dvnprod - Mathbox for Glauco Siliprandi

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Theorem dvnprod 44651

Description: The multinomial formula for the 𝑁-th derivative of a finite product. (Contributed by Glauco Siliprandi, 5-Apr-2020.)

**Hypotheses**
Ref	Expression
dvnprod.s	⊢ (𝜑 → 𝑆 ∈ {ℝ, ℂ})
dvnprod.x	⊢ (𝜑 → 𝑋 ∈ ((TopOpen‘ℂ_fld) ↾_t 𝑆))
dvnprod.t	⊢ (𝜑 → 𝑇 ∈ Fin)
dvnprod.h	⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → (𝐻‘𝑡):𝑋⟶ℂ)
dvnprod.n	⊢ (𝜑 → 𝑁 ∈ ℕ₀)
dvnprod.dvnh	⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇 ∧ 𝑘 ∈ (0...𝑁)) → ((𝑆 D^𝑛 (𝐻‘𝑡))‘𝑘):𝑋⟶ℂ)
dvnprod.f	⊢ 𝐹 = (𝑥 ∈ 𝑋 ↦ ∏𝑡 ∈ 𝑇 ((𝐻‘𝑡)‘𝑥))
dvnprod.c	⊢ 𝐶 = (𝑛 ∈ ℕ₀ ↦ {𝑐 ∈ ((0...𝑛) ↑_m 𝑇) ∣ Σ𝑡 ∈ 𝑇 (𝑐‘𝑡) = 𝑛})

**Assertion**
Ref	Expression
dvnprod	⊢ (𝜑 → ((𝑆 D^𝑛 𝐹)‘𝑁) = (𝑥 ∈ 𝑋 ↦ Σ𝑐 ∈ (𝐶‘𝑁)(((!‘𝑁) / ∏𝑡 ∈ 𝑇 (!‘(𝑐‘𝑡))) · ∏𝑡 ∈ 𝑇 (((𝑆 D^𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥))))

Distinct variable groups: 𝐶,𝑐 𝐻,𝑐,𝑛,𝑡,𝑥 𝑘,𝐻,𝑛,𝑡,𝑥 𝑁,𝑐,𝑛,𝑡,𝑥 𝑘,𝑁 𝑆,𝑐,𝑛,𝑡,𝑥 𝑆,𝑘 𝑇,𝑐,𝑛,𝑡,𝑥 𝑇,𝑘 𝑘,𝑋,𝑛,𝑡,𝑥 𝜑,𝑘,𝑛,𝑡,𝑥 Allowed substitution hints: 𝜑(𝑐) 𝐶(𝑥,𝑡,𝑘,𝑛) 𝐹(𝑥,𝑡,𝑘,𝑛,𝑐) 𝑋(𝑐)

Proof of Theorem dvnprod Dummy variables 𝑒 𝑠 𝑟 𝑑 𝑚 𝑢 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		dvnprod.s	. . 3 ⊢ (𝜑 → 𝑆 ∈ {ℝ, ℂ})
2		dvnprod.x	. . 3 ⊢ (𝜑 → 𝑋 ∈ ((TopOpen‘ℂ_fld) ↾_t 𝑆))
3		dvnprod.t	. . 3 ⊢ (𝜑 → 𝑇 ∈ Fin)
4		dvnprod.h	. . 3 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → (𝐻‘𝑡):𝑋⟶ℂ)
5		dvnprod.n	. . 3 ⊢ (𝜑 → 𝑁 ∈ ℕ₀)
6		dvnprod.dvnh	. . 3 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇 ∧ 𝑘 ∈ (0...𝑁)) → ((𝑆 D^𝑛 (𝐻‘𝑡))‘𝑘):𝑋⟶ℂ)
7		dvnprod.f	. . 3 ⊢ 𝐹 = (𝑥 ∈ 𝑋 ↦ ∏𝑡 ∈ 𝑇 ((𝐻‘𝑡)‘𝑥))
