lgamcvglem - Metamath Proof Explorer

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

Theorem lgamcvglem 26983

Description: Lemma for lgamf 26985 and lgamcvg 26997. (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 26982	. 2 ⊢ (𝜑 → ∃𝑟 ∈ ℕ 𝐴 ∈ 𝑈)
4		fveq2 6828	. . . . 5 ⊢ (𝑧 = 𝐴 → (log Γ‘𝑧) = (log Γ‘𝐴))
5	4	eleq1d 2816	. . . 4 ⊢ (𝑧 = 𝐴 → ((log Γ‘𝑧) ∈ ℂ ↔ (log Γ‘𝐴) ∈ ℂ))
6		simprl 770	. . . . . 6 ⊢ ((𝜑 ∧ (𝑟 ∈ ℕ ∧ 𝐴 ∈ 𝑈)) → 𝑟 ∈ ℕ)
7		fveq2 6828	. . . . . . . . . 10 ⊢ (𝑥 = 𝑡 → (abs‘𝑥) = (abs‘𝑡))
8	7	breq1d 5103	. . . . . . . . 9 ⊢ (𝑥 = 𝑡 → ((abs‘𝑥) ≤ 𝑟 ↔ (abs‘𝑡) ≤ 𝑟))
9		fvoveq1 7375	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑡 → (abs‘(𝑥 + 𝑘)) = (abs‘(𝑡 + 𝑘)))
10	9	breq2d 5105	. . . . . . . . . 10 ⊢ (𝑥 = 𝑡 → ((1 / 𝑟) ≤ (abs‘(𝑥 + 𝑘)) ↔ (1 / 𝑟) ≤ (abs‘(𝑡 + 𝑘))))
11	10	ralbidv 3155	. . . . . . . . 9 ⊢ (𝑥 = 𝑡 → (∀𝑘 ∈ ℕ₀ (1 / 𝑟) ≤ (abs‘(𝑥 + 𝑘)) ↔ ∀𝑘 ∈ ℕ₀ (1 / 𝑟) ≤ (abs‘(𝑡 + 𝑘))))
12	8, 11	anbi12d 632	. . . . . . . 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 3405	. . . . . . 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 2754	. . . . . 6 ⊢ 𝑈 = {𝑡 ∈ ℂ ∣ ((abs‘𝑡) ≤ 𝑟 ∧ ∀𝑘 ∈ ℕ₀ (1 / 𝑟) ≤ (abs‘(𝑡 + 𝑘)))}
15		eqid 2731	. . . . . 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 26979	. . . . 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 494	. . . 4 ⊢ ((𝜑 ∧ (𝑟 ∈ ℕ ∧ 𝐴 ∈ 𝑈)) → ∀𝑧 ∈ 𝑈 (log Γ‘𝑧) ∈ ℂ)
18		simprr 772	. . . 4 ⊢ ((𝜑 ∧ (𝑟 ∈ ℕ ∧ 𝐴 ∈ 𝑈)) → 𝐴 ∈ 𝑈)
19	5, 17, 18	rspcdva 3573	. . 3 ⊢ ((𝜑 ∧ (𝑟 ∈ ℕ ∧ 𝐴 ∈ 𝑈)) → (log Γ‘𝐴) ∈ ℂ)
20		nnuz 12781	. . . . 5 ⊢ ℕ = (ℤ_≥‘1)
21		1zzd 12509	. . . . 5 ⊢ ((𝜑 ∧ (𝑟 ∈ ℕ ∧ 𝐴 ∈ 𝑈)) → 1 ∈ ℤ)
22		1z 12508	. . . . . . . 8 ⊢ 1 ∈ ℤ
23		seqfn 13926	. . . . . . . 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 6585	. . . . . . 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 231	. . . . . 6 ⊢ seq1( ∘_f + , (𝑚 ∈ ℕ ↦ (𝑧 ∈ 𝑈 ↦ ((𝑧 · (log‘((𝑚 + 1) / 𝑚))) − (log‘((𝑧 / 𝑚) + 1)))))) Fn ℕ
