abelthlem5 - Metamath Proof Explorer

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Theorem abelthlem5 26635

Description: Lemma for abelth 26641. (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 12918	. . . 4 ⊢ ℕ₀ = (ℤ_≥‘0)
2		0zd 12621	. . . 4 ⊢ (𝜑 → 0 ∈ ℤ)
3		1rp 13038	. . . . 5 ⊢ 1 ∈ ℝ⁺
4	3	a1i 11	. . . 4 ⊢ (𝜑 → 1 ∈ ℝ⁺)
5		eqidd 2767	. . . 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 15589	. . 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 7431	. . . . . 6 ⊢ (𝑛 = 𝑖 → ((abs‘𝑋)↑𝑛) = ((abs‘𝑋)↑𝑖))
11		eqid 2766	. . . . . 6 ⊢ (𝑛 ∈ ℕ₀ ↦ ((abs‘𝑋)↑𝑛)) = (𝑛 ∈ ℕ₀ ↦ ((abs‘𝑋)↑𝑛))
12		ovex 7456	. . . . . 6 ⊢ ((abs‘𝑋)↑𝑖) ∈ V
13	10, 11, 12	fvmpt 6996	. . . . 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 24966	. . . . . . . 8 ⊢ (abs ∘ − ) ∈ (∞Met‘ℂ)
16		0cn 11216	. . . . . . . 8 ⊢ 0 ∈ ℂ
17		1xr 11286	. . . . . . . 8 ⊢ 1 ∈ ℝ^*
18		blssm 24612	. . . . . . . 8 ⊢ (((abs ∘ − ) ∈ (∞Met‘ℂ) ∧ 0 ∈ ℂ ∧ 1 ∈ ℝ^*) → (0(ball‘(abs ∘ − ))1) ⊆ ℂ)
19	15, 16, 17, 18	mp3an 1490	. . . . . . 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 3938	. . . . . 6 ⊢ (((𝜑 ∧ 𝑋 ∈ (0(ball‘(abs ∘ − ))1)) ∧ (𝑗 ∈ ℕ₀ ∧ ∀𝑚 ∈ (ℤ_≥‘𝑗)(abs‘(seq0( + , 𝐴)‘𝑚)) < 1)) → 𝑋 ∈ ℂ)
22	21	abscld 15516	. . . . 5 ⊢ (((𝜑 ∧ 𝑋 ∈ (0(ball‘(abs ∘ − ))1)) ∧ (𝑗 ∈ ℕ₀ ∧ ∀𝑚 ∈ (ℤ_≥‘𝑗)(abs‘(seq0( + , 𝐴)‘𝑚)) < 1)) → (abs‘𝑋) ∈ ℝ)
23		reexpcl 14134	. . . . 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 2866	. . 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 6888	. . . . . . 7 ⊢ (𝑘 = 𝑖 → (seq0( + , 𝐴)‘𝑘) = (seq0( + , 𝐴)‘𝑖))
27		oveq2 7431	. . . . . . 7 ⊢ (𝑘 = 𝑖 → (𝑋↑𝑘) = (𝑋↑𝑖))
28	26, 27	oveq12d 7441	. . . . . 6 ⊢ (𝑘 = 𝑖 → ((seq0( + , 𝐴)‘𝑘) · (𝑋↑𝑘)) = ((seq0( + , 𝐴)‘𝑖) · (𝑋↑𝑖)))
29		eqid 2766	. . . . . 6 ⊢ (𝑘 ∈ ℕ₀ ↦ ((seq0( + , 𝐴)‘𝑘) · (𝑋↑𝑘))) = (𝑘 ∈ ℕ₀ ↦ ((seq0( + , 𝐴)‘𝑘) · (𝑋↑𝑘)))
30		ovex 7456	. . . . . 6 ⊢ ((seq0( + , 𝐴)‘𝑖) · (𝑋↑𝑖)) ∈ V
31	28, 29, 30	fvmpt 6996	. . . . 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 7086	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ℕ₀) → (𝐴‘𝑥) ∈ ℂ)
35	1, 2, 34	serf 14086	. . . . . . 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 7086	. . . . 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 14135	. . . . . 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 11247	. . . 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 2866	. . 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 11255	. . . . 5 ⊢ (((𝜑 ∧ 𝑋 ∈ (0(ball‘(abs ∘ − ))1)) ∧ (𝑗 ∈ ℕ₀ ∧ ∀𝑚 ∈ (ℤ_≥‘𝑗)(abs‘(seq0( + , 𝐴)‘𝑚)) < 1)) → (abs‘𝑋) ∈ ℂ)
43		absidm 15401	. . . . . . 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 2766	. . . . . . . . . 10 ⊢ (abs ∘ − ) = (abs ∘ − )
46	45	cnmetdval 24964	. . . . . . . . 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 11576	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑋 ∈ (0(ball‘(abs ∘ − ))1)) ∧ (𝑗 ∈ ℕ₀ ∧ ∀𝑚 ∈ (ℤ_≥‘𝑗)(abs‘(seq0( + , 𝐴)‘𝑚)) < 1)) → (𝑋 − 0) = 𝑋)
49	48	fveq2d 6892	. . . . . . . 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 2801	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑋 ∈ (0(ball‘(abs ∘ − ))1)) ∧ (𝑗 ∈ ℕ₀ ∧ ∀𝑚 ∈ (ℤ_≥‘𝑗)(abs‘(seq0( + , 𝐴)‘𝑚)) < 1)) → (𝑋(abs ∘ − )0) = (abs‘𝑋))
51		elbl3 24586	. . . . . . . . . 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 5140	. . . . . 6 ⊢ (((𝜑 ∧ 𝑋 ∈ (0(ball‘(abs ∘ − ))1)) ∧ (𝑗 ∈ ℕ₀ ∧ ∀𝑚 ∈ (ℤ_≥‘𝑗)(abs‘(seq0( + , 𝐴)‘𝑚)) < 1)) → (abs‘𝑋) < 1)
56	44, 55	eqbrtrd 5138	. . . . 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 15950	. . . 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 15569	. . . . 5 ⊢ Rel ⇝
59	58	releldmi 5943	. . . 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 11227	. . 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 12959	. . . . . . . . 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 7087	. . . . . . 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 15534	. . . . . 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 15532	. . . . . . 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 7439	. . . . . 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 2801	. . . . 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 15516	. . . . . 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 11227	. . . . . 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 15524	. . . . . . 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 5142	. . . . . 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 6893	. . . . . . . . . 10 ⊢ (𝑚 = 𝑖 → (abs‘(seq0( + , 𝐴)‘𝑚)) = (abs‘(seq0( + , 𝐴)‘𝑖)))
79	78	breq1d 5124	. . . . . . . . 9 ⊢ (𝑚 = 𝑖 → ((abs‘(seq0( + , 𝐴)‘𝑚)) < 1 ↔ (abs‘(seq0( + , 𝐴)‘𝑖)) < 1))
80	79	rspccva 3583	. . . . . . . 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 11226	. . . . . . . 8 ⊢ 1 ∈ ℝ
83		ltle 11316	. . . . . . . 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 12172	. . . . 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 5138	. . . 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 6892	. . . 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 7439	. . . 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 5148	. . 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 15896	. 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 3175	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 2146 ∀wral 3082 ∃wrex 3092 {crab 3419 ⊆ wss 3908 class class class wbr 5114 ↦ cmpt 5197 dom cdm 5666 ∘ ccom 5670 ⟶wf 6539 ‘cfv 6543 (class class class)co 7423 ℂcc 11116 ℝcr 11117 0cc0 11118 1c1 11119 + caddc 11121 · cmul 11123 ℝ^*cxr 11260 < clt 11261 ≤ cle 11262 − cmin 11459 / cdiv 11889 ℕ₀cn0 12522 ℤ_≥cuz 12880 ℝ⁺crp 13034 seqcseq 14057 ↑cexp 14117 abscabs 15311 ⇝ cli 15561 Σcsu 15763 ∞Metcxmet 21544 ballcbl 21546

