zetacvg - Metamath Proof Explorer

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

Theorem zetacvg 26519

Description: The zeta series is convergent. (Contributed by Mario Carneiro, 18-Jul-2014.)

**Hypotheses**
Ref	Expression
zetacvg.1	⊢ (𝜑 → 𝑆 ∈ ℂ)
zetacvg.2	⊢ (𝜑 → 1 < (ℜ‘𝑆))
zetacvg.3	⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝐹‘𝑘) = (𝑘↑_𝑐-𝑆))

**Assertion**
Ref	Expression
zetacvg	⊢ (𝜑 → seq1( + , 𝐹) ∈ dom ⇝ )

Distinct variable groups: 𝑆,𝑘 𝑘,𝐹 𝜑,𝑘

Proof of Theorem zetacvg Dummy variables 𝑚 𝑛 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		nnuz 12865	. 2 ⊢ ℕ = (ℤ_≥‘1)
2		1zzd 12593	. 2 ⊢ (𝜑 → 1 ∈ ℤ)
3		oveq1 7416	. . . . 5 ⊢ (𝑛 = 𝑘 → (𝑛↑_𝑐-(ℜ‘𝑆)) = (𝑘↑_𝑐-(ℜ‘𝑆)))
4		eqid 2733	. . . . 5 ⊢ (𝑛 ∈ ℕ ↦ (𝑛↑_𝑐-(ℜ‘𝑆))) = (𝑛 ∈ ℕ ↦ (𝑛↑_𝑐-(ℜ‘𝑆)))
5		ovex 7442	. . . . 5 ⊢ (𝑘↑_𝑐-(ℜ‘𝑆)) ∈ V
6	3, 4, 5	fvmpt 6999	. . . 4 ⊢ (𝑘 ∈ ℕ → ((𝑛 ∈ ℕ ↦ (𝑛↑_𝑐-(ℜ‘𝑆)))‘𝑘) = (𝑘↑_𝑐-(ℜ‘𝑆)))
7	6	adantl 483	. . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ (𝑛↑_𝑐-(ℜ‘𝑆)))‘𝑘) = (𝑘↑_𝑐-(ℜ‘𝑆)))
8		zetacvg.3	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝐹‘𝑘) = (𝑘↑_𝑐-𝑆))
9		nncn 12220	. . . . . . . 8 ⊢ (𝑘 ∈ ℕ → 𝑘 ∈ ℂ)
10	9	adantl 483	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → 𝑘 ∈ ℂ)
11		nnne0 12246	. . . . . . . 8 ⊢ (𝑘 ∈ ℕ → 𝑘 ≠ 0)
12	11	adantl 483	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → 𝑘 ≠ 0)
13		zetacvg.1	. . . . . . . . 9 ⊢ (𝜑 → 𝑆 ∈ ℂ)
14	13	negcld 11558	. . . . . . . 8 ⊢ (𝜑 → -𝑆 ∈ ℂ)
15	14	adantr 482	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → -𝑆 ∈ ℂ)
16	10, 12, 15	cxpefd 26220	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝑘↑_𝑐-𝑆) = (exp‘(-𝑆 · (log‘𝑘))))
17	8, 16	eqtrd 2773	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝐹‘𝑘) = (exp‘(-𝑆 · (log‘𝑘))))
18	17	fveq2d 6896	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (abs‘(𝐹‘𝑘)) = (abs‘(exp‘(-𝑆 · (log‘𝑘)))))
19		nnrp 12985	. . . . . . . 8 ⊢ (𝑘 ∈ ℕ → 𝑘 ∈ ℝ⁺)
20	19	relogcld 26131	. . . . . . 7 ⊢ (𝑘 ∈ ℕ → (log‘𝑘) ∈ ℝ)
21	20	recnd 11242	. . . . . 6 ⊢ (𝑘 ∈ ℕ → (log‘𝑘) ∈ ℂ)
22		mulcl 11194	. . . . . 6 ⊢ ((-𝑆 ∈ ℂ ∧ (log‘𝑘) ∈ ℂ) → (-𝑆 · (log‘𝑘)) ∈ ℂ)
23	14, 21, 22	syl2an 597	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (-𝑆 · (log‘𝑘)) ∈ ℂ)
24		absef 16140	. . . . 5 ⊢ ((-𝑆 · (log‘𝑘)) ∈ ℂ → (abs‘(exp‘(-𝑆 · (log‘𝑘)))) = (exp‘(ℜ‘(-𝑆 · (log‘𝑘)))))
25	23, 24	syl 17	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (abs‘(exp‘(-𝑆 · (log‘𝑘)))) = (exp‘(ℜ‘(-𝑆 · (log‘𝑘)))))
26		remul 15076	. . . . . . . 8 ⊢ ((-𝑆 ∈ ℂ ∧ (log‘𝑘) ∈ ℂ) → (ℜ‘(-𝑆 · (log‘𝑘))) = (((ℜ‘-𝑆) · (ℜ‘(log‘𝑘))) − ((ℑ‘-𝑆) · (ℑ‘(log‘𝑘)))))
27	14, 21, 26	syl2an 597	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (ℜ‘(-𝑆 · (log‘𝑘))) = (((ℜ‘-𝑆) · (ℜ‘(log‘𝑘))) − ((ℑ‘-𝑆) · (ℑ‘(log‘𝑘)))))
28	13	renegd 15156	. . . . . . . . 9 ⊢ (𝜑 → (ℜ‘-𝑆) = -(ℜ‘𝑆))
29	20	rered 15171	. . . . . . . . 9 ⊢ (𝑘 ∈ ℕ → (ℜ‘(log‘𝑘)) = (log‘𝑘))
