perfectALTVlem1 - Mathbox for Alexander van der Vekens

	Mathbox for Alexander van der Vekens		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > Mathboxes > perfectALTVlem1		Structured version Visualization version GIF version

Theorem perfectALTVlem1 45061

Description: Lemma for perfectALTV 45063. (Contributed by Mario Carneiro, 7-Jun-2016.) (Revised by AV, 1-Jul-2020.)

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

**Assertion**
Ref	Expression
perfectALTVlem1	⊢ (𝜑 → ((2↑(𝐴 + 1)) ∈ ℕ ∧ ((2↑(𝐴 + 1)) − 1) ∈ ℕ ∧ (𝐵 / ((2↑(𝐴 + 1)) − 1)) ∈ ℕ))

**Proof of Theorem perfectALTVlem1**
Step	Hyp	Ref	Expression
1		2nn 11976	. . 3 ⊢ 2 ∈ ℕ
2		perfectALTVlem.1	. . . . 5 ⊢ (𝜑 → 𝐴 ∈ ℕ)
3	2	nnnn0d 12223	. . . 4 ⊢ (𝜑 → 𝐴 ∈ ℕ₀)
4		peano2nn0 12203	. . . 4 ⊢ (𝐴 ∈ ℕ₀ → (𝐴 + 1) ∈ ℕ₀)
5	3, 4	syl 17	. . 3 ⊢ (𝜑 → (𝐴 + 1) ∈ ℕ₀)
6		nnexpcl 13723	. . 3 ⊢ ((2 ∈ ℕ ∧ (𝐴 + 1) ∈ ℕ₀) → (2↑(𝐴 + 1)) ∈ ℕ)
7	1, 5, 6	sylancr 586	. 2 ⊢ (𝜑 → (2↑(𝐴 + 1)) ∈ ℕ)
8		2re 11977	. . . . 5 ⊢ 2 ∈ ℝ
9	8	a1i 11	. . . 4 ⊢ (𝜑 → 2 ∈ ℝ)
10	2	peano2nnd 11920	. . . 4 ⊢ (𝜑 → (𝐴 + 1) ∈ ℕ)
11		1lt2 12074	. . . . 5 ⊢ 1 < 2
12	11	a1i 11	. . . 4 ⊢ (𝜑 → 1 < 2)
13		expgt1 13749	. . . 4 ⊢ ((2 ∈ ℝ ∧ (𝐴 + 1) ∈ ℕ ∧ 1 < 2) → 1 < (2↑(𝐴 + 1)))
14	9, 10, 12, 13	syl3anc 1369	. . 3 ⊢ (𝜑 → 1 < (2↑(𝐴 + 1)))
15		1nn 11914	. . . 4 ⊢ 1 ∈ ℕ
16		nnsub 11947	. . . 4 ⊢ ((1 ∈ ℕ ∧ (2↑(𝐴 + 1)) ∈ ℕ) → (1 < (2↑(𝐴 + 1)) ↔ ((2↑(𝐴 + 1)) − 1) ∈ ℕ))
17	15, 7, 16	sylancr 586	. . 3 ⊢ (𝜑 → (1 < (2↑(𝐴 + 1)) ↔ ((2↑(𝐴 + 1)) − 1) ∈ ℕ))
18	14, 17	mpbid 231	. 2 ⊢ (𝜑 → ((2↑(𝐴 + 1)) − 1) ∈ ℕ)
19	7	nnzd 12354	. . . . . . 7 ⊢ (𝜑 → (2↑(𝐴 + 1)) ∈ ℤ)
20		peano2zm 12293	. . . . . . 7 ⊢ ((2↑(𝐴 + 1)) ∈ ℤ → ((2↑(𝐴 + 1)) − 1) ∈ ℤ)
21	19, 20	syl 17	. . . . . 6 ⊢ (𝜑 → ((2↑(𝐴 + 1)) − 1) ∈ ℤ)
22		1nn0 12179	. . . . . . . 8 ⊢ 1 ∈ ℕ₀
23		perfectALTVlem.2	. . . . . . . 8 ⊢ (𝜑 → 𝐵 ∈ ℕ)
