etransclem31 - Mathbox for Glauco Siliprandi

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Theorem etransclem31 46186

Description: The 𝑁-th derivative of 𝐻 applied to 𝑌. (Contributed by Glauco Siliprandi, 5-Apr-2020.)

**Hypotheses**
Ref	Expression
etransclem31.s	⊢ (𝜑 → 𝑆 ∈ {ℝ, ℂ})
etransclem31.x	⊢ (𝜑 → 𝑋 ∈ ((TopOpen‘ℂ_fld) ↾_t 𝑆))
etransclem31.p	⊢ (𝜑 → 𝑃 ∈ ℕ)
etransclem31.m	⊢ (𝜑 → 𝑀 ∈ ℕ₀)
etransclem31.f	⊢ 𝐹 = (𝑥 ∈ 𝑋 ↦ ((𝑥↑(𝑃 − 1)) · ∏𝑗 ∈ (1...𝑀)((𝑥 − 𝑗)↑𝑃)))
etransclem31.n	⊢ (𝜑 → 𝑁 ∈ ℕ₀)
etransclem31.h	⊢ 𝐻 = (𝑗 ∈ (0...𝑀) ↦ (𝑥 ∈ 𝑋 ↦ ((𝑥 − 𝑗)↑if(𝑗 = 0, (𝑃 − 1), 𝑃))))
etransclem31.c	⊢ 𝐶 = (𝑛 ∈ ℕ₀ ↦ {𝑐 ∈ ((0...𝑛) ↑_m (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐‘𝑗) = 𝑛})
etransclem31.y	⊢ (𝜑 → 𝑌 ∈ 𝑋)

**Assertion**
Ref	Expression
etransclem31	⊢ (𝜑 → (((𝑆 D^𝑛 𝐹)‘𝑁)‘𝑌) = Σ𝑐 ∈ (𝐶‘𝑁)(((!‘𝑁) / ∏𝑗 ∈ (0...𝑀)(!‘(𝑐‘𝑗))) · (if((𝑃 − 1) < (𝑐‘0), 0, (((!‘(𝑃 − 1)) / (!‘((𝑃 − 1) − (𝑐‘0)))) · (𝑌↑((𝑃 − 1) − (𝑐‘0))))) · ∏𝑗 ∈ (1...𝑀)if(𝑃 < (𝑐‘𝑗), 0, (((!‘𝑃) / (!‘(𝑃 − (𝑐‘𝑗)))) · ((𝑌 − 𝑗)↑(𝑃 − (𝑐‘𝑗))))))))

Distinct variable groups: 𝐶,𝑐,𝑗,𝑥 𝐻,𝑐,𝑗,𝑛,𝑥 𝑀,𝑐,𝑗,𝑥,𝑛 𝑁,𝑐,𝑗,𝑥,𝑛 𝑃,𝑗,𝑥 𝑆,𝑐,𝑗,𝑛,𝑥 𝑗,𝑋,𝑥,𝑛 𝑌,𝑐,𝑗,𝑥 𝜑,𝑐,𝑗,𝑥,𝑛 Allowed substitution hints: 𝐶(𝑛) 𝑃(𝑛,𝑐) 𝐹(𝑥,𝑗,𝑛,𝑐) 𝑋(𝑐) 𝑌(𝑛)

Proof of Theorem etransclem31 Dummy variables 𝑘 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		etransclem31.s	. . . 4 ⊢ (𝜑 → 𝑆 ∈ {ℝ, ℂ})
2		etransclem31.x	. . . 4 ⊢ (𝜑 → 𝑋 ∈ ((TopOpen‘ℂ_fld) ↾_t 𝑆))
3		etransclem31.p	. . . 4 ⊢ (𝜑 → 𝑃 ∈ ℕ)
4		etransclem31.m	. . . 4 ⊢ (𝜑 → 𝑀 ∈ ℕ₀)
5		etransclem31.f	. . . 4 ⊢ 𝐹 = (𝑥 ∈ 𝑋 ↦ ((𝑥↑(𝑃 − 1)) · ∏𝑗 ∈ (1...𝑀)((𝑥 − 𝑗)↑𝑃)))
6		etransclem31.n	. . . 4 ⊢ (𝜑 → 𝑁 ∈ ℕ₀)
7		etransclem31.h	. . . 4 ⊢ 𝐻 = (𝑗 ∈ (0...𝑀) ↦ (𝑥 ∈ 𝑋 ↦ ((𝑥 − 𝑗)↑if(𝑗 = 0, (𝑃 − 1), 𝑃))))
8		etransclem31.c	. . . 4 ⊢ 𝐶 = (𝑛 ∈ ℕ₀ ↦ {𝑐 ∈ ((0...𝑛) ↑_m (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐‘𝑗) = 𝑛})
9	1, 2, 3, 4, 5, 6, 7, 8	etransclem30 46185	. . 3 ⊢ (𝜑 → ((𝑆 D^𝑛 𝐹)‘𝑁) = (𝑥 ∈ 𝑋 ↦ Σ𝑐 ∈ (𝐶‘𝑁)(((!‘𝑁) / ∏𝑗 ∈ (0...𝑀)(!‘(𝑐‘𝑗))) · ∏𝑗 ∈ (0...𝑀)(((𝑆 D^𝑛 (𝐻‘𝑗))‘(𝑐‘𝑗))‘𝑥))))
10		fveq2 6920	. . . . . . 7 ⊢ (𝑥 = 𝑌 → (((𝑆 D^𝑛 (𝐻‘𝑗))‘(𝑐‘𝑗))‘𝑥) = (((𝑆 D^𝑛 (𝐻‘𝑗))‘(𝑐‘𝑗))‘𝑌))
11	10	prodeq2ad 45513	. . . . . 6 ⊢ (𝑥 = 𝑌 → ∏𝑗 ∈ (0...𝑀)(((𝑆 D^𝑛 (𝐻‘𝑗))‘(𝑐‘𝑗))‘𝑥) = ∏𝑗 ∈ (0...𝑀)(((𝑆 D^𝑛 (𝐻‘𝑗))‘(𝑐‘𝑗))‘𝑌))
12	11	oveq2d 7464	. . . . 5 ⊢ (𝑥 = 𝑌 → (((!‘𝑁) / ∏𝑗 ∈ (0...𝑀)(!‘(𝑐‘𝑗))) · ∏𝑗 ∈ (0...𝑀)(((𝑆 D^𝑛 (𝐻‘𝑗))‘(𝑐‘𝑗))‘𝑥)) = (((!‘𝑁) / ∏𝑗 ∈ (0...𝑀)(!‘(𝑐‘𝑗))) · ∏𝑗 ∈ (0...𝑀)(((𝑆 D^𝑛 (𝐻‘𝑗))‘(𝑐‘𝑗))‘𝑌)))
13	12	sumeq2sdv 15751	. . . 4 ⊢ (𝑥 = 𝑌 → Σ𝑐 ∈ (𝐶‘𝑁)(((!‘𝑁) / ∏𝑗 ∈ (0...𝑀)(!‘(𝑐‘𝑗))) · ∏𝑗 ∈ (0...𝑀)(((𝑆 D^𝑛 (𝐻‘𝑗))‘(𝑐‘𝑗))‘𝑥)) = Σ𝑐 ∈ (𝐶‘𝑁)(((!‘𝑁) / ∏𝑗 ∈ (0...𝑀)(!‘(𝑐‘𝑗))) · ∏𝑗 ∈ (0...𝑀)(((𝑆 D^𝑛 (𝐻‘𝑗))‘(𝑐‘𝑗))‘𝑌)))
