etransclem47 - Mathbox for Glauco Siliprandi

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Theorem etransclem47 46236

Description: e is transcendental. Section *5 of [Juillerat] p. 11 can be used as a reference for this proof. (Contributed by Glauco Siliprandi, 5-Apr-2020.)

**Hypotheses**
Ref	Expression
etransclem47.q	⊢ (𝜑 → 𝑄 ∈ ((Poly‘ℤ) ∖ {0_𝑝}))
etransclem47.qe0	⊢ (𝜑 → (𝑄‘e) = 0)
etransclem47.a	⊢ 𝐴 = (coeff‘𝑄)
etransclem47.a0	⊢ (𝜑 → (𝐴‘0) ≠ 0)
etransclem47.m	⊢ 𝑀 = (deg‘𝑄)
etransclem47.p	⊢ (𝜑 → 𝑃 ∈ ℙ)
etransclem47.ap	⊢ (𝜑 → (abs‘(𝐴‘0)) < 𝑃)
etransclem47.mp	⊢ (𝜑 → (!‘𝑀) < 𝑃)
etransclem47.9	⊢ (𝜑 → (Σ𝑗 ∈ (0...𝑀)((abs‘((𝐴‘𝑗) · (e↑_𝑐𝑗))) · (𝑀 · (𝑀↑(𝑀 + 1)))) · (((𝑀↑(𝑀 + 1))↑(𝑃 − 1)) / (!‘(𝑃 − 1)))) < 1)
etransclem47.f	⊢ 𝐹 = (𝑥 ∈ ℝ ↦ ((𝑥↑(𝑃 − 1)) · ∏𝑗 ∈ (1...𝑀)((𝑥 − 𝑗)↑𝑃)))
etransclem47.l	⊢ 𝐿 = Σ𝑗 ∈ (0...𝑀)(((𝐴‘𝑗) · (e↑_𝑐𝑗)) · ∫(0(,)𝑗)((e↑_𝑐-𝑥) · (𝐹‘𝑥)) d𝑥)
etransclem47.k	⊢ 𝐾 = (𝐿 / (!‘(𝑃 − 1)))

**Assertion**
Ref	Expression
etransclem47	⊢ (𝜑 → ∃𝑘 ∈ ℤ (𝑘 ≠ 0 ∧ (abs‘𝑘) < 1))

Distinct variable groups: 𝐴,𝑗,𝑘 𝑗,𝐹,𝑘,𝑥 𝑘,𝐾 𝑗,𝑀,𝑘,𝑥 𝑃,𝑗,𝑘,𝑥 𝑄,𝑗 𝜑,𝑗,𝑘,𝑥 Allowed substitution hints: 𝐴(𝑥) 𝑄(𝑥,𝑘) 𝐾(𝑥,𝑗) 𝐿(𝑥,𝑗,𝑘)

Proof of Theorem etransclem47 Dummy variables 𝑖 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		etransclem47.k	. . . . 5 ⊢ 𝐾 = (𝐿 / (!‘(𝑃 − 1)))
2	1	a1i 11	. . . 4 ⊢ (𝜑 → 𝐾 = (𝐿 / (!‘(𝑃 − 1))))
3		etransclem47.q	. . . . 5 ⊢ (𝜑 → 𝑄 ∈ ((Poly‘ℤ) ∖ {0_𝑝}))
4		etransclem47.qe0	. . . . 5 ⊢ (𝜑 → (𝑄‘e) = 0)
5		etransclem47.a	. . . . 5 ⊢ 𝐴 = (coeff‘𝑄)
6		etransclem47.m	. . . . 5 ⊢ 𝑀 = (deg‘𝑄)
7		ssid 4017	. . . . . 6 ⊢ ℝ ⊆ ℝ
8	7	a1i 11	. . . . 5 ⊢ (𝜑 → ℝ ⊆ ℝ)
9		reelprrecn 11244	. . . . . 6 ⊢ ℝ ∈ {ℝ, ℂ}
10	9	a1i 11	. . . . 5 ⊢ (𝜑 → ℝ ∈ {ℝ, ℂ})
11		reopn 45239	. . . . . . 7 ⊢ ℝ ∈ (topGen‘ran (,))
12		eqid 2734	. . . . . . . 8 ⊢ (TopOpen‘ℂ_fld) = (TopOpen‘ℂ_fld)
13	12	tgioo2 24838	. . . . . . 7 ⊢ (topGen‘ran (,)) = ((TopOpen‘ℂ_fld) ↾_t ℝ)
14	11, 13	eleqtri 2836	. . . . . 6 ⊢ ℝ ∈ ((TopOpen‘ℂ_fld) ↾_t ℝ)
