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Theorem lgamgulmlem4 27341

Description: Lemma for lgamgulm 27344. (Contributed by Mario Carneiro, 3-Jul-2017.)

**Hypotheses**
Ref	Expression
lgamgulm.r	⊢ (𝜑 → 𝑅 ∈ ℕ)
lgamgulm.u	⊢ 𝑈 = {𝑥 ∈ ℂ ∣ ((abs‘𝑥) ≤ 𝑅 ∧ ∀𝑘 ∈ ℕ₀ (1 / 𝑅) ≤ (abs‘(𝑥 + 𝑘)))}
lgamgulm.g	⊢ 𝐺 = (𝑚 ∈ ℕ ↦ (𝑧 ∈ 𝑈 ↦ ((𝑧 · (log‘((𝑚 + 1) / 𝑚))) − (log‘((𝑧 / 𝑚) + 1)))))
lgamgulm.t	⊢ 𝑇 = (𝑚 ∈ ℕ ↦ if((2 · 𝑅) ≤ 𝑚, (𝑅 · ((2 · (𝑅 + 1)) / (𝑚↑2))), ((𝑅 · (log‘((𝑚 + 1) / 𝑚))) + ((log‘((𝑅 + 1) · 𝑚)) + π))))

**Assertion**
Ref	Expression
lgamgulmlem4	⊢ (𝜑 → seq1( + , 𝑇) ∈ dom ⇝ )

Distinct variable groups: 𝑘,𝑚,𝑥,𝑧,𝑅 𝑈,𝑚,𝑧 𝜑,𝑚,𝑥,𝑧 Allowed substitution hints: 𝜑(𝑘) 𝑇(𝑥, 𝑧, 𝑘, 𝑚) 𝑈(𝑥, 𝑘) 𝐺(𝑥, 𝑧, 𝑘, 𝑚)

Proof of Theorem lgamgulmlem4 Dummy variable 𝑛 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		2nn 12397	. . . . . . 7 ⊢ 2 ∈ ℕ
2	1	a1i 11	. . . . . 6 ⊢ (𝜑 → 2 ∈ ℕ)
3		lgamgulm.r	. . . . . 6 ⊢ (𝜑 → 𝑅 ∈ ℕ)
4	2, 3	nnmulcld 12372	. . . . 5 ⊢ (𝜑 → (2 · 𝑅) ∈ ℕ)
5	4	nnzd 12700	. . . 4 ⊢ (𝜑 → (2 · 𝑅) ∈ ℤ)
6		eluzle 12959	. . . . . . 7 ⊢ (𝑛 ∈ (ℤ_≥‘(2 · 𝑅)) → (2 · 𝑅) ≤ 𝑛)
7	6	adantl 487	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘(2 · 𝑅))) → (2 · 𝑅) ≤ 𝑛)
8	7	iftrued 4490	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘(2 · 𝑅))) → if((2 · 𝑅) ≤ 𝑛, (𝑅 · ((2 · (𝑅 + 1)) / (𝑛↑2))), ((𝑅 · (log‘((𝑛 + 1) / 𝑛))) + ((log‘((𝑅 + 1) · 𝑛)) + π))) = (𝑅 · ((2 · (𝑅 + 1)) / (𝑛↑2))))
9		eluznn 13026	. . . . . . 7 ⊢ (((2 · 𝑅) ∈ ℕ ∧ 𝑛 ∈ (ℤ_≥‘(2 · 𝑅))) → 𝑛 ∈ ℕ)
10	4, 9	sylan 592	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘(2 · 𝑅))) → 𝑛 ∈ ℕ)
11		breq2 5107	. . . . . . . 8 ⊢ (𝑚 = 𝑛 → ((2 · 𝑅) ≤ 𝑚 ↔ (2 · 𝑅) ≤ 𝑛))
12		oveq1 7419	. . . . . . . . . 10 ⊢ (𝑚 = 𝑛 → (𝑚↑2) = (𝑛↑2))
13	12	oveq2d 7428	. . . . . . . . 9 ⊢ (𝑚 = 𝑛 → ((2 · (𝑅 + 1)) / (𝑚↑2)) = ((2 · (𝑅 + 1)) / (𝑛↑2)))
14	13	oveq2d 7428	. . . . . . . 8 ⊢ (𝑚 = 𝑛 → (𝑅 · ((2 · (𝑅 + 1)) / (𝑚↑2))) = (𝑅 · ((2 · (𝑅 + 1)) / (𝑛↑2))))
15		oveq1 7419	. . . . . . . . . . . 12 ⊢ (𝑚 = 𝑛 → (𝑚 + 1) = (𝑛 + 1))
16		id 23	. . . . . . . . . . . 12 ⊢ (𝑚 = 𝑛 → 𝑚 = 𝑛)
17	15, 16	oveq12d 7430	. . . . . . . . . . 11 ⊢ (𝑚 = 𝑛 → ((𝑚 + 1) / 𝑚) = ((𝑛 + 1) / 𝑛))
18	17	fveq2d 6881	. . . . . . . . . 10 ⊢ (𝑚 = 𝑛 → (log‘((𝑚 + 1) / 𝑚)) = (log‘((𝑛 + 1) / 𝑛)))
19	18	oveq2d 7428	. . . . . . . . 9 ⊢ (𝑚 = 𝑛 → (𝑅 · (log‘((𝑚 + 1) / 𝑚))) = (𝑅 · (log‘((𝑛 + 1) / 𝑛))))
20		oveq2 7420	. . . . . . . . . . 11 ⊢ (𝑚 = 𝑛 → ((𝑅 + 1) · 𝑚) = ((𝑅 + 1) · 𝑛))
