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Theorem abelthlem8 26396

Description: Lemma for abelth 26398. (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 12780	. . 3 ⊢ ℕ₀ = (ℤ_≥‘0)
2		0zd 12491	. . 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 12970	. . . 4 ⊢ (𝜑 → (𝑀 + 1) ∈ ℝ⁺)
7		rpdivcl 12923	. . . 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 2734	. . 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 15426	. 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 13887	. . . . . . 7 ⊢ (𝜑 → (0...(𝑗 − 1)) ∈ Fin)
15		0zd 12491	. . . . . . . . . 10 ⊢ (𝜑 → 0 ∈ ℤ)
16		abelth.1	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐴:ℕ₀⟶ℂ)
17	16	ffvelcdmda 7026	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑤 ∈ ℕ₀) → (𝐴‘𝑤) ∈ ℂ)
18	1, 15, 17	serf 13944	. . . . . . . . 9 ⊢ (𝜑 → seq0( + , 𝐴):ℕ₀⟶ℂ)
19		elfznn0 13527	. . . . . . . . 9 ⊢ (𝑖 ∈ (0...(𝑗 − 1)) → 𝑖 ∈ ℕ₀)
20		ffvelcdm 7023	. . . . . . . . 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 15353	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...(𝑗 − 1))) → (abs‘(seq0( + , 𝐴)‘𝑖)) ∈ ℝ)
23	14, 22	fsumrecl 15648	. . . . . 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 15361	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...(𝑗 − 1))) → 0 ≤ (abs‘(seq0( + , 𝐴)‘𝑖)))
26	14, 22, 25	fsumge0 15709	. . . . . 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 12970	. . . 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 12957	. . 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 26389	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (1 ∈ 𝑆 ∧ (𝑆 ∖ {1}) ⊆ (0(ball‘(abs ∘ − ))1)))
33	32	simpld 494	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 1 ∈ 𝑆)
34		oveq1 7362	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 = 1 → (𝑥↑𝑛) = (1↑𝑛))
35		nn0z 12503	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑛 ∈ ℕ₀ → 𝑛 ∈ ℤ)
36		1exp 14005	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑛 ∈ ℤ → (1↑𝑛) = 1)
37	35, 36	syl 17	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑛 ∈ ℕ₀ → (1↑𝑛) = 1)
38	34, 37	sylan9eq 2788	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑥 = 1 ∧ 𝑛 ∈ ℕ₀) → (𝑥↑𝑛) = 1)
39	38	oveq2d 7371	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 = 1 ∧ 𝑛 ∈ ℕ₀) → ((𝐴‘𝑛) · (𝑥↑𝑛)) = ((𝐴‘𝑛) · 1))
40	39	sumeq2dv 15616	. . . . . . . . . . . . . . . 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 15602	. . . . . . . . . . . . . . . 16 ⊢ Σ𝑛 ∈ ℕ₀ ((𝐴‘𝑛) · 1) ∈ V
43	40, 41, 42	fvmpt 6938	. . . . . . . . . . . . . . 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 7026	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ₀) → (𝐴‘𝑛) ∈ ℂ)
46	45	mulridd 11140	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ₀) → ((𝐴‘𝑛) · 1) = (𝐴‘𝑛))
47	46	eqcomd 2739	. . . . . . . . . . . . . . 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 15671	. . . . . . . . . . . . . 14 ⊢ (𝜑 → Σ𝑛 ∈ ℕ₀ ((𝐴‘𝑛) · 1) = 0)
50	44, 49	eqtrd 2768	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐹‘1) = 0)
51	50	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑆) → (𝐹‘1) = 0)
52	51	oveq1d 7370	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑆) → ((𝐹‘1) − (𝐹‘𝑦)) = (0 − (𝐹‘𝑦)))
53		df-neg 11358	. . . . . . . . . . 11 ⊢ -(𝐹‘𝑦) = (0 − (𝐹‘𝑦))
54	52, 53	eqtr4di 2786	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑆) → ((𝐹‘1) − (𝐹‘𝑦)) = -(𝐹‘𝑦))
55	54	fveq2d 6835	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑆) → (abs‘((𝐹‘1) − (𝐹‘𝑦))) = (abs‘-(𝐹‘𝑦)))
56	16, 30, 4, 5, 31, 41	abelthlem4 26391	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐹:𝑆⟶ℂ)
57	56	ffvelcdmda 7026	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑆) → (𝐹‘𝑦) ∈ ℂ)
58	57	absnegd 15366	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑆) → (abs‘-(𝐹‘𝑦)) = (abs‘(𝐹‘𝑦)))
59	55, 58	eqtrd 2768	. . . . . . . 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 6831	. . . . . . . . . . 11 ⊢ (𝑦 = 1 → (𝐹‘𝑦) = (𝐹‘1))
63	62, 50	sylan9eqr 2790	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 = 1) → (𝐹‘𝑦) = 0)
64	63	abs00bd 15205	. . . . . . . . 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 12943	. . . . . . . . 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 5117	. . . . . . 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 4739	. . . . . . . . . . 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 6836	. . . . . . . . . . . . 13 ⊢ (𝑘 = 𝑚 → (abs‘(seq0( + , 𝐴)‘𝑘)) = (abs‘(seq0( + , 𝐴)‘𝑚)))
83	82	breq1d 5105	. . . . . . . . . . . 12 ⊢ (𝑘 = 𝑚 → ((abs‘(seq0( + , 𝐴)‘𝑘)) < (𝑅 / (𝑀 + 1)) ↔ (abs‘(seq0( + , 𝐴)‘𝑚)) < (𝑅 / (𝑀 + 1))))
84	83	cbvralvw 3211	. . . . . . . . . . 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 6836	. . . . . . . . . . . . . 14 ⊢ (𝑖 = 𝑛 → (abs‘(seq0( + , 𝐴)‘𝑖)) = (abs‘(seq0( + , 𝐴)‘𝑛)))
88	87	cbvsumv 15610	. . . . . . . . . . . . 13 ⊢ Σ𝑖 ∈ (0...(𝑗 − 1))(abs‘(seq0( + , 𝐴)‘𝑖)) = Σ𝑛 ∈ (0...(𝑗 − 1))(abs‘(seq0( + , 𝐴)‘𝑛))
89	88	oveq1i 7365	. . . . . . . . . . . 12 ⊢ (Σ𝑖 ∈ (0...(𝑗 − 1))(abs‘(seq0( + , 𝐴)‘𝑖)) + 1) = (Σ𝑛 ∈ (0...(𝑗 − 1))(abs‘(seq0( + , 𝐴)‘𝑛)) + 1)
90	89	oveq2i 7366	. . . . . . . . . . 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 5136	. . . . . . . . . 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 26395	. . . . . . . . 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 12907	. . . . . . . . . . . 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 12942	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑅 ∈ ℝ⁺) → (𝑀 + 1) ∈ ℂ)
97	95	rpne0d 12945	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑅 ∈ ℝ⁺) → (𝑀 + 1) ≠ 0)
98	94, 96, 97	divcan2d 11910	. . . . . . . . . 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 5121	. . . . . . . 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 3016	. . . . . 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 5117	. . . . 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 5099	. . . 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 3579	. . 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 1541 ∈ wcel 2113 ≠ wne 2929 ∀wral 3048 ∃wrex 3057 {crab 3396 ∖ cdif 3895 ⊆ wss 3898 {csn 4577 class class class wbr 5095 ↦ cmpt 5176 dom cdm 5621 ∘ ccom 5625 ⟶wf 6485 ‘cfv 6489 (class class class)co 7355 ℂcc 11015 ℝcr 11016 0cc0 11017 1c1 11018 + caddc 11020 · cmul 11022 < clt 11157 ≤ cle 11158 − cmin 11355 -cneg 11356 / cdiv 11785 ℕ₀cn0 12392 ℤcz 12479 ℤ_≥cuz 12742 ℝ⁺crp 12896 ...cfz 13414 seqcseq 13915 ↑cexp 13975 abscabs 15148 ⇝ cli 15398 Σcsu 15600 ballcbl 21287

