etransclem44 - Mathbox for Glauco Siliprandi

	Mathbox for Glauco Siliprandi		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > Mathboxes > etransclem44		Structured version Visualization version GIF version

Theorem etransclem44 46199

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 1913	. . . . 5 ⊢ Ⅎ𝑘𝜑
4		nfcv 2908	. . . . 5 ⊢ Ⅎ𝑘((𝐴‘0) · (((ℝ D^𝑛 𝐹)‘(𝑃 − 1))‘0))
5		fzfi 14023	. . . . . . 7 ⊢ (0...𝑀) ∈ Fin
6		fzfi 14023	. . . . . . 7 ⊢ (0...((𝑀 · 𝑃) + (𝑃 − 1))) ∈ Fin
7		xpfi 9386	. . . . . . 7 ⊢ (((0...𝑀) ∈ Fin ∧ (0...((𝑀 · 𝑃) + (𝑃 − 1))) ∈ Fin) → ((0...𝑀) × (0...((𝑀 · 𝑃) + (𝑃 − 1)))) ∈ Fin)
8	5, 6, 7	mp2an 691	. . . . . 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 45232	. . . . . . . . . 10 ⊢ (0...𝑀) ⊆ ℕ₀
13		xp1st 8062	. . . . . . . . . 10 ⊢ (𝑘 ∈ ((0...𝑀) × (0...((𝑀 · 𝑃) + (𝑃 − 1)))) → (1^st ‘𝑘) ∈ (0...𝑀))
14	12, 13	sselid 4006	. . . . . . . . 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 7119	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ ((0...𝑀) × (0...((𝑀 · 𝑃) + (𝑃 − 1))))) → (𝐴‘(1^st ‘𝑘)) ∈ ℤ)
17		reelprrecn 11276	. . . . . . . . 9 ⊢ ℝ ∈ {ℝ, ℂ}
18	17	a1i 11	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ ((0...𝑀) × (0...((𝑀 · 𝑃) + (𝑃 − 1))))) → ℝ ∈ {ℝ, ℂ})
19		reopn 45204	. . . . . . . . . 10 ⊢ ℝ ∈ (topGen‘ran (,))
20		eqid 2740	. . . . . . . . . . 11 ⊢ (TopOpen‘ℂ_fld) = (TopOpen‘ℂ_fld)
21	20	tgioo2 24844	. . . . . . . . . 10 ⊢ (topGen‘ran (,)) = ((TopOpen‘ℂ_fld) ↾_t ℝ)
22	19, 21	eleqtri 2842	. . . . . . . . 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 16721	. . . . . . . . . 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 8063	. . . . . . . . . 10 ⊢ (𝑘 ∈ ((0...𝑀) × (0...((𝑀 · 𝑃) + (𝑃 − 1)))) → (2^nd ‘𝑘) ∈ (0...((𝑀 · 𝑃) + (𝑃 − 1))))
32		elfznn0 13677	. . . . . . . . . 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 12614	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ ((0...𝑀) × (0...((𝑀 · 𝑃) + (𝑃 − 1))))) → (1^st ‘𝑘) ∈ ℝ)
36	15	nn0zd 12665	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ ((0...𝑀) × (0...((𝑀 · 𝑃) + (𝑃 − 1))))) → (1^st ‘𝑘) ∈ ℤ)
37	18, 23, 27, 29, 30, 34, 35, 36	etransclem42 46197	. . . . . . 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 12753	. . . . . 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 12748	. . . . 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 12945	. . . . . . . 8 ⊢ ℕ₀ = (ℤ_≥‘0)
41	28, 40	eleqtrdi 2854	. . . . . . 7 ⊢ (𝜑 → 𝑀 ∈ (ℤ_≥‘0))
42		eluzfz1 13591	. . . . . . 7 ⊢ (𝑀 ∈ (ℤ_≥‘0) → 0 ∈ (0...𝑀))
43	41, 42	syl 17	. . . . . 6 ⊢ (𝜑 → 0 ∈ (0...𝑀))
44		0zd 12651	. . . . . . 7 ⊢ (𝜑 → 0 ∈ ℤ)
45	28	nn0zd 12665	. . . . . . . . 9 ⊢ (𝜑 → 𝑀 ∈ ℤ)
46	26	nnzd 12666	. . . . . . . . 9 ⊢ (𝜑 → 𝑃 ∈ ℤ)
47	45, 46	zmulcld 12753	. . . . . . . 8 ⊢ (𝜑 → (𝑀 · 𝑃) ∈ ℤ)
48		nnm1nn0 12594	. . . . . . . . . 10 ⊢ (𝑃 ∈ ℕ → (𝑃 − 1) ∈ ℕ₀)
49	26, 48	syl 17	. . . . . . . . 9 ⊢ (𝜑 → (𝑃 − 1) ∈ ℕ₀)
