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Theorem eirrlem 16143

Description: Lemma for eirr 16144. (Contributed by Paul Chapman, 9-Feb-2008.) (Revised by Mario Carneiro, 29-Apr-2014.)

**Hypotheses**
Ref	Expression
eirr.1	⊢ 𝐹 = (𝑛 ∈ ℕ₀ ↦ (1 / (!‘𝑛)))
eirr.2	⊢ (𝜑 → 𝑃 ∈ ℤ)
eirr.3	⊢ (𝜑 → 𝑄 ∈ ℕ)
eirr.4	⊢ (𝜑 → e = (𝑃 / 𝑄))

**Assertion**
Ref	Expression
eirrlem	⊢ ¬ 𝜑

Distinct variable group: 𝑄,𝑛 Allowed substitution hints: 𝜑(𝑛) 𝑃(𝑛) 𝐹(𝑛)

Proof of Theorem eirrlem Dummy variable 𝑘 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fzfid 13934	. . . . . . 7 ⊢ (𝜑 → (0...𝑄) ∈ Fin)
2		elfznn0 13590	. . . . . . . 8 ⊢ (𝑘 ∈ (0...𝑄) → 𝑘 ∈ ℕ₀)
3		eirr.1	. . . . . . . . . . . 12 ⊢ 𝐹 = (𝑛 ∈ ℕ₀ ↦ (1 / (!‘𝑛)))
4		nn0z 12579	. . . . . . . . . . . . . . 15 ⊢ (𝑛 ∈ ℕ₀ → 𝑛 ∈ ℤ)
5		1exp 14053	. . . . . . . . . . . . . . 15 ⊢ (𝑛 ∈ ℤ → (1↑𝑛) = 1)
6	4, 5	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝑛 ∈ ℕ₀ → (1↑𝑛) = 1)
7	6	oveq1d 7420	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ ℕ₀ → ((1↑𝑛) / (!‘𝑛)) = (1 / (!‘𝑛)))
8	7	mpteq2ia 5250	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ ℕ₀ ↦ ((1↑𝑛) / (!‘𝑛))) = (𝑛 ∈ ℕ₀ ↦ (1 / (!‘𝑛)))
9	3, 8	eqtr4i 2763	. . . . . . . . . . 11 ⊢ 𝐹 = (𝑛 ∈ ℕ₀ ↦ ((1↑𝑛) / (!‘𝑛)))
10	9	eftval 16016	. . . . . . . . . 10 ⊢ (𝑘 ∈ ℕ₀ → (𝐹‘𝑘) = ((1↑𝑘) / (!‘𝑘)))
11	10	adantl 482	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ₀) → (𝐹‘𝑘) = ((1↑𝑘) / (!‘𝑘)))
12		ax-1cn 11164	. . . . . . . . . . 11 ⊢ 1 ∈ ℂ
13	12	a1i 11	. . . . . . . . . 10 ⊢ (𝜑 → 1 ∈ ℂ)
14		eftcl 16013	. . . . . . . . . 10 ⊢ ((1 ∈ ℂ ∧ 𝑘 ∈ ℕ₀) → ((1↑𝑘) / (!‘𝑘)) ∈ ℂ)
15	13, 14	sylan 580	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ₀) → ((1↑𝑘) / (!‘𝑘)) ∈ ℂ)
16	11, 15	eqeltrd 2833	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ₀) → (𝐹‘𝑘) ∈ ℂ)
17	2, 16	sylan2 593	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑄)) → (𝐹‘𝑘) ∈ ℂ)
18	1, 17	fsumcl 15675	. . . . . 6 ⊢ (𝜑 → Σ𝑘 ∈ (0...𝑄)(𝐹‘𝑘) ∈ ℂ)
19		nn0uz 12860	. . . . . . . . 9 ⊢ ℕ₀ = (ℤ_≥‘0)
20		eqid 2732	. . . . . . . . 9 ⊢ (ℤ_≥‘(𝑄 + 1)) = (ℤ_≥‘(𝑄 + 1))
21		eirr.3	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑄 ∈ ℕ)
