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Theorem lgamgulm2 27154

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 27147	. . . . . 6 ⊢ (𝜑 → 𝑈 ⊆ (ℂ ∖ (ℤ ∖ ℕ)))
4	3	sselda 3939	. . . . 5 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑈) → 𝑧 ∈ (ℂ ∖ (ℤ ∖ ℕ)))
5		ovex 7433	. . . . 5 ⊢ (Σ𝑛 ∈ ℕ ((𝑧 · (log‘((𝑛 + 1) / 𝑛))) − (log‘((𝑧 / 𝑛) + 1))) − (log‘𝑧)) ∈ V
6		df-lgam 27137	. . . . . 6 ⊢ log Γ = (𝑧 ∈ (ℂ ∖ (ℤ ∖ ℕ)) ↦ (Σ𝑛 ∈ ℕ ((𝑧 · (log‘((𝑛 + 1) / 𝑛))) − (log‘((𝑧 / 𝑛) + 1))) − (log‘𝑧)))
7	6	fvmpt2 6991	. . . . 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 597	. . . 4 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑈) → (log Γ‘𝑧) = (Σ𝑛 ∈ ℕ ((𝑧 · (log‘((𝑛 + 1) / 𝑛))) − (log‘((𝑧 / 𝑛) + 1))) − (log‘𝑧)))
9		nnuz 12889	. . . . . . 7 ⊢ ℕ = (ℤ_≥‘1)
10		1zzd 12613	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑈) → 1 ∈ ℤ)
11		oveq1 7407	. . . . . . . . . . . . 13 ⊢ (𝑚 = 𝑛 → (𝑚 + 1) = (𝑛 + 1))
12		id 23	. . . . . . . . . . . . 13 ⊢ (𝑚 = 𝑛 → 𝑚 = 𝑛)
13	11, 12	oveq12d 7418	. . . . . . . . . . . 12 ⊢ (𝑚 = 𝑛 → ((𝑚 + 1) / 𝑚) = ((𝑛 + 1) / 𝑛))
14	13	fveq2d 6875	. . . . . . . . . . 11 ⊢ (𝑚 = 𝑛 → (log‘((𝑚 + 1) / 𝑚)) = (log‘((𝑛 + 1) / 𝑛)))
15	14	oveq2d 7416	. . . . . . . . . 10 ⊢ (𝑚 = 𝑛 → (𝑧 · (log‘((𝑚 + 1) / 𝑚))) = (𝑧 · (log‘((𝑛 + 1) / 𝑛))))
16		oveq2 7408	. . . . . . . . . . 11 ⊢ (𝑚 = 𝑛 → (𝑧 / 𝑚) = (𝑧 / 𝑛))
17	16	fvoveq1d 7422	. . . . . . . . . 10 ⊢ (𝑚 = 𝑛 → (log‘((𝑧 / 𝑚) + 1)) = (log‘((𝑧 / 𝑛) + 1)))
18	15, 17	oveq12d 7418	. . . . . . . . 9 ⊢ (𝑚 = 𝑛 → ((𝑧 · (log‘((𝑚 + 1) / 𝑚))) − (log‘((𝑧 / 𝑚) + 1))) = ((𝑧 · (log‘((𝑛 + 1) / 𝑛))) − (log‘((𝑧 / 𝑛) + 1))))
19		eqid 2765	. . . . . . . . 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 7433	. . . . . . . . 9 ⊢ ((𝑧 · (log‘((𝑛 + 1) / 𝑛))) − (log‘((𝑧 / 𝑛) + 1))) ∈ V
21	18, 19, 20	fvmpt 6979	. . . . . . . 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 486	. . . . . . 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 3919	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑈) → 𝑧 ∈ ℂ)
24	23	adantr 485	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑈) ∧ 𝑛 ∈ ℕ) → 𝑧 ∈ ℂ)
25		simpr 489	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑈) ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ ℕ)
26	25	peano2nnd 12238	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑈) ∧ 𝑛 ∈ ℕ) → (𝑛 + 1) ∈ ℕ)
27	26	nnrpd 13046	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑈) ∧ 𝑛 ∈ ℕ) → (𝑛 + 1) ∈ ℝ⁺)
28	25	nnrpd 13046	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑈) ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ ℝ⁺)
29	27, 28	rpdivcld 13065	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑈) ∧ 𝑛 ∈ ℕ) → ((𝑛 + 1) / 𝑛) ∈ ℝ⁺)
30	29	relogcld 26742	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑈) ∧ 𝑛 ∈ ℕ) → (log‘((𝑛 + 1) / 𝑛)) ∈ ℝ)