50	49	nn0zd 12665	. . . . . . . 8 ⊢ (𝜑 → (𝑃 − 1) ∈ ℤ)
51	47, 50	zaddcld 12751	. . . . . . 7 ⊢ (𝜑 → ((𝑀 · 𝑃) + (𝑃 − 1)) ∈ ℤ)
52	49	nn0ge0d 12616	. . . . . . 7 ⊢ (𝜑 → 0 ≤ (𝑃 − 1))
53	26	nnnn0d 12613	. . . . . . . . . 10 ⊢ (𝜑 → 𝑃 ∈ ℕ₀)
54	28, 53	nn0mulcld 12618	. . . . . . . . 9 ⊢ (𝜑 → (𝑀 · 𝑃) ∈ ℕ₀)
55	54	nn0ge0d 12616	. . . . . . . 8 ⊢ (𝜑 → 0 ≤ (𝑀 · 𝑃))
56	49	nn0red 12614	. . . . . . . . 9 ⊢ (𝜑 → (𝑃 − 1) ∈ ℝ)
57	47	zred 12747	. . . . . . . . 9 ⊢ (𝜑 → (𝑀 · 𝑃) ∈ ℝ)
58	56, 57	addge02d 11879	. . . . . . . 8 ⊢ (𝜑 → (0 ≤ (𝑀 · 𝑃) ↔ (𝑃 − 1) ≤ ((𝑀 · 𝑃) + (𝑃 − 1))))
59	55, 58	mpbid 232	. . . . . . 7 ⊢ (𝜑 → (𝑃 − 1) ≤ ((𝑀 · 𝑃) + (𝑃 − 1)))
60	44, 51, 50, 52, 59	elfzd 13575	. . . . . 6 ⊢ (𝜑 → (𝑃 − 1) ∈ (0...((𝑀 · 𝑃) + (𝑃 − 1))))
61		opelxp 5736	. . . . . 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 6920	. . . . . . . 8 ⊢ (𝑘 = ⟨0, (𝑃 − 1)⟩ → (1^st ‘𝑘) = (1^st ‘⟨0, (𝑃 − 1)⟩))
64		0re 11292	. . . . . . . . 9 ⊢ 0 ∈ ℝ
65		ovex 7481	. . . . . . . . 9 ⊢ (𝑃 − 1) ∈ V
66		op1stg 8042	. . . . . . . . 9 ⊢ ((0 ∈ ℝ ∧ (𝑃 − 1) ∈ V) → (1^st ‘⟨0, (𝑃 − 1)⟩) = 0)
67	64, 65, 66	mp2an 691	. . . . . . . 8 ⊢ (1^st ‘⟨0, (𝑃 − 1)⟩) = 0
68	63, 67	eqtrdi 2796	. . . . . . 7 ⊢ (𝑘 = ⟨0, (𝑃 − 1)⟩ → (1^st ‘𝑘) = 0)
69	68	fveq2d 6924	. . . . . 6 ⊢ (𝑘 = ⟨0, (𝑃 − 1)⟩ → (𝐴‘(1^st ‘𝑘)) = (𝐴‘0))
70		fveq2 6920	. . . . . . . . 9 ⊢ (𝑘 = ⟨0, (𝑃 − 1)⟩ → (2^nd ‘𝑘) = (2^nd ‘⟨0, (𝑃 − 1)⟩))
71		op2ndg 8043	. . . . . . . . . 10 ⊢ ((0 ∈ ℝ ∧ (𝑃 − 1) ∈ V) → (2^nd ‘⟨0, (𝑃 − 1)⟩) = (𝑃 − 1))
72	64, 65, 71	mp2an 691	. . . . . . . . 9 ⊢ (2^nd ‘⟨0, (𝑃 − 1)⟩) = (𝑃 − 1)
73	70, 72	eqtrdi 2796	. . . . . . . 8 ⊢ (𝑘 = ⟨0, (𝑃 − 1)⟩ → (2^nd ‘𝑘) = (𝑃 − 1))
74	73	fveq2d 6924	. . . . . . 7 ⊢ (𝑘 = ⟨0, (𝑃 − 1)⟩ → ((ℝ D^𝑛 𝐹)‘(2^nd ‘𝑘)) = ((ℝ D^𝑛 𝐹)‘(𝑃 − 1)))
75	74, 68	fveq12d 6927	. . . . . 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 7466	. . . . 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 15793	. . . 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 7463	. . 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 4006	. . . . . . 7 ⊢ (𝜑 → 0 ∈ ℕ₀)
80	10, 79	ffvelcdmd 7119	. . . . . 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 46197	. . . . . 6 ⊢ (𝜑 → (((ℝ D^𝑛 𝐹)‘(𝑃 − 1))‘0) ∈ ℤ)
85	80, 84	zmulcld 12753	. . . . 5 ⊢ (𝜑 → ((𝐴‘0) · (((ℝ D^𝑛 𝐹)‘(𝑃 − 1))‘0)) ∈ ℤ)
86	85	zcnd 12748	. . . 4 ⊢ (𝜑 → ((𝐴‘0) · (((ℝ D^𝑛 𝐹)‘(𝑃 − 1))‘0)) ∈ ℂ)
87		difss 4159	. . . . . . . 8 ⊢ (((0...𝑀) × (0...((𝑀 · 𝑃) + (𝑃 − 1)))) ∖ {⟨0, (𝑃 − 1)⟩}) ⊆ ((0...𝑀) × (0...((𝑀 · 𝑃) + (𝑃 − 1))))
88		ssfi 9240	. . . . . . . 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 691	. . . . . . 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 4154	. . . . . . 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 15783	. . . . 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 12748	. . . 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 14333	. . . . 5 ⊢ (𝜑 → (!‘(𝑃 − 1)) ∈ ℕ)
