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Theorem lgamgulm2 27013

Description: Rewrite the limit of the sequence 𝐺 in terms of the log-Gamma function. (Contributed by Mario Carneiro, 6-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)))))

**Assertion**
Ref	Expression
lgamgulm2	⊢ (𝜑 → (∀𝑧 ∈ 𝑈 (log Γ‘𝑧) ∈ ℂ ∧ seq1( ∘_f + , 𝐺)(⇝_𝑢‘𝑈)(𝑧 ∈ 𝑈 ↦ ((log Γ‘𝑧) + (log‘𝑧)))))

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

Proof of Theorem lgamgulm2 Dummy variable 𝑛 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		lgamgulm.r	. . . . . . 7 ⊢ (𝜑 → 𝑅 ∈ ℕ)
2		lgamgulm.u	. . . . . . 7 ⊢ 𝑈 = {𝑥 ∈ ℂ ∣ ((abs‘𝑥) ≤ 𝑅 ∧ ∀𝑘 ∈ ℕ₀ (1 / 𝑅) ≤ (abs‘(𝑥 + 𝑘)))}
3	1, 2	lgamgulmlem1 27006	. . . . . 6 ⊢ (𝜑 → 𝑈 ⊆ (ℂ ∖ (ℤ ∖ ℕ)))
4	3	sselda 3922	. . . . 5 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑈) → 𝑧 ∈ (ℂ ∖ (ℤ ∖ ℕ)))
5		ovex 7393	. . . . 5 ⊢ (Σ𝑛 ∈ ℕ ((𝑧 · (log‘((𝑛 + 1) / 𝑛))) − (log‘((𝑧 / 𝑛) + 1))) − (log‘𝑧)) ∈ V
6		df-lgam 26996	. . . . . 6 ⊢ log Γ = (𝑧 ∈ (ℂ ∖ (ℤ ∖ ℕ)) ↦ (Σ𝑛 ∈ ℕ ((𝑧 · (log‘((𝑛 + 1) / 𝑛))) − (log‘((𝑧 / 𝑛) + 1))) − (log‘𝑧)))
7	6	fvmpt2 6953	. . . . 5 ⊢ ((𝑧 ∈ (ℂ ∖ (ℤ ∖ ℕ)) ∧ (Σ𝑛 ∈ ℕ ((𝑧 · (log‘((𝑛 + 1) / 𝑛))) − (log‘((𝑧 / 𝑛) + 1))) − (log‘𝑧)) ∈ V) → (log Γ‘𝑧) = (Σ𝑛 ∈ ℕ ((𝑧 · (log‘((𝑛 + 1) / 𝑛))) − (log‘((𝑧 / 𝑛) + 1))) − (log‘𝑧)))
8	4, 5, 7	sylancl 587	. . . 4 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑈) → (log Γ‘𝑧) = (Σ𝑛 ∈ ℕ ((𝑧 · (log‘((𝑛 + 1) / 𝑛))) − (log‘((𝑧 / 𝑛) + 1))) − (log‘𝑧)))
9		nnuz 12818	. . . . . . 7 ⊢ ℕ = (ℤ_≥‘1)
10		1zzd 12549	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑈) → 1 ∈ ℤ)
11		oveq1 7367	. . . . . . . . . . . . 13 ⊢ (𝑚 = 𝑛 → (𝑚 + 1) = (𝑛 + 1))
12		id 22	. . . . . . . . . . . . 13 ⊢ (𝑚 = 𝑛 → 𝑚 = 𝑛)
13	11, 12	oveq12d 7378	. . . . . . . . . . . 12 ⊢ (𝑚 = 𝑛 → ((𝑚 + 1) / 𝑚) = ((𝑛 + 1) / 𝑛))
14	13	fveq2d 6838	. . . . . . . . . . 11 ⊢ (𝑚 = 𝑛 → (log‘((𝑚 + 1) / 𝑚)) = (log‘((𝑛 + 1) / 𝑛)))
15	14	oveq2d 7376	. . . . . . . . . 10 ⊢ (𝑚 = 𝑛 → (𝑧 · (log‘((𝑚 + 1) / 𝑚))) = (𝑧 · (log‘((𝑛 + 1) / 𝑛))))
16		oveq2 7368	. . . . . . . . . . 11 ⊢ (𝑚 = 𝑛 → (𝑧 / 𝑚) = (𝑧 / 𝑛))
17	16	fvoveq1d 7382	. . . . . . . . . 10 ⊢ (𝑚 = 𝑛 → (log‘((𝑧 / 𝑚) + 1)) = (log‘((𝑧 / 𝑛) + 1)))
