etransclem44 - Mathbox for Glauco Siliprandi

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Theorem etransclem44 46263

Description: The given finite sum is nonzero. This is the claim proved after equation (7) in [Juillerat] p. 12 . (Contributed by Glauco Siliprandi, 5-Apr-2020.)

**Hypotheses**
Ref	Expression
etransclem44.a	⊢ (𝜑 → 𝐴:ℕ₀⟶ℤ)
etransclem44.a0	⊢ (𝜑 → (𝐴‘0) ≠ 0)
etransclem44.m	⊢ (𝜑 → 𝑀 ∈ ℕ₀)
etransclem44.p	⊢ (𝜑 → 𝑃 ∈ ℙ)
etransclem44.ap	⊢ (𝜑 → (abs‘(𝐴‘0)) < 𝑃)
etransclem44.mp	⊢ (𝜑 → (!‘𝑀) < 𝑃)
etransclem44.f	⊢ 𝐹 = (𝑥 ∈ ℝ ↦ ((𝑥↑(𝑃 − 1)) · ∏𝑗 ∈ (1...𝑀)((𝑥 − 𝑗)↑𝑃)))
etransclem44.k	⊢ 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)))

**Assertion**
Ref	Expression
etransclem44	⊢ (𝜑 → 𝐾 ≠ 0)

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

Proof of Theorem etransclem44 Dummy variables 𝑐 𝑑 𝑛 𝑚 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		etransclem44.k	. . . 4 ⊢ 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)))
2	1	a1i 11	. . 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))))
3		nfv 1914	. . . . 5 ⊢ Ⅎ𝑘𝜑
4		nfcv 2891	. . . . 5 ⊢ Ⅎ𝑘((𝐴‘0) · (((ℝ D^𝑛 𝐹)‘(𝑃 − 1))‘0))
5		fzfi 13879	. . . . . . 7 ⊢ (0...𝑀) ∈ Fin
6		fzfi 13879	. . . . . . 7 ⊢ (0...((𝑀 · 𝑃) + (𝑃 − 1))) ∈ Fin
7		xpfi 9209	. . . . . . 7 ⊢ (((0...𝑀) ∈ Fin ∧ (0...((𝑀 · 𝑃) + (𝑃 − 1))) ∈ Fin) → ((0...𝑀) × (0...((𝑀 · 𝑃) + (𝑃 − 1)))) ∈ Fin)
8	5, 6, 7	mp2an 692	. . . . . 6 ⊢ ((0...𝑀) × (0...((𝑀 · 𝑃) + (𝑃 − 1)))) ∈ Fin
9	8	a1i 11	. . . . 5 ⊢ (𝜑 → ((0...𝑀) × (0...((𝑀 · 𝑃) + (𝑃 − 1)))) ∈ Fin)
10		etransclem44.a	. . . . . . . . 9 ⊢ (𝜑 → 𝐴:ℕ₀⟶ℤ)
11	10	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ ((0...𝑀) × (0...((𝑀 · 𝑃) + (𝑃 − 1))))) → 𝐴:ℕ₀⟶ℤ)
12		fzssnn0 45302	. . . . . . . . . 10 ⊢ (0...𝑀) ⊆ ℕ₀
13		xp1st 7956	. . . . . . . . . 10 ⊢ (𝑘 ∈ ((0...𝑀) × (0...((𝑀 · 𝑃) + (𝑃 − 1)))) → (1^st ‘𝑘) ∈ (0...𝑀))
14	12, 13	sselid 3933	. . . . . . . . 9 ⊢ (𝑘 ∈ ((0...𝑀) × (0...((𝑀 · 𝑃) + (𝑃 − 1)))) → (1^st ‘𝑘) ∈ ℕ₀)
15	14	adantl 481	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ ((0...𝑀) × (0...((𝑀 · 𝑃) + (𝑃 − 1))))) → (1^st ‘𝑘) ∈ ℕ₀)
16	11, 15	ffvelcdmd 7019	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ ((0...𝑀) × (0...((𝑀 · 𝑃) + (𝑃 − 1))))) → (𝐴‘(1^st ‘𝑘)) ∈ ℤ)
17		reelprrecn 11101	. . . . . . . . 9 ⊢ ℝ ∈ {ℝ, ℂ}
18	17	a1i 11	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ ((0...𝑀) × (0...((𝑀 · 𝑃) + (𝑃 − 1))))) → ℝ ∈ {ℝ, ℂ})
19		reopn 45275	. . . . . . . . . 10 ⊢ ℝ ∈ (topGen‘ran (,))
20		tgioo4 24691	. . . . . . . . . 10 ⊢ (topGen‘ran (,)) = ((TopOpen‘ℂ_fld) ↾_t ℝ)
21	19, 20	eleqtri 2826	. . . . . . . . 9 ⊢ ℝ ∈ ((TopOpen‘ℂ_fld) ↾_t ℝ)
22	21	a1i 11	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ ((0...𝑀) × (0...((𝑀 · 𝑃) + (𝑃 − 1))))) → ℝ ∈ ((TopOpen‘ℂ_fld) ↾_t ℝ))
23		etransclem44.p	. . . . . . . . . 10 ⊢ (𝜑 → 𝑃 ∈ ℙ)
24		prmnn 16585	. . . . . . . . . 10 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ ℕ)
25	23, 24	syl 17	. . . . . . . . 9 ⊢ (𝜑 → 𝑃 ∈ ℕ)
26	25	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ ((0...𝑀) × (0...((𝑀 · 𝑃) + (𝑃 − 1))))) → 𝑃 ∈ ℕ)
27		etransclem44.m	. . . . . . . . 9 ⊢ (𝜑 → 𝑀 ∈ ℕ₀)
28	27	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ ((0...𝑀) × (0...((𝑀 · 𝑃) + (𝑃 − 1))))) → 𝑀 ∈ ℕ₀)
29		etransclem44.f	. . . . . . . 8 ⊢ 𝐹 = (𝑥 ∈ ℝ ↦ ((𝑥↑(𝑃 − 1)) · ∏𝑗 ∈ (1...𝑀)((𝑥 − 𝑗)↑𝑃)))
30		xp2nd 7957	. . . . . . . . . 10 ⊢ (𝑘 ∈ ((0...𝑀) × (0...((𝑀 · 𝑃) + (𝑃 − 1)))) → (2^nd ‘𝑘) ∈ (0...((𝑀 · 𝑃) + (𝑃 − 1))))
