perfectALTV - Mathbox for Alexander van der Vekens

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Theorem perfectALTV 45440

Description: The Euclid-Euler theorem, or Perfect Number theorem. A positive even integer 𝑁 is a perfect number (that is, its divisor sum is 2𝑁) if and only if it is of the form 2↑(𝑝 − 1) · (2↑𝑝 − 1), where 2↑𝑝 − 1 is prime (a Mersenne prime). (It follows from this that 𝑝 is also prime.) This is Metamath 100 proof #70. (Contributed by Mario Carneiro, 17-May-2016.) (Revised by AV, 1-Jul-2020.) (Proof modification is discouraged.)

**Assertion**
Ref	Expression
perfectALTV	⊢ ((𝑁 ∈ ℕ ∧ 𝑁 ∈ Even ) → ((1 σ 𝑁) = (2 · 𝑁) ↔ ∃𝑝 ∈ ℤ (((2↑𝑝) − 1) ∈ ℙ ∧ 𝑁 = ((2↑(𝑝 − 1)) · ((2↑𝑝) − 1)))))

Distinct variable group: 𝑁,𝑝

**Proof of Theorem perfectALTV**
Step	Hyp	Ref	Expression
1		2dvdseven 45370	. . . . . . . 8 ⊢ (𝑁 ∈ Even → 2 ∥ 𝑁)
2	1	ad2antlr 724	. . . . . . 7 ⊢ (((𝑁 ∈ ℕ ∧ 𝑁 ∈ Even ) ∧ (1 σ 𝑁) = (2 · 𝑁)) → 2 ∥ 𝑁)
3		2prm 16474	. . . . . . . 8 ⊢ 2 ∈ ℙ
4		simpll 764	. . . . . . . 8 ⊢ (((𝑁 ∈ ℕ ∧ 𝑁 ∈ Even ) ∧ (1 σ 𝑁) = (2 · 𝑁)) → 𝑁 ∈ ℕ)
5		pcelnn 16648	. . . . . . . 8 ⊢ ((2 ∈ ℙ ∧ 𝑁 ∈ ℕ) → ((2 pCnt 𝑁) ∈ ℕ ↔ 2 ∥ 𝑁))
6	3, 4, 5	sylancr 587	. . . . . . 7 ⊢ (((𝑁 ∈ ℕ ∧ 𝑁 ∈ Even ) ∧ (1 σ 𝑁) = (2 · 𝑁)) → ((2 pCnt 𝑁) ∈ ℕ ↔ 2 ∥ 𝑁))
7	2, 6	mpbird 256	. . . . . 6 ⊢ (((𝑁 ∈ ℕ ∧ 𝑁 ∈ Even ) ∧ (1 σ 𝑁) = (2 · 𝑁)) → (2 pCnt 𝑁) ∈ ℕ)
8	7	nnzd 12505	. . . . 5 ⊢ (((𝑁 ∈ ℕ ∧ 𝑁 ∈ Even ) ∧ (1 σ 𝑁) = (2 · 𝑁)) → (2 pCnt 𝑁) ∈ ℤ)
9	8	peano2zd 12509	. . . 4 ⊢ (((𝑁 ∈ ℕ ∧ 𝑁 ∈ Even ) ∧ (1 σ 𝑁) = (2 · 𝑁)) → ((2 pCnt 𝑁) + 1) ∈ ℤ)
10		pcdvds 16642	. . . . . . . . 9 ⊢ ((2 ∈ ℙ ∧ 𝑁 ∈ ℕ) → (2↑(2 pCnt 𝑁)) ∥ 𝑁)
11	3, 4, 10	sylancr 587	. . . . . . . 8 ⊢ (((𝑁 ∈ ℕ ∧ 𝑁 ∈ Even ) ∧ (1 σ 𝑁) = (2 · 𝑁)) → (2↑(2 pCnt 𝑁)) ∥ 𝑁)
