etransclem44 - Mathbox for Glauco Siliprandi

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Theorem etransclem44 42553

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 1909	. . . . 5 ⊢ Ⅎ𝑘𝜑
4		nfcv 2975	. . . . 5 ⊢ Ⅎ𝑘((𝐴‘0) · (((ℝ D^𝑛 𝐹)‘(𝑃 − 1))‘0))
5		fzfi 13332	. . . . . . 7 ⊢ (0...𝑀) ∈ Fin
6		fzfi 13332	. . . . . . 7 ⊢ (0...((𝑀 · 𝑃) + (𝑃 − 1))) ∈ Fin
7		xpfi 8781	. . . . . . 7 ⊢ (((0...𝑀) ∈ Fin ∧ (0...((𝑀 · 𝑃) + (𝑃 − 1))) ∈ Fin) → ((0...𝑀) × (0...((𝑀 · 𝑃) + (𝑃 − 1)))) ∈ Fin)
8	5, 6, 7	mp2an 690	. . . . . 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 483	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ ((0...𝑀) × (0...((𝑀 · 𝑃) + (𝑃 − 1))))) → 𝐴:ℕ₀⟶ℤ)
12		fzssnn0 41574	. . . . . . . . . 10 ⊢ (0...𝑀) ⊆ ℕ₀
13		xp1st 7713	. . . . . . . . . 10 ⊢ (𝑘 ∈ ((0...𝑀) × (0...((𝑀 · 𝑃) + (𝑃 − 1)))) → (1^st ‘𝑘) ∈ (0...𝑀))
14	12, 13	sseldi 3963	. . . . . . . . 9 ⊢ (𝑘 ∈ ((0...𝑀) × (0...((𝑀 · 𝑃) + (𝑃 − 1)))) → (1^st ‘𝑘) ∈ ℕ₀)
15	14	adantl 484	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ ((0...𝑀) × (0...((𝑀 · 𝑃) + (𝑃 − 1))))) → (1^st ‘𝑘) ∈ ℕ₀)
16	11, 15	ffvelrnd 6845	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ ((0...𝑀) × (0...((𝑀 · 𝑃) + (𝑃 − 1))))) → (𝐴‘(1^st ‘𝑘)) ∈ ℤ)
17		reelprrecn 10621	. . . . . . . . 9 ⊢ ℝ ∈ {ℝ, ℂ}
18	17	a1i 11	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ ((0...𝑀) × (0...((𝑀 · 𝑃) + (𝑃 − 1))))) → ℝ ∈ {ℝ, ℂ})
19		reopn 41544	. . . . . . . . . 10 ⊢ ℝ ∈ (topGen‘ran (,))
20		eqid 2819	. . . . . . . . . . 11 ⊢ (TopOpen‘ℂ_fld) = (TopOpen‘ℂ_fld)
21	20	tgioo2 23403	. . . . . . . . . 10 ⊢ (topGen‘ran (,)) = ((TopOpen‘ℂ_fld) ↾_t ℝ)
22	19, 21	eleqtri 2909	. . . . . . . . 9 ⊢ ℝ ∈ ((TopOpen‘ℂ_fld) ↾_t ℝ)
23	22	a1i 11	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ ((0...𝑀) × (0...((𝑀 · 𝑃) + (𝑃 − 1))))) → ℝ ∈ ((TopOpen‘ℂ_fld) ↾_t ℝ))
24		etransclem44.p	. . . . . . . . . 10 ⊢ (𝜑 → 𝑃 ∈ ℙ)
25		prmnn 16010	. . . . . . . . . 10 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ ℕ)
26	24, 25	syl 17	. . . . . . . . 9 ⊢ (𝜑 → 𝑃 ∈ ℕ)
27	26	adantr 483	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ ((0...𝑀) × (0...((𝑀 · 𝑃) + (𝑃 − 1))))) → 𝑃 ∈ ℕ)
28		etransclem44.m	. . . . . . . . 9 ⊢ (𝜑 → 𝑀 ∈ ℕ₀)
29	28	adantr 483	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ ((0...𝑀) × (0...((𝑀 · 𝑃) + (𝑃 − 1))))) → 𝑀 ∈ ℕ₀)
30		etransclem44.f	. . . . . . . 8 ⊢ 𝐹 = (𝑥 ∈ ℝ ↦ ((𝑥↑(𝑃 − 1)) · ∏𝑗 ∈ (1...𝑀)((𝑥 − 𝑗)↑𝑃)))
31		xp2nd 7714	. . . . . . . . . 10 ⊢ (𝑘 ∈ ((0...𝑀) × (0...((𝑀 · 𝑃) + (𝑃 − 1)))) → (2^nd ‘𝑘) ∈ (0...((𝑀 · 𝑃) + (𝑃 − 1))))
32		elfznn0 12992	. . . . . . . . . 10 ⊢ ((2^nd ‘𝑘) ∈ (0...((𝑀 · 𝑃) + (𝑃 − 1))) → (2^nd ‘𝑘) ∈ ℕ₀)
33	31, 32	syl 17	. . . . . . . . 9 ⊢ (𝑘 ∈ ((0...𝑀) × (0...((𝑀 · 𝑃) + (𝑃 − 1)))) → (2^nd ‘𝑘) ∈ ℕ₀)
34	33	adantl 484	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ ((0...𝑀) × (0...((𝑀 · 𝑃) + (𝑃 − 1))))) → (2^nd ‘𝑘) ∈ ℕ₀)
