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Theorem abelthlem8 25942

Description: Lemma for abelth 25944. (Contributed by Mario Carneiro, 2-Apr-2015.)

**Hypotheses**
Ref	Expression
abelth.1	⊢ (𝜑 → 𝐴:ℕ₀⟶ℂ)
abelth.2	⊢ (𝜑 → seq0( + , 𝐴) ∈ dom ⇝ )
abelth.3	⊢ (𝜑 → 𝑀 ∈ ℝ)
abelth.4	⊢ (𝜑 → 0 ≤ 𝑀)
abelth.5	⊢ 𝑆 = {𝑧 ∈ ℂ ∣ (abs‘(1 − 𝑧)) ≤ (𝑀 · (1 − (abs‘𝑧)))}
abelth.6	⊢ 𝐹 = (𝑥 ∈ 𝑆 ↦ Σ𝑛 ∈ ℕ₀ ((𝐴‘𝑛) · (𝑥↑𝑛)))
abelth.7	⊢ (𝜑 → seq0( + , 𝐴) ⇝ 0)

**Assertion**
Ref	Expression
abelthlem8	⊢ ((𝜑 ∧ 𝑅 ∈ ℝ⁺) → ∃𝑤 ∈ ℝ⁺ ∀𝑦 ∈ 𝑆 ((abs‘(1 − 𝑦)) < 𝑤 → (abs‘((𝐹‘1) − (𝐹‘𝑦))) < 𝑅))

Distinct variable groups: 𝑤,𝑛,𝑥,𝑦,𝑧,𝑀 𝑅,𝑛,𝑤,𝑥,𝑦,𝑧 𝐴,𝑛,𝑤,𝑥,𝑦,𝑧 𝜑,𝑛,𝑤,𝑥,𝑦 𝑤,𝐹,𝑦 𝑆,𝑛,𝑤,𝑥,𝑦 Allowed substitution hints: 𝜑(𝑧) 𝑆(𝑧) 𝐹(𝑥,𝑧,𝑛)

Proof of Theorem abelthlem8 Dummy variables 𝑖 𝑗 𝑘 𝑚 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		nn0uz 12860	. . 3 ⊢ ℕ₀ = (ℤ_≥‘0)
2		0zd 12566	. . 3 ⊢ ((𝜑 ∧ 𝑅 ∈ ℝ⁺) → 0 ∈ ℤ)
3		id 22	. . . 4 ⊢ (𝑅 ∈ ℝ⁺ → 𝑅 ∈ ℝ⁺)
4		abelth.3	. . . . 5 ⊢ (𝜑 → 𝑀 ∈ ℝ)
5		abelth.4	. . . . 5 ⊢ (𝜑 → 0 ≤ 𝑀)
6	4, 5	ge0p1rpd 13042	. . . 4 ⊢ (𝜑 → (𝑀 + 1) ∈ ℝ⁺)
7		rpdivcl 12995	. . . 4 ⊢ ((𝑅 ∈ ℝ⁺ ∧ (𝑀 + 1) ∈ ℝ⁺) → (𝑅 / (𝑀 + 1)) ∈ ℝ⁺)
8	3, 6, 7	syl2anr 597	. . 3 ⊢ ((𝜑 ∧ 𝑅 ∈ ℝ⁺) → (𝑅 / (𝑀 + 1)) ∈ ℝ⁺)
9		eqidd 2733	. . 3 ⊢ (((𝜑 ∧ 𝑅 ∈ ℝ⁺) ∧ 𝑘 ∈ ℕ₀) → (seq0( + , 𝐴)‘𝑘) = (seq0( + , 𝐴)‘𝑘))
10		abelth.7	. . . 4 ⊢ (𝜑 → seq0( + , 𝐴) ⇝ 0)
11	10	adantr 481	. . 3 ⊢ ((𝜑 ∧ 𝑅 ∈ ℝ⁺) → seq0( + , 𝐴) ⇝ 0)
12	1, 2, 8, 9, 11	climi0 15452	. 2 ⊢ ((𝜑 ∧ 𝑅 ∈ ℝ⁺) → ∃𝑗 ∈ ℕ₀ ∀𝑘 ∈ (ℤ_≥‘𝑗)(abs‘(seq0( + , 𝐴)‘𝑘)) < (𝑅 / (𝑀 + 1)))
13	8	adantr 481	. . . 4 ⊢ (((𝜑 ∧ 𝑅 ∈ ℝ⁺) ∧ (𝑗 ∈ ℕ₀ ∧ ∀𝑘 ∈ (ℤ_≥‘𝑗)(abs‘(seq0( + , 𝐴)‘𝑘)) < (𝑅 / (𝑀 + 1)))) → (𝑅 / (𝑀 + 1)) ∈ ℝ⁺)
14		fzfid 13934	. . . . . . 7 ⊢ (𝜑 → (0...(𝑗 − 1)) ∈ Fin)
15		0zd 12566	. . . . . . . . . 10 ⊢ (𝜑 → 0 ∈ ℤ)
16		abelth.1	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐴:ℕ₀⟶ℂ)
17	16	ffvelcdmda 7083	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑤 ∈ ℕ₀) → (𝐴‘𝑤) ∈ ℂ)
18	1, 15, 17	serf 13992	. . . . . . . . 9 ⊢ (𝜑 → seq0( + , 𝐴):ℕ₀⟶ℂ)
19		elfznn0 13590	. . . . . . . . 9 ⊢ (𝑖 ∈ (0...(𝑗 − 1)) → 𝑖 ∈ ℕ₀)
20		ffvelcdm 7080	. . . . . . . . 9 ⊢ ((seq0( + , 𝐴):ℕ₀⟶ℂ ∧ 𝑖 ∈ ℕ₀) → (seq0( + , 𝐴)‘𝑖) ∈ ℂ)
