perfectALTVlem1 - Mathbox for Alexander van der Vekens

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Theorem perfectALTVlem1 45151

Description: Lemma for perfectALTV 45153. (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 12056	. . 3 ⊢ 2 ∈ ℕ
2		perfectALTVlem.1	. . . . 5 ⊢ (𝜑 → 𝐴 ∈ ℕ)
3	2	nnnn0d 12303	. . . 4 ⊢ (𝜑 → 𝐴 ∈ ℕ₀)
4		peano2nn0 12283	. . . 4 ⊢ (𝐴 ∈ ℕ₀ → (𝐴 + 1) ∈ ℕ₀)
5	3, 4	syl 17	. . 3 ⊢ (𝜑 → (𝐴 + 1) ∈ ℕ₀)
6		nnexpcl 13805	. . 3 ⊢ ((2 ∈ ℕ ∧ (𝐴 + 1) ∈ ℕ₀) → (2↑(𝐴 + 1)) ∈ ℕ)
7	1, 5, 6	sylancr 587	. 2 ⊢ (𝜑 → (2↑(𝐴 + 1)) ∈ ℕ)
8		2re 12057	. . . . 5 ⊢ 2 ∈ ℝ
9	8	a1i 11	. . . 4 ⊢ (𝜑 → 2 ∈ ℝ)
10	2	peano2nnd 12000	. . . 4 ⊢ (𝜑 → (𝐴 + 1) ∈ ℕ)
11		1lt2 12154	. . . . 5 ⊢ 1 < 2
12	11	a1i 11	. . . 4 ⊢ (𝜑 → 1 < 2)
13		expgt1 13831	. . . 4 ⊢ ((2 ∈ ℝ ∧ (𝐴 + 1) ∈ ℕ ∧ 1 < 2) → 1 < (2↑(𝐴 + 1)))
14	9, 10, 12, 13	syl3anc 1370	. . 3 ⊢ (𝜑 → 1 < (2↑(𝐴 + 1)))
15		1nn 11994	. . . 4 ⊢ 1 ∈ ℕ
16		nnsub 12027	. . . 4 ⊢ ((1 ∈ ℕ ∧ (2↑(𝐴 + 1)) ∈ ℕ) → (1 < (2↑(𝐴 + 1)) ↔ ((2↑(𝐴 + 1)) − 1) ∈ ℕ))
17	15, 7, 16	sylancr 587	. . 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 12435	. . . . . . 7 ⊢ (𝜑 → (2↑(𝐴 + 1)) ∈ ℤ)
20		peano2zm 12373	. . . . . . 7 ⊢ ((2↑(𝐴 + 1)) ∈ ℤ → ((2↑(𝐴 + 1)) − 1) ∈ ℤ)
21	19, 20	syl 17	. . . . . 6 ⊢ (𝜑 → ((2↑(𝐴 + 1)) − 1) ∈ ℤ)
22		1nn0 12259	. . . . . . . 8 ⊢ 1 ∈ ℕ₀
23		perfectALTVlem.2	. . . . . . . 8 ⊢ (𝜑 → 𝐵 ∈ ℕ)
24		sgmnncl 26306	. . . . . . . 8 ⊢ ((1 ∈ ℕ₀ ∧ 𝐵 ∈ ℕ) → (1 σ 𝐵) ∈ ℕ)
25	22, 23, 24	sylancr 587	. . . . . . 7 ⊢ (𝜑 → (1 σ 𝐵) ∈ ℕ)
26	25	nnzd 12435	. . . . . 6 ⊢ (𝜑 → (1 σ 𝐵) ∈ ℤ)
27		dvdsmul1 15997	. . . . . 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 584	. . . . 5 ⊢ (𝜑 → ((2↑(𝐴 + 1)) − 1) ∥ (((2↑(𝐴 + 1)) − 1) · (1 σ 𝐵)))
29		2cn 12058	. . . . . . . . 9 ⊢ 2 ∈ ℂ
30		expp1 13799	. . . . . . . . 9 ⊢ ((2 ∈ ℂ ∧ 𝐴 ∈ ℕ₀) → (2↑(𝐴 + 1)) = ((2↑𝐴) · 2))
31	29, 3, 30	sylancr 587	. . . . . . . 8 ⊢ (𝜑 → (2↑(𝐴 + 1)) = ((2↑𝐴) · 2))
32		nnexpcl 13805	. . . . . . . . . . 11 ⊢ ((2 ∈ ℕ ∧ 𝐴 ∈ ℕ₀) → (2↑𝐴) ∈ ℕ)
33	1, 3, 32	sylancr 587	. . . . . . . . . 10 ⊢ (𝜑 → (2↑𝐴) ∈ ℕ)
34	33	nncnd 11999	. . . . . . . . 9 ⊢ (𝜑 → (2↑𝐴) ∈ ℂ)
35		mulcom 10967	. . . . . . . . 9 ⊢ (((2↑𝐴) ∈ ℂ ∧ 2 ∈ ℂ) → ((2↑𝐴) · 2) = (2 · (2↑𝐴)))
36	34, 29, 35	sylancl 586	. . . . . . . 8 ⊢ (𝜑 → ((2↑𝐴) · 2) = (2 · (2↑𝐴)))