35	15	nn0red 11948	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ ((0...𝑀) × (0...((𝑀 · 𝑃) + (𝑃 − 1))))) → (1^st ‘𝑘) ∈ ℝ)
36	15	nn0zd 12077	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ ((0...𝑀) × (0...((𝑀 · 𝑃) + (𝑃 − 1))))) → (1^st ‘𝑘) ∈ ℤ)
37	18, 23, 27, 29, 30, 34, 35, 36	etransclem42 42551	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ ((0...𝑀) × (0...((𝑀 · 𝑃) + (𝑃 − 1))))) → (((ℝ D^𝑛 𝐹)‘(2^nd ‘𝑘))‘(1^st ‘𝑘)) ∈ ℤ)
38	16, 37	zmulcld 12085	. . . . . 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)
39	38	zcnd 12080	. . . . 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)
40		nn0uz 12272	. . . . . . . 8 ⊢ ℕ₀ = (ℤ_≥‘0)
41	28, 40	eleqtrdi 2921	. . . . . . 7 ⊢ (𝜑 → 𝑀 ∈ (ℤ_≥‘0))
42		eluzfz1 12906	. . . . . . 7 ⊢ (𝑀 ∈ (ℤ_≥‘0) → 0 ∈ (0...𝑀))
43	41, 42	syl 17	. . . . . 6 ⊢ (𝜑 → 0 ∈ (0...𝑀))
44		0zd 11985	. . . . . . . . 9 ⊢ (𝜑 → 0 ∈ ℤ)
45	28	nn0zd 12077	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑀 ∈ ℤ)
46	26	nnzd 12078	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑃 ∈ ℤ)
47	45, 46	zmulcld 12085	. . . . . . . . . 10 ⊢ (𝜑 → (𝑀 · 𝑃) ∈ ℤ)
48		nnm1nn0 11930	. . . . . . . . . . . 12 ⊢ (𝑃 ∈ ℕ → (𝑃 − 1) ∈ ℕ₀)
49	26, 48	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑃 − 1) ∈ ℕ₀)
50	49	nn0zd 12077	. . . . . . . . . 10 ⊢ (𝜑 → (𝑃 − 1) ∈ ℤ)
51	47, 50	zaddcld 12083	. . . . . . . . 9 ⊢ (𝜑 → ((𝑀 · 𝑃) + (𝑃 − 1)) ∈ ℤ)
52	44, 51, 50	3jca 1123	. . . . . . . 8 ⊢ (𝜑 → (0 ∈ ℤ ∧ ((𝑀 · 𝑃) + (𝑃 − 1)) ∈ ℤ ∧ (𝑃 − 1) ∈ ℤ))
53	49	nn0ge0d 11950	. . . . . . . 8 ⊢ (𝜑 → 0 ≤ (𝑃 − 1))
54	26	nnnn0d 11947	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑃 ∈ ℕ₀)
55	28, 54	nn0mulcld 11952	. . . . . . . . . 10 ⊢ (𝜑 → (𝑀 · 𝑃) ∈ ℕ₀)
56	55	nn0ge0d 11950	. . . . . . . . 9 ⊢ (𝜑 → 0 ≤ (𝑀 · 𝑃))
57	49	nn0red 11948	. . . . . . . . . 10 ⊢ (𝜑 → (𝑃 − 1) ∈ ℝ)
58	47	zred 12079	. . . . . . . . . 10 ⊢ (𝜑 → (𝑀 · 𝑃) ∈ ℝ)
59	57, 58	addge02d 11221	. . . . . . . . 9 ⊢ (𝜑 → (0 ≤ (𝑀 · 𝑃) ↔ (𝑃 − 1) ≤ ((𝑀 · 𝑃) + (𝑃 − 1))))
60	56, 59	mpbid 234	. . . . . . . 8 ⊢ (𝜑 → (𝑃 − 1) ≤ ((𝑀 · 𝑃) + (𝑃 − 1)))
61	52, 53, 60	jca32 518	. . . . . . 7 ⊢ (𝜑 → ((0 ∈ ℤ ∧ ((𝑀 · 𝑃) + (𝑃 − 1)) ∈ ℤ ∧ (𝑃 − 1) ∈ ℤ) ∧ (0 ≤ (𝑃 − 1) ∧ (𝑃 − 1) ≤ ((𝑀 · 𝑃) + (𝑃 − 1)))))
62		elfz2 12891	. . . . . . 7 ⊢ ((𝑃 − 1) ∈ (0...((𝑀 · 𝑃) + (𝑃 − 1))) ↔ ((0 ∈ ℤ ∧ ((𝑀 · 𝑃) + (𝑃 − 1)) ∈ ℤ ∧ (𝑃 − 1) ∈ ℤ) ∧ (0 ≤ (𝑃 − 1) ∧ (𝑃 − 1) ≤ ((𝑀 · 𝑃) + (𝑃 − 1)))))
63	61, 62	sylibr 236	. . . . . 6 ⊢ (𝜑 → (𝑃 − 1) ∈ (0...((𝑀 · 𝑃) + (𝑃 − 1))))
64		opelxp 5584	. . . . . 6 ⊢ (⟨0, (𝑃 − 1)⟩ ∈ ((0...𝑀) × (0...((𝑀 · 𝑃) + (𝑃 − 1)))) ↔ (0 ∈ (0...𝑀) ∧ (𝑃 − 1) ∈ (0...((𝑀 · 𝑃) + (𝑃 − 1)))))
65	43, 63, 64	sylanbrc 585	. . . . 5 ⊢ (𝜑 → ⟨0, (𝑃 − 1)⟩ ∈ ((0...𝑀) × (0...((𝑀 · 𝑃) + (𝑃 − 1)))))
66		fveq2 6663	. . . . . . . 8 ⊢ (𝑘 = ⟨0, (𝑃 − 1)⟩ → (1^st ‘𝑘) = (1^st ‘⟨0, (𝑃 − 1)⟩))
67		0re 10635	. . . . . . . . 9 ⊢ 0 ∈ ℝ
