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Theorem lgamgulmlem4 26503

Description: Lemma for lgamgulm 26506. (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 12272	. . . . . . 7 ⊢ 2 ∈ ℕ
2	1	a1i 11	. . . . . 6 ⊢ (𝜑 → 2 ∈ ℕ)
3		lgamgulm.r	. . . . . 6 ⊢ (𝜑 → 𝑅 ∈ ℕ)
4	2, 3	nnmulcld 12252	. . . . 5 ⊢ (𝜑 → (2 · 𝑅) ∈ ℕ)
5	4	nnzd 12572	. . . 4 ⊢ (𝜑 → (2 · 𝑅) ∈ ℤ)
6		eluzle 12822	. . . . . . 7 ⊢ (𝑛 ∈ (ℤ_≥‘(2 · 𝑅)) → (2 · 𝑅) ≤ 𝑛)
7	6	adantl 483	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘(2 · 𝑅))) → (2 · 𝑅) ≤ 𝑛)
8	7	iftrued 4532	. . . . 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 12889	. . . . . . 7 ⊢ (((2 · 𝑅) ∈ ℕ ∧ 𝑛 ∈ (ℤ_≥‘(2 · 𝑅))) → 𝑛 ∈ ℕ)
10	4, 9	sylan 581	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘(2 · 𝑅))) → 𝑛 ∈ ℕ)
11		breq2 5148	. . . . . . . 8 ⊢ (𝑚 = 𝑛 → ((2 · 𝑅) ≤ 𝑚 ↔ (2 · 𝑅) ≤ 𝑛))
12		oveq1 7403	. . . . . . . . . 10 ⊢ (𝑚 = 𝑛 → (𝑚↑2) = (𝑛↑2))
13	12	oveq2d 7412	. . . . . . . . 9 ⊢ (𝑚 = 𝑛 → ((2 · (𝑅 + 1)) / (𝑚↑2)) = ((2 · (𝑅 + 1)) / (𝑛↑2)))
14	13	oveq2d 7412	. . . . . . . 8 ⊢ (𝑚 = 𝑛 → (𝑅 · ((2 · (𝑅 + 1)) / (𝑚↑2))) = (𝑅 · ((2 · (𝑅 + 1)) / (𝑛↑2))))
15		oveq1 7403	. . . . . . . . . . . 12 ⊢ (𝑚 = 𝑛 → (𝑚 + 1) = (𝑛 + 1))
16		id 22	. . . . . . . . . . . 12 ⊢ (𝑚 = 𝑛 → 𝑚 = 𝑛)
17	15, 16	oveq12d 7414	. . . . . . . . . . 11 ⊢ (𝑚 = 𝑛 → ((𝑚 + 1) / 𝑚) = ((𝑛 + 1) / 𝑛))
18	17	fveq2d 6885	. . . . . . . . . 10 ⊢ (𝑚 = 𝑛 → (log‘((𝑚 + 1) / 𝑚)) = (log‘((𝑛 + 1) / 𝑛)))
19	18	oveq2d 7412	. . . . . . . . 9 ⊢ (𝑚 = 𝑛 → (𝑅 · (log‘((𝑚 + 1) / 𝑚))) = (𝑅 · (log‘((𝑛 + 1) / 𝑛))))
20		oveq2 7404	. . . . . . . . . . 11 ⊢ (𝑚 = 𝑛 → ((𝑅 + 1) · 𝑚) = ((𝑅 + 1) · 𝑛))
21	20	fveq2d 6885	. . . . . . . . . 10 ⊢ (𝑚 = 𝑛 → (log‘((𝑅 + 1) · 𝑚)) = (log‘((𝑅 + 1) · 𝑛)))
22	21	oveq1d 7411	. . . . . . . . 9 ⊢ (𝑚 = 𝑛 → ((log‘((𝑅 + 1) · 𝑚)) + π) = ((log‘((𝑅 + 1) · 𝑛)) + π))
23	19, 22	oveq12d 7414	. . . . . . . 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 4552	. . . . . . 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 7429	. . . . . . . 8 ⊢ (𝑅 · ((2 · (𝑅 + 1)) / (𝑛↑2))) ∈ V
27		ovex 7429	. . . . . . . 8 ⊢ ((𝑅 · (log‘((𝑛 + 1) / 𝑛))) + ((log‘((𝑅 + 1) · 𝑛)) + π)) ∈ V
28	26, 27	ifex 4574	. . . . . . 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 6987	. . . . . 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 2733	. . . . . . 7 ⊢ (𝑚 ∈ ℕ ↦ (𝑅 · ((2 · (𝑅 + 1)) / (𝑚↑2)))) = (𝑚 ∈ ℕ ↦ (𝑅 · ((2 · (𝑅 + 1)) / (𝑚↑2))))
32	14, 31, 26	fvmpt 6987	. . . . . 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 2783	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘(2 · 𝑅))) → (𝑇‘𝑛) = ((𝑚 ∈ ℕ ↦ (𝑅 · ((2 · (𝑅 + 1)) / (𝑚↑2))))‘𝑛))
35	5, 34	seqfeq 13980	. . 3 ⊢ (𝜑 → seq(2 · 𝑅)( + , 𝑇) = seq(2 · 𝑅)( + , (𝑚 ∈ ℕ ↦ (𝑅 · ((2 · (𝑅 + 1)) / (𝑚↑2))))))
