etransclem44 - Mathbox for Glauco Siliprandi

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Theorem etransclem44 45479

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 2895	. . . . 5 ⊢ Ⅎ𝑘((𝐴‘0) · (((ℝ D^𝑛 𝐹)‘(𝑃 − 1))‘0))
5		fzfi 13934	. . . . . . 7 ⊢ (0...𝑀) ∈ Fin
6		fzfi 13934	. . . . . . 7 ⊢ (0...((𝑀 · 𝑃) + (𝑃 − 1))) ∈ Fin
7		xpfi 9313	. . . . . . 7 ⊢ (((0...𝑀) ∈ Fin ∧ (0...((𝑀 · 𝑃) + (𝑃 − 1))) ∈ Fin) → ((0...𝑀) × (0...((𝑀 · 𝑃) + (𝑃 − 1)))) ∈ Fin)
8	5, 6, 7	mp2an 689	. . . . . 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 44512	. . . . . . . . . 10 ⊢ (0...𝑀) ⊆ ℕ₀
13		xp1st 8000	. . . . . . . . . 10 ⊢ (𝑘 ∈ ((0...𝑀) × (0...((𝑀 · 𝑃) + (𝑃 − 1)))) → (1^st ‘𝑘) ∈ (0...𝑀))
14	12, 13	sselid 3972	. . . . . . . . 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 7077	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ ((0...𝑀) × (0...((𝑀 · 𝑃) + (𝑃 − 1))))) → (𝐴‘(1^st ‘𝑘)) ∈ ℤ)
17		reelprrecn 11198	. . . . . . . . 9 ⊢ ℝ ∈ {ℝ, ℂ}
18	17	a1i 11	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ ((0...𝑀) × (0...((𝑀 · 𝑃) + (𝑃 − 1))))) → ℝ ∈ {ℝ, ℂ})
19		reopn 44484	. . . . . . . . . 10 ⊢ ℝ ∈ (topGen‘ran (,))
20		eqid 2724	. . . . . . . . . . 11 ⊢ (TopOpen‘ℂ_fld) = (TopOpen‘ℂ_fld)
21	20	tgioo2 24641	. . . . . . . . . 10 ⊢ (topGen‘ran (,)) = ((TopOpen‘ℂ_fld) ↾_t ℝ)
22	19, 21	eleqtri 2823	. . . . . . . . 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 16608	. . . . . . . . . 10 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ ℕ)
26	24, 25	syl 17	. . . . . . . . 9 ⊢ (𝜑 → 𝑃 ∈ ℕ)
27	26	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ ((0...𝑀) × (0...((𝑀 · 𝑃) + (𝑃 − 1))))) → 𝑃 ∈ ℕ)
28		etransclem44.m	. . . . . . . . 9 ⊢ (𝜑 → 𝑀 ∈ ℕ₀)
29	28	adantr 480	. . . . . . . 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 8001	. . . . . . . . . 10 ⊢ (𝑘 ∈ ((0...𝑀) × (0...((𝑀 · 𝑃) + (𝑃 − 1)))) → (2^nd ‘𝑘) ∈ (0...((𝑀 · 𝑃) + (𝑃 − 1))))
32		elfznn0 13591	. . . . . . . . . 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 481	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ ((0...𝑀) × (0...((𝑀 · 𝑃) + (𝑃 − 1))))) → (2^nd ‘𝑘) ∈ ℕ₀)
35	15	nn0red 12530	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ ((0...𝑀) × (0...((𝑀 · 𝑃) + (𝑃 − 1))))) → (1^st ‘𝑘) ∈ ℝ)
36	15	nn0zd 12581	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ ((0...𝑀) × (0...((𝑀 · 𝑃) + (𝑃 − 1))))) → (1^st ‘𝑘) ∈ ℤ)