14	13	adantl 481	. . 3 ⊢ ((𝜑 ∧ 𝑥 = 𝑌) → Σ𝑐 ∈ (𝐶‘𝑁)(((!‘𝑁) / ∏𝑗 ∈ (0...𝑀)(!‘(𝑐‘𝑗))) · ∏𝑗 ∈ (0...𝑀)(((𝑆 D^𝑛 (𝐻‘𝑗))‘(𝑐‘𝑗))‘𝑥)) = Σ𝑐 ∈ (𝐶‘𝑁)(((!‘𝑁) / ∏𝑗 ∈ (0...𝑀)(!‘(𝑐‘𝑗))) · ∏𝑗 ∈ (0...𝑀)(((𝑆 D^𝑛 (𝐻‘𝑗))‘(𝑐‘𝑗))‘𝑌)))
15		etransclem31.y	. . 3 ⊢ (𝜑 → 𝑌 ∈ 𝑋)
16	8, 6	etransclem16 46171	. . . 4 ⊢ (𝜑 → (𝐶‘𝑁) ∈ Fin)
17	6	faccld 14333	. . . . . . . 8 ⊢ (𝜑 → (!‘𝑁) ∈ ℕ)
18	17	nncnd 12309	. . . . . . 7 ⊢ (𝜑 → (!‘𝑁) ∈ ℂ)
19	18	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐶‘𝑁)) → (!‘𝑁) ∈ ℂ)
20		fzfid 14024	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐶‘𝑁)) → (0...𝑀) ∈ Fin)
21		fzssnn0 45232	. . . . . . . . . 10 ⊢ (0...𝑁) ⊆ ℕ₀
22		ssrab2 4103	. . . . . . . . . . . . . 14 ⊢ {𝑐 ∈ ((0...𝑁) ↑_m (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐‘𝑗) = 𝑁} ⊆ ((0...𝑁) ↑_m (0...𝑀))
23		simpr 484	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐶‘𝑁)) → 𝑐 ∈ (𝐶‘𝑁))
24	8, 6	etransclem12 46167	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝐶‘𝑁) = {𝑐 ∈ ((0...𝑁) ↑_m (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐‘𝑗) = 𝑁})
25	24	adantr 480	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐶‘𝑁)) → (𝐶‘𝑁) = {𝑐 ∈ ((0...𝑁) ↑_m (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐‘𝑗) = 𝑁})
26	23, 25	eleqtrd 2846	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐶‘𝑁)) → 𝑐 ∈ {𝑐 ∈ ((0...𝑁) ↑_m (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐‘𝑗) = 𝑁})
27	22, 26	sselid 4006	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐶‘𝑁)) → 𝑐 ∈ ((0...𝑁) ↑_m (0...𝑀)))
28	27	adantr 480	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑐 ∈ (𝐶‘𝑁)) ∧ 𝑗 ∈ (0...𝑀)) → 𝑐 ∈ ((0...𝑁) ↑_m (0...𝑀)))
29		elmapi 8907	. . . . . . . . . . . 12 ⊢ (𝑐 ∈ ((0...𝑁) ↑_m (0...𝑀)) → 𝑐:(0...𝑀)⟶(0...𝑁))
30	28, 29	syl 17	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑐 ∈ (𝐶‘𝑁)) ∧ 𝑗 ∈ (0...𝑀)) → 𝑐:(0...𝑀)⟶(0...𝑁))
31		simpr 484	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑐 ∈ (𝐶‘𝑁)) ∧ 𝑗 ∈ (0...𝑀)) → 𝑗 ∈ (0...𝑀))
32	30, 31	ffvelcdmd 7119	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑐 ∈ (𝐶‘𝑁)) ∧ 𝑗 ∈ (0...𝑀)) → (𝑐‘𝑗) ∈ (0...𝑁))
33	21, 32	sselid 4006	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑐 ∈ (𝐶‘𝑁)) ∧ 𝑗 ∈ (0...𝑀)) → (𝑐‘𝑗) ∈ ℕ₀)
34	33	faccld 14333	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑐 ∈ (𝐶‘𝑁)) ∧ 𝑗 ∈ (0...𝑀)) → (!‘(𝑐‘𝑗)) ∈ ℕ)
35	34	nncnd 12309	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑐 ∈ (𝐶‘𝑁)) ∧ 𝑗 ∈ (0...𝑀)) → (!‘(𝑐‘𝑗)) ∈ ℂ)
36	20, 35	fprodcl 16000	. . . . . 6 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐶‘𝑁)) → ∏𝑗 ∈ (0...𝑀)(!‘(𝑐‘𝑗)) ∈ ℂ)
