etransclem47 - Mathbox for Glauco Siliprandi

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Theorem etransclem47 46277

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 3986	. . . . . 6 ⊢ ℝ ⊆ ℝ
8	7	a1i 11	. . . . 5 ⊢ (𝜑 → ℝ ⊆ ℝ)
9		reelprrecn 11226	. . . . . 6 ⊢ ℝ ∈ {ℝ, ℂ}
10	9	a1i 11	. . . . 5 ⊢ (𝜑 → ℝ ∈ {ℝ, ℂ})
11		reopn 45285	. . . . . . 7 ⊢ ℝ ∈ (topGen‘ran (,))
12		tgioo4 24749	. . . . . . 7 ⊢ (topGen‘ran (,)) = ((TopOpen‘ℂ_fld) ↾_t ℝ)
13	11, 12	eleqtri 2833	. . . . . 6 ⊢ ℝ ∈ ((TopOpen‘ℂ_fld) ↾_t ℝ)
14	13	a1i 11	. . . . 5 ⊢ (𝜑 → ℝ ∈ ((TopOpen‘ℂ_fld) ↾_t ℝ))
15		etransclem47.p	. . . . . 6 ⊢ (𝜑 → 𝑃 ∈ ℙ)
16		prmnn 16698	. . . . . 6 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ ℕ)
17	15, 16	syl 17	. . . . 5 ⊢ (𝜑 → 𝑃 ∈ ℕ)
18		etransclem47.f	. . . . 5 ⊢ 𝐹 = (𝑥 ∈ ℝ ↦ ((𝑥↑(𝑃 − 1)) · ∏𝑗 ∈ (1...𝑀)((𝑥 − 𝑗)↑𝑃)))
19		etransclem47.l	. . . . 5 ⊢ 𝐿 = Σ𝑗 ∈ (0...𝑀)(((𝐴‘𝑗) · (e↑_𝑐𝑗)) · ∫(0(,)𝑗)((e↑_𝑐-𝑥) · (𝐹‘𝑥)) d𝑥)
20		eqid 2736	. . . . 5 ⊢ ((𝑀 · 𝑃) + (𝑃 − 1)) = ((𝑀 · 𝑃) + (𝑃 − 1))
21		fveq2 6881	. . . . . . 7 ⊢ (𝑦 = 𝑥 → (((ℝ D^𝑛 𝐹)‘𝑖)‘𝑦) = (((ℝ D^𝑛 𝐹)‘𝑖)‘𝑥))
22	21	sumeq2sdv 15724	. . . . . 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))
23	22	cbvmptv 5230	. . . . 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))
24		negeq 11479	. . . . . . . . 9 ⊢ (𝑧 = 𝑥 → -𝑧 = -𝑥)
25	24	oveq2d 7426	. . . . . . . 8 ⊢ (𝑧 = 𝑥 → (e↑_𝑐-𝑧) = (e↑_𝑐-𝑥))
26		fveq2 6881	. . . . . . . 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))
27	25, 26	oveq12d 7428	. . . . . . 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)))
28	27	negeqd 11481	. . . . . 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)))
29	28	cbvmptv 5230	. . . . 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)))
30	3, 4, 5, 6, 8, 10, 14, 17, 18, 19, 20, 23, 29	etransclem46 46276	. . . 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))))
31		fzfid 13996	. . . . . . . 8 ⊢ (𝜑 → (0...𝑀) ∈ Fin)
32		fzfid 13996	. . . . . . . 8 ⊢ (𝜑 → (0...((𝑀 · 𝑃) + (𝑃 − 1))) ∈ Fin)
33		xpfi 9335	. . . . . . . 8 ⊢ (((0...𝑀) ∈ Fin ∧ (0...((𝑀 · 𝑃) + (𝑃 − 1))) ∈ Fin) → ((0...𝑀) × (0...((𝑀 · 𝑃) + (𝑃 − 1)))) ∈ Fin)
34	31, 32, 33	syl2anc 584	. . . . . . 7 ⊢ (𝜑 → ((0...𝑀) × (0...((𝑀 · 𝑃) + (𝑃 − 1)))) ∈ Fin)
35	3	eldifad 3943	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑄 ∈ (Poly‘ℤ))
36		0zd 12605	. . . . . . . . . . . 12 ⊢ (𝜑 → 0 ∈ ℤ)
37	5	coef2 26193	. . . . . . . . . . . 12 ⊢ ((𝑄 ∈ (Poly‘ℤ) ∧ 0 ∈ ℤ) → 𝐴:ℕ₀⟶ℤ)
