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Theorem lgamgulm2 24762

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 24755	. . . . . 6 ⊢ (𝜑 → 𝑈 ⊆ (ℂ ∖ (ℤ ∖ ℕ)))
4	3	sselda 3603	. . . . 5 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑈) → 𝑧 ∈ (ℂ ∖ (ℤ ∖ ℕ)))
5		ovex 6678	. . . . 5 ⊢ (Σ𝑛 ∈ ℕ ((𝑧 · (log‘((𝑛 + 1) / 𝑛))) − (log‘((𝑧 / 𝑛) + 1))) − (log‘𝑧)) ∈ V
6		df-lgam 24745	. . . . . 6 ⊢ log Γ = (𝑧 ∈ (ℂ ∖ (ℤ ∖ ℕ)) ↦ (Σ𝑛 ∈ ℕ ((𝑧 · (log‘((𝑛 + 1) / 𝑛))) − (log‘((𝑧 / 𝑛) + 1))) − (log‘𝑧)))
7	6	fvmpt2 6291	. . . . 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 694	. . . 4 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑈) → (log Γ‘𝑧) = (Σ𝑛 ∈ ℕ ((𝑧 · (log‘((𝑛 + 1) / 𝑛))) − (log‘((𝑧 / 𝑛) + 1))) − (log‘𝑧)))
9		nnuz 11723	. . . . . . 7 ⊢ ℕ = (ℤ_≥‘1)
10		1zzd 11408	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑈) → 1 ∈ ℤ)
11		oveq1 6657	. . . . . . . . . . . . 13 ⊢ (𝑚 = 𝑛 → (𝑚 + 1) = (𝑛 + 1))
12		id 22	. . . . . . . . . . . . 13 ⊢ (𝑚 = 𝑛 → 𝑚 = 𝑛)
13	11, 12	oveq12d 6668	. . . . . . . . . . . 12 ⊢ (𝑚 = 𝑛 → ((𝑚 + 1) / 𝑚) = ((𝑛 + 1) / 𝑛))
14	13	fveq2d 6195	. . . . . . . . . . 11 ⊢ (𝑚 = 𝑛 → (log‘((𝑚 + 1) / 𝑚)) = (log‘((𝑛 + 1) / 𝑛)))
15	14	oveq2d 6666	. . . . . . . . . 10 ⊢ (𝑚 = 𝑛 → (𝑧 · (log‘((𝑚 + 1) / 𝑚))) = (𝑧 · (log‘((𝑛 + 1) / 𝑛))))
16		oveq2 6658	. . . . . . . . . . . 12 ⊢ (𝑚 = 𝑛 → (𝑧 / 𝑚) = (𝑧 / 𝑛))
17	16	oveq1d 6665	. . . . . . . . . . 11 ⊢ (𝑚 = 𝑛 → ((𝑧 / 𝑚) + 1) = ((𝑧 / 𝑛) + 1))
18	17	fveq2d 6195	. . . . . . . . . 10 ⊢ (𝑚 = 𝑛 → (log‘((𝑧 / 𝑚) + 1)) = (log‘((𝑧 / 𝑛) + 1)))
19	15, 18	oveq12d 6668	. . . . . . . . 9 ⊢ (𝑚 = 𝑛 → ((𝑧 · (log‘((𝑚 + 1) / 𝑚))) − (log‘((𝑧 / 𝑚) + 1))) = ((𝑧 · (log‘((𝑛 + 1) / 𝑛))) − (log‘((𝑧 / 𝑛) + 1))))
20		eqid 2622	. . . . . . . . 9 ⊢ (𝑚 ∈ ℕ ↦ ((𝑧 · (log‘((𝑚 + 1) / 𝑚))) − (log‘((𝑧 / 𝑚) + 1)))) = (𝑚 ∈ ℕ ↦ ((𝑧 · (log‘((𝑚 + 1) / 𝑚))) − (log‘((𝑧 / 𝑚) + 1))))
21		ovex 6678	. . . . . . . . 9 ⊢ ((𝑧 · (log‘((𝑛 + 1) / 𝑛))) − (log‘((𝑧 / 𝑛) + 1))) ∈ V
22	19, 20, 21	fvmpt 6282	. . . . . . . 8 ⊢ (𝑛 ∈ ℕ → ((𝑚 ∈ ℕ ↦ ((𝑧 · (log‘((𝑚 + 1) / 𝑚))) − (log‘((𝑧 / 𝑚) + 1))))‘𝑛) = ((𝑧 · (log‘((𝑛 + 1) / 𝑛))) − (log‘((𝑧 / 𝑛) + 1))))
23	22	adantl 482	. . . . . . 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))))
24	4	eldifad 3586	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑈) → 𝑧 ∈ ℂ)
25	24	adantr 481	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑈) ∧ 𝑛 ∈ ℕ) → 𝑧 ∈ ℂ)
