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Theorem lgamgulmlem4 26773

Description: Lemma for lgamgulm 26776. (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 12290	. . . . . . 7 ⊢ 2 ∈ ℕ
2	1	a1i 11	. . . . . 6 ⊢ (𝜑 → 2 ∈ ℕ)
3		lgamgulm.r	. . . . . 6 ⊢ (𝜑 → 𝑅 ∈ ℕ)
4	2, 3	nnmulcld 12270	. . . . 5 ⊢ (𝜑 → (2 · 𝑅) ∈ ℕ)
5	4	nnzd 12590	. . . 4 ⊢ (𝜑 → (2 · 𝑅) ∈ ℤ)
6		eluzle 12840	. . . . . . 7 ⊢ (𝑛 ∈ (ℤ_≥‘(2 · 𝑅)) → (2 · 𝑅) ≤ 𝑛)
7	6	adantl 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘(2 · 𝑅))) → (2 · 𝑅) ≤ 𝑛)
8	7	iftrued 4536	. . . . 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 12907	. . . . . . 7 ⊢ (((2 · 𝑅) ∈ ℕ ∧ 𝑛 ∈ (ℤ_≥‘(2 · 𝑅))) → 𝑛 ∈ ℕ)
10	4, 9	sylan 579	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘(2 · 𝑅))) → 𝑛 ∈ ℕ)
11		breq2 5152	. . . . . . . 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 22	. . . . . . . . . . . 12 ⊢ (𝑚 = 𝑛 → 𝑚 = 𝑛)
17	15, 16	oveq12d 7430	. . . . . . . . . . 11 ⊢ (𝑚 = 𝑛 → ((𝑚 + 1) / 𝑚) = ((𝑛 + 1) / 𝑛))
18	17	fveq2d 6895	. . . . . . . . . 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 6895	. . . . . . . . . 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 4556	. . . . . . 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 4578	. . . . . . 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 6998	. . . . . 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 2731	. . . . . . 7 ⊢ (𝑚 ∈ ℕ ↦ (𝑅 · ((2 · (𝑅 + 1)) / (𝑚↑2)))) = (𝑚 ∈ ℕ ↦ (𝑅 · ((2 · (𝑅 + 1)) / (𝑚↑2))))
32	14, 31, 26	fvmpt 6998	. . . . . 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 2781	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘(2 · 𝑅))) → (𝑇‘𝑛) = ((𝑚 ∈ ℕ ↦ (𝑅 · ((2 · (𝑅 + 1)) / (𝑚↑2))))‘𝑛))
35	5, 34	seqfeq 13998	. . 3 ⊢ (𝜑 → seq(2 · 𝑅)( + , 𝑇) = seq(2 · 𝑅)( + , (𝑚 ∈ ℕ ↦ (𝑅 · ((2 · (𝑅 + 1)) / (𝑚↑2))))))
36		nnuz 12870	. . . . . 6 ⊢ ℕ = (ℤ_≥‘1)
37		1zzd 12598	. . . . . 6 ⊢ (𝜑 → 1 ∈ ℤ)
38	3	nncnd 12233	. . . . . . 7 ⊢ (𝜑 → 𝑅 ∈ ℂ)
39		2cnd 12295	. . . . . . . 8 ⊢ (𝜑 → 2 ∈ ℂ)
40		1cnd 11214	. . . . . . . . 9 ⊢ (𝜑 → 1 ∈ ℂ)
41	38, 40	addcld 11238	. . . . . . . 8 ⊢ (𝜑 → (𝑅 + 1) ∈ ℂ)
42	39, 41	mulcld 11239	. . . . . . 7 ⊢ (𝜑 → (2 · (𝑅 + 1)) ∈ ℂ)
43	38, 42	mulcld 11239	. . . . . 6 ⊢ (𝜑 → (𝑅 · (2 · (𝑅 + 1))) ∈ ℂ)
44		1lt2 12388	. . . . . . . . . 10 ⊢ 1 < 2
45		2re 12291	. . . . . . . . . . 11 ⊢ 2 ∈ ℝ
46		rere 15074	. . . . . . . . . . 11 ⊢ (2 ∈ ℝ → (ℜ‘2) = 2)
47	45, 46	ax-mp 5	. . . . . . . . . 10 ⊢ (ℜ‘2) = 2
48	44, 47	breqtrri 5175	. . . . . . . . 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 2731	. . . . . . . . . 10 ⊢ (𝑚 ∈ ℕ ↦ (𝑚↑_𝑐-2)) = (𝑚 ∈ ℕ ↦ (𝑚↑_𝑐-2))
52		ovex 7445	. . . . . . . . . 10 ⊢ (𝑛↑_𝑐-2) ∈ V
53	50, 51, 52	fvmpt 6998	. . . . . . . . 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 26756	. . . . . . 7 ⊢ (𝜑 → seq1( + , (𝑚 ∈ ℕ ↦ (𝑚↑_𝑐-2))) ∈ dom ⇝ )
56		climdm 15503	. . . . . . 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 484	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ ℕ)
59	58	nncnd 12233	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ ℂ)
60		2cnd 12295	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 2 ∈ ℂ)
61	60	negcld 11563	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → -2 ∈ ℂ)
62	59, 61	cxpcld 26453	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑛↑_𝑐-2) ∈ ℂ)
63	54, 62	eqeltrd 2832	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((𝑚 ∈ ℕ ↦ (𝑚↑_𝑐-2))‘𝑛) ∈ ℂ)
64	38	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝑅 ∈ ℂ)
65		1cnd 11214	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 1 ∈ ℂ)
66	64, 65	addcld 11238	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑅 + 1) ∈ ℂ)
67	60, 66	mulcld 11239	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (2 · (𝑅 + 1)) ∈ ℂ)
68	64, 67	mulcld 11239	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑅 · (2 · (𝑅 + 1))) ∈ ℂ)
69	59	sqcld 14114	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑛↑2) ∈ ℂ)
70	58	nnne0d 12267	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝑛 ≠ 0)
71		2z 12599	. . . . . . . . . . 11 ⊢ 2 ∈ ℤ
72	71	a1i 11	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 2 ∈ ℤ)
73	59, 70, 72	expne0d 14122	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑛↑2) ≠ 0)
