etransclem47 - Mathbox for Glauco Siliprandi

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Theorem etransclem47 46730

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 3945	. . . . . 6 ⊢ ℝ ⊆ ℝ
8	7	a1i 11	. . . . 5 ⊢ (𝜑 → ℝ ⊆ ℝ)
9		reelprrecn 11124	. . . . . 6 ⊢ ℝ ∈ {ℝ, ℂ}
10	9	a1i 11	. . . . 5 ⊢ (𝜑 → ℝ ∈ {ℝ, ℂ})
11		reopn 45743	. . . . . . 7 ⊢ ℝ ∈ (topGen‘ran (,))
12		tgioo4 24783	. . . . . . 7 ⊢ (topGen‘ran (,)) = ((TopOpen‘ℂ_fld) ↾_t ℝ)
13	11, 12	eleqtri 2835	. . . . . 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 16637	. . . . . 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 2737	. . . . 5 ⊢ ((𝑀 · 𝑃) + (𝑃 − 1)) = ((𝑀 · 𝑃) + (𝑃 − 1))
21		fveq2 6835	. . . . . . 7 ⊢ (𝑦 = 𝑥 → (((ℝ D^𝑛 𝐹)‘𝑖)‘𝑦) = (((ℝ D^𝑛 𝐹)‘𝑖)‘𝑥))
22	21	sumeq2sdv 15659	. . . . . 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 5190	. . . . 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 11379	. . . . . . . . 9 ⊢ (𝑧 = 𝑥 → -𝑧 = -𝑥)
25	24	oveq2d 7377	. . . . . . . 8 ⊢ (𝑧 = 𝑥 → (e↑_𝑐-𝑧) = (e↑_𝑐-𝑥))
26		fveq2 6835	. . . . . . . 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 7379	. . . . . . 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 11381	. . . . . 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 5190	. . . . 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 46729	. . . 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 13929	. . . . . . . 8 ⊢ (𝜑 → (0...𝑀) ∈ Fin)
32		fzfid 13929	. . . . . . . 8 ⊢ (𝜑 → (0...((𝑀 · 𝑃) + (𝑃 − 1))) ∈ Fin)
33		xpfi 9224	. . . . . . . 8 ⊢ (((0...𝑀) ∈ Fin ∧ (0...((𝑀 · 𝑃) + (𝑃 − 1))) ∈ Fin) → ((0...𝑀) × (0...((𝑀 · 𝑃) + (𝑃 − 1)))) ∈ Fin)
34	31, 32, 33	syl2anc 585	. . . . . . 7 ⊢ (𝜑 → ((0...𝑀) × (0...((𝑀 · 𝑃) + (𝑃 − 1)))) ∈ Fin)
35	3	eldifad 3902	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑄 ∈ (Poly‘ℤ))
36		0zd 12530	. . . . . . . . . . . 12 ⊢ (𝜑 → 0 ∈ ℤ)
37	5	coef2 26209	. . . . . . . . . . . 12 ⊢ ((𝑄 ∈ (Poly‘ℤ) ∧ 0 ∈ ℤ) → 𝐴:ℕ₀⟶ℤ)
38	35, 36, 37	syl2anc 585	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐴:ℕ₀⟶ℤ)
39	38	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ ((0...𝑀) × (0...((𝑀 · 𝑃) + (𝑃 − 1))))) → 𝐴:ℕ₀⟶ℤ)
40		xp1st 7968	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ ((0...𝑀) × (0...((𝑀 · 𝑃) + (𝑃 − 1)))) → (1^st ‘𝑘) ∈ (0...𝑀))
41		elfznn0 13568	. . . . . . . . . . . 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 7032	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ ((0...𝑀) × (0...((𝑀 · 𝑃) + (𝑃 − 1))))) → (𝐴‘(1^st ‘𝑘)) ∈ ℤ)