21	20	fveq2d 6881	. . . . . . . . . 10 ⊢ (𝑚 = 𝑛 → (log‘((𝑅 + 1) · 𝑚)) = (log‘((𝑅 + 1) · 𝑛)))
22	21	oveq1d 7427	. . . . . . . . 9 ⊢ (𝑚 = 𝑛 → ((log‘((𝑅 + 1) · 𝑚)) + π) = ((log‘((𝑅 + 1) · 𝑛)) + π))
23	19, 22	oveq12d 7430	. . . . . . . 8 ⊢ (𝑚 = 𝑛 → ((𝑅 · (log‘((𝑚 + 1) / 𝑚))) + ((log‘((𝑅 + 1) · 𝑚)) + π)) = ((𝑅 · (log‘((𝑛 + 1) / 𝑛))) + ((log‘((𝑅 + 1) · 𝑛)) + π)))
24	11, 14, 23	ifbieq12d 4511	. . . . . . 7 ⊢ (𝑚 = 𝑛 → if((2 · 𝑅) ≤ 𝑚, (𝑅 · ((2 · (𝑅 + 1)) / (𝑚↑2))), ((𝑅 · (log‘((𝑚 + 1) / 𝑚))) + ((log‘((𝑅 + 1) · 𝑚)) + π))) = if((2 · 𝑅) ≤ 𝑛, (𝑅 · ((2 · (𝑅 + 1)) / (𝑛↑2))), ((𝑅 · (log‘((𝑛 + 1) / 𝑛))) + ((log‘((𝑅 + 1) · 𝑛)) + π))))
25		lgamgulm.t	. . . . . . 7 ⊢ 𝑇 = (𝑚 ∈ ℕ ↦ if((2 · 𝑅) ≤ 𝑚, (𝑅 · ((2 · (𝑅 + 1)) / (𝑚↑2))), ((𝑅 · (log‘((𝑚 + 1) / 𝑚))) + ((log‘((𝑅 + 1) · 𝑚)) + π))))
26		ovex 7445	. . . . . . . 8 ⊢ (𝑅 · ((2 · (𝑅 + 1)) / (𝑛↑2))) ∈ V
27		ovex 7445	. . . . . . . 8 ⊢ ((𝑅 · (log‘((𝑛 + 1) / 𝑛))) + ((log‘((𝑅 + 1) · 𝑛)) + π)) ∈ V
28	26, 27	ifex 4533	. . . . . . 7 ⊢ if((2 · 𝑅) ≤ 𝑛, (𝑅 · ((2 · (𝑅 + 1)) / (𝑛↑2))), ((𝑅 · (log‘((𝑛 + 1) / 𝑛))) + ((log‘((𝑅 + 1) · 𝑛)) + π))) ∈ V
29	24, 25, 28	fvmpt 6985	. . . . . 6 ⊢ (𝑛 ∈ ℕ → (𝑇‘𝑛) = if((2 · 𝑅) ≤ 𝑛, (𝑅 · ((2 · (𝑅 + 1)) / (𝑛↑2))), ((𝑅 · (log‘((𝑛 + 1) / 𝑛))) + ((log‘((𝑅 + 1) · 𝑛)) + π))))
30	10, 29	syl 18	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘(2 · 𝑅))) → (𝑇‘𝑛) = if((2 · 𝑅) ≤ 𝑛, (𝑅 · ((2 · (𝑅 + 1)) / (𝑛↑2))), ((𝑅 · (log‘((𝑛 + 1) / 𝑛))) + ((log‘((𝑅 + 1) · 𝑛)) + π))))
31		eqid 2761	. . . . . . 7 ⊢ (𝑚 ∈ ℕ ↦ (𝑅 · ((2 · (𝑅 + 1)) / (𝑚↑2)))) = (𝑚 ∈ ℕ ↦ (𝑅 · ((2 · (𝑅 + 1)) / (𝑚↑2))))
32	14, 31, 26	fvmpt 6985	. . . . . 6 ⊢ (𝑛 ∈ ℕ → ((𝑚 ∈ ℕ ↦ (𝑅 · ((2 · (𝑅 + 1)) / (𝑚↑2))))‘𝑛) = (𝑅 · ((2 · (𝑅 + 1)) / (𝑛↑2))))
33	10, 32	syl 18	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘(2 · 𝑅))) → ((𝑚 ∈ ℕ ↦ (𝑅 · ((2 · (𝑅 + 1)) / (𝑚↑2))))‘𝑛) = (𝑅 · ((2 · (𝑅 + 1)) / (𝑛↑2))))
34	8, 30, 33	3eqtr4d 2806	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘(2 · 𝑅))) → (𝑇‘𝑛) = ((𝑚 ∈ ℕ ↦ (𝑅 · ((2 · (𝑅 + 1)) / (𝑚↑2))))‘𝑛))
35	5, 34	seqfeq 14150	. . 3 ⊢ (𝜑 → seq(2 · 𝑅)( + , 𝑇) = seq(2 · 𝑅)( + , (𝑚 ∈ ℕ ↦ (𝑅 · ((2 · (𝑅 + 1)) / (𝑚↑2))))))
36		nnuz 12985	. . . . . 6 ⊢ ℕ = (ℤ_≥‘1)
37		1zzd 12708	. . . . . 6 ⊢ (𝜑 → 1 ∈ ℤ)
38	3	nncnd 12332	. . . . . . 7 ⊢ (𝜑 → 𝑅 ∈ ℂ)
39		2cnd 12402	. . . . . . . 8 ⊢ (𝜑 → 2 ∈ ℂ)
40		1cnd 11283	. . . . . . . . 9 ⊢ (𝜑 → 1 ∈ ℂ)
41	38, 40	addcld 11309	. . . . . . . 8 ⊢ (𝜑 → (𝑅 + 1) ∈ ℂ)
