Theorem perfect1 15234

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

Assertion

Ref	Expression
perfect1	|- ( ( P e. ZZ /\ ( ( 2 ^ P ) - 1 ) e. Prime ) -> ( 1 sigma ( ( 2 ^ ( P - 1 ) ) x. ( ( 2 ^ P ) - 1 ) ) ) = ( ( 2 ^ P ) x. ( ( 2 ^ P ) - 1 ) ) )

Proof of Theorem perfect1

Step	Hyp	Ref	Expression
1		mersenne 15233	. . . . 5 |- ( ( P e. ZZ /\ ( ( 2 ^ P ) - 1 ) e. Prime ) -> P e. Prime )
2		prmnn 12278	. . . . 5 |- ( P e. Prime -> P e. NN )
3	1, 2	syl 14	. . . 4 |- ( ( P e. ZZ /\ ( ( 2 ^ P ) - 1 ) e. Prime ) -> P e. NN )
4		1sgm2ppw 15231	. . . 4 |- ( P e. NN -> ( 1 sigma ( 2 ^ ( P - 1 ) ) ) = ( ( 2 ^ P ) - 1 ) )
5	3, 4	syl 14	. . 3 |- ( ( P e. ZZ /\ ( ( 2 ^ P ) - 1 ) e. Prime ) -> ( 1 sigma ( 2 ^ ( P - 1 ) ) ) = ( ( 2 ^ P ) - 1 ) )
6		1sgmprm 15230	. . . . 5 |- ( ( ( 2 ^ P ) - 1 ) e. Prime -> ( 1 sigma ( ( 2 ^ P ) - 1 ) ) = ( ( ( 2 ^ P ) - 1 ) + 1 ) )
7	6	adantl 277	. . . 4 |- ( ( P e. ZZ /\ ( ( 2 ^ P ) - 1 ) e. Prime ) -> ( 1 sigma ( ( 2 ^ P ) - 1 ) ) = ( ( ( 2 ^ P ) - 1 ) + 1 ) )
8		2nn 9152	. . . . . . 7 |- 2 e. NN
9	3	nnnn0d 9302	. . . . . . 7 |- ( ( P e. ZZ /\ ( ( 2 ^ P ) - 1 ) e. Prime ) -> P e. NN0 )
10		nnexpcl 10644	. . . . . . 7 |- ( ( 2 e. NN /\ P e. NN0 ) -> ( 2 ^ P ) e. NN )
11	8, 9, 10	sylancr 414	. . . . . 6 |- ( ( P e. ZZ /\ ( ( 2 ^ P ) - 1 ) e. Prime ) -> ( 2 ^ P ) e. NN )
12	11	nncnd 9004	. . . . 5 |- ( ( P e. ZZ /\ ( ( 2 ^ P ) - 1 ) e. Prime ) -> ( 2 ^ P ) e. CC )
13		ax-1cn 7972	. . . . 5 |- 1 e. CC
14		npcan 8235	. . . . 5 |- ( ( ( 2 ^ P ) e. CC /\ 1 e. CC ) -> ( ( ( 2 ^ P ) - 1 ) + 1 ) = ( 2 ^ P ) )
15	12, 13, 14	sylancl 413	. . . 4 |- ( ( P e. ZZ /\ ( ( 2 ^ P ) - 1 ) e. Prime ) -> ( ( ( 2 ^ P ) - 1 ) + 1 ) = ( 2 ^ P ) )
16	7, 15	eqtrd 2229	. . 3 |- ( ( P e. ZZ /\ ( ( 2 ^ P ) - 1 ) e. Prime ) -> ( 1 sigma ( ( 2 ^ P ) - 1 ) ) = ( 2 ^ P ) )
17	5, 16	oveq12d 5940	. 2 |- ( ( P e. ZZ /\ ( ( 2 ^ P ) - 1 ) e. Prime ) -> ( ( 1 sigma ( 2 ^ ( P - 1 ) ) ) x. ( 1 sigma ( ( 2 ^ P ) - 1 ) ) ) = ( ( ( 2 ^ P ) - 1 ) x. ( 2 ^ P ) ) )
18	13	a1i 9	. . 3 |- ( ( P e. ZZ /\ ( ( 2 ^ P ) - 1 ) e. Prime ) -> 1 e. CC )
19		nnm1nn0 9290	. . . . 5 |- ( P e. NN -> ( P - 1 ) e. NN0 )