22	21	peano2nnd 12225	. . . . . . . . . 10 ⊢ (𝜑 → (𝑄 + 1) ∈ ℕ)
23	22	nnnn0d 12528	. . . . . . . . 9 ⊢ (𝜑 → (𝑄 + 1) ∈ ℕ₀)
24		eqidd 2733	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ₀) → (𝐹‘𝑘) = (𝐹‘𝑘))
25		fveq2 6888	. . . . . . . . . . . . 13 ⊢ (𝑛 = 𝑘 → (!‘𝑛) = (!‘𝑘))
26	25	oveq2d 7421	. . . . . . . . . . . 12 ⊢ (𝑛 = 𝑘 → (1 / (!‘𝑛)) = (1 / (!‘𝑘)))
27		ovex 7438	. . . . . . . . . . . 12 ⊢ (1 / (!‘𝑘)) ∈ V
28	26, 3, 27	fvmpt 6995	. . . . . . . . . . 11 ⊢ (𝑘 ∈ ℕ₀ → (𝐹‘𝑘) = (1 / (!‘𝑘)))
29	28	adantl 482	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ₀) → (𝐹‘𝑘) = (1 / (!‘𝑘)))
30		faccl 14239	. . . . . . . . . . . . 13 ⊢ (𝑘 ∈ ℕ₀ → (!‘𝑘) ∈ ℕ)
31	30	adantl 482	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ₀) → (!‘𝑘) ∈ ℕ)
32	31	nnrpd 13010	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ₀) → (!‘𝑘) ∈ ℝ⁺)
33	32	rpreccld 13022	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ₀) → (1 / (!‘𝑘)) ∈ ℝ⁺)
34	29, 33	eqeltrd 2833	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ₀) → (𝐹‘𝑘) ∈ ℝ⁺)
35	9	efcllem 16017	. . . . . . . . . 10 ⊢ (1 ∈ ℂ → seq0( + , 𝐹) ∈ dom ⇝ )
36	13, 35	syl 17	. . . . . . . . 9 ⊢ (𝜑 → seq0( + , 𝐹) ∈ dom ⇝ )
37	19, 20, 23, 24, 34, 36	isumrpcl 15785	. . . . . . . 8 ⊢ (𝜑 → Σ𝑘 ∈ (ℤ_≥‘(𝑄 + 1))(𝐹‘𝑘) ∈ ℝ⁺)
38	37	rpred 13012	. . . . . . 7 ⊢ (𝜑 → Σ𝑘 ∈ (ℤ_≥‘(𝑄 + 1))(𝐹‘𝑘) ∈ ℝ)
39	38	recnd 11238	. . . . . 6 ⊢ (𝜑 → Σ𝑘 ∈ (ℤ_≥‘(𝑄 + 1))(𝐹‘𝑘) ∈ ℂ)
40		esum 16020	. . . . . . . . 9 ⊢ e = Σ𝑘 ∈ ℕ₀ (1 / (!‘𝑘))
41	28	sumeq2i 15641	. . . . . . . . 9 ⊢ Σ𝑘 ∈ ℕ₀ (𝐹‘𝑘) = Σ𝑘 ∈ ℕ₀ (1 / (!‘𝑘))
42	40, 41	eqtr4i 2763	. . . . . . . 8 ⊢ e = Σ𝑘 ∈ ℕ₀ (𝐹‘𝑘)
43	19, 20, 23, 24, 16, 36	isumsplit 15782	. . . . . . . 8 ⊢ (𝜑 → Σ𝑘 ∈ ℕ₀ (𝐹‘𝑘) = (Σ𝑘 ∈ (0...((𝑄 + 1) − 1))(𝐹‘𝑘) + Σ𝑘 ∈ (ℤ_≥‘(𝑄 + 1))(𝐹‘𝑘)))
44	42, 43	eqtrid 2784	. . . . . . 7 ⊢ (𝜑 → e = (Σ𝑘 ∈ (0...((𝑄 + 1) − 1))(𝐹‘𝑘) + Σ𝑘 ∈ (ℤ_≥‘(𝑄 + 1))(𝐹‘𝑘)))
45	21	nncnd 12224	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑄 ∈ ℂ)
46		pncan 11462	. . . . . . . . . . 11 ⊢ ((𝑄 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝑄 + 1) − 1) = 𝑄)
47	45, 12, 46	sylancl 586	. . . . . . . . . 10 ⊢ (𝜑 → ((𝑄 + 1) − 1) = 𝑄)
48	47	oveq2d 7421	. . . . . . . . 9 ⊢ (𝜑 → (0...((𝑄 + 1) − 1)) = (0...𝑄))
