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Theorem lgamcvglem 26189

Description: Lemma for lgamf 26191 and lgamcvg 26203. (Contributed by Mario Carneiro, 8-Jul-2017.)

**Hypotheses**
Ref	Expression
lgamucov.u	⊢ 𝑈 = {𝑥 ∈ ℂ ∣ ((abs‘𝑥) ≤ 𝑟 ∧ ∀𝑘 ∈ ℕ₀ (1 / 𝑟) ≤ (abs‘(𝑥 + 𝑘)))}
lgamucov.a	⊢ (𝜑 → 𝐴 ∈ (ℂ ∖ (ℤ ∖ ℕ)))
lgamcvglem.g	⊢ 𝐺 = (𝑚 ∈ ℕ ↦ ((𝐴 · (log‘((𝑚 + 1) / 𝑚))) − (log‘((𝐴 / 𝑚) + 1))))

**Assertion**
Ref	Expression
lgamcvglem	⊢ (𝜑 → ((log Γ‘𝐴) ∈ ℂ ∧ seq1( + , 𝐺) ⇝ ((log Γ‘𝐴) + (log‘𝐴))))

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

Proof of Theorem lgamcvglem Dummy variables 𝑛 𝑡 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		lgamucov.u	. . 3 ⊢ 𝑈 = {𝑥 ∈ ℂ ∣ ((abs‘𝑥) ≤ 𝑟 ∧ ∀𝑘 ∈ ℕ₀ (1 / 𝑟) ≤ (abs‘(𝑥 + 𝑘)))}
2		lgamucov.a	. . 3 ⊢ (𝜑 → 𝐴 ∈ (ℂ ∖ (ℤ ∖ ℕ)))
3	1, 2	lgamucov2 26188	. 2 ⊢ (𝜑 → ∃𝑟 ∈ ℕ 𝐴 ∈ 𝑈)
4		fveq2 6774	. . . . 5 ⊢ (𝑧 = 𝐴 → (log Γ‘𝑧) = (log Γ‘𝐴))
5	4	eleq1d 2823	. . . 4 ⊢ (𝑧 = 𝐴 → ((log Γ‘𝑧) ∈ ℂ ↔ (log Γ‘𝐴) ∈ ℂ))
6		simprl 768	. . . . . 6 ⊢ ((𝜑 ∧ (𝑟 ∈ ℕ ∧ 𝐴 ∈ 𝑈)) → 𝑟 ∈ ℕ)
7		fveq2 6774	. . . . . . . . . 10 ⊢ (𝑥 = 𝑡 → (abs‘𝑥) = (abs‘𝑡))
8	7	breq1d 5084	. . . . . . . . 9 ⊢ (𝑥 = 𝑡 → ((abs‘𝑥) ≤ 𝑟 ↔ (abs‘𝑡) ≤ 𝑟))
9		fvoveq1 7298	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑡 → (abs‘(𝑥 + 𝑘)) = (abs‘(𝑡 + 𝑘)))
10	9	breq2d 5086	. . . . . . . . . 10 ⊢ (𝑥 = 𝑡 → ((1 / 𝑟) ≤ (abs‘(𝑥 + 𝑘)) ↔ (1 / 𝑟) ≤ (abs‘(𝑡 + 𝑘))))
11	10	ralbidv 3112	. . . . . . . . 9 ⊢ (𝑥 = 𝑡 → (∀𝑘 ∈ ℕ₀ (1 / 𝑟) ≤ (abs‘(𝑥 + 𝑘)) ↔ ∀𝑘 ∈ ℕ₀ (1 / 𝑟) ≤ (abs‘(𝑡 + 𝑘))))
12	8, 11	anbi12d 631	. . . . . . . 8 ⊢ (𝑥 = 𝑡 → (((abs‘𝑥) ≤ 𝑟 ∧ ∀𝑘 ∈ ℕ₀ (1 / 𝑟) ≤ (abs‘(𝑥 + 𝑘))) ↔ ((abs‘𝑡) ≤ 𝑟 ∧ ∀𝑘 ∈ ℕ₀ (1 / 𝑟) ≤ (abs‘(𝑡 + 𝑘)))))