68		ovex 7181	. . . . . . . . 9 ⊢ (𝑃 − 1) ∈ V
69		op1stg 7693	. . . . . . . . 9 ⊢ ((0 ∈ ℝ ∧ (𝑃 − 1) ∈ V) → (1^st ‘⟨0, (𝑃 − 1)⟩) = 0)
70	67, 68, 69	mp2an 690	. . . . . . . 8 ⊢ (1^st ‘⟨0, (𝑃 − 1)⟩) = 0
71	66, 70	syl6eq 2870	. . . . . . 7 ⊢ (𝑘 = ⟨0, (𝑃 − 1)⟩ → (1^st ‘𝑘) = 0)
72	71	fveq2d 6667	. . . . . 6 ⊢ (𝑘 = ⟨0, (𝑃 − 1)⟩ → (𝐴‘(1^st ‘𝑘)) = (𝐴‘0))
73		fveq2 6663	. . . . . . . . 9 ⊢ (𝑘 = ⟨0, (𝑃 − 1)⟩ → (2^nd ‘𝑘) = (2^nd ‘⟨0, (𝑃 − 1)⟩))
74		op2ndg 7694	. . . . . . . . . 10 ⊢ ((0 ∈ ℝ ∧ (𝑃 − 1) ∈ V) → (2^nd ‘⟨0, (𝑃 − 1)⟩) = (𝑃 − 1))
75	67, 68, 74	mp2an 690	. . . . . . . . 9 ⊢ (2^nd ‘⟨0, (𝑃 − 1)⟩) = (𝑃 − 1)
76	73, 75	syl6eq 2870	. . . . . . . 8 ⊢ (𝑘 = ⟨0, (𝑃 − 1)⟩ → (2^nd ‘𝑘) = (𝑃 − 1))
77	76	fveq2d 6667	. . . . . . 7 ⊢ (𝑘 = ⟨0, (𝑃 − 1)⟩ → ((ℝ D^𝑛 𝐹)‘(2^nd ‘𝑘)) = ((ℝ D^𝑛 𝐹)‘(𝑃 − 1)))
78	77, 71	fveq12d 6670	. . . . . 6 ⊢ (𝑘 = ⟨0, (𝑃 − 1)⟩ → (((ℝ D^𝑛 𝐹)‘(2^nd ‘𝑘))‘(1^st ‘𝑘)) = (((ℝ D^𝑛 𝐹)‘(𝑃 − 1))‘0))
79	72, 78	oveq12d 7166	. . . . 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)))
80	3, 4, 9, 39, 65, 79	fsumsplit1 41842	. . . 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)))))
81	80	oveq1d 7163	. . 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))))
82	12, 43	sseldi 3963	. . . . . . 7 ⊢ (𝜑 → 0 ∈ ℕ₀)
83	10, 82	ffvelrnd 6845	. . . . . 6 ⊢ (𝜑 → (𝐴‘0) ∈ ℤ)
84	17	a1i 11	. . . . . . 7 ⊢ (𝜑 → ℝ ∈ {ℝ, ℂ})
85	22	a1i 11	. . . . . . 7 ⊢ (𝜑 → ℝ ∈ ((TopOpen‘ℂ_fld) ↾_t ℝ))
86	67	a1i 11	. . . . . . 7 ⊢ (𝜑 → 0 ∈ ℝ)
87	84, 85, 26, 28, 30, 49, 86, 44	etransclem42 42551	. . . . . 6 ⊢ (𝜑 → (((ℝ D^𝑛 𝐹)‘(𝑃 − 1))‘0) ∈ ℤ)
88	83, 87	zmulcld 12085	. . . . 5 ⊢ (𝜑 → ((𝐴‘0) · (((ℝ D^𝑛 𝐹)‘(𝑃 − 1))‘0)) ∈ ℤ)
89	88	zcnd 12080	. . . 4 ⊢ (𝜑 → ((𝐴‘0) · (((ℝ D^𝑛 𝐹)‘(𝑃 − 1))‘0)) ∈ ℂ)
90		difss 4106	. . . . . . . 8 ⊢ (((0...𝑀) × (0...((𝑀 · 𝑃) + (𝑃 − 1)))) ∖ {⟨0, (𝑃 − 1)⟩}) ⊆ ((0...𝑀) × (0...((𝑀 · 𝑃) + (𝑃 − 1))))
91		ssfi 8730	. . . . . . . 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)
92	8, 90, 91	mp2an 690	. . . . . . 7 ⊢ (((0...𝑀) × (0...((𝑀 · 𝑃) + (𝑃 − 1)))) ∖ {⟨0, (𝑃 − 1)⟩}) ∈ Fin
93	92	a1i 11	. . . . . 6 ⊢ (𝜑 → (((0...𝑀) × (0...((𝑀 · 𝑃) + (𝑃 − 1)))) ∖ {⟨0, (𝑃 − 1)⟩}) ∈ Fin)
94		eldifi 4101	. . . . . . 7 ⊢ (𝑘 ∈ (((0...𝑀) × (0...((𝑀 · 𝑃) + (𝑃 − 1)))) ∖ {⟨0, (𝑃 − 1)⟩}) → 𝑘 ∈ ((0...𝑀) × (0...((𝑀 · 𝑃) + (𝑃 − 1)))))
95	94, 38	sylan2 594	. . . . . 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)
96	93, 95	fsumzcl 15084	. . . . 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)
97	96	zcnd 12080	. . . 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)
98	49	faccld 13636	. . . . 5 ⊢ (𝜑 → (!‘(𝑃 − 1)) ∈ ℕ)
99	98	nncnd 11646	. . . 4 ⊢ (𝜑 → (!‘(𝑃 − 1)) ∈ ℂ)
100	98	nnne0d 11679	. . . 4 ⊢ (𝜑 → (!‘(𝑃 − 1)) ≠ 0)
101	89, 97, 99, 100	divdird 11446	. . 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)))))
102	2, 81, 101	3eqtrd 2858	. 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)))))