12		2nn 12126	. . . . . . . . . 10 ⊢ 2 ∈ ℕ
13	7	nnnn0d 12373	. . . . . . . . . 10 ⊢ (((𝑁 ∈ ℕ ∧ 𝑁 ∈ Even ) ∧ (1 σ 𝑁) = (2 · 𝑁)) → (2 pCnt 𝑁) ∈ ℕ₀)
14		nnexpcl 13875	. . . . . . . . . 10 ⊢ ((2 ∈ ℕ ∧ (2 pCnt 𝑁) ∈ ℕ₀) → (2↑(2 pCnt 𝑁)) ∈ ℕ)
15	12, 13, 14	sylancr 587	. . . . . . . . 9 ⊢ (((𝑁 ∈ ℕ ∧ 𝑁 ∈ Even ) ∧ (1 σ 𝑁) = (2 · 𝑁)) → (2↑(2 pCnt 𝑁)) ∈ ℕ)
16		nndivdvds 16051	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℕ ∧ (2↑(2 pCnt 𝑁)) ∈ ℕ) → ((2↑(2 pCnt 𝑁)) ∥ 𝑁 ↔ (𝑁 / (2↑(2 pCnt 𝑁))) ∈ ℕ))
17	4, 15, 16	syl2anc 584	. . . . . . . 8 ⊢ (((𝑁 ∈ ℕ ∧ 𝑁 ∈ Even ) ∧ (1 σ 𝑁) = (2 · 𝑁)) → ((2↑(2 pCnt 𝑁)) ∥ 𝑁 ↔ (𝑁 / (2↑(2 pCnt 𝑁))) ∈ ℕ))
18	11, 17	mpbid 231	. . . . . . 7 ⊢ (((𝑁 ∈ ℕ ∧ 𝑁 ∈ Even ) ∧ (1 σ 𝑁) = (2 · 𝑁)) → (𝑁 / (2↑(2 pCnt 𝑁))) ∈ ℕ)
19	18	nnzd 12505	. . . . . . . 8 ⊢ (((𝑁 ∈ ℕ ∧ 𝑁 ∈ Even ) ∧ (1 σ 𝑁) = (2 · 𝑁)) → (𝑁 / (2↑(2 pCnt 𝑁))) ∈ ℤ)
20		pcndvds2 16646	. . . . . . . . 9 ⊢ ((2 ∈ ℙ ∧ 𝑁 ∈ ℕ) → ¬ 2 ∥ (𝑁 / (2↑(2 pCnt 𝑁))))
21	3, 4, 20	sylancr 587	. . . . . . . 8 ⊢ (((𝑁 ∈ ℕ ∧ 𝑁 ∈ Even ) ∧ (1 σ 𝑁) = (2 · 𝑁)) → ¬ 2 ∥ (𝑁 / (2↑(2 pCnt 𝑁))))
22		isodd3 45369	. . . . . . . 8 ⊢ ((𝑁 / (2↑(2 pCnt 𝑁))) ∈ Odd ↔ ((𝑁 / (2↑(2 pCnt 𝑁))) ∈ ℤ ∧ ¬ 2 ∥ (𝑁 / (2↑(2 pCnt 𝑁)))))
23	19, 21, 22	sylanbrc 583	. . . . . . 7 ⊢ (((𝑁 ∈ ℕ ∧ 𝑁 ∈ Even ) ∧ (1 σ 𝑁) = (2 · 𝑁)) → (𝑁 / (2↑(2 pCnt 𝑁))) ∈ Odd )
24		simpr 485	. . . . . . . 8 ⊢ (((𝑁 ∈ ℕ ∧ 𝑁 ∈ Even ) ∧ (1 σ 𝑁) = (2 · 𝑁)) → (1 σ 𝑁) = (2 · 𝑁))
25		nncn 12061	. . . . . . . . . . 11 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℂ)
26	25	ad2antrr 723	. . . . . . . . . 10 ⊢ (((𝑁 ∈ ℕ ∧ 𝑁 ∈ Even ) ∧ (1 σ 𝑁) = (2 · 𝑁)) → 𝑁 ∈ ℂ)
27	15	nncnd 12069	. . . . . . . . . 10 ⊢ (((𝑁 ∈ ℕ ∧ 𝑁 ∈ Even ) ∧ (1 σ 𝑁) = (2 · 𝑁)) → (2↑(2 pCnt 𝑁)) ∈ ℂ)