38	35, 36, 37	syl2anc 584	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐴:ℕ₀⟶ℤ)
39	38	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ ((0...𝑀) × (0...((𝑀 · 𝑃) + (𝑃 − 1))))) → 𝐴:ℕ₀⟶ℤ)
40		xp1st 8025	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ ((0...𝑀) × (0...((𝑀 · 𝑃) + (𝑃 − 1)))) → (1^st ‘𝑘) ∈ (0...𝑀))
41		elfznn0 13642	. . . . . . . . . . . 12 ⊢ ((1^st ‘𝑘) ∈ (0...𝑀) → (1^st ‘𝑘) ∈ ℕ₀)
42	40, 41	syl 17	. . . . . . . . . . 11 ⊢ (𝑘 ∈ ((0...𝑀) × (0...((𝑀 · 𝑃) + (𝑃 − 1)))) → (1^st ‘𝑘) ∈ ℕ₀)
43	42	adantl 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ ((0...𝑀) × (0...((𝑀 · 𝑃) + (𝑃 − 1))))) → (1^st ‘𝑘) ∈ ℕ₀)
44	39, 43	ffvelcdmd 7080	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ ((0...𝑀) × (0...((𝑀 · 𝑃) + (𝑃 − 1))))) → (𝐴‘(1^st ‘𝑘)) ∈ ℤ)
45	44	zcnd 12703	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ ((0...𝑀) × (0...((𝑀 · 𝑃) + (𝑃 − 1))))) → (𝐴‘(1^st ‘𝑘)) ∈ ℂ)
46	9	a1i 11	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ ((0...𝑀) × (0...((𝑀 · 𝑃) + (𝑃 − 1))))) → ℝ ∈ {ℝ, ℂ})
47	13	a1i 11	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ ((0...𝑀) × (0...((𝑀 · 𝑃) + (𝑃 − 1))))) → ℝ ∈ ((TopOpen‘ℂ_fld) ↾_t ℝ))
48	17	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ ((0...𝑀) × (0...((𝑀 · 𝑃) + (𝑃 − 1))))) → 𝑃 ∈ ℕ)
49		dgrcl 26195	. . . . . . . . . . . . 13 ⊢ (𝑄 ∈ (Poly‘ℤ) → (deg‘𝑄) ∈ ℕ₀)
50	35, 49	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → (deg‘𝑄) ∈ ℕ₀)
51	6, 50	eqeltrid 2839	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑀 ∈ ℕ₀)
52	51	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ ((0...𝑀) × (0...((𝑀 · 𝑃) + (𝑃 − 1))))) → 𝑀 ∈ ℕ₀)
53		xp2nd 8026	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ ((0...𝑀) × (0...((𝑀 · 𝑃) + (𝑃 − 1)))) → (2^nd ‘𝑘) ∈ (0...((𝑀 · 𝑃) + (𝑃 − 1))))
54		elfznn0 13642	. . . . . . . . . . . 12 ⊢ ((2^nd ‘𝑘) ∈ (0...((𝑀 · 𝑃) + (𝑃 − 1))) → (2^nd ‘𝑘) ∈ ℕ₀)
55	53, 54	syl 17	. . . . . . . . . . 11 ⊢ (𝑘 ∈ ((0...𝑀) × (0...((𝑀 · 𝑃) + (𝑃 − 1)))) → (2^nd ‘𝑘) ∈ ℕ₀)
56	55	adantl 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ ((0...𝑀) × (0...((𝑀 · 𝑃) + (𝑃 − 1))))) → (2^nd ‘𝑘) ∈ ℕ₀)
57	46, 47, 48, 52, 18, 56	etransclem33 46263	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ ((0...𝑀) × (0...((𝑀 · 𝑃) + (𝑃 − 1))))) → ((ℝ D^𝑛 𝐹)‘(2^nd ‘𝑘)):ℝ⟶ℂ)
58	43	nn0red 12568	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ ((0...𝑀) × (0...((𝑀 · 𝑃) + (𝑃 − 1))))) → (1^st ‘𝑘) ∈ ℝ)
