Step |
Hyp |
Ref |
Expression |
1 |
|
dvnprod.s |
⊢ ( 𝜑 → 𝑆 ∈ { ℝ , ℂ } ) |
2 |
|
dvnprod.x |
⊢ ( 𝜑 → 𝑋 ∈ ( ( TopOpen ‘ ℂfld ) ↾t 𝑆 ) ) |
3 |
|
dvnprod.t |
⊢ ( 𝜑 → 𝑇 ∈ Fin ) |
4 |
|
dvnprod.h |
⊢ ( ( 𝜑 ∧ 𝑡 ∈ 𝑇 ) → ( 𝐻 ‘ 𝑡 ) : 𝑋 ⟶ ℂ ) |
5 |
|
dvnprod.n |
⊢ ( 𝜑 → 𝑁 ∈ ℕ0 ) |
6 |
|
dvnprod.dvnh |
⊢ ( ( 𝜑 ∧ 𝑡 ∈ 𝑇 ∧ 𝑘 ∈ ( 0 ... 𝑁 ) ) → ( ( 𝑆 D𝑛 ( 𝐻 ‘ 𝑡 ) ) ‘ 𝑘 ) : 𝑋 ⟶ ℂ ) |
7 |
|
dvnprod.f |
⊢ 𝐹 = ( 𝑥 ∈ 𝑋 ↦ ∏ 𝑡 ∈ 𝑇 ( ( 𝐻 ‘ 𝑡 ) ‘ 𝑥 ) ) |
8 |
|
dvnprod.c |
⊢ 𝐶 = ( 𝑛 ∈ ℕ0 ↦ { 𝑐 ∈ ( ( 0 ... 𝑛 ) ↑m 𝑇 ) ∣ Σ 𝑡 ∈ 𝑇 ( 𝑐 ‘ 𝑡 ) = 𝑛 } ) |
9 |
|
fveq2 |
⊢ ( 𝑢 = 𝑡 → ( 𝑑 ‘ 𝑢 ) = ( 𝑑 ‘ 𝑡 ) ) |
10 |
9
|
cbvsumv |
⊢ Σ 𝑢 ∈ 𝑟 ( 𝑑 ‘ 𝑢 ) = Σ 𝑡 ∈ 𝑟 ( 𝑑 ‘ 𝑡 ) |
11 |
10
|
eqeq1i |
⊢ ( Σ 𝑢 ∈ 𝑟 ( 𝑑 ‘ 𝑢 ) = 𝑚 ↔ Σ 𝑡 ∈ 𝑟 ( 𝑑 ‘ 𝑡 ) = 𝑚 ) |
12 |
11
|
rabbii |
⊢ { 𝑑 ∈ ( ( 0 ... 𝑚 ) ↑m 𝑟 ) ∣ Σ 𝑢 ∈ 𝑟 ( 𝑑 ‘ 𝑢 ) = 𝑚 } = { 𝑑 ∈ ( ( 0 ... 𝑚 ) ↑m 𝑟 ) ∣ Σ 𝑡 ∈ 𝑟 ( 𝑑 ‘ 𝑡 ) = 𝑚 } |
13 |
|
fveq1 |
⊢ ( 𝑑 = 𝑒 → ( 𝑑 ‘ 𝑡 ) = ( 𝑒 ‘ 𝑡 ) ) |
14 |
13
|
sumeq2sdv |
⊢ ( 𝑑 = 𝑒 → Σ 𝑡 ∈ 𝑟 ( 𝑑 ‘ 𝑡 ) = Σ 𝑡 ∈ 𝑟 ( 𝑒 ‘ 𝑡 ) ) |
15 |
14
|
eqeq1d |
⊢ ( 𝑑 = 𝑒 → ( Σ 𝑡 ∈ 𝑟 ( 𝑑 ‘ 𝑡 ) = 𝑚 ↔ Σ 𝑡 ∈ 𝑟 ( 𝑒 ‘ 𝑡 ) = 𝑚 ) ) |
16 |