28	15	nnne0d 12103	. . . . . . . . . 10 ⊢ (((𝑁 ∈ ℕ ∧ 𝑁 ∈ Even ) ∧ (1 σ 𝑁) = (2 · 𝑁)) → (2↑(2 pCnt 𝑁)) ≠ 0)
29	26, 27, 28	divcan2d 11833	. . . . . . . . 9 ⊢ (((𝑁 ∈ ℕ ∧ 𝑁 ∈ Even ) ∧ (1 σ 𝑁) = (2 · 𝑁)) → ((2↑(2 pCnt 𝑁)) · (𝑁 / (2↑(2 pCnt 𝑁)))) = 𝑁)
30	29	oveq2d 7333	. . . . . . . 8 ⊢ (((𝑁 ∈ ℕ ∧ 𝑁 ∈ Even ) ∧ (1 σ 𝑁) = (2 · 𝑁)) → (1 σ ((2↑(2 pCnt 𝑁)) · (𝑁 / (2↑(2 pCnt 𝑁))))) = (1 σ 𝑁))
31	29	oveq2d 7333	. . . . . . . 8 ⊢ (((𝑁 ∈ ℕ ∧ 𝑁 ∈ Even ) ∧ (1 σ 𝑁) = (2 · 𝑁)) → (2 · ((2↑(2 pCnt 𝑁)) · (𝑁 / (2↑(2 pCnt 𝑁))))) = (2 · 𝑁))
32	24, 30, 31	3eqtr4d 2787	. . . . . . 7 ⊢ (((𝑁 ∈ ℕ ∧ 𝑁 ∈ Even ) ∧ (1 σ 𝑁) = (2 · 𝑁)) → (1 σ ((2↑(2 pCnt 𝑁)) · (𝑁 / (2↑(2 pCnt 𝑁))))) = (2 · ((2↑(2 pCnt 𝑁)) · (𝑁 / (2↑(2 pCnt 𝑁))))))
33	7, 18, 23, 32	perfectALTVlem2 45439	. . . . . 6 ⊢ (((𝑁 ∈ ℕ ∧ 𝑁 ∈ Even ) ∧ (1 σ 𝑁) = (2 · 𝑁)) → ((𝑁 / (2↑(2 pCnt 𝑁))) ∈ ℙ ∧ (𝑁 / (2↑(2 pCnt 𝑁))) = ((2↑((2 pCnt 𝑁) + 1)) − 1)))
34	33	simprd 496	. . . . 5 ⊢ (((𝑁 ∈ ℕ ∧ 𝑁 ∈ Even ) ∧ (1 σ 𝑁) = (2 · 𝑁)) → (𝑁 / (2↑(2 pCnt 𝑁))) = ((2↑((2 pCnt 𝑁) + 1)) − 1))
35	33	simpld 495	. . . . 5 ⊢ (((𝑁 ∈ ℕ ∧ 𝑁 ∈ Even ) ∧ (1 σ 𝑁) = (2 · 𝑁)) → (𝑁 / (2↑(2 pCnt 𝑁))) ∈ ℙ)
36	34, 35	eqeltrrd 2839	. . . 4 ⊢ (((𝑁 ∈ ℕ ∧ 𝑁 ∈ Even ) ∧ (1 σ 𝑁) = (2 · 𝑁)) → ((2↑((2 pCnt 𝑁) + 1)) − 1) ∈ ℙ)
37	7	nncnd 12069	. . . . . . . . 9 ⊢ (((𝑁 ∈ ℕ ∧ 𝑁 ∈ Even ) ∧ (1 σ 𝑁) = (2 · 𝑁)) → (2 pCnt 𝑁) ∈ ℂ)
38		ax-1cn 11009	. . . . . . . . 9 ⊢ 1 ∈ ℂ
39		pncan 11307	. . . . . . . . 9 ⊢ (((2 pCnt 𝑁) ∈ ℂ ∧ 1 ∈ ℂ) → (((2 pCnt 𝑁) + 1) − 1) = (2 pCnt 𝑁))
40	37, 38, 39	sylancl 586	. . . . . . . 8 ⊢ (((𝑁 ∈ ℕ ∧ 𝑁 ∈ Even ) ∧ (1 σ 𝑁) = (2 · 𝑁)) → (((2 pCnt 𝑁) + 1) − 1) = (2 pCnt 𝑁))
41	40	eqcomd 2743	. . . . . . 7 ⊢ (((𝑁 ∈ ℕ ∧ 𝑁 ∈ Even ) ∧ (1 σ 𝑁) = (2 · 𝑁)) → (2 pCnt 𝑁) = (((2 pCnt 𝑁) + 1) − 1))