18	15, 17	oveq12d 7378	. . . . . . . . 9 ⊢ (𝑚 = 𝑛 → ((𝑧 · (log‘((𝑚 + 1) / 𝑚))) − (log‘((𝑧 / 𝑚) + 1))) = ((𝑧 · (log‘((𝑛 + 1) / 𝑛))) − (log‘((𝑧 / 𝑛) + 1))))
19		eqid 2737	. . . . . . . . 9 ⊢ (𝑚 ∈ ℕ ↦ ((𝑧 · (log‘((𝑚 + 1) / 𝑚))) − (log‘((𝑧 / 𝑚) + 1)))) = (𝑚 ∈ ℕ ↦ ((𝑧 · (log‘((𝑚 + 1) / 𝑚))) − (log‘((𝑧 / 𝑚) + 1))))
20		ovex 7393	. . . . . . . . 9 ⊢ ((𝑧 · (log‘((𝑛 + 1) / 𝑛))) − (log‘((𝑧 / 𝑛) + 1))) ∈ V
21	18, 19, 20	fvmpt 6941	. . . . . . . 8 ⊢ (𝑛 ∈ ℕ → ((𝑚 ∈ ℕ ↦ ((𝑧 · (log‘((𝑚 + 1) / 𝑚))) − (log‘((𝑧 / 𝑚) + 1))))‘𝑛) = ((𝑧 · (log‘((𝑛 + 1) / 𝑛))) − (log‘((𝑧 / 𝑛) + 1))))
22	21	adantl 481	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑈) ∧ 𝑛 ∈ ℕ) → ((𝑚 ∈ ℕ ↦ ((𝑧 · (log‘((𝑚 + 1) / 𝑚))) − (log‘((𝑧 / 𝑚) + 1))))‘𝑛) = ((𝑧 · (log‘((𝑛 + 1) / 𝑛))) − (log‘((𝑧 / 𝑛) + 1))))
23	4	eldifad 3902	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑈) → 𝑧 ∈ ℂ)
24	23	adantr 480	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑈) ∧ 𝑛 ∈ ℕ) → 𝑧 ∈ ℂ)
25		simpr 484	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑈) ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ ℕ)
26	25	peano2nnd 12182	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑈) ∧ 𝑛 ∈ ℕ) → (𝑛 + 1) ∈ ℕ)
27	26	nnrpd 12975	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑈) ∧ 𝑛 ∈ ℕ) → (𝑛 + 1) ∈ ℝ⁺)
28	25	nnrpd 12975	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑈) ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ ℝ⁺)
29	27, 28	rpdivcld 12994	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑈) ∧ 𝑛 ∈ ℕ) → ((𝑛 + 1) / 𝑛) ∈ ℝ⁺)
30	29	relogcld 26600	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑈) ∧ 𝑛 ∈ ℕ) → (log‘((𝑛 + 1) / 𝑛)) ∈ ℝ)
31	30	recnd 11164	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑈) ∧ 𝑛 ∈ ℕ) → (log‘((𝑛 + 1) / 𝑛)) ∈ ℂ)
32	24, 31	mulcld 11156	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑈) ∧ 𝑛 ∈ ℕ) → (𝑧 · (log‘((𝑛 + 1) / 𝑛))) ∈ ℂ)
33	25	nncnd 12181	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑈) ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ ℂ)
34	25	nnne0d 12218	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑈) ∧ 𝑛 ∈ ℕ) → 𝑛 ≠ 0)
