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Theorem abelthlem8 26349

Description: Lemma for abelth 26351. (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 12835	. . 3 ⊢ ℕ₀ = (ℤ_≥‘0)
2		0zd 12541	. . 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 13025	. . . 4 ⊢ (𝜑 → (𝑀 + 1) ∈ ℝ⁺)
7		rpdivcl 12978	. . . 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 2730	. . 3 ⊢ (((𝜑 ∧ 𝑅 ∈ ℝ⁺) ∧ 𝑘 ∈ ℕ₀) → (seq0( + , 𝐴)‘𝑘) = (seq0( + , 𝐴)‘𝑘))
10		abelth.7	. . . 4 ⊢ (𝜑 → seq0( + , 𝐴) ⇝ 0)
11	10	adantr 480	. . 3 ⊢ ((𝜑 ∧ 𝑅 ∈ ℝ⁺) → seq0( + , 𝐴) ⇝ 0)
12	1, 2, 8, 9, 11	climi0 15478	. 2 ⊢ ((𝜑 ∧ 𝑅 ∈ ℝ⁺) → ∃𝑗 ∈ ℕ₀ ∀𝑘 ∈ (ℤ_≥‘𝑗)(abs‘(seq0( + , 𝐴)‘𝑘)) < (𝑅 / (𝑀 + 1)))
13	8	adantr 480	. . . 4 ⊢ (((𝜑 ∧ 𝑅 ∈ ℝ⁺) ∧ (𝑗 ∈ ℕ₀ ∧ ∀𝑘 ∈ (ℤ_≥‘𝑗)(abs‘(seq0( + , 𝐴)‘𝑘)) < (𝑅 / (𝑀 + 1)))) → (𝑅 / (𝑀 + 1)) ∈ ℝ⁺)
14		fzfid 13938	. . . . . . 7 ⊢ (𝜑 → (0...(𝑗 − 1)) ∈ Fin)
15		0zd 12541	. . . . . . . . . 10 ⊢ (𝜑 → 0 ∈ ℤ)
16		abelth.1	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐴:ℕ₀⟶ℂ)
17	16	ffvelcdmda 7056	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑤 ∈ ℕ₀) → (𝐴‘𝑤) ∈ ℂ)
18	1, 15, 17	serf 13995	. . . . . . . . 9 ⊢ (𝜑 → seq0( + , 𝐴):ℕ₀⟶ℂ)
19		elfznn0 13581	. . . . . . . . 9 ⊢ (𝑖 ∈ (0...(𝑗 − 1)) → 𝑖 ∈ ℕ₀)
20		ffvelcdm 7053	. . . . . . . . 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 15405	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...(𝑗 − 1))) → (abs‘(seq0( + , 𝐴)‘𝑖)) ∈ ℝ)
23	14, 22	fsumrecl 15700	. . . . . 6 ⊢ (𝜑 → Σ𝑖 ∈ (0...(𝑗 − 1))(abs‘(seq0( + , 𝐴)‘𝑖)) ∈ ℝ)
24	23	ad2antrr 726	. . . . 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 15413	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...(𝑗 − 1))) → 0 ≤ (abs‘(seq0( + , 𝐴)‘𝑖)))
26	14, 22, 25	fsumge0 15761	. . . . . 6 ⊢ (𝜑 → 0 ≤ Σ𝑖 ∈ (0...(𝑗 − 1))(abs‘(seq0( + , 𝐴)‘𝑖)))
27	26	ad2antrr 726	. . . . 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 13025	. . . 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 13012	. . 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 26342	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (1 ∈ 𝑆 ∧ (𝑆 ∖ {1}) ⊆ (0(ball‘(abs ∘ − ))1)))
33	32	simpld 494	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 1 ∈ 𝑆)
34		oveq1 7394	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 = 1 → (𝑥↑𝑛) = (1↑𝑛))
35		nn0z 12554	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑛 ∈ ℕ₀ → 𝑛 ∈ ℤ)
36		1exp 14056	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑛 ∈ ℤ → (1↑𝑛) = 1)
37	35, 36	syl 17	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑛 ∈ ℕ₀ → (1↑𝑛) = 1)
38	34, 37	sylan9eq 2784	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑥 = 1 ∧ 𝑛 ∈ ℕ₀) → (𝑥↑𝑛) = 1)
39	38	oveq2d 7403	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 = 1 ∧ 𝑛 ∈ ℕ₀) → ((𝐴‘𝑛) · (𝑥↑𝑛)) = ((𝐴‘𝑛) · 1))
40	39	sumeq2dv 15668	. . . . . . . . . . . . . . . 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 15654	. . . . . . . . . . . . . . . 16 ⊢ Σ𝑛 ∈ ℕ₀ ((𝐴‘𝑛) · 1) ∈ V
43	40, 41, 42	fvmpt 6968	. . . . . . . . . . . . . . 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 7056	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ₀) → (𝐴‘𝑛) ∈ ℂ)
46	45	mulridd 11191	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ₀) → ((𝐴‘𝑛) · 1) = (𝐴‘𝑛))
47	46	eqcomd 2735	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ₀) → (𝐴‘𝑛) = ((𝐴‘𝑛) · 1))
48	46, 45	eqeltrd 2828	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ₀) → ((𝐴‘𝑛) · 1) ∈ ℂ)
49	1, 15, 47, 48, 10	isumclim 15723	. . . . . . . . . . . . . 14 ⊢ (𝜑 → Σ𝑛 ∈ ℕ₀ ((𝐴‘𝑛) · 1) = 0)
50	44, 49	eqtrd 2764	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐹‘1) = 0)
51	50	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑆) → (𝐹‘1) = 0)
52	51	oveq1d 7402	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑆) → ((𝐹‘1) − (𝐹‘𝑦)) = (0 − (𝐹‘𝑦)))
53		df-neg 11408	. . . . . . . . . . 11 ⊢ -(𝐹‘𝑦) = (0 − (𝐹‘𝑦))
54	52, 53	eqtr4di 2782	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑆) → ((𝐹‘1) − (𝐹‘𝑦)) = -(𝐹‘𝑦))
55	54	fveq2d 6862	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑆) → (abs‘((𝐹‘1) − (𝐹‘𝑦))) = (abs‘-(𝐹‘𝑦)))
56	16, 30, 4, 5, 31, 41	abelthlem4 26344	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐹:𝑆⟶ℂ)
57	56	ffvelcdmda 7056	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑆) → (𝐹‘𝑦) ∈ ℂ)
58	57	absnegd 15418	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑆) → (abs‘-(𝐹‘𝑦)) = (abs‘(𝐹‘𝑦)))
59	55, 58	eqtrd 2764	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑆) → (abs‘((𝐹‘1) − (𝐹‘𝑦))) = (abs‘(𝐹‘𝑦)))