15	14	a1i 11	. . . . 5 ⊢ (𝜑 → ℝ ∈ ((TopOpen‘ℂ_fld) ↾_t ℝ))
16		etransclem47.p	. . . . . 6 ⊢ (𝜑 → 𝑃 ∈ ℙ)
17		prmnn 16707	. . . . . 6 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ ℕ)
18	16, 17	syl 17	. . . . 5 ⊢ (𝜑 → 𝑃 ∈ ℕ)
19		etransclem47.f	. . . . 5 ⊢ 𝐹 = (𝑥 ∈ ℝ ↦ ((𝑥↑(𝑃 − 1)) · ∏𝑗 ∈ (1...𝑀)((𝑥 − 𝑗)↑𝑃)))
20		etransclem47.l	. . . . 5 ⊢ 𝐿 = Σ𝑗 ∈ (0...𝑀)(((𝐴‘𝑗) · (e↑_𝑐𝑗)) · ∫(0(,)𝑗)((e↑_𝑐-𝑥) · (𝐹‘𝑥)) d𝑥)
21		eqid 2734	. . . . 5 ⊢ ((𝑀 · 𝑃) + (𝑃 − 1)) = ((𝑀 · 𝑃) + (𝑃 − 1))
22		fveq2 6906	. . . . . . 7 ⊢ (𝑦 = 𝑥 → (((ℝ D^𝑛 𝐹)‘𝑖)‘𝑦) = (((ℝ D^𝑛 𝐹)‘𝑖)‘𝑥))
23	22	sumeq2sdv 15735	. . . . . 6 ⊢ (𝑦 = 𝑥 → Σ𝑖 ∈ (0...((𝑀 · 𝑃) + (𝑃 − 1)))(((ℝ D^𝑛 𝐹)‘𝑖)‘𝑦) = Σ𝑖 ∈ (0...((𝑀 · 𝑃) + (𝑃 − 1)))(((ℝ D^𝑛 𝐹)‘𝑖)‘𝑥))
24	23	cbvmptv 5260	. . . . 5 ⊢ (𝑦 ∈ ℝ ↦ Σ𝑖 ∈ (0...((𝑀 · 𝑃) + (𝑃 − 1)))(((ℝ D^𝑛 𝐹)‘𝑖)‘𝑦)) = (𝑥 ∈ ℝ ↦ Σ𝑖 ∈ (0...((𝑀 · 𝑃) + (𝑃 − 1)))(((ℝ D^𝑛 𝐹)‘𝑖)‘𝑥))
25		negeq 11497	. . . . . . . . 9 ⊢ (𝑧 = 𝑥 → -𝑧 = -𝑥)
26	25	oveq2d 7446	. . . . . . . 8 ⊢ (𝑧 = 𝑥 → (e↑_𝑐-𝑧) = (e↑_𝑐-𝑥))
27		fveq2 6906	. . . . . . . 8 ⊢ (𝑧 = 𝑥 → ((𝑦 ∈ ℝ ↦ Σ𝑖 ∈ (0...((𝑀 · 𝑃) + (𝑃 − 1)))(((ℝ D^𝑛 𝐹)‘𝑖)‘𝑦))‘𝑧) = ((𝑦 ∈ ℝ ↦ Σ𝑖 ∈ (0...((𝑀 · 𝑃) + (𝑃 − 1)))(((ℝ D^𝑛 𝐹)‘𝑖)‘𝑦))‘𝑥))
28	26, 27	oveq12d 7448	. . . . . . 7 ⊢ (𝑧 = 𝑥 → ((e↑_𝑐-𝑧) · ((𝑦 ∈ ℝ ↦ Σ𝑖 ∈ (0...((𝑀 · 𝑃) + (𝑃 − 1)))(((ℝ D^𝑛 𝐹)‘𝑖)‘𝑦))‘𝑧)) = ((e↑_𝑐-𝑥) · ((𝑦 ∈ ℝ ↦ Σ𝑖 ∈ (0...((𝑀 · 𝑃) + (𝑃 − 1)))(((ℝ D^𝑛 𝐹)‘𝑖)‘𝑦))‘𝑥)))
29	28	negeqd 11499	. . . . . 6 ⊢ (𝑧 = 𝑥 → -((e↑_𝑐-𝑧) · ((𝑦 ∈ ℝ ↦ Σ𝑖 ∈ (0...((𝑀 · 𝑃) + (𝑃 − 1)))(((ℝ D^𝑛 𝐹)‘𝑖)‘𝑦))‘𝑧)) = -((e↑_𝑐-𝑥) · ((𝑦 ∈ ℝ ↦ Σ𝑖 ∈ (0...((𝑀 · 𝑃) + (𝑃 − 1)))(((ℝ D^𝑛 𝐹)‘𝑖)‘𝑦))‘𝑥)))
30	29	cbvmptv 5260	. . . . 5 ⊢ (𝑧 ∈ (0[,]𝑗) ↦ -((e↑_𝑐-𝑧) · ((𝑦 ∈ ℝ ↦ Σ𝑖 ∈ (0...((𝑀 · 𝑃) + (𝑃 − 1)))(((ℝ D^𝑛 𝐹)‘𝑖)‘𝑦))‘𝑧))) = (𝑥 ∈ (0[,]𝑗) ↦ -((e↑_𝑐-𝑥) · ((𝑦 ∈ ℝ ↦ Σ𝑖 ∈ (0...((𝑀 · 𝑃) + (𝑃 − 1)))(((ℝ D^𝑛 𝐹)‘𝑖)‘𝑦))‘𝑥)))
31	3, 4, 5, 6, 8, 10, 15, 18, 19, 20, 21, 24, 30	etransclem46 46235	. . . 4 ⊢ (𝜑 → (𝐿 / (!‘(𝑃 − 1))) = (-Σ𝑘 ∈ ((0...𝑀) × (0...((𝑀 · 𝑃) + (𝑃 − 1))))((𝐴‘(1^st ‘𝑘)) · (((ℝ D^𝑛 𝐹)‘(2^nd ‘𝑘))‘(1^st ‘𝑘))) / (!‘(𝑃 − 1))))
