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Theorem lgamgulmlem4 26994

Description: Lemma for lgamgulm 26997. (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 12313	. . . . . . 7 ⊢ 2 ∈ ℕ
2	1	a1i 11	. . . . . 6 ⊢ (𝜑 → 2 ∈ ℕ)
3		lgamgulm.r	. . . . . 6 ⊢ (𝜑 → 𝑅 ∈ ℕ)
4	2, 3	nnmulcld 12293	. . . . 5 ⊢ (𝜑 → (2 · 𝑅) ∈ ℕ)
5	4	nnzd 12615	. . . 4 ⊢ (𝜑 → (2 · 𝑅) ∈ ℤ)
6		eluzle 12865	. . . . . . 7 ⊢ (𝑛 ∈ (ℤ_≥‘(2 · 𝑅)) → (2 · 𝑅) ≤ 𝑛)
7	6	adantl 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘(2 · 𝑅))) → (2 · 𝑅) ≤ 𝑛)
8	7	iftrued 4508	. . . . 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 12934	. . . . . . 7 ⊢ (((2 · 𝑅) ∈ ℕ ∧ 𝑛 ∈ (ℤ_≥‘(2 · 𝑅))) → 𝑛 ∈ ℕ)
10	4, 9	sylan 580	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘(2 · 𝑅))) → 𝑛 ∈ ℕ)
11		breq2 5123	. . . . . . . 8 ⊢ (𝑚 = 𝑛 → ((2 · 𝑅) ≤ 𝑚 ↔ (2 · 𝑅) ≤ 𝑛))
12		oveq1 7412	. . . . . . . . . 10 ⊢ (𝑚 = 𝑛 → (𝑚↑2) = (𝑛↑2))
13	12	oveq2d 7421	. . . . . . . . 9 ⊢ (𝑚 = 𝑛 → ((2 · (𝑅 + 1)) / (𝑚↑2)) = ((2 · (𝑅 + 1)) / (𝑛↑2)))
14	13	oveq2d 7421	. . . . . . . 8 ⊢ (𝑚 = 𝑛 → (𝑅 · ((2 · (𝑅 + 1)) / (𝑚↑2))) = (𝑅 · ((2 · (𝑅 + 1)) / (𝑛↑2))))
15		oveq1 7412	. . . . . . . . . . . 12 ⊢ (𝑚 = 𝑛 → (𝑚 + 1) = (𝑛 + 1))
16		id 22	. . . . . . . . . . . 12 ⊢ (𝑚 = 𝑛 → 𝑚 = 𝑛)
17	15, 16	oveq12d 7423	. . . . . . . . . . 11 ⊢ (𝑚 = 𝑛 → ((𝑚 + 1) / 𝑚) = ((𝑛 + 1) / 𝑛))
18	17	fveq2d 6880	. . . . . . . . . 10 ⊢ (𝑚 = 𝑛 → (log‘((𝑚 + 1) / 𝑚)) = (log‘((𝑛 + 1) / 𝑛)))
19	18	oveq2d 7421	. . . . . . . . 9 ⊢ (𝑚 = 𝑛 → (𝑅 · (log‘((𝑚 + 1) / 𝑚))) = (𝑅 · (log‘((𝑛 + 1) / 𝑛))))
20		oveq2 7413	. . . . . . . . . . 11 ⊢ (𝑚 = 𝑛 → ((𝑅 + 1) · 𝑚) = ((𝑅 + 1) · 𝑛))
21	20	fveq2d 6880	. . . . . . . . . 10 ⊢ (𝑚 = 𝑛 → (log‘((𝑅 + 1) · 𝑚)) = (log‘((𝑅 + 1) · 𝑛)))
22	21	oveq1d 7420	. . . . . . . . 9 ⊢ (𝑚 = 𝑛 → ((log‘((𝑅 + 1) · 𝑚)) + π) = ((log‘((𝑅 + 1) · 𝑛)) + π))
23	19, 22	oveq12d 7423	. . . . . . . 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 4529	. . . . . . 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 7438	. . . . . . . 8 ⊢ (𝑅 · ((2 · (𝑅 + 1)) / (𝑛↑2))) ∈ V
27		ovex 7438	. . . . . . . 8 ⊢ ((𝑅 · (log‘((𝑛 + 1) / 𝑛))) + ((log‘((𝑅 + 1) · 𝑛)) + π)) ∈ V
28	26, 27	ifex 4551	. . . . . . 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 6986	. . . . . 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 17	. . . . 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 2735	. . . . . . 7 ⊢ (𝑚 ∈ ℕ ↦ (𝑅 · ((2 · (𝑅 + 1)) / (𝑚↑2)))) = (𝑚 ∈ ℕ ↦ (𝑅 · ((2 · (𝑅 + 1)) / (𝑚↑2))))
32	14, 31, 26	fvmpt 6986	. . . . . 6 ⊢ (𝑛 ∈ ℕ → ((𝑚 ∈ ℕ ↦ (𝑅 · ((2 · (𝑅 + 1)) / (𝑚↑2))))‘𝑛) = (𝑅 · ((2 · (𝑅 + 1)) / (𝑛↑2))))
33	10, 32	syl 17	. . . . 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 2780	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘(2 · 𝑅))) → (𝑇‘𝑛) = ((𝑚 ∈ ℕ ↦ (𝑅 · ((2 · (𝑅 + 1)) / (𝑚↑2))))‘𝑛))
35	5, 34	seqfeq 14045	. . 3 ⊢ (𝜑 → seq(2 · 𝑅)( + , 𝑇) = seq(2 · 𝑅)( + , (𝑚 ∈ ℕ ↦ (𝑅 · ((2 · (𝑅 + 1)) / (𝑚↑2))))))