60	59	adantlr 715	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑅 ∈ ℝ⁺) ∧ 𝑦 ∈ 𝑆) → (abs‘((𝐹‘1) − (𝐹‘𝑦))) = (abs‘(𝐹‘𝑦)))
61	60	ad2ant2r 747	. . . . . 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 6858	. . . . . . . . . . 11 ⊢ (𝑦 = 1 → (𝐹‘𝑦) = (𝐹‘1))
63	62, 50	sylan9eqr 2786	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 = 1) → (𝐹‘𝑦) = 0)
64	63	abs00bd 15257	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 = 1) → (abs‘(𝐹‘𝑦)) = 0)
65	64	ad5ant15 758	. . . . . . . 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 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)))) → 𝑅 ∈ ℝ⁺)
67	66	rpgt0d 12998	. . . . . . . . 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 480	. . . . . . . 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 5129	. . . . . . 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 730	. . . . . . . . . 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 730	. . . . . . . . . 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 730	. . . . . . . . . 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 730	. . . . . . . . . 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 730	. . . . . . . . . 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 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)) → 𝑦 ∈ 𝑆)
76		simprr 772	. . . . . . . . . . 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 4750	. . . . . . . . . . 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 726	. . . . . . . . . 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 776	. . . . . . . . . 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 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)) → ∀𝑘 ∈ (ℤ_≥‘𝑗)(abs‘(seq0( + , 𝐴)‘𝑘)) < (𝑅 / (𝑀 + 1)))
82		2fveq3 6863	. . . . . . . . . . . . 13 ⊢ (𝑘 = 𝑚 → (abs‘(seq0( + , 𝐴)‘𝑘)) = (abs‘(seq0( + , 𝐴)‘𝑚)))
83	82	breq1d 5117	. . . . . . . . . . . 12 ⊢ (𝑘 = 𝑚 → ((abs‘(seq0( + , 𝐴)‘𝑘)) < (𝑅 / (𝑀 + 1)) ↔ (abs‘(seq0( + , 𝐴)‘𝑚)) < (𝑅 / (𝑀 + 1))))
84	83	cbvralvw 3215	. . . . . . . . . . 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 218	. . . . . . . . . 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 779	. . . . . . . . . . 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 6863	. . . . . . . . . . . . . 14 ⊢ (𝑖 = 𝑛 → (abs‘(seq0( + , 𝐴)‘𝑖)) = (abs‘(seq0( + , 𝐴)‘𝑛)))
88	87	cbvsumv 15662	. . . . . . . . . . . . 13 ⊢ Σ𝑖 ∈ (0...(𝑗 − 1))(abs‘(seq0( + , 𝐴)‘𝑖)) = Σ𝑛 ∈ (0...(𝑗 − 1))(abs‘(seq0( + , 𝐴)‘𝑛))
89	88	oveq1i 7397	. . . . . . . . . . . 12 ⊢ (Σ𝑖 ∈ (0...(𝑗 − 1))(abs‘(seq0( + , 𝐴)‘𝑖)) + 1) = (Σ𝑛 ∈ (0...(𝑗 − 1))(abs‘(seq0( + , 𝐴)‘𝑛)) + 1)
90	89	oveq2i 7398	. . . . . . . . . . 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 5148	. . . . . . . . . 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 26348	. . . . . . . . 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 12962	. . . . . . . . . . . 12 ⊢ (𝑅 ∈ ℝ⁺ → 𝑅 ∈ ℂ)
94	93	adantl 481	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑅 ∈ ℝ⁺) → 𝑅 ∈ ℂ)
95	6	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑅 ∈ ℝ⁺) → (𝑀 + 1) ∈ ℝ⁺)
96	95	rpcnd 12997	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑅 ∈ ℝ⁺) → (𝑀 + 1) ∈ ℂ)
97	95	rpne0d 13000	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑅 ∈ ℝ⁺) → (𝑀 + 1) ≠ 0)
98	94, 96, 97	divcan2d 11960	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑅 ∈ ℝ⁺) → ((𝑀 + 1) · (𝑅 / (𝑀 + 1))) = 𝑅)
99	98	ad2antrr 726	. . . . . . . . 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 5133	. . . . . . . 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 467	. . . . . . 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 3012	. . . . . 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 5129	. . . . 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 456	. . . 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 3125	. . 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 5111	. . . 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 3594	. . 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 3140	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 395 = wceq 1540 ∈ wcel 2109 ≠ wne 2925 ∀wral 3044 ∃wrex 3053 {crab 3405 ∖ cdif 3911 ⊆ wss 3914 {csn 4589 class class class wbr 5107 ↦ cmpt 5188 dom cdm 5638 ∘ ccom 5642 ⟶wf 6507 ‘cfv 6511 (class class class)co 7387 ℂcc 11066 ℝcr 11067 0cc0 11068 1c1 11069 + caddc 11071 · cmul 11073 < clt 11208 ≤ cle 11209 − cmin 11405 -cneg 11406 / cdiv 11835 ℕ₀cn0 12442 ℤcz 12529 ℤ_≥cuz 12793 ℝ⁺crp 12951 ...cfz 13468 seqcseq 13966 ↑cexp 14026 abscabs 15200 ⇝ cli 15450 Σcsu 15652 ballcbl 21251

