dvnprod - Mathbox for Glauco Siliprandi

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Theorem dvnprod 45940

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 6822	. . . . . . . . . . 11 ⊢ (𝑢 = 𝑡 → (𝑑‘𝑢) = (𝑑‘𝑡))
9	8	cbvsumv 15603	. . . . . . . . . 10 ⊢ Σ𝑢 ∈ 𝑟 (𝑑‘𝑢) = Σ𝑡 ∈ 𝑟 (𝑑‘𝑡)
10	9	eqeq1i 2734	. . . . . . . . 9 ⊢ (Σ𝑢 ∈ 𝑟 (𝑑‘𝑢) = 𝑚 ↔ Σ𝑡 ∈ 𝑟 (𝑑‘𝑡) = 𝑚)
11	10	rabbii 3400	. . . . . . . 8 ⊢ {𝑑 ∈ ((0...𝑚) ↑_m 𝑟) ∣ Σ𝑢 ∈ 𝑟 (𝑑‘𝑢) = 𝑚} = {𝑑 ∈ ((0...𝑚) ↑_m 𝑟) ∣ Σ𝑡 ∈ 𝑟 (𝑑‘𝑡) = 𝑚}
12		fveq1 6821	. . . . . . . . . . 11 ⊢ (𝑑 = 𝑒 → (𝑑‘𝑡) = (𝑒‘𝑡))
13	12	sumeq2sdv 15610	. . . . . . . . . 10 ⊢ (𝑑 = 𝑒 → Σ𝑡 ∈ 𝑟 (𝑑‘𝑡) = Σ𝑡 ∈ 𝑟 (𝑒‘𝑡))
14	13	eqeq1d 2731	. . . . . . . . 9 ⊢ (𝑑 = 𝑒 → (Σ𝑡 ∈ 𝑟 (𝑑‘𝑡) = 𝑚 ↔ Σ𝑡 ∈ 𝑟 (𝑒‘𝑡) = 𝑚))
15	14	cbvrabv 3405	. . . . . . . 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 2752	. . . . . . 7 ⊢ {𝑑 ∈ ((0...𝑚) ↑_m 𝑟) ∣ Σ𝑢 ∈ 𝑟 (𝑑‘𝑢) = 𝑚} = {𝑒 ∈ ((0...𝑚) ↑_m 𝑟) ∣ Σ𝑡 ∈ 𝑟 (𝑒‘𝑡) = 𝑚}
17	16	mpteq2i 5188	. . . . . 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 2741	. . . . . . . . 9 ⊢ (𝑚 = 𝑛 → (Σ𝑡 ∈ 𝑟 (𝑒‘𝑡) = 𝑚 ↔ Σ𝑡 ∈ 𝑟 (𝑒‘𝑡) = 𝑛))
19	18	rabbidv 3402	. . . . . . . 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 7357	. . . . . . . . . 10 ⊢ (𝑚 = 𝑛 → (0...𝑚) = (0...𝑛))
21	20	oveq1d 7364	. . . . . . . . 9 ⊢ (𝑚 = 𝑛 → ((0...𝑚) ↑_m 𝑟) = ((0...𝑛) ↑_m 𝑟))
22		rabeq 3409	. . . . . . . . 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 2764	. . . . . . 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 5196	. . . . . 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 2752	. . . . 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 5188	. . . 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 15596	. . . . . . . . 9 ⊢ (𝑟 = 𝑠 → Σ𝑡 ∈ 𝑟 (𝑒‘𝑡) = Σ𝑡 ∈ 𝑠 (𝑒‘𝑡))
29	28	eqeq1d 2731	. . . . . . . 8 ⊢ (𝑟 = 𝑠 → (Σ𝑡 ∈ 𝑟 (𝑒‘𝑡) = 𝑛 ↔ Σ𝑡 ∈ 𝑠 (𝑒‘𝑡) = 𝑛))
30	29	rabbidv 3402	. . . . . . 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 7357	. . . . . . . 8 ⊢ (𝑟 = 𝑠 → ((0...𝑛) ↑_m 𝑟) = ((0...𝑛) ↑_m 𝑠))