13	12	cbvrabv 3426	. . . . . . 7 ⊢ {𝑥 ∈ ℂ ∣ ((abs‘𝑥) ≤ 𝑟 ∧ ∀𝑘 ∈ ℕ₀ (1 / 𝑟) ≤ (abs‘(𝑥 + 𝑘)))} = {𝑡 ∈ ℂ ∣ ((abs‘𝑡) ≤ 𝑟 ∧ ∀𝑘 ∈ ℕ₀ (1 / 𝑟) ≤ (abs‘(𝑡 + 𝑘)))}
14	1, 13	eqtri 2766	. . . . . 6 ⊢ 𝑈 = {𝑡 ∈ ℂ ∣ ((abs‘𝑡) ≤ 𝑟 ∧ ∀𝑘 ∈ ℕ₀ (1 / 𝑟) ≤ (abs‘(𝑡 + 𝑘)))}
15		eqid 2738	. . . . . 6 ⊢ (𝑚 ∈ ℕ ↦ (𝑧 ∈ 𝑈 ↦ ((𝑧 · (log‘((𝑚 + 1) / 𝑚))) − (log‘((𝑧 / 𝑚) + 1))))) = (𝑚 ∈ ℕ ↦ (𝑧 ∈ 𝑈 ↦ ((𝑧 · (log‘((𝑚 + 1) / 𝑚))) − (log‘((𝑧 / 𝑚) + 1)))))
16	6, 14, 15	lgamgulm2 26185	. . . . 5 ⊢ ((𝜑 ∧ (𝑟 ∈ ℕ ∧ 𝐴 ∈ 𝑈)) → (∀𝑧 ∈ 𝑈 (log Γ‘𝑧) ∈ ℂ ∧ seq1( ∘_f + , (𝑚 ∈ ℕ ↦ (𝑧 ∈ 𝑈 ↦ ((𝑧 · (log‘((𝑚 + 1) / 𝑚))) − (log‘((𝑧 / 𝑚) + 1))))))(⇝_𝑢‘𝑈)(𝑧 ∈ 𝑈 ↦ ((log Γ‘𝑧) + (log‘𝑧)))))
17	16	simpld 495	. . . 4 ⊢ ((𝜑 ∧ (𝑟 ∈ ℕ ∧ 𝐴 ∈ 𝑈)) → ∀𝑧 ∈ 𝑈 (log Γ‘𝑧) ∈ ℂ)
18		simprr 770	. . . 4 ⊢ ((𝜑 ∧ (𝑟 ∈ ℕ ∧ 𝐴 ∈ 𝑈)) → 𝐴 ∈ 𝑈)
19	5, 17, 18	rspcdva 3562	. . 3 ⊢ ((𝜑 ∧ (𝑟 ∈ ℕ ∧ 𝐴 ∈ 𝑈)) → (log Γ‘𝐴) ∈ ℂ)
20		nnuz 12621	. . . . 5 ⊢ ℕ = (ℤ_≥‘1)
21		1zzd 12351	. . . . 5 ⊢ ((𝜑 ∧ (𝑟 ∈ ℕ ∧ 𝐴 ∈ 𝑈)) → 1 ∈ ℤ)
22		1z 12350	. . . . . . . 8 ⊢ 1 ∈ ℤ
23		seqfn 13733	. . . . . . . 8 ⊢ (1 ∈ ℤ → seq1( ∘_f + , (𝑚 ∈ ℕ ↦ (𝑧 ∈ 𝑈 ↦ ((𝑧 · (log‘((𝑚 + 1) / 𝑚))) − (log‘((𝑧 / 𝑚) + 1)))))) Fn (ℤ_≥‘1))
24	22, 23	ax-mp 5	. . . . . . 7 ⊢ seq1( ∘_f + , (𝑚 ∈ ℕ ↦ (𝑧 ∈ 𝑈 ↦ ((𝑧 · (log‘((𝑚 + 1) / 𝑚))) − (log‘((𝑧 / 𝑚) + 1)))))) Fn (ℤ_≥‘1)
25	20	fneq2i 6531	. . . . . . 7 ⊢ (seq1( ∘_f + , (𝑚 ∈ ℕ ↦ (𝑧 ∈ 𝑈 ↦ ((𝑧 · (log‘((𝑚 + 1) / 𝑚))) − (log‘((𝑧 / 𝑚) + 1)))))) Fn ℕ ↔ seq1( ∘_f + , (𝑚 ∈ ℕ ↦ (𝑧 ∈ 𝑈 ↦ ((𝑧 · (log‘((𝑚 + 1) / 𝑚))) − (log‘((𝑧 / 𝑚) + 1)))))) Fn (ℤ_≥‘1))
26	24, 25	mpbir 230	. . . . . 6 ⊢ seq1( ∘_f + , (𝑚 ∈ ℕ ↦ (𝑧 ∈ 𝑈 ↦ ((𝑧 · (log‘((𝑚 + 1) / 𝑚))) − (log‘((𝑧 / 𝑚) + 1)))))) Fn ℕ
27	16	simprd 496	. . . . . 6 ⊢ ((𝜑 ∧ (𝑟 ∈ ℕ ∧ 𝐴 ∈ 𝑈)) → seq1( ∘_f + , (𝑚 ∈ ℕ ↦ (𝑧 ∈ 𝑈 ↦ ((𝑧 · (log‘((𝑚 + 1) / 𝑚))) − (log‘((𝑧 / 𝑚) + 1))))))(⇝_𝑢‘𝑈)(𝑧 ∈ 𝑈 ↦ ((log Γ‘𝑧) + (log‘𝑧))))