42	41	oveq2d 7333	. . . . . 6 ⊢ (((𝑁 ∈ ℕ ∧ 𝑁 ∈ Even ) ∧ (1 σ 𝑁) = (2 · 𝑁)) → (2↑(2 pCnt 𝑁)) = (2↑(((2 pCnt 𝑁) + 1) − 1)))
43	42, 34	oveq12d 7335	. . . . 5 ⊢ (((𝑁 ∈ ℕ ∧ 𝑁 ∈ Even ) ∧ (1 σ 𝑁) = (2 · 𝑁)) → ((2↑(2 pCnt 𝑁)) · (𝑁 / (2↑(2 pCnt 𝑁)))) = ((2↑(((2 pCnt 𝑁) + 1) − 1)) · ((2↑((2 pCnt 𝑁) + 1)) − 1)))
44	29, 43	eqtr3d 2779	. . . 4 ⊢ (((𝑁 ∈ ℕ ∧ 𝑁 ∈ Even ) ∧ (1 σ 𝑁) = (2 · 𝑁)) → 𝑁 = ((2↑(((2 pCnt 𝑁) + 1) − 1)) · ((2↑((2 pCnt 𝑁) + 1)) − 1)))
45		oveq2 7325	. . . . . . . 8 ⊢ (𝑝 = ((2 pCnt 𝑁) + 1) → (2↑𝑝) = (2↑((2 pCnt 𝑁) + 1)))
46	45	oveq1d 7332	. . . . . . 7 ⊢ (𝑝 = ((2 pCnt 𝑁) + 1) → ((2↑𝑝) − 1) = ((2↑((2 pCnt 𝑁) + 1)) − 1))
47	46	eleq1d 2822	. . . . . 6 ⊢ (𝑝 = ((2 pCnt 𝑁) + 1) → (((2↑𝑝) − 1) ∈ ℙ ↔ ((2↑((2 pCnt 𝑁) + 1)) − 1) ∈ ℙ))
48		oveq1 7324	. . . . . . . . 9 ⊢ (𝑝 = ((2 pCnt 𝑁) + 1) → (𝑝 − 1) = (((2 pCnt 𝑁) + 1) − 1))
49	48	oveq2d 7333	. . . . . . . 8 ⊢ (𝑝 = ((2 pCnt 𝑁) + 1) → (2↑(𝑝 − 1)) = (2↑(((2 pCnt 𝑁) + 1) − 1)))
50	49, 46	oveq12d 7335	. . . . . . 7 ⊢ (𝑝 = ((2 pCnt 𝑁) + 1) → ((2↑(𝑝 − 1)) · ((2↑𝑝) − 1)) = ((2↑(((2 pCnt 𝑁) + 1) − 1)) · ((2↑((2 pCnt 𝑁) + 1)) − 1)))
51	50	eqeq2d 2748	. . . . . 6 ⊢ (𝑝 = ((2 pCnt 𝑁) + 1) → (𝑁 = ((2↑(𝑝 − 1)) · ((2↑𝑝) − 1)) ↔ 𝑁 = ((2↑(((2 pCnt 𝑁) + 1) − 1)) · ((2↑((2 pCnt 𝑁) + 1)) − 1))))
52	47, 51	anbi12d 631	. . . . 5 ⊢ (𝑝 = ((2 pCnt 𝑁) + 1) → ((((2↑𝑝) − 1) ∈ ℙ ∧ 𝑁 = ((2↑(𝑝 − 1)) · ((2↑𝑝) − 1))) ↔ (((2↑((2 pCnt 𝑁) + 1)) − 1) ∈ ℙ ∧ 𝑁 = ((2↑(((2 pCnt 𝑁) + 1) − 1)) · ((2↑((2 pCnt 𝑁) + 1)) − 1)))))
53	52	rspcev 3570	. . . 4 ⊢ ((((2 pCnt 𝑁) + 1) ∈ ℤ ∧ (((2↑((2 pCnt 𝑁) + 1)) − 1) ∈ ℙ ∧ 𝑁 = ((2↑(((2 pCnt 𝑁) + 1) − 1)) · ((2↑((2 pCnt 𝑁) + 1)) − 1)))) → ∃𝑝 ∈ ℤ (((2↑𝑝) − 1) ∈ ℙ ∧ 𝑁 = ((2↑(𝑝 − 1)) · ((2↑𝑝) − 1))))