24		sgmnncl 26201	. . . . . . . 8 ⊢ ((1 ∈ ℕ₀ ∧ 𝐵 ∈ ℕ) → (1 σ 𝐵) ∈ ℕ)
25	22, 23, 24	sylancr 586	. . . . . . 7 ⊢ (𝜑 → (1 σ 𝐵) ∈ ℕ)
26	25	nnzd 12354	. . . . . 6 ⊢ (𝜑 → (1 σ 𝐵) ∈ ℤ)
27		dvdsmul1 15915	. . . . . 6 ⊢ ((((2↑(𝐴 + 1)) − 1) ∈ ℤ ∧ (1 σ 𝐵) ∈ ℤ) → ((2↑(𝐴 + 1)) − 1) ∥ (((2↑(𝐴 + 1)) − 1) · (1 σ 𝐵)))
28	21, 26, 27	syl2anc 583	. . . . 5 ⊢ (𝜑 → ((2↑(𝐴 + 1)) − 1) ∥ (((2↑(𝐴 + 1)) − 1) · (1 σ 𝐵)))
29		2cn 11978	. . . . . . . . 9 ⊢ 2 ∈ ℂ
30		expp1 13717	. . . . . . . . 9 ⊢ ((2 ∈ ℂ ∧ 𝐴 ∈ ℕ₀) → (2↑(𝐴 + 1)) = ((2↑𝐴) · 2))
31	29, 3, 30	sylancr 586	. . . . . . . 8 ⊢ (𝜑 → (2↑(𝐴 + 1)) = ((2↑𝐴) · 2))
32		nnexpcl 13723	. . . . . . . . . . 11 ⊢ ((2 ∈ ℕ ∧ 𝐴 ∈ ℕ₀) → (2↑𝐴) ∈ ℕ)
33	1, 3, 32	sylancr 586	. . . . . . . . . 10 ⊢ (𝜑 → (2↑𝐴) ∈ ℕ)
34	33	nncnd 11919	. . . . . . . . 9 ⊢ (𝜑 → (2↑𝐴) ∈ ℂ)
35		mulcom 10888	. . . . . . . . 9 ⊢ (((2↑𝐴) ∈ ℂ ∧ 2 ∈ ℂ) → ((2↑𝐴) · 2) = (2 · (2↑𝐴)))
36	34, 29, 35	sylancl 585	. . . . . . . 8 ⊢ (𝜑 → ((2↑𝐴) · 2) = (2 · (2↑𝐴)))
37	31, 36	eqtrd 2778	. . . . . . 7 ⊢ (𝜑 → (2↑(𝐴 + 1)) = (2 · (2↑𝐴)))
38	37	oveq1d 7270	. . . . . 6 ⊢ (𝜑 → ((2↑(𝐴 + 1)) · 𝐵) = ((2 · (2↑𝐴)) · 𝐵))
39	29	a1i 11	. . . . . . 7 ⊢ (𝜑 → 2 ∈ ℂ)
40	23	nncnd 11919	. . . . . . 7 ⊢ (𝜑 → 𝐵 ∈ ℂ)
41	39, 34, 40	mulassd 10929	. . . . . 6 ⊢ (𝜑 → ((2 · (2↑𝐴)) · 𝐵) = (2 · ((2↑𝐴) · 𝐵)))
42		1cnd 10901	. . . . . . . 8 ⊢ (𝜑 → 1 ∈ ℂ)
43		perfectALTVlem.3	. . . . . . . . . 10 ⊢ (𝜑 → 𝐵 ∈ Odd )
44		isodd7 45005	. . . . . . . . . . 11 ⊢ (𝐵 ∈ Odd ↔ (𝐵 ∈ ℤ ∧ (2 gcd 𝐵) = 1))
45	44	simprbi 496	. . . . . . . . . 10 ⊢ (𝐵 ∈ Odd → (2 gcd 𝐵) = 1)
46	43, 45	syl 17	. . . . . . . . 9 ⊢ (𝜑 → (2 gcd 𝐵) = 1)
47		2z 12282	. . . . . . . . . . 11 ⊢ 2 ∈ ℤ
48	47	a1i 11	. . . . . . . . . 10 ⊢ (𝜑 → 2 ∈ ℤ)
49	23	nnzd 12354	. . . . . . . . . 10 ⊢ (𝜑 → 𝐵 ∈ ℤ)
50		rpexp1i 16356	. . . . . . . . . 10 ⊢ ((2 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐴 ∈ ℕ₀) → ((2 gcd 𝐵) = 1 → ((2↑𝐴) gcd 𝐵) = 1))
