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Theorem eirrlem 16188

Description: Lemma for eirr 16189. (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 13978	. . . . . . 7 ⊢ (𝜑 → (0...𝑄) ∈ Fin)
2		elfznn0 13634	. . . . . . . 8 ⊢ (𝑘 ∈ (0...𝑄) → 𝑘 ∈ ℕ₀)
3		eirr.1	. . . . . . . . . . . 12 ⊢ 𝐹 = (𝑛 ∈ ℕ₀ ↦ (1 / (!‘𝑛)))
4		nn0z 12621	. . . . . . . . . . . . . . 15 ⊢ (𝑛 ∈ ℕ₀ → 𝑛 ∈ ℤ)
5		1exp 14096	. . . . . . . . . . . . . . 15 ⊢ (𝑛 ∈ ℤ → (1↑𝑛) = 1)
6	4, 5	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝑛 ∈ ℕ₀ → (1↑𝑛) = 1)
7	6	oveq1d 7441	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ ℕ₀ → ((1↑𝑛) / (!‘𝑛)) = (1 / (!‘𝑛)))
8	7	mpteq2ia 5255	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ ℕ₀ ↦ ((1↑𝑛) / (!‘𝑛))) = (𝑛 ∈ ℕ₀ ↦ (1 / (!‘𝑛)))
9	3, 8	eqtr4i 2759	. . . . . . . . . . 11 ⊢ 𝐹 = (𝑛 ∈ ℕ₀ ↦ ((1↑𝑛) / (!‘𝑛)))
10	9	eftval 16060	. . . . . . . . . 10 ⊢ (𝑘 ∈ ℕ₀ → (𝐹‘𝑘) = ((1↑𝑘) / (!‘𝑘)))
11	10	adantl 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ₀) → (𝐹‘𝑘) = ((1↑𝑘) / (!‘𝑘)))
12		ax-1cn 11204	. . . . . . . . . . 11 ⊢ 1 ∈ ℂ
13	12	a1i 11	. . . . . . . . . 10 ⊢ (𝜑 → 1 ∈ ℂ)
14		eftcl 16057	. . . . . . . . . 10 ⊢ ((1 ∈ ℂ ∧ 𝑘 ∈ ℕ₀) → ((1↑𝑘) / (!‘𝑘)) ∈ ℂ)
15	13, 14	sylan 578	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ₀) → ((1↑𝑘) / (!‘𝑘)) ∈ ℂ)
16	11, 15	eqeltrd 2829	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ₀) → (𝐹‘𝑘) ∈ ℂ)
17	2, 16	sylan2 591	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑄)) → (𝐹‘𝑘) ∈ ℂ)
18	1, 17	fsumcl 15719	. . . . . 6 ⊢ (𝜑 → Σ𝑘 ∈ (0...𝑄)(𝐹‘𝑘) ∈ ℂ)
19		nn0uz 12902	. . . . . . . . 9 ⊢ ℕ₀ = (ℤ_≥‘0)
20		eqid 2728	. . . . . . . . 9 ⊢ (ℤ_≥‘(𝑄 + 1)) = (ℤ_≥‘(𝑄 + 1))
21		eirr.3	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑄 ∈ ℕ)
22	21	peano2nnd 12267	. . . . . . . . . 10 ⊢ (𝜑 → (𝑄 + 1) ∈ ℕ)
23	22	nnnn0d 12570	. . . . . . . . 9 ⊢ (𝜑 → (𝑄 + 1) ∈ ℕ₀)
24		eqidd 2729	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ₀) → (𝐹‘𝑘) = (𝐹‘𝑘))
