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Theorem abelthlem8 26406

Description: Lemma for abelth 26408. (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 12899	. . 3 ⊢ ℕ₀ = (ℤ_≥‘0)
2		0zd 12605	. . 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 13086	. . . 4 ⊢ (𝜑 → (𝑀 + 1) ∈ ℝ⁺)
7		rpdivcl 13039	. . . 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 2737	. . 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 15533	. 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 13996	. . . . . . 7 ⊢ (𝜑 → (0...(𝑗 − 1)) ∈ Fin)
15		0zd 12605	. . . . . . . . . 10 ⊢ (𝜑 → 0 ∈ ℤ)
16		abelth.1	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐴:ℕ₀⟶ℂ)
17	16	ffvelcdmda 7079	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑤 ∈ ℕ₀) → (𝐴‘𝑤) ∈ ℂ)
18	1, 15, 17	serf 14053	. . . . . . . . 9 ⊢ (𝜑 → seq0( + , 𝐴):ℕ₀⟶ℂ)
19		elfznn0 13642	. . . . . . . . 9 ⊢ (𝑖 ∈ (0...(𝑗 − 1)) → 𝑖 ∈ ℕ₀)
20		ffvelcdm 7076	. . . . . . . . 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 15460	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...(𝑗 − 1))) → (abs‘(seq0( + , 𝐴)‘𝑖)) ∈ ℝ)
23	14, 22	fsumrecl 15755	. . . . . 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 15468	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...(𝑗 − 1))) → 0 ≤ (abs‘(seq0( + , 𝐴)‘𝑖)))
26	14, 22, 25	fsumge0 15816	. . . . . 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 13086	. . . 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 13073	. . 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 26399	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (1 ∈ 𝑆 ∧ (𝑆 ∖ {1}) ⊆ (0(ball‘(abs ∘ − ))1)))
33	32	simpld 494	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 1 ∈ 𝑆)
34		oveq1 7417	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 = 1 → (𝑥↑𝑛) = (1↑𝑛))
35		nn0z 12618	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑛 ∈ ℕ₀ → 𝑛 ∈ ℤ)
36		1exp 14114	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑛 ∈ ℤ → (1↑𝑛) = 1)
37	35, 36	syl 17	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑛 ∈ ℕ₀ → (1↑𝑛) = 1)
38	34, 37	sylan9eq 2791	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑥 = 1 ∧ 𝑛 ∈ ℕ₀) → (𝑥↑𝑛) = 1)
39	38	oveq2d 7426	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 = 1 ∧ 𝑛 ∈ ℕ₀) → ((𝐴‘𝑛) · (𝑥↑𝑛)) = ((𝐴‘𝑛) · 1))
40	39	sumeq2dv 15723	. . . . . . . . . . . . . . . 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 15709	. . . . . . . . . . . . . . . 16 ⊢ Σ𝑛 ∈ ℕ₀ ((𝐴‘𝑛) · 1) ∈ V
43	40, 41, 42	fvmpt 6991	. . . . . . . . . . . . . . 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 7079	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ₀) → (𝐴‘𝑛) ∈ ℂ)
46	45	mulridd 11257	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ₀) → ((𝐴‘𝑛) · 1) = (𝐴‘𝑛))
47	46	eqcomd 2742	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ₀) → (𝐴‘𝑛) = ((𝐴‘𝑛) · 1))
48	46, 45	eqeltrd 2835	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ₀) → ((𝐴‘𝑛) · 1) ∈ ℂ)
49	1, 15, 47, 48, 10	isumclim 15778	. . . . . . . . . . . . . 14 ⊢ (𝜑 → Σ𝑛 ∈ ℕ₀ ((𝐴‘𝑛) · 1) = 0)
50	44, 49	eqtrd 2771	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐹‘1) = 0)
51	50	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑆) → (𝐹‘1) = 0)
52	51	oveq1d 7425	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑆) → ((𝐹‘1) − (𝐹‘𝑦)) = (0 − (𝐹‘𝑦)))
53		df-neg 11474	. . . . . . . . . . 11 ⊢ -(𝐹‘𝑦) = (0 − (𝐹‘𝑦))
54	52, 53	eqtr4di 2789	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑆) → ((𝐹‘1) − (𝐹‘𝑦)) = -(𝐹‘𝑦))
55	54	fveq2d 6885	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑆) → (abs‘((𝐹‘1) − (𝐹‘𝑦))) = (abs‘-(𝐹‘𝑦)))
56	16, 30, 4, 5, 31, 41	abelthlem4 26401	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐹:𝑆⟶ℂ)
57	56	ffvelcdmda 7079	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑆) → (𝐹‘𝑦) ∈ ℂ)
58	57	absnegd 15473	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑆) → (abs‘-(𝐹‘𝑦)) = (abs‘(𝐹‘𝑦)))
59	55, 58	eqtrd 2771	. . . . . . . 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 6881	. . . . . . . . . . 11 ⊢ (𝑦 = 1 → (𝐹‘𝑦) = (𝐹‘1))
63	62, 50	sylan9eqr 2793	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 = 1) → (𝐹‘𝑦) = 0)
64	63	abs00bd 15315	. . . . . . . . 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 13059	. . . . . . . . 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 5146	. . . . . . 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 4767	. . . . . . . . . . 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 6886	. . . . . . . . . . . . 13 ⊢ (𝑘 = 𝑚 → (abs‘(seq0( + , 𝐴)‘𝑘)) = (abs‘(seq0( + , 𝐴)‘𝑚)))
83	82	breq1d 5134	. . . . . . . . . . . 12 ⊢ (𝑘 = 𝑚 → ((abs‘(seq0( + , 𝐴)‘𝑘)) < (𝑅 / (𝑀 + 1)) ↔ (abs‘(seq0( + , 𝐴)‘𝑚)) < (𝑅 / (𝑀 + 1))))
84	83	cbvralvw 3224	. . . . . . . . . . 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 6886	. . . . . . . . . . . . . 14 ⊢ (𝑖 = 𝑛 → (abs‘(seq0( + , 𝐴)‘𝑖)) = (abs‘(seq0( + , 𝐴)‘𝑛)))
88	87	cbvsumv 15717	. . . . . . . . . . . . 13 ⊢ Σ𝑖 ∈ (0...(𝑗 − 1))(abs‘(seq0( + , 𝐴)‘𝑖)) = Σ𝑛 ∈ (0...(𝑗 − 1))(abs‘(seq0( + , 𝐴)‘𝑛))
89	88	oveq1i 7420	. . . . . . . . . . . 12 ⊢ (Σ𝑖 ∈ (0...(𝑗 − 1))(abs‘(seq0( + , 𝐴)‘𝑖)) + 1) = (Σ𝑛 ∈ (0...(𝑗 − 1))(abs‘(seq0( + , 𝐴)‘𝑛)) + 1)
90	89	oveq2i 7421	. . . . . . . . . . 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 5165	. . . . . . . . . 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 26405	. . . . . . . . 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 13024	. . . . . . . . . . . 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 13058	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑅 ∈ ℝ⁺) → (𝑀 + 1) ∈ ℂ)
97	95	rpne0d 13061	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑅 ∈ ℝ⁺) → (𝑀 + 1) ≠ 0)
98	94, 96, 97	divcan2d 12024	. . . . . . . . . 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 5150	. . . . . . . 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 3020	. . . . . 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 5146	. . . . 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 3133	. . 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 5128	. . . 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 3612	. . 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 3148	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 2933 ∀wral 3052 ∃wrex 3061 {crab 3420 ∖ cdif 3928 ⊆ wss 3931 {csn 4606 class class class wbr 5124 ↦ cmpt 5206 dom cdm 5659 ∘ ccom 5663 ⟶wf 6532 ‘cfv 6536 (class class class)co 7410 ℂcc 11132 ℝcr 11133 0cc0 11134 1c1 11135 + caddc 11137 · cmul 11139 < clt 11274 ≤ cle 11275 − cmin 11471 -cneg 11472 / cdiv 11899 ℕ₀cn0 12506 ℤcz 12593 ℤ_≥cuz 12857 ℝ⁺crp 13013 ...cfz 13529 seqcseq 14024 ↑cexp 14084 abscabs 15258 ⇝ cli 15505 Σcsu 15707 ballcbl 21307