49	48	sumeq1d 15643	. . . . . . . 8 ⊢ (𝜑 → Σ𝑘 ∈ (0...((𝑄 + 1) − 1))(𝐹‘𝑘) = Σ𝑘 ∈ (0...𝑄)(𝐹‘𝑘))
50	49	oveq1d 7420	. . . . . . 7 ⊢ (𝜑 → (Σ𝑘 ∈ (0...((𝑄 + 1) − 1))(𝐹‘𝑘) + Σ𝑘 ∈ (ℤ_≥‘(𝑄 + 1))(𝐹‘𝑘)) = (Σ𝑘 ∈ (0...𝑄)(𝐹‘𝑘) + Σ𝑘 ∈ (ℤ_≥‘(𝑄 + 1))(𝐹‘𝑘)))
51	44, 50	eqtrd 2772	. . . . . 6 ⊢ (𝜑 → e = (Σ𝑘 ∈ (0...𝑄)(𝐹‘𝑘) + Σ𝑘 ∈ (ℤ_≥‘(𝑄 + 1))(𝐹‘𝑘)))
52	18, 39, 51	mvrladdd 11623	. . . . 5 ⊢ (𝜑 → (e − Σ𝑘 ∈ (0...𝑄)(𝐹‘𝑘)) = Σ𝑘 ∈ (ℤ_≥‘(𝑄 + 1))(𝐹‘𝑘))
53	52	oveq2d 7421	. . . 4 ⊢ (𝜑 → ((!‘𝑄) · (e − Σ𝑘 ∈ (0...𝑄)(𝐹‘𝑘))) = ((!‘𝑄) · Σ𝑘 ∈ (ℤ_≥‘(𝑄 + 1))(𝐹‘𝑘)))
54	21	nnnn0d 12528	. . . . . . 7 ⊢ (𝜑 → 𝑄 ∈ ℕ₀)
55	54	faccld 14240	. . . . . 6 ⊢ (𝜑 → (!‘𝑄) ∈ ℕ)
56	55	nncnd 12224	. . . . 5 ⊢ (𝜑 → (!‘𝑄) ∈ ℂ)
57		ere 16028	. . . . . . 7 ⊢ e ∈ ℝ
58	57	recni 11224	. . . . . 6 ⊢ e ∈ ℂ
59	58	a1i 11	. . . . 5 ⊢ (𝜑 → e ∈ ℂ)
60	56, 59, 18	subdid 11666	. . . 4 ⊢ (𝜑 → ((!‘𝑄) · (e − Σ𝑘 ∈ (0...𝑄)(𝐹‘𝑘))) = (((!‘𝑄) · e) − ((!‘𝑄) · Σ𝑘 ∈ (0...𝑄)(𝐹‘𝑘))))
61	53, 60	eqtr3d 2774	. . 3 ⊢ (𝜑 → ((!‘𝑄) · Σ𝑘 ∈ (ℤ_≥‘(𝑄 + 1))(𝐹‘𝑘)) = (((!‘𝑄) · e) − ((!‘𝑄) · Σ𝑘 ∈ (0...𝑄)(𝐹‘𝑘))))
62		eirr.4	. . . . . . 7 ⊢ (𝜑 → e = (𝑃 / 𝑄))
63	62	oveq2d 7421	. . . . . 6 ⊢ (𝜑 → ((!‘𝑄) · e) = ((!‘𝑄) · (𝑃 / 𝑄)))
64		eirr.2	. . . . . . . 8 ⊢ (𝜑 → 𝑃 ∈ ℤ)
65	64	zcnd 12663	. . . . . . 7 ⊢ (𝜑 → 𝑃 ∈ ℂ)
66	21	nnne0d 12258	. . . . . . 7 ⊢ (𝜑 → 𝑄 ≠ 0)
67	56, 65, 45, 66	div12d 12022	. . . . . 6 ⊢ (𝜑 → ((!‘𝑄) · (𝑃 / 𝑄)) = (𝑃 · ((!‘𝑄) / 𝑄)))
68	63, 67	eqtrd 2772	. . . . 5 ⊢ (𝜑 → ((!‘𝑄) · e) = (𝑃 · ((!‘𝑄) / 𝑄)))
69	21	nnred 12223	. . . . . . . . 9 ⊢ (𝜑 → 𝑄 ∈ ℝ)
70	69	leidd 11776	. . . . . . . 8 ⊢ (𝜑 → 𝑄 ≤ 𝑄)
71		facdiv 14243	. . . . . . . 8 ⊢ ((𝑄 ∈ ℕ₀ ∧ 𝑄 ∈ ℕ ∧ 𝑄 ≤ 𝑄) → ((!‘𝑄) / 𝑄) ∈ ℕ)
72	54, 21, 70, 71	syl3anc 1371	. . . . . . 7 ⊢ (𝜑 → ((!‘𝑄) / 𝑄) ∈ ℕ)
73	72	nnzd 12581	. . . . . 6 ⊢ (𝜑 → ((!‘𝑄) / 𝑄) ∈ ℤ)
74	64, 73	zmulcld 12668	. . . . 5 ⊢ (𝜑 → (𝑃 · ((!‘𝑄) / 𝑄)) ∈ ℤ)
75	68, 74	eqeltrd 2833	. . . 4 ⊢ (𝜑 → ((!‘𝑄) · e) ∈ ℤ)
76	1, 56, 17	fsummulc2 15726	. . . . 5 ⊢ (𝜑 → ((!‘𝑄) · Σ𝑘 ∈ (0...𝑄)(𝐹‘𝑘)) = Σ𝑘 ∈ (0...𝑄)((!‘𝑄) · (𝐹‘𝑘)))
77	2	adantl 482	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑄)) → 𝑘 ∈ ℕ₀)
78	77, 28	syl 17	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑄)) → (𝐹‘𝑘) = (1 / (!‘𝑘)))
