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Theorem perfect1 27564

Description: Euclid's contribution to the Euclid-Euler theorem. A number of the form 2↑(𝑝 − 1) · (2↑𝑝 − 1) is a perfect number. (Contributed by Mario Carneiro, 17-May-2016.)

**Assertion**
Ref	Expression
perfect1	⊢ ((𝑃 ∈ ℤ ∧ ((2↑𝑃) − 1) ∈ ℙ) → (1 σ ((2↑(𝑃 − 1)) · ((2↑𝑃) − 1))) = ((2↑𝑃) · ((2↑𝑃) − 1)))

**Proof of Theorem perfect1**
Step	Hyp	Ref	Expression
1		mersenne 27563	. . . . 5 ⊢ ((𝑃 ∈ ℤ ∧ ((2↑𝑃) − 1) ∈ ℙ) → 𝑃 ∈ ℙ)
2		prmnn 16858	. . . . 5 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ ℕ)
3	1, 2	syl 18	. . . 4 ⊢ ((𝑃 ∈ ℤ ∧ ((2↑𝑃) − 1) ∈ ℙ) → 𝑃 ∈ ℕ)
4		1sgm2ppw 27536	. . . 4 ⊢ (𝑃 ∈ ℕ → (1 σ (2↑(𝑃 − 1))) = ((2↑𝑃) − 1))
5	3, 4	syl 18	. . 3 ⊢ ((𝑃 ∈ ℤ ∧ ((2↑𝑃) − 1) ∈ ℙ) → (1 σ (2↑(𝑃 − 1))) = ((2↑𝑃) − 1))
6		1sgmprm 27535	. . . . 5 ⊢ (((2↑𝑃) − 1) ∈ ℙ → (1 σ ((2↑𝑃) − 1)) = (((2↑𝑃) − 1) + 1))
7	6	adantl 487	. . . 4 ⊢ ((𝑃 ∈ ℤ ∧ ((2↑𝑃) − 1) ∈ ℙ) → (1 σ ((2↑𝑃) − 1)) = (((2↑𝑃) − 1) + 1))
8		2nn 12423	. . . . . . 7 ⊢ 2 ∈ ℕ
9	3	nnnn0d 12674	. . . . . . 7 ⊢ ((𝑃 ∈ ℤ ∧ ((2↑𝑃) − 1) ∈ ℙ) → 𝑃 ∈ ℕ₀)
10		nnexpcl 14224	. . . . . . 7 ⊢ ((2 ∈ ℕ ∧ 𝑃 ∈ ℕ₀) → (2↑𝑃) ∈ ℕ)
11	8, 9, 10	sylancr 599	. . . . . 6 ⊢ ((𝑃 ∈ ℤ ∧ ((2↑𝑃) − 1) ∈ ℙ) → (2↑𝑃) ∈ ℕ)
12	11	nncnd 12358	. . . . 5 ⊢ ((𝑃 ∈ ℤ ∧ ((2↑𝑃) − 1) ∈ ℙ) → (2↑𝑃) ∈ ℂ)
13		ax-1cn 11265	. . . . 5 ⊢ 1 ∈ ℂ
14		npcan 11573	. . . . 5 ⊢ (((2↑𝑃) ∈ ℂ ∧ 1 ∈ ℂ) → (((2↑𝑃) − 1) + 1) = (2↑𝑃))
15	12, 13, 14	sylancl 598	. . . 4 ⊢ ((𝑃 ∈ ℤ ∧ ((2↑𝑃) − 1) ∈ ℙ) → (((2↑𝑃) − 1) + 1) = (2↑𝑃))
16	7, 15	eqtrd 2795	. . 3 ⊢ ((𝑃 ∈ ℤ ∧ ((2↑𝑃) − 1) ∈ ℙ) → (1 σ ((2↑𝑃) − 1)) = (2↑𝑃))
17	5, 16	oveq12d 7442	. 2 ⊢ ((𝑃 ∈ ℤ ∧ ((2↑𝑃) − 1) ∈ ℙ) → ((1 σ (2↑(𝑃 − 1))) · (1 σ ((2↑𝑃) − 1))) = (((2↑𝑃) − 1) · (2↑𝑃)))
18	13	a1i 11	. . 3 ⊢ ((𝑃 ∈ ℤ ∧ ((2↑𝑃) − 1) ∈ ℙ) → 1 ∈ ℂ)
19		nnm1nn0 12654	. . . . 5 ⊢ (𝑃 ∈ ℕ → (𝑃 − 1) ∈ ℕ₀)
20	3, 19	syl 18	. . . 4 ⊢ ((𝑃 ∈ ℤ ∧ ((2↑𝑃) − 1) ∈ ℙ) → (𝑃 − 1) ∈ ℕ₀)