59	57, 58	ffvelcdmd 7080	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ ((0...𝑀) × (0...((𝑀 · 𝑃) + (𝑃 − 1))))) → (((ℝ D^𝑛 𝐹)‘(2^nd ‘𝑘))‘(1^st ‘𝑘)) ∈ ℂ)
60	45, 59	mulcld 11260	. . . . . . 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)
61	34, 60	fsumcl 15754	. . . . . 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)
62		nnm1nn0 12547	. . . . . . . . 9 ⊢ (𝑃 ∈ ℕ → (𝑃 − 1) ∈ ℕ₀)
63	17, 62	syl 17	. . . . . . . 8 ⊢ (𝜑 → (𝑃 − 1) ∈ ℕ₀)
64	63	faccld 14307	. . . . . . 7 ⊢ (𝜑 → (!‘(𝑃 − 1)) ∈ ℕ)
65	64	nncnd 12261	. . . . . 6 ⊢ (𝜑 → (!‘(𝑃 − 1)) ∈ ℂ)
66	64	nnne0d 12295	. . . . . 6 ⊢ (𝜑 → (!‘(𝑃 − 1)) ≠ 0)
67	61, 65, 66	divnegd 12035	. . . . 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))))
68	67	eqcomd 2742	. . . 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))))
69	2, 30, 68	3eqtrd 2775	. . 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))))
70		eqid 2736	. . . . 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)))
71	17, 51, 18, 38, 70	etransclem45 46275	. . . 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)
72	71	znegcld 12704	. . 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)
73	69, 72	eqeltrd 2835	. 2 ⊢ (𝜑 → 𝐾 ∈ ℤ)
74	1, 30	eqtrid 2783	. . 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))))
75	61, 65, 66	divcld 12022	. . . . 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)
76		etransclem47.a0	. . . . . 6 ⊢ (𝜑 → (𝐴‘0) ≠ 0)
77		etransclem47.ap	. . . . . 6 ⊢ (𝜑 → (abs‘(𝐴‘0)) < 𝑃)
78		etransclem47.mp	. . . . . 6 ⊢ (𝜑 → (!‘𝑀) < 𝑃)
79	38, 76, 51, 15, 77, 78, 18, 70	etransclem44 46274	. . . . 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)
80	75, 79	negne0d 11597	. . . 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)
81	68, 80	eqnetrd 3000	. . 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)
82	74, 81	eqnetrd 3000	. 2 ⊢ (𝜑 → 𝐾 ≠ 0)
83		eldifsni 4771	. . . . . 6 ⊢ (𝑄 ∈ ((Poly‘ℤ) ∖ {0_𝑝}) → 𝑄 ≠ 0_𝑝)
84	3, 83	syl 17	. . . . 5 ⊢ (𝜑 → 𝑄 ≠ 0_𝑝)
85		ere 16110	. . . . . . 7 ⊢ e ∈ ℝ
86	85	recni 11254	. . . . . 6 ⊢ e ∈ ℂ
87	86	a1i 11	. . . . 5 ⊢ (𝜑 → e ∈ ℂ)
88		dgrnznn 26209	. . . . 5 ⊢ (((𝑄 ∈ (Poly‘ℤ) ∧ 𝑄 ≠ 0_𝑝) ∧ (e ∈ ℂ ∧ (𝑄‘e) = 0)) → (deg‘𝑄) ∈ ℕ)
89	35, 84, 87, 4, 88	syl22anc 838	. . . 4 ⊢ (𝜑 → (deg‘𝑄) ∈ ℕ)
90	6, 89	eqeltrid 2839	. . 3 ⊢ (𝜑 → 𝑀 ∈ ℕ)
91		etransclem47.9	. . 3 ⊢ (𝜑 → (Σ𝑗 ∈ (0...𝑀)((abs‘((𝐴‘𝑗) · (e↑_𝑐𝑗))) · (𝑀 · (𝑀↑(𝑀 + 1)))) · (((𝑀↑(𝑀 + 1))↑(𝑃 − 1)) / (!‘(𝑃 − 1)))) < 1)
92	38, 19, 1, 17, 90, 18, 91	etransclem23 46253	. 2 ⊢ (𝜑 → (abs‘𝐾) < 1)
93		neeq1 2995	. . . 4 ⊢ (𝑘 = 𝐾 → (𝑘 ≠ 0 ↔ 𝐾 ≠ 0))