79	78	oveq2d 7421	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑄)) → ((!‘𝑄) · (𝐹‘𝑘)) = ((!‘𝑄) · (1 / (!‘𝑘))))
80	56	adantr 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑄)) → (!‘𝑄) ∈ ℂ)
81	2, 31	sylan2 593	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑄)) → (!‘𝑘) ∈ ℕ)
82	81	nncnd 12224	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑄)) → (!‘𝑘) ∈ ℂ)
83		facne0 14242	. . . . . . . . . . 11 ⊢ (𝑘 ∈ ℕ₀ → (!‘𝑘) ≠ 0)
84	77, 83	syl 17	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑄)) → (!‘𝑘) ≠ 0)
85	80, 82, 84	divrecd 11989	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑄)) → ((!‘𝑄) / (!‘𝑘)) = ((!‘𝑄) · (1 / (!‘𝑘))))
86	79, 85	eqtr4d 2775	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑄)) → ((!‘𝑄) · (𝐹‘𝑘)) = ((!‘𝑄) / (!‘𝑘)))
87		permnn 14282	. . . . . . . . 9 ⊢ (𝑘 ∈ (0...𝑄) → ((!‘𝑄) / (!‘𝑘)) ∈ ℕ)
88	87	adantl 482	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑄)) → ((!‘𝑄) / (!‘𝑘)) ∈ ℕ)
89	86, 88	eqeltrd 2833	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑄)) → ((!‘𝑄) · (𝐹‘𝑘)) ∈ ℕ)
90	89	nnzd 12581	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑄)) → ((!‘𝑄) · (𝐹‘𝑘)) ∈ ℤ)
91	1, 90	fsumzcl 15677	. . . . 5 ⊢ (𝜑 → Σ𝑘 ∈ (0...𝑄)((!‘𝑄) · (𝐹‘𝑘)) ∈ ℤ)
92	76, 91	eqeltrd 2833	. . . 4 ⊢ (𝜑 → ((!‘𝑄) · Σ𝑘 ∈ (0...𝑄)(𝐹‘𝑘)) ∈ ℤ)
93	75, 92	zsubcld 12667	. . 3 ⊢ (𝜑 → (((!‘𝑄) · e) − ((!‘𝑄) · Σ𝑘 ∈ (0...𝑄)(𝐹‘𝑘))) ∈ ℤ)
94	61, 93	eqeltrd 2833	. 2 ⊢ (𝜑 → ((!‘𝑄) · Σ𝑘 ∈ (ℤ_≥‘(𝑄 + 1))(𝐹‘𝑘)) ∈ ℤ)
95		0zd 12566	. . 3 ⊢ (𝜑 → 0 ∈ ℤ)
96	55	nnrpd 13010	. . . . 5 ⊢ (𝜑 → (!‘𝑄) ∈ ℝ⁺)
97	96, 37	rpmulcld 13028	. . . 4 ⊢ (𝜑 → ((!‘𝑄) · Σ𝑘 ∈ (ℤ_≥‘(𝑄 + 1))(𝐹‘𝑘)) ∈ ℝ⁺)
98	97	rpgt0d 13015	. . 3 ⊢ (𝜑 → 0 < ((!‘𝑄) · Σ𝑘 ∈ (ℤ_≥‘(𝑄 + 1))(𝐹‘𝑘)))
99	22	peano2nnd 12225	. . . . . . . 8 ⊢ (𝜑 → ((𝑄 + 1) + 1) ∈ ℕ)
100	99	nnred 12223	. . . . . . 7 ⊢ (𝜑 → ((𝑄 + 1) + 1) ∈ ℝ)
101	23	faccld 14240	. . . . . . . 8 ⊢ (𝜑 → (!‘(𝑄 + 1)) ∈ ℕ)
102	101, 22	nnmulcld 12261	. . . . . . 7 ⊢ (𝜑 → ((!‘(𝑄 + 1)) · (𝑄 + 1)) ∈ ℕ)
103	100, 102	nndivred 12262	. . . . . 6 ⊢ (𝜑 → (((𝑄 + 1) + 1) / ((!‘(𝑄 + 1)) · (𝑄 + 1))) ∈ ℝ)
104	55	nnrecred 12259	. . . . . 6 ⊢ (𝜑 → (1 / (!‘𝑄)) ∈ ℝ)