35	24, 33, 34	divcld 11922	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑈) ∧ 𝑛 ∈ ℕ) → (𝑧 / 𝑛) ∈ ℂ)
36		1cnd 11130	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑈) ∧ 𝑛 ∈ ℕ) → 1 ∈ ℂ)
37	35, 36	addcld 11155	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑈) ∧ 𝑛 ∈ ℕ) → ((𝑧 / 𝑛) + 1) ∈ ℂ)
38	4	adantr 480	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑈) ∧ 𝑛 ∈ ℕ) → 𝑧 ∈ (ℂ ∖ (ℤ ∖ ℕ)))
39	38, 25	dmgmdivn0 27005	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑈) ∧ 𝑛 ∈ ℕ) → ((𝑧 / 𝑛) + 1) ≠ 0)
40	37, 39	logcld 26547	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑈) ∧ 𝑛 ∈ ℕ) → (log‘((𝑧 / 𝑛) + 1)) ∈ ℂ)
41	32, 40	subcld 11496	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑈) ∧ 𝑛 ∈ ℕ) → ((𝑧 · (log‘((𝑛 + 1) / 𝑛))) − (log‘((𝑧 / 𝑛) + 1))) ∈ ℂ)
42		1z 12548	. . . . . . . . . . . 12 ⊢ 1 ∈ ℤ
43		seqfn 13966	. . . . . . . . . . . 12 ⊢ (1 ∈ ℤ → seq1( ∘_f + , 𝐺) Fn (ℤ_≥‘1))
44	42, 43	ax-mp 5	. . . . . . . . . . 11 ⊢ seq1( ∘_f + , 𝐺) Fn (ℤ_≥‘1)
45	9	fneq2i 6590	. . . . . . . . . . 11 ⊢ (seq1( ∘_f + , 𝐺) Fn ℕ ↔ seq1( ∘_f + , 𝐺) Fn (ℤ_≥‘1))
46	44, 45	mpbir 231	. . . . . . . . . 10 ⊢ seq1( ∘_f + , 𝐺) Fn ℕ
47		lgamgulm.g	. . . . . . . . . . . 12 ⊢ 𝐺 = (𝑚 ∈ ℕ ↦ (𝑧 ∈ 𝑈 ↦ ((𝑧 · (log‘((𝑚 + 1) / 𝑚))) − (log‘((𝑧 / 𝑚) + 1)))))
48	1, 2, 47	lgamgulm 27012	. . . . . . . . . . 11 ⊢ (𝜑 → seq1( ∘_f + , 𝐺) ∈ dom (⇝_𝑢‘𝑈))
49		ulmdm 26371	. . . . . . . . . . 11 ⊢ (seq1( ∘_f + , 𝐺) ∈ dom (⇝_𝑢‘𝑈) ↔ seq1( ∘_f + , 𝐺)(⇝_𝑢‘𝑈)((⇝_𝑢‘𝑈)‘seq1( ∘_f + , 𝐺)))
50	48, 49	sylib 218	. . . . . . . . . 10 ⊢ (𝜑 → seq1( ∘_f + , 𝐺)(⇝_𝑢‘𝑈)((⇝_𝑢‘𝑈)‘seq1( ∘_f + , 𝐺)))
51		ulmf2 26362	. . . . . . . . . 10 ⊢ ((seq1( ∘_f + , 𝐺) Fn ℕ ∧ seq1( ∘_f + , 𝐺)(⇝_𝑢‘𝑈)((⇝_𝑢‘𝑈)‘seq1( ∘_f + , 𝐺))) → seq1( ∘_f + , 𝐺):ℕ⟶(ℂ ↑_m 𝑈))
52	46, 50, 51	sylancr 588	. . . . . . . . 9 ⊢ (𝜑 → seq1( ∘_f + , 𝐺):ℕ⟶(ℂ ↑_m 𝑈))
53	52	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑈) → seq1( ∘_f + , 𝐺):ℕ⟶(ℂ ↑_m 𝑈))
54		simpr 484	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑈) → 𝑧 ∈ 𝑈)
55		seqex 13956	. . . . . . . . 9 ⊢ seq1( + , (𝑚 ∈ ℕ ↦ ((𝑧 · (log‘((𝑚 + 1) / 𝑚))) − (log‘((𝑧 / 𝑚) + 1))))) ∈ V
56	55	a1i 11	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑈) → seq1( + , (𝑚 ∈ ℕ ↦ ((𝑧 · (log‘((𝑚 + 1) / 𝑚))) − (log‘((𝑧 / 𝑚) + 1))))) ∈ V)