42	39, 41	mulcld 11310	. . . . . . 7 ⊢ (𝜑 → (2 · (𝑅 + 1)) ∈ ℂ)
43	38, 42	mulcld 11310	. . . . . 6 ⊢ (𝜑 → (𝑅 · (2 · (𝑅 + 1))) ∈ ℂ)
44		1lt2 12496	. . . . . . . . . 10 ⊢ 1 < 2
45		2re 12398	. . . . . . . . . . 11 ⊢ 2 ∈ ℝ
46		rere 15269	. . . . . . . . . . 11 ⊢ (2 ∈ ℝ → (ℜ‘2) = 2)
47	45, 46	ax-mp 5	. . . . . . . . . 10 ⊢ (ℜ‘2) = 2
48	44, 47	breqtrri 5132	. . . . . . . . 9 ⊢ 1 < (ℜ‘2)
49	48	a1i 11	. . . . . . . 8 ⊢ (𝜑 → 1 < (ℜ‘2))
50		oveq1 7419	. . . . . . . . . 10 ⊢ (𝑚 = 𝑛 → (𝑚↑_𝑐-2) = (𝑛↑_𝑐-2))
51		eqid 2761	. . . . . . . . . 10 ⊢ (𝑚 ∈ ℕ ↦ (𝑚↑_𝑐-2)) = (𝑚 ∈ ℕ ↦ (𝑚↑_𝑐-2))
52		ovex 7445	. . . . . . . . . 10 ⊢ (𝑛↑_𝑐-2) ∈ V
53	50, 51, 52	fvmpt 6985	. . . . . . . . 9 ⊢ (𝑛 ∈ ℕ → ((𝑚 ∈ ℕ ↦ (𝑚↑_𝑐-2))‘𝑛) = (𝑛↑_𝑐-2))
54	53	adantl 487	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((𝑚 ∈ ℕ ↦ (𝑚↑_𝑐-2))‘𝑛) = (𝑛↑_𝑐-2))
55	39, 49, 54	zetacvg 27324	. . . . . . 7 ⊢ (𝜑 → seq1( + , (𝑚 ∈ ℕ ↦ (𝑚↑_𝑐-2))) ∈ dom ⇝ )
56		climdm 15701	. . . . . . 7 ⊢ (seq1( + , (𝑚 ∈ ℕ ↦ (𝑚↑_𝑐-2))) ∈ dom ⇝ ↔ seq1( + , (𝑚 ∈ ℕ ↦ (𝑚↑_𝑐-2))) ⇝ ( ⇝ ‘seq1( + , (𝑚 ∈ ℕ ↦ (𝑚↑_𝑐-2)))))
57	55, 56	sylib 221	. . . . . 6 ⊢ (𝜑 → seq1( + , (𝑚 ∈ ℕ ↦ (𝑚↑_𝑐-2))) ⇝ ( ⇝ ‘seq1( + , (𝑚 ∈ ℕ ↦ (𝑚↑_𝑐-2)))))
58		simpr 490	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ ℕ)
59	58	nncnd 12332	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ ℂ)
60		2cnd 12402	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 2 ∈ ℂ)
61	60	negcld 11637	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → -2 ∈ ℂ)
62	59, 61	cxpcld 27018	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑛↑_𝑐-2) ∈ ℂ)
63	54, 62	eqeltrd 2861	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((𝑚 ∈ ℕ ↦ (𝑚↑_𝑐-2))‘𝑛) ∈ ℂ)
64	38	adantr 486	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝑅 ∈ ℂ)
65		1cnd 11283	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 1 ∈ ℂ)
66	64, 65	addcld 11309	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑅 + 1) ∈ ℂ)
67	60, 66	mulcld 11310	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (2 · (𝑅 + 1)) ∈ ℂ)
68	64, 67	mulcld 11310	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑅 · (2 · (𝑅 + 1))) ∈ ℂ)
69	59	sqcld 14267	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑛↑2) ∈ ℂ)
70	58	nnne0d 12369	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝑛 ≠ 0)
71		2z 12709	. . . . . . . . . . 11 ⊢ 2 ∈ ℤ
72	71	a1i 11	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 2 ∈ ℤ)
