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Theorem zetacvg 26967

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 12903	. 2 ⊢ ℕ = (ℤ_≥‘1)
2		1zzd 12631	. 2 ⊢ (𝜑 → 1 ∈ ℤ)
3		oveq1 7433	. . . . 5 ⊢ (𝑛 = 𝑘 → (𝑛↑_𝑐-(ℜ‘𝑆)) = (𝑘↑_𝑐-(ℜ‘𝑆)))
4		eqid 2728	. . . . 5 ⊢ (𝑛 ∈ ℕ ↦ (𝑛↑_𝑐-(ℜ‘𝑆))) = (𝑛 ∈ ℕ ↦ (𝑛↑_𝑐-(ℜ‘𝑆)))
5		ovex 7459	. . . . 5 ⊢ (𝑘↑_𝑐-(ℜ‘𝑆)) ∈ V
6	3, 4, 5	fvmpt 7010	. . . 4 ⊢ (𝑘 ∈ ℕ → ((𝑛 ∈ ℕ ↦ (𝑛↑_𝑐-(ℜ‘𝑆)))‘𝑘) = (𝑘↑_𝑐-(ℜ‘𝑆)))
7	6	adantl 480	. . 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 12258	. . . . . . . 8 ⊢ (𝑘 ∈ ℕ → 𝑘 ∈ ℂ)
10	9	adantl 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → 𝑘 ∈ ℂ)
11		nnne0 12284	. . . . . . . 8 ⊢ (𝑘 ∈ ℕ → 𝑘 ≠ 0)
12	11	adantl 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → 𝑘 ≠ 0)
13		zetacvg.1	. . . . . . . . 9 ⊢ (𝜑 → 𝑆 ∈ ℂ)
14	13	negcld 11596	. . . . . . . 8 ⊢ (𝜑 → -𝑆 ∈ ℂ)
15	14	adantr 479	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → -𝑆 ∈ ℂ)
16	10, 12, 15	cxpefd 26666	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝑘↑_𝑐-𝑆) = (exp‘(-𝑆 · (log‘𝑘))))
17	8, 16	eqtrd 2768	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝐹‘𝑘) = (exp‘(-𝑆 · (log‘𝑘))))
18	17	fveq2d 6906	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (abs‘(𝐹‘𝑘)) = (abs‘(exp‘(-𝑆 · (log‘𝑘)))))
19		nnrp 13025	. . . . . . . 8 ⊢ (𝑘 ∈ ℕ → 𝑘 ∈ ℝ⁺)
20	19	relogcld 26577	. . . . . . 7 ⊢ (𝑘 ∈ ℕ → (log‘𝑘) ∈ ℝ)
21	20	recnd 11280	. . . . . 6 ⊢ (𝑘 ∈ ℕ → (log‘𝑘) ∈ ℂ)
22		mulcl 11230	. . . . . 6 ⊢ ((-𝑆 ∈ ℂ ∧ (log‘𝑘) ∈ ℂ) → (-𝑆 · (log‘𝑘)) ∈ ℂ)
23	14, 21, 22	syl2an 594	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (-𝑆 · (log‘𝑘)) ∈ ℂ)
24		absef 16181	. . . . 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 15116	. . . . . . . 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 594	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (ℜ‘(-𝑆 · (log‘𝑘))) = (((ℜ‘-𝑆) · (ℜ‘(log‘𝑘))) − ((ℑ‘-𝑆) · (ℑ‘(log‘𝑘)))))
28	13	renegd 15196	. . . . . . . . 9 ⊢ (𝜑 → (ℜ‘-𝑆) = -(ℜ‘𝑆))
29	20	rered 15211	. . . . . . . . 9 ⊢ (𝑘 ∈ ℕ → (ℜ‘(log‘𝑘)) = (log‘𝑘))
30	28, 29	oveqan12d 7445	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ((ℜ‘-𝑆) · (ℜ‘(log‘𝑘))) = (-(ℜ‘𝑆) · (log‘𝑘)))
31	20	reim0d 15212	. . . . . . . . . 10 ⊢ (𝑘 ∈ ℕ → (ℑ‘(log‘𝑘)) = 0)
32	31	oveq2d 7442	. . . . . . . . 9 ⊢ (𝑘 ∈ ℕ → ((ℑ‘-𝑆) · (ℑ‘(log‘𝑘))) = ((ℑ‘-𝑆) · 0))
33		imcl 15098	. . . . . . . . . . . 12 ⊢ (-𝑆 ∈ ℂ → (ℑ‘-𝑆) ∈ ℝ)
34	33	recnd 11280	. . . . . . . . . . 11 ⊢ (-𝑆 ∈ ℂ → (ℑ‘-𝑆) ∈ ℂ)
35	14, 34	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → (ℑ‘-𝑆) ∈ ℂ)