57	47	a1i 11	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑈) ∧ 𝑛 ∈ ℕ) → 𝐺 = (𝑚 ∈ ℕ ↦ (𝑧 ∈ 𝑈 ↦ ((𝑧 · (log‘((𝑚 + 1) / 𝑚))) − (log‘((𝑧 / 𝑚) + 1))))))
58	57	seqeq3d 13962	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑈) ∧ 𝑛 ∈ ℕ) → seq1( ∘_f + , 𝐺) = seq1( ∘_f + , (𝑚 ∈ ℕ ↦ (𝑧 ∈ 𝑈 ↦ ((𝑧 · (log‘((𝑚 + 1) / 𝑚))) − (log‘((𝑧 / 𝑚) + 1)))))))
59	58	fveq1d 6836	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑈) ∧ 𝑛 ∈ ℕ) → (seq1( ∘_f + , 𝐺)‘𝑛) = (seq1( ∘_f + , (𝑚 ∈ ℕ ↦ (𝑧 ∈ 𝑈 ↦ ((𝑧 · (log‘((𝑚 + 1) / 𝑚))) − (log‘((𝑧 / 𝑚) + 1))))))‘𝑛))
60		cnex 11110	. . . . . . . . . . . . . . 15 ⊢ ℂ ∈ V
61	2, 60	rabex2 5278	. . . . . . . . . . . . . 14 ⊢ 𝑈 ∈ V
62	61	a1i 11	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝑈 ∈ V)
63		simpr 484	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ ℕ)
64	63, 9	eleqtrdi 2847	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ (ℤ_≥‘1))
65		fz1ssnn 13500	. . . . . . . . . . . . . 14 ⊢ (1...𝑛) ⊆ ℕ
66	65	a1i 11	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (1...𝑛) ⊆ ℕ)
67		ovexd 7395	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ (𝑚 ∈ ℕ ∧ 𝑧 ∈ 𝑈)) → ((𝑧 · (log‘((𝑚 + 1) / 𝑚))) − (log‘((𝑧 / 𝑚) + 1))) ∈ V)
68	62, 64, 66, 67	seqof2 14013	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (seq1( ∘_f + , (𝑚 ∈ ℕ ↦ (𝑧 ∈ 𝑈 ↦ ((𝑧 · (log‘((𝑚 + 1) / 𝑚))) − (log‘((𝑧 / 𝑚) + 1))))))‘𝑛) = (𝑧 ∈ 𝑈 ↦ (seq1( + , (𝑚 ∈ ℕ ↦ ((𝑧 · (log‘((𝑚 + 1) / 𝑚))) − (log‘((𝑧 / 𝑚) + 1)))))‘𝑛)))
69	68	adantlr 716	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑈) ∧ 𝑛 ∈ ℕ) → (seq1( ∘_f + , (𝑚 ∈ ℕ ↦ (𝑧 ∈ 𝑈 ↦ ((𝑧 · (log‘((𝑚 + 1) / 𝑚))) − (log‘((𝑧 / 𝑚) + 1))))))‘𝑛) = (𝑧 ∈ 𝑈 ↦ (seq1( + , (𝑚 ∈ ℕ ↦ ((𝑧 · (log‘((𝑚 + 1) / 𝑚))) − (log‘((𝑧 / 𝑚) + 1)))))‘𝑛)))
70	59, 69	eqtrd 2772	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑈) ∧ 𝑛 ∈ ℕ) → (seq1( ∘_f + , 𝐺)‘𝑛) = (𝑧 ∈ 𝑈 ↦ (seq1( + , (𝑚 ∈ ℕ ↦ ((𝑧 · (log‘((𝑚 + 1) / 𝑚))) − (log‘((𝑧 / 𝑚) + 1)))))‘𝑛)))
71	70	fveq1d 6836	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑈) ∧ 𝑛 ∈ ℕ) → ((seq1( ∘_f + , 𝐺)‘𝑛)‘𝑧) = ((𝑧 ∈ 𝑈 ↦ (seq1( + , (𝑚 ∈ ℕ ↦ ((𝑧 · (log‘((𝑚 + 1) / 𝑚))) − (log‘((𝑧 / 𝑚) + 1)))))‘𝑛))‘𝑧))
