etransclem31 - Mathbox for Glauco Siliprandi

	Mathbox for Glauco Siliprandi		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > Mathboxes > etransclem31		Structured version Visualization version GIF version

Theorem etransclem31 42570

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 42569	. . 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 6670	. . . . . . 7 ⊢ (𝑥 = 𝑌 → (((𝑆 D^𝑛 (𝐻‘𝑗))‘(𝑐‘𝑗))‘𝑥) = (((𝑆 D^𝑛 (𝐻‘𝑗))‘(𝑐‘𝑗))‘𝑌))
11	10	prodeq2ad 41893	. . . . . 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 7172	. . . . 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 15061	. . . 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 484	. . 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 42555	. . . 4 ⊢ (𝜑 → (𝐶‘𝑁) ∈ Fin)
17	6	faccld 13645	. . . . . . . 8 ⊢ (𝜑 → (!‘𝑁) ∈ ℕ)
18	17	nncnd 11654	. . . . . . 7 ⊢ (𝜑 → (!‘𝑁) ∈ ℂ)
19	18	adantr 483	. . . . . 6 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐶‘𝑁)) → (!‘𝑁) ∈ ℂ)
20		fzfid 13342	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐶‘𝑁)) → (0...𝑀) ∈ Fin)
21		fzssnn0 41605	. . . . . . . . . 10 ⊢ (0...𝑁) ⊆ ℕ₀
22		ssrab2 4056	. . . . . . . . . . . . . 14 ⊢ {𝑐 ∈ ((0...𝑁) ↑_m (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐‘𝑗) = 𝑁} ⊆ ((0...𝑁) ↑_m (0...𝑀))
23		simpr 487	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐶‘𝑁)) → 𝑐 ∈ (𝐶‘𝑁))
24	8, 6	etransclem12 42551	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝐶‘𝑁) = {𝑐 ∈ ((0...𝑁) ↑_m (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐‘𝑗) = 𝑁})
25	24	adantr 483	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐶‘𝑁)) → (𝐶‘𝑁) = {𝑐 ∈ ((0...𝑁) ↑_m (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐‘𝑗) = 𝑁})
26	23, 25	eleqtrd 2915	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐶‘𝑁)) → 𝑐 ∈ {𝑐 ∈ ((0...𝑁) ↑_m (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐‘𝑗) = 𝑁})
27	22, 26	sseldi 3965	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐶‘𝑁)) → 𝑐 ∈ ((0...𝑁) ↑_m (0...𝑀)))
28	27	adantr 483	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑐 ∈ (𝐶‘𝑁)) ∧ 𝑗 ∈ (0...𝑀)) → 𝑐 ∈ ((0...𝑁) ↑_m (0...𝑀)))
29		elmapi 8428	. . . . . . . . . . . 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 487	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑐 ∈ (𝐶‘𝑁)) ∧ 𝑗 ∈ (0...𝑀)) → 𝑗 ∈ (0...𝑀))
32	30, 31	ffvelrnd 6852	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑐 ∈ (𝐶‘𝑁)) ∧ 𝑗 ∈ (0...𝑀)) → (𝑐‘𝑗) ∈ (0...𝑁))
33	21, 32	sseldi 3965	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑐 ∈ (𝐶‘𝑁)) ∧ 𝑗 ∈ (0...𝑀)) → (𝑐‘𝑗) ∈ ℕ₀)
34	33	faccld 13645	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑐 ∈ (𝐶‘𝑁)) ∧ 𝑗 ∈ (0...𝑀)) → (!‘(𝑐‘𝑗)) ∈ ℕ)
35	34	nncnd 11654	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑐 ∈ (𝐶‘𝑁)) ∧ 𝑗 ∈ (0...𝑀)) → (!‘(𝑐‘𝑗)) ∈ ℂ)
36	20, 35	fprodcl 15306	. . . . . 6 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐶‘𝑁)) → ∏𝑗 ∈ (0...𝑀)(!‘(𝑐‘𝑗)) ∈ ℂ)
37	34	nnne0d 11688	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑐 ∈ (𝐶‘𝑁)) ∧ 𝑗 ∈ (0...𝑀)) → (!‘(𝑐‘𝑗)) ≠ 0)
38	20, 35, 37	fprodn0 15333	. . . . . 6 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐶‘𝑁)) → ∏𝑗 ∈ (0...𝑀)(!‘(𝑐‘𝑗)) ≠ 0)
