈ 𝑅 (((𝑆 D𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥)) · (((𝑆 D𝑛 (𝐻‘𝑍))‘(𝐽 − 𝑘))‘𝑥))) |
262 | 261 | oveq2d 6565 |
. . . . . 6
⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ (0...𝐽)) → ((𝐽C𝑘) · ((((𝑘 ∈ (0...𝐽) ↦ ((𝑆 D𝑛 (𝑦 ∈ 𝑋 ↦ ∏𝑡 ∈ 𝑅 ((𝐻‘𝑡)‘𝑦)))‘𝑘))‘𝑘)‘𝑥) · (((𝑘 ∈ (0...𝐽) ↦ ((𝑆 D𝑛 (𝑦 ∈ 𝑋 ↦ ((𝐻‘𝑍)‘𝑦)))‘𝑘))‘(𝐽 − 𝑘))‘𝑥))) = ((𝐽C𝑘) · (Σ𝑐 ∈ ((𝐶‘𝑅)‘𝑘)(((!‘𝑘) / ∏𝑡 ∈ 𝑅 (!‘(𝑐‘𝑡))) · ∏𝑡 ∈ 𝑅 (((𝑆 D𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥)) · (((𝑆 D𝑛 (𝐻‘𝑍))‘(𝐽 − 𝑘))‘𝑥)))) |
263 | 90 | adantlr 747 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ (0...𝐽)) → ((𝐶‘𝑅)‘𝑘) ∈ Fin) |
264 | | ovex 6577 |
. . . . . . . . . . . 12
⊢ (𝐽 − 𝑘) ∈ V |
265 | | eleq1 2676 |
. . . . . . . . . . . . . 14
⊢ (𝑗 = (𝐽 − 𝑘) → (𝑗 ∈ (0...𝐽) ↔ (𝐽 − 𝑘) ∈ (0...𝐽))) |
266 | 265 | anbi2d 736 |
. . . . . . . . . . . . 13
⊢ (𝑗 = (𝐽 − 𝑘) → ((𝜑 ∧ 𝑗 ∈ (0...𝐽)) ↔ (𝜑 ∧ (𝐽 − 𝑘) ∈ (0...𝐽)))) |
267 | 240 | feq1d 5943 |
. . . . . . . . . . . . 13
⊢ (𝑗 = (𝐽 − 𝑘) → (((𝑆 D𝑛 (𝐻‘𝑍))‘𝑗):𝑋⟶ℂ ↔ ((𝑆 D𝑛 (𝐻‘𝑍))‘(𝐽 − 𝑘)):𝑋⟶ℂ)) |
268 | 266, 267 | imbi12d 333 |
. . . . . . . . . . . 12
⊢ (𝑗 = (𝐽 − 𝑘) → (((𝜑 ∧ 𝑗 ∈ (0...𝐽)) → ((𝑆 D𝑛 (𝐻‘𝑍))‘𝑗):𝑋⟶ℂ) ↔ ((𝜑 ∧ (𝐽 − 𝑘) ∈ (0...𝐽)) → ((𝑆 D𝑛 (𝐻‘𝑍))‘(𝐽 − 𝑘)):𝑋⟶ℂ))) |
269 | | eleq1 2676 |
. . . . . . . . . . . . . . 15
⊢ (𝑘 = 𝑗 → (𝑘 ∈ (0...𝐽) ↔ 𝑗 ∈ (0...𝐽))) |
270 | 269 | anbi2d 736 |
. . . . . . . . . . . . . 14
⊢ (𝑘 = 𝑗 → ((𝜑 ∧ 𝑘 ∈ (0...𝐽)) ↔ (𝜑 ∧ 𝑗 ∈ (0...𝐽)))) |
271 | | fveq2 6103 |
. . . . . . . . . . . . . . 15
⊢ (𝑘 = 𝑗 → ((𝑆 D𝑛 (𝐻‘𝑍))‘𝑘) = ((𝑆 D𝑛 (𝐻‘𝑍))‘𝑗)) |
272 | 271 | feq1d 5943 |
. . . . . . . . . . . . . 14
⊢ (𝑘 = 𝑗 → (((𝑆 D𝑛 (𝐻‘𝑍))‘𝑘):𝑋⟶ℂ ↔ ((𝑆 D𝑛 (𝐻‘𝑍))‘𝑗):𝑋⟶ℂ)) |
273 | 270, 272 | imbi12d 333 |
. . . . . . . . . . . . 13
⊢ (𝑘 = 𝑗 → (((𝜑 ∧ 𝑘 ∈ (0...𝐽)) → ((𝑆 D𝑛 (𝐻‘𝑍))‘𝑘):𝑋⟶ℂ) ↔ ((𝜑 ∧ 𝑗 ∈ (0...𝐽)) → ((𝑆 D𝑛 (𝐻‘𝑍))‘𝑗):𝑋⟶ℂ))) |
274 | 273, 196 | chvarv 2251 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝐽)) → ((𝑆 D𝑛 (𝐻‘𝑍))‘𝑗):𝑋⟶ℂ) |
275 | 264, 268,
274 | vtocl 3232 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ (𝐽 − 𝑘) ∈ (0...𝐽)) → ((𝑆 D𝑛 (𝐻‘𝑍))‘(𝐽 − 𝑘)):𝑋⟶ℂ) |
276 | 182, 255,
275 | syl2anc 691 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽)) → ((𝑆 D𝑛 (𝐻‘𝑍))‘(𝐽 − 𝑘)):𝑋⟶ℂ) |
277 | 276 | adantlr 747 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ (0...𝐽)) → ((𝑆 D𝑛 (𝐻‘𝑍))‘(𝐽 − 𝑘)):𝑋⟶ℂ) |
278 | 277, 227 | ffvelrnd 6268 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ (0...𝐽)) → (((𝑆 D𝑛 (𝐻‘𝑍))‘(𝐽 − 𝑘))‘𝑥) ∈ ℂ) |
279 | | anass 679 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑘 ∈ (0...𝐽)) ∧ 𝑥 ∈ 𝑋) ↔ (𝜑 ∧ (𝑘 ∈ (0...𝐽) ∧ 𝑥 ∈ 𝑋))) |
280 | | ancom 465 |
. . . . . . . . . . . . . 14
⊢ ((𝑘 ∈ (0...𝐽) ∧ 𝑥 ∈ 𝑋) ↔ (𝑥 ∈ 𝑋 ∧ 𝑘 ∈ (0...𝐽))) |
281 | 280 | anbi2i 726 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ (𝑘 ∈ (0...𝐽) ∧ 𝑥 ∈ 𝑋)) ↔ (𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑘 ∈ (0...𝐽)))) |
282 | | anass 679 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ (0...𝐽)) ↔ (𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑘 ∈ (0...𝐽)))) |
283 | 282 | bicomi 213 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑘 ∈ (0...𝐽))) ↔ ((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ (0...𝐽))) |
284 | 281, 283 | bitri 263 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ (𝑘 ∈ (0...𝐽) ∧ 𝑥 ∈ 𝑋)) ↔ ((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ (0...𝐽))) |
285 | 279, 284 | bitri 263 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑘 ∈ (0...𝐽)) ∧ 𝑥 ∈ 𝑋) ↔ ((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ (0...𝐽))) |
286 | 285 | anbi1i 727 |
. . . . . . . . . 10
⊢ ((((𝜑 ∧ 𝑘 ∈ (0...𝐽)) ∧ 𝑥 ∈ 𝑋) ∧ 𝑐 ∈ ((𝐶‘𝑅)‘𝑘)) ↔ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ (0...𝐽)) ∧ 𝑐 ∈ ((𝐶‘𝑅)‘𝑘))) |
287 | 286 | imbi1i 338 |
. . . . . . . . 9
⊢
(((((𝜑 ∧ 𝑘 ∈ (0...𝐽)) ∧ 𝑥 ∈ 𝑋) ∧ 𝑐 ∈ ((𝐶‘𝑅)‘𝑘)) → (((!‘𝑘) / ∏𝑡 ∈ 𝑅 (!‘(𝑐‘𝑡))) · ∏𝑡 ∈ 𝑅 (((𝑆 D𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥)) ∈ ℂ) ↔ ((((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ (0...𝐽)) ∧ 𝑐 ∈ ((𝐶‘𝑅)‘𝑘)) → (((!‘𝑘) / ∏𝑡 ∈ 𝑅 (!‘(𝑐‘𝑡))) · ∏𝑡 ∈ 𝑅 (((𝑆 D𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥)) ∈ ℂ)) |
288 | 153, 287 | mpbi 219 |
. . . . . . . 8
⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ (0...𝐽)) ∧ 𝑐 ∈ ((𝐶‘𝑅)‘𝑘)) → (((!‘𝑘) / ∏𝑡 ∈ 𝑅 (!‘(𝑐‘𝑡))) · ∏𝑡 ∈ 𝑅 (((𝑆 D𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥)) ∈ ℂ) |
289 | 263, 278,
288 | fsummulc1 14359 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ (0...𝐽)) → (Σ𝑐 ∈ ((𝐶‘𝑅)‘𝑘)(((!‘𝑘) / ∏𝑡 ∈ 𝑅 (!‘(𝑐‘𝑡))) · ∏𝑡 ∈ 𝑅 (((𝑆 D𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥)) · (((𝑆 D𝑛 (𝐻‘𝑍))‘(𝐽 − 𝑘))‘𝑥)) = Σ𝑐 ∈ ((𝐶‘𝑅)‘𝑘)((((!‘𝑘) / ∏𝑡 ∈ 𝑅 (!‘(𝑐‘𝑡))) · ∏𝑡 ∈ 𝑅 (((𝑆 D𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥)) · (((𝑆 D𝑛 (𝐻‘𝑍))‘(𝐽 − 𝑘))‘𝑥))) |
290 | 289 | oveq2d 6565 |
. . . . . 6
⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ (0...𝐽)) → ((𝐽C𝑘) · (Σ𝑐 ∈ ((𝐶‘𝑅)‘𝑘)(((!‘𝑘) / ∏𝑡 ∈ 𝑅 (!‘(𝑐‘𝑡))) · ∏𝑡 ∈ 𝑅 (((𝑆 D𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥)) · (((𝑆 D𝑛 (𝐻‘𝑍))‘(𝐽 − 𝑘))‘𝑥))) = ((𝐽C𝑘) · Σ𝑐 ∈ ((𝐶‘𝑅)‘𝑘)((((!‘𝑘) / ∏𝑡 ∈ 𝑅 (!‘(𝑐‘𝑡))) · ∏𝑡 ∈ 𝑅 (((𝑆 D𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥)) · (((𝑆 D𝑛 (𝐻‘𝑍))‘(𝐽 − 𝑘))‘𝑥)))) |
291 | 182, 45 | syl 17 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽)) → 𝐽 ∈
ℕ0) |
292 | 291, 162 | bccld 38472 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽)) → (𝐽C𝑘) ∈
ℕ0) |
293 | 292 | nn0cnd 11230 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽)) → (𝐽C𝑘) ∈ ℂ) |
294 | 293 | adantlr 747 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ (0...𝐽)) → (𝐽C𝑘) ∈ ℂ) |
295 | 278 | adantr 480 |
. . . . . . . 8
⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ (0...𝐽)) ∧ 𝑐 ∈ ((𝐶‘𝑅)‘𝑘)) → (((𝑆 D𝑛 (𝐻‘𝑍))‘(𝐽 − 𝑘))‘𝑥) ∈ ℂ) |
296 | 288, 295 | mulcld 9939 |
. . . . . . 7
⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ (0...𝐽)) ∧ 𝑐 ∈ ((𝐶‘𝑅)‘𝑘)) → ((((!‘𝑘) / ∏𝑡 ∈ 𝑅 (!‘(𝑐‘𝑡))) · ∏𝑡 ∈ 𝑅 (((𝑆 D𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥)) · (((𝑆 D𝑛 (𝐻‘𝑍))‘(𝐽 − 𝑘))‘𝑥)) ∈ ℂ) |
297 | 263, 294,
296 | fsummulc2 14358 |
. . . . . 6
⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ (0...𝐽)) → ((𝐽C𝑘) · Σ𝑐 ∈ ((𝐶‘𝑅)‘𝑘)((((!‘𝑘) / ∏𝑡 ∈ 𝑅 (!‘(𝑐‘𝑡))) · ∏𝑡 ∈ 𝑅 (((𝑆 D𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥)) · (((𝑆 D𝑛 (𝐻‘𝑍))‘(𝐽 − 𝑘))‘𝑥))) = Σ𝑐 ∈ ((𝐶‘𝑅)‘𝑘)((𝐽C𝑘) · ((((!‘𝑘) / ∏𝑡 ∈ 𝑅 (!‘(𝑐‘𝑡))) · ∏𝑡 ∈ 𝑅 (((𝑆 D𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥)) · (((𝑆 D𝑛 (𝐻‘𝑍))‘(𝐽 − 𝑘))‘𝑥)))) |
298 | 262, 290,
297 | 3eqtrd 2648 |
. . . . 5
⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ (0...𝐽)) → ((𝐽C𝑘) · ((((𝑘 ∈ (0...𝐽) ↦ ((𝑆 D𝑛 (𝑦 ∈ 𝑋 ↦ ∏𝑡 ∈ 𝑅 ((𝐻‘𝑡)‘𝑦)))‘𝑘))‘𝑘)‘𝑥) · (((𝑘 ∈ (0...𝐽) ↦ ((𝑆 D𝑛 (𝑦 ∈ 𝑋 ↦ ((𝐻‘𝑍)‘𝑦)))‘𝑘))‘(𝐽 − 𝑘))‘𝑥))) = Σ𝑐 ∈ ((𝐶‘𝑅)‘𝑘)((𝐽C𝑘) · ((((!‘𝑘) / ∏𝑡 ∈ 𝑅 (!‘(𝑐‘𝑡))) · ∏𝑡 ∈ 𝑅 (((𝑆 D𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥)) · (((𝑆 D𝑛 (𝐻‘𝑍))‘(𝐽 − 𝑘))‘𝑥)))) |
299 | 298 | sumeq2dv 14281 |
. . . 4
⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → Σ𝑘 ∈ (0...𝐽)((𝐽C𝑘) · ((((𝑘 ∈ (0...𝐽) ↦ ((𝑆 D𝑛 (𝑦 ∈ 𝑋 ↦ ∏𝑡 ∈ 𝑅 ((𝐻‘𝑡)‘𝑦)))‘𝑘))‘𝑘)‘𝑥) · (((𝑘 ∈ (0...𝐽) ↦ ((𝑆 D𝑛 (𝑦 ∈ 𝑋 ↦ ((𝐻‘𝑍)‘𝑦)))‘𝑘))‘(𝐽 − 𝑘))‘𝑥))) = Σ𝑘 ∈ (0...𝐽)Σ𝑐 ∈ ((𝐶‘𝑅)‘𝑘)((𝐽C𝑘) · ((((!‘𝑘) / ∏𝑡 ∈ 𝑅 (!‘(𝑐‘𝑡))) · ∏𝑡 ∈ 𝑅 (((𝑆 D𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥)) · (((𝑆 D𝑛 (𝐻‘𝑍))‘(𝐽 − 𝑘))‘𝑥)))) |
300 | | vex 3176 |
. . . . . . . 8
⊢ 𝑘 ∈ V |
301 | | vex 3176 |
. . . . . . . 8
⊢ 𝑐 ∈ V |
302 | 300, 301 | op1std 7069 |
. . . . . . 7
⊢ (𝑝 = 〈𝑘, 𝑐〉 → (1st ‘𝑝) = 𝑘) |
303 | 302 | oveq2d 6565 |
. . . . . 6
⊢ (𝑝 = 〈𝑘, 𝑐〉 → (𝐽C(1st ‘𝑝)) = (𝐽C𝑘)) |
304 | 302 | fveq2d 6107 |
. . . . . . . . 9
⊢ (𝑝 = 〈𝑘, 𝑐〉 → (!‘(1st
‘𝑝)) = (!‘𝑘)) |
305 | 300, 301 | op2ndd 7070 |
. . . . . . . . . . . 12
⊢ (𝑝 = 〈𝑘, 𝑐〉 → (2nd ‘𝑝) = 𝑐) |
306 | 305 | fveq1d 6105 |
. . . . . . . . . . 11
⊢ (𝑝 = 〈𝑘, 𝑐〉 → ((2nd ‘𝑝)‘𝑡) = (𝑐‘𝑡)) |
307 | 306 | fveq2d 6107 |
. . . . . . . . . 10
⊢ (𝑝 = 〈𝑘, 𝑐〉 → (!‘((2nd
‘𝑝)‘𝑡)) = (!‘(𝑐‘𝑡))) |
308 | 307 | prodeq2ad 38659 |
. . . . . . . . 9
⊢ (𝑝 = 〈𝑘, 𝑐〉 → ∏𝑡 ∈ 𝑅 (!‘((2nd ‘𝑝)‘𝑡)) = ∏𝑡 ∈ 𝑅 (!‘(𝑐‘𝑡))) |
309 | 304, 308 | oveq12d 6567 |
. . . . . . . 8
⊢ (𝑝 = 〈𝑘, 𝑐〉 → ((!‘(1st
‘𝑝)) / ∏𝑡 ∈ 𝑅 (!‘((2nd ‘𝑝)‘𝑡))) = ((!‘𝑘) / ∏𝑡 ∈ 𝑅 (!‘(𝑐‘𝑡)))) |
310 | 306 | fveq2d 6107 |
. . . . . . . . . 10
⊢ (𝑝 = 〈𝑘, 𝑐〉 → ((𝑆 D𝑛 (𝐻‘𝑡))‘((2nd ‘𝑝)‘𝑡)) = ((𝑆 D𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))) |
311 | 310 | fveq1d 6105 |
. . . . . . . . 9
⊢ (𝑝 = 〈𝑘, 𝑐〉 → (((𝑆 D𝑛 (𝐻‘𝑡))‘((2nd ‘𝑝)‘𝑡))‘𝑥) = (((𝑆 D𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥)) |
312 | 311 | prodeq2ad 38659 |
. . . . . . . 8
⊢ (𝑝 = 〈𝑘, 𝑐〉 → ∏𝑡 ∈ 𝑅 (((𝑆 D𝑛 (𝐻‘𝑡))‘((2nd ‘𝑝)‘𝑡))‘𝑥) = ∏𝑡 ∈ 𝑅 (((𝑆 D𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥)) |