36		nnuz 12852	. . . . . 6 ⊢ ℕ = (ℤ_≥‘1)
37		1zzd 12580	. . . . . 6 ⊢ (𝜑 → 1 ∈ ℤ)
38	3	nncnd 12215	. . . . . . 7 ⊢ (𝜑 → 𝑅 ∈ ℂ)
39		2cnd 12277	. . . . . . . 8 ⊢ (𝜑 → 2 ∈ ℂ)
40		1cnd 11196	. . . . . . . . 9 ⊢ (𝜑 → 1 ∈ ℂ)
41	38, 40	addcld 11220	. . . . . . . 8 ⊢ (𝜑 → (𝑅 + 1) ∈ ℂ)
42	39, 41	mulcld 11221	. . . . . . 7 ⊢ (𝜑 → (2 · (𝑅 + 1)) ∈ ℂ)
43	38, 42	mulcld 11221	. . . . . 6 ⊢ (𝜑 → (𝑅 · (2 · (𝑅 + 1))) ∈ ℂ)
44		1lt2 12370	. . . . . . . . . 10 ⊢ 1 < 2
45		2re 12273	. . . . . . . . . . 11 ⊢ 2 ∈ ℝ
46		rere 15056	. . . . . . . . . . 11 ⊢ (2 ∈ ℝ → (ℜ‘2) = 2)
47	45, 46	ax-mp 5	. . . . . . . . . 10 ⊢ (ℜ‘2) = 2
48	44, 47	breqtrri 5171	. . . . . . . . 9 ⊢ 1 < (ℜ‘2)
49	48	a1i 11	. . . . . . . 8 ⊢ (𝜑 → 1 < (ℜ‘2))
50		oveq1 7403	. . . . . . . . . 10 ⊢ (𝑚 = 𝑛 → (𝑚↑_𝑐-2) = (𝑛↑_𝑐-2))
51		eqid 2733	. . . . . . . . . 10 ⊢ (𝑚 ∈ ℕ ↦ (𝑚↑_𝑐-2)) = (𝑚 ∈ ℕ ↦ (𝑚↑_𝑐-2))
52		ovex 7429	. . . . . . . . . 10 ⊢ (𝑛↑_𝑐-2) ∈ V
53	50, 51, 52	fvmpt 6987	. . . . . . . . 9 ⊢ (𝑛 ∈ ℕ → ((𝑚 ∈ ℕ ↦ (𝑚↑_𝑐-2))‘𝑛) = (𝑛↑_𝑐-2))
54	53	adantl 483	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((𝑚 ∈ ℕ ↦ (𝑚↑_𝑐-2))‘𝑛) = (𝑛↑_𝑐-2))
55	39, 49, 54	zetacvg 26486	. . . . . . 7 ⊢ (𝜑 → seq1( + , (𝑚 ∈ ℕ ↦ (𝑚↑_𝑐-2))) ∈ dom ⇝ )
56		climdm 15485	. . . . . . 7 ⊢ (seq1( + , (𝑚 ∈ ℕ ↦ (𝑚↑_𝑐-2))) ∈ dom ⇝ ↔ seq1( + , (𝑚 ∈ ℕ ↦ (𝑚↑_𝑐-2))) ⇝ ( ⇝ ‘seq1( + , (𝑚 ∈ ℕ ↦ (𝑚↑_𝑐-2)))))
57	55, 56	sylib 217	. . . . . 6 ⊢ (𝜑 → seq1( + , (𝑚 ∈ ℕ ↦ (𝑚↑_𝑐-2))) ⇝ ( ⇝ ‘seq1( + , (𝑚 ∈ ℕ ↦ (𝑚↑_𝑐-2)))))
58		simpr 486	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ ℕ)
59	58	nncnd 12215	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ ℂ)
60		2cnd 12277	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 2 ∈ ℂ)
61	60	negcld 11545	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → -2 ∈ ℂ)
62	59, 61	cxpcld 26185	. . . . . . 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 482	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝑅 ∈ ℂ)
65		1cnd 11196	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 1 ∈ ℂ)
66	64, 65	addcld 11220	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑅 + 1) ∈ ℂ)
67	60, 66	mulcld 11221	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (2 · (𝑅 + 1)) ∈ ℂ)
68	64, 67	mulcld 11221	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑅 · (2 · (𝑅 + 1))) ∈ ℂ)
69	59	sqcld 14096	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑛↑2) ∈ ℂ)
70	58	nnne0d 12249	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝑛 ≠ 0)
71		2z 12581	. . . . . . . . . . 11 ⊢ 2 ∈ ℤ
72	71	a1i 11	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 2 ∈ ℤ)
73	59, 70, 72	expne0d 14104	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑛↑2) ≠ 0)