32		fzfid 14010	. . . . . . . 8 ⊢ (𝜑 → (0...𝑀) ∈ Fin)
33		fzfid 14010	. . . . . . . 8 ⊢ (𝜑 → (0...((𝑀 · 𝑃) + (𝑃 − 1))) ∈ Fin)
34		xpfi 9355	. . . . . . . 8 ⊢ (((0...𝑀) ∈ Fin ∧ (0...((𝑀 · 𝑃) + (𝑃 − 1))) ∈ Fin) → ((0...𝑀) × (0...((𝑀 · 𝑃) + (𝑃 − 1)))) ∈ Fin)
35	32, 33, 34	syl2anc 584	. . . . . . 7 ⊢ (𝜑 → ((0...𝑀) × (0...((𝑀 · 𝑃) + (𝑃 − 1)))) ∈ Fin)
36	3	eldifad 3974	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑄 ∈ (Poly‘ℤ))
37		0zd 12622	. . . . . . . . . . . 12 ⊢ (𝜑 → 0 ∈ ℤ)
38	5	coef2 26284	. . . . . . . . . . . 12 ⊢ ((𝑄 ∈ (Poly‘ℤ) ∧ 0 ∈ ℤ) → 𝐴:ℕ₀⟶ℤ)
39	36, 37, 38	syl2anc 584	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐴:ℕ₀⟶ℤ)
40	39	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ ((0...𝑀) × (0...((𝑀 · 𝑃) + (𝑃 − 1))))) → 𝐴:ℕ₀⟶ℤ)
41		xp1st 8044	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ ((0...𝑀) × (0...((𝑀 · 𝑃) + (𝑃 − 1)))) → (1^st ‘𝑘) ∈ (0...𝑀))
42		elfznn0 13656	. . . . . . . . . . . 12 ⊢ ((1^st ‘𝑘) ∈ (0...𝑀) → (1^st ‘𝑘) ∈ ℕ₀)
43	41, 42	syl 17	. . . . . . . . . . 11 ⊢ (𝑘 ∈ ((0...𝑀) × (0...((𝑀 · 𝑃) + (𝑃 − 1)))) → (1^st ‘𝑘) ∈ ℕ₀)
44	43	adantl 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ ((0...𝑀) × (0...((𝑀 · 𝑃) + (𝑃 − 1))))) → (1^st ‘𝑘) ∈ ℕ₀)
45	40, 44	ffvelcdmd 7104	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ ((0...𝑀) × (0...((𝑀 · 𝑃) + (𝑃 − 1))))) → (𝐴‘(1^st ‘𝑘)) ∈ ℤ)
46	45	zcnd 12720	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ ((0...𝑀) × (0...((𝑀 · 𝑃) + (𝑃 − 1))))) → (𝐴‘(1^st ‘𝑘)) ∈ ℂ)
47	9	a1i 11	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ ((0...𝑀) × (0...((𝑀 · 𝑃) + (𝑃 − 1))))) → ℝ ∈ {ℝ, ℂ})
48	14	a1i 11	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ ((0...𝑀) × (0...((𝑀 · 𝑃) + (𝑃 − 1))))) → ℝ ∈ ((TopOpen‘ℂ_fld) ↾_t ℝ))
49	18	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ ((0...𝑀) × (0...((𝑀 · 𝑃) + (𝑃 − 1))))) → 𝑃 ∈ ℕ)