36	35	mul01d 11451	. . . . . . . . 9 ⊢ (𝜑 → ((ℑ‘-𝑆) · 0) = 0)
37	32, 36	sylan9eqr 2790	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ((ℑ‘-𝑆) · (ℑ‘(log‘𝑘))) = 0)
38	30, 37	oveq12d 7444	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (((ℜ‘-𝑆) · (ℜ‘(log‘𝑘))) − ((ℑ‘-𝑆) · (ℑ‘(log‘𝑘)))) = ((-(ℜ‘𝑆) · (log‘𝑘)) − 0))
39	13	recld 15181	. . . . . . . . . . 11 ⊢ (𝜑 → (ℜ‘𝑆) ∈ ℝ)
40	39	renegcld 11679	. . . . . . . . . 10 ⊢ (𝜑 → -(ℜ‘𝑆) ∈ ℝ)
41	40	recnd 11280	. . . . . . . . 9 ⊢ (𝜑 → -(ℜ‘𝑆) ∈ ℂ)
42		mulcl 11230	. . . . . . . . 9 ⊢ ((-(ℜ‘𝑆) ∈ ℂ ∧ (log‘𝑘) ∈ ℂ) → (-(ℜ‘𝑆) · (log‘𝑘)) ∈ ℂ)
43	41, 21, 42	syl2an 594	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (-(ℜ‘𝑆) · (log‘𝑘)) ∈ ℂ)
44	43	subid1d 11598	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ((-(ℜ‘𝑆) · (log‘𝑘)) − 0) = (-(ℜ‘𝑆) · (log‘𝑘)))
45	27, 38, 44	3eqtrd 2772	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (ℜ‘(-𝑆 · (log‘𝑘))) = (-(ℜ‘𝑆) · (log‘𝑘)))
46	45	fveq2d 6906	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (exp‘(ℜ‘(-𝑆 · (log‘𝑘)))) = (exp‘(-(ℜ‘𝑆) · (log‘𝑘))))
47	41	adantr 479	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → -(ℜ‘𝑆) ∈ ℂ)
48	10, 12, 47	cxpefd 26666	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝑘↑_𝑐-(ℜ‘𝑆)) = (exp‘(-(ℜ‘𝑆) · (log‘𝑘))))
49	46, 48	eqtr4d 2771	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (exp‘(ℜ‘(-𝑆 · (log‘𝑘)))) = (𝑘↑_𝑐-(ℜ‘𝑆)))
50	18, 25, 49	3eqtrd 2772	. . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (abs‘(𝐹‘𝑘)) = (𝑘↑_𝑐-(ℜ‘𝑆)))
51	7, 50	eqtr4d 2771	. 2 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ (𝑛↑_𝑐-(ℜ‘𝑆)))‘𝑘) = (abs‘(𝐹‘𝑘)))
52	10, 15	cxpcld 26662	. . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝑘↑_𝑐-𝑆) ∈ ℂ)
53	8, 52	eqeltrd 2829	. 2 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝐹‘𝑘) ∈ ℂ)
54		2rp 13019	. . . . . . 7 ⊢ 2 ∈ ℝ⁺
55		1re 11252	. . . . . . . 8 ⊢ 1 ∈ ℝ
56		resubcl 11562	. . . . . . . 8 ⊢ ((1 ∈ ℝ ∧ (ℜ‘𝑆) ∈ ℝ) → (1 − (ℜ‘𝑆)) ∈ ℝ)
57	55, 39, 56	sylancr 585	. . . . . . 7 ⊢ (𝜑 → (1 − (ℜ‘𝑆)) ∈ ℝ)
58		rpcxpcl 26630	. . . . . . 7 ⊢ ((2 ∈ ℝ⁺ ∧ (1 − (ℜ‘𝑆)) ∈ ℝ) → (2↑_𝑐(1 − (ℜ‘𝑆))) ∈ ℝ⁺)
59	54, 57, 58	sylancr 585	. . . . . 6 ⊢ (𝜑 → (2↑_𝑐(1 − (ℜ‘𝑆))) ∈ ℝ⁺)
60	59	rpcnd 13058	. . . . 5 ⊢ (𝜑 → (2↑_𝑐(1 − (ℜ‘𝑆))) ∈ ℂ)
61		zetacvg.2	. . . . . . . . 9 ⊢ (𝜑 → 1 < (ℜ‘𝑆))
62		recl 15097	. . . . . . . . . . . 12 ⊢ (𝑆 ∈ ℂ → (ℜ‘𝑆) ∈ ℝ)
63	62	recnd 11280	. . . . . . . . . . 11 ⊢ (𝑆 ∈ ℂ → (ℜ‘𝑆) ∈ ℂ)
64	13, 63	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → (ℜ‘𝑆) ∈ ℂ)
