abelthlem5 - Metamath Proof Explorer

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Theorem abelthlem5 26488

Description: Lemma for abelth 26494. (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 12926	. . . 4 ⊢ ℕ₀ = (ℤ_≥‘0)
2		0zd 12632	. . . 4 ⊢ (𝜑 → 0 ∈ ℤ)
3		1rp 13042	. . . . 5 ⊢ 1 ∈ ℝ⁺
4	3	a1i 11	. . . 4 ⊢ (𝜑 → 1 ∈ ℝ⁺)
5		eqidd 2730	. . . 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 15535	. . 3 ⊢ (𝜑 → ∃𝑗 ∈ ℕ₀ ∀𝑚 ∈ (ℤ_≥‘𝑗)(abs‘(seq0( + , 𝐴)‘𝑚)) < 1)
8	7	adantr 479	. 2 ⊢ ((𝜑 ∧ 𝑋 ∈ (0(ball‘(abs ∘ − ))1)) → ∃𝑗 ∈ ℕ₀ ∀𝑚 ∈ (ℤ_≥‘𝑗)(abs‘(seq0( + , 𝐴)‘𝑚)) < 1)
9		simprl 769	. . 3 ⊢ (((𝜑 ∧ 𝑋 ∈ (0(ball‘(abs ∘ − ))1)) ∧ (𝑗 ∈ ℕ₀ ∧ ∀𝑚 ∈ (ℤ_≥‘𝑗)(abs‘(seq0( + , 𝐴)‘𝑚)) < 1)) → 𝑗 ∈ ℕ₀)
10		oveq2 7436	. . . . . 6 ⊢ (𝑛 = 𝑖 → ((abs‘𝑋)↑𝑛) = ((abs‘𝑋)↑𝑖))
11		eqid 2729	. . . . . 6 ⊢ (𝑛 ∈ ℕ₀ ↦ ((abs‘𝑋)↑𝑛)) = (𝑛 ∈ ℕ₀ ↦ ((abs‘𝑋)↑𝑛))
12		ovex 7461	. . . . . 6 ⊢ ((abs‘𝑋)↑𝑖) ∈ V
13	10, 11, 12	fvmpt 7012	. . . . 5 ⊢ (𝑖 ∈ ℕ₀ → ((𝑛 ∈ ℕ₀ ↦ ((abs‘𝑋)↑𝑛))‘𝑖) = ((abs‘𝑋)↑𝑖))
14	13	adantl 480	. . . 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 24803	. . . . . . . 8 ⊢ (abs ∘ − ) ∈ (∞Met‘ℂ)
16		0cn 11263	. . . . . . . 8 ⊢ 0 ∈ ℂ
17		1xr 11330	. . . . . . . 8 ⊢ 1 ∈ ℝ^*
18		blssm 24438	. . . . . . . 8 ⊢ (((abs ∘ − ) ∈ (∞Met‘ℂ) ∧ 0 ∈ ℂ ∧ 1 ∈ ℝ^*) → (0(ball‘(abs ∘ − ))1) ⊆ ℂ)
19	15, 16, 17, 18	mp3an 1458	. . . . . . 7 ⊢ (0(ball‘(abs ∘ − ))1) ⊆ ℂ
20		simplr 767	. . . . . . 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 3987	. . . . . 6 ⊢ (((𝜑 ∧ 𝑋 ∈ (0(ball‘(abs ∘ − ))1)) ∧ (𝑗 ∈ ℕ₀ ∧ ∀𝑚 ∈ (ℤ_≥‘𝑗)(abs‘(seq0( + , 𝐴)‘𝑚)) < 1)) → 𝑋 ∈ ℂ)
22	21	abscld 15462	. . . . 5 ⊢ (((𝜑 ∧ 𝑋 ∈ (0(ball‘(abs ∘ − ))1)) ∧ (𝑗 ∈ ℕ₀ ∧ ∀𝑚 ∈ (ℤ_≥‘𝑗)(abs‘(seq0( + , 𝐴)‘𝑚)) < 1)) → (abs‘𝑋) ∈ ℝ)
23		reexpcl 14109	. . . . 5 ⊢ (((abs‘𝑋) ∈ ℝ ∧ 𝑖 ∈ ℕ₀) → ((abs‘𝑋)↑𝑖) ∈ ℝ)
24	22, 23	sylan 578	. . . 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 2829	. . 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 6903	. . . . . . 7 ⊢ (𝑘 = 𝑖 → (seq0( + , 𝐴)‘𝑘) = (seq0( + , 𝐴)‘𝑖))
27		oveq2 7436	. . . . . . 7 ⊢ (𝑘 = 𝑖 → (𝑋↑𝑘) = (𝑋↑𝑖))
28	26, 27	oveq12d 7446	. . . . . 6 ⊢ (𝑘 = 𝑖 → ((seq0( + , 𝐴)‘𝑘) · (𝑋↑𝑘)) = ((seq0( + , 𝐴)‘𝑖) · (𝑋↑𝑖)))
29		eqid 2729	. . . . . 6 ⊢ (𝑘 ∈ ℕ₀ ↦ ((seq0( + , 𝐴)‘𝑘) · (𝑋↑𝑘))) = (𝑘 ∈ ℕ₀ ↦ ((seq0( + , 𝐴)‘𝑘) · (𝑋↑𝑘)))
30		ovex 7461	. . . . . 6 ⊢ ((seq0( + , 𝐴)‘𝑖) · (𝑋↑𝑖)) ∈ V
31	28, 29, 30	fvmpt 7012	. . . . 5 ⊢ (𝑖 ∈ ℕ₀ → ((𝑘 ∈ ℕ₀ ↦ ((seq0( + , 𝐴)‘𝑘) · (𝑋↑𝑘)))‘𝑖) = ((seq0( + , 𝐴)‘𝑖) · (𝑋↑𝑖)))