50		dgrcl 26286	. . . . . . . . . . . . 13 ⊢ (𝑄 ∈ (Poly‘ℤ) → (deg‘𝑄) ∈ ℕ₀)
51	36, 50	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → (deg‘𝑄) ∈ ℕ₀)
52	6, 51	eqeltrid 2842	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑀 ∈ ℕ₀)
53	52	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ ((0...𝑀) × (0...((𝑀 · 𝑃) + (𝑃 − 1))))) → 𝑀 ∈ ℕ₀)
54		xp2nd 8045	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ ((0...𝑀) × (0...((𝑀 · 𝑃) + (𝑃 − 1)))) → (2^nd ‘𝑘) ∈ (0...((𝑀 · 𝑃) + (𝑃 − 1))))
55		elfznn0 13656	. . . . . . . . . . . 12 ⊢ ((2^nd ‘𝑘) ∈ (0...((𝑀 · 𝑃) + (𝑃 − 1))) → (2^nd ‘𝑘) ∈ ℕ₀)
56	54, 55	syl 17	. . . . . . . . . . 11 ⊢ (𝑘 ∈ ((0...𝑀) × (0...((𝑀 · 𝑃) + (𝑃 − 1)))) → (2^nd ‘𝑘) ∈ ℕ₀)
57	56	adantl 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ ((0...𝑀) × (0...((𝑀 · 𝑃) + (𝑃 − 1))))) → (2^nd ‘𝑘) ∈ ℕ₀)
58	47, 48, 49, 53, 19, 57	etransclem33 46222	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ ((0...𝑀) × (0...((𝑀 · 𝑃) + (𝑃 − 1))))) → ((ℝ D^𝑛 𝐹)‘(2^nd ‘𝑘)):ℝ⟶ℂ)
59	44	nn0red 12585	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ ((0...𝑀) × (0...((𝑀 · 𝑃) + (𝑃 − 1))))) → (1^st ‘𝑘) ∈ ℝ)
60	58, 59	ffvelcdmd 7104	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ ((0...𝑀) × (0...((𝑀 · 𝑃) + (𝑃 − 1))))) → (((ℝ D^𝑛 𝐹)‘(2^nd ‘𝑘))‘(1^st ‘𝑘)) ∈ ℂ)
61	46, 60	mulcld 11278	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ ((0...𝑀) × (0...((𝑀 · 𝑃) + (𝑃 − 1))))) → ((𝐴‘(1^st ‘𝑘)) · (((ℝ D^𝑛 𝐹)‘(2^nd ‘𝑘))‘(1^st ‘𝑘))) ∈ ℂ)
62	35, 61	fsumcl 15765	. . . . . 6 ⊢ (𝜑 → Σ𝑘 ∈ ((0...𝑀) × (0...((𝑀 · 𝑃) + (𝑃 − 1))))((𝐴‘(1^st ‘𝑘)) · (((ℝ D^𝑛 𝐹)‘(2^nd ‘𝑘))‘(1^st ‘𝑘))) ∈ ℂ)
63		nnm1nn0 12564	. . . . . . . . 9 ⊢ (𝑃 ∈ ℕ → (𝑃 − 1) ∈ ℕ₀)
64	18, 63	syl 17	. . . . . . . 8 ⊢ (𝜑 → (𝑃 − 1) ∈ ℕ₀)
65	64	faccld 14319	. . . . . . 7 ⊢ (𝜑 → (!‘(𝑃 − 1)) ∈ ℕ)
66	65	nncnd 12279	. . . . . 6 ⊢ (𝜑 → (!‘(𝑃 − 1)) ∈ ℂ)
67	65	nnne0d 12313	. . . . . 6 ⊢ (𝜑 → (!‘(𝑃 − 1)) ≠ 0)