21	18, 19, 20	syl2an 596	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...(𝑗 − 1))) → (seq0( + , 𝐴)‘𝑖) ∈ ℂ)
22	21	abscld 15379	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...(𝑗 − 1))) → (abs‘(seq0( + , 𝐴)‘𝑖)) ∈ ℝ)
23	14, 22	fsumrecl 15676	. . . . . 6 ⊢ (𝜑 → Σ𝑖 ∈ (0...(𝑗 − 1))(abs‘(seq0( + , 𝐴)‘𝑖)) ∈ ℝ)
24	23	ad2antrr 724	. . . . 5 ⊢ (((𝜑 ∧ 𝑅 ∈ ℝ⁺) ∧ (𝑗 ∈ ℕ₀ ∧ ∀𝑘 ∈ (ℤ_≥‘𝑗)(abs‘(seq0( + , 𝐴)‘𝑘)) < (𝑅 / (𝑀 + 1)))) → Σ𝑖 ∈ (0...(𝑗 − 1))(abs‘(seq0( + , 𝐴)‘𝑖)) ∈ ℝ)
25	21	absge0d 15387	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...(𝑗 − 1))) → 0 ≤ (abs‘(seq0( + , 𝐴)‘𝑖)))
26	14, 22, 25	fsumge0 15737	. . . . . 6 ⊢ (𝜑 → 0 ≤ Σ𝑖 ∈ (0...(𝑗 − 1))(abs‘(seq0( + , 𝐴)‘𝑖)))
27	26	ad2antrr 724	. . . . 5 ⊢ (((𝜑 ∧ 𝑅 ∈ ℝ⁺) ∧ (𝑗 ∈ ℕ₀ ∧ ∀𝑘 ∈ (ℤ_≥‘𝑗)(abs‘(seq0( + , 𝐴)‘𝑘)) < (𝑅 / (𝑀 + 1)))) → 0 ≤ Σ𝑖 ∈ (0...(𝑗 − 1))(abs‘(seq0( + , 𝐴)‘𝑖)))
28	24, 27	ge0p1rpd 13042	. . . 4 ⊢ (((𝜑 ∧ 𝑅 ∈ ℝ⁺) ∧ (𝑗 ∈ ℕ₀ ∧ ∀𝑘 ∈ (ℤ_≥‘𝑗)(abs‘(seq0( + , 𝐴)‘𝑘)) < (𝑅 / (𝑀 + 1)))) → (Σ𝑖 ∈ (0...(𝑗 − 1))(abs‘(seq0( + , 𝐴)‘𝑖)) + 1) ∈ ℝ⁺)
29	13, 28	rpdivcld 13029	. . 3 ⊢ (((𝜑 ∧ 𝑅 ∈ ℝ⁺) ∧ (𝑗 ∈ ℕ₀ ∧ ∀𝑘 ∈ (ℤ_≥‘𝑗)(abs‘(seq0( + , 𝐴)‘𝑘)) < (𝑅 / (𝑀 + 1)))) → ((𝑅 / (𝑀 + 1)) / (Σ𝑖 ∈ (0...(𝑗 − 1))(abs‘(seq0( + , 𝐴)‘𝑖)) + 1)) ∈ ℝ⁺)
30		abelth.2	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → seq0( + , 𝐴) ∈ dom ⇝ )
31		abelth.5	. . . . . . . . . . . . . . . . 17 ⊢ 𝑆 = {𝑧 ∈ ℂ ∣ (abs‘(1 − 𝑧)) ≤ (𝑀 · (1 − (abs‘𝑧)))}
32	16, 30, 4, 5, 31	abelthlem2 25935	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (1 ∈ 𝑆 ∧ (𝑆 ∖ {1}) ⊆ (0(ball‘(abs ∘ − ))1)))
33	32	simpld 495	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 1 ∈ 𝑆)
34		oveq1 7412	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 = 1 → (𝑥↑𝑛) = (1↑𝑛))
35		nn0z 12579	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑛 ∈ ℕ₀ → 𝑛 ∈ ℤ)
36		1exp 14053	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑛 ∈ ℤ → (1↑𝑛) = 1)
37	35, 36	syl 17	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑛 ∈ ℕ₀ → (1↑𝑛) = 1)
38	34, 37	sylan9eq 2792	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑥 = 1 ∧ 𝑛 ∈ ℕ₀) → (𝑥↑𝑛) = 1)
39	38	oveq2d 7421	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 = 1 ∧ 𝑛 ∈ ℕ₀) → ((𝐴‘𝑛) · (𝑥↑𝑛)) = ((𝐴‘𝑛) · 1))
