etransclem44 - Mathbox for Glauco Siliprandi

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Theorem etransclem44 44489

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 1917	. . . . 5 ⊢ Ⅎ𝑘𝜑
4		nfcv 2906	. . . . 5 ⊢ Ⅎ𝑘((𝐴‘0) · (((ℝ D^𝑛 𝐹)‘(𝑃 − 1))‘0))
5		fzfi 13874	. . . . . . 7 ⊢ (0...𝑀) ∈ Fin
6		fzfi 13874	. . . . . . 7 ⊢ (0...((𝑀 · 𝑃) + (𝑃 − 1))) ∈ Fin
7		xpfi 9258	. . . . . . 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 481	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ ((0...𝑀) × (0...((𝑀 · 𝑃) + (𝑃 − 1))))) → 𝐴:ℕ₀⟶ℤ)
12		fzssnn0 43525	. . . . . . . . . 10 ⊢ (0...𝑀) ⊆ ℕ₀
13		xp1st 7950	. . . . . . . . . 10 ⊢ (𝑘 ∈ ((0...𝑀) × (0...((𝑀 · 𝑃) + (𝑃 − 1)))) → (1^st ‘𝑘) ∈ (0...𝑀))
14	12, 13	sselid 3941	. . . . . . . . 9 ⊢ (𝑘 ∈ ((0...𝑀) × (0...((𝑀 · 𝑃) + (𝑃 − 1)))) → (1^st ‘𝑘) ∈ ℕ₀)
15	14	adantl 482	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ ((0...𝑀) × (0...((𝑀 · 𝑃) + (𝑃 − 1))))) → (1^st ‘𝑘) ∈ ℕ₀)
16	11, 15	ffvelcdmd 7033	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ ((0...𝑀) × (0...((𝑀 · 𝑃) + (𝑃 − 1))))) → (𝐴‘(1^st ‘𝑘)) ∈ ℤ)
17		reelprrecn 11140	. . . . . . . . 9 ⊢ ℝ ∈ {ℝ, ℂ}
18	17	a1i 11	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ ((0...𝑀) × (0...((𝑀 · 𝑃) + (𝑃 − 1))))) → ℝ ∈ {ℝ, ℂ})
19		reopn 43497	. . . . . . . . . 10 ⊢ ℝ ∈ (topGen‘ran (,))
20		eqid 2736	. . . . . . . . . . 11 ⊢ (TopOpen‘ℂ_fld) = (TopOpen‘ℂ_fld)
21	20	tgioo2 24162	. . . . . . . . . 10 ⊢ (topGen‘ran (,)) = ((TopOpen‘ℂ_fld) ↾_t ℝ)
22	19, 21	eleqtri 2836	. . . . . . . . 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 16547	. . . . . . . . . 10 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ ℕ)
26	24, 25	syl 17	. . . . . . . . 9 ⊢ (𝜑 → 𝑃 ∈ ℕ)
27	26	adantr 481	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ ((0...𝑀) × (0...((𝑀 · 𝑃) + (𝑃 − 1))))) → 𝑃 ∈ ℕ)
28		etransclem44.m	. . . . . . . . 9 ⊢ (𝜑 → 𝑀 ∈ ℕ₀)
29	28	adantr 481	. . . . . . . 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 7951	. . . . . . . . . 10 ⊢ (𝑘 ∈ ((0...𝑀) × (0...((𝑀 · 𝑃) + (𝑃 − 1)))) → (2^nd ‘𝑘) ∈ (0...((𝑀 · 𝑃) + (𝑃 − 1))))
32		elfznn0 13531	. . . . . . . . . 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 482	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ ((0...𝑀) × (0...((𝑀 · 𝑃) + (𝑃 − 1))))) → (2^nd ‘𝑘) ∈ ℕ₀)
35	15	nn0red 12471	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ ((0...𝑀) × (0...((𝑀 · 𝑃) + (𝑃 − 1))))) → (1^st ‘𝑘) ∈ ℝ)
36	15	nn0zd 12522	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ ((0...𝑀) × (0...((𝑀 · 𝑃) + (𝑃 − 1))))) → (1^st ‘𝑘) ∈ ℤ)
