perfectALTV - Mathbox for Alexander van der Vekens

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Theorem perfectALTV 48565

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 48495	. . . . . . . 8 ⊢ (𝑁 ∈ Even → 2 ∥ 𝑁)
2	1	ad2antlr 740	. . . . . . 7 ⊢ (((𝑁 ∈ ℕ ∧ 𝑁 ∈ Even ) ∧ (1 σ 𝑁) = (2 · 𝑁)) → 2 ∥ 𝑁)
3		2prm 16776	. . . . . . . 8 ⊢ 2 ∈ ℙ
4		simpll 779	. . . . . . . 8 ⊢ (((𝑁 ∈ ℕ ∧ 𝑁 ∈ Even ) ∧ (1 σ 𝑁) = (2 · 𝑁)) → 𝑁 ∈ ℕ)
5		pcelnn 16956	. . . . . . . 8 ⊢ ((2 ∈ ℙ ∧ 𝑁 ∈ ℕ) → ((2 pCnt 𝑁) ∈ ℕ ↔ 2 ∥ 𝑁))
6	3, 4, 5	sylancr 599	. . . . . . 7 ⊢ (((𝑁 ∈ ℕ ∧ 𝑁 ∈ Even ) ∧ (1 σ 𝑁) = (2 · 𝑁)) → ((2 pCnt 𝑁) ∈ ℕ ↔ 2 ∥ 𝑁))
7	2, 6	mpbird 260	. . . . . 6 ⊢ (((𝑁 ∈ ℕ ∧ 𝑁 ∈ Even ) ∧ (1 σ 𝑁) = (2 · 𝑁)) → (2 pCnt 𝑁) ∈ ℕ)
8	7	nnzd 12636	. . . . 5 ⊢ (((𝑁 ∈ ℕ ∧ 𝑁 ∈ Even ) ∧ (1 σ 𝑁) = (2 · 𝑁)) → (2 pCnt 𝑁) ∈ ℤ)
9	8	peano2zd 12723	. . . 4 ⊢ (((𝑁 ∈ ℕ ∧ 𝑁 ∈ Even ) ∧ (1 σ 𝑁) = (2 · 𝑁)) → ((2 pCnt 𝑁) + 1) ∈ ℤ)
10		pcdvds 16950	. . . . . . . . 9 ⊢ ((2 ∈ ℙ ∧ 𝑁 ∈ ℕ) → (2↑(2 pCnt 𝑁)) ∥ 𝑁)
11	3, 4, 10	sylancr 599	. . . . . . . 8 ⊢ (((𝑁 ∈ ℕ ∧ 𝑁 ∈ Even ) ∧ (1 σ 𝑁) = (2 · 𝑁)) → (2↑(2 pCnt 𝑁)) ∥ 𝑁)
12		2nn 12333	. . . . . . . . . 10 ⊢ 2 ∈ ℕ
13	7	nnnn0d 12584	. . . . . . . . . 10 ⊢ (((𝑁 ∈ ℕ ∧ 𝑁 ∈ Even ) ∧ (1 σ 𝑁) = (2 · 𝑁)) → (2 pCnt 𝑁) ∈ ℕ₀)
14		nnexpcl 14132	. . . . . . . . . 10 ⊢ ((2 ∈ ℕ ∧ (2 pCnt 𝑁) ∈ ℕ₀) → (2↑(2 pCnt 𝑁)) ∈ ℕ)
15	12, 13, 14	sylancr 599	. . . . . . . . 9 ⊢ (((𝑁 ∈ ℕ ∧ 𝑁 ∈ Even ) ∧ (1 σ 𝑁) = (2 · 𝑁)) → (2↑(2 pCnt 𝑁)) ∈ ℕ)
16		nndivdvds 16345	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℕ ∧ (2↑(2 pCnt 𝑁)) ∈ ℕ) → ((2↑(2 pCnt 𝑁)) ∥ 𝑁 ↔ (𝑁 / (2↑(2 pCnt 𝑁))) ∈ ℕ))
17	4, 15, 16	syl2anc 596	. . . . . . . 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 235	. . . . . . 7 ⊢ (((𝑁 ∈ ℕ ∧ 𝑁 ∈ Even ) ∧ (1 σ 𝑁) = (2 · 𝑁)) → (𝑁 / (2↑(2 pCnt 𝑁))) ∈ ℕ)
19	18	nnzd 12636	. . . . . . . 8 ⊢ (((𝑁 ∈ ℕ ∧ 𝑁 ∈ Even ) ∧ (1 σ 𝑁) = (2 · 𝑁)) → (𝑁 / (2↑(2 pCnt 𝑁))) ∈ ℤ)
20		pcndvds2 16954	. . . . . . . . 9 ⊢ ((2 ∈ ℙ ∧ 𝑁 ∈ ℕ) → ¬ 2 ∥ (𝑁 / (2↑(2 pCnt 𝑁))))