40	39	sumeq2dv 15645	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 1 → Σ𝑛 ∈ ℕ₀ ((𝐴‘𝑛) · (𝑥↑𝑛)) = Σ𝑛 ∈ ℕ₀ ((𝐴‘𝑛) · 1))
41		abelth.6	. . . . . . . . . . . . . . . 16 ⊢ 𝐹 = (𝑥 ∈ 𝑆 ↦ Σ𝑛 ∈ ℕ₀ ((𝐴‘𝑛) · (𝑥↑𝑛)))
42		sumex 15630	. . . . . . . . . . . . . . . 16 ⊢ Σ𝑛 ∈ ℕ₀ ((𝐴‘𝑛) · 1) ∈ V
43	40, 41, 42	fvmpt 6995	. . . . . . . . . . . . . . 15 ⊢ (1 ∈ 𝑆 → (𝐹‘1) = Σ𝑛 ∈ ℕ₀ ((𝐴‘𝑛) · 1))
44	33, 43	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝐹‘1) = Σ𝑛 ∈ ℕ₀ ((𝐴‘𝑛) · 1))
45	16	ffvelcdmda 7083	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ₀) → (𝐴‘𝑛) ∈ ℂ)
46	45	mulridd 11227	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ₀) → ((𝐴‘𝑛) · 1) = (𝐴‘𝑛))
47	46	eqcomd 2738	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ₀) → (𝐴‘𝑛) = ((𝐴‘𝑛) · 1))
48	46, 45	eqeltrd 2833	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ₀) → ((𝐴‘𝑛) · 1) ∈ ℂ)
49	1, 15, 47, 48, 10	isumclim 15699	. . . . . . . . . . . . . 14 ⊢ (𝜑 → Σ𝑛 ∈ ℕ₀ ((𝐴‘𝑛) · 1) = 0)
50	44, 49	eqtrd 2772	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐹‘1) = 0)
51	50	adantr 481	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑆) → (𝐹‘1) = 0)
52	51	oveq1d 7420	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑆) → ((𝐹‘1) − (𝐹‘𝑦)) = (0 − (𝐹‘𝑦)))
53		df-neg 11443	. . . . . . . . . . 11 ⊢ -(𝐹‘𝑦) = (0 − (𝐹‘𝑦))
54	52, 53	eqtr4di 2790	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑆) → ((𝐹‘1) − (𝐹‘𝑦)) = -(𝐹‘𝑦))
55	54	fveq2d 6892	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑆) → (abs‘((𝐹‘1) − (𝐹‘𝑦))) = (abs‘-(𝐹‘𝑦)))
56	16, 30, 4, 5, 31, 41	abelthlem4 25937	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐹:𝑆⟶ℂ)
57	56	ffvelcdmda 7083	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑆) → (𝐹‘𝑦) ∈ ℂ)
58	57	absnegd 15392	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑆) → (abs‘-(𝐹‘𝑦)) = (abs‘(𝐹‘𝑦)))
59	55, 58	eqtrd 2772	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑆) → (abs‘((𝐹‘1) − (𝐹‘𝑦))) = (abs‘(𝐹‘𝑦)))
60	59	adantlr 713	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑅 ∈ ℝ⁺) ∧ 𝑦 ∈ 𝑆) → (abs‘((𝐹‘1) − (𝐹‘𝑦))) = (abs‘(𝐹‘𝑦)))