37	18, 23, 27, 29, 30, 34, 35, 36	etransclem42 44487	. . . . . . 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 12610	. . . . . 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 12605	. . . . 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 12802	. . . . . . . 8 ⊢ ℕ₀ = (ℤ_≥‘0)
41	28, 40	eleqtrdi 2848	. . . . . . 7 ⊢ (𝜑 → 𝑀 ∈ (ℤ_≥‘0))
42		eluzfz1 13445	. . . . . . 7 ⊢ (𝑀 ∈ (ℤ_≥‘0) → 0 ∈ (0...𝑀))
43	41, 42	syl 17	. . . . . 6 ⊢ (𝜑 → 0 ∈ (0...𝑀))
44		0zd 12508	. . . . . . 7 ⊢ (𝜑 → 0 ∈ ℤ)
45	28	nn0zd 12522	. . . . . . . . 9 ⊢ (𝜑 → 𝑀 ∈ ℤ)
46	26	nnzd 12523	. . . . . . . . 9 ⊢ (𝜑 → 𝑃 ∈ ℤ)
47	45, 46	zmulcld 12610	. . . . . . . 8 ⊢ (𝜑 → (𝑀 · 𝑃) ∈ ℤ)
48		nnm1nn0 12451	. . . . . . . . . 10 ⊢ (𝑃 ∈ ℕ → (𝑃 − 1) ∈ ℕ₀)
49	26, 48	syl 17	. . . . . . . . 9 ⊢ (𝜑 → (𝑃 − 1) ∈ ℕ₀)
50	49	nn0zd 12522	. . . . . . . 8 ⊢ (𝜑 → (𝑃 − 1) ∈ ℤ)
51	47, 50	zaddcld 12608	. . . . . . 7 ⊢ (𝜑 → ((𝑀 · 𝑃) + (𝑃 − 1)) ∈ ℤ)
52	49	nn0ge0d 12473	. . . . . . 7 ⊢ (𝜑 → 0 ≤ (𝑃 − 1))
53	26	nnnn0d 12470	. . . . . . . . . 10 ⊢ (𝜑 → 𝑃 ∈ ℕ₀)
54	28, 53	nn0mulcld 12475	. . . . . . . . 9 ⊢ (𝜑 → (𝑀 · 𝑃) ∈ ℕ₀)
55	54	nn0ge0d 12473	. . . . . . . 8 ⊢ (𝜑 → 0 ≤ (𝑀 · 𝑃))
56	49	nn0red 12471	. . . . . . . . 9 ⊢ (𝜑 → (𝑃 − 1) ∈ ℝ)
57	47	zred 12604	. . . . . . . . 9 ⊢ (𝜑 → (𝑀 · 𝑃) ∈ ℝ)
58	56, 57	addge02d 11741	. . . . . . . 8 ⊢ (𝜑 → (0 ≤ (𝑀 · 𝑃) ↔ (𝑃 − 1) ≤ ((𝑀 · 𝑃) + (𝑃 − 1))))
59	55, 58	mpbid 231	. . . . . . 7 ⊢ (𝜑 → (𝑃 − 1) ≤ ((𝑀 · 𝑃) + (𝑃 − 1)))
60	44, 51, 50, 52, 59	elfzd 13429	. . . . . 6 ⊢ (𝜑 → (𝑃 − 1) ∈ (0...((𝑀 · 𝑃) + (𝑃 − 1))))
61		opelxp 5668	. . . . . 6 ⊢ (⟨0, (𝑃 − 1)⟩ ∈ ((0...𝑀) × (0...((𝑀 · 𝑃) + (𝑃 − 1)))) ↔ (0 ∈ (0...𝑀) ∧ (𝑃 − 1) ∈ (0...((𝑀 · 𝑃) + (𝑃 − 1)))))
62	43, 60, 61	sylanbrc 583	. . . . 5 ⊢ (𝜑 → ⟨0, (𝑃 − 1)⟩ ∈ ((0...𝑀) × (0...((𝑀 · 𝑃) + (𝑃 − 1)))))
63		fveq2 6840	. . . . . . . 8 ⊢ (𝑘 = ⟨0, (𝑃 − 1)⟩ → (1^st ‘𝑘) = (1^st ‘⟨0, (𝑃 − 1)⟩))
64		0re 11154	. . . . . . . . 9 ⊢ 0 ∈ ℝ
65		ovex 7387	. . . . . . . . 9 ⊢ (𝑃 − 1) ∈ V
66		op1stg 7930	. . . . . . . . 9 ⊢ ((0 ∈ ℝ ∧ (𝑃 − 1) ∈ V) → (1^st ‘⟨0, (𝑃 − 1)⟩) = 0)
67	64, 65, 66	mp2an 690	. . . . . . . 8 ⊢ (1^st ‘⟨0, (𝑃 − 1)⟩) = 0
68	63, 67	eqtrdi 2792	. . . . . . 7 ⊢ (𝑘 = ⟨0, (𝑃 − 1)⟩ → (1^st ‘𝑘) = 0)
69	68	fveq2d 6844	. . . . . 6 ⊢ (𝑘 = ⟨0, (𝑃 − 1)⟩ → (𝐴‘(1^st ‘𝑘)) = (𝐴‘0))
70		fveq2 6840	. . . . . . . . 9 ⊢ (𝑘 = ⟨0, (𝑃 − 1)⟩ → (2^nd ‘𝑘) = (2^nd ‘⟨0, (𝑃 − 1)⟩))
71		op2ndg 7931	. . . . . . . . . 10 ⊢ ((0 ∈ ℝ ∧ (𝑃 − 1) ∈ V) → (2^nd ‘⟨0, (𝑃 − 1)⟩) = (𝑃 − 1))
72	64, 65, 71	mp2an 690	. . . . . . . . 9 ⊢ (2^nd ‘⟨0, (𝑃 − 1)⟩) = (𝑃 − 1)
73	70, 72	eqtrdi 2792	. . . . . . . 8 ⊢ (𝑘 = ⟨0, (𝑃 − 1)⟩ → (2^nd ‘𝑘) = (𝑃 − 1))
74	73	fveq2d 6844	. . . . . . 7 ⊢ (𝑘 = ⟨0, (𝑃 − 1)⟩ → ((ℝ D^𝑛 𝐹)‘(2^nd ‘𝑘)) = ((ℝ D^𝑛 𝐹)‘(𝑃 − 1)))