72	54	adantr 480	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑈) ∧ 𝑛 ∈ ℕ) → 𝑧 ∈ 𝑈)
73		fvex 6847	. . . . . . . . . 10 ⊢ (seq1( + , (𝑚 ∈ ℕ ↦ ((𝑧 · (log‘((𝑚 + 1) / 𝑚))) − (log‘((𝑧 / 𝑚) + 1)))))‘𝑛) ∈ V
74		eqid 2737	. . . . . . . . . . 11 ⊢ (𝑧 ∈ 𝑈 ↦ (seq1( + , (𝑚 ∈ ℕ ↦ ((𝑧 · (log‘((𝑚 + 1) / 𝑚))) − (log‘((𝑧 / 𝑚) + 1)))))‘𝑛)) = (𝑧 ∈ 𝑈 ↦ (seq1( + , (𝑚 ∈ ℕ ↦ ((𝑧 · (log‘((𝑚 + 1) / 𝑚))) − (log‘((𝑧 / 𝑚) + 1)))))‘𝑛))
75	74	fvmpt2 6953	. . . . . . . . . 10 ⊢ ((𝑧 ∈ 𝑈 ∧ (seq1( + , (𝑚 ∈ ℕ ↦ ((𝑧 · (log‘((𝑚 + 1) / 𝑚))) − (log‘((𝑧 / 𝑚) + 1)))))‘𝑛) ∈ V) → ((𝑧 ∈ 𝑈 ↦ (seq1( + , (𝑚 ∈ ℕ ↦ ((𝑧 · (log‘((𝑚 + 1) / 𝑚))) − (log‘((𝑧 / 𝑚) + 1)))))‘𝑛))‘𝑧) = (seq1( + , (𝑚 ∈ ℕ ↦ ((𝑧 · (log‘((𝑚 + 1) / 𝑚))) − (log‘((𝑧 / 𝑚) + 1)))))‘𝑛))
76	72, 73, 75	sylancl 587	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑈) ∧ 𝑛 ∈ ℕ) → ((𝑧 ∈ 𝑈 ↦ (seq1( + , (𝑚 ∈ ℕ ↦ ((𝑧 · (log‘((𝑚 + 1) / 𝑚))) − (log‘((𝑧 / 𝑚) + 1)))))‘𝑛))‘𝑧) = (seq1( + , (𝑚 ∈ ℕ ↦ ((𝑧 · (log‘((𝑚 + 1) / 𝑚))) − (log‘((𝑧 / 𝑚) + 1)))))‘𝑛))
77	71, 76	eqtrd 2772	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑈) ∧ 𝑛 ∈ ℕ) → ((seq1( ∘_f + , 𝐺)‘𝑛)‘𝑧) = (seq1( + , (𝑚 ∈ ℕ ↦ ((𝑧 · (log‘((𝑚 + 1) / 𝑚))) − (log‘((𝑧 / 𝑚) + 1)))))‘𝑛))
78	50	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑈) → seq1( ∘_f + , 𝐺)(⇝_𝑢‘𝑈)((⇝_𝑢‘𝑈)‘seq1( ∘_f + , 𝐺)))
79	9, 10, 53, 54, 56, 77, 78	ulmclm 26365	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑈) → seq1( + , (𝑚 ∈ ℕ ↦ ((𝑧 · (log‘((𝑚 + 1) / 𝑚))) − (log‘((𝑧 / 𝑚) + 1))))) ⇝ (((⇝_𝑢‘𝑈)‘seq1( ∘_f + , 𝐺))‘𝑧))
80	9, 10, 22, 41, 79	isumclim 15710	. . . . . 6 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑈) → Σ𝑛 ∈ ℕ ((𝑧 · (log‘((𝑛 + 1) / 𝑛))) − (log‘((𝑧 / 𝑛) + 1))) = (((⇝_𝑢‘𝑈)‘seq1( ∘_f + , 𝐺))‘𝑧))
81		ulmcl 26359	. . . . . . . 8 ⊢ (seq1( ∘_f + , 𝐺)(⇝_𝑢‘𝑈)((⇝_𝑢‘𝑈)‘seq1( ∘_f + , 𝐺)) → ((⇝_𝑢‘𝑈)‘seq1( ∘_f + , 𝐺)):𝑈⟶ℂ)
82	50, 81	syl 17	. . . . . . 7 ⊢ (𝜑 → ((⇝_𝑢‘𝑈)‘seq1( ∘_f + , 𝐺)):𝑈⟶ℂ)
83	82	ffvelcdmda 7030	. . . . . 6 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑈) → (((⇝_𝑢‘𝑈)‘seq1( ∘_f + , 𝐺))‘𝑧) ∈ ℂ)
84	80, 83	eqeltrd 2837	. . . . 5 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑈) → Σ𝑛 ∈ ℕ ((𝑧 · (log‘((𝑛 + 1) / 𝑛))) − (log‘((𝑧 / 𝑛) + 1))) ∈ ℂ)