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 2708 ax-rep 5254 ax-sep 5271 ax-nul 5281 ax-pow 5340 ax-pr 5407 ax-un 7734 ax-inf2 9660 ax-cnex 11190 ax-resscn 11191 ax-1cn 11192 ax-icn 11193 ax-addcl 11194 ax-addrcl 11195 ax-mulcl 11196 ax-mulrcl 11197 ax-mulcom 11198 ax-addass 11199 ax-mulass 11200 ax-distr 11201 ax-i2m1 11202 ax-1ne0 11203 ax-1rid 11204 ax-rnegex 11205 ax-rrecex 11206 ax-cnre 11207 ax-pre-lttri 11208 ax-pre-lttrn 11209 ax-pre-ltadd 11210 ax-pre-mulgt0 11211 ax-pre-sup 11212

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 2540 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2810 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3062 df-rmo 3364 df-reu 3365 df-rab 3421 df-v 3466 df-sbc 3771 df-csb 3880 df-dif 3934 df-un 3936 df-in 3938 df-ss 3948 df-pss 3951 df-nul 4314 df-if 4506 df-pw 4582 df-sn 4607 df-pr 4609 df-op 4613 df-uni 4889 df-int 4928 df-iun 4974 df-br 5125 df-opab 5187 df-mpt 5207 df-tr 5235 df-id 5553 df-eprel 5558 df-po 5566 df-so 5567 df-fr 5611 df-se 5612 df-we 5613 df-xp 5665 df-rel 5666 df-cnv 5667 df-co 5668 df-dm 5669 df-rn 5670 df-res 5671 df-ima 5672 df-pred 6295 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6489 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-isom 6545 df-riota 7367 df-ov 7413 df-oprab 7414 df-mpo 7415 df-om 7867 df-1st 7993 df-2nd 7994 df-frecs 8285 df-wrecs 8316 df-recs 8390 df-rdg 8429 df-1o 8485 df-er 8724 df-map 8847 df-pm 8848 df-en 8965 df-dom 8966 df-sdom 8967 df-fin 8968 df-sup 9459 df-inf 9460 df-oi 9529 df-card 9958 df-pnf 11276 df-mnf 11277 df-xr 11278 df-ltxr 11279 df-le 11280 df-sub 11473 df-neg 11474 df-div 11900 df-nn 12246 df-2 12308 df-3 12309 df-n0 12507 df-z 12594 df-uz 12858 df-rp 13014 df-xadd 13134 df-ico 13373 df-icc 13374 df-fz 13530 df-fzo 13677 df-fl 13814 df-seq 14025 df-exp 14085 df-hash 14354 df-shft 15091 df-cj 15123 df-re 15124 df-im 15125 df-sqrt 15259 df-abs 15260 df-limsup 15492 df-clim 15509 df-rlim 15510 df-sum 15708 df-psmet 21312 df-xmet 21313 df-met 21314 df-bl 21315

This theorem is referenced by: abelthlem9 26407

W3C validator