65	64	addlidd 11453	. . . . . . . . 9 ⊢ (𝜑 → (0 + (ℜ‘𝑆)) = (ℜ‘𝑆))
66	61, 65	breqtrrd 5180	. . . . . . . 8 ⊢ (𝜑 → 1 < (0 + (ℜ‘𝑆)))
67		0re 11254	. . . . . . . . . 10 ⊢ 0 ∈ ℝ
68		ltsubadd 11722	. . . . . . . . . 10 ⊢ ((1 ∈ ℝ ∧ (ℜ‘𝑆) ∈ ℝ ∧ 0 ∈ ℝ) → ((1 − (ℜ‘𝑆)) < 0 ↔ 1 < (0 + (ℜ‘𝑆))))
69	55, 67, 68	mp3an13 1448	. . . . . . . . 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 256	. . . . . . 7 ⊢ (𝜑 → (1 − (ℜ‘𝑆)) < 0)
72		2re 12324	. . . . . . . . 9 ⊢ 2 ∈ ℝ
73		1lt2 12421	. . . . . . . . 9 ⊢ 1 < 2
74		cxplt 26648	. . . . . . . . 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 700	. . . . . . . 8 ⊢ (((1 − (ℜ‘𝑆)) ∈ ℝ ∧ 0 ∈ ℝ) → ((1 − (ℜ‘𝑆)) < 0 ↔ (2↑_𝑐(1 − (ℜ‘𝑆))) < (2↑_𝑐0)))
76	57, 67, 75	sylancl 584	. . . . . . 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 13063	. . . . . . 7 ⊢ (𝜑 → ((2↑_𝑐(1 − (ℜ‘𝑆))) ∈ ℝ ∧ 0 ≤ (2↑_𝑐(1 − (ℜ‘𝑆)))))
79		absid 15283	. . . . . . 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 12325	. . . . . . . . 9 ⊢ 2 ∈ ℂ
82		cxp0 26624	. . . . . . . . 9 ⊢ (2 ∈ ℂ → (2↑_𝑐0) = 1)
83	81, 82	ax-mp 5	. . . . . . . 8 ⊢ (2↑_𝑐0) = 1
84	83	eqcomi 2737	. . . . . . 7 ⊢ 1 = (2↑_𝑐0)
85	84	a1i 11	. . . . . 6 ⊢ (𝜑 → 1 = (2↑_𝑐0))
86	77, 80, 85	3brtr4d 5184	. . . . 5 ⊢ (𝜑 → (abs‘(2↑_𝑐(1 − (ℜ‘𝑆)))) < 1)
87		oveq2 7434	. . . . . . 7 ⊢ (𝑛 = 𝑚 → ((2↑_𝑐(1 − (ℜ‘𝑆)))↑𝑛) = ((2↑_𝑐(1 − (ℜ‘𝑆)))↑𝑚))
88		eqid 2728	. . . . . . 7 ⊢ (𝑛 ∈ ℕ₀ ↦ ((2↑_𝑐(1 − (ℜ‘𝑆)))↑𝑛)) = (𝑛 ∈ ℕ₀ ↦ ((2↑_𝑐(1 − (ℜ‘𝑆)))↑𝑛))
89		ovex 7459	. . . . . . 7 ⊢ ((2↑_𝑐(1 − (ℜ‘𝑆)))↑𝑚) ∈ V
90	87, 88, 89	fvmpt 7010	. . . . . 6 ⊢ (𝑚 ∈ ℕ₀ → ((𝑛 ∈ ℕ₀ ↦ ((2↑_𝑐(1 − (ℜ‘𝑆)))↑𝑛))‘𝑚) = ((2↑_𝑐(1 − (ℜ‘𝑆)))↑𝑚))
91	90	adantl 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ₀) → ((𝑛 ∈ ℕ₀ ↦ ((2↑_𝑐(1 − (ℜ‘𝑆)))↑𝑛))‘𝑚) = ((2↑_𝑐(1 − (ℜ‘𝑆)))↑𝑚))
92	60, 86, 91	geolim 15856	. . . 4 ⊢ (𝜑 → seq0( + , (𝑛 ∈ ℕ₀ ↦ ((2↑_𝑐(1 − (ℜ‘𝑆)))↑𝑛))) ⇝ (1 / (1 − (2↑_𝑐(1 − (ℜ‘𝑆))))))
93		seqex 14008	. . . . 5 ⊢ seq0( + , (𝑛 ∈ ℕ₀ ↦ ((2↑_𝑐(1 − (ℜ‘𝑆)))↑𝑛))) ∈ V
94		ovex 7459	. . . . 5 ⊢ (1 / (1 − (2↑_𝑐(1 − (ℜ‘𝑆))))) ∈ V
95	93, 94	breldm 5915	. . . 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 26630	. . . . . . 7 ⊢ ((𝑘 ∈ ℝ⁺ ∧ -(ℜ‘𝑆) ∈ ℝ) → (𝑘↑_𝑐-(ℜ‘𝑆)) ∈ ℝ⁺)
98	19, 40, 97	syl2anr 595	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝑘↑_𝑐-(ℜ‘𝑆)) ∈ ℝ⁺)
99	98	rpred 13056	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝑘↑_𝑐-(ℜ‘𝑆)) ∈ ℝ)
100	7, 99	eqeltrd 2829	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ (𝑛↑_𝑐-(ℜ‘𝑆)))‘𝑘) ∈ ℝ)
101	98	rpge0d 13060	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → 0 ≤ (𝑘↑_𝑐-(ℜ‘𝑆)))