105		abs1 15240	. . . . . . . . . . . 12 ⊢ (abs‘1) = 1
106	105	oveq1i 7415	. . . . . . . . . . 11 ⊢ ((abs‘1)↑𝑛) = (1↑𝑛)
107	106	oveq1i 7415	. . . . . . . . . 10 ⊢ (((abs‘1)↑𝑛) / (!‘𝑛)) = ((1↑𝑛) / (!‘𝑛))
108	107	mpteq2i 5252	. . . . . . . . 9 ⊢ (𝑛 ∈ ℕ₀ ↦ (((abs‘1)↑𝑛) / (!‘𝑛))) = (𝑛 ∈ ℕ₀ ↦ ((1↑𝑛) / (!‘𝑛)))
109	9, 108	eqtr4i 2763	. . . . . . . 8 ⊢ 𝐹 = (𝑛 ∈ ℕ₀ ↦ (((abs‘1)↑𝑛) / (!‘𝑛)))
110		eqid 2732	. . . . . . . 8 ⊢ (𝑛 ∈ ℕ₀ ↦ ((((abs‘1)↑(𝑄 + 1)) / (!‘(𝑄 + 1))) · ((1 / ((𝑄 + 1) + 1))↑𝑛))) = (𝑛 ∈ ℕ₀ ↦ ((((abs‘1)↑(𝑄 + 1)) / (!‘(𝑄 + 1))) · ((1 / ((𝑄 + 1) + 1))↑𝑛)))
111		1le1 11838	. . . . . . . . . 10 ⊢ 1 ≤ 1
112	105, 111	eqbrtri 5168	. . . . . . . . 9 ⊢ (abs‘1) ≤ 1
113	112	a1i 11	. . . . . . . 8 ⊢ (𝜑 → (abs‘1) ≤ 1)
114	9, 109, 110, 22, 13, 113	eftlub 16048	. . . . . . 7 ⊢ (𝜑 → (abs‘Σ𝑘 ∈ (ℤ_≥‘(𝑄 + 1))(𝐹‘𝑘)) ≤ (((abs‘1)↑(𝑄 + 1)) · (((𝑄 + 1) + 1) / ((!‘(𝑄 + 1)) · (𝑄 + 1)))))
115	37	rprege0d 13019	. . . . . . . 8 ⊢ (𝜑 → (Σ𝑘 ∈ (ℤ_≥‘(𝑄 + 1))(𝐹‘𝑘) ∈ ℝ ∧ 0 ≤ Σ𝑘 ∈ (ℤ_≥‘(𝑄 + 1))(𝐹‘𝑘)))
116		absid 15239	. . . . . . . 8 ⊢ ((Σ𝑘 ∈ (ℤ_≥‘(𝑄 + 1))(𝐹‘𝑘) ∈ ℝ ∧ 0 ≤ Σ𝑘 ∈ (ℤ_≥‘(𝑄 + 1))(𝐹‘𝑘)) → (abs‘Σ𝑘 ∈ (ℤ_≥‘(𝑄 + 1))(𝐹‘𝑘)) = Σ𝑘 ∈ (ℤ_≥‘(𝑄 + 1))(𝐹‘𝑘))
117	115, 116	syl 17	. . . . . . 7 ⊢ (𝜑 → (abs‘Σ𝑘 ∈ (ℤ_≥‘(𝑄 + 1))(𝐹‘𝑘)) = Σ𝑘 ∈ (ℤ_≥‘(𝑄 + 1))(𝐹‘𝑘))
118	105	oveq1i 7415	. . . . . . . . . 10 ⊢ ((abs‘1)↑(𝑄 + 1)) = (1↑(𝑄 + 1))
119	22	nnzd 12581	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑄 + 1) ∈ ℤ)
120		1exp 14053	. . . . . . . . . . 11 ⊢ ((𝑄 + 1) ∈ ℤ → (1↑(𝑄 + 1)) = 1)
121	119, 120	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → (1↑(𝑄 + 1)) = 1)
122	118, 121	eqtrid 2784	. . . . . . . . 9 ⊢ (𝜑 → ((abs‘1)↑(𝑄 + 1)) = 1)
123	122	oveq1d 7420	. . . . . . . 8 ⊢ (𝜑 → (((abs‘1)↑(𝑄 + 1)) · (((𝑄 + 1) + 1) / ((!‘(𝑄 + 1)) · (𝑄 + 1)))) = (1 · (((𝑄 + 1) + 1) / ((!‘(𝑄 + 1)) · (𝑄 + 1)))))
124	103	recnd 11238	. . . . . . . . 9 ⊢ (𝜑 → (((𝑄 + 1) + 1) / ((!‘(𝑄 + 1)) · (𝑄 + 1))) ∈ ℂ)
125	124	mullidd 11228	. . . . . . . 8 ⊢ (𝜑 → (1 · (((𝑄 + 1) + 1) / ((!‘(𝑄 + 1)) · (𝑄 + 1)))) = (((𝑄 + 1) + 1) / ((!‘(𝑄 + 1)) · (𝑄 + 1))))
126	123, 125	eqtrd 2772	. . . . . . 7 ⊢ (𝜑 → (((abs‘1)↑(𝑄 + 1)) · (((𝑄 + 1) + 1) / ((!‘(𝑄 + 1)) · (𝑄 + 1)))) = (((𝑄 + 1) + 1) / ((!‘(𝑄 + 1)) · (𝑄 + 1))))
