perfectlem1 - Intuitionistic Logic Explorer

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Theorem perfectlem1 15752

Description: Lemma for perfect 15754. (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 9310	. . 3 ⊢ 2 ∈ ℕ
2		perfectlem.1	. . . . 5 ⊢ (𝜑 → 𝐴 ∈ ℕ)
3	2	nnnn0d 9460	. . . 4 ⊢ (𝜑 → 𝐴 ∈ ℕ₀)
4		peano2nn0 9447	. . . 4 ⊢ (𝐴 ∈ ℕ₀ → (𝐴 + 1) ∈ ℕ₀)
5	3, 4	syl 14	. . 3 ⊢ (𝜑 → (𝐴 + 1) ∈ ℕ₀)
6		nnexpcl 10820	. . 3 ⊢ ((2 ∈ ℕ ∧ (𝐴 + 1) ∈ ℕ₀) → (2↑(𝐴 + 1)) ∈ ℕ)
7	1, 5, 6	sylancr 414	. 2 ⊢ (𝜑 → (2↑(𝐴 + 1)) ∈ ℕ)
8		2re 9218	. . . 4 ⊢ 2 ∈ ℝ
9	2	peano2nnd 9163	. . . 4 ⊢ (𝜑 → (𝐴 + 1) ∈ ℕ)
10		1lt2 9318	. . . . 5 ⊢ 1 < 2
11	10	a1i 9	. . . 4 ⊢ (𝜑 → 1 < 2)
12		expgt1 10845	. . . 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 9159	. . . 4 ⊢ 1 ∈ ℕ
15		nnsub 9187	. . . 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 9606	. . . . . . 7 ⊢ (𝜑 → (2↑(𝐴 + 1)) ∈ ℤ)
19		peano2zm 9522	. . . . . . 7 ⊢ ((2↑(𝐴 + 1)) ∈ ℤ → ((2↑(𝐴 + 1)) − 1) ∈ ℤ)
20	18, 19	syl 14	. . . . . 6 ⊢ (𝜑 → ((2↑(𝐴 + 1)) − 1) ∈ ℤ)
21		1nn0 9423	. . . . . . . 8 ⊢ 1 ∈ ℕ₀
22		perfectlem.2	. . . . . . . 8 ⊢ (𝜑 → 𝐵 ∈ ℕ)
23		sgmnncl 15741	. . . . . . . 8 ⊢ ((1 ∈ ℕ₀ ∧ 𝐵 ∈ ℕ) → (1 σ 𝐵) ∈ ℕ)
24	21, 22, 23	sylancr 414	. . . . . . 7 ⊢ (𝜑 → (1 σ 𝐵) ∈ ℕ)
25	24	nnzd 9606	. . . . . 6 ⊢ (𝜑 → (1 σ 𝐵) ∈ ℤ)
26		dvdsmul1 12397	. . . . . 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 9219	. . . . . . . . 9 ⊢ 2 ∈ ℂ
29		expp1 10814	. . . . . . . . 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 10820	. . . . . . . . . . 11 ⊢ ((2 ∈ ℕ ∧ 𝐴 ∈ ℕ₀) → (2↑𝐴) ∈ ℕ)
32	1, 3, 31	sylancr 414	. . . . . . . . . 10 ⊢ (𝜑 → (2↑𝐴) ∈ ℕ)
33	32	nncnd 9162	. . . . . . . . 9 ⊢ (𝜑 → (2↑𝐴) ∈ ℂ)
34		mulcom 8166	. . . . . . . . 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 2263	. . . . . . 7 ⊢ (𝜑 → (2↑(𝐴 + 1)) = (2 · (2↑𝐴)))
37	36	oveq1d 6038	. . . . . 6 ⊢ (𝜑 → ((2↑(𝐴 + 1)) · 𝐵) = ((2 · (2↑𝐴)) · 𝐵))
38	28	a1i 9	. . . . . . 7 ⊢ (𝜑 → 2 ∈ ℂ)
39	22	nncnd 9162	. . . . . . 7 ⊢ (𝜑 → 𝐵 ∈ ℂ)
40	38, 33, 39	mulassd 8208	. . . . . 6 ⊢ (𝜑 → ((2 · (2↑𝐴)) · 𝐵) = (2 · ((2↑𝐴) · 𝐵)))
41		ax-1cn 8130	. . . . . . . . 9 ⊢ 1 ∈ ℂ
42	41	a1i 9	. . . . . . . 8 ⊢ (𝜑 → 1 ∈ ℂ)
43		perfectlem.3	. . . . . . . . . 10 ⊢ (𝜑 → ¬ 2 ∥ 𝐵)
44		2prm 12722	. . . . . . . . . . 11 ⊢ 2 ∈ ℙ
45	22	nnzd 9606	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐵 ∈ ℤ)