37	18, 23, 27, 29, 30, 34, 35, 36	etransclem42 45477	. . . . . . 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 12669	. . . . . 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 12664	. . . . 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 12861	. . . . . . . 8 ⊢ ℕ₀ = (ℤ_≥‘0)
41	28, 40	eleqtrdi 2835	. . . . . . 7 ⊢ (𝜑 → 𝑀 ∈ (ℤ_≥‘0))
42		eluzfz1 13505	. . . . . . 7 ⊢ (𝑀 ∈ (ℤ_≥‘0) → 0 ∈ (0...𝑀))
43	41, 42	syl 17	. . . . . 6 ⊢ (𝜑 → 0 ∈ (0...𝑀))
44		0zd 12567	. . . . . . 7 ⊢ (𝜑 → 0 ∈ ℤ)
45	28	nn0zd 12581	. . . . . . . . 9 ⊢ (𝜑 → 𝑀 ∈ ℤ)
46	26	nnzd 12582	. . . . . . . . 9 ⊢ (𝜑 → 𝑃 ∈ ℤ)
47	45, 46	zmulcld 12669	. . . . . . . 8 ⊢ (𝜑 → (𝑀 · 𝑃) ∈ ℤ)
48		nnm1nn0 12510	. . . . . . . . . 10 ⊢ (𝑃 ∈ ℕ → (𝑃 − 1) ∈ ℕ₀)
49	26, 48	syl 17	. . . . . . . . 9 ⊢ (𝜑 → (𝑃 − 1) ∈ ℕ₀)
50	49	nn0zd 12581	. . . . . . . 8 ⊢ (𝜑 → (𝑃 − 1) ∈ ℤ)
51	47, 50	zaddcld 12667	. . . . . . 7 ⊢ (𝜑 → ((𝑀 · 𝑃) + (𝑃 − 1)) ∈ ℤ)
52	49	nn0ge0d 12532	. . . . . . 7 ⊢ (𝜑 → 0 ≤ (𝑃 − 1))
53	26	nnnn0d 12529	. . . . . . . . . 10 ⊢ (𝜑 → 𝑃 ∈ ℕ₀)
54	28, 53	nn0mulcld 12534	. . . . . . . . 9 ⊢ (𝜑 → (𝑀 · 𝑃) ∈ ℕ₀)
55	54	nn0ge0d 12532	. . . . . . . 8 ⊢ (𝜑 → 0 ≤ (𝑀 · 𝑃))
56	49	nn0red 12530	. . . . . . . . 9 ⊢ (𝜑 → (𝑃 − 1) ∈ ℝ)
57	47	zred 12663	. . . . . . . . 9 ⊢ (𝜑 → (𝑀 · 𝑃) ∈ ℝ)
58	56, 57	addge02d 11800	. . . . . . . 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 13489	. . . . . 6 ⊢ (𝜑 → (𝑃 − 1) ∈ (0...((𝑀 · 𝑃) + (𝑃 − 1))))
61		opelxp 5702	. . . . . 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 582	. . . . 5 ⊢ (𝜑 → ⟨0, (𝑃 − 1)⟩ ∈ ((0...𝑀) × (0...((𝑀 · 𝑃) + (𝑃 − 1)))))
63		fveq2 6881	. . . . . . . 8 ⊢ (𝑘 = ⟨0, (𝑃 − 1)⟩ → (1^st ‘𝑘) = (1^st ‘⟨0, (𝑃 − 1)⟩))
64		0re 11213	. . . . . . . . 9 ⊢ 0 ∈ ℝ
65		ovex 7434	. . . . . . . . 9 ⊢ (𝑃 − 1) ∈ V
66		op1stg 7980	. . . . . . . . 9 ⊢ ((0 ∈ ℝ ∧ (𝑃 − 1) ∈ V) → (1^st ‘⟨0, (𝑃 − 1)⟩) = 0)
67	64, 65, 66	mp2an 689	. . . . . . . 8 ⊢ (1^st ‘⟨0, (𝑃 − 1)⟩) = 0
68	63, 67	eqtrdi 2780	. . . . . . 7 ⊢ (𝑘 = ⟨0, (𝑃 − 1)⟩ → (1^st ‘𝑘) = 0)
69	68	fveq2d 6885	. . . . . 6 ⊢ (𝑘 = ⟨0, (𝑃 − 1)⟩ → (𝐴‘(1^st ‘𝑘)) = (𝐴‘0))
70		fveq2 6881	. . . . . . . . 9 ⊢ (𝑘 = ⟨0, (𝑃 − 1)⟩ → (2^nd ‘𝑘) = (2^nd ‘⟨0, (𝑃 − 1)⟩))
71		op2ndg 7981	. . . . . . . . . 10 ⊢ ((0 ∈ ℝ ∧ (𝑃 − 1) ∈ V) → (2^nd ‘⟨0, (𝑃 − 1)⟩) = (𝑃 − 1))
72	64, 65, 71	mp2an 689	. . . . . . . . 9 ⊢ (2^nd ‘⟨0, (𝑃 − 1)⟩) = (𝑃 − 1)
73	70, 72	eqtrdi 2780	. . . . . . . 8 ⊢ (𝑘 = ⟨0, (𝑃 − 1)⟩ → (2^nd ‘𝑘) = (𝑃 − 1))
74	73	fveq2d 6885	. . . . . . 7 ⊢ (𝑘 = ⟨0, (𝑃 − 1)⟩ → ((ℝ D^𝑛 𝐹)‘(2^nd ‘𝑘)) = ((ℝ D^𝑛 𝐹)‘(𝑃 − 1)))
