abelthlem5 - Metamath Proof Explorer

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Theorem abelthlem5 26795

Description: Lemma for abelth 26801. (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 13033	. . . 4 ⊢ ℕ₀ = (ℤ_≥‘0)
2		0zd 12735	. . . 4 ⊢ (𝜑 → 0 ∈ ℤ)
3		1rp 13154	. . . . 5 ⊢ 1 ∈ ℝ⁺
4	3	a1i 11	. . . 4 ⊢ (𝜑 → 1 ∈ ℝ⁺)
5		eqidd 2764	. . . 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 15711	. . 3 ⊢ (𝜑 → ∃𝑗 ∈ ℕ₀ ∀𝑚 ∈ (ℤ_≥‘𝑗)(abs‘(seq0( + , 𝐴)‘𝑚)) < 1)
8	7	adantr 486	. 2 ⊢ ((𝜑 ∧ 𝑋 ∈ (0(ball‘(abs ∘ − ))1)) → ∃𝑗 ∈ ℕ₀ ∀𝑚 ∈ (ℤ_≥‘𝑗)(abs‘(seq0( + , 𝐴)‘𝑚)) < 1)
9		simprl 783	. . 3 ⊢ (((𝜑 ∧ 𝑋 ∈ (0(ball‘(abs ∘ − ))1)) ∧ (𝑗 ∈ ℕ₀ ∧ ∀𝑚 ∈ (ℤ_≥‘𝑗)(abs‘(seq0( + , 𝐴)‘𝑚)) < 1)) → 𝑗 ∈ ℕ₀)
10		oveq2 7454	. . . . . 6 ⊢ (𝑛 = 𝑖 → ((abs‘𝑋)↑𝑛) = ((abs‘𝑋)↑𝑖))
11		eqid 2763	. . . . . 6 ⊢ (𝑛 ∈ ℕ₀ ↦ ((abs‘𝑋)↑𝑛)) = (𝑛 ∈ ℕ₀ ↦ ((abs‘𝑋)↑𝑛))
12		ovex 7479	. . . . . 6 ⊢ ((abs‘𝑋)↑𝑖) ∈ V
13	10, 11, 12	fvmpt 7019	. . . . 5 ⊢ (𝑖 ∈ ℕ₀ → ((𝑛 ∈ ℕ₀ ↦ ((abs‘𝑋)↑𝑛))‘𝑖) = ((abs‘𝑋)↑𝑖))
14	13	adantl 487	. . . 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 25124	. . . . . . . 8 ⊢ (abs ∘ − ) ∈ (∞Met‘ℂ)
16		0cn 11328	. . . . . . . 8 ⊢ 0 ∈ ℂ
17		1xr 11398	. . . . . . . 8 ⊢ 1 ∈ ℝ^*
18		blssm 24770	. . . . . . . 8 ⊢ (((abs ∘ − ) ∈ (∞Met‘ℂ) ∧ 0 ∈ ℂ ∧ 1 ∈ ℝ^*) → (0(ball‘(abs ∘ − ))1) ⊆ ℂ)
19	15, 16, 17, 18	mp3an 1491	. . . . . . 7 ⊢ (0(ball‘(abs ∘ − ))1) ⊆ ℂ
20		simplr 781	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑋 ∈ (0(ball‘(abs ∘ − ))1)) ∧ (𝑗 ∈ ℕ₀ ∧ ∀𝑚 ∈ (ℤ_≥‘𝑗)(abs‘(seq0( + , 𝐴)‘𝑚)) < 1)) → 𝑋 ∈ (0(ball‘(abs ∘ − ))1))
21	19, 20	sselid 3911	. . . . . 6 ⊢ (((𝜑 ∧ 𝑋 ∈ (0(ball‘(abs ∘ − ))1)) ∧ (𝑗 ∈ ℕ₀ ∧ ∀𝑚 ∈ (ℤ_≥‘𝑗)(abs‘(seq0( + , 𝐴)‘𝑚)) < 1)) → 𝑋 ∈ ℂ)
22	21	abscld 15638	. . . . 5 ⊢ (((𝜑 ∧ 𝑋 ∈ (0(ball‘(abs ∘ − ))1)) ∧ (𝑗 ∈ ℕ₀ ∧ ∀𝑚 ∈ (ℤ_≥‘𝑗)(abs‘(seq0( + , 𝐴)‘𝑚)) < 1)) → (abs‘𝑋) ∈ ℝ)
23		reexpcl 14251	. . . . 5 ⊢ (((abs‘𝑋) ∈ ℝ ∧ 𝑖 ∈ ℕ₀) → ((abs‘𝑋)↑𝑖) ∈ ℝ)
24	22, 23	sylan 592	. . . 4 ⊢ ((((𝜑 ∧ 𝑋 ∈ (0(ball‘(abs ∘ − ))1)) ∧ (𝑗 ∈ ℕ₀ ∧ ∀𝑚 ∈ (ℤ_≥‘𝑗)(abs‘(seq0( + , 𝐴)‘𝑚)) < 1)) ∧ 𝑖 ∈ ℕ₀) → ((abs‘𝑋)↑𝑖) ∈ ℝ)
25	14, 24	eqeltrd 2864	. . 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)