51	48, 49, 3, 50	syl3anc 1369	. . . . . . . . 9 ⊢ (𝜑 → ((2 gcd 𝐵) = 1 → ((2↑𝐴) gcd 𝐵) = 1))
52	46, 51	mpd 15	. . . . . . . 8 ⊢ (𝜑 → ((2↑𝐴) gcd 𝐵) = 1)
53		sgmmul 26254	. . . . . . . 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, 33, 23, 52, 53	syl13anc 1370	. . . . . . 7 ⊢ (𝜑 → (1 σ ((2↑𝐴) · 𝐵)) = ((1 σ (2↑𝐴)) · (1 σ 𝐵)))
55		perfectALTVlem.4	. . . . . . 7 ⊢ (𝜑 → (1 σ ((2↑𝐴) · 𝐵)) = (2 · ((2↑𝐴) · 𝐵)))
56	2	nncnd 11919	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐴 ∈ ℂ)
57		pncan1 11329	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ ℂ → ((𝐴 + 1) − 1) = 𝐴)
58	56, 57	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → ((𝐴 + 1) − 1) = 𝐴)
59	58	oveq2d 7271	. . . . . . . . . 10 ⊢ (𝜑 → (2↑((𝐴 + 1) − 1)) = (2↑𝐴))
60	59	oveq2d 7271	. . . . . . . . 9 ⊢ (𝜑 → (1 σ (2↑((𝐴 + 1) − 1))) = (1 σ (2↑𝐴)))
61		1sgm2ppw 26253	. . . . . . . . . 10 ⊢ ((𝐴 + 1) ∈ ℕ → (1 σ (2↑((𝐴 + 1) − 1))) = ((2↑(𝐴 + 1)) − 1))
62	10, 61	syl 17	. . . . . . . . 9 ⊢ (𝜑 → (1 σ (2↑((𝐴 + 1) − 1))) = ((2↑(𝐴 + 1)) − 1))
63	60, 62	eqtr3d 2780	. . . . . . . 8 ⊢ (𝜑 → (1 σ (2↑𝐴)) = ((2↑(𝐴 + 1)) − 1))
64	63	oveq1d 7270	. . . . . . 7 ⊢ (𝜑 → ((1 σ (2↑𝐴)) · (1 σ 𝐵)) = (((2↑(𝐴 + 1)) − 1) · (1 σ 𝐵)))
65	54, 55, 64	3eqtr3d 2786	. . . . . 6 ⊢ (𝜑 → (2 · ((2↑𝐴) · 𝐵)) = (((2↑(𝐴 + 1)) − 1) · (1 σ 𝐵)))
66	38, 41, 65	3eqtrd 2782	. . . . 5 ⊢ (𝜑 → ((2↑(𝐴 + 1)) · 𝐵) = (((2↑(𝐴 + 1)) − 1) · (1 σ 𝐵)))
67	28, 66	breqtrrd 5098	. . . 4 ⊢ (𝜑 → ((2↑(𝐴 + 1)) − 1) ∥ ((2↑(𝐴 + 1)) · 𝐵))
68	21, 19	gcdcomd 16149	. . . . 5 ⊢ (𝜑 → (((2↑(𝐴 + 1)) − 1) gcd (2↑(𝐴 + 1))) = ((2↑(𝐴 + 1)) gcd ((2↑(𝐴 + 1)) − 1)))
69		nnpw2evenALTV 45042	. . . . . . . . 9 ⊢ ((𝐴 + 1) ∈ ℕ → (2↑(𝐴 + 1)) ∈ Even )
70	10, 69	syl 17	. . . . . . . 8 ⊢ (𝜑 → (2↑(𝐴 + 1)) ∈ Even )