127	114, 117, 126	3brtr3d 5178	. . . . . 6 ⊢ (𝜑 → Σ𝑘 ∈ (ℤ_≥‘(𝑄 + 1))(𝐹‘𝑘) ≤ (((𝑄 + 1) + 1) / ((!‘(𝑄 + 1)) · (𝑄 + 1))))
128	22	nnred 12223	. . . . . . . . . 10 ⊢ (𝜑 → (𝑄 + 1) ∈ ℝ)
129	128, 128	readdcld 11239	. . . . . . . . 9 ⊢ (𝜑 → ((𝑄 + 1) + (𝑄 + 1)) ∈ ℝ)
130	128, 128	remulcld 11240	. . . . . . . . 9 ⊢ (𝜑 → ((𝑄 + 1) · (𝑄 + 1)) ∈ ℝ)
131		1red 11211	. . . . . . . . . 10 ⊢ (𝜑 → 1 ∈ ℝ)
132	21	nnge1d 12256	. . . . . . . . . . 11 ⊢ (𝜑 → 1 ≤ 𝑄)
133		1nn 12219	. . . . . . . . . . . 12 ⊢ 1 ∈ ℕ
134		nnleltp1 12613	. . . . . . . . . . . 12 ⊢ ((1 ∈ ℕ ∧ 𝑄 ∈ ℕ) → (1 ≤ 𝑄 ↔ 1 < (𝑄 + 1)))
135	133, 21, 134	sylancr 587	. . . . . . . . . . 11 ⊢ (𝜑 → (1 ≤ 𝑄 ↔ 1 < (𝑄 + 1)))
136	132, 135	mpbid 231	. . . . . . . . . 10 ⊢ (𝜑 → 1 < (𝑄 + 1))
137	131, 128, 128, 136	ltadd2dd 11369	. . . . . . . . 9 ⊢ (𝜑 → ((𝑄 + 1) + 1) < ((𝑄 + 1) + (𝑄 + 1)))
138	22	nncnd 12224	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑄 + 1) ∈ ℂ)
139	138	2timesd 12451	. . . . . . . . . 10 ⊢ (𝜑 → (2 · (𝑄 + 1)) = ((𝑄 + 1) + (𝑄 + 1)))
140		df-2 12271	. . . . . . . . . . . 12 ⊢ 2 = (1 + 1)
141	131, 69, 131, 132	leadd1dd 11824	. . . . . . . . . . . 12 ⊢ (𝜑 → (1 + 1) ≤ (𝑄 + 1))
142	140, 141	eqbrtrid 5182	. . . . . . . . . . 11 ⊢ (𝜑 → 2 ≤ (𝑄 + 1))
143		2re 12282	. . . . . . . . . . . . 13 ⊢ 2 ∈ ℝ
144	143	a1i 11	. . . . . . . . . . . 12 ⊢ (𝜑 → 2 ∈ ℝ)
145	22	nngt0d 12257	. . . . . . . . . . . 12 ⊢ (𝜑 → 0 < (𝑄 + 1))
146		lemul1 12062	. . . . . . . . . . . 12 ⊢ ((2 ∈ ℝ ∧ (𝑄 + 1) ∈ ℝ ∧ ((𝑄 + 1) ∈ ℝ ∧ 0 < (𝑄 + 1))) → (2 ≤ (𝑄 + 1) ↔ (2 · (𝑄 + 1)) ≤ ((𝑄 + 1) · (𝑄 + 1))))
147	144, 128, 128, 145, 146	syl112anc 1374	. . . . . . . . . . 11 ⊢ (𝜑 → (2 ≤ (𝑄 + 1) ↔ (2 · (𝑄 + 1)) ≤ ((𝑄 + 1) · (𝑄 + 1))))
148	142, 147	mpbid 231	. . . . . . . . . 10 ⊢ (𝜑 → (2 · (𝑄 + 1)) ≤ ((𝑄 + 1) · (𝑄 + 1)))
149	139, 148	eqbrtrrd 5171	. . . . . . . . 9 ⊢ (𝜑 → ((𝑄 + 1) + (𝑄 + 1)) ≤ ((𝑄 + 1) · (𝑄 + 1)))
150	100, 129, 130, 137, 149	ltletrd 11370	. . . . . . . 8 ⊢ (𝜑 → ((𝑄 + 1) + 1) < ((𝑄 + 1) · (𝑄 + 1)))
151		facp1 14234	. . . . . . . . . . . . 13 ⊢ (𝑄 ∈ ℕ₀ → (!‘(𝑄 + 1)) = ((!‘𝑄) · (𝑄 + 1)))