26		fveq2 6911	. . . . . . 7 ⊢ (𝑘 = 𝑖 → (seq0( + , 𝐴)‘𝑘) = (seq0( + , 𝐴)‘𝑖))
27		oveq2 7454	. . . . . . 7 ⊢ (𝑘 = 𝑖 → (𝑋↑𝑘) = (𝑋↑𝑖))
28	26, 27	oveq12d 7464	. . . . . 6 ⊢ (𝑘 = 𝑖 → ((seq0( + , 𝐴)‘𝑘) · (𝑋↑𝑘)) = ((seq0( + , 𝐴)‘𝑖) · (𝑋↑𝑖)))
29		eqid 2763	. . . . . 6 ⊢ (𝑘 ∈ ℕ₀ ↦ ((seq0( + , 𝐴)‘𝑘) · (𝑋↑𝑘))) = (𝑘 ∈ ℕ₀ ↦ ((seq0( + , 𝐴)‘𝑘) · (𝑋↑𝑘)))
30		ovex 7479	. . . . . 6 ⊢ ((seq0( + , 𝐴)‘𝑖) · (𝑋↑𝑖)) ∈ V
31	28, 29, 30	fvmpt 7019	. . . . 5 ⊢ (𝑖 ∈ ℕ₀ → ((𝑘 ∈ ℕ₀ ↦ ((seq0( + , 𝐴)‘𝑘) · (𝑋↑𝑘)))‘𝑖) = ((seq0( + , 𝐴)‘𝑖) · (𝑋↑𝑖)))
32	31	adantl 487	. . . 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)))
33		abelth.1	. . . . . . . . 9 ⊢ (𝜑 → 𝐴:ℕ₀⟶ℂ)
34	33	ffvelcdmda 7111	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ℕ₀) → (𝐴‘𝑥) ∈ ℂ)
35	1, 2, 34	serf 14203	. . . . . . 7 ⊢ (𝜑 → seq0( + , 𝐴):ℕ₀⟶ℂ)
36	35	ad2antrr 739	. . . . . 6 ⊢ (((𝜑 ∧ 𝑋 ∈ (0(ball‘(abs ∘ − ))1)) ∧ (𝑗 ∈ ℕ₀ ∧ ∀𝑚 ∈ (ℤ_≥‘𝑗)(abs‘(seq0( + , 𝐴)‘𝑚)) < 1)) → seq0( + , 𝐴):ℕ₀⟶ℂ)
37	36	ffvelcdmda 7111	. . . . 5 ⊢ ((((𝜑 ∧ 𝑋 ∈ (0(ball‘(abs ∘ − ))1)) ∧ (𝑗 ∈ ℕ₀ ∧ ∀𝑚 ∈ (ℤ_≥‘𝑗)(abs‘(seq0( + , 𝐴)‘𝑚)) < 1)) ∧ 𝑖 ∈ ℕ₀) → (seq0( + , 𝐴)‘𝑖) ∈ ℂ)
38		expcl 14252	. . . . . 6 ⊢ ((𝑋 ∈ ℂ ∧ 𝑖 ∈ ℕ₀) → (𝑋↑𝑖) ∈ ℂ)
39	21, 38	sylan 592	. . . . 5 ⊢ ((((𝜑 ∧ 𝑋 ∈ (0(ball‘(abs ∘ − ))1)) ∧ (𝑗 ∈ ℕ₀ ∧ ∀𝑚 ∈ (ℤ_≥‘𝑗)(abs‘(seq0( + , 𝐴)‘𝑚)) < 1)) ∧ 𝑖 ∈ ℕ₀) → (𝑋↑𝑖) ∈ ℂ)
40	37, 39	mulcld 11359	. . . 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)
41	32, 40	eqeltrd 2864	. . 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)
42	22	recnd 11367	. . . . 5 ⊢ (((𝜑 ∧ 𝑋 ∈ (0(ball‘(abs ∘ − ))1)) ∧ (𝑗 ∈ ℕ₀ ∧ ∀𝑚 ∈ (ℤ_≥‘𝑗)(abs‘(seq0( + , 𝐴)‘𝑚)) < 1)) → (abs‘𝑋) ∈ ℂ)
43		absidm 15523	. . . . . . 7 ⊢ (𝑋 ∈ ℂ → (abs‘(abs‘𝑋)) = (abs‘𝑋))
44	21, 43	syl 18	. . . . . 6 ⊢ (((𝜑 ∧ 𝑋 ∈ (0(ball‘(abs ∘ − ))1)) ∧ (𝑗 ∈ ℕ₀ ∧ ∀𝑚 ∈ (ℤ_≥‘𝑗)(abs‘(seq0( + , 𝐴)‘𝑚)) < 1)) → (abs‘(abs‘𝑋)) = (abs‘𝑋))
45		eqid 2763	. . . . . . . . . 10 ⊢ (abs ∘ − ) = (abs ∘ − )
46	45	cnmetdval 25122	. . . . . . . . 9 ⊢ ((𝑋 ∈ ℂ ∧ 0 ∈ ℂ) → (𝑋(abs ∘ − )0) = (abs‘(𝑋 − 0)))
47	21, 16, 46	sylancl 598	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑋 ∈ (0(ball‘(abs ∘ − ))1)) ∧ (𝑗 ∈ ℕ₀ ∧ ∀𝑚 ∈ (ℤ_≥‘𝑗)(abs‘(seq0( + , 𝐴)‘𝑚)) < 1)) → (𝑋(abs ∘ − )0) = (abs‘(𝑋 − 0)))