21		nnexpcl 14224	. . . 4 ⊢ ((2 ∈ ℕ ∧ (𝑃 − 1) ∈ ℕ₀) → (2↑(𝑃 − 1)) ∈ ℕ)
22	8, 20, 21	sylancr 599	. . 3 ⊢ ((𝑃 ∈ ℤ ∧ ((2↑𝑃) − 1) ∈ ℙ) → (2↑(𝑃 − 1)) ∈ ℕ)
23		prmnn 16858	. . . 4 ⊢ (((2↑𝑃) − 1) ∈ ℙ → ((2↑𝑃) − 1) ∈ ℕ)
24	23	adantl 487	. . 3 ⊢ ((𝑃 ∈ ℤ ∧ ((2↑𝑃) − 1) ∈ ℙ) → ((2↑𝑃) − 1) ∈ ℕ)
25	22	nnzd 12726	. . . . 5 ⊢ ((𝑃 ∈ ℤ ∧ ((2↑𝑃) − 1) ∈ ℙ) → (2↑(𝑃 − 1)) ∈ ℤ)
26		prmz 16859	. . . . . 6 ⊢ (((2↑𝑃) − 1) ∈ ℙ → ((2↑𝑃) − 1) ∈ ℤ)
27	26	adantl 487	. . . . 5 ⊢ ((𝑃 ∈ ℤ ∧ ((2↑𝑃) − 1) ∈ ℙ) → ((2↑𝑃) − 1) ∈ ℤ)
28	25, 27	gcdcomd 16695	. . . 4 ⊢ ((𝑃 ∈ ℤ ∧ ((2↑𝑃) − 1) ∈ ℙ) → ((2↑(𝑃 − 1)) gcd ((2↑𝑃) − 1)) = (((2↑𝑃) − 1) gcd (2↑(𝑃 − 1))))
29		iddvds 16448	. . . . . . . 8 ⊢ (((2↑𝑃) − 1) ∈ ℤ → ((2↑𝑃) − 1) ∥ ((2↑𝑃) − 1))
30	27, 29	syl 18	. . . . . . 7 ⊢ ((𝑃 ∈ ℤ ∧ ((2↑𝑃) − 1) ∈ ℙ) → ((2↑𝑃) − 1) ∥ ((2↑𝑃) − 1))
31		prmuz2 16880	. . . . . . . . . 10 ⊢ (((2↑𝑃) − 1) ∈ ℙ → ((2↑𝑃) − 1) ∈ (ℤ_≥‘2))
32	31	adantl 487	. . . . . . . . 9 ⊢ ((𝑃 ∈ ℤ ∧ ((2↑𝑃) − 1) ∈ ℙ) → ((2↑𝑃) − 1) ∈ (ℤ_≥‘2))
33		eluz2gt1 13054	. . . . . . . . 9 ⊢ (((2↑𝑃) − 1) ∈ (ℤ_≥‘2) → 1 < ((2↑𝑃) − 1))
34	32, 33	syl 18	. . . . . . . 8 ⊢ ((𝑃 ∈ ℤ ∧ ((2↑𝑃) − 1) ∈ ℙ) → 1 < ((2↑𝑃) − 1))
35		ndvdsp1 16590	. . . . . . . 8 ⊢ ((((2↑𝑃) − 1) ∈ ℤ ∧ ((2↑𝑃) − 1) ∈ ℕ ∧ 1 < ((2↑𝑃) − 1)) → (((2↑𝑃) − 1) ∥ ((2↑𝑃) − 1) → ¬ ((2↑𝑃) − 1) ∥ (((2↑𝑃) − 1) + 1)))
36	27, 24, 34, 35	syl3anc 1398	. . . . . . 7 ⊢ ((𝑃 ∈ ℤ ∧ ((2↑𝑃) − 1) ∈ ℙ) → (((2↑𝑃) − 1) ∥ ((2↑𝑃) − 1) → ¬ ((2↑𝑃) − 1) ∥ (((2↑𝑃) − 1) + 1)))
37	30, 36	mpd 16	. . . . . 6 ⊢ ((𝑃 ∈ ℤ ∧ ((2↑𝑃) − 1) ∈ ℙ) → ¬ ((2↑𝑃) − 1) ∥ (((2↑𝑃) − 1) + 1))
38		2z 12735	. . . . . . . . 9 ⊢ 2 ∈ ℤ
39	38	a1i 11	. . . . . . . 8 ⊢ ((𝑃 ∈ ℤ ∧ ((2↑𝑃) − 1) ∈ ℙ) → 2 ∈ ℤ)
40		dvdsmultr1 16475	. . . . . . . 8 ⊢ ((((2↑𝑃) − 1) ∈ ℤ ∧ (2↑(𝑃 − 1)) ∈ ℤ ∧ 2 ∈ ℤ) → (((2↑𝑃) − 1) ∥ (2↑(𝑃 − 1)) → ((2↑𝑃) − 1) ∥ ((2↑(𝑃 − 1)) · 2)))
41	27, 25, 39, 40	syl3anc 1398	. . . . . . 7 ⊢ ((𝑃 ∈ ℤ ∧ ((2↑𝑃) − 1) ∈ ℙ) → (((2↑𝑃) − 1) ∥ (2↑(𝑃 − 1)) → ((2↑𝑃) − 1) ∥ ((2↑(𝑃 − 1)) · 2)))
