Theorem prmorcht 16248

Description: Relate the primorial (product of the primes up to 𝐴) to the Chebyshev function. (Contributed by Mario Carneiro, 22-Sep-2014.)

Hypothesis

Ref	Expression
prmorcht.1	⊢ 𝐹 = (𝑛 ∈ ℕ ↦ if(𝑛 ∈ ℙ, 𝑛, 1))

Assertion

Ref	Expression
prmorcht	⊢ (𝐴 ∈ ℕ → (exp‘(θ‘𝐴)) = (seq1( · , 𝐹)‘𝐴))

Proof of Theorem prmorcht

Dummy variables 𝑘 𝑝 𝑗 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		nnq 10043	. . . . . . 7 ⊢ (𝐴 ∈ ℕ → 𝐴 ∈ ℚ)
2		chtqval 16211	. . . . . . 7 ⊢ (𝐴 ∈ ℚ → (θ‘𝐴) = Σ𝑘 ∈ ((0[,]𝐴) ∩ ℙ)(log‘𝑘))
3	1, 2	syl 14	. . . . . 6 ⊢ (𝐴 ∈ ℕ → (θ‘𝐴) = Σ𝑘 ∈ ((0[,]𝐴) ∩ ℙ)(log‘𝑘))
4		2eluzge1 9986	. . . . . . . . . 10 ⊢ 2 ∈ (ℤ≥‘1)
5		ppiqsval2 16207	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℚ ∧ 2 ∈ (ℤ≥‘1)) → ((0[,]𝐴) ∩ ℙ) = ((1...(⌊‘𝐴)) ∩ ℙ))
6	1, 4, 5	sylancl 417	. . . . . . . . 9 ⊢ (𝐴 ∈ ℕ → ((0[,]𝐴) ∩ ℙ) = ((1...(⌊‘𝐴)) ∩ ℙ))
7		nnz 9668	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ ℕ → 𝐴 ∈ ℤ)
8		flid 10734	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ ℤ → (⌊‘𝐴) = 𝐴)
9	7, 8	syl 14	. . . . . . . . . . 11 ⊢ (𝐴 ∈ ℕ → (⌊‘𝐴) = 𝐴)
10	9	oveq2d 6101	. . . . . . . . . 10 ⊢ (𝐴 ∈ ℕ → (1...(⌊‘𝐴)) = (1...𝐴))
11	10	ineq1d 3431	. . . . . . . . 9 ⊢ (𝐴 ∈ ℕ → ((1...(⌊‘𝐴)) ∩ ℙ) = ((1...𝐴) ∩ ℙ))
12	6, 11	eqtrd 2271	. . . . . . . 8 ⊢ (𝐴 ∈ ℕ → ((0[,]𝐴) ∩ ℙ) = ((1...𝐴) ∩ ℙ))
13	12	sumeq1d 12151	. . . . . . 7 ⊢ (𝐴 ∈ ℕ → Σ𝑘 ∈ ((0[,]𝐴) ∩ ℙ)(log‘𝑘) = Σ𝑘 ∈ ((1...𝐴) ∩ ℙ)(log‘𝑘))
14		inss1 3451	. . . . . . . . 9 ⊢ ((1...𝐴) ∩ ℙ) ⊆ (1...𝐴)
15	14	a1i 9	. . . . . . . 8 ⊢ (𝐴 ∈ ℕ → ((1...𝐴) ∩ ℙ) ⊆ (1...𝐴))
16		animorrl 838	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℕ ∧ 𝑗 ∈ (1...𝐴)) → (𝑗 ∈ (1...𝐴) ∨ ¬ 𝑗 ∈ (1...𝐴)))
17		df-dc 847	. . . . . . . . . . . 12 ⊢ (DECID 𝑗 ∈ (1...𝐴) ↔ (𝑗 ∈ (1...𝐴) ∨ ¬ 𝑗 ∈ (1...𝐴)))