28		ulmf2 25543	. . . . . 6 ⊢ ((seq1( ∘_f + , (𝑚 ∈ ℕ ↦ (𝑧 ∈ 𝑈 ↦ ((𝑧 · (log‘((𝑚 + 1) / 𝑚))) − (log‘((𝑧 / 𝑚) + 1)))))) Fn ℕ ∧ seq1( ∘_f + , (𝑚 ∈ ℕ ↦ (𝑧 ∈ 𝑈 ↦ ((𝑧 · (log‘((𝑚 + 1) / 𝑚))) − (log‘((𝑧 / 𝑚) + 1))))))(⇝_𝑢‘𝑈)(𝑧 ∈ 𝑈 ↦ ((log Γ‘𝑧) + (log‘𝑧)))) → seq1( ∘_f + , (𝑚 ∈ ℕ ↦ (𝑧 ∈ 𝑈 ↦ ((𝑧 · (log‘((𝑚 + 1) / 𝑚))) − (log‘((𝑧 / 𝑚) + 1)))))):ℕ⟶(ℂ ↑_m 𝑈))
29	26, 27, 28	sylancr 587	. . . . 5 ⊢ ((𝜑 ∧ (𝑟 ∈ ℕ ∧ 𝐴 ∈ 𝑈)) → seq1( ∘_f + , (𝑚 ∈ ℕ ↦ (𝑧 ∈ 𝑈 ↦ ((𝑧 · (log‘((𝑚 + 1) / 𝑚))) − (log‘((𝑧 / 𝑚) + 1)))))):ℕ⟶(ℂ ↑_m 𝑈))
30		seqex 13723	. . . . . 6 ⊢ seq1( + , 𝐺) ∈ V
31	30	a1i 11	. . . . 5 ⊢ ((𝜑 ∧ (𝑟 ∈ ℕ ∧ 𝐴 ∈ 𝑈)) → seq1( + , 𝐺) ∈ V)
32		cnex 10952	. . . . . . . . 9 ⊢ ℂ ∈ V
33	1, 32	rabex2 5258	. . . . . . . 8 ⊢ 𝑈 ∈ V
34	33	a1i 11	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑟 ∈ ℕ ∧ 𝐴 ∈ 𝑈)) ∧ 𝑛 ∈ ℕ) → 𝑈 ∈ V)
35		simpr 485	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑟 ∈ ℕ ∧ 𝐴 ∈ 𝑈)) ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ ℕ)
36	35, 20	eleqtrdi 2849	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑟 ∈ ℕ ∧ 𝐴 ∈ 𝑈)) ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ (ℤ_≥‘1))
37		fz1ssnn 13287	. . . . . . . 8 ⊢ (1...𝑛) ⊆ ℕ
38	37	a1i 11	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑟 ∈ ℕ ∧ 𝐴 ∈ 𝑈)) ∧ 𝑛 ∈ ℕ) → (1...𝑛) ⊆ ℕ)
39		ovexd 7310	. . . . . . 7 ⊢ ((((𝜑 ∧ (𝑟 ∈ ℕ ∧ 𝐴 ∈ 𝑈)) ∧ 𝑛 ∈ ℕ) ∧ (𝑚 ∈ ℕ ∧ 𝑧 ∈ 𝑈)) → ((𝑧 · (log‘((𝑚 + 1) / 𝑚))) − (log‘((𝑧 / 𝑚) + 1))) ∈ V)
40	34, 36, 38, 39	seqof2 13781	. . . . . 6 ⊢ (((𝜑 ∧ (𝑟 ∈ ℕ ∧ 𝐴 ∈ 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)))
41		simplr 766	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ (𝑟 ∈ ℕ ∧ 𝐴 ∈ 𝑈)) ∧ 𝑛 ∈ ℕ) ∧ 𝑧 = 𝐴) ∧ 𝑚 ∈ ℕ) → 𝑧 = 𝐴)
