etransclem31 - Mathbox for Glauco Siliprandi

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Theorem etransclem31 46236

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 46235	. . 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 6840	. . . . . . 7 ⊢ (𝑥 = 𝑌 → (((𝑆 D^𝑛 (𝐻‘𝑗))‘(𝑐‘𝑗))‘𝑥) = (((𝑆 D^𝑛 (𝐻‘𝑗))‘(𝑐‘𝑗))‘𝑌))
11	10	prodeq2ad 45563	. . . . . 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 7385	. . . . 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 15645	. . . 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 46221	. . . 4 ⊢ (𝜑 → (𝐶‘𝑁) ∈ Fin)
17	6	faccld 14225	. . . . . . . 8 ⊢ (𝜑 → (!‘𝑁) ∈ ℕ)
18	17	nncnd 12178	. . . . . . 7 ⊢ (𝜑 → (!‘𝑁) ∈ ℂ)
19	18	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐶‘𝑁)) → (!‘𝑁) ∈ ℂ)
20		fzfid 13914	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐶‘𝑁)) → (0...𝑀) ∈ Fin)
21		fzssnn0 45287	. . . . . . . . . 10 ⊢ (0...𝑁) ⊆ ℕ₀
22		ssrab2 4039	. . . . . . . . . . . . . 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 46217	. . . . . . . . . . . . . . . 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 2830	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐶‘𝑁)) → 𝑐 ∈ {𝑐 ∈ ((0...𝑁) ↑_m (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐‘𝑗) = 𝑁})
27	22, 26	sselid 3941	. . . . . . . . . . . . 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 8799	. . . . . . . . . . . 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 7039	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑐 ∈ (𝐶‘𝑁)) ∧ 𝑗 ∈ (0...𝑀)) → (𝑐‘𝑗) ∈ (0...𝑁))
33	21, 32	sselid 3941	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑐 ∈ (𝐶‘𝑁)) ∧ 𝑗 ∈ (0...𝑀)) → (𝑐‘𝑗) ∈ ℕ₀)
34	33	faccld 14225	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑐 ∈ (𝐶‘𝑁)) ∧ 𝑗 ∈ (0...𝑀)) → (!‘(𝑐‘𝑗)) ∈ ℕ)
35	34	nncnd 12178	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑐 ∈ (𝐶‘𝑁)) ∧ 𝑗 ∈ (0...𝑀)) → (!‘(𝑐‘𝑗)) ∈ ℂ)
36	20, 35	fprodcl 15894	. . . . . 6 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐶‘𝑁)) → ∏𝑗 ∈ (0...𝑀)(!‘(𝑐‘𝑗)) ∈ ℂ)
37	34	nnne0d 12212	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑐 ∈ (𝐶‘𝑁)) ∧ 𝑗 ∈ (0...𝑀)) → (!‘(𝑐‘𝑗)) ≠ 0)
38	20, 35, 37	fprodn0 15921	. . . . . 6 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐶‘𝑁)) → ∏𝑗 ∈ (0...𝑀)(!‘(𝑐‘𝑗)) ≠ 0)
39	19, 36, 38	divcld 11934	. . . . 5 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐶‘𝑁)) → ((!‘𝑁) / ∏𝑗 ∈ (0...𝑀)(!‘(𝑐‘𝑗))) ∈ ℂ)
40	1	ad2antrr 726	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑐 ∈ (𝐶‘𝑁)) ∧ 𝑗 ∈ (0...𝑀)) → 𝑆 ∈ {ℝ, ℂ})
41	2	ad2antrr 726	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑐 ∈ (𝐶‘𝑁)) ∧ 𝑗 ∈ (0...𝑀)) → 𝑋 ∈ ((TopOpen‘ℂ_fld) ↾_t 𝑆))
42	3	ad2antrr 726	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑐 ∈ (𝐶‘𝑁)) ∧ 𝑗 ∈ (0...𝑀)) → 𝑃 ∈ ℕ)
43		etransclem5 46210	. . . . . . . . 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 2752	. . . . . . . 8 ⊢ 𝐻 = (𝑘 ∈ (0...𝑀) ↦ (𝑦 ∈ 𝑋 ↦ ((𝑦 − 𝑘)↑if(𝑘 = 0, (𝑃 − 1), 𝑃))))