18	16, 17	sylibr 134	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℕ ∧ 𝑗 ∈ (1...𝐴)) → DECID 𝑗 ∈ (1...𝐴))
19		elfzelz 10439	. . . . . . . . . . . . 13 ⊢ (𝑗 ∈ (1...𝐴) → 𝑗 ∈ ℤ)
20	19	adantl 277	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℕ ∧ 𝑗 ∈ (1...𝐴)) → 𝑗 ∈ ℤ)
21		prmdcz 12928	. . . . . . . . . . . 12 ⊢ (𝑗 ∈ ℤ → DECID 𝑗 ∈ ℙ)
22	20, 21	syl 14	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℕ ∧ 𝑗 ∈ (1...𝐴)) → DECID 𝑗 ∈ ℙ)
23	18, 22	dcand 945	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℕ ∧ 𝑗 ∈ (1...𝐴)) → DECID (𝑗 ∈ (1...𝐴) ∧ 𝑗 ∈ ℙ))
24		elin 3412	. . . . . . . . . . 11 ⊢ (𝑗 ∈ ((1...𝐴) ∩ ℙ) ↔ (𝑗 ∈ (1...𝐴) ∧ 𝑗 ∈ ℙ))
25	24	dcbii 852	. . . . . . . . . 10 ⊢ (DECID 𝑗 ∈ ((1...𝐴) ∩ ℙ) ↔ DECID (𝑗 ∈ (1...𝐴) ∧ 𝑗 ∈ ℙ))
26	23, 25	sylibr 134	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℕ ∧ 𝑗 ∈ (1...𝐴)) → DECID 𝑗 ∈ ((1...𝐴) ∩ ℙ))
27	26	ralrimiva 2623	. . . . . . . 8 ⊢ (𝐴 ∈ ℕ → ∀𝑗 ∈ (1...𝐴)DECID 𝑗 ∈ ((1...𝐴) ∩ ℙ))
28		elinel1 3415	. . . . . . . . . 10 ⊢ (𝑘 ∈ ((1...𝐴) ∩ ℙ) → 𝑘 ∈ (1...𝐴))
29		elfznn 10471	. . . . . . . . . . . . . 14 ⊢ (𝑘 ∈ (1...𝐴) → 𝑘 ∈ ℕ)
30	29	adantl 277	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℕ ∧ 𝑘 ∈ (1...𝐴)) → 𝑘 ∈ ℕ)
31	30	nnrpd 10106	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℕ ∧ 𝑘 ∈ (1...𝐴)) → 𝑘 ∈ ℝ+)
32	31	relogcld 16078	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℕ ∧ 𝑘 ∈ (1...𝐴)) → (log‘𝑘) ∈ ℝ)
33	32	recnd 8355	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℕ ∧ 𝑘 ∈ (1...𝐴)) → (log‘𝑘) ∈ ℂ)
34	28, 33	sylan2 286	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℕ ∧ 𝑘 ∈ ((1...𝐴) ∩ ℙ)) → (log‘𝑘) ∈ ℂ)
35	34	ralrimiva 2623	. . . . . . . 8 ⊢ (𝐴 ∈ ℕ → ∀𝑘 ∈ ((1...𝐴) ∩ ℙ)(log‘𝑘) ∈ ℂ)
36		1zzd 9676	. . . . . . . . . 10 ⊢ (𝐴 ∈ ℕ → 1 ∈ ℤ)
37	36, 7	fzfigd 10883	. . . . . . . . 9 ⊢ (𝐴 ∈ ℕ → (1...𝐴) ∈ Fin)