21	3, 4, 20	sylancr 599	. . . . . . . 8 ⊢ (((𝑁 ∈ ℕ ∧ 𝑁 ∈ Even ) ∧ (1 σ 𝑁) = (2 · 𝑁)) → ¬ 2 ∥ (𝑁 / (2↑(2 pCnt 𝑁))))
22		isodd3 48494	. . . . . . . 8 ⊢ ((𝑁 / (2↑(2 pCnt 𝑁))) ∈ Odd ↔ ((𝑁 / (2↑(2 pCnt 𝑁))) ∈ ℤ ∧ ¬ 2 ∥ (𝑁 / (2↑(2 pCnt 𝑁)))))
23	19, 21, 22	sylanbrc 595	. . . . . . 7 ⊢ (((𝑁 ∈ ℕ ∧ 𝑁 ∈ Even ) ∧ (1 σ 𝑁) = (2 · 𝑁)) → (𝑁 / (2↑(2 pCnt 𝑁))) ∈ Odd )
24		simpr 490	. . . . . . . 8 ⊢ (((𝑁 ∈ ℕ ∧ 𝑁 ∈ Even ) ∧ (1 σ 𝑁) = (2 · 𝑁)) → (1 σ 𝑁) = (2 · 𝑁))
25		nncn 12260	. . . . . . . . . . 11 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℂ)
26	25	ad2antrr 739	. . . . . . . . . 10 ⊢ (((𝑁 ∈ ℕ ∧ 𝑁 ∈ Even ) ∧ (1 σ 𝑁) = (2 · 𝑁)) → 𝑁 ∈ ℂ)
27	15	nncnd 12268	. . . . . . . . . 10 ⊢ (((𝑁 ∈ ℕ ∧ 𝑁 ∈ Even ) ∧ (1 σ 𝑁) = (2 · 𝑁)) → (2↑(2 pCnt 𝑁)) ∈ ℂ)
28	15	nnne0d 12305	. . . . . . . . . 10 ⊢ (((𝑁 ∈ ℕ ∧ 𝑁 ∈ Even ) ∧ (1 σ 𝑁) = (2 · 𝑁)) → (2↑(2 pCnt 𝑁)) ≠ 0)
29	26, 27, 28	divcan2d 12012	. . . . . . . . 9 ⊢ (((𝑁 ∈ ℕ ∧ 𝑁 ∈ Even ) ∧ (1 σ 𝑁) = (2 · 𝑁)) → ((2↑(2 pCnt 𝑁)) · (𝑁 / (2↑(2 pCnt 𝑁)))) = 𝑁)
30	29	oveq2d 7435	. . . . . . . 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 7435	. . . . . . . 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 2810	. . . . . . 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 48564	. . . . . 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 501	. . . . 5 ⊢ (((𝑁 ∈ ℕ ∧ 𝑁 ∈ Even ) ∧ (1 σ 𝑁) = (2 · 𝑁)) → (𝑁 / (2↑(2 pCnt 𝑁))) = ((2↑((2 pCnt 𝑁) + 1)) − 1))
35	33	simpld 500	. . . . 5 ⊢ (((𝑁 ∈ ℕ ∧ 𝑁 ∈ Even ) ∧ (1 σ 𝑁) = (2 · 𝑁)) → (𝑁 / (2↑(2 pCnt 𝑁))) ∈ ℙ)
36	34, 35	eqeltrrd 2866	. . . 4 ⊢ (((𝑁 ∈ ℕ ∧ 𝑁 ∈ Even ) ∧ (1 σ 𝑁) = (2 · 𝑁)) → ((2↑((2 pCnt 𝑁) + 1)) − 1) ∈ ℙ)
37	7	nncnd 12268	. . . . . . . . 9 ⊢ (((𝑁 ∈ ℕ ∧ 𝑁 ∈ Even ) ∧ (1 σ 𝑁) = (2 · 𝑁)) → (2 pCnt 𝑁) ∈ ℂ)
38		ax-1cn 11177	. . . . . . . . 9 ⊢ 1 ∈ ℂ
39		pncan 11482	. . . . . . . . 9 ⊢ (((2 pCnt 𝑁) ∈ ℂ ∧ 1 ∈ ℂ) → (((2 pCnt 𝑁) + 1) − 1) = (2 pCnt 𝑁))
40	37, 38, 39	sylancl 598	. . . . . . . 8 ⊢ (((𝑁 ∈ ℕ ∧ 𝑁 ∈ Even ) ∧ (1 σ 𝑁) = (2 · 𝑁)) → (((2 pCnt 𝑁) + 1) − 1) = (2 pCnt 𝑁))