27	16	simprd 495	. . . . . 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 26326	. . . . . 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 13916	. . . . . 6 ⊢ seq1( + , 𝐺) ∈ V
31	30	a1i 11	. . . . 5 ⊢ ((𝜑 ∧ (𝑟 ∈ ℕ ∧ 𝐴 ∈ 𝑈)) → seq1( + , 𝐺) ∈ V)
32		cnex 11093	. . . . . . . . 9 ⊢ ℂ ∈ V
33	1, 32	rabex2 5281	. . . . . . . 8 ⊢ 𝑈 ∈ V
34	33	a1i 11	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑟 ∈ ℕ ∧ 𝐴 ∈ 𝑈)) ∧ 𝑛 ∈ ℕ) → 𝑈 ∈ V)
35		simpr 484	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑟 ∈ ℕ ∧ 𝐴 ∈ 𝑈)) ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ ℕ)
36	35, 20	eleqtrdi 2841	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑟 ∈ ℕ ∧ 𝐴 ∈ 𝑈)) ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ (ℤ_≥‘1))
37		fz1ssnn 13461	. . . . . . . 8 ⊢ (1...𝑛) ⊆ ℕ
38	37	a1i 11	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑟 ∈ ℕ ∧ 𝐴 ∈ 𝑈)) ∧ 𝑛 ∈ ℕ) → (1...𝑛) ⊆ ℕ)
39		ovexd 7387	. . . . . . 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 13973	. . . . . 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 768	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ (𝑟 ∈ ℕ ∧ 𝐴 ∈ 𝑈)) ∧ 𝑛 ∈ ℕ) ∧ 𝑧 = 𝐴) ∧ 𝑚 ∈ ℕ) → 𝑧 = 𝐴)
42	41	oveq1d 7367	. . . . . . . . . . 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 7367	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ (𝑟 ∈ ℕ ∧ 𝐴 ∈ 𝑈)) ∧ 𝑛 ∈ ℕ) ∧ 𝑧 = 𝐴) ∧ 𝑚 ∈ ℕ) → (𝑧 / 𝑚) = (𝐴 / 𝑚))
44	43	fvoveq1d 7374	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ (𝑟 ∈ ℕ ∧ 𝐴 ∈ 𝑈)) ∧ 𝑛 ∈ ℕ) ∧ 𝑧 = 𝐴) ∧ 𝑚 ∈ ℕ) → (log‘((𝑧 / 𝑚) + 1)) = (log‘((𝐴 / 𝑚) + 1)))