75	74, 68	fveq12d 6847	. . . . . 6 ⊢ (𝑘 = ⟨0, (𝑃 − 1)⟩ → (((ℝ D^𝑛 𝐹)‘(2^nd ‘𝑘))‘(1^st ‘𝑘)) = (((ℝ D^𝑛 𝐹)‘(𝑃 − 1))‘0))
76	69, 75	oveq12d 7372	. . . . 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)))
77	3, 4, 9, 39, 62, 76	fsumsplit1 15627	. . . 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)))))
78	77	oveq1d 7369	. . 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))))
79	12, 43	sselid 3941	. . . . . . 7 ⊢ (𝜑 → 0 ∈ ℕ₀)
80	10, 79	ffvelcdmd 7033	. . . . . 6 ⊢ (𝜑 → (𝐴‘0) ∈ ℤ)
81	17	a1i 11	. . . . . . 7 ⊢ (𝜑 → ℝ ∈ {ℝ, ℂ})
82	22	a1i 11	. . . . . . 7 ⊢ (𝜑 → ℝ ∈ ((TopOpen‘ℂ_fld) ↾_t ℝ))
83	64	a1i 11	. . . . . . 7 ⊢ (𝜑 → 0 ∈ ℝ)
84	81, 82, 26, 28, 30, 49, 83, 44	etransclem42 44487	. . . . . 6 ⊢ (𝜑 → (((ℝ D^𝑛 𝐹)‘(𝑃 − 1))‘0) ∈ ℤ)
85	80, 84	zmulcld 12610	. . . . 5 ⊢ (𝜑 → ((𝐴‘0) · (((ℝ D^𝑛 𝐹)‘(𝑃 − 1))‘0)) ∈ ℤ)
86	85	zcnd 12605	. . . 4 ⊢ (𝜑 → ((𝐴‘0) · (((ℝ D^𝑛 𝐹)‘(𝑃 − 1))‘0)) ∈ ℂ)
87		difss 4090	. . . . . . . 8 ⊢ (((0...𝑀) × (0...((𝑀 · 𝑃) + (𝑃 − 1)))) ∖ {⟨0, (𝑃 − 1)⟩}) ⊆ ((0...𝑀) × (0...((𝑀 · 𝑃) + (𝑃 − 1))))
88		ssfi 9114	. . . . . . . 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)
89	8, 87, 88	mp2an 690	. . . . . . 7 ⊢ (((0...𝑀) × (0...((𝑀 · 𝑃) + (𝑃 − 1)))) ∖ {⟨0, (𝑃 − 1)⟩}) ∈ Fin
90	89	a1i 11	. . . . . 6 ⊢ (𝜑 → (((0...𝑀) × (0...((𝑀 · 𝑃) + (𝑃 − 1)))) ∖ {⟨0, (𝑃 − 1)⟩}) ∈ Fin)
91		eldifi 4085	. . . . . . 7 ⊢ (𝑘 ∈ (((0...𝑀) × (0...((𝑀 · 𝑃) + (𝑃 − 1)))) ∖ {⟨0, (𝑃 − 1)⟩}) → 𝑘 ∈ ((0...𝑀) × (0...((𝑀 · 𝑃) + (𝑃 − 1)))))
92	91, 38	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)
93	90, 92	fsumzcl 15617	. . . . 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)
94	93	zcnd 12605	. . . 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)
95	49	faccld 14181	. . . . 5 ⊢ (𝜑 → (!‘(𝑃 − 1)) ∈ ℕ)
96	95	nncnd 12166	. . . 4 ⊢ (𝜑 → (!‘(𝑃 − 1)) ∈ ℂ)
97	95	nnne0d 12200	. . . 4 ⊢ (𝜑 → (!‘(𝑃 − 1)) ≠ 0)
98	86, 94, 96, 97	divdird 11966	. . 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)))))
99	2, 78, 98	3eqtrd 2780	. 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)))))
100	26	nnne0d 12200	. . 3 ⊢ (𝜑 → 𝑃 ≠ 0)
101	80	zcnd 12605	. . . . 5 ⊢ (𝜑 → (𝐴‘0) ∈ ℂ)
102	84	zcnd 12605	. . . . 5 ⊢ (𝜑 → (((ℝ D^𝑛 𝐹)‘(𝑃 − 1))‘0) ∈ ℂ)
103	101, 102, 96, 97	divassd 11963	. . . 4 ⊢ (𝜑 → (((𝐴‘0) · (((ℝ D^𝑛 𝐹)‘(𝑃 − 1))‘0)) / (!‘(𝑃 − 1))) = ((𝐴‘0) · ((((ℝ D^𝑛 𝐹)‘(𝑃 − 1))‘0) / (!‘(𝑃 − 1)))))
104		etransclem5 44450	. . . . . . 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))))
105		etransclem11 44456	. . . . . . 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})
106	81, 82, 26, 28, 30, 49, 104, 105, 43, 83	etransclem37 44482	. . . . . 6 ⊢ (𝜑 → (!‘(𝑃 − 1)) ∥ (((ℝ D^𝑛 𝐹)‘(𝑃 − 1))‘0))