85	4	dmgmn0 27003	. . . . . 6 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑈) → 𝑧 ≠ 0)
86	23, 85	logcld 26547	. . . . 5 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑈) → (log‘𝑧) ∈ ℂ)
87	84, 86	subcld 11496	. . . 4 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑈) → (Σ𝑛 ∈ ℕ ((𝑧 · (log‘((𝑛 + 1) / 𝑛))) − (log‘((𝑧 / 𝑛) + 1))) − (log‘𝑧)) ∈ ℂ)
88	8, 87	eqeltrd 2837	. . 3 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑈) → (log Γ‘𝑧) ∈ ℂ)
89	88	ralrimiva 3130	. 2 ⊢ (𝜑 → ∀𝑧 ∈ 𝑈 (log Γ‘𝑧) ∈ ℂ)
90		ffn 6662	. . . . . 6 ⊢ (((⇝_𝑢‘𝑈)‘seq1( ∘_f + , 𝐺)):𝑈⟶ℂ → ((⇝_𝑢‘𝑈)‘seq1( ∘_f + , 𝐺)) Fn 𝑈)
91	50, 81, 90	3syl 18	. . . . 5 ⊢ (𝜑 → ((⇝_𝑢‘𝑈)‘seq1( ∘_f + , 𝐺)) Fn 𝑈)
92		nfcv 2899	. . . . . . 7 ⊢ Ⅎ𝑧(⇝_𝑢‘𝑈)
93		nfcv 2899	. . . . . . . 8 ⊢ Ⅎ𝑧1
94		nfcv 2899	. . . . . . . 8 ⊢ Ⅎ𝑧 ∘_f +
95		nfcv 2899	. . . . . . . . . 10 ⊢ Ⅎ𝑧ℕ
96		nfmpt1 5185	. . . . . . . . . 10 ⊢ Ⅎ𝑧(𝑧 ∈ 𝑈 ↦ ((𝑧 · (log‘((𝑚 + 1) / 𝑚))) − (log‘((𝑧 / 𝑚) + 1))))
97	95, 96	nfmpt 5184	. . . . . . . . 9 ⊢ Ⅎ𝑧(𝑚 ∈ ℕ ↦ (𝑧 ∈ 𝑈 ↦ ((𝑧 · (log‘((𝑚 + 1) / 𝑚))) − (log‘((𝑧 / 𝑚) + 1)))))
98	47, 97	nfcxfr 2897	. . . . . . . 8 ⊢ Ⅎ𝑧𝐺
99	93, 94, 98	nfseq 13964	. . . . . . 7 ⊢ Ⅎ𝑧seq1( ∘_f + , 𝐺)
100	92, 99	nffv 6844	. . . . . 6 ⊢ Ⅎ𝑧((⇝_𝑢‘𝑈)‘seq1( ∘_f + , 𝐺))
101	100	dffn5f 6905	. . . . 5 ⊢ (((⇝_𝑢‘𝑈)‘seq1( ∘_f + , 𝐺)) Fn 𝑈 ↔ ((⇝_𝑢‘𝑈)‘seq1( ∘_f + , 𝐺)) = (𝑧 ∈ 𝑈 ↦ (((⇝_𝑢‘𝑈)‘seq1( ∘_f + , 𝐺))‘𝑧)))
102	91, 101	sylib 218	. . . 4 ⊢ (𝜑 → ((⇝_𝑢‘𝑈)‘seq1( ∘_f + , 𝐺)) = (𝑧 ∈ 𝑈 ↦ (((⇝_𝑢‘𝑈)‘seq1( ∘_f + , 𝐺))‘𝑧)))
103	8	oveq1d 7375	. . . . . 6 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑈) → ((log Γ‘𝑧) + (log‘𝑧)) = ((Σ𝑛 ∈ ℕ ((𝑧 · (log‘((𝑛 + 1) / 𝑛))) − (log‘((𝑧 / 𝑛) + 1))) − (log‘𝑧)) + (log‘𝑧)))
104	84, 86	npcand 11500	. . . . . 6 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑈) → ((Σ𝑛 ∈ ℕ ((𝑧 · (log‘((𝑛 + 1) / 𝑛))) − (log‘((𝑧 / 𝑛) + 1))) − (log‘𝑧)) + (log‘𝑧)) = Σ𝑛 ∈ ℕ ((𝑧 · (log‘((𝑛 + 1) / 𝑛))) − (log‘((𝑧 / 𝑛) + 1))))
105	103, 104, 80	3eqtrrd 2777	. . . . 5 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑈) → (((⇝_𝑢‘𝑈)‘seq1( ∘_f + , 𝐺))‘𝑧) = ((log Γ‘𝑧) + (log‘𝑧)))
