dvnprod - Mathbox for Glauco Siliprandi

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Theorem dvnprod 46400

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 6828	. . . . . . . . . . 11 ⊢ (𝑢 = 𝑡 → (𝑑‘𝑢) = (𝑑‘𝑡))
9	8	cbvsumv 15650	. . . . . . . . . 10 ⊢ Σ𝑢 ∈ 𝑟 (𝑑‘𝑢) = Σ𝑡 ∈ 𝑟 (𝑑‘𝑡)
10	9	eqeq1i 2744	. . . . . . . . 9 ⊢ (Σ𝑢 ∈ 𝑟 (𝑑‘𝑢) = 𝑚 ↔ Σ𝑡 ∈ 𝑟 (𝑑‘𝑡) = 𝑚)
11	10	rabbii 3396	. . . . . . . 8 ⊢ {𝑑 ∈ ((0...𝑚) ↑_m 𝑟) ∣ Σ𝑢 ∈ 𝑟 (𝑑‘𝑢) = 𝑚} = {𝑑 ∈ ((0...𝑚) ↑_m 𝑟) ∣ Σ𝑡 ∈ 𝑟 (𝑑‘𝑡) = 𝑚}
12		fveq1 6827	. . . . . . . . . . 11 ⊢ (𝑑 = 𝑒 → (𝑑‘𝑡) = (𝑒‘𝑡))
13	12	sumeq2sdv 15657	. . . . . . . . . 10 ⊢ (𝑑 = 𝑒 → Σ𝑡 ∈ 𝑟 (𝑑‘𝑡) = Σ𝑡 ∈ 𝑟 (𝑒‘𝑡))
14	13	eqeq1d 2741	. . . . . . . . 9 ⊢ (𝑑 = 𝑒 → (Σ𝑡 ∈ 𝑟 (𝑑‘𝑡) = 𝑚 ↔ Σ𝑡 ∈ 𝑟 (𝑒‘𝑡) = 𝑚))
15	14	cbvrabv 3401	. . . . . . . 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 2762	. . . . . . 7 ⊢ {𝑑 ∈ ((0...𝑚) ↑_m 𝑟) ∣ Σ𝑢 ∈ 𝑟 (𝑑‘𝑢) = 𝑚} = {𝑒 ∈ ((0...𝑚) ↑_m 𝑟) ∣ Σ𝑡 ∈ 𝑟 (𝑒‘𝑡) = 𝑚}
17	16	mpteq2i 5169	. . . . . 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 2751	. . . . . . . . 9 ⊢ (𝑚 = 𝑛 → (Σ𝑡 ∈ 𝑟 (𝑒‘𝑡) = 𝑚 ↔ Σ𝑡 ∈ 𝑟 (𝑒‘𝑡) = 𝑛))
19	18	rabbidv 3398	. . . . . . . 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 7365	. . . . . . . . . 10 ⊢ (𝑚 = 𝑛 → (0...𝑚) = (0...𝑛))
21	20	oveq1d 7372	. . . . . . . . 9 ⊢ (𝑚 = 𝑛 → ((0...𝑚) ↑_m 𝑟) = ((0...𝑛) ↑_m 𝑟))
22		rabeq 3405	. . . . . . . . 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 2774	. . . . . . 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 5177	. . . . . 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 2762	. . . . 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 5169	. . . 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 15643	. . . . . . . . 9 ⊢ (𝑟 = 𝑠 → Σ𝑡 ∈ 𝑟 (𝑒‘𝑡) = Σ𝑡 ∈ 𝑠 (𝑒‘𝑡))
29	28	eqeq1d 2741	. . . . . . . 8 ⊢ (𝑟 = 𝑠 → (Σ𝑡 ∈ 𝑟 (𝑒‘𝑡) = 𝑛 ↔ Σ𝑡 ∈ 𝑠 (𝑒‘𝑡) = 𝑛))
30	29	rabbidv 3398	. . . . . . 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 7365	. . . . . . . 8 ⊢ (𝑟 = 𝑠 → ((0...𝑛) ↑_m 𝑟) = ((0...𝑛) ↑_m 𝑠))