102	101, 7	breqtrrd 5180	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → 0 ≤ ((𝑛 ∈ ℕ ↦ (𝑛↑_𝑐-(ℜ‘𝑆)))‘𝑘))
103		nnre 12257	. . . . . . . . . . . . 13 ⊢ (𝑘 ∈ ℕ → 𝑘 ∈ ℝ)
104	103	lep1d 12183	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ ℕ → 𝑘 ≤ (𝑘 + 1))
105	19	reeflogd 26578	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ ℕ → (exp‘(log‘𝑘)) = 𝑘)
106		peano2nn 12262	. . . . . . . . . . . . . 14 ⊢ (𝑘 ∈ ℕ → (𝑘 + 1) ∈ ℕ)
107	106	nnrpd 13054	. . . . . . . . . . . . 13 ⊢ (𝑘 ∈ ℕ → (𝑘 + 1) ∈ ℝ⁺)
108	107	reeflogd 26578	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ ℕ → (exp‘(log‘(𝑘 + 1))) = (𝑘 + 1))
109	104, 105, 108	3brtr4d 5184	. . . . . . . . . . 11 ⊢ (𝑘 ∈ ℕ → (exp‘(log‘𝑘)) ≤ (exp‘(log‘(𝑘 + 1))))
110	107	relogcld 26577	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ ℕ → (log‘(𝑘 + 1)) ∈ ℝ)
111		efle 16102	. . . . . . . . . . . 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 582	. . . . . . . . . . 11 ⊢ (𝑘 ∈ ℕ → ((log‘𝑘) ≤ (log‘(𝑘 + 1)) ↔ (exp‘(log‘𝑘)) ≤ (exp‘(log‘(𝑘 + 1)))))
113	109, 112	mpbird 256	. . . . . . . . . 10 ⊢ (𝑘 ∈ ℕ → (log‘𝑘) ≤ (log‘(𝑘 + 1)))
114	113	adantl 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (log‘𝑘) ≤ (log‘(𝑘 + 1)))
115	20	adantl 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (log‘𝑘) ∈ ℝ)
116	106	adantl 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝑘 + 1) ∈ ℕ)
117	116	nnrpd 13054	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝑘 + 1) ∈ ℝ⁺)
118	117	relogcld 26577	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (log‘(𝑘 + 1)) ∈ ℝ)
119	39	adantr 479	. . . . . . . . . 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 11774	. . . . . . . . . . . . 13 ⊢ 0 < 1
123	122	a1i 11	. . . . . . . . . . . 12 ⊢ (𝜑 → 0 < 1)
124	120, 121, 39, 123, 61	lttrd 11413	. . . . . . . . . . 11 ⊢ (𝜑 → 0 < (ℜ‘𝑆))
125	124	adantr 479	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → 0 < (ℜ‘𝑆))
126		lemul2 12105	. . . . . . . . . 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 1371	. . . . . . . . 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 11231	. . . . . . . . . 10 ⊢ (((ℜ‘𝑆) ∈ ℝ ∧ (log‘𝑘) ∈ ℝ) → ((ℜ‘𝑆) · (log‘𝑘)) ∈ ℝ)
130	39, 20, 129	syl2an 594	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ((ℜ‘𝑆) · (log‘𝑘)) ∈ ℝ)
131		remulcl 11231	. . . . . . . . . 10 ⊢ (((ℜ‘𝑆) ∈ ℝ ∧ (log‘(𝑘 + 1)) ∈ ℝ) → ((ℜ‘𝑆) · (log‘(𝑘 + 1))) ∈ ℝ)
132	39, 110, 131	syl2an 594	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ((ℜ‘𝑆) · (log‘(𝑘 + 1))) ∈ ℝ)