37	31, 36	eqtrd 2778	. . . . . . 7 ⊢ (𝜑 → (2↑(𝐴 + 1)) = (2 · (2↑𝐴)))
38	37	oveq1d 7282	. . . . . 6 ⊢ (𝜑 → ((2↑(𝐴 + 1)) · 𝐵) = ((2 · (2↑𝐴)) · 𝐵))
39	29	a1i 11	. . . . . . 7 ⊢ (𝜑 → 2 ∈ ℂ)
40	23	nncnd 11999	. . . . . . 7 ⊢ (𝜑 → 𝐵 ∈ ℂ)
41	39, 34, 40	mulassd 11008	. . . . . 6 ⊢ (𝜑 → ((2 · (2↑𝐴)) · 𝐵) = (2 · ((2↑𝐴) · 𝐵)))
42		1cnd 10980	. . . . . . . 8 ⊢ (𝜑 → 1 ∈ ℂ)
43		perfectALTVlem.3	. . . . . . . . . 10 ⊢ (𝜑 → 𝐵 ∈ Odd )
44		isodd7 45095	. . . . . . . . . . 11 ⊢ (𝐵 ∈ Odd ↔ (𝐵 ∈ ℤ ∧ (2 gcd 𝐵) = 1))
45	44	simprbi 497	. . . . . . . . . 10 ⊢ (𝐵 ∈ Odd → (2 gcd 𝐵) = 1)
46	43, 45	syl 17	. . . . . . . . 9 ⊢ (𝜑 → (2 gcd 𝐵) = 1)
47		2z 12362	. . . . . . . . . . 11 ⊢ 2 ∈ ℤ
48	47	a1i 11	. . . . . . . . . 10 ⊢ (𝜑 → 2 ∈ ℤ)
49	23	nnzd 12435	. . . . . . . . . 10 ⊢ (𝜑 → 𝐵 ∈ ℤ)
50		rpexp1i 16438	. . . . . . . . . 10 ⊢ ((2 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐴 ∈ ℕ₀) → ((2 gcd 𝐵) = 1 → ((2↑𝐴) gcd 𝐵) = 1))
51	48, 49, 3, 50	syl3anc 1370	. . . . . . . . 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 26359	. . . . . . . 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 1371	. . . . . . 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 11999	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐴 ∈ ℂ)
57		pncan1 11409	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ ℂ → ((𝐴 + 1) − 1) = 𝐴)
58	56, 57	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → ((𝐴 + 1) − 1) = 𝐴)
59	58	oveq2d 7283	. . . . . . . . . 10 ⊢ (𝜑 → (2↑((𝐴 + 1) − 1)) = (2↑𝐴))
60	59	oveq2d 7283	. . . . . . . . 9 ⊢ (𝜑 → (1 σ (2↑((𝐴 + 1) − 1))) = (1 σ (2↑𝐴)))
61		1sgm2ppw 26358	. . . . . . . . . 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 7282	. . . . . . 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 5101	. . . 4 ⊢ (𝜑 → ((2↑(𝐴 + 1)) − 1) ∥ ((2↑(𝐴 + 1)) · 𝐵))
68	21, 19	gcdcomd 16231	. . . . 5 ⊢ (𝜑 → (((2↑(𝐴 + 1)) − 1) gcd (2↑(𝐴 + 1))) = ((2↑(𝐴 + 1)) gcd ((2↑(𝐴 + 1)) − 1)))
69		nnpw2evenALTV 45132	. . . . . . . . 9 ⊢ ((𝐴 + 1) ∈ ℕ → (2↑(𝐴 + 1)) ∈ Even )
70	10, 69	syl 17	. . . . . . . 8 ⊢ (𝜑 → (2↑(𝐴 + 1)) ∈ Even )
71		evenm1odd 45069	. . . . . . . 8 ⊢ ((2↑(𝐴 + 1)) ∈ Even → ((2↑(𝐴 + 1)) − 1) ∈ Odd )
72	70, 71	syl 17	. . . . . . 7 ⊢ (𝜑 → ((2↑(𝐴 + 1)) − 1) ∈ Odd )