31		elfznn0 13523	. . . . . . . . . 10 ⊢ ((2^nd ‘𝑘) ∈ (0...((𝑀 · 𝑃) + (𝑃 − 1))) → (2^nd ‘𝑘) ∈ ℕ₀)
32	30, 31	syl 17	. . . . . . . . 9 ⊢ (𝑘 ∈ ((0...𝑀) × (0...((𝑀 · 𝑃) + (𝑃 − 1)))) → (2^nd ‘𝑘) ∈ ℕ₀)
33	32	adantl 481	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ ((0...𝑀) × (0...((𝑀 · 𝑃) + (𝑃 − 1))))) → (2^nd ‘𝑘) ∈ ℕ₀)
34	15	nn0red 12446	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ ((0...𝑀) × (0...((𝑀 · 𝑃) + (𝑃 − 1))))) → (1^st ‘𝑘) ∈ ℝ)
35	15	nn0zd 12497	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ ((0...𝑀) × (0...((𝑀 · 𝑃) + (𝑃 − 1))))) → (1^st ‘𝑘) ∈ ℤ)
36	18, 22, 26, 28, 29, 33, 34, 35	etransclem42 46261	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ ((0...𝑀) × (0...((𝑀 · 𝑃) + (𝑃 − 1))))) → (((ℝ D^𝑛 𝐹)‘(2^nd ‘𝑘))‘(1^st ‘𝑘)) ∈ ℤ)
37	16, 36	zmulcld 12586	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ ((0...𝑀) × (0...((𝑀 · 𝑃) + (𝑃 − 1))))) → ((𝐴‘(1^st ‘𝑘)) · (((ℝ D^𝑛 𝐹)‘(2^nd ‘𝑘))‘(1^st ‘𝑘))) ∈ ℤ)
38	37	zcnd 12581	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ ((0...𝑀) × (0...((𝑀 · 𝑃) + (𝑃 − 1))))) → ((𝐴‘(1^st ‘𝑘)) · (((ℝ D^𝑛 𝐹)‘(2^nd ‘𝑘))‘(1^st ‘𝑘))) ∈ ℂ)
39		nn0uz 12777	. . . . . . . 8 ⊢ ℕ₀ = (ℤ_≥‘0)
40	27, 39	eleqtrdi 2838	. . . . . . 7 ⊢ (𝜑 → 𝑀 ∈ (ℤ_≥‘0))
41		eluzfz1 13434	. . . . . . 7 ⊢ (𝑀 ∈ (ℤ_≥‘0) → 0 ∈ (0...𝑀))
42	40, 41	syl 17	. . . . . 6 ⊢ (𝜑 → 0 ∈ (0...𝑀))
43		0zd 12483	. . . . . . 7 ⊢ (𝜑 → 0 ∈ ℤ)
44	27	nn0zd 12497	. . . . . . . . 9 ⊢ (𝜑 → 𝑀 ∈ ℤ)
45	25	nnzd 12498	. . . . . . . . 9 ⊢ (𝜑 → 𝑃 ∈ ℤ)
46	44, 45	zmulcld 12586	. . . . . . . 8 ⊢ (𝜑 → (𝑀 · 𝑃) ∈ ℤ)
47		nnm1nn0 12425	. . . . . . . . . 10 ⊢ (𝑃 ∈ ℕ → (𝑃 − 1) ∈ ℕ₀)
48	25, 47	syl 17	. . . . . . . . 9 ⊢ (𝜑 → (𝑃 − 1) ∈ ℕ₀)
49	48	nn0zd 12497	. . . . . . . 8 ⊢ (𝜑 → (𝑃 − 1) ∈ ℤ)
50	46, 49	zaddcld 12584	. . . . . . 7 ⊢ (𝜑 → ((𝑀 · 𝑃) + (𝑃 − 1)) ∈ ℤ)
51	48	nn0ge0d 12448	. . . . . . 7 ⊢ (𝜑 → 0 ≤ (𝑃 − 1))
52	25	nnnn0d 12445	. . . . . . . . . 10 ⊢ (𝜑 → 𝑃 ∈ ℕ₀)
53	27, 52	nn0mulcld 12450	. . . . . . . . 9 ⊢ (𝜑 → (𝑀 · 𝑃) ∈ ℕ₀)
54	53	nn0ge0d 12448	. . . . . . . 8 ⊢ (𝜑 → 0 ≤ (𝑀 · 𝑃))
55	48	nn0red 12446	. . . . . . . . 9 ⊢ (𝜑 → (𝑃 − 1) ∈ ℝ)
56	46	zred 12580	. . . . . . . . 9 ⊢ (𝜑 → (𝑀 · 𝑃) ∈ ℝ)
57	55, 56	addge02d 11709	. . . . . . . 8 ⊢ (𝜑 → (0 ≤ (𝑀 · 𝑃) ↔ (𝑃 − 1) ≤ ((𝑀 · 𝑃) + (𝑃 − 1))))
58	54, 57	mpbid 232	. . . . . . 7 ⊢ (𝜑 → (𝑃 − 1) ≤ ((𝑀 · 𝑃) + (𝑃 − 1)))
59	43, 50, 49, 51, 58	elfzd 13418	. . . . . 6 ⊢ (𝜑 → (𝑃 − 1) ∈ (0...((𝑀 · 𝑃) + (𝑃 − 1))))
60		opelxp 5655	. . . . . 6 ⊢ (⟨0, (𝑃 − 1)⟩ ∈ ((0...𝑀) × (0...((𝑀 · 𝑃) + (𝑃 − 1)))) ↔ (0 ∈ (0...𝑀) ∧ (𝑃 − 1) ∈ (0...((𝑀 · 𝑃) + (𝑃 − 1)))))
61	42, 59, 60	sylanbrc 583	. . . . 5 ⊢ (𝜑 → ⟨0, (𝑃 − 1)⟩ ∈ ((0...𝑀) × (0...((𝑀 · 𝑃) + (𝑃 − 1)))))
62		fveq2 6822	. . . . . . . 8 ⊢ (𝑘 = ⟨0, (𝑃 − 1)⟩ → (1^st ‘𝑘) = (1^st ‘⟨0, (𝑃 − 1)⟩))
63		0re 11117	. . . . . . . . 9 ⊢ 0 ∈ ℝ
64		ovex 7382	. . . . . . . . 9 ⊢ (𝑃 − 1) ∈ V
65		op1stg 7936	. . . . . . . . 9 ⊢ ((0 ∈ ℝ ∧ (𝑃 − 1) ∈ V) → (1^st ‘⟨0, (𝑃 − 1)⟩) = 0)
66	63, 64, 65	mp2an 692	. . . . . . . 8 ⊢ (1^st ‘⟨0, (𝑃 − 1)⟩) = 0
67	62, 66	eqtrdi 2780	. . . . . . 7 ⊢ (𝑘 = ⟨0, (𝑃 − 1)⟩ → (1^st ‘𝑘) = 0)
68	67	fveq2d 6826	. . . . . 6 ⊢ (𝑘 = ⟨0, (𝑃 − 1)⟩ → (𝐴‘(1^st ‘𝑘)) = (𝐴‘0))
69		fveq2 6822	. . . . . . . . 9 ⊢ (𝑘 = ⟨0, (𝑃 − 1)⟩ → (2^nd ‘𝑘) = (2^nd ‘⟨0, (𝑃 − 1)⟩))
70		op2ndg 7937	. . . . . . . . . 10 ⊢ ((0 ∈ ℝ ∧ (𝑃 − 1) ∈ V) → (2^nd ‘⟨0, (𝑃 − 1)⟩) = (𝑃 − 1))
71	63, 64, 70	mp2an 692	. . . . . . . . 9 ⊢ (2^nd ‘⟨0, (𝑃 − 1)⟩) = (𝑃 − 1)
72	69, 71	eqtrdi 2780	. . . . . . . 8 ⊢ (𝑘 = ⟨0, (𝑃 − 1)⟩ → (2^nd ‘𝑘) = (𝑃 − 1))