61	60	ad2ant2r 745	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑅 ∈ ℝ⁺) ∧ (𝑗 ∈ ℕ₀ ∧ ∀𝑘 ∈ (ℤ_≥‘𝑗)(abs‘(seq0( + , 𝐴)‘𝑘)) < (𝑅 / (𝑀 + 1)))) ∧ (𝑦 ∈ 𝑆 ∧ (abs‘(1 − 𝑦)) < ((𝑅 / (𝑀 + 1)) / (Σ𝑖 ∈ (0...(𝑗 − 1))(abs‘(seq0( + , 𝐴)‘𝑖)) + 1)))) → (abs‘((𝐹‘1) − (𝐹‘𝑦))) = (abs‘(𝐹‘𝑦)))
62		fveq2 6888	. . . . . . . . . . 11 ⊢ (𝑦 = 1 → (𝐹‘𝑦) = (𝐹‘1))
63	62, 50	sylan9eqr 2794	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 = 1) → (𝐹‘𝑦) = 0)
64	63	abs00bd 15234	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 = 1) → (abs‘(𝐹‘𝑦)) = 0)
65	64	ad5ant15 757	. . . . . . . 8 ⊢ (((((𝜑 ∧ 𝑅 ∈ ℝ⁺) ∧ (𝑗 ∈ ℕ₀ ∧ ∀𝑘 ∈ (ℤ_≥‘𝑗)(abs‘(seq0( + , 𝐴)‘𝑘)) < (𝑅 / (𝑀 + 1)))) ∧ (𝑦 ∈ 𝑆 ∧ (abs‘(1 − 𝑦)) < ((𝑅 / (𝑀 + 1)) / (Σ𝑖 ∈ (0...(𝑗 − 1))(abs‘(seq0( + , 𝐴)‘𝑖)) + 1)))) ∧ 𝑦 = 1) → (abs‘(𝐹‘𝑦)) = 0)
66		simpllr 774	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑅 ∈ ℝ⁺) ∧ (𝑗 ∈ ℕ₀ ∧ ∀𝑘 ∈ (ℤ_≥‘𝑗)(abs‘(seq0( + , 𝐴)‘𝑘)) < (𝑅 / (𝑀 + 1)))) ∧ (𝑦 ∈ 𝑆 ∧ (abs‘(1 − 𝑦)) < ((𝑅 / (𝑀 + 1)) / (Σ𝑖 ∈ (0...(𝑗 − 1))(abs‘(seq0( + , 𝐴)‘𝑖)) + 1)))) → 𝑅 ∈ ℝ⁺)
67	66	rpgt0d 13015	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑅 ∈ ℝ⁺) ∧ (𝑗 ∈ ℕ₀ ∧ ∀𝑘 ∈ (ℤ_≥‘𝑗)(abs‘(seq0( + , 𝐴)‘𝑘)) < (𝑅 / (𝑀 + 1)))) ∧ (𝑦 ∈ 𝑆 ∧ (abs‘(1 − 𝑦)) < ((𝑅 / (𝑀 + 1)) / (Σ𝑖 ∈ (0...(𝑗 − 1))(abs‘(seq0( + , 𝐴)‘𝑖)) + 1)))) → 0 < 𝑅)
68	67	adantr 481	. . . . . . . 8 ⊢ (((((𝜑 ∧ 𝑅 ∈ ℝ⁺) ∧ (𝑗 ∈ ℕ₀ ∧ ∀𝑘 ∈ (ℤ_≥‘𝑗)(abs‘(seq0( + , 𝐴)‘𝑘)) < (𝑅 / (𝑀 + 1)))) ∧ (𝑦 ∈ 𝑆 ∧ (abs‘(1 − 𝑦)) < ((𝑅 / (𝑀 + 1)) / (Σ𝑖 ∈ (0...(𝑗 − 1))(abs‘(seq0( + , 𝐴)‘𝑖)) + 1)))) ∧ 𝑦 = 1) → 0 < 𝑅)
69	65, 68	eqbrtrd 5169	. . . . . . 7 ⊢ (((((𝜑 ∧ 𝑅 ∈ ℝ⁺) ∧ (𝑗 ∈ ℕ₀ ∧ ∀𝑘 ∈ (ℤ_≥‘𝑗)(abs‘(seq0( + , 𝐴)‘𝑘)) < (𝑅 / (𝑀 + 1)))) ∧ (𝑦 ∈ 𝑆 ∧ (abs‘(1 − 𝑦)) < ((𝑅 / (𝑀 + 1)) / (Σ𝑖 ∈ (0...(𝑗 − 1))(abs‘(seq0( + , 𝐴)‘𝑖)) + 1)))) ∧ 𝑦 = 1) → (abs‘(𝐹‘𝑦)) < 𝑅)
70	16	ad3antrrr 728	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑅 ∈ ℝ⁺) ∧ (𝑗 ∈ ℕ₀ ∧ ∀𝑘 ∈ (ℤ_≥‘𝑗)(abs‘(seq0( + , 𝐴)‘𝑘)) < (𝑅 / (𝑀 + 1)))) ∧ ((𝑦 ∈ 𝑆 ∧ (abs‘(1 − 𝑦)) < ((𝑅 / (𝑀 + 1)) / (Σ𝑖 ∈ (0...(𝑗 − 1))(abs‘(seq0( + , 𝐴)‘𝑖)) + 1))) ∧ 𝑦 ≠ 1)) → 𝐴:ℕ₀⟶ℂ)
71	30	ad3antrrr 728	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑅 ∈ ℝ⁺) ∧ (𝑗 ∈ ℕ₀ ∧ ∀𝑘 ∈ (ℤ_≥‘𝑗)(abs‘(seq0( + , 𝐴)‘𝑘)) < (𝑅 / (𝑀 + 1)))) ∧ ((𝑦 ∈ 𝑆 ∧ (abs‘(1 − 𝑦)) < ((𝑅 / (𝑀 + 1)) / (Σ𝑖 ∈ (0...(𝑗 − 1))(abs‘(seq0( + , 𝐴)‘𝑖)) + 1))) ∧ 𝑦 ≠ 1)) → seq0( + , 𝐴) ∈ dom ⇝ )