37	34	nnne0d 12343	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑐 ∈ (𝐶‘𝑁)) ∧ 𝑗 ∈ (0...𝑀)) → (!‘(𝑐‘𝑗)) ≠ 0)
38	20, 35, 37	fprodn0 16027	. . . . . 6 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐶‘𝑁)) → ∏𝑗 ∈ (0...𝑀)(!‘(𝑐‘𝑗)) ≠ 0)
39	19, 36, 38	divcld 12070	. . . . 5 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐶‘𝑁)) → ((!‘𝑁) / ∏𝑗 ∈ (0...𝑀)(!‘(𝑐‘𝑗))) ∈ ℂ)
40	1	ad2antrr 725	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑐 ∈ (𝐶‘𝑁)) ∧ 𝑗 ∈ (0...𝑀)) → 𝑆 ∈ {ℝ, ℂ})
41	2	ad2antrr 725	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑐 ∈ (𝐶‘𝑁)) ∧ 𝑗 ∈ (0...𝑀)) → 𝑋 ∈ ((TopOpen‘ℂ_fld) ↾_t 𝑆))
42	3	ad2antrr 725	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑐 ∈ (𝐶‘𝑁)) ∧ 𝑗 ∈ (0...𝑀)) → 𝑃 ∈ ℕ)
43		etransclem5 46160	. . . . . . . . 9 ⊢ (𝑗 ∈ (0...𝑀) ↦ (𝑥 ∈ 𝑋 ↦ ((𝑥 − 𝑗)↑if(𝑗 = 0, (𝑃 − 1), 𝑃)))) = (𝑘 ∈ (0...𝑀) ↦ (𝑦 ∈ 𝑋 ↦ ((𝑦 − 𝑘)↑if(𝑘 = 0, (𝑃 − 1), 𝑃))))
44	7, 43	eqtri 2768	. . . . . . . 8 ⊢ 𝐻 = (𝑘 ∈ (0...𝑀) ↦ (𝑦 ∈ 𝑋 ↦ ((𝑦 − 𝑘)↑if(𝑘 = 0, (𝑃 − 1), 𝑃))))
45	40, 41, 42, 44, 31, 33	etransclem20 46175	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑐 ∈ (𝐶‘𝑁)) ∧ 𝑗 ∈ (0...𝑀)) → ((𝑆 D^𝑛 (𝐻‘𝑗))‘(𝑐‘𝑗)):𝑋⟶ℂ)
46	15	ad2antrr 725	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑐 ∈ (𝐶‘𝑁)) ∧ 𝑗 ∈ (0...𝑀)) → 𝑌 ∈ 𝑋)
47	45, 46	ffvelcdmd 7119	. . . . . 6 ⊢ (((𝜑 ∧ 𝑐 ∈ (𝐶‘𝑁)) ∧ 𝑗 ∈ (0...𝑀)) → (((𝑆 D^𝑛 (𝐻‘𝑗))‘(𝑐‘𝑗))‘𝑌) ∈ ℂ)
48	20, 47	fprodcl 16000	. . . . 5 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐶‘𝑁)) → ∏𝑗 ∈ (0...𝑀)(((𝑆 D^𝑛 (𝐻‘𝑗))‘(𝑐‘𝑗))‘𝑌) ∈ ℂ)
49	39, 48	mulcld 11310	. . . 4 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐶‘𝑁)) → (((!‘𝑁) / ∏𝑗 ∈ (0...𝑀)(!‘(𝑐‘𝑗))) · ∏𝑗 ∈ (0...𝑀)(((𝑆 D^𝑛 (𝐻‘𝑗))‘(𝑐‘𝑗))‘𝑌)) ∈ ℂ)
50	16, 49	fsumcl 15781	. . 3 ⊢ (𝜑 → Σ𝑐 ∈ (𝐶‘𝑁)(((!‘𝑁) / ∏𝑗 ∈ (0...𝑀)(!‘(𝑐‘𝑗))) · ∏𝑗 ∈ (0...𝑀)(((𝑆 D^𝑛 (𝐻‘𝑗))‘(𝑐‘𝑗))‘𝑌)) ∈ ℂ)
51	9, 14, 15, 50	fvmptd 7036	. 2 ⊢ (𝜑 → (((𝑆 D^𝑛 𝐹)‘𝑁)‘𝑌) = Σ𝑐 ∈ (𝐶‘𝑁)(((!‘𝑁) / ∏𝑗 ∈ (0...𝑀)(!‘(𝑐‘𝑗))) · ∏𝑗 ∈ (0...𝑀)(((𝑆 D^𝑛 (𝐻‘𝑗))‘(𝑐‘𝑗))‘𝑌)))
52	40, 41, 42, 44, 31, 33, 46	etransclem21 46176	. . . . . 6 ⊢ (((𝜑 ∧ 𝑐 ∈ (𝐶‘𝑁)) ∧ 𝑗 ∈ (0...𝑀)) → (((𝑆 D^𝑛 (𝐻‘𝑗))‘(𝑐‘𝑗))‘𝑌) = if(if(𝑗 = 0, (𝑃 − 1), 𝑃) < (𝑐‘𝑗), 0, (((!‘if(𝑗 = 0, (𝑃 − 1), 𝑃)) / (!‘(if(𝑗 = 0, (𝑃 − 1), 𝑃) − (𝑐‘𝑗)))) · ((𝑌 − 𝑗)↑(if(𝑗 = 0, (𝑃 − 1), 𝑃) − (𝑐‘𝑗))))))
53	52	prodeq2dv 15970	. . . . 5 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐶‘𝑁)) → ∏𝑗 ∈ (0...𝑀)(((𝑆 D^𝑛 (𝐻‘𝑗))‘(𝑐‘𝑗))‘𝑌) = ∏𝑗 ∈ (0...𝑀)if(if(𝑗 = 0, (𝑃 − 1), 𝑃) < (𝑐‘𝑗), 0, (((!‘if(𝑗 = 0, (𝑃 − 1), 𝑃)) / (!‘(if(𝑗 = 0, (𝑃 − 1), 𝑃) − (𝑐‘𝑗)))) · ((𝑌 − 𝑗)↑(if(𝑗 = 0, (𝑃 − 1), 𝑃) − (𝑐‘𝑗))))))