38	37	olcd 746	. . . . . . . 8 ⊢ (𝐴 ∈ ℕ → ((1 ∈ ℤ ∧ (1...𝐴) ⊆ (ℤ≥‘1) ∧ ∀𝑗 ∈ (ℤ≥‘1)DECID 𝑗 ∈ (1...𝐴)) ∨ (1...𝐴) ∈ Fin))
39	15, 27, 35, 38	isumss2 12179	. . . . . . 7 ⊢ (𝐴 ∈ ℕ → Σ𝑘 ∈ ((1...𝐴) ∩ ℙ)(log‘𝑘) = Σ𝑘 ∈ (1...𝐴)if(𝑘 ∈ ((1...𝐴) ∩ ℙ), (log‘𝑘), 0))
40	13, 39	eqtrd 2271	. . . . . 6 ⊢ (𝐴 ∈ ℕ → Σ𝑘 ∈ ((0[,]𝐴) ∩ ℙ)(log‘𝑘) = Σ𝑘 ∈ (1...𝐴)if(𝑘 ∈ ((1...𝐴) ∩ ℙ), (log‘𝑘), 0))
41	3, 40	eqtrd 2271	. . . . 5 ⊢ (𝐴 ∈ ℕ → (θ‘𝐴) = Σ𝑘 ∈ (1...𝐴)if(𝑘 ∈ ((1...𝐴) ∩ ℙ), (log‘𝑘), 0))
42		elin 3412	. . . . . . . 8 ⊢ (𝑘 ∈ ((1...𝐴) ∩ ℙ) ↔ (𝑘 ∈ (1...𝐴) ∧ 𝑘 ∈ ℙ))
43	42	baibr 932	. . . . . . 7 ⊢ (𝑘 ∈ (1...𝐴) → (𝑘 ∈ ℙ ↔ 𝑘 ∈ ((1...𝐴) ∩ ℙ)))
44	43	ifbid 3662	. . . . . 6 ⊢ (𝑘 ∈ (1...𝐴) → if(𝑘 ∈ ℙ, (log‘𝑘), 0) = if(𝑘 ∈ ((1...𝐴) ∩ ℙ), (log‘𝑘), 0))
45	44	sumeq2i 12149	. . . . 5 ⊢ Σ𝑘 ∈ (1...𝐴)if(𝑘 ∈ ℙ, (log‘𝑘), 0) = Σ𝑘 ∈ (1...𝐴)if(𝑘 ∈ ((1...𝐴) ∩ ℙ), (log‘𝑘), 0)
46	41, 45	eqtr4di 2289	. . . 4 ⊢ (𝐴 ∈ ℕ → (θ‘𝐴) = Σ𝑘 ∈ (1...𝐴)if(𝑘 ∈ ℙ, (log‘𝑘), 0))
47		eqid 2238	. . . . . 6 ⊢ (𝑛 ∈ ℕ ↦ if(𝑛 ∈ ℙ, (log‘𝑛), 0)) = (𝑛 ∈ ℕ ↦ if(𝑛 ∈ ℙ, (log‘𝑛), 0))
48		eleq1w 2299	. . . . . . 7 ⊢ (𝑛 = 𝑘 → (𝑛 ∈ ℙ ↔ 𝑘 ∈ ℙ))
49		fveq2 5695	. . . . . . 7 ⊢ (𝑛 = 𝑘 → (log‘𝑛) = (log‘𝑘))
50	48, 49	ifbieq1d 3663	. . . . . 6 ⊢ (𝑛 = 𝑘 → if(𝑛 ∈ ℙ, (log‘𝑛), 0) = if(𝑘 ∈ ℙ, (log‘𝑘), 0))
51		elnnuz 9969	. . . . . . 7 ⊢ (𝑘 ∈ ℕ ↔ 𝑘 ∈ (ℤ≥‘1))
52	51	bilanri 389	. . . . . 6 ⊢ ((𝐴 ∈ ℕ ∧ 𝑘 ∈ (ℤ≥‘1)) → 𝑘 ∈ ℕ)
53	52	nnrpd 10106	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℕ ∧ 𝑘 ∈ (ℤ≥‘1)) → 𝑘 ∈ ℝ+)
54	53	relogcld 16078	. . . . . . . 8 ⊢ ((𝐴 ∈ ℕ ∧ 𝑘 ∈ (ℤ≥‘1)) → (log‘𝑘) ∈ ℝ)
55	54	recnd 8355	. . . . . . 7 ⊢ ((𝐴 ∈ ℕ ∧ 𝑘 ∈ (ℤ≥‘1)) → (log‘𝑘) ∈ ℂ)
56		0cnd 8320	. . . . . . 7 ⊢ ((𝐴 ∈ ℕ ∧ 𝑘 ∈ (ℤ≥‘1)) → 0 ∈ ℂ)
57		eluzelz 9941	. . . . . . . . 9 ⊢ (𝑘 ∈ (ℤ≥‘1) → 𝑘 ∈ ℤ)
58	57	adantl 277	. . . . . . . 8 ⊢ ((𝐴 ∈ ℕ ∧ 𝑘 ∈ (ℤ≥‘1)) → 𝑘 ∈ ℤ)