75	74, 68	fveq12d 6888	. . . . . 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 7419	. . . . 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 15688	. . . 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 7416	. . 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 3972	. . . . . . 7 ⊢ (𝜑 → 0 ∈ ℕ₀)
80	10, 79	ffvelcdmd 7077	. . . . . 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 45477	. . . . . 6 ⊢ (𝜑 → (((ℝ D^𝑛 𝐹)‘(𝑃 − 1))‘0) ∈ ℤ)
85	80, 84	zmulcld 12669	. . . . 5 ⊢ (𝜑 → ((𝐴‘0) · (((ℝ D^𝑛 𝐹)‘(𝑃 − 1))‘0)) ∈ ℤ)
86	85	zcnd 12664	. . . 4 ⊢ (𝜑 → ((𝐴‘0) · (((ℝ D^𝑛 𝐹)‘(𝑃 − 1))‘0)) ∈ ℂ)
87		difss 4123	. . . . . . . 8 ⊢ (((0...𝑀) × (0...((𝑀 · 𝑃) + (𝑃 − 1)))) ∖ {⟨0, (𝑃 − 1)⟩}) ⊆ ((0...𝑀) × (0...((𝑀 · 𝑃) + (𝑃 − 1))))
88		ssfi 9169	. . . . . . . 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 689	. . . . . . 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 4118	. . . . . . 7 ⊢ (𝑘 ∈ (((0...𝑀) × (0...((𝑀 · 𝑃) + (𝑃 − 1)))) ∖ {⟨0, (𝑃 − 1)⟩}) → 𝑘 ∈ ((0...𝑀) × (0...((𝑀 · 𝑃) + (𝑃 − 1)))))
92	91, 38	sylan2 592	. . . . . 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 15678	. . . . 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 12664	. . . 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 14241	. . . . 5 ⊢ (𝜑 → (!‘(𝑃 − 1)) ∈ ℕ)
96	95	nncnd 12225	. . . 4 ⊢ (𝜑 → (!‘(𝑃 − 1)) ∈ ℂ)
97	95	nnne0d 12259	. . . 4 ⊢ (𝜑 → (!‘(𝑃 − 1)) ≠ 0)
98	86, 94, 96, 97	divdird 12025	. . 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 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)))))
100	26	nnne0d 12259	. . 3 ⊢ (𝜑 → 𝑃 ≠ 0)
101	80	zcnd 12664	. . . . 5 ⊢ (𝜑 → (𝐴‘0) ∈ ℂ)
102	84	zcnd 12664	. . . . 5 ⊢ (𝜑 → (((ℝ D^𝑛 𝐹)‘(𝑃 − 1))‘0) ∈ ℂ)
103	101, 102, 96, 97	divassd 12022	. . . 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 45440	. . . . . . 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 45446	. . . . . . 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 45472	. . . . . 6 ⊢ (𝜑 → (!‘(𝑃 − 1)) ∥ (((ℝ D^𝑛 𝐹)‘(𝑃 − 1))‘0))
107	95	nnzd 12582	. . . . . . 7 ⊢ (𝜑 → (!‘(𝑃 − 1)) ∈ ℤ)
108		dvdsval2 16197	. . . . . . 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 1368	. . . . . 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 12669	. . . 4 ⊢ (𝜑 → ((𝐴‘0) · ((((ℝ D^𝑛 𝐹)‘(𝑃 − 1))‘0) / (!‘(𝑃 − 1)))) ∈ ℤ)
112	103, 111	eqeltrd 2825	. . 3 ⊢ (𝜑 → (((𝐴‘0) · (((ℝ D^𝑛 𝐹)‘(𝑃 − 1))‘0)) / (!‘(𝑃 − 1))) ∈ ℤ)