20	3, 19	syl 14	. . . 4 |- ( ( P e. ZZ /\ ( ( 2 ^ P ) - 1 ) e. Prime ) -> ( P - 1 ) e. NN0 )
21		nnexpcl 10644	. . . 4 |- ( ( 2 e. NN /\ ( P - 1 ) e. NN0 ) -> ( 2 ^ ( P - 1 ) ) e. NN )
22	8, 20, 21	sylancr 414	. . 3 |- ( ( P e. ZZ /\ ( ( 2 ^ P ) - 1 ) e. Prime ) -> ( 2 ^ ( P - 1 ) ) e. NN )
23		prmnn 12278	. . . 4 |- ( ( ( 2 ^ P ) - 1 ) e. Prime -> ( ( 2 ^ P ) - 1 ) e. NN )
24	23	adantl 277	. . 3 |- ( ( P e. ZZ /\ ( ( 2 ^ P ) - 1 ) e. Prime ) -> ( ( 2 ^ P ) - 1 ) e. NN )
25	22	nnzd 9447	. . . . 5 |- ( ( P e. ZZ /\ ( ( 2 ^ P ) - 1 ) e. Prime ) -> ( 2 ^ ( P - 1 ) ) e. ZZ )
26		prmz 12279	. . . . . 6 |- ( ( ( 2 ^ P ) - 1 ) e. Prime -> ( ( 2 ^ P ) - 1 ) e. ZZ )
27	26	adantl 277	. . . . 5 |- ( ( P e. ZZ /\ ( ( 2 ^ P ) - 1 ) e. Prime ) -> ( ( 2 ^ P ) - 1 ) e. ZZ )
28	25, 27	gcdcomd 12141	. . . 4 |- ( ( P e. ZZ /\ ( ( 2 ^ P ) - 1 ) e. Prime ) -> ( ( 2 ^ ( P - 1 ) ) gcd ( ( 2 ^ P ) - 1 ) ) = ( ( ( 2 ^ P ) - 1 ) gcd ( 2 ^ ( P - 1 ) ) ) )
29		iddvds 11969	. . . . . . . 8 |- ( ( ( 2 ^ P ) - 1 ) e. ZZ -> ( ( 2 ^ P ) - 1 ) || ( ( 2 ^ P ) - 1 ) )
30	27, 29	syl 14	. . . . . . 7 |- ( ( P e. ZZ /\ ( ( 2 ^ P ) - 1 ) e. Prime ) -> ( ( 2 ^ P ) - 1 ) || ( ( 2 ^ P ) - 1 ) )
31		prmuz2 12299	. . . . . . . . . 10 |- ( ( ( 2 ^ P ) - 1 ) e. Prime -> ( ( 2 ^ P ) - 1 ) e. ( ZZ>= ` 2 ) )
32	31	adantl 277	. . . . . . . . 9 |- ( ( P e. ZZ /\ ( ( 2 ^ P ) - 1 ) e. Prime ) -> ( ( 2 ^ P ) - 1 ) e. ( ZZ>= ` 2 ) )
33		eluz2gt1 9676	. . . . . . . . 9 |- ( ( ( 2 ^ P ) - 1 ) e. ( ZZ>= ` 2 ) -> 1 < ( ( 2 ^ P ) - 1 ) )
34	32, 33	syl 14	. . . . . . . 8 |- ( ( P e. ZZ /\ ( ( 2 ^ P ) - 1 ) e. Prime ) -> 1 < ( ( 2 ^ P ) - 1 ) )
35		ndvdsp1 12097	. . . . . . . 8 |- ( ( ( ( 2 ^ P ) - 1 ) e. ZZ /\ ( ( 2 ^ P ) - 1 ) e. NN /\ 1 < ( ( 2 ^ P ) - 1 ) ) -> ( ( ( 2 ^ P ) - 1 ) || ( ( 2 ^ P ) - 1 ) -> -. ( ( 2 ^ P ) - 1 ) || ( ( ( 2 ^ P ) - 1 ) + 1 ) ) )
36	27, 24, 34, 35	syl3anc 1249	. . . . . . 7 |- ( ( P e. ZZ /\ ( ( 2 ^ P ) - 1 ) e. Prime ) -> ( ( ( 2 ^ P ) - 1 ) || ( ( 2 ^ P ) - 1 ) -> -. ( ( 2 ^ P ) - 1 ) || ( ( ( 2 ^ P ) - 1 ) + 1 ) ) )
37	30, 36	mpd 13	. . . . . 6 |- ( ( P e. ZZ /\ ( ( 2 ^ P ) - 1 ) e. Prime ) -> -. ( ( 2 ^ P ) - 1 ) || ( ( ( 2 ^ P ) - 1 ) + 1 ) )