42	41	oveq1d 7290	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ (𝑟 ∈ ℕ ∧ 𝐴 ∈ 𝑈)) ∧ 𝑛 ∈ ℕ) ∧ 𝑧 = 𝐴) ∧ 𝑚 ∈ ℕ) → (𝑧 · (log‘((𝑚 + 1) / 𝑚))) = (𝐴 · (log‘((𝑚 + 1) / 𝑚))))
43	41	oveq1d 7290	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ (𝑟 ∈ ℕ ∧ 𝐴 ∈ 𝑈)) ∧ 𝑛 ∈ ℕ) ∧ 𝑧 = 𝐴) ∧ 𝑚 ∈ ℕ) → (𝑧 / 𝑚) = (𝐴 / 𝑚))
44	43	fvoveq1d 7297	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ (𝑟 ∈ ℕ ∧ 𝐴 ∈ 𝑈)) ∧ 𝑛 ∈ ℕ) ∧ 𝑧 = 𝐴) ∧ 𝑚 ∈ ℕ) → (log‘((𝑧 / 𝑚) + 1)) = (log‘((𝐴 / 𝑚) + 1)))
45	42, 44	oveq12d 7293	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ (𝑟 ∈ ℕ ∧ 𝐴 ∈ 𝑈)) ∧ 𝑛 ∈ ℕ) ∧ 𝑧 = 𝐴) ∧ 𝑚 ∈ ℕ) → ((𝑧 · (log‘((𝑚 + 1) / 𝑚))) − (log‘((𝑧 / 𝑚) + 1))) = ((𝐴 · (log‘((𝑚 + 1) / 𝑚))) − (log‘((𝐴 / 𝑚) + 1))))
46	45	mpteq2dva 5174	. . . . . . . . 9 ⊢ ((((𝜑 ∧ (𝑟 ∈ ℕ ∧ 𝐴 ∈ 𝑈)) ∧ 𝑛 ∈ ℕ) ∧ 𝑧 = 𝐴) → (𝑚 ∈ ℕ ↦ ((𝑧 · (log‘((𝑚 + 1) / 𝑚))) − (log‘((𝑧 / 𝑚) + 1)))) = (𝑚 ∈ ℕ ↦ ((𝐴 · (log‘((𝑚 + 1) / 𝑚))) − (log‘((𝐴 / 𝑚) + 1)))))
47		lgamcvglem.g	. . . . . . . . 9 ⊢ 𝐺 = (𝑚 ∈ ℕ ↦ ((𝐴 · (log‘((𝑚 + 1) / 𝑚))) − (log‘((𝐴 / 𝑚) + 1))))
48	46, 47	eqtr4di 2796	. . . . . . . 8 ⊢ ((((𝜑 ∧ (𝑟 ∈ ℕ ∧ 𝐴 ∈ 𝑈)) ∧ 𝑛 ∈ ℕ) ∧ 𝑧 = 𝐴) → (𝑚 ∈ ℕ ↦ ((𝑧 · (log‘((𝑚 + 1) / 𝑚))) − (log‘((𝑧 / 𝑚) + 1)))) = 𝐺)
49	48	seqeq3d 13729	. . . . . . 7 ⊢ ((((𝜑 ∧ (𝑟 ∈ ℕ ∧ 𝐴 ∈ 𝑈)) ∧ 𝑛 ∈ ℕ) ∧ 𝑧 = 𝐴) → seq1( + , (𝑚 ∈ ℕ ↦ ((𝑧 · (log‘((𝑚 + 1) / 𝑚))) − (log‘((𝑧 / 𝑚) + 1))))) = seq1( + , 𝐺))
50	49	fveq1d 6776	. . . . . 6 ⊢ ((((𝜑 ∧ (𝑟 ∈ ℕ ∧ 𝐴 ∈ 𝑈)) ∧ 𝑛 ∈ ℕ) ∧ 𝑧 = 𝐴) → (seq1( + , (𝑚 ∈ ℕ ↦ ((𝑧 · (log‘((𝑚 + 1) / 𝑚))) − (log‘((𝑧 / 𝑚) + 1)))))‘𝑛) = (seq1( + , 𝐺)‘𝑛))
51		simplrr 775	. . . . . 6 ⊢ (((𝜑 ∧ (𝑟 ∈ ℕ ∧ 𝐴 ∈ 𝑈)) ∧ 𝑛 ∈ ℕ) → 𝐴 ∈ 𝑈)