45	44	zcnd 12628	. . . . . . . 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 26211	. . . . . . . . . . . . 13 ⊢ (𝑄 ∈ (Poly‘ℤ) → (deg‘𝑄) ∈ ℕ₀)
50	35, 49	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → (deg‘𝑄) ∈ ℕ₀)
51	6, 50	eqeltrid 2841	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑀 ∈ ℕ₀)
52	51	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ ((0...𝑀) × (0...((𝑀 · 𝑃) + (𝑃 − 1))))) → 𝑀 ∈ ℕ₀)
53		xp2nd 7969	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ ((0...𝑀) × (0...((𝑀 · 𝑃) + (𝑃 − 1)))) → (2^nd ‘𝑘) ∈ (0...((𝑀 · 𝑃) + (𝑃 − 1))))
54		elfznn0 13568	. . . . . . . . . . . 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 46716	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ ((0...𝑀) × (0...((𝑀 · 𝑃) + (𝑃 − 1))))) → ((ℝ D^𝑛 𝐹)‘(2^nd ‘𝑘)):ℝ⟶ℂ)
58	43	nn0red 12493	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ ((0...𝑀) × (0...((𝑀 · 𝑃) + (𝑃 − 1))))) → (1^st ‘𝑘) ∈ ℝ)
59	57, 58	ffvelcdmd 7032	. . . . . . . 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 11159	. . . . . . 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 15689	. . . . . 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 12472	. . . . . . . . 9 ⊢ (𝑃 ∈ ℕ → (𝑃 − 1) ∈ ℕ₀)
63	17, 62	syl 17	. . . . . . . 8 ⊢ (𝜑 → (𝑃 − 1) ∈ ℕ₀)
64	63	faccld 14240	. . . . . . 7 ⊢ (𝜑 → (!‘(𝑃 − 1)) ∈ ℕ)
65	64	nncnd 12184	. . . . . 6 ⊢ (𝜑 → (!‘(𝑃 − 1)) ∈ ℂ)
66	64	nnne0d 12221	. . . . . 6 ⊢ (𝜑 → (!‘(𝑃 − 1)) ≠ 0)
67	61, 65, 66	divnegd 11938	. . . . 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 2743	. . . 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 2776	. . 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 2737	. . . . 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 46728	. . . 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 12629	. . 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 2837	. 2 ⊢ (𝜑 → 𝐾 ∈ ℤ)
74	1, 30	eqtrid 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))))
75	61, 65, 66	divcld 11925	. . . . 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 46727	. . . . 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 11497	. . . 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 4734	. . . . . 6 ⊢ (𝑄 ∈ ((Poly‘ℤ) ∖ {0_𝑝}) → 𝑄 ≠ 0_𝑝)
84	3, 83	syl 17	. . . . 5 ⊢ (𝜑 → 𝑄 ≠ 0_𝑝)
85		ere 16048	. . . . . . 7 ⊢ e ∈ ℝ
86	85	recni 11153	. . . . . 6 ⊢ e ∈ ℂ
87	86	a1i 11	. . . . 5 ⊢ (𝜑 → e ∈ ℂ)
88		dgrnznn 26225	. . . . 5 ⊢ (((𝑄 ∈ (Poly‘ℤ) ∧ 𝑄 ≠ 0_𝑝) ∧ (e ∈ ℂ ∧ (𝑄‘e) = 0)) → (deg‘𝑄) ∈ ℕ)
89	35, 84, 87, 4, 88	syl22anc 839	. . . 4 ⊢ (𝜑 → (deg‘𝑄) ∈ ℕ)