25		fveq2 6902	. . . . . . . . . . . . 13 ⊢ (𝑛 = 𝑘 → (!‘𝑛) = (!‘𝑘))
26	25	oveq2d 7442	. . . . . . . . . . . 12 ⊢ (𝑛 = 𝑘 → (1 / (!‘𝑛)) = (1 / (!‘𝑘)))
27		ovex 7459	. . . . . . . . . . . 12 ⊢ (1 / (!‘𝑘)) ∈ V
28	26, 3, 27	fvmpt 7010	. . . . . . . . . . 11 ⊢ (𝑘 ∈ ℕ₀ → (𝐹‘𝑘) = (1 / (!‘𝑘)))
29	28	adantl 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ₀) → (𝐹‘𝑘) = (1 / (!‘𝑘)))
30		faccl 14282	. . . . . . . . . . . . 13 ⊢ (𝑘 ∈ ℕ₀ → (!‘𝑘) ∈ ℕ)
31	30	adantl 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ₀) → (!‘𝑘) ∈ ℕ)
32	31	nnrpd 13054	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ₀) → (!‘𝑘) ∈ ℝ⁺)
33	32	rpreccld 13066	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ₀) → (1 / (!‘𝑘)) ∈ ℝ⁺)
34	29, 33	eqeltrd 2829	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ₀) → (𝐹‘𝑘) ∈ ℝ⁺)
35	9	efcllem 16061	. . . . . . . . . 10 ⊢ (1 ∈ ℂ → seq0( + , 𝐹) ∈ dom ⇝ )
36	13, 35	syl 17	. . . . . . . . 9 ⊢ (𝜑 → seq0( + , 𝐹) ∈ dom ⇝ )
37	19, 20, 23, 24, 34, 36	isumrpcl 15829	. . . . . . . 8 ⊢ (𝜑 → Σ𝑘 ∈ (ℤ_≥‘(𝑄 + 1))(𝐹‘𝑘) ∈ ℝ⁺)
38	37	rpred 13056	. . . . . . 7 ⊢ (𝜑 → Σ𝑘 ∈ (ℤ_≥‘(𝑄 + 1))(𝐹‘𝑘) ∈ ℝ)
39	38	recnd 11280	. . . . . 6 ⊢ (𝜑 → Σ𝑘 ∈ (ℤ_≥‘(𝑄 + 1))(𝐹‘𝑘) ∈ ℂ)
40		esum 16064	. . . . . . . . 9 ⊢ e = Σ𝑘 ∈ ℕ₀ (1 / (!‘𝑘))
41	28	sumeq2i 15685	. . . . . . . . 9 ⊢ Σ𝑘 ∈ ℕ₀ (𝐹‘𝑘) = Σ𝑘 ∈ ℕ₀ (1 / (!‘𝑘))
42	40, 41	eqtr4i 2759	. . . . . . . 8 ⊢ e = Σ𝑘 ∈ ℕ₀ (𝐹‘𝑘)
43	19, 20, 23, 24, 16, 36	isumsplit 15826	. . . . . . . 8 ⊢ (𝜑 → Σ𝑘 ∈ ℕ₀ (𝐹‘𝑘) = (Σ𝑘 ∈ (0...((𝑄 + 1) − 1))(𝐹‘𝑘) + Σ𝑘 ∈ (ℤ_≥‘(𝑄 + 1))(𝐹‘𝑘)))
44	42, 43	eqtrid 2780	. . . . . . 7 ⊢ (𝜑 → e = (Σ𝑘 ∈ (0...((𝑄 + 1) − 1))(𝐹‘𝑘) + Σ𝑘 ∈ (ℤ_≥‘(𝑄 + 1))(𝐹‘𝑘)))
45	21	nncnd 12266	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑄 ∈ ℂ)
46		pncan 11504	. . . . . . . . . . 11 ⊢ ((𝑄 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝑄 + 1) − 1) = 𝑄)
47	45, 12, 46	sylancl 584	. . . . . . . . . 10 ⊢ (𝜑 → ((𝑄 + 1) − 1) = 𝑄)
48	47	oveq2d 7442	. . . . . . . . 9 ⊢ (𝜑 → (0...((𝑄 + 1) − 1)) = (0...𝑄))