54		nn0uz 12945	. . . . . . . 8 ⊢ ℕ₀ = (ℤ_≥‘0)
55	4, 54	eleqtrdi 2854	. . . . . . 7 ⊢ (𝜑 → 𝑀 ∈ (ℤ_≥‘0))
56	55	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐶‘𝑁)) → 𝑀 ∈ (ℤ_≥‘0))
57	52, 47	eqeltrrd 2845	. . . . . 6 ⊢ (((𝜑 ∧ 𝑐 ∈ (𝐶‘𝑁)) ∧ 𝑗 ∈ (0...𝑀)) → if(if(𝑗 = 0, (𝑃 − 1), 𝑃) < (𝑐‘𝑗), 0, (((!‘if(𝑗 = 0, (𝑃 − 1), 𝑃)) / (!‘(if(𝑗 = 0, (𝑃 − 1), 𝑃) − (𝑐‘𝑗)))) · ((𝑌 − 𝑗)↑(if(𝑗 = 0, (𝑃 − 1), 𝑃) − (𝑐‘𝑗))))) ∈ ℂ)
58		iftrue 4554	. . . . . . . 8 ⊢ (𝑗 = 0 → if(𝑗 = 0, (𝑃 − 1), 𝑃) = (𝑃 − 1))
59		fveq2 6920	. . . . . . . 8 ⊢ (𝑗 = 0 → (𝑐‘𝑗) = (𝑐‘0))
60	58, 59	breq12d 5179	. . . . . . 7 ⊢ (𝑗 = 0 → (if(𝑗 = 0, (𝑃 − 1), 𝑃) < (𝑐‘𝑗) ↔ (𝑃 − 1) < (𝑐‘0)))
61	58	fveq2d 6924	. . . . . . . . 9 ⊢ (𝑗 = 0 → (!‘if(𝑗 = 0, (𝑃 − 1), 𝑃)) = (!‘(𝑃 − 1)))
62	58, 59	oveq12d 7466	. . . . . . . . . 10 ⊢ (𝑗 = 0 → (if(𝑗 = 0, (𝑃 − 1), 𝑃) − (𝑐‘𝑗)) = ((𝑃 − 1) − (𝑐‘0)))
63	62	fveq2d 6924	. . . . . . . . 9 ⊢ (𝑗 = 0 → (!‘(if(𝑗 = 0, (𝑃 − 1), 𝑃) − (𝑐‘𝑗))) = (!‘((𝑃 − 1) − (𝑐‘0))))
64	61, 63	oveq12d 7466	. . . . . . . 8 ⊢ (𝑗 = 0 → ((!‘if(𝑗 = 0, (𝑃 − 1), 𝑃)) / (!‘(if(𝑗 = 0, (𝑃 − 1), 𝑃) − (𝑐‘𝑗)))) = ((!‘(𝑃 − 1)) / (!‘((𝑃 − 1) − (𝑐‘0)))))
65		oveq2 7456	. . . . . . . . 9 ⊢ (𝑗 = 0 → (𝑌 − 𝑗) = (𝑌 − 0))
66	65, 62	oveq12d 7466	. . . . . . . 8 ⊢ (𝑗 = 0 → ((𝑌 − 𝑗)↑(if(𝑗 = 0, (𝑃 − 1), 𝑃) − (𝑐‘𝑗))) = ((𝑌 − 0)↑((𝑃 − 1) − (𝑐‘0))))
67	64, 66	oveq12d 7466	. . . . . . 7 ⊢ (𝑗 = 0 → (((!‘if(𝑗 = 0, (𝑃 − 1), 𝑃)) / (!‘(if(𝑗 = 0, (𝑃 − 1), 𝑃) − (𝑐‘𝑗)))) · ((𝑌 − 𝑗)↑(if(𝑗 = 0, (𝑃 − 1), 𝑃) − (𝑐‘𝑗)))) = (((!‘(𝑃 − 1)) / (!‘((𝑃 − 1) − (𝑐‘0)))) · ((𝑌 − 0)↑((𝑃 − 1) − (𝑐‘0)))))
68	60, 67	ifbieq2d 4574	. . . . . 6 ⊢ (𝑗 = 0 → if(if(𝑗 = 0, (𝑃 − 1), 𝑃) < (𝑐‘𝑗), 0, (((!‘if(𝑗 = 0, (𝑃 − 1), 𝑃)) / (!‘(if(𝑗 = 0, (𝑃 − 1), 𝑃) − (𝑐‘𝑗)))) · ((𝑌 − 𝑗)↑(if(𝑗 = 0, (𝑃 − 1), 𝑃) − (𝑐‘𝑗))))) = if((𝑃 − 1) < (𝑐‘0), 0, (((!‘(𝑃 − 1)) / (!‘((𝑃 − 1) − (𝑐‘0)))) · ((𝑌 − 0)↑((𝑃 − 1) − (𝑐‘0))))))
69	56, 57, 68	fprod1p 16016	. . . . 5 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐶‘𝑁)) → ∏𝑗 ∈ (0...𝑀)if(if(𝑗 = 0, (𝑃 − 1), 𝑃) < (𝑐‘𝑗), 0, (((!‘if(𝑗 = 0, (𝑃 − 1), 𝑃)) / (!‘(if(𝑗 = 0, (𝑃 − 1), 𝑃) − (𝑐‘𝑗)))) · ((𝑌 − 𝑗)↑(if(𝑗 = 0, (𝑃 − 1), 𝑃) − (𝑐‘𝑗))))) = (if((𝑃 − 1) < (𝑐‘0), 0, (((!‘(𝑃 − 1)) / (!‘((𝑃 − 1) − (𝑐‘0)))) · ((𝑌 − 0)↑((𝑃 − 1) − (𝑐‘0))))) · ∏𝑗 ∈ ((0 + 1)...𝑀)if(if(𝑗 = 0, (𝑃 − 1), 𝑃) < (𝑐‘𝑗), 0, (((!‘if(𝑗 = 0, (𝑃 − 1), 𝑃)) / (!‘(if(𝑗 = 0, (𝑃 − 1), 𝑃) − (𝑐‘𝑗)))) · ((𝑌 − 𝑗)↑(if(𝑗 = 0, (𝑃 − 1), 𝑃) − (𝑐‘𝑗)))))))
70	1, 2	dvdmsscn 45857	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑋 ⊆ ℂ)
71	70, 15	sseldd 4009	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑌 ∈ ℂ)
