abelthlem5 - Metamath Proof Explorer

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Theorem abelthlem5 26402

Description: Lemma for abelth 26408. (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 12899	. . . 4 ⊢ ℕ₀ = (ℤ_≥‘0)
2		0zd 12605	. . . 4 ⊢ (𝜑 → 0 ∈ ℤ)
3		1rp 13017	. . . . 5 ⊢ 1 ∈ ℝ⁺
4	3	a1i 11	. . . 4 ⊢ (𝜑 → 1 ∈ ℝ⁺)
5		eqidd 2737	. . . 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 15533	. . 3 ⊢ (𝜑 → ∃𝑗 ∈ ℕ₀ ∀𝑚 ∈ (ℤ_≥‘𝑗)(abs‘(seq0( + , 𝐴)‘𝑚)) < 1)
8	7	adantr 480	. 2 ⊢ ((𝜑 ∧ 𝑋 ∈ (0(ball‘(abs ∘ − ))1)) → ∃𝑗 ∈ ℕ₀ ∀𝑚 ∈ (ℤ_≥‘𝑗)(abs‘(seq0( + , 𝐴)‘𝑚)) < 1)
9		simprl 770	. . 3 ⊢ (((𝜑 ∧ 𝑋 ∈ (0(ball‘(abs ∘ − ))1)) ∧ (𝑗 ∈ ℕ₀ ∧ ∀𝑚 ∈ (ℤ_≥‘𝑗)(abs‘(seq0( + , 𝐴)‘𝑚)) < 1)) → 𝑗 ∈ ℕ₀)
10		oveq2 7418	. . . . . 6 ⊢ (𝑛 = 𝑖 → ((abs‘𝑋)↑𝑛) = ((abs‘𝑋)↑𝑖))
11		eqid 2736	. . . . . 6 ⊢ (𝑛 ∈ ℕ₀ ↦ ((abs‘𝑋)↑𝑛)) = (𝑛 ∈ ℕ₀ ↦ ((abs‘𝑋)↑𝑛))
12		ovex 7443	. . . . . 6 ⊢ ((abs‘𝑋)↑𝑖) ∈ V
13	10, 11, 12	fvmpt 6991	. . . . 5 ⊢ (𝑖 ∈ ℕ₀ → ((𝑛 ∈ ℕ₀ ↦ ((abs‘𝑋)↑𝑛))‘𝑖) = ((abs‘𝑋)↑𝑖))
14	13	adantl 481	. . . 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 24716	. . . . . . . 8 ⊢ (abs ∘ − ) ∈ (∞Met‘ℂ)
16		0cn 11232	. . . . . . . 8 ⊢ 0 ∈ ℂ
17		1xr 11299	. . . . . . . 8 ⊢ 1 ∈ ℝ^*
18		blssm 24362	. . . . . . . 8 ⊢ (((abs ∘ − ) ∈ (∞Met‘ℂ) ∧ 0 ∈ ℂ ∧ 1 ∈ ℝ^*) → (0(ball‘(abs ∘ − ))1) ⊆ ℂ)
19	15, 16, 17, 18	mp3an 1463	. . . . . . 7 ⊢ (0(ball‘(abs ∘ − ))1) ⊆ ℂ
20		simplr 768	. . . . . . 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 3961	. . . . . 6 ⊢ (((𝜑 ∧ 𝑋 ∈ (0(ball‘(abs ∘ − ))1)) ∧ (𝑗 ∈ ℕ₀ ∧ ∀𝑚 ∈ (ℤ_≥‘𝑗)(abs‘(seq0( + , 𝐴)‘𝑚)) < 1)) → 𝑋 ∈ ℂ)
22	21	abscld 15460	. . . . 5 ⊢ (((𝜑 ∧ 𝑋 ∈ (0(ball‘(abs ∘ − ))1)) ∧ (𝑗 ∈ ℕ₀ ∧ ∀𝑚 ∈ (ℤ_≥‘𝑗)(abs‘(seq0( + , 𝐴)‘𝑚)) < 1)) → (abs‘𝑋) ∈ ℝ)
23		reexpcl 14101	. . . . 5 ⊢ (((abs‘𝑋) ∈ ℝ ∧ 𝑖 ∈ ℕ₀) → ((abs‘𝑋)↑𝑖) ∈ ℝ)
24	22, 23	sylan 580	. . . 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 2835	. . 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 6881	. . . . . . 7 ⊢ (𝑘 = 𝑖 → (seq0( + , 𝐴)‘𝑘) = (seq0( + , 𝐴)‘𝑖))
27		oveq2 7418	. . . . . . 7 ⊢ (𝑘 = 𝑖 → (𝑋↑𝑘) = (𝑋↑𝑖))
28	26, 27	oveq12d 7428	. . . . . 6 ⊢ (𝑘 = 𝑖 → ((seq0( + , 𝐴)‘𝑘) · (𝑋↑𝑘)) = ((seq0( + , 𝐴)‘𝑖) · (𝑋↑𝑖)))
29		eqid 2736	. . . . . 6 ⊢ (𝑘 ∈ ℕ₀ ↦ ((seq0( + , 𝐴)‘𝑘) · (𝑋↑𝑘))) = (𝑘 ∈ ℕ₀ ↦ ((seq0( + , 𝐴)‘𝑘) · (𝑋↑𝑘)))
30		ovex 7443	. . . . . 6 ⊢ ((seq0( + , 𝐴)‘𝑖) · (𝑋↑𝑖)) ∈ V
31	28, 29, 30	fvmpt 6991	. . . . 5 ⊢ (𝑖 ∈ ℕ₀ → ((𝑘 ∈ ℕ₀ ↦ ((seq0( + , 𝐴)‘𝑘) · (𝑋↑𝑘)))‘𝑖) = ((seq0( + , 𝐴)‘𝑖) · (𝑋↑𝑖)))
32	31	adantl 481	. . . 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 7079	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ℕ₀) → (𝐴‘𝑥) ∈ ℂ)
35	1, 2, 34	serf 14053	. . . . . . 7 ⊢ (𝜑 → seq0( + , 𝐴):ℕ₀⟶ℂ)
36	35	ad2antrr 726	. . . . . 6 ⊢ (((𝜑 ∧ 𝑋 ∈ (0(ball‘(abs ∘ − ))1)) ∧ (𝑗 ∈ ℕ₀ ∧ ∀𝑚 ∈ (ℤ_≥‘𝑗)(abs‘(seq0( + , 𝐴)‘𝑚)) < 1)) → seq0( + , 𝐴):ℕ₀⟶ℂ)