103	26	nnne0d 11679	. . 3 ⊢ (𝜑 → 𝑃 ≠ 0)
104	83	zcnd 12080	. . . . 5 ⊢ (𝜑 → (𝐴‘0) ∈ ℂ)
105	87	zcnd 12080	. . . . 5 ⊢ (𝜑 → (((ℝ D^𝑛 𝐹)‘(𝑃 − 1))‘0) ∈ ℂ)
106	104, 105, 99, 100	divassd 11443	. . . 4 ⊢ (𝜑 → (((𝐴‘0) · (((ℝ D^𝑛 𝐹)‘(𝑃 − 1))‘0)) / (!‘(𝑃 − 1))) = ((𝐴‘0) · ((((ℝ D^𝑛 𝐹)‘(𝑃 − 1))‘0) / (!‘(𝑃 − 1)))))
107		etransclem5 42514	. . . . . . 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))))
108		etransclem11 42520	. . . . . . 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})
109	84, 85, 26, 28, 30, 49, 107, 108, 43, 86	etransclem37 42546	. . . . . 6 ⊢ (𝜑 → (!‘(𝑃 − 1)) ∥ (((ℝ D^𝑛 𝐹)‘(𝑃 − 1))‘0))
110	98	nnzd 12078	. . . . . . 7 ⊢ (𝜑 → (!‘(𝑃 − 1)) ∈ ℤ)
111		dvdsval2 15602	. . . . . . 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))
112	110, 100, 87, 111	syl3anc 1366	. . . . . 6 ⊢ (𝜑 → ((!‘(𝑃 − 1)) ∥ (((ℝ D^𝑛 𝐹)‘(𝑃 − 1))‘0) ↔ ((((ℝ D^𝑛 𝐹)‘(𝑃 − 1))‘0) / (!‘(𝑃 − 1))) ∈ ℤ))
113	109, 112	mpbid 234	. . . . 5 ⊢ (𝜑 → ((((ℝ D^𝑛 𝐹)‘(𝑃 − 1))‘0) / (!‘(𝑃 − 1))) ∈ ℤ)
114	83, 113	zmulcld 12085	. . . 4 ⊢ (𝜑 → ((𝐴‘0) · ((((ℝ D^𝑛 𝐹)‘(𝑃 − 1))‘0) / (!‘(𝑃 − 1)))) ∈ ℤ)
115	106, 114	eqeltrd 2911	. . 3 ⊢ (𝜑 → (((𝐴‘0) · (((ℝ D^𝑛 𝐹)‘(𝑃 − 1))‘0)) / (!‘(𝑃 − 1))) ∈ ℤ)
116	94, 39	sylan2 594	. . . . 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)
117	93, 99, 116, 100	fsumdivc 15133	. . . 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))))
118	16	zcnd 12080	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ ((0...𝑀) × (0...((𝑀 · 𝑃) + (𝑃 − 1))))) → (𝐴‘(1^st ‘𝑘)) ∈ ℂ)
119	94, 118	sylan2 594	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (((0...𝑀) × (0...((𝑀 · 𝑃) + (𝑃 − 1)))) ∖ {⟨0, (𝑃 − 1)⟩})) → (𝐴‘(1^st ‘𝑘)) ∈ ℂ)
120	94, 37	sylan2 594	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (((0...𝑀) × (0...((𝑀 · 𝑃) + (𝑃 − 1)))) ∖ {⟨0, (𝑃 − 1)⟩})) → (((ℝ D^𝑛 𝐹)‘(2^nd ‘𝑘))‘(1^st ‘𝑘)) ∈ ℤ)
121	120	zcnd 12080	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (((0...𝑀) × (0...((𝑀 · 𝑃) + (𝑃 − 1)))) ∖ {⟨0, (𝑃 − 1)⟩})) → (((ℝ D^𝑛 𝐹)‘(2^nd ‘𝑘))‘(1^st ‘𝑘)) ∈ ℂ)
122	99	adantr 483	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (((0...𝑀) × (0...((𝑀 · 𝑃) + (𝑃 − 1)))) ∖ {⟨0, (𝑃 − 1)⟩})) → (!‘(𝑃 − 1)) ∈ ℂ)
123	100	adantr 483	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (((0...𝑀) × (0...((𝑀 · 𝑃) + (𝑃 − 1)))) ∖ {⟨0, (𝑃 − 1)⟩})) → (!‘(𝑃 − 1)) ≠ 0)
124	119, 121, 122, 123	divassd 11443	. . . . . 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)))))
125	94, 16	sylan2 594	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (((0...𝑀) × (0...((𝑀 · 𝑃) + (𝑃 − 1)))) ∖ {⟨0, (𝑃 − 1)⟩})) → (𝐴‘(1^st ‘𝑘)) ∈ ℤ)
126	17	a1i 11	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (((0...𝑀) × (0...((𝑀 · 𝑃) + (𝑃 − 1)))) ∖ {⟨0, (𝑃 − 1)⟩})) → ℝ ∈ {ℝ, ℂ})
127	22	a1i 11	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (((0...𝑀) × (0...((𝑀 · 𝑃) + (𝑃 − 1)))) ∖ {⟨0, (𝑃 − 1)⟩})) → ℝ ∈ ((TopOpen‘ℂ_fld) ↾_t ℝ))
