etransclem47 - Mathbox for Glauco Siliprandi

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Theorem etransclem47 46202

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 4031	. . . . . 6 ⊢ ℝ ⊆ ℝ
8	7	a1i 11	. . . . 5 ⊢ (𝜑 → ℝ ⊆ ℝ)
9		reelprrecn 11276	. . . . . 6 ⊢ ℝ ∈ {ℝ, ℂ}
10	9	a1i 11	. . . . 5 ⊢ (𝜑 → ℝ ∈ {ℝ, ℂ})
11		reopn 45204	. . . . . . 7 ⊢ ℝ ∈ (topGen‘ran (,))
12		eqid 2740	. . . . . . . 8 ⊢ (TopOpen‘ℂ_fld) = (TopOpen‘ℂ_fld)
13	12	tgioo2 24844	. . . . . . 7 ⊢ (topGen‘ran (,)) = ((TopOpen‘ℂ_fld) ↾_t ℝ)
14	11, 13	eleqtri 2842	. . . . . 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 16721	. . . . . 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 2740	. . . . 5 ⊢ ((𝑀 · 𝑃) + (𝑃 − 1)) = ((𝑀 · 𝑃) + (𝑃 − 1))
22		fveq2 6920	. . . . . . 7 ⊢ (𝑦 = 𝑥 → (((ℝ D^𝑛 𝐹)‘𝑖)‘𝑦) = (((ℝ D^𝑛 𝐹)‘𝑖)‘𝑥))
23	22	sumeq2sdv 15751	. . . . . 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 5279	. . . . 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 11528	. . . . . . . . 9 ⊢ (𝑧 = 𝑥 → -𝑧 = -𝑥)
26	25	oveq2d 7464	. . . . . . . 8 ⊢ (𝑧 = 𝑥 → (e↑_𝑐-𝑧) = (e↑_𝑐-𝑥))
27		fveq2 6920	. . . . . . . 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 7466	. . . . . . 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 11530	. . . . . 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 5279	. . . . 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 46201	. . . 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 14024	. . . . . . . 8 ⊢ (𝜑 → (0...𝑀) ∈ Fin)
33		fzfid 14024	. . . . . . . 8 ⊢ (𝜑 → (0...((𝑀 · 𝑃) + (𝑃 − 1))) ∈ Fin)
34		xpfi 9386	. . . . . . . 8 ⊢ (((0...𝑀) ∈ Fin ∧ (0...((𝑀 · 𝑃) + (𝑃 − 1))) ∈ Fin) → ((0...𝑀) × (0...((𝑀 · 𝑃) + (𝑃 − 1)))) ∈ Fin)
35	32, 33, 34	syl2anc 583	. . . . . . 7 ⊢ (𝜑 → ((0...𝑀) × (0...((𝑀 · 𝑃) + (𝑃 − 1)))) ∈ Fin)
36	3	eldifad 3988	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑄 ∈ (Poly‘ℤ))
37		0zd 12651	. . . . . . . . . . . 12 ⊢ (𝜑 → 0 ∈ ℤ)
38	5	coef2 26290	. . . . . . . . . . . 12 ⊢ ((𝑄 ∈ (Poly‘ℤ) ∧ 0 ∈ ℤ) → 𝐴:ℕ₀⟶ℤ)
39	36, 37, 38	syl2anc 583	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐴:ℕ₀⟶ℤ)
40	39	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ ((0...𝑀) × (0...((𝑀 · 𝑃) + (𝑃 − 1))))) → 𝐴:ℕ₀⟶ℤ)
41		xp1st 8062	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ ((0...𝑀) × (0...((𝑀 · 𝑃) + (𝑃 − 1)))) → (1^st ‘𝑘) ∈ (0...𝑀))
42		elfznn0 13677	. . . . . . . . . . . 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 7119	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ ((0...𝑀) × (0...((𝑀 · 𝑃) + (𝑃 − 1))))) → (𝐴‘(1^st ‘𝑘)) ∈ ℤ)
46	45	zcnd 12748	. . . . . . . 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 26292	. . . . . . . . . . . . 13 ⊢ (𝑄 ∈ (Poly‘ℤ) → (deg‘𝑄) ∈ ℕ₀)
51	36, 50	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → (deg‘𝑄) ∈ ℕ₀)
52	6, 51	eqeltrid 2848	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑀 ∈ ℕ₀)
53	52	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ ((0...𝑀) × (0...((𝑀 · 𝑃) + (𝑃 − 1))))) → 𝑀 ∈ ℕ₀)
54		xp2nd 8063	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ ((0...𝑀) × (0...((𝑀 · 𝑃) + (𝑃 − 1)))) → (2^nd ‘𝑘) ∈ (0...((𝑀 · 𝑃) + (𝑃 − 1))))
55		elfznn0 13677	. . . . . . . . . . . 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 46188	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ ((0...𝑀) × (0...((𝑀 · 𝑃) + (𝑃 − 1))))) → ((ℝ D^𝑛 𝐹)‘(2^nd ‘𝑘)):ℝ⟶ℂ)
59	44	nn0red 12614	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ ((0...𝑀) × (0...((𝑀 · 𝑃) + (𝑃 − 1))))) → (1^st ‘𝑘) ∈ ℝ)
60	58, 59	ffvelcdmd 7119	. . . . . . . 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 11310	. . . . . . 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 15781	. . . . . 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 12594	. . . . . . . . 9 ⊢ (𝑃 ∈ ℕ → (𝑃 − 1) ∈ ℕ₀)
64	18, 63	syl 17	. . . . . . . 8 ⊢ (𝜑 → (𝑃 − 1) ∈ ℕ₀)
65	64	faccld 14333	. . . . . . 7 ⊢ (𝜑 → (!‘(𝑃 − 1)) ∈ ℕ)