8		fveq2 6888	. . . . . . . . . . 11 ⊢ (𝑢 = 𝑡 → (𝑑‘𝑢) = (𝑑‘𝑡))
9	8	cbvsumv 15638	. . . . . . . . . 10 ⊢ Σ𝑢 ∈ 𝑟 (𝑑‘𝑢) = Σ𝑡 ∈ 𝑟 (𝑑‘𝑡)
10	9	eqeq1i 2737	. . . . . . . . 9 ⊢ (Σ𝑢 ∈ 𝑟 (𝑑‘𝑢) = 𝑚 ↔ Σ𝑡 ∈ 𝑟 (𝑑‘𝑡) = 𝑚)
11	10	rabbii 3438	. . . . . . . 8 ⊢ {𝑑 ∈ ((0...𝑚) ↑_m 𝑟) ∣ Σ𝑢 ∈ 𝑟 (𝑑‘𝑢) = 𝑚} = {𝑑 ∈ ((0...𝑚) ↑_m 𝑟) ∣ Σ𝑡 ∈ 𝑟 (𝑑‘𝑡) = 𝑚}
12		fveq1 6887	. . . . . . . . . . 11 ⊢ (𝑑 = 𝑒 → (𝑑‘𝑡) = (𝑒‘𝑡))
13	12	sumeq2sdv 15646	. . . . . . . . . 10 ⊢ (𝑑 = 𝑒 → Σ𝑡 ∈ 𝑟 (𝑑‘𝑡) = Σ𝑡 ∈ 𝑟 (𝑒‘𝑡))
14	13	eqeq1d 2734	. . . . . . . . 9 ⊢ (𝑑 = 𝑒 → (Σ𝑡 ∈ 𝑟 (𝑑‘𝑡) = 𝑚 ↔ Σ𝑡 ∈ 𝑟 (𝑒‘𝑡) = 𝑚))
15	14	cbvrabv 3442	. . . . . . . 8 ⊢ {𝑑 ∈ ((0...𝑚) ↑_m 𝑟) ∣ Σ𝑡 ∈ 𝑟 (𝑑‘𝑡) = 𝑚} = {𝑒 ∈ ((0...𝑚) ↑_m 𝑟) ∣ Σ𝑡 ∈ 𝑟 (𝑒‘𝑡) = 𝑚}
16	11, 15	eqtri 2760	. . . . . . 7 ⊢ {𝑑 ∈ ((0...𝑚) ↑_m 𝑟) ∣ Σ𝑢 ∈ 𝑟 (𝑑‘𝑢) = 𝑚} = {𝑒 ∈ ((0...𝑚) ↑_m 𝑟) ∣ Σ𝑡 ∈ 𝑟 (𝑒‘𝑡) = 𝑚}
17	16	mpteq2i 5252	. . . . . 6 ⊢ (𝑚 ∈ ℕ₀ ↦ {𝑑 ∈ ((0...𝑚) ↑_m 𝑟) ∣ Σ𝑢 ∈ 𝑟 (𝑑‘𝑢) = 𝑚}) = (𝑚 ∈ ℕ₀ ↦ {𝑒 ∈ ((0...𝑚) ↑_m 𝑟) ∣ Σ𝑡 ∈ 𝑟 (𝑒‘𝑡) = 𝑚})
18		eqeq2 2744	. . . . . . . . 9 ⊢ (𝑚 = 𝑛 → (Σ𝑡 ∈ 𝑟 (𝑒‘𝑡) = 𝑚 ↔ Σ𝑡 ∈ 𝑟 (𝑒‘𝑡) = 𝑛))
19	18	rabbidv 3440	. . . . . . . 8 ⊢ (𝑚 = 𝑛 → {𝑒 ∈ ((0...𝑚) ↑_m 𝑟) ∣ Σ𝑡 ∈ 𝑟 (𝑒‘𝑡) = 𝑚} = {𝑒 ∈ ((0...𝑚) ↑_m 𝑟) ∣ Σ𝑡 ∈ 𝑟 (𝑒‘𝑡) = 𝑛})
20		oveq2 7413	. . . . . . . . . 10 ⊢ (𝑚 = 𝑛 → (0...𝑚) = (0...𝑛))
21	20	oveq1d 7420	. . . . . . . . 9 ⊢ (𝑚 = 𝑛 → ((0...𝑚) ↑_m 𝑟) = ((0...𝑛) ↑_m 𝑟))
22		rabeq 3446	. . . . . . . . 9 ⊢ (((0...𝑚) ↑_m 𝑟) = ((0...𝑛) ↑_m 𝑟) → {𝑒 ∈ ((0...𝑚) ↑_m 𝑟) ∣ Σ𝑡 ∈ 𝑟 (𝑒‘𝑡) = 𝑛} = {𝑒 ∈ ((0...𝑛) ↑_m 𝑟) ∣ Σ𝑡 ∈ 𝑟 (𝑒‘𝑡) = 𝑛})
23	21, 22	syl 17	. . . . . . . 8 ⊢ (𝑚 = 𝑛 → {𝑒 ∈ ((0...𝑚) ↑_m 𝑟) ∣ Σ𝑡 ∈ 𝑟 (𝑒‘𝑡) = 𝑛} = {𝑒 ∈ ((0...𝑛) ↑_m 𝑟) ∣ Σ𝑡 ∈ 𝑟 (𝑒‘𝑡) = 𝑛})