73	72	fveq2d 6826	. . . . . . 7 ⊢ (𝑘 = ⟨0, (𝑃 − 1)⟩ → ((ℝ D^𝑛 𝐹)‘(2^nd ‘𝑘)) = ((ℝ D^𝑛 𝐹)‘(𝑃 − 1)))
74	73, 67	fveq12d 6829	. . . . . 6 ⊢ (𝑘 = ⟨0, (𝑃 − 1)⟩ → (((ℝ D^𝑛 𝐹)‘(2^nd ‘𝑘))‘(1^st ‘𝑘)) = (((ℝ D^𝑛 𝐹)‘(𝑃 − 1))‘0))
75	68, 74	oveq12d 7367	. . . . 5 ⊢ (𝑘 = ⟨0, (𝑃 − 1)⟩ → ((𝐴‘(1^st ‘𝑘)) · (((ℝ D^𝑛 𝐹)‘(2^nd ‘𝑘))‘(1^st ‘𝑘))) = ((𝐴‘0) · (((ℝ D^𝑛 𝐹)‘(𝑃 − 1))‘0)))
76	3, 4, 9, 38, 61, 75	fsumsplit1 15652	. . . 4 ⊢ (𝜑 → Σ𝑘 ∈ ((0...𝑀) × (0...((𝑀 · 𝑃) + (𝑃 − 1))))((𝐴‘(1^st ‘𝑘)) · (((ℝ D^𝑛 𝐹)‘(2^nd ‘𝑘))‘(1^st ‘𝑘))) = (((𝐴‘0) · (((ℝ D^𝑛 𝐹)‘(𝑃 − 1))‘0)) + Σ𝑘 ∈ (((0...𝑀) × (0...((𝑀 · 𝑃) + (𝑃 − 1)))) ∖ {⟨0, (𝑃 − 1)⟩})((𝐴‘(1^st ‘𝑘)) · (((ℝ D^𝑛 𝐹)‘(2^nd ‘𝑘))‘(1^st ‘𝑘)))))
77	76	oveq1d 7364	. . 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))) = ((((𝐴‘0) · (((ℝ D^𝑛 𝐹)‘(𝑃 − 1))‘0)) + Σ𝑘 ∈ (((0...𝑀) × (0...((𝑀 · 𝑃) + (𝑃 − 1)))) ∖ {⟨0, (𝑃 − 1)⟩})((𝐴‘(1^st ‘𝑘)) · (((ℝ D^𝑛 𝐹)‘(2^nd ‘𝑘))‘(1^st ‘𝑘)))) / (!‘(𝑃 − 1))))
78	12, 42	sselid 3933	. . . . . . 7 ⊢ (𝜑 → 0 ∈ ℕ₀)
79	10, 78	ffvelcdmd 7019	. . . . . 6 ⊢ (𝜑 → (𝐴‘0) ∈ ℤ)
80	17	a1i 11	. . . . . . 7 ⊢ (𝜑 → ℝ ∈ {ℝ, ℂ})
81	21	a1i 11	. . . . . . 7 ⊢ (𝜑 → ℝ ∈ ((TopOpen‘ℂ_fld) ↾_t ℝ))
82	63	a1i 11	. . . . . . 7 ⊢ (𝜑 → 0 ∈ ℝ)
83	80, 81, 25, 27, 29, 48, 82, 43	etransclem42 46261	. . . . . 6 ⊢ (𝜑 → (((ℝ D^𝑛 𝐹)‘(𝑃 − 1))‘0) ∈ ℤ)
84	79, 83	zmulcld 12586	. . . . 5 ⊢ (𝜑 → ((𝐴‘0) · (((ℝ D^𝑛 𝐹)‘(𝑃 − 1))‘0)) ∈ ℤ)
85	84	zcnd 12581	. . . 4 ⊢ (𝜑 → ((𝐴‘0) · (((ℝ D^𝑛 𝐹)‘(𝑃 − 1))‘0)) ∈ ℂ)
86		difss 4087	. . . . . . . 8 ⊢ (((0...𝑀) × (0...((𝑀 · 𝑃) + (𝑃 − 1)))) ∖ {⟨0, (𝑃 − 1)⟩}) ⊆ ((0...𝑀) × (0...((𝑀 · 𝑃) + (𝑃 − 1))))
87		ssfi 9087	. . . . . . . 8 ⊢ ((((0...𝑀) × (0...((𝑀 · 𝑃) + (𝑃 − 1)))) ∈ Fin ∧ (((0...𝑀) × (0...((𝑀 · 𝑃) + (𝑃 − 1)))) ∖ {⟨0, (𝑃 − 1)⟩}) ⊆ ((0...𝑀) × (0...((𝑀 · 𝑃) + (𝑃 − 1))))) → (((0...𝑀) × (0...((𝑀 · 𝑃) + (𝑃 − 1)))) ∖ {⟨0, (𝑃 − 1)⟩}) ∈ Fin)
88	8, 86, 87	mp2an 692	. . . . . . 7 ⊢ (((0...𝑀) × (0...((𝑀 · 𝑃) + (𝑃 − 1)))) ∖ {⟨0, (𝑃 − 1)⟩}) ∈ Fin
89	88	a1i 11	. . . . . 6 ⊢ (𝜑 → (((0...𝑀) × (0...((𝑀 · 𝑃) + (𝑃 − 1)))) ∖ {⟨0, (𝑃 − 1)⟩}) ∈ Fin)
90		eldifi 4082	. . . . . . 7 ⊢ (𝑘 ∈ (((0...𝑀) × (0...((𝑀 · 𝑃) + (𝑃 − 1)))) ∖ {⟨0, (𝑃 − 1)⟩}) → 𝑘 ∈ ((0...𝑀) × (0...((𝑀 · 𝑃) + (𝑃 − 1)))))
91	90, 37	sylan2 593	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (((0...𝑀) × (0...((𝑀 · 𝑃) + (𝑃 − 1)))) ∖ {⟨0, (𝑃 − 1)⟩})) → ((𝐴‘(1^st ‘𝑘)) · (((ℝ D^𝑛 𝐹)‘(2^nd ‘𝑘))‘(1^st ‘𝑘))) ∈ ℤ)
92	89, 91	fsumzcl 15642	. . . . 5 ⊢ (𝜑 → Σ𝑘 ∈ (((0...𝑀) × (0...((𝑀 · 𝑃) + (𝑃 − 1)))) ∖ {⟨0, (𝑃 − 1)⟩})((𝐴‘(1^st ‘𝑘)) · (((ℝ D^𝑛 𝐹)‘(2^nd ‘𝑘))‘(1^st ‘𝑘))) ∈ ℤ)
93	92	zcnd 12581	. . . 4 ⊢ (𝜑 → Σ𝑘 ∈ (((0...𝑀) × (0...((𝑀 · 𝑃) + (𝑃 − 1)))) ∖ {⟨0, (𝑃 − 1)⟩})((𝐴‘(1^st ‘𝑘)) · (((ℝ D^𝑛 𝐹)‘(2^nd ‘𝑘))‘(1^st ‘𝑘))) ∈ ℂ)
94	48	faccld 14191	. . . . 5 ⊢ (𝜑 → (!‘(𝑃 − 1)) ∈ ℕ)
95	94	nncnd 12144	. . . 4 ⊢ (𝜑 → (!‘(𝑃 − 1)) ∈ ℂ)
96	94	nnne0d 12178	. . . 4 ⊢ (𝜑 → (!‘(𝑃 − 1)) ≠ 0)
97	85, 93, 95, 96	divdird 11938	. . 3 ⊢ (𝜑 → ((((𝐴‘0) · (((ℝ D^𝑛 𝐹)‘(𝑃 − 1))‘0)) + Σ𝑘 ∈ (((0...𝑀) × (0...((𝑀 · 𝑃) + (𝑃 − 1)))) ∖ {⟨0, (𝑃 − 1)⟩})((𝐴‘(1^st ‘𝑘)) · (((ℝ D^𝑛 𝐹)‘(2^nd ‘𝑘))‘(1^st ‘𝑘)))) / (!‘(𝑃 − 1))) = ((((𝐴‘0) · (((ℝ D^𝑛 𝐹)‘(𝑃 − 1))‘0)) / (!‘(𝑃 − 1))) + (Σ𝑘 ∈ (((0...𝑀) × (0...((𝑀 · 𝑃) + (𝑃 − 1)))) ∖ {⟨0, (𝑃 − 1)⟩})((𝐴‘(1^st ‘𝑘)) · (((ℝ D^𝑛 𝐹)‘(2^nd ‘𝑘))‘(1^st ‘𝑘))) / (!‘(𝑃 − 1)))))
98	2, 77, 97	3eqtrd 2768	. 2 ⊢ (𝜑 → 𝐾 = ((((𝐴‘0) · (((ℝ D^𝑛 𝐹)‘(𝑃 − 1))‘0)) / (!‘(𝑃 − 1))) + (Σ𝑘 ∈ (((0...𝑀) × (0...((𝑀 · 𝑃) + (𝑃 − 1)))) ∖ {⟨0, (𝑃 − 1)⟩})((𝐴‘(1^st ‘𝑘)) · (((ℝ D^𝑛 𝐹)‘(2^nd ‘𝑘))‘(1^st ‘𝑘))) / (!‘(𝑃 − 1)))))