71		evenm1odd 44979	. . . . . . . 8 ⊢ ((2↑(𝐴 + 1)) ∈ Even → ((2↑(𝐴 + 1)) − 1) ∈ Odd )
72	70, 71	syl 17	. . . . . . 7 ⊢ (𝜑 → ((2↑(𝐴 + 1)) − 1) ∈ Odd )
73		isodd7 45005	. . . . . . . 8 ⊢ (((2↑(𝐴 + 1)) − 1) ∈ Odd ↔ (((2↑(𝐴 + 1)) − 1) ∈ ℤ ∧ (2 gcd ((2↑(𝐴 + 1)) − 1)) = 1))
74	73	simprbi 496	. . . . . . 7 ⊢ (((2↑(𝐴 + 1)) − 1) ∈ Odd → (2 gcd ((2↑(𝐴 + 1)) − 1)) = 1)
75	72, 74	syl 17	. . . . . 6 ⊢ (𝜑 → (2 gcd ((2↑(𝐴 + 1)) − 1)) = 1)
76		rpexp1i 16356	. . . . . . 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))
77	48, 21, 5, 76	syl3anc 1369	. . . . . 6 ⊢ (𝜑 → ((2 gcd ((2↑(𝐴 + 1)) − 1)) = 1 → ((2↑(𝐴 + 1)) gcd ((2↑(𝐴 + 1)) − 1)) = 1))
78	75, 77	mpd 15	. . . . 5 ⊢ (𝜑 → ((2↑(𝐴 + 1)) gcd ((2↑(𝐴 + 1)) − 1)) = 1)
79	68, 78	eqtrd 2778	. . . 4 ⊢ (𝜑 → (((2↑(𝐴 + 1)) − 1) gcd (2↑(𝐴 + 1))) = 1)
80		coprmdvds 16286	. . . . 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))
81	21, 19, 49, 80	syl3anc 1369	. . . 4 ⊢ (𝜑 → ((((2↑(𝐴 + 1)) − 1) ∥ ((2↑(𝐴 + 1)) · 𝐵) ∧ (((2↑(𝐴 + 1)) − 1) gcd (2↑(𝐴 + 1))) = 1) → ((2↑(𝐴 + 1)) − 1) ∥ 𝐵))
82	67, 79, 81	mp2and 695	. . 3 ⊢ (𝜑 → ((2↑(𝐴 + 1)) − 1) ∥ 𝐵)
83		nndivdvds 15900	. . . 4 ⊢ ((𝐵 ∈ ℕ ∧ ((2↑(𝐴 + 1)) − 1) ∈ ℕ) → (((2↑(𝐴 + 1)) − 1) ∥ 𝐵 ↔ (𝐵 / ((2↑(𝐴 + 1)) − 1)) ∈ ℕ))
84	23, 18, 83	syl2anc 583	. . 3 ⊢ (𝜑 → (((2↑(𝐴 + 1)) − 1) ∥ 𝐵 ↔ (𝐵 / ((2↑(𝐴 + 1)) − 1)) ∈ ℕ))
85	82, 84	mpbid 231	. 2 ⊢ (𝜑 → (𝐵 / ((2↑(𝐴 + 1)) − 1)) ∈ ℕ)
86	7, 18, 85	3jca 1126	1 ⊢ (𝜑 → ((2↑(𝐴 + 1)) ∈ ℕ ∧ ((2↑(𝐴 + 1)) − 1) ∈ ℕ ∧ (𝐵 / ((2↑(𝐴 + 1)) − 1)) ∈ ℕ))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 ∧ w3a 1085 = wceq 1539 ∈ wcel 2108 class class class wbr 5070 (class class class)co 7255 ℂcc 10800 ℝcr 10801 1c1 10803 + caddc 10805 · cmul 10807 < clt 10940 − cmin 11135 / cdiv 11562 ℕcn 11903 2c2 11958 ℕ₀cn0 12163 ℤcz 12249 ↑cexp 13710 ∥ cdvds 15891 gcd cgcd 16129 σ csgm 26150 Even ceven 44964 Odd codd 44965