152	54, 151	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → (!‘(𝑄 + 1)) = ((!‘𝑄) · (𝑄 + 1)))
153	152	oveq1d 7420	. . . . . . . . . . 11 ⊢ (𝜑 → ((!‘(𝑄 + 1)) / (!‘𝑄)) = (((!‘𝑄) · (𝑄 + 1)) / (!‘𝑄)))
154	101	nncnd 12224	. . . . . . . . . . . 12 ⊢ (𝜑 → (!‘(𝑄 + 1)) ∈ ℂ)
155	55	nnne0d 12258	. . . . . . . . . . . 12 ⊢ (𝜑 → (!‘𝑄) ≠ 0)
156	154, 56, 155	divrecd 11989	. . . . . . . . . . 11 ⊢ (𝜑 → ((!‘(𝑄 + 1)) / (!‘𝑄)) = ((!‘(𝑄 + 1)) · (1 / (!‘𝑄))))
157	138, 56, 155	divcan3d 11991	. . . . . . . . . . 11 ⊢ (𝜑 → (((!‘𝑄) · (𝑄 + 1)) / (!‘𝑄)) = (𝑄 + 1))
158	153, 156, 157	3eqtr3rd 2781	. . . . . . . . . 10 ⊢ (𝜑 → (𝑄 + 1) = ((!‘(𝑄 + 1)) · (1 / (!‘𝑄))))
159	158	oveq1d 7420	. . . . . . . . 9 ⊢ (𝜑 → ((𝑄 + 1) · (𝑄 + 1)) = (((!‘(𝑄 + 1)) · (1 / (!‘𝑄))) · (𝑄 + 1)))
160	104	recnd 11238	. . . . . . . . . 10 ⊢ (𝜑 → (1 / (!‘𝑄)) ∈ ℂ)
161	154, 160, 138	mul32d 11420	. . . . . . . . 9 ⊢ (𝜑 → (((!‘(𝑄 + 1)) · (1 / (!‘𝑄))) · (𝑄 + 1)) = (((!‘(𝑄 + 1)) · (𝑄 + 1)) · (1 / (!‘𝑄))))
162	159, 161	eqtrd 2772	. . . . . . . 8 ⊢ (𝜑 → ((𝑄 + 1) · (𝑄 + 1)) = (((!‘(𝑄 + 1)) · (𝑄 + 1)) · (1 / (!‘𝑄))))
163	150, 162	breqtrd 5173	. . . . . . 7 ⊢ (𝜑 → ((𝑄 + 1) + 1) < (((!‘(𝑄 + 1)) · (𝑄 + 1)) · (1 / (!‘𝑄))))
164	102	nnred 12223	. . . . . . . 8 ⊢ (𝜑 → ((!‘(𝑄 + 1)) · (𝑄 + 1)) ∈ ℝ)
165	102	nngt0d 12257	. . . . . . . 8 ⊢ (𝜑 → 0 < ((!‘(𝑄 + 1)) · (𝑄 + 1)))
166		ltdivmul 12085	. . . . . . . 8 ⊢ ((((𝑄 + 1) + 1) ∈ ℝ ∧ (1 / (!‘𝑄)) ∈ ℝ ∧ (((!‘(𝑄 + 1)) · (𝑄 + 1)) ∈ ℝ ∧ 0 < ((!‘(𝑄 + 1)) · (𝑄 + 1)))) → ((((𝑄 + 1) + 1) / ((!‘(𝑄 + 1)) · (𝑄 + 1))) < (1 / (!‘𝑄)) ↔ ((𝑄 + 1) + 1) < (((!‘(𝑄 + 1)) · (𝑄 + 1)) · (1 / (!‘𝑄)))))
167	100, 104, 164, 165, 166	syl112anc 1374	. . . . . . 7 ⊢ (𝜑 → ((((𝑄 + 1) + 1) / ((!‘(𝑄 + 1)) · (𝑄 + 1))) < (1 / (!‘𝑄)) ↔ ((𝑄 + 1) + 1) < (((!‘(𝑄 + 1)) · (𝑄 + 1)) · (1 / (!‘𝑄)))))
168	163, 167	mpbird 256	. . . . . 6 ⊢ (𝜑 → (((𝑄 + 1) + 1) / ((!‘(𝑄 + 1)) · (𝑄 + 1))) < (1 / (!‘𝑄)))
169	38, 103, 104, 127, 168	lelttrd 11368	. . . . 5 ⊢ (𝜑 → Σ𝑘 ∈ (ℤ_≥‘(𝑄 + 1))(𝐹‘𝑘) < (1 / (!‘𝑄)))
170	38, 131, 96	ltmuldiv2d 13060	. . . . 5 ⊢ (𝜑 → (((!‘𝑄) · Σ𝑘 ∈ (ℤ_≥‘(𝑄 + 1))(𝐹‘𝑘)) < 1 ↔ Σ𝑘 ∈ (ℤ_≥‘(𝑄 + 1))(𝐹‘𝑘) < (1 / (!‘𝑄))))
171	169, 170	mpbird 256	. . . 4 ⊢ (𝜑 → ((!‘𝑄) · Σ𝑘 ∈ (ℤ_≥‘(𝑄 + 1))(𝐹‘𝑘)) < 1)
