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Theorem lgamgulm2 27237

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 27230	. . . . . 6 ⊢ (𝜑 → 𝑈 ⊆ (ℂ ∖ (ℤ ∖ ℕ)))
4	3	sselda 3940	. . . . 5 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑈) → 𝑧 ∈ (ℂ ∖ (ℤ ∖ ℕ)))
5		ovex 7456	. . . . 5 ⊢ (Σ𝑛 ∈ ℕ ((𝑧 · (log‘((𝑛 + 1) / 𝑛))) − (log‘((𝑧 / 𝑛) + 1))) − (log‘𝑧)) ∈ V
6		df-lgam 27220	. . . . . 6 ⊢ log Γ = (𝑧 ∈ (ℂ ∖ (ℤ ∖ ℕ)) ↦ (Σ𝑛 ∈ ℕ ((𝑧 · (log‘((𝑛 + 1) / 𝑛))) − (log‘((𝑧 / 𝑛) + 1))) − (log‘𝑧)))
7	6	fvmpt2 7008	. . . . 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 598	. . . 4 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑈) → (log Γ‘𝑧) = (Σ𝑛 ∈ ℕ ((𝑧 · (log‘((𝑛 + 1) / 𝑛))) − (log‘((𝑧 / 𝑛) + 1))) − (log‘𝑧)))
9		nnuz 12919	. . . . . . 7 ⊢ ℕ = (ℤ_≥‘1)
10		1zzd 12643	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑈) → 1 ∈ ℤ)
11		oveq1 7430	. . . . . . . . . . . . 13 ⊢ (𝑚 = 𝑛 → (𝑚 + 1) = (𝑛 + 1))
12		id 23	. . . . . . . . . . . . 13 ⊢ (𝑚 = 𝑛 → 𝑚 = 𝑛)
13	11, 12	oveq12d 7441	. . . . . . . . . . . 12 ⊢ (𝑚 = 𝑛 → ((𝑚 + 1) / 𝑚) = ((𝑛 + 1) / 𝑛))
14	13	fveq2d 6892	. . . . . . . . . . 11 ⊢ (𝑚 = 𝑛 → (log‘((𝑚 + 1) / 𝑚)) = (log‘((𝑛 + 1) / 𝑛)))
15	14	oveq2d 7439	. . . . . . . . . 10 ⊢ (𝑚 = 𝑛 → (𝑧 · (log‘((𝑚 + 1) / 𝑚))) = (𝑧 · (log‘((𝑛 + 1) / 𝑛))))
16		oveq2 7431	. . . . . . . . . . 11 ⊢ (𝑚 = 𝑛 → (𝑧 / 𝑚) = (𝑧 / 𝑛))
17	16	fvoveq1d 7445	. . . . . . . . . 10 ⊢ (𝑚 = 𝑛 → (log‘((𝑧 / 𝑚) + 1)) = (log‘((𝑧 / 𝑛) + 1)))
18	15, 17	oveq12d 7441	. . . . . . . . 9 ⊢ (𝑚 = 𝑛 → ((𝑧 · (log‘((𝑚 + 1) / 𝑚))) − (log‘((𝑧 / 𝑚) + 1))) = ((𝑧 · (log‘((𝑛 + 1) / 𝑛))) − (log‘((𝑧 / 𝑛) + 1))))
19		eqid 2766	. . . . . . . . 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 7456	. . . . . . . . 9 ⊢ ((𝑧 · (log‘((𝑛 + 1) / 𝑛))) − (log‘((𝑧 / 𝑛) + 1))) ∈ V
21	18, 19, 20	fvmpt 6996	. . . . . . . 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 487	. . . . . . 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 3920	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑈) → 𝑧 ∈ ℂ)
24	23	adantr 486	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑈) ∧ 𝑛 ∈ ℕ) → 𝑧 ∈ ℂ)
25		simpr 490	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑈) ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ ℕ)
26	25	peano2nnd 12268	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑈) ∧ 𝑛 ∈ ℕ) → (𝑛 + 1) ∈ ℕ)
27	26	nnrpd 13076	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑈) ∧ 𝑛 ∈ ℕ) → (𝑛 + 1) ∈ ℝ⁺)
28	25	nnrpd 13076	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑈) ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ ℝ⁺)
29	27, 28	rpdivcld 13095	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑈) ∧ 𝑛 ∈ ℕ) → ((𝑛 + 1) / 𝑛) ∈ ℝ⁺)