133	130, 132	lenegd 11831	. . . . . . . 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 11280	. . . . . . . 8 ⊢ (𝑘 ∈ ℕ → (log‘(𝑘 + 1)) ∈ ℂ)
136		mulneg1 11688	. . . . . . . 8 ⊢ (((ℜ‘𝑆) ∈ ℂ ∧ (log‘(𝑘 + 1)) ∈ ℂ) → (-(ℜ‘𝑆) · (log‘(𝑘 + 1))) = -((ℜ‘𝑆) · (log‘(𝑘 + 1))))
137	64, 135, 136	syl2an 594	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (-(ℜ‘𝑆) · (log‘(𝑘 + 1))) = -((ℜ‘𝑆) · (log‘(𝑘 + 1))))
138		mulneg1 11688	. . . . . . . 8 ⊢ (((ℜ‘𝑆) ∈ ℂ ∧ (log‘𝑘) ∈ ℂ) → (-(ℜ‘𝑆) · (log‘𝑘)) = -((ℜ‘𝑆) · (log‘𝑘)))
139	64, 21, 138	syl2an 594	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (-(ℜ‘𝑆) · (log‘𝑘)) = -((ℜ‘𝑆) · (log‘𝑘)))
140	134, 137, 139	3brtr4d 5184	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (-(ℜ‘𝑆) · (log‘(𝑘 + 1))) ≤ (-(ℜ‘𝑆) · (log‘𝑘)))
141		remulcl 11231	. . . . . . . 8 ⊢ ((-(ℜ‘𝑆) ∈ ℝ ∧ (log‘(𝑘 + 1)) ∈ ℝ) → (-(ℜ‘𝑆) · (log‘(𝑘 + 1))) ∈ ℝ)
142	40, 110, 141	syl2an 594	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (-(ℜ‘𝑆) · (log‘(𝑘 + 1))) ∈ ℝ)
143		remulcl 11231	. . . . . . . 8 ⊢ ((-(ℜ‘𝑆) ∈ ℝ ∧ (log‘𝑘) ∈ ℝ) → (-(ℜ‘𝑆) · (log‘𝑘)) ∈ ℝ)
144	40, 20, 143	syl2an 594	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (-(ℜ‘𝑆) · (log‘𝑘)) ∈ ℝ)
145		efle 16102	. . . . . . 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 582	. . . . . 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 7433	. . . . . . . 8 ⊢ (𝑛 = (𝑘 + 1) → (𝑛↑_𝑐-(ℜ‘𝑆)) = ((𝑘 + 1)↑_𝑐-(ℜ‘𝑆)))
149		ovex 7459	. . . . . . . 8 ⊢ ((𝑘 + 1)↑_𝑐-(ℜ‘𝑆)) ∈ V
150	148, 4, 149	fvmpt 7010	. . . . . . 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 12266	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝑘 + 1) ∈ ℂ)
153	116	nnne0d 12300	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝑘 + 1) ≠ 0)
154	152, 153, 47	cxpefd 26666	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ((𝑘 + 1)↑_𝑐-(ℜ‘𝑆)) = (exp‘(-(ℜ‘𝑆) · (log‘(𝑘 + 1)))))
155	151, 154	eqtrd 2768	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ (𝑛↑_𝑐-(ℜ‘𝑆)))‘(𝑘 + 1)) = (exp‘(-(ℜ‘𝑆) · (log‘(𝑘 + 1)))))
156	7, 48	eqtrd 2768	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ (𝑛↑_𝑐-(ℜ‘𝑆)))‘𝑘) = (exp‘(-(ℜ‘𝑆) · (log‘𝑘))))
157	147, 155, 156	3brtr4d 5184	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ (𝑛↑_𝑐-(ℜ‘𝑆)))‘(𝑘 + 1)) ≤ ((𝑛 ∈ ℕ ↦ (𝑛↑_𝑐-(ℜ‘𝑆)))‘𝑘))
158	57	recnd 11280	. . . . . . . . . . 11 ⊢ (𝜑 → (1 − (ℜ‘𝑆)) ∈ ℂ)
159	158	adantr 479	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ₀) → (1 − (ℜ‘𝑆)) ∈ ℂ)
160		nn0re 12519	. . . . . . . . . . . 12 ⊢ (𝑚 ∈ ℕ₀ → 𝑚 ∈ ℝ)
161	160	adantl 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ₀) → 𝑚 ∈ ℝ)