30	28, 29	oveqan12d 7428	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ((ℜ‘-𝑆) · (ℜ‘(log‘𝑘))) = (-(ℜ‘𝑆) · (log‘𝑘)))
31	20	reim0d 15172	. . . . . . . . . 10 ⊢ (𝑘 ∈ ℕ → (ℑ‘(log‘𝑘)) = 0)
32	31	oveq2d 7425	. . . . . . . . 9 ⊢ (𝑘 ∈ ℕ → ((ℑ‘-𝑆) · (ℑ‘(log‘𝑘))) = ((ℑ‘-𝑆) · 0))
33		imcl 15058	. . . . . . . . . . . 12 ⊢ (-𝑆 ∈ ℂ → (ℑ‘-𝑆) ∈ ℝ)
34	33	recnd 11242	. . . . . . . . . . 11 ⊢ (-𝑆 ∈ ℂ → (ℑ‘-𝑆) ∈ ℂ)
35	14, 34	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → (ℑ‘-𝑆) ∈ ℂ)
36	35	mul01d 11413	. . . . . . . . 9 ⊢ (𝜑 → ((ℑ‘-𝑆) · 0) = 0)
37	32, 36	sylan9eqr 2795	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ((ℑ‘-𝑆) · (ℑ‘(log‘𝑘))) = 0)
38	30, 37	oveq12d 7427	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (((ℜ‘-𝑆) · (ℜ‘(log‘𝑘))) − ((ℑ‘-𝑆) · (ℑ‘(log‘𝑘)))) = ((-(ℜ‘𝑆) · (log‘𝑘)) − 0))
39	13	recld 15141	. . . . . . . . . . 11 ⊢ (𝜑 → (ℜ‘𝑆) ∈ ℝ)
40	39	renegcld 11641	. . . . . . . . . 10 ⊢ (𝜑 → -(ℜ‘𝑆) ∈ ℝ)
41	40	recnd 11242	. . . . . . . . 9 ⊢ (𝜑 → -(ℜ‘𝑆) ∈ ℂ)
42		mulcl 11194	. . . . . . . . 9 ⊢ ((-(ℜ‘𝑆) ∈ ℂ ∧ (log‘𝑘) ∈ ℂ) → (-(ℜ‘𝑆) · (log‘𝑘)) ∈ ℂ)
43	41, 21, 42	syl2an 597	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (-(ℜ‘𝑆) · (log‘𝑘)) ∈ ℂ)
44	43	subid1d 11560	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ((-(ℜ‘𝑆) · (log‘𝑘)) − 0) = (-(ℜ‘𝑆) · (log‘𝑘)))
45	27, 38, 44	3eqtrd 2777	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (ℜ‘(-𝑆 · (log‘𝑘))) = (-(ℜ‘𝑆) · (log‘𝑘)))
46	45	fveq2d 6896	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (exp‘(ℜ‘(-𝑆 · (log‘𝑘)))) = (exp‘(-(ℜ‘𝑆) · (log‘𝑘))))
47	41	adantr 482	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → -(ℜ‘𝑆) ∈ ℂ)
48	10, 12, 47	cxpefd 26220	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝑘↑_𝑐-(ℜ‘𝑆)) = (exp‘(-(ℜ‘𝑆) · (log‘𝑘))))
49	46, 48	eqtr4d 2776	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (exp‘(ℜ‘(-𝑆 · (log‘𝑘)))) = (𝑘↑_𝑐-(ℜ‘𝑆)))
50	18, 25, 49	3eqtrd 2777	. . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (abs‘(𝐹‘𝑘)) = (𝑘↑_𝑐-(ℜ‘𝑆)))
51	7, 50	eqtr4d 2776	. 2 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ (𝑛↑_𝑐-(ℜ‘𝑆)))‘𝑘) = (abs‘(𝐹‘𝑘)))
52	10, 15	cxpcld 26216	. . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝑘↑_𝑐-𝑆) ∈ ℂ)
53	8, 52	eqeltrd 2834	. 2 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝐹‘𝑘) ∈ ℂ)
54		2rp 12979	. . . . . . 7 ⊢ 2 ∈ ℝ⁺
55		1re 11214	. . . . . . . 8 ⊢ 1 ∈ ℝ
56		resubcl 11524	. . . . . . . 8 ⊢ ((1 ∈ ℝ ∧ (ℜ‘𝑆) ∈ ℝ) → (1 − (ℜ‘𝑆)) ∈ ℝ)
57	55, 39, 56	sylancr 588	. . . . . . 7 ⊢ (𝜑 → (1 − (ℜ‘𝑆)) ∈ ℝ)
58		rpcxpcl 26184	. . . . . . 7 ⊢ ((2 ∈ ℝ⁺ ∧ (1 − (ℜ‘𝑆)) ∈ ℝ) → (2↑_𝑐(1 − (ℜ‘𝑆))) ∈ ℝ⁺)
59	54, 57, 58	sylancr 588	. . . . . 6 ⊢ (𝜑 → (2↑_𝑐(1 − (ℜ‘𝑆))) ∈ ℝ⁺)
60	59	rpcnd 13018	. . . . 5 ⊢ (𝜑 → (2↑_𝑐(1 − (ℜ‘𝑆))) ∈ ℂ)
61		zetacvg.2	. . . . . . . . 9 ⊢ (𝜑 → 1 < (ℜ‘𝑆))
62		recl 15057	. . . . . . . . . . . 12 ⊢ (𝑆 ∈ ℂ → (ℜ‘𝑆) ∈ ℝ)
63	62	recnd 11242	. . . . . . . . . . 11 ⊢ (𝑆 ∈ ℂ → (ℜ‘𝑆) ∈ ℂ)
64	13, 63	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → (ℜ‘𝑆) ∈ ℂ)