74	68, 69, 73	divrecd 11980	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((𝑅 · (2 · (𝑅 + 1))) / (𝑛↑2)) = ((𝑅 · (2 · (𝑅 + 1))) · (1 / (𝑛↑2))))
75	64, 67, 69, 73	divassd 12012	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((𝑅 · (2 · (𝑅 + 1))) / (𝑛↑2)) = (𝑅 · ((2 · (𝑅 + 1)) / (𝑛↑2))))
76	59, 70, 60	cxpnegd 26192	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑛↑_𝑐-2) = (1 / (𝑛↑_𝑐2)))
77	59, 70, 72	cxpexpzd 26188	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑛↑_𝑐2) = (𝑛↑2))
78	77	oveq2d 7412	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (1 / (𝑛↑_𝑐2)) = (1 / (𝑛↑2)))
79	76, 78	eqtr2d 2774	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (1 / (𝑛↑2)) = (𝑛↑_𝑐-2))
80	79	oveq2d 7412	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((𝑅 · (2 · (𝑅 + 1))) · (1 / (𝑛↑2))) = ((𝑅 · (2 · (𝑅 + 1))) · (𝑛↑_𝑐-2)))
81	74, 75, 80	3eqtr3d 2781	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑅 · ((2 · (𝑅 + 1)) / (𝑛↑2))) = ((𝑅 · (2 · (𝑅 + 1))) · (𝑛↑_𝑐-2)))
82	32	adantl 483	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((𝑚 ∈ ℕ ↦ (𝑅 · ((2 · (𝑅 + 1)) / (𝑚↑2))))‘𝑛) = (𝑅 · ((2 · (𝑅 + 1)) / (𝑛↑2))))
83	54	oveq2d 7412	. . . . . . 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 2783	. . . . . 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 15591	. . . . 5 ⊢ (𝜑 → seq1( + , (𝑚 ∈ ℕ ↦ (𝑅 · ((2 · (𝑅 + 1)) / (𝑚↑2))))) ⇝ ((𝑅 · (2 · (𝑅 + 1))) · ( ⇝ ‘seq1( + , (𝑚 ∈ ℕ ↦ (𝑚↑_𝑐-2))))))
86		climrel 15423	. . . . . 6 ⊢ Rel ⇝
87	86	releldmi 5942	. . . . 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 11977	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((2 · (𝑅 + 1)) / (𝑛↑2)) ∈ ℂ)
90	64, 89	mulcld 11221	. . . . . 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 15590	. . . 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 231	. . 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 483	. . . . 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 482	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝑅 ∈ ℕ)
97	96	nnred 12214	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝑅 ∈ ℝ)
98	45	a1i 11	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 2 ∈ ℝ)
99		1red 11202	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 1 ∈ ℝ)
100	97, 99	readdcld 11230	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑅 + 1) ∈ ℝ)
101	98, 100	remulcld 11231	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (2 · (𝑅 + 1)) ∈ ℝ)
102	58	nnsqcld 14194	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑛↑2) ∈ ℕ)
103	101, 102	nndivred 12253	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((2 · (𝑅 + 1)) / (𝑛↑2)) ∈ ℝ)