72	4	ad3antrrr 728	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑅 ∈ ℝ⁺) ∧ (𝑗 ∈ ℕ₀ ∧ ∀𝑘 ∈ (ℤ_≥‘𝑗)(abs‘(seq0( + , 𝐴)‘𝑘)) < (𝑅 / (𝑀 + 1)))) ∧ ((𝑦 ∈ 𝑆 ∧ (abs‘(1 − 𝑦)) < ((𝑅 / (𝑀 + 1)) / (Σ𝑖 ∈ (0...(𝑗 − 1))(abs‘(seq0( + , 𝐴)‘𝑖)) + 1))) ∧ 𝑦 ≠ 1)) → 𝑀 ∈ ℝ)
73	5	ad3antrrr 728	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑅 ∈ ℝ⁺) ∧ (𝑗 ∈ ℕ₀ ∧ ∀𝑘 ∈ (ℤ_≥‘𝑗)(abs‘(seq0( + , 𝐴)‘𝑘)) < (𝑅 / (𝑀 + 1)))) ∧ ((𝑦 ∈ 𝑆 ∧ (abs‘(1 − 𝑦)) < ((𝑅 / (𝑀 + 1)) / (Σ𝑖 ∈ (0...(𝑗 − 1))(abs‘(seq0( + , 𝐴)‘𝑖)) + 1))) ∧ 𝑦 ≠ 1)) → 0 ≤ 𝑀)
74	10	ad3antrrr 728	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑅 ∈ ℝ⁺) ∧ (𝑗 ∈ ℕ₀ ∧ ∀𝑘 ∈ (ℤ_≥‘𝑗)(abs‘(seq0( + , 𝐴)‘𝑘)) < (𝑅 / (𝑀 + 1)))) ∧ ((𝑦 ∈ 𝑆 ∧ (abs‘(1 − 𝑦)) < ((𝑅 / (𝑀 + 1)) / (Σ𝑖 ∈ (0...(𝑗 − 1))(abs‘(seq0( + , 𝐴)‘𝑖)) + 1))) ∧ 𝑦 ≠ 1)) → seq0( + , 𝐴) ⇝ 0)
75		simprll 777	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑅 ∈ ℝ⁺) ∧ (𝑗 ∈ ℕ₀ ∧ ∀𝑘 ∈ (ℤ_≥‘𝑗)(abs‘(seq0( + , 𝐴)‘𝑘)) < (𝑅 / (𝑀 + 1)))) ∧ ((𝑦 ∈ 𝑆 ∧ (abs‘(1 − 𝑦)) < ((𝑅 / (𝑀 + 1)) / (Σ𝑖 ∈ (0...(𝑗 − 1))(abs‘(seq0( + , 𝐴)‘𝑖)) + 1))) ∧ 𝑦 ≠ 1)) → 𝑦 ∈ 𝑆)
76		simprr 771	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑅 ∈ ℝ⁺) ∧ (𝑗 ∈ ℕ₀ ∧ ∀𝑘 ∈ (ℤ_≥‘𝑗)(abs‘(seq0( + , 𝐴)‘𝑘)) < (𝑅 / (𝑀 + 1)))) ∧ ((𝑦 ∈ 𝑆 ∧ (abs‘(1 − 𝑦)) < ((𝑅 / (𝑀 + 1)) / (Σ𝑖 ∈ (0...(𝑗 − 1))(abs‘(seq0( + , 𝐴)‘𝑖)) + 1))) ∧ 𝑦 ≠ 1)) → 𝑦 ≠ 1)
77		eldifsn 4789	. . . . . . . . . . 11 ⊢ (𝑦 ∈ (𝑆 ∖ {1}) ↔ (𝑦 ∈ 𝑆 ∧ 𝑦 ≠ 1))
78	75, 76, 77	sylanbrc 583	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑅 ∈ ℝ⁺) ∧ (𝑗 ∈ ℕ₀ ∧ ∀𝑘 ∈ (ℤ_≥‘𝑗)(abs‘(seq0( + , 𝐴)‘𝑘)) < (𝑅 / (𝑀 + 1)))) ∧ ((𝑦 ∈ 𝑆 ∧ (abs‘(1 − 𝑦)) < ((𝑅 / (𝑀 + 1)) / (Σ𝑖 ∈ (0...(𝑗 − 1))(abs‘(seq0( + , 𝐴)‘𝑖)) + 1))) ∧ 𝑦 ≠ 1)) → 𝑦 ∈ (𝑆 ∖ {1}))
79	8	ad2antrr 724	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑅 ∈ ℝ⁺) ∧ (𝑗 ∈ ℕ₀ ∧ ∀𝑘 ∈ (ℤ_≥‘𝑗)(abs‘(seq0( + , 𝐴)‘𝑘)) < (𝑅 / (𝑀 + 1)))) ∧ ((𝑦 ∈ 𝑆 ∧ (abs‘(1 − 𝑦)) < ((𝑅 / (𝑀 + 1)) / (Σ𝑖 ∈ (0...(𝑗 − 1))(abs‘(seq0( + , 𝐴)‘𝑖)) + 1))) ∧ 𝑦 ≠ 1)) → (𝑅 / (𝑀 + 1)) ∈ ℝ⁺)
80		simplrl 775	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑅 ∈ ℝ⁺) ∧ (𝑗 ∈ ℕ₀ ∧ ∀𝑘 ∈ (ℤ_≥‘𝑗)(abs‘(seq0( + , 𝐴)‘𝑘)) < (𝑅 / (𝑀 + 1)))) ∧ ((𝑦 ∈ 𝑆 ∧ (abs‘(1 − 𝑦)) < ((𝑅 / (𝑀 + 1)) / (Σ𝑖 ∈ (0...(𝑗 − 1))(abs‘(seq0( + , 𝐴)‘𝑖)) + 1))) ∧ 𝑦 ≠ 1)) → 𝑗 ∈ ℕ₀)
