etransclem47 - Mathbox for Glauco Siliprandi

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Theorem etransclem47 45295

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 4003	. . . . . 6 ⊢ ℝ ⊆ ℝ
8	7	a1i 11	. . . . 5 ⊢ (𝜑 → ℝ ⊆ ℝ)
9		reelprrecn 11204	. . . . . 6 ⊢ ℝ ∈ {ℝ, ℂ}
10	9	a1i 11	. . . . 5 ⊢ (𝜑 → ℝ ∈ {ℝ, ℂ})
11		reopn 44297	. . . . . . 7 ⊢ ℝ ∈ (topGen‘ran (,))
12		eqid 2730	. . . . . . . 8 ⊢ (TopOpen‘ℂ_fld) = (TopOpen‘ℂ_fld)
13	12	tgioo2 24539	. . . . . . 7 ⊢ (topGen‘ran (,)) = ((TopOpen‘ℂ_fld) ↾_t ℝ)
14	11, 13	eleqtri 2829	. . . . . 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 16615	. . . . . 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 2730	. . . . 5 ⊢ ((𝑀 · 𝑃) + (𝑃 − 1)) = ((𝑀 · 𝑃) + (𝑃 − 1))
22		fveq2 6890	. . . . . . 7 ⊢ (𝑦 = 𝑥 → (((ℝ D^𝑛 𝐹)‘𝑖)‘𝑦) = (((ℝ D^𝑛 𝐹)‘𝑖)‘𝑥))
23	22	sumeq2sdv 15654	. . . . . 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 11456	. . . . . . . . 9 ⊢ (𝑧 = 𝑥 → -𝑧 = -𝑥)
26	25	oveq2d 7427	. . . . . . . 8 ⊢ (𝑧 = 𝑥 → (e↑_𝑐-𝑧) = (e↑_𝑐-𝑥))
27		fveq2 6890	. . . . . . . 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 7429	. . . . . . 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 11458	. . . . . 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 45294	. . . 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 13942	. . . . . . . 8 ⊢ (𝜑 → (0...𝑀) ∈ Fin)
33		fzfid 13942	. . . . . . . 8 ⊢ (𝜑 → (0...((𝑀 · 𝑃) + (𝑃 − 1))) ∈ Fin)
34		xpfi 9319	. . . . . . . 8 ⊢ (((0...𝑀) ∈ Fin ∧ (0...((𝑀 · 𝑃) + (𝑃 − 1))) ∈ Fin) → ((0...𝑀) × (0...((𝑀 · 𝑃) + (𝑃 − 1)))) ∈ Fin)
35	32, 33, 34	syl2anc 582	. . . . . . 7 ⊢ (𝜑 → ((0...𝑀) × (0...((𝑀 · 𝑃) + (𝑃 − 1)))) ∈ Fin)
36	3	eldifad 3959	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑄 ∈ (Poly‘ℤ))
37		0zd 12574	. . . . . . . . . . . 12 ⊢ (𝜑 → 0 ∈ ℤ)
38	5	coef2 25980	. . . . . . . . . . . 12 ⊢ ((𝑄 ∈ (Poly‘ℤ) ∧ 0 ∈ ℤ) → 𝐴:ℕ₀⟶ℤ)
39	36, 37, 38	syl2anc 582	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐴:ℕ₀⟶ℤ)
40	39	adantr 479	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ ((0...𝑀) × (0...((𝑀 · 𝑃) + (𝑃 − 1))))) → 𝐴:ℕ₀⟶ℤ)
41		xp1st 8009	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ ((0...𝑀) × (0...((𝑀 · 𝑃) + (𝑃 − 1)))) → (1^st ‘𝑘) ∈ (0...𝑀))
42		elfznn0 13598	. . . . . . . . . . . 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 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ ((0...𝑀) × (0...((𝑀 · 𝑃) + (𝑃 − 1))))) → (1^st ‘𝑘) ∈ ℕ₀)
45	40, 44	ffvelcdmd 7086	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ ((0...𝑀) × (0...((𝑀 · 𝑃) + (𝑃 − 1))))) → (𝐴‘(1^st ‘𝑘)) ∈ ℤ)
46	45	zcnd 12671	. . . . . . . 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 479	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ ((0...𝑀) × (0...((𝑀 · 𝑃) + (𝑃 − 1))))) → 𝑃 ∈ ℕ)
50		dgrcl 25982	. . . . . . . . . . . . 13 ⊢ (𝑄 ∈ (Poly‘ℤ) → (deg‘𝑄) ∈ ℕ₀)
51	36, 50	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → (deg‘𝑄) ∈ ℕ₀)
52	6, 51	eqeltrid 2835	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑀 ∈ ℕ₀)
53	52	adantr 479	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ ((0...𝑀) × (0...((𝑀 · 𝑃) + (𝑃 − 1))))) → 𝑀 ∈ ℕ₀)
54		xp2nd 8010	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ ((0...𝑀) × (0...((𝑀 · 𝑃) + (𝑃 − 1)))) → (2^nd ‘𝑘) ∈ (0...((𝑀 · 𝑃) + (𝑃 − 1))))
55		elfznn0 13598	. . . . . . . . . . . 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 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ ((0...𝑀) × (0...((𝑀 · 𝑃) + (𝑃 − 1))))) → (2^nd ‘𝑘) ∈ ℕ₀)