45	42, 44	oveq12d 7370	. . . . . . . . . 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 5186	. . . . . . . . 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 2784	. . . . . . . 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 13922	. . . . . . 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 6830	. . . . . 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 777	. . . . . 6 ⊢ (((𝜑 ∧ (𝑟 ∈ ℕ ∧ 𝐴 ∈ 𝑈)) ∧ 𝑛 ∈ ℕ) → 𝐴 ∈ 𝑈)
52		fvexd 6843	. . . . . 6 ⊢ (((𝜑 ∧ (𝑟 ∈ ℕ ∧ 𝐴 ∈ 𝑈)) ∧ 𝑛 ∈ ℕ) → (seq1( + , 𝐺)‘𝑛) ∈ V)
53	40, 50, 51, 52	fvmptd 6942	. . . . 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 26329	. . . 4 ⊢ ((𝜑 ∧ (𝑟 ∈ ℕ ∧ 𝐴 ∈ 𝑈)) → seq1( + , 𝐺) ⇝ ((𝑧 ∈ 𝑈 ↦ ((log Γ‘𝑧) + (log‘𝑧)))‘𝐴))
55		fveq2 6828	. . . . . . 7 ⊢ (𝑧 = 𝐴 → (log‘𝑧) = (log‘𝐴))
56	4, 55	oveq12d 7370	. . . . . 6 ⊢ (𝑧 = 𝐴 → ((log Γ‘𝑧) + (log‘𝑧)) = ((log Γ‘𝐴) + (log‘𝐴)))
57		eqid 2731	. . . . . 6 ⊢ (𝑧 ∈ 𝑈 ↦ ((log Γ‘𝑧) + (log‘𝑧))) = (𝑧 ∈ 𝑈 ↦ ((log Γ‘𝑧) + (log‘𝑧)))
58		ovex 7385	. . . . . 6 ⊢ ((log Γ‘𝐴) + (log‘𝐴)) ∈ V
59	56, 57, 58	fvmpt 6935	. . . . 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 5119	. . 3 ⊢ ((𝜑 ∧ (𝑟 ∈ ℕ ∧ 𝐴 ∈ 𝑈)) → seq1( + , 𝐺) ⇝ ((log Γ‘𝐴) + (log‘𝐴)))
62	19, 61	jca 511	. 2 ⊢ ((𝜑 ∧ (𝑟 ∈ ℕ ∧ 𝐴 ∈ 𝑈)) → ((log Γ‘𝐴) ∈ ℂ ∧ seq1( + , 𝐺) ⇝ ((log Γ‘𝐴) + (log‘𝐴))))
63	3, 62	rexlimddv 3139	1 ⊢ (𝜑 → ((log Γ‘𝐴) ∈ ℂ ∧ seq1( + , 𝐺) ⇝ ((log Γ‘𝐴) + (log‘𝐴))))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1541 ∈ wcel 2111 ∀wral 3047 {crab 3395 Vcvv 3436 ∖ cdif 3894 ⊆ wss 3897 class class class wbr 5093 ↦ cmpt 5174 Fn wfn 6482 ⟶wf 6483 ‘cfv 6487 (class class class)co 7352 ∘_f cof 7614 ↑_m cmap 8756 ℂcc 11010 1c1 11013 + caddc 11015 · cmul 11017 ≤ cle 11153 − cmin 11350 / cdiv 11780 ℕcn 12131 ℕ₀cn0 12387 ℤcz 12474 ℤ_≥cuz 12738 ...cfz 13413 seqcseq 13914 abscabs 15147 ⇝ cli 15397 ⇝_𝑢culm 26318 logclog 26496 log Γclgam 26959