37	36	ffvelcdmda 7079	. . . . 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 14102	. . . . . 6 ⊢ ((𝑋 ∈ ℂ ∧ 𝑖 ∈ ℕ₀) → (𝑋↑𝑖) ∈ ℂ)
39	21, 38	sylan 580	. . . . 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 11260	. . . 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 2835	. . 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 11268	. . . . 5 ⊢ (((𝜑 ∧ 𝑋 ∈ (0(ball‘(abs ∘ − ))1)) ∧ (𝑗 ∈ ℕ₀ ∧ ∀𝑚 ∈ (ℤ_≥‘𝑗)(abs‘(seq0( + , 𝐴)‘𝑚)) < 1)) → (abs‘𝑋) ∈ ℂ)
43		absidm 15347	. . . . . . 7 ⊢ (𝑋 ∈ ℂ → (abs‘(abs‘𝑋)) = (abs‘𝑋))
44	21, 43	syl 17	. . . . . 6 ⊢ (((𝜑 ∧ 𝑋 ∈ (0(ball‘(abs ∘ − ))1)) ∧ (𝑗 ∈ ℕ₀ ∧ ∀𝑚 ∈ (ℤ_≥‘𝑗)(abs‘(seq0( + , 𝐴)‘𝑚)) < 1)) → (abs‘(abs‘𝑋)) = (abs‘𝑋))
45		eqid 2736	. . . . . . . . . 10 ⊢ (abs ∘ − ) = (abs ∘ − )
46	45	cnmetdval 24714	. . . . . . . . 9 ⊢ ((𝑋 ∈ ℂ ∧ 0 ∈ ℂ) → (𝑋(abs ∘ − )0) = (abs‘(𝑋 − 0)))
47	21, 16, 46	sylancl 586	. . . . . . . 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 11588	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑋 ∈ (0(ball‘(abs ∘ − ))1)) ∧ (𝑗 ∈ ℕ₀ ∧ ∀𝑚 ∈ (ℤ_≥‘𝑗)(abs‘(seq0( + , 𝐴)‘𝑚)) < 1)) → (𝑋 − 0) = 𝑋)
49	48	fveq2d 6885	. . . . . . . 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 2771	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑋 ∈ (0(ball‘(abs ∘ − ))1)) ∧ (𝑗 ∈ ℕ₀ ∧ ∀𝑚 ∈ (ℤ_≥‘𝑗)(abs‘(seq0( + , 𝐴)‘𝑚)) < 1)) → (𝑋(abs ∘ − )0) = (abs‘𝑋))
51		elbl3 24336	. . . . . . . . . 10 ⊢ ((((abs ∘ − ) ∈ (∞Met‘ℂ) ∧ 1 ∈ ℝ^*) ∧ (0 ∈ ℂ ∧ 𝑋 ∈ ℂ)) → (𝑋 ∈ (0(ball‘(abs ∘ − ))1) ↔ (𝑋(abs ∘ − )0) < 1))
52	15, 17, 51	mpanl12 702	. . . . . . . . 9 ⊢ ((0 ∈ ℂ ∧ 𝑋 ∈ ℂ) → (𝑋 ∈ (0(ball‘(abs ∘ − ))1) ↔ (𝑋(abs ∘ − )0) < 1))
53	16, 21, 52	sylancr 587	. . . . . . . 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 232	. . . . . . 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 5148	. . . . . 6 ⊢ (((𝜑 ∧ 𝑋 ∈ (0(ball‘(abs ∘ − ))1)) ∧ (𝑗 ∈ ℕ₀ ∧ ∀𝑚 ∈ (ℤ_≥‘𝑗)(abs‘(seq0( + , 𝐴)‘𝑚)) < 1)) → (abs‘𝑋) < 1)
56	44, 55	eqbrtrd 5146	. . . . 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 15891	. . . 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 15513	. . . . 5 ⊢ Rel ⇝
59	58	releldmi 5933	. . . 4 ⊢ (seq0( + , (𝑛 ∈ ℕ₀ ↦ ((abs‘𝑋)↑𝑛))) ⇝ (1 / (1 − (abs‘𝑋))) → seq0( + , (𝑛 ∈ ℕ₀ ↦ ((abs‘𝑋)↑𝑛))) ∈ dom ⇝ )
60	57, 59	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 ⇝ )
61		1red 11241	. . 3 ⊢ (((𝜑 ∧ 𝑋 ∈ (0(ball‘(abs ∘ − ))1)) ∧ (𝑗 ∈ ℕ₀ ∧ ∀𝑚 ∈ (ℤ_≥‘𝑗)(abs‘(seq0( + , 𝐴)‘𝑚)) < 1)) → 1 ∈ ℝ)
62	36	adantr 480	. . . . . . . 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 12938	. . . . . . . . 9 ⊢ ((𝑗 ∈ ℕ₀ ∧ 𝑖 ∈ (ℤ_≥‘𝑗)) → 𝑖 ∈ ℕ₀)