313 | 309, 312 | oveq12d 6567 |
. . . . . . 7
⊢ (𝑝 = 〈𝑘, 𝑐〉 → (((!‘(1st
‘𝑝)) / ∏𝑡 ∈ 𝑅 (!‘((2nd ‘𝑝)‘𝑡))) · ∏𝑡 ∈ 𝑅 (((𝑆 D𝑛 (𝐻‘𝑡))‘((2nd ‘𝑝)‘𝑡))‘𝑥)) = (((!‘𝑘) / ∏𝑡 ∈ 𝑅 (!‘(𝑐‘𝑡))) · ∏𝑡 ∈ 𝑅 (((𝑆 D𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥))) |
314 | 302 | oveq2d 6565 |
. . . . . . . . 9
⊢ (𝑝 = 〈𝑘, 𝑐〉 → (𝐽 − (1st ‘𝑝)) = (𝐽 − 𝑘)) |
315 | 314 | fveq2d 6107 |
. . . . . . . 8
⊢ (𝑝 = 〈𝑘, 𝑐〉 → ((𝑆 D𝑛 (𝐻‘𝑍))‘(𝐽 − (1st ‘𝑝))) = ((𝑆 D𝑛 (𝐻‘𝑍))‘(𝐽 − 𝑘))) |
316 | 315 | fveq1d 6105 |
. . . . . . 7
⊢ (𝑝 = 〈𝑘, 𝑐〉 → (((𝑆 D𝑛 (𝐻‘𝑍))‘(𝐽 − (1st ‘𝑝)))‘𝑥) = (((𝑆 D𝑛 (𝐻‘𝑍))‘(𝐽 − 𝑘))‘𝑥)) |
317 | 313, 316 | oveq12d 6567 |
. . . . . 6
⊢ (𝑝 = 〈𝑘, 𝑐〉 → ((((!‘(1st
‘𝑝)) / ∏𝑡 ∈ 𝑅 (!‘((2nd ‘𝑝)‘𝑡))) · ∏𝑡 ∈ 𝑅 (((𝑆 D𝑛 (𝐻‘𝑡))‘((2nd ‘𝑝)‘𝑡))‘𝑥)) · (((𝑆 D𝑛 (𝐻‘𝑍))‘(𝐽 − (1st ‘𝑝)))‘𝑥)) = ((((!‘𝑘) / ∏𝑡 ∈ 𝑅 (!‘(𝑐‘𝑡))) · ∏𝑡 ∈ 𝑅 (((𝑆 D𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥)) · (((𝑆 D𝑛 (𝐻‘𝑍))‘(𝐽 − 𝑘))‘𝑥))) |
318 | 303, 317 | oveq12d 6567 |
. . . . 5
⊢ (𝑝 = 〈𝑘, 𝑐〉 → ((𝐽C(1st ‘𝑝)) · ((((!‘(1st
‘𝑝)) / ∏𝑡 ∈ 𝑅 (!‘((2nd ‘𝑝)‘𝑡))) · ∏𝑡 ∈ 𝑅 (((𝑆 D𝑛 (𝐻‘𝑡))‘((2nd ‘𝑝)‘𝑡))‘𝑥)) · (((𝑆 D𝑛 (𝐻‘𝑍))‘(𝐽 − (1st ‘𝑝)))‘𝑥))) = ((𝐽C𝑘) · ((((!‘𝑘) / ∏𝑡 ∈ 𝑅 (!‘(𝑐‘𝑡))) · ∏𝑡 ∈ 𝑅 (((𝑆 D𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥)) · (((𝑆 D𝑛 (𝐻‘𝑍))‘(𝐽 − 𝑘))‘𝑥)))) |
319 | | fzfid 12634 |
. . . . 5
⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (0...𝐽) ∈ Fin) |
320 | 294 | adantrr 749 |
. . . . . 6
⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ (𝑘 ∈ (0...𝐽) ∧ 𝑐 ∈ ((𝐶‘𝑅)‘𝑘))) → (𝐽C𝑘) ∈ ℂ) |
321 | 296 | anasss 677 |
. . . . . 6
⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ (𝑘 ∈ (0...𝐽) ∧ 𝑐 ∈ ((𝐶‘𝑅)‘𝑘))) → ((((!‘𝑘) / ∏𝑡 ∈ 𝑅 (!‘(𝑐‘𝑡))) · ∏𝑡 ∈ 𝑅 (((𝑆 D𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥)) · (((𝑆 D𝑛 (𝐻‘𝑍))‘(𝐽 − 𝑘))‘𝑥)) ∈ ℂ) |
322 | 320, 321 | mulcld 9939 |
. . . . 5
⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ (𝑘 ∈ (0...𝐽) ∧ 𝑐 ∈ ((𝐶‘𝑅)‘𝑘))) → ((𝐽C𝑘) · ((((!‘𝑘) / ∏𝑡 ∈ 𝑅 (!‘(𝑐‘𝑡))) · ∏𝑡 ∈ 𝑅 (((𝑆 D𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥)) · (((𝑆 D𝑛 (𝐻‘𝑍))‘(𝐽 − 𝑘))‘𝑥))) ∈ ℂ) |
323 | 318, 319,
263, 322 | fsum2d 14344 |
. . . 4
⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → Σ𝑘 ∈ (0...𝐽)Σ𝑐 ∈ ((𝐶‘𝑅)‘𝑘)((𝐽C𝑘) · ((((!‘𝑘) / ∏𝑡 ∈ 𝑅 (!‘(𝑐‘𝑡))) · ∏𝑡 ∈ 𝑅 (((𝑆 D𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥)) · (((𝑆 D𝑛 (𝐻‘𝑍))‘(𝐽 − 𝑘))‘𝑥))) = Σ𝑝 ∈ ∪
𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘))((𝐽C(1st ‘𝑝)) · ((((!‘(1st
‘𝑝)) / ∏𝑡 ∈ 𝑅 (!‘((2nd ‘𝑝)‘𝑡))) · ∏𝑡 ∈ 𝑅 (((𝑆 D𝑛 (𝐻‘𝑡))‘((2nd ‘𝑝)‘𝑡))‘𝑥)) · (((𝑆 D𝑛 (𝐻‘𝑍))‘(𝐽 − (1st ‘𝑝)))‘𝑥)))) |
324 | | ovex 6577 |
. . . . . . . . 9
⊢ (𝐽 − (𝑐‘𝑍)) ∈ V |
325 | 301 | resex 5363 |
. . . . . . . . 9
⊢ (𝑐 ↾ 𝑅) ∈ V |
326 | 324, 325 | op1std 7069 |
. . . . . . . 8
⊢ (𝑝 = 〈(𝐽 − (𝑐‘𝑍)), (𝑐 ↾ 𝑅)〉 → (1st ‘𝑝) = (𝐽 − (𝑐‘𝑍))) |
327 | 326 | oveq2d 6565 |
. . . . . . 7
⊢ (𝑝 = 〈(𝐽 − (𝑐‘𝑍)), (𝑐 ↾ 𝑅)〉 → (𝐽C(1st ‘𝑝)) = (𝐽C(𝐽 − (𝑐‘𝑍)))) |
328 | 326 | fveq2d 6107 |
. . . . . . . . . 10
⊢ (𝑝 = 〈(𝐽 − (𝑐‘𝑍)), (𝑐 ↾ 𝑅)〉 → (!‘(1st
‘𝑝)) =
(!‘(𝐽 − (𝑐‘𝑍)))) |
329 | 324, 325 | op2ndd 7070 |
. . . . . . . . . . . . 13
⊢ (𝑝 = 〈(𝐽 − (𝑐‘𝑍)), (𝑐 ↾ 𝑅)〉 → (2nd ‘𝑝) = (𝑐 ↾ 𝑅)) |
330 | 329 | fveq1d 6105 |
. . . . . . . . . . . 12
⊢ (𝑝 = 〈(𝐽 − (𝑐‘𝑍)), (𝑐 ↾ 𝑅)〉 → ((2nd ‘𝑝)‘𝑡) = ((𝑐 ↾ 𝑅)‘𝑡)) |
331 | 330 | fveq2d 6107 |
. . . . . . . . . . 11
⊢ (𝑝 = 〈(𝐽 − (𝑐‘𝑍)), (𝑐 ↾ 𝑅)〉 → (!‘((2nd
‘𝑝)‘𝑡)) = (!‘((𝑐 ↾ 𝑅)‘𝑡))) |
332 | 331 | prodeq2ad 38659 |
. . . . . . . . . 10
⊢ (𝑝 = 〈(𝐽 − (𝑐‘𝑍)), (𝑐 ↾ 𝑅)〉 → ∏𝑡 ∈ 𝑅 (!‘((2nd ‘𝑝)‘𝑡)) = ∏𝑡 ∈ 𝑅 (!‘((𝑐 ↾ 𝑅)‘𝑡))) |
333 | 328, 332 | oveq12d 6567 |
. . . . . . . . 9
⊢ (𝑝 = 〈(𝐽 − (𝑐‘𝑍)), (𝑐 ↾ 𝑅)〉 → ((!‘(1st
‘𝑝)) / ∏𝑡 ∈ 𝑅 (!‘((2nd ‘𝑝)‘𝑡))) = ((!‘(𝐽 − (𝑐‘𝑍))) / ∏𝑡 ∈ 𝑅 (!‘((𝑐 ↾ 𝑅)‘𝑡)))) |
334 | 330 | fveq2d 6107 |
. . . . . . . . . . 11
⊢ (𝑝 = 〈(𝐽 − (𝑐‘𝑍)), (𝑐 ↾ 𝑅)〉 → ((𝑆 D𝑛 (𝐻‘𝑡))‘((2nd ‘𝑝)‘𝑡)) = ((𝑆 D𝑛 (𝐻‘𝑡))‘((𝑐 ↾ 𝑅)‘𝑡))) |
335 | 334 | fveq1d 6105 |
. . . . . . . . . 10
⊢ (𝑝 = 〈(𝐽 − (𝑐‘𝑍)), (𝑐 ↾ 𝑅)〉 → (((𝑆 D𝑛 (𝐻‘𝑡))‘((2nd ‘𝑝)‘𝑡))‘𝑥) = (((𝑆 D𝑛 (𝐻‘𝑡))‘((𝑐 ↾ 𝑅)‘𝑡))‘𝑥)) |
336 | 335 | prodeq2ad 38659 |
. . . . . . . . 9
⊢ (𝑝 = 〈(𝐽 − (𝑐‘𝑍)), (𝑐 ↾ 𝑅)〉 → ∏𝑡 ∈ 𝑅 (((𝑆 D𝑛 (𝐻‘𝑡))‘((2nd ‘𝑝)‘𝑡))‘𝑥) = ∏𝑡 ∈ 𝑅 (((𝑆 D𝑛 (𝐻‘𝑡))‘((𝑐 ↾ 𝑅)‘𝑡))‘𝑥)) |
337 | 333, 336 | oveq12d 6567 |
. . . . . . . 8
⊢ (𝑝 = 〈(𝐽 − (𝑐‘𝑍)), (𝑐 ↾ 𝑅)〉 → (((!‘(1st
‘𝑝)) / ∏𝑡 ∈ 𝑅 (!‘((2nd ‘𝑝)‘𝑡))) · ∏𝑡 ∈ 𝑅 (((𝑆 D𝑛 (𝐻‘𝑡))‘((2nd ‘𝑝)‘𝑡))‘𝑥)) = (((!‘(𝐽 − (𝑐‘𝑍))) / ∏𝑡 ∈ 𝑅 (!‘((𝑐 ↾ 𝑅)‘𝑡))) · ∏𝑡 ∈ 𝑅 (((𝑆 D𝑛 (𝐻‘𝑡))‘((𝑐 ↾ 𝑅)‘𝑡))‘𝑥))) |
338 | 326 | oveq2d 6565 |
. . . . . . . . . 10
⊢ (𝑝 = 〈(𝐽 − (𝑐‘𝑍)), (𝑐 ↾ 𝑅)〉 → (𝐽 − (1st ‘𝑝)) = (𝐽 − (𝐽 − (𝑐‘𝑍)))) |
339 | 338 | fveq2d 6107 |
. . . . . . . . 9
⊢ (𝑝 = 〈(𝐽 − (𝑐‘𝑍)), (𝑐 ↾ 𝑅)〉 → ((𝑆 D𝑛 (𝐻‘𝑍))‘(𝐽 − (1st ‘𝑝))) = ((𝑆 D𝑛 (𝐻‘𝑍))‘(𝐽 − (𝐽 − (𝑐‘𝑍))))) |
340 | 339 | fveq1d 6105 |
. . . . . . . 8
⊢ (𝑝 = 〈(𝐽 − (𝑐‘𝑍)), (𝑐 ↾ 𝑅)〉 → (((𝑆 D𝑛 (𝐻‘𝑍))‘(𝐽 − (1st ‘𝑝)))‘𝑥) = (((𝑆 D𝑛 (𝐻‘𝑍))‘(𝐽 − (𝐽 − (𝑐‘𝑍))))‘𝑥)) |
341 | 337, 340 | oveq12d 6567 |
. . . . . . 7
⊢ (𝑝 = 〈(𝐽 − (𝑐‘𝑍)), (𝑐 ↾ 𝑅)〉 → ((((!‘(1st
‘𝑝)) / ∏𝑡 ∈ 𝑅 (!‘((2nd ‘𝑝)‘𝑡))) · ∏𝑡 ∈ 𝑅 (((𝑆 D𝑛 (𝐻‘𝑡))‘((2nd ‘𝑝)‘𝑡))‘𝑥)) · (((𝑆 D𝑛 (𝐻‘𝑍))‘(𝐽 − (1st ‘𝑝)))‘𝑥)) = ((((!‘(𝐽 − (𝑐‘𝑍))) / ∏𝑡 ∈ 𝑅 (!‘((𝑐 ↾ 𝑅)‘𝑡))) · ∏𝑡 ∈ 𝑅 (((𝑆 D𝑛 (𝐻‘𝑡))‘((𝑐 ↾ 𝑅)‘𝑡))‘𝑥)) · (((𝑆 D𝑛 (𝐻‘𝑍))‘(𝐽 − (𝐽 − (𝑐‘𝑍))))‘𝑥))) |
342 | 327, 341 | oveq12d 6567 |
. . . . . 6
⊢ (𝑝 = 〈(𝐽 − (𝑐‘𝑍)), (𝑐 ↾ 𝑅)〉 → ((𝐽C(1st ‘𝑝)) · ((((!‘(1st
‘𝑝)) / ∏𝑡 ∈ 𝑅 (!‘((2nd ‘𝑝)‘𝑡))) · ∏𝑡 ∈ 𝑅 (((𝑆 D𝑛 (𝐻‘𝑡))‘((2nd ‘𝑝)‘𝑡))‘𝑥)) · (((𝑆 D𝑛 (𝐻‘𝑍))‘(𝐽 − (1st ‘𝑝)))‘𝑥))) = ((𝐽C(𝐽 − (𝑐‘𝑍))) · ((((!‘(𝐽 − (𝑐‘𝑍))) / ∏𝑡 ∈ 𝑅 (!‘((𝑐 ↾ 𝑅)‘𝑡))) · ∏𝑡 ∈ 𝑅 (((𝑆 D𝑛 (𝐻‘𝑡))‘((𝑐 ↾ 𝑅)‘𝑡))‘𝑥)) · (((𝑆 D𝑛 (𝐻‘𝑍))‘(𝐽 − (𝐽 − (𝑐‘𝑍))))‘𝑥)))) |
343 | 48 | a1i 11 |
. . . . . . . . . 10
⊢ (𝜑 → 𝐶 = (𝑠 ∈ 𝒫 𝑇 ↦ (𝑛 ∈ ℕ0 ↦ {𝑐 ∈ ((0...𝑛) ↑𝑚 𝑠) ∣ Σ𝑡 ∈ 𝑠 (𝑐‘𝑡) = 𝑛}))) |
344 | | oveq2 6557 |
. . . . . . . . . . . . . 14
⊢ (𝑠 = (𝑅 ∪ {𝑍}) → ((0...𝑛) ↑𝑚 𝑠) = ((0...𝑛) ↑𝑚 (𝑅 ∪ {𝑍}))) |
345 | | rabeq 3166 |
. . . . . . . . . . . . . 14
⊢
(((0...𝑛)
↑𝑚 𝑠) = ((0...𝑛) ↑𝑚 (𝑅 ∪ {𝑍})) → {𝑐 ∈ ((0...𝑛) ↑𝑚 𝑠) ∣ Σ𝑡 ∈ 𝑠 (𝑐‘𝑡) = 𝑛} = {𝑐 ∈ ((0...𝑛) ↑𝑚 (𝑅 ∪ {𝑍})) ∣ Σ𝑡 ∈ 𝑠 (𝑐‘𝑡) = 𝑛}) |
346 | 344, 345 | syl 17 |
. . . . . . . . . . . . 13
⊢ (𝑠 = (𝑅 ∪ {𝑍}) → {𝑐 ∈ ((0...𝑛) ↑𝑚 𝑠) ∣ Σ𝑡 ∈ 𝑠 (𝑐‘𝑡) = 𝑛} = {𝑐 ∈ ((0...𝑛) ↑𝑚 (𝑅 ∪ {𝑍})) ∣ Σ𝑡 ∈ 𝑠 (𝑐‘𝑡) = 𝑛}) |
347 | | sumeq1 14267 |
. . . . . . . . . . . . . . 15
⊢ (𝑠 = (𝑅 ∪ {𝑍}) → Σ𝑡 ∈ 𝑠 (𝑐‘𝑡) = Σ𝑡 ∈ (𝑅 ∪ {𝑍})(𝑐‘𝑡)) |
348 | 347 | eqeq1d 2612 |
. . . . . . . . . . . . . 14
⊢ (𝑠 = (𝑅 ∪ {𝑍}) → (Σ𝑡 ∈ 𝑠 (𝑐‘𝑡) = 𝑛 ↔ Σ𝑡 ∈ (𝑅 ∪ {𝑍})(𝑐‘𝑡) = 𝑛)) |
349 | 348 | rabbidv 3164 |
. . . . . . . . . . . . 13
⊢ (𝑠 = (𝑅 ∪ {𝑍}) → {𝑐 ∈ ((0...𝑛) ↑𝑚 (𝑅 ∪ {𝑍})) ∣ Σ𝑡 ∈ 𝑠 (𝑐‘𝑡) = 𝑛} = {𝑐 ∈ ((0...𝑛) ↑𝑚 (𝑅 ∪ {𝑍})) ∣ Σ𝑡 ∈ (𝑅 ∪ {𝑍})(𝑐‘𝑡) = 𝑛}) |
350 | 346, 349 | eqtrd 2644 |
. . . . . . . . . . . 12
⊢ (𝑠 = (𝑅 ∪ {𝑍}) → {𝑐 ∈ ((0...𝑛) ↑𝑚 𝑠) ∣ Σ𝑡 ∈ 𝑠 (𝑐‘𝑡) = 𝑛} = {𝑐 ∈ ((0...𝑛) ↑𝑚 (𝑅 ∪ {𝑍})) ∣ Σ𝑡 ∈ (𝑅 ∪ {𝑍})(𝑐‘𝑡) = 𝑛}) |
351 | 350 | mpteq2dv 4673 |
. . . . . . . . . . 11
⊢ (𝑠 = (𝑅 ∪ {𝑍}) → (𝑛 ∈ ℕ0 ↦ {𝑐 ∈ ((0...𝑛) ↑𝑚 𝑠) ∣ Σ𝑡 ∈ 𝑠 (𝑐‘𝑡) = 𝑛}) = (𝑛 ∈ ℕ0 ↦ {𝑐 ∈ ((0...𝑛) ↑𝑚 (𝑅 ∪ {𝑍})) ∣ Σ𝑡 ∈ (𝑅 ∪ {𝑍})(𝑐‘𝑡) = 𝑛})) |
352 | 351 | adantl 481 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑠 = (𝑅 ∪ {𝑍})) → (𝑛 ∈ ℕ0 ↦ {𝑐 ∈ ((0...𝑛) ↑𝑚 𝑠) ∣ Σ𝑡 ∈ 𝑠 (𝑐‘𝑡) = 𝑛}) = (𝑛 ∈ ℕ0 ↦ {𝑐 ∈ ((0...𝑛) ↑𝑚 (𝑅 ∪ {𝑍})) ∣ Σ𝑡 ∈ (𝑅 ∪ {𝑍})(𝑐‘𝑡) = 𝑛})) |
353 | 23 | snssd 4281 |
. . . . . . . . . . . 12
⊢ (𝜑 → {𝑍} ⊆ 𝑇) |
354 | 4, 353 | unssd 3751 |
. . . . . . . . . . 11
⊢ (𝜑 → (𝑅 ∪ {𝑍}) ⊆ 𝑇) |
355 | 3, 354 | ssexd 4733 |
. . . . . . . . . . . 12
⊢ (𝜑 → (𝑅 ∪ {𝑍}) ∈ V) |
356 | | elpwg 4116 |
. . . . . . . . . . . 12
⊢ ((𝑅 ∪ {𝑍}) ∈ V → ((𝑅 ∪ {𝑍}) ∈ 𝒫 𝑇 ↔ (𝑅 ∪ {𝑍}) ⊆ 𝑇)) |
357 | 355, 356 | syl 17 |
. . . . . . . . . . 11
⊢ (𝜑 → ((𝑅 ∪ {𝑍}) ∈ 𝒫 𝑇 ↔ (𝑅 ∪ {𝑍}) ⊆ 𝑇)) |
358 | 354, 357 | mpbird 246 |
. . . . . . . . . 10
⊢ (𝜑 → (𝑅 ∪ {𝑍}) ∈ 𝒫 𝑇) |
359 | 65 | mptex 6390 |
. . . . . . . . . . 11
⊢ (𝑛 ∈ ℕ0
↦ {𝑐 ∈
((0...𝑛)
↑𝑚 (𝑅 ∪ {𝑍})) ∣ Σ𝑡 ∈ (𝑅 ∪ {𝑍})(𝑐‘𝑡) = 𝑛}) ∈ V |
360 | 359 | a1i 11 |
. . . . . . . . . 10
⊢ (𝜑 → (𝑛 ∈ ℕ0 ↦ {𝑐 ∈ ((0...𝑛) ↑𝑚 (𝑅 ∪ {𝑍})) ∣ Σ𝑡 ∈ (𝑅 ∪ {𝑍})(𝑐‘𝑡) = 𝑛}) ∈ V) |
361 | 343, 352,
358, 360 | fvmptd 6197 |
. . . . . . . . 9
⊢ (𝜑 → (𝐶‘(𝑅 ∪ {𝑍})) = (𝑛 ∈ ℕ0 ↦ {𝑐 ∈ ((0...𝑛) ↑𝑚 (𝑅 ∪ {𝑍})) ∣ Σ𝑡 ∈ (𝑅 ∪ {𝑍})(𝑐‘𝑡) = 𝑛})) |
362 | | oveq2 6557 |
. . . . . . . . . . . . 13
⊢ (𝑛 = 𝐽 → (0...𝑛) = (0...𝐽)) |
363 | 362 | oveq1d 6564 |
. . . . . . . . . . . 12
⊢ (𝑛 = 𝐽 → ((0...𝑛) ↑𝑚 (𝑅 ∪ {𝑍})) = ((0...𝐽) ↑𝑚 (𝑅 ∪ {𝑍}))) |
364 | | rabeq 3166 |
. . . . . . . . . . . 12
⊢
(((0...𝑛)
↑𝑚 (𝑅 ∪ {𝑍})) = ((0...𝐽) ↑𝑚 (𝑅 ∪ {𝑍})) → {𝑐 ∈ ((0...𝑛) ↑𝑚 (𝑅 ∪ {𝑍})) ∣ Σ𝑡 ∈ (𝑅 ∪ {𝑍})(𝑐‘𝑡) = 𝑛} = {𝑐 ∈ ((0...𝐽) ↑𝑚 (𝑅 ∪ {𝑍})) ∣ Σ𝑡 ∈ (𝑅 ∪ {𝑍})(𝑐‘𝑡) = 𝑛}) |
365 | 363, 364 | syl 17 |
. . . . . . . . . . 11
⊢ (𝑛 = 𝐽 → {𝑐 ∈ ((0...𝑛) ↑𝑚 (𝑅 ∪ {𝑍})) ∣ Σ𝑡 ∈ (𝑅 ∪ {𝑍})(𝑐‘𝑡) = 𝑛} = {𝑐 ∈ ((0...𝐽) ↑𝑚 (𝑅 ∪ {𝑍})) ∣ Σ𝑡 ∈ (𝑅 ∪ {𝑍})(𝑐‘𝑡) = 𝑛}) |