42		2cn 12425	. . . . . . . . . 10 ⊢ 2 ∈ ℂ
43		expm1t 14240	. . . . . . . . . 10 ⊢ ((2 ∈ ℂ ∧ 𝑃 ∈ ℕ) → (2↑𝑃) = ((2↑(𝑃 − 1)) · 2))
44	42, 3, 43	sylancr 599	. . . . . . . . 9 ⊢ ((𝑃 ∈ ℤ ∧ ((2↑𝑃) − 1) ∈ ℙ) → (2↑𝑃) = ((2↑(𝑃 − 1)) · 2))
45	15, 44	eqtr2d 2796	. . . . . . . 8 ⊢ ((𝑃 ∈ ℤ ∧ ((2↑𝑃) − 1) ∈ ℙ) → ((2↑(𝑃 − 1)) · 2) = (((2↑𝑃) − 1) + 1))
46	45	breq2d 5114	. . . . . . 7 ⊢ ((𝑃 ∈ ℤ ∧ ((2↑𝑃) − 1) ∈ ℙ) → (((2↑𝑃) − 1) ∥ ((2↑(𝑃 − 1)) · 2) ↔ ((2↑𝑃) − 1) ∥ (((2↑𝑃) − 1) + 1)))
47	41, 46	sylibd 242	. . . . . 6 ⊢ ((𝑃 ∈ ℤ ∧ ((2↑𝑃) − 1) ∈ ℙ) → (((2↑𝑃) − 1) ∥ (2↑(𝑃 − 1)) → ((2↑𝑃) − 1) ∥ (((2↑𝑃) − 1) + 1)))
48	37, 47	mtod 201	. . . . 5 ⊢ ((𝑃 ∈ ℤ ∧ ((2↑𝑃) − 1) ∈ ℙ) → ¬ ((2↑𝑃) − 1) ∥ (2↑(𝑃 − 1)))
49		simpr 490	. . . . . 6 ⊢ ((𝑃 ∈ ℤ ∧ ((2↑𝑃) − 1) ∈ ℙ) → ((2↑𝑃) − 1) ∈ ℙ)
50		coprm 16896	. . . . . 6 ⊢ ((((2↑𝑃) − 1) ∈ ℙ ∧ (2↑(𝑃 − 1)) ∈ ℤ) → (¬ ((2↑𝑃) − 1) ∥ (2↑(𝑃 − 1)) ↔ (((2↑𝑃) − 1) gcd (2↑(𝑃 − 1))) = 1))
51	49, 25, 50	syl2anc 596	. . . . 5 ⊢ ((𝑃 ∈ ℤ ∧ ((2↑𝑃) − 1) ∈ ℙ) → (¬ ((2↑𝑃) − 1) ∥ (2↑(𝑃 − 1)) ↔ (((2↑𝑃) − 1) gcd (2↑(𝑃 − 1))) = 1))
52	48, 51	mpbid 235	. . . 4 ⊢ ((𝑃 ∈ ℤ ∧ ((2↑𝑃) − 1) ∈ ℙ) → (((2↑𝑃) − 1) gcd (2↑(𝑃 − 1))) = 1)
53	28, 52	eqtrd 2795	. . 3 ⊢ ((𝑃 ∈ ℤ ∧ ((2↑𝑃) − 1) ∈ ℙ) → ((2↑(𝑃 − 1)) gcd ((2↑𝑃) − 1)) = 1)
54		sgmmul 27537	. . 3 ⊢ ((1 ∈ ℂ ∧ ((2↑(𝑃 − 1)) ∈ ℕ ∧ ((2↑𝑃) − 1) ∈ ℕ ∧ ((2↑(𝑃 − 1)) gcd ((2↑𝑃) − 1)) = 1)) → (1 σ ((2↑(𝑃 − 1)) · ((2↑𝑃) − 1))) = ((1 σ (2↑(𝑃 − 1))) · (1 σ ((2↑𝑃) − 1))))
55	18, 22, 24, 53, 54	syl13anc 1399	. 2 ⊢ ((𝑃 ∈ ℤ ∧ ((2↑𝑃) − 1) ∈ ℙ) → (1 σ ((2↑(𝑃 − 1)) · ((2↑𝑃) − 1))) = ((1 σ (2↑(𝑃 − 1))) · (1 σ ((2↑𝑃) − 1))))
56		subcl 11563	. . . 4 ⊢ (((2↑𝑃) ∈ ℂ ∧ 1 ∈ ℂ) → ((2↑𝑃) − 1) ∈ ℂ)
57	12, 13, 56	sylancl 598	. . 3 ⊢ ((𝑃 ∈ ℤ ∧ ((2↑𝑃) − 1) ∈ ℙ) → ((2↑𝑃) − 1) ∈ ℂ)
58	12, 57	mulcomd 11337	. 2 ⊢ ((𝑃 ∈ ℤ ∧ ((2↑𝑃) − 1) ∈ ℙ) → ((2↑𝑃) · ((2↑𝑃) − 1)) = (((2↑𝑃) − 1) · (2↑𝑃)))
59	17, 55, 58	3eqtr4d 2805	1 ⊢ ((𝑃 ∈ ℤ ∧ ((2↑𝑃) − 1) ∈ ℙ) → (1 σ ((2↑(𝑃 − 1)) · ((2↑𝑃) − 1))) = ((2↑𝑃) · ((2↑𝑃) − 1)))