41	40	eqcomd 2771	. . . . . . 7 ⊢ (((𝑁 ∈ ℕ ∧ 𝑁 ∈ Even ) ∧ (1 σ 𝑁) = (2 · 𝑁)) → (2 pCnt 𝑁) = (((2 pCnt 𝑁) + 1) − 1))
42	41	oveq2d 7435	. . . . . 6 ⊢ (((𝑁 ∈ ℕ ∧ 𝑁 ∈ Even ) ∧ (1 σ 𝑁) = (2 · 𝑁)) → (2↑(2 pCnt 𝑁)) = (2↑(((2 pCnt 𝑁) + 1) − 1)))
43	42, 34	oveq12d 7437	. . . . 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 2802	. . . 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 7427	. . . . . . . 8 ⊢ (𝑝 = ((2 pCnt 𝑁) + 1) → (2↑𝑝) = (2↑((2 pCnt 𝑁) + 1)))
46	45	oveq1d 7434	. . . . . . 7 ⊢ (𝑝 = ((2 pCnt 𝑁) + 1) → ((2↑𝑝) − 1) = ((2↑((2 pCnt 𝑁) + 1)) − 1))
47	46	eleq1d 2850	. . . . . 6 ⊢ (𝑝 = ((2 pCnt 𝑁) + 1) → (((2↑𝑝) − 1) ∈ ℙ ↔ ((2↑((2 pCnt 𝑁) + 1)) − 1) ∈ ℙ))
48		oveq1 7426	. . . . . . . . 9 ⊢ (𝑝 = ((2 pCnt 𝑁) + 1) → (𝑝 − 1) = (((2 pCnt 𝑁) + 1) − 1))
49	48	oveq2d 7435	. . . . . . . 8 ⊢ (𝑝 = ((2 pCnt 𝑁) + 1) → (2↑(𝑝 − 1)) = (2↑(((2 pCnt 𝑁) + 1) − 1)))
50	49, 46	oveq12d 7437	. . . . . . 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 2776	. . . . . 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 644	. . . . 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 3583	. . . 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 850	. . 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 418	. 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 27447	. . . . . 6 ⊢ ((𝑝 ∈ ℤ ∧ ((2↑𝑝) − 1) ∈ ℙ) → (1 σ ((2↑(𝑝 − 1)) · ((2↑𝑝) − 1))) = ((2↑𝑝) · ((2↑𝑝) − 1)))
57		2cn 12335	. . . . . . . . 9 ⊢ 2 ∈ ℂ
58		mersenne 27446	. . . . . . . . . 10 ⊢ ((𝑝 ∈ ℤ ∧ ((2↑𝑝) − 1) ∈ ℙ) → 𝑝 ∈ ℙ)
59		prmnn 16758	. . . . . . . . . 10 ⊢ (𝑝 ∈ ℙ → 𝑝 ∈ ℕ)
60	58, 59	syl 18	. . . . . . . . 9 ⊢ ((𝑝 ∈ ℤ ∧ ((2↑𝑝) − 1) ∈ ℙ) → 𝑝 ∈ ℕ)
61		expm1t 14148	. . . . . . . . 9 ⊢ ((2 ∈ ℂ ∧ 𝑝 ∈ ℕ) → (2↑𝑝) = ((2↑(𝑝 − 1)) · 2))
62	57, 60, 61	sylancr 599	. . . . . . . 8 ⊢ ((𝑝 ∈ ℤ ∧ ((2↑𝑝) − 1) ∈ ℙ) → (2↑𝑝) = ((2↑(𝑝 − 1)) · 2))
63		nnm1nn0 12564	. . . . . . . . . . 11 ⊢ (𝑝 ∈ ℕ → (𝑝 − 1) ∈ ℕ₀)