59		prmdcz 12928	. . . . . . . 8 ⊢ (𝑘 ∈ ℤ → DECID 𝑘 ∈ ℙ)
60	58, 59	syl 14	. . . . . . 7 ⊢ ((𝐴 ∈ ℕ ∧ 𝑘 ∈ (ℤ≥‘1)) → DECID 𝑘 ∈ ℙ)
61	55, 56, 60	ifcldcd 3678	. . . . . 6 ⊢ ((𝐴 ∈ ℕ ∧ 𝑘 ∈ (ℤ≥‘1)) → if(𝑘 ∈ ℙ, (log‘𝑘), 0) ∈ ℂ)
62	47, 50, 52, 61	fvmptd3 5799	. . . . 5 ⊢ ((𝐴 ∈ ℕ ∧ 𝑘 ∈ (ℤ≥‘1)) → ((𝑛 ∈ ℕ ↦ if(𝑛 ∈ ℙ, (log‘𝑛), 0))‘𝑘) = if(𝑘 ∈ ℙ, (log‘𝑘), 0))
63		elnnuz 9969	. . . . . 6 ⊢ (𝐴 ∈ ℕ ↔ 𝐴 ∈ (ℤ≥‘1))
64	63	biimpi 120	. . . . 5 ⊢ (𝐴 ∈ ℕ → 𝐴 ∈ (ℤ≥‘1))
65	62, 64, 61	fsum3ser 12183	. . . 4 ⊢ (𝐴 ∈ ℕ → Σ𝑘 ∈ (1...𝐴)if(𝑘 ∈ ℙ, (log‘𝑘), 0) = (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛 ∈ ℙ, (log‘𝑛), 0)))‘𝐴))
66	46, 65	eqtrd 2271	. . 3 ⊢ (𝐴 ∈ ℕ → (θ‘𝐴) = (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛 ∈ ℙ, (log‘𝑛), 0)))‘𝐴))
67	66	fveq2d 5699	. 2 ⊢ (𝐴 ∈ ℕ → (exp‘(θ‘𝐴)) = (exp‘(seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛 ∈ ℙ, (log‘𝑛), 0)))‘𝐴)))
68		addcl 8305	. . . 4 ⊢ ((𝑘 ∈ ℂ ∧ 𝑝 ∈ ℂ) → (𝑘 + 𝑝) ∈ ℂ)
69	68	adantl 277	. . 3 ⊢ ((𝐴 ∈ ℕ ∧ (𝑘 ∈ ℂ ∧ 𝑝 ∈ ℂ)) → (𝑘 + 𝑝) ∈ ℂ)
70	62, 61	eqeltrd 2315	. . 3 ⊢ ((𝐴 ∈ ℕ ∧ 𝑘 ∈ (ℤ≥‘1)) → ((𝑛 ∈ ℕ ↦ if(𝑛 ∈ ℙ, (log‘𝑛), 0))‘𝑘) ∈ ℂ)
71		efadd 12461	. . . 4 ⊢ ((𝑘 ∈ ℂ ∧ 𝑝 ∈ ℂ) → (exp‘(𝑘 + 𝑝)) = ((exp‘𝑘) · (exp‘𝑝)))
72	71	adantl 277	. . 3 ⊢ ((𝐴 ∈ ℕ ∧ (𝑘 ∈ ℂ ∧ 𝑝 ∈ ℂ)) → (exp‘(𝑘 + 𝑝)) = ((exp‘𝑘) · (exp‘𝑝)))
73		simpr 110	. . . . . . . 8 ⊢ ((𝐴 ∈ ℕ ∧ 𝑘 ∈ ℕ) → 𝑘 ∈ ℕ)
74		1nn 9318	. . . . . . . . 9 ⊢ 1 ∈ ℕ
75	74	a1i 9	. . . . . . . 8 ⊢ ((𝐴 ∈ ℕ ∧ 𝑘 ∈ ℕ) → 1 ∈ ℕ)
76	51, 60	sylan2b 287	. . . . . . . 8 ⊢ ((𝐴 ∈ ℕ ∧ 𝑘 ∈ ℕ) → DECID 𝑘 ∈ ℙ)
77	73, 75, 76	ifcldcd 3678	. . . . . . 7 ⊢ ((𝐴 ∈ ℕ ∧ 𝑘 ∈ ℕ) → if(𝑘 ∈ ℙ, 𝑘, 1) ∈ ℕ)
78	77	nnrpd 10106	. . . . . 6 ⊢ ((𝐴 ∈ ℕ ∧ 𝑘 ∈ ℕ) → if(𝑘 ∈ ℙ, 𝑘, 1) ∈ ℝ+)
79	78	reeflogd 16079	. . . . 5 ⊢ ((𝐴 ∈ ℕ ∧ 𝑘 ∈ ℕ) → (exp‘(log‘if(𝑘 ∈ ℙ, 𝑘, 1))) = if(𝑘 ∈ ℙ, 𝑘, 1))
