Theorem perfect1 16264

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 16263	. . . . 5 |- ( ( P e. ZZ /\ ( ( 2 ^ P ) - 1 ) e. Prime ) -> P e. Prime )
2		prmnn 12907	. . . . 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 16255	. . . 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 16254	. . . . 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 9471	. . . . . . 7 |- 2 e. NN
9	3	nnnn0d 9625	. . . . . . 7 |- ( ( P e. ZZ /\ ( ( 2 ^ P ) - 1 ) e. Prime ) -> P e. NN0 )
10		nnexpcl 11004	. . . . . . 7 |- ( ( 2 e. NN /\ P e. NN0 ) -> ( 2 ^ P ) e. NN )
11	8, 9, 10	sylancr 418	. . . . . 6 |- ( ( P e. ZZ /\ ( ( 2 ^ P ) - 1 ) e. Prime ) -> ( 2 ^ P ) e. NN )
12	11	nncnd 9321	. . . . 5 |- ( ( P e. ZZ /\ ( ( 2 ^ P ) - 1 ) e. Prime ) -> ( 2 ^ P ) e. CC )
13		ax-1cn 8273	. . . . 5 |- 1 e. CC
14		npcan 8537	. . . . 5 |- ( ( ( 2 ^ P ) e. CC /\ 1 e. CC ) -> ( ( ( 2 ^ P ) - 1 ) + 1 ) = ( 2 ^ P ) )
15	12, 13, 14	sylancl 417	. . . 4 |- ( ( P e. ZZ /\ ( ( 2 ^ P ) - 1 ) e. Prime ) -> ( ( ( 2 ^ P ) - 1 ) + 1 ) = ( 2 ^ P ) )
16	7, 15	eqtrd 2271	. . 3 |- ( ( P e. ZZ /\ ( ( 2 ^ P ) - 1 ) e. Prime ) -> ( 1 sigma ( ( 2 ^ P ) - 1 ) ) = ( 2 ^ P ) )
17	5, 16	oveq12d 6103	. 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 9609	. . . . 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 11004	. . . 4 |- ( ( 2 e. NN /\ ( P - 1 ) e. NN0 ) -> ( 2 ^ ( P - 1 ) ) e. NN )
22	8, 20, 21	sylancr 418	. . 3 |- ( ( P e. ZZ /\ ( ( 2 ^ P ) - 1 ) e. Prime ) -> ( 2 ^ ( P - 1 ) ) e. NN )
23		prmnn 12907	. . . 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 9772	. . . . 5 |- ( ( P e. ZZ /\ ( ( 2 ^ P ) - 1 ) e. Prime ) -> ( 2 ^ ( P - 1 ) ) e. ZZ )
26		prmz 12908	. . . . . 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 12770	. . . 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 12590	. . . . . . . 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 12929	. . . . . . . . . 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 10012	. . . . . . . . 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 12718	. . . . . . . 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 1278	. . . . . . 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 9677	. . . . . . . . 9 |- 2 e. ZZ
39	38	a1i 9	. . . . . . . 8 |- ( ( P e. ZZ /\ ( ( 2 ^ P ) - 1 ) e. Prime ) -> 2 e. ZZ )
40		dvdsmultr1 12617	. . . . . . . 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 1278	. . . . . . 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 9378	. . . . . . . . . 10 |- 2 e. CC
43		expm1t 11019	. . . . . . . . . 10 |- ( ( 2 e. CC /\ P e. NN ) -> ( 2 ^ P ) = ( ( 2 ^ ( P - 1 ) ) x. 2 ) )
44	42, 3, 43	sylancr 418	. . . . . . . . 9 |- ( ( P e. ZZ /\ ( ( 2 ^ P ) - 1 ) e. Prime ) -> ( 2 ^ P ) = ( ( 2 ^ ( P - 1 ) ) x. 2 ) )
45	15, 44	eqtr2d 2272	. . . . . . . 8 |- ( ( P e. ZZ /\ ( ( 2 ^ P ) - 1 ) e. Prime ) -> ( ( 2 ^ ( P - 1 ) ) x. 2 ) = ( ( ( 2 ^ P ) - 1 ) + 1 ) )
46	45	breq2d 4142	. . . . . . 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 673	. . . . 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 12942	. . . . . 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 415	. . . . 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 2271	. . 3 |- ( ( P e. ZZ /\ ( ( 2 ^ P ) - 1 ) e. Prime ) -> ( ( 2 ^ ( P - 1 ) ) gcd ( ( 2 ^ P ) - 1 ) ) = 1 )
54		sgmmul 16256	. . 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 1280	. 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 8527	. . . 4 |- ( ( ( 2 ^ P ) e. CC /\ 1 e. CC ) -> ( ( 2 ^ P ) - 1 ) e. CC )
57	12, 13, 56	sylancl 417	. . 3 |- ( ( P e. ZZ /\ ( ( 2 ^ P ) - 1 ) e. Prime ) -> ( ( 2 ^ P ) - 1 ) e. CC )
58	12, 57	mulcomd 8348	. 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 2281	1 |- ( ( P e. ZZ /\ ( ( 2 ^ P ) - 1 ) e. Prime ) -> ( 1 sigma ( ( 2 ^ ( P - 1 ) ) x. ( ( 2 ^ P ) - 1 ) ) ) = ( ( 2 ^ P ) x. ( ( 2 ^ P ) - 1 ) ) )