81		simplrr 776	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑅 ∈ ℝ⁺) ∧ (𝑗 ∈ ℕ₀ ∧ ∀𝑘 ∈ (ℤ_≥‘𝑗)(abs‘(seq0( + , 𝐴)‘𝑘)) < (𝑅 / (𝑀 + 1)))) ∧ ((𝑦 ∈ 𝑆 ∧ (abs‘(1 − 𝑦)) < ((𝑅 / (𝑀 + 1)) / (Σ𝑖 ∈ (0...(𝑗 − 1))(abs‘(seq0( + , 𝐴)‘𝑖)) + 1))) ∧ 𝑦 ≠ 1)) → ∀𝑘 ∈ (ℤ_≥‘𝑗)(abs‘(seq0( + , 𝐴)‘𝑘)) < (𝑅 / (𝑀 + 1)))
82		2fveq3 6893	. . . . . . . . . . . . 13 ⊢ (𝑘 = 𝑚 → (abs‘(seq0( + , 𝐴)‘𝑘)) = (abs‘(seq0( + , 𝐴)‘𝑚)))
83	82	breq1d 5157	. . . . . . . . . . . 12 ⊢ (𝑘 = 𝑚 → ((abs‘(seq0( + , 𝐴)‘𝑘)) < (𝑅 / (𝑀 + 1)) ↔ (abs‘(seq0( + , 𝐴)‘𝑚)) < (𝑅 / (𝑀 + 1))))
84	83	cbvralvw 3234	. . . . . . . . . . 11 ⊢ (∀𝑘 ∈ (ℤ_≥‘𝑗)(abs‘(seq0( + , 𝐴)‘𝑘)) < (𝑅 / (𝑀 + 1)) ↔ ∀𝑚 ∈ (ℤ_≥‘𝑗)(abs‘(seq0( + , 𝐴)‘𝑚)) < (𝑅 / (𝑀 + 1)))
85	81, 84	sylib 217	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑅 ∈ ℝ⁺) ∧ (𝑗 ∈ ℕ₀ ∧ ∀𝑘 ∈ (ℤ_≥‘𝑗)(abs‘(seq0( + , 𝐴)‘𝑘)) < (𝑅 / (𝑀 + 1)))) ∧ ((𝑦 ∈ 𝑆 ∧ (abs‘(1 − 𝑦)) < ((𝑅 / (𝑀 + 1)) / (Σ𝑖 ∈ (0...(𝑗 − 1))(abs‘(seq0( + , 𝐴)‘𝑖)) + 1))) ∧ 𝑦 ≠ 1)) → ∀𝑚 ∈ (ℤ_≥‘𝑗)(abs‘(seq0( + , 𝐴)‘𝑚)) < (𝑅 / (𝑀 + 1)))
86		simprlr 778	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑅 ∈ ℝ⁺) ∧ (𝑗 ∈ ℕ₀ ∧ ∀𝑘 ∈ (ℤ_≥‘𝑗)(abs‘(seq0( + , 𝐴)‘𝑘)) < (𝑅 / (𝑀 + 1)))) ∧ ((𝑦 ∈ 𝑆 ∧ (abs‘(1 − 𝑦)) < ((𝑅 / (𝑀 + 1)) / (Σ𝑖 ∈ (0...(𝑗 − 1))(abs‘(seq0( + , 𝐴)‘𝑖)) + 1))) ∧ 𝑦 ≠ 1)) → (abs‘(1 − 𝑦)) < ((𝑅 / (𝑀 + 1)) / (Σ𝑖 ∈ (0...(𝑗 − 1))(abs‘(seq0( + , 𝐴)‘𝑖)) + 1)))
87		2fveq3 6893	. . . . . . . . . . . . . 14 ⊢ (𝑖 = 𝑛 → (abs‘(seq0( + , 𝐴)‘𝑖)) = (abs‘(seq0( + , 𝐴)‘𝑛)))
88	87	cbvsumv 15638	. . . . . . . . . . . . 13 ⊢ Σ𝑖 ∈ (0...(𝑗 − 1))(abs‘(seq0( + , 𝐴)‘𝑖)) = Σ𝑛 ∈ (0...(𝑗 − 1))(abs‘(seq0( + , 𝐴)‘𝑛))
89	88	oveq1i 7415	. . . . . . . . . . . 12 ⊢ (Σ𝑖 ∈ (0...(𝑗 − 1))(abs‘(seq0( + , 𝐴)‘𝑖)) + 1) = (Σ𝑛 ∈ (0...(𝑗 − 1))(abs‘(seq0( + , 𝐴)‘𝑛)) + 1)
90	89	oveq2i 7416	. . . . . . . . . . 11 ⊢ ((𝑅 / (𝑀 + 1)) / (Σ𝑖 ∈ (0...(𝑗 − 1))(abs‘(seq0( + , 𝐴)‘𝑖)) + 1)) = ((𝑅 / (𝑀 + 1)) / (Σ𝑛 ∈ (0...(𝑗 − 1))(abs‘(seq0( + , 𝐴)‘𝑛)) + 1))
91	86, 90	breqtrdi 5188	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑅 ∈ ℝ⁺) ∧ (𝑗 ∈ ℕ₀ ∧ ∀𝑘 ∈ (ℤ_≥‘𝑗)(abs‘(seq0( + , 𝐴)‘𝑘)) < (𝑅 / (𝑀 + 1)))) ∧ ((𝑦 ∈ 𝑆 ∧ (abs‘(1 − 𝑦)) < ((𝑅 / (𝑀 + 1)) / (Σ𝑖 ∈ (0...(𝑗 − 1))(abs‘(seq0( + , 𝐴)‘𝑖)) + 1))) ∧ 𝑦 ≠ 1)) → (abs‘(1 − 𝑦)) < ((𝑅 / (𝑀 + 1)) / (Σ𝑛 ∈ (0...(𝑗 − 1))(abs‘(seq0( + , 𝐴)‘𝑛)) + 1)))