31	30	recnd 11225	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑈) ∧ 𝑛 ∈ ℕ) → (log‘((𝑛 + 1) / 𝑛)) ∈ ℂ)
32	24, 31	mulcld 11217	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑈) ∧ 𝑛 ∈ ℕ) → (𝑧 · (log‘((𝑛 + 1) / 𝑛))) ∈ ℂ)
33	25	nncnd 12237	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑈) ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ ℂ)
34	25	nnne0d 12274	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑈) ∧ 𝑛 ∈ ℕ) → 𝑛 ≠ 0)
35	24, 33, 34	divcld 11979	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑈) ∧ 𝑛 ∈ ℕ) → (𝑧 / 𝑛) ∈ ℂ)
36		1cnd 11190	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑈) ∧ 𝑛 ∈ ℕ) → 1 ∈ ℂ)
37	35, 36	addcld 11216	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑈) ∧ 𝑛 ∈ ℕ) → ((𝑧 / 𝑛) + 1) ∈ ℂ)
38	4	adantr 485	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑈) ∧ 𝑛 ∈ ℕ) → 𝑧 ∈ (ℂ ∖ (ℤ ∖ ℕ)))
39	38, 25	dmgmdivn0 27146	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑈) ∧ 𝑛 ∈ ℕ) → ((𝑧 / 𝑛) + 1) ≠ 0)
40	37, 39	logcld 26689	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑈) ∧ 𝑛 ∈ ℕ) → (log‘((𝑧 / 𝑛) + 1)) ∈ ℂ)
41	32, 40	subcld 11557	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑈) ∧ 𝑛 ∈ ℕ) → ((𝑧 · (log‘((𝑛 + 1) / 𝑛))) − (log‘((𝑧 / 𝑛) + 1))) ∈ ℂ)
42		1z 12612	. . . . . . . . . . . 12 ⊢ 1 ∈ ℤ
43		seqfn 14037	. . . . . . . . . . . 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 6623	. . . . . . . . . . 11 ⊢ (seq1( ∘_f + , 𝐺) Fn ℕ ↔ seq1( ∘_f + , 𝐺) Fn (ℤ_≥‘1))
46	44, 45	mpbir 234	. . . . . . . . . 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 27153	. . . . . . . . . . 11 ⊢ (𝜑 → seq1( ∘_f + , 𝐺) ∈ dom (⇝_𝑢‘𝑈))
49		ulmdm 26510	. . . . . . . . . . 11 ⊢ (seq1( ∘_f + , 𝐺) ∈ dom (⇝_𝑢‘𝑈) ↔ seq1( ∘_f + , 𝐺)(⇝_𝑢‘𝑈)((⇝_𝑢‘𝑈)‘seq1( ∘_f + , 𝐺)))
50	48, 49	sylib 221	. . . . . . . . . 10 ⊢ (𝜑 → seq1( ∘_f + , 𝐺)(⇝_𝑢‘𝑈)((⇝_𝑢‘𝑈)‘seq1( ∘_f + , 𝐺)))
51		ulmf2 26501	. . . . . . . . . 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 598	. . . . . . . . 9 ⊢ (𝜑 → seq1( ∘_f + , 𝐺):ℕ⟶(ℂ ↑_m 𝑈))
53	52	adantr 485	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑈) → seq1( ∘_f + , 𝐺):ℕ⟶(ℂ ↑_m 𝑈))
54		simpr 489	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑈) → 𝑧 ∈ 𝑈)
55		seqex 14027	. . . . . . . . 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 14033	. . . . . . . . . . . 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 6873	. . . . . . . . . . 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 11169	. . . . . . . . . . . . . . 15 ⊢ ℂ ∈ V
61	2, 60	rabex2 5301	. . . . . . . . . . . . . 14 ⊢ 𝑈 ∈ V
62	61	a1i 11	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝑈 ∈ V)
63		simpr 489	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ ℕ)
