perfectlem1 - Intuitionistic Logic Explorer

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Theorem perfectlem1 15726

Description: Lemma for perfect 15728. (Contributed by Mario Carneiro, 7-Jun-2016.)

**Hypotheses**
Ref	Expression
perfectlem.1	⊢ (𝜑 → 𝐴 ∈ ℕ)
perfectlem.2	⊢ (𝜑 → 𝐵 ∈ ℕ)
perfectlem.3	⊢ (𝜑 → ¬ 2 ∥ 𝐵)
perfectlem.4	⊢ (𝜑 → (1 σ ((2↑𝐴) · 𝐵)) = (2 · ((2↑𝐴) · 𝐵)))

**Assertion**
Ref	Expression
perfectlem1	⊢ (𝜑 → ((2↑(𝐴 + 1)) ∈ ℕ ∧ ((2↑(𝐴 + 1)) − 1) ∈ ℕ ∧ (𝐵 / ((2↑(𝐴 + 1)) − 1)) ∈ ℕ))

**Proof of Theorem perfectlem1**
Step	Hyp	Ref	Expression
1		2nn 9305	. . 3 ⊢ 2 ∈ ℕ
2		perfectlem.1	. . . . 5 ⊢ (𝜑 → 𝐴 ∈ ℕ)
3	2	nnnn0d 9455	. . . 4 ⊢ (𝜑 → 𝐴 ∈ ℕ₀)
4		peano2nn0 9442	. . . 4 ⊢ (𝐴 ∈ ℕ₀ → (𝐴 + 1) ∈ ℕ₀)
5	3, 4	syl 14	. . 3 ⊢ (𝜑 → (𝐴 + 1) ∈ ℕ₀)
6		nnexpcl 10815	. . 3 ⊢ ((2 ∈ ℕ ∧ (𝐴 + 1) ∈ ℕ₀) → (2↑(𝐴 + 1)) ∈ ℕ)
7	1, 5, 6	sylancr 414	. 2 ⊢ (𝜑 → (2↑(𝐴 + 1)) ∈ ℕ)
8		2re 9213	. . . 4 ⊢ 2 ∈ ℝ
9	2	peano2nnd 9158	. . . 4 ⊢ (𝜑 → (𝐴 + 1) ∈ ℕ)
10		1lt2 9313	. . . . 5 ⊢ 1 < 2
11	10	a1i 9	. . . 4 ⊢ (𝜑 → 1 < 2)
12		expgt1 10840	. . . 4 ⊢ ((2 ∈ ℝ ∧ (𝐴 + 1) ∈ ℕ ∧ 1 < 2) → 1 < (2↑(𝐴 + 1)))
13	8, 9, 11, 12	mp3an2i 1378	. . 3 ⊢ (𝜑 → 1 < (2↑(𝐴 + 1)))
14		1nn 9154	. . . 4 ⊢ 1 ∈ ℕ
15		nnsub 9182	. . . 4 ⊢ ((1 ∈ ℕ ∧ (2↑(𝐴 + 1)) ∈ ℕ) → (1 < (2↑(𝐴 + 1)) ↔ ((2↑(𝐴 + 1)) − 1) ∈ ℕ))
16	14, 7, 15	sylancr 414	. . 3 ⊢ (𝜑 → (1 < (2↑(𝐴 + 1)) ↔ ((2↑(𝐴 + 1)) − 1) ∈ ℕ))
17	13, 16	mpbid 147	. 2 ⊢ (𝜑 → ((2↑(𝐴 + 1)) − 1) ∈ ℕ)
18	7	nnzd 9601	. . . . . . 7 ⊢ (𝜑 → (2↑(𝐴 + 1)) ∈ ℤ)
19		peano2zm 9517	. . . . . . 7 ⊢ ((2↑(𝐴 + 1)) ∈ ℤ → ((2↑(𝐴 + 1)) − 1) ∈ ℤ)
20	18, 19	syl 14	. . . . . 6 ⊢ (𝜑 → ((2↑(𝐴 + 1)) − 1) ∈ ℤ)
21		1nn0 9418	. . . . . . . 8 ⊢ 1 ∈ ℕ₀
22		perfectlem.2	. . . . . . . 8 ⊢ (𝜑 → 𝐵 ∈ ℕ)
23		sgmnncl 15715	. . . . . . . 8 ⊢ ((1 ∈ ℕ₀ ∧ 𝐵 ∈ ℕ) → (1 σ 𝐵) ∈ ℕ)
24	21, 22, 23	sylancr 414	. . . . . . 7 ⊢ (𝜑 → (1 σ 𝐵) ∈ ℕ)