39	19, 36, 38	divcld 11416	. . . . 5 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐶‘𝑁)) → ((!‘𝑁) / ∏𝑗 ∈ (0...𝑀)(!‘(𝑐‘𝑗))) ∈ ℂ)
40	1	ad2antrr 724	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑐 ∈ (𝐶‘𝑁)) ∧ 𝑗 ∈ (0...𝑀)) → 𝑆 ∈ {ℝ, ℂ})
41	2	ad2antrr 724	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑐 ∈ (𝐶‘𝑁)) ∧ 𝑗 ∈ (0...𝑀)) → 𝑋 ∈ ((TopOpen‘ℂ_fld) ↾_t 𝑆))
42	3	ad2antrr 724	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑐 ∈ (𝐶‘𝑁)) ∧ 𝑗 ∈ (0...𝑀)) → 𝑃 ∈ ℕ)
43		etransclem5 42544	. . . . . . . . 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 2844	. . . . . . . 8 ⊢ 𝐻 = (𝑘 ∈ (0...𝑀) ↦ (𝑦 ∈ 𝑋 ↦ ((𝑦 − 𝑘)↑if(𝑘 = 0, (𝑃 − 1), 𝑃))))
45	40, 41, 42, 44, 31, 33	etransclem20 42559	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑐 ∈ (𝐶‘𝑁)) ∧ 𝑗 ∈ (0...𝑀)) → ((𝑆 D^𝑛 (𝐻‘𝑗))‘(𝑐‘𝑗)):𝑋⟶ℂ)
46	15	ad2antrr 724	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑐 ∈ (𝐶‘𝑁)) ∧ 𝑗 ∈ (0...𝑀)) → 𝑌 ∈ 𝑋)
47	45, 46	ffvelrnd 6852	. . . . . 6 ⊢ (((𝜑 ∧ 𝑐 ∈ (𝐶‘𝑁)) ∧ 𝑗 ∈ (0...𝑀)) → (((𝑆 D^𝑛 (𝐻‘𝑗))‘(𝑐‘𝑗))‘𝑌) ∈ ℂ)
48	20, 47	fprodcl 15306	. . . . 5 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐶‘𝑁)) → ∏𝑗 ∈ (0...𝑀)(((𝑆 D^𝑛 (𝐻‘𝑗))‘(𝑐‘𝑗))‘𝑌) ∈ ℂ)
49	39, 48	mulcld 10661	. . . 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 15090	. . 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 6775	. 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 42560	. . . . . 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 15277	. . . . 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 12281	. . . . . . . 8 ⊢ ℕ₀ = (ℤ_≥‘0)
55	4, 54	eleqtrdi 2923	. . . . . . 7 ⊢ (𝜑 → 𝑀 ∈ (ℤ_≥‘0))
56	55	adantr 483	. . . . . 6 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐶‘𝑁)) → 𝑀 ∈ (ℤ_≥‘0))
57	52, 47	eqeltrrd 2914	. . . . . 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 4473	. . . . . . . 8 ⊢ (𝑗 = 0 → if(𝑗 = 0, (𝑃 − 1), 𝑃) = (𝑃 − 1))
59		fveq2 6670	. . . . . . . 8 ⊢ (𝑗 = 0 → (𝑐‘𝑗) = (𝑐‘0))
60	58, 59	breq12d 5079	. . . . . . 7 ⊢ (𝑗 = 0 → (if(𝑗 = 0, (𝑃 − 1), 𝑃) < (𝑐‘𝑗) ↔ (𝑃 − 1) < (𝑐‘0)))
61	58	fveq2d 6674	. . . . . . . . 9 ⊢ (𝑗 = 0 → (!‘if(𝑗 = 0, (𝑃 − 1), 𝑃)) = (!‘(𝑃 − 1)))
62	58, 59	oveq12d 7174	. . . . . . . . . 10 ⊢ (𝑗 = 0 → (if(𝑗 = 0, (𝑃 − 1), 𝑃) − (𝑐‘𝑗)) = ((𝑃 − 1) − (𝑐‘0)))
63	62	fveq2d 6674	. . . . . . . . 9 ⊢ (𝑗 = 0 → (!‘(if(𝑗 = 0, (𝑃 − 1), 𝑃) − (𝑐‘𝑗))) = (!‘((𝑃 − 1) − (𝑐‘0))))
64	61, 63	oveq12d 7174	. . . . . . . 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 7164	. . . . . . . . 9 ⊢ (𝑗 = 0 → (𝑌 − 𝑗) = (𝑌 − 0))
66	65, 62	oveq12d 7174	. . . . . . . 8 ⊢ (𝑗 = 0 → ((𝑌 − 𝑗)↑(if(𝑗 = 0, (𝑃 − 1), 𝑃) − (𝑐‘𝑗))) = ((𝑌 − 0)↑((𝑃 − 1) − (𝑐‘0))))
67	64, 66	oveq12d 7174	. . . . . . 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 4492	. . . . . 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 15322	. . . . 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 42241	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑋 ⊆ ℂ)
71	70, 15	sseldd 3968	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑌 ∈ ℂ)
72	71	subid1d 10986	. . . . . . . . . 10 ⊢ (𝜑 → (𝑌 − 0) = 𝑌)