15
|
cbvrabv |
⊢ { 𝑑 ∈ ( ( 0 ... 𝑚 ) ↑m 𝑟 ) ∣ Σ 𝑡 ∈ 𝑟 ( 𝑑 ‘ 𝑡 ) = 𝑚 } = { 𝑒 ∈ ( ( 0 ... 𝑚 ) ↑m 𝑟 ) ∣ Σ 𝑡 ∈ 𝑟 ( 𝑒 ‘ 𝑡 ) = 𝑚 } |
17 |
12 16
|
eqtri |
⊢ { 𝑑 ∈ ( ( 0 ... 𝑚 ) ↑m 𝑟 ) ∣ Σ 𝑢 ∈ 𝑟 ( 𝑑 ‘ 𝑢 ) = 𝑚 } = { 𝑒 ∈ ( ( 0 ... 𝑚 ) ↑m 𝑟 ) ∣ Σ 𝑡 ∈ 𝑟 ( 𝑒 ‘ 𝑡 ) = 𝑚 } |
18 |
17
|
mpteq2i |
⊢ ( 𝑚 ∈ ℕ0 ↦ { 𝑑 ∈ ( ( 0 ... 𝑚 ) ↑m 𝑟 ) ∣ Σ 𝑢 ∈ 𝑟 ( 𝑑 ‘ 𝑢 ) = 𝑚 } ) = ( 𝑚 ∈ ℕ0 ↦ { 𝑒 ∈ ( ( 0 ... 𝑚 ) ↑m 𝑟 ) ∣ Σ 𝑡 ∈ 𝑟 ( 𝑒 ‘ 𝑡 ) = 𝑚 } ) |
19 |
|
eqeq2 |
⊢ ( 𝑚 = 𝑛 → ( Σ 𝑡 ∈ 𝑟 ( 𝑒 ‘ 𝑡 ) = 𝑚 ↔ Σ 𝑡 ∈ 𝑟 ( 𝑒 ‘ 𝑡 ) = 𝑛 ) ) |
20 |
19
|
rabbidv |
⊢ ( 𝑚 = 𝑛 → { 𝑒 ∈ ( ( 0 ... 𝑚 ) ↑m 𝑟 ) ∣ Σ 𝑡 ∈ 𝑟 ( 𝑒 ‘ 𝑡 ) = 𝑚 } = { 𝑒 ∈ ( ( 0 ... 𝑚 ) ↑m 𝑟 ) ∣ Σ 𝑡 ∈ 𝑟 ( 𝑒 ‘ 𝑡 ) = 𝑛 } ) |
21 |
|
oveq2 |
⊢ ( 𝑚 = 𝑛 → ( 0 ... 𝑚 ) = ( 0 ... 𝑛 ) ) |
22 |
21
|
oveq1d |
⊢ ( 𝑚 = 𝑛 → ( ( 0 ... 𝑚 ) ↑m 𝑟 ) = ( ( 0 ... 𝑛 ) ↑m 𝑟 ) ) |
23 |
|
rabeq |
⊢ ( ( ( 0 ... 𝑚 ) ↑m 𝑟 ) = ( ( 0 ... 𝑛 ) ↑m 𝑟 ) → { 𝑒 ∈ ( ( 0 ... 𝑚 ) ↑m 𝑟 ) ∣ Σ 𝑡 ∈ 𝑟 ( 𝑒 ‘ 𝑡 ) = 𝑛 } = { 𝑒 ∈ ( ( 0 ... 𝑛 ) ↑m 𝑟 ) ∣ Σ 𝑡 ∈ 𝑟 ( 𝑒 ‘ 𝑡 ) = 𝑛 } ) |
24 |
22 23
|
syl |
⊢ ( 𝑚 = 𝑛 → { 𝑒 ∈ ( ( 0 ... 𝑚 ) ↑m 𝑟 ) ∣ Σ 𝑡 ∈ 𝑟 ( 𝑒 ‘ 𝑡 ) = 𝑛 } = { 𝑒 ∈ ( ( 0 ... 𝑛 ) ↑m 𝑟 ) ∣ Σ 𝑡 ∈ 𝑟 ( 𝑒 ‘ 𝑡 ) = 𝑛 } ) |