25	24	nnzd 9601	. . . . . 6 ⊢ (𝜑 → (1 σ 𝐵) ∈ ℤ)
26		dvdsmul1 12376	. . . . . 6 ⊢ ((((2↑(𝐴 + 1)) − 1) ∈ ℤ ∧ (1 σ 𝐵) ∈ ℤ) → ((2↑(𝐴 + 1)) − 1) ∥ (((2↑(𝐴 + 1)) − 1) · (1 σ 𝐵)))
27	20, 25, 26	syl2anc 411	. . . . 5 ⊢ (𝜑 → ((2↑(𝐴 + 1)) − 1) ∥ (((2↑(𝐴 + 1)) − 1) · (1 σ 𝐵)))
28		2cn 9214	. . . . . . . . 9 ⊢ 2 ∈ ℂ
29		expp1 10809	. . . . . . . . 9 ⊢ ((2 ∈ ℂ ∧ 𝐴 ∈ ℕ₀) → (2↑(𝐴 + 1)) = ((2↑𝐴) · 2))
30	28, 3, 29	sylancr 414	. . . . . . . 8 ⊢ (𝜑 → (2↑(𝐴 + 1)) = ((2↑𝐴) · 2))
31		nnexpcl 10815	. . . . . . . . . . 11 ⊢ ((2 ∈ ℕ ∧ 𝐴 ∈ ℕ₀) → (2↑𝐴) ∈ ℕ)
32	1, 3, 31	sylancr 414	. . . . . . . . . 10 ⊢ (𝜑 → (2↑𝐴) ∈ ℕ)
33	32	nncnd 9157	. . . . . . . . 9 ⊢ (𝜑 → (2↑𝐴) ∈ ℂ)
34		mulcom 8161	. . . . . . . . 9 ⊢ (((2↑𝐴) ∈ ℂ ∧ 2 ∈ ℂ) → ((2↑𝐴) · 2) = (2 · (2↑𝐴)))
35	33, 28, 34	sylancl 413	. . . . . . . 8 ⊢ (𝜑 → ((2↑𝐴) · 2) = (2 · (2↑𝐴)))
36	30, 35	eqtrd 2264	. . . . . . 7 ⊢ (𝜑 → (2↑(𝐴 + 1)) = (2 · (2↑𝐴)))
37	36	oveq1d 6033	. . . . . 6 ⊢ (𝜑 → ((2↑(𝐴 + 1)) · 𝐵) = ((2 · (2↑𝐴)) · 𝐵))
38	28	a1i 9	. . . . . . 7 ⊢ (𝜑 → 2 ∈ ℂ)
39	22	nncnd 9157	. . . . . . 7 ⊢ (𝜑 → 𝐵 ∈ ℂ)
40	38, 33, 39	mulassd 8203	. . . . . 6 ⊢ (𝜑 → ((2 · (2↑𝐴)) · 𝐵) = (2 · ((2↑𝐴) · 𝐵)))
41		ax-1cn 8125	. . . . . . . . 9 ⊢ 1 ∈ ℂ
42	41	a1i 9	. . . . . . . 8 ⊢ (𝜑 → 1 ∈ ℂ)
43		perfectlem.3	. . . . . . . . . 10 ⊢ (𝜑 → ¬ 2 ∥ 𝐵)
44		2prm 12701	. . . . . . . . . . 11 ⊢ 2 ∈ ℙ
45	22	nnzd 9601	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐵 ∈ ℤ)
46		coprm 12718	. . . . . . . . . . 11 ⊢ ((2 ∈ ℙ ∧ 𝐵 ∈ ℤ) → (¬ 2 ∥ 𝐵 ↔ (2 gcd 𝐵) = 1))
47	44, 45, 46	sylancr 414	. . . . . . . . . 10 ⊢ (𝜑 → (¬ 2 ∥ 𝐵 ↔ (2 gcd 𝐵) = 1))
48	43, 47	mpbid 147	. . . . . . . . 9 ⊢ (𝜑 → (2 gcd 𝐵) = 1)
49		2z 9507	. . . . . . . . . 10 ⊢ 2 ∈ ℤ
50		rpexp1i 12728	. . . . . . . . . 10 ⊢ ((2 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐴 ∈ ℕ₀) → ((2 gcd 𝐵) = 1 → ((2↑𝐴) gcd 𝐵) = 1))