46		coprm 12739	. . . . . . . . . . 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 9512	. . . . . . . . . 10 ⊢ 2 ∈ ℤ
50		rpexp1i 12749	. . . . . . . . . 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 15749	. . . . . . . 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 9162	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐴 ∈ ℂ)
57		pncan 8390	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝐴 + 1) − 1) = 𝐴)
58	56, 41, 57	sylancl 413	. . . . . . . . . . 11 ⊢ (𝜑 → ((𝐴 + 1) − 1) = 𝐴)
59	58	oveq2d 6039	. . . . . . . . . 10 ⊢ (𝜑 → (2↑((𝐴 + 1) − 1)) = (2↑𝐴))
60	59	oveq2d 6039	. . . . . . . . 9 ⊢ (𝜑 → (1 σ (2↑((𝐴 + 1) − 1))) = (1 σ (2↑𝐴)))
61		1sgm2ppw 15748	. . . . . . . . . 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 2265	. . . . . . . 8 ⊢ (𝜑 → (1 σ (2↑𝐴)) = ((2↑(𝐴 + 1)) − 1))
64	63	oveq1d 6038	. . . . . . 7 ⊢ (𝜑 → ((1 σ (2↑𝐴)) · (1 σ 𝐵)) = (((2↑(𝐴 + 1)) − 1) · (1 σ 𝐵)))
65	54, 55, 64	3eqtr3d 2271	. . . . . 6 ⊢ (𝜑 → (2 · ((2↑𝐴) · 𝐵)) = (((2↑(𝐴 + 1)) − 1) · (1 σ 𝐵)))
66	37, 40, 65	3eqtrd 2267	. . . . 5 ⊢ (𝜑 → ((2↑(𝐴 + 1)) · 𝐵) = (((2↑(𝐴 + 1)) − 1) · (1 σ 𝐵)))
67	27, 66	breqtrrd 4117	. . . 4 ⊢ (𝜑 → ((2↑(𝐴 + 1)) − 1) ∥ ((2↑(𝐴 + 1)) · 𝐵))
68	20, 18	gcdcomd 12568	. . . . 5 ⊢ (𝜑 → (((2↑(𝐴 + 1)) − 1) gcd (2↑(𝐴 + 1))) = ((2↑(𝐴 + 1)) gcd ((2↑(𝐴 + 1)) − 1)))
69		iddvdsexp 12399	. . . . . . . . 9 ⊢ ((2 ∈ ℤ ∧ (𝐴 + 1) ∈ ℕ) → 2 ∥ (2↑(𝐴 + 1)))
70	49, 9, 69	sylancr 414	. . . . . . . 8 ⊢ (𝜑 → 2 ∥ (2↑(𝐴 + 1)))
71		n2dvds1 12496	. . . . . . . . . 10 ⊢ ¬ 2 ∥ 1
72	49	a1i 9	. . . . . . . . . . . 12 ⊢ (𝜑 → 2 ∈ ℤ)
73		1zzd 9511	. . . . . . . . . . . 12 ⊢ (𝜑 → 1 ∈ ℤ)
74	72, 18, 73	3jca 1203	. . . . . . . . . . 11 ⊢ (𝜑 → (2 ∈ ℤ ∧ (2↑(𝐴 + 1)) ∈ ℤ ∧ 1 ∈ ℤ))
75		dvdssub2 12419	. . . . . . . . . . 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 12739	. . . . . . . 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 12749	. . . . . . 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 2263	. . . 4 ⊢ (𝜑 → (((2↑(𝐴 + 1)) − 1) gcd (2↑(𝐴 + 1))) = 1)
87		coprmdvds 12687	. . . . 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 12380	. . . 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 2201 class class class wbr 4089 (class class class)co 6023 ℂcc 8035 ℝcr 8036 1c1 8038 + caddc 8040 · cmul 8042 < clt 8219 − cmin 8355 / cdiv 8857 ℕcn 9148 2c2 9199 ℕ₀cn0 9407 ℤcz 9484 ↑cexp 10806 ∥ cdvds 12371 gcd cgcd 12547 ℙcprime 12702 σ csgm 15734