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-rep 5234 ax-sep 5251 ax-nul 5261 ax-pow 5320 ax-pr 5387 ax-un 7711 ax-inf2 9594 ax-cnex 11124 ax-resscn 11125 ax-1cn 11126 ax-icn 11127 ax-addcl 11128 ax-addrcl 11129 ax-mulcl 11130 ax-mulrcl 11131 ax-mulcom 11132 ax-addass 11133 ax-mulass 11134 ax-distr 11135 ax-i2m1 11136 ax-1ne0 11137 ax-1rid 11138 ax-rnegex 11139 ax-rrecex 11140 ax-cnre 11141 ax-pre-lttri 11142 ax-pre-lttrn 11143 ax-pre-ltadd 11144 ax-pre-mulgt0 11145 ax-pre-sup 11146

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-rmo 3354 df-reu 3355 df-rab 3406 df-v 3449 df-sbc 3754 df-csb 3863 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-pss 3934 df-nul 4297 df-if 4489 df-pw 4565 df-sn 4590 df-pr 4592 df-op 4596 df-uni 4872 df-int 4911 df-iun 4957 df-br 5108 df-opab 5170 df-mpt 5189 df-tr 5215 df-id 5533 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5591 df-se 5592 df-we 5593 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-pred 6274 df-ord 6335 df-on 6336 df-lim 6337 df-suc 6338 df-iota 6464 df-fun 6513 df-fn 6514 df-f 6515 df-f1 6516 df-fo 6517 df-f1o 6518 df-fv 6519 df-isom 6520 df-riota 7344 df-ov 7390 df-oprab 7391 df-mpo 7392 df-om 7843 df-1st 7968 df-2nd 7969 df-frecs 8260 df-wrecs 8291 df-recs 8340 df-rdg 8378 df-1o 8434 df-er 8671 df-map 8801 df-pm 8802 df-en 8919 df-dom 8920 df-sdom 8921 df-fin 8922 df-sup 9393 df-inf 9394 df-oi 9463 df-card 9892 df-pnf 11210 df-mnf 11211 df-xr 11212 df-ltxr 11213 df-le 11214 df-sub 11407 df-neg 11408 df-div 11836 df-nn 12187 df-2 12249 df-3 12250 df-n0 12443 df-z 12530 df-uz 12794 df-rp 12952 df-xadd 13073 df-ico 13312 df-icc 13313 df-fz 13469 df-fzo 13616 df-fl 13754 df-seq 13967 df-exp 14027 df-hash 14296 df-shft 15033 df-cj 15065 df-re 15066 df-im 15067 df-sqrt 15201 df-abs 15202 df-limsup 15437 df-clim 15454 df-rlim 15455 df-sum 15653 df-psmet 21256 df-xmet 21257 df-met 21258 df-bl 21259

This theorem is referenced by: abelthlem9 26350

W3C validator