This proof depends on syntax axioms:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1402   e. wcel 2209   class class class wbr 4130   ` cfv 5377  (class class class)co 6085   CCcc 8178   1c1 8181   + caddc 8183   x. cmul 8185   < clt 8361   - cmin 8499   NNcn 9307   2c2 9358   NN0cn0 9568   ZZcz 9649   ZZ>=cuz 9931   ^cexp 10990   || cdvds 12573   gcd cgcd 12749   Primecprime 12904   sigma csgm 16201

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4246  ax-sep 4249  ax-nul 4259  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684  ax-iinf 4735  ax-cnex 8271  ax-resscn 8272  ax-1cn 8273  ax-1re 8274  ax-icn 8275  ax-addcl 8276  ax-addrcl 8277  ax-mulcl 8278  ax-mulrcl 8279  ax-addcom 8280  ax-mulcom 8281  ax-addass 8282  ax-mulass 8283  ax-distr 8284  ax-i2m1 8285  ax-0lt1 8286  ax-1rid 8287  ax-0id 8288  ax-rnegex 8289  ax-precex 8290  ax-cnre 8291  ax-pre-ltirr 8292  ax-pre-ltwlin 8293  ax-pre-lttrn 8294  ax-pre-apti 8295  ax-pre-ltadd 8296  ax-pre-mulgt0 8297  ax-pre-mulext 8298  ax-arch 8299  ax-caucvg 8300  ax-pre-suploc 8301  ax-addf 8302  ax-mulf 8303

This proof depends on definitions:  df-bi 117  df-stab 843  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-if 3639  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-disj 4107  df-br 4131  df-opab 4193  df-mpt 4194  df-tr 4230  df-id 4438  df-po 4441  df-iso 4442  df-iord 4511  df-on 4513  df-ilim 4514  df-suc 4516  df-iom 4738  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-f1 5382  df-fo 5383  df-f1o 5384  df-fv 5385  df-isom 5386  df-riota 6038  df-ov 6088  df-oprab 6089  df-mpo 6090  df-of 6302  df-1st 6374  df-2nd 6375  df-recs 6576  df-irdg 6641  df-frec 6662  df-1o 6687  df-2o 6688  df-oadd 6691  df-er 6807  df-map 6924  df-pm 6925  df-en 7023  df-dom 7024  df-fin 7025  df-sup 7325  df-inf 7326  df-pnf 8363  df-mnf 8364  df-xr 8365  df-ltxr 8366  df-le 8367  df-sub 8501  df-neg 8502  df-reap 8906  df-ap 8913  df-div 9006  df-inn 9308  df-2 9366  df-3 9367  df-4 9368  df-n0 9569  df-xnn0 9636  df-z 9650  df-uz 9932  df-q 10030  df-rp 10066  df-xneg 10185  df-xadd 10186  df-ioo 10305  df-ico 10307  df-icc 10308  df-fz 10423  df-fzo 10561  df-fl 10716  df-mod 10775  df-seqfrec 10900  df-exp 10991  df-fac 11180  df-bc 11202  df-ihash 11231  df-shft 11596  df-cj 11623  df-re 11624  df-im 11625  df-rsqrt 11780  df-abs 11781  df-clim 12064  df-sumdc 12139  df-ef 12434  df-e 12435  df-dvds 12574  df-gcd 12750  df-prm 12905  df-pc 13087  df-rest 13648  df-topgen 13667  df-psmet 14964  df-xmet 14965  df-met 14966  df-bl 14967  df-mopn 14968  df-top 15190  df-topon 15203  df-bases 15235  df-ntr 15288  df-cn 15380  df-cnp 15381  df-tx 15445  df-cncf 15763  df-limced 15848  df-dvap 15849  df-relog 16053  df-rpcxp 16054  df-sgm 16204

This theorem is used by:  perfect  16267