36		nnuz 12895	. . . . . 6 ⊢ ℕ = (ℤ_≥‘1)
37		1zzd 12623	. . . . . 6 ⊢ (𝜑 → 1 ∈ ℤ)
38	3	nncnd 12256	. . . . . . 7 ⊢ (𝜑 → 𝑅 ∈ ℂ)
39		2cnd 12318	. . . . . . . 8 ⊢ (𝜑 → 2 ∈ ℂ)
40		1cnd 11230	. . . . . . . . 9 ⊢ (𝜑 → 1 ∈ ℂ)
41	38, 40	addcld 11254	. . . . . . . 8 ⊢ (𝜑 → (𝑅 + 1) ∈ ℂ)
42	39, 41	mulcld 11255	. . . . . . 7 ⊢ (𝜑 → (2 · (𝑅 + 1)) ∈ ℂ)
43	38, 42	mulcld 11255	. . . . . 6 ⊢ (𝜑 → (𝑅 · (2 · (𝑅 + 1))) ∈ ℂ)
44		1lt2 12411	. . . . . . . . . 10 ⊢ 1 < 2
45		2re 12314	. . . . . . . . . . 11 ⊢ 2 ∈ ℝ
46		rere 15141	. . . . . . . . . . 11 ⊢ (2 ∈ ℝ → (ℜ‘2) = 2)
47	45, 46	ax-mp 5	. . . . . . . . . 10 ⊢ (ℜ‘2) = 2
48	44, 47	breqtrri 5146	. . . . . . . . 9 ⊢ 1 < (ℜ‘2)
49	48	a1i 11	. . . . . . . 8 ⊢ (𝜑 → 1 < (ℜ‘2))
50		oveq1 7412	. . . . . . . . . 10 ⊢ (𝑚 = 𝑛 → (𝑚↑_𝑐-2) = (𝑛↑_𝑐-2))
51		eqid 2735	. . . . . . . . . 10 ⊢ (𝑚 ∈ ℕ ↦ (𝑚↑_𝑐-2)) = (𝑚 ∈ ℕ ↦ (𝑚↑_𝑐-2))
52		ovex 7438	. . . . . . . . . 10 ⊢ (𝑛↑_𝑐-2) ∈ V
53	50, 51, 52	fvmpt 6986	. . . . . . . . 9 ⊢ (𝑛 ∈ ℕ → ((𝑚 ∈ ℕ ↦ (𝑚↑_𝑐-2))‘𝑛) = (𝑛↑_𝑐-2))
54	53	adantl 481	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((𝑚 ∈ ℕ ↦ (𝑚↑_𝑐-2))‘𝑛) = (𝑛↑_𝑐-2))
55	39, 49, 54	zetacvg 26977	. . . . . . 7 ⊢ (𝜑 → seq1( + , (𝑚 ∈ ℕ ↦ (𝑚↑_𝑐-2))) ∈ dom ⇝ )
56		climdm 15570	. . . . . . 7 ⊢ (seq1( + , (𝑚 ∈ ℕ ↦ (𝑚↑_𝑐-2))) ∈ dom ⇝ ↔ seq1( + , (𝑚 ∈ ℕ ↦ (𝑚↑_𝑐-2))) ⇝ ( ⇝ ‘seq1( + , (𝑚 ∈ ℕ ↦ (𝑚↑_𝑐-2)))))
57	55, 56	sylib 218	. . . . . 6 ⊢ (𝜑 → seq1( + , (𝑚 ∈ ℕ ↦ (𝑚↑_𝑐-2))) ⇝ ( ⇝ ‘seq1( + , (𝑚 ∈ ℕ ↦ (𝑚↑_𝑐-2)))))
58		simpr 484	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ ℕ)
59	58	nncnd 12256	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ ℂ)
60		2cnd 12318	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 2 ∈ ℂ)
61	60	negcld 11581	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → -2 ∈ ℂ)
62	59, 61	cxpcld 26669	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑛↑_𝑐-2) ∈ ℂ)
63	54, 62	eqeltrd 2834	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((𝑚 ∈ ℕ ↦ (𝑚↑_𝑐-2))‘𝑛) ∈ ℂ)
64	38	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝑅 ∈ ℂ)
65		1cnd 11230	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 1 ∈ ℂ)
66	64, 65	addcld 11254	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑅 + 1) ∈ ℂ)
67	60, 66	mulcld 11255	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (2 · (𝑅 + 1)) ∈ ℂ)
68	64, 67	mulcld 11255	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑅 · (2 · (𝑅 + 1))) ∈ ℂ)
69	59	sqcld 14162	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑛↑2) ∈ ℂ)
70	58	nnne0d 12290	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝑛 ≠ 0)
71		2z 12624	. . . . . . . . . . 11 ⊢ 2 ∈ ℤ
72	71	a1i 11	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 2 ∈ ℤ)
73	59, 70, 72	expne0d 14170	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑛↑2) ≠ 0)
74	68, 69, 73	divrecd 12020	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((𝑅 · (2 · (𝑅 + 1))) / (𝑛↑2)) = ((𝑅 · (2 · (𝑅 + 1))) · (1 / (𝑛↑2))))
