abelthlem5 - Metamath Proof Explorer

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Theorem abelthlem5 24189

Description: Lemma for abelth 24195. (Contributed by Mario Carneiro, 1-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
abelthlem5	⊢ ((𝜑 ∧ 𝑋 ∈ (0(ball‘(abs ∘ − ))1)) → seq0( + , (𝑘 ∈ ℕ₀ ↦ ((seq0( + , 𝐴)‘𝑘) · (𝑋↑𝑘)))) ∈ dom ⇝ )

Distinct variable groups: 𝑘,𝑛,𝑥,𝑧,𝑀 𝑘,𝑋,𝑛,𝑥,𝑧 𝐴,𝑘,𝑛,𝑥,𝑧 𝜑,𝑘,𝑛,𝑥 𝑆,𝑘,𝑛,𝑥 Allowed substitution hints: 𝜑(𝑧) 𝑆(𝑧) 𝐹(𝑥,𝑧,𝑘,𝑛)

Proof of Theorem abelthlem5 Dummy variables 𝑖 𝑗 𝑚 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		nn0uz 11722	. . . 4 ⊢ ℕ₀ = (ℤ_≥‘0)
2		0zd 11389	. . . 4 ⊢ (𝜑 → 0 ∈ ℤ)
3		1rp 11836	. . . . 5 ⊢ 1 ∈ ℝ⁺
4	3	a1i 11	. . . 4 ⊢ (𝜑 → 1 ∈ ℝ⁺)
5		eqidd 2623	. . . 4 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ₀) → (seq0( + , 𝐴)‘𝑚) = (seq0( + , 𝐴)‘𝑚))
6		abelth.7	. . . 4 ⊢ (𝜑 → seq0( + , 𝐴) ⇝ 0)
7	1, 2, 4, 5, 6	climi0 14243	. . 3 ⊢ (𝜑 → ∃𝑗 ∈ ℕ₀ ∀𝑚 ∈ (ℤ_≥‘𝑗)(abs‘(seq0( + , 𝐴)‘𝑚)) < 1)
8	7	adantr 481	. 2 ⊢ ((𝜑 ∧ 𝑋 ∈ (0(ball‘(abs ∘ − ))1)) → ∃𝑗 ∈ ℕ₀ ∀𝑚 ∈ (ℤ_≥‘𝑗)(abs‘(seq0( + , 𝐴)‘𝑚)) < 1)
9		simprl 794	. . 3 ⊢ (((𝜑 ∧ 𝑋 ∈ (0(ball‘(abs ∘ − ))1)) ∧ (𝑗 ∈ ℕ₀ ∧ ∀𝑚 ∈ (ℤ_≥‘𝑗)(abs‘(seq0( + , 𝐴)‘𝑚)) < 1)) → 𝑗 ∈ ℕ₀)
10		oveq2 6658	. . . . . 6 ⊢ (𝑛 = 𝑖 → ((abs‘𝑋)↑𝑛) = ((abs‘𝑋)↑𝑖))
11		eqid 2622	. . . . . 6 ⊢ (𝑛 ∈ ℕ₀ ↦ ((abs‘𝑋)↑𝑛)) = (𝑛 ∈ ℕ₀ ↦ ((abs‘𝑋)↑𝑛))
12		ovex 6678	. . . . . 6 ⊢ ((abs‘𝑋)↑𝑖) ∈ V
13	10, 11, 12	fvmpt 6282	. . . . 5 ⊢ (𝑖 ∈ ℕ₀ → ((𝑛 ∈ ℕ₀ ↦ ((abs‘𝑋)↑𝑛))‘𝑖) = ((abs‘𝑋)↑𝑖))
14	13	adantl 482	. . . 4 ⊢ ((((𝜑 ∧ 𝑋 ∈ (0(ball‘(abs ∘ − ))1)) ∧ (𝑗 ∈ ℕ₀ ∧ ∀𝑚 ∈ (ℤ_≥‘𝑗)(abs‘(seq0( + , 𝐴)‘𝑚)) < 1)) ∧ 𝑖 ∈ ℕ₀) → ((𝑛 ∈ ℕ₀ ↦ ((abs‘𝑋)↑𝑛))‘𝑖) = ((abs‘𝑋)↑𝑖))
15		cnxmet 22576	. . . . . . . 8 ⊢ (abs ∘ − ) ∈ (∞Met‘ℂ)
16		0cn 10032	. . . . . . . 8 ⊢ 0 ∈ ℂ
17		rpxr 11840	. . . . . . . . 9 ⊢ (1 ∈ ℝ⁺ → 1 ∈ ℝ^*)
18	3, 17	ax-mp 5	. . . . . . . 8 ⊢ 1 ∈ ℝ^*