48	21	subid1d 11688	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑋 ∈ (0(ball‘(abs ∘ − ))1)) ∧ (𝑗 ∈ ℕ₀ ∧ ∀𝑚 ∈ (ℤ_≥‘𝑗)(abs‘(seq0( + , 𝐴)‘𝑚)) < 1)) → (𝑋 − 0) = 𝑋)
49	48	fveq2d 6915	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑋 ∈ (0(ball‘(abs ∘ − ))1)) ∧ (𝑗 ∈ ℕ₀ ∧ ∀𝑚 ∈ (ℤ_≥‘𝑗)(abs‘(seq0( + , 𝐴)‘𝑚)) < 1)) → (abs‘(𝑋 − 0)) = (abs‘𝑋))
50	47, 49	eqtrd 2798	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑋 ∈ (0(ball‘(abs ∘ − ))1)) ∧ (𝑗 ∈ ℕ₀ ∧ ∀𝑚 ∈ (ℤ_≥‘𝑗)(abs‘(seq0( + , 𝐴)‘𝑚)) < 1)) → (𝑋(abs ∘ − )0) = (abs‘𝑋))
51		elbl3 24744	. . . . . . . . . 10 ⊢ ((((abs ∘ − ) ∈ (∞Met‘ℂ) ∧ 1 ∈ ℝ^*) ∧ (0 ∈ ℂ ∧ 𝑋 ∈ ℂ)) → (𝑋 ∈ (0(ball‘(abs ∘ − ))1) ↔ (𝑋(abs ∘ − )0) < 1))
52	15, 17, 51	mpanl12 715	. . . . . . . . 9 ⊢ ((0 ∈ ℂ ∧ 𝑋 ∈ ℂ) → (𝑋 ∈ (0(ball‘(abs ∘ − ))1) ↔ (𝑋(abs ∘ − )0) < 1))
53	16, 21, 52	sylancr 599	. . . . . . . 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))
54	20, 53	mpbid 235	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑋 ∈ (0(ball‘(abs ∘ − ))1)) ∧ (𝑗 ∈ ℕ₀ ∧ ∀𝑚 ∈ (ℤ_≥‘𝑗)(abs‘(seq0( + , 𝐴)‘𝑚)) < 1)) → (𝑋(abs ∘ − )0) < 1)
55	50, 54	eqbrtrrd 5133	. . . . . 6 ⊢ (((𝜑 ∧ 𝑋 ∈ (0(ball‘(abs ∘ − ))1)) ∧ (𝑗 ∈ ℕ₀ ∧ ∀𝑚 ∈ (ℤ_≥‘𝑗)(abs‘(seq0( + , 𝐴)‘𝑚)) < 1)) → (abs‘𝑋) < 1)
56	44, 55	eqbrtrd 5131	. . . . 5 ⊢ (((𝜑 ∧ 𝑋 ∈ (0(ball‘(abs ∘ − ))1)) ∧ (𝑗 ∈ ℕ₀ ∧ ∀𝑚 ∈ (ℤ_≥‘𝑗)(abs‘(seq0( + , 𝐴)‘𝑚)) < 1)) → (abs‘(abs‘𝑋)) < 1)
57	42, 56, 14	geolim 16071	. . . 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))))
58		climrel 15691	. . . . 5 ⊢ Rel ⇝
59	58	releldmi 5938	. . . 4 ⊢ (seq0( + , (𝑛 ∈ ℕ₀ ↦ ((abs‘𝑋)↑𝑛))) ⇝ (1 / (1 − (abs‘𝑋))) → seq0( + , (𝑛 ∈ ℕ₀ ↦ ((abs‘𝑋)↑𝑛))) ∈ dom ⇝ )
60	57, 59	syl 18	. . 3 ⊢ (((𝜑 ∧ 𝑋 ∈ (0(ball‘(abs ∘ − ))1)) ∧ (𝑗 ∈ ℕ₀ ∧ ∀𝑚 ∈ (ℤ_≥‘𝑗)(abs‘(seq0( + , 𝐴)‘𝑚)) < 1)) → seq0( + , (𝑛 ∈ ℕ₀ ↦ ((abs‘𝑋)↑𝑛))) ∈ dom ⇝ )
61		1red 11339	. . 3 ⊢ (((𝜑 ∧ 𝑋 ∈ (0(ball‘(abs ∘ − ))1)) ∧ (𝑗 ∈ ℕ₀ ∧ ∀𝑚 ∈ (ℤ_≥‘𝑗)(abs‘(seq0( + , 𝐴)‘𝑚)) < 1)) → 1 ∈ ℝ)
62	36	adantr 486	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑋 ∈ (0(ball‘(abs ∘ − ))1)) ∧ (𝑗 ∈ ℕ₀ ∧ ∀𝑚 ∈ (ℤ_≥‘𝑗)(abs‘(seq0( + , 𝐴)‘𝑚)) < 1)) ∧ 𝑖 ∈ (ℤ_≥‘𝑗)) → seq0( + , 𝐴):ℕ₀⟶ℂ)