32	31	adantl 480	. . . 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 7101	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ℕ₀) → (𝐴‘𝑥) ∈ ℂ)
35	1, 2, 34	serf 14061	. . . . . . 7 ⊢ (𝜑 → seq0( + , 𝐴):ℕ₀⟶ℂ)
36	35	ad2antrr 724	. . . . . 6 ⊢ (((𝜑 ∧ 𝑋 ∈ (0(ball‘(abs ∘ − ))1)) ∧ (𝑗 ∈ ℕ₀ ∧ ∀𝑚 ∈ (ℤ_≥‘𝑗)(abs‘(seq0( + , 𝐴)‘𝑚)) < 1)) → seq0( + , 𝐴):ℕ₀⟶ℂ)
37	36	ffvelcdmda 7101	. . . . 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 14110	. . . . . 6 ⊢ ((𝑋 ∈ ℂ ∧ 𝑖 ∈ ℕ₀) → (𝑋↑𝑖) ∈ ℂ)
39	21, 38	sylan 578	. . . . 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 11291	. . . 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 2829	. . 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 11299	. . . . 5 ⊢ (((𝜑 ∧ 𝑋 ∈ (0(ball‘(abs ∘ − ))1)) ∧ (𝑗 ∈ ℕ₀ ∧ ∀𝑚 ∈ (ℤ_≥‘𝑗)(abs‘(seq0( + , 𝐴)‘𝑚)) < 1)) → (abs‘𝑋) ∈ ℂ)
43		absidm 15349	. . . . . . 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 2729	. . . . . . . . . 10 ⊢ (abs ∘ − ) = (abs ∘ − )
46	45	cnmetdval 24801	. . . . . . . . 9 ⊢ ((𝑋 ∈ ℂ ∧ 0 ∈ ℂ) → (𝑋(abs ∘ − )0) = (abs‘(𝑋 − 0)))
47	21, 16, 46	sylancl 584	. . . . . . . 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 11617	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑋 ∈ (0(ball‘(abs ∘ − ))1)) ∧ (𝑗 ∈ ℕ₀ ∧ ∀𝑚 ∈ (ℤ_≥‘𝑗)(abs‘(seq0( + , 𝐴)‘𝑚)) < 1)) → (𝑋 − 0) = 𝑋)
49	48	fveq2d 6907	. . . . . . . 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 2769	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑋 ∈ (0(ball‘(abs ∘ − ))1)) ∧ (𝑗 ∈ ℕ₀ ∧ ∀𝑚 ∈ (ℤ_≥‘𝑗)(abs‘(seq0( + , 𝐴)‘𝑚)) < 1)) → (𝑋(abs ∘ − )0) = (abs‘𝑋))
51		elbl3 24412	. . . . . . . . . 10 ⊢ ((((abs ∘ − ) ∈ (∞Met‘ℂ) ∧ 1 ∈ ℝ^*) ∧ (0 ∈ ℂ ∧ 𝑋 ∈ ℂ)) → (𝑋 ∈ (0(ball‘(abs ∘ − ))1) ↔ (𝑋(abs ∘ − )0) < 1))
52	15, 17, 51	mpanl12 700	. . . . . . . . 9 ⊢ ((0 ∈ ℂ ∧ 𝑋 ∈ ℂ) → (𝑋 ∈ (0(ball‘(abs ∘ − ))1) ↔ (𝑋(abs ∘ − )0) < 1))
53	16, 21, 52	sylancr 585	. . . . . . . 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 231	. . . . . . 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 5178	. . . . . 6 ⊢ (((𝜑 ∧ 𝑋 ∈ (0(ball‘(abs ∘ − ))1)) ∧ (𝑗 ∈ ℕ₀ ∧ ∀𝑚 ∈ (ℤ_≥‘𝑗)(abs‘(seq0( + , 𝐴)‘𝑚)) < 1)) → (abs‘𝑋) < 1)
56	44, 55	eqbrtrd 5176	. . . . 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 15895	. . . 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 15515	. . . . 5 ⊢ Rel ⇝
59	58	releldmi 5956	. . . 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 11272	. . 3 ⊢ (((𝜑 ∧ 𝑋 ∈ (0(ball‘(abs ∘ − ))1)) ∧ (𝑗 ∈ ℕ₀ ∧ ∀𝑚 ∈ (ℤ_≥‘𝑗)(abs‘(seq0( + , 𝐴)‘𝑚)) < 1)) → 1 ∈ ℝ)