64	60, 63	syl 18	. . . . . . . . . 10 ⊢ ((𝑝 ∈ ℤ ∧ ((2↑𝑝) − 1) ∈ ℙ) → (𝑝 − 1) ∈ ℕ₀)
65		expcl 14137	. . . . . . . . . 10 ⊢ ((2 ∈ ℂ ∧ (𝑝 − 1) ∈ ℕ₀) → (2↑(𝑝 − 1)) ∈ ℂ)
66	57, 64, 65	sylancr 599	. . . . . . . . 9 ⊢ ((𝑝 ∈ ℤ ∧ ((2↑𝑝) − 1) ∈ ℙ) → (2↑(𝑝 − 1)) ∈ ℂ)
67		mulcom 11205	. . . . . . . . 9 ⊢ (((2↑(𝑝 − 1)) ∈ ℂ ∧ 2 ∈ ℂ) → ((2↑(𝑝 − 1)) · 2) = (2 · (2↑(𝑝 − 1))))
68	66, 57, 67	sylancl 598	. . . . . . . 8 ⊢ ((𝑝 ∈ ℤ ∧ ((2↑𝑝) − 1) ∈ ℙ) → ((2↑(𝑝 − 1)) · 2) = (2 · (2↑(𝑝 − 1))))
69	62, 68	eqtrd 2800	. . . . . . 7 ⊢ ((𝑝 ∈ ℤ ∧ ((2↑𝑝) − 1) ∈ ℙ) → (2↑𝑝) = (2 · (2↑(𝑝 − 1))))
70	69	oveq1d 7434	. . . . . 6 ⊢ ((𝑝 ∈ ℤ ∧ ((2↑𝑝) − 1) ∈ ℙ) → ((2↑𝑝) · ((2↑𝑝) − 1)) = ((2 · (2↑(𝑝 − 1))) · ((2↑𝑝) − 1)))
71		2cnd 12338	. . . . . . 7 ⊢ ((𝑝 ∈ ℤ ∧ ((2↑𝑝) − 1) ∈ ℙ) → 2 ∈ ℂ)
72		prmnn 16758	. . . . . . . . 9 ⊢ (((2↑𝑝) − 1) ∈ ℙ → ((2↑𝑝) − 1) ∈ ℕ)
73	72	adantl 487	. . . . . . . 8 ⊢ ((𝑝 ∈ ℤ ∧ ((2↑𝑝) − 1) ∈ ℙ) → ((2↑𝑝) − 1) ∈ ℕ)
74	73	nncnd 12268	. . . . . . 7 ⊢ ((𝑝 ∈ ℤ ∧ ((2↑𝑝) − 1) ∈ ℙ) → ((2↑𝑝) − 1) ∈ ℂ)
75	71, 66, 74	mulassd 11251	. . . . . 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 2804	. . . . 5 ⊢ ((𝑝 ∈ ℤ ∧ ((2↑𝑝) − 1) ∈ ℙ) → (1 σ ((2↑(𝑝 − 1)) · ((2↑𝑝) − 1))) = (2 · ((2↑(𝑝 − 1)) · ((2↑𝑝) − 1))))
77		oveq2 7427	. . . . . 6 ⊢ (𝑁 = ((2↑(𝑝 − 1)) · ((2↑𝑝) − 1)) → (1 σ 𝑁) = (1 σ ((2↑(𝑝 − 1)) · ((2↑𝑝) − 1))))
78		oveq2 7427	. . . . . 6 ⊢ (𝑁 = ((2↑(𝑝 − 1)) · ((2↑𝑝) − 1)) → (2 · 𝑁) = (2 · ((2↑(𝑝 − 1)) · ((2↑𝑝) − 1))))
79	77, 78	eqeq12d 2781	. . . . 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 250	. . . 4 ⊢ ((𝑝 ∈ ℤ ∧ ((2↑𝑝) − 1) ∈ ℙ) → (𝑁 = ((2↑(𝑝 − 1)) · ((2↑𝑝) − 1)) → (1 σ 𝑁) = (2 · 𝑁)))
81	80	impr 460	. . 3 ⊢ ((𝑝 ∈ ℤ ∧ (((2↑𝑝) − 1) ∈ ℙ ∧ 𝑁 = ((2↑(𝑝 − 1)) · ((2↑𝑝) − 1)))) → (1 σ 𝑁) = (2 · 𝑁))
82	81	rexlimiva 3160	. 2 ⊢ (∃𝑝 ∈ ℤ (((2↑𝑝) − 1) ∈ ℙ ∧ 𝑁 = ((2↑(𝑝 − 1)) · ((2↑𝑝) − 1))) → (1 σ 𝑁) = (2 · 𝑁))
83	55, 82	impbid1 228	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