54	9, 36, 44, 53	syl12anc 834	. . 3 ⊢ (((𝑁 ∈ ℕ ∧ 𝑁 ∈ Even ) ∧ (1 σ 𝑁) = (2 · 𝑁)) → ∃𝑝 ∈ ℤ (((2↑𝑝) − 1) ∈ ℙ ∧ 𝑁 = ((2↑(𝑝 − 1)) · ((2↑𝑝) − 1))))
55	54	ex 413	. 2 ⊢ ((𝑁 ∈ ℕ ∧ 𝑁 ∈ Even ) → ((1 σ 𝑁) = (2 · 𝑁) → ∃𝑝 ∈ ℤ (((2↑𝑝) − 1) ∈ ℙ ∧ 𝑁 = ((2↑(𝑝 − 1)) · ((2↑𝑝) − 1)))))
56		perfect1 26459	. . . . . 6 ⊢ ((𝑝 ∈ ℤ ∧ ((2↑𝑝) − 1) ∈ ℙ) → (1 σ ((2↑(𝑝 − 1)) · ((2↑𝑝) − 1))) = ((2↑𝑝) · ((2↑𝑝) − 1)))
57		2cn 12128	. . . . . . . . 9 ⊢ 2 ∈ ℂ
58		mersenne 26458	. . . . . . . . . 10 ⊢ ((𝑝 ∈ ℤ ∧ ((2↑𝑝) − 1) ∈ ℙ) → 𝑝 ∈ ℙ)
59		prmnn 16456	. . . . . . . . . 10 ⊢ (𝑝 ∈ ℙ → 𝑝 ∈ ℕ)
60	58, 59	syl 17	. . . . . . . . 9 ⊢ ((𝑝 ∈ ℤ ∧ ((2↑𝑝) − 1) ∈ ℙ) → 𝑝 ∈ ℕ)
61		expm1t 13891	. . . . . . . . 9 ⊢ ((2 ∈ ℂ ∧ 𝑝 ∈ ℕ) → (2↑𝑝) = ((2↑(𝑝 − 1)) · 2))
62	57, 60, 61	sylancr 587	. . . . . . . 8 ⊢ ((𝑝 ∈ ℤ ∧ ((2↑𝑝) − 1) ∈ ℙ) → (2↑𝑝) = ((2↑(𝑝 − 1)) · 2))
63		nnm1nn0 12354	. . . . . . . . . . 11 ⊢ (𝑝 ∈ ℕ → (𝑝 − 1) ∈ ℕ₀)
64	60, 63	syl 17	. . . . . . . . . 10 ⊢ ((𝑝 ∈ ℤ ∧ ((2↑𝑝) − 1) ∈ ℙ) → (𝑝 − 1) ∈ ℕ₀)
65		expcl 13880	. . . . . . . . . 10 ⊢ ((2 ∈ ℂ ∧ (𝑝 − 1) ∈ ℕ₀) → (2↑(𝑝 − 1)) ∈ ℂ)
66	57, 64, 65	sylancr 587	. . . . . . . . 9 ⊢ ((𝑝 ∈ ℤ ∧ ((2↑𝑝) − 1) ∈ ℙ) → (2↑(𝑝 − 1)) ∈ ℂ)
67		mulcom 11037	. . . . . . . . 9 ⊢ (((2↑(𝑝 − 1)) ∈ ℂ ∧ 2 ∈ ℂ) → ((2↑(𝑝 − 1)) · 2) = (2 · (2↑(𝑝 − 1))))
68	66, 57, 67	sylancl 586	. . . . . . . 8 ⊢ ((𝑝 ∈ ℤ ∧ ((2↑𝑝) − 1) ∈ ℙ) → ((2↑(𝑝 − 1)) · 2) = (2 · (2↑(𝑝 − 1))))
69	62, 68	eqtrd 2777	. . . . . . 7 ⊢ ((𝑝 ∈ ℤ ∧ ((2↑𝑝) − 1) ∈ ℙ) → (2↑𝑝) = (2 · (2↑(𝑝 − 1))))
70	69	oveq1d 7332	. . . . . 6 ⊢ ((𝑝 ∈ ℤ ∧ ((2↑𝑝) − 1) ∈ ℙ) → ((2↑𝑝) · ((2↑𝑝) − 1)) = ((2 · (2↑(𝑝 − 1))) · ((2↑𝑝) − 1)))
71		2cnd 12131	. . . . . . 7 ⊢ ((𝑝 ∈ ℤ ∧ ((2↑𝑝) − 1) ∈ ℙ) → 2 ∈ ℂ)
72		prmnn 16456	. . . . . . . . 9 ⊢ (((2↑𝑝) − 1) ∈ ℙ → ((2↑𝑝) − 1) ∈ ℕ)
73	72	adantl 482	. . . . . . . 8 ⊢ ((𝑝 ∈ ℤ ∧ ((2↑𝑝) − 1) ∈ ℙ) → ((2↑𝑝) − 1) ∈ ℕ)