106	105	mpteq2dva 5179	. . . 4 ⊢ (𝜑 → (𝑧 ∈ 𝑈 ↦ (((⇝_𝑢‘𝑈)‘seq1( ∘_f + , 𝐺))‘𝑧)) = (𝑧 ∈ 𝑈 ↦ ((log Γ‘𝑧) + (log‘𝑧))))
107	102, 106	eqtrd 2772	. . 3 ⊢ (𝜑 → ((⇝_𝑢‘𝑈)‘seq1( ∘_f + , 𝐺)) = (𝑧 ∈ 𝑈 ↦ ((log Γ‘𝑧) + (log‘𝑧))))
108	50, 107	breqtrd 5112	. 2 ⊢ (𝜑 → seq1( ∘_f + , 𝐺)(⇝_𝑢‘𝑈)(𝑧 ∈ 𝑈 ↦ ((log Γ‘𝑧) + (log‘𝑧))))
109	89, 108	jca 511	1 ⊢ (𝜑 → (∀𝑧 ∈ 𝑈 (log Γ‘𝑧) ∈ ℂ ∧ seq1( ∘_f + , 𝐺)(⇝_𝑢‘𝑈)(𝑧 ∈ 𝑈 ↦ ((log Γ‘𝑧) + (log‘𝑧)))))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1542 ∈ wcel 2114 ∀wral 3052 {crab 3390 Vcvv 3430 ∖ cdif 3887 ⊆ wss 3890 class class class wbr 5086 ↦ cmpt 5167 dom cdm 5624 Fn wfn 6487 ⟶wf 6488 ‘cfv 6492 (class class class)co 7360 ∘_f cof 7622 ↑_m cmap 8766 ℂcc 11027 1c1 11030 + caddc 11032 · cmul 11034 ≤ cle 11171 − cmin 11368 / cdiv 11798 ℕcn 12165 ℕ₀cn0 12428 ℤcz 12515 ℤ_≥cuz 12779 ...cfz 13452 seqcseq 13954 abscabs 15187 Σcsu 15639 ⇝_𝑢culm 26354 logclog 26531 log Γclgam 26993

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-rep 5212 ax-sep 5231 ax-nul 5241 ax-pow 5302 ax-pr 5370 ax-un 7682 ax-inf2 9553 ax-cnex 11085 ax-resscn 11086 ax-1cn 11087 ax-icn 11088 ax-addcl 11089 ax-addrcl 11090 ax-mulcl 11091 ax-mulrcl 11092 ax-mulcom 11093 ax-addass 11094 ax-mulass 11095 ax-distr 11096 ax-i2m1 11097 ax-1ne0 11098 ax-1rid 11099 ax-rnegex 11100 ax-rrecex 11101 ax-cnre 11102 ax-pre-lttri 11103 ax-pre-lttrn 11104 ax-pre-ltadd 11105 ax-pre-mulgt0 11106 ax-pre-sup 11107 ax-addf 11108

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3063 df-rmo 3343 df-reu 3344 df-rab 3391 df-v 3432 df-sbc 3730 df-csb 3839 df-dif 3893 df-un 3895 df-in 3897 df-ss 3907 df-pss 3910 df-nul 4275 df-if 4468 df-pw 4544 df-sn 4569 df-pr 4571 df-tp 4573 df-op 4575 df-uni 4852 df-int 4891 df-iun 4936 df-iin 4937 df-br 5087 df-opab 5149 df-mpt 5168 df-tr 5194 df-id 5519 df-eprel 5524 df-po 5532 df-so 5533 df-fr 5577 df-se 5578 df-we 5579 df-xp 5630 df-rel 5631 df-cnv 5632 df-co 5633 df-dm 5634 df-rn 5635 df-res 5636 df-ima 5637 df-pred 6259 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6494 df-fn 6495 df-f 6496 df-f1 6497 df-fo 6498 df-f1o 6499 df-fv 6500 df-isom 6501 df-riota 7317 df-ov 7363 df-oprab 7364 df-mpo 7365 df-of 7624 df-om 7811 