366 | | eqeq2 2621 |
. . . . . . . . . . . 12
⊢ (𝑛 = 𝐽 → (Σ𝑡 ∈ (𝑅 ∪ {𝑍})(𝑐‘𝑡) = 𝑛 ↔ Σ𝑡 ∈ (𝑅 ∪ {𝑍})(𝑐‘𝑡) = 𝐽)) |
367 | 366 | rabbidv 3164 |
. . . . . . . . . . 11
⊢ (𝑛 = 𝐽 → {𝑐 ∈ ((0...𝐽) ↑𝑚 (𝑅 ∪ {𝑍})) ∣ Σ𝑡 ∈ (𝑅 ∪ {𝑍})(𝑐‘𝑡) = 𝑛} = {𝑐 ∈ ((0...𝐽) ↑𝑚 (𝑅 ∪ {𝑍})) ∣ Σ𝑡 ∈ (𝑅 ∪ {𝑍})(𝑐‘𝑡) = 𝐽}) |
368 | 365, 367 | eqtrd 2644 |
. . . . . . . . . 10
⊢ (𝑛 = 𝐽 → {𝑐 ∈ ((0...𝑛) ↑𝑚 (𝑅 ∪ {𝑍})) ∣ Σ𝑡 ∈ (𝑅 ∪ {𝑍})(𝑐‘𝑡) = 𝑛} = {𝑐 ∈ ((0...𝐽) ↑𝑚 (𝑅 ∪ {𝑍})) ∣ Σ𝑡 ∈ (𝑅 ∪ {𝑍})(𝑐‘𝑡) = 𝐽}) |
369 | 368 | adantl 481 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑛 = 𝐽) → {𝑐 ∈ ((0...𝑛) ↑𝑚 (𝑅 ∪ {𝑍})) ∣ Σ𝑡 ∈ (𝑅 ∪ {𝑍})(𝑐‘𝑡) = 𝑛} = {𝑐 ∈ ((0...𝐽) ↑𝑚 (𝑅 ∪ {𝑍})) ∣ Σ𝑡 ∈ (𝑅 ∪ {𝑍})(𝑐‘𝑡) = 𝐽}) |
370 | | ovex 6577 |
. . . . . . . . . . 11
⊢
((0...𝐽)
↑𝑚 (𝑅 ∪ {𝑍})) ∈ V |
371 | 370 | rabex 4740 |
. . . . . . . . . 10
⊢ {𝑐 ∈ ((0...𝐽) ↑𝑚 (𝑅 ∪ {𝑍})) ∣ Σ𝑡 ∈ (𝑅 ∪ {𝑍})(𝑐‘𝑡) = 𝐽} ∈ V |
372 | 371 | a1i 11 |
. . . . . . . . 9
⊢ (𝜑 → {𝑐 ∈ ((0...𝐽) ↑𝑚 (𝑅 ∪ {𝑍})) ∣ Σ𝑡 ∈ (𝑅 ∪ {𝑍})(𝑐‘𝑡) = 𝐽} ∈ V) |
373 | 361, 369,
45, 372 | fvmptd 6197 |
. . . . . . . 8
⊢ (𝜑 → ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽) = {𝑐 ∈ ((0...𝐽) ↑𝑚 (𝑅 ∪ {𝑍})) ∣ Σ𝑡 ∈ (𝑅 ∪ {𝑍})(𝑐‘𝑡) = 𝐽}) |
374 | | fzfid 12634 |
. . . . . . . . . 10
⊢ (𝜑 → (0...𝐽) ∈ Fin) |
375 | | snfi 7923 |
. . . . . . . . . . . 12
⊢ {𝑍} ∈ Fin |
376 | 375 | a1i 11 |
. . . . . . . . . . 11
⊢ (𝜑 → {𝑍} ∈ Fin) |
377 | | unfi 8112 |
. . . . . . . . . . 11
⊢ ((𝑅 ∈ Fin ∧ {𝑍} ∈ Fin) → (𝑅 ∪ {𝑍}) ∈ Fin) |
378 | 6, 376, 377 | syl2anc 691 |
. . . . . . . . . 10
⊢ (𝜑 → (𝑅 ∪ {𝑍}) ∈ Fin) |
379 | | mapfi 8145 |
. . . . . . . . . 10
⊢
(((0...𝐽) ∈ Fin
∧ (𝑅 ∪ {𝑍}) ∈ Fin) →
((0...𝐽)
↑𝑚 (𝑅 ∪ {𝑍})) ∈ Fin) |
380 | 374, 378,
379 | syl2anc 691 |
. . . . . . . . 9
⊢ (𝜑 → ((0...𝐽) ↑𝑚 (𝑅 ∪ {𝑍})) ∈ Fin) |
381 | | ssrab2 3650 |
. . . . . . . . . 10
⊢ {𝑐 ∈ ((0...𝐽) ↑𝑚 (𝑅 ∪ {𝑍})) ∣ Σ𝑡 ∈ (𝑅 ∪ {𝑍})(𝑐‘𝑡) = 𝐽} ⊆ ((0...𝐽) ↑𝑚 (𝑅 ∪ {𝑍})) |
382 | 381 | a1i 11 |
. . . . . . . . 9
⊢ (𝜑 → {𝑐 ∈ ((0...𝐽) ↑𝑚 (𝑅 ∪ {𝑍})) ∣ Σ𝑡 ∈ (𝑅 ∪ {𝑍})(𝑐‘𝑡) = 𝐽} ⊆ ((0...𝐽) ↑𝑚 (𝑅 ∪ {𝑍}))) |
383 | | ssfi 8065 |
. . . . . . . . 9
⊢
((((0...𝐽)
↑𝑚 (𝑅 ∪ {𝑍})) ∈ Fin ∧ {𝑐 ∈ ((0...𝐽) ↑𝑚 (𝑅 ∪ {𝑍})) ∣ Σ𝑡 ∈ (𝑅 ∪ {𝑍})(𝑐‘𝑡) = 𝐽} ⊆ ((0...𝐽) ↑𝑚 (𝑅 ∪ {𝑍}))) → {𝑐 ∈ ((0...𝐽) ↑𝑚 (𝑅 ∪ {𝑍})) ∣ Σ𝑡 ∈ (𝑅 ∪ {𝑍})(𝑐‘𝑡) = 𝐽} ∈ Fin) |
384 | 380, 382,
383 | syl2anc 691 |
. . . . . . . 8
⊢ (𝜑 → {𝑐 ∈ ((0...𝐽) ↑𝑚 (𝑅 ∪ {𝑍})) ∣ Σ𝑡 ∈ (𝑅 ∪ {𝑍})(𝑐‘𝑡) = 𝐽} ∈ Fin) |
385 | 373, 384 | eqeltrd 2688 |
. . . . . . 7
⊢ (𝜑 → ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽) ∈ Fin) |
386 | 385 | adantr 480 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽) ∈ Fin) |
387 | | dvnprodlem2.d |
. . . . . . . 8
⊢ 𝐷 = (𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽) ↦ 〈(𝐽 − (𝑐‘𝑍)), (𝑐 ↾ 𝑅)〉) |
388 | 48, 45, 387, 3, 23, 10, 354 | dvnprodlem1 38836 |
. . . . . . 7
⊢ (𝜑 → 𝐷:((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)–1-1-onto→∪ 𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘))) |
389 | 388 | adantr 480 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐷:((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)–1-1-onto→∪ 𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘))) |
390 | | simpr 476 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) |
391 | | opex 4859 |
. . . . . . . . 9
⊢
〈(𝐽 −
(𝑐‘𝑍)), (𝑐 ↾ 𝑅)〉 ∈ V |
392 | 391 | a1i 11 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → 〈(𝐽 − (𝑐‘𝑍)), (𝑐 ↾ 𝑅)〉 ∈ V) |
393 | 387 | fvmpt2 6200 |
. . . . . . . 8
⊢ ((𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽) ∧ 〈(𝐽 − (𝑐‘𝑍)), (𝑐 ↾ 𝑅)〉 ∈ V) → (𝐷‘𝑐) = 〈(𝐽 − (𝑐‘𝑍)), (𝑐 ↾ 𝑅)〉) |
394 | 390, 392,
393 | syl2anc 691 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → (𝐷‘𝑐) = 〈(𝐽 − (𝑐‘𝑍)), (𝑐 ↾ 𝑅)〉) |
395 | 394 | adantlr 747 |
. . . . . 6
⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → (𝐷‘𝑐) = 〈(𝐽 − (𝑐‘𝑍)), (𝑐 ↾ 𝑅)〉) |
396 | 45 | adantr 480 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑝 ∈ ∪
𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘))) → 𝐽 ∈
ℕ0) |
397 | | eliun 4460 |
. . . . . . . . . . . . . 14
⊢ (𝑝 ∈ ∪ 𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘)) ↔ ∃𝑘 ∈ (0...𝐽)𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) |
398 | 397 | biimpi 205 |
. . . . . . . . . . . . 13
⊢ (𝑝 ∈ ∪ 𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘)) → ∃𝑘 ∈ (0...𝐽)𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) |
399 | 398 | adantl 481 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑝 ∈ ∪
𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘))) → ∃𝑘 ∈ (0...𝐽)𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) |
400 | | nfv 1830 |
. . . . . . . . . . . . . 14
⊢
Ⅎ𝑘𝜑 |
401 | | nfcv 2751 |
. . . . . . . . . . . . . . 15
⊢
Ⅎ𝑘𝑝 |
402 | | nfiu1 4486 |
. . . . . . . . . . . . . . 15
⊢
Ⅎ𝑘∪ 𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘)) |
403 | 401, 402 | nfel 2763 |
. . . . . . . . . . . . . 14
⊢
Ⅎ𝑘 𝑝 ∈ ∪ 𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘)) |
404 | 400, 403 | nfan 1816 |
. . . . . . . . . . . . 13
⊢
Ⅎ𝑘(𝜑 ∧ 𝑝 ∈ ∪
𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘))) |
405 | | nfv 1830 |
. . . . . . . . . . . . 13
⊢
Ⅎ𝑘(1st ‘𝑝) ∈ (0...𝐽) |
406 | | xp1st 7089 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘)) → (1st ‘𝑝) ∈ {𝑘}) |
407 | | elsni 4142 |
. . . . . . . . . . . . . . . . . 18
⊢
((1st ‘𝑝) ∈ {𝑘} → (1st ‘𝑝) = 𝑘) |
408 | 406, 407 | syl 17 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘)) → (1st ‘𝑝) = 𝑘) |
409 | 408 | adantl 481 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) → (1st ‘𝑝) = 𝑘) |
410 | | simpl 472 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) → 𝑘 ∈ (0...𝐽)) |
411 | 409, 410 | eqeltrd 2688 |
. . . . . . . . . . . . . . 15
⊢ ((𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) → (1st ‘𝑝) ∈ (0...𝐽)) |
412 | 411 | ex 449 |
. . . . . . . . . . . . . 14
⊢ (𝑘 ∈ (0...𝐽) → (𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘)) → (1st ‘𝑝) ∈ (0...𝐽))) |
413 | 412 | a1i 11 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑝 ∈ ∪
𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘))) → (𝑘 ∈ (0...𝐽) → (𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘)) → (1st ‘𝑝) ∈ (0...𝐽)))) |
414 | 404, 405,
413 | rexlimd 3008 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑝 ∈ ∪
𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘))) → (∃𝑘 ∈ (0...𝐽)𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘)) → (1st ‘𝑝) ∈ (0...𝐽))) |
415 | 399, 414 | mpd 15 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑝 ∈ ∪
𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘))) → (1st ‘𝑝) ∈ (0...𝐽)) |
416 | | elfzelz 12213 |
. . . . . . . . . . 11
⊢
((1st ‘𝑝) ∈ (0...𝐽) → (1st ‘𝑝) ∈
ℤ) |
417 | 415, 416 | syl 17 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑝 ∈ ∪
𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘))) → (1st ‘𝑝) ∈
ℤ) |
418 | 396, 417 | bccld 38472 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑝 ∈ ∪
𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘))) → (𝐽C(1st ‘𝑝)) ∈
ℕ0) |
419 | 418 | nn0cnd 11230 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑝 ∈ ∪
𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘))) → (𝐽C(1st ‘𝑝)) ∈ ℂ) |
420 | 419 | adantlr 747 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑝 ∈ ∪
𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘))) → (𝐽C(1st ‘𝑝)) ∈ ℂ) |
421 | | elfznn0 12302 |
. . . . . . . . . . . . . 14
⊢
((1st ‘𝑝) ∈ (0...𝐽) → (1st ‘𝑝) ∈
ℕ0) |
422 | 415, 421 | syl 17 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑝 ∈ ∪
𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘))) → (1st ‘𝑝) ∈
ℕ0) |
423 | 422 | faccld 12933 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑝 ∈ ∪
𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘))) → (!‘(1st
‘𝑝)) ∈
ℕ) |
424 | 423 | nncnd 10913 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑝 ∈ ∪
𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘))) → (!‘(1st
‘𝑝)) ∈
ℂ) |
425 | 424 | adantlr 747 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑝 ∈ ∪
𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘))) → (!‘(1st
‘𝑝)) ∈
ℂ) |
426 | 6 | adantr 480 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑝 ∈ ∪
𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘))) → 𝑅 ∈ Fin) |
427 | | nfv 1830 |
. . . . . . . . . . . . . . . 16
⊢
Ⅎ𝑘(2nd ‘𝑝):𝑅⟶(0...𝐽) |
428 | 86, 84 | eqsstrd 3602 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽)) → ((𝐶‘𝑅)‘𝑘) ⊆ ((0...𝑘) ↑𝑚 𝑅)) |
429 | | ovex 6577 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢
(0...𝐽) ∈
V |
430 | 429 | a1i 11 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝑘 ∈ (0...𝐽) → (0...𝐽) ∈ V) |