Colors of variables: wff setvar class

This proof depends on syntax axioms: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 401 = wceq 1569 ∈ wcel 2144 class class class wbr 5102 ‘cfv 6543 (class class class)co 7424 ℂcc 11205 1c1 11208 + caddc 11210 · cmul 11212 < clt 11350 − cmin 11548 ℕcn 12342 2c2 12404 ℕ₀cn0 12613 ℤcz 12700 ℤ_≥cuz 12972 ↑cexp 14211 ∥ cdvds 16431 gcd cgcd 16673 ℙcprime 16855 σ csgm 27432

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1827 ax-4 1841 ax-5 1942 ax-6 1999 ax-7 2040 ax-8 2146 ax-9 2154 ax-10 2177 ax-11 2193 ax-12 2212 ax-ext 2732 ax-rep 5231 ax-sep 5248 ax-nul 5259 ax-pow 5327 ax-pr 5391 ax-un 7755 ax-inf2 9649 ax-cnex 11263 ax-resscn 11264 ax-1cn 11265 ax-icn 11266 ax-addcl 11267 ax-addrcl 11268 ax-mulcl 11269 ax-mulrcl 11270 ax-mulcom 11271 ax-addass 11272 ax-mulass 11273 ax-distr 11274 ax-i2m1 11275 ax-1ne0 11276 ax-1rid 11277 ax-rnegex 11278 ax-rrecex 11279 ax-cnre 11280 ax-pre-lttri 11281 ax-pre-lttrn 11282 ax-pre-ltadd 11283 ax-pre-mulgt0 11284 ax-pre-sup 11285 ax-addf 11286