32		rabeq 3409	. . . . . . . 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 2764	. . . . . 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 5186	. . . . 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 5196	. . . 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 2752	. . 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 6821	. . . . . . . 8 ⊢ (𝑐 = 𝑒 → (𝑐‘𝑡) = (𝑒‘𝑡))
40	39	sumeq2sdv 15610	. . . . . . 7 ⊢ (𝑐 = 𝑒 → Σ𝑡 ∈ 𝑇 (𝑐‘𝑡) = Σ𝑡 ∈ 𝑇 (𝑒‘𝑡))
41	40	eqeq1d 2731	. . . . . 6 ⊢ (𝑐 = 𝑒 → (Σ𝑡 ∈ 𝑇 (𝑐‘𝑡) = 𝑛 ↔ Σ𝑡 ∈ 𝑇 (𝑒‘𝑡) = 𝑛))
42	41	cbvrabv 3405	. . . . 5 ⊢ {𝑐 ∈ ((0...𝑛) ↑_m 𝑇) ∣ Σ𝑡 ∈ 𝑇 (𝑐‘𝑡) = 𝑛} = {𝑒 ∈ ((0...𝑛) ↑_m 𝑇) ∣ Σ𝑡 ∈ 𝑇 (𝑒‘𝑡) = 𝑛}
43	42	mpteq2i 5188	. . . 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 2752	. . 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 45939	. 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 6821	. . . . . . . . . 10 ⊢ (𝑒 = 𝑐 → (𝑒‘𝑡) = (𝑐‘𝑡))
47	46	fveq2d 6826	. . . . . . . . 9 ⊢ (𝑒 = 𝑐 → (!‘(𝑒‘𝑡)) = (!‘(𝑐‘𝑡)))
48	47	prodeq2ad 45583	. . . . . . . 8 ⊢ (𝑒 = 𝑐 → ∏𝑡 ∈ 𝑇 (!‘(𝑒‘𝑡)) = ∏𝑡 ∈ 𝑇 (!‘(𝑐‘𝑡)))
49	48	oveq2d 7365	. . . . . . 7 ⊢ (𝑒 = 𝑐 → ((!‘𝑁) / ∏𝑡 ∈ 𝑇 (!‘(𝑒‘𝑡))) = ((!‘𝑁) / ∏𝑡 ∈ 𝑇 (!‘(𝑐‘𝑡))))
50	46	fveq2d 6826	. . . . . . . . 9 ⊢ (𝑒 = 𝑐 → ((𝑆 D^𝑛 (𝐻‘𝑡))‘(𝑒‘𝑡)) = ((𝑆 D^𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡)))
51	50	fveq1d 6824	. . . . . . . 8 ⊢ (𝑒 = 𝑐 → (((𝑆 D^𝑛 (𝐻‘𝑡))‘(𝑒‘𝑡))‘𝑥) = (((𝑆 D^𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥))
52	51	prodeq2ad 45583	. . . . . . 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 7367	. . . . . 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 15603	. . . . 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 2729	. . . . 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 2752	. . . 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 5188	. . 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 2764	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 395 ∧ w3a 1086 = wceq 1540 ∈ wcel 2109 {crab 3394 𝒫 cpw 4551 {cpr 4579 ↦ cmpt 5173 ⟶wf 6478 ‘cfv 6482 (class class class)co 7349 ↑_m cmap 8753 Fincfn 8872 ℂcc 11007 ℝcr 11008 0cc0 11009 · cmul 11014 / cdiv 11777 ℕ₀cn0 12384 ...cfz 13410 !cfa 14180 Σcsu 15593 ∏cprod 15810 ↾_t crest 17324 TopOpenctopn 17325 ℂ_fldccnfld 21261 D^𝑛 cdvn 25763