107	95	nnzd 12523	. . . . . . 7 ⊢ (𝜑 → (!‘(𝑃 − 1)) ∈ ℤ)
108		dvdsval2 16136	. . . . . . 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))
109	107, 97, 84, 108	syl3anc 1371	. . . . . 6 ⊢ (𝜑 → ((!‘(𝑃 − 1)) ∥ (((ℝ D^𝑛 𝐹)‘(𝑃 − 1))‘0) ↔ ((((ℝ D^𝑛 𝐹)‘(𝑃 − 1))‘0) / (!‘(𝑃 − 1))) ∈ ℤ))
110	106, 109	mpbid 231	. . . . 5 ⊢ (𝜑 → ((((ℝ D^𝑛 𝐹)‘(𝑃 − 1))‘0) / (!‘(𝑃 − 1))) ∈ ℤ)
111	80, 110	zmulcld 12610	. . . 4 ⊢ (𝜑 → ((𝐴‘0) · ((((ℝ D^𝑛 𝐹)‘(𝑃 − 1))‘0) / (!‘(𝑃 − 1)))) ∈ ℤ)
112	103, 111	eqeltrd 2838	. . 3 ⊢ (𝜑 → (((𝐴‘0) · (((ℝ D^𝑛 𝐹)‘(𝑃 − 1))‘0)) / (!‘(𝑃 − 1))) ∈ ℤ)
113	91, 39	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)
114	90, 96, 113, 97	fsumdivc 15668	. . . 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))))
115	16	zcnd 12605	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ ((0...𝑀) × (0...((𝑀 · 𝑃) + (𝑃 − 1))))) → (𝐴‘(1^st ‘𝑘)) ∈ ℂ)
116	91, 115	sylan2 593	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (((0...𝑀) × (0...((𝑀 · 𝑃) + (𝑃 − 1)))) ∖ {⟨0, (𝑃 − 1)⟩})) → (𝐴‘(1^st ‘𝑘)) ∈ ℂ)
117	91, 37	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)
118	117	zcnd 12605	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (((0...𝑀) × (0...((𝑀 · 𝑃) + (𝑃 − 1)))) ∖ {⟨0, (𝑃 − 1)⟩})) → (((ℝ D^𝑛 𝐹)‘(2^nd ‘𝑘))‘(1^st ‘𝑘)) ∈ ℂ)
119	96	adantr 481	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (((0...𝑀) × (0...((𝑀 · 𝑃) + (𝑃 − 1)))) ∖ {⟨0, (𝑃 − 1)⟩})) → (!‘(𝑃 − 1)) ∈ ℂ)
120	97	adantr 481	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (((0...𝑀) × (0...((𝑀 · 𝑃) + (𝑃 − 1)))) ∖ {⟨0, (𝑃 − 1)⟩})) → (!‘(𝑃 − 1)) ≠ 0)
121	116, 118, 119, 120	divassd 11963	. . . . . 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)))))
122	91, 16	sylan2 593	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (((0...𝑀) × (0...((𝑀 · 𝑃) + (𝑃 − 1)))) ∖ {⟨0, (𝑃 − 1)⟩})) → (𝐴‘(1^st ‘𝑘)) ∈ ℤ)
123	17	a1i 11	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (((0...𝑀) × (0...((𝑀 · 𝑃) + (𝑃 − 1)))) ∖ {⟨0, (𝑃 − 1)⟩})) → ℝ ∈ {ℝ, ℂ})
124	22	a1i 11	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (((0...𝑀) × (0...((𝑀 · 𝑃) + (𝑃 − 1)))) ∖ {⟨0, (𝑃 − 1)⟩})) → ℝ ∈ ((TopOpen‘ℂ_fld) ↾_t ℝ))
125	26	adantr 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (((0...𝑀) × (0...((𝑀 · 𝑃) + (𝑃 − 1)))) ∖ {⟨0, (𝑃 − 1)⟩})) → 𝑃 ∈ ℕ)
126	28	adantr 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (((0...𝑀) × (0...((𝑀 · 𝑃) + (𝑃 − 1)))) ∖ {⟨0, (𝑃 − 1)⟩})) → 𝑀 ∈ ℕ₀)
127	91	adantl 482	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ (((0...𝑀) × (0...((𝑀 · 𝑃) + (𝑃 − 1)))) ∖ {⟨0, (𝑃 − 1)⟩})) → 𝑘 ∈ ((0...𝑀) × (0...((𝑀 · 𝑃) + (𝑃 − 1)))))