96	95	nncnd 12309	. . . 4 ⊢ (𝜑 → (!‘(𝑃 − 1)) ∈ ℂ)
97	95	nnne0d 12343	. . . 4 ⊢ (𝜑 → (!‘(𝑃 − 1)) ≠ 0)
98	86, 94, 96, 97	divdird 12108	. . 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 2784	. 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 12343	. . 3 ⊢ (𝜑 → 𝑃 ≠ 0)
101	80	zcnd 12748	. . . . 5 ⊢ (𝜑 → (𝐴‘0) ∈ ℂ)
102	84	zcnd 12748	. . . . 5 ⊢ (𝜑 → (((ℝ D^𝑛 𝐹)‘(𝑃 − 1))‘0) ∈ ℂ)
103	101, 102, 96, 97	divassd 12105	. . . 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 46160	. . . . . . 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 46166	. . . . . . 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 46192	. . . . . 6 ⊢ (𝜑 → (!‘(𝑃 − 1)) ∥ (((ℝ D^𝑛 𝐹)‘(𝑃 − 1))‘0))
107	95	nnzd 12666	. . . . . . 7 ⊢ (𝜑 → (!‘(𝑃 − 1)) ∈ ℤ)
108		dvdsval2 16305	. . . . . . 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 232	. . . . 5 ⊢ (𝜑 → ((((ℝ D^𝑛 𝐹)‘(𝑃 − 1))‘0) / (!‘(𝑃 − 1))) ∈ ℤ)
111	80, 110	zmulcld 12753	. . . 4 ⊢ (𝜑 → ((𝐴‘0) · ((((ℝ D^𝑛 𝐹)‘(𝑃 − 1))‘0) / (!‘(𝑃 − 1)))) ∈ ℤ)
112	103, 111	eqeltrd 2844	. . 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 15834	. . . 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 12748	. . . . . . . 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 12748	. . . . . . 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 12105	. . . . . 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 46192	. . . . . . . 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 16305	. . . . . . . . 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 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)
136	122, 135	zmulcld 12753	. . . . . 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 2844	. . . . 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 15783	. . . 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 2844	. . 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 12674	. . . . . . . . . 10 ⊢ (𝜑 → 1 ∈ ℤ)
141		zabscl 15362	. . . . . . . . . . 11 ⊢ ((𝐴‘0) ∈ ℤ → (abs‘(𝐴‘0)) ∈ ℤ)
142	80, 141	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → (abs‘(𝐴‘0)) ∈ ℤ)
143		nn0abscl 15361	. . . . . . . . . . . . 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 15496	. . . . . . . . . . . 12 ⊢ (𝜑 → (abs‘(𝐴‘0)) ≠ 0)
147		elnnne0 12567	. . . . . . . . . . . 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 12341	. . . . . . . . . 10 ⊢ (𝜑 → 1 ≤ (abs‘(𝐴‘0)))
150		etransclem44.ap	. . . . . . . . . . 11 ⊢ (𝜑 → (abs‘(𝐴‘0)) < 𝑃)
151		zltlem1 12696	. . . . . . . . . . . 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 232	. . . . . . . . . 10 ⊢ (𝜑 → (abs‘(𝐴‘0)) ≤ (𝑃 − 1))
154	140, 50, 142, 149, 153	elfzd 13575	. . . . . . . . 9 ⊢ (𝜑 → (abs‘(𝐴‘0)) ∈ (1...(𝑃 − 1)))