62	36	adantr 479	. . . . . . . 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 12963	. . . . . . . . 9 ⊢ ((𝑗 ∈ ℕ₀ ∧ 𝑖 ∈ (ℤ_≥‘𝑗)) → 𝑖 ∈ ℕ₀)
64	9, 63	sylan 578	. . . . . . . 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 7102	. . . . . . 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 589	. . . . . . 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 15480	. . . . . 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 479	. . . . . . . 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 15478	. . . . . . 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 7444	. . . . . 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 2769	. . . . 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 15462	. . . . . 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 11272	. . . . . 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 589	. . . . . 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 15470	. . . . . . 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 5180	. . . . . 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 771	. . . . . . . 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 6908	. . . . . . . . . 10 ⊢ (𝑚 = 𝑖 → (abs‘(seq0( + , 𝐴)‘𝑚)) = (abs‘(seq0( + , 𝐴)‘𝑖)))
79	78	breq1d 5164	. . . . . . . . 9 ⊢ (𝑚 = 𝑖 → ((abs‘(seq0( + , 𝐴)‘𝑚)) < 1 ↔ (abs‘(seq0( + , 𝐴)‘𝑖)) < 1))
80	79	rspccva 3617	. . . . . . . 8 ⊢ ((∀𝑚 ∈ (ℤ_≥‘𝑗)(abs‘(seq0( + , 𝐴)‘𝑚)) < 1 ∧ 𝑖 ∈ (ℤ_≥‘𝑗)) → (abs‘(seq0( + , 𝐴)‘𝑖)) < 1)
81	77, 80	sylan 578	. . . . . . 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 11271	. . . . . . . 8 ⊢ 1 ∈ ℝ
83		ltle 11359	. . . . . . . 8 ⊢ (((abs‘(seq0( + , 𝐴)‘𝑖)) ∈ ℝ ∧ 1 ∈ ℝ) → ((abs‘(seq0( + , 𝐴)‘𝑖)) < 1 → (abs‘(seq0( + , 𝐴)‘𝑖)) ≤ 1))
84	72, 82, 83	sylancl 584	. . . . . . 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 12215	. . . . 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 5176	. . . 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 6907	. . . 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 7444	. . . 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 5186	. . 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 15843	. 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 3154	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 205 ∧ wa 394 = wceq 1534 ∈ wcel 2100 ∀wral 3054 ∃wrex 3063 {crab 3428 ⊆ wss 3957 class class class wbr 5154 ↦ cmpt 5237 dom cdm 5684 ∘ ccom 5688 ⟶wf 6552 ‘cfv 6556 (class class class)co 7428 ℂcc 11163 ℝcr 11164 0cc0 11165 1c1 11166 + caddc 11168 · cmul 11170 ℝ^*cxr 11304 < clt 11305 ≤ cle 11306 − cmin 11501 / cdiv 11928 ℕ₀cn0 12534 ℤ_≥cuz 12884 ℝ⁺crp 13038 seqcseq 14032 ↑cexp 14092 abscabs 15260 ⇝ cli 15507 Σcsu 15711 ∞Metcxmet 21350 ballcbl 21352