74	68, 69, 73	divrecd 11998	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((𝑅 · (2 · (𝑅 + 1))) / (𝑛↑2)) = ((𝑅 · (2 · (𝑅 + 1))) · (1 / (𝑛↑2))))
75	64, 67, 69, 73	divassd 12030	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((𝑅 · (2 · (𝑅 + 1))) / (𝑛↑2)) = (𝑅 · ((2 · (𝑅 + 1)) / (𝑛↑2))))
76	59, 70, 60	cxpnegd 26460	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑛↑_𝑐-2) = (1 / (𝑛↑_𝑐2)))
77	59, 70, 72	cxpexpzd 26456	. . . . . . . . . . 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 2772	. . . . . . . . 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 2779	. . . . . . 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 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 2781	. . . . . 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 15609	. . . . 5 ⊢ (𝜑 → seq1( + , (𝑚 ∈ ℕ ↦ (𝑅 · ((2 · (𝑅 + 1)) / (𝑚↑2))))) ⇝ ((𝑅 · (2 · (𝑅 + 1))) · ( ⇝ ‘seq1( + , (𝑚 ∈ ℕ ↦ (𝑚↑_𝑐-2))))))
86		climrel 15441	. . . . . 6 ⊢ Rel ⇝
87	86	releldmi 5947	. . . . 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 11995	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((2 · (𝑅 + 1)) / (𝑛↑2)) ∈ ℂ)
90	64, 89	mulcld 11239	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑅 · ((2 · (𝑅 + 1)) / (𝑛↑2))) ∈ ℂ)
91	82, 90	eqeltrd 2832	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((𝑚 ∈ ℕ ↦ (𝑅 · ((2 · (𝑅 + 1)) / (𝑚↑2))))‘𝑛) ∈ ℂ)
92	36, 4, 91	iserex 15608	. . . 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 2832	. 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 12232	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝑅 ∈ ℝ)
98	45	a1i 11	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 2 ∈ ℝ)
99		1red 11220	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 1 ∈ ℝ)
100	97, 99	readdcld 11248	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑅 + 1) ∈ ℝ)
101	98, 100	remulcld 11249	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (2 · (𝑅 + 1)) ∈ ℝ)
102	58	nnsqcld 14212	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑛↑2) ∈ ℕ)
103	101, 102	nndivred 12271	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((2 · (𝑅 + 1)) / (𝑛↑2)) ∈ ℝ)
104	97, 103	remulcld 11249	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑅 · ((2 · (𝑅 + 1)) / (𝑛↑2))) ∈ ℝ)
105	58	peano2nnd 12234	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑛 + 1) ∈ ℕ)
106	105	nnrpd 13019	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑛 + 1) ∈ ℝ⁺)
107	58	nnrpd 13019	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ ℝ⁺)
108	106, 107	rpdivcld 13038	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((𝑛 + 1) / 𝑛) ∈ ℝ⁺)
109	108	relogcld 26368	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (log‘((𝑛 + 1) / 𝑛)) ∈ ℝ)
110	97, 109	remulcld 11249	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑅 · (log‘((𝑛 + 1) / 𝑛))) ∈ ℝ)
111	96	peano2nnd 12234	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑅 + 1) ∈ ℕ)
112	111	nnrpd 13019	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑅 + 1) ∈ ℝ⁺)
113	112, 107	rpmulcld 13037	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((𝑅 + 1) · 𝑛) ∈ ℝ⁺)
114	113	relogcld 26368	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (log‘((𝑅 + 1) · 𝑛)) ∈ ℝ)
115		pire 26205	. . . . . . . . 9 ⊢ π ∈ ℝ
116	115	a1i 11	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → π ∈ ℝ)
117	114, 116	readdcld 11248	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((log‘((𝑅 + 1) · 𝑛)) + π) ∈ ℝ)
118	110, 117	readdcld 11248	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((𝑅 · (log‘((𝑛 + 1) / 𝑛))) + ((log‘((𝑅 + 1) · 𝑛)) + π)) ∈ ℝ)
119	104, 118	ifcld 4574	. . . . 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 2832	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑇‘𝑛) ∈ ℝ)
121	120	recnd 11247	. . 3 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑇‘𝑛) ∈ ℂ)
122	36, 4, 121	iserex 15608	. 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 2105 ∀wral 3060 {crab 3431 ifcif 4528 class class class wbr 5148 ↦ cmpt 5231 dom cdm 5676 ‘cfv 6543 (class class class)co 7412 ℂcc 11112 ℝcr 11113 1c1 11115 + caddc 11117 · cmul 11119 < clt 11253 ≤ cle 11254 − cmin 11449 -cneg 11450 / cdiv 11876 ℕcn 12217 2c2 12272 ℕ₀cn0 12477 ℤcz 12563 ℤ_≥cuz 12827 seqcseq 13971 ↑cexp 14032 ℜcre 15049 abscabs 15186 ⇝ cli 15433 πcpi 16015 logclog 26300 ↑_𝑐ccxp 26301