19		blssm 22223	. . . . . . . 8 ⊢ (((abs ∘ − ) ∈ (∞Met‘ℂ) ∧ 0 ∈ ℂ ∧ 1 ∈ ℝ^*) → (0(ball‘(abs ∘ − ))1) ⊆ ℂ)
20	15, 16, 18, 19	mp3an 1424	. . . . . . 7 ⊢ (0(ball‘(abs ∘ − ))1) ⊆ ℂ
21		simplr 792	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑋 ∈ (0(ball‘(abs ∘ − ))1)) ∧ (𝑗 ∈ ℕ₀ ∧ ∀𝑚 ∈ (ℤ_≥‘𝑗)(abs‘(seq0( + , 𝐴)‘𝑚)) < 1)) → 𝑋 ∈ (0(ball‘(abs ∘ − ))1))
22	20, 21	sseldi 3601	. . . . . 6 ⊢ (((𝜑 ∧ 𝑋 ∈ (0(ball‘(abs ∘ − ))1)) ∧ (𝑗 ∈ ℕ₀ ∧ ∀𝑚 ∈ (ℤ_≥‘𝑗)(abs‘(seq0( + , 𝐴)‘𝑚)) < 1)) → 𝑋 ∈ ℂ)
23	22	abscld 14175	. . . . 5 ⊢ (((𝜑 ∧ 𝑋 ∈ (0(ball‘(abs ∘ − ))1)) ∧ (𝑗 ∈ ℕ₀ ∧ ∀𝑚 ∈ (ℤ_≥‘𝑗)(abs‘(seq0( + , 𝐴)‘𝑚)) < 1)) → (abs‘𝑋) ∈ ℝ)
24		reexpcl 12877	. . . . 5 ⊢ (((abs‘𝑋) ∈ ℝ ∧ 𝑖 ∈ ℕ₀) → ((abs‘𝑋)↑𝑖) ∈ ℝ)
25	23, 24	sylan 488	. . . 4 ⊢ ((((𝜑 ∧ 𝑋 ∈ (0(ball‘(abs ∘ − ))1)) ∧ (𝑗 ∈ ℕ₀ ∧ ∀𝑚 ∈ (ℤ_≥‘𝑗)(abs‘(seq0( + , 𝐴)‘𝑚)) < 1)) ∧ 𝑖 ∈ ℕ₀) → ((abs‘𝑋)↑𝑖) ∈ ℝ)
26	14, 25	eqeltrd 2701	. . 3 ⊢ ((((𝜑 ∧ 𝑋 ∈ (0(ball‘(abs ∘ − ))1)) ∧ (𝑗 ∈ ℕ₀ ∧ ∀𝑚 ∈ (ℤ_≥‘𝑗)(abs‘(seq0( + , 𝐴)‘𝑚)) < 1)) ∧ 𝑖 ∈ ℕ₀) → ((𝑛 ∈ ℕ₀ ↦ ((abs‘𝑋)↑𝑛))‘𝑖) ∈ ℝ)
27		fveq2 6191	. . . . . . 7 ⊢ (𝑘 = 𝑖 → (seq0( + , 𝐴)‘𝑘) = (seq0( + , 𝐴)‘𝑖))
28		oveq2 6658	. . . . . . 7 ⊢ (𝑘 = 𝑖 → (𝑋↑𝑘) = (𝑋↑𝑖))
29	27, 28	oveq12d 6668	. . . . . 6 ⊢ (𝑘 = 𝑖 → ((seq0( + , 𝐴)‘𝑘) · (𝑋↑𝑘)) = ((seq0( + , 𝐴)‘𝑖) · (𝑋↑𝑖)))
30		eqid 2622	. . . . . 6 ⊢ (𝑘 ∈ ℕ₀ ↦ ((seq0( + , 𝐴)‘𝑘) · (𝑋↑𝑘))) = (𝑘 ∈ ℕ₀ ↦ ((seq0( + , 𝐴)‘𝑘) · (𝑋↑𝑘)))
31		ovex 6678	. . . . . 6 ⊢ ((seq0( + , 𝐴)‘𝑖) · (𝑋↑𝑖)) ∈ V
32	29, 30, 31	fvmpt 6282	. . . . 5 ⊢ (𝑖 ∈ ℕ₀ → ((𝑘 ∈ ℕ₀ ↦ ((seq0( + , 𝐴)‘𝑘) · (𝑋↑𝑘)))‘𝑖) = ((seq0( + , 𝐴)‘𝑖) · (𝑋↑𝑖)))
33	32	adantl 482	. . . 4 ⊢ ((((𝜑 ∧ 𝑋 ∈ (0(ball‘(abs ∘ − ))1)) ∧ (𝑗 ∈ ℕ₀ ∧ ∀𝑚 ∈ (ℤ_≥‘𝑗)(abs‘(seq0( + , 𝐴)‘𝑚)) < 1)) ∧ 𝑖 ∈ ℕ₀) → ((𝑘 ∈ ℕ₀ ↦ ((seq0( + , 𝐴)‘𝑘) · (𝑋↑𝑘)))‘𝑖) = ((seq0( + , 𝐴)‘𝑖) · (𝑋↑𝑖)))
34		abelth.1	. . . . . . . . 9 ⊢ (𝜑 → 𝐴:ℕ₀⟶ℂ)
35	34	ffvelrnda 6359	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ℕ₀) → (𝐴‘𝑥) ∈ ℂ)
36	1, 2, 35	serf 12829	. . . . . . 7 ⊢ (𝜑 → seq0( + , 𝐴):ℕ₀⟶ℂ)