113	91, 39	sylan2 592	. . . . 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 15729	. . . 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 12664	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ ((0...𝑀) × (0...((𝑀 · 𝑃) + (𝑃 − 1))))) → (𝐴‘(1^st ‘𝑘)) ∈ ℂ)
116	91, 115	sylan2 592	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (((0...𝑀) × (0...((𝑀 · 𝑃) + (𝑃 − 1)))) ∖ {⟨0, (𝑃 − 1)⟩})) → (𝐴‘(1^st ‘𝑘)) ∈ ℂ)
117	91, 37	sylan2 592	. . . . . . . 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 12664	. . . . . . 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 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (((0...𝑀) × (0...((𝑀 · 𝑃) + (𝑃 − 1)))) ∖ {⟨0, (𝑃 − 1)⟩})) → (!‘(𝑃 − 1)) ∈ ℂ)
120	97	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (((0...𝑀) × (0...((𝑀 · 𝑃) + (𝑃 − 1)))) ∖ {⟨0, (𝑃 − 1)⟩})) → (!‘(𝑃 − 1)) ≠ 0)
121	116, 118, 119, 120	divassd 12022	. . . . . 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 592	. . . . . . 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 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (((0...𝑀) × (0...((𝑀 · 𝑃) + (𝑃 − 1)))) ∖ {⟨0, (𝑃 − 1)⟩})) → 𝑃 ∈ ℕ)
126	28	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (((0...𝑀) × (0...((𝑀 · 𝑃) + (𝑃 − 1)))) ∖ {⟨0, (𝑃 − 1)⟩})) → 𝑀 ∈ ℕ₀)
127	91	adantl 481	. . . . . . . . . 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 592	. . . . . . . . 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 45472	. . . . . . . 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 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (((0...𝑀) × (0...((𝑀 · 𝑃) + (𝑃 − 1)))) ∖ {⟨0, (𝑃 − 1)⟩})) → (!‘(𝑃 − 1)) ∈ ℤ)
133		dvdsval2 16197	. . . . . . . . 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 1368	. . . . . . . 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 12669	. . . . . 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 2825	. . . . 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 15678	. . . 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 2825	. . 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 12590	. . . . . . . . . 10 ⊢ (𝜑 → 1 ∈ ℤ)
141		zabscl 15257	. . . . . . . . . . 11 ⊢ ((𝐴‘0) ∈ ℤ → (abs‘(𝐴‘0)) ∈ ℤ)
142	80, 141	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → (abs‘(𝐴‘0)) ∈ ℤ)
143		nn0abscl 15256	. . . . . . . . . . . . 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 15391	. . . . . . . . . . . 12 ⊢ (𝜑 → (abs‘(𝐴‘0)) ≠ 0)