65	64	addlidd 11415	. . . . . . . . 9 ⊢ (𝜑 → (0 + (ℜ‘𝑆)) = (ℜ‘𝑆))
66	61, 65	breqtrrd 5177	. . . . . . . 8 ⊢ (𝜑 → 1 < (0 + (ℜ‘𝑆)))
67		0re 11216	. . . . . . . . . 10 ⊢ 0 ∈ ℝ
68		ltsubadd 11684	. . . . . . . . . 10 ⊢ ((1 ∈ ℝ ∧ (ℜ‘𝑆) ∈ ℝ ∧ 0 ∈ ℝ) → ((1 − (ℜ‘𝑆)) < 0 ↔ 1 < (0 + (ℜ‘𝑆))))
69	55, 67, 68	mp3an13 1453	. . . . . . . . 9 ⊢ ((ℜ‘𝑆) ∈ ℝ → ((1 − (ℜ‘𝑆)) < 0 ↔ 1 < (0 + (ℜ‘𝑆))))
70	39, 69	syl 17	. . . . . . . 8 ⊢ (𝜑 → ((1 − (ℜ‘𝑆)) < 0 ↔ 1 < (0 + (ℜ‘𝑆))))
71	66, 70	mpbird 257	. . . . . . 7 ⊢ (𝜑 → (1 − (ℜ‘𝑆)) < 0)
72		2re 12286	. . . . . . . . 9 ⊢ 2 ∈ ℝ
73		1lt2 12383	. . . . . . . . 9 ⊢ 1 < 2
74		cxplt 26202	. . . . . . . . 9 ⊢ (((2 ∈ ℝ ∧ 1 < 2) ∧ ((1 − (ℜ‘𝑆)) ∈ ℝ ∧ 0 ∈ ℝ)) → ((1 − (ℜ‘𝑆)) < 0 ↔ (2↑_𝑐(1 − (ℜ‘𝑆))) < (2↑_𝑐0)))
75	72, 73, 74	mpanl12 701	. . . . . . . 8 ⊢ (((1 − (ℜ‘𝑆)) ∈ ℝ ∧ 0 ∈ ℝ) → ((1 − (ℜ‘𝑆)) < 0 ↔ (2↑_𝑐(1 − (ℜ‘𝑆))) < (2↑_𝑐0)))
76	57, 67, 75	sylancl 587	. . . . . . 7 ⊢ (𝜑 → ((1 − (ℜ‘𝑆)) < 0 ↔ (2↑_𝑐(1 − (ℜ‘𝑆))) < (2↑_𝑐0)))
77	71, 76	mpbid 231	. . . . . 6 ⊢ (𝜑 → (2↑_𝑐(1 − (ℜ‘𝑆))) < (2↑_𝑐0))
78	59	rprege0d 13023	. . . . . . 7 ⊢ (𝜑 → ((2↑_𝑐(1 − (ℜ‘𝑆))) ∈ ℝ ∧ 0 ≤ (2↑_𝑐(1 − (ℜ‘𝑆)))))
79		absid 15243	. . . . . . 7 ⊢ (((2↑_𝑐(1 − (ℜ‘𝑆))) ∈ ℝ ∧ 0 ≤ (2↑_𝑐(1 − (ℜ‘𝑆)))) → (abs‘(2↑_𝑐(1 − (ℜ‘𝑆)))) = (2↑_𝑐(1 − (ℜ‘𝑆))))
80	78, 79	syl 17	. . . . . 6 ⊢ (𝜑 → (abs‘(2↑_𝑐(1 − (ℜ‘𝑆)))) = (2↑_𝑐(1 − (ℜ‘𝑆))))
81		2cn 12287	. . . . . . . . 9 ⊢ 2 ∈ ℂ
82		cxp0 26178	. . . . . . . . 9 ⊢ (2 ∈ ℂ → (2↑_𝑐0) = 1)
83	81, 82	ax-mp 5	. . . . . . . 8 ⊢ (2↑_𝑐0) = 1
84	83	eqcomi 2742	. . . . . . 7 ⊢ 1 = (2↑_𝑐0)
85	84	a1i 11	. . . . . 6 ⊢ (𝜑 → 1 = (2↑_𝑐0))
86	77, 80, 85	3brtr4d 5181	. . . . 5 ⊢ (𝜑 → (abs‘(2↑_𝑐(1 − (ℜ‘𝑆)))) < 1)
87		oveq2 7417	. . . . . . 7 ⊢ (𝑛 = 𝑚 → ((2↑_𝑐(1 − (ℜ‘𝑆)))↑𝑛) = ((2↑_𝑐(1 − (ℜ‘𝑆)))↑𝑚))
88		eqid 2733	. . . . . . 7 ⊢ (𝑛 ∈ ℕ₀ ↦ ((2↑_𝑐(1 − (ℜ‘𝑆)))↑𝑛)) = (𝑛 ∈ ℕ₀ ↦ ((2↑_𝑐(1 − (ℜ‘𝑆)))↑𝑛))
89		ovex 7442	. . . . . . 7 ⊢ ((2↑_𝑐(1 − (ℜ‘𝑆)))↑𝑚) ∈ V
90	87, 88, 89	fvmpt 6999	. . . . . 6 ⊢ (𝑚 ∈ ℕ₀ → ((𝑛 ∈ ℕ₀ ↦ ((2↑_𝑐(1 − (ℜ‘𝑆)))↑𝑛))‘𝑚) = ((2↑_𝑐(1 − (ℜ‘𝑆)))↑𝑚))
91	90	adantl 483	. . . . 5 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ₀) → ((𝑛 ∈ ℕ₀ ↦ ((2↑_𝑐(1 − (ℜ‘𝑆)))↑𝑛))‘𝑚) = ((2↑_𝑐(1 − (ℜ‘𝑆)))↑𝑚))
92	60, 86, 91	geolim 15816	. . . 4 ⊢ (𝜑 → seq0( + , (𝑛 ∈ ℕ₀ ↦ ((2↑_𝑐(1 − (ℜ‘𝑆)))↑𝑛))) ⇝ (1 / (1 − (2↑_𝑐(1 − (ℜ‘𝑆))))))
93		seqex 13968	. . . . 5 ⊢ seq0( + , (𝑛 ∈ ℕ₀ ↦ ((2↑_𝑐(1 − (ℜ‘𝑆)))↑𝑛))) ∈ V
94		ovex 7442	. . . . 5 ⊢ (1 / (1 − (2↑_𝑐(1 − (ℜ‘𝑆))))) ∈ V
95	93, 94	breldm 5909	. . . 4 ⊢ (seq0( + , (𝑛 ∈ ℕ₀ ↦ ((2↑_𝑐(1 − (ℜ‘𝑆)))↑𝑛))) ⇝ (1 / (1 − (2↑_𝑐(1 − (ℜ‘𝑆))))) → seq0( + , (𝑛 ∈ ℕ₀ ↦ ((2↑_𝑐(1 − (ℜ‘𝑆)))↑𝑛))) ∈ dom ⇝ )
96	92, 95	syl 17	. . 3 ⊢ (𝜑 → seq0( + , (𝑛 ∈ ℕ₀ ↦ ((2↑_𝑐(1 − (ℜ‘𝑆)))↑𝑛))) ∈ dom ⇝ )
97		rpcxpcl 26184	. . . . . . 7 ⊢ ((𝑘 ∈ ℝ⁺ ∧ -(ℜ‘𝑆) ∈ ℝ) → (𝑘↑_𝑐-(ℜ‘𝑆)) ∈ ℝ⁺)
98	19, 40, 97	syl2anr 598	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝑘↑_𝑐-(ℜ‘𝑆)) ∈ ℝ⁺)
99	98	rpred 13016	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝑘↑_𝑐-(ℜ‘𝑆)) ∈ ℝ)
