lgamgulm2 - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > lgamgulm2		Structured version Visualization version GIF version

Theorem lgamgulm2 27079

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 27072	. . . . . 6 ⊢ (𝜑 → 𝑈 ⊆ (ℂ ∖ (ℤ ∖ ℕ)))
4	3	sselda 3983	. . . . 5 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑈) → 𝑧 ∈ (ℂ ∖ (ℤ ∖ ℕ)))
5		ovex 7464	. . . . 5 ⊢ (Σ𝑛 ∈ ℕ ((𝑧 · (log‘((𝑛 + 1) / 𝑛))) − (log‘((𝑧 / 𝑛) + 1))) − (log‘𝑧)) ∈ V
6		df-lgam 27062	. . . . . 6 ⊢ log Γ = (𝑧 ∈ (ℂ ∖ (ℤ ∖ ℕ)) ↦ (Σ𝑛 ∈ ℕ ((𝑧 · (log‘((𝑛 + 1) / 𝑛))) − (log‘((𝑧 / 𝑛) + 1))) − (log‘𝑧)))
7	6	fvmpt2 7027	. . . . 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 586	. . . 4 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑈) → (log Γ‘𝑧) = (Σ𝑛 ∈ ℕ ((𝑧 · (log‘((𝑛 + 1) / 𝑛))) − (log‘((𝑧 / 𝑛) + 1))) − (log‘𝑧)))
9		nnuz 12921	. . . . . . 7 ⊢ ℕ = (ℤ_≥‘1)
10		1zzd 12648	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑈) → 1 ∈ ℤ)
11		oveq1 7438	. . . . . . . . . . . . 13 ⊢ (𝑚 = 𝑛 → (𝑚 + 1) = (𝑛 + 1))
12		id 22	. . . . . . . . . . . . 13 ⊢ (𝑚 = 𝑛 → 𝑚 = 𝑛)
13	11, 12	oveq12d 7449	. . . . . . . . . . . 12 ⊢ (𝑚 = 𝑛 → ((𝑚 + 1) / 𝑚) = ((𝑛 + 1) / 𝑛))
14	13	fveq2d 6910	. . . . . . . . . . 11 ⊢ (𝑚 = 𝑛 → (log‘((𝑚 + 1) / 𝑚)) = (log‘((𝑛 + 1) / 𝑛)))
15	14	oveq2d 7447	. . . . . . . . . 10 ⊢ (𝑚 = 𝑛 → (𝑧 · (log‘((𝑚 + 1) / 𝑚))) = (𝑧 · (log‘((𝑛 + 1) / 𝑛))))
16		oveq2 7439	. . . . . . . . . . 11 ⊢ (𝑚 = 𝑛 → (𝑧 / 𝑚) = (𝑧 / 𝑛))
17	16	fvoveq1d 7453	. . . . . . . . . 10 ⊢ (𝑚 = 𝑛 → (log‘((𝑧 / 𝑚) + 1)) = (log‘((𝑧 / 𝑛) + 1)))
18	15, 17	oveq12d 7449	. . . . . . . . 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 7464	. . . . . . . . 9 ⊢ ((𝑧 · (log‘((𝑛 + 1) / 𝑛))) − (log‘((𝑧 / 𝑛) + 1))) ∈ V
21	18, 19, 20	fvmpt 7016	. . . . . . . 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 3963	. . . . . . . . . 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 12283	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑈) ∧ 𝑛 ∈ ℕ) → (𝑛 + 1) ∈ ℕ)
27	26	nnrpd 13075	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑈) ∧ 𝑛 ∈ ℕ) → (𝑛 + 1) ∈ ℝ⁺)
28	25	nnrpd 13075	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑈) ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ ℝ⁺)
29	27, 28	rpdivcld 13094	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑈) ∧ 𝑛 ∈ ℕ) → ((𝑛 + 1) / 𝑛) ∈ ℝ⁺)
30	29	relogcld 26665	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑈) ∧ 𝑛 ∈ ℕ) → (log‘((𝑛 + 1) / 𝑛)) ∈ ℝ)
31	30	recnd 11289	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑈) ∧ 𝑛 ∈ ℕ) → (log‘((𝑛 + 1) / 𝑛)) ∈ ℂ)
32	24, 31	mulcld 11281	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑈) ∧ 𝑛 ∈ ℕ) → (𝑧 · (log‘((𝑛 + 1) / 𝑛))) ∈ ℂ)
33	25	nncnd 12282	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑈) ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ ℂ)
34	25	nnne0d 12316	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑈) ∧ 𝑛 ∈ ℕ) → 𝑛 ≠ 0)
35	24, 33, 34	divcld 12043	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑈) ∧ 𝑛 ∈ ℕ) → (𝑧 / 𝑛) ∈ ℂ)
36		1cnd 11256	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑈) ∧ 𝑛 ∈ ℕ) → 1 ∈ ℂ)
37	35, 36	addcld 11280	. . . . . . . . 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 27071	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑈) ∧ 𝑛 ∈ ℕ) → ((𝑧 / 𝑛) + 1) ≠ 0)
40	37, 39	logcld 26612	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑈) ∧ 𝑛 ∈ ℕ) → (log‘((𝑧 / 𝑛) + 1)) ∈ ℂ)
41	32, 40	subcld 11620	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑈) ∧ 𝑛 ∈ ℕ) → ((𝑧 · (log‘((𝑛 + 1) / 𝑛))) − (log‘((𝑧 / 𝑛) + 1))) ∈ ℂ)
42		1z 12647	. . . . . . . . . . . 12 ⊢ 1 ∈ ℤ
43		seqfn 14054	. . . . . . . . . . . 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 6666	. . . . . . . . . . 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 27078	. . . . . . . . . . 11 ⊢ (𝜑 → seq1( ∘_f + , 𝐺) ∈ dom (⇝_𝑢‘𝑈))
49		ulmdm 26436	. . . . . . . . . . 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 26427	. . . . . . . . . 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 587	. . . . . . . . 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 14044	. . . . . . . . 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 14050	. . . . . . . . . . . 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 6908	. . . . . . . . . . 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 11236	. . . . . . . . . . . . . . 15 ⊢ ℂ ∈ V