26		simpr 477	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑈) ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ ℕ)
27	26	peano2nnd 11037	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑈) ∧ 𝑛 ∈ ℕ) → (𝑛 + 1) ∈ ℕ)
28	27	nnrpd 11870	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑈) ∧ 𝑛 ∈ ℕ) → (𝑛 + 1) ∈ ℝ⁺)
29	26	nnrpd 11870	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑈) ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ ℝ⁺)
30	28, 29	rpdivcld 11889	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑈) ∧ 𝑛 ∈ ℕ) → ((𝑛 + 1) / 𝑛) ∈ ℝ⁺)
31	30	relogcld 24369	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑈) ∧ 𝑛 ∈ ℕ) → (log‘((𝑛 + 1) / 𝑛)) ∈ ℝ)
32	31	recnd 10068	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑈) ∧ 𝑛 ∈ ℕ) → (log‘((𝑛 + 1) / 𝑛)) ∈ ℂ)
33	25, 32	mulcld 10060	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑈) ∧ 𝑛 ∈ ℕ) → (𝑧 · (log‘((𝑛 + 1) / 𝑛))) ∈ ℂ)
34	26	nncnd 11036	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑈) ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ ℂ)
35	26	nnne0d 11065	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑈) ∧ 𝑛 ∈ ℕ) → 𝑛 ≠ 0)
36	25, 34, 35	divcld 10801	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑈) ∧ 𝑛 ∈ ℕ) → (𝑧 / 𝑛) ∈ ℂ)
37		1cnd 10056	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑈) ∧ 𝑛 ∈ ℕ) → 1 ∈ ℂ)
38	36, 37	addcld 10059	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑈) ∧ 𝑛 ∈ ℕ) → ((𝑧 / 𝑛) + 1) ∈ ℂ)
39	4	adantr 481	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑈) ∧ 𝑛 ∈ ℕ) → 𝑧 ∈ (ℂ ∖ (ℤ ∖ ℕ)))
40	39, 26	dmgmdivn0 24754	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑈) ∧ 𝑛 ∈ ℕ) → ((𝑧 / 𝑛) + 1) ≠ 0)
41	38, 40	logcld 24317	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑈) ∧ 𝑛 ∈ ℕ) → (log‘((𝑧 / 𝑛) + 1)) ∈ ℂ)
42	33, 41	subcld 10392	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑈) ∧ 𝑛 ∈ ℕ) → ((𝑧 · (log‘((𝑛 + 1) / 𝑛))) − (log‘((𝑧 / 𝑛) + 1))) ∈ ℂ)
43		1z 11407	. . . . . . . . . . . 12 ⊢ 1 ∈ ℤ
44		seqfn 12813	. . . . . . . . . . . 12 ⊢ (1 ∈ ℤ → seq1( ∘_𝑓 + , 𝐺) Fn (ℤ_≥‘1))
45	43, 44	ax-mp 5	. . . . . . . . . . 11 ⊢ seq1( ∘_𝑓 + , 𝐺) Fn (ℤ_≥‘1)
46	9	fneq2i 5986	. . . . . . . . . . 11 ⊢ (seq1( ∘_𝑓 + , 𝐺) Fn ℕ ↔ seq1( ∘_𝑓 + , 𝐺) Fn (ℤ_≥‘1))
47	45, 46	mpbir 221	. . . . . . . . . 10 ⊢ seq1( ∘_𝑓 + , 𝐺) Fn ℕ
48		lgamgulm.g	. . . . . . . . . . . 12 ⊢ 𝐺 = (𝑚 ∈ ℕ ↦ (𝑧 ∈ 𝑈 ↦ ((𝑧 · (log‘((𝑚 + 1) / 𝑚))) − (log‘((𝑧 / 𝑚) + 1)))))
