etransclem31 - Mathbox for Glauco Siliprandi

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Theorem etransclem31 43696

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 43695	. . 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 6756	. . . . . . 7 ⊢ (𝑥 = 𝑌 → (((𝑆 D^𝑛 (𝐻‘𝑗))‘(𝑐‘𝑗))‘𝑥) = (((𝑆 D^𝑛 (𝐻‘𝑗))‘(𝑐‘𝑗))‘𝑌))
11	10	prodeq2ad 43023	. . . . . 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 7271	. . . . 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 15344	. . . 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 43681	. . . 4 ⊢ (𝜑 → (𝐶‘𝑁) ∈ Fin)
17	6	faccld 13926	. . . . . . . 8 ⊢ (𝜑 → (!‘𝑁) ∈ ℕ)
18	17	nncnd 11919	. . . . . . 7 ⊢ (𝜑 → (!‘𝑁) ∈ ℂ)
19	18	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐶‘𝑁)) → (!‘𝑁) ∈ ℂ)
20		fzfid 13621	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐶‘𝑁)) → (0...𝑀) ∈ Fin)
21		fzssnn0 42746	. . . . . . . . . 10 ⊢ (0...𝑁) ⊆ ℕ₀
22		ssrab2 4009	. . . . . . . . . . . . . 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 43677	. . . . . . . . . . . . . . . 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 2841	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐶‘𝑁)) → 𝑐 ∈ {𝑐 ∈ ((0...𝑁) ↑_m (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐‘𝑗) = 𝑁})
27	22, 26	sselid 3915	. . . . . . . . . . . . 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 8595	. . . . . . . . . . . 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	ffvelrnd 6944	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑐 ∈ (𝐶‘𝑁)) ∧ 𝑗 ∈ (0...𝑀)) → (𝑐‘𝑗) ∈ (0...𝑁))
33	21, 32	sselid 3915	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑐 ∈ (𝐶‘𝑁)) ∧ 𝑗 ∈ (0...𝑀)) → (𝑐‘𝑗) ∈ ℕ₀)
34	33	faccld 13926	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑐 ∈ (𝐶‘𝑁)) ∧ 𝑗 ∈ (0...𝑀)) → (!‘(𝑐‘𝑗)) ∈ ℕ)
35	34	nncnd 11919	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑐 ∈ (𝐶‘𝑁)) ∧ 𝑗 ∈ (0...𝑀)) → (!‘(𝑐‘𝑗)) ∈ ℂ)
36	20, 35	fprodcl 15590	. . . . . 6 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐶‘𝑁)) → ∏𝑗 ∈ (0...𝑀)(!‘(𝑐‘𝑗)) ∈ ℂ)
37	34	nnne0d 11953	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑐 ∈ (𝐶‘𝑁)) ∧ 𝑗 ∈ (0...𝑀)) → (!‘(𝑐‘𝑗)) ≠ 0)
38	20, 35, 37	fprodn0 15617	. . . . . 6 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐶‘𝑁)) → ∏𝑗 ∈ (0...𝑀)(!‘(𝑐‘𝑗)) ≠ 0)
39	19, 36, 38	divcld 11681	. . . . 5 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐶‘𝑁)) → ((!‘𝑁) / ∏𝑗 ∈ (0...𝑀)(!‘(𝑐‘𝑗))) ∈ ℂ)
40	1	ad2antrr 722	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑐 ∈ (𝐶‘𝑁)) ∧ 𝑗 ∈ (0...𝑀)) → 𝑆 ∈ {ℝ, ℂ})
41	2	ad2antrr 722	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑐 ∈ (𝐶‘𝑁)) ∧ 𝑗 ∈ (0...𝑀)) → 𝑋 ∈ ((TopOpen‘ℂ_fld) ↾_t 𝑆))
42	3	ad2antrr 722	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑐 ∈ (𝐶‘𝑁)) ∧ 𝑗 ∈ (0...𝑀)) → 𝑃 ∈ ℕ)
43		etransclem5 43670	. . . . . . . . 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 2766	. . . . . . . 8 ⊢ 𝐻 = (𝑘 ∈ (0...𝑀) ↦ (𝑦 ∈ 𝑋 ↦ ((𝑦 − 𝑘)↑if(𝑘 = 0, (𝑃 − 1), 𝑃))))