49	48	sumeq1d 15687	. . . . . . . 8 ⊢ (𝜑 → Σ𝑘 ∈ (0...((𝑄 + 1) − 1))(𝐹‘𝑘) = Σ𝑘 ∈ (0...𝑄)(𝐹‘𝑘))
50	49	oveq1d 7441	. . . . . . 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 2768	. . . . . 6 ⊢ (𝜑 → e = (Σ𝑘 ∈ (0...𝑄)(𝐹‘𝑘) + Σ𝑘 ∈ (ℤ_≥‘(𝑄 + 1))(𝐹‘𝑘)))
52	18, 39, 51	mvrladdd 11665	. . . . 5 ⊢ (𝜑 → (e − Σ𝑘 ∈ (0...𝑄)(𝐹‘𝑘)) = Σ𝑘 ∈ (ℤ_≥‘(𝑄 + 1))(𝐹‘𝑘))
53	52	oveq2d 7442	. . . 4 ⊢ (𝜑 → ((!‘𝑄) · (e − Σ𝑘 ∈ (0...𝑄)(𝐹‘𝑘))) = ((!‘𝑄) · Σ𝑘 ∈ (ℤ_≥‘(𝑄 + 1))(𝐹‘𝑘)))
54	21	nnnn0d 12570	. . . . . . 7 ⊢ (𝜑 → 𝑄 ∈ ℕ₀)
55	54	faccld 14283	. . . . . 6 ⊢ (𝜑 → (!‘𝑄) ∈ ℕ)
56	55	nncnd 12266	. . . . 5 ⊢ (𝜑 → (!‘𝑄) ∈ ℂ)
57		ere 16073	. . . . . . 7 ⊢ e ∈ ℝ
58	57	recni 11266	. . . . . 6 ⊢ e ∈ ℂ
59	58	a1i 11	. . . . 5 ⊢ (𝜑 → e ∈ ℂ)
60	56, 59, 18	subdid 11708	. . . 4 ⊢ (𝜑 → ((!‘𝑄) · (e − Σ𝑘 ∈ (0...𝑄)(𝐹‘𝑘))) = (((!‘𝑄) · e) − ((!‘𝑄) · Σ𝑘 ∈ (0...𝑄)(𝐹‘𝑘))))
61	53, 60	eqtr3d 2770	. . 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 7442	. . . . . 6 ⊢ (𝜑 → ((!‘𝑄) · e) = ((!‘𝑄) · (𝑃 / 𝑄)))
64		eirr.2	. . . . . . . 8 ⊢ (𝜑 → 𝑃 ∈ ℤ)
65	64	zcnd 12705	. . . . . . 7 ⊢ (𝜑 → 𝑃 ∈ ℂ)
66	21	nnne0d 12300	. . . . . . 7 ⊢ (𝜑 → 𝑄 ≠ 0)
67	56, 65, 45, 66	div12d 12064	. . . . . 6 ⊢ (𝜑 → ((!‘𝑄) · (𝑃 / 𝑄)) = (𝑃 · ((!‘𝑄) / 𝑄)))
68	63, 67	eqtrd 2768	. . . . 5 ⊢ (𝜑 → ((!‘𝑄) · e) = (𝑃 · ((!‘𝑄) / 𝑄)))
69	21	nnred 12265	. . . . . . . . 9 ⊢ (𝜑 → 𝑄 ∈ ℝ)
70	69	leidd 11818	. . . . . . . 8 ⊢ (𝜑 → 𝑄 ≤ 𝑄)
71		facdiv 14286	. . . . . . . 8 ⊢ ((𝑄 ∈ ℕ₀ ∧ 𝑄 ∈ ℕ ∧ 𝑄 ≤ 𝑄) → ((!‘𝑄) / 𝑄) ∈ ℕ)
72	54, 21, 70, 71	syl3anc 1368	. . . . . . 7 ⊢ (𝜑 → ((!‘𝑄) / 𝑄) ∈ ℕ)
73	72	nnzd 12623	. . . . . 6 ⊢ (𝜑 → ((!‘𝑄) / 𝑄) ∈ ℤ)
74	64, 73	zmulcld 12710	. . . . 5 ⊢ (𝜑 → (𝑃 · ((!‘𝑄) / 𝑄)) ∈ ℤ)
75	68, 74	eqeltrd 2829	. . . 4 ⊢ (𝜑 → ((!‘𝑄) · e) ∈ ℤ)
76	1, 56, 17	fsummulc2 15770	. . . . 5 ⊢ (𝜑 → ((!‘𝑄) · Σ𝑘 ∈ (0...𝑄)(𝐹‘𝑘)) = Σ𝑘 ∈ (0...𝑄)((!‘𝑄) · (𝐹‘𝑘)))
77	2	adantl 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑄)) → 𝑘 ∈ ℕ₀)
78	77, 28	syl 17	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑄)) → (𝐹‘𝑘) = (1 / (!‘𝑘)))