75	64, 67, 69, 73	divassd 12052	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((𝑅 · (2 · (𝑅 + 1))) / (𝑛↑2)) = (𝑅 · ((2 · (𝑅 + 1)) / (𝑛↑2))))
76	59, 70, 60	cxpnegd 26676	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑛↑_𝑐-2) = (1 / (𝑛↑_𝑐2)))
77	59, 70, 72	cxpexpzd 26672	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑛↑_𝑐2) = (𝑛↑2))
78	77	oveq2d 7421	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (1 / (𝑛↑_𝑐2)) = (1 / (𝑛↑2)))
79	76, 78	eqtr2d 2771	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (1 / (𝑛↑2)) = (𝑛↑_𝑐-2))
80	79	oveq2d 7421	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((𝑅 · (2 · (𝑅 + 1))) · (1 / (𝑛↑2))) = ((𝑅 · (2 · (𝑅 + 1))) · (𝑛↑_𝑐-2)))
81	74, 75, 80	3eqtr3d 2778	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑅 · ((2 · (𝑅 + 1)) / (𝑛↑2))) = ((𝑅 · (2 · (𝑅 + 1))) · (𝑛↑_𝑐-2)))
82	32	adantl 481	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((𝑚 ∈ ℕ ↦ (𝑅 · ((2 · (𝑅 + 1)) / (𝑚↑2))))‘𝑛) = (𝑅 · ((2 · (𝑅 + 1)) / (𝑛↑2))))
83	54	oveq2d 7421	. . . . . . 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 2780	. . . . . 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 15674	. . . . 5 ⊢ (𝜑 → seq1( + , (𝑚 ∈ ℕ ↦ (𝑅 · ((2 · (𝑅 + 1)) / (𝑚↑2))))) ⇝ ((𝑅 · (2 · (𝑅 + 1))) · ( ⇝ ‘seq1( + , (𝑚 ∈ ℕ ↦ (𝑚↑_𝑐-2))))))
86		climrel 15508	. . . . . 6 ⊢ Rel ⇝
87	86	releldmi 5928	. . . . 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 17	. . . 4 ⊢ (𝜑 → seq1( + , (𝑚 ∈ ℕ ↦ (𝑅 · ((2 · (𝑅 + 1)) / (𝑚↑2))))) ∈ dom ⇝ )
89	67, 69, 73	divcld 12017	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((2 · (𝑅 + 1)) / (𝑛↑2)) ∈ ℂ)
90	64, 89	mulcld 11255	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑅 · ((2 · (𝑅 + 1)) / (𝑛↑2))) ∈ ℂ)
91	82, 90	eqeltrd 2834	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((𝑚 ∈ ℕ ↦ (𝑅 · ((2 · (𝑅 + 1)) / (𝑚↑2))))‘𝑛) ∈ ℂ)
92	36, 4, 91	iserex 15673	. . . 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 232	. . 3 ⊢ (𝜑 → seq(2 · 𝑅)( + , (𝑚 ∈ ℕ ↦ (𝑅 · ((2 · (𝑅 + 1)) / (𝑚↑2))))) ∈ dom ⇝ )
94	35, 93	eqeltrd 2834	. 2 ⊢ (𝜑 → seq(2 · 𝑅)( + , 𝑇) ∈ dom ⇝ )
95	29	adantl 481	. . . . 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 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝑅 ∈ ℕ)
97	96	nnred 12255	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝑅 ∈ ℝ)
98	45	a1i 11	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 2 ∈ ℝ)
99		1red 11236	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 1 ∈ ℝ)
100	97, 99	readdcld 11264	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑅 + 1) ∈ ℝ)
101	98, 100	remulcld 11265	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (2 · (𝑅 + 1)) ∈ ℝ)
102	58	nnsqcld 14262	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑛↑2) ∈ ℕ)
103	101, 102	nndivred 12294	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((2 · (𝑅 + 1)) / (𝑛↑2)) ∈ ℝ)