128	127, 33	syl 17	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (((0...𝑀) × (0...((𝑀 · 𝑃) + (𝑃 − 1)))) ∖ {⟨0, (𝑃 − 1)⟩})) → (2^nd ‘𝑘) ∈ ℕ₀)
129	127, 13	syl 17	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (((0...𝑀) × (0...((𝑀 · 𝑃) + (𝑃 − 1)))) ∖ {⟨0, (𝑃 − 1)⟩})) → (1^st ‘𝑘) ∈ (0...𝑀))
130	91, 35	sylan2 593	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (((0...𝑀) × (0...((𝑀 · 𝑃) + (𝑃 − 1)))) ∖ {⟨0, (𝑃 − 1)⟩})) → (1^st ‘𝑘) ∈ ℝ)
131	123, 124, 125, 126, 30, 128, 104, 105, 129, 130	etransclem37 44482	. . . . . . . 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)))
132	107	adantr 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (((0...𝑀) × (0...((𝑀 · 𝑃) + (𝑃 − 1)))) ∖ {⟨0, (𝑃 − 1)⟩})) → (!‘(𝑃 − 1)) ∈ ℤ)
133		dvdsval2 16136	. . . . . . . . 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))
134	132, 120, 117, 133	syl3anc 1371	. . . . . . . 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))
135	131, 134	mpbid 231	. . . . . . 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)
136	122, 135	zmulcld 12610	. . . . . 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)
137	121, 136	eqeltrd 2838	. . . . 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)
138	90, 137	fsumzcl 15617	. . . 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)
139	114, 138	eqeltrd 2838	. . 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)
140		1zzd 12531	. . . . . . . . . 10 ⊢ (𝜑 → 1 ∈ ℤ)
141		zabscl 15195	. . . . . . . . . . 11 ⊢ ((𝐴‘0) ∈ ℤ → (abs‘(𝐴‘0)) ∈ ℤ)
142	80, 141	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → (abs‘(𝐴‘0)) ∈ ℤ)
143		nn0abscl 15194	. . . . . . . . . . . . 13 ⊢ ((𝐴‘0) ∈ ℤ → (abs‘(𝐴‘0)) ∈ ℕ₀)
144	80, 143	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → (abs‘(𝐴‘0)) ∈ ℕ₀)
145		etransclem44.a0	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐴‘0) ≠ 0)
146	101, 145	absne0d 15329	. . . . . . . . . . . 12 ⊢ (𝜑 → (abs‘(𝐴‘0)) ≠ 0)
147		elnnne0 12424	. . . . . . . . . . . 12 ⊢ ((abs‘(𝐴‘0)) ∈ ℕ ↔ ((abs‘(𝐴‘0)) ∈ ℕ₀ ∧ (abs‘(𝐴‘0)) ≠ 0))
148	144, 146, 147	sylanbrc 583	. . . . . . . . . . 11 ⊢ (𝜑 → (abs‘(𝐴‘0)) ∈ ℕ)
149	148	nnge1d 12198	. . . . . . . . . 10 ⊢ (𝜑 → 1 ≤ (abs‘(𝐴‘0)))
150		etransclem44.ap	. . . . . . . . . . 11 ⊢ (𝜑 → (abs‘(𝐴‘0)) < 𝑃)
151		zltlem1 12553	. . . . . . . . . . . 12 ⊢ (((abs‘(𝐴‘0)) ∈ ℤ ∧ 𝑃 ∈ ℤ) → ((abs‘(𝐴‘0)) < 𝑃 ↔ (abs‘(𝐴‘0)) ≤ (𝑃 − 1)))
152	142, 46, 151	syl2anc 584	. . . . . . . . . . 11 ⊢ (𝜑 → ((abs‘(𝐴‘0)) < 𝑃 ↔ (abs‘(𝐴‘0)) ≤ (𝑃 − 1)))
153	150, 152	mpbid 231	. . . . . . . . . 10 ⊢ (𝜑 → (abs‘(𝐴‘0)) ≤ (𝑃 − 1))
154	140, 50, 142, 149, 153	elfzd 13429	. . . . . . . . 9 ⊢ (𝜑 → (abs‘(𝐴‘0)) ∈ (1...(𝑃 − 1)))