37	36	ad2antrr 762	. . . . . 6 ⊢ (((𝜑 ∧ 𝑋 ∈ (0(ball‘(abs ∘ − ))1)) ∧ (𝑗 ∈ ℕ₀ ∧ ∀𝑚 ∈ (ℤ_≥‘𝑗)(abs‘(seq0( + , 𝐴)‘𝑚)) < 1)) → seq0( + , 𝐴):ℕ₀⟶ℂ)
38	37	ffvelrnda 6359	. . . . 5 ⊢ ((((𝜑 ∧ 𝑋 ∈ (0(ball‘(abs ∘ − ))1)) ∧ (𝑗 ∈ ℕ₀ ∧ ∀𝑚 ∈ (ℤ_≥‘𝑗)(abs‘(seq0( + , 𝐴)‘𝑚)) < 1)) ∧ 𝑖 ∈ ℕ₀) → (seq0( + , 𝐴)‘𝑖) ∈ ℂ)
39		expcl 12878	. . . . . 6 ⊢ ((𝑋 ∈ ℂ ∧ 𝑖 ∈ ℕ₀) → (𝑋↑𝑖) ∈ ℂ)
40	22, 39	sylan 488	. . . . 5 ⊢ ((((𝜑 ∧ 𝑋 ∈ (0(ball‘(abs ∘ − ))1)) ∧ (𝑗 ∈ ℕ₀ ∧ ∀𝑚 ∈ (ℤ_≥‘𝑗)(abs‘(seq0( + , 𝐴)‘𝑚)) < 1)) ∧ 𝑖 ∈ ℕ₀) → (𝑋↑𝑖) ∈ ℂ)
41	38, 40	mulcld 10060	. . . 4 ⊢ ((((𝜑 ∧ 𝑋 ∈ (0(ball‘(abs ∘ − ))1)) ∧ (𝑗 ∈ ℕ₀ ∧ ∀𝑚 ∈ (ℤ_≥‘𝑗)(abs‘(seq0( + , 𝐴)‘𝑚)) < 1)) ∧ 𝑖 ∈ ℕ₀) → ((seq0( + , 𝐴)‘𝑖) · (𝑋↑𝑖)) ∈ ℂ)
42	33, 41	eqeltrd 2701	. . 3 ⊢ ((((𝜑 ∧ 𝑋 ∈ (0(ball‘(abs ∘ − ))1)) ∧ (𝑗 ∈ ℕ₀ ∧ ∀𝑚 ∈ (ℤ_≥‘𝑗)(abs‘(seq0( + , 𝐴)‘𝑚)) < 1)) ∧ 𝑖 ∈ ℕ₀) → ((𝑘 ∈ ℕ₀ ↦ ((seq0( + , 𝐴)‘𝑘) · (𝑋↑𝑘)))‘𝑖) ∈ ℂ)
43	23	recnd 10068	. . . . 5 ⊢ (((𝜑 ∧ 𝑋 ∈ (0(ball‘(abs ∘ − ))1)) ∧ (𝑗 ∈ ℕ₀ ∧ ∀𝑚 ∈ (ℤ_≥‘𝑗)(abs‘(seq0( + , 𝐴)‘𝑚)) < 1)) → (abs‘𝑋) ∈ ℂ)
44		absidm 14063	. . . . . . 7 ⊢ (𝑋 ∈ ℂ → (abs‘(abs‘𝑋)) = (abs‘𝑋))
45	22, 44	syl 17	. . . . . 6 ⊢ (((𝜑 ∧ 𝑋 ∈ (0(ball‘(abs ∘ − ))1)) ∧ (𝑗 ∈ ℕ₀ ∧ ∀𝑚 ∈ (ℤ_≥‘𝑗)(abs‘(seq0( + , 𝐴)‘𝑚)) < 1)) → (abs‘(abs‘𝑋)) = (abs‘𝑋))
46		eqid 2622	. . . . . . . . . 10 ⊢ (abs ∘ − ) = (abs ∘ − )
47	46	cnmetdval 22574	. . . . . . . . 9 ⊢ ((𝑋 ∈ ℂ ∧ 0 ∈ ℂ) → (𝑋(abs ∘ − )0) = (abs‘(𝑋 − 0)))
48	22, 16, 47	sylancl 694	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑋 ∈ (0(ball‘(abs ∘ − ))1)) ∧ (𝑗 ∈ ℕ₀ ∧ ∀𝑚 ∈ (ℤ_≥‘𝑗)(abs‘(seq0( + , 𝐴)‘𝑚)) < 1)) → (𝑋(abs ∘ − )0) = (abs‘(𝑋 − 0)))
49	22	subid1d 10381	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑋 ∈ (0(ball‘(abs ∘ − ))1)) ∧ (𝑗 ∈ ℕ₀ ∧ ∀𝑚 ∈ (ℤ_≥‘𝑗)(abs‘(seq0( + , 𝐴)‘𝑚)) < 1)) → (𝑋 − 0) = 𝑋)
50	49	fveq2d 6195	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑋 ∈ (0(ball‘(abs ∘ − ))1)) ∧ (𝑗 ∈ ℕ₀ ∧ ∀𝑚 ∈ (ℤ_≥‘𝑗)(abs‘(seq0( + , 𝐴)‘𝑚)) < 1)) → (abs‘(𝑋 − 0)) = (abs‘𝑋))