94		fveq2 6881	. . . . 5 ⊢ (𝑘 = 𝐾 → (abs‘𝑘) = (abs‘𝐾))
95	94	breq1d 5134	. . . 4 ⊢ (𝑘 = 𝐾 → ((abs‘𝑘) < 1 ↔ (abs‘𝐾) < 1))
96	93, 95	anbi12d 632	. . 3 ⊢ (𝑘 = 𝐾 → ((𝑘 ≠ 0 ∧ (abs‘𝑘) < 1) ↔ (𝐾 ≠ 0 ∧ (abs‘𝐾) < 1)))
97	96	rspcev 3606	. 2 ⊢ ((𝐾 ∈ ℤ ∧ (𝐾 ≠ 0 ∧ (abs‘𝐾) < 1)) → ∃𝑘 ∈ ℤ (𝑘 ≠ 0 ∧ (abs‘𝑘) < 1))
98	73, 82, 92, 97	syl12anc 836	1 ⊢ (𝜑 → ∃𝑘 ∈ ℤ (𝑘 ≠ 0 ∧ (abs‘𝑘) < 1))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2109 ≠ wne 2933 ∃wrex 3061 ∖ cdif 3928 ⊆ wss 3931 {csn 4606 {cpr 4608 class class class wbr 5124 ↦ cmpt 5206 × cxp 5657 ran crn 5660 ⟶wf 6532 ‘cfv 6536 (class class class)co 7410 1^st c1st 7991 2^nd c2nd 7992 Fincfn 8964 ℂcc 11132 ℝcr 11133 0cc0 11134 1c1 11135 + caddc 11137 · cmul 11139 < clt 11274 − cmin 11471 -cneg 11472 / cdiv 11899 ℕcn 12245 ℕ₀cn0 12506 ℤcz 12593 (,)cioo 13367 [,]cicc 13370 ...cfz 13529 ↑cexp 14084 !cfa 14296 abscabs 15258 Σcsu 15707 ∏cprod 15924 eceu 16083 ℙcprime 16695 ↾_t crest 17439 TopOpenctopn 17440 topGenctg 17456 ℂ_fldccnfld 21320 ∫citg 25576 0_𝑝c0p 25627 D^𝑛 cdvn 25822 Polycply 26146 coeffccoe 26148 degcdgr 26149 ↑_𝑐ccxp 26521

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 2708 ax-rep 5254 ax-sep 5271 ax-nul 5281 ax-pow 5340 ax-pr 5407 ax-un 7734 ax-inf2 9660 ax-cc 10454 ax-cnex 11190 ax-resscn 11191 ax-1cn 11192 ax-icn 11193 ax-addcl 11194 ax-addrcl 11195 ax-mulcl 11196 ax-mulrcl 11197 ax-mulcom 11198 ax-addass 11199 ax-mulass 11200 ax-distr 11201 ax-i2m1 11202 ax-1ne0 11203 ax-1rid 11204 ax-rnegex 11205 ax-rrecex 11206 ax-cnre 11207 ax-pre-lttri 11208 ax-pre-lttrn 11209 ax-pre-ltadd 11210 ax-pre-mulgt0 11211 ax-pre-sup 11212 ax-addf 11213

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 2540 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2810 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3062 df-rmo 3364 df-reu 3365 df-rab 3421 df-v 3466 df-sbc 3771 df-csb 3880 df-dif 3934 df-un 3936 df-in 3938 df-ss 3948 df-pss 3951 df-symdif 4233 df-nul 4314 df-if 4506 df-pw 4582 df-sn 4607 df-pr 4609 df-tp 4611 df-op 4613 df-uni 4889 df-int 4928 df-iun 4974 df-iin 4975 df-disj 5092 df-br 5125 df-opab 5187 df-mpt 5207 df-tr 5235 df-id 5553 df-eprel 5558 df-po 5566 df-so 5567 df-fr 5611 df-se 5612 df-we 5613 df-xp 5665 df-rel 5666 df-cnv 5667 df-co 5668 df-dm 5669 df-rn 5670 df-res 5671 df-ima 5672 df-pred 6295 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6489 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-isom 