104	97, 103	remulcld 11265	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑅 · ((2 · (𝑅 + 1)) / (𝑛↑2))) ∈ ℝ)
105	58	peano2nnd 12257	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑛 + 1) ∈ ℕ)
106	105	nnrpd 13049	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑛 + 1) ∈ ℝ⁺)
107	58	nnrpd 13049	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ ℝ⁺)
108	106, 107	rpdivcld 13068	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((𝑛 + 1) / 𝑛) ∈ ℝ⁺)
109	108	relogcld 26584	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (log‘((𝑛 + 1) / 𝑛)) ∈ ℝ)
110	97, 109	remulcld 11265	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑅 · (log‘((𝑛 + 1) / 𝑛))) ∈ ℝ)
111	96	peano2nnd 12257	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑅 + 1) ∈ ℕ)
112	111	nnrpd 13049	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑅 + 1) ∈ ℝ⁺)
113	112, 107	rpmulcld 13067	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((𝑅 + 1) · 𝑛) ∈ ℝ⁺)
114	113	relogcld 26584	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (log‘((𝑅 + 1) · 𝑛)) ∈ ℝ)
115		pire 26418	. . . . . . . . 9 ⊢ π ∈ ℝ
116	115	a1i 11	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → π ∈ ℝ)
117	114, 116	readdcld 11264	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((log‘((𝑅 + 1) · 𝑛)) + π) ∈ ℝ)
118	110, 117	readdcld 11264	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((𝑅 · (log‘((𝑛 + 1) / 𝑛))) + ((log‘((𝑅 + 1) · 𝑛)) + π)) ∈ ℝ)
119	104, 118	ifcld 4547	. . . . 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 2834	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑇‘𝑛) ∈ ℝ)
121	120	recnd 11263	. . 3 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑇‘𝑛) ∈ ℂ)
122	36, 4, 121	iserex 15673	. 2 ⊢ (𝜑 → (seq1( + , 𝑇) ∈ dom ⇝ ↔ seq(2 · 𝑅)( + , 𝑇) ∈ dom ⇝ ))
123	94, 122	mpbird 257	1 ⊢ (𝜑 → seq1( + , 𝑇) ∈ dom ⇝ )

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2108 ∀wral 3051 {crab 3415 ifcif 4500 class class class wbr 5119 ↦ cmpt 5201 dom cdm 5654 ‘cfv 6531 (class class class)co 7405 ℂcc 11127 ℝcr 11128 1c1 11130 + caddc 11132 · cmul 11134 < clt 11269 ≤ cle 11270 − cmin 11466 -cneg 11467 / cdiv 11894 ℕcn 12240 2c2 12295 ℕ₀cn0 12501 ℤcz 12588 ℤ_≥cuz 12852 seqcseq 14019 ↑cexp 14079 ℜcre 15116 abscabs 15253 ⇝ cli 15500 πcpi 16082 logclog 26515 ↑_𝑐ccxp 26516

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 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2707 ax-rep 5249 ax-sep 5266 ax-nul 5276 ax-pow 5335 ax-pr 5402 ax-un 7729 ax-inf2 9655 ax-cnex 11185 ax-resscn 11186 ax-1cn 11187 ax-icn 11188 ax-addcl 11189 ax-addrcl 11190 ax-mulcl 11191 ax-mulrcl 11192 ax-mulcom 11193 ax-addass 11194 ax-mulass 11195 ax-distr 11196 ax-i2m1 11197 ax-1ne0 11198 ax-1rid 11199 ax-rnegex 11200 ax-rrecex 11201 ax-cnre 11202 ax-pre-lttri 11203 ax-pre-lttrn 11204 ax-pre-ltadd 11205 ax-pre-mulgt0 11206 ax-pre-sup 11207 ax-addf 11208