51	48, 50	eqtrd 2656	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑋 ∈ (0(ball‘(abs ∘ − ))1)) ∧ (𝑗 ∈ ℕ₀ ∧ ∀𝑚 ∈ (ℤ_≥‘𝑗)(abs‘(seq0( + , 𝐴)‘𝑚)) < 1)) → (𝑋(abs ∘ − )0) = (abs‘𝑋))
52		elbl3 22197	. . . . . . . . . 10 ⊢ ((((abs ∘ − ) ∈ (∞Met‘ℂ) ∧ 1 ∈ ℝ^*) ∧ (0 ∈ ℂ ∧ 𝑋 ∈ ℂ)) → (𝑋 ∈ (0(ball‘(abs ∘ − ))1) ↔ (𝑋(abs ∘ − )0) < 1))
53	15, 18, 52	mpanl12 718	. . . . . . . . 9 ⊢ ((0 ∈ ℂ ∧ 𝑋 ∈ ℂ) → (𝑋 ∈ (0(ball‘(abs ∘ − ))1) ↔ (𝑋(abs ∘ − )0) < 1))
54	16, 22, 53	sylancr 695	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑋 ∈ (0(ball‘(abs ∘ − ))1)) ∧ (𝑗 ∈ ℕ₀ ∧ ∀𝑚 ∈ (ℤ_≥‘𝑗)(abs‘(seq0( + , 𝐴)‘𝑚)) < 1)) → (𝑋 ∈ (0(ball‘(abs ∘ − ))1) ↔ (𝑋(abs ∘ − )0) < 1))
55	21, 54	mpbid 222	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑋 ∈ (0(ball‘(abs ∘ − ))1)) ∧ (𝑗 ∈ ℕ₀ ∧ ∀𝑚 ∈ (ℤ_≥‘𝑗)(abs‘(seq0( + , 𝐴)‘𝑚)) < 1)) → (𝑋(abs ∘ − )0) < 1)
56	51, 55	eqbrtrrd 4677	. . . . . 6 ⊢ (((𝜑 ∧ 𝑋 ∈ (0(ball‘(abs ∘ − ))1)) ∧ (𝑗 ∈ ℕ₀ ∧ ∀𝑚 ∈ (ℤ_≥‘𝑗)(abs‘(seq0( + , 𝐴)‘𝑚)) < 1)) → (abs‘𝑋) < 1)
57	45, 56	eqbrtrd 4675	. . . . 5 ⊢ (((𝜑 ∧ 𝑋 ∈ (0(ball‘(abs ∘ − ))1)) ∧ (𝑗 ∈ ℕ₀ ∧ ∀𝑚 ∈ (ℤ_≥‘𝑗)(abs‘(seq0( + , 𝐴)‘𝑚)) < 1)) → (abs‘(abs‘𝑋)) < 1)
58	43, 57, 14	geolim 14601	. . . 4 ⊢ (((𝜑 ∧ 𝑋 ∈ (0(ball‘(abs ∘ − ))1)) ∧ (𝑗 ∈ ℕ₀ ∧ ∀𝑚 ∈ (ℤ_≥‘𝑗)(abs‘(seq0( + , 𝐴)‘𝑚)) < 1)) → seq0( + , (𝑛 ∈ ℕ₀ ↦ ((abs‘𝑋)↑𝑛))) ⇝ (1 / (1 − (abs‘𝑋))))
59		climrel 14223	. . . . 5 ⊢ Rel ⇝
60	59	releldmi 5362	. . . 4 ⊢ (seq0( + , (𝑛 ∈ ℕ₀ ↦ ((abs‘𝑋)↑𝑛))) ⇝ (1 / (1 − (abs‘𝑋))) → seq0( + , (𝑛 ∈ ℕ₀ ↦ ((abs‘𝑋)↑𝑛))) ∈ dom ⇝ )
61	58, 60	syl 17	. . 3 ⊢ (((𝜑 ∧ 𝑋 ∈ (0(ball‘(abs ∘ − ))1)) ∧ (𝑗 ∈ ℕ₀ ∧ ∀𝑚 ∈ (ℤ_≥‘𝑗)(abs‘(seq0( + , 𝐴)‘𝑚)) < 1)) → seq0( + , (𝑛 ∈ ℕ₀ ↦ ((abs‘𝑋)↑𝑛))) ∈ dom ⇝ )
62		1red 10055	. . 3 ⊢ (((𝜑 ∧ 𝑋 ∈ (0(ball‘(abs ∘ − ))1)) ∧ (𝑗 ∈ ℕ₀ ∧ ∀𝑚 ∈ (ℤ_≥‘𝑗)(abs‘(seq0( + , 𝐴)‘𝑚)) < 1)) → 1 ∈ ℝ)