172		0p1e1 12330	. . . 4 ⊢ (0 + 1) = 1
173	171, 172	breqtrrdi 5189	. . 3 ⊢ (𝜑 → ((!‘𝑄) · Σ𝑘 ∈ (ℤ_≥‘(𝑄 + 1))(𝐹‘𝑘)) < (0 + 1))
174		btwnnz 12634	. . 3 ⊢ ((0 ∈ ℤ ∧ 0 < ((!‘𝑄) · Σ𝑘 ∈ (ℤ_≥‘(𝑄 + 1))(𝐹‘𝑘)) ∧ ((!‘𝑄) · Σ𝑘 ∈ (ℤ_≥‘(𝑄 + 1))(𝐹‘𝑘)) < (0 + 1)) → ¬ ((!‘𝑄) · Σ𝑘 ∈ (ℤ_≥‘(𝑄 + 1))(𝐹‘𝑘)) ∈ ℤ)
175	95, 98, 173, 174	syl3anc 1371	. 2 ⊢ (𝜑 → ¬ ((!‘𝑄) · Σ𝑘 ∈ (ℤ_≥‘(𝑄 + 1))(𝐹‘𝑘)) ∈ ℤ)
176	94, 175	pm2.65i 193	1 ⊢ ¬ 𝜑

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 396 = wceq 1541 ∈ wcel 2106 ≠ wne 2940 class class class wbr 5147 ↦ cmpt 5230 dom cdm 5675 ‘cfv 6540 (class class class)co 7405 ℂcc 11104 ℝcr 11105 0cc0 11106 1c1 11107 + caddc 11109 · cmul 11111 < clt 11244 ≤ cle 11245 − cmin 11440 / cdiv 11867 ℕcn 12208 2c2 12263 ℕ₀cn0 12468 ℤcz 12554 ℤ_≥cuz 12818 ℝ⁺crp 12970 ...cfz 13480 seqcseq 13962 ↑cexp 14023 !cfa 14229 abscabs 15177 ⇝ cli 15424 Σcsu 15628 eceu 16002

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 2703 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7721 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

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 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-int 4950 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-se 5631 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6297 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-isom 6549 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7852 df-1st 7971 df-2nd 7972 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-1o 8462 df-er 8699 df-pm 8819 df-en 8936 df-dom 8937 df-sdom 8938 df-fin 8939 df-sup 9433 df-inf 9434 df-oi 9501 df-card 9930 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11442 df-neg 11443 df-div 11868 df-nn 12209 df-2 12271 df-3 12272 df-n0 12469 df-z 12555 df-uz 12819 df-rp 12971 df-ico 13326 df-fz 13481 df-fzo 13624 df-fl 13753 df-seq 13963 df-exp 14024 df-fac 14230 df-bc 14259 df-hash 14287 df-shft 15010 df-cj 15042 df-re 15043 df-im 15044 df-sqrt 15178 df-abs 15179 df-limsup 15411 df-clim 15428 df-rlim 15429 df-sum 15629 df-ef 16007 df-e 16008

This theorem is referenced by: eirr 16144

W3C validator