24	19, 23	eqtrd 2772	. . . . . . 7 ⊢ (𝑚 = 𝑛 → {𝑒 ∈ ((0...𝑚) ↑_m 𝑟) ∣ Σ𝑡 ∈ 𝑟 (𝑒‘𝑡) = 𝑚} = {𝑒 ∈ ((0...𝑛) ↑_m 𝑟) ∣ Σ𝑡 ∈ 𝑟 (𝑒‘𝑡) = 𝑛})
25	24	cbvmptv 5260	. . . . . 6 ⊢ (𝑚 ∈ ℕ₀ ↦ {𝑒 ∈ ((0...𝑚) ↑_m 𝑟) ∣ Σ𝑡 ∈ 𝑟 (𝑒‘𝑡) = 𝑚}) = (𝑛 ∈ ℕ₀ ↦ {𝑒 ∈ ((0...𝑛) ↑_m 𝑟) ∣ Σ𝑡 ∈ 𝑟 (𝑒‘𝑡) = 𝑛})
26	17, 25	eqtri 2760	. . . . 5 ⊢ (𝑚 ∈ ℕ₀ ↦ {𝑑 ∈ ((0...𝑚) ↑_m 𝑟) ∣ Σ𝑢 ∈ 𝑟 (𝑑‘𝑢) = 𝑚}) = (𝑛 ∈ ℕ₀ ↦ {𝑒 ∈ ((0...𝑛) ↑_m 𝑟) ∣ Σ𝑡 ∈ 𝑟 (𝑒‘𝑡) = 𝑛})
27	26	mpteq2i 5252	. . . 4 ⊢ (𝑟 ∈ 𝒫 𝑇 ↦ (𝑚 ∈ ℕ₀ ↦ {𝑑 ∈ ((0...𝑚) ↑_m 𝑟) ∣ Σ𝑢 ∈ 𝑟 (𝑑‘𝑢) = 𝑚})) = (𝑟 ∈ 𝒫 𝑇 ↦ (𝑛 ∈ ℕ₀ ↦ {𝑒 ∈ ((0...𝑛) ↑_m 𝑟) ∣ Σ𝑡 ∈ 𝑟 (𝑒‘𝑡) = 𝑛}))
28		sumeq1 15631	. . . . . . . . 9 ⊢ (𝑟 = 𝑠 → Σ𝑡 ∈ 𝑟 (𝑒‘𝑡) = Σ𝑡 ∈ 𝑠 (𝑒‘𝑡))
29	28	eqeq1d 2734	. . . . . . . 8 ⊢ (𝑟 = 𝑠 → (Σ𝑡 ∈ 𝑟 (𝑒‘𝑡) = 𝑛 ↔ Σ𝑡 ∈ 𝑠 (𝑒‘𝑡) = 𝑛))
30	29	rabbidv 3440	. . . . . . 7 ⊢ (𝑟 = 𝑠 → {𝑒 ∈ ((0...𝑛) ↑_m 𝑟) ∣ Σ𝑡 ∈ 𝑟 (𝑒‘𝑡) = 𝑛} = {𝑒 ∈ ((0...𝑛) ↑_m 𝑟) ∣ Σ𝑡 ∈ 𝑠 (𝑒‘𝑡) = 𝑛})
31		oveq2 7413	. . . . . . . 8 ⊢ (𝑟 = 𝑠 → ((0...𝑛) ↑_m 𝑟) = ((0...𝑛) ↑_m 𝑠))
32		rabeq 3446	. . . . . . . 8 ⊢ (((0...𝑛) ↑_m 𝑟) = ((0...𝑛) ↑_m 𝑠) → {𝑒 ∈ ((0...𝑛) ↑_m 𝑟) ∣ Σ𝑡 ∈ 𝑠 (𝑒‘𝑡) = 𝑛} = {𝑒 ∈ ((0...𝑛) ↑_m 𝑠) ∣ Σ𝑡 ∈ 𝑠 (𝑒‘𝑡) = 𝑛})
33	31, 32	syl 17	. . . . . . 7 ⊢ (𝑟 = 𝑠 → {𝑒 ∈ ((0...𝑛) ↑_m 𝑟) ∣ Σ𝑡 ∈ 𝑠 (𝑒‘𝑡) = 𝑛} = {𝑒 ∈ ((0...𝑛) ↑_m 𝑠) ∣ Σ𝑡 ∈ 𝑠 (𝑒‘𝑡) = 𝑛})
34	30, 33	eqtrd 2772	. . . . . 6 ⊢ (𝑟 = 𝑠 → {𝑒 ∈ ((0...𝑛) ↑_m 𝑟) ∣ Σ𝑡 ∈ 𝑟 (𝑒‘𝑡) = 𝑛} = {𝑒 ∈ ((0...𝑛) ↑_m 𝑠) ∣ Σ𝑡 ∈ 𝑠 (𝑒‘𝑡) = 𝑛})
35	34	mpteq2dv 5249	. . . . 5 ⊢ (𝑟 = 𝑠 → (𝑛 ∈ ℕ₀ ↦ {𝑒 ∈ ((0...𝑛) ↑_m 𝑟) ∣ Σ𝑡 ∈ 𝑟 (𝑒‘𝑡) = 𝑛}) = (𝑛 ∈ ℕ₀ ↦ {𝑒 ∈ ((0...𝑛) ↑_m 𝑠) ∣ Σ𝑡 ∈ 𝑠 (𝑒‘𝑡) = 𝑛}))