79	78	oveq2d 7442	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑄)) → ((!‘𝑄) · (𝐹‘𝑘)) = ((!‘𝑄) · (1 / (!‘𝑘))))
80	56	adantr 479	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑄)) → (!‘𝑄) ∈ ℂ)
81	2, 31	sylan2 591	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑄)) → (!‘𝑘) ∈ ℕ)
82	81	nncnd 12266	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑄)) → (!‘𝑘) ∈ ℂ)
83		facne0 14285	. . . . . . . . . . 11 ⊢ (𝑘 ∈ ℕ₀ → (!‘𝑘) ≠ 0)
84	77, 83	syl 17	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑄)) → (!‘𝑘) ≠ 0)
85	80, 82, 84	divrecd 12031	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑄)) → ((!‘𝑄) / (!‘𝑘)) = ((!‘𝑄) · (1 / (!‘𝑘))))
86	79, 85	eqtr4d 2771	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑄)) → ((!‘𝑄) · (𝐹‘𝑘)) = ((!‘𝑄) / (!‘𝑘)))
87		permnn 14325	. . . . . . . . 9 ⊢ (𝑘 ∈ (0...𝑄) → ((!‘𝑄) / (!‘𝑘)) ∈ ℕ)
88	87	adantl 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑄)) → ((!‘𝑄) / (!‘𝑘)) ∈ ℕ)
89	86, 88	eqeltrd 2829	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑄)) → ((!‘𝑄) · (𝐹‘𝑘)) ∈ ℕ)
90	89	nnzd 12623	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑄)) → ((!‘𝑄) · (𝐹‘𝑘)) ∈ ℤ)
91	1, 90	fsumzcl 15721	. . . . 5 ⊢ (𝜑 → Σ𝑘 ∈ (0...𝑄)((!‘𝑄) · (𝐹‘𝑘)) ∈ ℤ)
92	76, 91	eqeltrd 2829	. . . 4 ⊢ (𝜑 → ((!‘𝑄) · Σ𝑘 ∈ (0...𝑄)(𝐹‘𝑘)) ∈ ℤ)
93	75, 92	zsubcld 12709	. . 3 ⊢ (𝜑 → (((!‘𝑄) · e) − ((!‘𝑄) · Σ𝑘 ∈ (0...𝑄)(𝐹‘𝑘))) ∈ ℤ)
94	61, 93	eqeltrd 2829	. 2 ⊢ (𝜑 → ((!‘𝑄) · Σ𝑘 ∈ (ℤ_≥‘(𝑄 + 1))(𝐹‘𝑘)) ∈ ℤ)
95		0zd 12608	. . 3 ⊢ (𝜑 → 0 ∈ ℤ)
96	55	nnrpd 13054	. . . . 5 ⊢ (𝜑 → (!‘𝑄) ∈ ℝ⁺)
97	96, 37	rpmulcld 13072	. . . 4 ⊢ (𝜑 → ((!‘𝑄) · Σ𝑘 ∈ (ℤ_≥‘(𝑄 + 1))(𝐹‘𝑘)) ∈ ℝ⁺)
98	97	rpgt0d 13059	. . 3 ⊢ (𝜑 → 0 < ((!‘𝑄) · Σ𝑘 ∈ (ℤ_≥‘(𝑄 + 1))(𝐹‘𝑘)))
99	22	peano2nnd 12267	. . . . . . . 8 ⊢ (𝜑 → ((𝑄 + 1) + 1) ∈ ℕ)
100	99	nnred 12265	. . . . . . 7 ⊢ (𝜑 → ((𝑄 + 1) + 1) ∈ ℝ)
101	23	faccld 14283	. . . . . . . 8 ⊢ (𝜑 → (!‘(𝑄 + 1)) ∈ ℕ)
102	101, 22	nnmulcld 12303	. . . . . . 7 ⊢ (𝜑 → ((!‘(𝑄 + 1)) · (𝑄 + 1)) ∈ ℕ)