52		fvexd 6789	. . . . . 6 ⊢ (((𝜑 ∧ (𝑟 ∈ ℕ ∧ 𝐴 ∈ 𝑈)) ∧ 𝑛 ∈ ℕ) → (seq1( + , 𝐺)‘𝑛) ∈ V)
53	40, 50, 51, 52	fvmptd 6882	. . . . 5 ⊢ (((𝜑 ∧ (𝑟 ∈ ℕ ∧ 𝐴 ∈ 𝑈)) ∧ 𝑛 ∈ ℕ) → ((seq1( ∘_f + , (𝑚 ∈ ℕ ↦ (𝑧 ∈ 𝑈 ↦ ((𝑧 · (log‘((𝑚 + 1) / 𝑚))) − (log‘((𝑧 / 𝑚) + 1))))))‘𝑛)‘𝐴) = (seq1( + , 𝐺)‘𝑛))
54	20, 21, 29, 18, 31, 53, 27	ulmclm 25546	. . . 4 ⊢ ((𝜑 ∧ (𝑟 ∈ ℕ ∧ 𝐴 ∈ 𝑈)) → seq1( + , 𝐺) ⇝ ((𝑧 ∈ 𝑈 ↦ ((log Γ‘𝑧) + (log‘𝑧)))‘𝐴))
55		fveq2 6774	. . . . . . 7 ⊢ (𝑧 = 𝐴 → (log‘𝑧) = (log‘𝐴))
56	4, 55	oveq12d 7293	. . . . . 6 ⊢ (𝑧 = 𝐴 → ((log Γ‘𝑧) + (log‘𝑧)) = ((log Γ‘𝐴) + (log‘𝐴)))
57		eqid 2738	. . . . . 6 ⊢ (𝑧 ∈ 𝑈 ↦ ((log Γ‘𝑧) + (log‘𝑧))) = (𝑧 ∈ 𝑈 ↦ ((log Γ‘𝑧) + (log‘𝑧)))
58		ovex 7308	. . . . . 6 ⊢ ((log Γ‘𝐴) + (log‘𝐴)) ∈ V
59	56, 57, 58	fvmpt 6875	. . . . 5 ⊢ (𝐴 ∈ 𝑈 → ((𝑧 ∈ 𝑈 ↦ ((log Γ‘𝑧) + (log‘𝑧)))‘𝐴) = ((log Γ‘𝐴) + (log‘𝐴)))
60	18, 59	syl 17	. . . 4 ⊢ ((𝜑 ∧ (𝑟 ∈ ℕ ∧ 𝐴 ∈ 𝑈)) → ((𝑧 ∈ 𝑈 ↦ ((log Γ‘𝑧) + (log‘𝑧)))‘𝐴) = ((log Γ‘𝐴) + (log‘𝐴)))
61	54, 60	breqtrd 5100	. . 3 ⊢ ((𝜑 ∧ (𝑟 ∈ ℕ ∧ 𝐴 ∈ 𝑈)) → seq1( + , 𝐺) ⇝ ((log Γ‘𝐴) + (log‘𝐴)))
62	19, 61	jca 512	. 2 ⊢ ((𝜑 ∧ (𝑟 ∈ ℕ ∧ 𝐴 ∈ 𝑈)) → ((log Γ‘𝐴) ∈ ℂ ∧ seq1( + , 𝐺) ⇝ ((log Γ‘𝐴) + (log‘𝐴))))
63	3, 62	rexlimddv 3220	1 ⊢ (𝜑 → ((log Γ‘𝐴) ∈ ℂ ∧ seq1( + , 𝐺) ⇝ ((log Γ‘𝐴) + (log‘𝐴))))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 396 = wceq 1539 ∈ wcel 2106 ∀wral 3064 {crab 3068 Vcvv 3432 ∖ cdif 3884 ⊆ wss 3887 class class class wbr 5074 ↦ cmpt 5157 Fn wfn 6428 ⟶wf 6429 ‘cfv 6433 (class class class)co 7275 ∘_f cof 7531 ↑_m cmap 8615 ℂcc 10869 1c1 10872 + caddc 10874 · cmul 10876 ≤ cle 11010 − cmin 11205 / cdiv 11632 ℕcn 11973 ℕ₀cn0 12233 ℤcz 12319 ℤ_≥cuz 12582 ...cfz 13239 seqcseq 13721 abscabs 14945 ⇝ cli 15193 ⇝_𝑢culm 25535 logclog 25710 log Γclgam 26165

