dvnprod - Mathbox for Glauco Siliprandi

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Theorem dvnprod 44180

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 6842	. . . . . . . . . . 11 ⊢ (𝑢 = 𝑡 → (𝑑‘𝑢) = (𝑑‘𝑡))
9	8	cbvsumv 15581	. . . . . . . . . 10 ⊢ Σ𝑢 ∈ 𝑟 (𝑑‘𝑢) = Σ𝑡 ∈ 𝑟 (𝑑‘𝑡)
10	9	eqeq1i 2741	. . . . . . . . 9 ⊢ (Σ𝑢 ∈ 𝑟 (𝑑‘𝑢) = 𝑚 ↔ Σ𝑡 ∈ 𝑟 (𝑑‘𝑡) = 𝑚)
11	10	rabbii 3413	. . . . . . . 8 ⊢ {𝑑 ∈ ((0...𝑚) ↑_m 𝑟) ∣ Σ𝑢 ∈ 𝑟 (𝑑‘𝑢) = 𝑚} = {𝑑 ∈ ((0...𝑚) ↑_m 𝑟) ∣ Σ𝑡 ∈ 𝑟 (𝑑‘𝑡) = 𝑚}
12		fveq1 6841	. . . . . . . . . . 11 ⊢ (𝑑 = 𝑒 → (𝑑‘𝑡) = (𝑒‘𝑡))
13	12	sumeq2sdv 15589	. . . . . . . . . 10 ⊢ (𝑑 = 𝑒 → Σ𝑡 ∈ 𝑟 (𝑑‘𝑡) = Σ𝑡 ∈ 𝑟 (𝑒‘𝑡))
14	13	eqeq1d 2738	. . . . . . . . 9 ⊢ (𝑑 = 𝑒 → (Σ𝑡 ∈ 𝑟 (𝑑‘𝑡) = 𝑚 ↔ Σ𝑡 ∈ 𝑟 (𝑒‘𝑡) = 𝑚))
15	14	cbvrabv 3417	. . . . . . . 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 2764	. . . . . . 7 ⊢ {𝑑 ∈ ((0...𝑚) ↑_m 𝑟) ∣ Σ𝑢 ∈ 𝑟 (𝑑‘𝑢) = 𝑚} = {𝑒 ∈ ((0...𝑚) ↑_m 𝑟) ∣ Σ𝑡 ∈ 𝑟 (𝑒‘𝑡) = 𝑚}
17	16	mpteq2i 5210	. . . . . 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 2748	. . . . . . . . 9 ⊢ (𝑚 = 𝑛 → (Σ𝑡 ∈ 𝑟 (𝑒‘𝑡) = 𝑚 ↔ Σ𝑡 ∈ 𝑟 (𝑒‘𝑡) = 𝑛))
19	18	rabbidv 3415	. . . . . . . 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 3421	. . . . . . . . 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 2776	. . . . . . 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 5218	. . . . . 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 2764	. . . . 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 5210	. . . 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 15573	. . . . . . . . 9 ⊢ (𝑟 = 𝑠 → Σ𝑡 ∈ 𝑟 (𝑒‘𝑡) = Σ𝑡 ∈ 𝑠 (𝑒‘𝑡))
29	28	eqeq1d 2738	. . . . . . . 8 ⊢ (𝑟 = 𝑠 → (Σ𝑡 ∈ 𝑟 (𝑒‘𝑡) = 𝑛 ↔ Σ𝑡 ∈ 𝑠 (𝑒‘𝑡) = 𝑛))
30	29	rabbidv 3415	. . . . . . 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 3421	. . . . . . . 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 2776	. . . . . 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 5207	. . . . 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 5218	. . . 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 2764	. . 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 6841	. . . . . . . 8 ⊢ (𝑐 = 𝑒 → (𝑐‘𝑡) = (𝑒‘𝑡))
40	39	sumeq2sdv 15589	. . . . . . 7 ⊢ (𝑐 = 𝑒 → Σ𝑡 ∈ 𝑇 (𝑐‘𝑡) = Σ𝑡 ∈ 𝑇 (𝑒‘𝑡))
41	40	eqeq1d 2738	. . . . . 6 ⊢ (𝑐 = 𝑒 → (Σ𝑡 ∈ 𝑇 (𝑐‘𝑡) = 𝑛 ↔ Σ𝑡 ∈ 𝑇 (𝑒‘𝑡) = 𝑛))
42	41	cbvrabv 3417	. . . . 5 ⊢ {𝑐 ∈ ((0...𝑛) ↑_m 𝑇) ∣ Σ𝑡 ∈ 𝑇 (𝑐‘𝑡) = 𝑛} = {𝑒 ∈ ((0...𝑛) ↑_m 𝑇) ∣ Σ𝑡 ∈ 𝑇 (𝑒‘𝑡) = 𝑛}
43	42	mpteq2i 5210	. . . 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 2764	. . 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 44179	. 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 6841	. . . . . . . . . 10 ⊢ (𝑒 = 𝑐 → (𝑒‘𝑡) = (𝑐‘𝑡))
47	46	fveq2d 6846	. . . . . . . . 9 ⊢ (𝑒 = 𝑐 → (!‘(𝑒‘𝑡)) = (!‘(𝑐‘𝑡)))
48	47	prodeq2ad 43823	. . . . . . . 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 6846	. . . . . . . . 9 ⊢ (𝑒 = 𝑐 → ((𝑆 D^𝑛 (𝐻‘𝑡))‘(𝑒‘𝑡)) = ((𝑆 D^𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡)))
51	50	fveq1d 6844	. . . . . . . 8 ⊢ (𝑒 = 𝑐 → (((𝑆 D^𝑛 (𝐻‘𝑡))‘(𝑒‘𝑡))‘𝑥) = (((𝑆 D^𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥))
52	51	prodeq2ad 43823	. . . . . . 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 15581	. . . . 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 2736	. . . . 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 2764	. . . 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 5210	. . 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 2776	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 3407 𝒫 cpw 4560 {cpr 4588 ↦ cmpt 5188 ⟶wf 6492 ‘cfv 6496 (class class class)co 7357 ↑_m cmap 8765 Fincfn 8883 ℂcc 11049 ℝcr 11050 0cc0 11051 · cmul 11056 / cdiv 11812 ℕ₀cn0 12413 ...cfz 13424 !cfa 14173 Σcsu 15570 ∏cprod 15788 ↾_t crest 17302 TopOpenctopn 17303 ℂ_fldccnfld 20796 D^𝑛 cdvn 25228