92	70, 71, 72, 73, 31, 41, 74, 78, 79, 80, 85, 91	abelthlem7 25941	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑅 ∈ ℝ⁺) ∧ (𝑗 ∈ ℕ₀ ∧ ∀𝑘 ∈ (ℤ_≥‘𝑗)(abs‘(seq0( + , 𝐴)‘𝑘)) < (𝑅 / (𝑀 + 1)))) ∧ ((𝑦 ∈ 𝑆 ∧ (abs‘(1 − 𝑦)) < ((𝑅 / (𝑀 + 1)) / (Σ𝑖 ∈ (0...(𝑗 − 1))(abs‘(seq0( + , 𝐴)‘𝑖)) + 1))) ∧ 𝑦 ≠ 1)) → (abs‘(𝐹‘𝑦)) < ((𝑀 + 1) · (𝑅 / (𝑀 + 1))))
93		rpcn 12980	. . . . . . . . . . . 12 ⊢ (𝑅 ∈ ℝ⁺ → 𝑅 ∈ ℂ)
94	93	adantl 482	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑅 ∈ ℝ⁺) → 𝑅 ∈ ℂ)
95	6	adantr 481	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑅 ∈ ℝ⁺) → (𝑀 + 1) ∈ ℝ⁺)
96	95	rpcnd 13014	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑅 ∈ ℝ⁺) → (𝑀 + 1) ∈ ℂ)
97	95	rpne0d 13017	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑅 ∈ ℝ⁺) → (𝑀 + 1) ≠ 0)
98	94, 96, 97	divcan2d 11988	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑅 ∈ ℝ⁺) → ((𝑀 + 1) · (𝑅 / (𝑀 + 1))) = 𝑅)
99	98	ad2antrr 724	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑅 ∈ ℝ⁺) ∧ (𝑗 ∈ ℕ₀ ∧ ∀𝑘 ∈ (ℤ_≥‘𝑗)(abs‘(seq0( + , 𝐴)‘𝑘)) < (𝑅 / (𝑀 + 1)))) ∧ ((𝑦 ∈ 𝑆 ∧ (abs‘(1 − 𝑦)) < ((𝑅 / (𝑀 + 1)) / (Σ𝑖 ∈ (0...(𝑗 − 1))(abs‘(seq0( + , 𝐴)‘𝑖)) + 1))) ∧ 𝑦 ≠ 1)) → ((𝑀 + 1) · (𝑅 / (𝑀 + 1))) = 𝑅)
100	92, 99	breqtrd 5173	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑅 ∈ ℝ⁺) ∧ (𝑗 ∈ ℕ₀ ∧ ∀𝑘 ∈ (ℤ_≥‘𝑗)(abs‘(seq0( + , 𝐴)‘𝑘)) < (𝑅 / (𝑀 + 1)))) ∧ ((𝑦 ∈ 𝑆 ∧ (abs‘(1 − 𝑦)) < ((𝑅 / (𝑀 + 1)) / (Σ𝑖 ∈ (0...(𝑗 − 1))(abs‘(seq0( + , 𝐴)‘𝑖)) + 1))) ∧ 𝑦 ≠ 1)) → (abs‘(𝐹‘𝑦)) < 𝑅)
101	100	anassrs 468	. . . . . . 7 ⊢ (((((𝜑 ∧ 𝑅 ∈ ℝ⁺) ∧ (𝑗 ∈ ℕ₀ ∧ ∀𝑘 ∈ (ℤ_≥‘𝑗)(abs‘(seq0( + , 𝐴)‘𝑘)) < (𝑅 / (𝑀 + 1)))) ∧ (𝑦 ∈ 𝑆 ∧ (abs‘(1 − 𝑦)) < ((𝑅 / (𝑀 + 1)) / (Σ𝑖 ∈ (0...(𝑗 − 1))(abs‘(seq0( + , 𝐴)‘𝑖)) + 1)))) ∧ 𝑦 ≠ 1) → (abs‘(𝐹‘𝑦)) < 𝑅)
102	69, 101	pm2.61dane 3029	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑅 ∈ ℝ⁺) ∧ (𝑗 ∈ ℕ₀ ∧ ∀𝑘 ∈ (ℤ_≥‘𝑗)(abs‘(seq0( + , 𝐴)‘𝑘)) < (𝑅 / (𝑀 + 1)))) ∧ (𝑦 ∈ 𝑆 ∧ (abs‘(1 − 𝑦)) < ((𝑅 / (𝑀 + 1)) / (Σ𝑖 ∈ (0...(𝑗 − 1))(abs‘(seq0( + , 𝐴)‘𝑖)) + 1)))) → (abs‘(𝐹‘𝑦)) < 𝑅)
103	61, 102	eqbrtrd 5169	. . . . 5 ⊢ ((((𝜑 ∧ 𝑅 ∈ ℝ⁺) ∧ (𝑗 ∈ ℕ₀ ∧ ∀𝑘 ∈ (ℤ_≥‘𝑗)(abs‘(seq0( + , 𝐴)‘𝑘)) < (𝑅 / (𝑀 + 1)))) ∧ (𝑦 ∈ 𝑆 ∧ (abs‘(1 − 𝑦)) < ((𝑅 / (𝑀 + 1)) / (Σ𝑖 ∈ (0...(𝑗 − 1))(abs‘(seq0( + , 𝐴)‘𝑖)) + 1)))) → (abs‘((𝐹‘1) − (𝐹‘𝑦))) < 𝑅)