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 2065 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2727 df-clel 2809 df-nfc 2885 df-ne 2933 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3359 df-reu 3360 df-rab 3416 df-v 3461 df-sbc 3766 df-csb 3875 df-dif 3929 df-un 3931 df-in 3933 df-ss 3943 df-pss 3946 df-nul 4309 df-if 4501 df-pw 4577 df-sn 4602 df-pr 4604 df-tp 4606 df-op 4608 df-uni 4884 df-int 4923 df-iun 4969 df-iin 4970 df-br 5120 df-opab 5182 df-mpt 5202 df-tr 5230 df-id 5548 df-eprel 5553 df-po 5561 df-so 5562 df-fr 5606 df-se 5607 df-we 5608 df-xp 5660 df-rel 5661 df-cnv 5662 df-co 5663 df-dm 5664 df-rn 5665 df-res 5666 df-ima 5667 df-pred 6290 df-ord 6355 df-on 6356 df-lim 6357 df-suc 6358 df-iota 6484 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 7362 df-ov 7408 df-oprab 7409 df-mpo 7410 df-of 7671 df-om 7862 df-1st 7988 df-2nd 7989 df-supp 8160 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-rdg 8424 df-1o 8480 df-2o 8481 df-oadd 8484 df-er 8719 df-map 8842 df-pm 8843 df-ixp 8912 df-en 8960 df-dom 8961 df-sdom 8962 df-fin 8963 df-fsupp 9374 df-fi 9423 df-sup 9454 df-inf 9455 df-oi 9524 df-dju 9915 df-card 9953 df-pnf 11271 df-mnf 11272 df-xr 11273 df-ltxr 11274 df-le 11275 df-sub 11468 df-neg 11469 df-div 11895 df-nn 12241 df-2 12303 df-3 12304 df-4 12305 df-5 12306 df-6 12307 df-7 12308 df-8 12309 df-9 12310 df-n0 12502 df-z 12589 df-dec 12709 df-uz 12853 df-q 12965 df-rp 13009 df-xneg 13128 df-xadd 13129 df-xmul 13130 df-ioo 13366 df-ioc 13367 df-ico 13368 df-icc 13369 df-fz 13525 df-fzo 13672 df-fl 13809 df-mod 13887 df-seq 14020 df-exp 14080 df-fac 14292 df-bc 14321 df-hash 14349 df-shft 15086 df-cj 15118 df-re 15119 df-im 15120 df-sqrt 15254 df-abs 15255 df-limsup 15487 df-clim 15504 df-rlim 15505 df-sum 15703 df-ef 16083 df-sin 16085 df-cos 16086 df-pi 16088 df-struct 17166 df-sets 17183 df-slot 17201 df-ndx 17213 df-base 17229 df-ress 17252 df-plusg 17284 df-mulr 17285 df-starv 17286 df-sca 17287 df-vsca 17288 df-ip 17289 df-tset 17290 df-ple 17291 df-ds 17293 df-unif 17294 df-hom 17295 df-cco 17296 df-rest 17436 df-topn 17437 df-0g 17455 df-gsum 17456 df-topgen 17457 df-pt 17458 df-prds 17461 df-xrs 17516 df-qtop 17521 df-imas 17522 df-xps 17524 df-mre 17598 df-mrc 17599 df-acs 17601 df-mgm 18618 df-sgrp 18697 df-mnd 18713 df-submnd 18762 df-mulg 19051 df-cntz 19300 df-cmn 19763 df-psmet 21307 df-xmet 21308 df-met 21309 df-bl 21310 df-mopn 21311 df-fbas 21312 df-fg 21313 df-cnfld 21316 df-top 22832 df-topon 22849 df-topsp 22871 df-bases 22884 df-cld 22957 df-ntr 22958 df-cls 22959 df-nei 23036 df-lp 23074 df-perf 23075 df-cn 23165 df-cnp 23166 df-haus 23253 df-tx 23500 df-hmeo 23693 df-fil 23784 df-fm 23876 df-flim 23877 df-flf 23878 df-xms 24259 df-ms 24260 df-tms 24261 df-cncf 24822 df-limc 25819 df-dv 25820 df-log 26517 df-cxp 26518

This theorem is referenced by: lgamgulmlem6 26996

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