100	7, 99	eqeltrd 2834	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ (𝑛↑_𝑐-(ℜ‘𝑆)))‘𝑘) ∈ ℝ)
101	98	rpge0d 13020	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → 0 ≤ (𝑘↑_𝑐-(ℜ‘𝑆)))
102	101, 7	breqtrrd 5177	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → 0 ≤ ((𝑛 ∈ ℕ ↦ (𝑛↑_𝑐-(ℜ‘𝑆)))‘𝑘))
103		nnre 12219	. . . . . . . . . . . . 13 ⊢ (𝑘 ∈ ℕ → 𝑘 ∈ ℝ)
104	103	lep1d 12145	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ ℕ → 𝑘 ≤ (𝑘 + 1))
105	19	reeflogd 26132	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ ℕ → (exp‘(log‘𝑘)) = 𝑘)
106		peano2nn 12224	. . . . . . . . . . . . . 14 ⊢ (𝑘 ∈ ℕ → (𝑘 + 1) ∈ ℕ)
107	106	nnrpd 13014	. . . . . . . . . . . . 13 ⊢ (𝑘 ∈ ℕ → (𝑘 + 1) ∈ ℝ⁺)
108	107	reeflogd 26132	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ ℕ → (exp‘(log‘(𝑘 + 1))) = (𝑘 + 1))
109	104, 105, 108	3brtr4d 5181	. . . . . . . . . . 11 ⊢ (𝑘 ∈ ℕ → (exp‘(log‘𝑘)) ≤ (exp‘(log‘(𝑘 + 1))))
110	107	relogcld 26131	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ ℕ → (log‘(𝑘 + 1)) ∈ ℝ)
111		efle 16061	. . . . . . . . . . . 12 ⊢ (((log‘𝑘) ∈ ℝ ∧ (log‘(𝑘 + 1)) ∈ ℝ) → ((log‘𝑘) ≤ (log‘(𝑘 + 1)) ↔ (exp‘(log‘𝑘)) ≤ (exp‘(log‘(𝑘 + 1)))))
112	20, 110, 111	syl2anc 585	. . . . . . . . . . 11 ⊢ (𝑘 ∈ ℕ → ((log‘𝑘) ≤ (log‘(𝑘 + 1)) ↔ (exp‘(log‘𝑘)) ≤ (exp‘(log‘(𝑘 + 1)))))
113	109, 112	mpbird 257	. . . . . . . . . 10 ⊢ (𝑘 ∈ ℕ → (log‘𝑘) ≤ (log‘(𝑘 + 1)))
114	113	adantl 483	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (log‘𝑘) ≤ (log‘(𝑘 + 1)))
115	20	adantl 483	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (log‘𝑘) ∈ ℝ)
116	106	adantl 483	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝑘 + 1) ∈ ℕ)
117	116	nnrpd 13014	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝑘 + 1) ∈ ℝ⁺)
118	117	relogcld 26131	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (log‘(𝑘 + 1)) ∈ ℝ)
119	39	adantr 482	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (ℜ‘𝑆) ∈ ℝ)
120	67	a1i 11	. . . . . . . . . . . 12 ⊢ (𝜑 → 0 ∈ ℝ)
121	55	a1i 11	. . . . . . . . . . . 12 ⊢ (𝜑 → 1 ∈ ℝ)
122		0lt1 11736	. . . . . . . . . . . . 13 ⊢ 0 < 1
123	122	a1i 11	. . . . . . . . . . . 12 ⊢ (𝜑 → 0 < 1)
124	120, 121, 39, 123, 61	lttrd 11375	. . . . . . . . . . 11 ⊢ (𝜑 → 0 < (ℜ‘𝑆))
125	124	adantr 482	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → 0 < (ℜ‘𝑆))
126		lemul2 12067	. . . . . . . . . 10 ⊢ (((log‘𝑘) ∈ ℝ ∧ (log‘(𝑘 + 1)) ∈ ℝ ∧ ((ℜ‘𝑆) ∈ ℝ ∧ 0 < (ℜ‘𝑆))) → ((log‘𝑘) ≤ (log‘(𝑘 + 1)) ↔ ((ℜ‘𝑆) · (log‘𝑘)) ≤ ((ℜ‘𝑆) · (log‘(𝑘 + 1)))))
127	115, 118, 119, 125, 126	syl112anc 1375	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ((log‘𝑘) ≤ (log‘(𝑘 + 1)) ↔ ((ℜ‘𝑆) · (log‘𝑘)) ≤ ((ℜ‘𝑆) · (log‘(𝑘 + 1)))))
128	114, 127	mpbid 231	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ((ℜ‘𝑆) · (log‘𝑘)) ≤ ((ℜ‘𝑆) · (log‘(𝑘 + 1))))
129		remulcl 11195	. . . . . . . . . 10 ⊢ (((ℜ‘𝑆) ∈ ℝ ∧ (log‘𝑘) ∈ ℝ) → ((ℜ‘𝑆) · (log‘𝑘)) ∈ ℝ)