74	73	nncnd 12069	. . . . . . 7 ⊢ ((𝑝 ∈ ℤ ∧ ((2↑𝑝) − 1) ∈ ℙ) → ((2↑𝑝) − 1) ∈ ℂ)
75	71, 66, 74	mulassd 11078	. . . . . 6 ⊢ ((𝑝 ∈ ℤ ∧ ((2↑𝑝) − 1) ∈ ℙ) → ((2 · (2↑(𝑝 − 1))) · ((2↑𝑝) − 1)) = (2 · ((2↑(𝑝 − 1)) · ((2↑𝑝) − 1))))
76	56, 70, 75	3eqtrd 2781	. . . . 5 ⊢ ((𝑝 ∈ ℤ ∧ ((2↑𝑝) − 1) ∈ ℙ) → (1 σ ((2↑(𝑝 − 1)) · ((2↑𝑝) − 1))) = (2 · ((2↑(𝑝 − 1)) · ((2↑𝑝) − 1))))
77		oveq2 7325	. . . . . 6 ⊢ (𝑁 = ((2↑(𝑝 − 1)) · ((2↑𝑝) − 1)) → (1 σ 𝑁) = (1 σ ((2↑(𝑝 − 1)) · ((2↑𝑝) − 1))))
78		oveq2 7325	. . . . . 6 ⊢ (𝑁 = ((2↑(𝑝 − 1)) · ((2↑𝑝) − 1)) → (2 · 𝑁) = (2 · ((2↑(𝑝 − 1)) · ((2↑𝑝) − 1))))
79	77, 78	eqeq12d 2753	. . . . 5 ⊢ (𝑁 = ((2↑(𝑝 − 1)) · ((2↑𝑝) − 1)) → ((1 σ 𝑁) = (2 · 𝑁) ↔ (1 σ ((2↑(𝑝 − 1)) · ((2↑𝑝) − 1))) = (2 · ((2↑(𝑝 − 1)) · ((2↑𝑝) − 1)))))
80	76, 79	syl5ibrcom 246	. . . 4 ⊢ ((𝑝 ∈ ℤ ∧ ((2↑𝑝) − 1) ∈ ℙ) → (𝑁 = ((2↑(𝑝 − 1)) · ((2↑𝑝) − 1)) → (1 σ 𝑁) = (2 · 𝑁)))
81	80	impr 455	. . 3 ⊢ ((𝑝 ∈ ℤ ∧ (((2↑𝑝) − 1) ∈ ℙ ∧ 𝑁 = ((2↑(𝑝 − 1)) · ((2↑𝑝) − 1)))) → (1 σ 𝑁) = (2 · 𝑁))
82	81	rexlimiva 3141	. 2 ⊢ (∃𝑝 ∈ ℤ (((2↑𝑝) − 1) ∈ ℙ ∧ 𝑁 = ((2↑(𝑝 − 1)) · ((2↑𝑝) − 1))) → (1 σ 𝑁) = (2 · 𝑁))
83	55, 82	impbid1 224	1 ⊢ ((𝑁 ∈ ℕ ∧ 𝑁 ∈ Even ) → ((1 σ 𝑁) = (2 · 𝑁) ↔ ∃𝑝 ∈ ℤ (((2↑𝑝) − 1) ∈ ℙ ∧ 𝑁 = ((2↑(𝑝 − 1)) · ((2↑𝑝) − 1)))))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 396 = wceq 1540 ∈ wcel 2105 ∃wrex 3071 class class class wbr 5087 (class class class)co 7317 ℂcc 10949 1c1 10952 + caddc 10954 · cmul 10956 − cmin 11285 / cdiv 11712 ℕcn 12053 2c2 12108 ℕ₀cn0 12313 ℤcz 12399 ↑cexp 13862 ∥ cdvds 16042 ℙcprime 16453 pCnt cpc 16614 σ csgm 26328 Even ceven 45341 Odd codd 45342