103	100, 102	nndivred 12304	. . . . . 6 ⊢ (𝜑 → (((𝑄 + 1) + 1) / ((!‘(𝑄 + 1)) · (𝑄 + 1))) ∈ ℝ)
104	55	nnrecred 12301	. . . . . 6 ⊢ (𝜑 → (1 / (!‘𝑄)) ∈ ℝ)
105		abs1 15284	. . . . . . . . . . . 12 ⊢ (abs‘1) = 1
106	105	oveq1i 7436	. . . . . . . . . . 11 ⊢ ((abs‘1)↑𝑛) = (1↑𝑛)
107	106	oveq1i 7436	. . . . . . . . . 10 ⊢ (((abs‘1)↑𝑛) / (!‘𝑛)) = ((1↑𝑛) / (!‘𝑛))
108	107	mpteq2i 5257	. . . . . . . . 9 ⊢ (𝑛 ∈ ℕ₀ ↦ (((abs‘1)↑𝑛) / (!‘𝑛))) = (𝑛 ∈ ℕ₀ ↦ ((1↑𝑛) / (!‘𝑛)))
109	9, 108	eqtr4i 2759	. . . . . . . 8 ⊢ 𝐹 = (𝑛 ∈ ℕ₀ ↦ (((abs‘1)↑𝑛) / (!‘𝑛)))
110		eqid 2728	. . . . . . . 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 11880	. . . . . . . . . 10 ⊢ 1 ≤ 1
112	105, 111	eqbrtri 5173	. . . . . . . . 9 ⊢ (abs‘1) ≤ 1
113	112	a1i 11	. . . . . . . 8 ⊢ (𝜑 → (abs‘1) ≤ 1)
114	9, 109, 110, 22, 13, 113	eftlub 16093	. . . . . . 7 ⊢ (𝜑 → (abs‘Σ𝑘 ∈ (ℤ_≥‘(𝑄 + 1))(𝐹‘𝑘)) ≤ (((abs‘1)↑(𝑄 + 1)) · (((𝑄 + 1) + 1) / ((!‘(𝑄 + 1)) · (𝑄 + 1)))))
115	37	rprege0d 13063	. . . . . . . 8 ⊢ (𝜑 → (Σ𝑘 ∈ (ℤ_≥‘(𝑄 + 1))(𝐹‘𝑘) ∈ ℝ ∧ 0 ≤ Σ𝑘 ∈ (ℤ_≥‘(𝑄 + 1))(𝐹‘𝑘)))
116		absid 15283	. . . . . . . 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 7436	. . . . . . . . . 10 ⊢ ((abs‘1)↑(𝑄 + 1)) = (1↑(𝑄 + 1))
119	22	nnzd 12623	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑄 + 1) ∈ ℤ)
120		1exp 14096	. . . . . . . . . . 11 ⊢ ((𝑄 + 1) ∈ ℤ → (1↑(𝑄 + 1)) = 1)
121	119, 120	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → (1↑(𝑄 + 1)) = 1)
122	118, 121	eqtrid 2780	. . . . . . . . 9 ⊢ (𝜑 → ((abs‘1)↑(𝑄 + 1)) = 1)
123	122	oveq1d 7441	. . . . . . . 8 ⊢ (𝜑 → (((abs‘1)↑(𝑄 + 1)) · (((𝑄 + 1) + 1) / ((!‘(𝑄 + 1)) · (𝑄 + 1)))) = (1 · (((𝑄 + 1) + 1) / ((!‘(𝑄 + 1)) · (𝑄 + 1)))))
124	103	recnd 11280	. . . . . . . . 9 ⊢ (𝜑 → (((𝑄 + 1) + 1) / ((!‘(𝑄 + 1)) · (𝑄 + 1))) ∈ ℂ)
125	124	mullidd 11270	. . . . . . . 8 ⊢ (𝜑 → (1 · (((𝑄 + 1) + 1) / ((!‘(𝑄 + 1)) · (𝑄 + 1)))) = (((𝑄 + 1) + 1) / ((!‘(𝑄 + 1)) · (𝑄 + 1))))