68	62, 66, 67	divnegd 12053	. . . . 5 ⊢ (𝜑 → -(Σ𝑘 ∈ ((0...𝑀) × (0...((𝑀 · 𝑃) + (𝑃 − 1))))((𝐴‘(1^st ‘𝑘)) · (((ℝ D^𝑛 𝐹)‘(2^nd ‘𝑘))‘(1^st ‘𝑘))) / (!‘(𝑃 − 1))) = (-Σ𝑘 ∈ ((0...𝑀) × (0...((𝑀 · 𝑃) + (𝑃 − 1))))((𝐴‘(1^st ‘𝑘)) · (((ℝ D^𝑛 𝐹)‘(2^nd ‘𝑘))‘(1^st ‘𝑘))) / (!‘(𝑃 − 1))))
69	68	eqcomd 2740	. . . 4 ⊢ (𝜑 → (-Σ𝑘 ∈ ((0...𝑀) × (0...((𝑀 · 𝑃) + (𝑃 − 1))))((𝐴‘(1^st ‘𝑘)) · (((ℝ D^𝑛 𝐹)‘(2^nd ‘𝑘))‘(1^st ‘𝑘))) / (!‘(𝑃 − 1))) = -(Σ𝑘 ∈ ((0...𝑀) × (0...((𝑀 · 𝑃) + (𝑃 − 1))))((𝐴‘(1^st ‘𝑘)) · (((ℝ D^𝑛 𝐹)‘(2^nd ‘𝑘))‘(1^st ‘𝑘))) / (!‘(𝑃 − 1))))
70	2, 31, 69	3eqtrd 2778	. . 3 ⊢ (𝜑 → 𝐾 = -(Σ𝑘 ∈ ((0...𝑀) × (0...((𝑀 · 𝑃) + (𝑃 − 1))))((𝐴‘(1^st ‘𝑘)) · (((ℝ D^𝑛 𝐹)‘(2^nd ‘𝑘))‘(1^st ‘𝑘))) / (!‘(𝑃 − 1))))
71		eqid 2734	. . . . 5 ⊢ (Σ𝑘 ∈ ((0...𝑀) × (0...((𝑀 · 𝑃) + (𝑃 − 1))))((𝐴‘(1^st ‘𝑘)) · (((ℝ D^𝑛 𝐹)‘(2^nd ‘𝑘))‘(1^st ‘𝑘))) / (!‘(𝑃 − 1))) = (Σ𝑘 ∈ ((0...𝑀) × (0...((𝑀 · 𝑃) + (𝑃 − 1))))((𝐴‘(1^st ‘𝑘)) · (((ℝ D^𝑛 𝐹)‘(2^nd ‘𝑘))‘(1^st ‘𝑘))) / (!‘(𝑃 − 1)))
72	18, 52, 19, 39, 71	etransclem45 46234	. . . 4 ⊢ (𝜑 → (Σ𝑘 ∈ ((0...𝑀) × (0...((𝑀 · 𝑃) + (𝑃 − 1))))((𝐴‘(1^st ‘𝑘)) · (((ℝ D^𝑛 𝐹)‘(2^nd ‘𝑘))‘(1^st ‘𝑘))) / (!‘(𝑃 − 1))) ∈ ℤ)
73	72	znegcld 12721	. . 3 ⊢ (𝜑 → -(Σ𝑘 ∈ ((0...𝑀) × (0...((𝑀 · 𝑃) + (𝑃 − 1))))((𝐴‘(1^st ‘𝑘)) · (((ℝ D^𝑛 𝐹)‘(2^nd ‘𝑘))‘(1^st ‘𝑘))) / (!‘(𝑃 − 1))) ∈ ℤ)
74	70, 73	eqeltrd 2838	. 2 ⊢ (𝜑 → 𝐾 ∈ ℤ)
75	1, 31	eqtrid 2786	. . 3 ⊢ (𝜑 → 𝐾 = (-Σ𝑘 ∈ ((0...𝑀) × (0...((𝑀 · 𝑃) + (𝑃 − 1))))((𝐴‘(1^st ‘𝑘)) · (((ℝ D^𝑛 𝐹)‘(2^nd ‘𝑘))‘(1^st ‘𝑘))) / (!‘(𝑃 − 1))))
76	62, 66, 67	divcld 12040	. . . . 5 ⊢ (𝜑 → (Σ𝑘 ∈ ((0...𝑀) × (0...((𝑀 · 𝑃) + (𝑃 − 1))))((𝐴‘(1^st ‘𝑘)) · (((ℝ D^𝑛 𝐹)‘(2^nd ‘𝑘))‘(1^st ‘𝑘))) / (!‘(𝑃 − 1))) ∈ ℂ)
77		etransclem47.a0	. . . . . 6 ⊢ (𝜑 → (𝐴‘0) ≠ 0)
78		etransclem47.ap	. . . . . 6 ⊢ (𝜑 → (abs‘(𝐴‘0)) < 𝑃)
79		etransclem47.mp	. . . . . 6 ⊢ (𝜑 → (!‘𝑀) < 𝑃)
80	39, 77, 52, 16, 78, 79, 19, 71	etransclem44 46233	. . . . 5 ⊢ (𝜑 → (Σ𝑘 ∈ ((0...𝑀) × (0...((𝑀 · 𝑃) + (𝑃 − 1))))((𝐴‘(1^st ‘𝑘)) · (((ℝ D^𝑛 𝐹)‘(2^nd ‘𝑘))‘(1^st ‘𝑘))) / (!‘(𝑃 − 1))) ≠ 0)