128	26	adantr 483	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (((0...𝑀) × (0...((𝑀 · 𝑃) + (𝑃 − 1)))) ∖ {⟨0, (𝑃 − 1)⟩})) → 𝑃 ∈ ℕ)
129	28	adantr 483	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (((0...𝑀) × (0...((𝑀 · 𝑃) + (𝑃 − 1)))) ∖ {⟨0, (𝑃 − 1)⟩})) → 𝑀 ∈ ℕ₀)
130	94	adantl 484	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ (((0...𝑀) × (0...((𝑀 · 𝑃) + (𝑃 − 1)))) ∖ {⟨0, (𝑃 − 1)⟩})) → 𝑘 ∈ ((0...𝑀) × (0...((𝑀 · 𝑃) + (𝑃 − 1)))))
131	130, 33	syl 17	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (((0...𝑀) × (0...((𝑀 · 𝑃) + (𝑃 − 1)))) ∖ {⟨0, (𝑃 − 1)⟩})) → (2^nd ‘𝑘) ∈ ℕ₀)
132	130, 13	syl 17	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (((0...𝑀) × (0...((𝑀 · 𝑃) + (𝑃 − 1)))) ∖ {⟨0, (𝑃 − 1)⟩})) → (1^st ‘𝑘) ∈ (0...𝑀))
133	94, 35	sylan2 594	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (((0...𝑀) × (0...((𝑀 · 𝑃) + (𝑃 − 1)))) ∖ {⟨0, (𝑃 − 1)⟩})) → (1^st ‘𝑘) ∈ ℝ)
134	126, 127, 128, 129, 30, 131, 107, 108, 132, 133	etransclem37 42546	. . . . . . . 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)))
135	110	adantr 483	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (((0...𝑀) × (0...((𝑀 · 𝑃) + (𝑃 − 1)))) ∖ {⟨0, (𝑃 − 1)⟩})) → (!‘(𝑃 − 1)) ∈ ℤ)
136		dvdsval2 15602	. . . . . . . . 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))
137	135, 123, 120, 136	syl3anc 1366	. . . . . . . 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))
138	134, 137	mpbid 234	. . . . . . 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)
139	125, 138	zmulcld 12085	. . . . . 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)
140	124, 139	eqeltrd 2911	. . . . 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)
141	93, 140	fsumzcl 15084	. . . 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)
142	117, 141	eqeltrd 2911	. . 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)
143		1zzd 12005	. . . . . . . . . . . 12 ⊢ (𝜑 → 1 ∈ ℤ)
144		zabscl 14665	. . . . . . . . . . . . 13 ⊢ ((𝐴‘0) ∈ ℤ → (abs‘(𝐴‘0)) ∈ ℤ)
145	83, 144	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → (abs‘(𝐴‘0)) ∈ ℤ)
146	143, 50, 145	3jca 1123	. . . . . . . . . . 11 ⊢ (𝜑 → (1 ∈ ℤ ∧ (𝑃 − 1) ∈ ℤ ∧ (abs‘(𝐴‘0)) ∈ ℤ))
147		nn0abscl 14664	. . . . . . . . . . . . . 14 ⊢ ((𝐴‘0) ∈ ℤ → (abs‘(𝐴‘0)) ∈ ℕ₀)
148	83, 147	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → (abs‘(𝐴‘0)) ∈ ℕ₀)
149		etransclem44.a0	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝐴‘0) ≠ 0)
150	104, 149	absne0d 14799	. . . . . . . . . . . . 13 ⊢ (𝜑 → (abs‘(𝐴‘0)) ≠ 0)
151		elnnne0 11903	. . . . . . . . . . . . 13 ⊢ ((abs‘(𝐴‘0)) ∈ ℕ ↔ ((abs‘(𝐴‘0)) ∈ ℕ₀ ∧ (abs‘(𝐴‘0)) ≠ 0))
152	148, 150, 151	sylanbrc 585	. . . . . . . . . . . 12 ⊢ (𝜑 → (abs‘(𝐴‘0)) ∈ ℕ)
153	152	nnge1d 11677	. . . . . . . . . . 11 ⊢ (𝜑 → 1 ≤ (abs‘(𝐴‘0)))
154		etransclem44.ap	. . . . . . . . . . . 12 ⊢ (𝜑 → (abs‘(𝐴‘0)) < 𝑃)