155		fzm1ndvds 16370	. . . . . . . . 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 16324	. . . . . . . . 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 46196	. . . . . . 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		pm4.56 989	. . . . . 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 218	. . . . 5 ⊢ (𝜑 → ¬ (𝑃 ∥ (𝐴‘0) ∨ 𝑃 ∥ ((((ℝ D^𝑛 𝐹)‘(𝑃 − 1))‘0) / (!‘(𝑃 − 1)))))
165		euclemma 16760	. . . . . 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 325	. . . 4 ⊢ (𝜑 → ¬ 𝑃 ∥ ((𝐴‘0) · ((((ℝ D^𝑛 𝐹)‘(𝑃 − 1))‘0) / (!‘(𝑃 − 1)))))
168	103	breq2d 5178	. . . 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 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 4155	. . . . . . . . . 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 8069	. . . . . . . . . . . . 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 4905	. . . . . . . . . . . . 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 2780	. . . . . . . . . . 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 4664	. . . . . . . . . . 11 ⊢ (𝑘 ∈ {⟨0, (𝑃 − 1)⟩} ↔ 𝑘 = ⟨0, (𝑃 − 1)⟩)
182	180, 181	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)⟩})
183	172, 182	mtand 815	. . . . . . . . 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 46193	. . . . . . 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 16346	. . . . . . 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 5194	. . . . 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 16356	. . . 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 5194	. . 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 46164	. 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 3014	1 ⊢ (𝜑 → 𝐾 ≠ 0)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 ∨ wo 846 ∧ w3a 1087 = wceq 1537 ∈ wcel 2108 ≠ wne 2946 {crab 3443 Vcvv 3488 ∖ cdif 3973 ⊆ wss 3976 ifcif 4548 {csn 4648 {cpr 4650 ⟨cop 4654 class class class wbr 5166 ↦ cmpt 5249 × cxp 5698 ran crn 5701 ⟶wf 6569 ‘cfv 6573 (class class class)co 7448 1^st c1st 8028 2^nd c2nd 8029 ↑_m cmap 8884 Fincfn 9003 ℂcc 11182 ℝcr 11183 0cc0 11184 1c1 11185 + caddc 11187 · cmul 11189 < clt 11324 ≤ cle 11325 − cmin 11520 / cdiv 11947 ℕcn 12293 ℕ₀cn0 12553 ℤcz 12639 ℤ_≥cuz 12903 (,)cioo 13407 ...cfz 13567 ↑cexp 14112 !cfa 14322 abscabs 15283 Σcsu 15734 ∏cprod 15951 ∥ cdvds 16302 ℙcprime 16718 ↾_t crest 17480 TopOpenctopn 17481 topGenctg 17497 ℂ_fldccnfld 21387 D^𝑛 cdvn 25919

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2158 ax-12 2178 ax-ext 2711 ax-rep 5303 ax-sep 5317 ax-nul 5324 ax-pow 5383 ax-pr 5447 ax-un 7770 ax-inf2 9710 ax-cnex 11240 ax-resscn 11241 ax-1cn 11242 ax-icn 11243 ax-addcl 11244 ax-addrcl 11245 ax-mulcl 11246 ax-mulrcl 11247 ax-mulcom 11248 ax-addass 11249 ax-mulass 11250 ax-distr 11251 ax-i2m1 11252 ax-1ne0 11253 ax-1rid 11254 ax-rnegex 11255 ax-rrecex 11256 ax-cnre 11257 ax-pre-lttri 11258 ax-pre-lttrn 11259 ax-pre-ltadd 11260 ax-pre-mulgt0 11261 ax-pre-sup 11262 ax-addf 11263