38		2z 9354	. . . . . . . . 9 |- 2 e. ZZ
39	38	a1i 9	. . . . . . . 8 |- ( ( P e. ZZ /\ ( ( 2 ^ P ) - 1 ) e. Prime ) -> 2 e. ZZ )
40		dvdsmultr1 11996	. . . . . . . 8 |- ( ( ( ( 2 ^ P ) - 1 ) e. ZZ /\ ( 2 ^ ( P - 1 ) ) e. ZZ /\ 2 e. ZZ ) -> ( ( ( 2 ^ P ) - 1 ) || ( 2 ^ ( P - 1 ) ) -> ( ( 2 ^ P ) - 1 ) || ( ( 2 ^ ( P - 1 ) ) x. 2 ) ) )
41	27, 25, 39, 40	syl3anc 1249	. . . . . . 7 |- ( ( P e. ZZ /\ ( ( 2 ^ P ) - 1 ) e. Prime ) -> ( ( ( 2 ^ P ) - 1 ) || ( 2 ^ ( P - 1 ) ) -> ( ( 2 ^ P ) - 1 ) || ( ( 2 ^ ( P - 1 ) ) x. 2 ) ) )
42		2cn 9061	. . . . . . . . . 10 |- 2 e. CC
43		expm1t 10659	. . . . . . . . . 10 |- ( ( 2 e. CC /\ P e. NN ) -> ( 2 ^ P ) = ( ( 2 ^ ( P - 1 ) ) x. 2 ) )
44	42, 3, 43	sylancr 414	. . . . . . . . 9 |- ( ( P e. ZZ /\ ( ( 2 ^ P ) - 1 ) e. Prime ) -> ( 2 ^ P ) = ( ( 2 ^ ( P - 1 ) ) x. 2 ) )
45	15, 44	eqtr2d 2230	. . . . . . . 8 |- ( ( P e. ZZ /\ ( ( 2 ^ P ) - 1 ) e. Prime ) -> ( ( 2 ^ ( P - 1 ) ) x. 2 ) = ( ( ( 2 ^ P ) - 1 ) + 1 ) )
46	45	breq2d 4045	. . . . . . 7 |- ( ( P e. ZZ /\ ( ( 2 ^ P ) - 1 ) e. Prime ) -> ( ( ( 2 ^ P ) - 1 ) || ( ( 2 ^ ( P - 1 ) ) x. 2 ) <-> ( ( 2 ^ P ) - 1 ) || ( ( ( 2 ^ P ) - 1 ) + 1 ) ) )
47	41, 46	sylibd 149	. . . . . 6 |- ( ( P e. ZZ /\ ( ( 2 ^ P ) - 1 ) e. Prime ) -> ( ( ( 2 ^ P ) - 1 ) || ( 2 ^ ( P - 1 ) ) -> ( ( 2 ^ P ) - 1 ) || ( ( ( 2 ^ P ) - 1 ) + 1 ) ) )
48	37, 47	mtod 664	. . . . 5 |- ( ( P e. ZZ /\ ( ( 2 ^ P ) - 1 ) e. Prime ) -> -. ( ( 2 ^ P ) - 1 ) || ( 2 ^ ( P - 1 ) ) )
49		simpr 110	. . . . . 6 |- ( ( P e. ZZ /\ ( ( 2 ^ P ) - 1 ) e. Prime ) -> ( ( 2 ^ P ) - 1 ) e. Prime )
50		coprm 12312	. . . . . 6 |- ( ( ( ( 2 ^ P ) - 1 ) e. Prime /\ ( 2 ^ ( P - 1 ) ) e. ZZ ) -> ( -. ( ( 2 ^ P ) - 1 ) || ( 2 ^ ( P - 1 ) ) <-> ( ( ( 2 ^ P ) - 1 ) gcd ( 2 ^ ( P - 1 ) ) ) = 1 ) )
51	49, 25, 50	syl2anc 411	. . . . 5 |- ( ( P e. ZZ /\ ( ( 2 ^ P ) - 1 ) e. Prime ) -> ( -. ( ( 2 ^ P ) - 1 ) || ( 2 ^ ( P - 1 ) ) <-> ( ( ( 2 ^ P ) - 1 ) gcd ( 2 ^ ( P - 1 ) ) ) = 1 ) )
52	48, 51	mpbid 147	. . . 4 |- ( ( P e. ZZ /\ ( ( 2 ^ P ) - 1 ) e. Prime ) -> ( ( ( 2 ^ P ) - 1 ) gcd ( 2 ^ ( P - 1 ) ) ) = 1 )
