Theorem perfect1 15995

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 15994	. . . . 5 |- ( ( P e. ZZ /\ ( ( 2 ^ P ) - 1 ) e. Prime ) -> P e. Prime )
2		prmnn 12835	. . . . 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 15992	. . . 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 15991	. . . . 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 9419	. . . . . . 7 |- 2 e. NN
9	3	nnnn0d 9573	. . . . . . 7 |- ( ( P e. ZZ /\ ( ( 2 ^ P ) - 1 ) e. Prime ) -> P e. NN0 )
10		nnexpcl 10941	. . . . . . 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 9271	. . . . 5 |- ( ( P e. ZZ /\ ( ( 2 ^ P ) - 1 ) e. Prime ) -> ( 2 ^ P ) e. CC )
13		ax-1cn 8236	. . . . 5 |- 1 e. CC
14		npcan 8499	. . . . 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 2267	. . 3 |- ( ( P e. ZZ /\ ( ( 2 ^ P ) - 1 ) e. Prime ) -> ( 1 sigma ( ( 2 ^ P ) - 1 ) ) = ( 2 ^ P ) )
17	5, 16	oveq12d 6076	. 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 9557	. . . . 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 10941	. . . 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 12835	. . . 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 9720	. . . . 5 |- ( ( P e. ZZ /\ ( ( 2 ^ P ) - 1 ) e. Prime ) -> ( 2 ^ ( P - 1 ) ) e. ZZ )
26		prmz 12836	. . . . . 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 12698	. . . 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 12518	. . . . . . . 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 12856	. . . . . . . . . 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 9955	. . . . . . . . 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 12646	. . . . . . . 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 1274	. . . . . . 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 9625	. . . . . . . . 9 |- 2 e. ZZ
39	38	a1i 9	. . . . . . . 8 |- ( ( P e. ZZ /\ ( ( 2 ^ P ) - 1 ) e. Prime ) -> 2 e. ZZ )
40		dvdsmultr1 12545	. . . . . . . 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 1274	. . . . . . 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 9328	. . . . . . . . . 10 |- 2 e. CC
43		expm1t 10956	. . . . . . . . . 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 2268	. . . . . . . 8 |- ( ( P e. ZZ /\ ( ( 2 ^ P ) - 1 ) e. Prime ) -> ( ( 2 ^ ( P - 1 ) ) x. 2 ) = ( ( ( 2 ^ P ) - 1 ) + 1 ) )
46	45	breq2d 4126	. . . . . . 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 669	. . . . 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 12869	. . . . . 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 2267	. . 3 |- ( ( P e. ZZ /\ ( ( 2 ^ P ) - 1 ) e. Prime ) -> ( ( 2 ^ ( P - 1 ) ) gcd ( ( 2 ^ P ) - 1 ) ) = 1 )
54		sgmmul 15993	. . 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 1276	. 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 8489	. . . 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 8311	. 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 2277	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 1398   e. wcel 2205   class class class wbr 4114   ` cfv 5357  (class class class)co 6058   CCcc 8141   1c1 8144   + caddc 8146   x. cmul 8148   < clt 8324   - cmin 8461   NNcn 9257   2c2 9308   NN0cn0 9516   ZZcz 9597   ZZ>=cuz 9874   ^cexp 10927   || cdvds 12501   gcd cgcd 12677   Primecprime 12832   sigma csgm 15978

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-coll 4230  ax-sep 4233  ax-nul 4241  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-setind 4664  ax-iinf 4715  ax-cnex 8234  ax-resscn 8235  ax-1cn 8236  ax-1re 8237  ax-icn 8238  ax-addcl 8239  ax-addrcl 8240  ax-mulcl 8241  ax-mulrcl 8242  ax-addcom 8243  ax-mulcom 8244  ax-addass 8245  ax-mulass 8246  ax-distr 8247  ax-i2m1 8248  ax-0lt1 8249  ax-1rid 8250  ax-0id 8251  ax-rnegex 8252  ax-precex 8253  ax-cnre 8254  ax-pre-ltirr 8255  ax-pre-ltwlin 8256  ax-pre-lttrn 8257  ax-pre-apti 8258  ax-pre-ltadd 8259  ax-pre-mulgt0 8260  ax-pre-mulext 8261  ax-arch 8262  ax-caucvg 8263  ax-pre-suploc 8264  ax-addf 8265  ax-mulf 8266

This theorem depends on definitions:  df-bi 117  df-stab 839  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-nel 2510  df-ral 2527  df-rex 2528  df-reu 2529  df-rmo 2530  df-rab 2531  df-v 2817  df-sbc 3046  df-csb 3142  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-nul 3513  df-if 3625  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-int 3955  df-iun 3998  df-disj 4091  df-br 4115  df-opab 4177  df-mpt 4178  df-tr 4214  df-id 4419  df-po 4422  df-iso 4423  df-iord 4492  df-on 4494  df-ilim 4495  df-suc 4497  df-iom 4718  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-ima 4767  df-iota 5317  df-fun 5359  df-fn 5360  df-f 5361  df-f1 5362  df-fo 5363  df-f1o 5364  df-fv 5365  df-isom 5366  df-riota 6011  df-ov 6061  df-oprab 6062  df-mpo 6063  df-of 6275  df-1st 6347  df-2nd 6348  df-recs 6549  df-irdg 6614  df-frec 6635  df-1o 6660  df-2o 6661  df-oadd 6664  df-er 6780  df-map 6897  df-pm 6898  df-en 6989  df-dom 6990  df-fin 6991  df-sup 7288  df-inf 7289  df-pnf 8326  df-mnf 8327  df-xr 8328  df-ltxr 8329  df-le 8330  df-sub 8463  df-neg 8464  df-reap 8867  df-ap 8874  df-div 8967  df-inn 9258  df-2 9316  df-3 9317  df-4 9318  df-n0 9517  df-xnn0 9584  df-z 9598  df-uz 9875  df-q 9973  df-rp 10008  df-xneg 10127  df-xadd 10128  df-ioo 10247  df-ico 10249  df-icc 10250  df-fz 10365  df-fzo 10502  df-fl 10657  df-mod 10712  df-seqfrec 10837  df-exp 10928  df-fac 11116  df-bc 11138  df-ihash 11167  df-shft 11528  df-cj 11555  df-re 11556  df-im 11557  df-rsqrt 11711  df-abs 11712  df-clim 11992  df-sumdc 12067  df-ef 12362  df-e 12363  df-dvds 12502  df-gcd 12678  df-prm 12833  df-pc 13011  df-rest 13541  df-topgen 13560  df-psmet 14820  df-xmet 14821  df-met 14822  df-bl 14823  df-mopn 14824  df-top 14992  df-topon 15005  df-bases 15037  df-ntr 15090  df-cn 15182  df-cnp 15183  df-tx 15247  df-cncf 15565  df-limced 15650  df-dvap 15651  df-relog 15852  df-rpcxp 15853  df-sgm 15979

This theorem is referenced by:  perfect  15998