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2102 ax-9 2110 ax-10 2133 ax-11 2150 ax-12 2170 ax-ext 2700 ax-rep 5291 ax-sep 5305 ax-nul 5312 ax-pow 5371 ax-pr 5435 ax-un 7749 ax-inf2 9691 ax-cnex 11221 ax-resscn 11222 ax-1cn 11223 ax-icn 11224 ax-addcl 11225 ax-addrcl 11226 ax-mulcl 11227 ax-mulrcl 11228 ax-mulcom 11229 ax-addass 11230 ax-mulass 11231 ax-distr 11232 ax-i2m1 11233 ax-1ne0 11234 ax-1rid 11235 ax-rnegex 11236 ax-rrecex 11237 ax-cnre 11238 ax-pre-lttri 11239 ax-pre-lttrn 11240 ax-pre-ltadd 11241 ax-pre-mulgt0 11242 ax-pre-sup 11243

This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2062 df-mo 2532 df-eu 2561 df-clab 2707 df-cleq 2721 df-clel 2806 df-nfc 2881 df-ne 2934 df-nel 3040 df-ral 3055 df-rex 3064 df-rmo 3373 df-reu 3374 df-rab 3429 df-v 3474 df-sbc 3787 df-csb 3903 df-dif 3960 df-un 3962 df-in 3964 df-ss 3974 df-pss 3977 df-nul 4334 df-if 4535 df-pw 4610 df-sn 4635 df-pr 4637 df-op 4641 df-uni 4917 df-int 4958 df-iun 5006 df-br 5155 df-opab 5217 df-mpt 5238 df-tr 5272 df-id 5582 df-eprel 5588 df-po 5596 df-so 5597 df-fr 5639 df-se 5640 df-we 5641 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-pred 6315 df-ord 6381 df-on 6382 df-lim 6383 df-suc 6384 df-iota 6508 df-fun 6558 df-fn 6559 df-f 6560 df-f1 6561 df-fo 6562 df-f1o 6563 df-fv 6564 df-isom 6565 df-riota 7384 df-ov 7431 df-oprab 7432 df-mpo 7433 df-om 7883 df-1st 8009 df-2nd 8010 df-frecs 8302 df-wrecs 8333 df-recs 8407 df-rdg 8446 df-1o 8502 df-er 8740 df-map 8863 df-pm 8864 df-en 8981 df-dom 8982 df-sdom 8983 df-fin 8984 df-sup 9492 df-inf 9493 df-oi 9560 df-card 9989 df-pnf 11307 df-mnf 11308 df-xr 11309 df-ltxr 11310 df-le 11311 df-sub 11503 df-neg 11504 df-div 11929 df-nn 12275 df-2 12337 df-3 12338 df-n0 12535 df-z 12621 df-uz 12885 df-rp 13039 df-xadd 13157 df-ico 13394 df-fz 13549 df-fzo 13692 df-fl 13823 df-seq 14033 df-exp 14093 df-hash 14360 df-cj 15125 df-re 15126 df-im 15127 df-sqrt 15261 df-abs 15262 df-limsup 15494 df-clim 15511 df-rlim 15512 df-sum 15712 df-psmet 21357 df-xmet 21358 df-met 21359 df-bl 21360

This theorem is referenced by: abelthlem6 26489 abelthlem7 26491

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