130	39, 20, 129	syl2an 597	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ((ℜ‘𝑆) · (log‘𝑘)) ∈ ℝ)
131		remulcl 11195	. . . . . . . . . 10 ⊢ (((ℜ‘𝑆) ∈ ℝ ∧ (log‘(𝑘 + 1)) ∈ ℝ) → ((ℜ‘𝑆) · (log‘(𝑘 + 1))) ∈ ℝ)
132	39, 110, 131	syl2an 597	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ((ℜ‘𝑆) · (log‘(𝑘 + 1))) ∈ ℝ)
133	130, 132	lenegd 11793	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (((ℜ‘𝑆) · (log‘𝑘)) ≤ ((ℜ‘𝑆) · (log‘(𝑘 + 1))) ↔ -((ℜ‘𝑆) · (log‘(𝑘 + 1))) ≤ -((ℜ‘𝑆) · (log‘𝑘))))
134	128, 133	mpbid 231	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → -((ℜ‘𝑆) · (log‘(𝑘 + 1))) ≤ -((ℜ‘𝑆) · (log‘𝑘)))
135	110	recnd 11242	. . . . . . . 8 ⊢ (𝑘 ∈ ℕ → (log‘(𝑘 + 1)) ∈ ℂ)
136		mulneg1 11650	. . . . . . . 8 ⊢ (((ℜ‘𝑆) ∈ ℂ ∧ (log‘(𝑘 + 1)) ∈ ℂ) → (-(ℜ‘𝑆) · (log‘(𝑘 + 1))) = -((ℜ‘𝑆) · (log‘(𝑘 + 1))))
137	64, 135, 136	syl2an 597	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (-(ℜ‘𝑆) · (log‘(𝑘 + 1))) = -((ℜ‘𝑆) · (log‘(𝑘 + 1))))
138		mulneg1 11650	. . . . . . . 8 ⊢ (((ℜ‘𝑆) ∈ ℂ ∧ (log‘𝑘) ∈ ℂ) → (-(ℜ‘𝑆) · (log‘𝑘)) = -((ℜ‘𝑆) · (log‘𝑘)))
139	64, 21, 138	syl2an 597	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (-(ℜ‘𝑆) · (log‘𝑘)) = -((ℜ‘𝑆) · (log‘𝑘)))
140	134, 137, 139	3brtr4d 5181	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (-(ℜ‘𝑆) · (log‘(𝑘 + 1))) ≤ (-(ℜ‘𝑆) · (log‘𝑘)))
141		remulcl 11195	. . . . . . . 8 ⊢ ((-(ℜ‘𝑆) ∈ ℝ ∧ (log‘(𝑘 + 1)) ∈ ℝ) → (-(ℜ‘𝑆) · (log‘(𝑘 + 1))) ∈ ℝ)
142	40, 110, 141	syl2an 597	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (-(ℜ‘𝑆) · (log‘(𝑘 + 1))) ∈ ℝ)
143		remulcl 11195	. . . . . . . 8 ⊢ ((-(ℜ‘𝑆) ∈ ℝ ∧ (log‘𝑘) ∈ ℝ) → (-(ℜ‘𝑆) · (log‘𝑘)) ∈ ℝ)
144	40, 20, 143	syl2an 597	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (-(ℜ‘𝑆) · (log‘𝑘)) ∈ ℝ)
145		efle 16061	. . . . . . 7 ⊢ (((-(ℜ‘𝑆) · (log‘(𝑘 + 1))) ∈ ℝ ∧ (-(ℜ‘𝑆) · (log‘𝑘)) ∈ ℝ) → ((-(ℜ‘𝑆) · (log‘(𝑘 + 1))) ≤ (-(ℜ‘𝑆) · (log‘𝑘)) ↔ (exp‘(-(ℜ‘𝑆) · (log‘(𝑘 + 1)))) ≤ (exp‘(-(ℜ‘𝑆) · (log‘𝑘)))))
146	142, 144, 145	syl2anc 585	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ((-(ℜ‘𝑆) · (log‘(𝑘 + 1))) ≤ (-(ℜ‘𝑆) · (log‘𝑘)) ↔ (exp‘(-(ℜ‘𝑆) · (log‘(𝑘 + 1)))) ≤ (exp‘(-(ℜ‘𝑆) · (log‘𝑘)))))
147	140, 146	mpbid 231	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (exp‘(-(ℜ‘𝑆) · (log‘(𝑘 + 1)))) ≤ (exp‘(-(ℜ‘𝑆) · (log‘𝑘))))
148		oveq1 7416	. . . . . . . 8 ⊢ (𝑛 = (𝑘 + 1) → (𝑛↑_𝑐-(ℜ‘𝑆)) = ((𝑘 + 1)↑_𝑐-(ℜ‘𝑆)))
149		ovex 7442	. . . . . . . 8 ⊢ ((𝑘 + 1)↑_𝑐-(ℜ‘𝑆)) ∈ V
150	148, 4, 149	fvmpt 6999	. . . . . . 7 ⊢ ((𝑘 + 1) ∈ ℕ → ((𝑛 ∈ ℕ ↦ (𝑛↑_𝑐-(ℜ‘𝑆)))‘(𝑘 + 1)) = ((𝑘 + 1)↑_𝑐-(ℜ‘𝑆)))
151	116, 150	syl 17	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ (𝑛↑_𝑐-(ℜ‘𝑆)))‘(𝑘 + 1)) = ((𝑘 + 1)↑_𝑐-(ℜ‘𝑆)))
152	116	nncnd 12228	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝑘 + 1) ∈ ℂ)
153	116	nnne0d 12262	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝑘 + 1) ≠ 0)
154	152, 153, 47	cxpefd 26220	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ((𝑘 + 1)↑_𝑐-(ℜ‘𝑆)) = (exp‘(-(ℜ‘𝑆) · (log‘(𝑘 + 1)))))