df-1st 7935 df-2nd 7936 df-supp 8104 df-frecs 8224 df-wrecs 8255 df-recs 8304 df-rdg 8342 df-1o 8398 df-2o 8399 df-oadd 8402 df-er 8636 df-map 8768 df-pm 8769 df-ixp 8839 df-en 8887 df-dom 8888 df-sdom 8889 df-fin 8890 df-fsupp 9268 df-fi 9317 df-sup 9348 df-inf 9349 df-oi 9418 df-dju 9816 df-card 9854 df-pnf 11172 df-mnf 11173 df-xr 11174 df-ltxr 11175 df-le 11176 df-sub 11370 df-neg 11371 df-div 11799 df-nn 12166 df-2 12235 df-3 12236 df-4 12237 df-5 12238 df-6 12239 df-7 12240 df-8 12241 df-9 12242 df-n0 12429 df-z 12516 df-dec 12636 df-uz 12780 df-q 12890 df-rp 12934 df-xneg 13054 df-xadd 13055 df-xmul 13056 df-ioo 13293 df-ioc 13294 df-ico 13295 df-icc 13296 df-fz 13453 df-fzo 13600 df-fl 13742 df-mod 13820 df-seq 13955 df-exp 14015 df-fac 14227 df-bc 14256 df-hash 14284 df-shft 15020 df-cj 15052 df-re 15053 df-im 15054 df-sqrt 15188 df-abs 15189 df-limsup 15424 df-clim 15441 df-rlim 15442 df-sum 15640 df-ef 16023 df-sin 16025 df-cos 16026 df-tan 16027 df-pi 16028 df-struct 17108 df-sets 17125 df-slot 17143 df-ndx 17155 df-base 17171 df-ress 17192 df-plusg 17224 df-mulr 17225 df-starv 17226 df-sca 17227 df-vsca 17228 df-ip 17229 df-tset 17230 df-ple 17231 df-ds 17233 df-unif 17234 df-hom 17235 df-cco 17236 df-rest 17376 df-topn 17377 df-0g 17395 df-gsum 17396 df-topgen 17397 df-pt 17398 df-prds 17401 df-xrs 17457 df-qtop 17462 df-imas 17463 df-xps 17465 df-mre 17539 df-mrc 17540 df-acs 17542 df-mgm 18599 df-sgrp 18678 df-mnd 18694 df-submnd 18743 df-mulg 19035 df-cntz 19283 df-cmn 19748 df-psmet 21336 df-xmet 21337 df-met 21338 df-bl 21339 df-mopn 21340 df-fbas 21341 df-fg 21342 df-cnfld 21345 df-top 22869 df-topon 22886 df-topsp 22908 df-bases 22921 df-cld 22994 df-ntr 22995 df-cls 22996 df-nei 23073 df-lp 23111 df-perf 23112 df-cn 23202 df-cnp 23203 df-haus 23290 df-cmp 23362 df-tx 23537 df-hmeo 23730 df-fil 23821 df-fm 23913 df-flim 23914 df-flf 23915 df-xms 24295 df-ms 24296 df-tms 24297 df-cncf 24855 df-limc 25843 df-dv 25844 df-ulm 26355 df-log 26533 df-cxp 26534 df-lgam 26996

This theorem is referenced by: lgambdd 27014 lgamcvglem 27017

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