51	49, 45, 3, 50	mp3an2i 1378	. . . . . . . . 9 ⊢ (𝜑 → ((2 gcd 𝐵) = 1 → ((2↑𝐴) gcd 𝐵) = 1))
52	48, 51	mpd 13	. . . . . . . 8 ⊢ (𝜑 → ((2↑𝐴) gcd 𝐵) = 1)
53		sgmmul 15723	. . . . . . . 8 ⊢ ((1 ∈ ℂ ∧ ((2↑𝐴) ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ ((2↑𝐴) gcd 𝐵) = 1)) → (1 σ ((2↑𝐴) · 𝐵)) = ((1 σ (2↑𝐴)) · (1 σ 𝐵)))
54	42, 32, 22, 52, 53	syl13anc 1275	. . . . . . 7 ⊢ (𝜑 → (1 σ ((2↑𝐴) · 𝐵)) = ((1 σ (2↑𝐴)) · (1 σ 𝐵)))
55		perfectlem.4	. . . . . . 7 ⊢ (𝜑 → (1 σ ((2↑𝐴) · 𝐵)) = (2 · ((2↑𝐴) · 𝐵)))
56	2	nncnd 9157	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐴 ∈ ℂ)
57		pncan 8385	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝐴 + 1) − 1) = 𝐴)
58	56, 41, 57	sylancl 413	. . . . . . . . . . 11 ⊢ (𝜑 → ((𝐴 + 1) − 1) = 𝐴)
59	58	oveq2d 6034	. . . . . . . . . 10 ⊢ (𝜑 → (2↑((𝐴 + 1) − 1)) = (2↑𝐴))
60	59	oveq2d 6034	. . . . . . . . 9 ⊢ (𝜑 → (1 σ (2↑((𝐴 + 1) − 1))) = (1 σ (2↑𝐴)))
61		1sgm2ppw 15722	. . . . . . . . . 10 ⊢ ((𝐴 + 1) ∈ ℕ → (1 σ (2↑((𝐴 + 1) − 1))) = ((2↑(𝐴 + 1)) − 1))
62	9, 61	syl 14	. . . . . . . . 9 ⊢ (𝜑 → (1 σ (2↑((𝐴 + 1) − 1))) = ((2↑(𝐴 + 1)) − 1))
63	60, 62	eqtr3d 2266	. . . . . . . 8 ⊢ (𝜑 → (1 σ (2↑𝐴)) = ((2↑(𝐴 + 1)) − 1))
64	63	oveq1d 6033	. . . . . . 7 ⊢ (𝜑 → ((1 σ (2↑𝐴)) · (1 σ 𝐵)) = (((2↑(𝐴 + 1)) − 1) · (1 σ 𝐵)))
65	54, 55, 64	3eqtr3d 2272	. . . . . 6 ⊢ (𝜑 → (2 · ((2↑𝐴) · 𝐵)) = (((2↑(𝐴 + 1)) − 1) · (1 σ 𝐵)))
66	37, 40, 65	3eqtrd 2268	. . . . 5 ⊢ (𝜑 → ((2↑(𝐴 + 1)) · 𝐵) = (((2↑(𝐴 + 1)) − 1) · (1 σ 𝐵)))
67	27, 66	breqtrrd 4116	. . . 4 ⊢ (𝜑 → ((2↑(𝐴 + 1)) − 1) ∥ ((2↑(𝐴 + 1)) · 𝐵))
68	20, 18	gcdcomd 12547	. . . . 5 ⊢ (𝜑 → (((2↑(𝐴 + 1)) − 1) gcd (2↑(𝐴 + 1))) = ((2↑(𝐴 + 1)) gcd ((2↑(𝐴 + 1)) − 1)))
69		iddvdsexp 12378	. . . . . . . . 9 ⊢ ((2 ∈ ℤ ∧ (𝐴 + 1) ∈ ℕ) → 2 ∥ (2↑(𝐴 + 1)))
70	49, 9, 69	sylancr 414	. . . . . . . 8 ⊢ (𝜑 → 2 ∥ (2↑(𝐴 + 1)))
71		n2dvds1 12475	. . . . . . . . . 10 ⊢ ¬ 2 ∥ 1