45	40, 41, 42, 44, 31, 33	etransclem20 43685	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑐 ∈ (𝐶‘𝑁)) ∧ 𝑗 ∈ (0...𝑀)) → ((𝑆 D^𝑛 (𝐻‘𝑗))‘(𝑐‘𝑗)):𝑋⟶ℂ)
46	15	ad2antrr 722	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑐 ∈ (𝐶‘𝑁)) ∧ 𝑗 ∈ (0...𝑀)) → 𝑌 ∈ 𝑋)
47	45, 46	ffvelrnd 6944	. . . . . 6 ⊢ (((𝜑 ∧ 𝑐 ∈ (𝐶‘𝑁)) ∧ 𝑗 ∈ (0...𝑀)) → (((𝑆 D^𝑛 (𝐻‘𝑗))‘(𝑐‘𝑗))‘𝑌) ∈ ℂ)
48	20, 47	fprodcl 15590	. . . . 5 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐶‘𝑁)) → ∏𝑗 ∈ (0...𝑀)(((𝑆 D^𝑛 (𝐻‘𝑗))‘(𝑐‘𝑗))‘𝑌) ∈ ℂ)
49	39, 48	mulcld 10926	. . . 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 15373	. . 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 6864	. 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 43686	. . . . . 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 15561	. . . . 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 12549	. . . . . . . 8 ⊢ ℕ₀ = (ℤ_≥‘0)
55	4, 54	eleqtrdi 2849	. . . . . . 7 ⊢ (𝜑 → 𝑀 ∈ (ℤ_≥‘0))
56	55	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐶‘𝑁)) → 𝑀 ∈ (ℤ_≥‘0))
57	52, 47	eqeltrrd 2840	. . . . . 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 4462	. . . . . . . 8 ⊢ (𝑗 = 0 → if(𝑗 = 0, (𝑃 − 1), 𝑃) = (𝑃 − 1))
59		fveq2 6756	. . . . . . . 8 ⊢ (𝑗 = 0 → (𝑐‘𝑗) = (𝑐‘0))
60	58, 59	breq12d 5083	. . . . . . 7 ⊢ (𝑗 = 0 → (if(𝑗 = 0, (𝑃 − 1), 𝑃) < (𝑐‘𝑗) ↔ (𝑃 − 1) < (𝑐‘0)))
61	58	fveq2d 6760	. . . . . . . . 9 ⊢ (𝑗 = 0 → (!‘if(𝑗 = 0, (𝑃 − 1), 𝑃)) = (!‘(𝑃 − 1)))
62	58, 59	oveq12d 7273	. . . . . . . . . 10 ⊢ (𝑗 = 0 → (if(𝑗 = 0, (𝑃 − 1), 𝑃) − (𝑐‘𝑗)) = ((𝑃 − 1) − (𝑐‘0)))
63	62	fveq2d 6760	. . . . . . . . 9 ⊢ (𝑗 = 0 → (!‘(if(𝑗 = 0, (𝑃 − 1), 𝑃) − (𝑐‘𝑗))) = (!‘((𝑃 − 1) − (𝑐‘0))))
64	61, 63	oveq12d 7273	. . . . . . . 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 7263	. . . . . . . . 9 ⊢ (𝑗 = 0 → (𝑌 − 𝑗) = (𝑌 − 0))
66	65, 62	oveq12d 7273	. . . . . . . 8 ⊢ (𝑗 = 0 → ((𝑌 − 𝑗)↑(if(𝑗 = 0, (𝑃 − 1), 𝑃) − (𝑐‘𝑗))) = ((𝑌 − 0)↑((𝑃 − 1) − (𝑐‘0))))
67	64, 66	oveq12d 7273	. . . . . . 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 4482	. . . . . 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 15606	. . . . 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 43367	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑋 ⊆ ℂ)
71	70, 15	sseldd 3918	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑌 ∈ ℂ)
72	71	subid1d 11251	. . . . . . . . . 10 ⊢ (𝜑 → (𝑌 − 0) = 𝑌)
73	72	oveq1d 7270	. . . . . . . . 9 ⊢ (𝜑 → ((𝑌 − 0)↑((𝑃 − 1) − (𝑐‘0))) = (𝑌↑((𝑃 − 1) − (𝑐‘0))))
74	73	oveq2d 7271	. . . . . . . 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 4476	. . . . . . 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 12025	. . . . . . . . . . 11 ⊢ (0 + 1) = 1
77	76	oveq1i 7265	. . . . . . . . . 10 ⊢ ((0 + 1)...𝑀) = (1...𝑀)
78	77	prodeq1i 15556	. . . . . . . . 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 10909	. . . . . . . . . . . . . . . 16 ⊢ (𝑗 ∈ (1...𝑀) → 0 ∈ ℝ)
80		1red 10907	. . . . . . . . . . . . . . . 16 ⊢ (𝑗 ∈ (1...𝑀) → 1 ∈ ℝ)