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 2203 ax-14 2204 ax-ext 2212 ax-coll 4205 ax-sep 4208 ax-nul 4216 ax-pow 4266 ax-pr 4301 ax-un 4532 ax-setind 4637 ax-iinf 4688 ax-cnex 8128 ax-resscn 8129 ax-1cn 8130 ax-1re 8131 ax-icn 8132 ax-addcl 8133 ax-addrcl 8134 ax-mulcl 8135 ax-mulrcl 8136 ax-addcom 8137 ax-mulcom 8138 ax-addass 8139 ax-mulass 8140 ax-distr 8141 ax-i2m1 8142 ax-0lt1 8143 ax-1rid 8144 ax-0id 8145 ax-rnegex 8146 ax-precex 8147 ax-cnre 8148 ax-pre-ltirr 8149 ax-pre-ltwlin 8150 ax-pre-lttrn 8151 ax-pre-apti 8152 ax-pre-ltadd 8153 ax-pre-mulgt0 8154 ax-pre-mulext 8155 ax-arch 8156 ax-caucvg 8157 ax-pre-suploc 8158 ax-addf 8159 ax-mulf 8160

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 1810 df-eu 2081 df-mo 2082 df-clab 2217 df-cleq 2223 df-clel 2226 df-nfc 2362 df-ne 2402 df-nel 2497 df-ral 2514 df-rex 2515 df-reu 2516 df-rmo 2517 df-rab 2518 df-v 2803 df-sbc 3031 df-csb 3127 df-dif 3201 df-un 3203 df-in 3205 df-ss 3212 df-nul 3494 df-if 3605 df-pw 3655 df-sn 3676 df-pr 3677 df-op 3679 df-uni 3895 df-int 3930 df-iun 3973 df-disj 4066 df-br 4090 df-opab 4152 df-mpt 4153 df-tr 4189 df-id 4392 df-po 4395 df-iso 4396 df-iord 4465 df-on 4467 df-ilim 4468 df-suc 4470 df-iom 4691 df-xp 4733 df-rel 4734 df-cnv 4735 df-co 4736 df-dm 4737 df-rn 4738 df-res 4739 df-ima 4740 df-iota 5288 df-fun 5330 df-fn 5331 df-f 5332 df-f1 5333 df-fo 5334 df-f1o 5335 df-fv 5336 df-isom 5337 df-riota 5976 df-ov 6026 df-oprab 6027 df-mpo 6028 df-of 6240 df-1st 6308 df-2nd 6309 df-recs 6476 df-irdg 6541 df-frec 6562 df-1o 6587 df-2o 6588 df-oadd 6591 df-er 6707 df-map 6824 df-pm 6825 df-en 6915 df-dom 6916 df-fin 6917 df-sup 7188 df-inf 7189 df-pnf 8221 df-mnf 8222 df-xr 8223 df-ltxr 8224 df-le 8225 df-sub 8357 df-neg 8358 df-reap 8760 df-ap 8767 df-div 8858 df-inn 9149 df-2 9207 df-3 9208 df-4 9209 df-n0 9408 df-xnn0 9471 df-z 9485 df-uz 9761 df-q 9859 df-rp 9894 df-xneg 10012 df-xadd 10013 df-ioo 10132 df-ico 10134 df-icc 10135 df-fz 10249 df-fzo 10383 df-fl 10536 df-mod 10591 df-seqfrec 10716 df-exp 10807 df-fac 10994 df-bc 11016 df-ihash 11044 df-shft 11398 df-cj 11425 df-re 11426 df-im 11427 df-rsqrt 11581 df-abs 11582 df-clim 11862 df-sumdc 11937 df-ef 12232 df-e 12233 df-dvds 12372 df-gcd 12548 df-prm 12703 df-pc 12881 df-rest 13347 df-topgen 13366 df-psmet 14581 df-xmet 14582 df-met 14583 df-bl 14584 df-mopn 14585 df-top 14751 df-topon 14764 df-bases 14796 df-ntr 14849 df-cn 14941 df-cnp 14942 df-tx 15006 df-cncf 15324 df-limced 15409 df-dvap 15410 df-relog 15611 df-rpcxp 15612 df-sgm 15735

This theorem is referenced by: perfectlem2 15753

W3C validator