49	1, 2, 48	lgamgulm 24761	. . . . . . . . . . 11 ⊢ (𝜑 → seq1( ∘_𝑓 + , 𝐺) ∈ dom (⇝_𝑢‘𝑈))
50		ulmdm 24147	. . . . . . . . . . 11 ⊢ (seq1( ∘_𝑓 + , 𝐺) ∈ dom (⇝_𝑢‘𝑈) ↔ seq1( ∘_𝑓 + , 𝐺)(⇝_𝑢‘𝑈)((⇝_𝑢‘𝑈)‘seq1( ∘_𝑓 + , 𝐺)))
51	49, 50	sylib 208	. . . . . . . . . 10 ⊢ (𝜑 → seq1( ∘_𝑓 + , 𝐺)(⇝_𝑢‘𝑈)((⇝_𝑢‘𝑈)‘seq1( ∘_𝑓 + , 𝐺)))
52		ulmf2 24138	. . . . . . . . . 10 ⊢ ((seq1( ∘_𝑓 + , 𝐺) Fn ℕ ∧ seq1( ∘_𝑓 + , 𝐺)(⇝_𝑢‘𝑈)((⇝_𝑢‘𝑈)‘seq1( ∘_𝑓 + , 𝐺))) → seq1( ∘_𝑓 + , 𝐺):ℕ⟶(ℂ ↑_𝑚 𝑈))
53	47, 51, 52	sylancr 695	. . . . . . . . 9 ⊢ (𝜑 → seq1( ∘_𝑓 + , 𝐺):ℕ⟶(ℂ ↑_𝑚 𝑈))
54	53	adantr 481	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑈) → seq1( ∘_𝑓 + , 𝐺):ℕ⟶(ℂ ↑_𝑚 𝑈))
55		simpr 477	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑈) → 𝑧 ∈ 𝑈)
56		seqex 12803	. . . . . . . . 9 ⊢ seq1( + , (𝑚 ∈ ℕ ↦ ((𝑧 · (log‘((𝑚 + 1) / 𝑚))) − (log‘((𝑧 / 𝑚) + 1))))) ∈ V
57	56	a1i 11	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑈) → seq1( + , (𝑚 ∈ ℕ ↦ ((𝑧 · (log‘((𝑚 + 1) / 𝑚))) − (log‘((𝑧 / 𝑚) + 1))))) ∈ V)
58	48	a1i 11	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑈) ∧ 𝑛 ∈ ℕ) → 𝐺 = (𝑚 ∈ ℕ ↦ (𝑧 ∈ 𝑈 ↦ ((𝑧 · (log‘((𝑚 + 1) / 𝑚))) − (log‘((𝑧 / 𝑚) + 1))))))
59	58	seqeq3d 12809	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑈) ∧ 𝑛 ∈ ℕ) → seq1( ∘_𝑓 + , 𝐺) = seq1( ∘_𝑓 + , (𝑚 ∈ ℕ ↦ (𝑧 ∈ 𝑈 ↦ ((𝑧 · (log‘((𝑚 + 1) / 𝑚))) − (log‘((𝑧 / 𝑚) + 1)))))))
60	59	fveq1d 6193	. . . . . . . . . . 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))
61		cnex 10017	. . . . . . . . . . . . . . 15 ⊢ ℂ ∈ V
62	2, 61	rabex2 4815	. . . . . . . . . . . . . 14 ⊢ 𝑈 ∈ V
63	62	a1i 11	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝑈 ∈ V)
64		simpr 477	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ ℕ)
65	64, 9	syl6eleq 2711	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ (ℤ_≥‘1))
66		fz1ssnn 12372	. . . . . . . . . . . . . 14 ⊢ (1...𝑛) ⊆ ℕ
67	66	a1i 11	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (1...𝑛) ⊆ ℕ)
68		ovexd 6680	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ (𝑚 ∈ ℕ ∧ 𝑧 ∈ 𝑈)) → ((𝑧 · (log‘((𝑚 + 1) / 𝑚))) − (log‘((𝑧 / 𝑚) + 1))) ∈ V)
69	63, 65, 67, 68	seqof2 12859	. . . . . . . . . . . 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)))
70	69	adantlr 751	. . . . . . . . . . 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)))
71	60, 70	eqtrd 2656	. . . . . . . . . 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)))