63	37	adantr 481	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑋 ∈ (0(ball‘(abs ∘ − ))1)) ∧ (𝑗 ∈ ℕ₀ ∧ ∀𝑚 ∈ (ℤ_≥‘𝑗)(abs‘(seq0( + , 𝐴)‘𝑚)) < 1)) ∧ 𝑖 ∈ (ℤ_≥‘𝑗)) → seq0( + , 𝐴):ℕ₀⟶ℂ)
64		eluznn0 11757	. . . . . . . . 9 ⊢ ((𝑗 ∈ ℕ₀ ∧ 𝑖 ∈ (ℤ_≥‘𝑗)) → 𝑖 ∈ ℕ₀)
65	9, 64	sylan 488	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑋 ∈ (0(ball‘(abs ∘ − ))1)) ∧ (𝑗 ∈ ℕ₀ ∧ ∀𝑚 ∈ (ℤ_≥‘𝑗)(abs‘(seq0( + , 𝐴)‘𝑚)) < 1)) ∧ 𝑖 ∈ (ℤ_≥‘𝑗)) → 𝑖 ∈ ℕ₀)
66	63, 65	ffvelrnd 6360	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑋 ∈ (0(ball‘(abs ∘ − ))1)) ∧ (𝑗 ∈ ℕ₀ ∧ ∀𝑚 ∈ (ℤ_≥‘𝑗)(abs‘(seq0( + , 𝐴)‘𝑚)) < 1)) ∧ 𝑖 ∈ (ℤ_≥‘𝑗)) → (seq0( + , 𝐴)‘𝑖) ∈ ℂ)
67	65, 40	syldan 487	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑋 ∈ (0(ball‘(abs ∘ − ))1)) ∧ (𝑗 ∈ ℕ₀ ∧ ∀𝑚 ∈ (ℤ_≥‘𝑗)(abs‘(seq0( + , 𝐴)‘𝑚)) < 1)) ∧ 𝑖 ∈ (ℤ_≥‘𝑗)) → (𝑋↑𝑖) ∈ ℂ)
68	66, 67	absmuld 14193	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑋 ∈ (0(ball‘(abs ∘ − ))1)) ∧ (𝑗 ∈ ℕ₀ ∧ ∀𝑚 ∈ (ℤ_≥‘𝑗)(abs‘(seq0( + , 𝐴)‘𝑚)) < 1)) ∧ 𝑖 ∈ (ℤ_≥‘𝑗)) → (abs‘((seq0( + , 𝐴)‘𝑖) · (𝑋↑𝑖))) = ((abs‘(seq0( + , 𝐴)‘𝑖)) · (abs‘(𝑋↑𝑖))))
69	22	adantr 481	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑋 ∈ (0(ball‘(abs ∘ − ))1)) ∧ (𝑗 ∈ ℕ₀ ∧ ∀𝑚 ∈ (ℤ_≥‘𝑗)(abs‘(seq0( + , 𝐴)‘𝑚)) < 1)) ∧ 𝑖 ∈ (ℤ_≥‘𝑗)) → 𝑋 ∈ ℂ)
70	69, 65	absexpd 14191	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑋 ∈ (0(ball‘(abs ∘ − ))1)) ∧ (𝑗 ∈ ℕ₀ ∧ ∀𝑚 ∈ (ℤ_≥‘𝑗)(abs‘(seq0( + , 𝐴)‘𝑚)) < 1)) ∧ 𝑖 ∈ (ℤ_≥‘𝑗)) → (abs‘(𝑋↑𝑖)) = ((abs‘𝑋)↑𝑖))
71	70	oveq2d 6666	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑋 ∈ (0(ball‘(abs ∘ − ))1)) ∧ (𝑗 ∈ ℕ₀ ∧ ∀𝑚 ∈ (ℤ_≥‘𝑗)(abs‘(seq0( + , 𝐴)‘𝑚)) < 1)) ∧ 𝑖 ∈ (ℤ_≥‘𝑗)) → ((abs‘(seq0( + , 𝐴)‘𝑖)) · (abs‘(𝑋↑𝑖))) = ((abs‘(seq0( + , 𝐴)‘𝑖)) · ((abs‘𝑋)↑𝑖)))
72	68, 71	eqtrd 2656	. . . . 5 ⊢ ((((𝜑 ∧ 𝑋 ∈ (0(ball‘(abs ∘ − ))1)) ∧ (𝑗 ∈ ℕ₀ ∧ ∀𝑚 ∈ (ℤ_≥‘𝑗)(abs‘(seq0( + , 𝐴)‘𝑚)) < 1)) ∧ 𝑖 ∈ (ℤ_≥‘𝑗)) → (abs‘((seq0( + , 𝐴)‘𝑖) · (𝑋↑𝑖))) = ((abs‘(seq0( + , 𝐴)‘𝑖)) · ((abs‘𝑋)↑𝑖)))