25 |
20 24
|
eqtrd |
⊢ ( 𝑚 = 𝑛 → { 𝑒 ∈ ( ( 0 ... 𝑚 ) ↑m 𝑟 ) ∣ Σ 𝑡 ∈ 𝑟 ( 𝑒 ‘ 𝑡 ) = 𝑚 } = { 𝑒 ∈ ( ( 0 ... 𝑛 ) ↑m 𝑟 ) ∣ Σ 𝑡 ∈ 𝑟 ( 𝑒 ‘ 𝑡 ) = 𝑛 } ) |
26 |
25
|
cbvmptv |
⊢ ( 𝑚 ∈ ℕ0 ↦ { 𝑒 ∈ ( ( 0 ... 𝑚 ) ↑m 𝑟 ) ∣ Σ 𝑡 ∈ 𝑟 ( 𝑒 ‘ 𝑡 ) = 𝑚 } ) = ( 𝑛 ∈ ℕ0 ↦ { 𝑒 ∈ ( ( 0 ... 𝑛 ) ↑m 𝑟 ) ∣ Σ 𝑡 ∈ 𝑟 ( 𝑒 ‘ 𝑡 ) = 𝑛 } ) |
27 |
18 26
|
eqtri |
⊢ ( 𝑚 ∈ ℕ0 ↦ { 𝑑 ∈ ( ( 0 ... 𝑚 ) ↑m 𝑟 ) ∣ Σ 𝑢 ∈ 𝑟 ( 𝑑 ‘ 𝑢 ) = 𝑚 } ) = ( 𝑛 ∈ ℕ0 ↦ { 𝑒 ∈ ( ( 0 ... 𝑛 ) ↑m 𝑟 ) ∣ Σ 𝑡 ∈ 𝑟 ( 𝑒 ‘ 𝑡 ) = 𝑛 } ) |
28 |
27
|
mpteq2i |
⊢ ( 𝑟 ∈ 𝒫 𝑇 ↦ ( 𝑚 ∈ ℕ0 ↦ { 𝑑 ∈ ( ( 0 ... 𝑚 ) ↑m 𝑟 ) ∣ Σ 𝑢 ∈ 𝑟 ( 𝑑 ‘ 𝑢 ) = 𝑚 } ) ) = ( 𝑟 ∈ 𝒫 𝑇 ↦ ( 𝑛 ∈ ℕ0 ↦ { 𝑒 ∈ ( ( 0 ... 𝑛 ) ↑m 𝑟 ) ∣ Σ 𝑡 ∈ 𝑟 ( 𝑒 ‘ 𝑡 ) = 𝑛 } ) ) |
29 |
|
sumeq1 |
⊢ ( 𝑟 = 𝑠 → Σ 𝑡 ∈ 𝑟 ( 𝑒 ‘ 𝑡 ) = Σ 𝑡 ∈ 𝑠 ( 𝑒 ‘ 𝑡 ) ) |
30 |
29
|
eqeq1d |
⊢ ( 𝑟 = 𝑠 → ( Σ 𝑡 ∈ 𝑟 ( 𝑒 ‘ 𝑡 ) = 𝑛 ↔ Σ 𝑡 ∈ 𝑠 ( 𝑒 ‘ 𝑡 ) = 𝑛 ) ) |
31 |
30
|
rabbidv |
⊢ ( 𝑟 = 𝑠 → { 𝑒 ∈ ( ( 0 ... 𝑛 ) ↑m 𝑟 ) ∣ Σ 𝑡 ∈ 𝑟 ( 𝑒 ‘ 𝑡 ) = 𝑛 } = { 𝑒 ∈ ( ( 0 ... 𝑛 ) ↑m 𝑟 ) ∣ Σ 𝑡 ∈ 𝑠 ( 𝑒 ‘ 𝑡 ) = 𝑛 } ) |
32 |
|
oveq2 |
⊢ ( 𝑟 = 𝑠 → ( ( 0 ... 𝑛 ) ↑m 𝑟 ) = ( ( 0 ... 𝑛 ) ↑m 𝑠 ) ) |
33 |
|