72	71	fveq1d 6193	. . . . . . . . 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))
73	55	adantr 481	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑈) ∧ 𝑛 ∈ ℕ) → 𝑧 ∈ 𝑈)
74		fvex 6201	. . . . . . . . . 10 ⊢ (seq1( + , (𝑚 ∈ ℕ ↦ ((𝑧 · (log‘((𝑚 + 1) / 𝑚))) − (log‘((𝑧 / 𝑚) + 1)))))‘𝑛) ∈ V
75		eqid 2622	. . . . . . . . . . 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))
76	75	fvmpt2 6291	. . . . . . . . . 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))
77	73, 74, 76	sylancl 694	. . . . . . . . 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))
78	72, 77	eqtrd 2656	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑈) ∧ 𝑛 ∈ ℕ) → ((seq1( ∘_𝑓 + , 𝐺)‘𝑛)‘𝑧) = (seq1( + , (𝑚 ∈ ℕ ↦ ((𝑧 · (log‘((𝑚 + 1) / 𝑚))) − (log‘((𝑧 / 𝑚) + 1)))))‘𝑛))
79	51	adantr 481	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑈) → seq1( ∘_𝑓 + , 𝐺)(⇝_𝑢‘𝑈)((⇝_𝑢‘𝑈)‘seq1( ∘_𝑓 + , 𝐺)))
80	9, 10, 54, 55, 57, 78, 79	ulmclm 24141	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑈) → seq1( + , (𝑚 ∈ ℕ ↦ ((𝑧 · (log‘((𝑚 + 1) / 𝑚))) − (log‘((𝑧 / 𝑚) + 1))))) ⇝ (((⇝_𝑢‘𝑈)‘seq1( ∘_𝑓 + , 𝐺))‘𝑧))
81	9, 10, 23, 42, 80	isumclim 14488	. . . . . 6 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑈) → Σ𝑛 ∈ ℕ ((𝑧 · (log‘((𝑛 + 1) / 𝑛))) − (log‘((𝑧 / 𝑛) + 1))) = (((⇝_𝑢‘𝑈)‘seq1( ∘_𝑓 + , 𝐺))‘𝑧))
82		ulmcl 24135	. . . . . . . 8 ⊢ (seq1( ∘_𝑓 + , 𝐺)(⇝_𝑢‘𝑈)((⇝_𝑢‘𝑈)‘seq1( ∘_𝑓 + , 𝐺)) → ((⇝_𝑢‘𝑈)‘seq1( ∘_𝑓 + , 𝐺)):𝑈⟶ℂ)
83	51, 82	syl 17	. . . . . . 7 ⊢ (𝜑 → ((⇝_𝑢‘𝑈)‘seq1( ∘_𝑓 + , 𝐺)):𝑈⟶ℂ)
84	83	ffvelrnda 6359	. . . . . 6 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑈) → (((⇝_𝑢‘𝑈)‘seq1( ∘_𝑓 + , 𝐺))‘𝑧) ∈ ℂ)
85	81, 84	eqeltrd 2701	. . . . 5 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑈) → Σ𝑛 ∈ ℕ ((𝑧 · (log‘((𝑛 + 1) / 𝑛))) − (log‘((𝑧 / 𝑛) + 1))) ∈ ℂ)
86	4	dmgmn0 24752	. . . . . 6 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑈) → 𝑧 ≠ 0)
87	24, 86	logcld 24317	. . . . 5 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑈) → (log‘𝑧) ∈ ℂ)
88	85, 87	subcld 10392	. . . 4 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑈) → (Σ𝑛 ∈ ℕ ((𝑧 · (log‘((𝑛 + 1) / 𝑛))) − (log‘((𝑧 / 𝑛) + 1))) − (log‘𝑧)) ∈ ℂ)
89	8, 88	eqeltrd 2701	. . 3 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑈) → (log Γ‘𝑧) ∈ ℂ)
90	89	ralrimiva 2966	. 2 ⊢ (𝜑 → ∀𝑧 ∈ 𝑈 (log Γ‘𝑧) ∈ ℂ)
91		ffn 6045	. . . . . 6 ⊢ (((⇝_𝑢‘𝑈)‘seq1( ∘_𝑓 + , 𝐺)):𝑈⟶ℂ → ((⇝_𝑢‘𝑈)‘seq1( ∘_𝑓 + , 𝐺)) Fn 𝑈)