73	66	abscld 14175	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑋 ∈ (0(ball‘(abs ∘ − ))1)) ∧ (𝑗 ∈ ℕ₀ ∧ ∀𝑚 ∈ (ℤ_≥‘𝑗)(abs‘(seq0( + , 𝐴)‘𝑚)) < 1)) ∧ 𝑖 ∈ (ℤ_≥‘𝑗)) → (abs‘(seq0( + , 𝐴)‘𝑖)) ∈ ℝ)
74		1red 10055	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑋 ∈ (0(ball‘(abs ∘ − ))1)) ∧ (𝑗 ∈ ℕ₀ ∧ ∀𝑚 ∈ (ℤ_≥‘𝑗)(abs‘(seq0( + , 𝐴)‘𝑚)) < 1)) ∧ 𝑖 ∈ (ℤ_≥‘𝑗)) → 1 ∈ ℝ)
75	65, 25	syldan 487	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑋 ∈ (0(ball‘(abs ∘ − ))1)) ∧ (𝑗 ∈ ℕ₀ ∧ ∀𝑚 ∈ (ℤ_≥‘𝑗)(abs‘(seq0( + , 𝐴)‘𝑚)) < 1)) ∧ 𝑖 ∈ (ℤ_≥‘𝑗)) → ((abs‘𝑋)↑𝑖) ∈ ℝ)
76	67	absge0d 14183	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑋 ∈ (0(ball‘(abs ∘ − ))1)) ∧ (𝑗 ∈ ℕ₀ ∧ ∀𝑚 ∈ (ℤ_≥‘𝑗)(abs‘(seq0( + , 𝐴)‘𝑚)) < 1)) ∧ 𝑖 ∈ (ℤ_≥‘𝑗)) → 0 ≤ (abs‘(𝑋↑𝑖)))
77	76, 70	breqtrd 4679	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑋 ∈ (0(ball‘(abs ∘ − ))1)) ∧ (𝑗 ∈ ℕ₀ ∧ ∀𝑚 ∈ (ℤ_≥‘𝑗)(abs‘(seq0( + , 𝐴)‘𝑚)) < 1)) ∧ 𝑖 ∈ (ℤ_≥‘𝑗)) → 0 ≤ ((abs‘𝑋)↑𝑖))
78		simprr 796	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑋 ∈ (0(ball‘(abs ∘ − ))1)) ∧ (𝑗 ∈ ℕ₀ ∧ ∀𝑚 ∈ (ℤ_≥‘𝑗)(abs‘(seq0( + , 𝐴)‘𝑚)) < 1)) → ∀𝑚 ∈ (ℤ_≥‘𝑗)(abs‘(seq0( + , 𝐴)‘𝑚)) < 1)
79		fveq2 6191	. . . . . . . . . . 11 ⊢ (𝑚 = 𝑖 → (seq0( + , 𝐴)‘𝑚) = (seq0( + , 𝐴)‘𝑖))
80	79	fveq2d 6195	. . . . . . . . . 10 ⊢ (𝑚 = 𝑖 → (abs‘(seq0( + , 𝐴)‘𝑚)) = (abs‘(seq0( + , 𝐴)‘𝑖)))
81	80	breq1d 4663	. . . . . . . . 9 ⊢ (𝑚 = 𝑖 → ((abs‘(seq0( + , 𝐴)‘𝑚)) < 1 ↔ (abs‘(seq0( + , 𝐴)‘𝑖)) < 1))
82	81	rspccva 3308	. . . . . . . 8 ⊢ ((∀𝑚 ∈ (ℤ_≥‘𝑗)(abs‘(seq0( + , 𝐴)‘𝑚)) < 1 ∧ 𝑖 ∈ (ℤ_≥‘𝑗)) → (abs‘(seq0( + , 𝐴)‘𝑖)) < 1)
83	78, 82	sylan 488	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑋 ∈ (0(ball‘(abs ∘ − ))1)) ∧ (𝑗 ∈ ℕ₀ ∧ ∀𝑚 ∈ (ℤ_≥‘𝑗)(abs‘(seq0( + , 𝐴)‘𝑚)) < 1)) ∧ 𝑖 ∈ (ℤ_≥‘𝑗)) → (abs‘(seq0( + , 𝐴)‘𝑖)) < 1)
84		1re 10039	. . . . . . . 8 ⊢ 1 ∈ ℝ
85		ltle 10126	. . . . . . . 8 ⊢ (((abs‘(seq0( + , 𝐴)‘𝑖)) ∈ ℝ ∧ 1 ∈ ℝ) → ((abs‘(seq0( + , 𝐴)‘𝑖)) < 1 → (abs‘(seq0( + , 𝐴)‘𝑖)) ≤ 1))