73		isodd7 45095	. . . . . . . 8 ⊢ (((2↑(𝐴 + 1)) − 1) ∈ Odd ↔ (((2↑(𝐴 + 1)) − 1) ∈ ℤ ∧ (2 gcd ((2↑(𝐴 + 1)) − 1)) = 1))
74	73	simprbi 497	. . . . . . 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 16438	. . . . . . 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 1370	. . . . . 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 16368	. . . . 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 1370	. . . 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 696	. . 3 ⊢ (𝜑 → ((2↑(𝐴 + 1)) − 1) ∥ 𝐵)
83		nndivdvds 15982	. . . 4 ⊢ ((𝐵 ∈ ℕ ∧ ((2↑(𝐴 + 1)) − 1) ∈ ℕ) → (((2↑(𝐴 + 1)) − 1) ∥ 𝐵 ↔ (𝐵 / ((2↑(𝐴 + 1)) − 1)) ∈ ℕ))
84	23, 18, 83	syl2anc 584	. . 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 1127	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 396 ∧ w3a 1086 = wceq 1539 ∈ wcel 2106 class class class wbr 5073 (class class class)co 7267 ℂcc 10879 ℝcr 10880 1c1 10882 + caddc 10884 · cmul 10886 < clt 11019 − cmin 11215 / cdiv 11642 ℕcn 11983 2c2 12038 ℕ₀cn0 12243 ℤcz 12329 ↑cexp 13792 ∥ cdvds 15973 gcd cgcd 16211 σ csgm 26255 Even ceven 45054 Odd codd 45055

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-rep 5208 ax-sep 5221 ax-nul 5228 ax-pow 5286 ax-pr 5350 ax-un 7578 ax-inf2 9386 ax-cnex 10937 ax-resscn 10938 ax-1cn 10939 ax-icn 10940 ax-addcl 10941 ax-addrcl 10942 ax-mulcl 10943 ax-mulrcl 10944 ax-mulcom 10945 ax-addass 10946 ax-mulass 10947 ax-distr 10948 ax-i2m1 10949 ax-1ne0 10950 ax-1rid 10951 ax-rnegex 10952 ax-rrecex 10953 ax-cnre 10954 ax-pre-lttri 10955 ax-pre-lttrn 10956 ax-pre-ltadd 10957 ax-pre-mulgt0 10958 ax-pre-sup 10959 ax-addf 10960 ax-mulf 10961

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-reu 3071 df-rmo 3072 df-rab 3073 df-v 3431 df-sbc 3716 df-csb 3832 df-dif 3889 df-un 3891 df-in 3893 df-ss 3903 df-pss 3905 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-tp 4566 df-op 4568 df-uni 4840 df-int 4880 df-iun 4926 df-iin 4927 df-br 5074 df-opab 5136 df-mpt 5157 df-tr 5191 df-id 5484 df-eprel 5490 df-po 5498 df-so 5499 df-fr 5539 df-se 5540 df-we 5541 df-xp 5590 df-rel 5591 df-cnv 5592 df-co 5593 df-dm 5594 df-rn 5595 df-res 5596 df-ima 5597 df-pred 6195 df-ord 6262 df-on 6263 df-lim 6264 df-suc 6265 df-iota 6384 df-fun 6428 df-fn 6429 df-f 6430 df-f1 6431 df-fo 6432 df-f1o 6433 df-fv 