104	97, 103	remulcld 11231	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑅 · ((2 · (𝑅 + 1)) / (𝑛↑2))) ∈ ℝ)
105	58	peano2nnd 12216	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑛 + 1) ∈ ℕ)
106	105	nnrpd 13001	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑛 + 1) ∈ ℝ⁺)
107	58	nnrpd 13001	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ ℝ⁺)
108	106, 107	rpdivcld 13020	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((𝑛 + 1) / 𝑛) ∈ ℝ⁺)
109	108	relogcld 26100	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (log‘((𝑛 + 1) / 𝑛)) ∈ ℝ)
110	97, 109	remulcld 11231	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑅 · (log‘((𝑛 + 1) / 𝑛))) ∈ ℝ)
111	96	peano2nnd 12216	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑅 + 1) ∈ ℕ)
112	111	nnrpd 13001	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑅 + 1) ∈ ℝ⁺)
113	112, 107	rpmulcld 13019	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((𝑅 + 1) · 𝑛) ∈ ℝ⁺)
114	113	relogcld 26100	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (log‘((𝑅 + 1) · 𝑛)) ∈ ℝ)
115		pire 25937	. . . . . . . . 9 ⊢ π ∈ ℝ
116	115	a1i 11	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → π ∈ ℝ)
117	114, 116	readdcld 11230	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((log‘((𝑅 + 1) · 𝑛)) + π) ∈ ℝ)
118	110, 117	readdcld 11230	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((𝑅 · (log‘((𝑛 + 1) / 𝑛))) + ((log‘((𝑅 + 1) · 𝑛)) + π)) ∈ ℝ)
119	104, 118	ifcld 4570	. . . . 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 11229	. . 3 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑇‘𝑛) ∈ ℂ)
122	36, 4, 121	iserex 15590	. 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 397 = wceq 1542 ∈ wcel 2107 ∀wral 3062 {crab 3433 ifcif 4524 class class class wbr 5144 ↦ cmpt 5227 dom cdm 5672 ‘cfv 6535 (class class class)co 7396 ℂcc 11095 ℝcr 11096 1c1 11098 + caddc 11100 · cmul 11102 < clt 11235 ≤ cle 11236 − cmin 11431 -cneg 11432 / cdiv 11858 ℕcn 12199 2c2 12254 ℕ₀cn0 12459 ℤcz 12545 ℤ_≥cuz 12809 seqcseq 13953 ↑cexp 14014 ℜcre 15031 abscabs 15168 ⇝ cli 15415 πcpi 15997 logclog 26032 ↑_𝑐ccxp 26033

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5281 ax-sep 5295 ax-nul 5302 ax-pow 5359 ax-pr 5423 ax-un 7712 ax-inf2 9623 ax-cnex 11153 ax-resscn 11154 ax-1cn 11155 ax-icn 11156 ax-addcl 11157 ax-addrcl 11158 ax-mulcl 11159 ax-mulrcl 11160 ax-mulcom 11161 ax-addass 11162 ax-mulass 11163 ax-distr 11164 ax-i2m1 11165 ax-1ne0 11166 ax-1rid 11167 ax-rnegex 11168 ax-rrecex 11169 ax-cnre 11170 ax-pre-lttri 11171 ax-pre-lttrn 11172 ax-pre-ltadd 11173 ax-pre-mulgt0 11174 ax-pre-sup 11175 ax-addf 11176 ax-mulf 11177