72	49	a1i 9	. . . . . . . . . . . 12 ⊢ (𝜑 → 2 ∈ ℤ)
73		1zzd 9506	. . . . . . . . . . . 12 ⊢ (𝜑 → 1 ∈ ℤ)
74	72, 18, 73	3jca 1203	. . . . . . . . . . 11 ⊢ (𝜑 → (2 ∈ ℤ ∧ (2↑(𝐴 + 1)) ∈ ℤ ∧ 1 ∈ ℤ))
75		dvdssub2 12398	. . . . . . . . . . 11 ⊢ (((2 ∈ ℤ ∧ (2↑(𝐴 + 1)) ∈ ℤ ∧ 1 ∈ ℤ) ∧ 2 ∥ ((2↑(𝐴 + 1)) − 1)) → (2 ∥ (2↑(𝐴 + 1)) ↔ 2 ∥ 1))
76	74, 75	sylan 283	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 2 ∥ ((2↑(𝐴 + 1)) − 1)) → (2 ∥ (2↑(𝐴 + 1)) ↔ 2 ∥ 1))
77	71, 76	mtbiri 681	. . . . . . . . 9 ⊢ ((𝜑 ∧ 2 ∥ ((2↑(𝐴 + 1)) − 1)) → ¬ 2 ∥ (2↑(𝐴 + 1)))
78	77	ex 115	. . . . . . . 8 ⊢ (𝜑 → (2 ∥ ((2↑(𝐴 + 1)) − 1) → ¬ 2 ∥ (2↑(𝐴 + 1))))
79	70, 78	mt2d 630	. . . . . . 7 ⊢ (𝜑 → ¬ 2 ∥ ((2↑(𝐴 + 1)) − 1))
80		coprm 12718	. . . . . . . 8 ⊢ ((2 ∈ ℙ ∧ ((2↑(𝐴 + 1)) − 1) ∈ ℤ) → (¬ 2 ∥ ((2↑(𝐴 + 1)) − 1) ↔ (2 gcd ((2↑(𝐴 + 1)) − 1)) = 1))
81	44, 20, 80	sylancr 414	. . . . . . 7 ⊢ (𝜑 → (¬ 2 ∥ ((2↑(𝐴 + 1)) − 1) ↔ (2 gcd ((2↑(𝐴 + 1)) − 1)) = 1))
82	79, 81	mpbid 147	. . . . . 6 ⊢ (𝜑 → (2 gcd ((2↑(𝐴 + 1)) − 1)) = 1)
83		rpexp1i 12728	. . . . . . 7 ⊢ ((2 ∈ ℤ ∧ ((2↑(𝐴 + 1)) − 1) ∈ ℤ ∧ (𝐴 + 1) ∈ ℕ₀) → ((2 gcd ((2↑(𝐴 + 1)) − 1)) = 1 → ((2↑(𝐴 + 1)) gcd ((2↑(𝐴 + 1)) − 1)) = 1))
84	49, 20, 5, 83	mp3an2i 1378	. . . . . 6 ⊢ (𝜑 → ((2 gcd ((2↑(𝐴 + 1)) − 1)) = 1 → ((2↑(𝐴 + 1)) gcd ((2↑(𝐴 + 1)) − 1)) = 1))
85	82, 84	mpd 13	. . . . 5 ⊢ (𝜑 → ((2↑(𝐴 + 1)) gcd ((2↑(𝐴 + 1)) − 1)) = 1)
86	68, 85	eqtrd 2264	. . . 4 ⊢ (𝜑 → (((2↑(𝐴 + 1)) − 1) gcd (2↑(𝐴 + 1))) = 1)
87		coprmdvds 12666	. . . . 5 ⊢ ((((2↑(𝐴 + 1)) − 1) ∈ ℤ ∧ (2↑(𝐴 + 1)) ∈ ℤ ∧ 𝐵 ∈ ℤ) → ((((2↑(𝐴 + 1)) − 1) ∥ ((2↑(𝐴 + 1)) · 𝐵) ∧ (((2↑(𝐴 + 1)) − 1) gcd (2↑(𝐴 + 1))) = 1) → ((2↑(𝐴 + 1)) − 1) ∥ 𝐵))
88	20, 18, 45, 87	syl3anc 1273	. . . 4 ⊢ (𝜑 → ((((2↑(𝐴 + 1)) − 1) ∥ ((2↑(𝐴 + 1)) · 𝐵) ∧ (((2↑(𝐴 + 1)) − 1) gcd (2↑(𝐴 + 1))) = 1) → ((2↑(𝐴 + 1)) − 1) ∥ 𝐵))
89	67, 86, 88	mp2and 433	. . 3 ⊢ (𝜑 → ((2↑(𝐴 + 1)) − 1) ∥ 𝐵)