64	63, 9	eleqtrdi 2875	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ (ℤ_≥‘1))
65		fz1ssnn 13571	. . . . . . . . . . . . . 14 ⊢ (1...𝑛) ⊆ ℕ
66	65	a1i 11	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (1...𝑛) ⊆ ℕ)
67		ovexd 7435	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ (𝑚 ∈ ℕ ∧ 𝑧 ∈ 𝑈)) → ((𝑧 · (log‘((𝑚 + 1) / 𝑚))) − (log‘((𝑧 / 𝑚) + 1))) ∈ V)
68	62, 64, 66, 67	seqof2 14084	. . . . . . . . . . . 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 727	. . . . . . . . . . 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 2800	. . . . . . . . . 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 6873	. . . . . . . . 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 485	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑈) ∧ 𝑛 ∈ ℕ) → 𝑧 ∈ 𝑈)
73		fvex 6884	. . . . . . . . . 10 ⊢ (seq1( + , (𝑚 ∈ ℕ ↦ ((𝑧 · (log‘((𝑚 + 1) / 𝑚))) − (log‘((𝑧 / 𝑚) + 1)))))‘𝑛) ∈ V
74		eqid 2765	. . . . . . . . . . 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 6991	. . . . . . . . . 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 597	. . . . . . . . 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 2800	. . . . . . . 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 485	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑈) → seq1( ∘_f + , 𝐺)(⇝_𝑢‘𝑈)((⇝_𝑢‘𝑈)‘seq1( ∘_f + , 𝐺)))
79	9, 10, 53, 54, 56, 77, 78	ulmclm 26504	. . . . . . 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 15796	. . . . . 6 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑈) → Σ𝑛 ∈ ℕ ((𝑧 · (log‘((𝑛 + 1) / 𝑛))) − (log‘((𝑧 / 𝑛) + 1))) = (((⇝_𝑢‘𝑈)‘seq1( ∘_f + , 𝐺))‘𝑧))
81		ulmcl 26498	. . . . . . . 8 ⊢ (seq1( ∘_f + , 𝐺)(⇝_𝑢‘𝑈)((⇝_𝑢‘𝑈)‘seq1( ∘_f + , 𝐺)) → ((⇝_𝑢‘𝑈)‘seq1( ∘_f + , 𝐺)):𝑈⟶ℂ)
82	50, 81	syl 18	. . . . . . 7 ⊢ (𝜑 → ((⇝_𝑢‘𝑈)‘seq1( ∘_f + , 𝐺)):𝑈⟶ℂ)
83	82	ffvelcdmda 7069	. . . . . 6 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑈) → (((⇝_𝑢‘𝑈)‘seq1( ∘_f + , 𝐺))‘𝑧) ∈ ℂ)
84	80, 83	eqeltrd 2865	. . . . 5 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑈) → Σ𝑛 ∈ ℕ ((𝑧 · (log‘((𝑛 + 1) / 𝑛))) − (log‘((𝑧 / 𝑛) + 1))) ∈ ℂ)
85	4	dmgmn0 27144	. . . . . 6 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑈) → 𝑧 ≠ 0)
86	23, 85	logcld 26689	. . . . 5 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑈) → (log‘𝑧) ∈ ℂ)
87	84, 86	subcld 11557	. . . 4 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑈) → (Σ𝑛 ∈ ℕ ((𝑧 · (log‘((𝑛 + 1) / 𝑛))) − (log‘((𝑧 / 𝑛) + 1))) − (log‘𝑧)) ∈ ℂ)
88	8, 87	eqeltrd 2865	. . 3 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑈) → (log Γ‘𝑧) ∈ ℂ)
89	88	ralrimiva 3157	. 2 ⊢ (𝜑 → ∀𝑧 ∈ 𝑈 (log Γ‘𝑧) ∈ ℂ)
90		ffn 6695	. . . . . 6 ⊢ (((⇝_𝑢‘𝑈)‘seq1( ∘_f + , 𝐺)):𝑈⟶ℂ → ((⇝_𝑢‘𝑈)‘seq1( ∘_f + , 𝐺)) Fn 𝑈)
91	50, 81, 90	3syl 19	. . . . 5 ⊢ (𝜑 → ((⇝_𝑢‘𝑈)‘seq1( ∘_f + , 𝐺)) Fn 𝑈)