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-rep 5209 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7588 ax-inf2 9399 ax-cnex 10927 ax-resscn 10928 ax-1cn 10929 ax-icn 10930 ax-addcl 10931 ax-addrcl 10932 ax-mulcl 10933 ax-mulrcl 10934 ax-mulcom 10935 ax-addass 10936 ax-mulass 10937 ax-distr 10938 ax-i2m1 10939 ax-1ne0 10940 ax-1rid 10941 ax-rnegex 10942 ax-rrecex 10943 ax-cnre 10944 ax-pre-lttri 10945 ax-pre-lttrn 10946 ax-pre-ltadd 10947 ax-pre-mulgt0 10948 ax-pre-sup 10949 ax-addf 10950 ax-mulf 10951

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-rmo 3071 df-reu 3072 df-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-pss 3906 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-tp 4566 df-op 4568 df-uni 4840 df-int 4880 df-iun 4926 df-iin 4927 df-br 5075 df-opab 5137 df-mpt 5158 df-tr 5192 df-id 5489 df-eprel 5495 df-po 5503 df-so 5504 df-fr 5544 df-se 5545 df-we 5546 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-pred 6202 df-ord 6269 df-on 6270 df-lim 6271 df-suc 6272 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-f1 6438 df-fo 6439 df-f1o 6440 df-fv 6441 df-isom 6442 df-riota 7232 df-ov 7278 df-oprab 7279 df-mpo 7280 df-of 7533 df-om 7713 df-1st 7831 df-2nd 7832 df-supp 7978 df-frecs 8097 df-wrecs 8128 df-recs 8202 df-rdg 8241 df-1o 8297 df-2o 8298 df-oadd 8301 df-er 8498 df-map 8617 df-pm 8618 df-ixp 8686 df-en 8734 df-dom 8735 df-sdom 8736 df-fin 8737 df-fsupp 9129 df-fi 9170 df-sup 9201 df-inf 9202 df-oi 9269 df-dju 9659 df-card 9697 df-pnf 11011 df-mnf 11012 df-xr 11013 df-ltxr 11014 df-le 11015 df-sub 11207 df-neg 11208 df-div 11633 df-nn 11974 df-2 12036 df-3 12037 df-4 12038 df-5 12039 df-6 