90		nndivdvds 12359	. . . 4 ⊢ ((𝐵 ∈ ℕ ∧ ((2↑(𝐴 + 1)) − 1) ∈ ℕ) → (((2↑(𝐴 + 1)) − 1) ∥ 𝐵 ↔ (𝐵 / ((2↑(𝐴 + 1)) − 1)) ∈ ℕ))
91	22, 17, 90	syl2anc 411	. . 3 ⊢ (𝜑 → (((2↑(𝐴 + 1)) − 1) ∥ 𝐵 ↔ (𝐵 / ((2↑(𝐴 + 1)) − 1)) ∈ ℕ))
92	89, 91	mpbid 147	. 2 ⊢ (𝜑 → (𝐵 / ((2↑(𝐴 + 1)) − 1)) ∈ ℕ)
93	7, 17, 92	3jca 1203	1 ⊢ (𝜑 → ((2↑(𝐴 + 1)) ∈ ℕ ∧ ((2↑(𝐴 + 1)) − 1) ∈ ℕ ∧ (𝐵 / ((2↑(𝐴 + 1)) − 1)) ∈ ℕ))

Colors of variables: wff set class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 104 ↔ wb 105 ∧ w3a 1004 = wceq 1397 ∈ wcel 2202 class class class wbr 4088 (class class class)co 6018 ℂcc 8030 ℝcr 8031 1c1 8033 + caddc 8035 · cmul 8037 < clt 8214 − cmin 8350 / cdiv 8852 ℕcn 9143 2c2 9194 ℕ₀cn0 9402 ℤcz 9479 ↑cexp 10801 ∥ cdvds 12350 gcd cgcd 12526 ℙcprime 12681 σ csgm 15708

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 619 ax-in2 620 ax-io 716 ax-5 1495 ax-7 1496 ax-gen 1497 ax-ie1 1541 ax-ie2 1542 ax-8 1552 ax-10 1553 ax-11 1554 ax-i12 1555 ax-bndl 1557 ax-4 1558 ax-17 1574 ax-i9 1578 ax-ial 1582 ax-i5r 1583 ax-13 2204 ax-14 2205 ax-ext 2213 ax-coll 4204 ax-sep 4207 ax-nul 4215 ax-pow 4264 ax-pr 4299 ax-un 4530 ax-setind 4635 ax-iinf 4686 ax-cnex 8123 ax-resscn 8124 ax-1cn 8125 ax-1re 8126 ax-icn 8127 ax-addcl 8128 ax-addrcl 8129 ax-mulcl 8130 ax-mulrcl 8131 ax-addcom 8132 ax-mulcom 8133 ax-addass 8134 ax-mulass 8135 ax-distr 8136 ax-i2m1 8137 ax-0lt1 8138 ax-1rid 8139 ax-0id 8140 ax-rnegex 8141 ax-precex 8142 ax-cnre 8143 ax-pre-ltirr 8144 ax-pre-ltwlin 8145 ax-pre-lttrn 8146 ax-pre-apti 8147 ax-pre-ltadd 8148 ax-pre-mulgt0 8149 ax-pre-mulext 8150 ax-arch 8151 ax-caucvg 8152 ax-pre-suploc 8153 ax-addf 8154 ax-mulf 8155