36	35	cbvmptv 5260	. . . 4 ⊢ (𝑟 ∈ 𝒫 𝑇 ↦ (𝑛 ∈ ℕ₀ ↦ {𝑒 ∈ ((0...𝑛) ↑_m 𝑟) ∣ Σ𝑡 ∈ 𝑟 (𝑒‘𝑡) = 𝑛})) = (𝑠 ∈ 𝒫 𝑇 ↦ (𝑛 ∈ ℕ₀ ↦ {𝑒 ∈ ((0...𝑛) ↑_m 𝑠) ∣ Σ𝑡 ∈ 𝑠 (𝑒‘𝑡) = 𝑛}))
37	27, 36	eqtri 2760	. . 3 ⊢ (𝑟 ∈ 𝒫 𝑇 ↦ (𝑚 ∈ ℕ₀ ↦ {𝑑 ∈ ((0...𝑚) ↑_m 𝑟) ∣ Σ𝑢 ∈ 𝑟 (𝑑‘𝑢) = 𝑚})) = (𝑠 ∈ 𝒫 𝑇 ↦ (𝑛 ∈ ℕ₀ ↦ {𝑒 ∈ ((0...𝑛) ↑_m 𝑠) ∣ Σ𝑡 ∈ 𝑠 (𝑒‘𝑡) = 𝑛}))
38		dvnprod.c	. . . 4 ⊢ 𝐶 = (𝑛 ∈ ℕ₀ ↦ {𝑐 ∈ ((0...𝑛) ↑_m 𝑇) ∣ Σ𝑡 ∈ 𝑇 (𝑐‘𝑡) = 𝑛})
39		fveq1 6887	. . . . . . . 8 ⊢ (𝑐 = 𝑒 → (𝑐‘𝑡) = (𝑒‘𝑡))
40	39	sumeq2sdv 15646	. . . . . . 7 ⊢ (𝑐 = 𝑒 → Σ𝑡 ∈ 𝑇 (𝑐‘𝑡) = Σ𝑡 ∈ 𝑇 (𝑒‘𝑡))
41	40	eqeq1d 2734	. . . . . 6 ⊢ (𝑐 = 𝑒 → (Σ𝑡 ∈ 𝑇 (𝑐‘𝑡) = 𝑛 ↔ Σ𝑡 ∈ 𝑇 (𝑒‘𝑡) = 𝑛))
42	41	cbvrabv 3442	. . . . 5 ⊢ {𝑐 ∈ ((0...𝑛) ↑_m 𝑇) ∣ Σ𝑡 ∈ 𝑇 (𝑐‘𝑡) = 𝑛} = {𝑒 ∈ ((0...𝑛) ↑_m 𝑇) ∣ Σ𝑡 ∈ 𝑇 (𝑒‘𝑡) = 𝑛}
43	42	mpteq2i 5252	. . . 4 ⊢ (𝑛 ∈ ℕ₀ ↦ {𝑐 ∈ ((0...𝑛) ↑_m 𝑇) ∣ Σ𝑡 ∈ 𝑇 (𝑐‘𝑡) = 𝑛}) = (𝑛 ∈ ℕ₀ ↦ {𝑒 ∈ ((0...𝑛) ↑_m 𝑇) ∣ Σ𝑡 ∈ 𝑇 (𝑒‘𝑡) = 𝑛})
44	38, 43	eqtri 2760	. . 3 ⊢ 𝐶 = (𝑛 ∈ ℕ₀ ↦ {𝑒 ∈ ((0...𝑛) ↑_m 𝑇) ∣ Σ𝑡 ∈ 𝑇 (𝑒‘𝑡) = 𝑛})
45	1, 2, 3, 4, 5, 6, 7, 37, 44	dvnprodlem3 44650	. 2 ⊢ (𝜑 → ((𝑆 D^𝑛 𝐹)‘𝑁) = (𝑥 ∈ 𝑋 ↦ Σ𝑒 ∈ (𝐶‘𝑁)(((!‘𝑁) / ∏𝑡 ∈ 𝑇 (!‘(𝑒‘𝑡))) · ∏𝑡 ∈ 𝑇 (((𝑆 D^𝑛 (𝐻‘𝑡))‘(𝑒‘𝑡))‘𝑥))))
46		fveq1 6887	. . . . . . . . . 10 ⊢ (𝑒 = 𝑐 → (𝑒‘𝑡) = (𝑐‘𝑡))
47	46	fveq2d 6892	. . . . . . . . 9 ⊢ (𝑒 = 𝑐 → (!‘(𝑒‘𝑡)) = (!‘(𝑐‘𝑡)))
48	47	prodeq2ad 44294	. . . . . . . 8 ⊢ (𝑒 = 𝑐 → ∏𝑡 ∈ 𝑇 (!‘(𝑒‘𝑡)) = ∏𝑡 ∈ 𝑇 (!‘(𝑐‘𝑡)))
49	48	oveq2d 7421	. . . . . . 7 ⊢ (𝑒 = 𝑐 → ((!‘𝑁) / ∏𝑡 ∈ 𝑇 (!‘(𝑒‘𝑡))) = ((!‘𝑁) / ∏𝑡 ∈ 𝑇 (!‘(𝑐‘𝑡))))
50	46	fveq2d 6892	. . . . . . . . 9 ⊢ (𝑒 = 𝑐 → ((𝑆 D^𝑛 (𝐻‘𝑡))‘(𝑒‘𝑡)) = ((𝑆 D^𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡)))