99	25	nnne0d 12178	. . 3 ⊢ (𝜑 → 𝑃 ≠ 0)
100	79	zcnd 12581	. . . . 5 ⊢ (𝜑 → (𝐴‘0) ∈ ℂ)
101	83	zcnd 12581	. . . . 5 ⊢ (𝜑 → (((ℝ D^𝑛 𝐹)‘(𝑃 − 1))‘0) ∈ ℂ)
102	100, 101, 95, 96	divassd 11935	. . . 4 ⊢ (𝜑 → (((𝐴‘0) · (((ℝ D^𝑛 𝐹)‘(𝑃 − 1))‘0)) / (!‘(𝑃 − 1))) = ((𝐴‘0) · ((((ℝ D^𝑛 𝐹)‘(𝑃 − 1))‘0) / (!‘(𝑃 − 1)))))
103		etransclem5 46224	. . . . . . 7 ⊢ (𝑘 ∈ (0...𝑀) ↦ (𝑦 ∈ ℝ ↦ ((𝑦 − 𝑘)↑if(𝑘 = 0, (𝑃 − 1), 𝑃)))) = (𝑗 ∈ (0...𝑀) ↦ (𝑥 ∈ ℝ ↦ ((𝑥 − 𝑗)↑if(𝑗 = 0, (𝑃 − 1), 𝑃))))
104		etransclem11 46230	. . . . . . 7 ⊢ (𝑚 ∈ ℕ₀ ↦ {𝑑 ∈ ((0...𝑚) ↑_m (0...𝑀)) ∣ Σ𝑘 ∈ (0...𝑀)(𝑑‘𝑘) = 𝑚}) = (𝑛 ∈ ℕ₀ ↦ {𝑐 ∈ ((0...𝑛) ↑_m (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐‘𝑗) = 𝑛})
105	80, 81, 25, 27, 29, 48, 103, 104, 42, 82	etransclem37 46256	. . . . . 6 ⊢ (𝜑 → (!‘(𝑃 − 1)) ∥ (((ℝ D^𝑛 𝐹)‘(𝑃 − 1))‘0))
106	94	nnzd 12498	. . . . . . 7 ⊢ (𝜑 → (!‘(𝑃 − 1)) ∈ ℤ)
107		dvdsval2 16166	. . . . . . 7 ⊢ (((!‘(𝑃 − 1)) ∈ ℤ ∧ (!‘(𝑃 − 1)) ≠ 0 ∧ (((ℝ D^𝑛 𝐹)‘(𝑃 − 1))‘0) ∈ ℤ) → ((!‘(𝑃 − 1)) ∥ (((ℝ D^𝑛 𝐹)‘(𝑃 − 1))‘0) ↔ ((((ℝ D^𝑛 𝐹)‘(𝑃 − 1))‘0) / (!‘(𝑃 − 1))) ∈ ℤ))
108	106, 96, 83, 107	syl3anc 1373	. . . . . 6 ⊢ (𝜑 → ((!‘(𝑃 − 1)) ∥ (((ℝ D^𝑛 𝐹)‘(𝑃 − 1))‘0) ↔ ((((ℝ D^𝑛 𝐹)‘(𝑃 − 1))‘0) / (!‘(𝑃 − 1))) ∈ ℤ))
109	105, 108	mpbid 232	. . . . 5 ⊢ (𝜑 → ((((ℝ D^𝑛 𝐹)‘(𝑃 − 1))‘0) / (!‘(𝑃 − 1))) ∈ ℤ)
110	79, 109	zmulcld 12586	. . . 4 ⊢ (𝜑 → ((𝐴‘0) · ((((ℝ D^𝑛 𝐹)‘(𝑃 − 1))‘0) / (!‘(𝑃 − 1)))) ∈ ℤ)
111	102, 110	eqeltrd 2828	. . 3 ⊢ (𝜑 → (((𝐴‘0) · (((ℝ D^𝑛 𝐹)‘(𝑃 − 1))‘0)) / (!‘(𝑃 − 1))) ∈ ℤ)
112	90, 38	sylan2 593	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ (((0...𝑀) × (0...((𝑀 · 𝑃) + (𝑃 − 1)))) ∖ {⟨0, (𝑃 − 1)⟩})) → ((𝐴‘(1^st ‘𝑘)) · (((ℝ D^𝑛 𝐹)‘(2^nd ‘𝑘))‘(1^st ‘𝑘))) ∈ ℂ)
113	89, 95, 112, 96	fsumdivc 15693	. . . 4 ⊢ (𝜑 → (Σ𝑘 ∈ (((0...𝑀) × (0...((𝑀 · 𝑃) + (𝑃 − 1)))) ∖ {⟨0, (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)))) ∖ {⟨0, (𝑃 − 1)⟩})(((𝐴‘(1^st ‘𝑘)) · (((ℝ D^𝑛 𝐹)‘(2^nd ‘𝑘))‘(1^st ‘𝑘))) / (!‘(𝑃 − 1))))
114	16	zcnd 12581	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ ((0...𝑀) × (0...((𝑀 · 𝑃) + (𝑃 − 1))))) → (𝐴‘(1^st ‘𝑘)) ∈ ℂ)
115	90, 114	sylan2 593	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (((0...𝑀) × (0...((𝑀 · 𝑃) + (𝑃 − 1)))) ∖ {⟨0, (𝑃 − 1)⟩})) → (𝐴‘(1^st ‘𝑘)) ∈ ℂ)
116	90, 36	sylan2 593	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (((0...𝑀) × (0...((𝑀 · 𝑃) + (𝑃 − 1)))) ∖ {⟨0, (𝑃 − 1)⟩})) → (((ℝ D^𝑛 𝐹)‘(2^nd ‘𝑘))‘(1^st ‘𝑘)) ∈ ℤ)
117	116	zcnd 12581	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (((0...𝑀) × (0...((𝑀 · 𝑃) + (𝑃 − 1)))) ∖ {⟨0, (𝑃 − 1)⟩})) → (((ℝ D^𝑛 𝐹)‘(2^nd ‘𝑘))‘(1^st ‘𝑘)) ∈ ℂ)
118	95	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (((0...𝑀) × (0...((𝑀 · 𝑃) + (𝑃 − 1)))) ∖ {⟨0, (𝑃 − 1)⟩})) → (!‘(𝑃 − 1)) ∈ ℂ)
119	96	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (((0...𝑀) × (0...((𝑀 · 𝑃) + (𝑃 − 1)))) ∖ {⟨0, (𝑃 − 1)⟩})) → (!‘(𝑃 − 1)) ≠ 0)
120	115, 117, 118, 119	divassd 11935	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (((0...𝑀) × (0...((𝑀 · 𝑃) + (𝑃 − 1)))) ∖ {⟨0, (𝑃 − 1)⟩})) → (((𝐴‘(1^st ‘𝑘)) · (((ℝ D^𝑛 𝐹)‘(2^nd ‘𝑘))‘(1^st ‘𝑘))) / (!‘(𝑃 − 1))) = ((𝐴‘(1^st ‘𝑘)) · ((((ℝ D^𝑛 𝐹)‘(2^nd ‘𝑘))‘(1^st ‘𝑘)) / (!‘(𝑃 − 1)))))
121	90, 16	sylan2 593	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (((0...𝑀) × (0...((𝑀 · 𝑃) + (𝑃 − 1)))) ∖ {⟨0, (𝑃 − 1)⟩})) → (𝐴‘(1^st ‘𝑘)) ∈ ℤ)
122	17	a1i 11	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (((0...𝑀) × (0...((𝑀 · 𝑃) + (𝑃 − 1)))) ∖ {⟨0, (𝑃 − 1)⟩})) → ℝ ∈ {ℝ, ℂ})