64	9, 63	sylan 580	. . . . . . . 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 7080	. . . . . . 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 591	. . . . . . 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 15478	. . . . . 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 480	. . . . . . . 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 15476	. . . . . . 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 7426	. . . . . 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 2771	. . . . 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 15460	. . . . . 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 11241	. . . . . 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 591	. . . . . 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 15468	. . . . . . 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 5150	. . . . . 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 772	. . . . . . . 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 6886	. . . . . . . . . 10 ⊢ (𝑚 = 𝑖 → (abs‘(seq0( + , 𝐴)‘𝑚)) = (abs‘(seq0( + , 𝐴)‘𝑖)))
79	78	breq1d 5134	. . . . . . . . 9 ⊢ (𝑚 = 𝑖 → ((abs‘(seq0( + , 𝐴)‘𝑚)) < 1 ↔ (abs‘(seq0( + , 𝐴)‘𝑖)) < 1))
80	79	rspccva 3605	. . . . . . . 8 ⊢ ((∀𝑚 ∈ (ℤ_≥‘𝑗)(abs‘(seq0( + , 𝐴)‘𝑚)) < 1 ∧ 𝑖 ∈ (ℤ_≥‘𝑗)) → (abs‘(seq0( + , 𝐴)‘𝑖)) < 1)
81	77, 80	sylan 580	. . . . . . 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 11240	. . . . . . . 8 ⊢ 1 ∈ ℝ
83		ltle 11328	. . . . . . . 8 ⊢ (((abs‘(seq0( + , 𝐴)‘𝑖)) ∈ ℝ ∧ 1 ∈ ℝ) → ((abs‘(seq0( + , 𝐴)‘𝑖)) < 1 → (abs‘(seq0( + , 𝐴)‘𝑖)) ≤ 1))
84	72, 82, 83	sylancl 586	. . . . . . 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 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)
86	72, 73, 74, 76, 85	lemul1ad 12186	. . . . 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 5146	. . . 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 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)))
89	88	fveq2d 6885	. . . 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 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))
91	90	oveq2d 7426	. . . 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 5156	. . 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 15839	. 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 3148	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 206 ∧ wa 395 = wceq 1540 ∈ wcel 2109 ∀wral 3052 ∃wrex 3061 {crab 3420 ⊆ wss 3931 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 ℝ^*cxr 11273 < clt 11274 ≤ cle 11275 − cmin 11471 / cdiv 11899 ℕ₀cn0 12506 ℤ_≥cuz 12857 ℝ⁺crp 13013 seqcseq 14024 ↑cexp 14084 abscabs 15258 ⇝ cli 15505 Σcsu 15707 ∞Metcxmet 21305 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-fz 13530 df-fzo 13677 df-fl 13814 df-seq 14025 df-exp 14085 df-hash 14354 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: abelthlem6 26403 abelthlem7 26405

W3C validator