81		elfzelz 13185	. . . . . . . . . . . . . . . . 17 ⊢ (𝑗 ∈ (1...𝑀) → 𝑗 ∈ ℤ)
82	81	zred 12355	. . . . . . . . . . . . . . . 16 ⊢ (𝑗 ∈ (1...𝑀) → 𝑗 ∈ ℝ)
83		0lt1 11427	. . . . . . . . . . . . . . . . 17 ⊢ 0 < 1
84	83	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ (𝑗 ∈ (1...𝑀) → 0 < 1)
85		elfzle1 13188	. . . . . . . . . . . . . . . 16 ⊢ (𝑗 ∈ (1...𝑀) → 1 ≤ 𝑗)
86	79, 80, 82, 84, 85	ltletrd 11065	. . . . . . . . . . . . . . 15 ⊢ (𝑗 ∈ (1...𝑀) → 0 < 𝑗)
87	86	gt0ne0d 11469	. . . . . . . . . . . . . 14 ⊢ (𝑗 ∈ (1...𝑀) → 𝑗 ≠ 0)
88	87	neneqd 2947	. . . . . . . . . . . . 13 ⊢ (𝑗 ∈ (1...𝑀) → ¬ 𝑗 = 0)
89	88	iffalsed 4467	. . . . . . . . . . . 12 ⊢ (𝑗 ∈ (1...𝑀) → if(𝑗 = 0, (𝑃 − 1), 𝑃) = 𝑃)
90	89	breq1d 5080	. . . . . . . . . . 11 ⊢ (𝑗 ∈ (1...𝑀) → (if(𝑗 = 0, (𝑃 − 1), 𝑃) < (𝑐‘𝑗) ↔ 𝑃 < (𝑐‘𝑗)))
91	89	fveq2d 6760	. . . . . . . . . . . . 13 ⊢ (𝑗 ∈ (1...𝑀) → (!‘if(𝑗 = 0, (𝑃 − 1), 𝑃)) = (!‘𝑃))
92	89	oveq1d 7270	. . . . . . . . . . . . . 14 ⊢ (𝑗 ∈ (1...𝑀) → (if(𝑗 = 0, (𝑃 − 1), 𝑃) − (𝑐‘𝑗)) = (𝑃 − (𝑐‘𝑗)))
93	92	fveq2d 6760	. . . . . . . . . . . . 13 ⊢ (𝑗 ∈ (1...𝑀) → (!‘(if(𝑗 = 0, (𝑃 − 1), 𝑃) − (𝑐‘𝑗))) = (!‘(𝑃 − (𝑐‘𝑗))))
94	91, 93	oveq12d 7273	. . . . . . . . . . . 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 7271	. . . . . . . . . . . 12 ⊢ (𝑗 ∈ (1...𝑀) → ((𝑌 − 𝑗)↑(if(𝑗 = 0, (𝑃 − 1), 𝑃) − (𝑐‘𝑗))) = ((𝑌 − 𝑗)↑(𝑃 − (𝑐‘𝑗))))
96	94, 95	oveq12d 7273	. . . . . . . . . . 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 4482	. . . . . . . . . 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 15557	. . . . . . . . 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 2766	. . . . . . . 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 7273	. . . . . 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 2782	. . . 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 7271	. . 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 15343	. 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 2778	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 1539 ∈ wcel 2108 {crab 3067 ifcif 4456 {cpr 4560 class class class wbr 5070 ↦ cmpt 5153 ⟶wf 6414 ‘cfv 6418 (class class class)co 7255 ↑_m cmap 8573 ℂcc 10800 ℝcr 10801 0cc0 10802 1c1 10803 + caddc 10805 · cmul 10807 < clt 10940 − cmin 11135 / cdiv 11562 ℕcn 11903 ℕ₀cn0 12163 ℤ_≥cuz 12511 ...cfz 13168 ↑cexp 13710 !cfa 13915 Σcsu 15325 ∏cprod 15543 ↾_t crest 17048 TopOpenctopn 17049 ℂ_fldccnfld 20510 D^𝑛 cdvn 24933

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1799 ax-4 1813 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2156 ax-12 2173 ax-ext 2709 ax-rep 5205 ax-sep 5218 ax-nul 5225 ax-pow 5283 ax-pr 5347 ax-un 7566 ax-inf2 9329 ax-cnex 10858 ax-resscn 10859 ax-1cn 10860 ax-icn 10861 ax-addcl 10862 ax-addrcl 10863 ax-mulcl 10864 ax-mulrcl 10865 ax-mulcom 10866 ax-addass 10867 ax-mulass 10868 ax-distr 10869 ax-i2m1 10870 ax-1ne0 10871 ax-1rid 10872 ax-rnegex 10873 ax-rrecex 10874 ax-cnre 10875 ax-pre-lttri 10876 ax-pre-lttrn 10877 ax-pre-ltadd 10878 ax-pre-mulgt0 10879 ax-pre-sup 10880 ax-addf 10881 ax-mulf 10882