126	123, 125	eqtrd 2768	. . . . . . 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 5183	. . . . . 6 ⊢ (𝜑 → Σ𝑘 ∈ (ℤ_≥‘(𝑄 + 1))(𝐹‘𝑘) ≤ (((𝑄 + 1) + 1) / ((!‘(𝑄 + 1)) · (𝑄 + 1))))
128	22	nnred 12265	. . . . . . . . . 10 ⊢ (𝜑 → (𝑄 + 1) ∈ ℝ)
129	128, 128	readdcld 11281	. . . . . . . . 9 ⊢ (𝜑 → ((𝑄 + 1) + (𝑄 + 1)) ∈ ℝ)
130	128, 128	remulcld 11282	. . . . . . . . 9 ⊢ (𝜑 → ((𝑄 + 1) · (𝑄 + 1)) ∈ ℝ)
131		1red 11253	. . . . . . . . . 10 ⊢ (𝜑 → 1 ∈ ℝ)
132	21	nnge1d 12298	. . . . . . . . . . 11 ⊢ (𝜑 → 1 ≤ 𝑄)
133		1nn 12261	. . . . . . . . . . . 12 ⊢ 1 ∈ ℕ
134		nnleltp1 12655	. . . . . . . . . . . 12 ⊢ ((1 ∈ ℕ ∧ 𝑄 ∈ ℕ) → (1 ≤ 𝑄 ↔ 1 < (𝑄 + 1)))
135	133, 21, 134	sylancr 585	. . . . . . . . . . 11 ⊢ (𝜑 → (1 ≤ 𝑄 ↔ 1 < (𝑄 + 1)))
136	132, 135	mpbid 231	. . . . . . . . . 10 ⊢ (𝜑 → 1 < (𝑄 + 1))
137	131, 128, 128, 136	ltadd2dd 11411	. . . . . . . . 9 ⊢ (𝜑 → ((𝑄 + 1) + 1) < ((𝑄 + 1) + (𝑄 + 1)))
138	22	nncnd 12266	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑄 + 1) ∈ ℂ)
139	138	2timesd 12493	. . . . . . . . . 10 ⊢ (𝜑 → (2 · (𝑄 + 1)) = ((𝑄 + 1) + (𝑄 + 1)))
140		df-2 12313	. . . . . . . . . . . 12 ⊢ 2 = (1 + 1)
141	131, 69, 131, 132	leadd1dd 11866	. . . . . . . . . . . 12 ⊢ (𝜑 → (1 + 1) ≤ (𝑄 + 1))
142	140, 141	eqbrtrid 5187	. . . . . . . . . . 11 ⊢ (𝜑 → 2 ≤ (𝑄 + 1))
143		2re 12324	. . . . . . . . . . . . 13 ⊢ 2 ∈ ℝ
144	143	a1i 11	. . . . . . . . . . . 12 ⊢ (𝜑 → 2 ∈ ℝ)
145	22	nngt0d 12299	. . . . . . . . . . . 12 ⊢ (𝜑 → 0 < (𝑄 + 1))
146		lemul1 12104	. . . . . . . . . . . 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 1371	. . . . . . . . . . 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 5176	. . . . . . . . 9 ⊢ (𝜑 → ((𝑄 + 1) + (𝑄 + 1)) ≤ ((𝑄 + 1) · (𝑄 + 1)))
150	100, 129, 130, 137, 149	ltletrd 11412	. . . . . . . 8 ⊢ (𝜑 → ((𝑄 + 1) + 1) < ((𝑄 + 1) · (𝑄 + 1)))
151		facp1 14277	. . . . . . . . . . . . 13 ⊢ (𝑄 ∈ ℕ₀ → (!‘(𝑄 + 1)) = ((!‘𝑄) · (𝑄 + 1)))
152	54, 151	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → (!‘(𝑄 + 1)) = ((!‘𝑄) · (𝑄 + 1)))