155	151, 154	eqtrd 2773	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ (𝑛↑_𝑐-(ℜ‘𝑆)))‘(𝑘 + 1)) = (exp‘(-(ℜ‘𝑆) · (log‘(𝑘 + 1)))))
156	7, 48	eqtrd 2773	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ (𝑛↑_𝑐-(ℜ‘𝑆)))‘𝑘) = (exp‘(-(ℜ‘𝑆) · (log‘𝑘))))
157	147, 155, 156	3brtr4d 5181	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ (𝑛↑_𝑐-(ℜ‘𝑆)))‘(𝑘 + 1)) ≤ ((𝑛 ∈ ℕ ↦ (𝑛↑_𝑐-(ℜ‘𝑆)))‘𝑘))
158	57	recnd 11242	. . . . . . . . . . 11 ⊢ (𝜑 → (1 − (ℜ‘𝑆)) ∈ ℂ)
159	158	adantr 482	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ₀) → (1 − (ℜ‘𝑆)) ∈ ℂ)
160		nn0re 12481	. . . . . . . . . . . 12 ⊢ (𝑚 ∈ ℕ₀ → 𝑚 ∈ ℝ)
161	160	adantl 483	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ₀) → 𝑚 ∈ ℝ)
162	161	recnd 11242	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ₀) → 𝑚 ∈ ℂ)
163	159, 162	mulcomd 11235	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ₀) → ((1 − (ℜ‘𝑆)) · 𝑚) = (𝑚 · (1 − (ℜ‘𝑆))))
164	163	oveq2d 7425	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ₀) → (2↑_𝑐((1 − (ℜ‘𝑆)) · 𝑚)) = (2↑_𝑐(𝑚 · (1 − (ℜ‘𝑆)))))
165	54	a1i 11	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ₀) → 2 ∈ ℝ⁺)
166	165, 161, 159	cxpmuld 26245	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ₀) → (2↑_𝑐(𝑚 · (1 − (ℜ‘𝑆)))) = ((2↑_𝑐𝑚)↑_𝑐(1 − (ℜ‘𝑆))))
167		simpr 486	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ₀) → 𝑚 ∈ ℕ₀)
168		cxpexp 26176	. . . . . . . . . 10 ⊢ ((2 ∈ ℂ ∧ 𝑚 ∈ ℕ₀) → (2↑_𝑐𝑚) = (2↑𝑚))
169	81, 167, 168	sylancr 588	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ₀) → (2↑_𝑐𝑚) = (2↑𝑚))
170		ax-1cn 11168	. . . . . . . . . . 11 ⊢ 1 ∈ ℂ
171	64	adantr 482	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ₀) → (ℜ‘𝑆) ∈ ℂ)
172		negsub 11508	. . . . . . . . . . 11 ⊢ ((1 ∈ ℂ ∧ (ℜ‘𝑆) ∈ ℂ) → (1 + -(ℜ‘𝑆)) = (1 − (ℜ‘𝑆)))
173	170, 171, 172	sylancr 588	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ₀) → (1 + -(ℜ‘𝑆)) = (1 − (ℜ‘𝑆)))
174	173	eqcomd 2739	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ₀) → (1 − (ℜ‘𝑆)) = (1 + -(ℜ‘𝑆)))
175	169, 174	oveq12d 7427	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ₀) → ((2↑_𝑐𝑚)↑_𝑐(1 − (ℜ‘𝑆))) = ((2↑𝑚)↑_𝑐(1 + -(ℜ‘𝑆))))
176	164, 166, 175	3eqtrd 2777	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ₀) → (2↑_𝑐((1 − (ℜ‘𝑆)) · 𝑚)) = ((2↑𝑚)↑_𝑐(1 + -(ℜ‘𝑆))))
177	57	adantr 482	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ₀) → (1 − (ℜ‘𝑆)) ∈ ℝ)
178	165, 177, 162	cxpmuld 26245	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ₀) → (2↑_𝑐((1 − (ℜ‘𝑆)) · 𝑚)) = ((2↑_𝑐(1 − (ℜ‘𝑆)))↑_𝑐𝑚))
179		2nn 12285	. . . . . . . . . . 11 ⊢ 2 ∈ ℕ
180		nnexpcl 14040	. . . . . . . . . . 11 ⊢ ((2 ∈ ℕ ∧ 𝑚 ∈ ℕ₀) → (2↑𝑚) ∈ ℕ)
181	179, 180	mpan 689	. . . . . . . . . 10 ⊢ (𝑚 ∈ ℕ₀ → (2↑𝑚) ∈ ℕ)
182	181	adantl 483	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ₀) → (2↑𝑚) ∈ ℕ)
183	182	nncnd 12228	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ₀) → (2↑𝑚) ∈ ℂ)
184	182	nnne0d 12262	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ₀) → (2↑𝑚) ≠ 0)