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1572 df-fal 1582 df-ex 1812 df-nf 1816 df-sb 2099 df-mo 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ne 2956 df-nel 3062 df-ral 3077 df-rex 3087 df-rmo 3365 df-reu 3366 df-rab 3413 df-v 3452 df-sbc 3739 df-csb 3848 df-ss 3897 df-pss 4013 df-dif 4046 df-in 4090 df-un 4150 df-nul 4278 df-if 4482 df-pw 4558 df-sn 4585 df-pr 4587 df-tp 4589 df-op 4835 df-uni 4867 df-int 4907 df-iun 4952 df-iin 4953 df-br 5103 df-opab 5167 df-mpt 5186 df-tr 5212 df-id 5546 df-eprel 5551 df-po 5559 df-so 5560 df-fr 5604 df-se 5605 df-we 5606 df-xp 5657 df-rel 5658 df-cnv 5659 df-co 5660 df-dm 5661 df-rn 5662 df-res 5663 df-ima 5664 df-pred 6307 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6497 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-isom 6552 df-riota 7381 df-ov 7427 df-oprab 7428 df-mpo 7429 df-of 7697 df-om 7882 df-1st 8005 df-2nd 8006 df-supp 8185 df-frecs 8307 df-wrecs 8338 df-recs 8387 df-rdg 8426 df-1o 8484 df-2o 8485 df-er 8725 df-map 8857 df-pm 8858 df-ixp 8934 df-en 8982 df-dom 8983 df-sdom 8984 df-fin 8985 df-fsupp 9362 df-fi 9410 df-sup 9441 df-inf 9442 df-oi 9511 df-card 10027 df-pnf 11352 df-mnf 11353 df-xr 11354 df-ltxr 11355 df-le 11356 df-sub 11550 df-neg 11551 df-div 11981 df-nn 12343 df-2 12412 df-3 12413 df-4 12414 df-5 12415 df-6 12416 df-7 12417 df-8 12418 df-9 12419 df-n0 12614 df-z 12701 df-dec 12822 df-uz 12973 df-q 13083 df-rp 13128 df-xneg 13248 df-xadd 13249 df-xmul 13250 df-ioo 13487 df-ioc 13488 df-ico 13489 df-icc 13490 df-fz 13647 df-fzo 13796 df-fl 13939 df-mod 14017 df-seq 14152 df-exp 14212 df-fac 14425 df-bc 14454 df-hash 14482 df-shft 15229 df-cj 15275 df-re 15276 df-im 15277 df-sqrt 15411 df-abs 15412 df-limsup 15647 df-clim 15664 df-rlim 15665 df-sum 15863 df-ef 16242 df-sin 16244 df-cos 16245 df-pi 16247 df-dvds 16432 df-gcd 16674 df-prm 16856 df-pc 17024 df-struct 17334 df-sets 17351 df-slot 17369 df-ndx 17381 df-base 17397 df-ress 17418 df-plusg 17450 df-mulr 17451 df-starv 17452 df-sca 17453 df-vsca 17454 df-ip 17455 df-tset 17456 df-ple 17457 df-ds 17459 df-unif 17460 df-hom 17461 df-cco 17462 df-rest 17602 df-topn 17603 df-0g 17621 df-gsum 17622 df-topgen 17623 df-pt 17624 df-prds 17627 df-xrs 17683 df-qtop 17688 df-imas 17689 df-xps 17691 df-mre 17765 df-mrc 17766 df-acs 17768 df-mgm 18825 df-sgrp 18917 df-mnd 18933 df-submnd 18988 df-mulg 19287 df-cntz 19540 df-cmn 20005 df-psmet 21679 df-xmet 21680 df-met 21681 df-bl 21682 df-mopn 21683 df-fbas 21684 df-fg 21685 df-cnfld 21688 df-top 23221 df-topon 23238 df-topsp 23260 df-bases 23273 df-cld 23346 df-ntr 23347 df-cls 23348 df-nei 23425 df-lp 23463 df-perf 23464 df-cn 23554 df-cnp 23555 df-haus 23642 df-tx 23890 df-hmeo 24083 df-fil 24174 df-fm 24266 df-flim 24267 df-flf 24268 df-xms 24648 df-ms 24649 df-tms 24650 df-cncf 25208 df-limc 26195 df-dv 26196 df-log 26893 df-cxp 26894 df-sgm 27438

This theorem is used by: perfect 27567 perfectALTV 48957

W3C validator