72	71	subid1d 11636	. . . . . . . . . 10 ⊢ (𝜑 → (𝑌 − 0) = 𝑌)
73	72	oveq1d 7463	. . . . . . . . 9 ⊢ (𝜑 → ((𝑌 − 0)↑((𝑃 − 1) − (𝑐‘0))) = (𝑌↑((𝑃 − 1) − (𝑐‘0))))
74	73	oveq2d 7464	. . . . . . . 8 ⊢ (𝜑 → (((!‘(𝑃 − 1)) / (!‘((𝑃 − 1) − (𝑐‘0)))) · ((𝑌 − 0)↑((𝑃 − 1) − (𝑐‘0)))) = (((!‘(𝑃 − 1)) / (!‘((𝑃 − 1) − (𝑐‘0)))) · (𝑌↑((𝑃 − 1) − (𝑐‘0)))))
75	74	ifeq2d 4568	. . . . . . 7 ⊢ (𝜑 → if((𝑃 − 1) < (𝑐‘0), 0, (((!‘(𝑃 − 1)) / (!‘((𝑃 − 1) − (𝑐‘0)))) · ((𝑌 − 0)↑((𝑃 − 1) − (𝑐‘0))))) = if((𝑃 − 1) < (𝑐‘0), 0, (((!‘(𝑃 − 1)) / (!‘((𝑃 − 1) − (𝑐‘0)))) · (𝑌↑((𝑃 − 1) − (𝑐‘0))))))
76		0p1e1 12415	. . . . . . . . . . 11 ⊢ (0 + 1) = 1
77	76	oveq1i 7458	. . . . . . . . . 10 ⊢ ((0 + 1)...𝑀) = (1...𝑀)
78	77	prodeq1i 15964	. . . . . . . . 9 ⊢ ∏𝑗 ∈ ((0 + 1)...𝑀)if(if(𝑗 = 0, (𝑃 − 1), 𝑃) < (𝑐‘𝑗), 0, (((!‘if(𝑗 = 0, (𝑃 − 1), 𝑃)) / (!‘(if(𝑗 = 0, (𝑃 − 1), 𝑃) − (𝑐‘𝑗)))) · ((𝑌 − 𝑗)↑(if(𝑗 = 0, (𝑃 − 1), 𝑃) − (𝑐‘𝑗))))) = ∏𝑗 ∈ (1...𝑀)if(if(𝑗 = 0, (𝑃 − 1), 𝑃) < (𝑐‘𝑗), 0, (((!‘if(𝑗 = 0, (𝑃 − 1), 𝑃)) / (!‘(if(𝑗 = 0, (𝑃 − 1), 𝑃) − (𝑐‘𝑗)))) · ((𝑌 − 𝑗)↑(if(𝑗 = 0, (𝑃 − 1), 𝑃) − (𝑐‘𝑗)))))
79		0red 11293	. . . . . . . . . . . . . . . 16 ⊢ (𝑗 ∈ (1...𝑀) → 0 ∈ ℝ)
80		1red 11291	. . . . . . . . . . . . . . . 16 ⊢ (𝑗 ∈ (1...𝑀) → 1 ∈ ℝ)
81		elfzelz 13584	. . . . . . . . . . . . . . . . 17 ⊢ (𝑗 ∈ (1...𝑀) → 𝑗 ∈ ℤ)
82	81	zred 12747	. . . . . . . . . . . . . . . 16 ⊢ (𝑗 ∈ (1...𝑀) → 𝑗 ∈ ℝ)
83		0lt1 11812	. . . . . . . . . . . . . . . . 17 ⊢ 0 < 1
84	83	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ (𝑗 ∈ (1...𝑀) → 0 < 1)
85		elfzle1 13587	. . . . . . . . . . . . . . . 16 ⊢ (𝑗 ∈ (1...𝑀) → 1 ≤ 𝑗)
86	79, 80, 82, 84, 85	ltletrd 11450	. . . . . . . . . . . . . . 15 ⊢ (𝑗 ∈ (1...𝑀) → 0 < 𝑗)
87	86	gt0ne0d 11854	. . . . . . . . . . . . . 14 ⊢ (𝑗 ∈ (1...𝑀) → 𝑗 ≠ 0)
88	87	neneqd 2951	. . . . . . . . . . . . 13 ⊢ (𝑗 ∈ (1...𝑀) → ¬ 𝑗 = 0)
89	88	iffalsed 4559	. . . . . . . . . . . 12 ⊢ (𝑗 ∈ (1...𝑀) → if(𝑗 = 0, (𝑃 − 1), 𝑃) = 𝑃)
90	89	breq1d 5176	. . . . . . . . . . 11 ⊢ (𝑗 ∈ (1...𝑀) → (if(𝑗 = 0, (𝑃 − 1), 𝑃) < (𝑐‘𝑗) ↔ 𝑃 < (𝑐‘𝑗)))
91	89	fveq2d 6924	. . . . . . . . . . . . 13 ⊢ (𝑗 ∈ (1...𝑀) → (!‘if(𝑗 = 0, (𝑃 − 1), 𝑃)) = (!‘𝑃))
92	89	oveq1d 7463	. . . . . . . . . . . . . 14 ⊢ (𝑗 ∈ (1...𝑀) → (if(𝑗 = 0, (𝑃 − 1), 𝑃) − (𝑐‘𝑗)) = (𝑃 − (𝑐‘𝑗)))
93	92	fveq2d 6924	. . . . . . . . . . . . 13 ⊢ (𝑗 ∈ (1...𝑀) → (!‘(if(𝑗 = 0, (𝑃 − 1), 𝑃) − (𝑐‘𝑗))) = (!‘(𝑃 − (𝑐‘𝑗))))
94	91, 93	oveq12d 7466	. . . . . . . . . . . 12 ⊢ (𝑗 ∈ (1...𝑀) → ((!‘if(𝑗 = 0, (𝑃 − 1), 𝑃)) / (!‘(if(𝑗 = 0, (𝑃 − 1), 𝑃) − (𝑐‘𝑗)))) = ((!‘𝑃) / (!‘(𝑃 − (𝑐‘𝑗)))))
95	92	oveq2d 7464	. . . . . . . . . . . 12 ⊢ (𝑗 ∈ (1...𝑀) → ((𝑌 − 𝑗)↑(if(𝑗 = 0, (𝑃 − 1), 𝑃) − (𝑐‘𝑗))) = ((𝑌 − 𝑗)↑(𝑃 − (𝑐‘𝑗))))