104	103	expr 457	. . . 4 ⊢ ((((𝜑 ∧ 𝑅 ∈ ℝ⁺) ∧ (𝑗 ∈ ℕ₀ ∧ ∀𝑘 ∈ (ℤ_≥‘𝑗)(abs‘(seq0( + , 𝐴)‘𝑘)) < (𝑅 / (𝑀 + 1)))) ∧ 𝑦 ∈ 𝑆) → ((abs‘(1 − 𝑦)) < ((𝑅 / (𝑀 + 1)) / (Σ𝑖 ∈ (0...(𝑗 − 1))(abs‘(seq0( + , 𝐴)‘𝑖)) + 1)) → (abs‘((𝐹‘1) − (𝐹‘𝑦))) < 𝑅))
105	104	ralrimiva 3146	. . 3 ⊢ (((𝜑 ∧ 𝑅 ∈ ℝ⁺) ∧ (𝑗 ∈ ℕ₀ ∧ ∀𝑘 ∈ (ℤ_≥‘𝑗)(abs‘(seq0( + , 𝐴)‘𝑘)) < (𝑅 / (𝑀 + 1)))) → ∀𝑦 ∈ 𝑆 ((abs‘(1 − 𝑦)) < ((𝑅 / (𝑀 + 1)) / (Σ𝑖 ∈ (0...(𝑗 − 1))(abs‘(seq0( + , 𝐴)‘𝑖)) + 1)) → (abs‘((𝐹‘1) − (𝐹‘𝑦))) < 𝑅))
106		breq2 5151	. . . 4 ⊢ (𝑤 = ((𝑅 / (𝑀 + 1)) / (Σ𝑖 ∈ (0...(𝑗 − 1))(abs‘(seq0( + , 𝐴)‘𝑖)) + 1)) → ((abs‘(1 − 𝑦)) < 𝑤 ↔ (abs‘(1 − 𝑦)) < ((𝑅 / (𝑀 + 1)) / (Σ𝑖 ∈ (0...(𝑗 − 1))(abs‘(seq0( + , 𝐴)‘𝑖)) + 1))))
107	106	rspceaimv 3616	. . 3 ⊢ ((((𝑅 / (𝑀 + 1)) / (Σ𝑖 ∈ (0...(𝑗 − 1))(abs‘(seq0( + , 𝐴)‘𝑖)) + 1)) ∈ ℝ⁺ ∧ ∀𝑦 ∈ 𝑆 ((abs‘(1 − 𝑦)) < ((𝑅 / (𝑀 + 1)) / (Σ𝑖 ∈ (0...(𝑗 − 1))(abs‘(seq0( + , 𝐴)‘𝑖)) + 1)) → (abs‘((𝐹‘1) − (𝐹‘𝑦))) < 𝑅)) → ∃𝑤 ∈ ℝ⁺ ∀𝑦 ∈ 𝑆 ((abs‘(1 − 𝑦)) < 𝑤 → (abs‘((𝐹‘1) − (𝐹‘𝑦))) < 𝑅))
108	29, 105, 107	syl2anc 584	. 2 ⊢ (((𝜑 ∧ 𝑅 ∈ ℝ⁺) ∧ (𝑗 ∈ ℕ₀ ∧ ∀𝑘 ∈ (ℤ_≥‘𝑗)(abs‘(seq0( + , 𝐴)‘𝑘)) < (𝑅 / (𝑀 + 1)))) → ∃𝑤 ∈ ℝ⁺ ∀𝑦 ∈ 𝑆 ((abs‘(1 − 𝑦)) < 𝑤 → (abs‘((𝐹‘1) − (𝐹‘𝑦))) < 𝑅))
109	12, 108	rexlimddv 3161	1 ⊢ ((𝜑 ∧ 𝑅 ∈ ℝ⁺) → ∃𝑤 ∈ ℝ⁺ ∀𝑦 ∈ 𝑆 ((abs‘(1 − 𝑦)) < 𝑤 → (abs‘((𝐹‘1) − (𝐹‘𝑦))) < 𝑅))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 396 = wceq 1541 ∈ wcel 2106 ≠ wne 2940 ∀wral 3061 ∃wrex 3070 {crab 3432 ∖ cdif 3944 ⊆ wss 3947 {csn 4627 class class class wbr 5147 ↦ cmpt 5230 dom cdm 5675 ∘ ccom 5679 ⟶wf 6536 ‘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 -cneg 11441 / cdiv 11867 ℕ₀cn0 12468 ℤcz 12554 ℤ_≥cuz 12818 ℝ⁺crp 12970 ...cfz 13480 seqcseq 13962 ↑cexp 14023 abscabs 15177 ⇝ cli 15424 Σcsu 15628 ballcbl 20923

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-map 8818 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-xadd 13089 df-ico 13326 df-icc 13327 df-fz 13481 df-fzo 13624 df-fl 13753 df-seq 13963 df-exp 14024 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-psmet 20928 df-xmet 20929 df-met 20930 df-bl 20931

This theorem is referenced by: abelthlem9 25943

W3C validator