90	6, 89	eqeltrid 2841	. . 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 46706	. 2 ⊢ (𝜑 → (abs‘𝐾) < 1)
93		neeq1 2995	. . . 4 ⊢ (𝑘 = 𝐾 → (𝑘 ≠ 0 ↔ 𝐾 ≠ 0))
94		fveq2 6835	. . . . 5 ⊢ (𝑘 = 𝐾 → (abs‘𝑘) = (abs‘𝐾))
95	94	breq1d 5096	. . . 4 ⊢ (𝑘 = 𝐾 → ((abs‘𝑘) < 1 ↔ (abs‘𝐾) < 1))
96	93, 95	anbi12d 633	. . 3 ⊢ (𝑘 = 𝐾 → ((𝑘 ≠ 0 ∧ (abs‘𝑘) < 1) ↔ (𝐾 ≠ 0 ∧ (abs‘𝐾) < 1)))
97	96	rspcev 3565	. 2 ⊢ ((𝐾 ∈ ℤ ∧ (𝐾 ≠ 0 ∧ (abs‘𝐾) < 1)) → ∃𝑘 ∈ ℤ (𝑘 ≠ 0 ∧ (abs‘𝑘) < 1))
98	73, 82, 92, 97	syl12anc 837	1 ⊢ (𝜑 → ∃𝑘 ∈ ℤ (𝑘 ≠ 0 ∧ (abs‘𝑘) < 1))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1542 ∈ wcel 2114 ≠ wne 2933 ∃wrex 3062 ∖ cdif 3887 ⊆ wss 3890 {csn 4568 {cpr 4570 class class class wbr 5086 ↦ cmpt 5167 × cxp 5623 ran crn 5626 ⟶wf 6489 ‘cfv 6493 (class class class)co 7361 1^st c1st 7934 2^nd c2nd 7935 Fincfn 8887 ℂcc 11030 ℝcr 11031 0cc0 11032 1c1 11033 + caddc 11035 · cmul 11037 < clt 11173 − cmin 11371 -cneg 11372 / cdiv 11801 ℕcn 12168 ℕ₀cn0 12431 ℤcz 12518 (,)cioo 13292 [,]cicc 13295 ...cfz 13455 ↑cexp 14017 !cfa 14229 abscabs 15190 Σcsu 15642 ∏cprod 15862 eceu 16021 ℙcprime 16634 ↾_t crest 17377 TopOpenctopn 17378 topGenctg 17394 ℂ_fldccnfld 21347 ∫citg 25598 0_𝑝c0p 25649 D^𝑛 cdvn 25844 Polycply 26162 coeffccoe 26164 degcdgr 26165 ↑_𝑐ccxp 26535

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-rep 5213 ax-sep 5232 ax-nul 5242 ax-pow 5303 ax-pr 5371 ax-un 7683 ax-inf2 9556 ax-cc 10351 ax-cnex 11088 ax-resscn 11089 ax-1cn 11090 ax-icn 11091 ax-addcl 11092 ax-addrcl 11093 ax-mulcl 11094 ax-mulrcl 11095 ax-mulcom 11096 ax-addass 11097 ax-mulass 11098 ax-distr 11099 ax-i2m1 11100 ax-1ne0 11101 ax-1rid 11102 ax-rnegex 11103 ax-rrecex 11104 ax-cnre 11105 ax-pre-lttri 11106 ax-pre-lttrn 11107 ax-pre-ltadd 11108 ax-pre-mulgt0 11109 ax-pre-sup 11110 ax-addf 11111

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3063 df-rmo 3343 df-reu 3344 df-rab 3391 df-v 3432 df-sbc 3730 df-csb 3839 df-dif 3893 df-un 3895 df-in 3897 df-ss 3907 df-pss 3910 df-symdif 4194 df-nul 4275 df-if 4468 df-pw 4544 df-sn 4569 df-pr 4571 df-tp 4573 df-op 4575 df-uni 4852 df-int 4891 df-iun 4936 df-iin 4937 df-disj 5054 df-br 5087 df-opab 5149 df-mpt 5168 df-tr 5194 df-id 5520 df-eprel 5525 df-po 5533 df-so 5534 df-fr 5578 df-se 5579 df-we 5580 df-xp 5631 df-rel 5632 df-cnv 5633 df-co 5634 df-dm 5635 df-rn 5636 df-res 5637 df-ima 5638 df-pred 6260 df-ord 6321 df-on 6322 df-lim 6323 df-suc 6324 df-iota 6449 df-fun 6495 df-fn 6496 df-f 6497 df-f1 6498 