This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1784 df-nf 1788 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2817 df-nfc 2888 df-ne 2943 df-nel 3049 df-ral 3068 df-rex 3069 df-reu 3070 df-rmo 3071 df-rab 3072 df-v 3424 df-sbc 3712 df-csb 3829 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-pss 3902 df-nul 4254 df-if 4457 df-pw 4532 df-sn 4559 df-pr 4561 df-tp 4563 df-op 4565 df-uni 4837 df-int 4877 df-iun 4923 df-iin 4924 df-br 5071 df-opab 5133 df-mpt 5154 df-tr 5188 df-id 5480 df-eprel 5486 df-po 5494 df-so 5495 df-fr 5535 df-se 5536 df-we 5537 df-xp 5586 df-rel 5587 df-cnv 5588 df-co 5589 df-dm 5590 df-rn 5591 df-res 5592 df-ima 5593 df-pred 6191 df-ord 6254 df-on 6255 df-lim 6256 df-suc 6257 df-iota 6376 df-fun 6420 df-fn 6421 df-f 6422 df-f1 6423 df-fo 6424 df-f1o 6425 df-fv 6426 df-isom 6427 df-riota 7212 df-ov 7258 df-oprab 7259 df-mpo 7260 df-of 7511 df-om 7688 df-1st 7804 df-2nd 7805 df-supp 7949 df-frecs 8068 df-wrecs 8099 df-recs 8173 df-rdg 8212 df-1o 8267 df-2o 8268 df-er 8456 df-map 8575 df-pm 8576 df-ixp 8644 df-en 8692 df-dom 8693 df-sdom 8694 df-fin 8695 df-fsupp 9059 df-fi 9100 df-sup 9131 df-inf 9132 df-oi 9199 df-card 9628 df-pnf 10942 df-mnf 10943 df-xr 10944 df-ltxr 10945 df-le 10946 df-sub 11137 df-neg 11138 df-div 11563 df-nn 11904 df-2 11966 df-3 11967 df-4 11968 df-5 11969 df-6 11970 df-7 11971 df-8 11972 df-9 11973 df-n0 12164 df-z 12250 df-dec 12367 df-uz 12512 df-q 12618 df-rp 12660 df-xneg 12777 df-xadd 12778 df-xmul 12779 df-ico 13014 df-icc 13015 df-fz 13169 df-fzo 13312 df-seq 13650 df-exp 13711 df-fac 13916 df-bc 13945 df-hash 13973 df-cj 14738 df-re 14739 df-im 14740 df-sqrt 14874 df-abs 14875 df-clim 15125 df-sum 15326 df-prod 15544 df-struct 16776 df-sets 16793 df-slot 16811 df-ndx 16823 df-base 16841 df-ress 16868 df-plusg 16901 df-mulr 16902 df-starv 16903 df-sca 16904 df-vsca 16905 df-ip 16906 df-tset 16907 df-ple 16908 df-ds 16910 df-unif 16911 df-hom 16912 df-cco 16913 df-rest 17050 df-topn 17051 df-0g 17069 df-gsum 17070 df-topgen 17071 df-pt 17072 df-prds 17075 df-xrs 17130 df-qtop 17135 df-imas 17136 df-xps 17138 df-mre 17212 df-mrc 17213 df-acs 17215 df-mgm 18241 df-sgrp 18290 df-mnd 18301 df-submnd 18346 df-mulg 18616 df-cntz 18838 df-cmn 19303 df-psmet 20502 df-xmet 20503 df-met 20504 df-bl 20505 df-mopn 20506 df-fbas 20507 df-fg 20508 df-cnfld 20511 df-top 21951 df-topon 21968 df-topsp 21990 df-bases 22004 df-cld 22078 df-ntr 22079 df-cls 22080 df-nei 22157 df-lp 22195 df-perf 22196 df-cn 22286 df-cnp 22287 df-haus 22374 df-tx 22621 df-hmeo 22814 df-fil 22905 df-fm 22997 df-flim 22998 df-flf 22999 df-xms 23381 df-ms 23382 df-tms 23383 df-cncf 23947 df-limc 24935 df-dv 24936 df-dvn 24937

This theorem is referenced by: etransclem35 43700 etransclem36 43701 etransclem37 43702 etransclem38 43703

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