32		rabeq 3405	. . . . . . . 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 2774	. . . . . 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 5167	. . . . 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 5177	. . . 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 2762	. . 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 6827	. . . . . . . 8 ⊢ (𝑐 = 𝑒 → (𝑐‘𝑡) = (𝑒‘𝑡))
40	39	sumeq2sdv 15657	. . . . . . 7 ⊢ (𝑐 = 𝑒 → Σ𝑡 ∈ 𝑇 (𝑐‘𝑡) = Σ𝑡 ∈ 𝑇 (𝑒‘𝑡))
41	40	eqeq1d 2741	. . . . . 6 ⊢ (𝑐 = 𝑒 → (Σ𝑡 ∈ 𝑇 (𝑐‘𝑡) = 𝑛 ↔ Σ𝑡 ∈ 𝑇 (𝑒‘𝑡) = 𝑛))
42	41	cbvrabv 3401	. . . . 5 ⊢ {𝑐 ∈ ((0...𝑛) ↑_m 𝑇) ∣ Σ𝑡 ∈ 𝑇 (𝑐‘𝑡) = 𝑛} = {𝑒 ∈ ((0...𝑛) ↑_m 𝑇) ∣ Σ𝑡 ∈ 𝑇 (𝑒‘𝑡) = 𝑛}
43	42	mpteq2i 5169	. . . 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 2762	. . 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 46399	. 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 6827	. . . . . . . . . 10 ⊢ (𝑒 = 𝑐 → (𝑒‘𝑡) = (𝑐‘𝑡))
47	46	fveq2d 6832	. . . . . . . . 9 ⊢ (𝑒 = 𝑐 → (!‘(𝑒‘𝑡)) = (!‘(𝑐‘𝑡)))
48	47	prodeq2ad 46045	. . . . . . . 8 ⊢ (𝑒 = 𝑐 → ∏𝑡 ∈ 𝑇 (!‘(𝑒‘𝑡)) = ∏𝑡 ∈ 𝑇 (!‘(𝑐‘𝑡)))
49	48	oveq2d 7373	. . . . . . 7 ⊢ (𝑒 = 𝑐 → ((!‘𝑁) / ∏𝑡 ∈ 𝑇 (!‘(𝑒‘𝑡))) = ((!‘𝑁) / ∏𝑡 ∈ 𝑇 (!‘(𝑐‘𝑡))))
50	46	fveq2d 6832	. . . . . . . . 9 ⊢ (𝑒 = 𝑐 → ((𝑆 D^𝑛 (𝐻‘𝑡))‘(𝑒‘𝑡)) = ((𝑆 D^𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡)))
51	50	fveq1d 6830	. . . . . . . 8 ⊢ (𝑒 = 𝑐 → (((𝑆 D^𝑛 (𝐻‘𝑡))‘(𝑒‘𝑡))‘𝑥) = (((𝑆 D^𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥))
52	51	prodeq2ad 46045	. . . . . . 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 7375	. . . . . 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 15650	. . . . 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 2739	. . . . 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 2762	. . . 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 5169	. . 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 2774	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 1092 = wceq 1547 ∈ wcel 2119 {crab 3391 𝒫 cpw 4530 {cpr 4558 ↦ cmpt 5154 ⟶wf 6482 ‘cfv 6486 (class class class)co 7357 ↑_m cmap 8764 Fincfn 8884 ℂcc 11028 ℝcr 11029 0cc0 11030 · cmul 11035 / cdiv 11799 ℕ₀cn0 12429 ...cfz 13453 !cfa 14227 Σcsu 15640 ∏cprod 15860 ↾_t crest 17375 TopOpenctopn 17376 ℂ_fldccnfld 21348 D^𝑛 cdvn 25850