155		fzm1ndvds 16201	. . . . . . . . 9 ⊢ ((𝑃 ∈ ℕ ∧ (abs‘(𝐴‘0)) ∈ (1...(𝑃 − 1))) → ¬ 𝑃 ∥ (abs‘(𝐴‘0)))
156	26, 154, 155	syl2anc 584	. . . . . . . 8 ⊢ (𝜑 → ¬ 𝑃 ∥ (abs‘(𝐴‘0)))
157		dvdsabsb 16155	. . . . . . . . 9 ⊢ ((𝑃 ∈ ℤ ∧ (𝐴‘0) ∈ ℤ) → (𝑃 ∥ (𝐴‘0) ↔ 𝑃 ∥ (abs‘(𝐴‘0))))
158	46, 80, 157	syl2anc 584	. . . . . . . 8 ⊢ (𝜑 → (𝑃 ∥ (𝐴‘0) ↔ 𝑃 ∥ (abs‘(𝐴‘0))))
159	156, 158	mtbird 324	. . . . . . 7 ⊢ (𝜑 → ¬ 𝑃 ∥ (𝐴‘0))
160		etransclem44.mp	. . . . . . . 8 ⊢ (𝜑 → (!‘𝑀) < 𝑃)
161	28, 24, 160, 30	etransclem41 44486	. . . . . . 7 ⊢ (𝜑 → ¬ 𝑃 ∥ ((((ℝ D^𝑛 𝐹)‘(𝑃 − 1))‘0) / (!‘(𝑃 − 1))))
162	159, 161	jca 512	. . . . . 6 ⊢ (𝜑 → (¬ 𝑃 ∥ (𝐴‘0) ∧ ¬ 𝑃 ∥ ((((ℝ D^𝑛 𝐹)‘(𝑃 − 1))‘0) / (!‘(𝑃 − 1)))))
163		pm4.56 987	. . . . . 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)))))
164	162, 163	sylib 217	. . . . 5 ⊢ (𝜑 → ¬ (𝑃 ∥ (𝐴‘0) ∨ 𝑃 ∥ ((((ℝ D^𝑛 𝐹)‘(𝑃 − 1))‘0) / (!‘(𝑃 − 1)))))
165		euclemma 16586	. . . . . 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))))))
166	24, 80, 110, 165	syl3anc 1371	. . . . 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))))))
167	164, 166	mtbird 324	. . . 4 ⊢ (𝜑 → ¬ 𝑃 ∥ ((𝐴‘0) · ((((ℝ D^𝑛 𝐹)‘(𝑃 − 1))‘0) / (!‘(𝑃 − 1)))))
168	103	breq2d 5116	. . . 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))))))
169	167, 168	mtbird 324	. . 3 ⊢ (𝜑 → ¬ 𝑃 ∥ (((𝐴‘0) · (((ℝ D^𝑛 𝐹)‘(𝑃 − 1))‘0)) / (!‘(𝑃 − 1))))
170	46	adantr 481	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (((0...𝑀) × (0...((𝑀 · 𝑃) + (𝑃 − 1)))) ∖ {⟨0, (𝑃 − 1)⟩})) → 𝑃 ∈ ℤ)
171	170, 122, 135	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))
172		eldifn 4086	. . . . . . . . . 10 ⊢ (𝑘 ∈ (((0...𝑀) × (0...((𝑀 · 𝑃) + (𝑃 − 1)))) ∖ {⟨0, (𝑃 − 1)⟩}) → ¬ 𝑘 ∈ {⟨0, (𝑃 − 1)⟩})
173	91	adantr 481	. . . . . . . . . . . . 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)))))
174		1st2nd2 7957	. . . . . . . . . . . . 13 ⊢ (𝑘 ∈ ((0...𝑀) × (0...((𝑀 · 𝑃) + (𝑃 − 1)))) → 𝑘 = ⟨(1^st ‘𝑘), (2^nd ‘𝑘)⟩)
175	173, 174	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)⟩)
176		simpr 485	. . . . . . . . . . . . . 14 ⊢ (((2^nd ‘𝑘) = (𝑃 − 1) ∧ (1^st ‘𝑘) = 0) → (1^st ‘𝑘) = 0)
177		simpl 483	. . . . . . . . . . . . . 14 ⊢ (((2^nd ‘𝑘) = (𝑃 − 1) ∧ (1^st ‘𝑘) = 0) → (2^nd ‘𝑘) = (𝑃 − 1))
178	176, 177	opeq12d 4837	. . . . . . . . . . . . 13 ⊢ (((2^nd ‘𝑘) = (𝑃 − 1) ∧ (1^st ‘𝑘) = 0) → ⟨(1^st ‘𝑘), (2^nd ‘𝑘)⟩ = ⟨0, (𝑃 − 1)⟩)
179	178	adantl 482	. . . . . . . . . . . 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)⟩)
180	175, 179	eqtrd 2776	. . . . . . . . . . 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)⟩)
181		velsn 4601	. . . . . . . . . . 11 ⊢ (𝑘 ∈ {⟨0, (𝑃 − 1)⟩} ↔ 𝑘 = ⟨0, (𝑃 − 1)⟩)
182	180, 181	sylibr 233	. . . . . . . . . 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)⟩})