45	40, 41, 42, 44, 31, 33	etransclem20 46225	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑐 ∈ (𝐶‘𝑁)) ∧ 𝑗 ∈ (0...𝑀)) → ((𝑆 D^𝑛 (𝐻‘𝑗))‘(𝑐‘𝑗)):𝑋⟶ℂ)
46	15	ad2antrr 726	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑐 ∈ (𝐶‘𝑁)) ∧ 𝑗 ∈ (0...𝑀)) → 𝑌 ∈ 𝑋)
47	45, 46	ffvelcdmd 7039	. . . . . 6 ⊢ (((𝜑 ∧ 𝑐 ∈ (𝐶‘𝑁)) ∧ 𝑗 ∈ (0...𝑀)) → (((𝑆 D^𝑛 (𝐻‘𝑗))‘(𝑐‘𝑗))‘𝑌) ∈ ℂ)
48	20, 47	fprodcl 15894	. . . . 5 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐶‘𝑁)) → ∏𝑗 ∈ (0...𝑀)(((𝑆 D^𝑛 (𝐻‘𝑗))‘(𝑐‘𝑗))‘𝑌) ∈ ℂ)
49	39, 48	mulcld 11170	. . . 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 15675	. . 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 6957	. 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 46226	. . . . . 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 15864	. . . . 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 12811	. . . . . . . 8 ⊢ ℕ₀ = (ℤ_≥‘0)
55	4, 54	eleqtrdi 2838	. . . . . . 7 ⊢ (𝜑 → 𝑀 ∈ (ℤ_≥‘0))
56	55	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐶‘𝑁)) → 𝑀 ∈ (ℤ_≥‘0))
57	52, 47	eqeltrrd 2829	. . . . . 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 4490	. . . . . . . 8 ⊢ (𝑗 = 0 → if(𝑗 = 0, (𝑃 − 1), 𝑃) = (𝑃 − 1))
59		fveq2 6840	. . . . . . . 8 ⊢ (𝑗 = 0 → (𝑐‘𝑗) = (𝑐‘0))
60	58, 59	breq12d 5115	. . . . . . 7 ⊢ (𝑗 = 0 → (if(𝑗 = 0, (𝑃 − 1), 𝑃) < (𝑐‘𝑗) ↔ (𝑃 − 1) < (𝑐‘0)))
61	58	fveq2d 6844	. . . . . . . . 9 ⊢ (𝑗 = 0 → (!‘if(𝑗 = 0, (𝑃 − 1), 𝑃)) = (!‘(𝑃 − 1)))
62	58, 59	oveq12d 7387	. . . . . . . . . 10 ⊢ (𝑗 = 0 → (if(𝑗 = 0, (𝑃 − 1), 𝑃) − (𝑐‘𝑗)) = ((𝑃 − 1) − (𝑐‘0)))
63	62	fveq2d 6844	. . . . . . . . 9 ⊢ (𝑗 = 0 → (!‘(if(𝑗 = 0, (𝑃 − 1), 𝑃) − (𝑐‘𝑗))) = (!‘((𝑃 − 1) − (𝑐‘0))))
64	61, 63	oveq12d 7387	. . . . . . . 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 7377	. . . . . . . . 9 ⊢ (𝑗 = 0 → (𝑌 − 𝑗) = (𝑌 − 0))
66	65, 62	oveq12d 7387	. . . . . . . 8 ⊢ (𝑗 = 0 → ((𝑌 − 𝑗)↑(if(𝑗 = 0, (𝑃 − 1), 𝑃) − (𝑐‘𝑗))) = ((𝑌 − 0)↑((𝑃 − 1) − (𝑐‘0))))
67	64, 66	oveq12d 7387	. . . . . . 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 4511	. . . . . 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 15910	. . . . 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 45907	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑋 ⊆ ℂ)
71	70, 15	sseldd 3944	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑌 ∈ ℂ)
72	71	subid1d 11498	. . . . . . . . . 10 ⊢ (𝜑 → (𝑌 − 0) = 𝑌)
73	72	oveq1d 7384	. . . . . . . . 9 ⊢ (𝜑 → ((𝑌 − 0)↑((𝑃 − 1) − (𝑐‘0))) = (𝑌↑((𝑃 − 1) − (𝑐‘0))))
74	73	oveq2d 7385	. . . . . . . 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 4505	. . . . . . 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 12279	. . . . . . . . . . 11 ⊢ (0 + 1) = 1
77	76	oveq1i 7379	. . . . . . . . . 10 ⊢ ((0 + 1)...𝑀) = (1...𝑀)