51	50	fveq1d 6890	. . . . . . . 8 ⊢ (𝑒 = 𝑐 → (((𝑆 D^𝑛 (𝐻‘𝑡))‘(𝑒‘𝑡))‘𝑥) = (((𝑆 D^𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥))
52	51	prodeq2ad 44294	. . . . . . 7 ⊢ (𝑒 = 𝑐 → ∏𝑡 ∈ 𝑇 (((𝑆 D^𝑛 (𝐻‘𝑡))‘(𝑒‘𝑡))‘𝑥) = ∏𝑡 ∈ 𝑇 (((𝑆 D^𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥))
53	49, 52	oveq12d 7423	. . . . . 6 ⊢ (𝑒 = 𝑐 → (((!‘𝑁) / ∏𝑡 ∈ 𝑇 (!‘(𝑒‘𝑡))) · ∏𝑡 ∈ 𝑇 (((𝑆 D^𝑛 (𝐻‘𝑡))‘(𝑒‘𝑡))‘𝑥)) = (((!‘𝑁) / ∏𝑡 ∈ 𝑇 (!‘(𝑐‘𝑡))) · ∏𝑡 ∈ 𝑇 (((𝑆 D^𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥)))
54	53	cbvsumv 15638	. . . . 5 ⊢ Σ𝑒 ∈ (𝐶‘𝑁)(((!‘𝑁) / ∏𝑡 ∈ 𝑇 (!‘(𝑒‘𝑡))) · ∏𝑡 ∈ 𝑇 (((𝑆 D^𝑛 (𝐻‘𝑡))‘(𝑒‘𝑡))‘𝑥)) = Σ𝑐 ∈ (𝐶‘𝑁)(((!‘𝑁) / ∏𝑡 ∈ 𝑇 (!‘(𝑐‘𝑡))) · ∏𝑡 ∈ 𝑇 (((𝑆 D^𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥))
55		eqid 2732	. . . . 5 ⊢ Σ𝑐 ∈ (𝐶‘𝑁)(((!‘𝑁) / ∏𝑡 ∈ 𝑇 (!‘(𝑐‘𝑡))) · ∏𝑡 ∈ 𝑇 (((𝑆 D^𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥)) = Σ𝑐 ∈ (𝐶‘𝑁)(((!‘𝑁) / ∏𝑡 ∈ 𝑇 (!‘(𝑐‘𝑡))) · ∏𝑡 ∈ 𝑇 (((𝑆 D^𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥))
56	54, 55	eqtri 2760	. . . 4 ⊢ Σ𝑒 ∈ (𝐶‘𝑁)(((!‘𝑁) / ∏𝑡 ∈ 𝑇 (!‘(𝑒‘𝑡))) · ∏𝑡 ∈ 𝑇 (((𝑆 D^𝑛 (𝐻‘𝑡))‘(𝑒‘𝑡))‘𝑥)) = Σ𝑐 ∈ (𝐶‘𝑁)(((!‘𝑁) / ∏𝑡 ∈ 𝑇 (!‘(𝑐‘𝑡))) · ∏𝑡 ∈ 𝑇 (((𝑆 D^𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥))
57	56	mpteq2i 5252	. . 3 ⊢ (𝑥 ∈ 𝑋 ↦ Σ𝑒 ∈ (𝐶‘𝑁)(((!‘𝑁) / ∏𝑡 ∈ 𝑇 (!‘(𝑒‘𝑡))) · ∏𝑡 ∈ 𝑇 (((𝑆 D^𝑛 (𝐻‘𝑡))‘(𝑒‘𝑡))‘𝑥))) = (𝑥 ∈ 𝑋 ↦ Σ𝑐 ∈ (𝐶‘𝑁)(((!‘𝑁) / ∏𝑡 ∈ 𝑇 (!‘(𝑐‘𝑡))) · ∏𝑡 ∈ 𝑇 (((𝑆 D^𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥)))
58	57	a1i 11	. 2 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ Σ𝑒 ∈ (𝐶‘𝑁)(((!‘𝑁) / ∏𝑡 ∈ 𝑇 (!‘(𝑒‘𝑡))) · ∏𝑡 ∈ 𝑇 (((𝑆 D^𝑛 (𝐻‘𝑡))‘(𝑒‘𝑡))‘𝑥))) = (𝑥 ∈ 𝑋 ↦ Σ𝑐 ∈ (𝐶‘𝑁)(((!‘𝑁) / ∏𝑡 ∈ 𝑇 (!‘(𝑐‘𝑡))) · ∏𝑡 ∈ 𝑇 (((𝑆 D^𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥))))