73	59, 70, 72	expne0d 14275	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑛↑2) ≠ 0)
74	68, 69, 73	divrecd 12077	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((𝑅 · (2 · (𝑅 + 1))) / (𝑛↑2)) = ((𝑅 · (2 · (𝑅 + 1))) · (1 / (𝑛↑2))))
75	64, 67, 69, 73	divassd 12109	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((𝑅 · (2 · (𝑅 + 1))) / (𝑛↑2)) = (𝑅 · ((2 · (𝑅 + 1)) / (𝑛↑2))))
76	59, 70, 60	cxpnegd 27025	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑛↑_𝑐-2) = (1 / (𝑛↑_𝑐2)))
77	59, 70, 72	cxpexpzd 27021	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑛↑_𝑐2) = (𝑛↑2))
78	77	oveq2d 7428	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (1 / (𝑛↑_𝑐2)) = (1 / (𝑛↑2)))
79	76, 78	eqtr2d 2797	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (1 / (𝑛↑2)) = (𝑛↑_𝑐-2))
80	79	oveq2d 7428	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((𝑅 · (2 · (𝑅 + 1))) · (1 / (𝑛↑2))) = ((𝑅 · (2 · (𝑅 + 1))) · (𝑛↑_𝑐-2)))
81	74, 75, 80	3eqtr3d 2804	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑅 · ((2 · (𝑅 + 1)) / (𝑛↑2))) = ((𝑅 · (2 · (𝑅 + 1))) · (𝑛↑_𝑐-2)))
82	32	adantl 487	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((𝑚 ∈ ℕ ↦ (𝑅 · ((2 · (𝑅 + 1)) / (𝑚↑2))))‘𝑛) = (𝑅 · ((2 · (𝑅 + 1)) / (𝑛↑2))))
83	54	oveq2d 7428	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((𝑅 · (2 · (𝑅 + 1))) · ((𝑚 ∈ ℕ ↦ (𝑚↑_𝑐-2))‘𝑛)) = ((𝑅 · (2 · (𝑅 + 1))) · (𝑛↑_𝑐-2)))
84	81, 82, 83	3eqtr4d 2806	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((𝑚 ∈ ℕ ↦ (𝑅 · ((2 · (𝑅 + 1)) / (𝑚↑2))))‘𝑛) = ((𝑅 · (2 · (𝑅 + 1))) · ((𝑚 ∈ ℕ ↦ (𝑚↑_𝑐-2))‘𝑛)))
85	36, 37, 43, 57, 63, 84	isermulc2 15805	. . . . 5 ⊢ (𝜑 → seq1( + , (𝑚 ∈ ℕ ↦ (𝑅 · ((2 · (𝑅 + 1)) / (𝑚↑2))))) ⇝ ((𝑅 · (2 · (𝑅 + 1))) · ( ⇝ ‘seq1( + , (𝑚 ∈ ℕ ↦ (𝑚↑_𝑐-2))))))
86		climrel 15639	. . . . . 6 ⊢ Rel ⇝
87	86	releldmi 5930	. . . . 5 ⊢ (seq1( + , (𝑚 ∈ ℕ ↦ (𝑅 · ((2 · (𝑅 + 1)) / (𝑚↑2))))) ⇝ ((𝑅 · (2 · (𝑅 + 1))) · ( ⇝ ‘seq1( + , (𝑚 ∈ ℕ ↦ (𝑚↑_𝑐-2))))) → seq1( + , (𝑚 ∈ ℕ ↦ (𝑅 · ((2 · (𝑅 + 1)) / (𝑚↑2))))) ∈ dom ⇝ )
88	85, 87	syl 18	. . . 4 ⊢ (𝜑 → seq1( + , (𝑚 ∈ ℕ ↦ (𝑅 · ((2 · (𝑅 + 1)) / (𝑚↑2))))) ∈ dom ⇝ )
89	67, 69, 73	divcld 12074	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((2 · (𝑅 + 1)) / (𝑛↑2)) ∈ ℂ)
90	64, 89	mulcld 11310	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑅 · ((2 · (𝑅 + 1)) / (𝑛↑2))) ∈ ℂ)
91	82, 90	eqeltrd 2861	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((𝑚 ∈ ℕ ↦ (𝑅 · ((2 · (𝑅 + 1)) / (𝑚↑2))))‘𝑛) ∈ ℂ)