431 | | mapss 7786 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢
(((0...𝐽) ∈ V
∧ (0...𝑘) ⊆
(0...𝐽)) → ((0...𝑘) ↑𝑚
𝑅) ⊆ ((0...𝐽) ↑𝑚
𝑅)) |
432 | 430, 124,
431 | syl2anc 691 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑘 ∈ (0...𝐽) → ((0...𝑘) ↑𝑚 𝑅) ⊆ ((0...𝐽) ↑𝑚
𝑅)) |
433 | 432 | adantl 481 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽)) → ((0...𝑘) ↑𝑚 𝑅) ⊆ ((0...𝐽) ↑𝑚
𝑅)) |
434 | 428, 433 | sstrd 3578 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽)) → ((𝐶‘𝑅)‘𝑘) ⊆ ((0...𝐽) ↑𝑚 𝑅)) |
435 | 434 | 3adant3 1074 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) → ((𝐶‘𝑅)‘𝑘) ⊆ ((0...𝐽) ↑𝑚 𝑅)) |
436 | | xp2nd 7090 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘)) → (2nd ‘𝑝) ∈ ((𝐶‘𝑅)‘𝑘)) |
437 | 436 | 3ad2ant3 1077 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) → (2nd ‘𝑝) ∈ ((𝐶‘𝑅)‘𝑘)) |
438 | 435, 437 | sseldd 3569 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) → (2nd ‘𝑝) ∈ ((0...𝐽) ↑𝑚 𝑅)) |
439 | | elmapi 7765 |
. . . . . . . . . . . . . . . . . . 19
⊢
((2nd ‘𝑝) ∈ ((0...𝐽) ↑𝑚 𝑅) → (2nd
‘𝑝):𝑅⟶(0...𝐽)) |
440 | 438, 439 | syl 17 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) → (2nd ‘𝑝):𝑅⟶(0...𝐽)) |
441 | 440 | 3exp 1256 |
. . . . . . . . . . . . . . . . 17
⊢ (𝜑 → (𝑘 ∈ (0...𝐽) → (𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘)) → (2nd ‘𝑝):𝑅⟶(0...𝐽)))) |
442 | 441 | adantr 480 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑝 ∈ ∪
𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘))) → (𝑘 ∈ (0...𝐽) → (𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘)) → (2nd ‘𝑝):𝑅⟶(0...𝐽)))) |
443 | 404, 427,
442 | rexlimd 3008 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑝 ∈ ∪
𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘))) → (∃𝑘 ∈ (0...𝐽)𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘)) → (2nd ‘𝑝):𝑅⟶(0...𝐽))) |
444 | 399, 443 | mpd 15 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑝 ∈ ∪
𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘))) → (2nd ‘𝑝):𝑅⟶(0...𝐽)) |
445 | 444 | ffvelrnda 6267 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑝 ∈ ∪
𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘))) ∧ 𝑡 ∈ 𝑅) → ((2nd ‘𝑝)‘𝑡) ∈ (0...𝐽)) |
446 | | elfznn0 12302 |
. . . . . . . . . . . . . . 15
⊢
(((2nd ‘𝑝)‘𝑡) ∈ (0...𝐽) → ((2nd ‘𝑝)‘𝑡) ∈
ℕ0) |
447 | 446 | faccld 12933 |
. . . . . . . . . . . . . 14
⊢
(((2nd ‘𝑝)‘𝑡) ∈ (0...𝐽) → (!‘((2nd
‘𝑝)‘𝑡)) ∈
ℕ) |
448 | 447 | nncnd 10913 |
. . . . . . . . . . . . 13
⊢
(((2nd ‘𝑝)‘𝑡) ∈ (0...𝐽) → (!‘((2nd
‘𝑝)‘𝑡)) ∈
ℂ) |
449 | 445, 448 | syl 17 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑝 ∈ ∪
𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘))) ∧ 𝑡 ∈ 𝑅) → (!‘((2nd
‘𝑝)‘𝑡)) ∈
ℂ) |
450 | 426, 449 | fprodcl 14521 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑝 ∈ ∪
𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘))) → ∏𝑡 ∈ 𝑅 (!‘((2nd ‘𝑝)‘𝑡)) ∈ ℂ) |
451 | 450 | adantlr 747 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑝 ∈ ∪
𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘))) → ∏𝑡 ∈ 𝑅 (!‘((2nd ‘𝑝)‘𝑡)) ∈ ℂ) |
452 | 445, 447 | syl 17 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑝 ∈ ∪
𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘))) ∧ 𝑡 ∈ 𝑅) → (!‘((2nd
‘𝑝)‘𝑡)) ∈
ℕ) |
453 | | nnne0 10930 |
. . . . . . . . . . . . 13
⊢
((!‘((2nd ‘𝑝)‘𝑡)) ∈ ℕ →
(!‘((2nd ‘𝑝)‘𝑡)) ≠ 0) |
454 | 452, 453 | syl 17 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑝 ∈ ∪
𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘))) ∧ 𝑡 ∈ 𝑅) → (!‘((2nd
‘𝑝)‘𝑡)) ≠ 0) |
455 | 426, 449,
454 | fprodn0 14548 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑝 ∈ ∪
𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘))) → ∏𝑡 ∈ 𝑅 (!‘((2nd ‘𝑝)‘𝑡)) ≠ 0) |
456 | 455 | adantlr 747 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑝 ∈ ∪
𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘))) → ∏𝑡 ∈ 𝑅 (!‘((2nd ‘𝑝)‘𝑡)) ≠ 0) |
457 | 425, 451,
456 | divcld 10680 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑝 ∈ ∪
𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘))) → ((!‘(1st
‘𝑝)) / ∏𝑡 ∈ 𝑅 (!‘((2nd ‘𝑝)‘𝑡))) ∈ ℂ) |
458 | 7 | adantr 480 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑝 ∈ ∪
𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘))) → 𝑅 ∈ Fin) |
459 | | simpll 786 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑝 ∈ ∪
𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘))) ∧ 𝑡 ∈ 𝑅) → 𝜑) |
460 | 459, 13 | sylancom 698 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑝 ∈ ∪
𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘))) ∧ 𝑡 ∈ 𝑅) → 𝑡 ∈ 𝑇) |
461 | 459, 135 | syl 17 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑝 ∈ ∪
𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘))) ∧ 𝑡 ∈ 𝑅) → (0...𝐽) ⊆ (0...𝑁)) |
462 | 461, 445 | sseldd 3569 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑝 ∈ ∪
𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘))) ∧ 𝑡 ∈ 𝑅) → ((2nd ‘𝑝)‘𝑡) ∈ (0...𝑁)) |
463 | 459, 460,
462 | 3jca 1235 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑝 ∈ ∪
𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘))) ∧ 𝑡 ∈ 𝑅) → (𝜑 ∧ 𝑡 ∈ 𝑇 ∧ ((2nd ‘𝑝)‘𝑡) ∈ (0...𝑁))) |
464 | | eleq1 2676 |
. . . . . . . . . . . . . . . 16
⊢ (𝑗 = ((2nd ‘𝑝)‘𝑡) → (𝑗 ∈ (0...𝑁) ↔ ((2nd ‘𝑝)‘𝑡) ∈ (0...𝑁))) |
465 | 464 | 3anbi3d 1397 |
. . . . . . . . . . . . . . 15
⊢ (𝑗 = ((2nd ‘𝑝)‘𝑡) → ((𝜑 ∧ 𝑡 ∈ 𝑇 ∧ 𝑗 ∈ (0...𝑁)) ↔ (𝜑 ∧ 𝑡 ∈ 𝑇 ∧ ((2nd ‘𝑝)‘𝑡) ∈ (0...𝑁)))) |
466 | | fveq2 6103 |
. . . . . . . . . . . . . . . 16
⊢ (𝑗 = ((2nd ‘𝑝)‘𝑡) → ((𝑆 D𝑛 (𝐻‘𝑡))‘𝑗) = ((𝑆 D𝑛 (𝐻‘𝑡))‘((2nd ‘𝑝)‘𝑡))) |
467 | 466 | feq1d 5943 |
. . . . . . . . . . . . . . 15
⊢ (𝑗 = ((2nd ‘𝑝)‘𝑡) → (((𝑆 D𝑛 (𝐻‘𝑡))‘𝑗):𝑋⟶ℂ ↔ ((𝑆 D𝑛 (𝐻‘𝑡))‘((2nd ‘𝑝)‘𝑡)):𝑋⟶ℂ)) |
468 | 465, 467 | imbi12d 333 |
. . . . . . . . . . . . . 14
⊢ (𝑗 = ((2nd ‘𝑝)‘𝑡) → (((𝜑 ∧ 𝑡 ∈ 𝑇 ∧ 𝑗 ∈ (0...𝑁)) → ((𝑆 D𝑛 (𝐻‘𝑡))‘𝑗):𝑋⟶ℂ) ↔ ((𝜑 ∧ 𝑡 ∈ 𝑇 ∧ ((2nd ‘𝑝)‘𝑡) ∈ (0...𝑁)) → ((𝑆 D𝑛 (𝐻‘𝑡))‘((2nd ‘𝑝)‘𝑡)):𝑋⟶ℂ))) |
469 | 468, 147 | vtoclg 3239 |
. . . . . . . . . . . . 13
⊢
(((2nd ‘𝑝)‘𝑡) ∈ (0...𝐽) → ((𝜑 ∧ 𝑡 ∈ 𝑇 ∧ ((2nd ‘𝑝)‘𝑡) ∈ (0...𝑁)) → ((𝑆 D𝑛 (𝐻‘𝑡))‘((2nd ‘𝑝)‘𝑡)):𝑋⟶ℂ)) |
470 | 445, 463,
469 | sylc 63 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑝 ∈ ∪
𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘))) ∧ 𝑡 ∈ 𝑅) → ((𝑆 D𝑛 (𝐻‘𝑡))‘((2nd ‘𝑝)‘𝑡)):𝑋⟶ℂ) |
471 | 470 | adantllr 751 |
. . . . . . . . . . 11
⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑝 ∈ ∪
𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘))) ∧ 𝑡 ∈ 𝑅) → ((𝑆 D𝑛 (𝐻‘𝑡))‘((2nd ‘𝑝)‘𝑡)):𝑋⟶ℂ) |
472 | 17 | adantlr 747 |
. . . . . . . . . . 11
⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑝 ∈ ∪
𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘))) ∧ 𝑡 ∈ 𝑅) → 𝑥 ∈ 𝑋) |
473 | 471, 472 | ffvelrnd 6268 |
. . . . . . . . . 10
⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑝 ∈ ∪
𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘))) ∧ 𝑡 ∈ 𝑅) → (((𝑆 D𝑛 (𝐻‘𝑡))‘((2nd ‘𝑝)‘𝑡))‘𝑥) ∈ ℂ) |
474 | 458, 473 | fprodcl 14521 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑝 ∈ ∪
𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘))) → ∏𝑡 ∈ 𝑅 (((𝑆 D𝑛 (𝐻‘𝑡))‘((2nd ‘𝑝)‘𝑡))‘𝑥) ∈ ℂ) |
475 | 457, 474 | mulcld 9939 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑝 ∈ ∪
𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘))) → (((!‘(1st
‘𝑝)) / ∏𝑡 ∈ 𝑅 (!‘((2nd ‘𝑝)‘𝑡))) · ∏𝑡 ∈ 𝑅 (((𝑆 D𝑛 (𝐻‘𝑡))‘((2nd ‘𝑝)‘𝑡))‘𝑥)) ∈ ℂ) |
476 | | nfv 1830 |
. . . . . . . . . . . . 13
⊢
Ⅎ𝑘(𝐽 − (1st
‘𝑝)) ∈
(0...𝐽) |
477 | | simp1 1054 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) → 𝜑) |
478 | 411 | 3adant1 1072 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) → (1st ‘𝑝) ∈ (0...𝐽)) |
479 | | fznn0sub2 12315 |
. . . . . . . . . . . . . . . . 17
⊢
((1st ‘𝑝) ∈ (0...𝐽) → (𝐽 − (1st ‘𝑝)) ∈ (0...𝐽)) |
480 | 479 | adantl 481 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ (1st
‘𝑝) ∈ (0...𝐽)) → (𝐽 − (1st ‘𝑝)) ∈ (0...𝐽)) |
481 | 477, 478,
480 | syl2anc 691 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) → (𝐽 − (1st ‘𝑝)) ∈ (0...𝐽)) |
482 | 481 | 3exp 1256 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → (𝑘 ∈ (0...𝐽) → (𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘)) → (𝐽 − (1st ‘𝑝)) ∈ (0...𝐽)))) |
483 | 482 | adantr 480 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑝 ∈ ∪
𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘))) → (𝑘 ∈ (0...𝐽) → (𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘)) → (𝐽 − (1st ‘𝑝)) ∈ (0...𝐽)))) |
484 | 404, 476,
483 | rexlimd 3008 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑝 ∈ ∪
𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘))) → (∃𝑘 ∈ (0...𝐽)𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘)) → (𝐽 − (1st ‘𝑝)) ∈ (0...𝐽))) |
485 | 399, 484 | mpd 15 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑝 ∈ ∪
𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘))) → (𝐽 − (1st ‘𝑝)) ∈ (0...𝐽)) |
486 | | simpl 472 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑝 ∈ ∪
𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘))) → 𝜑) |
487 | 486, 23 | syl 17 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑝 ∈ ∪
𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘))) → 𝑍 ∈ 𝑇) |
488 | 486, 135 | syl 17 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑝 ∈ ∪
𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘))) → (0...𝐽) ⊆ (0...𝑁)) |
489 | 488, 485 | sseldd 3569 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑝 ∈ ∪
𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘))) → (𝐽 − (1st ‘𝑝)) ∈ (0...𝑁)) |
490 | 486, 487,
489 | 3jca 1235 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑝 ∈ ∪
𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘))) → (𝜑 ∧ 𝑍 ∈ 𝑇 ∧ (𝐽 − (1st ‘𝑝)) ∈ (0...𝑁))) |
491 | | eleq1 2676 |
. . . . . . . . . . . . . 14
⊢ (𝑗 = (𝐽 − (1st ‘𝑝)) → (𝑗 ∈ (0...𝑁) ↔ (𝐽 − (1st ‘𝑝)) ∈ (0...𝑁))) |
492 | 491 | 3anbi3d 1397 |
. . . . . . . . . . . . 13
⊢ (𝑗 = (𝐽 − (1st ‘𝑝)) → ((𝜑 ∧ 𝑍 ∈ 𝑇 ∧ 𝑗 ∈ (0...𝑁)) ↔ (𝜑 ∧ 𝑍 ∈ 𝑇 ∧ (𝐽 − (1st ‘𝑝)) ∈ (0...𝑁)))) |
493 | | fveq2 6103 |
. . . . . . . . . . . . . 14
⊢ (𝑗 = (𝐽 − (1st ‘𝑝)) → ((𝑆 D𝑛 (𝐻‘𝑍))‘𝑗) = ((𝑆 D𝑛 (𝐻‘𝑍))‘(𝐽 − (1st ‘𝑝)))) |
494 | 493 | feq1d 5943 |
. . . . . . . . . . . . 13
⊢ (𝑗 = (𝐽 − (1st ‘𝑝)) → (((𝑆 D𝑛 (𝐻‘𝑍))‘𝑗):𝑋⟶ℂ ↔ ((𝑆 D𝑛 (𝐻‘𝑍))‘(𝐽 − (1st ‘𝑝))):𝑋⟶ℂ)) |
495 | 492, 494 | imbi12d 333 |
. . . . . . . . . . . 12
⊢ (𝑗 = (𝐽 − (1st ‘𝑝)) → (((𝜑 ∧ 𝑍 ∈ 𝑇 ∧ 𝑗 ∈ (0...𝑁)) → ((𝑆 D𝑛 (𝐻‘𝑍))‘𝑗):𝑋⟶ℂ) ↔ ((𝜑 ∧ 𝑍 ∈ 𝑇 ∧ (𝐽 − (1st ‘𝑝)) ∈ (0...𝑁)) → ((𝑆 D𝑛 (𝐻‘𝑍))‘(𝐽 − (1st ‘𝑝))):𝑋⟶ℂ))) |
496 | | simp2 1055 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑍 ∈ 𝑇 ∧ 𝑗 ∈ (0...𝑁)) → 𝑍 ∈ 𝑇) |
497 | | id 22 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑍 ∈ 𝑇 ∧ 𝑗 ∈ (0...𝑁)) → (𝜑 ∧ 𝑍 ∈ 𝑇 ∧ 𝑗 ∈ (0...𝑁))) |
498 | 26 | 3anbi2d 1396 |
. . . . . . . . . . . . . . 15
⊢ (𝑡 = 𝑍 → ((𝜑 ∧ 𝑡 ∈ 𝑇 ∧ 𝑗 ∈ (0...𝑁)) ↔ (𝜑 ∧ 𝑍 ∈ 𝑇 ∧ 𝑗 ∈ (0...𝑁)))) |
499 | 185 | fveq1d 6105 |
. . . . . . . . . . . . . . . 16
⊢ (𝑡 = 𝑍 → ((𝑆 D𝑛 (𝐻‘𝑡))‘𝑗) = ((𝑆 D𝑛 (𝐻‘𝑍))‘𝑗)) |
500 | 499 | feq1d 5943 |
. . . . . . . . . . . . . . 15
⊢ (𝑡 = 𝑍 → (((𝑆 D𝑛 (𝐻‘𝑡))‘𝑗):𝑋⟶ℂ ↔ ((𝑆 D𝑛 (𝐻‘𝑍))‘𝑗):𝑋⟶ℂ)) |
501 | 498, 500 | imbi12d 333 |
. . . . . . . . . . . . . 14
⊢ (𝑡 = 𝑍 → (((𝜑 ∧ 𝑡 ∈ 𝑇 ∧ 𝑗 ∈ (0...𝑁)) → ((𝑆 D𝑛 (𝐻‘𝑡))‘𝑗):𝑋⟶ℂ) ↔ ((𝜑 ∧ 𝑍 ∈ 𝑇 ∧ 𝑗 ∈ (0...𝑁)) → ((𝑆 D𝑛 (𝐻‘𝑍))‘𝑗):𝑋⟶ℂ))) |
502 | 501, 147 | vtoclg 3239 |
. . . . . . . . . . . . 13
⊢ (𝑍 ∈ 𝑇 → ((𝜑 ∧ 𝑍 ∈ 𝑇 ∧ 𝑗 ∈ (0...𝑁)) → ((𝑆 D𝑛 (𝐻‘𝑍))‘𝑗):𝑋⟶ℂ)) |
503 | 496, 497,
502 | sylc 63 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑍 ∈ 𝑇 ∧ 𝑗 ∈ (0...𝑁)) → ((𝑆 D𝑛 (𝐻‘𝑍))‘𝑗):𝑋⟶ℂ) |
504 | 495, 503 | vtoclg 3239 |
. . . . . . . . . . 11
⊢ ((𝐽 − (1st
‘𝑝)) ∈
(0...𝐽) → ((𝜑 ∧ 𝑍 ∈ 𝑇 ∧ (𝐽 − (1st ‘𝑝)) ∈ (0...𝑁)) → ((𝑆 D𝑛 (𝐻‘𝑍))‘(𝐽 − (1st ‘𝑝))):𝑋⟶ℂ)) |
505 | 485, 490,
504 | sylc 63 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑝 ∈ ∪
𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘))) → ((𝑆 D𝑛 (𝐻‘𝑍))‘(𝐽 − (1st ‘𝑝))):𝑋⟶ℂ) |
506 | 505 | adantlr 747 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑝 ∈ ∪
𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘))) → ((𝑆 D𝑛 (𝐻‘𝑍))‘(𝐽 − (1st ‘𝑝))):𝑋⟶ℂ) |
507 | 34 | adantr 480 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑝 ∈ ∪
𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘))) → 𝑥 ∈ 𝑋) |
508 | 506, 507 | ffvelrnd 6268 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑝 ∈ ∪
𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘))) → (((𝑆 D𝑛 (𝐻‘𝑍))‘(𝐽 − (1st ‘𝑝)))‘𝑥) ∈ ℂ) |
509 | 475, 508 | mulcld 9939 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑝 ∈ ∪
𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘))) → ((((!‘(1st
‘𝑝)) / ∏𝑡 ∈ 𝑅 (!‘((2nd ‘𝑝)‘𝑡))) · ∏𝑡 ∈ 𝑅 (((𝑆 D𝑛 (𝐻‘𝑡))‘((2nd ‘𝑝)‘𝑡))‘𝑥)) · (((𝑆 D𝑛 (𝐻‘𝑍))‘(𝐽 − (1st ‘𝑝)))‘𝑥)) ∈ ℂ) |
510 | 420, 509 | mulcld 9939 |
. . . . . 6
⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑝 ∈ ∪
𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘))) → ((𝐽C(1st ‘𝑝)) · ((((!‘(1st
‘𝑝)) / ∏𝑡 ∈ 𝑅 (!‘((2nd ‘𝑝)‘𝑡))) · ∏𝑡 ∈ 𝑅 (((𝑆 D𝑛 (𝐻‘𝑡))‘((2nd ‘𝑝)‘𝑡))‘𝑥)) · (((𝑆 D𝑛 (𝐻‘𝑍))‘(𝐽 − (1st ‘𝑝)))‘𝑥))) ∈ ℂ) |
511 | 342, 386,
389, 395, 510 | fsumf1o 14301 |
. . . . 5
⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → Σ𝑝 ∈ ∪
𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘))((𝐽C(1st ‘𝑝)) · ((((!‘(1st
‘𝑝)) / ∏𝑡 ∈ 𝑅 (!‘((2nd ‘𝑝)‘𝑡))) · ∏𝑡 ∈ 𝑅 (((𝑆 D𝑛 (𝐻‘𝑡))‘((2nd ‘𝑝)‘𝑡))‘𝑥)) · (((𝑆 D𝑛 (𝐻‘𝑍))‘(𝐽 − (1st ‘𝑝)))‘𝑥))) = Σ𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)((𝐽C(𝐽 − (𝑐‘𝑍))) · ((((!‘(𝐽 − (𝑐‘𝑍))) / ∏𝑡 ∈ 𝑅 (!‘((𝑐 ↾ 𝑅)‘𝑡))) · ∏𝑡 ∈ 𝑅 (((𝑆 D𝑛 (𝐻‘𝑡))‘((𝑐 ↾ 𝑅)‘𝑡))‘𝑥)) · (((𝑆 D𝑛 (𝐻‘𝑍))‘(𝐽 − (𝐽 − (𝑐‘𝑍))))‘𝑥)))) |
512 | | simpl 472 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → 𝜑) |
513 | 373 | adantr 480 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽) = {𝑐 ∈ ((0...𝐽) ↑𝑚 (𝑅 ∪ {𝑍})) ∣ Σ𝑡 ∈ (𝑅 ∪ {𝑍})(𝑐‘𝑡) = 𝐽}) |
514 | 390, 513 | eleqtrd 2690 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → 𝑐 ∈ {𝑐 ∈ ((0...𝐽) ↑𝑚 (𝑅 ∪ {𝑍})) ∣ Σ𝑡 ∈ (𝑅 ∪ {𝑍})(𝑐‘𝑡) = 𝐽}) |
515 | 381 | sseli 3564 |
. . . . . . . . . . . . . . 15
⊢ (𝑐 ∈ {𝑐 ∈ ((0...𝐽) ↑𝑚 (𝑅 ∪ {𝑍})) ∣ Σ𝑡 ∈ (𝑅 ∪ {𝑍})(𝑐‘𝑡) = 𝐽} → 𝑐 ∈ ((0...𝐽) ↑𝑚 (𝑅 ∪ {𝑍}))) |
516 | 514, 515 | syl 17 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → 𝑐 ∈ ((0...𝐽) ↑𝑚 (𝑅 ∪ {𝑍}))) |
517 | | elmapi 7765 |
. . . . . . . . . . . . . 14
⊢ (𝑐 ∈ ((0...𝐽) ↑𝑚 (𝑅 ∪ {𝑍})) → 𝑐:(𝑅 ∪ {𝑍})⟶(0...𝐽)) |
518 | 516, 517 | syl 17 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → 𝑐:(𝑅 ∪ {𝑍})⟶(0...𝐽)) |
519 | | snidg 4153 |
. . . . . . . . . . . . . . . 16
⊢ (𝑍 ∈ 𝑇 → 𝑍 ∈ {𝑍}) |
520 | 23, 519 | syl 17 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → 𝑍 ∈ {𝑍}) |
521 | | elun2 3743 |
. . . . . . . . . . . . . . 15
⊢ (𝑍 ∈ {𝑍} → 𝑍 ∈ (𝑅 ∪ {𝑍})) |
522 | 520, 521 | syl 17 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → 𝑍 ∈ (𝑅 ∪ {𝑍})) |
523 | 522 | adantr 480 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → 𝑍 ∈ (𝑅 ∪ {𝑍})) |
524 | 518, 523 | ffvelrnd 6268 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → (𝑐‘𝑍) ∈ (0...𝐽)) |
525 | | 0zd 11266 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ (𝑐‘𝑍) ∈ (0...𝐽)) → 0 ∈ ℤ) |
526 | 126 | adantr 480 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ (𝑐‘𝑍) ∈ (0...𝐽)) → 𝐽 ∈ ℤ) |
527 | | fzssz 12214 |
. . . . . . . . . . . . . . . . . 18
⊢
(0...𝐽) ⊆
ℤ |
528 | 527 | sseli 3564 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝑐‘𝑍) ∈ (0...𝐽) → (𝑐‘𝑍) ∈ ℤ) |
529 | 528 | adantl 481 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ (𝑐‘𝑍) ∈ (0...𝐽)) → (𝑐‘𝑍) ∈ ℤ) |
530 | 526, 529 | zsubcld 11363 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ (𝑐‘𝑍) ∈ (0...𝐽)) → (𝐽 − (𝑐‘𝑍)) ∈ ℤ) |
531 | 525, 526,
530 | 3jca 1235 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ (𝑐‘𝑍) ∈ (0...𝐽)) → (0 ∈ ℤ ∧ 𝐽 ∈ ℤ ∧ (𝐽 − (𝑐‘𝑍)) ∈ ℤ)) |
532 | | elfzle2 12216 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑐‘𝑍) ∈ (0...𝐽) → (𝑐‘𝑍) ≤ 𝐽) |
533 | 532 | adantl 481 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ (𝑐‘𝑍) ∈ (0...𝐽)) → (𝑐‘𝑍) ≤ 𝐽) |
534 | 167 | adantr 480 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ (𝑐‘𝑍) ∈ (0...𝐽)) → 𝐽 ∈ ℝ) |
535 | 529 | zred 11358 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ (𝑐‘𝑍) ∈ (0...𝐽)) → (𝑐‘𝑍) ∈ ℝ) |
536 | 534, 535 | subge0d 10496 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ (𝑐‘𝑍) ∈ (0...𝐽)) → (0 ≤ (𝐽 − (𝑐‘𝑍)) ↔ (𝑐‘𝑍) ≤ 𝐽)) |
537 | 533, 536 | mpbird 246 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ (𝑐‘𝑍) ∈ (0...𝐽)) → 0 ≤ (𝐽 − (𝑐‘𝑍))) |
538 | | elfzle1 12215 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑐‘𝑍) ∈ (0...𝐽) → 0 ≤ (𝑐‘𝑍)) |
539 | 538 | adantl 481 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ (𝑐‘𝑍) ∈ (0...𝐽)) → 0 ≤ (𝑐‘𝑍)) |
540 | 534, 535 | subge02d 10498 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ (𝑐‘𝑍) ∈ (0...𝐽)) → (0 ≤ (𝑐‘𝑍) ↔ (𝐽 − (𝑐‘𝑍)) ≤ 𝐽)) |
541 | 539, 540 | mpbid 221 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ (𝑐‘𝑍) ∈ (0...𝐽)) → (𝐽 − (𝑐‘𝑍)) ≤ 𝐽) |
542 | 531, 537,
541 | jca32 556 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ (𝑐‘𝑍) ∈ (0...𝐽)) → ((0 ∈ ℤ ∧ 𝐽 ∈ ℤ ∧ (𝐽 − (𝑐‘𝑍)) ∈ ℤ) ∧ (0 ≤ (𝐽 − (𝑐‘𝑍)) ∧ (𝐽 − (𝑐‘𝑍)) ≤ 𝐽))) |
543 | | elfz2 12204 |
. . . . . . . . . . . . 13