81	76, 80	negne0d 11615	. . . 4 ⊢ (𝜑 → -(Σ𝑘 ∈ ((0...𝑀) × (0...((𝑀 · 𝑃) + (𝑃 − 1))))((𝐴‘(1^st ‘𝑘)) · (((ℝ D^𝑛 𝐹)‘(2^nd ‘𝑘))‘(1^st ‘𝑘))) / (!‘(𝑃 − 1))) ≠ 0)
82	69, 81	eqnetrd 3005	. . 3 ⊢ (𝜑 → (-Σ𝑘 ∈ ((0...𝑀) × (0...((𝑀 · 𝑃) + (𝑃 − 1))))((𝐴‘(1^st ‘𝑘)) · (((ℝ D^𝑛 𝐹)‘(2^nd ‘𝑘))‘(1^st ‘𝑘))) / (!‘(𝑃 − 1))) ≠ 0)
83	75, 82	eqnetrd 3005	. 2 ⊢ (𝜑 → 𝐾 ≠ 0)
84		eldifsni 4794	. . . . . 6 ⊢ (𝑄 ∈ ((Poly‘ℤ) ∖ {0_𝑝}) → 𝑄 ≠ 0_𝑝)
85	3, 84	syl 17	. . . . 5 ⊢ (𝜑 → 𝑄 ≠ 0_𝑝)
86		ere 16121	. . . . . . 7 ⊢ e ∈ ℝ
87	86	recni 11272	. . . . . 6 ⊢ e ∈ ℂ
88	87	a1i 11	. . . . 5 ⊢ (𝜑 → e ∈ ℂ)
89		dgrnznn 26300	. . . . 5 ⊢ (((𝑄 ∈ (Poly‘ℤ) ∧ 𝑄 ≠ 0_𝑝) ∧ (e ∈ ℂ ∧ (𝑄‘e) = 0)) → (deg‘𝑄) ∈ ℕ)
90	36, 85, 88, 4, 89	syl22anc 839	. . . 4 ⊢ (𝜑 → (deg‘𝑄) ∈ ℕ)
91	6, 90	eqeltrid 2842	. . 3 ⊢ (𝜑 → 𝑀 ∈ ℕ)
92		etransclem47.9	. . 3 ⊢ (𝜑 → (Σ𝑗 ∈ (0...𝑀)((abs‘((𝐴‘𝑗) · (e↑_𝑐𝑗))) · (𝑀 · (𝑀↑(𝑀 + 1)))) · (((𝑀↑(𝑀 + 1))↑(𝑃 − 1)) / (!‘(𝑃 − 1)))) < 1)
93	39, 20, 1, 18, 91, 19, 92	etransclem23 46212	. 2 ⊢ (𝜑 → (abs‘𝐾) < 1)
94		neeq1 3000	. . . 4 ⊢ (𝑘 = 𝐾 → (𝑘 ≠ 0 ↔ 𝐾 ≠ 0))
95		fveq2 6906	. . . . 5 ⊢ (𝑘 = 𝐾 → (abs‘𝑘) = (abs‘𝐾))
96	95	breq1d 5157	. . . 4 ⊢ (𝑘 = 𝐾 → ((abs‘𝑘) < 1 ↔ (abs‘𝐾) < 1))
97	94, 96	anbi12d 632	. . 3 ⊢ (𝑘 = 𝐾 → ((𝑘 ≠ 0 ∧ (abs‘𝑘) < 1) ↔ (𝐾 ≠ 0 ∧ (abs‘𝐾) < 1)))
98	97	rspcev 3621	. 2 ⊢ ((𝐾 ∈ ℤ ∧ (𝐾 ≠ 0 ∧ (abs‘𝐾) < 1)) → ∃𝑘 ∈ ℤ (𝑘 ≠ 0 ∧ (abs‘𝑘) < 1))
99	74, 83, 93, 98	syl12anc 837	1 ⊢ (𝜑 → ∃𝑘 ∈ ℤ (𝑘 ≠ 0 ∧ (abs‘𝑘) < 1))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1536 ∈ wcel 2105 ≠ wne 2937 ∃wrex 3067 ∖ cdif 3959 ⊆ wss 3962 {csn 4630 {cpr 4632 class class class wbr 5147 ↦ cmpt 5230 × cxp 5686 ran crn 5689 ⟶wf 6558 ‘cfv 6562 (class class class)co 7430 1^st c1st 8010 2^nd c2nd 8011 Fincfn 8983 ℂcc 11150 ℝcr 11151 0cc0 11152 1c1 11153 + caddc 11155 · cmul 11157 < clt 11292 − cmin 11489 -cneg 11490 / cdiv 11917 ℕcn 12263 ℕ₀cn0 12523 ℤcz 12610 (,)cioo 13383 [,]cicc 13386 ...cfz 13543 ↑cexp 14098 !cfa 14308 abscabs 15269 Σcsu 15718 ∏cprod 15935 eceu 16094 ℙcprime 16704 ↾_t crest 17466 TopOpenctopn 17467 topGenctg 17483 ℂ_fldccnfld 21381 ∫citg 25666 0_𝑝c0p 25717 D^𝑛 cdvn 25913 Polycply 26237 coeffccoe 26239 degcdgr 26240 ↑_𝑐ccxp 26611