92	36, 4, 91	iserex 15804	. . . 4 ⊢ (𝜑 → (seq1( + , (𝑚 ∈ ℕ ↦ (𝑅 · ((2 · (𝑅 + 1)) / (𝑚↑2))))) ∈ dom ⇝ ↔ seq(2 · 𝑅)( + , (𝑚 ∈ ℕ ↦ (𝑅 · ((2 · (𝑅 + 1)) / (𝑚↑2))))) ∈ dom ⇝ ))
93	88, 92	mpbid 235	. . 3 ⊢ (𝜑 → seq(2 · 𝑅)( + , (𝑚 ∈ ℕ ↦ (𝑅 · ((2 · (𝑅 + 1)) / (𝑚↑2))))) ∈ dom ⇝ )
94	35, 93	eqeltrd 2861	. 2 ⊢ (𝜑 → seq(2 · 𝑅)( + , 𝑇) ∈ dom ⇝ )
95	29	adantl 487	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑇‘𝑛) = if((2 · 𝑅) ≤ 𝑛, (𝑅 · ((2 · (𝑅 + 1)) / (𝑛↑2))), ((𝑅 · (log‘((𝑛 + 1) / 𝑛))) + ((log‘((𝑅 + 1) · 𝑛)) + π))))
96	3	adantr 486	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝑅 ∈ ℕ)
97	96	nnred 12331	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝑅 ∈ ℝ)
98	45	a1i 11	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 2 ∈ ℝ)
99		1red 11290	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 1 ∈ ℝ)
100	97, 99	readdcld 11319	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑅 + 1) ∈ ℝ)
101	98, 100	remulcld 11320	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (2 · (𝑅 + 1)) ∈ ℝ)
102	58	nnsqcld 14368	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑛↑2) ∈ ℕ)
103	101, 102	nndivred 12373	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((2 · (𝑅 + 1)) / (𝑛↑2)) ∈ ℝ)
104	97, 103	remulcld 11320	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑅 · ((2 · (𝑅 + 1)) / (𝑛↑2))) ∈ ℝ)
105	58	peano2nnd 12333	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑛 + 1) ∈ ℕ)
106	105	nnrpd 13143	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑛 + 1) ∈ ℝ⁺)
107	58	nnrpd 13143	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ ℝ⁺)
108	106, 107	rpdivcld 13162	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((𝑛 + 1) / 𝑛) ∈ ℝ⁺)
109	108	relogcld 26933	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (log‘((𝑛 + 1) / 𝑛)) ∈ ℝ)
110	97, 109	remulcld 11320	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑅 · (log‘((𝑛 + 1) / 𝑛))) ∈ ℝ)
111	96	peano2nnd 12333	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑅 + 1) ∈ ℕ)
112	111	nnrpd 13143	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑅 + 1) ∈ ℝ⁺)
113	112, 107	rpmulcld 13161	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((𝑅 + 1) · 𝑛) ∈ ℝ⁺)
114	113	relogcld 26933	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (log‘((𝑅 + 1) · 𝑛)) ∈ ℝ)
115		pire 26765	. . . . . . . . 9 ⊢ π ∈ ℝ
116	115	a1i 11	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → π ∈ ℝ)