This proof depends on syntax axioms: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 401 = wceq 1570 ∈ wcel 2146 ∃wrex 3091 class class class wbr 5111 (class class class)co 7419 ℂcc 11117 1c1 11120 + caddc 11122 · cmul 11124 − cmin 11460 / cdiv 11890 ℕcn 12252 2c2 12314 ℕ₀cn0 12523 ℤcz 12610 ↑cexp 14119 ∥ cdvds 16336 ℙcprime 16755 pCnt cpc 16922 σ csgm 27315 Even ceven 48466 Odd codd 48467

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2148 ax-9 2156 ax-10 2179 ax-11 2195 ax-12 2216 ax-ext 2737 ax-rep 5240 ax-sep 5259 ax-nul 5271 ax-pow 5338 ax-pr 5406 ax-un 7742 ax-inf2 9617 ax-cnex 11175 ax-resscn 11176 ax-1cn 11177 ax-icn 11178 ax-addcl 11179 ax-addrcl 11180 ax-mulcl 11181 ax-mulrcl 11182 ax-mulcom 11183 ax-addass 11184 ax-mulass 11185 ax-distr 11186 ax-i2m1 11187 ax-1ne0 11188 ax-1rid 11189 ax-rnegex 11190 ax-rrecex 11191 ax-cnre 11192 ax-pre-lttri 11193 ax-pre-lttrn 11194 ax-pre-ltadd 11195 ax-pre-mulgt0 11196 ax-pre-sup 11197 ax-addf 11198

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2569 df-eu 2599 df-clab 2744 df-cleq 2757 df-clel 2840 df-nfc 2914 df-ne 2961 df-nel 3067 df-ral 3082 df-rex 3092 df-rmo 3371 df-reu 3372 df-rab 3419 df-v 3459 df-sbc 3747 df-csb 3855 df-dif 3909 df-un 3911 df-in 3913 df-ss 3923 df-pss 3926 df-nul 4287 df-if 4490 df-pw 4566 df-sn 4592 df-pr 4594 df-tp 4596 df-op 4598 df-uni 4875 df-int 4915 df-iun 4960 df-iin 4961 df-br 5112 df-opab 5176 df-mpt 5195 df-tr 5221 df-id 5558 df-eprel 5563 df-po 5571 df-so 5572 df-fr 5616 df-se 5617 df-we 5618 df-xp 5669 df-rel 5670 df-cnv 5671 df-co 5672 df-dm 5673 df-rn 5674 df-res 5675 df-ima 5676 df-pred 6306 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6496 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-isom 6549 df-riota 7376 df-ov 7422 df-oprab 7423 df-mpo 7424 df-of 7684 df-om 7869 df-1st 7992 df-2nd 7993 df-supp 8163 df-frecs 8284 df-wrecs 8315 df-recs 8364 df-rdg 8403 df-1o 8459 df-2o 8460 df-er 8700 df-map 8832 df-pm 8833 df-ixp 8902 df-en 8950 df-dom 8951 