80	51, 62	sylan2b 287	. . . . . . 7 ⊢ ((𝐴 ∈ ℕ ∧ 𝑘 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ if(𝑛 ∈ ℙ, (log‘𝑛), 0))‘𝑘) = if(𝑘 ∈ ℙ, (log‘𝑘), 0))
81		prmdc 12927	. . . . . . . . . 10 ⊢ (𝑘 ∈ ℕ → DECID 𝑘 ∈ ℙ)
82		fvifdc 5717	. . . . . . . . . 10 ⊢ (DECID 𝑘 ∈ ℙ → (log‘if(𝑘 ∈ ℙ, 𝑘, 1)) = if(𝑘 ∈ ℙ, (log‘𝑘), (log‘1)))
83	81, 82	syl 14	. . . . . . . . 9 ⊢ (𝑘 ∈ ℕ → (log‘if(𝑘 ∈ ℙ, 𝑘, 1)) = if(𝑘 ∈ ℙ, (log‘𝑘), (log‘1)))
84	83	adantl 277	. . . . . . . 8 ⊢ ((𝐴 ∈ ℕ ∧ 𝑘 ∈ ℕ) → (log‘if(𝑘 ∈ ℙ, 𝑘, 1)) = if(𝑘 ∈ ℙ, (log‘𝑘), (log‘1)))
85		log1 16061	. . . . . . . . 9 ⊢ (log‘1) = 0
86		ifeq2 3644	. . . . . . . . 9 ⊢ ((log‘1) = 0 → if(𝑘 ∈ ℙ, (log‘𝑘), (log‘1)) = if(𝑘 ∈ ℙ, (log‘𝑘), 0))
87	85, 86	ax-mp 5	. . . . . . . 8 ⊢ if(𝑘 ∈ ℙ, (log‘𝑘), (log‘1)) = if(𝑘 ∈ ℙ, (log‘𝑘), 0)
88	84, 87	eqtrdi 2287	. . . . . . 7 ⊢ ((𝐴 ∈ ℕ ∧ 𝑘 ∈ ℕ) → (log‘if(𝑘 ∈ ℙ, 𝑘, 1)) = if(𝑘 ∈ ℙ, (log‘𝑘), 0))
89	80, 88	eqtr4d 2274	. . . . . 6 ⊢ ((𝐴 ∈ ℕ ∧ 𝑘 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ if(𝑛 ∈ ℙ, (log‘𝑛), 0))‘𝑘) = (log‘if(𝑘 ∈ ℙ, 𝑘, 1)))
90	89	fveq2d 5699	. . . . 5 ⊢ ((𝐴 ∈ ℕ ∧ 𝑘 ∈ ℕ) → (exp‘((𝑛 ∈ ℕ ↦ if(𝑛 ∈ ℙ, (log‘𝑛), 0))‘𝑘)) = (exp‘(log‘if(𝑘 ∈ ℙ, 𝑘, 1))))
91		id 19	. . . . . . . 8 ⊢ (𝑛 = 𝑘 → 𝑛 = 𝑘)
92	48, 91	ifbieq1d 3663	. . . . . . 7 ⊢ (𝑛 = 𝑘 → if(𝑛 ∈ ℙ, 𝑛, 1) = if(𝑘 ∈ ℙ, 𝑘, 1))
93		prmorcht.1	. . . . . . 7 ⊢ 𝐹 = (𝑛 ∈ ℕ ↦ if(𝑛 ∈ ℙ, 𝑛, 1))
94		vex 2824	. . . . . . . 8 ⊢ 𝑘 ∈ V
95		1ex 8322	. . . . . . . 8 ⊢ 1 ∈ V
96	94, 95	ifex 4632	. . . . . . 7 ⊢ if(𝑘 ∈ ℙ, 𝑘, 1) ∈ V
97	92, 93, 96	fvmpt 5782	. . . . . 6 ⊢ (𝑘 ∈ ℕ → (𝐹‘𝑘) = if(𝑘 ∈ ℙ, 𝑘, 1))
98	97	adantl 277	. . . . 5 ⊢ ((𝐴 ∈ ℕ ∧ 𝑘 ∈ ℕ) → (𝐹‘𝑘) = if(𝑘 ∈ ℙ, 𝑘, 1))
99	79, 90, 98	3eqtr4d 2281	. . . 4 ⊢ ((𝐴 ∈ ℕ ∧ 𝑘 ∈ ℕ) → (exp‘((𝑛 ∈ ℕ ↦ if(𝑛 ∈ ℙ, (log‘𝑛), 0))‘𝑘)) = (𝐹‘𝑘))
100	52, 99	syldan 282	. . 3 ⊢ ((𝐴 ∈ ℕ ∧ 𝑘 ∈ (ℤ≥‘1)) → (exp‘((𝑛 ∈ ℕ ↦ if(𝑛 ∈ ℙ, (log‘𝑛), 0))‘𝑘)) = (𝐹‘𝑘))