This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3776 df-csb 3892 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-pss 3965 df-nul 4321 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-tp 4629 df-op 4631 df-uni 4905 df-int 4947 df-iun 4995 df-iin 4996 df-br 5145 df-opab 5207 df-mpt 5228 df-tr 5262 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-se 5628 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6292 df-ord 6359 df-on 6360 df-lim 6361 df-suc 6362 df-iota 6487 df-fun 6537 df-fn 6538 df-f 6539 df-f1 6540 df-fo 6541 df-f1o 6542 df-fv 6543 df-isom 6544 df-riota 7352 df-ov 7399 df-oprab 7400 df-mpo 7401 df-of 7657 df-om 7843 df-1st 7962 df-2nd 7963 df-supp 8134 df-frecs 8253 df-wrecs 8284 df-recs 8358 df-rdg 8397 df-1o 8453 df-2o 8454 df-oadd 8457 df-er 8691 df-map 8810 df-pm 8811 df-ixp 8880 df-en 8928 df-dom 8929 df-sdom 8930 df-fin 8931 df-fsupp 9350 df-fi 9393 df-sup 9424 df-inf 9425 df-oi 9492 df-dju 9883 df-card 9921 df-pnf 11237 df-mnf 11238 df-xr 11239 df-ltxr 11240 df-le 11241 df-sub 11433 df-neg 11434 df-div 11859 df-nn 12200 df-2 12262 df-3 12263 df-4 12264 df-5 12265 df-6 12266 df-7 12267 df-8 12268 df-9 12269 df-n0 12460 df-z 12546 df-dec 12665 df-uz 12810 df-q 12920 df-rp 12962 df-xneg 13079 df-xadd 13080 df-xmul 13081 df-ioo 13315 df-ioc 13316 df-ico 13317 df-icc 13318 df-fz 13472 df-fzo 13615 df-fl 13744 df-mod 13822 df-seq 13954 df-exp 14015 df-fac 14221 df-bc 14250 df-hash 14278 df-shft 15001 df-cj 15033 df-re 15034 df-im 15035 df-sqrt 15169 df-abs 15170 df-limsup 15402 df-clim 15419 df-rlim 15420 df-sum 15620 df-ef 15998 df-sin 16000 df-cos 16001 df-pi 16003 df-struct 17067 df-sets 17084 df-slot 17102 df-ndx 17114 df-base 17132 df-ress 17161 df-plusg 17197 df-mulr 17198 df-starv 17199 df-sca 17200 df-vsca 17201 df-ip 17202 df-tset 17203 df-ple 17204 df-ds 17206 df-unif 17207 df-hom 17208 df-cco 17209 df-rest 17355 df-topn 17356 df-0g 17374 df-gsum 17375 df-topgen 17376 df-pt 17377 df-prds 17380 df-xrs 17435 df-qtop 17440 df-imas 17441 df-xps 17443 df-mre 17517 df-mrc 17518 df-acs 17520 df-mgm 18548 df-sgrp 18597 df-mnd 18613 df-submnd 18659 df-mulg 18936 df-cntz 19166 df-cmn 19634 df-psmet 20910 df-xmet 20911 df-met 20912 df-bl 20913 df-mopn 20914 df-fbas 20915 df-fg 20916 df-cnfld 20919 df-top 22365 df-topon 22382 df-topsp 22404 df-bases 22418 df-cld 22492 df-ntr 22493 df-cls 22494 df-nei 22571 df-lp 22609 df-perf 22610 df-cn 22700 df-cnp 22701 df-haus 22788 df-tx 23035 df-hmeo 23228 df-fil 23319 df-fm 23411 df-flim 23412 df-flf 23413 df-xms 23795 df-ms 23796 df-tms 23797 df-cncf 24363 df-limc 25352 df-dv 25353 df-log 26034 df-cxp 26035

This theorem is referenced by: lgamgulmlem6 26505

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