86	73, 84, 85	sylancl 694	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑋 ∈ (0(ball‘(abs ∘ − ))1)) ∧ (𝑗 ∈ ℕ₀ ∧ ∀𝑚 ∈ (ℤ_≥‘𝑗)(abs‘(seq0( + , 𝐴)‘𝑚)) < 1)) ∧ 𝑖 ∈ (ℤ_≥‘𝑗)) → ((abs‘(seq0( + , 𝐴)‘𝑖)) < 1 → (abs‘(seq0( + , 𝐴)‘𝑖)) ≤ 1))
87	83, 86	mpd 15	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑋 ∈ (0(ball‘(abs ∘ − ))1)) ∧ (𝑗 ∈ ℕ₀ ∧ ∀𝑚 ∈ (ℤ_≥‘𝑗)(abs‘(seq0( + , 𝐴)‘𝑚)) < 1)) ∧ 𝑖 ∈ (ℤ_≥‘𝑗)) → (abs‘(seq0( + , 𝐴)‘𝑖)) ≤ 1)
88	73, 74, 75, 77, 87	lemul1ad 10963	. . . . 5 ⊢ ((((𝜑 ∧ 𝑋 ∈ (0(ball‘(abs ∘ − ))1)) ∧ (𝑗 ∈ ℕ₀ ∧ ∀𝑚 ∈ (ℤ_≥‘𝑗)(abs‘(seq0( + , 𝐴)‘𝑚)) < 1)) ∧ 𝑖 ∈ (ℤ_≥‘𝑗)) → ((abs‘(seq0( + , 𝐴)‘𝑖)) · ((abs‘𝑋)↑𝑖)) ≤ (1 · ((abs‘𝑋)↑𝑖)))
89	72, 88	eqbrtrd 4675	. . . 4 ⊢ ((((𝜑 ∧ 𝑋 ∈ (0(ball‘(abs ∘ − ))1)) ∧ (𝑗 ∈ ℕ₀ ∧ ∀𝑚 ∈ (ℤ_≥‘𝑗)(abs‘(seq0( + , 𝐴)‘𝑚)) < 1)) ∧ 𝑖 ∈ (ℤ_≥‘𝑗)) → (abs‘((seq0( + , 𝐴)‘𝑖) · (𝑋↑𝑖))) ≤ (1 · ((abs‘𝑋)↑𝑖)))
90	65, 32	syl 17	. . . . 5 ⊢ ((((𝜑 ∧ 𝑋 ∈ (0(ball‘(abs ∘ − ))1)) ∧ (𝑗 ∈ ℕ₀ ∧ ∀𝑚 ∈ (ℤ_≥‘𝑗)(abs‘(seq0( + , 𝐴)‘𝑚)) < 1)) ∧ 𝑖 ∈ (ℤ_≥‘𝑗)) → ((𝑘 ∈ ℕ₀ ↦ ((seq0( + , 𝐴)‘𝑘) · (𝑋↑𝑘)))‘𝑖) = ((seq0( + , 𝐴)‘𝑖) · (𝑋↑𝑖)))
91	90	fveq2d 6195	. . . 4 ⊢ ((((𝜑 ∧ 𝑋 ∈ (0(ball‘(abs ∘ − ))1)) ∧ (𝑗 ∈ ℕ₀ ∧ ∀𝑚 ∈ (ℤ_≥‘𝑗)(abs‘(seq0( + , 𝐴)‘𝑚)) < 1)) ∧ 𝑖 ∈ (ℤ_≥‘𝑗)) → (abs‘((𝑘 ∈ ℕ₀ ↦ ((seq0( + , 𝐴)‘𝑘) · (𝑋↑𝑘)))‘𝑖)) = (abs‘((seq0( + , 𝐴)‘𝑖) · (𝑋↑𝑖))))
92	65, 13	syl 17	. . . . 5 ⊢ ((((𝜑 ∧ 𝑋 ∈ (0(ball‘(abs ∘ − ))1)) ∧ (𝑗 ∈ ℕ₀ ∧ ∀𝑚 ∈ (ℤ_≥‘𝑗)(abs‘(seq0( + , 𝐴)‘𝑚)) < 1)) ∧ 𝑖 ∈ (ℤ_≥‘𝑗)) → ((𝑛 ∈ ℕ₀ ↦ ((abs‘𝑋)↑𝑛))‘𝑖) = ((abs‘𝑋)↑𝑖))
93	92	oveq2d 6666	. . . 4 ⊢ ((((𝜑 ∧ 𝑋 ∈ (0(ball‘(abs ∘ − ))1)) ∧ (𝑗 ∈ ℕ₀ ∧ ∀𝑚 ∈ (ℤ_≥‘𝑗)(abs‘(seq0( + , 𝐴)‘𝑚)) < 1)) ∧ 𝑖 ∈ (ℤ_≥‘𝑗)) → (1 · ((𝑛 ∈ ℕ₀ ↦ ((abs‘𝑋)↑𝑛))‘𝑖)) = (1 · ((abs‘𝑋)↑𝑖)))
94	89, 91, 93	3brtr4d 4685	. . 3 ⊢ ((((𝜑 ∧ 𝑋 ∈ (0(ball‘(abs ∘ − ))1)) ∧ (𝑗 ∈ ℕ₀ ∧ ∀𝑚 ∈ (ℤ_≥‘𝑗)(abs‘(seq0( + , 𝐴)‘𝑚)) < 1)) ∧ 𝑖 ∈ (ℤ_≥‘𝑗)) → (abs‘((𝑘 ∈ ℕ₀ ↦ ((seq0( + , 𝐴)‘𝑘) · (𝑋↑𝑘)))‘𝑖)) ≤ (1 · ((𝑛 ∈ ℕ₀ ↦ ((abs‘𝑋)↑𝑛))‘𝑖)))