58	47, 48, 49, 53, 19, 57	etransclem33 45281	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ ((0...𝑀) × (0...((𝑀 · 𝑃) + (𝑃 − 1))))) → ((ℝ D^𝑛 𝐹)‘(2^nd ‘𝑘)):ℝ⟶ℂ)
59	44	nn0red 12537	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ ((0...𝑀) × (0...((𝑀 · 𝑃) + (𝑃 − 1))))) → (1^st ‘𝑘) ∈ ℝ)
60	58, 59	ffvelcdmd 7086	. . . . . . . 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 11238	. . . . . . 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 15683	. . . . . 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 12517	. . . . . . . . 9 ⊢ (𝑃 ∈ ℕ → (𝑃 − 1) ∈ ℕ₀)
64	18, 63	syl 17	. . . . . . . 8 ⊢ (𝜑 → (𝑃 − 1) ∈ ℕ₀)
65	64	faccld 14248	. . . . . . 7 ⊢ (𝜑 → (!‘(𝑃 − 1)) ∈ ℕ)
66	65	nncnd 12232	. . . . . 6 ⊢ (𝜑 → (!‘(𝑃 − 1)) ∈ ℂ)
67	65	nnne0d 12266	. . . . . 6 ⊢ (𝜑 → (!‘(𝑃 − 1)) ≠ 0)
68	62, 66, 67	divnegd 12007	. . . . 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 2736	. . . 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 2774	. . 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 2730	. . . . 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 45293	. . . 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 12672	. . 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 2831	. 2 ⊢ (𝜑 → 𝐾 ∈ ℤ)
75	1, 31	eqtrid 2782	. . 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 11994	. . . . 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 45292	. . . . 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 11573	. . . 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 3006	. . 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 3006	. 2 ⊢ (𝜑 → 𝐾 ≠ 0)
84		eldifsni 4792	. . . . . 6 ⊢ (𝑄 ∈ ((Poly‘ℤ) ∖ {0_𝑝}) → 𝑄 ≠ 0_𝑝)
85	3, 84	syl 17	. . . . 5 ⊢ (𝜑 → 𝑄 ≠ 0_𝑝)
86		ere 16036	. . . . . . 7 ⊢ e ∈ ℝ
87	86	recni 11232	. . . . . 6 ⊢ e ∈ ℂ
88	87	a1i 11	. . . . 5 ⊢ (𝜑 → e ∈ ℂ)
89		dgrnznn 25996	. . . . 5 ⊢ (((𝑄 ∈ (Poly‘ℤ) ∧ 𝑄 ≠ 0_𝑝) ∧ (e ∈ ℂ ∧ (𝑄‘e) = 0)) → (deg‘𝑄) ∈ ℕ)
90	36, 85, 88, 4, 89	syl22anc 835	. . . 4 ⊢ (𝜑 → (deg‘𝑄) ∈ ℕ)
91	6, 90	eqeltrid 2835	. . 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 45271	. 2 ⊢ (𝜑 → (abs‘𝐾) < 1)
94		neeq1 3001	. . . 4 ⊢ (𝑘 = 𝐾 → (𝑘 ≠ 0 ↔ 𝐾 ≠ 0))
95		fveq2 6890	. . . . 5 ⊢ (𝑘 = 𝐾 → (abs‘𝑘) = (abs‘𝐾))
96	95	breq1d 5157	. . . 4 ⊢ (𝑘 = 𝐾 → ((abs‘𝑘) < 1 ↔ (abs‘𝐾) < 1))
97	94, 96	anbi12d 629	. . 3 ⊢ (𝑘 = 𝐾 → ((𝑘 ≠ 0 ∧ (abs‘𝑘) < 1) ↔ (𝐾 ≠ 0 ∧ (abs‘𝐾) < 1)))
98	97	rspcev 3611	. 2 ⊢ ((𝐾 ∈ ℤ ∧ (𝐾 ≠ 0 ∧ (abs‘𝐾) < 1)) → ∃𝑘 ∈ ℤ (𝑘 ≠ 0 ∧ (abs‘𝑘) < 1))
99	74, 83, 93, 98	syl12anc 833	1 ⊢ (𝜑 → ∃𝑘 ∈ ℤ (𝑘 ≠ 0 ∧ (abs‘𝑘) < 1))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 394 = wceq 1539 ∈ wcel 2104 ≠ wne 2938 ∃wrex 3068 ∖ cdif 3944 ⊆ wss 3947 {csn 4627 {cpr 4629 class class class wbr 5147 ↦ cmpt 5230 × cxp 5673 ran crn 5676 ⟶wf 6538 ‘cfv 6542 (class class class)co 7411 1^st c1st 7975 2^nd c2nd 7976 Fincfn 8941 ℂcc 11110 ℝcr 11111 0cc0 11112 1c1 11113 + caddc 11115 · cmul 11117 < clt 11252 − cmin 11448 -cneg 11449 / cdiv 11875 ℕcn 12216 ℕ₀cn0 12476 ℤcz 12562 (,)cioo 13328 [,]cicc 13331 ...cfz 13488 ↑cexp 14031 !cfa 14237 abscabs 15185 Σcsu 15636 ∏cprod 15853 eceu 16010 ℙcprime 16612 ↾_t crest 17370 TopOpenctopn 17371 topGenctg 17387 ℂ_fldccnfld 21144 ∫citg 25367 0_𝑝c0p 25418 D^𝑛 cdvn 25613 Polycply 25933 coeffccoe 25935 degcdgr 25936 ↑_𝑐ccxp 26300