53	28, 52	eqtrd 2229	. . 3 |- ( ( P e. ZZ /\ ( ( 2 ^ P ) - 1 ) e. Prime ) -> ( ( 2 ^ ( P - 1 ) ) gcd ( ( 2 ^ P ) - 1 ) ) = 1 )
54		sgmmul 15232	. . 3 |- ( ( 1 e. CC /\ ( ( 2 ^ ( P - 1 ) ) e. NN /\ ( ( 2 ^ P ) - 1 ) e. NN /\ ( ( 2 ^ ( P - 1 ) ) gcd ( ( 2 ^ P ) - 1 ) ) = 1 ) ) -> ( 1 sigma ( ( 2 ^ ( P - 1 ) ) x. ( ( 2 ^ P ) - 1 ) ) ) = ( ( 1 sigma ( 2 ^ ( P - 1 ) ) ) x. ( 1 sigma ( ( 2 ^ P ) - 1 ) ) ) )
55	18, 22, 24, 53, 54	syl13anc 1251	. 2 |- ( ( P e. ZZ /\ ( ( 2 ^ P ) - 1 ) e. Prime ) -> ( 1 sigma ( ( 2 ^ ( P - 1 ) ) x. ( ( 2 ^ P ) - 1 ) ) ) = ( ( 1 sigma ( 2 ^ ( P - 1 ) ) ) x. ( 1 sigma ( ( 2 ^ P ) - 1 ) ) ) )
56		subcl 8225	. . . 4 |- ( ( ( 2 ^ P ) e. CC /\ 1 e. CC ) -> ( ( 2 ^ P ) - 1 ) e. CC )
57	12, 13, 56	sylancl 413	. . 3 |- ( ( P e. ZZ /\ ( ( 2 ^ P ) - 1 ) e. Prime ) -> ( ( 2 ^ P ) - 1 ) e. CC )
58	12, 57	mulcomd 8048	. 2 |- ( ( P e. ZZ /\ ( ( 2 ^ P ) - 1 ) e. Prime ) -> ( ( 2 ^ P ) x. ( ( 2 ^ P ) - 1 ) ) = ( ( ( 2 ^ P ) - 1 ) x. ( 2 ^ P ) ) )
59	17, 55, 58	3eqtr4d 2239	1 |- ( ( P e. ZZ /\ ( ( 2 ^ P ) - 1 ) e. Prime ) -> ( 1 sigma ( ( 2 ^ ( P - 1 ) ) x. ( ( 2 ^ P ) - 1 ) ) ) = ( ( 2 ^ P ) x. ( ( 2 ^ P ) - 1 ) ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1364   e. wcel 2167   class class class wbr 4033   ` cfv 5258  (class class class)co 5922   CCcc 7877   1c1 7880   + caddc 7882   x. cmul 7884   < clt 8061   - cmin 8197   NNcn 8990   2c2 9041   NN0cn0 9249   ZZcz 9326   ZZ>=cuz 9601   ^cexp 10630   || cdvds 11952   gcd cgcd 12120   Primecprime 12275   sigma csgm 15217

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 615  ax-in2 616  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-coll 4148  ax-sep 4151  ax-nul 4159  ax-pow 4207  ax-pr 4242  ax-un 4468  ax-setind 4573  ax-iinf 4624  ax-cnex 7970  ax-resscn 7971  ax-1cn 7972  ax-1re 7973  ax-icn 7974  ax-addcl 7975  ax-addrcl 7976  ax-mulcl 7977  ax-mulrcl 7978  ax-addcom 7979  ax-mulcom 7980  ax-addass 7981  ax-mulass 7982  ax-distr 7983  ax-i2m1 7984  ax-0lt1 7985  ax-1rid 7986  ax-0id 7987  ax-rnegex 7988  ax-precex 7989  ax-cnre 7990  ax-pre-ltirr 7991  ax-pre-ltwlin 7992  ax-pre-lttrn 7993  ax-pre-apti 7994  ax-pre-ltadd 7995  ax-pre-mulgt0 7996  ax-pre-mulext 7997  ax-arch 7998  ax-caucvg 7999  ax-pre-suploc 8000  ax-addf 8001  ax-mulf 8002