95	1, 9, 26, 42, 61, 62, 94	cvgcmpce 14550	. 2 ⊢ (((𝜑 ∧ 𝑋 ∈ (0(ball‘(abs ∘ − ))1)) ∧ (𝑗 ∈ ℕ₀ ∧ ∀𝑚 ∈ (ℤ_≥‘𝑗)(abs‘(seq0( + , 𝐴)‘𝑚)) < 1)) → seq0( + , (𝑘 ∈ ℕ₀ ↦ ((seq0( + , 𝐴)‘𝑘) · (𝑋↑𝑘)))) ∈ dom ⇝ )
96	8, 95	rexlimddv 3035	1 ⊢ ((𝜑 ∧ 𝑋 ∈ (0(ball‘(abs ∘ − ))1)) → seq0( + , (𝑘 ∈ ℕ₀ ↦ ((seq0( + , 𝐴)‘𝑘) · (𝑋↑𝑘)))) ∈ dom ⇝ )

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 196 ∧ wa 384 = wceq 1483 ∈ wcel 1990 ∀wral 2912 ∃wrex 2913 {crab 2916 ⊆ wss 3574 class class class wbr 4653 ↦ cmpt 4729 dom cdm 5114 ∘ ccom 5118 ⟶wf 5884 ‘cfv 5888 (class class class)co 6650 ℂcc 9934 ℝcr 9935 0cc0 9936 1c1 9937 + caddc 9939 · cmul 9941 ℝ^*cxr 10073 < clt 10074 ≤ cle 10075 − cmin 10266 / cdiv 10684 ℕ₀cn0 11292 ℤ_≥cuz 11687 ℝ⁺crp 11832 seqcseq 12801 ↑cexp 12860 abscabs 13974 ⇝ cli 14215 Σcsu 14416 ∞Metcxmt 19731 ballcbl 19733

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-inf2 8538 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 ax-pre-sup 10014 ax-addf 10015 ax-mulf 10016

This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-fal 1489 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-se 5074 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-isom 5897 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-oadd 7564 df-er 7742 df-map 7859 df-pm 7860 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-sup 8348 df-inf 8349 df-oi 8415 df-card 8765 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-div 10685 df-nn 11021 df-2 11079 df-3 11080 df-n0 11293 df-z 11378 df-uz 11688 df-rp 11833 df-xadd 11947 df-ico 12181 df-fz 12327 df-fzo 12466 df-fl 12593 df-seq 12802 df-exp 12861 df-hash 13118 df-cj 13839 df-re 13840 df-im 13841 df-sqrt 13975 df-abs 13976 df-limsup 14202 df-clim 14219 df-rlim 14220 df-sum 14417 df-psmet 19738 df-xmet 19739 df-met 19740 df-bl 19741

This theorem is referenced by: abelthlem6 24190 abelthlem7 24192

W3C validator