123	21	a1i 11	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (((0...𝑀) × (0...((𝑀 · 𝑃) + (𝑃 − 1)))) ∖ {⟨0, (𝑃 − 1)⟩})) → ℝ ∈ ((TopOpen‘ℂ_fld) ↾_t ℝ))
124	25	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (((0...𝑀) × (0...((𝑀 · 𝑃) + (𝑃 − 1)))) ∖ {⟨0, (𝑃 − 1)⟩})) → 𝑃 ∈ ℕ)
125	27	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (((0...𝑀) × (0...((𝑀 · 𝑃) + (𝑃 − 1)))) ∖ {⟨0, (𝑃 − 1)⟩})) → 𝑀 ∈ ℕ₀)
126	90	adantl 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ (((0...𝑀) × (0...((𝑀 · 𝑃) + (𝑃 − 1)))) ∖ {⟨0, (𝑃 − 1)⟩})) → 𝑘 ∈ ((0...𝑀) × (0...((𝑀 · 𝑃) + (𝑃 − 1)))))
127	126, 32	syl 17	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (((0...𝑀) × (0...((𝑀 · 𝑃) + (𝑃 − 1)))) ∖ {⟨0, (𝑃 − 1)⟩})) → (2^nd ‘𝑘) ∈ ℕ₀)
128	126, 13	syl 17	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (((0...𝑀) × (0...((𝑀 · 𝑃) + (𝑃 − 1)))) ∖ {⟨0, (𝑃 − 1)⟩})) → (1^st ‘𝑘) ∈ (0...𝑀))
129	90, 34	sylan2 593	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (((0...𝑀) × (0...((𝑀 · 𝑃) + (𝑃 − 1)))) ∖ {⟨0, (𝑃 − 1)⟩})) → (1^st ‘𝑘) ∈ ℝ)
130	122, 123, 124, 125, 29, 127, 103, 104, 128, 129	etransclem37 46256	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (((0...𝑀) × (0...((𝑀 · 𝑃) + (𝑃 − 1)))) ∖ {⟨0, (𝑃 − 1)⟩})) → (!‘(𝑃 − 1)) ∥ (((ℝ D^𝑛 𝐹)‘(2^nd ‘𝑘))‘(1^st ‘𝑘)))
131	106	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (((0...𝑀) × (0...((𝑀 · 𝑃) + (𝑃 − 1)))) ∖ {⟨0, (𝑃 − 1)⟩})) → (!‘(𝑃 − 1)) ∈ ℤ)
132		dvdsval2 16166	. . . . . . . . 9 ⊢ (((!‘(𝑃 − 1)) ∈ ℤ ∧ (!‘(𝑃 − 1)) ≠ 0 ∧ (((ℝ D^𝑛 𝐹)‘(2^nd ‘𝑘))‘(1^st ‘𝑘)) ∈ ℤ) → ((!‘(𝑃 − 1)) ∥ (((ℝ D^𝑛 𝐹)‘(2^nd ‘𝑘))‘(1^st ‘𝑘)) ↔ ((((ℝ D^𝑛 𝐹)‘(2^nd ‘𝑘))‘(1^st ‘𝑘)) / (!‘(𝑃 − 1))) ∈ ℤ))
133	131, 119, 116, 132	syl3anc 1373	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (((0...𝑀) × (0...((𝑀 · 𝑃) + (𝑃 − 1)))) ∖ {⟨0, (𝑃 − 1)⟩})) → ((!‘(𝑃 − 1)) ∥ (((ℝ D^𝑛 𝐹)‘(2^nd ‘𝑘))‘(1^st ‘𝑘)) ↔ ((((ℝ D^𝑛 𝐹)‘(2^nd ‘𝑘))‘(1^st ‘𝑘)) / (!‘(𝑃 − 1))) ∈ ℤ))
134	130, 133	mpbid 232	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (((0...𝑀) × (0...((𝑀 · 𝑃) + (𝑃 − 1)))) ∖ {⟨0, (𝑃 − 1)⟩})) → ((((ℝ D^𝑛 𝐹)‘(2^nd ‘𝑘))‘(1^st ‘𝑘)) / (!‘(𝑃 − 1))) ∈ ℤ)
135	121, 134	zmulcld 12586	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (((0...𝑀) × (0...((𝑀 · 𝑃) + (𝑃 − 1)))) ∖ {⟨0, (𝑃 − 1)⟩})) → ((𝐴‘(1^st ‘𝑘)) · ((((ℝ D^𝑛 𝐹)‘(2^nd ‘𝑘))‘(1^st ‘𝑘)) / (!‘(𝑃 − 1)))) ∈ ℤ)
136	120, 135	eqeltrd 2828	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ (((0...𝑀) × (0...((𝑀 · 𝑃) + (𝑃 − 1)))) ∖ {⟨0, (𝑃 − 1)⟩})) → (((𝐴‘(1^st ‘𝑘)) · (((ℝ D^𝑛 𝐹)‘(2^nd ‘𝑘))‘(1^st ‘𝑘))) / (!‘(𝑃 − 1))) ∈ ℤ)
137	89, 136	fsumzcl 15642	. . . 4 ⊢ (𝜑 → Σ𝑘 ∈ (((0...𝑀) × (0...((𝑀 · 𝑃) + (𝑃 − 1)))) ∖ {⟨0, (𝑃 − 1)⟩})(((𝐴‘(1^st ‘𝑘)) · (((ℝ D^𝑛 𝐹)‘(2^nd ‘𝑘))‘(1^st ‘𝑘))) / (!‘(𝑃 − 1))) ∈ ℤ)
138	113, 137	eqeltrd 2828	. . 3 ⊢ (𝜑 → (Σ𝑘 ∈ (((0...𝑀) × (0...((𝑀 · 𝑃) + (𝑃 − 1)))) ∖ {⟨0, (𝑃 − 1)⟩})((𝐴‘(1^st ‘𝑘)) · (((ℝ D^𝑛 𝐹)‘(2^nd ‘𝑘))‘(1^st ‘𝑘))) / (!‘(𝑃 − 1))) ∈ ℤ)
139		1zzd 12506	. . . . . . . . . 10 ⊢ (𝜑 → 1 ∈ ℤ)
140		zabscl 15220	. . . . . . . . . . 11 ⊢ ((𝐴‘0) ∈ ℤ → (abs‘(𝐴‘0)) ∈ ℤ)
141	79, 140	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → (abs‘(𝐴‘0)) ∈ ℤ)
142		nn0abscl 15219	. . . . . . . . . . . . 13 ⊢ ((𝐴‘0) ∈ ℤ → (abs‘(𝐴‘0)) ∈ ℕ₀)
143	79, 142	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → (abs‘(𝐴‘0)) ∈ ℕ₀)
144		etransclem44.a0	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐴‘0) ≠ 0)
145	100, 144	absne0d 15357	. . . . . . . . . . . 12 ⊢ (𝜑 → (abs‘(𝐴‘0)) ≠ 0)