101		efcl 12450	. . . . 5 ⊢ (((𝑛 ∈ ℕ ↦ if(𝑛 ∈ ℙ, (log‘𝑛), 0))‘𝑘) ∈ ℂ → (exp‘((𝑛 ∈ ℕ ↦ if(𝑛 ∈ ℙ, (log‘𝑛), 0))‘𝑘)) ∈ ℂ)
102	70, 101	syl 14	. . . 4 ⊢ ((𝐴 ∈ ℕ ∧ 𝑘 ∈ (ℤ≥‘1)) → (exp‘((𝑛 ∈ ℕ ↦ if(𝑛 ∈ ℙ, (log‘𝑛), 0))‘𝑘)) ∈ ℂ)
103	100, 102	eqeltrrd 2316	. . 3 ⊢ ((𝐴 ∈ ℕ ∧ 𝑘 ∈ (ℤ≥‘1)) → (𝐹‘𝑘) ∈ ℂ)
104		mulcl 8307	. . . 4 ⊢ ((𝑘 ∈ ℂ ∧ 𝑝 ∈ ℂ) → (𝑘 · 𝑝) ∈ ℂ)
105	104	adantl 277	. . 3 ⊢ ((𝐴 ∈ ℕ ∧ (𝑘 ∈ ℂ ∧ 𝑝 ∈ ℂ)) → (𝑘 · 𝑝) ∈ ℂ)
106	69, 70, 64, 72, 100, 103, 105	seq3homo 10979	. 2 ⊢ (𝐴 ∈ ℕ → (exp‘(seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛 ∈ ℙ, (log‘𝑛), 0)))‘𝐴)) = (seq1( · , 𝐹)‘𝐴))
107	67, 106	eqtrd 2271	1 ⊢ (𝐴 ∈ ℕ → (exp‘(θ‘𝐴)) = (seq1( · , 𝐹)‘𝐴))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 104   ∨ wo 720  DECID wdc 846   ∧ w3a 1009   = wceq 1402   ∈ wcel 2209  ∀wral 2528   ∩ cin 3219   ⊆ wss 3220  ifcif 3638   ↦ cmpt 4192  ‘cfv 5377  (class class class)co 6085  Fincfn 7022  ℂcc 8178  0cc0 8180  1c1 8181   + caddc 8183   · cmul 8185  ℕcn 9307  2c2 9358  ℤcz 9649  ℤ_≥cuz 9931  ℚcq 10029  [,]cicc 10304  ...cfz 10422  ⌊cfl 10714  seqcseq 10899  Σcsu 12138  expce 12428  ℙcprime 12904  logclog 16051  θccht 16199

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4246  ax-sep 4249  ax-nul 4259  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684  ax-iinf 4735  ax-cnex 8271  ax-resscn 8272  ax-1cn 8273  ax-1re 8274  ax-icn 8275  ax-addcl 8276  ax-addrcl 8277  ax-mulcl 8278  ax-mulrcl 8279  ax-addcom 8280  ax-mulcom 8281  ax-addass 8282  ax-mulass 8283  ax-distr 8284  ax-i2m1 8285  ax-0lt1 8286  ax-1rid 8287  ax-0id 8288  ax-rnegex 8289  ax-precex 8290  ax-cnre 8291  ax-pre-ltirr 8292  ax-pre-ltwlin 8293  ax-pre-lttrn 8294  ax-pre-apti 8295  ax-pre-ltadd 8296  ax-pre-mulgt0 8297  ax-pre-mulext 8298  ax-arch 8299  ax-caucvg 8300  ax-pre-suploc 8301  ax-addf 8302  ax-mulf 8303