30	29	relogcld 26825	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑈) ∧ 𝑛 ∈ ℕ) → (log‘((𝑛 + 1) / 𝑛)) ∈ ℝ)
31	30	recnd 11255	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑈) ∧ 𝑛 ∈ ℕ) → (log‘((𝑛 + 1) / 𝑛)) ∈ ℂ)
32	24, 31	mulcld 11247	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑈) ∧ 𝑛 ∈ ℕ) → (𝑧 · (log‘((𝑛 + 1) / 𝑛))) ∈ ℂ)
33	25	nncnd 12267	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑈) ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ ℂ)
34	25	nnne0d 12304	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑈) ∧ 𝑛 ∈ ℕ) → 𝑛 ≠ 0)
35	24, 33, 34	divcld 12009	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑈) ∧ 𝑛 ∈ ℕ) → (𝑧 / 𝑛) ∈ ℂ)
36		1cnd 11220	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑈) ∧ 𝑛 ∈ ℕ) → 1 ∈ ℂ)
37	35, 36	addcld 11246	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑈) ∧ 𝑛 ∈ ℕ) → ((𝑧 / 𝑛) + 1) ∈ ℂ)
38	4	adantr 486	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑈) ∧ 𝑛 ∈ ℕ) → 𝑧 ∈ (ℂ ∖ (ℤ ∖ ℕ)))
39	38, 25	dmgmdivn0 27229	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑈) ∧ 𝑛 ∈ ℕ) → ((𝑧 / 𝑛) + 1) ≠ 0)
40	37, 39	logcld 26772	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑈) ∧ 𝑛 ∈ ℕ) → (log‘((𝑧 / 𝑛) + 1)) ∈ ℂ)
41	32, 40	subcld 11587	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑈) ∧ 𝑛 ∈ ℕ) → ((𝑧 · (log‘((𝑛 + 1) / 𝑛))) − (log‘((𝑧 / 𝑛) + 1))) ∈ ℂ)
42		1z 12642	. . . . . . . . . . . 12 ⊢ 1 ∈ ℤ
43		seqfn 14069	. . . . . . . . . . . 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 6640	. . . . . . . . . . 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 27236	. . . . . . . . . . 11 ⊢ (𝜑 → seq1( ∘_f + , 𝐺) ∈ dom (⇝_𝑢‘𝑈))
49		ulmdm 26593	. . . . . . . . . . 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 26584	. . . . . . . . . 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 599	. . . . . . . . 9 ⊢ (𝜑 → seq1( ∘_f + , 𝐺):ℕ⟶(ℂ ↑_m 𝑈))
53	52	adantr 486	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑈) → seq1( ∘_f + , 𝐺):ℕ⟶(ℂ ↑_m 𝑈))
54		simpr 490	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑈) → 𝑧 ∈ 𝑈)
55		seqex 14059	. . . . . . . . 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 14065	. . . . . . . . . . . 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 6890	. . . . . . . . . . 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 11199	. . . . . . . . . . . . . . 15 ⊢ ℂ ∈ V
61	2, 60	rabex2 5316	. . . . . . . . . . . . . 14 ⊢ 𝑈 ∈ V
62	61	a1i 11	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝑈 ∈ V)
63		simpr 490	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ ℕ)