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 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2708 ax-rep 5224 ax-sep 5238 ax-nul 5245 ax-pow 5303 ax-pr 5367 ax-un 7630 ax-inf2 9477 ax-cnex 11007 ax-resscn 11008 ax-1cn 11009 ax-icn 11010 ax-addcl 11011 ax-addrcl 11012 ax-mulcl 11013 ax-mulrcl 11014 ax-mulcom 11015 ax-addass 11016 ax-mulass 11017 ax-distr 11018 ax-i2m1 11019 ax-1ne0 11020 ax-1rid 11021 ax-rnegex 11022 ax-rrecex 11023 ax-cnre 11024 ax-pre-lttri 11025 ax-pre-lttrn 11026 ax-pre-ltadd 11027 ax-pre-mulgt0 11028 ax-pre-sup 11029 ax-addf 11030 ax-mulf 11031

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2887 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3350 df-reu 3351 df-rab 3405 df-v 3443 df-sbc 3727 df-csb 3843 df-dif 3900 df-un 3902 df-in 3904 df-ss 3914 df-pss 3916 df-nul 4268 df-if 4472 df-pw 4547 df-sn 4572 df-pr 4574 df-tp 4576 df-op 4578 df-uni 4851 df-int 4893 df-iun 4939 df-iin 4940 df-br 5088 df-opab 5150 df-mpt 5171 df-tr 5205 df-id 5507 df-eprel 5513 df-po 5521 df-so 5522 df-fr 5563 df-se 5564 df-we 5565 df-xp 5614 df-rel 5615 df-cnv 5616 df-co 5617 df-dm 5618 df-rn 5619 df-res 5620 df-ima 5621 df-pred 6225 df-ord 6292 df-on 6293 df-lim 6294 df-suc 6295 df-iota 6418 df-fun 6468 df-fn 6469 df-f 6470 df-f1 6471 df-fo 6472 df-f1o 6473 df-fv 6474 df-isom 6475 df-riota 7274 df-ov 7320 df-oprab 7321 df-mpo 7322 df-of 7575 df-om 7760 df-1st 7878 df-2nd 7879 df-supp 8027 df-frecs 8146 df-wrecs 8177 df-recs 8251 df-rdg 8290 df-1o 8346 df-2o 8347 df-er 8548 df-map 8667 df-pm 8668 df-ixp 8736 df-en 8784 df-dom 8785 df-sdom 8786 df-fin 8787 df-fsupp 9206 df-fi 9247 df-sup 9278 df-inf 9279 df-oi 9346 df-card 9775 df-pnf 11091 df-mnf 11092 df-xr 11093 df-ltxr 11094 df-le 11095 df-sub 11287 df-neg 11288 df-div 11713 df-nn 12054 df-2 12116 df-3 12117 df-4 12118 