⊢ ((𝐽 − (𝑐‘𝑍)) ∈ (0...𝐽) ↔ ((0 ∈ ℤ ∧ 𝐽 ∈ ℤ ∧ (𝐽 − (𝑐‘𝑍)) ∈ ℤ) ∧ (0 ≤ (𝐽 − (𝑐‘𝑍)) ∧ (𝐽 − (𝑐‘𝑍)) ≤ 𝐽))) |
544 | 542, 543 | sylibr 223 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ (𝑐‘𝑍) ∈ (0...𝐽)) → (𝐽 − (𝑐‘𝑍)) ∈ (0...𝐽)) |
545 | 512, 524,
544 | syl2anc 691 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → (𝐽 − (𝑐‘𝑍)) ∈ (0...𝐽)) |
546 | | bcval2 12954 |
. . . . . . . . . . 11
⊢ ((𝐽 − (𝑐‘𝑍)) ∈ (0...𝐽) → (𝐽C(𝐽 − (𝑐‘𝑍))) = ((!‘𝐽) / ((!‘(𝐽 − (𝐽 − (𝑐‘𝑍)))) · (!‘(𝐽 − (𝑐‘𝑍)))))) |
547 | 545, 546 | syl 17 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → (𝐽C(𝐽 − (𝑐‘𝑍))) = ((!‘𝐽) / ((!‘(𝐽 − (𝐽 − (𝑐‘𝑍)))) · (!‘(𝐽 − (𝑐‘𝑍)))))) |
548 | 167 | recnd 9947 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → 𝐽 ∈ ℂ) |
549 | 548 | adantr 480 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → 𝐽 ∈ ℂ) |
550 | | zsscn 11262 |
. . . . . . . . . . . . . . . . 17
⊢ ℤ
⊆ ℂ |
551 | 527, 550 | sstri 3577 |
. . . . . . . . . . . . . . . 16
⊢
(0...𝐽) ⊆
ℂ |
552 | 551 | a1i 11 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → (0...𝐽) ⊆ ℂ) |
553 | 552, 524 | sseldd 3569 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → (𝑐‘𝑍) ∈ ℂ) |
554 | 549, 553 | nncand 10276 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → (𝐽 − (𝐽 − (𝑐‘𝑍))) = (𝑐‘𝑍)) |
555 | 554 | fveq2d 6107 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → (!‘(𝐽 − (𝐽 − (𝑐‘𝑍)))) = (!‘(𝑐‘𝑍))) |
556 | 555 | oveq1d 6564 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → ((!‘(𝐽 − (𝐽 − (𝑐‘𝑍)))) · (!‘(𝐽 − (𝑐‘𝑍)))) = ((!‘(𝑐‘𝑍)) · (!‘(𝐽 − (𝑐‘𝑍))))) |
557 | 556 | oveq2d 6565 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → ((!‘𝐽) / ((!‘(𝐽 − (𝐽 − (𝑐‘𝑍)))) · (!‘(𝐽 − (𝑐‘𝑍))))) = ((!‘𝐽) / ((!‘(𝑐‘𝑍)) · (!‘(𝐽 − (𝑐‘𝑍)))))) |
558 | 45 | faccld 12933 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → (!‘𝐽) ∈ ℕ) |
559 | 558 | nncnd 10913 |
. . . . . . . . . . . . 13
⊢ (𝜑 → (!‘𝐽) ∈ ℂ) |
560 | 559 | adantr 480 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → (!‘𝐽) ∈ ℂ) |
561 | | elfznn0 12302 |
. . . . . . . . . . . . . . 15
⊢ ((𝑐‘𝑍) ∈ (0...𝐽) → (𝑐‘𝑍) ∈
ℕ0) |
562 | 524, 561 | syl 17 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → (𝑐‘𝑍) ∈
ℕ0) |
563 | 562 | faccld 12933 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → (!‘(𝑐‘𝑍)) ∈ ℕ) |
564 | 563 | nncnd 10913 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → (!‘(𝑐‘𝑍)) ∈ ℂ) |
565 | | elfznn0 12302 |
. . . . . . . . . . . . . . 15
⊢ ((𝐽 − (𝑐‘𝑍)) ∈ (0...𝐽) → (𝐽 − (𝑐‘𝑍)) ∈
ℕ0) |
566 | 545, 565 | syl 17 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → (𝐽 − (𝑐‘𝑍)) ∈
ℕ0) |
567 | 566 | faccld 12933 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → (!‘(𝐽 − (𝑐‘𝑍))) ∈ ℕ) |
568 | 567 | nncnd 10913 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → (!‘(𝐽 − (𝑐‘𝑍))) ∈ ℂ) |
569 | 563 | nnne0d 10942 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → (!‘(𝑐‘𝑍)) ≠ 0) |
570 | 567 | nnne0d 10942 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → (!‘(𝐽 − (𝑐‘𝑍))) ≠ 0) |
571 | 560, 564,
568, 569, 570 | divdiv1d 10711 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → (((!‘𝐽) / (!‘(𝑐‘𝑍))) / (!‘(𝐽 − (𝑐‘𝑍)))) = ((!‘𝐽) / ((!‘(𝑐‘𝑍)) · (!‘(𝐽 − (𝑐‘𝑍)))))) |
572 | 571 | eqcomd 2616 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → ((!‘𝐽) / ((!‘(𝑐‘𝑍)) · (!‘(𝐽 − (𝑐‘𝑍))))) = (((!‘𝐽) / (!‘(𝑐‘𝑍))) / (!‘(𝐽 − (𝑐‘𝑍))))) |
573 | 547, 557,
572 | 3eqtrd 2648 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → (𝐽C(𝐽 − (𝑐‘𝑍))) = (((!‘𝐽) / (!‘(𝑐‘𝑍))) / (!‘(𝐽 − (𝑐‘𝑍))))) |
574 | 573 | adantlr 747 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → (𝐽C(𝐽 − (𝑐‘𝑍))) = (((!‘𝐽) / (!‘(𝑐‘𝑍))) / (!‘(𝐽 − (𝑐‘𝑍))))) |
575 | | fvres 6117 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑡 ∈ 𝑅 → ((𝑐 ↾ 𝑅)‘𝑡) = (𝑐‘𝑡)) |
576 | 575 | fveq2d 6107 |
. . . . . . . . . . . . . . . 16
⊢ (𝑡 ∈ 𝑅 → (!‘((𝑐 ↾ 𝑅)‘𝑡)) = (!‘(𝑐‘𝑡))) |
577 | 576 | prodeq2i 14488 |
. . . . . . . . . . . . . . 15
⊢
∏𝑡 ∈
𝑅 (!‘((𝑐 ↾ 𝑅)‘𝑡)) = ∏𝑡 ∈ 𝑅 (!‘(𝑐‘𝑡)) |
578 | 577 | oveq2i 6560 |
. . . . . . . . . . . . . 14
⊢
((!‘(𝐽 −
(𝑐‘𝑍))) / ∏𝑡 ∈ 𝑅 (!‘((𝑐 ↾ 𝑅)‘𝑡))) = ((!‘(𝐽 − (𝑐‘𝑍))) / ∏𝑡 ∈ 𝑅 (!‘(𝑐‘𝑡))) |
579 | 575 | fveq2d 6107 |
. . . . . . . . . . . . . . . 16
⊢ (𝑡 ∈ 𝑅 → ((𝑆 D𝑛 (𝐻‘𝑡))‘((𝑐 ↾ 𝑅)‘𝑡)) = ((𝑆 D𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))) |
580 | 579 | fveq1d 6105 |
. . . . . . . . . . . . . . 15
⊢ (𝑡 ∈ 𝑅 → (((𝑆 D𝑛 (𝐻‘𝑡))‘((𝑐 ↾ 𝑅)‘𝑡))‘𝑥) = (((𝑆 D𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥)) |
581 | 580 | prodeq2i 14488 |
. . . . . . . . . . . . . 14
⊢
∏𝑡 ∈
𝑅 (((𝑆 D𝑛 (𝐻‘𝑡))‘((𝑐 ↾ 𝑅)‘𝑡))‘𝑥) = ∏𝑡 ∈ 𝑅 (((𝑆 D𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥) |
582 | 578, 581 | oveq12i 6561 |
. . . . . . . . . . . . 13
⊢
(((!‘(𝐽
− (𝑐‘𝑍))) / ∏𝑡 ∈ 𝑅 (!‘((𝑐 ↾ 𝑅)‘𝑡))) · ∏𝑡 ∈ 𝑅 (((𝑆 D𝑛 (𝐻‘𝑡))‘((𝑐 ↾ 𝑅)‘𝑡))‘𝑥)) = (((!‘(𝐽 − (𝑐‘𝑍))) / ∏𝑡 ∈ 𝑅 (!‘(𝑐‘𝑡))) · ∏𝑡 ∈ 𝑅 (((𝑆 D𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥)) |
583 | 582 | a1i 11 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → (((!‘(𝐽 − (𝑐‘𝑍))) / ∏𝑡 ∈ 𝑅 (!‘((𝑐 ↾ 𝑅)‘𝑡))) · ∏𝑡 ∈ 𝑅 (((𝑆 D𝑛 (𝐻‘𝑡))‘((𝑐 ↾ 𝑅)‘𝑡))‘𝑥)) = (((!‘(𝐽 − (𝑐‘𝑍))) / ∏𝑡 ∈ 𝑅 (!‘(𝑐‘𝑡))) · ∏𝑡 ∈ 𝑅 (((𝑆 D𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥))) |
584 | 583 | adantlr 747 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → (((!‘(𝐽 − (𝑐‘𝑍))) / ∏𝑡 ∈ 𝑅 (!‘((𝑐 ↾ 𝑅)‘𝑡))) · ∏𝑡 ∈ 𝑅 (((𝑆 D𝑛 (𝐻‘𝑡))‘((𝑐 ↾ 𝑅)‘𝑡))‘𝑥)) = (((!‘(𝐽 − (𝑐‘𝑍))) / ∏𝑡 ∈ 𝑅 (!‘(𝑐‘𝑡))) · ∏𝑡 ∈ 𝑅 (((𝑆 D𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥))) |
585 | 568 | adantlr 747 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → (!‘(𝐽 − (𝑐‘𝑍))) ∈ ℂ) |
586 | 512, 6 | syl 17 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → 𝑅 ∈ Fin) |
587 | 77 | ssriv 3572 |
. . . . . . . . . . . . . . . . . 18
⊢
(0...𝐽) ⊆
ℕ0 |
588 | 587 | a1i 11 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) ∧ 𝑡 ∈ 𝑅) → (0...𝐽) ⊆
ℕ0) |
589 | 518 | adantr 480 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) ∧ 𝑡 ∈ 𝑅) → 𝑐:(𝑅 ∪ {𝑍})⟶(0...𝐽)) |
590 | | elun1 3742 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑡 ∈ 𝑅 → 𝑡 ∈ (𝑅 ∪ {𝑍})) |
591 | 590 | adantl 481 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) ∧ 𝑡 ∈ 𝑅) → 𝑡 ∈ (𝑅 ∪ {𝑍})) |
592 | 589, 591 | ffvelrnd 6268 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) ∧ 𝑡 ∈ 𝑅) → (𝑐‘𝑡) ∈ (0...𝐽)) |
593 | 588, 592 | sseldd 3569 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) ∧ 𝑡 ∈ 𝑅) → (𝑐‘𝑡) ∈
ℕ0) |
594 | 593 | faccld 12933 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) ∧ 𝑡 ∈ 𝑅) → (!‘(𝑐‘𝑡)) ∈ ℕ) |
595 | 594 | nncnd 10913 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) ∧ 𝑡 ∈ 𝑅) → (!‘(𝑐‘𝑡)) ∈ ℂ) |
596 | 586, 595 | fprodcl 14521 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → ∏𝑡 ∈ 𝑅 (!‘(𝑐‘𝑡)) ∈ ℂ) |
597 | 596 | adantlr 747 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → ∏𝑡 ∈ 𝑅 (!‘(𝑐‘𝑡)) ∈ ℂ) |
598 | 7 | adantr 480 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → 𝑅 ∈ Fin) |
599 | 512 | adantr 480 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) ∧ 𝑡 ∈ 𝑅) → 𝜑) |
600 | 512, 13 | sylan 487 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) ∧ 𝑡 ∈ 𝑅) → 𝑡 ∈ 𝑇) |
601 | 599, 135 | syl 17 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) ∧ 𝑡 ∈ 𝑅) → (0...𝐽) ⊆ (0...𝑁)) |
602 | 601, 592 | sseldd 3569 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) ∧ 𝑡 ∈ 𝑅) → (𝑐‘𝑡) ∈ (0...𝑁)) |
603 | 599, 600,
602, 148 | syl3anc 1318 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) ∧ 𝑡 ∈ 𝑅) → ((𝑆 D𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡)):𝑋⟶ℂ) |
604 | 603 | adantllr 751 |
. . . . . . . . . . . . . 14
⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) ∧ 𝑡 ∈ 𝑅) → ((𝑆 D𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡)):𝑋⟶ℂ) |
605 | 17 | adantlr 747 |
. . . . . . . . . . . . . 14
⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) ∧ 𝑡 ∈ 𝑅) → 𝑥 ∈ 𝑋) |
606 | 604, 605 | ffvelrnd 6268 |
. . . . . . . . . . . . 13
⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) ∧ 𝑡 ∈ 𝑅) → (((𝑆 D𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥) ∈ ℂ) |
607 | 598, 606 | fprodcl 14521 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → ∏𝑡 ∈ 𝑅 (((𝑆 D𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥) ∈ ℂ) |
608 | 586, 594 | fprodnncl 14524 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → ∏𝑡 ∈ 𝑅 (!‘(𝑐‘𝑡)) ∈ ℕ) |
609 | | nnne0 10930 |
. . . . . . . . . . . . . 14
⊢
(∏𝑡 ∈
𝑅 (!‘(𝑐‘𝑡)) ∈ ℕ → ∏𝑡 ∈ 𝑅 (!‘(𝑐‘𝑡)) ≠ 0) |
610 | 608, 609 | syl 17 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → ∏𝑡 ∈ 𝑅 (!‘(𝑐‘𝑡)) ≠ 0) |
611 | 610 | adantlr 747 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → ∏𝑡 ∈ 𝑅 (!‘(𝑐‘𝑡)) ≠ 0) |
612 | 585, 597,