162	161	recnd 11280	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ₀) → 𝑚 ∈ ℂ)
163	159, 162	mulcomd 11273	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ₀) → ((1 − (ℜ‘𝑆)) · 𝑚) = (𝑚 · (1 − (ℜ‘𝑆))))
164	163	oveq2d 7442	. . . . . . . 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 26691	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ₀) → (2↑_𝑐(𝑚 · (1 − (ℜ‘𝑆)))) = ((2↑_𝑐𝑚)↑_𝑐(1 − (ℜ‘𝑆))))
167		simpr 483	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ₀) → 𝑚 ∈ ℕ₀)
168		cxpexp 26622	. . . . . . . . . 10 ⊢ ((2 ∈ ℂ ∧ 𝑚 ∈ ℕ₀) → (2↑_𝑐𝑚) = (2↑𝑚))
169	81, 167, 168	sylancr 585	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ₀) → (2↑_𝑐𝑚) = (2↑𝑚))
170		ax-1cn 11204	. . . . . . . . . . 11 ⊢ 1 ∈ ℂ
171	64	adantr 479	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ₀) → (ℜ‘𝑆) ∈ ℂ)
172		negsub 11546	. . . . . . . . . . 11 ⊢ ((1 ∈ ℂ ∧ (ℜ‘𝑆) ∈ ℂ) → (1 + -(ℜ‘𝑆)) = (1 − (ℜ‘𝑆)))
173	170, 171, 172	sylancr 585	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ₀) → (1 + -(ℜ‘𝑆)) = (1 − (ℜ‘𝑆)))
174	173	eqcomd 2734	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ₀) → (1 − (ℜ‘𝑆)) = (1 + -(ℜ‘𝑆)))
175	169, 174	oveq12d 7444	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ₀) → ((2↑_𝑐𝑚)↑_𝑐(1 − (ℜ‘𝑆))) = ((2↑𝑚)↑_𝑐(1 + -(ℜ‘𝑆))))
176	164, 166, 175	3eqtrd 2772	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ₀) → (2↑_𝑐((1 − (ℜ‘𝑆)) · 𝑚)) = ((2↑𝑚)↑_𝑐(1 + -(ℜ‘𝑆))))
177	57	adantr 479	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ₀) → (1 − (ℜ‘𝑆)) ∈ ℝ)
178	165, 177, 162	cxpmuld 26691	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ₀) → (2↑_𝑐((1 − (ℜ‘𝑆)) · 𝑚)) = ((2↑_𝑐(1 − (ℜ‘𝑆)))↑_𝑐𝑚))
179		2nn 12323	. . . . . . . . . . 11 ⊢ 2 ∈ ℕ
180		nnexpcl 14079	. . . . . . . . . . 11 ⊢ ((2 ∈ ℕ ∧ 𝑚 ∈ ℕ₀) → (2↑𝑚) ∈ ℕ)
181	179, 180	mpan 688	. . . . . . . . . 10 ⊢ (𝑚 ∈ ℕ₀ → (2↑𝑚) ∈ ℕ)
182	181	adantl 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ₀) → (2↑𝑚) ∈ ℕ)
183	182	nncnd 12266	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ₀) → (2↑𝑚) ∈ ℂ)
184	182	nnne0d 12300	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ₀) → (2↑𝑚) ≠ 0)
185		1cnd 11247	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ₀) → 1 ∈ ℂ)
186	41	adantr 479	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ₀) → -(ℜ‘𝑆) ∈ ℂ)
187	183, 184, 185, 186	cxpaddd 26671	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ₀) → ((2↑𝑚)↑_𝑐(1 + -(ℜ‘𝑆))) = (((2↑𝑚)↑_𝑐1) · ((2↑𝑚)↑_𝑐-(ℜ‘𝑆))))
188	176, 178, 187	3eqtr3d 2776	. . . . . 6 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ₀) → ((2↑_𝑐(1 − (ℜ‘𝑆)))↑_𝑐𝑚) = (((2↑𝑚)↑_𝑐1) · ((2↑𝑚)↑_𝑐-(ℜ‘𝑆))))