117	114, 116	readdcld 11319	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((log‘((𝑅 + 1) · 𝑛)) + π) ∈ ℝ)
118	110, 117	readdcld 11319	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((𝑅 · (log‘((𝑛 + 1) / 𝑛))) + ((log‘((𝑅 + 1) · 𝑛)) + π)) ∈ ℝ)
119	104, 118	ifcld 4529	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → if((2 · 𝑅) ≤ 𝑛, (𝑅 · ((2 · (𝑅 + 1)) / (𝑛↑2))), ((𝑅 · (log‘((𝑛 + 1) / 𝑛))) + ((log‘((𝑅 + 1) · 𝑛)) + π))) ∈ ℝ)
120	95, 119	eqeltrd 2861	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑇‘𝑛) ∈ ℝ)
121	120	recnd 11318	. . 3 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑇‘𝑛) ∈ ℂ)
122	36, 4, 121	iserex 15804	. 2 ⊢ (𝜑 → (seq1( + , 𝑇) ∈ dom ⇝ ↔ seq(2 · 𝑅)( + , 𝑇) ∈ dom ⇝ ))
123	94, 122	mpbird 260	1 ⊢ (𝜑 → seq1( + , 𝑇) ∈ dom ⇝ )

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ∧ wa 401 = wceq 1570 ∈ wcel 2145 ∀wral 3077 {crab 3413 ifcif 4482 class class class wbr 5103 ↦ cmpt 5186 dom cdm 5651 ‘cfv 6531 (class class class)co 7412 ℂcc 11179 ℝcr 11180 1c1 11182 + caddc 11184 · cmul 11186 < clt 11324 ≤ cle 11325 − cmin 11522 -cneg 11523 / cdiv 11954 ℕcn 12316 2c2 12378 ℕ₀cn0 12587 ℤcz 12674 ℤ_≥cuz 12946 seqcseq 14124 ↑cexp 14184 ℜcre 15244 abscabs 15381 ⇝ cli 15631 πcpi 16212 logclog 26864 ↑_𝑐ccxp 26865

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 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2213 ax-ext 2733 ax-rep 5232 ax-sep 5249 ax-nul 5260 ax-pow 5327 ax-pr 5391 ax-un 7740 ax-inf2 9626 ax-cnex 11237 ax-resscn 11238 ax-1cn 11239 ax-icn 11240 ax-addcl 11241 ax-addrcl 11242 ax-mulcl 11243 ax-mulrcl 11244 ax-mulcom 11245 ax-addass 11246 ax-mulass 11247 ax-distr 11248 ax-i2m1 11249 ax-1ne0 11250 ax-1rid 11251 ax-rnegex 11252 ax-rrecex 11253 ax-cnre 11254 ax-pre-lttri 11255 ax-pre-lttrn 11256 ax-pre-ltadd 11257 ax-pre-mulgt0 11258 ax-pre-sup 11259 ax-addf 11260

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 2565 df-eu 2595 df-clab 2740 df-cleq 2753 df-clel 2836 df-nfc 2910 df-ne 2957 df-nel 3063 df-ral 3078 df-rex 3088 df-rmo 3366 df-reu 3367 df-rab 3414 df-v 3453 df-sbc 3740 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-pss 3919 df-nul 4280 df-if 4483 df-pw 4559 df-sn 4585 df-pr 4587 df-tp 4589 df-op 4591 df-uni 4868 df-int 4908 df-iun 4953 df-iin 4954 df-br 5104 df-opab 5168 df-mpt 5187 df-tr 5213 df-id 5546 df-eprel 5551 df-po 5559 df-so 5560 df-fr 5604 df-se 5605 df-we 5606 df-xp 5657 