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-rep 5284 ax-sep 5301 ax-nul 5311 ax-pow 5370 ax-pr 5437 ax-un 7753 ax-inf2 9678 ax-cc 10472 ax-cnex 11208 ax-resscn 11209 ax-1cn 11210 ax-icn 11211 ax-addcl 11212 ax-addrcl 11213 ax-mulcl 11214 ax-mulrcl 11215 ax-mulcom 11216 ax-addass 11217 ax-mulass 11218 ax-distr 11219 ax-i2m1 11220 ax-1ne0 11221 ax-1rid 11222 ax-rnegex 11223 ax-rrecex 11224 ax-cnre 11225 ax-pre-lttri 11226 ax-pre-lttrn 11227 ax-pre-ltadd 11228 ax-pre-mulgt0 11229 ax-pre-sup 11230 ax-addf 11231

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3377 df-reu 3378 df-rab 3433 df-v 3479 df-sbc 3791 df-csb 3908 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-pss 3982 df-symdif 4258 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-tp 4635 df-op 4637 df-uni 4912 df-int 4951 df-iun 4997 df-iin 4998 df-disj 5115 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5582 df-eprel 5588 df-po 5596 df-so 5597 df-fr 5640 df-se 5641 df-we 5642 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-res 5700 df-ima 5701 df-pred 6322 df-ord 6388 df-on 6389 df-lim 6390 df-suc 6391 df-iota 6515 df-fun 6564 df-fn 6565 df-f 6566 df-f1 6567 df-fo 6568 df-f1o 6569 df-fv 6570 df-isom 6571 df-riota 7387 df-ov 7433 df-oprab 7434 df-mpo 7435 df-of 7696 df-ofr 7697 df-om 7887 df-1st 8012 df-2nd 8013 df-supp 8184 df-frecs 8304 df-wrecs 8335 df-recs 8409 df-rdg 8448 df-1o 8504 df-2o 8505 df-oadd 8508 df-omul 8509 df-er 8743 df-map 8866 df-pm 8867 df-ixp 8936 df-en 8984 df-dom 8985 df-sdom 8986 df-fin 8987 df-fsupp 9399 df-fi 9448 df-sup 9479 df-inf 9480 df-oi 9547 df-dju 9938 df-card 9976 df-acn 9979 df-pnf 11294 df-mnf 11295 df-xr 11296 df-ltxr 11297 df-le 11298 df-sub 11491 df-neg 11492 df-div 11918 