This theorem depends on definitions:  df-bi 117  df-stab 832  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-nel 2463  df-ral 2480  df-rex 2481  df-reu 2482  df-rmo 2483  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3451  df-if 3562  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-int 3875  df-iun 3918  df-disj 4011  df-br 4034  df-opab 4095  df-mpt 4096  df-tr 4132  df-id 4328  df-po 4331  df-iso 4332  df-iord 4401  df-on 4403  df-ilim 4404  df-suc 4406  df-iom 4627  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-rn 4674  df-res 4675  df-ima 4676  df-iota 5219  df-fun 5260  df-fn 5261  df-f 5262  df-f1 5263  df-fo 5264  df-f1o 5265  df-fv 5266  df-isom 5267  df-riota 5877  df-ov 5925  df-oprab 5926  df-mpo 5927  df-of 6135  df-1st 6198  df-2nd 6199  df-recs 6363  df-irdg 6428  df-frec 6449  df-1o 6474  df-2o 6475  df-oadd 6478  df-er 6592  df-map 6709  df-pm 6710  df-en 6800  df-dom 6801  df-fin 6802  df-sup 7050  df-inf 7051  df-pnf 8063  df-mnf 8064  df-xr 8065  df-ltxr 8066  df-le 8067  df-sub 8199  df-neg 8200  df-reap 8602  df-ap 8609  df-div 8700  df-inn 8991  df-2 9049  df-3 9050  df-4 9051  df-n0 9250  df-xnn0 9313  df-z 9327  df-uz 9602  df-q 9694  df-rp 9729  df-xneg 9847  df-xadd 9848  df-ioo 9967  df-ico 9969  df-icc 9970  df-fz 10084  df-fzo 10218  df-fl 10360  df-mod 10415  df-seqfrec 10540  df-exp 10631  df-fac 10818  df-bc 10840  df-ihash 10868  df-shft 10980  df-cj 11007  df-re 11008  df-im 11009  df-rsqrt 11163  df-abs 11164  df-clim 11444  df-sumdc 11519  df-ef 11813  df-e 11814  df-dvds 11953  df-gcd 12121  df-prm 12276  df-pc 12454  df-rest 12912  df-topgen 12931  df-psmet 14099  df-xmet 14100  df-met 14101  df-bl 14102  df-mopn 14103  df-top 14234  df-topon 14247  df-bases 14279  df-ntr 14332  df-cn 14424  df-cnp 14425  df-tx 14489  df-cncf 14807  df-limced 14892  df-dvap 14893  df-relog 15094  df-rpcxp 15095  df-sgm 15218

This theorem is referenced by:  perfect  15237