147		elnnne0 12483	. . . . . . . . . . . 12 ⊢ ((abs‘(𝐴‘0)) ∈ ℕ ↔ ((abs‘(𝐴‘0)) ∈ ℕ₀ ∧ (abs‘(𝐴‘0)) ≠ 0))
148	144, 146, 147	sylanbrc 582	. . . . . . . . . . 11 ⊢ (𝜑 → (abs‘(𝐴‘0)) ∈ ℕ)
149	148	nnge1d 12257	. . . . . . . . . 10 ⊢ (𝜑 → 1 ≤ (abs‘(𝐴‘0)))
150		etransclem44.ap	. . . . . . . . . . 11 ⊢ (𝜑 → (abs‘(𝐴‘0)) < 𝑃)
151		zltlem1 12612	. . . . . . . . . . . 12 ⊢ (((abs‘(𝐴‘0)) ∈ ℤ ∧ 𝑃 ∈ ℤ) → ((abs‘(𝐴‘0)) < 𝑃 ↔ (abs‘(𝐴‘0)) ≤ (𝑃 − 1)))
152	142, 46, 151	syl2anc 583	. . . . . . . . . . 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 13489	. . . . . . . . 9 ⊢ (𝜑 → (abs‘(𝐴‘0)) ∈ (1...(𝑃 − 1)))
155		fzm1ndvds 16262	. . . . . . . . 9 ⊢ ((𝑃 ∈ ℕ ∧ (abs‘(𝐴‘0)) ∈ (1...(𝑃 − 1))) → ¬ 𝑃 ∥ (abs‘(𝐴‘0)))
156	26, 154, 155	syl2anc 583	. . . . . . . 8 ⊢ (𝜑 → ¬ 𝑃 ∥ (abs‘(𝐴‘0)))
157		dvdsabsb 16216	. . . . . . . . 9 ⊢ ((𝑃 ∈ ℤ ∧ (𝐴‘0) ∈ ℤ) → (𝑃 ∥ (𝐴‘0) ↔ 𝑃 ∥ (abs‘(𝐴‘0))))
158	46, 80, 157	syl2anc 583	. . . . . . . 8 ⊢ (𝜑 → (𝑃 ∥ (𝐴‘0) ↔ 𝑃 ∥ (abs‘(𝐴‘0))))
159	156, 158	mtbird 325	. . . . . . 7 ⊢ (𝜑 → ¬ 𝑃 ∥ (𝐴‘0))
160		etransclem44.mp	. . . . . . . 8 ⊢ (𝜑 → (!‘𝑀) < 𝑃)
161	28, 24, 160, 30	etransclem41 45476	. . . . . . 7 ⊢ (𝜑 → ¬ 𝑃 ∥ ((((ℝ D^𝑛 𝐹)‘(𝑃 − 1))‘0) / (!‘(𝑃 − 1))))
162	159, 161	jca 511	. . . . . 6 ⊢ (𝜑 → (¬ 𝑃 ∥ (𝐴‘0) ∧ ¬ 𝑃 ∥ ((((ℝ D^𝑛 𝐹)‘(𝑃 − 1))‘0) / (!‘(𝑃 − 1)))))
163		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)))))
164	162, 163	sylib 217	. . . . 5 ⊢ (𝜑 → ¬ (𝑃 ∥ (𝐴‘0) ∨ 𝑃 ∥ ((((ℝ D^𝑛 𝐹)‘(𝑃 − 1))‘0) / (!‘(𝑃 − 1)))))
165		euclemma 16647	. . . . . 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 1368	. . . . 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 325	. . . 4 ⊢ (𝜑 → ¬ 𝑃 ∥ ((𝐴‘0) · ((((ℝ D^𝑛 𝐹)‘(𝑃 − 1))‘0) / (!‘(𝑃 − 1)))))
168	103	breq2d 5150	. . . 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 325	. . 3 ⊢ (𝜑 → ¬ 𝑃 ∥ (((𝐴‘0) · (((ℝ D^𝑛 𝐹)‘(𝑃 − 1))‘0)) / (!‘(𝑃 − 1))))
170	46	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (((0...𝑀) × (0...((𝑀 · 𝑃) + (𝑃 − 1)))) ∖ {⟨0, (𝑃 − 1)⟩})) → 𝑃 ∈ ℤ)
171	170, 122, 135	3jca 1125	. . . . . . 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 4119	. . . . . . . . . 10 ⊢ (𝑘 ∈ (((0...𝑀) × (0...((𝑀 · 𝑃) + (𝑃 − 1)))) ∖ {⟨0, (𝑃 − 1)⟩}) → ¬ 𝑘 ∈ {⟨0, (𝑃 − 1)⟩})
173	91	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)))))
174		1st2nd2 8007	. . . . . . . . . . . . 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 484	. . . . . . . . . . . . . 14 ⊢ (((2^nd ‘𝑘) = (𝑃 − 1) ∧ (1^st ‘𝑘) = 0) → (1^st ‘𝑘) = 0)