189		cxpexp 26622	. . . . . . 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 578	. . . . . 6 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ₀) → ((2↑_𝑐(1 − (ℜ‘𝑆)))↑_𝑐𝑚) = ((2↑_𝑐(1 − (ℜ‘𝑆)))↑𝑚))
191	183	cxp1d 26660	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ₀) → ((2↑𝑚)↑_𝑐1) = (2↑𝑚))
192	191	oveq1d 7441	. . . . . 6 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ₀) → (((2↑𝑚)↑_𝑐1) · ((2↑𝑚)↑_𝑐-(ℜ‘𝑆))) = ((2↑𝑚) · ((2↑𝑚)↑_𝑐-(ℜ‘𝑆))))
193	188, 190, 192	3eqtr3d 2776	. . . . 5 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ₀) → ((2↑_𝑐(1 − (ℜ‘𝑆)))↑𝑚) = ((2↑𝑚) · ((2↑𝑚)↑_𝑐-(ℜ‘𝑆))))
194	179, 167, 180	sylancr 585	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ₀) → (2↑𝑚) ∈ ℕ)
195		oveq1 7433	. . . . . . . 8 ⊢ (𝑛 = (2↑𝑚) → (𝑛↑_𝑐-(ℜ‘𝑆)) = ((2↑𝑚)↑_𝑐-(ℜ‘𝑆)))
196		ovex 7459	. . . . . . . 8 ⊢ ((2↑𝑚)↑_𝑐-(ℜ‘𝑆)) ∈ V
197	195, 4, 196	fvmpt 7010	. . . . . . 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 7442	. . . . 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 2778	. . . 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 15837	. . 3 ⊢ (𝜑 → (seq1( + , (𝑛 ∈ ℕ ↦ (𝑛↑_𝑐-(ℜ‘𝑆)))) ∈ dom ⇝ ↔ seq0( + , (𝑛 ∈ ℕ₀ ↦ ((2↑_𝑐(1 − (ℜ‘𝑆)))↑𝑛))) ∈ dom ⇝ ))
202	96, 201	mpbird 256	. 2 ⊢ (𝜑 → seq1( + , (𝑛 ∈ ℕ ↦ (𝑛↑_𝑐-(ℜ‘𝑆)))) ∈ dom ⇝ )
203	1, 2, 51, 53, 202	abscvgcvg 15805	1 ⊢ (𝜑 → seq1( + , 𝐹) ∈ dom ⇝ )

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 394 = wceq 1533 ∈ wcel 2098 ≠ wne 2937 class class class wbr 5152 ↦ cmpt 5235 dom cdm 5682 ‘cfv 6553 (class class class)co 7426 ℂcc 11144 ℝcr 11145 0cc0 11146 1c1 11147 + caddc 11149 · cmul 11151 < clt 11286 ≤ cle 11287 − cmin 11482 -cneg 11483 / cdiv 11909 ℕcn 12250 2c2 12305 ℕ₀cn0 12510 ℝ⁺crp 13014 seqcseq 14006 ↑cexp 14066 ℜcre 15084 ℑcim 15085 abscabs 15221 ⇝ cli 15468 expce 16045 logclog 26508 ↑_𝑐ccxp 26509

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2699 ax-rep 5289 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-inf2 9672 ax-cnex 11202 ax-resscn 11203 ax-1cn 11204 ax-icn 11205 ax-addcl 11206 ax-addrcl 11207 ax-mulcl 11208 ax-mulrcl 11209 ax-mulcom 11210 ax-addass 11211 ax-mulass 11212 ax-distr 11213 ax-i2m1 11214 ax-1ne0 11215 ax-1rid 11216 ax-rnegex 11217 ax-rrecex 11218 ax-cnre 11219 ax-pre-lttri 11220 ax-pre-lttrn 11221 ax-pre-ltadd 11222 ax-pre-mulgt0 11223 ax-pre-sup 11224 ax-addf 11225