153	152	oveq1d 7441	. . . . . . . . . . 11 ⊢ (𝜑 → ((!‘(𝑄 + 1)) / (!‘𝑄)) = (((!‘𝑄) · (𝑄 + 1)) / (!‘𝑄)))
154	101	nncnd 12266	. . . . . . . . . . . 12 ⊢ (𝜑 → (!‘(𝑄 + 1)) ∈ ℂ)
155	55	nnne0d 12300	. . . . . . . . . . . 12 ⊢ (𝜑 → (!‘𝑄) ≠ 0)
156	154, 56, 155	divrecd 12031	. . . . . . . . . . 11 ⊢ (𝜑 → ((!‘(𝑄 + 1)) / (!‘𝑄)) = ((!‘(𝑄 + 1)) · (1 / (!‘𝑄))))
157	138, 56, 155	divcan3d 12033	. . . . . . . . . . 11 ⊢ (𝜑 → (((!‘𝑄) · (𝑄 + 1)) / (!‘𝑄)) = (𝑄 + 1))
158	153, 156, 157	3eqtr3rd 2777	. . . . . . . . . 10 ⊢ (𝜑 → (𝑄 + 1) = ((!‘(𝑄 + 1)) · (1 / (!‘𝑄))))
159	158	oveq1d 7441	. . . . . . . . 9 ⊢ (𝜑 → ((𝑄 + 1) · (𝑄 + 1)) = (((!‘(𝑄 + 1)) · (1 / (!‘𝑄))) · (𝑄 + 1)))
160	104	recnd 11280	. . . . . . . . . 10 ⊢ (𝜑 → (1 / (!‘𝑄)) ∈ ℂ)
161	154, 160, 138	mul32d 11462	. . . . . . . . 9 ⊢ (𝜑 → (((!‘(𝑄 + 1)) · (1 / (!‘𝑄))) · (𝑄 + 1)) = (((!‘(𝑄 + 1)) · (𝑄 + 1)) · (1 / (!‘𝑄))))
162	159, 161	eqtrd 2768	. . . . . . . 8 ⊢ (𝜑 → ((𝑄 + 1) · (𝑄 + 1)) = (((!‘(𝑄 + 1)) · (𝑄 + 1)) · (1 / (!‘𝑄))))
163	150, 162	breqtrd 5178	. . . . . . 7 ⊢ (𝜑 → ((𝑄 + 1) + 1) < (((!‘(𝑄 + 1)) · (𝑄 + 1)) · (1 / (!‘𝑄))))
164	102	nnred 12265	. . . . . . . 8 ⊢ (𝜑 → ((!‘(𝑄 + 1)) · (𝑄 + 1)) ∈ ℝ)
165	102	nngt0d 12299	. . . . . . . 8 ⊢ (𝜑 → 0 < ((!‘(𝑄 + 1)) · (𝑄 + 1)))
166		ltdivmul 12127	. . . . . . . 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 1371	. . . . . . 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 11410	. . . . 5 ⊢ (𝜑 → Σ𝑘 ∈ (ℤ_≥‘(𝑄 + 1))(𝐹‘𝑘) < (1 / (!‘𝑄)))
170	38, 131, 96	ltmuldiv2d 13104	. . . . 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 12372	. . . 4 ⊢ (0 + 1) = 1
173	171, 172	breqtrrdi 5194	. . 3 ⊢ (𝜑 → ((!‘𝑄) · Σ𝑘 ∈ (ℤ_≥‘(𝑄 + 1))(𝐹‘𝑘)) < (0 + 1))
174		btwnnz 12676	. . 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 1368	. 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 394 = wceq 1533 ∈ wcel 2098 ≠ wne 2937 class class class wbr 5152 ↦ cmpt 5235 dom cdm 5682 ‘cfv 6553 (class class class)co 7426 ℂcc 11144 ℝcr 11145 0cc0 11146 1c1 11147 + caddc 11149 · cmul 11151 < clt 11286 ≤ cle 11287 − cmin 11482 / cdiv 11909 ℕcn 12250 2c2 12305 ℕ₀cn0 12510 ℤcz 12596 ℤ_≥cuz 12860 ℝ⁺crp 13014 ...cfz 13524 seqcseq 14006 ↑cexp 14066 !cfa 14272 abscabs 15221 ⇝ cli 15468 Σcsu 15672 eceu 16046