146		elnnne0 12398	. . . . . . . . . . . 12 ⊢ ((abs‘(𝐴‘0)) ∈ ℕ ↔ ((abs‘(𝐴‘0)) ∈ ℕ₀ ∧ (abs‘(𝐴‘0)) ≠ 0))
147	143, 145, 146	sylanbrc 583	. . . . . . . . . . 11 ⊢ (𝜑 → (abs‘(𝐴‘0)) ∈ ℕ)
148	147	nnge1d 12176	. . . . . . . . . 10 ⊢ (𝜑 → 1 ≤ (abs‘(𝐴‘0)))
149		etransclem44.ap	. . . . . . . . . . 11 ⊢ (𝜑 → (abs‘(𝐴‘0)) < 𝑃)
150		zltlem1 12528	. . . . . . . . . . . 12 ⊢ (((abs‘(𝐴‘0)) ∈ ℤ ∧ 𝑃 ∈ ℤ) → ((abs‘(𝐴‘0)) < 𝑃 ↔ (abs‘(𝐴‘0)) ≤ (𝑃 − 1)))
151	141, 45, 150	syl2anc 584	. . . . . . . . . . 11 ⊢ (𝜑 → ((abs‘(𝐴‘0)) < 𝑃 ↔ (abs‘(𝐴‘0)) ≤ (𝑃 − 1)))
152	149, 151	mpbid 232	. . . . . . . . . 10 ⊢ (𝜑 → (abs‘(𝐴‘0)) ≤ (𝑃 − 1))
153	139, 49, 141, 148, 152	elfzd 13418	. . . . . . . . 9 ⊢ (𝜑 → (abs‘(𝐴‘0)) ∈ (1...(𝑃 − 1)))
154		fzm1ndvds 16233	. . . . . . . . 9 ⊢ ((𝑃 ∈ ℕ ∧ (abs‘(𝐴‘0)) ∈ (1...(𝑃 − 1))) → ¬ 𝑃 ∥ (abs‘(𝐴‘0)))
155	25, 153, 154	syl2anc 584	. . . . . . . 8 ⊢ (𝜑 → ¬ 𝑃 ∥ (abs‘(𝐴‘0)))
156		dvdsabsb 16186	. . . . . . . . 9 ⊢ ((𝑃 ∈ ℤ ∧ (𝐴‘0) ∈ ℤ) → (𝑃 ∥ (𝐴‘0) ↔ 𝑃 ∥ (abs‘(𝐴‘0))))
157	45, 79, 156	syl2anc 584	. . . . . . . 8 ⊢ (𝜑 → (𝑃 ∥ (𝐴‘0) ↔ 𝑃 ∥ (abs‘(𝐴‘0))))
158	155, 157	mtbird 325	. . . . . . 7 ⊢ (𝜑 → ¬ 𝑃 ∥ (𝐴‘0))
159		etransclem44.mp	. . . . . . . 8 ⊢ (𝜑 → (!‘𝑀) < 𝑃)
160	27, 23, 159, 29	etransclem41 46260	. . . . . . 7 ⊢ (𝜑 → ¬ 𝑃 ∥ ((((ℝ D^𝑛 𝐹)‘(𝑃 − 1))‘0) / (!‘(𝑃 − 1))))
161	158, 160	jca 511	. . . . . 6 ⊢ (𝜑 → (¬ 𝑃 ∥ (𝐴‘0) ∧ ¬ 𝑃 ∥ ((((ℝ D^𝑛 𝐹)‘(𝑃 − 1))‘0) / (!‘(𝑃 − 1)))))
162		pm4.56 990	. . . . . 6 ⊢ ((¬ 𝑃 ∥ (𝐴‘0) ∧ ¬ 𝑃 ∥ ((((ℝ D^𝑛 𝐹)‘(𝑃 − 1))‘0) / (!‘(𝑃 − 1)))) ↔ ¬ (𝑃 ∥ (𝐴‘0) ∨ 𝑃 ∥ ((((ℝ D^𝑛 𝐹)‘(𝑃 − 1))‘0) / (!‘(𝑃 − 1)))))
163	161, 162	sylib 218	. . . . 5 ⊢ (𝜑 → ¬ (𝑃 ∥ (𝐴‘0) ∨ 𝑃 ∥ ((((ℝ D^𝑛 𝐹)‘(𝑃 − 1))‘0) / (!‘(𝑃 − 1)))))
164		euclemma 16624	. . . . . 6 ⊢ ((𝑃 ∈ ℙ ∧ (𝐴‘0) ∈ ℤ ∧ ((((ℝ D^𝑛 𝐹)‘(𝑃 − 1))‘0) / (!‘(𝑃 − 1))) ∈ ℤ) → (𝑃 ∥ ((𝐴‘0) · ((((ℝ D^𝑛 𝐹)‘(𝑃 − 1))‘0) / (!‘(𝑃 − 1)))) ↔ (𝑃 ∥ (𝐴‘0) ∨ 𝑃 ∥ ((((ℝ D^𝑛 𝐹)‘(𝑃 − 1))‘0) / (!‘(𝑃 − 1))))))
165	23, 79, 109, 164	syl3anc 1373	. . . . 5 ⊢ (𝜑 → (𝑃 ∥ ((𝐴‘0) · ((((ℝ D^𝑛 𝐹)‘(𝑃 − 1))‘0) / (!‘(𝑃 − 1)))) ↔ (𝑃 ∥ (𝐴‘0) ∨ 𝑃 ∥ ((((ℝ D^𝑛 𝐹)‘(𝑃 − 1))‘0) / (!‘(𝑃 − 1))))))
166	163, 165	mtbird 325	. . . 4 ⊢ (𝜑 → ¬ 𝑃 ∥ ((𝐴‘0) · ((((ℝ D^𝑛 𝐹)‘(𝑃 − 1))‘0) / (!‘(𝑃 − 1)))))
167	102	breq2d 5104	. . . 4 ⊢ (𝜑 → (𝑃 ∥ (((𝐴‘0) · (((ℝ D^𝑛 𝐹)‘(𝑃 − 1))‘0)) / (!‘(𝑃 − 1))) ↔ 𝑃 ∥ ((𝐴‘0) · ((((ℝ D^𝑛 𝐹)‘(𝑃 − 1))‘0) / (!‘(𝑃 − 1))))))
168	166, 167	mtbird 325	. . 3 ⊢ (𝜑 → ¬ 𝑃 ∥ (((𝐴‘0) · (((ℝ D^𝑛 𝐹)‘(𝑃 − 1))‘0)) / (!‘(𝑃 − 1))))
169	45	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (((0...𝑀) × (0...((𝑀 · 𝑃) + (𝑃 − 1)))) ∖ {⟨0, (𝑃 − 1)⟩})) → 𝑃 ∈ ℤ)
170	169, 121, 134	3jca 1128	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (((0...𝑀) × (0...((𝑀 · 𝑃) + (𝑃 − 1)))) ∖ {⟨0, (𝑃 − 1)⟩})) → (𝑃 ∈ ℤ ∧ (𝐴‘(1^st ‘𝑘)) ∈ ℤ ∧ ((((ℝ D^𝑛 𝐹)‘(2^nd ‘𝑘))‘(1^st ‘𝑘)) / (!‘(𝑃 − 1))) ∈ ℤ))
171		eldifn 4083	. . . . . . . . . 10 ⊢ (𝑘 ∈ (((0...𝑀) × (0...((𝑀 · 𝑃) + (𝑃 − 1)))) ∖ {⟨0, (𝑃 − 1)⟩}) → ¬ 𝑘 ∈ {⟨0, (𝑃 − 1)⟩})
172	90	adantr 480	. . . . . . . . . . . . 13 ⊢ ((𝑘 ∈ (((0...𝑀) × (0...((𝑀 · 𝑃) + (𝑃 − 1)))) ∖ {⟨0, (𝑃 − 1)⟩}) ∧ ((2^nd ‘𝑘) = (𝑃 − 1) ∧ (1^st ‘𝑘) = 0)) → 𝑘 ∈ ((0...𝑀) × (0...((𝑀 · 𝑃) + (𝑃 − 1)))))
173		1st2nd2 7963	. . . . . . . . . . . . 13 ⊢ (𝑘 ∈ ((0...𝑀) × (0...((𝑀 · 𝑃) + (𝑃 − 1)))) → 𝑘 = ⟨(1^st ‘𝑘), (2^nd ‘𝑘)⟩)
174	172, 173	syl 17	. . . . . . . . . . . 12 ⊢ ((𝑘 ∈ (((0...𝑀) × (0...((𝑀 · 𝑃) + (𝑃 − 1)))) ∖ {⟨0, (𝑃 − 1)⟩}) ∧ ((2^nd ‘𝑘) = (𝑃 − 1) ∧ (1^st ‘𝑘) = 0)) → 𝑘 = ⟨(1^st ‘𝑘), (2^nd ‘𝑘)⟩)
175		simpr 484	. . . . . . . . . . . . . 14 ⊢ (((2^nd ‘𝑘) = (𝑃 − 1) ∧ (1^st ‘𝑘) = 0) → (1^st ‘𝑘) = 0)