This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3374 df-reu 3375 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-tp 4637 df-op 4639 df-uni 4913 df-int 4954 df-iun 5002 df-iin 5003 df-br 5153 df-opab 5215 df-mpt 5236 df-tr 5270 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-se 5638 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6310 df-ord 6377 df-on 6378 df-lim 6379 df-suc 6380 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-isom 6562 df-riota 7382 df-ov 7429 df-oprab 7430 df-mpo 7431 df-of 7691 df-om 7877 df-1st 7999 df-2nd 8000 df-supp 8172 df-frecs 8293 df-wrecs 8324 df-recs 8398 df-rdg 8437 df-1o 8493 df-2o 8494 df-oadd 8497 df-er 8731 df-map 8853 df-pm 8854 df-ixp 8923 df-en 8971 df-dom 8972 df-sdom 8973 df-fin 8974 df-fsupp 9394 df-fi 9442 df-sup 9473 df-inf 9474 df-oi 9541 df-dju 9932 df-card 9970 df-pnf 11288 df-mnf 11289 df-xr 11290 df-ltxr 11291 df-le 11292 df-sub 11484 df-neg 11485 df-div 11910 df-nn 12251 df-2 12313 df-3 12314 df-4 12315 df-5 12316 df-6 12317 df-7 12318 df-8 12319 df-9 12320 df-n0 12511 df-z 12597 df-dec 12716 df-uz 12861 df-q 12971 df-rp 13015 df-xneg 13132 df-xadd 13133 df-xmul 13134 df-ioo 13368 df-ioc 13369 df-ico 13370 df-icc 13371 df-fz 13525 df-fzo 13668 df-fl 13797 df-mod 13875 df-seq 14007 df-exp 14067 df-fac 14273 df-bc 14302 df-hash 14330 df-shft 15054 df-cj 15086 df-re 15087 df-im 15088 df-sqrt 15222 df-abs 15223 df-limsup 15455 df-clim 15472 df-rlim 15473 df-sum 15673 df-ef 16051 df-sin 16053 df-cos 16054 df-pi 16056 df-struct 17123 df-sets 17140 df-slot 17158 df-ndx 17170 df-base 17188 df-ress 17217 df-plusg 17253 df-mulr 17254 df-starv 17255 df-sca 17256 df-vsca 17257 df-ip 17258 df-tset 17259 df-ple 17260 df-ds 17262 df-unif 17263 df-hom 17264 df-cco 17265 df-rest 17411 df-topn 17412 df-0g 17430 df-gsum 17431 df-topgen 17432 df-pt 17433 df-prds 17436 df-xrs 17491 df-qtop 17496 df-imas 17497 df-xps 17499 df-mre 17573 df-mrc 17574 df-acs 17576 df-mgm 18607 df-sgrp 18686 df-mnd 18702 df-submnd 18748 df-mulg 19031 df-cntz 19275 df-cmn 19744 df-psmet 21278 df-xmet 21279 df-met 21280 df-bl 21281 df-mopn 21282 df-fbas 21283 df-fg 21284 df-cnfld 21287 df-top 22816 df-topon 22833 df-topsp 22855 df-bases 22869 df-cld 22943 df-ntr 22944 df-cls 22945 df-nei 23022 df-lp 23060 df-perf 23061 df-cn 23151 df-cnp 23152 df-haus 23239 df-tx 23486 df-hmeo 23679 df-fil 23770 df-fm 23862 df-flim 23863 df-flf 23864 df-xms 24246 df-ms 24247 df-tms 24248 df-cncf 24818 df-limc 25815 df-dv 25816 df-log 26510 df-cxp 26511

This theorem is referenced by: lgamgulmlem4 26984

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