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2115 ax-9 2123 ax-10 2146 ax-11 2162 ax-12 2182 ax-ext 2705 ax-rep 5221 ax-sep 5238 ax-nul 5248 ax-pow 5307 ax-pr 5374 ax-un 7677 ax-inf2 9542 ax-cnex 11073 ax-resscn 11074 ax-1cn 11075 ax-icn 11076 ax-addcl 11077 ax-addrcl 11078 ax-mulcl 11079 ax-mulrcl 11080 ax-mulcom 11081 ax-addass 11082 ax-mulass 11083 ax-distr 11084 ax-i2m1 11085 ax-1ne0 11086 ax-1rid 11087 ax-rnegex 11088 ax-rrecex 11089 ax-cnre 11090 ax-pre-lttri 11091 ax-pre-lttrn 11092 ax-pre-ltadd 11093 ax-pre-mulgt0 11094 ax-pre-sup 11095

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2725 df-clel 2808 df-nfc 2882 df-ne 2930 df-nel 3034 df-ral 3049 df-rex 3058 df-rmo 3347 df-reu 3348 df-rab 3397 df-v 3439 df-sbc 3738 df-csb 3847 df-dif 3901 df-un 3903 df-in 3905 df-ss 3915 df-pss 3918 df-nul 4283 df-if 4477 df-pw 4553 df-sn 4578 df-pr 4580 df-op 4584 df-uni 4861 df-int 4900 df-iun 4945 df-br 5096 df-opab 5158 df-mpt 5177 df-tr 5203 df-id 5516 df-eprel 5521 df-po 5529 df-so 5530 df-fr 5574 df-se 5575 df-we 5576 df-xp 5627 df-rel 5628 df-cnv 5629 df-co 5630 df-dm 5631 df-rn 5632 df-res 5633 df-ima 5634 df-pred 6256 df-ord 6317 df-on 6318 df-lim 6319 df-suc 6320 df-iota 6445 df-fun 6491 df-fn 6492 df-f 6493 df-f1 6494 df-fo 6495 df-f1o 6496 df-fv 6497 df-isom 6498 df-riota 7312 df-ov 7358 df-oprab 7359 df-mpo 7360 df-om 7806 df-1st 7930 df-2nd 7931 df-frecs 8220 df-wrecs 8251 df-recs 8300 df-rdg 8338 df-1o 8394 df-er 8631 df-map 8761 df-pm 8762 df-en 8880 df-dom 8881 df-sdom 8882 df-fin 8883 df-sup 9337 df-inf 9338 df-oi 9407 df-card 9843 df-pnf 11159 df-mnf 11160 df-xr 11161 df-ltxr 11162 df-le 11163 df-sub 11357 df-neg 11358 df-div 11786 df-nn 12137 df-2 12199 df-3 12200 df-n0 12393 df-z 12480 df-uz 12743 df-rp 12897 df-xadd 13018 df-ico 13258 df-icc 13259 df-fz 13415 df-fzo 13562 df-fl 13703 df-seq 13916 df-exp 13976 df-hash 14245 df-shft 14981 df-cj 15013 df-re 15014 df-im 15015 df-sqrt 15149 df-abs 15150 df-limsup 15385 df-clim 15402 df-rlim 15403 df-sum 15601 df-psmet 21292 df-xmet 21293 df-met 21294 df-bl 21295

This theorem is referenced by: abelthlem9 26397

W3C validator