66	65	nncnd 12309	. . . . . 6 ⊢ (𝜑 → (!‘(𝑃 − 1)) ∈ ℂ)
67	65	nnne0d 12343	. . . . . 6 ⊢ (𝜑 → (!‘(𝑃 − 1)) ≠ 0)
68	62, 66, 67	divnegd 12083	. . . . 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 2746	. . . 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 2784	. . 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 2740	. . . . 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 46200	. . . 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 12749	. . 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 2844	. 2 ⊢ (𝜑 → 𝐾 ∈ ℤ)
75	1, 31	eqtrid 2792	. . 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 12070	. . . . 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 46199	. . . . 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 11645	. . . 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 3014	. . 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 3014	. 2 ⊢ (𝜑 → 𝐾 ≠ 0)
84		eldifsni 4815	. . . . . 6 ⊢ (𝑄 ∈ ((Poly‘ℤ) ∖ {0_𝑝}) → 𝑄 ≠ 0_𝑝)
85	3, 84	syl 17	. . . . 5 ⊢ (𝜑 → 𝑄 ≠ 0_𝑝)
86		ere 16137	. . . . . . 7 ⊢ e ∈ ℝ
87	86	recni 11304	. . . . . 6 ⊢ e ∈ ℂ
88	87	a1i 11	. . . . 5 ⊢ (𝜑 → e ∈ ℂ)
89		dgrnznn 26306	. . . . 5 ⊢ (((𝑄 ∈ (Poly‘ℤ) ∧ 𝑄 ≠ 0_𝑝) ∧ (e ∈ ℂ ∧ (𝑄‘e) = 0)) → (deg‘𝑄) ∈ ℕ)
90	36, 85, 88, 4, 89	syl22anc 838	. . . 4 ⊢ (𝜑 → (deg‘𝑄) ∈ ℕ)
91	6, 90	eqeltrid 2848	. . 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 46178	. 2 ⊢ (𝜑 → (abs‘𝐾) < 1)
94		neeq1 3009	. . . 4 ⊢ (𝑘 = 𝐾 → (𝑘 ≠ 0 ↔ 𝐾 ≠ 0))
95		fveq2 6920	. . . . 5 ⊢ (𝑘 = 𝐾 → (abs‘𝑘) = (abs‘𝐾))
96	95	breq1d 5176	. . . 4 ⊢ (𝑘 = 𝐾 → ((abs‘𝑘) < 1 ↔ (abs‘𝐾) < 1))
97	94, 96	anbi12d 631	. . 3 ⊢ (𝑘 = 𝐾 → ((𝑘 ≠ 0 ∧ (abs‘𝑘) < 1) ↔ (𝐾 ≠ 0 ∧ (abs‘𝐾) < 1)))
98	97	rspcev 3635	. 2 ⊢ ((𝐾 ∈ ℤ ∧ (𝐾 ≠ 0 ∧ (abs‘𝐾) < 1)) → ∃𝑘 ∈ ℤ (𝑘 ≠ 0 ∧ (abs‘𝑘) < 1))
99	74, 83, 93, 98	syl12anc 836	1 ⊢ (𝜑 → ∃𝑘 ∈ ℤ (𝑘 ≠ 0 ∧ (abs‘𝑘) < 1))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1537 ∈ wcel 2108 ≠ wne 2946 ∃wrex 3076 ∖ cdif 3973 ⊆ wss 3976 {csn 4648 {cpr 4650 class class class wbr 5166 ↦ cmpt 5249 × cxp 5698 ran crn 5701 ⟶wf 6569 ‘cfv 6573 (class class class)co 7448 1^st c1st 8028 2^nd c2nd 8029 Fincfn 9003 ℂcc 11182 ℝcr 11183 0cc0 11184 1c1 11185 + caddc 11187 · cmul 11189 < clt 11324 − cmin 11520 -cneg 11521 / cdiv 11947 ℕcn 12293 ℕ₀cn0 12553 ℤcz 12639 (,)cioo 13407 [,]cicc 13410 ...cfz 13567 ↑cexp 14112 !cfa 14322 abscabs 15283 Σcsu 15734 ∏cprod 15951 eceu 16110 ℙcprime 16718 ↾_t crest 17480 TopOpenctopn 17481 topGenctg 17497 ℂ_fldccnfld 21387 ∫citg 25672 0_𝑝c0p 25723 D^𝑛 cdvn 25919 Polycply 26243 coeffccoe 26245 degcdgr 26246 ↑_𝑐ccxp 26615

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-cc 10504 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-symdif 4272 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-disj 5134 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-ofr 7715 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-oadd 8526 df-omul 8527 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-dju 9970 df-card 10008 df-acn 10011 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-ioo 13411 df-ioc 13412 df-ico 13413 df-icc 13414 df-fz 13568 df-fzo 13712 df-fl 13843 df-mod 13921 df-seq 14053 df-exp 14113 df-fac 14323 df-bc 14352 df-hash 14380 df-shft 15116 df-cj 15148 df-re 15149 df-im 15150 df-sqrt 15284 df-abs 15285 df-limsup 15517 df-clim 15534 df-rlim 15535 df-sum 15735 df-prod 15952 df-ef 16115 df-e 16116 df-sin 16117 df-cos 16118 df-tan 16119 df-pi 16120 df-dvds 16303 df-gcd 16541 df-prm 16719 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-cmp 23416 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-ovol 25518 df-vol 25519 df-mbf 25673 df-itg1 25674 df-itg2 25675 df-ibl 25676 df-itg 25677 df-0p 25724 df-limc 25921 df-dv 25922 df-dvn 25923 df-ply 26247 df-coe 26249 df-dgr 26250 df-log 26616 df-cxp 26617

This theorem is referenced by: etransclem48 46203

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