607, 611 | div32d 10703 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → (((!‘(𝐽 − (𝑐‘𝑍))) / ∏𝑡 ∈ 𝑅 (!‘(𝑐‘𝑡))) · ∏𝑡 ∈ 𝑅 (((𝑆 D𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥)) = ((!‘(𝐽 − (𝑐‘𝑍))) · (∏𝑡 ∈ 𝑅 (((𝑆 D𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥) / ∏𝑡 ∈ 𝑅 (!‘(𝑐‘𝑡))))) |
613 | 584, 612 | eqtrd 2644 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → (((!‘(𝐽 − (𝑐‘𝑍))) / ∏𝑡 ∈ 𝑅 (!‘((𝑐 ↾ 𝑅)‘𝑡))) · ∏𝑡 ∈ 𝑅 (((𝑆 D𝑛 (𝐻‘𝑡))‘((𝑐 ↾ 𝑅)‘𝑡))‘𝑥)) = ((!‘(𝐽 − (𝑐‘𝑍))) · (∏𝑡 ∈ 𝑅 (((𝑆 D𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥) / ∏𝑡 ∈ 𝑅 (!‘(𝑐‘𝑡))))) |
614 | 554 | fveq2d 6107 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → ((𝑆 D𝑛 (𝐻‘𝑍))‘(𝐽 − (𝐽 − (𝑐‘𝑍)))) = ((𝑆 D𝑛 (𝐻‘𝑍))‘(𝑐‘𝑍))) |
615 | 614 | fveq1d 6105 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → (((𝑆 D𝑛 (𝐻‘𝑍))‘(𝐽 − (𝐽 − (𝑐‘𝑍))))‘𝑥) = (((𝑆 D𝑛 (𝐻‘𝑍))‘(𝑐‘𝑍))‘𝑥)) |
616 | 615 | adantlr 747 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → (((𝑆 D𝑛 (𝐻‘𝑍))‘(𝐽 − (𝐽 − (𝑐‘𝑍))))‘𝑥) = (((𝑆 D𝑛 (𝐻‘𝑍))‘(𝑐‘𝑍))‘𝑥)) |
617 | 613, 616 | oveq12d 6567 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → ((((!‘(𝐽 − (𝑐‘𝑍))) / ∏𝑡 ∈ 𝑅 (!‘((𝑐 ↾ 𝑅)‘𝑡))) · ∏𝑡 ∈ 𝑅 (((𝑆 D𝑛 (𝐻‘𝑡))‘((𝑐 ↾ 𝑅)‘𝑡))‘𝑥)) · (((𝑆 D𝑛 (𝐻‘𝑍))‘(𝐽 − (𝐽 − (𝑐‘𝑍))))‘𝑥)) = (((!‘(𝐽 − (𝑐‘𝑍))) · (∏𝑡 ∈ 𝑅 (((𝑆 D𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥) / ∏𝑡 ∈ 𝑅 (!‘(𝑐‘𝑡)))) · (((𝑆 D𝑛 (𝐻‘𝑍))‘(𝑐‘𝑍))‘𝑥))) |
618 | 607, 597,
611 | divcld 10680 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → (∏𝑡 ∈ 𝑅 (((𝑆 D𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥) / ∏𝑡 ∈ 𝑅 (!‘(𝑐‘𝑡))) ∈ ℂ) |
619 | 512, 23 | syl 17 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → 𝑍 ∈ 𝑇) |
620 | 512, 135 | syl 17 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → (0...𝐽) ⊆ (0...𝑁)) |
621 | 620, 524 | sseldd 3569 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → (𝑐‘𝑍) ∈ (0...𝑁)) |
622 | 512, 619,
621 | 3jca 1235 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → (𝜑 ∧ 𝑍 ∈ 𝑇 ∧ (𝑐‘𝑍) ∈ (0...𝑁))) |
623 | | eleq1 2676 |
. . . . . . . . . . . . . . . 16
⊢ (𝑗 = (𝑐‘𝑍) → (𝑗 ∈ (0...𝑁) ↔ (𝑐‘𝑍) ∈ (0...𝑁))) |
624 | 623 | 3anbi3d 1397 |
. . . . . . . . . . . . . . 15
⊢ (𝑗 = (𝑐‘𝑍) → ((𝜑 ∧ 𝑍 ∈ 𝑇 ∧ 𝑗 ∈ (0...𝑁)) ↔ (𝜑 ∧ 𝑍 ∈ 𝑇 ∧ (𝑐‘𝑍) ∈ (0...𝑁)))) |
625 | | fveq2 6103 |
. . . . . . . . . . . . . . . 16
⊢ (𝑗 = (𝑐‘𝑍) → ((𝑆 D𝑛 (𝐻‘𝑍))‘𝑗) = ((𝑆 D𝑛 (𝐻‘𝑍))‘(𝑐‘𝑍))) |
626 | 625 | feq1d 5943 |
. . . . . . . . . . . . . . 15
⊢ (𝑗 = (𝑐‘𝑍) → (((𝑆 D𝑛 (𝐻‘𝑍))‘𝑗):𝑋⟶ℂ ↔ ((𝑆 D𝑛 (𝐻‘𝑍))‘(𝑐‘𝑍)):𝑋⟶ℂ)) |
627 | 624, 626 | imbi12d 333 |
. . . . . . . . . . . . . 14
⊢ (𝑗 = (𝑐‘𝑍) → (((𝜑 ∧ 𝑍 ∈ 𝑇 ∧ 𝑗 ∈ (0...𝑁)) → ((𝑆 D𝑛 (𝐻‘𝑍))‘𝑗):𝑋⟶ℂ) ↔ ((𝜑 ∧ 𝑍 ∈ 𝑇 ∧ (𝑐‘𝑍) ∈ (0...𝑁)) → ((𝑆 D𝑛 (𝐻‘𝑍))‘(𝑐‘𝑍)):𝑋⟶ℂ))) |
628 | 627, 503 | vtoclg 3239 |
. . . . . . . . . . . . 13
⊢ ((𝑐‘𝑍) ∈ (0...𝐽) → ((𝜑 ∧ 𝑍 ∈ 𝑇 ∧ (𝑐‘𝑍) ∈ (0...𝑁)) → ((𝑆 D𝑛 (𝐻‘𝑍))‘(𝑐‘𝑍)):𝑋⟶ℂ)) |
629 | 524, 622,
628 | sylc 63 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → ((𝑆 D𝑛 (𝐻‘𝑍))‘(𝑐‘𝑍)):𝑋⟶ℂ) |
630 | 629 | adantlr 747 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → ((𝑆 D𝑛 (𝐻‘𝑍))‘(𝑐‘𝑍)):𝑋⟶ℂ) |
631 | 34 | adantr 480 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → 𝑥 ∈ 𝑋) |
632 | 630, 631 | ffvelrnd 6268 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → (((𝑆 D𝑛 (𝐻‘𝑍))‘(𝑐‘𝑍))‘𝑥) ∈ ℂ) |
633 | 585, 618,
632 | mulassd 9942 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → (((!‘(𝐽 − (𝑐‘𝑍))) · (∏𝑡 ∈ 𝑅 (((𝑆 D𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥) / ∏𝑡 ∈ 𝑅 (!‘(𝑐‘𝑡)))) · (((𝑆 D𝑛 (𝐻‘𝑍))‘(𝑐‘𝑍))‘𝑥)) = ((!‘(𝐽 − (𝑐‘𝑍))) · ((∏𝑡 ∈ 𝑅 (((𝑆 D𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥) / ∏𝑡 ∈ 𝑅 (!‘(𝑐‘𝑡))) · (((𝑆 D𝑛 (𝐻‘𝑍))‘(𝑐‘𝑍))‘𝑥)))) |
634 | 617, 633 | eqtrd 2644 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → ((((!‘(𝐽 − (𝑐‘𝑍))) / ∏𝑡 ∈ 𝑅 (!‘((𝑐 ↾ 𝑅)‘𝑡))) · ∏𝑡 ∈ 𝑅 (((𝑆 D𝑛 (𝐻‘𝑡))‘((𝑐 ↾ 𝑅)‘𝑡))‘𝑥)) · (((𝑆 D𝑛 (𝐻‘𝑍))‘(𝐽 − (𝐽 − (𝑐‘𝑍))))‘𝑥)) = ((!‘(𝐽 − (𝑐‘𝑍))) · ((∏𝑡 ∈ 𝑅 (((𝑆 D𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥) / ∏𝑡 ∈ 𝑅 (!‘(𝑐‘𝑡))) · (((𝑆 D𝑛 (𝐻‘𝑍))‘(𝑐‘𝑍))‘𝑥)))) |
635 | 574, 634 | oveq12d 6567 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → ((𝐽C(𝐽 − (𝑐‘𝑍))) · ((((!‘(𝐽 − (𝑐‘𝑍))) / ∏𝑡 ∈ 𝑅 (!‘((𝑐 ↾ 𝑅)‘𝑡))) · ∏𝑡 ∈ 𝑅 (((𝑆 D𝑛 (𝐻‘𝑡))‘((𝑐 ↾ 𝑅)‘𝑡))‘𝑥)) · (((𝑆 D𝑛 (𝐻‘𝑍))‘(𝐽 − (𝐽 − (𝑐‘𝑍))))‘𝑥))) = ((((!‘𝐽) / (!‘(𝑐‘𝑍))) / (!‘(𝐽 − (𝑐‘𝑍)))) · ((!‘(𝐽 − (𝑐‘𝑍))) · ((∏𝑡 ∈ 𝑅 (((𝑆 D𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥) / ∏𝑡 ∈ 𝑅 (!‘(𝑐‘𝑡))) · (((𝑆 D𝑛 (𝐻‘𝑍))‘(𝑐‘𝑍))‘𝑥))))) |
636 | 559 | ad2antrr 758 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → (!‘𝐽) ∈ ℂ) |
637 | 564 | adantlr 747 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → (!‘(𝑐‘𝑍)) ∈ ℂ) |
638 | 569 | adantlr 747 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → (!‘(𝑐‘𝑍)) ≠ 0) |
639 | 636, 637,
638 | divcld 10680 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → ((!‘𝐽) / (!‘(𝑐‘𝑍))) ∈ ℂ) |
640 | 618, 632 | mulcld 9939 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → ((∏𝑡 ∈ 𝑅 (((𝑆 D𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥) / ∏𝑡 ∈ 𝑅 (!‘(𝑐‘𝑡))) · (((𝑆 D𝑛 (𝐻‘𝑍))‘(𝑐‘𝑍))‘𝑥)) ∈ ℂ) |
641 | 570 | adantlr 747 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → (!‘(𝐽 − (𝑐‘𝑍))) ≠ 0) |
642 | 639, 585,
640, 641 | dmmcand 38469 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → ((((!‘𝐽) / (!‘(𝑐‘𝑍))) / (!‘(𝐽 − (𝑐‘𝑍)))) · ((!‘(𝐽 − (𝑐‘𝑍))) · ((∏𝑡 ∈ 𝑅 (((𝑆 D𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥) / ∏𝑡 ∈ 𝑅 (!‘(𝑐‘𝑡))) · (((𝑆 D𝑛 (𝐻‘𝑍))‘(𝑐‘𝑍))‘𝑥)))) = (((!‘𝐽) / (!‘(𝑐‘𝑍))) · ((∏𝑡 ∈ 𝑅 (((𝑆 D𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥) / ∏𝑡 ∈ 𝑅 (!‘(𝑐‘𝑡))) · (((𝑆 D𝑛 (𝐻‘𝑍))‘(𝑐‘𝑍))‘𝑥)))) |
643 | 607, 632,
597, 611 | div23d 10717 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → ((∏𝑡 ∈ 𝑅 (((𝑆 D𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥) · (((𝑆 D𝑛 (𝐻‘𝑍))‘(𝑐‘𝑍))‘𝑥)) / ∏𝑡 ∈ 𝑅 (!‘(𝑐‘𝑡))) = ((∏𝑡 ∈ 𝑅 (((𝑆 D𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥) / ∏𝑡 ∈ 𝑅 (!‘(𝑐‘𝑡))) · (((𝑆 D𝑛 (𝐻‘𝑍))‘(𝑐‘𝑍))‘𝑥))) |
644 | 643 | eqcomd 2616 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → ((∏𝑡 ∈ 𝑅 (((𝑆 D𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥) / ∏𝑡 ∈ 𝑅 (!‘(𝑐‘𝑡))) · (((𝑆 D𝑛 (𝐻‘𝑍))‘(𝑐‘𝑍))‘𝑥)) = ((∏𝑡 ∈ 𝑅 (((𝑆 D𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥) · (((𝑆 D𝑛 (𝐻‘𝑍))‘(𝑐‘𝑍))‘𝑥)) / ∏𝑡 ∈ 𝑅 (!‘(𝑐‘𝑡)))) |
645 | | nfv 1830 |
. . . . . . . . . . . . 13
⊢
Ⅎ𝑡((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) |
646 | | nfcv 2751 |
. . . . . . . . . . . . 13
⊢
Ⅎ𝑡(((𝑆 D𝑛 (𝐻‘𝑍))‘(𝑐‘𝑍))‘𝑥) |
647 | 619 | adantlr 747 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → 𝑍 ∈ 𝑇) |
648 | 11 | adantr 480 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → ¬ 𝑍 ∈ 𝑅) |
649 | | fveq2 6103 |
. . . . . . . . . . . . . . 15
⊢ (𝑡 = 𝑍 → (𝑐‘𝑡) = (𝑐‘𝑍)) |
650 | 185, 649 | fveq12d 6109 |
. . . . . . . . . . . . . 14
⊢ (𝑡 = 𝑍 → ((𝑆 D𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡)) = ((𝑆 D𝑛 (𝐻‘𝑍))‘(𝑐‘𝑍))) |
651 | 650 | fveq1d 6105 |
. . . . . . . . . . . . 13
⊢ (𝑡 = 𝑍 → (((𝑆 D𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥) = (((𝑆 D𝑛 (𝐻‘𝑍))‘(𝑐‘𝑍))‘𝑥)) |
652 | 645, 646,
598, 647, 648, 606, 651, 632 | fprodsplitsn 14559 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → ∏𝑡 ∈ (𝑅 ∪ {𝑍})(((𝑆 D𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥) = (∏𝑡 ∈ 𝑅 (((𝑆 D𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥) · (((𝑆 D𝑛 (𝐻‘𝑍))‘(𝑐‘𝑍))‘𝑥))) |
653 | 652 | eqcomd 2616 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → (∏𝑡 ∈ 𝑅 (((𝑆 D𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥) · (((𝑆 D𝑛 (𝐻‘𝑍))‘(𝑐‘𝑍))‘𝑥)) = ∏𝑡 ∈ (𝑅 ∪ {𝑍})(((𝑆 D𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥)) |
654 | 653 | oveq1d 6564 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → ((∏𝑡 ∈ 𝑅 (((𝑆 D𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥) · (((𝑆 D𝑛 (𝐻‘𝑍))‘(𝑐‘𝑍))‘𝑥)) / ∏𝑡 ∈ 𝑅 (!‘(𝑐‘𝑡))) = (∏𝑡 ∈ (𝑅 ∪ {𝑍})(((𝑆 D𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥) / ∏𝑡 ∈ 𝑅 (!‘(𝑐‘𝑡)))) |
655 | 644, 654 | eqtrd 2644 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → ((∏𝑡 ∈ 𝑅 (((𝑆 D𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥) / ∏𝑡 ∈ 𝑅 (!‘(𝑐‘𝑡))) · (((𝑆 D𝑛 (𝐻‘𝑍))‘(𝑐‘𝑍))‘𝑥)) = (∏𝑡 ∈ (𝑅 ∪ {𝑍})(((𝑆 D𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥) / ∏𝑡 ∈ 𝑅 (!‘(𝑐‘𝑡)))) |
656 | 655 | oveq2d 6565 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → (((!‘𝐽) / (!‘(𝑐‘𝑍))) · ((∏𝑡 ∈ 𝑅 (((𝑆 D𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥) / ∏𝑡 ∈ 𝑅 (!‘(𝑐‘𝑡))) · (((𝑆 D𝑛 (𝐻‘𝑍))‘(𝑐‘𝑍))‘𝑥))) = (((!‘𝐽) / (!‘(𝑐‘𝑍))) · (∏𝑡 ∈ (𝑅 ∪ {𝑍})(((𝑆 D𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥) / ∏𝑡 ∈ 𝑅 (!‘(𝑐‘𝑡))))) |
657 | 598, 375,