63		eluznn0 13074	. . . . . . . . 9 ⊢ ((𝑗 ∈ ℕ₀ ∧ 𝑖 ∈ (ℤ_≥‘𝑗)) → 𝑖 ∈ ℕ₀)
64	9, 63	sylan 592	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑋 ∈ (0(ball‘(abs ∘ − ))1)) ∧ (𝑗 ∈ ℕ₀ ∧ ∀𝑚 ∈ (ℤ_≥‘𝑗)(abs‘(seq0( + , 𝐴)‘𝑚)) < 1)) ∧ 𝑖 ∈ (ℤ_≥‘𝑗)) → 𝑖 ∈ ℕ₀)
65	62, 64	ffvelcdmd 7112	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑋 ∈ (0(ball‘(abs ∘ − ))1)) ∧ (𝑗 ∈ ℕ₀ ∧ ∀𝑚 ∈ (ℤ_≥‘𝑗)(abs‘(seq0( + , 𝐴)‘𝑚)) < 1)) ∧ 𝑖 ∈ (ℤ_≥‘𝑗)) → (seq0( + , 𝐴)‘𝑖) ∈ ℂ)
66	64, 39	syldan 603	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑋 ∈ (0(ball‘(abs ∘ − ))1)) ∧ (𝑗 ∈ ℕ₀ ∧ ∀𝑚 ∈ (ℤ_≥‘𝑗)(abs‘(seq0( + , 𝐴)‘𝑚)) < 1)) ∧ 𝑖 ∈ (ℤ_≥‘𝑗)) → (𝑋↑𝑖) ∈ ℂ)
67	65, 66	absmuld 15656	. . . . . 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))))
68	21	adantr 486	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑋 ∈ (0(ball‘(abs ∘ − ))1)) ∧ (𝑗 ∈ ℕ₀ ∧ ∀𝑚 ∈ (ℤ_≥‘𝑗)(abs‘(seq0( + , 𝐴)‘𝑚)) < 1)) ∧ 𝑖 ∈ (ℤ_≥‘𝑗)) → 𝑋 ∈ ℂ)
69	68, 64	absexpd 15654	. . . . . . 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))
70	69	oveq2d 7462	. . . . . 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)))
71	67, 70	eqtrd 2798	. . . . 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)))
72	65	abscld 15638	. . . . . 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)
73		1red 11339	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑋 ∈ (0(ball‘(abs ∘ − ))1)) ∧ (𝑗 ∈ ℕ₀ ∧ ∀𝑚 ∈ (ℤ_≥‘𝑗)(abs‘(seq0( + , 𝐴)‘𝑚)) < 1)) ∧ 𝑖 ∈ (ℤ_≥‘𝑗)) → 1 ∈ ℝ)
74	64, 24	syldan 603	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑋 ∈ (0(ball‘(abs ∘ − ))1)) ∧ (𝑗 ∈ ℕ₀ ∧ ∀𝑚 ∈ (ℤ_≥‘𝑗)(abs‘(seq0( + , 𝐴)‘𝑚)) < 1)) ∧ 𝑖 ∈ (ℤ_≥‘𝑗)) → ((abs‘𝑋)↑𝑖) ∈ ℝ)
75	66	absge0d 15646	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑋 ∈ (0(ball‘(abs ∘ − ))1)) ∧ (𝑗 ∈ ℕ₀ ∧ ∀𝑚 ∈ (ℤ_≥‘𝑗)(abs‘(seq0( + , 𝐴)‘𝑚)) < 1)) ∧ 𝑖 ∈ (ℤ_≥‘𝑗)) → 0 ≤ (abs‘(𝑋↑𝑖)))
76	75, 69	breqtrd 5135	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑋 ∈ (0(ball‘(abs ∘ − ))1)) ∧ (𝑗 ∈ ℕ₀ ∧ ∀𝑚 ∈ (ℤ_≥‘𝑗)(abs‘(seq0( + , 𝐴)‘𝑚)) < 1)) ∧ 𝑖 ∈ (ℤ_≥‘𝑗)) → 0 ≤ ((abs‘𝑋)↑𝑖))
77		simprr 785	. . . . . . . 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)