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 1943 ax-6 2000 ax-7 2041 ax-8 2148 ax-9 2156 ax-10 2179 ax-11 2195 ax-12 2216 ax-ext 2738 ax-rep 5243 ax-sep 5262 ax-nul 5274 ax-pow 5341 ax-pr 5409 ax-un 7745 ax-inf2 9620 ax-cnex 11174 ax-resscn 11175 ax-1cn 11176 ax-icn 11177 ax-addcl 11178 ax-addrcl 11179 ax-mulcl 11180 ax-mulrcl 11181 ax-mulcom 11182 ax-addass 11183 ax-mulass 11184 ax-distr 11185 ax-i2m1 11186 ax-1ne0 11187 ax-1rid 11188 ax-rnegex 11189 ax-rrecex 11190 ax-cnre 11191 ax-pre-lttri 11192 ax-pre-lttrn 11193 ax-pre-ltadd 11194 ax-pre-mulgt0 11195 ax-pre-sup 11196

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 2100 df-mo 2570 df-eu 2600 df-clab 2745 df-cleq 2758 df-clel 2841 df-nfc 2915 df-ne 2962 df-nel 3068 df-ral 3083 df-rex 3093 df-rmo 3372 df-reu 3373 df-rab 3420 df-v 3460 df-sbc 3748 df-csb 3857 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-pss 3928 df-nul 4290 df-if 4493 df-pw 4569 df-sn 4595 df-pr 4597 df-op 4601 df-uni 4878 df-int 4918 df-iun 4963 df-br 5115 df-opab 5179 df-mpt 5198 df-tr 5224 df-id 5561 df-eprel 5566 df-po 5574 df-so 5575 df-fr 5619 df-se 5620 df-we 5621 df-xp 5672 df-rel 5673 df-cnv 5674 df-co 5675 df-dm 5676 df-rn 5677 df-res 5678 df-ima 5679 df-pred 6309 df-ord 6370 df-on 6371 df-lim 6372 df-suc 6373 df-iota 6499 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-isom 6552 df-riota 7380 df-ov 7426 df-oprab 7427 df-mpo 7428 df-om 7872 df-1st 7995 df-2nd 7996 df-frecs 8287 df-wrecs 8318 df-recs 8367 df-rdg 8406 df-1o 8462 df-er 8703 df-map 8835 df-pm 8836 df-en 8953 df-dom 8954 df-sdom 8955 df-fin 8956 df-sup 9412 df-inf 9413 df-oi 9482 df-card 9944 df-pnf 11263 df-mnf 11264 df-xr 11265 df-ltxr 11266 df-le 11267 df-sub 11461 df-neg 11462 df-div 11890 df-nn 12252 df-2 12321 df-3 12322 df-n0 12523 df-z 12610 df-uz 12881 df-rp 13035 df-xadd 13156 df-ico 13396 df-fz 13554 df-fzo 13702 df-fl 13845 df-seq 14058 df-exp 14118 df-hash 14387 df-cj 15176 df-re 15177 df-im 15178 df-sqrt 15312 df-abs 15313 df-limsup 15548 df-clim 15565 df-rlim 15566 df-sum 15764 df-psmet 21551 df-xmet 21552 df-met 21553 df-bl 21554

This theorem is used by: abelthlem6 26636 abelthlem7 26638

W3C validator