rabeq |
⊢ ( ( ( 0 ... 𝑛 ) ↑m 𝑟 ) = ( ( 0 ... 𝑛 ) ↑m 𝑠 ) → { 𝑒 ∈ ( ( 0 ... 𝑛 ) ↑m 𝑟 ) ∣ Σ 𝑡 ∈ 𝑠 ( 𝑒 ‘ 𝑡 ) = 𝑛 } = { 𝑒 ∈ ( ( 0 ... 𝑛 ) ↑m 𝑠 ) ∣ Σ 𝑡 ∈ 𝑠 ( 𝑒 ‘ 𝑡 ) = 𝑛 } ) |
34 |
32 33
|
syl |
⊢ ( 𝑟 = 𝑠 → { 𝑒 ∈ ( ( 0 ... 𝑛 ) ↑m 𝑟 ) ∣ Σ 𝑡 ∈ 𝑠 ( 𝑒 ‘ 𝑡 ) = 𝑛 } = { 𝑒 ∈ ( ( 0 ... 𝑛 ) ↑m 𝑠 ) ∣ Σ 𝑡 ∈ 𝑠 ( 𝑒 ‘ 𝑡 ) = 𝑛 } ) |
35 |
31 34
|
eqtrd |
⊢ ( 𝑟 = 𝑠 → { 𝑒 ∈ ( ( 0 ... 𝑛 ) ↑m 𝑟 ) ∣ Σ 𝑡 ∈ 𝑟 ( 𝑒 ‘ 𝑡 ) = 𝑛 } = { 𝑒 ∈ ( ( 0 ... 𝑛 ) ↑m 𝑠 ) ∣ Σ 𝑡 ∈ 𝑠 ( 𝑒 ‘ 𝑡 ) = 𝑛 } ) |
36 |
35
|
mpteq2dv |
⊢ ( 𝑟 = 𝑠 → ( 𝑛 ∈ ℕ0 ↦ { 𝑒 ∈ ( ( 0 ... 𝑛 ) ↑m 𝑟 ) ∣ Σ 𝑡 ∈ 𝑟 ( 𝑒 ‘ 𝑡 ) = 𝑛 } ) = ( 𝑛 ∈ ℕ0 ↦ { 𝑒 ∈ ( ( 0 ... 𝑛 ) ↑m 𝑠 ) ∣ Σ 𝑡 ∈ 𝑠 ( 𝑒 ‘ 𝑡 ) = 𝑛 } ) ) |
37 |
36
|
cbvmptv |
⊢ ( 𝑟 ∈ 𝒫 𝑇 ↦ ( 𝑛 ∈ ℕ0 ↦ { 𝑒 ∈ ( ( 0 ... 𝑛 ) ↑m 𝑟 ) ∣ Σ 𝑡 ∈ 𝑟 ( 𝑒 ‘ 𝑡 ) = 𝑛 } ) ) = ( 𝑠 ∈ 𝒫 𝑇 ↦ ( 𝑛 ∈ ℕ0 ↦ { 𝑒 ∈ ( ( 0 ... 𝑛 ) ↑m 𝑠 ) ∣ Σ 𝑡 ∈ 𝑠 ( 𝑒 ‘ 𝑡 ) = 𝑛 } ) ) |
38 |
28 37
|
eqtri |
⊢ ( 𝑟 ∈ 𝒫 𝑇 ↦ ( 𝑚 ∈ ℕ0 ↦ { 𝑑 ∈ ( ( 0 ... 𝑚 ) ↑m 𝑟 ) ∣ Σ 𝑢 ∈ 𝑟 ( 𝑑 ‘ 𝑢 ) = 𝑚 } ) ) = ( 𝑠 ∈ 𝒫 𝑇 ↦ ( 𝑛 ∈ ℕ0 ↦ { 𝑒 ∈ ( ( 0 ... 𝑛 ) ↑m 𝑠 ) ∣ Σ 𝑡 ∈ 𝑠 ( 𝑒 ‘ 𝑡 ) = 𝑛 } ) ) |
39 |
|
fveq1 |
⊢ ( 𝑐 = 𝑒 → ( 𝑐 ‘ 𝑡 ) = ( 𝑒 ‘ 𝑡 ) ) |
40 |
39
|