155		zltlem1 12027	. . . . . . . . . . . . 13 ⊢ (((abs‘(𝐴‘0)) ∈ ℤ ∧ 𝑃 ∈ ℤ) → ((abs‘(𝐴‘0)) < 𝑃 ↔ (abs‘(𝐴‘0)) ≤ (𝑃 − 1)))
156	145, 46, 155	syl2anc 586	. . . . . . . . . . . 12 ⊢ (𝜑 → ((abs‘(𝐴‘0)) < 𝑃 ↔ (abs‘(𝐴‘0)) ≤ (𝑃 − 1)))
157	154, 156	mpbid 234	. . . . . . . . . . 11 ⊢ (𝜑 → (abs‘(𝐴‘0)) ≤ (𝑃 − 1))
158	146, 153, 157	jca32 518	. . . . . . . . . 10 ⊢ (𝜑 → ((1 ∈ ℤ ∧ (𝑃 − 1) ∈ ℤ ∧ (abs‘(𝐴‘0)) ∈ ℤ) ∧ (1 ≤ (abs‘(𝐴‘0)) ∧ (abs‘(𝐴‘0)) ≤ (𝑃 − 1))))
159		elfz2 12891	. . . . . . . . . 10 ⊢ ((abs‘(𝐴‘0)) ∈ (1...(𝑃 − 1)) ↔ ((1 ∈ ℤ ∧ (𝑃 − 1) ∈ ℤ ∧ (abs‘(𝐴‘0)) ∈ ℤ) ∧ (1 ≤ (abs‘(𝐴‘0)) ∧ (abs‘(𝐴‘0)) ≤ (𝑃 − 1))))
160	158, 159	sylibr 236	. . . . . . . . 9 ⊢ (𝜑 → (abs‘(𝐴‘0)) ∈ (1...(𝑃 − 1)))
161		fzm1ndvds 15664	. . . . . . . . 9 ⊢ ((𝑃 ∈ ℕ ∧ (abs‘(𝐴‘0)) ∈ (1...(𝑃 − 1))) → ¬ 𝑃 ∥ (abs‘(𝐴‘0)))
162	26, 160, 161	syl2anc 586	. . . . . . . 8 ⊢ (𝜑 → ¬ 𝑃 ∥ (abs‘(𝐴‘0)))
163		dvdsabsb 15621	. . . . . . . . 9 ⊢ ((𝑃 ∈ ℤ ∧ (𝐴‘0) ∈ ℤ) → (𝑃 ∥ (𝐴‘0) ↔ 𝑃 ∥ (abs‘(𝐴‘0))))
164	46, 83, 163	syl2anc 586	. . . . . . . 8 ⊢ (𝜑 → (𝑃 ∥ (𝐴‘0) ↔ 𝑃 ∥ (abs‘(𝐴‘0))))
165	162, 164	mtbird 327	. . . . . . 7 ⊢ (𝜑 → ¬ 𝑃 ∥ (𝐴‘0))
166		etransclem44.mp	. . . . . . . 8 ⊢ (𝜑 → (!‘𝑀) < 𝑃)
167	28, 24, 166, 30	etransclem41 42550	. . . . . . 7 ⊢ (𝜑 → ¬ 𝑃 ∥ ((((ℝ D^𝑛 𝐹)‘(𝑃 − 1))‘0) / (!‘(𝑃 − 1))))
168	165, 167	jca 514	. . . . . 6 ⊢ (𝜑 → (¬ 𝑃 ∥ (𝐴‘0) ∧ ¬ 𝑃 ∥ ((((ℝ D^𝑛 𝐹)‘(𝑃 − 1))‘0) / (!‘(𝑃 − 1)))))
169		pm4.56 985	. . . . . 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)))))
170	168, 169	sylib 220	. . . . 5 ⊢ (𝜑 → ¬ (𝑃 ∥ (𝐴‘0) ∨ 𝑃 ∥ ((((ℝ D^𝑛 𝐹)‘(𝑃 − 1))‘0) / (!‘(𝑃 − 1)))))
171		euclemma 16049	. . . . . 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))))))
172	24, 83, 113, 171	syl3anc 1366	. . . . 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))))))
173	170, 172	mtbird 327	. . . 4 ⊢ (𝜑 → ¬ 𝑃 ∥ ((𝐴‘0) · ((((ℝ D^𝑛 𝐹)‘(𝑃 − 1))‘0) / (!‘(𝑃 − 1)))))
174	106	breq2d 5069	. . . 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))))))
175	173, 174	mtbird 327	. . 3 ⊢ (𝜑 → ¬ 𝑃 ∥ (((𝐴‘0) · (((ℝ D^𝑛 𝐹)‘(𝑃 − 1))‘0)) / (!‘(𝑃 − 1))))
176	46	adantr 483	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (((0...𝑀) × (0...((𝑀 · 𝑃) + (𝑃 − 1)))) ∖ {⟨0, (𝑃 − 1)⟩})) → 𝑃 ∈ ℤ)
177	176, 125, 138	3jca 1123	. . . . . . 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))
178		eldifn 4102	. . . . . . . . . 10 ⊢ (𝑘 ∈ (((0...𝑀) × (0...((𝑀 · 𝑃) + (𝑃 − 1)))) ∖ {⟨0, (𝑃 − 1)⟩}) → ¬ 𝑘 ∈ {⟨0, (𝑃 − 1)⟩})
179	94	adantr 483	. . . . . . . . . . . . 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)))))
180		1st2nd2 7720	. . . . . . . . . . . . 13 ⊢ (𝑘 ∈ ((0...𝑀) × (0...((𝑀 · 𝑃) + (𝑃 − 1)))) → 𝑘 = ⟨(1^st ‘𝑘), (2^nd ‘𝑘)⟩)
181	179, 180	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)⟩)
182		simpr 487	. . . . . . . . . . . . . 14 ⊢ (((2^nd ‘𝑘) = (𝑃 − 1) ∧ (1^st ‘𝑘) = 0) → (1^st ‘𝑘) = 0)