df-sdom 8952 df-fin 8953 df-fsupp 9329 df-fi 9378 df-sup 9409 df-inf 9410 df-oi 9479 df-card 9941 df-pnf 11264 df-mnf 11265 df-xr 11266 df-ltxr 11267 df-le 11268 df-sub 11462 df-neg 11463 df-div 11891 df-nn 12253 df-2 12322 df-3 12323 df-4 12324 df-5 12325 df-6 12326 df-7 12327 df-8 12328 df-9 12329 df-n0 12524 df-z 12611 df-dec 12732 df-uz 12883 df-q 12993 df-rp 13037 df-xneg 13157 df-xadd 13158 df-xmul 13159 df-ioo 13396 df-ioc 13397 df-ico 13398 df-icc 13399 df-fz 13556 df-fzo 13704 df-fl 13847 df-mod 13925 df-seq 14060 df-exp 14120 df-fac 14332 df-bc 14361 df-hash 14389 df-shft 15132 df-cj 15178 df-re 15179 df-im 15180 df-sqrt 15314 df-abs 15315 df-limsup 15550 df-clim 15567 df-rlim 15568 df-sum 15766 df-ef 16147 df-sin 16149 df-cos 16150 df-pi 16152 df-dvds 16337 df-gcd 16579 df-prm 16756 df-pc 16923 df-struct 17233 df-sets 17250 df-slot 17268 df-ndx 17280 df-base 17296 df-ress 17317 df-plusg 17349 df-mulr 17350 df-starv 17351 df-sca 17352 df-vsca 17353 df-ip 17354 df-tset 17355 df-ple 17356 df-ds 17358 df-unif 17359 df-hom 17360 df-cco 17361 df-rest 17501 df-topn 17502 df-0g 17520 df-gsum 17521 df-topgen 17522 df-pt 17523 df-prds 17526 df-xrs 17582 df-qtop 17587 df-imas 17588 df-xps 17590 df-mre 17664 df-mrc 17665 df-acs 17667 df-mgm 18724 df-sgrp 18813 df-mnd 18829 df-submnd 18883 df-mulg 19182 df-cntz 19435 df-cmn 19900 df-psmet 21568 df-xmet 21569 df-met 21570 df-bl 21571 df-mopn 21572 df-fbas 21573 df-fg 21574 df-cnfld 21577 df-top 23105 df-topon 23122 df-topsp 23144 df-bases 23157 df-cld 23230 df-ntr 23231 df-cls 23232 df-nei 23309 df-lp 23347 df-perf 23348 df-cn 23438 df-cnp 23439 df-haus 23526 df-tx 23774 df-hmeo 23967 df-fil 24058 df-fm 24150 df-flim 24151 df-flf 24152 df-xms 24532 df-ms 24533 df-tms 24534 df-cncf 25092 df-limc 26080 df-dv 26081 df-log 26776 df-cxp 26777 df-sgm 27321 df-even 48468 df-odd 48469

This theorem is used by: (None)

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