This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3or 1088 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2543 df-eu 2572 df-clab 2718 df-cleq 2732 df-clel 2819 df-nfc 2895 df-ne 2947 df-nel 3053 df-ral 3068 df-rex 3077 df-rmo 3388 df-reu 3389 df-rab 3444 df-v 3490 df-sbc 3805 df-csb 3922 df-dif 3979 df-un 3981 df-in 3983 df-ss 3993 df-pss 3996 df-nul 4353 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-tp 4653 df-op 4655 df-uni 4932 df-int 4971 df-iun 5017 df-iin 5018 df-br 5167 df-opab 5229 df-mpt 5250 df-tr 5284 df-id 5593 df-eprel 5599 df-po 5607 df-so 5608 df-fr 5652 df-se 5653 df-we 5654 df-xp 5706 df-rel 5707 df-cnv 5708 df-co 5709 df-dm 5710 df-rn 5711 df-res 5712 df-ima 5713 df-pred 6332 df-ord 6398 df-on 6399 df-lim 6400 df-suc 6401 df-iota 6525 df-fun 6575 df-fn 6576 df-f 6577 df-f1 6578 df-fo 6579 df-f1o 6580 df-fv 6581 df-isom 6582 df-riota 7404 df-ov 7451 df-oprab 7452 df-mpo 7453 df-of 7714 df-om 7904 df-1st 8030 df-2nd 8031 df-supp 8202 df-frecs 8322 df-wrecs 8353 df-recs 8427 df-rdg 8466 df-1o 8522 df-2o 8523 df-er 8763 df-map 8886 df-pm 8887 df-ixp 8956 df-en 9004 df-dom 9005 df-sdom 9006 df-fin 9007 df-fsupp 9432 df-fi 9480 df-sup 9511 df-inf 9512 df-oi 9579 df-card 10008 df-pnf 11326 df-mnf 11327 df-xr 11328 df-ltxr 11329 df-le 11330 df-sub 11522 df-neg 11523 df-div 11948 df-nn 12294 df-2 12356 df-3 12357 df-4 12358 df-5 12359 df-6 12360 df-7 12361 df-8 12362 df-9 12363 df-n0 12554 df-z 12640 df-dec 12759 df-uz 12904 df-q 13014 df-rp 13058 df-xneg 13175 df-xadd 13176 df-xmul 13177 df-ioo 13411 df-ico 13413 df-icc 13414 df-fz 13568 df-fzo 13712 df-fl 13843 df-mod 13921 df-seq 14053 df-exp 14113 df-fac 14323 df-bc 14352 df-hash 14380 df-cj 15148 df-re 15149 df-im 15150 df-sqrt 15284 df-abs 15285 df-clim 15534 df-sum 15735 df-prod 15952 df-dvds 16303 df-gcd 16541 df-prm 16719 df-struct 17194 df-sets 17211 df-slot 17229 df-ndx 17241 df-base 17259 df-ress 17288 df-plusg 17324 df-mulr 17325 df-starv 17326 df-sca 17327 df-vsca 17328 df-ip 17329 df-tset 17330 df-ple 17331 df-ds 17333 df-unif 17334 df-hom 17335 df-cco 17336 df-rest 17482 df-topn 17483 df-0g 17501 df-gsum 17502 df-topgen 17503 df-pt 17504 df-prds 17507 df-xrs 17562 df-qtop 17567 df-imas 17568 df-xps 17570 df-mre 17644 df-mrc 17645 df-acs 17647 df-mgm 18678 df-sgrp 18757 df-mnd 18773 df-submnd 18819 df-mulg 19108 df-cntz 19357 df-cmn 19824 df-psmet 21379 df-xmet 21380 df-met 21381 df-bl 21382 df-mopn 21383 df-fbas 21384 df-fg 21385 df-cnfld 21388 df-top 22921 df-topon 22938 df-topsp 22960 df-bases 22974 df-cld 23048 df-ntr 23049 df-cls 23050 df-nei 23127 df-lp 23165 df-perf 23166 df-cn 23256 df-cnp 23257 df-haus 23344 df-tx 23591 df-hmeo 23784 df-fil 23875 df-fm 23967 df-flim 23968 df-flf 23969 df-xms 24351 df-ms 24352 df-tms 24353 df-cncf 24923 df-limc 25921 df-dv 25922 df-dvn 25923

This theorem is referenced by: etransclem47 46202

W3C validator