61	2, 60	rabex2 5341	. . . . . . . . . . . . . 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 2851	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ (ℤ_≥‘1))
65		fz1ssnn 13595	. . . . . . . . . . . . . 14 ⊢ (1...𝑛) ⊆ ℕ
66	65	a1i 11	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (1...𝑛) ⊆ ℕ)
67		ovexd 7466	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ (𝑚 ∈ ℕ ∧ 𝑧 ∈ 𝑈)) → ((𝑧 · (log‘((𝑚 + 1) / 𝑚))) − (log‘((𝑧 / 𝑚) + 1))) ∈ V)
68	62, 64, 66, 67	seqof2 14101	. . . . . . . . . . . 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 715	. . . . . . . . . . 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 2777	. . . . . . . . . 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 6908	. . . . . . . . 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 6919	. . . . . . . . . 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 7027	. . . . . . . . . 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 586	. . . . . . . . 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 2777	. . . . . . . 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 26430	. . . . . . 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 15793	. . . . . 6 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑈) → Σ𝑛 ∈ ℕ ((𝑧 · (log‘((𝑛 + 1) / 𝑛))) − (log‘((𝑧 / 𝑛) + 1))) = (((⇝_𝑢‘𝑈)‘seq1( ∘_f + , 𝐺))‘𝑧))
81		ulmcl 26424	. . . . . . . 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 7104	. . . . . 6 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑈) → (((⇝_𝑢‘𝑈)‘seq1( ∘_f + , 𝐺))‘𝑧) ∈ ℂ)
84	80, 83	eqeltrd 2841	. . . . 5 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑈) → Σ𝑛 ∈ ℕ ((𝑧 · (log‘((𝑛 + 1) / 𝑛))) − (log‘((𝑧 / 𝑛) + 1))) ∈ ℂ)
85	4	dmgmn0 27069	. . . . . 6 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑈) → 𝑧 ≠ 0)
86	23, 85	logcld 26612	. . . . 5 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑈) → (log‘𝑧) ∈ ℂ)
87	84, 86	subcld 11620	. . . 4 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑈) → (Σ𝑛 ∈ ℕ ((𝑧 · (log‘((𝑛 + 1) / 𝑛))) − (log‘((𝑧 / 𝑛) + 1))) − (log‘𝑧)) ∈ ℂ)
88	8, 87	eqeltrd 2841	. . 3 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑈) → (log Γ‘𝑧) ∈ ℂ)
89	88	ralrimiva 3146	. 2 ⊢ (𝜑 → ∀𝑧 ∈ 𝑈 (log Γ‘𝑧) ∈ ℂ)
90		ffn 6736	. . . . . 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 2905	. . . . . . 7 ⊢ Ⅎ𝑧(⇝_𝑢‘𝑈)
93		nfcv 2905	. . . . . . . 8 ⊢ Ⅎ𝑧1
94		nfcv 2905	. . . . . . . 8 ⊢ Ⅎ𝑧 ∘_f +
95		nfcv 2905	. . . . . . . . . 10 ⊢ Ⅎ𝑧ℕ
96		nfmpt1 5250	. . . . . . . . . 10 ⊢ Ⅎ𝑧(𝑧 ∈ 𝑈 ↦ ((𝑧 · (log‘((𝑚 + 1) / 𝑚))) − (log‘((𝑧 / 𝑚) + 1))))
97	95, 96	nfmpt 5249	. . . . . . . . 9 ⊢ Ⅎ𝑧(𝑚 ∈ ℕ ↦ (𝑧 ∈ 𝑈 ↦ ((𝑧 · (log‘((𝑚 + 1) / 𝑚))) − (log‘((𝑧 / 𝑚) + 1)))))
98	47, 97	nfcxfr 2903	. . . . . . . 8 ⊢ Ⅎ𝑧𝐺
99	93, 94, 98	nfseq 14052	. . . . . . 7 ⊢ Ⅎ𝑧seq1( ∘_f + , 𝐺)
100	92, 99	nffv 6916	. . . . . 6 ⊢ Ⅎ𝑧((⇝_𝑢‘𝑈)‘seq1( ∘_f + , 𝐺))
101	100	dffn5f 6980	. . . . 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 7446	. . . . . 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 11624	. . . . . 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 2782	. . . . 5 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑈) → (((⇝_𝑢‘𝑈)‘seq1( ∘_f + , 𝐺))‘𝑧) = ((log Γ‘𝑧) + (log‘𝑧)))
106	105	mpteq2dva 5242	. . . 4 ⊢ (𝜑 → (𝑧 ∈ 𝑈 ↦ (((⇝_𝑢‘𝑈)‘seq1( ∘_f + , 𝐺))‘𝑧)) = (𝑧 ∈ 𝑈 ↦ ((log Γ‘𝑧) + (log‘𝑧))))
107	102, 106	eqtrd 2777	. . 3 ⊢ (𝜑 → ((⇝_𝑢‘𝑈)‘seq1( ∘_f + , 𝐺)) = (𝑧 ∈ 𝑈 ↦ ((log Γ‘𝑧) + (log‘𝑧))))
108	50, 107	breqtrd 5169	. 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 1540 ∈ wcel 2108 ∀wral 3061 {crab 3436 Vcvv 3480 ∖ cdif 3948 ⊆ wss 3951 class class class wbr 5143 ↦ cmpt 5225 dom cdm 5685 Fn wfn 6556 ⟶wf 6557 ‘cfv 6561 (class class class)co 7431 ∘_f cof 7695 ↑_m cmap 8866 ℂcc 11153 1c1 11156 + caddc 11158 · cmul 11160 ≤ cle 11296 − cmin 11492 / cdiv 11920 ℕcn 12266 ℕ₀cn0 12526 ℤcz 12613 ℤ_≥cuz 12878 ...cfz 13547 seqcseq 14042 abscabs 15273 Σcsu 15722 ⇝_𝑢culm 26419 logclog 26596 log Γclgam 27059