176		simpl 482	. . . . . . . . . . . . . 14 ⊢ (((2^nd ‘𝑘) = (𝑃 − 1) ∧ (1^st ‘𝑘) = 0) → (2^nd ‘𝑘) = (𝑃 − 1))
177	175, 176	opeq12d 4832	. . . . . . . . . . . . 13 ⊢ (((2^nd ‘𝑘) = (𝑃 − 1) ∧ (1^st ‘𝑘) = 0) → ⟨(1^st ‘𝑘), (2^nd ‘𝑘)⟩ = ⟨0, (𝑃 − 1)⟩)
178	177	adantl 481	. . . . . . . . . . . 12 ⊢ ((𝑘 ∈ (((0...𝑀) × (0...((𝑀 · 𝑃) + (𝑃 − 1)))) ∖ {⟨0, (𝑃 − 1)⟩}) ∧ ((2^nd ‘𝑘) = (𝑃 − 1) ∧ (1^st ‘𝑘) = 0)) → ⟨(1^st ‘𝑘), (2^nd ‘𝑘)⟩ = ⟨0, (𝑃 − 1)⟩)
179	174, 178	eqtrd 2764	. . . . . . . . . . 11 ⊢ ((𝑘 ∈ (((0...𝑀) × (0...((𝑀 · 𝑃) + (𝑃 − 1)))) ∖ {⟨0, (𝑃 − 1)⟩}) ∧ ((2^nd ‘𝑘) = (𝑃 − 1) ∧ (1^st ‘𝑘) = 0)) → 𝑘 = ⟨0, (𝑃 − 1)⟩)
180		velsn 4593	. . . . . . . . . . 11 ⊢ (𝑘 ∈ {⟨0, (𝑃 − 1)⟩} ↔ 𝑘 = ⟨0, (𝑃 − 1)⟩)
181	179, 180	sylibr 234	. . . . . . . . . 10 ⊢ ((𝑘 ∈ (((0...𝑀) × (0...((𝑀 · 𝑃) + (𝑃 − 1)))) ∖ {⟨0, (𝑃 − 1)⟩}) ∧ ((2^nd ‘𝑘) = (𝑃 − 1) ∧ (1^st ‘𝑘) = 0)) → 𝑘 ∈ {⟨0, (𝑃 − 1)⟩})
182	171, 181	mtand 815	. . . . . . . . 9 ⊢ (𝑘 ∈ (((0...𝑀) × (0...((𝑀 · 𝑃) + (𝑃 − 1)))) ∖ {⟨0, (𝑃 − 1)⟩}) → ¬ ((2^nd ‘𝑘) = (𝑃 − 1) ∧ (1^st ‘𝑘) = 0))
183	182	adantl 481	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (((0...𝑀) × (0...((𝑀 · 𝑃) + (𝑃 − 1)))) ∖ {⟨0, (𝑃 − 1)⟩})) → ¬ ((2^nd ‘𝑘) = (𝑃 − 1) ∧ (1^st ‘𝑘) = 0))
184	124, 125, 29, 127, 128, 183, 104	etransclem38 46257	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (((0...𝑀) × (0...((𝑀 · 𝑃) + (𝑃 − 1)))) ∖ {⟨0, (𝑃 − 1)⟩})) → 𝑃 ∥ ((((ℝ D^𝑛 𝐹)‘(2^nd ‘𝑘))‘(1^st ‘𝑘)) / (!‘(𝑃 − 1))))
185		dvdsmultr2 16209	. . . . . . 7 ⊢ ((𝑃 ∈ ℤ ∧ (𝐴‘(1^st ‘𝑘)) ∈ ℤ ∧ ((((ℝ D^𝑛 𝐹)‘(2^nd ‘𝑘))‘(1^st ‘𝑘)) / (!‘(𝑃 − 1))) ∈ ℤ) → (𝑃 ∥ ((((ℝ D^𝑛 𝐹)‘(2^nd ‘𝑘))‘(1^st ‘𝑘)) / (!‘(𝑃 − 1))) → 𝑃 ∥ ((𝐴‘(1^st ‘𝑘)) · ((((ℝ D^𝑛 𝐹)‘(2^nd ‘𝑘))‘(1^st ‘𝑘)) / (!‘(𝑃 − 1))))))
186	170, 184, 185	sylc 65	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (((0...𝑀) × (0...((𝑀 · 𝑃) + (𝑃 − 1)))) ∖ {⟨0, (𝑃 − 1)⟩})) → 𝑃 ∥ ((𝐴‘(1^st ‘𝑘)) · ((((ℝ D^𝑛 𝐹)‘(2^nd ‘𝑘))‘(1^st ‘𝑘)) / (!‘(𝑃 − 1)))))
187	186, 120	breqtrrd 5120	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ (((0...𝑀) × (0...((𝑀 · 𝑃) + (𝑃 − 1)))) ∖ {⟨0, (𝑃 − 1)⟩})) → 𝑃 ∥ (((𝐴‘(1^st ‘𝑘)) · (((ℝ D^𝑛 𝐹)‘(2^nd ‘𝑘))‘(1^st ‘𝑘))) / (!‘(𝑃 − 1))))
188	89, 45, 136, 187	fsumdvds 16219	. . . 4 ⊢ (𝜑 → 𝑃 ∥ Σ𝑘 ∈ (((0...𝑀) × (0...((𝑀 · 𝑃) + (𝑃 − 1)))) ∖ {⟨0, (𝑃 − 1)⟩})(((𝐴‘(1^st ‘𝑘)) · (((ℝ D^𝑛 𝐹)‘(2^nd ‘𝑘))‘(1^st ‘𝑘))) / (!‘(𝑃 − 1))))
189	188, 113	breqtrrd 5120	. . 3 ⊢ (𝜑 → 𝑃 ∥ (Σ𝑘 ∈ (((0...𝑀) × (0...((𝑀 · 𝑃) + (𝑃 − 1)))) ∖ {⟨0, (𝑃 − 1)⟩})((𝐴‘(1^st ‘𝑘)) · (((ℝ D^𝑛 𝐹)‘(2^nd ‘𝑘))‘(1^st ‘𝑘))) / (!‘(𝑃 − 1))))
190	45, 99, 111, 138, 168, 189	etransclem9 46228	. 2 ⊢ (𝜑 → ((((𝐴‘0) · (((ℝ D^𝑛 𝐹)‘(𝑃 − 1))‘0)) / (!‘(𝑃 − 1))) + (Σ𝑘 ∈ (((0...𝑀) × (0...((𝑀 · 𝑃) + (𝑃 − 1)))) ∖ {⟨0, (𝑃 − 1)⟩})((𝐴‘(1^st ‘𝑘)) · (((ℝ D^𝑛 𝐹)‘(2^nd ‘𝑘))‘(1^st ‘𝑘))) / (!‘(𝑃 − 1)))) ≠ 0)
191	98, 190	eqnetrd 2992	1 ⊢ (𝜑 → 𝐾 ≠ 0)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 ∨ wo 847 ∧ w3a 1086 = wceq 1540 ∈ wcel 2109 ≠ wne 2925 {crab 3394 Vcvv 3436 ∖ cdif 3900 ⊆ wss 3903 ifcif 4476 {csn 4577 {cpr 4579 ⟨cop 4583 class class class wbr 5092 ↦ cmpt 5173 × cxp 5617 ran crn 5620 ⟶wf 6478 ‘cfv 6482 (class class class)co 7349 1^st c1st 7922 2^nd c2nd 7923 ↑_m cmap 8753 Fincfn 8872 ℂcc 11007 ℝcr 11008 0cc0 11009 1c1 11010 + caddc 11012 · cmul 11014 < clt 11149 ≤ cle 11150 − cmin 11347 / cdiv 11777 ℕcn 12128 ℕ₀cn0 12384 ℤcz 12471 ℤ_≥cuz 12735 (,)cioo 13248 ...cfz 13410 ↑cexp 13968 !cfa 14180 abscabs 15141 Σcsu 15593 ∏cprod 15810 ∥ cdvds 16163 ℙcprime 16582 ↾_t crest 17324 TopOpenctopn 17325 topGenctg 17341 ℂ_fldccnfld 21261 D^𝑛 cdvn 25763