This theorem depends on definitions: df-bi 117 df-stab 838 df-dc 842 df-3or 1005 df-3an 1006 df-tru 1400 df-fal 1403 df-nf 1509 df-sb 1811 df-eu 2082 df-mo 2083 df-clab 2218 df-cleq 2224 df-clel 2227 df-nfc 2363 df-ne 2403 df-nel 2498 df-ral 2515 df-rex 2516 df-reu 2517 df-rmo 2518 df-rab 2519 df-v 2804 df-sbc 3032 df-csb 3128 df-dif 3202 df-un 3204 df-in 3206 df-ss 3213 df-nul 3495 df-if 3606 df-pw 3654 df-sn 3675 df-pr 3676 df-op 3678 df-uni 3894 df-int 3929 df-iun 3972 df-disj 4065 df-br 4089 df-opab 4151 df-mpt 4152 df-tr 4188 df-id 4390 df-po 4393 df-iso 4394 df-iord 4463 df-on 4465 df-ilim 4466 df-suc 4468 df-iom 4689 df-xp 4731 df-rel 4732 df-cnv 4733 df-co 4734 df-dm 4735 df-rn 4736 df-res 4737 df-ima 4738 df-iota 5286 df-fun 5328 df-fn 5329 df-f 5330 df-f1 5331 df-fo 5332 df-f1o 5333 df-fv 5334 df-isom 5335 df-riota 5971 df-ov 6021 df-oprab 6022 df-mpo 6023 df-of 6235 df-1st 6303 df-2nd 6304 df-recs 6471 df-irdg 6536 df-frec 6557 df-1o 6582 df-2o 6583 df-oadd 6586 df-er 6702 df-map 6819 df-pm 6820 df-en 6910 df-dom 6911 df-fin 6912 df-sup 7183 df-inf 7184 df-pnf 8216 df-mnf 8217 df-xr 8218 df-ltxr 8219 df-le 8220 df-sub 8352 df-neg 8353 df-reap 8755 df-ap 8762 df-div 8853 df-inn 9144 df-2 9202 df-3 9203 df-4 9204 df-n0 9403 df-xnn0 9466 df-z 9480 df-uz 9756 df-q 9854 df-rp 9889 df-xneg 10007 df-xadd 10008 df-ioo 10127 df-ico 10129 df-icc 10130 df-fz 10244 df-fzo 10378 df-fl 10531 df-mod 10586 df-seqfrec 10711 df-exp 10802 df-fac 10989 df-bc 11011 df-ihash 11039 df-shft 11377 df-cj 11404 df-re 11405 df-im 11406 df-rsqrt 11560 df-abs 11561 df-clim 11841 df-sumdc 11916 df-ef 12211 df-e 12212 df-dvds 12351 df-gcd 12527 df-prm 12682 df-pc 12860 df-rest 13326 df-topgen 13345 df-psmet 14560 df-xmet 14561 df-met 14562 df-bl 14563 df-mopn 14564 df-top 14725 df-topon 14738 df-bases 14770 df-ntr 14823 df-cn 14915 df-cnp 14916 df-tx 14980 df-cncf 15298 df-limced 15383 df-dvap 15384 df-relog 15585 df-rpcxp 15586 df-sgm 15709

This theorem is referenced by: perfectlem2 15727

W3C validator