6545 df-riota 7367 df-ov 7413 df-oprab 7414 df-mpo 7415 df-of 7676 df-ofr 7677 df-om 7867 df-1st 7993 df-2nd 7994 df-supp 8165 df-frecs 8285 df-wrecs 8316 df-recs 8390 df-rdg 8429 df-1o 8485 df-2o 8486 df-oadd 8489 df-omul 8490 df-er 8724 df-map 8847 df-pm 8848 df-ixp 8917 df-en 8965 df-dom 8966 df-sdom 8967 df-fin 8968 df-fsupp 9379 df-fi 9428 df-sup 9459 df-inf 9460 df-oi 9529 df-dju 9920 df-card 9958 df-acn 9961 df-pnf 11276 df-mnf 11277 df-xr 11278 df-ltxr 11279 df-le 11280 df-sub 11473 df-neg 11474 df-div 11900 df-nn 12246 df-2 12308 df-3 12309 df-4 12310 df-5 12311 df-6 12312 df-7 12313 df-8 12314 df-9 12315 df-n0 12507 df-z 12594 df-dec 12714 df-uz 12858 df-q 12970 df-rp 13014 df-xneg 13133 df-xadd 13134 df-xmul 13135 df-ioo 13371 df-ioc 13372 df-ico 13373 df-icc 13374 df-fz 13530 df-fzo 13677 df-fl 13814 df-mod 13892 df-seq 14025 df-exp 14085 df-fac 14297 df-bc 14326 df-hash 14354 df-shft 15091 df-cj 15123 df-re 15124 df-im 15125 df-sqrt 15259 df-abs 15260 df-limsup 15492 df-clim 15509 df-rlim 15510 df-sum 15708 df-prod 15925 df-ef 16088 df-e 16089 df-sin 16090 df-cos 16091 df-tan 16092 df-pi 16093 df-dvds 16278 df-gcd 16519 df-prm 16696 df-struct 17171 df-sets 17188 df-slot 17206 df-ndx 17218 df-base 17234 df-ress 17257 df-plusg 17289 df-mulr 17290 df-starv 17291 df-sca 17292 df-vsca 17293 df-ip 17294 df-tset 17295 df-ple 17296 df-ds 17298 df-unif 17299 df-hom 17300 df-cco 17301 df-rest 17441 df-topn 17442 df-0g 17460 df-gsum 17461 df-topgen 17462 df-pt 17463 df-prds 17466 df-xrs 17521 df-qtop 17526 df-imas 17527 df-xps 17529 df-mre 17603 df-mrc 17604 df-acs 17606 df-mgm 18623 df-sgrp 18702 df-mnd 18718 df-submnd 18767 df-mulg 19056 df-cntz 19305 df-cmn 19768 df-psmet 21312 df-xmet 21313 df-met 21314 df-bl 21315 df-mopn 21316 df-fbas 21317 df-fg 21318 df-cnfld 21321 df-top 22837 df-topon 22854 df-topsp 22876 df-bases 22889 df-cld 22962 df-ntr 22963 df-cls 22964 df-nei 23041 df-lp 23079 df-perf 23080 df-cn 23170 df-cnp 23171 df-haus 23258 df-cmp 23330 df-tx 23505 df-hmeo 23698 df-fil 23789 df-fm 23881 df-flim 23882 df-flf 23883 df-xms 24264 df-ms 24265 df-tms 24266 df-cncf 24827 df-ovol 25422 df-vol 25423 df-mbf 25577 df-itg1 25578 df-itg2 25579 df-ibl 25580 df-itg 25581 df-0p 25628 df-limc 25824 df-dv 25825 df-dvn 25826 df-ply 26150 df-coe 26152 df-dgr 26153 df-log 26522 df-cxp 26523

This theorem is referenced by: etransclem48 46278

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