78		2fveq3 6916	. . . . . . . . . 10 ⊢ (𝑚 = 𝑖 → (abs‘(seq0( + , 𝐴)‘𝑚)) = (abs‘(seq0( + , 𝐴)‘𝑖)))
79	78	breq1d 5116	. . . . . . . . 9 ⊢ (𝑚 = 𝑖 → ((abs‘(seq0( + , 𝐴)‘𝑚)) < 1 ↔ (abs‘(seq0( + , 𝐴)‘𝑖)) < 1))
80	79	rspccva 3579	. . . . . . . 8 ⊢ ((∀𝑚 ∈ (ℤ_≥‘𝑗)(abs‘(seq0( + , 𝐴)‘𝑚)) < 1 ∧ 𝑖 ∈ (ℤ_≥‘𝑗)) → (abs‘(seq0( + , 𝐴)‘𝑖)) < 1)
81	77, 80	sylan 592	. . . . . . 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)
82		1re 11338	. . . . . . . 8 ⊢ 1 ∈ ℝ
83		ltle 11428	. . . . . . . 8 ⊢ (((abs‘(seq0( + , 𝐴)‘𝑖)) ∈ ℝ ∧ 1 ∈ ℝ) → ((abs‘(seq0( + , 𝐴)‘𝑖)) < 1 → (abs‘(seq0( + , 𝐴)‘𝑖)) ≤ 1))
84	72, 82, 83	sylancl 598	. . . . . . 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))
85	81, 84	mpd 16	. . . . . 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)
86	72, 73, 74, 76, 85	lemul1ad 12286	. . . . 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)))
87	71, 86	eqbrtrd 5131	. . . 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)))
88	64, 31	syl 18	. . . . 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)))
89	88	fveq2d 6915	. . . 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))))
90	64, 13	syl 18	. . . . 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))
91	90	oveq2d 7462	. . . 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)))
92	87, 89, 91	3brtr4d 5141	. . 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)))
93	1, 9, 25, 41, 60, 61, 92	cvgcmpce 16017	. 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 ⇝ )
94	8, 93	rexlimddv 3173	1 ⊢ ((𝜑 ∧ 𝑋 ∈ (0(ball‘(abs ∘ − ))1)) → seq0( + , (𝑘 ∈ ℕ₀ ↦ ((seq0( + , 𝐴)‘𝑘) · (𝑋↑𝑘)))) ∈ dom ⇝ )

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ↔ wb 209 ∧ wa 401 = wceq 1570 ∈ wcel 2147 ∀wral 3080 ∃wrex 3090 {crab 3416 ⊆ wss 3900 class class class wbr 5106 ↦ cmpt 5190 dom cdm 5656 ∘ ccom 5660 ⟶wf 6557 ‘cfv 6561 (class class class)co 7446 ℂcc 11228 ℝcr 11229 0cc0 11230 1c1 11231 + caddc 11233 · cmul 11235 ℝ^*cxr 11372 < clt 11373 ≤ cle 11374 − cmin 11571 / cdiv 12003 ℕ₀cn0 12636 ℤ_≥cuz 12995 ℝ⁺crp 13150 seqcseq 14174 ↑cexp 14234 abscabs 15433 ⇝ cli 15683 Σcsu 15885 ∞Metcxmet 21696 ballcbl 21698