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1974 ax-7 2015 ax-8 2121 ax-9 2129 ax-10 2152 ax-11 2168 ax-12 2189 ax-ext 2711 ax-rep 5200 ax-sep 5219 ax-nul 5229 ax-pow 5295 ax-pr 5363 ax-un 7679 ax-inf2 9554 ax-cnex 11086 ax-resscn 11087 ax-1cn 11088 ax-icn 11089 ax-addcl 11090 ax-addrcl 11091 ax-mulcl 11092 ax-mulrcl 11093 ax-mulcom 11094 ax-addass 11095 ax-mulass 11096 ax-distr 11097 ax-i2m1 11098 ax-1ne0 11099 ax-1rid 11100 ax-rnegex 11101 ax-rrecex 11102 ax-cnre 11103 ax-pre-lttri 11104 ax-pre-lttrn 11105 ax-pre-ltadd 11106 ax-pre-mulgt0 11107 ax-pre-sup 11108 ax-addf 11109

This theorem depends on definitions: df-bi 208 df-an 397 df-or 854 df-3or 1093 df-3an 1094 df-tru 1550 df-fal 1560 df-ex 1787 df-nf 1791 df-sb 2074 df-mo 2543 df-eu 2573 df-clab 2718 df-cleq 2731 df-clel 2814 df-nfc 2888 df-ne 2935 df-nel 3039 df-ral 3054 df-rex 3064 df-rmo 3344 df-reu 3345 df-rab 3392 df-v 3433 df-sbc 3724 df-csb 3832 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-pss 3903 df-nul 4263 df-if 4456 df-pw 4532 df-sn 4557 df-pr 4559 df-tp 4561 df-op 4563 df-uni 4840 df-int 4879 df-iun 4924 df-iin 4925 df-br 5074 df-opab 5136 df-mpt 5155 df-tr 5181 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 6253 df-ord 6314 df-on 6315 df-lim 6316 df-suc 6317 df-iota 6442 df-fun 6488 df-fn 6489 df-f 6490 df-f1 6491 df-fo 6492 df-f1o 6493 df-fv 6494 df-isom 6495 df-riota 7314 df-ov 7360 df-oprab 7361 df-mpo 7362 df-of 7621 df-om 7808 df-1st 7932 df-2nd 7933 df-supp 8102 df-frecs 8222 df-wrecs 8253 df-recs 8302 df-rdg 8340 df-1o 8396 df-2o 8397 df-er 8634 df-map 8766 df-pm 8767 df-ixp 8837 df-en 8885 df-dom 8886 df-sdom 8887 df-fin 8888 df-fsupp 9266 df-fi 9315 df-sup 9346 df-inf 9347 df-oi 9416 df-card 9855 df-pnf 11173 df-mnf 11174 df-xr 11175 df-ltxr 11176 df-le 11177 df-sub 11371 df-neg 11372 df-div 11800 df-nn 12167 df-2 12236 df-3 12237 df-4 12238 df-5 12239 df-6 12240 df-7 12241 df-8 12242 df-9 12243 df-n0 12430 df-z 12517 df-dec 12637 df-uz 12781 df-q 12891 df-rp 12935 df-xneg 13055 df-xadd 13056 df-xmul 13057 df-ico 13296 df-icc 13297 df-fz 13454 df-fzo 13601 df-seq 13956 df-exp 14016 df-fac 14228 df-bc 14257 df-hash 14285 df-cj 15053 df-re 15054 df-im 15055 df-sqrt 15189 df-abs 15190 df-clim 15442 df-sum 15641 df-prod 15861 df-struct 17109 df-sets 17126 df-slot 17144 df-ndx 17156 df-base 17172 df-ress 17193 df-plusg 17225 df-mulr 17226 df-starv 17227 df-sca 17228 df-vsca 17229 df-ip 17230 df-tset 17231 df-ple 17232 df-ds 17234 df-unif 17235 df-hom 17236 df-cco 17237 df-rest 17377 df-topn 17378 df-0g 17396 df-gsum 17397 df-topgen 17398 df-pt 17399 df-prds 17402 df-xrs 17458 df-qtop 17463 df-imas 17464 df-xps 17466 df-mre 17540 df-mrc 17541 df-acs 17543 df-mgm 18600 df-sgrp 18679 df-mnd 18695 df-submnd 18744 df-mulg 19036 df-cntz 19284 df-cmn 19749 df-psmet 21340 df-xmet 21341 df-met 21342 df-bl 21343 df-mopn 21344 df-fbas 21345 df-fg 21346 df-cnfld 21349 df-top 22878 df-topon 22895 df-topsp 22917 df-bases 22930 df-cld 23003 df-ntr 23004 df-cls 23005 df-nei 23082 df-lp 23120 df-perf 23121 df-cn 23211 df-cnp 23212 df-haus 23299 df-tx 23546 df-hmeo 23739 df-fil 23830 df-fm 23922 df-flim 23923 df-flf 23924 df-xms 24304 df-ms 24305 df-tms 24306 df-cncf 24864 df-limc 25852 df-dv 25853 df-dvn 25854

This theorem is referenced by: etransclem29 46714

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