96	94, 95	oveq12d 7466	. . . . . . . . . . 11 ⊢ (𝑗 ∈ (1...𝑀) → (((!‘if(𝑗 = 0, (𝑃 − 1), 𝑃)) / (!‘(if(𝑗 = 0, (𝑃 − 1), 𝑃) − (𝑐‘𝑗)))) · ((𝑌 − 𝑗)↑(if(𝑗 = 0, (𝑃 − 1), 𝑃) − (𝑐‘𝑗)))) = (((!‘𝑃) / (!‘(𝑃 − (𝑐‘𝑗)))) · ((𝑌 − 𝑗)↑(𝑃 − (𝑐‘𝑗)))))
97	90, 96	ifbieq2d 4574	. . . . . . . . . 10 ⊢ (𝑗 ∈ (1...𝑀) → if(if(𝑗 = 0, (𝑃 − 1), 𝑃) < (𝑐‘𝑗), 0, (((!‘if(𝑗 = 0, (𝑃 − 1), 𝑃)) / (!‘(if(𝑗 = 0, (𝑃 − 1), 𝑃) − (𝑐‘𝑗)))) · ((𝑌 − 𝑗)↑(if(𝑗 = 0, (𝑃 − 1), 𝑃) − (𝑐‘𝑗))))) = if(𝑃 < (𝑐‘𝑗), 0, (((!‘𝑃) / (!‘(𝑃 − (𝑐‘𝑗)))) · ((𝑌 − 𝑗)↑(𝑃 − (𝑐‘𝑗))))))
98	97	prodeq2i 15966	. . . . . . . . 9 ⊢ ∏𝑗 ∈ (1...𝑀)if(if(𝑗 = 0, (𝑃 − 1), 𝑃) < (𝑐‘𝑗), 0, (((!‘if(𝑗 = 0, (𝑃 − 1), 𝑃)) / (!‘(if(𝑗 = 0, (𝑃 − 1), 𝑃) − (𝑐‘𝑗)))) · ((𝑌 − 𝑗)↑(if(𝑗 = 0, (𝑃 − 1), 𝑃) − (𝑐‘𝑗))))) = ∏𝑗 ∈ (1...𝑀)if(𝑃 < (𝑐‘𝑗), 0, (((!‘𝑃) / (!‘(𝑃 − (𝑐‘𝑗)))) · ((𝑌 − 𝑗)↑(𝑃 − (𝑐‘𝑗)))))
99	78, 98	eqtri 2768	. . . . . . . 8 ⊢ ∏𝑗 ∈ ((0 + 1)...𝑀)if(if(𝑗 = 0, (𝑃 − 1), 𝑃) < (𝑐‘𝑗), 0, (((!‘if(𝑗 = 0, (𝑃 − 1), 𝑃)) / (!‘(if(𝑗 = 0, (𝑃 − 1), 𝑃) − (𝑐‘𝑗)))) · ((𝑌 − 𝑗)↑(if(𝑗 = 0, (𝑃 − 1), 𝑃) − (𝑐‘𝑗))))) = ∏𝑗 ∈ (1...𝑀)if(𝑃 < (𝑐‘𝑗), 0, (((!‘𝑃) / (!‘(𝑃 − (𝑐‘𝑗)))) · ((𝑌 − 𝑗)↑(𝑃 − (𝑐‘𝑗)))))
100	99	a1i 11	. . . . . . 7 ⊢ (𝜑 → ∏𝑗 ∈ ((0 + 1)...𝑀)if(if(𝑗 = 0, (𝑃 − 1), 𝑃) < (𝑐‘𝑗), 0, (((!‘if(𝑗 = 0, (𝑃 − 1), 𝑃)) / (!‘(if(𝑗 = 0, (𝑃 − 1), 𝑃) − (𝑐‘𝑗)))) · ((𝑌 − 𝑗)↑(if(𝑗 = 0, (𝑃 − 1), 𝑃) − (𝑐‘𝑗))))) = ∏𝑗 ∈ (1...𝑀)if(𝑃 < (𝑐‘𝑗), 0, (((!‘𝑃) / (!‘(𝑃 − (𝑐‘𝑗)))) · ((𝑌 − 𝑗)↑(𝑃 − (𝑐‘𝑗))))))
101	75, 100	oveq12d 7466	. . . . . 6 ⊢ (𝜑 → (if((𝑃 − 1) < (𝑐‘0), 0, (((!‘(𝑃 − 1)) / (!‘((𝑃 − 1) − (𝑐‘0)))) · ((𝑌 − 0)↑((𝑃 − 1) − (𝑐‘0))))) · ∏𝑗 ∈ ((0 + 1)...𝑀)if(if(𝑗 = 0, (𝑃 − 1), 𝑃) < (𝑐‘𝑗), 0, (((!‘if(𝑗 = 0, (𝑃 − 1), 𝑃)) / (!‘(if(𝑗 = 0, (𝑃 − 1), 𝑃) − (𝑐‘𝑗)))) · ((𝑌 − 𝑗)↑(if(𝑗 = 0, (𝑃 − 1), 𝑃) − (𝑐‘𝑗)))))) = (if((𝑃 − 1) < (𝑐‘0), 0, (((!‘(𝑃 − 1)) / (!‘((𝑃 − 1) − (𝑐‘0)))) · (𝑌↑((𝑃 − 1) − (𝑐‘0))))) · ∏𝑗 ∈ (1...𝑀)if(𝑃 < (𝑐‘𝑗), 0, (((!‘𝑃) / (!‘(𝑃 − (𝑐‘𝑗)))) · ((𝑌 − 𝑗)↑(𝑃 − (𝑐‘𝑗)))))))
102	101	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐶‘𝑁)) → (if((𝑃 − 1) < (𝑐‘0), 0, (((!‘(𝑃 − 1)) / (!‘((𝑃 − 1) − (𝑐‘0)))) · ((𝑌 − 0)↑((𝑃 − 1) − (𝑐‘0))))) · ∏𝑗 ∈ ((0 + 1)...𝑀)if(if(𝑗 = 0, (𝑃 − 1), 𝑃) < (𝑐‘𝑗), 0, (((!‘if(𝑗 = 0, (𝑃 − 1), 𝑃)) / (!‘(if(𝑗 = 0, (𝑃 − 1), 𝑃) − (𝑐‘𝑗)))) · ((𝑌 − 𝑗)↑(if(𝑗 = 0, (𝑃 − 1), 𝑃) − (𝑐‘𝑗)))))) = (if((𝑃 − 1) < (𝑐‘0), 0, (((!‘(𝑃 − 1)) / (!‘((𝑃 − 1) − (𝑐‘0)))) · (𝑌↑((𝑃 − 1) − (𝑐‘0))))) · ∏𝑗 ∈ (1...𝑀)if(𝑃 < (𝑐‘𝑗), 0, (((!‘𝑃) / (!‘(𝑃 − (𝑐‘𝑗)))) · ((𝑌 − 𝑗)↑(𝑃 − (𝑐‘𝑗)))))))