377 | sylancl 693 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → (𝑅 ∪ {𝑍}) ∈ Fin) |
658 | 512 | adantr 480 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) ∧ 𝑡 ∈ (𝑅 ∪ {𝑍})) → 𝜑) |
659 | 354 | sselda 3568 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑡 ∈ (𝑅 ∪ {𝑍})) → 𝑡 ∈ 𝑇) |
660 | 659 | adantlr 747 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) ∧ 𝑡 ∈ (𝑅 ∪ {𝑍})) → 𝑡 ∈ 𝑇) |
661 | 518, 620 | fssd 5970 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → 𝑐:(𝑅 ∪ {𝑍})⟶(0...𝑁)) |
662 | 661 | ffvelrnda 6267 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) ∧ 𝑡 ∈ (𝑅 ∪ {𝑍})) → (𝑐‘𝑡) ∈ (0...𝑁)) |
663 | 658, 660,
662, 148 | syl3anc 1318 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) ∧ 𝑡 ∈ (𝑅 ∪ {𝑍})) → ((𝑆 D𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡)):𝑋⟶ℂ) |
664 | 663 | adantllr 751 |
. . . . . . . . . . 11
⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) ∧ 𝑡 ∈ (𝑅 ∪ {𝑍})) → ((𝑆 D𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡)):𝑋⟶ℂ) |
665 | 631 | adantr 480 |
. . . . . . . . . . 11
⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) ∧ 𝑡 ∈ (𝑅 ∪ {𝑍})) → 𝑥 ∈ 𝑋) |
666 | 664, 665 | ffvelrnd 6268 |
. . . . . . . . . 10
⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) ∧ 𝑡 ∈ (𝑅 ∪ {𝑍})) → (((𝑆 D𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥) ∈ ℂ) |
667 | 657, 666 | fprodcl 14521 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → ∏𝑡 ∈ (𝑅 ∪ {𝑍})(((𝑆 D𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥) ∈ ℂ) |
668 | 636, 637,
667, 597, 638, 611 | divmuldivd 10721 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → (((!‘𝐽) / (!‘(𝑐‘𝑍))) · (∏𝑡 ∈ (𝑅 ∪ {𝑍})(((𝑆 D𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥) / ∏𝑡 ∈ 𝑅 (!‘(𝑐‘𝑡)))) = (((!‘𝐽) · ∏𝑡 ∈ (𝑅 ∪ {𝑍})(((𝑆 D𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥)) / ((!‘(𝑐‘𝑍)) · ∏𝑡 ∈ 𝑅 (!‘(𝑐‘𝑡))))) |
669 | 564, 596 | mulcomd 9940 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → ((!‘(𝑐‘𝑍)) · ∏𝑡 ∈ 𝑅 (!‘(𝑐‘𝑡))) = (∏𝑡 ∈ 𝑅 (!‘(𝑐‘𝑡)) · (!‘(𝑐‘𝑍)))) |
670 | | nfv 1830 |
. . . . . . . . . . . . . 14
⊢
Ⅎ𝑡(𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) |
671 | | nfcv 2751 |
. . . . . . . . . . . . . 14
⊢
Ⅎ𝑡(!‘(𝑐‘𝑍)) |
672 | 512, 10 | syl 17 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → ¬ 𝑍 ∈ 𝑅) |
673 | 649 | fveq2d 6107 |
. . . . . . . . . . . . . 14
⊢ (𝑡 = 𝑍 → (!‘(𝑐‘𝑡)) = (!‘(𝑐‘𝑍))) |
674 | 670, 671,
586, 619, 672, 595, 673, 564 | fprodsplitsn 14559 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → ∏𝑡 ∈ (𝑅 ∪ {𝑍})(!‘(𝑐‘𝑡)) = (∏𝑡 ∈ 𝑅 (!‘(𝑐‘𝑡)) · (!‘(𝑐‘𝑍)))) |
675 | 674 | eqcomd 2616 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → (∏𝑡 ∈ 𝑅 (!‘(𝑐‘𝑡)) · (!‘(𝑐‘𝑍))) = ∏𝑡 ∈ (𝑅 ∪ {𝑍})(!‘(𝑐‘𝑡))) |
676 | 669, 675 | eqtrd 2644 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → ((!‘(𝑐‘𝑍)) · ∏𝑡 ∈ 𝑅 (!‘(𝑐‘𝑡))) = ∏𝑡 ∈ (𝑅 ∪ {𝑍})(!‘(𝑐‘𝑡))) |
677 | 676 | oveq2d 6565 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → (((!‘𝐽) · ∏𝑡 ∈ (𝑅 ∪ {𝑍})(((𝑆 D𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥)) / ((!‘(𝑐‘𝑍)) · ∏𝑡 ∈ 𝑅 (!‘(𝑐‘𝑡)))) = (((!‘𝐽) · ∏𝑡 ∈ (𝑅 ∪ {𝑍})(((𝑆 D𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥)) / ∏𝑡 ∈ (𝑅 ∪ {𝑍})(!‘(𝑐‘𝑡)))) |
678 | 677 | adantlr 747 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → (((!‘𝐽) · ∏𝑡 ∈ (𝑅 ∪ {𝑍})(((𝑆 D𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥)) / ((!‘(𝑐‘𝑍)) · ∏𝑡 ∈ 𝑅 (!‘(𝑐‘𝑡)))) = (((!‘𝐽) · ∏𝑡 ∈ (𝑅 ∪ {𝑍})(((𝑆 D𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥)) / ∏𝑡 ∈ (𝑅 ∪ {𝑍})(!‘(𝑐‘𝑡)))) |
679 | 512, 378 | syl 17 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → (𝑅 ∪ {𝑍}) ∈ Fin) |
680 | 587 | a1i 11 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) ∧ 𝑡 ∈ (𝑅 ∪ {𝑍})) → (0...𝐽) ⊆
ℕ0) |
681 | 518 | ffvelrnda 6267 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) ∧ 𝑡 ∈ (𝑅 ∪ {𝑍})) → (𝑐‘𝑡) ∈ (0...𝐽)) |
682 | 680, 681 | sseldd 3569 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) ∧ 𝑡 ∈ (𝑅 ∪ {𝑍})) → (𝑐‘𝑡) ∈
ℕ0) |
683 | 682 | faccld 12933 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) ∧ 𝑡 ∈ (𝑅 ∪ {𝑍})) → (!‘(𝑐‘𝑡)) ∈ ℕ) |
684 | 683 | nncnd 10913 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) ∧ 𝑡 ∈ (𝑅 ∪ {𝑍})) → (!‘(𝑐‘𝑡)) ∈ ℂ) |
685 | 679, 684 | fprodcl 14521 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → ∏𝑡 ∈ (𝑅 ∪ {𝑍})(!‘(𝑐‘𝑡)) ∈ ℂ) |
686 | 685 | adantlr 747 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → ∏𝑡 ∈ (𝑅 ∪ {𝑍})(!‘(𝑐‘𝑡)) ∈ ℂ) |
687 | 683 | nnne0d 10942 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) ∧ 𝑡 ∈ (𝑅 ∪ {𝑍})) → (!‘(𝑐‘𝑡)) ≠ 0) |
688 | 679, 684,
687 | fprodn0 14548 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → ∏𝑡 ∈ (𝑅 ∪ {𝑍})(!‘(𝑐‘𝑡)) ≠ 0) |
689 | 688 | adantlr 747 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → ∏𝑡 ∈ (𝑅 ∪ {𝑍})(!‘(𝑐‘𝑡)) ≠ 0) |
690 | 636, 667,
686, 689 | div23d 10717 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → (((!‘𝐽) · ∏𝑡 ∈ (𝑅 ∪ {𝑍})(((𝑆 D𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥)) / ∏𝑡 ∈ (𝑅 ∪ {𝑍})(!‘(𝑐‘𝑡))) = (((!‘𝐽) / ∏𝑡 ∈ (𝑅 ∪ {𝑍})(!‘(𝑐‘𝑡))) · ∏𝑡 ∈ (𝑅 ∪ {𝑍})(((𝑆 D𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥))) |
691 | | eqidd 2611 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → (((!‘𝐽) / ∏𝑡 ∈ (𝑅 ∪ {𝑍})(!‘(𝑐‘𝑡))) · ∏𝑡 ∈ (𝑅 ∪ {𝑍})(((𝑆 D𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥)) = (((!‘𝐽) / ∏𝑡 ∈ (𝑅 ∪ {𝑍})(!‘(𝑐‘𝑡))) · ∏𝑡 ∈ (𝑅 ∪ {𝑍})(((𝑆 D𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥))) |
692 | 678, 690,
691 | 3eqtrd 2648 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → (((!‘𝐽) · ∏𝑡 ∈ (𝑅 ∪ {𝑍})(((𝑆 D𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥)) / ((!‘(𝑐‘𝑍)) · ∏𝑡 ∈ 𝑅 (!‘(𝑐‘𝑡)))) = (((!‘𝐽) / ∏𝑡 ∈ (𝑅 ∪ {𝑍})(!‘(𝑐‘𝑡))) · ∏𝑡 ∈ (𝑅 ∪ {𝑍})(((𝑆 D𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥))) |
693 | 656, 668,
692 | 3eqtrd 2648 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → (((!‘𝐽) / (!‘(𝑐‘𝑍))) · ((∏𝑡 ∈ 𝑅 (((𝑆 D𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥) / ∏𝑡 ∈ 𝑅 (!‘(𝑐‘𝑡))) · (((𝑆 D𝑛 (𝐻‘𝑍))‘(𝑐‘𝑍))‘𝑥))) = (((!‘𝐽) / ∏𝑡 ∈ (𝑅 ∪ {𝑍})(!‘(𝑐‘𝑡))) · ∏𝑡 ∈ (𝑅 ∪ {𝑍})(((𝑆 D𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥))) |
694 | 635, 642,
693 | 3eqtrd 2648 |
. . . . . 6
⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → ((𝐽C(𝐽 − (𝑐‘𝑍))) · ((((!‘(𝐽 − (𝑐‘𝑍))) / ∏𝑡 ∈ 𝑅 (!‘((𝑐 ↾ 𝑅)‘𝑡))) · ∏𝑡 ∈ 𝑅 (((𝑆 D𝑛 (𝐻‘𝑡))‘((𝑐 ↾ 𝑅)‘𝑡))‘𝑥)) · (((𝑆 D𝑛 (𝐻‘𝑍))‘(𝐽 − (𝐽 − (𝑐‘𝑍))))‘𝑥))) = (((!‘𝐽) / ∏𝑡 ∈ (𝑅 ∪ {𝑍})(!‘(𝑐‘𝑡))) · ∏𝑡 ∈ (𝑅 ∪ {𝑍})(((𝑆 D𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥))) |
695 | 694 | sumeq2dv 14281 |
. . . . 5
⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → Σ𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)((𝐽C(𝐽 − (𝑐‘𝑍))) · ((((!‘(𝐽 − (𝑐‘𝑍))) / ∏𝑡 ∈ 𝑅 (!‘((𝑐 ↾ 𝑅)‘𝑡))) · ∏𝑡 ∈ 𝑅 (((𝑆 D𝑛 (𝐻‘𝑡))‘((𝑐 ↾ 𝑅)‘𝑡))‘𝑥)) · (((𝑆 D𝑛 (𝐻‘𝑍))‘(𝐽 − (𝐽 − (𝑐‘𝑍))))‘𝑥))) = Σ𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)(((!‘𝐽) / ∏𝑡 ∈ (𝑅 ∪ {𝑍})(!‘(𝑐‘𝑡))) · ∏𝑡 ∈ (𝑅 ∪ {𝑍})(((𝑆 D𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥))) |
696 | 511, 695 | eqtrd 2644 |
. . . 4
⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → Σ𝑝 ∈ ∪
𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘))((𝐽C(1st ‘𝑝)) · ((((!‘(1st
‘𝑝)) / ∏𝑡 ∈ 𝑅 (!‘((2nd ‘𝑝)‘𝑡))) · ∏𝑡 ∈ 𝑅 (((𝑆 D𝑛 (𝐻‘𝑡))‘((2nd ‘𝑝)‘𝑡))‘𝑥)) · (((𝑆 D𝑛 (𝐻‘𝑍))‘(𝐽 − (1st ‘𝑝)))‘𝑥))) = Σ𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)(((!‘𝐽) / ∏𝑡 ∈ (𝑅 ∪ {𝑍})(!‘(𝑐‘𝑡))) · ∏𝑡 ∈ (𝑅 ∪ {𝑍})(((𝑆 D𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥))) |
697 | 299, 323,
696 | 3eqtrd 2648 |
. . 3
⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → Σ𝑘 ∈ (0...𝐽)((𝐽C𝑘) · ((((𝑘 ∈ (0...𝐽) ↦ ((𝑆 D𝑛 (𝑦 ∈ 𝑋 ↦ ∏𝑡 ∈ 𝑅 ((𝐻‘𝑡)‘𝑦)))‘𝑘))‘𝑘)‘𝑥) · (((𝑘 ∈ (0...𝐽) ↦ ((𝑆 D𝑛 (𝑦 ∈ 𝑋 ↦ ((𝐻‘𝑍)‘𝑦)))‘𝑘))‘(𝐽 − 𝑘))‘𝑥))) = Σ𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)(((!‘𝐽) / ∏𝑡 ∈ (𝑅 ∪ {𝑍})(!‘(𝑐‘𝑡))) · ∏𝑡 ∈ (𝑅 ∪ {𝑍})(((𝑆 D𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥))) |
698 | 697 | mpteq2dva 4672 |
. 2
⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ (0...𝐽)((𝐽C𝑘) · ((((𝑘 ∈ (0...𝐽) ↦ ((𝑆 D𝑛 (𝑦 ∈ 𝑋 ↦ ∏𝑡 ∈ 𝑅 ((𝐻‘𝑡)‘𝑦)))‘𝑘))‘𝑘)‘𝑥) · (((𝑘 ∈ (0...𝐽) ↦ ((𝑆 D𝑛 (𝑦 ∈ 𝑋 ↦ ((𝐻‘𝑍)‘𝑦)))‘𝑘))‘(𝐽 − 𝑘))‘𝑥)))) = (𝑥 ∈ 𝑋 ↦ Σ𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)(((!‘𝐽) / ∏𝑡 ∈ (𝑅 ∪ {𝑍})(!‘(𝑐‘𝑡))) · ∏𝑡 ∈ (𝑅 ∪ {𝑍})(((𝑆 D𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥)))) |
699 | 39, 215, 698 | 3eqtrd 2648 |
1
⊢ (𝜑 → ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ ∏𝑡 ∈ (𝑅 ∪ {𝑍})((𝐻‘𝑡)‘𝑥)))‘𝐽) = (𝑥 ∈ 𝑋 ↦ Σ𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)(((!‘𝐽) / ∏𝑡 ∈ (𝑅 ∪ {𝑍})(!‘(𝑐‘𝑡))) · ∏𝑡 ∈ (𝑅 ∪ {𝑍})(((𝑆 D𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥)))) |