78	77	prodeq1i 15858	. . . . . . . . 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 11153	. . . . . . . . . . . . . . . 16 ⊢ (𝑗 ∈ (1...𝑀) → 0 ∈ ℝ)
80		1red 11151	. . . . . . . . . . . . . . . 16 ⊢ (𝑗 ∈ (1...𝑀) → 1 ∈ ℝ)
81		elfzelz 13461	. . . . . . . . . . . . . . . . 17 ⊢ (𝑗 ∈ (1...𝑀) → 𝑗 ∈ ℤ)
82	81	zred 12614	. . . . . . . . . . . . . . . 16 ⊢ (𝑗 ∈ (1...𝑀) → 𝑗 ∈ ℝ)
83		0lt1 11676	. . . . . . . . . . . . . . . . 17 ⊢ 0 < 1
84	83	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ (𝑗 ∈ (1...𝑀) → 0 < 1)
85		elfzle1 13464	. . . . . . . . . . . . . . . 16 ⊢ (𝑗 ∈ (1...𝑀) → 1 ≤ 𝑗)
86	79, 80, 82, 84, 85	ltletrd 11310	. . . . . . . . . . . . . . 15 ⊢ (𝑗 ∈ (1...𝑀) → 0 < 𝑗)
87	86	gt0ne0d 11718	. . . . . . . . . . . . . 14 ⊢ (𝑗 ∈ (1...𝑀) → 𝑗 ≠ 0)
88	87	neneqd 2930	. . . . . . . . . . . . 13 ⊢ (𝑗 ∈ (1...𝑀) → ¬ 𝑗 = 0)
89	88	iffalsed 4495	. . . . . . . . . . . 12 ⊢ (𝑗 ∈ (1...𝑀) → if(𝑗 = 0, (𝑃 − 1), 𝑃) = 𝑃)
90	89	breq1d 5112	. . . . . . . . . . 11 ⊢ (𝑗 ∈ (1...𝑀) → (if(𝑗 = 0, (𝑃 − 1), 𝑃) < (𝑐‘𝑗) ↔ 𝑃 < (𝑐‘𝑗)))
91	89	fveq2d 6844	. . . . . . . . . . . . 13 ⊢ (𝑗 ∈ (1...𝑀) → (!‘if(𝑗 = 0, (𝑃 − 1), 𝑃)) = (!‘𝑃))
92	89	oveq1d 7384	. . . . . . . . . . . . . 14 ⊢ (𝑗 ∈ (1...𝑀) → (if(𝑗 = 0, (𝑃 − 1), 𝑃) − (𝑐‘𝑗)) = (𝑃 − (𝑐‘𝑗)))
93	92	fveq2d 6844	. . . . . . . . . . . . 13 ⊢ (𝑗 ∈ (1...𝑀) → (!‘(if(𝑗 = 0, (𝑃 − 1), 𝑃) − (𝑐‘𝑗))) = (!‘(𝑃 − (𝑐‘𝑗))))
94	91, 93	oveq12d 7387	. . . . . . . . . . . 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 7385	. . . . . . . . . . . 12 ⊢ (𝑗 ∈ (1...𝑀) → ((𝑌 − 𝑗)↑(if(𝑗 = 0, (𝑃 − 1), 𝑃) − (𝑐‘𝑗))) = ((𝑌 − 𝑗)↑(𝑃 − (𝑐‘𝑗))))
96	94, 95	oveq12d 7387	. . . . . . . . . . 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 4511	. . . . . . . . . 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 15860	. . . . . . . . 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 2752	. . . . . . . 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 7387	. . . . . 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 2768	. . . 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 7385	. . 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 15644	. 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 2764	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 1540 ∈ wcel 2109 {crab 3402 ifcif 4484 {cpr 4587 class class class wbr 5102 ↦ cmpt 5183 ⟶wf 6495 ‘cfv 6499 (class class class)co 7369 ↑_m cmap 8776 ℂcc 11042 ℝcr 11043 0cc0 11044 1c1 11045 + caddc 11047 · cmul 11049 < clt 11184 − cmin 11381 / cdiv 11811 ℕcn 12162 ℕ₀cn0 12418 ℤ_≥cuz 12769 ...cfz 13444 ↑cexp 14002 !cfa 14214 Σcsu 15628 ∏cprod 15845 ↾_t crest 17359 TopOpenctopn 17360 ℂ_fldccnfld 21240 D^𝑛 cdvn 25741

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 5229 ax-sep 5246 ax-nul 5256 ax-pow 5315 ax-pr 5382 ax-un 7691 ax-inf2 9570 ax-cnex 11100 ax-resscn 11101 ax-1cn 11102 ax-icn 11103 ax-addcl 11104 ax-addrcl 11105 ax-mulcl 11106 ax-mulrcl 11107 ax-mulcom 11108 ax-addass 11109 ax-mulass 11110 ax-distr 11111 ax-i2m1 11112 ax-1ne0 11113 ax-1rid 11114 ax-rnegex 11115 ax-rrecex 11116 ax-cnre 11117 ax-pre-lttri 11118 ax-pre-lttrn 11119 ax-pre-ltadd 11120 ax-pre-mulgt0 11121 ax-pre-sup 11122 ax-addf 11123