183	172, 182	mtand 814	. . . . . . . . 9 ⊢ (𝑘 ∈ (((0...𝑀) × (0...((𝑀 · 𝑃) + (𝑃 − 1)))) ∖ {⟨0, (𝑃 − 1)⟩}) → ¬ ((2^nd ‘𝑘) = (𝑃 − 1) ∧ (1^st ‘𝑘) = 0))
184	183	adantl 482	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (((0...𝑀) × (0...((𝑀 · 𝑃) + (𝑃 − 1)))) ∖ {⟨0, (𝑃 − 1)⟩})) → ¬ ((2^nd ‘𝑘) = (𝑃 − 1) ∧ (1^st ‘𝑘) = 0))
185	125, 126, 30, 128, 129, 184, 105	etransclem38 44483	. . . . . . 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))))
186		dvdsmultr2 16177	. . . . . . 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))))))
187	171, 185, 186	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)))))
188	187, 121	breqtrrd 5132	. . . . 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))))
189	90, 46, 137, 188	fsumdvds 16187	. . . 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))))
190	189, 114	breqtrrd 5132	. . 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))))
191	46, 100, 112, 139, 169, 190	etransclem9 44454	. 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)
192	99, 191	eqnetrd 3010	1 ⊢ (𝜑 → 𝐾 ≠ 0)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 396 ∨ wo 845 ∧ w3a 1087 = wceq 1541 ∈ wcel 2106 ≠ wne 2942 {crab 3406 Vcvv 3444 ∖ cdif 3906 ⊆ wss 3909 ifcif 4485 {csn 4585 {cpr 4587 ⟨cop 4591 class class class wbr 5104 ↦ cmpt 5187 × cxp 5630 ran crn 5633 ⟶wf 6490 ‘cfv 6494 (class class class)co 7354 1^st c1st 7916 2^nd c2nd 7917 ↑_m cmap 8762 Fincfn 8880 ℂcc 11046 ℝcr 11047 0cc0 11048 1c1 11049 + caddc 11051 · cmul 11053 < clt 11186 ≤ cle 11187 − cmin 11382 / cdiv 11809 ℕcn 12150 ℕ₀cn0 12410 ℤcz 12496 ℤ_≥cuz 12760 (,)cioo 13261 ...cfz 13421 ↑cexp 13964 !cfa 14170 abscabs 15116 Σcsu 15567 ∏cprod 15785 ∥ cdvds 16133 ℙcprime 16544 ↾_t crest 17299 TopOpenctopn 17300 topGenctg 17316 ℂ_fldccnfld 20792 D^𝑛 cdvn 25224

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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-rep 5241 ax-sep 5255 ax-nul 5262 ax-pow 5319 ax-pr 5383 ax-un 7669 ax-inf2 9574 ax-cnex 11104 ax-resscn 11105 ax-1cn 11106 ax-icn 11107 ax-addcl 11108 ax-addrcl 11109 ax-mulcl 11110 ax-mulrcl 11111 ax-mulcom 11112 ax-addass 11113 ax-mulass 11114 ax-distr 11115 ax-i2m1 11116 ax-1ne0 11117 ax-1rid 11118 ax-rnegex 11119 ax-rrecex 11120 ax-cnre 11121 ax-pre-lttri 11122 ax-pre-lttrn 11123 ax-pre-ltadd 11124 ax-pre-mulgt0 11125 ax-pre-sup 11126 ax-addf 11127 ax-mulf 11128