64	63, 9	eleqtrdi 2876	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ (ℤ_≥‘1))
65		fz1ssnn 13602	. . . . . . . . . . . . . 14 ⊢ (1...𝑛) ⊆ ℕ
66	65	a1i 11	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (1...𝑛) ⊆ ℕ)
67		ovexd 7458	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ (𝑚 ∈ ℕ ∧ 𝑧 ∈ 𝑈)) → ((𝑧 · (log‘((𝑚 + 1) / 𝑚))) − (log‘((𝑧 / 𝑚) + 1))) ∈ V)
68	62, 64, 66, 67	seqof2 14116	. . . . . . . . . . . 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 728	. . . . . . . . . . 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 2801	. . . . . . . . . 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 6890	. . . . . . . . 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 486	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑈) ∧ 𝑛 ∈ ℕ) → 𝑧 ∈ 𝑈)
73		fvex 6901	. . . . . . . . . 10 ⊢ (seq1( + , (𝑚 ∈ ℕ ↦ ((𝑧 · (log‘((𝑚 + 1) / 𝑚))) − (log‘((𝑧 / 𝑚) + 1)))))‘𝑛) ∈ V
74		eqid 2766	. . . . . . . . . . 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 7008	. . . . . . . . . 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 598	. . . . . . . . 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 2801	. . . . . . . 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 486	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑈) → seq1( ∘_f + , 𝐺)(⇝_𝑢‘𝑈)((⇝_𝑢‘𝑈)‘seq1( ∘_f + , 𝐺)))
79	9, 10, 53, 54, 56, 77, 78	ulmclm 26587	. . . . . . 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 15834	. . . . . 6 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑈) → Σ𝑛 ∈ ℕ ((𝑧 · (log‘((𝑛 + 1) / 𝑛))) − (log‘((𝑧 / 𝑛) + 1))) = (((⇝_𝑢‘𝑈)‘seq1( ∘_f + , 𝐺))‘𝑧))
81		ulmcl 26581	. . . . . . . 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 7086	. . . . . 6 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑈) → (((⇝_𝑢‘𝑈)‘seq1( ∘_f + , 𝐺))‘𝑧) ∈ ℂ)
84	80, 83	eqeltrd 2866	. . . . 5 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑈) → Σ𝑛 ∈ ℕ ((𝑧 · (log‘((𝑛 + 1) / 𝑛))) − (log‘((𝑧 / 𝑛) + 1))) ∈ ℂ)
85	4	dmgmn0 27227	. . . . . 6 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑈) → 𝑧 ≠ 0)
86	23, 85	logcld 26772	. . . . 5 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑈) → (log‘𝑧) ∈ ℂ)
87	84, 86	subcld 11587	. . . 4 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑈) → (Σ𝑛 ∈ ℕ ((𝑧 · (log‘((𝑛 + 1) / 𝑛))) − (log‘((𝑧 / 𝑛) + 1))) − (log‘𝑧)) ∈ ℂ)
88	8, 87	eqeltrd 2866	. . 3 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑈) → (log Γ‘𝑧) ∈ ℂ)
89	88	ralrimiva 3160	. 2 ⊢ (𝜑 → ∀𝑧 ∈ 𝑈 (log Γ‘𝑧) ∈ ℂ)
90		ffn 6712	. . . . . 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 2928	. . . . . . 7 ⊢ Ⅎ𝑧(⇝_𝑢‘𝑈)
93		nfcv 2928	. . . . . . . 8 ⊢ Ⅎ𝑧1