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 3351 df-reu 3352 df-rab 3403 df-v 3446 df-sbc 3751 df-csb 3860 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3931 df-nul 4293 df-if 4485 df-pw 4561 df-sn 4586 df-pr 4588 df-tp 4590 df-op 4592 df-uni 4868 df-int 4907 df-iun 4953 df-iin 4954 df-br 5103 df-opab 5165 df-mpt 5184 df-tr 5210 df-id 5526 df-eprel 5531 df-po 5539 df-so 5540 df-fr 5584 df-se 5585 df-we 5586 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6262 df-ord 6323 df-on 6324 df-lim 6325 df-suc 6326 df-iota 6452 df-fun 6501 df-fn 6502 df-f 6503 df-f1 6504 df-fo 6505 df-f1o 6506 df-fv 6507 df-isom 6508 df-riota 7326 df-ov 7372 df-oprab 7373 df-mpo 7374 df-of 7633 df-om 7823 df-1st 7947 df-2nd 7948 df-supp 8117 df-frecs 8237 df-wrecs 8268 df-recs 8317 df-rdg 8355 df-1o 8411 df-2o 8412 df-er 8648 df-map 8778 df-pm 8779 df-ixp 8848 df-en 8896 df-dom 8897 df-sdom 8898 df-fin 8899 df-fsupp 9289 df-fi 9338 df-sup 9369 df-inf 9370 df-oi 9439 df-card 9868 df-pnf 11186 df-mnf 11187 df-xr 11188 df-ltxr 11189 df-le 11190 df-sub 11383 df-neg 11384 df-div 11812 df-nn 12163 df-2 12225 df-3 12226 df-4 12227 df-5 12228 df-6 12229 df-7 12230 df-8 12231 df-9 12232 df-n0 12419 df-z 12506 df-dec 12626 df-uz 12770 df-q 12884 df-rp 12928 df-xneg 13048 df-xadd 13049 df-xmul 13050 df-ico 13288 df-icc 13289 df-fz 13445 df-fzo 13592 df-seq 13943 df-exp 14003 df-fac 14215 df-bc 14244 df-hash 14272 df-cj 15041 df-re 15042 df-im 15043 df-sqrt 15177 df-abs 15178 df-clim 15430 df-sum 15629 df-prod 15846 df-struct 17093 df-sets 17110 df-slot 17128 df-ndx 17140 df-base 17156 df-ress 17177 df-plusg 17209 df-mulr 17210 df-starv 17211 df-sca 17212 df-vsca 17213 df-ip 17214 df-tset 17215 df-ple 17216 df-ds 17218 df-unif 17219 df-hom 17220 df-cco 17221 df-rest 17361 df-topn 17362 df-0g 17380 df-gsum 17381 df-topgen 17382 df-pt 17383 df-prds 17386 df-xrs 17441 df-qtop 17446 df-imas 17447 df-xps 17449 df-mre 17523 df-mrc 17524 df-acs 17526 df-mgm 18543 df-sgrp 18622 df-mnd 18638 df-submnd 18687 df-mulg 18976 df-cntz 19225 df-cmn 19688 df-psmet 21232 df-xmet 21233 df-met 21234 df-bl 21235 df-mopn 21236 df-fbas 21237 df-fg 21238 df-cnfld 21241 df-top 22757 df-topon 22774 df-topsp 22796 df-bases 22809 df-cld 22882 df-ntr 22883 df-cls 22884 df-nei 22961 df-lp 22999 df-perf 23000 df-cn 23090 df-cnp 23091 df-haus 23178 df-tx 23425 df-hmeo 23618 df-fil 23709 df-fm 23801 df-flim 23802 df-flf 23803 df-xms 24184 df-ms 24185 df-tms 24186 df-cncf 24747 df-limc 25743 df-dv 25744 df-dvn 25745

This theorem is referenced by: etransclem35 46240 etransclem36 46241 etransclem37 46242 etransclem38 46243

W3C validator