103	53, 69, 102	3eqtrd 2784	. . . 4 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐶‘𝑁)) → ∏𝑗 ∈ (0...𝑀)(((𝑆 D^𝑛 (𝐻‘𝑗))‘(𝑐‘𝑗))‘𝑌) = (if((𝑃 − 1) < (𝑐‘0), 0, (((!‘(𝑃 − 1)) / (!‘((𝑃 − 1) − (𝑐‘0)))) · (𝑌↑((𝑃 − 1) − (𝑐‘0))))) · ∏𝑗 ∈ (1...𝑀)if(𝑃 < (𝑐‘𝑗), 0, (((!‘𝑃) / (!‘(𝑃 − (𝑐‘𝑗)))) · ((𝑌 − 𝑗)↑(𝑃 − (𝑐‘𝑗)))))))
104	103	oveq2d 7464	. . 3 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐶‘𝑁)) → (((!‘𝑁) / ∏𝑗 ∈ (0...𝑀)(!‘(𝑐‘𝑗))) · ∏𝑗 ∈ (0...𝑀)(((𝑆 D^𝑛 (𝐻‘𝑗))‘(𝑐‘𝑗))‘𝑌)) = (((!‘𝑁) / ∏𝑗 ∈ (0...𝑀)(!‘(𝑐‘𝑗))) · (if((𝑃 − 1) < (𝑐‘0), 0, (((!‘(𝑃 − 1)) / (!‘((𝑃 − 1) − (𝑐‘0)))) · (𝑌↑((𝑃 − 1) − (𝑐‘0))))) · ∏𝑗 ∈ (1...𝑀)if(𝑃 < (𝑐‘𝑗), 0, (((!‘𝑃) / (!‘(𝑃 − (𝑐‘𝑗)))) · ((𝑌 − 𝑗)↑(𝑃 − (𝑐‘𝑗))))))))
105	104	sumeq2dv 15750	. 2 ⊢ (𝜑 → Σ𝑐 ∈ (𝐶‘𝑁)(((!‘𝑁) / ∏𝑗 ∈ (0...𝑀)(!‘(𝑐‘𝑗))) · ∏𝑗 ∈ (0...𝑀)(((𝑆 D^𝑛 (𝐻‘𝑗))‘(𝑐‘𝑗))‘𝑌)) = Σ𝑐 ∈ (𝐶‘𝑁)(((!‘𝑁) / ∏𝑗 ∈ (0...𝑀)(!‘(𝑐‘𝑗))) · (if((𝑃 − 1) < (𝑐‘0), 0, (((!‘(𝑃 − 1)) / (!‘((𝑃 − 1) − (𝑐‘0)))) · (𝑌↑((𝑃 − 1) − (𝑐‘0))))) · ∏𝑗 ∈ (1...𝑀)if(𝑃 < (𝑐‘𝑗), 0, (((!‘𝑃) / (!‘(𝑃 − (𝑐‘𝑗)))) · ((𝑌 − 𝑗)↑(𝑃 − (𝑐‘𝑗))))))))
106	51, 105	eqtrd 2780	1 ⊢ (𝜑 → (((𝑆 D^𝑛 𝐹)‘𝑁)‘𝑌) = Σ𝑐 ∈ (𝐶‘𝑁)(((!‘𝑁) / ∏𝑗 ∈ (0...𝑀)(!‘(𝑐‘𝑗))) · (if((𝑃 − 1) < (𝑐‘0), 0, (((!‘(𝑃 − 1)) / (!‘((𝑃 − 1) − (𝑐‘0)))) · (𝑌↑((𝑃 − 1) − (𝑐‘0))))) · ∏𝑗 ∈ (1...𝑀)if(𝑃 < (𝑐‘𝑗), 0, (((!‘𝑃) / (!‘(𝑃 − (𝑐‘𝑗)))) · ((𝑌 − 𝑗)↑(𝑃 − (𝑐‘𝑗))))))))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1537 ∈ wcel 2108 {crab 3443 ifcif 4548 {cpr 4650 class class class wbr 5166 ↦ cmpt 5249 ⟶wf 6569 ‘cfv 6573 (class class class)co 7448 ↑_m cmap 8884 ℂcc 11182 ℝcr 11183 0cc0 11184 1c1 11185 + caddc 11187 · cmul 11189 < clt 11324 − cmin 11520 / cdiv 11947 ℕcn 12293 ℕ₀cn0 12553 ℤ_≥cuz 12903 ...cfz 13567 ↑cexp 14112 !cfa 14322 Σcsu 15734 ∏cprod 15951 ↾_t crest 17480 TopOpenctopn 17481 ℂ_fldccnfld 21387 D^𝑛 cdvn 25919

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2158 ax-12 2178 ax-ext 2711 ax-rep 5303 ax-sep 5317 ax-nul 5324 ax-pow 5383 ax-pr 5447 ax-un 7770 ax-inf2 9710 ax-cnex 11240 ax-resscn 11241 ax-1cn 11242 ax-icn 11243 ax-addcl 11244 ax-addrcl 11245 ax-mulcl 11246 ax-mulrcl 11247 ax-mulcom 11248 ax-addass 11249 ax-mulass 11250 ax-distr 11251 ax-i2m1 11252 ax-1ne0 11253 ax-1rid 11254 ax-rnegex 11255 ax-rrecex 11256 ax-cnre 11257 ax-pre-lttri 11258 ax-pre-lttrn 11259 ax-pre-ltadd 11260 ax-pre-mulgt0 11261 ax-pre-sup 11262 ax-addf 11263