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2888 df-ne 2943 df-nel 3049 df-ral 3064 df-rex 3073 df-rmo 3352 df-reu 3353 df-rab 3407 df-v 3446 df-sbc 3739 df-csb 3855 df-dif 3912 df-un 3914 df-in 3916 df-ss 3926 df-pss 3928 df-nul 4282 df-if 4486 df-pw 4561 df-sn 4586 df-pr 4588 df-tp 4590 df-op 4592 df-uni 4865 df-int 4907 df-iun 4955 df-iin 4956 df-br 5105 df-opab 5167 df-mpt 5188 df-tr 5222 df-id 5530 df-eprel 5536 df-po 5544 df-so 5545 df-fr 5587 df-se 5588 df-we 5589 df-xp 5638 df-rel 5639 df-cnv 5640 df-co 5641 df-dm 5642 df-rn 5643 df-res 5644 df-ima 5645 df-pred 6252 df-ord 6319 df-on 6320 df-lim 6321 df-suc 6322 df-iota 6446 df-fun 6496 df-fn 6497 df-f 6498 df-f1 6499 df-fo 6500 df-f1o 6501 df-fv 6502 df-isom 6503 df-riota 7310 df-ov 7357 df-oprab 7358 df-mpo 7359 df-of 7614 df-om 7800 df-1st 7918 df-2nd 7919 df-supp 8090 df-frecs 8209 df-wrecs 8240 df-recs 8314 df-rdg 8353 df-1o 8409 df-2o 8410 df-er 8645 df-map 8764 df-pm 8765 df-ixp 8833 df-en 8881 df-dom 8882 df-sdom 8883 df-fin 8884 df-fsupp 9303 df-fi 9344 df-sup 9375 df-inf 9376 df-oi 9443 df-card 9872 df-pnf 11188 df-mnf 11189 df-xr 11190 df-ltxr 11191 df-le 11192 df-sub 11384 df-neg 11385 df-div 11810 df-nn 12151 df-2 12213 df-3 12214 df-4 12215 df-5 12216 df-6 12217 df-7 12218 df-8 12219 df-9 12220 df-n0 12411 df-z 12497 df-dec 12616 df-uz 12761 df-q 12871 df-rp 12913 df-xneg 13030 df-xadd 13031 df-xmul 13032 df-ioo 13265 df-ico 13267 df-icc 13268 df-fz 13422 df-fzo 13565 df-fl 13694 df-mod 13772 df-seq 13904 df-exp 13965 df-fac 14171 df-bc 14200 df-hash 14228 df-cj 14981 df-re 14982 df-im 14983 df-sqrt 15117 df-abs 15118 df-clim 15367 df-sum 15568 df-prod 15786 df-dvds 16134 df-gcd 16372 df-prm 16545 df-struct 17016 df-sets 17033 df-slot 17051 df-ndx 17063 df-base 17081 df-ress 17110 df-plusg 17143 df-mulr 17144 df-starv 17145 df-sca 17146 df-vsca 17147 df-ip 17148 df-tset 17149 df-ple 17150 df-ds 17152 df-unif 17153 df-hom 17154 df-cco 17155 df-rest 17301 df-topn 17302 df-0g 17320 df-gsum 17321 df-topgen 17322 df-pt 17323 df-prds 17326 df-xrs 17381 df-qtop 17386 df-imas 17387 df-xps 17389 df-mre 17463 df-mrc 17464 df-acs 17466 df-mgm 18494 df-sgrp 18543 df-mnd 18554 df-submnd 18599 df-mulg 18869 df-cntz 19093 df-cmn 19560 df-psmet 20784 df-xmet 20785 df-met 20786 df-bl 20787 df-mopn 20788 df-fbas 20789 df-fg 20790 df-cnfld 20793 df-top 22239 df-topon 22256 df-topsp 22278 df-bases 22292 df-cld 22366 df-ntr 22367 df-cls 22368 df-nei 22445 df-lp 22483 df-perf 22484 df-cn 22574 df-cnp 22575 df-haus 22662 df-tx 22909 df-hmeo 23102 df-fil 23193 df-fm 23285 df-flim 23286 df-flf 23287 df-xms 23669 df-ms 23670 df-tms 23671 df-cncf 24237 df-limc 25226 df-dv 25227 df-dvn 25228

This theorem is referenced by: etransclem47 44492

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