94		nfcv 2928	. . . . . . . 8 ⊢ Ⅎ𝑧 ∘_f +
95		nfcv 2928	. . . . . . . . . 10 ⊢ Ⅎ𝑧ℕ
96		nfmpt1 5215	. . . . . . . . . 10 ⊢ Ⅎ𝑧(𝑧 ∈ 𝑈 ↦ ((𝑧 · (log‘((𝑚 + 1) / 𝑚))) − (log‘((𝑧 / 𝑚) + 1))))
97	95, 96	nfmpt 5214	. . . . . . . . 9 ⊢ Ⅎ𝑧(𝑚 ∈ ℕ ↦ (𝑧 ∈ 𝑈 ↦ ((𝑧 · (log‘((𝑚 + 1) / 𝑚))) − (log‘((𝑧 / 𝑚) + 1)))))
98	47, 97	nfcxfr 2926	. . . . . . . 8 ⊢ Ⅎ𝑧𝐺
99	93, 94, 98	nfseq 14067	. . . . . . 7 ⊢ Ⅎ𝑧seq1( ∘_f + , 𝐺)
100	92, 99	nffv 6898	. . . . . 6 ⊢ Ⅎ𝑧((⇝_𝑢‘𝑈)‘seq1( ∘_f + , 𝐺))
101	100	dffn5f 6959	. . . . 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 7438	. . . . . 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 11591	. . . . . 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 2806	. . . . 5 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑈) → (((⇝_𝑢‘𝑈)‘seq1( ∘_f + , 𝐺))‘𝑧) = ((log Γ‘𝑧) + (log‘𝑧)))
106	105	mpteq2dva 5209	. . . 4 ⊢ (𝜑 → (𝑧 ∈ 𝑈 ↦ (((⇝_𝑢‘𝑈)‘seq1( ∘_f + , 𝐺))‘𝑧)) = (𝑧 ∈ 𝑈 ↦ ((log Γ‘𝑧) + (log‘𝑧))))
107	102, 106	eqtrd 2801	. . 3 ⊢ (𝜑 → ((⇝_𝑢‘𝑈)‘seq1( ∘_f + , 𝐺)) = (𝑧 ∈ 𝑈 ↦ ((log Γ‘𝑧) + (log‘𝑧))))
108	50, 107	breqtrd 5142	. 2 ⊢ (𝜑 → seq1( ∘_f + , 𝐺)(⇝_𝑢‘𝑈)(𝑧 ∈ 𝑈 ↦ ((log Γ‘𝑧) + (log‘𝑧))))
109	89, 108	jca 521	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

This proof depends on syntax axioms: → wi 4 ∧ wa 401 = wceq 1570 ∈ wcel 2146 ∀wral 3082 {crab 3419 Vcvv 3458 ∖ cdif 3905 ⊆ wss 3908 class class class wbr 5114 ↦ cmpt 5197 dom cdm 5666 Fn wfn 6538 ⟶wf 6539 ‘cfv 6543 (class class class)co 7423 ∘_f cof 7685 ↑_m cmap 8833 ℂcc 11116 1c1 11119 + caddc 11121 · cmul 11123 ≤ cle 11262 − cmin 11459 / cdiv 11889 ℕcn 12251 ℕ₀cn0 12522 ℤcz 12609 ℤ_≥cuz 12880 ...cfz 13553 seqcseq 14057 abscabs 15311 Σcsu 15763 ⇝_𝑢culm 26576 logclog 26756 log Γclgam 27217

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2148 ax-9 2156 ax-10 2179 ax-11 2195 ax-12 2216 ax-ext 2738 ax-rep 5243 ax-sep 5262 ax-nul 5274 ax-pow 5341 ax-pr 5409 ax-un 7745 ax-inf2 9620 ax-cnex 11174 ax-resscn 11175 ax-1cn 11176 ax-icn 11177 ax-addcl 11178 ax-addrcl 11179 ax-mulcl 11180 ax-mulrcl 11181 ax-mulcom 11182 ax-addass 11183 ax-mulass 11184 ax-distr 11185 ax-i2m1 11186 ax-1ne0 11187 ax-1rid 11188 ax-rnegex 11189 ax-rrecex 11190 ax-cnre 11191 ax-pre-lttri 11192 ax-pre-lttrn 11193 ax-pre-ltadd 11194 ax-pre-mulgt0 11195 ax-pre-sup 11196 ax-addf 11197