177		simpl 482	. . . . . . . . . . . . . 14 ⊢ (((2^nd ‘𝑘) = (𝑃 − 1) ∧ (1^st ‘𝑘) = 0) → (2^nd ‘𝑘) = (𝑃 − 1))
178	176, 177	opeq12d 4873	. . . . . . . . . . . . 13 ⊢ (((2^nd ‘𝑘) = (𝑃 − 1) ∧ (1^st ‘𝑘) = 0) → ⟨(1^st ‘𝑘), (2^nd ‘𝑘)⟩ = ⟨0, (𝑃 − 1)⟩)
179	178	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)⟩)
180	175, 179	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)⟩)
181		velsn 4636	. . . . . . . . . . 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 813	. . . . . . . . 9 ⊢ (𝑘 ∈ (((0...𝑀) × (0...((𝑀 · 𝑃) + (𝑃 − 1)))) ∖ {⟨0, (𝑃 − 1)⟩}) → ¬ ((2^nd ‘𝑘) = (𝑃 − 1) ∧ (1^st ‘𝑘) = 0))
184	183	adantl 481	. . . . . . . 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 45473	. . . . . . 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 16238	. . . . . . 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 5166	. . . . 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 16248	. . . 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 5166	. . 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 45444	. 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 3000	1 ⊢ (𝜑 → 𝐾 ≠ 0)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 395 ∨ wo 844 ∧ w3a 1084 = wceq 1533 ∈ wcel 2098 ≠ wne 2932 {crab 3424 Vcvv 3466 ∖ cdif 3937 ⊆ wss 3940 ifcif 4520 {csn 4620 {cpr 4622 ⟨cop 4626 class class class wbr 5138 ↦ cmpt 5221 × cxp 5664 ran crn 5667 ⟶wf 6529 ‘cfv 6533 (class class class)co 7401 1^st c1st 7966 2^nd c2nd 7967 ↑_m cmap 8816 Fincfn 8935 ℂcc 11104 ℝcr 11105 0cc0 11106 1c1 11107 + caddc 11109 · cmul 11111 < clt 11245 ≤ cle 11246 − cmin 11441 / cdiv 11868 ℕcn 12209 ℕ₀cn0 12469 ℤcz 12555 ℤ_≥cuz 12819 (,)cioo 13321 ...cfz 13481 ↑cexp 14024 !cfa 14230 abscabs 15178 Σcsu 15629 ∏cprod 15846 ∥ cdvds 16194 ℙcprime 16605 ↾_t crest 17365 TopOpenctopn 17366 topGenctg 17382 ℂ_fldccnfld 21228 D^𝑛 cdvn 25715

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-rep 5275 ax-sep 5289 ax-nul 5296 ax-pow 5353 ax-pr 5417 ax-un 7718 ax-inf2 9632 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183 ax-pre-sup 11184 ax-addf 11185