185		1cnd 11209	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ₀) → 1 ∈ ℂ)
186	41	adantr 482	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ₀) → -(ℜ‘𝑆) ∈ ℂ)
187	183, 184, 185, 186	cxpaddd 26225	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ₀) → ((2↑𝑚)↑_𝑐(1 + -(ℜ‘𝑆))) = (((2↑𝑚)↑_𝑐1) · ((2↑𝑚)↑_𝑐-(ℜ‘𝑆))))
188	176, 178, 187	3eqtr3d 2781	. . . . . 6 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ₀) → ((2↑_𝑐(1 − (ℜ‘𝑆)))↑_𝑐𝑚) = (((2↑𝑚)↑_𝑐1) · ((2↑𝑚)↑_𝑐-(ℜ‘𝑆))))
189		cxpexp 26176	. . . . . . 7 ⊢ (((2↑_𝑐(1 − (ℜ‘𝑆))) ∈ ℂ ∧ 𝑚 ∈ ℕ₀) → ((2↑_𝑐(1 − (ℜ‘𝑆)))↑_𝑐𝑚) = ((2↑_𝑐(1 − (ℜ‘𝑆)))↑𝑚))
190	60, 189	sylan 581	. . . . . 6 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ₀) → ((2↑_𝑐(1 − (ℜ‘𝑆)))↑_𝑐𝑚) = ((2↑_𝑐(1 − (ℜ‘𝑆)))↑𝑚))
191	183	cxp1d 26214	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ₀) → ((2↑𝑚)↑_𝑐1) = (2↑𝑚))
192	191	oveq1d 7424	. . . . . 6 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ₀) → (((2↑𝑚)↑_𝑐1) · ((2↑𝑚)↑_𝑐-(ℜ‘𝑆))) = ((2↑𝑚) · ((2↑𝑚)↑_𝑐-(ℜ‘𝑆))))
193	188, 190, 192	3eqtr3d 2781	. . . . 5 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ₀) → ((2↑_𝑐(1 − (ℜ‘𝑆)))↑𝑚) = ((2↑𝑚) · ((2↑𝑚)↑_𝑐-(ℜ‘𝑆))))
194	179, 167, 180	sylancr 588	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ₀) → (2↑𝑚) ∈ ℕ)
195		oveq1 7416	. . . . . . . 8 ⊢ (𝑛 = (2↑𝑚) → (𝑛↑_𝑐-(ℜ‘𝑆)) = ((2↑𝑚)↑_𝑐-(ℜ‘𝑆)))
196		ovex 7442	. . . . . . . 8 ⊢ ((2↑𝑚)↑_𝑐-(ℜ‘𝑆)) ∈ V
197	195, 4, 196	fvmpt 6999	. . . . . . 7 ⊢ ((2↑𝑚) ∈ ℕ → ((𝑛 ∈ ℕ ↦ (𝑛↑_𝑐-(ℜ‘𝑆)))‘(2↑𝑚)) = ((2↑𝑚)↑_𝑐-(ℜ‘𝑆)))
198	194, 197	syl 17	. . . . . 6 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ₀) → ((𝑛 ∈ ℕ ↦ (𝑛↑_𝑐-(ℜ‘𝑆)))‘(2↑𝑚)) = ((2↑𝑚)↑_𝑐-(ℜ‘𝑆)))
199	198	oveq2d 7425	. . . . 5 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ₀) → ((2↑𝑚) · ((𝑛 ∈ ℕ ↦ (𝑛↑_𝑐-(ℜ‘𝑆)))‘(2↑𝑚))) = ((2↑𝑚) · ((2↑𝑚)↑_𝑐-(ℜ‘𝑆))))
200	193, 91, 199	3eqtr4d 2783	. . . 4 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ₀) → ((𝑛 ∈ ℕ₀ ↦ ((2↑_𝑐(1 − (ℜ‘𝑆)))↑𝑛))‘𝑚) = ((2↑𝑚) · ((𝑛 ∈ ℕ ↦ (𝑛↑_𝑐-(ℜ‘𝑆)))‘(2↑𝑚))))
201	100, 102, 157, 200	climcnds 15797	. . 3 ⊢ (𝜑 → (seq1( + , (𝑛 ∈ ℕ ↦ (𝑛↑_𝑐-(ℜ‘𝑆)))) ∈ dom ⇝ ↔ seq0( + , (𝑛 ∈ ℕ₀ ↦ ((2↑_𝑐(1 − (ℜ‘𝑆)))↑𝑛))) ∈ dom ⇝ ))
202	96, 201	mpbird 257	. 2 ⊢ (𝜑 → seq1( + , (𝑛 ∈ ℕ ↦ (𝑛↑_𝑐-(ℜ‘𝑆)))) ∈ dom ⇝ )
203	1, 2, 51, 53, 202	abscvgcvg 15765	1 ⊢ (𝜑 → seq1( + , 𝐹) ∈ dom ⇝ )

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 397 = wceq 1542 ∈ wcel 2107 ≠ wne 2941 class class class wbr 5149 ↦ cmpt 5232 dom cdm 5677 ‘cfv 6544 (class class class)co 7409 ℂcc 11108 ℝcr 11109 0cc0 11110 1c1 11111 + caddc 11113 · cmul 11115 < clt 11248 ≤ cle 11249 − cmin 11444 -cneg 11445 / cdiv 11871 ℕcn 12212 2c2 12267 ℕ₀cn0 12472 ℝ⁺crp 12974 seqcseq 13966 ↑cexp 14027 ℜcre 15044 ℑcim 15045 abscabs 15181 ⇝ cli 15428 expce 16005 logclog 26063 ↑_𝑐ccxp 26064