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 2707 ax-rep 5242 ax-sep 5256 ax-nul 5263 ax-pow 5320 ax-pr 5384 ax-un 7672 ax-inf2 9577 ax-cnex 11107 ax-resscn 11108 ax-1cn 11109 ax-icn 11110 ax-addcl 11111 ax-addrcl 11112 ax-mulcl 11113 ax-mulrcl 11114 ax-mulcom 11115 ax-addass 11116 ax-mulass 11117 ax-distr 11118 ax-i2m1 11119 ax-1ne0 11120 ax-1rid 11121 ax-rnegex 11122 ax-rrecex 11123 ax-cnre 11124 ax-pre-lttri 11125 ax-pre-lttrn 11126 ax-pre-ltadd 11127 ax-pre-mulgt0 11128 ax-pre-sup 11129 ax-addf 11130 ax-mulf 11131

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 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3065 df-rex 3074 df-rmo 3353 df-reu 3354 df-rab 3408 df-v 3447 df-sbc 3740 df-csb 3856 df-dif 3913 df-un 3915 df-in 3917 df-ss 3927 df-pss 3929 df-nul 4283 df-if 4487 df-pw 4562 df-sn 4587 df-pr 4589 df-tp 4591 df-op 4593 df-uni 4866 df-int 4908 df-iun 4956 df-iin 4957 df-br 5106 df-opab 5168 df-mpt 5189 df-tr 5223 df-id 5531 df-eprel 5537 df-po 5545 df-so 5546 df-fr 5588 df-se 5589 df-we 5590 df-xp 5639 df-rel 5640 df-cnv 5641 df-co 5642 df-dm 5643 df-rn 5644 df-res 5645 df-ima 5646 df-pred 6253 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6498 df-fn 6499 df-f 6500 df-f1 6501 df-fo 6502 df-f1o 6503 df-fv 6504 df-isom 6505 df-riota 7313 df-ov 7360 df-oprab 7361 df-mpo 7362 df-of 7617 df-om 7803 df-1st 7921 df-2nd 7922 df-supp 8093 df-frecs 8212 df-wrecs 8243 df-recs 8317 df-rdg 8356 df-1o 8412 df-2o 8413 df-er 8648 df-map 8767 df-pm 8768 df-ixp 8836 df-en 8884 df-dom 8885 df-sdom 8886 df-fin 8887 df-fsupp 9306 df-fi 9347 df-sup 9378 df-inf 9379 df-oi 9446 df-card 9875 df-pnf 11191 df-mnf 11192 df-xr 11193 df-ltxr 11194 df-le 11195 df-sub 11387 df-neg 11388 df-div 11813 df-nn 12154 df-2 12216 df-3 12217 df-4 12218 df-5 12219 df-6 12220 df-7 12221 df-8 12222 df-9 12223 df-n0 12414 df-z 12500 df-dec 12619 df-uz 12764 df-q 12874 df-rp 12916 df-xneg 13033 df-xadd 13034 df-xmul 13035 df-ico 13270 df-icc 13271 df-fz 13425 df-fzo 13568 df-seq 13907 df-exp 13968 df-fac 14174 df-bc 14203 df-hash 14231 df-cj 14984 df-re 14985 df-im 14986 df-sqrt 15120 df-abs 15121 df-clim 15370 df-sum 15571 df-prod 15789 df-struct 17019 df-sets 17036 df-slot 17054 df-ndx 17066 df-base 17084 df-ress 17113 df-plusg 17146 df-mulr 17147 df-starv 17148 df-sca 17149 df-vsca 17150 df-ip 17151 df-tset 17152 df-ple 17153 df-ds 17155 df-unif 17156 df-hom 17157 df-cco 17158 df-rest 17304 df-topn 17305 df-0g 17323 df-gsum 17324 df-topgen 17325 df-pt 17326 df-prds 17329 df-xrs 17384 df-qtop 17389 df-imas 17390 df-xps 17392 df-mre 17466 df-mrc 17467 df-acs 17469 df-mgm 18497 df-sgrp 18546 df-mnd 18557 df-submnd 18602 df-mulg 18873 df-cntz 19097 df-cmn 19564 df-psmet 20788 df-xmet 20789 df-met 20790 df-bl 20791 df-mopn 20792 df-fbas 20793 df-fg 20794 df-cnfld 20797 df-top 22243 df-topon 22260 df-topsp 22282 df-bases 22296 df-cld 22370 df-ntr 22371 df-cls 22372 df-nei 22449 df-lp 22487 df-perf 22488 df-cn 22578 df-cnp 22579 df-haus 22666 df-tx 22913 df-hmeo 23106 df-fil 23197 df-fm 23289 df-flim 23290 df-flf 23291 df-xms 23673 df-ms 23674 df-tms 23675 df-cncf 24241 df-limc 25230 df-dv 25231 df-dvn 25232

This theorem is referenced by: etransclem29 44494

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