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-inf2 9681 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 ax-pre-sup 11233 ax-addf 11234

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-tp 4631 df-op 4633 df-uni 4908 df-int 4947 df-iun 4993 df-iin 4994 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-se 5638 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-isom 6570 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-of 7697 df-om 7888 df-1st 8014 df-2nd 8015 df-supp 8186 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-2o 8507 df-oadd 8510 df-er 8745 df-map 8868 df-pm 8869 df-ixp 8938 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-fsupp 9402 df-fi 9451 df-sup 9482 df-inf 9483 df-oi 9550 df-dju 9941 df-card 9979 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-div 11921 df-nn 12267 df-2 12329 df-3 12330 df-4 12331 df-5 12332 df-6 12333 df-7 12334 df-8 12335 df-9 12336 df-n0 12527 df-z 12614 df-dec 12734 df-uz 12879 df-q 12991 df-rp 13035 df-xneg 13154 df-xadd 13155 df-xmul 13156 df-ioo 13391 df-ioc 13392 df-ico 13393 df-icc 13394 df-fz 13548 df-fzo 13695 df-fl 13832 df-mod 13910 df-seq 14043 df-exp 14103 df-fac 14313 df-bc 14342 df-hash 14370 df-shft 15106 df-cj 15138 df-re 15139 df-im 15140 df-sqrt 15274 df-abs 15275 df-limsup 15507 df-clim 15524 df-rlim 15525 df-sum 15723 df-ef 16103 df-sin 16105 df-cos 16106 df-tan 16107 df-pi 16108 df-struct 17184 df-sets 17201 df-slot 17219 df-ndx 17231 df-base 17248 df-ress 17275 df-plusg 17310 df-mulr 17311 df-starv 17312 df-sca 17313 df-vsca 17314 df-ip 17315 df-tset 17316 df-ple 17317 df-ds 17319 df-unif 17320 df-hom 17321 df-cco 17322 df-rest 17467 df-topn 17468 df-0g 17486 df-gsum 17487 df-topgen 17488 df-pt 17489 df-prds 17492 df-xrs 17547 df-qtop 17552 df-imas 17553 df-xps 17555 df-mre 17629 df-mrc 17630 df-acs 17632 df-mgm 18653 df-sgrp 18732 df-mnd 18748 df-submnd 18797 df-mulg 19086 df-cntz 19335 df-cmn 19800 df-psmet 21356 df-xmet 21357 df-met 21358 df-bl 21359 df-mopn 21360 df-fbas 21361 df-fg 21362 df-cnfld 21365 df-top 22900 df-topon 22917 df-topsp 22939 df-bases 22953 df-cld 23027 df-ntr 23028 df-cls 23029 df-nei 23106 df-lp 23144 df-perf 23145 df-cn 23235 df-cnp 23236 df-haus 23323 df-cmp 23395 df-tx 23570 df-hmeo 23763 df-fil 23854 df-fm 23946 df-flim 23947 df-flf 23948 df-xms 24330 df-ms 24331 df-tms 24332 df-cncf 24904 df-limc 25901 df-dv 25902 df-ulm 26420 df-log 26598 df-cxp 26599 df-lgam 27062

This theorem is referenced by: lgambdd 27080 lgamcvglem 27083

W3C validator