sumeq2sdv |
⊢ ( 𝑐 = 𝑒 → Σ 𝑡 ∈ 𝑇 ( 𝑐 ‘ 𝑡 ) = Σ 𝑡 ∈ 𝑇 ( 𝑒 ‘ 𝑡 ) ) |
41 |
40
|
eqeq1d |
⊢ ( 𝑐 = 𝑒 → ( Σ 𝑡 ∈ 𝑇 ( 𝑐 ‘ 𝑡 ) = 𝑛 ↔ Σ 𝑡 ∈ 𝑇 ( 𝑒 ‘ 𝑡 ) = 𝑛 ) ) |
42 |
41
|
cbvrabv |
⊢ { 𝑐 ∈ ( ( 0 ... 𝑛 ) ↑m 𝑇 ) ∣ Σ 𝑡 ∈ 𝑇 ( 𝑐 ‘ 𝑡 ) = 𝑛 } = { 𝑒 ∈ ( ( 0 ... 𝑛 ) ↑m 𝑇 ) ∣ Σ 𝑡 ∈ 𝑇 ( 𝑒 ‘ 𝑡 ) = 𝑛 } |
43 |
42
|
mpteq2i |
⊢ ( 𝑛 ∈ ℕ0 ↦ { 𝑐 ∈ ( ( 0 ... 𝑛 ) ↑m 𝑇 ) ∣ Σ 𝑡 ∈ 𝑇 ( 𝑐 ‘ 𝑡 ) = 𝑛 } ) = ( 𝑛 ∈ ℕ0 ↦ { 𝑒 ∈ ( ( 0 ... 𝑛 ) ↑m 𝑇 ) ∣ Σ 𝑡 ∈ 𝑇 ( 𝑒 ‘ 𝑡 ) = 𝑛 } ) |
44 |
8 43
|
eqtri |
⊢ 𝐶 = ( 𝑛 ∈ ℕ0 ↦ { 𝑒 ∈ ( ( 0 ... 𝑛 ) ↑m 𝑇 ) ∣ Σ 𝑡 ∈ 𝑇 ( 𝑒 ‘ 𝑡 ) = 𝑛 } ) |
45 |
1 2 3 4 5 6 7 38 44
|
dvnprodlem3 |
⊢ ( 𝜑 → ( ( 𝑆 D𝑛 𝐹 ) ‘ 𝑁 ) = ( 𝑥 ∈ 𝑋 ↦ Σ 𝑒 ∈ ( 𝐶 ‘ 𝑁 ) ( ( ( ! ‘ 𝑁 ) / ∏ 𝑡 ∈ 𝑇 ( ! ‘ ( 𝑒 ‘ 𝑡 ) ) ) · ∏ 𝑡 ∈ 𝑇 ( ( ( 𝑆 D𝑛 ( 𝐻 ‘ 𝑡 ) ) ‘ ( 𝑒 ‘ 𝑡 ) ) ‘ 𝑥 ) ) ) ) |
46 |
|
fveq1 |
⊢ ( 𝑒 = 𝑐 → ( 𝑒 ‘ 𝑡 ) = ( 𝑐 ‘ 𝑡 ) ) |
47 |
46
|
fveq2d |
⊢ ( 𝑒 = 𝑐 → ( ! ‘ ( 𝑒 ‘ 𝑡 ) ) = ( ! ‘ ( 𝑐 ‘ 𝑡 ) ) ) |
48 |
47
|
prodeq2ad |
⊢ ( 𝑒 = 𝑐 → ∏ 𝑡 ∈ 𝑇 ( ! ‘ ( 𝑒 ‘ 𝑡 ) ) = ∏ 𝑡 ∈ 𝑇 ( ! ‘ ( 𝑐 ‘ 𝑡 ) ) ) |
49 |
48
|
oveq2d |
⊢ ( 𝑒 = 𝑐 → ( ( ! ‘ 𝑁 ) / ∏ 𝑡 ∈ 𝑇 ( ! ‘ ( 𝑒 ‘ 𝑡 ) ) ) = ( ( ! ‘ 𝑁 ) / ∏ 𝑡 ∈ 𝑇 ( ! ‘ ( 𝑐 ‘ 𝑡 ) ) ) ) |