df-nn 12264 df-2 12326 df-3 12327 df-4 12328 df-5 12329 df-6 12330 df-7 12331 df-8 12332 df-9 12333 df-n0 12524 df-z 12611 df-dec 12731 df-uz 12876 df-q 12988 df-rp 13032 df-xneg 13151 df-xadd 13152 df-xmul 13153 df-ioo 13387 df-ioc 13388 df-ico 13389 df-icc 13390 df-fz 13544 df-fzo 13691 df-fl 13828 df-mod 13906 df-seq 14039 df-exp 14099 df-fac 14309 df-bc 14338 df-hash 14366 df-shft 15102 df-cj 15134 df-re 15135 df-im 15136 df-sqrt 15270 df-abs 15271 df-limsup 15503 df-clim 15520 df-rlim 15521 df-sum 15719 df-prod 15936 df-ef 16099 df-e 16100 df-sin 16101 df-cos 16102 df-tan 16103 df-pi 16104 df-dvds 16287 df-gcd 16528 df-prm 16705 df-struct 17180 df-sets 17197 df-slot 17215 df-ndx 17227 df-base 17245 df-ress 17274 df-plusg 17310 df-mulr 17311 df-starv 17312 df-sca 17313 df-vsca 17314 df-ip 17315 df-tset 17316 df-ple 17317 df-ds 17319 df-unif 17320 df-hom 17321 df-cco 17322 df-rest 17468 df-topn 17469 df-0g 17487 df-gsum 17488 df-topgen 17489 df-pt 17490 df-prds 17493 df-xrs 17548 df-qtop 17553 df-imas 17554 df-xps 17556 df-mre 17630 df-mrc 17631 df-acs 17633 df-mgm 18665 df-sgrp 18744 df-mnd 18760 df-submnd 18809 df-mulg 19098 df-cntz 19347 df-cmn 19814 df-psmet 21373 df-xmet 21374 df-met 21375 df-bl 21376 df-mopn 21377 df-fbas 21378 df-fg 21379 df-cnfld 21382 df-top 22915 df-topon 22932 df-topsp 22954 df-bases 22968 df-cld 23042 df-ntr 23043 df-cls 23044 df-nei 23121 df-lp 23159 df-perf 23160 df-cn 23250 df-cnp 23251 df-haus 23338 df-cmp 23410 df-tx 23585 df-hmeo 23778 df-fil 23869 df-fm 23961 df-flim 23962 df-flf 23963 df-xms 24345 df-ms 24346 df-tms 24347 df-cncf 24917 df-ovol 25512 df-vol 25513 df-mbf 25667 df-itg1 25668 df-itg2 25669 df-ibl 25670 df-itg 25671 df-0p 25718 df-limc 25915 df-dv 25916 df-dvn 25917 df-ply 26241 df-coe 26243 df-dgr 26244 df-log 26612 df-cxp 26613

This theorem is referenced by: etransclem48 46237

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