df-rel 5658 df-cnv 5659 df-co 5660 df-dm 5661 df-rn 5662 df-res 5663 df-ima 5664 df-pred 6297 df-ord 6358 df-on 6359 df-lim 6360 df-suc 6361 df-iota 6487 df-fun 6533 df-fn 6534 df-f 6535 df-f1 6536 df-fo 6537 df-f1o 6538 df-fv 6539 df-isom 6540 df-riota 7369 df-ov 7415 df-oprab 7416 df-mpo 7417 df-of 7682 df-om 7867 df-1st 7990 df-2nd 7991 df-supp 8162 df-frecs 8283 df-wrecs 8314 df-recs 8363 df-rdg 8402 df-1o 8460 df-2o 8461 df-oadd 8464 df-er 8701 df-map 8833 df-pm 8834 df-ixp 8910 df-en 8958 df-dom 8959 df-sdom 8960 df-fin 8961 df-fsupp 9338 df-fi 9387 df-sup 9418 df-inf 9419 df-oi 9488 df-dju 9963 df-card 10001 df-pnf 11326 df-mnf 11327 df-xr 11328 df-ltxr 11329 df-le 11330 df-sub 11524 df-neg 11525 df-div 11955 df-nn 12317 df-2 12386 df-3 12387 df-4 12388 df-5 12389 df-6 12390 df-7 12391 df-8 12392 df-9 12393 df-n0 12588 df-z 12675 df-dec 12796 df-uz 12947 df-q 13057 df-rp 13102 df-xneg 13222 df-xadd 13223 df-xmul 13224 df-ioo 13461 df-ioc 13462 df-ico 13463 df-icc 13464 df-fz 13621 df-fzo 13769 df-fl 13912 df-mod 13990 df-seq 14125 df-exp 14185 df-fac 14398 df-bc 14427 df-hash 14455 df-shft 15200 df-cj 15246 df-re 15247 df-im 15248 df-sqrt 15382 df-abs 15383 df-limsup 15618 df-clim 15635 df-rlim 15636 df-sum 15834 df-ef 16213 df-sin 16215 df-cos 16216 df-pi 16218 df-struct 17305 df-sets 17322 df-slot 17340 df-ndx 17352 df-base 17368 df-ress 17389 df-plusg 17421 df-mulr 17422 df-starv 17423 df-sca 17424 df-vsca 17425 df-ip 17426 df-tset 17427 df-ple 17428 df-ds 17430 df-unif 17431 df-hom 17432 df-cco 17433 df-rest 17573 df-topn 17574 df-0g 17592 df-gsum 17593 df-topgen 17594 df-pt 17595 df-prds 17598 df-xrs 17654 df-qtop 17659 df-imas 17660 df-xps 17662 df-mre 17736 df-mrc 17737 df-acs 17739 df-mgm 18796 df-sgrp 18888 df-mnd 18904 df-submnd 18959 df-mulg 19258 df-cntz 19511 df-cmn 19976 df-psmet 21650 df-xmet 21651 df-met 21652 df-bl 21653 df-mopn 21654 df-fbas 21655 df-fg 21656 df-cnfld 21659 df-top 23192 df-topon 23209 df-topsp 23231 df-bases 23244 df-cld 23317 df-ntr 23318 df-cls 23319 df-nei 23396 df-lp 23434 df-perf 23435 df-cn 23525 df-cnp 23526 df-haus 23613 df-tx 23861 df-hmeo 24054 df-fil 24145 df-fm 24237 df-flim 24238 df-flf 24239 df-xms 24619 df-ms 24620 df-tms 24621 df-cncf 25179 df-limc 26166 df-dv 26167 df-log 26866 df-cxp 26867

This theorem is used by: lgamgulmlem6 27343

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