50 |
46
|
fveq2d |
⊢ ( 𝑒 = 𝑐 → ( ( 𝑆 D𝑛 ( 𝐻 ‘ 𝑡 ) ) ‘ ( 𝑒 ‘ 𝑡 ) ) = ( ( 𝑆 D𝑛 ( 𝐻 ‘ 𝑡 ) ) ‘ ( 𝑐 ‘ 𝑡 ) ) ) |
51 |
50
|
fveq1d |
⊢ ( 𝑒 = 𝑐 → ( ( ( 𝑆 D𝑛 ( 𝐻 ‘ 𝑡 ) ) ‘ ( 𝑒 ‘ 𝑡 ) ) ‘ 𝑥 ) = ( ( ( 𝑆 D𝑛 ( 𝐻 ‘ 𝑡 ) ) ‘ ( 𝑐 ‘ 𝑡 ) ) ‘ 𝑥 ) ) |
52 |
51
|
prodeq2ad |
⊢ ( 𝑒 = 𝑐 → ∏ 𝑡 ∈ 𝑇 ( ( ( 𝑆 D𝑛 ( 𝐻 ‘ 𝑡 ) ) ‘ ( 𝑒 ‘ 𝑡 ) ) ‘ 𝑥 ) = ∏ 𝑡 ∈ 𝑇 ( ( ( 𝑆 D𝑛 ( 𝐻 ‘ 𝑡 ) ) ‘ ( 𝑐 ‘ 𝑡 ) ) ‘ 𝑥 ) ) |
53 |
49 52
|
oveq12d |
⊢ ( 𝑒 = 𝑐 → ( ( ( ! ‘ 𝑁 ) / ∏ 𝑡 ∈ 𝑇 ( ! ‘ ( 𝑒 ‘ 𝑡 ) ) ) · ∏ 𝑡 ∈ 𝑇 ( ( ( 𝑆 D𝑛 ( 𝐻 ‘ 𝑡 ) ) ‘ ( 𝑒 ‘ 𝑡 ) ) ‘ 𝑥 ) ) = ( ( ( ! ‘ 𝑁 ) / ∏ 𝑡 ∈ 𝑇 ( ! ‘ ( 𝑐 ‘ 𝑡 ) ) ) · ∏ 𝑡 ∈ 𝑇 ( ( ( 𝑆 D𝑛 ( 𝐻 ‘ 𝑡 ) ) ‘ ( 𝑐 ‘ 𝑡 ) ) ‘ 𝑥 ) ) ) |
54 |
53
|
cbvsumv |
⊢ Σ 𝑒 ∈ ( 𝐶 ‘ 𝑁 ) ( ( ( ! ‘ 𝑁 ) / ∏ 𝑡 ∈ 𝑇 ( ! ‘ ( 𝑒 ‘ 𝑡 ) ) ) · ∏ 𝑡 ∈ 𝑇 ( ( ( 𝑆 D𝑛 ( 𝐻 ‘ 𝑡 ) ) ‘ ( 𝑒 ‘ 𝑡 ) ) ‘ 𝑥 ) ) = Σ 𝑐 ∈ ( 𝐶 ‘ 𝑁 ) ( ( ( ! ‘ 𝑁 ) / ∏ 𝑡 ∈ 𝑇 ( ! ‘ ( 𝑐 ‘ 𝑡 ) ) ) · ∏ 𝑡 ∈ 𝑇 ( ( ( 𝑆 D𝑛 ( 𝐻 ‘ 𝑡 ) ) ‘ ( 𝑐 ‘ 𝑡 ) ) ‘ 𝑥 ) ) |
55 |
|
eqid |
⊢ Σ 𝑐 ∈ ( 𝐶 ‘ 𝑁 ) ( ( ( ! ‘ 𝑁 ) / ∏ 𝑡 ∈ 𝑇 ( ! ‘ ( 𝑐 ‘ 𝑡 ) ) ) · ∏ 𝑡 ∈ 𝑇 ( ( ( 𝑆 D𝑛 ( 𝐻 ‘ 𝑡 ) ) ‘ ( 𝑐 ‘ 𝑡 ) ) ‘ 𝑥 ) ) = Σ 𝑐 ∈ ( 𝐶 ‘ 𝑁 ) ( ( ( ! ‘ 𝑁 ) / ∏ 𝑡 ∈ 𝑇 ( ! ‘ ( 𝑐 ‘ 𝑡 ) ) ) · ∏ 𝑡 ∈ 𝑇 ( ( ( 𝑆 D𝑛 ( 𝐻 ‘ 𝑡 ) ) ‘ ( 𝑐 ‘ 𝑡 ) ) ‘ 𝑥 ) ) |