92		nfcv 2927	. . . . . . 7 ⊢ Ⅎ𝑧(⇝_𝑢‘𝑈)
93		nfcv 2927	. . . . . . . 8 ⊢ Ⅎ𝑧1
94		nfcv 2927	. . . . . . . 8 ⊢ Ⅎ𝑧 ∘_f +
95		nfcv 2927	. . . . . . . . . 10 ⊢ Ⅎ𝑧ℕ
96		nfmpt1 5203	. . . . . . . . . 10 ⊢ Ⅎ𝑧(𝑧 ∈ 𝑈 ↦ ((𝑧 · (log‘((𝑚 + 1) / 𝑚))) − (log‘((𝑧 / 𝑚) + 1))))
97	95, 96	nfmpt 5202	. . . . . . . . 9 ⊢ Ⅎ𝑧(𝑚 ∈ ℕ ↦ (𝑧 ∈ 𝑈 ↦ ((𝑧 · (log‘((𝑚 + 1) / 𝑚))) − (log‘((𝑧 / 𝑚) + 1)))))
98	47, 97	nfcxfr 2925	. . . . . . . 8 ⊢ Ⅎ𝑧𝐺
99	93, 94, 98	nfseq 14035	. . . . . . 7 ⊢ Ⅎ𝑧seq1( ∘_f + , 𝐺)
100	92, 99	nffv 6881	. . . . . 6 ⊢ Ⅎ𝑧((⇝_𝑢‘𝑈)‘seq1( ∘_f + , 𝐺))
101	100	dffn5f 6942	. . . . 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 221	. . . 4 ⊢ (𝜑 → ((⇝_𝑢‘𝑈)‘seq1( ∘_f + , 𝐺)) = (𝑧 ∈ 𝑈 ↦ (((⇝_𝑢‘𝑈)‘seq1( ∘_f + , 𝐺))‘𝑧)))
103	8	oveq1d 7415	. . . . . 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 11561	. . . . . 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 2805	. . . . 5 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑈) → (((⇝_𝑢‘𝑈)‘seq1( ∘_f + , 𝐺))‘𝑧) = ((log Γ‘𝑧) + (log‘𝑧)))
106	105	mpteq2dva 5197	. . . 4 ⊢ (𝜑 → (𝑧 ∈ 𝑈 ↦ (((⇝_𝑢‘𝑈)‘seq1( ∘_f + , 𝐺))‘𝑧)) = (𝑧 ∈ 𝑈 ↦ ((log Γ‘𝑧) + (log‘𝑧))))
107	102, 106	eqtrd 2800	. . 3 ⊢ (𝜑 → ((⇝_𝑢‘𝑈)‘seq1( ∘_f + , 𝐺)) = (𝑧 ∈ 𝑈 ↦ ((log Γ‘𝑧) + (log‘𝑧))))
108	50, 107	breqtrd 5130	. 2 ⊢ (𝜑 → seq1( ∘_f + , 𝐺)(⇝_𝑢‘𝑈)(𝑧 ∈ 𝑈 ↦ ((log Γ‘𝑧) + (log‘𝑧))))
109	89, 108	jca 520	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 400 = wceq 1563 ∈ wcel 2145 ∀wral 3079 {crab 3417 Vcvv 3457 ∖ cdif 3904 ⊆ wss 3907 class class class wbr 5104 ↦ cmpt 5185 dom cdm 5651 Fn wfn 6520 ⟶wf 6521 ‘cfv 6525 (class class class)co 7400 ∘_f cof 7662 ↑_m cmap 8812 ℂcc 11086 1c1 11089 + caddc 11091 · cmul 11093 ≤ cle 11232 − cmin 11429 / cdiv 11859 ℕcn 12221 ℕ₀cn0 12492 ℤcz 12579 ℤ_≥cuz 12850 ...cfz 13523 seqcseq 14025 abscabs 15273 Σcsu 15725 ⇝_𝑢culm 26493 logclog 26673 log Γclgam 27134

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1818 ax-4 1832 ax-5 1933 ax-6 1990 ax-7 2031 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2215 ax-ext 2737 ax-rep 5231 ax-sep 5250 ax-nul 5260 ax-pow 5326 ax-pr 5394 ax-un 7722 ax-inf2 9598 ax-cnex 11144 ax-resscn 11145 ax-1cn 11146 ax-icn 11147 ax-addcl 11148 ax-addrcl 11149 ax-mulcl 11150 ax-mulrcl 11151 ax-mulcom 11152 ax-addass 11153 ax-mulass 11154 ax-distr 11155 ax-i2m1 11156 ax-1ne0 11157 ax-1rid 11158 ax-rnegex 11159 ax-rrecex 11160 ax-cnre 11161 ax-pre-lttri 11162 ax-pre-lttrn 11163 ax-pre-ltadd 11164 ax-pre-mulgt0 11165 ax-pre-sup 11166 ax-addf 11167