12040 df-7 12041 df-8 12042 df-9 12043 df-n0 12234 df-z 12320 df-dec 12438 df-uz 12583 df-q 12689 df-rp 12731 df-xneg 12848 df-xadd 12849 df-xmul 12850 df-ioo 13083 df-ioc 13084 df-ico 13085 df-icc 13086 df-fz 13240 df-fzo 13383 df-fl 13512 df-mod 13590 df-seq 13722 df-exp 13783 df-fac 13988 df-bc 14017 df-hash 14045 df-shft 14778 df-cj 14810 df-re 14811 df-im 14812 df-sqrt 14946 df-abs 14947 df-limsup 15180 df-clim 15197 df-rlim 15198 df-sum 15398 df-ef 15777 df-sin 15779 df-cos 15780 df-tan 15781 df-pi 15782 df-struct 16848 df-sets 16865 df-slot 16883 df-ndx 16895 df-base 16913 df-ress 16942 df-plusg 16975 df-mulr 16976 df-starv 16977 df-sca 16978 df-vsca 16979 df-ip 16980 df-tset 16981 df-ple 16982 df-ds 16984 df-unif 16985 df-hom 16986 df-cco 16987 df-rest 17133 df-topn 17134 df-0g 17152 df-gsum 17153 df-topgen 17154 df-pt 17155 df-prds 17158 df-xrs 17213 df-qtop 17218 df-imas 17219 df-xps 17221 df-mre 17295 df-mrc 17296 df-acs 17298 df-mgm 18326 df-sgrp 18375 df-mnd 18386 df-submnd 18431 df-mulg 18701 df-cntz 18923 df-cmn 19388 df-psmet 20589 df-xmet 20590 df-met 20591 df-bl 20592 df-mopn 20593 df-fbas 20594 df-fg 20595 df-cnfld 20598 df-top 22043 df-topon 22060 df-topsp 22082 df-bases 22096 df-cld 22170 df-ntr 22171 df-cls 22172 df-nei 22249 df-lp 22287 df-perf 22288 df-cn 22378 df-cnp 22379 df-haus 22466 df-cmp 22538 df-tx 22713 df-hmeo 22906 df-fil 22997 df-fm 23089 df-flim 23090 df-flf 23091 df-xms 23473 df-ms 23474 df-tms 23475 df-cncf 24041 df-limc 25030 df-dv 25031 df-ulm 25536 df-log 25712 df-cxp 25713 df-lgam 26168

This theorem is referenced by: lgamcl 26190 lgamcvg 26203

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