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2570 df-eu 2600 df-clab 2745 df-cleq 2758 df-clel 2841 df-nfc 2915 df-ne 2962 df-nel 3068 df-ral 3083 df-rex 3093 df-rmo 3372 df-reu 3373 df-rab 3420 df-v 3460 df-sbc 3748 df-csb 3857 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-pss 3928 df-nul 4290 df-if 4493 df-pw 4569 df-sn 4595 df-pr 4597 df-tp 4599 df-op 4601 df-uni 4878 df-int 4918 df-iun 4963 df-iin 4964 df-br 5115 df-opab 5179 df-mpt 5198 df-tr 5224 df-id 5561 df-eprel 5566 df-po 5574 df-so 5575 df-fr 5619 df-se 5620 df-we 5621 df-xp 5672 df-rel 5673 df-cnv 5674 df-co 5675 df-dm 5676 df-rn 5677 df-res 5678 df-ima 5679 df-pred 6309 df-ord 6370 df-on 6371 df-lim 6372 df-suc 6373 df-iota 6499 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 7380 df-ov 7426 df-oprab 7427 df-mpo 7428 df-of 7687 df-om 7872 df-1st 7995 df-2nd 7996 df-supp 8166 df-frecs 8287 df-wrecs 8318 df-recs 8367 df-rdg 8406 df-1o 8462 df-2o 8463 df-oadd 8466 df-er 8703 df-map 8835 df-pm 8836 df-ixp 8905 df-en 8953 df-dom 8954 df-sdom 8955 df-fin 8956 df-fsupp 9332 df-fi 9381 df-sup 9412 df-inf 9413 df-oi 9482 df-dju 9906 df-card 9944 df-pnf 11263 df-mnf 11264 df-xr 11265 df-ltxr 11266 df-le 11267 df-sub 11461 df-neg 11462 df-div 11890 df-nn 12252 df-2 12321 df-3 12322 df-4 12323 df-5 12324 df-6 12325 df-7 12326 df-8 12327 df-9 12328 df-n0 12523 df-z 12610 df-dec 12730 df-uz 12881 df-q 12991 df-rp 13035 df-xneg 13155 df-xadd 13156 df-xmul 13157 df-ioo 13394 df-ioc 13395 df-ico 13396 df-icc 13397 df-fz 13554 df-fzo 13702 df-fl 13845 df-mod 13923 df-seq 14058 df-exp 14118 df-fac 14330 df-bc 14359 df-hash 14387 df-shft 15130 df-cj 15176 df-re 15177 df-im 15178 df-sqrt 15312 df-abs 15313 df-limsup 15548 df-clim 15565 df-rlim 15566 df-sum 15764 df-ef 16146 df-sin 16148 df-cos 16149 df-tan 16150 df-pi 16151 df-struct 17232 df-sets 17249 df-slot 17267 df-ndx 17279 df-base 17295 df-ress 17316 df-plusg 17348 df-mulr 17349 df-starv 17350 df-sca 17351 df-vsca 17352 df-ip 17353 df-tset 17354 df-ple 17355 df-ds 17357 df-unif 17358 df-hom 17359 df-cco 17360 df-rest 17500 df-topn 17501 df-0g 17519 df-gsum 17520 df-topgen 17521 df-pt 17522 df-prds 17525 df-xrs 17581 df-qtop 17586 df-imas 17587 df-xps 17589 df-mre 17663 df-mrc 17664 df-acs 17666 df-mgm 18723 df-sgrp 18806 df-mnd 18822 df-submnd 18873 df-mulg 19165 df-cntz 19418 df-cmn 19883 df-psmet 21551 df-xmet 21552 df-met 21553 df-bl 21554 df-mopn 21555 df-fbas 21556 df-fg 21557 df-cnfld 21560 df-top 23088 df-topon 23105 df-topsp 23127 df-bases 23140 df-cld 23213 df-ntr 23214 df-cls 23215 df-nei 23292 df-lp 23330 df-perf 23331 df-cn 23421 df-cnp 23422 df-haus 23509 df-cmp 23581 df-tx 23756 df-hmeo 23949 df-fil 24040 df-fm 24132 df-flim 24133 df-flf 24134 df-xms 24514 df-ms 24515 df-tms 24516 df-cncf 25074 df-limc 26062 df-dv 26063 df-ulm 26577 df-log 26758 df-cxp 26759 df-lgam 27220

This theorem is used by: lgambdd 27238 lgamcvglem 27241

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