73	72	oveq1d 7171	. . . . . . . . 9 ⊢ (𝜑 → ((𝑌 − 0)↑((𝑃 − 1) − (𝑐‘0))) = (𝑌↑((𝑃 − 1) − (𝑐‘0))))
74	73	oveq2d 7172	. . . . . . . 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 4486	. . . . . . 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 11760	. . . . . . . . . . 11 ⊢ (0 + 1) = 1
77	76	oveq1i 7166	. . . . . . . . . 10 ⊢ ((0 + 1)...𝑀) = (1...𝑀)
78	77	prodeq1i 15272	. . . . . . . . 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 10644	. . . . . . . . . . . . . . . 16 ⊢ (𝑗 ∈ (1...𝑀) → 0 ∈ ℝ)
80		1red 10642	. . . . . . . . . . . . . . . 16 ⊢ (𝑗 ∈ (1...𝑀) → 1 ∈ ℝ)
81		elfzelz 12909	. . . . . . . . . . . . . . . . 17 ⊢ (𝑗 ∈ (1...𝑀) → 𝑗 ∈ ℤ)
82	81	zred 12088	. . . . . . . . . . . . . . . 16 ⊢ (𝑗 ∈ (1...𝑀) → 𝑗 ∈ ℝ)
83		0lt1 11162	. . . . . . . . . . . . . . . . 17 ⊢ 0 < 1
84	83	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ (𝑗 ∈ (1...𝑀) → 0 < 1)
85		elfzle1 12911	. . . . . . . . . . . . . . . 16 ⊢ (𝑗 ∈ (1...𝑀) → 1 ≤ 𝑗)
86	79, 80, 82, 84, 85	ltletrd 10800	. . . . . . . . . . . . . . 15 ⊢ (𝑗 ∈ (1...𝑀) → 0 < 𝑗)
87	86	gt0ne0d 11204	. . . . . . . . . . . . . 14 ⊢ (𝑗 ∈ (1...𝑀) → 𝑗 ≠ 0)
88	87	neneqd 3021	. . . . . . . . . . . . 13 ⊢ (𝑗 ∈ (1...𝑀) → ¬ 𝑗 = 0)
89	88	iffalsed 4478	. . . . . . . . . . . 12 ⊢ (𝑗 ∈ (1...𝑀) → if(𝑗 = 0, (𝑃 − 1), 𝑃) = 𝑃)
90	89	breq1d 5076	. . . . . . . . . . 11 ⊢ (𝑗 ∈ (1...𝑀) → (if(𝑗 = 0, (𝑃 − 1), 𝑃) < (𝑐‘𝑗) ↔ 𝑃 < (𝑐‘𝑗)))
91	89	fveq2d 6674	. . . . . . . . . . . . 13 ⊢ (𝑗 ∈ (1...𝑀) → (!‘if(𝑗 = 0, (𝑃 − 1), 𝑃)) = (!‘𝑃))
92	89	oveq1d 7171	. . . . . . . . . . . . . 14 ⊢ (𝑗 ∈ (1...𝑀) → (if(𝑗 = 0, (𝑃 − 1), 𝑃) − (𝑐‘𝑗)) = (𝑃 − (𝑐‘𝑗)))
93	92	fveq2d 6674	. . . . . . . . . . . . 13 ⊢ (𝑗 ∈ (1...𝑀) → (!‘(if(𝑗 = 0, (𝑃 − 1), 𝑃) − (𝑐‘𝑗))) = (!‘(𝑃 − (𝑐‘𝑗))))
94	91, 93	oveq12d 7174	. . . . . . . . . . . 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 7172	. . . . . . . . . . . 12 ⊢ (𝑗 ∈ (1...𝑀) → ((𝑌 − 𝑗)↑(if(𝑗 = 0, (𝑃 − 1), 𝑃) − (𝑐‘𝑗))) = ((𝑌 − 𝑗)↑(𝑃 − (𝑐‘𝑗))))
96	94, 95	oveq12d 7174	. . . . . . . . . . 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 4492	. . . . . . . . . 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 15273	. . . . . . . . 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 2844	. . . . . . . 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 7174	. . . . . 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 483	. . . . 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 2860	. . . 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 7172	. . 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 15060	. 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 2856	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 398 = wceq 1537 ∈ wcel 2114 {crab 3142 ifcif 4467 {cpr 4569 class class class wbr 5066 ↦ cmpt 5146 ⟶wf 6351 ‘cfv 6355 (class class class)co 7156 ↑_m cmap 8406 ℂcc 10535 ℝcr 10536 0cc0 10537 1c1 10538 + caddc 10540 · cmul 10542 < clt 10675 − cmin 10870 / cdiv 11297 ℕcn 11638 ℕ₀cn0 11898 ℤ_≥cuz 12244 ...cfz 12893 ↑cexp 13430 !cfa 13634 Σcsu 15042 ∏cprod 15259 ↾_t crest 16694 TopOpenctopn 16695 ℂ_fldccnfld 20545 D^𝑛 cdvn 24462