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1799 ax-4 1813 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2156 ax-12 2173 ax-ext 2709 ax-rep 5205 ax-sep 5218 ax-nul 5225 ax-pow 5283 ax-pr 5347 ax-un 7566 ax-inf2 9329 ax-cnex 10858 ax-resscn 10859 ax-1cn 10860 ax-icn 10861 ax-addcl 10862 ax-addrcl 10863 ax-mulcl 10864 ax-mulrcl 10865 ax-mulcom 10866 ax-addass 10867 ax-mulass 10868 ax-distr 10869 ax-i2m1 10870 ax-1ne0 10871 ax-1rid 10872 ax-rnegex 10873 ax-rrecex 10874 ax-cnre 10875 ax-pre-lttri 10876 ax-pre-lttrn 10877 ax-pre-ltadd 10878 ax-pre-mulgt0 10879 ax-pre-sup 10880 ax-addf 10881 ax-mulf 10882

This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1784 df-nf 1788 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2817 df-nfc 2888 df-ne 2943 df-nel 3049 df-ral 3068 df-rex 3069 df-reu 3070 df-rmo 3071 df-rab 3072 df-v 3424 df-sbc 3712 df-csb 3829 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-pss 3902 df-nul 4254 df-if 4457 df-pw 4532 df-sn 4559 df-pr 4561 df-tp 4563 df-op 4565 df-uni 4837 df-int 4877 df-iun 4923 df-iin 4924 df-br 5071 df-opab 5133 df-mpt 5154 df-tr 5188 df-id 5480 df-eprel 5486 df-po 5494 df-so 5495 df-fr 5535 df-se 5536 df-we 5537 df-xp 5586 df-rel 5587 df-cnv 5588 df-co 5589 df-dm 5590 df-rn 5591 df-res 5592 df-ima 5593 df-pred 6191 df-ord 6254 df-on 6255 df-lim 6256 df-suc 6257 df-iota 6376 df-fun 6420 df-fn 6421 df-f 6422 df-f1 6423 df-fo 6424 df-f1o 6425 df-fv 6426 df-isom 6427 df-riota 7212 df-ov 7258 df-oprab 7259 df-mpo 7260 df-of 7511 df-om 7688 df-1st 7804 df-2nd 7805 df-supp 7949 df-frecs 8068 df-wrecs 8099 df-recs 8173 df-rdg 8212 df-1o 8267 df-2o 8268 df-er 8456 df-map 8575 df-pm 8576 df-ixp 8644 df-en 8692 df-dom 8693 df-sdom 8694 df-fin 8695 df-fsupp 9059 df-fi 9100 df-sup 9131 df-inf 9132 df-oi 9199 df-card 9628 df-pnf 10942 df-mnf 10943 df-xr 10944 df-ltxr 10945 df-le 10946 df-sub 11137 df-neg 11138 df-div 11563 df-nn 11904 df-2 11966 df-3 11967 df-4 11968 df-5 11969 df-6 11970 df-7 11971 df-8 11972 df-9 11973 df-n0 12164 df-z 12250 df-dec 12367 df-uz 12512 df-q 12618 df-rp 12660 df-xneg 12777 df-xadd 12778 df-xmul 12779 df-ioo 13012 df-ioc 13013 df-ico 13014 df-icc 13015 df-fz 13169 df-fzo 13312 df-fl 13440 df-mod 13518 df-seq 13650 df-exp 13711 df-fac 13916 df-bc 13945 df-hash 13973 df-shft 14706 df-cj 14738 df-re 14739 df-im 14740 df-sqrt 14874 df-abs 14875 df-limsup 15108 df-clim 15125 df-rlim 15126 df-sum 15326 df-ef 15705 df-sin 15707 df-cos 15708 df-pi 15710 df-dvds 15892 df-gcd 16130 df-prm 16305 df-pc 16466 df-struct 16776 df-sets 16793 df-slot 16811 df-ndx 16823 df-base 16841 df-ress 16868 df-plusg 16901 df-mulr 16902 df-starv 16903 df-sca 16904 df-vsca 16905 df-ip 16906 df-tset 16907 df-ple 16908 df-ds 16910 df-unif 16911 df-hom 16912 df-cco 16913 df-rest 17050 df-topn 17051 df-0g 17069 df-gsum 17070 df-topgen 17071 df-pt 17072 df-prds 17075 df-xrs 17130 df-qtop 17135 df-imas 17136 df-xps 17138 df-mre 17212 df-mrc 17213 df-acs 17215 df-mgm 18241 df-sgrp 18290 df-mnd 18301 df-submnd 18346 df-mulg 18616 df-cntz 18838 df-cmn 19303 df-psmet 20502 df-xmet 20503 df-met 20504 df-bl 20505 df-mopn 20506 df-fbas 20507 df-fg 20508 df-cnfld 20511 df-top 21951 df-topon 21968 df-topsp 21990 df-bases 22004 df-cld 22078 df-ntr 22079 df-cls 22080 df-nei 22157 df-lp 22195 df-perf 22196 df-cn 22286 df-cnp 22287 df-haus 22374 df-tx 22621 df-hmeo 22814 df-fil 22905 df-fm 22997 df-flim 22998 df-flf 22999 df-xms 23381 df-ms 23382 df-tms 23383 df-cncf 23947 df-limc 24935 df-dv 24936 df-log 25617 df-cxp 25618 df-sgm 26156 df-even 44966 df-odd 44967

This theorem is referenced by: perfectALTVlem2 45062

W3C validator