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 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-10 2135 ax-11 2152 ax-12 2169 ax-ext 2701 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7727 ax-inf2 9638 ax-cc 10432 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189 ax-pre-sup 11190 ax-addf 11191 ax-mulf 11192

This theorem depends on definitions: df-bi 206 df-an 395 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2532 df-eu 2561 df-clab 2708 df-cleq 2722 df-clel 2808 df-nfc 2883 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3374 df-reu 3375 df-rab 3431 df-v 3474 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-symdif 4241 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-tp 4632 df-op 4634 df-uni 4908 df-int 4950 df-iun 4998 df-iin 4999 df-disj 5113 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-se 5631 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-isom 6551 df-riota 7367 df-ov 7414 df-oprab 7415 df-mpo 7416 df-of 7672 df-ofr 7673 df-om 7858 df-1st 7977 df-2nd 7978 df-supp 8149 df-frecs 8268 df-wrecs 8299 df-recs 8373 df-rdg 8412 df-1o 8468 df-2o 8469 df-oadd 8472 df-omul 8473 df-er 8705 df-map 8824 df-pm 8825 df-ixp 8894 df-en 8942 df-dom 8943 df-sdom 8944 df-fin 8945 df-fsupp 9364 df-fi 9408 df-sup 9439 df-inf 9440 df-oi 9507 df-dju 9898 df-card 9936 df-acn 9939 df-pnf 11254 df-mnf 11255 df-xr 11256 df-ltxr 11257 df-le 11258 df-sub 11450 df-neg 11451 df-div 11876 df-nn 12217 df-2 12279 df-3 12280 df-4 12281 df-5 12282 df-6 12283 df-7 12284 df-8 12285 df-9 12286 df-n0 12477 df-z 12563 df-dec 12682 df-uz 12827 df-q 12937 df-rp 12979 df-xneg 13096 df-xadd 13097 df-xmul 13098 df-ioo 13332 df-ioc 13333 df-ico 13334 df-icc 13335 df-fz 13489 df-fzo 13632 df-fl 13761 df-mod 13839 df-seq 13971 df-exp 14032 df-fac 14238 df-bc 14267 df-hash 14295 df-shft 15018 df-cj 15050 df-re 15051 df-im 15052 df-sqrt 15186 df-abs 15187 df-limsup 15419 df-clim 15436 df-rlim 15437 df-sum 15637 df-prod 15854 df-ef 16015 df-e 16016 df-sin 16017 df-cos 16018 df-tan 16019 df-pi 16020 df-dvds 16202 df-gcd 16440 df-prm 16613 df-struct 17084 df-sets 17101 df-slot 17119 df-ndx 17131 df-base 17149 df-ress 17178 df-plusg 17214 df-mulr 17215 df-starv 17216 df-sca 17217 df-vsca 17218 df-ip 17219 df-tset 17220 df-ple 17221 df-ds 17223 df-unif 17224 df-hom 17225 df-cco 17226 df-rest 17372 df-topn 17373 df-0g 17391 df-gsum 17392 df-topgen 17393 df-pt 17394 df-prds 17397 df-xrs 17452 df-qtop 17457 df-imas 17458 df-xps 17460 df-mre 17534 df-mrc 17535 df-acs 17537 df-mgm 18565 df-sgrp 18644 df-mnd 18660 df-submnd 18706 df-mulg 18987 df-cntz 19222 df-cmn 19691 df-psmet 21136 df-xmet 21137 df-met 21138 df-bl 21139 df-mopn 21140 df-fbas 21141 df-fg 21142 df-cnfld 21145 df-top 22616 df-topon 22633 df-topsp 22655 df-bases 22669 df-cld 22743 df-ntr 22744 df-cls 22745 df-nei 22822 df-lp 22860 df-perf 22861 df-cn 22951 df-cnp 22952 df-haus 23039 df-cmp 23111 df-tx 23286 df-hmeo 23479 df-fil 23570 df-fm 23662 df-flim 23663 df-flf 23664 df-xms 24046 df-ms 24047 df-tms 24048 df-cncf 24618 df-ovol 25213 df-vol 25214 df-mbf 25368 df-itg1 25369 df-itg2 25370 df-ibl 25371 df-itg 25372 df-0p 25419 df-limc 25615 df-dv 25616 df-dvn 25617 df-ply 25937 df-coe 25939 df-dgr 25940 df-log 26301 df-cxp 26302

This theorem is referenced by: etransclem48 45296

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