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7729 ax-inf2 9640 ax-cnex 11170 ax-resscn 11171 ax-1cn 11172 ax-icn 11173 ax-addcl 11174 ax-addrcl 11175 ax-mulcl 11176 ax-mulrcl 11177 ax-mulcom 11178 ax-addass 11179 ax-mulass 11180 ax-distr 11181 ax-i2m1 11182 ax-1ne0 11183 ax-1rid 11184 ax-rnegex 11185 ax-rrecex 11186 ax-cnre 11187 ax-pre-lttri 11188 ax-pre-lttrn 11189 ax-pre-ltadd 11190 ax-pre-mulgt0 11191 ax-pre-sup 11192 ax-addf 11193

This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3375 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-tp 4633 df-op 4635 df-uni 4909 df-int 4951 df-iun 4999 df-iin 5000 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-se 5632 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 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 7368 df-ov 7415 df-oprab 7416 df-mpo 7417 df-of 7674 df-om 7860 df-1st 7979 df-2nd 7980 df-supp 8151 df-frecs 8270 df-wrecs 8301 df-recs 8375 df-rdg 8414 df-1o 8470 df-2o 8471 df-oadd 8474 df-er 8707 df-map 8826 df-pm 8827 df-ixp 8896 df-en 8944 df-dom 8945 df-sdom 8946 df-fin 8947 df-fsupp 9366 df-fi 9410 df-sup 9441 df-inf 9442 df-oi 9509 df-dju 9900 df-card 9938 df-pnf 11255 df-mnf 11256 df-xr 11257 df-ltxr 11258 df-le 11259 df-sub 11451 df-neg 11452 df-div 11877 df-nn 12218 df-2 12280 df-3 12281 df-4 12282 df-5 12283 df-6 12284 df-7 12285 df-8 12286 df-9 12287 df-n0 12478 df-z 12564 df-dec 12683 df-uz 12828 df-q 12938 df-rp 12980 df-xneg 13097 df-xadd 13098 df-xmul 13099 df-ioo 13333 df-ioc 13334 df-ico 13335 df-icc 13336 df-fz 13490 df-fzo 13633 df-fl 13762 df-mod 13840 df-seq 13972 df-exp 14033 df-fac 14239 df-bc 14268 df-hash 14296 df-shft 15019 df-cj 15051 df-re 15052 df-im 15053 df-sqrt 15187 df-abs 15188 df-limsup 15420 df-clim 15437 df-rlim 15438 df-sum 15638 df-ef 16016 df-sin 16018 df-cos 16019 df-pi 16021 df-struct 17085 df-sets 17102 df-slot 17120 df-ndx 17132 df-base 17150 df-ress 17179 df-plusg 17215 df-mulr 17216 df-starv 17217 df-sca 17218 df-vsca 17219 df-ip 17220 df-tset 17221 df-ple 17222 df-ds 17224 df-unif 17225 df-hom 17226 df-cco 17227 df-rest 17373 df-topn 17374 df-0g 17392 df-gsum 17393 df-topgen 17394 df-pt 17395 df-prds 17398 df-xrs 17453 df-qtop 17458 df-imas 17459 df-xps 17461 df-mre 17535 df-mrc 17536 df-acs 17538 df-mgm 18566 df-sgrp 18645 df-mnd 18661 df-submnd 18707 df-mulg 18988 df-cntz 19223 df-cmn 19692 df-psmet 21137 df-xmet 21138 df-met 21139 df-bl 21140 df-mopn 21141 df-fbas 21142 df-fg 21143 df-cnfld 21146 df-top 22617 df-topon 22634 df-topsp 22656 df-bases 22670 df-cld 22744 df-ntr 22745 df-cls 22746 df-nei 22823 df-lp 22861 df-perf 22862 df-cn 22952 df-cnp 22953 df-haus 23040 df-tx 23287 df-hmeo 23480 df-fil 23571 df-fm 23663 df-flim 23664 df-flf 23665 df-xms 24047 df-ms 24048 df-tms 24049 df-cncf 24619 df-limc 25616 df-dv 25617 df-log 26302 df-cxp 26303

This theorem is referenced by: lgamgulmlem6 26775

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