df-5 12119 df-6 12120 df-7 12121 df-8 12122 df-9 12123 df-n0 12314 df-z 12400 df-dec 12518 df-uz 12663 df-q 12769 df-rp 12811 df-xneg 12928 df-xadd 12929 df-xmul 12930 df-ioo 13163 df-ioc 13164 df-ico 13165 df-icc 13166 df-fz 13320 df-fzo 13463 df-fl 13592 df-mod 13670 df-seq 13802 df-exp 13863 df-fac 14068 df-bc 14097 df-hash 14125 df-shft 14857 df-cj 14889 df-re 14890 df-im 14891 df-sqrt 15025 df-abs 15026 df-limsup 15259 df-clim 15276 df-rlim 15277 df-sum 15477 df-ef 15856 df-sin 15858 df-cos 15859 df-pi 15861 df-dvds 16043 df-gcd 16281 df-prm 16454 df-pc 16615 df-struct 16925 df-sets 16942 df-slot 16960 df-ndx 16972 df-base 16990 df-ress 17019 df-plusg 17052 df-mulr 17053 df-starv 17054 df-sca 17055 df-vsca 17056 df-ip 17057 df-tset 17058 df-ple 17059 df-ds 17061 df-unif 17062 df-hom 17063 df-cco 17064 df-rest 17210 df-topn 17211 df-0g 17229 df-gsum 17230 df-topgen 17231 df-pt 17232 df-prds 17235 df-xrs 17290 df-qtop 17295 df-imas 17296 df-xps 17298 df-mre 17372 df-mrc 17373 df-acs 17375 df-mgm 18403 df-sgrp 18452 df-mnd 18463 df-submnd 18508 df-mulg 18777 df-cntz 18999 df-cmn 19463 df-psmet 20672 df-xmet 20673 df-met 20674 df-bl 20675 df-mopn 20676 df-fbas 20677 df-fg 20678 df-cnfld 20681 df-top 22126 df-topon 22143 df-topsp 22165 df-bases 22179 df-cld 22253 df-ntr 22254 df-cls 22255 df-nei 22332 df-lp 22370 df-perf 22371 df-cn 22461 df-cnp 22462 df-haus 22549 df-tx 22796 df-hmeo 22989 df-fil 23080 df-fm 23172 df-flim 23173 df-flf 23174 df-xms 23556 df-ms 23557 df-tms 23558 df-cncf 24124 df-limc 25113 df-dv 25114 df-log 25795 df-cxp 25796 df-sgm 26334 df-even 45343 df-odd 45344

This theorem is referenced by: (None)

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