df-fo 6499 df-f1o 6500 df-fv 6501 df-isom 6502 df-riota 7318 df-ov 7364 df-oprab 7365 df-mpo 7366 df-of 7625 df-ofr 7626 df-om 7812 df-1st 7936 df-2nd 7937 df-supp 8105 df-frecs 8225 df-wrecs 8256 df-recs 8305 df-rdg 8343 df-1o 8399 df-2o 8400 df-oadd 8403 df-omul 8404 df-er 8637 df-map 8769 df-pm 8770 df-ixp 8840 df-en 8888 df-dom 8889 df-sdom 8890 df-fin 8891 df-fsupp 9269 df-fi 9318 df-sup 9349 df-inf 9350 df-oi 9419 df-dju 9819 df-card 9857 df-acn 9860 df-pnf 11175 df-mnf 11176 df-xr 11177 df-ltxr 11178 df-le 11179 df-sub 11373 df-neg 11374 df-div 11802 df-nn 12169 df-2 12238 df-3 12239 df-4 12240 df-5 12241 df-6 12242 df-7 12243 df-8 12244 df-9 12245 df-n0 12432 df-z 12519 df-dec 12639 df-uz 12783 df-q 12893 df-rp 12937 df-xneg 13057 df-xadd 13058 df-xmul 13059 df-ioo 13296 df-ioc 13297 df-ico 13298 df-icc 13299 df-fz 13456 df-fzo 13603 df-fl 13745 df-mod 13823 df-seq 13958 df-exp 14018 df-fac 14230 df-bc 14259 df-hash 14287 df-shft 15023 df-cj 15055 df-re 15056 df-im 15057 df-sqrt 15191 df-abs 15192 df-limsup 15427 df-clim 15444 df-rlim 15445 df-sum 15643 df-prod 15863 df-ef 16026 df-e 16027 df-sin 16028 df-cos 16029 df-tan 16030 df-pi 16031 df-dvds 16216 df-gcd 16458 df-prm 16635 df-struct 17111 df-sets 17128 df-slot 17146 df-ndx 17158 df-base 17174 df-ress 17195 df-plusg 17227 df-mulr 17228 df-starv 17229 df-sca 17230 df-vsca 17231 df-ip 17232 df-tset 17233 df-ple 17234 df-ds 17236 df-unif 17237 df-hom 17238 df-cco 17239 df-rest 17379 df-topn 17380 df-0g 17398 df-gsum 17399 df-topgen 17400 df-pt 17401 df-prds 17404 df-xrs 17460 df-qtop 17465 df-imas 17466 df-xps 17468 df-mre 17542 df-mrc 17543 df-acs 17545 df-mgm 18602 df-sgrp 18681 df-mnd 18697 df-submnd 18746 df-mulg 19038 df-cntz 19286 df-cmn 19751 df-psmet 21339 df-xmet 21340 df-met 21341 df-bl 21342 df-mopn 21343 df-fbas 21344 df-fg 21345 df-cnfld 21348 df-top 22872 df-topon 22889 df-topsp 22911 df-bases 22924 df-cld 22997 df-ntr 22998 df-cls 22999 df-nei 23076 df-lp 23114 df-perf 23115 df-cn 23205 df-cnp 23206 df-haus 23293 df-cmp 23365 df-tx 23540 df-hmeo 23733 df-fil 23824 df-fm 23916 df-flim 23917 df-flf 23918 df-xms 24298 df-ms 24299 df-tms 24300 df-cncf 24858 df-ovol 25444 df-vol 25445 df-mbf 25599 df-itg1 25600 df-itg2 25601 df-ibl 25602 df-itg 25603 df-0p 25650 df-limc 25846 df-dv 25847 df-dvn 25848 df-ply 26166 df-coe 26168 df-dgr 26169 df-log 26536 df-cxp 26537

This theorem is referenced by: etransclem48 46731

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