59	45, 58	eqtrd 2772	1 ⊢ (𝜑 → ((𝑆 D^𝑛 𝐹)‘𝑁) = (𝑥 ∈ 𝑋 ↦ Σ𝑐 ∈ (𝐶‘𝑁)(((!‘𝑁) / ∏𝑡 ∈ 𝑇 (!‘(𝑐‘𝑡))) · ∏𝑡 ∈ 𝑇 (((𝑆 D^𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥))))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 396 ∧ w3a 1087 = wceq 1541 ∈ wcel 2106 {crab 3432 𝒫 cpw 4601 {cpr 4629 ↦ cmpt 5230 ⟶wf 6536 ‘cfv 6540 (class class class)co 7405 ↑_m cmap 8816 Fincfn 8935 ℂcc 11104 ℝcr 11105 0cc0 11106 · cmul 11111 / cdiv 11867 ℕ₀cn0 12468 ...cfz 13480 !cfa 14229 Σcsu 15628 ∏cprod 15845 ↾_t crest 17362 TopOpenctopn 17363 ℂ_fldccnfld 20936 D^𝑛 cdvn 25372

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7721 ax-inf2 9632 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183 ax-pre-sup 11184 ax-addf 11185 ax-mulf 11186

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-tp 4632 df-op 4634 df-uni 4908 df-int 4950 df-iun 4998 df-iin 4999 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-se 5631 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6297 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-isom 6549 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-of 7666 df-om 7852 df-1st 7971 df-2nd 7972 df-supp 8143 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-1o 8462 df-2o 8463 df-er 8699 df-map 8818 df-pm 8819 df-ixp 8888 df-en 8936 df-dom 8937 df-sdom 8938 df-fin 8939 df-fsupp 9358 df-fi 9402 df-sup 9433 df-inf 9434 df-oi 9501 df-card 9930 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11442 