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1102 df-3an 1103 df-tru 1566 df-fal 1576 df-ex 1803 df-nf 1807 df-sb 2094 df-mo 2569 df-eu 2599 df-clab 2744 df-cleq 2757 df-clel 2840 df-nfc 2914 df-ne 2961 df-nel 3065 df-ral 3080 df-rex 3090 df-rmo 3370 df-reu 3371 df-rab 3418 df-v 3459 df-sbc 3748 df-csb 3856 df-dif 3910 df-un 3912 df-in 3914 df-ss 3924 df-pss 3927 df-nul 4289 df-if 4484 df-pw 4560 df-sn 4586 df-pr 4588 df-tp 4590 df-op 4592 df-uni 4868 df-int 4908 df-iun 4953 df-iin 4954 df-br 5105 df-opab 5167 df-mpt 5186 df-tr 5212 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 6291 df-ord 6352 df-on 6353 df-lim 6354 df-suc 6355 df-iota 6481 df-fun 6527 df-fn 6528 df-f 6529 df-f1 6530 df-fo 6531 df-f1o 6532 df-fv 6533 df-isom 6534 df-riota 7357 df-ov 7403 df-oprab 7404 df-mpo 7405 df-of 7664 df-om 7851 df-1st 7974 df-2nd 7975 df-supp 8145 df-frecs 8266 df-wrecs 8297 df-recs 8346 df-rdg 8385 df-1o 8441 df-2o 8442 df-oadd 8445 df-er 8682 df-map 8814 df-pm 8815 df-ixp 8884 df-en 8932 df-dom 8933 df-sdom 8934 df-fin 8935 df-fsupp 9310 df-fi 9359 df-sup 9390 df-inf 9391 df-oi 9460 df-dju 9875 df-card 9913 df-pnf 11233 df-mnf 11234 df-xr 11235 df-ltxr 11236 df-le 11237 df-sub 11431 df-neg 11432 df-div 11860 df-nn 12222 df-2 12291 df-3 12292 df-4 12293 df-5 12294 df-6 12295 df-7 12296 df-8 12297 df-9 12298 df-n0 12493 df-z 12580 df-dec 12700 df-uz 12851 df-q 12961 df-rp 13005 df-xneg 13125 df-xadd 13126 df-xmul 13127 df-ioo 13364 df-ioc 13365 df-ico 13366 df-icc 13367 df-fz 13524 df-fzo 13671 df-fl 13813 df-mod 13891 df-seq 14026 df-exp 14086 df-fac 14298 df-bc 14327 df-hash 14355 df-shft 15092 df-cj 15138 df-re 15139 df-im 15140 df-sqrt 15274 df-abs 15275 df-limsup 15510 df-clim 15527 df-rlim 15528 df-sum 15726 df-ef 16109 df-sin 16111 df-cos 16112 df-tan 16113 df-pi 16114 df-struct 17195 df-sets 17212 df-slot 17230 df-ndx 17242 df-base 17258 df-ress 17279 df-plusg 17311 df-mulr 17312 df-starv 17313 df-sca 17314 df-vsca 17315 df-ip 17316 df-tset 17317 df-ple 17318 df-ds 17320 df-unif 17321 df-hom 17322 df-cco 17323 df-rest 17463 df-topn 17464 df-0g 17482 df-gsum 17483 df-topgen 17484 df-pt 17485 df-prds 17488 df-xrs 17544 df-qtop 17549 df-imas 17550 df-xps 17552 df-mre 17626 df-mrc 17627 df-acs 17629 df-mgm 18686 df-sgrp 18765 df-mnd 18781 df-submnd 18830 df-mulg 19122 df-cntz 19375 df-cmn 19840 df-psmet 21471 df-xmet 21472 df-met 21473 df-bl 21474 df-mopn 21475 df-fbas 21476 df-fg 21477 df-cnfld 21480 df-top 23008 df-topon 23025 df-topsp 23047 df-bases 23060 df-cld 23133 df-ntr 23134 df-cls 23135 df-nei 23212 df-lp 23250 df-perf 23251 df-cn 23341 df-cnp 23342 df-haus 23429 df-cmp 23501 df-tx 23676 df-hmeo 23869 df-fil 23960 df-fm 24052 df-flim 24053 df-flf 24054 df-xms 24434 df-ms 24435 df-tms 24436 df-cncf 24994 df-limc 25982 df-dv 25983 df-ulm 26494 df-log 26675 df-cxp 26676 df-lgam 27137

This theorem is referenced by: lgambdd 27155 lgamcvglem 27158

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