183		simpl 485	. . . . . . . . . . . . . 14 ⊢ (((2^nd ‘𝑘) = (𝑃 − 1) ∧ (1^st ‘𝑘) = 0) → (2^nd ‘𝑘) = (𝑃 − 1))
184	182, 183	opeq12d 4803	. . . . . . . . . . . . 13 ⊢ (((2^nd ‘𝑘) = (𝑃 − 1) ∧ (1^st ‘𝑘) = 0) → ⟨(1^st ‘𝑘), (2^nd ‘𝑘)⟩ = ⟨0, (𝑃 − 1)⟩)
185	184	adantl 484	. . . . . . . . . . . 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)⟩)
186	181, 185	eqtrd 2854	. . . . . . . . . . 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)⟩)
187		velsn 4575	. . . . . . . . . . 11 ⊢ (𝑘 ∈ {⟨0, (𝑃 − 1)⟩} ↔ 𝑘 = ⟨0, (𝑃 − 1)⟩)
188	186, 187	sylibr 236	. . . . . . . . . 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)⟩})
189	178, 188	mtand 814	. . . . . . . . 9 ⊢ (𝑘 ∈ (((0...𝑀) × (0...((𝑀 · 𝑃) + (𝑃 − 1)))) ∖ {⟨0, (𝑃 − 1)⟩}) → ¬ ((2^nd ‘𝑘) = (𝑃 − 1) ∧ (1^st ‘𝑘) = 0))
190	189	adantl 484	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (((0...𝑀) × (0...((𝑀 · 𝑃) + (𝑃 − 1)))) ∖ {⟨0, (𝑃 − 1)⟩})) → ¬ ((2^nd ‘𝑘) = (𝑃 − 1) ∧ (1^st ‘𝑘) = 0))
191	128, 129, 30, 131, 132, 190, 108	etransclem38 42547	. . . . . . 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))))
192		dvdsmultr2 15641	. . . . . . 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))))))
193	177, 191, 192	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)))))
194	193, 124	breqtrrd 5085	. . . . 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))))
195	93, 46, 140, 194	fsumdvds 15650	. . . 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))))
196	195, 117	breqtrrd 5085	. . 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))))
197	46, 103, 115, 142, 175, 196	etransclem9 42518	. 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)
198	102, 197	eqnetrd 3081	1 ⊢ (𝜑 → 𝐾 ≠ 0)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 208 ∧ wa 398 ∨ wo 843 ∧ w3a 1082 = wceq 1531 ∈ wcel 2108 ≠ wne 3014 {crab 3140 Vcvv 3493 ∖ cdif 3931 ⊆ wss 3934 ifcif 4465 {csn 4559 {cpr 4561 ⟨cop 4565 class class class wbr 5057 ↦ cmpt 5137 × cxp 5546 ran crn 5549 ⟶wf 6344 ‘cfv 6348 (class class class)co 7148 1^st c1st 7679 2^nd c2nd 7680 ↑_m cmap 8398 Fincfn 8501 ℂcc 10527 ℝcr 10528 0cc0 10529 1c1 10530 + caddc 10532 · cmul 10534 < clt 10667 ≤ cle 10668 − cmin 10862 / cdiv 11289 ℕcn 11630 ℕ₀cn0 11889 ℤcz 11973 ℤ_≥cuz 12235 (,)cioo 12730 ...cfz 12884 ↑cexp 13421 !cfa 13625 abscabs 14585 Σcsu 15034 ∏cprod 15251 ∥ cdvds 15599 ℙcprime 16007 ↾_t crest 16686 TopOpenctopn 16687 topGenctg 16703 ℂ_fldccnfld 20537 D^𝑛 cdvn 24454

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1905 ax-6 1964 ax-7 2009 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2154 ax-12 2170 ax-ext 2791 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7453 ax-inf2 9096 ax-cnex 10585 ax-resscn 10586 ax-1cn 10587 ax-icn 10588 ax-addcl 10589 ax-addrcl 10590 ax-mulcl 10591 ax-mulrcl 10592 ax-mulcom 10593 ax-addass 10594 ax-mulass 10595 ax-distr 10596 ax-i2m1 10597 ax-1ne0 10598 ax-1rid 10599 ax-rnegex 10600 ax-rrecex 10601 ax-cnre 10602 ax-pre-lttri 10603 ax-pre-lttrn 10604 ax-pre-ltadd 10605 ax-pre-mulgt0 10606 ax-pre-sup 10607 ax-addf 10608 ax-mulf 10609