df-neg 11443 df-div 11868 df-nn 12209 df-2 12271 df-3 12272 df-4 12273 df-5 12274 df-6 12275 df-7 12276 df-8 12277 df-9 12278 df-n0 12469 df-z 12555 df-dec 12674 df-uz 12819 df-q 12929 df-rp 12971 df-xneg 13088 df-xadd 13089 df-xmul 13090 df-ico 13326 df-icc 13327 df-fz 13481 df-fzo 13624 df-seq 13963 df-exp 14024 df-fac 14230 df-bc 14259 df-hash 14287 df-cj 15042 df-re 15043 df-im 15044 df-sqrt 15178 df-abs 15179 df-clim 15428 df-sum 15629 df-prod 15846 df-struct 17076 df-sets 17093 df-slot 17111 df-ndx 17123 df-base 17141 df-ress 17170 df-plusg 17206 df-mulr 17207 df-starv 17208 df-sca 17209 df-vsca 17210 df-ip 17211 df-tset 17212 df-ple 17213 df-ds 17215 df-unif 17216 df-hom 17217 df-cco 17218 df-rest 17364 df-topn 17365 df-0g 17383 df-gsum 17384 df-topgen 17385 df-pt 17386 df-prds 17389 df-xrs 17444 df-qtop 17449 df-imas 17450 df-xps 17452 df-mre 17526 df-mrc 17527 df-acs 17529 df-mgm 18557 df-sgrp 18606 df-mnd 18622 df-submnd 18668 df-mulg 18945 df-cntz 19175 df-cmn 19644 df-psmet 20928 df-xmet 20929 df-met 20930 df-bl 20931 df-mopn 20932 df-fbas 20933 df-fg 20934 df-cnfld 20937 df-top 22387 df-topon 22404 df-topsp 22426 df-bases 22440 df-cld 22514 df-ntr 22515 df-cls 22516 df-nei 22593 df-lp 22631 df-perf 22632 df-cn 22722 df-cnp 22723 df-haus 22810 df-tx 23057 df-hmeo 23250 df-fil 23341 df-fm 23433 df-flim 23434 df-flf 23435 df-xms 23817 df-ms 23818 df-tms 23819 df-cncf 24385 df-limc 25374 df-dv 25375 df-dvn 25376

This theorem is referenced by: etransclem29 44965

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