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-inf2 9104 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614 ax-pre-sup 10615 ax-addf 10616 ax-mulf 10617

This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-fal 1550 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-int 4877 df-iun 4921 df-iin 4922 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-se 5515 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-isom 6364 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-of 7409 df-om 7581 df-1st 7689 df-2nd 7690 df-supp 7831 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-1o 8102 df-2o 8103 df-oadd 8106 df-er 8289 df-map 8408 df-pm 8409 df-ixp 8462 df-en 8510 df-dom 8511 df-sdom 8512 df-fin 8513 df-fsupp 8834 df-fi 8875 df-sup 8906 df-inf 8907 df-oi 8974 df-card 9368 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-div 11298 df-nn 11639 df-2 11701 df-3 11702 df-4 11703 df-5 11704 df-6 11705 df-7 11706 df-8 11707 df-9 11708 df-n0 11899 df-z 11983 df-dec 12100 df-uz 12245 df-q 12350 df-rp 12391 df-xneg 12508 df-xadd 12509 df-xmul 12510 df-ico 12745 df-icc 12746 df-fz 12894 df-fzo 13035 df-seq 13371 df-exp 13431 df-fac 13635 df-bc 13664 df-hash 13692 df-cj 14458 df-re 14459 df-im 14460 df-sqrt 14594 df-abs 14595 df-clim 14845 df-sum 15043 df-prod 15260 df-struct 16485 df-ndx 16486 df-slot 16487 df-base 16489 df-sets 16490 df-ress 16491 df-plusg 16578 df-mulr 16579 df-starv 16580 df-sca 16581 df-vsca 16582 df-ip 16583 df-tset 16584 df-ple 16585 df-ds 16587 df-unif 16588 df-hom 16589 df-cco 16590 df-rest 16696 df-topn 16697 df-0g 16715 df-gsum 16716 df-topgen 16717 df-pt 16718 df-prds 16721 df-xrs 16775 df-qtop 16780 df-imas 16781 df-xps 16783 df-mre 16857 df-mrc 16858 df-acs 16860 df-mgm 17852 df-sgrp 17901 df-mnd 17912 df-submnd 17957 df-mulg 18225 df-cntz 18447 df-cmn 18908 df-psmet 20537 df-xmet 20538 df-met 20539 df-bl 20540 df-mopn 20541 df-fbas 20542 df-fg 20543 df-cnfld 20546 df-top 21502 df-topon 21519 df-topsp 21541 df-bases 21554 df-cld 21627 df-ntr 21628 df-cls 21629 df-nei 21706 df-lp 21744 df-perf 21745 df-cn 21835 df-cnp 21836 df-haus 21923 df-tx 22170 df-hmeo 22363 df-fil 22454 df-fm 22546 df-flim 22547 df-flf 22548 df-xms 22930 df-ms 22931 df-tms 22932 df-cncf 23486 df-limc 24464 df-dv 24465 df-dvn 24466

This theorem is referenced by: etransclem35 42574 etransclem36 42575 etransclem37 42576 etransclem38 42577

W3C validator