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 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7725 ax-inf2 9636 ax-cnex 11166 ax-resscn 11167 ax-1cn 11168 ax-icn 11169 ax-addcl 11170 ax-addrcl 11171 ax-mulcl 11172 ax-mulrcl 11173 ax-mulcom 11174 ax-addass 11175 ax-mulass 11176 ax-distr 11177 ax-i2m1 11178 ax-1ne0 11179 ax-1rid 11180 ax-rnegex 11181 ax-rrecex 11182 ax-cnre 11183 ax-pre-lttri 11184 ax-pre-lttrn 11185 ax-pre-ltadd 11186 ax-pre-mulgt0 11187 ax-pre-sup 11188 ax-addf 11189 ax-mulf 11190

This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-tp 4634 df-op 4636 df-uni 4910 df-int 4952 df-iun 5000 df-iin 5001 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5575 df-eprel 5581 df-po 5589 df-so 5590 df-fr 5632 df-se 5633 df-we 5634 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-pred 6301 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-isom 6553 df-riota 7365 df-ov 7412 df-oprab 7413 df-mpo 7414 df-of 7670 df-om 7856 df-1st 7975 df-2nd 7976 df-supp 8147 df-frecs 8266 df-wrecs 8297 df-recs 8371 df-rdg 8410 df-1o 8466 df-2o 8467 df-oadd 8470 df-er 8703 df-map 8822 df-pm 8823 df-ixp 8892 df-en 8940 df-dom 8941 df-sdom 8942 df-fin 8943 df-fsupp 9362 df-fi 9406 df-sup 9437 df-inf 9438 df-oi 9505 df-dju 9896 df-card 9934 df-pnf 11250 df-mnf 11251 df-xr 11252 df-ltxr 11253 df-le 11254 df-sub 11446 df-neg 11447 df-div 11872 df-nn 12213 df-2 12275 df-3 12276 df-4 12277 df-5 12278 df-6 12279 df-7 12280 df-8 12281 df-9 12282 df-n0 12473 df-z 12559 df-dec 12678 df-uz 12823 df-q 12933 df-rp 12975 df-xneg 13092 df-xadd 13093 df-xmul 13094 df-ioo 13328 df-ioc 13329 df-ico 13330 df-icc 13331 df-fz 13485 df-fzo 13628 df-fl 13757 df-mod 13835 df-seq 13967 df-exp 14028 df-fac 14234 df-bc 14263 df-hash 14291 df-shft 15014 df-cj 15046 df-re 15047 df-im 15048 df-sqrt 15182 df-abs 15183 df-limsup 15415 df-clim 15432 df-rlim 15433 df-sum 15633 df-ef 16011 df-sin 16013 df-cos 16014 df-pi 16016 df-struct 17080 df-sets 17097 df-slot 17115 df-ndx 17127 df-base 17145 df-ress 17174 df-plusg 17210 df-mulr 17211 df-starv 17212 df-sca 17213 df-vsca 17214 df-ip 17215 df-tset 17216 df-ple 17217 df-ds 17219 df-unif 17220 df-hom 17221 df-cco 17222 df-rest 17368 df-topn 17369 df-0g 17387 df-gsum 17388 df-topgen 17389 df-pt 17390 df-prds 17393 df-xrs 17448 df-qtop 17453 df-imas 17454 df-xps 17456 df-mre 17530 df-mrc 17531 df-acs 17533 df-mgm 18561 df-sgrp 18610 df-mnd 18626 df-submnd 18672 df-mulg 18951 df-cntz 19181 df-cmn 19650 df-psmet 20936 df-xmet 20937 df-met 20938 df-bl 20939 df-mopn 20940 df-fbas 20941 df-fg 20942 df-cnfld 20945 df-top 22396 df-topon 22413 df-topsp 22435 df-bases 22449 df-cld 22523 df-ntr 22524 df-cls 22525 df-nei 22602 df-lp 22640 df-perf 22641 df-cn 22731 df-cnp 22732 df-haus 22819 df-tx 23066 df-hmeo 23259 df-fil 23350 df-fm 23442 df-flim 23443 df-flf 23444 df-xms 23826 df-ms 23827 df-tms 23828 df-cncf 24394 df-limc 25383 df-dv 25384 df-log 26065 df-cxp 26066

This theorem is referenced by: lgamgulmlem4 26536

W3C validator