This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1083 df-3an 1084 df-tru 1534 df-fal 1544 df-ex 1775 df-nf 1779 df-sb 2064 df-mo 2616 df-eu 2648 df-clab 2798 df-cleq 2812 df-clel 2891 df-nfc 2961 df-ne 3015 df-nel 3122 df-ral 3141 df-rex 3142 df-reu 3143 df-rmo 3144 df-rab 3145 df-v 3495 df-sbc 3771 df-csb 3882 df-dif 3937 df-un 3939 df-in 3941 df-ss 3950 df-pss 3952 df-nul 4290 df-if 4466 df-pw 4539 df-sn 4560 df-pr 4562 df-tp 4564 df-op 4566 df-uni 4831 df-int 4868 df-iun 4912 df-iin 4913 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-se 5508 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-isom 6357 df-riota 7106 df-ov 7151 df-oprab 7152 df-mpo 7153 df-of 7401 df-om 7573 df-1st 7681 df-2nd 7682 df-supp 7823 df-wrecs 7939 df-recs 8000 df-rdg 8038 df-1o 8094 df-2o 8095 df-oadd 8098 df-er 8281 df-map 8400 df-pm 8401 df-ixp 8454 df-en 8502 df-dom 8503 df-sdom 8504 df-fin 8505 df-fsupp 8826 df-fi 8867 df-sup 8898 df-inf 8899 df-oi 8966 df-card 9360 df-pnf 10669 df-mnf 10670 df-xr 10671 df-ltxr 10672 df-le 10673 df-sub 10864 df-neg 10865 df-div 11290 df-nn 11631 df-2 11692 df-3 11693 df-4 11694 df-5 11695 df-6 11696 df-7 11697 df-8 11698 df-9 11699 df-n0 11890 df-z 11974 df-dec 12091 df-uz 12236 df-q 12341 df-rp 12382 df-xneg 12499 df-xadd 12500 df-xmul 12501 df-ioo 12734 df-ico 12736 df-icc 12737 df-fz 12885 df-fzo 13026 df-fl 13154 df-mod 13230 df-seq 13362 df-exp 13422 df-fac 13626 df-bc 13655 df-hash 13683 df-cj 14450 df-re 14451 df-im 14452 df-sqrt 14586 df-abs 14587 df-clim 14837 df-sum 15035 df-prod 15252 df-dvds 15600 df-gcd 15836 df-prm 16008 df-struct 16477 df-ndx 16478 df-slot 16479 df-base 16481 df-sets 16482 df-ress 16483 df-plusg 16570 df-mulr 16571 df-starv 16572 df-sca 16573 df-vsca 16574 df-ip 16575 df-tset 16576 df-ple 16577 df-ds 16579 df-unif 16580 df-hom 16581 df-cco 16582 df-rest 16688 df-topn 16689 df-0g 16707 df-gsum 16708 df-topgen 16709 df-pt 16710 df-prds 16713 df-xrs 16767 df-qtop 16772 df-imas 16773 df-xps 16775 df-mre 16849 df-mrc 16850 df-acs 16852 df-mgm 17844 df-sgrp 17893 df-mnd 17904 df-submnd 17949 df-mulg 18217 df-cntz 18439 df-cmn 18900 df-psmet 20529 df-xmet 20530 df-met 20531 df-bl 20532 df-mopn 20533 df-fbas 20534 df-fg 20535 df-cnfld 20538 df-top 21494 df-topon 21511 df-topsp 21533 df-bases 21546 df-cld 21619 df-ntr 21620 df-cls 21621 df-nei 21698 df-lp 21736 df-perf 21737 df-cn 21827 df-cnp 21828 df-haus 21915 df-tx 22162 df-hmeo 22355 df-fil 22446 df-fm 22538 df-flim 22539 df-flf 22540 df-xms 22922 df-ms 22923 df-tms 22924 df-cncf 23478 df-limc 24456 df-dv 24457 df-dvn 24458

This theorem is referenced by: etransclem47 42556

W3C validator