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 2166 ax-ext 2699 ax-rep 5289 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-inf2 9672 ax-cnex 11202 ax-resscn 11203 ax-1cn 11204 ax-icn 11205 ax-addcl 11206 ax-addrcl 11207 ax-mulcl 11208 ax-mulrcl 11209 ax-mulcom 11210 ax-addass 11211 ax-mulass 11212 ax-distr 11213 ax-i2m1 11214 ax-1ne0 11215 ax-1rid 11216 ax-rnegex 11217 ax-rrecex 11218 ax-cnre 11219 ax-pre-lttri 11220 ax-pre-lttrn 11221 ax-pre-ltadd 11222 ax-pre-mulgt0 11223 ax-pre-sup 11224

This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3374 df-reu 3375 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-int 4954 df-iun 5002 df-br 5153 df-opab 5215 df-mpt 5236 df-tr 5270 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-se 5638 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6310 df-ord 6377 df-on 6378 df-lim 6379 df-suc 6380 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-isom 6562 df-riota 7382 df-ov 7429 df-oprab 7430 df-mpo 7431 df-om 7877 df-1st 7999 df-2nd 8000 df-frecs 8293 df-wrecs 8324 df-recs 8398 df-rdg 8437 df-1o 8493 df-er 8731 df-pm 8854 df-en 8971 df-dom 8972 df-sdom 8973 df-fin 8974 df-sup 9473 df-inf 9474 df-oi 9541 df-card 9970 df-pnf 11288 df-mnf 11289 df-xr 11290 df-ltxr 11291 df-le 11292 df-sub 11484 df-neg 11485 df-div 11910 df-nn 12251 df-2 12313 df-3 12314 df-n0 12511 df-z 12597 df-uz 12861 df-rp 13015 df-ico 13370 df-fz 13525 df-fzo 13668 df-fl 13797 df-seq 14007 df-exp 14067 df-fac 14273 df-bc 14302 df-hash 14330 df-shft 15054 df-cj 15086 df-re 15087 df-im 15088 df-sqrt 15222 df-abs 15223 df-limsup 15455 df-clim 15472 df-rlim 15473 df-sum 15673 df-ef 16051 df-e 16052

This theorem is referenced by: eirr 16189

W3C validator