92	51, 82, 91	3syl 18	. . . . 5 ⊢ (𝜑 → ((⇝_𝑢‘𝑈)‘seq1( ∘_𝑓 + , 𝐺)) Fn 𝑈)
93		nfcv 2764	. . . . . . 7 ⊢ Ⅎ𝑧(⇝_𝑢‘𝑈)
94		nfcv 2764	. . . . . . . 8 ⊢ Ⅎ𝑧1
95		nfcv 2764	. . . . . . . 8 ⊢ Ⅎ𝑧 ∘_𝑓 +
96		nfcv 2764	. . . . . . . . . 10 ⊢ Ⅎ𝑧ℕ
97		nfmpt1 4747	. . . . . . . . . 10 ⊢ Ⅎ𝑧(𝑧 ∈ 𝑈 ↦ ((𝑧 · (log‘((𝑚 + 1) / 𝑚))) − (log‘((𝑧 / 𝑚) + 1))))
98	96, 97	nfmpt 4746	. . . . . . . . 9 ⊢ Ⅎ𝑧(𝑚 ∈ ℕ ↦ (𝑧 ∈ 𝑈 ↦ ((𝑧 · (log‘((𝑚 + 1) / 𝑚))) − (log‘((𝑧 / 𝑚) + 1)))))
99	48, 98	nfcxfr 2762	. . . . . . . 8 ⊢ Ⅎ𝑧𝐺
100	94, 95, 99	nfseq 12811	. . . . . . 7 ⊢ Ⅎ𝑧seq1( ∘_𝑓 + , 𝐺)
101	93, 100	nffv 6198	. . . . . 6 ⊢ Ⅎ𝑧((⇝_𝑢‘𝑈)‘seq1( ∘_𝑓 + , 𝐺))
102	101	dffn5f 6252	. . . . 5 ⊢ (((⇝_𝑢‘𝑈)‘seq1( ∘_𝑓 + , 𝐺)) Fn 𝑈 ↔ ((⇝_𝑢‘𝑈)‘seq1( ∘_𝑓 + , 𝐺)) = (𝑧 ∈ 𝑈 ↦ (((⇝_𝑢‘𝑈)‘seq1( ∘_𝑓 + , 𝐺))‘𝑧)))
103	92, 102	sylib 208	. . . 4 ⊢ (𝜑 → ((⇝_𝑢‘𝑈)‘seq1( ∘_𝑓 + , 𝐺)) = (𝑧 ∈ 𝑈 ↦ (((⇝_𝑢‘𝑈)‘seq1( ∘_𝑓 + , 𝐺))‘𝑧)))
104	8	oveq1d 6665	. . . . . 6 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑈) → ((log Γ‘𝑧) + (log‘𝑧)) = ((Σ𝑛 ∈ ℕ ((𝑧 · (log‘((𝑛 + 1) / 𝑛))) − (log‘((𝑧 / 𝑛) + 1))) − (log‘𝑧)) + (log‘𝑧)))
105	85, 87	npcand 10396	. . . . . 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))))
106	104, 105, 81	3eqtrrd 2661	. . . . 5 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑈) → (((⇝_𝑢‘𝑈)‘seq1( ∘_𝑓 + , 𝐺))‘𝑧) = ((log Γ‘𝑧) + (log‘𝑧)))
107	106	mpteq2dva 4744	. . . 4 ⊢ (𝜑 → (𝑧 ∈ 𝑈 ↦ (((⇝_𝑢‘𝑈)‘seq1( ∘_𝑓 + , 𝐺))‘𝑧)) = (𝑧 ∈ 𝑈 ↦ ((log Γ‘𝑧) + (log‘𝑧))))
108	103, 107	eqtrd 2656	. . 3 ⊢ (𝜑 → ((⇝_𝑢‘𝑈)‘seq1( ∘_𝑓 + , 𝐺)) = (𝑧 ∈ 𝑈 ↦ ((log Γ‘𝑧) + (log‘𝑧))))
109	51, 108	breqtrd 4679	. 2 ⊢ (𝜑 → seq1( ∘_𝑓 + , 𝐺)(⇝_𝑢‘𝑈)(𝑧 ∈ 𝑈 ↦ ((log Γ‘𝑧) + (log‘𝑧))))
110	90, 109	jca 554	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 384 = wceq 1483 ∈ wcel 1990 ∀wral 2912 {crab 2916 Vcvv 3200 ∖ cdif 3571 ⊆ wss 3574 class class class wbr 4653 ↦ cmpt 4729 dom cdm 5114 Fn wfn 5883 ⟶wf 5884 ‘cfv 5888 (class class class)co 6650 ∘_𝑓 cof 6895 ↑_𝑚 cmap 7857 ℂcc 9934 1c1 9937 + caddc 9939 · cmul 9941 ≤ cle 10075 − cmin 10266 / cdiv 10684 ℕcn 11020 ℕ₀cn0 11292 ℤcz 11377 ℤ_≥cuz 11687 ...cfz 12326 seqcseq 12801 abscabs 13974 Σcsu 14416 ⇝_𝑢culm 24130 logclog 24301 log Γclgam 24742

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-inf2 8538 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 ax-pre-sup 10014 ax-addf 10015 ax-mulf 10016