6434 df-isom 6435 df-riota 7224 df-ov 7270 df-oprab 7271 df-mpo 7272 df-of 7523 df-om 7703 df-1st 7820 df-2nd 7821 df-supp 7965 df-frecs 8084 df-wrecs 8115 df-recs 8189 df-rdg 8228 df-1o 8284 df-2o 8285 df-er 8485 df-map 8604 df-pm 8605 df-ixp 8673 df-en 8721 df-dom 8722 df-sdom 8723 df-fin 8724 df-fsupp 9116 df-fi 9157 df-sup 9188 df-inf 9189 df-oi 9256 df-card 9707 df-pnf 11021 df-mnf 11022 df-xr 11023 df-ltxr 11024 df-le 11025 df-sub 11217 df-neg 11218 df-div 11643 df-nn 11984 df-2 12046 df-3 12047 df-4 12048 df-5 12049 df-6 12050 df-7 12051 df-8 12052 df-9 12053 df-n0 12244 df-z 12330 df-dec 12448 df-uz 12593 df-q 12699 df-rp 12741 df-xneg 12858 df-xadd 12859 df-xmul 12860 df-ioo 13093 df-ioc 13094 df-ico 13095 df-icc 13096 df-fz 13250 df-fzo 13393 df-fl 13522 df-mod 13600 df-seq 13732 df-exp 13793 df-fac 13998 df-bc 14027 df-hash 14055 df-shft 14788 df-cj 14820 df-re 14821 df-im 14822 df-sqrt 14956 df-abs 14957 df-limsup 15190 df-clim 15207 df-rlim 15208 df-sum 15408 df-ef 15787 df-sin 15789 df-cos 15790 df-pi 15792 df-dvds 15974 df-gcd 16212 df-prm 16387 df-pc 16548 df-struct 16858 df-sets 16875 df-slot 16893 df-ndx 16905 df-base 16923 df-ress 16952 df-plusg 16985 df-mulr 16986 df-starv 16987 df-sca 16988 df-vsca 16989 df-ip 16990 df-tset 16991 df-ple 16992 df-ds 16994 df-unif 16995 df-hom 16996 df-cco 16997 df-rest 17143 df-topn 17144 df-0g 17162 df-gsum 17163 df-topgen 17164 df-pt 17165 df-prds 17168 df-xrs 17223 df-qtop 17228 df-imas 17229 df-xps 17231 df-mre 17305 df-mrc 17306 df-acs 17308 df-mgm 18336 df-sgrp 18385 df-mnd 18396 df-submnd 18441 df-mulg 18711 df-cntz 18933 df-cmn 19398 df-psmet 20599 df-xmet 20600 df-met 20601 df-bl 20602 df-mopn 20603 df-fbas 20604 df-fg 20605 df-cnfld 20608 df-top 22053 df-topon 22070 df-topsp 22092 df-bases 22106 df-cld 22180 df-ntr 22181 df-cls 22182 df-nei 22259 df-lp 22297 df-perf 22298 df-cn 22388 df-cnp 22389 df-haus 22476 df-tx 22723 df-hmeo 22916 df-fil 23007 df-fm 23099 df-flim 23100 df-flf 23101 df-xms 23483 df-ms 23484 df-tms 23485 df-cncf 24051 df-limc 25040 df-dv 25041 df-log 25722 df-cxp 25723 df-sgm 26261 df-even 45056 df-odd 45057

This theorem is referenced by: perfectALTVlem2 45152

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