This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-nel 3039 df-ral 3054 df-rex 3063 df-rmo 3368 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3770 df-csb 3886 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-pss 3959 df-nul 4315 df-if 4521 df-pw 4596 df-sn 4621 df-pr 4623 df-tp 4625 df-op 4627 df-uni 4900 df-int 4941 df-iun 4989 df-iin 4990 df-br 5139 df-opab 5201 df-mpt 5222 df-tr 5256 df-id 5564 df-eprel 5570 df-po 5578 df-so 5579 df-fr 5621 df-se 5622 df-we 5623 df-xp 5672 df-rel 5673 df-cnv 5674 df-co 5675 df-dm 5676 df-rn 5677 df-res 5678 df-ima 5679 df-pred 6290 df-ord 6357 df-on 6358 df-lim 6359 df-suc 6360 df-iota 6485 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-isom 6542 df-riota 7357 df-ov 7404 df-oprab 7405 df-mpo 7406 df-of 7663 df-om 7849 df-1st 7968 df-2nd 7969 df-supp 8141 df-frecs 8261 df-wrecs 8292 df-recs 8366 df-rdg 8405 df-1o 8461 df-2o 8462 df-er 8699 df-map 8818 df-pm 8819 df-ixp 8888 df-en 8936 df-dom 8937 df-sdom 8938 df-fin 8939 df-fsupp 9358 df-fi 9402 df-sup 9433 df-inf 9434 df-oi 9501 df-card 9930 df-pnf 11247 df-mnf 11248 df-xr 11249 df-ltxr 11250 df-le 11251 df-sub 11443 df-neg 11444 df-div 11869 df-nn 12210 df-2 12272 df-3 12273 df-4 12274 df-5 12275 df-6 12276 df-7 12277 df-8 12278 df-9 12279 df-n0 12470 df-z 12556 df-dec 12675 df-uz 12820 df-q 12930 df-rp 12972 df-xneg 13089 df-xadd 13090 df-xmul 13091 df-ioo 13325 df-ico 13327 df-icc 13328 df-fz 13482 df-fzo 13625 df-fl 13754 df-mod 13832 df-seq 13964 df-exp 14025 df-fac 14231 df-bc 14260 df-hash 14288 df-cj 15043 df-re 15044 df-im 15045 df-sqrt 15179 df-abs 15180 df-clim 15429 df-sum 15630 df-prod 15847 df-dvds 16195 df-gcd 16433 df-prm 16606 df-struct 17079 df-sets 17096 df-slot 17114 df-ndx 17126 df-base 17144 df-ress 17173 df-plusg 17209 df-mulr 17210 df-starv 17211 df-sca 17212 df-vsca 17213 df-ip 17214 df-tset 17215 df-ple 17216 df-ds 17218 df-unif 17219 df-hom 17220 df-cco 17221 df-rest 17367 df-topn 17368 df-0g 17386 df-gsum 17387 df-topgen 17388 df-pt 17389 df-prds 17392 df-xrs 17447 df-qtop 17452 df-imas 17453 df-xps 17455 df-mre 17529 df-mrc 17530 df-acs 17532 df-mgm 18563 df-sgrp 18642 df-mnd 18658 df-submnd 18704 df-mulg 18986 df-cntz 19223 df-cmn 19692 df-psmet 21220 df-xmet 21221 df-met 21222 df-bl 21223 df-mopn 21224 df-fbas 21225 df-fg 21226 df-cnfld 21229 df-top 22718 df-topon 22735 df-topsp 22757 df-bases 22771 df-cld 22845 df-ntr 22846 df-cls 22847 df-nei 22924 df-lp 22962 df-perf 22963 df-cn 23053 df-cnp 23054 df-haus 23141 df-tx 23388 df-hmeo 23581 df-fil 23672 df-fm 23764 df-flim 23765 df-flf 23766 df-xms 24148 df-ms 24149 df-tms 24150 df-cncf 24720 df-limc 25717 df-dv 25718 df-dvn 25719

This theorem is referenced by: etransclem47 45482

W3C validator