56 |
54 55
|
eqtri |
⊢ Σ 𝑒 ∈ ( 𝐶 ‘ 𝑁 ) ( ( ( ! ‘ 𝑁 ) / ∏ 𝑡 ∈ 𝑇 ( ! ‘ ( 𝑒 ‘ 𝑡 ) ) ) · ∏ 𝑡 ∈ 𝑇 ( ( ( 𝑆 D𝑛 ( 𝐻 ‘ 𝑡 ) ) ‘ ( 𝑒 ‘ 𝑡 ) ) ‘ 𝑥 ) ) = Σ 𝑐 ∈ ( 𝐶 ‘ 𝑁 ) ( ( ( ! ‘ 𝑁 ) / ∏ 𝑡 ∈ 𝑇 ( ! ‘ ( 𝑐 ‘ 𝑡 ) ) ) · ∏ 𝑡 ∈ 𝑇 ( ( ( 𝑆 D𝑛 ( 𝐻 ‘ 𝑡 ) ) ‘ ( 𝑐 ‘ 𝑡 ) ) ‘ 𝑥 ) ) |
57 |
56
|
mpteq2i |
⊢ ( 𝑥 ∈ 𝑋 ↦ Σ 𝑒 ∈ ( 𝐶 ‘ 𝑁 ) ( ( ( ! ‘ 𝑁 ) / ∏ 𝑡 ∈ 𝑇 ( ! ‘ ( 𝑒 ‘ 𝑡 ) ) ) · ∏ 𝑡 ∈ 𝑇 ( ( ( 𝑆 D𝑛 ( 𝐻 ‘ 𝑡 ) ) ‘ ( 𝑒 ‘ 𝑡 ) ) ‘ 𝑥 ) ) ) = ( 𝑥 ∈ 𝑋 ↦ Σ 𝑐 ∈ ( 𝐶 ‘ 𝑁 ) ( ( ( ! ‘ 𝑁 ) / ∏ 𝑡 ∈ 𝑇 ( ! ‘ ( 𝑐 ‘ 𝑡 ) ) ) · ∏ 𝑡 ∈ 𝑇 ( ( ( 𝑆 D𝑛 ( 𝐻 ‘ 𝑡 ) ) ‘ ( 𝑐 ‘ 𝑡 ) ) ‘ 𝑥 ) ) ) |
58 |
57
|
a1i |
⊢ ( 𝜑 → ( 𝑥 ∈ 𝑋 ↦ Σ 𝑒 ∈ ( 𝐶 ‘ 𝑁 ) ( ( ( ! ‘ 𝑁 ) / ∏ 𝑡 ∈ 𝑇 ( ! ‘ ( 𝑒 ‘ 𝑡 ) ) ) · ∏ 𝑡 ∈ 𝑇 ( ( ( 𝑆 D𝑛 ( 𝐻 ‘ 𝑡 ) ) ‘ ( 𝑒 ‘ 𝑡 ) ) ‘ 𝑥 ) ) ) = ( 𝑥 ∈ 𝑋 ↦ Σ 𝑐 ∈ ( 𝐶 ‘ 𝑁 ) ( ( ( ! ‘ 𝑁 ) / ∏ 𝑡 ∈ 𝑇 ( ! ‘ ( 𝑐 ‘ 𝑡 ) ) ) · ∏ 𝑡 ∈ 𝑇 ( ( ( 𝑆 D𝑛 ( 𝐻 ‘ 𝑡 ) ) ‘ ( 𝑐 ‘ 𝑡 ) ) ‘ 𝑥 ) ) ) ) |
59 |
45 58
|
eqtrd |
⊢ ( 𝜑 → ( ( 𝑆 D𝑛 𝐹 ) ‘ 𝑁 ) = ( 𝑥 ∈ 𝑋 ↦ Σ 𝑐 ∈ ( 𝐶 ‘ 𝑁 ) ( ( ( ! ‘ 𝑁 ) / ∏ 𝑡 ∈ 𝑇 ( ! ‘ ( 𝑐 ‘ 𝑡 ) ) ) · ∏ 𝑡 ∈ 𝑇 ( ( ( 𝑆 D𝑛 ( 𝐻 ‘ 𝑡 ) ) ‘ ( 𝑐 ‘ 𝑡 ) ) ‘ 𝑥 ) ) ) ) |