This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3or 1088 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2543 df-eu 2572 df-clab 2718 df-cleq 2732 df-clel 2819 df-nfc 2895 df-ne 2947 df-nel 3053 df-ral 3068 df-rex 3077 df-rmo 3388 df-reu 3389 df-rab 3444 df-v 3490 df-sbc 3805 df-csb 3922 df-dif 3979 df-un 3981 df-in 3983 df-ss 3993 df-pss 3996 df-nul 4353 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-tp 4653 df-op 4655 df-uni 4932 df-int 4971 df-iun 5017 df-iin 5018 df-br 5167 df-opab 5229 df-mpt 5250 df-tr 5284 df-id 5593 df-eprel 5599 df-po 5607 df-so 5608 df-fr 5652 df-se 5653 df-we 5654 df-xp 5706 df-rel 5707 df-cnv 5708 df-co 5709 df-dm 5710 df-rn 5711 df-res 5712 df-ima 5713 df-pred 6332 df-ord 6398 df-on 6399 df-lim 6400 df-suc 6401 df-iota 6525 df-fun 6575 df-fn 6576 df-f 6577 df-f1 6578 df-fo 6579 df-f1o 6580 df-fv 6581 df-isom 6582 df-riota 7404 df-ov 7451 df-oprab 7452 df-mpo 7453 df-of 7714 df-om 7904 df-1st 8030 df-2nd 8031 df-supp 8202 df-frecs 8322 df-wrecs 8353 df-recs 8427 df-rdg 8466 df-1o 8522 df-2o 8523 df-er 8763 df-map 8886 df-pm 8887 df-ixp 8956 df-en 9004 df-dom 9005 df-sdom 9006 df-fin 9007 df-fsupp 9432 df-fi 9480 df-sup 9511 df-inf 9512 df-oi 9579 df-card 10008 df-pnf 11326 df-mnf 11327 df-xr 11328 df-ltxr 11329 df-le 11330 df-sub 11522 df-neg 11523 df-div 11948 df-nn 12294 df-2 12356 df-3 12357 df-4 12358 df-5 12359 df-6 12360 df-7 12361 df-8 12362 df-9 12363 df-n0 12554 df-z 12640 df-dec 12759 df-uz 12904 df-q 13014 df-rp 13058 df-xneg 13175 df-xadd 13176 df-xmul 13177 df-ico 13413 df-icc 13414 df-fz 13568 df-fzo 13712 df-seq 14053 df-exp 14113 df-fac 14323 df-bc 14352 df-hash 14380 df-cj 15148 df-re 15149 df-im 15150 df-sqrt 15284 df-abs 15285 df-clim 15534 df-sum 15735 df-prod 15952 df-struct 17194 df-sets 17211 df-slot 17229 df-ndx 17241 df-base 17259 df-ress 17288 df-plusg 17324 df-mulr 17325 df-starv 17326 df-sca 17327 df-vsca 17328 df-ip 17329 df-tset 17330 df-ple 17331 df-ds 17333 df-unif 17334 df-hom 17335 df-cco 17336 df-rest 17482 df-topn 17483 df-0g 17501 df-gsum 17502 df-topgen 17503 df-pt 17504 df-prds 17507 df-xrs 17562 df-qtop 17567 df-imas 17568 df-xps 17570 df-mre 17644 df-mrc 17645 df-acs 17647 df-mgm 18678 df-sgrp 18757 df-mnd 18773 df-submnd 18819 df-mulg 19108 df-cntz 19357 df-cmn 19824 df-psmet 21379 df-xmet 21380 df-met 21381 df-bl 21382 df-mopn 21383 df-fbas 21384 df-fg 21385 df-cnfld 21388 df-top 22921 df-topon 22938 df-topsp 22960 df-bases 22974 df-cld 23048 df-ntr 23049 df-cls 23050 df-nei 23127 df-lp 23165 df-perf 23166 df-cn 23256 df-cnp 23257 df-haus 23344 df-tx 23591 df-hmeo 23784 df-fil 23875 df-fm 23967 df-flim 23968 df-flf 23969 df-xms 24351 df-ms 24352 df-tms 24353 df-cncf 24923 df-limc 25921 df-dv 25922 df-dvn 25923

This theorem is referenced by: etransclem35 46190 etransclem36 46191 etransclem37 46192 etransclem38 46193

W3C validator