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1945 ax-6 2002 ax-7 2043 ax-8 2149 ax-9 2157 ax-10 2180 ax-11 2197 ax-12 2216 ax-ext 2735 ax-rep 5236 ax-sep 5253 ax-nul 5264 ax-pow 5332 ax-pr 5396 ax-un 7778 ax-inf2 9672 ax-cnex 11286 ax-resscn 11287 ax-1cn 11288 ax-icn 11289 ax-addcl 11290 ax-addrcl 11291 ax-mulcl 11292 ax-mulrcl 11293 ax-mulcom 11294 ax-addass 11295 ax-mulass 11296 ax-distr 11297 ax-i2m1 11298 ax-1ne0 11299 ax-1rid 11300 ax-rnegex 11301 ax-rrecex 11302 ax-cnre 11303 ax-pre-lttri 11304 ax-pre-lttrn 11305 ax-pre-ltadd 11306 ax-pre-mulgt0 11307 ax-pre-sup 11308

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2102 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2839 df-nfc 2913 df-ne 2960 df-nel 3066 df-ral 3081 df-rex 3091 df-rmo 3369 df-reu 3370 df-rab 3417 df-v 3456 df-sbc 3743 df-csb 3852 df-ss 3901 df-pss 4017 df-dif 4050 df-in 4094 df-un 4154 df-nul 4282 df-if 4486 df-pw 4562 df-sn 4589 df-pr 4591 df-op 4839 df-uni 4871 df-int 4911 df-iun 4956 df-br 5107 df-opab 5172 df-mpt 5191 df-tr 5217 df-id 5551 df-eprel 5556 df-po 5564 df-so 5565 df-fr 5609 df-se 5610 df-we 5611 df-xp 5662 df-rel 5663 df-cnv 5664 df-co 5665 df-dm 5666 df-rn 5667 df-res 5668 df-ima 5669 df-pred 6325 df-ord 6386 df-on 6387 df-lim 6388 df-suc 6389 df-iota 6515 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-isom 6570 df-riota 7403 df-ov 7449 df-oprab 7450 df-mpo 7451 df-om 7905 df-1st 8028 df-2nd 8029 df-frecs 8330 df-wrecs 8361 df-recs 8410 df-rdg 8449 df-1o 8507 df-er 8748 df-map 8880 df-pm 8881 df-en 9005 df-dom 9006 df-sdom 9007 df-fin 9008 df-sup 9464 df-inf 9465 df-oi 9534 df-card 10050 df-pnf 11375 df-mnf 11376 df-xr 11377 df-ltxr 11378 df-le 11379 df-sub 11573 df-neg 11574 df-div 12004 df-nn 12366 df-2 12435 df-3 12436 df-n0 12637 df-z 12724 df-uz 12996 df-rp 13151 df-xadd 13272 df-ico 13512 df-fz 13670 df-fzo 13819 df-fl 13962 df-seq 14175 df-exp 14235 df-hash 14505 df-cj 15298 df-re 15299 df-im 15300 df-sqrt 15434 df-abs 15435 df-limsup 15670 df-clim 15687 df-rlim 15688 df-sum 15886 df-psmet 21703 df-xmet 21704 df-met 21705 df-bl 21706

This theorem is used by: abelthlem6 26796 abelthlem7 26798

W3C validator