This proof depends on definitions:  df-bi 117  df-stab 843  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-if 3639  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-disj 4107  df-br 4131  df-opab 4193  df-mpt 4194  df-tr 4230  df-id 4438  df-po 4441  df-iso 4442  df-iord 4511  df-on 4513  df-ilim 4514  df-suc 4516  df-iom 4738  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-f1 5382  df-fo 5383  df-f1o 5384  df-fv 5385  df-isom 5386  df-riota 6038  df-ov 6088  df-oprab 6089  df-mpo 6090  df-of 6302  df-1st 6374  df-2nd 6375  df-recs 6576  df-irdg 6641  df-frec 6662  df-1o 6687  df-2o 6688  df-oadd 6691  df-er 6807  df-map 6924  df-pm 6925  df-en 7023  df-dom 7024  df-fin 7025  df-sup 7325  df-inf 7326  df-pnf 8363  df-mnf 8364  df-xr 8365  df-ltxr 8366  df-le 8367  df-sub 8501  df-neg 8502  df-reap 8906  df-ap 8913  df-div 9006  df-inn 9308  df-2 9366  df-3 9367  df-4 9368  df-n0 9569  df-z 9650  df-uz 9932  df-q 10030  df-rp 10066  df-xneg 10185  df-xadd 10186  df-ioo 10305  df-ico 10307  df-icc 10308  df-fz 10423  df-fzo 10561  df-fl 10716  df-mod 10775  df-seqfrec 10900  df-exp 10991  df-fac 11180  df-bc 11202  df-ihash 11231  df-shft 11596  df-cj 11623  df-re 11624  df-im 11625  df-rsqrt 11780  df-abs 11781  df-clim 12064  df-sumdc 12139  df-ef 12434  df-e 12435  df-dvds 12574  df-prm 12905  df-rest 13648  df-topgen 13667  df-psmet 14964  df-xmet 14965  df-met 14966  df-bl 14967  df-mopn 14968  df-top 15190  df-topon 15203  df-bases 15235  df-ntr 15288  df-cn 15380  df-cnp 15381  df-tx 15445  df-cncf 15763  df-limced 15848  df-dvap 15849  df-relog 16053  df-cht 16202

This theorem is used by:  chtublem  16261  bposlem6  16282