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 2701 ax-rep 5218 ax-sep 5235 ax-nul 5245 ax-pow 5304 ax-pr 5371 ax-un 7671 ax-inf2 9537 ax-cnex 11065 ax-resscn 11066 ax-1cn 11067 ax-icn 11068 ax-addcl 11069 ax-addrcl 11070 ax-mulcl 11071 ax-mulrcl 11072 ax-mulcom 11073 ax-addass 11074 ax-mulass 11075 ax-distr 11076 ax-i2m1 11077 ax-1ne0 11078 ax-1rid 11079 ax-rnegex 11080 ax-rrecex 11081 ax-cnre 11082 ax-pre-lttri 11083 ax-pre-lttrn 11084 ax-pre-ltadd 11085 ax-pre-mulgt0 11086 ax-pre-sup 11087 ax-addf 11088

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 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-rmo 3343 df-reu 3344 df-rab 3395 df-v 3438 df-sbc 3743 df-csb 3852 df-dif 3906 df-un 3908 df-in 3910 df-ss 3920 df-pss 3923 df-nul 4285 df-if 4477 df-pw 4553 df-sn 4578 df-pr 4580 df-tp 4582 df-op 4584 df-uni 4859 df-int 4897 df-iun 4943 df-iin 4944 df-br 5093 df-opab 5155 df-mpt 5174 df-tr 5200 df-id 5514 df-eprel 5519 df-po 5527 df-so 5528 df-fr 5572 df-se 5573 df-we 5574 df-xp 5625 df-rel 5626 df-cnv 5627 df-co 5628 df-dm 5629 df-rn 5630 df-res 5631 df-ima 5632 df-pred 6249 df-ord 6310 df-on 6311 df-lim 6312 df-suc 6313 df-iota 6438 df-fun 6484 df-fn 6485 df-f 6486 df-f1 6487 df-fo 6488 df-f1o 6489 df-fv 6490 df-isom 6491 df-riota 7306 df-ov 7352 df-oprab 7353 df-mpo 7354 df-of 7613 df-om 7800 df-1st 7924 df-2nd 7925 df-supp 8094 df-frecs 8214 df-wrecs 8245 df-recs 8294 df-rdg 8332 df-1o 8388 df-2o 8389 df-er 8625 df-map 8755 df-pm 8756 df-ixp 8825 df-en 8873 df-dom 8874 df-sdom 8875 df-fin 8876 df-fsupp 9252 df-fi 9301 df-sup 9332 df-inf 9333 df-oi 9402 df-card 9835 df-pnf 11151 df-mnf 11152 df-xr 11153 df-ltxr 11154 df-le 11155 df-sub 11349 df-neg 11350 df-div 11778 df-nn 12129 df-2 12191 df-3 12192 df-4 12193 df-5 12194 df-6 12195 df-7 12196 df-8 12197 df-9 12198 df-n0 12385 df-z 12472 df-dec 12592 df-uz 12736 df-q 12850 df-rp 12894 df-xneg 13014 df-xadd 13015 df-xmul 13016 df-ioo 13252 df-ico 13254 df-icc 13255 df-fz 13411 df-fzo 13558 df-fl 13696 df-mod 13774 df-seq 13909 df-exp 13969 df-fac 14181 df-bc 14210 df-hash 14238 df-cj 15006 df-re 15007 df-im 15008 df-sqrt 15142 df-abs 15143 df-clim 15395 df-sum 15594 df-prod 15811 df-dvds 16164 df-gcd 16406 df-prm 16583 df-struct 17058 df-sets 17075 df-slot 17093 df-ndx 17105 df-base 17121 df-ress 17142 df-plusg 17174 df-mulr 17175 df-starv 17176 df-sca 17177 df-vsca 17178 df-ip 17179 df-tset 17180 df-ple 17181 df-ds 17183 df-unif 17184 df-hom 17185 df-cco 17186 df-rest 17326 df-topn 17327 df-0g 17345 df-gsum 17346 df-topgen 17347 df-pt 17348 df-prds 17351 df-xrs 17406 df-qtop 17411 df-imas 17412 df-xps 17414 df-mre 17488 df-mrc 17489 df-acs 17491 df-mgm 18514 df-sgrp 18593 df-mnd 18609 df-submnd 18658 df-mulg 18947 df-cntz 19196 df-cmn 19661 df-psmet 21253 df-xmet 21254 df-met 21255 df-bl 21256 df-mopn 21257 df-fbas 21258 df-fg 21259 df-cnfld 21262 df-top 22779 df-topon 22796 df-topsp 22818 df-bases 22831 df-cld 22904 df-ntr 22905 df-cls 22906 df-nei 22983 df-lp 23021 df-perf 23022 df-cn 23112 df-cnp 23113 df-haus 23200 df-tx 23447 df-hmeo 23640 df-fil 23731 df-fm 23823 df-flim 23824 df-flf 23825 df-xms 24206 df-ms 24207 df-tms 24208 df-cncf 24769 df-limc 25765 df-dv 25766 df-dvn 25767

This theorem is referenced by: etransclem47 46266

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