This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-fal 1489 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-iin 4523 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-se 5074 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-isom 5897 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-of 6897 df-om 7066 df-1st 7168 df-2nd 7169 df-supp 7296 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-2o 7561 df-oadd 7564 df-er 7742 df-map 7859 df-pm 7860 df-ixp 7909 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-fsupp 8276 df-fi 8317 df-sup 8348 df-inf 8349 df-oi 8415 df-card 8765 df-cda 8990 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-div 10685 df-nn 11021 df-2 11079 df-3 11080 df-4 11081 df-5 11082 df-6 11083 df-7 11084 df-8 11085 df-9 11086 df-n0 11293 df-z 11378 df-dec 11494 df-uz 11688 df-q 11789 df-rp 11833 df-xneg 11946 df-xadd 11947 df-xmul 11948 df-ioo 12179 df-ioc 12180 df-ico 12181 df-icc 12182 df-fz 12327 df-fzo 12466 df-fl 12593 df-mod 12669 df-seq 12802 df-exp 12861 df-fac 13061 df-bc 13090 df-hash 13118 df-shft 13807 df-cj 13839 df-re 13840 df-im 13841 df-sqrt 13975 df-abs 13976 df-limsup 14202 df-clim 14219 df-rlim 14220 df-sum 14417 df-ef 14798 df-sin 14800 df-cos 14801 df-tan 14802 df-pi 14803 df-struct 15859 df-ndx 15860 df-slot 15861 df-base 15863 df-sets 15864 df-ress 15865 df-plusg 15954 df-mulr 15955 df-starv 15956 df-sca 15957 df-vsca 15958 df-ip 15959 df-tset 15960 df-ple 15961 df-ds 15964 df-unif 15965 df-hom 15966 df-cco 15967 df-rest 16083 df-topn 16084 df-0g 16102 df-gsum 16103 df-topgen 16104 df-pt 16105 df-prds 16108 df-xrs 16162 df-qtop 16167 df-imas 16168 df-xps 16170 df-mre 16246 df-mrc 16247 df-acs 16249 df-mgm 17242 df-sgrp 17284 df-mnd 17295 df-submnd 17336 df-mulg 17541 df-cntz 17750 df-cmn 18195 df-psmet 19738 df-xmet 19739 df-met 19740 df-bl 19741 df-mopn 19742 df-fbas 19743 df-fg 19744 df-cnfld 19747 df-top 20699 df-topon 20716 df-topsp 20737 df-bases 20750 df-cld 20823 df-ntr 20824 df-cls 20825 df-nei 20902 df-lp 20940 df-perf 20941 df-cn 21031 df-cnp 21032 df-haus 21119 df-cmp 21190 df-tx 21365 df-hmeo 21558 df-fil 21650 df-fm 21742 df-flim 21743 df-flf 21744 df-xms 22125 df-ms 22126 df-tms 22127 df-cncf 22681 df-limc 23630 df-dv 23631 df-ulm 24131 df-log 24303 df-cxp 24304 df-lgam 24745

This theorem is referenced by: lgambdd 24763 lgamcvglem 24766

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