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2113 ax-9 2121 ax-10 2144 ax-11 2160 ax-12 2180 ax-ext 2703 ax-rep 5219 ax-sep 5236 ax-nul 5246 ax-pow 5305 ax-pr 5372 ax-un 7674 ax-inf2 9537 ax-cnex 11068 ax-resscn 11069 ax-1cn 11070 ax-icn 11071 ax-addcl 11072 ax-addrcl 11073 ax-mulcl 11074 ax-mulrcl 11075 ax-mulcom 11076 ax-addass 11077 ax-mulass 11078 ax-distr 11079 ax-i2m1 11080 ax-1ne0 11081 ax-1rid 11082 ax-rnegex 11083 ax-rrecex 11084 ax-cnre 11085 ax-pre-lttri 11086 ax-pre-lttrn 11087 ax-pre-ltadd 11088 ax-pre-mulgt0 11089 ax-pre-sup 11090 ax-addf 11091

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2535 df-eu 2564 df-clab 2710 df-cleq 2723 df-clel 2806 df-nfc 2881 df-ne 2929 df-nel 3033 df-ral 3048 df-rex 3057 df-rmo 3346 df-reu 3347 df-rab 3396 df-v 3438 df-sbc 3737 df-csb 3846 df-dif 3900 df-un 3902 df-in 3904 df-ss 3914 df-pss 3917 df-nul 4283 df-if 4475 df-pw 4551 df-sn 4576 df-pr 4578 df-tp 4580 df-op 4582 df-uni 4859 df-int 4898 df-iun 4943 df-iin 4944 df-br 5094 df-opab 5156 df-mpt 5175 df-tr 5201 df-id 5514 df-eprel 5519 df-po 5527 df-so 5528 df-fr 5572 df-se 5573 df-we 5574 df-xp 5625 df-rel 5626 df-cnv 5627 df-co 5628 df-dm 5629 df-rn 5630 df-res 5631 df-ima 5632 df-pred 6254 df-ord 6315 df-on 6316 df-lim 6317 df-suc 6318 df-iota 6443 df-fun 6489 df-fn 6490 df-f 6491 df-f1 6492 df-fo 6493 df-f1o 6494 df-fv 6495 df-isom 6496 df-riota 7309 df-ov 7355 df-oprab 7356 df-mpo 7357 df-of 7616 df-om 7803 df-1st 7927 df-2nd 7928 df-supp 8097 df-frecs 8217 df-wrecs 8248 df-recs 8297 df-rdg 8335 df-1o 8391 df-2o 8392 df-oadd 8395 df-er 8628 df-map 8758 df-pm 8759 df-ixp 8828 df-en 8876 df-dom 8877 df-sdom 8878 df-fin 8879 df-fsupp 9252 df-fi 9301 df-sup 9332 df-inf 9333 df-oi 9402 df-dju 9800 df-card 9838 df-pnf 11154 df-mnf 11155 df-xr 11156 df-ltxr 11157 df-le 11158 df-sub 11352 df-neg 11353 df-div 11781 df-nn 12132 df-2 12194 df-3 12195 df-4 12196 df-5 12197 df-6 12198 df-7 12199 df-8 12200 df-9 12201 df-n0 12388 df-z 12475 df-dec 12595 df-uz 12739 df-q 12853 df-rp 12897 df-xneg 13017 df-xadd 13018 df-xmul 13019 df-ioo 13255 df-ioc 13256 df-ico 13257 df-icc 13258 df-fz 13414 df-fzo 13561 df-fl 13702 df-mod 13780 df-seq 13915 df-exp 13975 df-fac 14187 df-bc 14216 df-hash 14244 df-shft 14980 df-cj 15012 df-re 15013 df-im 15014 df-sqrt 15148 df-abs 15149 df-limsup 15384 df-clim 15401 df-rlim 15402 df-sum 15600 df-ef 15980 df-sin 15982 df-cos 15983 df-tan 15984 df-pi 15985 df-struct 17064 df-sets 17081 df-slot 17099 df-ndx 17111 df-base 17127 df-ress 17148 df-plusg 17180 df-mulr 17181 df-starv 17182 df-sca 17183 df-vsca 17184 df-ip 17185 df-tset 17186 df-ple 17187 df-ds 17189 df-unif 17190 df-hom 17191 df-cco 17192 df-rest 17332 df-topn 17333 df-0g 17351 df-gsum 17352 df-topgen 17353 df-pt 17354 df-prds 17357 df-xrs 17412 df-qtop 17417 df-imas 17418 df-xps 17420 df-mre 17494 df-mrc 17495 df-acs 17497 df-mgm 18554 df-sgrp 18633 df-mnd 18649 df-submnd 18698 df-mulg 18987 df-cntz 19235 df-cmn 19700 df-psmet 21289 df-xmet 21290 df-met 21291 df-bl 21292 df-mopn 21293 df-fbas 21294 df-fg 21295 df-cnfld 21298 df-top 22815 df-topon 22832 df-topsp 22854 df-bases 22867 df-cld 22940 df-ntr 22941 df-cls 22942 df-nei 23019 df-lp 23057 df-perf 23058 df-cn 23148 df-cnp 23149 df-haus 23236 df-cmp 23308 df-tx 23483 df-hmeo 23676 df-fil 23767 df-fm 23859 df-flim 23860 df-flf 23861 df-xms 24241 df-ms 24242 df-tms 24243 df-cncf 24804 df-limc 25800 df-dv 25801 df-ulm 26319 df-log 26498 df-cxp 26499 df-lgam 26962

This theorem is referenced by: lgamcl 26984 lgamcvg 26997

W3C validator