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-rep 5218 ax-sep 5235 ax-nul 5245 ax-pow 5304 ax-pr 5371 ax-un 7671 ax-inf2 9537 ax-cnex 11065 ax-resscn 11066 ax-1cn 11067 ax-icn 11068 ax-addcl 11069 ax-addrcl 11070 ax-mulcl 11071 ax-mulrcl 11072 ax-mulcom 11073 ax-addass 11074 ax-mulass 11075 ax-distr 11076 ax-i2m1 11077 ax-1ne0 11078 ax-1rid 11079 ax-rnegex 11080 ax-rrecex 11081 ax-cnre 11082 ax-pre-lttri 11083 ax-pre-lttrn 11084 ax-pre-ltadd 11085 ax-pre-mulgt0 11086 ax-pre-sup 11087 ax-addf 11088

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-rmo 3343 df-reu 3344 df-rab 3395 df-v 3438 df-sbc 3743 df-csb 3852 df-dif 3906 df-un 3908 df-in 3910 df-ss 3920 df-pss 3923 df-nul 4285 df-if 4477 df-pw 4553 df-sn 4578 df-pr 4580 df-tp 4582 df-op 4584 df-uni 4859 df-int 4897 df-iun 4943 df-iin 4944 df-br 5093 df-opab 5155 df-mpt 5174 df-tr 5200 df-id 5514 df-eprel 5519 df-po 5527 df-so 5528 df-fr 5572 df-se 5573 df-we 5574 df-xp 5625 df-rel 5626 df-cnv 5627 df-co 5628 df-dm 5629 df-rn 5630 df-res 5631 df-ima 5632 df-pred 6249 df-ord 6310 df-on 6311 df-lim 6312 df-suc 6313 df-iota 6438 df-fun 6484 df-fn 6485 df-f 6486 df-f1 6487 df-fo 6488 df-f1o 6489 df-fv 6490 df-isom 6491 df-riota 7306 df-ov 7352 df-oprab 7353 df-mpo 7354 df-of 7613 df-om 7800 df-1st 7924 df-2nd 7925 df-supp 8094 df-frecs 8214 df-wrecs 8245 df-recs 8294 df-rdg 8332 df-1o 8388 df-2o 8389 df-er 8625 df-map 8755 df-pm 8756 df-ixp 8825 df-en 8873 df-dom 8874 df-sdom 8875 df-fin 8876 df-fsupp 9252 df-fi 9301 df-sup 9332 df-inf 9333 df-oi 9402 df-card 9835 df-pnf 11151 df-mnf 11152 df-xr 11153 df-ltxr 11154 df-le 11155 df-sub 11349 df-neg 11350 df-div 11778 df-nn 12129 df-2 12191 df-3 12192 df-4 12193 df-5 12194 df-6 12195 df-7 12196 df-8 12197 df-9 12198 df-n0 12385 df-z 12472 df-dec 12592 df-uz 12736 df-q 12850 df-rp 12894 df-xneg 13014 df-xadd 13015 df-xmul 13016 df-ico 13254 df-icc 13255 df-fz 13411 df-fzo 13558 df-seq 13909 df-exp 13969 df-fac 14181 df-bc 14210 df-hash 14238 df-cj 15006 df-re 15007 df-im 15008 df-sqrt 15142 df-abs 15143 df-clim 15395 df-sum 15594 df-prod 15811 df-struct 17058 df-sets 17075 df-slot 17093 df-ndx 17105 df-base 17121 df-ress 17142 df-plusg 17174 df-mulr 17175 df-starv 17176 df-sca 17177 df-vsca 17178 df-ip 17179 df-tset 17180 df-ple 17181 df-ds 17183 df-unif 17184 df-hom 17185 df-cco 17186 df-rest 17326 df-topn 17327 df-0g 17345 df-gsum 17346 df-topgen 17347 df-pt 17348 df-prds 17351 df-xrs 17406 df-qtop 17411 df-imas 17412 df-xps 17414 df-mre 17488 df-mrc 17489 df-acs 17491 df-mgm 18514 df-sgrp 18593 df-mnd 18609 df-submnd 18658 df-mulg 18947 df-cntz 19196 df-cmn 19661 df-psmet 21253 df-xmet 21254 df-met 21255 df-bl 21256 df-mopn 21257 df-fbas 21258 df-fg 21259 df-cnfld 21262 df-top 22779 df-topon 22796 df-topsp 22818 df-bases 22831 df-cld 22904 df-ntr 22905 df-cls 22906 df-nei 22983 df-lp 23021 df-perf 23022 df-cn 23112 df-cnp 23113 df-haus 23200 df-tx 23447 df-hmeo 23640 df-fil 23731 df-fm 23823 df-flim 23824 df-flf 23825 df-xms 24206 df-ms 24207 df-tms 24208 df-cncf 24769 df-limc 25765 df-dv 25766 df-dvn 25767

This theorem is referenced by: etransclem29 46254

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