Theorem perfect1 15795

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 15794	. . . . 5 |- ( ( P e. ZZ /\ ( ( 2 ^ P ) - 1 ) e. Prime ) -> P e. Prime )
2		prmnn 12745	. . . . 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 15792	. . . 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 15791	. . . . 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 9347	. . . . . . 7 |- 2 e. NN
9	3	nnnn0d 9499	. . . . . . 7 |- ( ( P e. ZZ /\ ( ( 2 ^ P ) - 1 ) e. Prime ) -> P e. NN0 )
10		nnexpcl 10860	. . . . . . 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 9199	. . . . 5 |- ( ( P e. ZZ /\ ( ( 2 ^ P ) - 1 ) e. Prime ) -> ( 2 ^ P ) e. CC )
13		ax-1cn 8168	. . . . 5 |- 1 e. CC
14		npcan 8430	. . . . 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 2264	. . 3 |- ( ( P e. ZZ /\ ( ( 2 ^ P ) - 1 ) e. Prime ) -> ( 1 sigma ( ( 2 ^ P ) - 1 ) ) = ( 2 ^ P ) )
17	5, 16	oveq12d 6046	. 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 9485	. . . . 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 10860	. . . 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 12745	. . . 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 9645	. . . . 5 |- ( ( P e. ZZ /\ ( ( 2 ^ P ) - 1 ) e. Prime ) -> ( 2 ^ ( P - 1 ) ) e. ZZ )
26		prmz 12746	. . . . . 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 12608	. . . 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 12428	. . . . . . . 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 12766	. . . . . . . . . 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 9880	. . . . . . . . 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 12556	. . . . . . . 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 9551	. . . . . . . . 9 |- 2 e. ZZ
39	38	a1i 9	. . . . . . . 8 |- ( ( P e. ZZ /\ ( ( 2 ^ P ) - 1 ) e. Prime ) -> 2 e. ZZ )
40		dvdsmultr1 12455	. . . . . . . 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 9256	. . . . . . . . . 10 |- 2 e. CC
43		expm1t 10875	. . . . . . . . . 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 2265	. . . . . . . 8 |- ( ( P e. ZZ /\ ( ( 2 ^ P ) - 1 ) e. Prime ) -> ( ( 2 ^ ( P - 1 ) ) x. 2 ) = ( ( ( 2 ^ P ) - 1 ) + 1 ) )
46	45	breq2d 4105	. . . . . . 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 12779	. . . . . 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 2264	. . 3 |- ( ( P e. ZZ /\ ( ( 2 ^ P ) - 1 ) e. Prime ) -> ( ( 2 ^ ( P - 1 ) ) gcd ( ( 2 ^ P ) - 1 ) ) = 1 )
54		sgmmul 15793	. . 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 8420	. . . 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 8243	. 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 2274	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 2202   class class class wbr 4093   ` cfv 5333  (class class class)co 6028   CCcc 8073   1c1 8076   + caddc 8078   x. cmul 8080   < clt 8256   - cmin 8392   NNcn 9185   2c2 9236   NN0cn0 9444   ZZcz 9523   ZZ>=cuz 9799   ^cexp 10846   || cdvds 12411   gcd cgcd 12587   Primecprime 12742   sigma csgm 15778

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 2204  ax-14 2205  ax-ext 2213  ax-coll 4209  ax-sep 4212  ax-nul 4220  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-setind 4641  ax-iinf 4692  ax-cnex 8166  ax-resscn 8167  ax-1cn 8168  ax-1re 8169  ax-icn 8170  ax-addcl 8171  ax-addrcl 8172  ax-mulcl 8173  ax-mulrcl 8174  ax-addcom 8175  ax-mulcom 8176  ax-addass 8177  ax-mulass 8178  ax-distr 8179  ax-i2m1 8180  ax-0lt1 8181  ax-1rid 8182  ax-0id 8183  ax-rnegex 8184  ax-precex 8185  ax-cnre 8186  ax-pre-ltirr 8187  ax-pre-ltwlin 8188  ax-pre-lttrn 8189  ax-pre-apti 8190  ax-pre-ltadd 8191  ax-pre-mulgt0 8192  ax-pre-mulext 8193  ax-arch 8194  ax-caucvg 8195  ax-pre-suploc 8196  ax-addf 8197  ax-mulf 8198

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 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-nel 2499  df-ral 2516  df-rex 2517  df-reu 2518  df-rmo 2519  df-rab 2520  df-v 2805  df-sbc 3033  df-csb 3129  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-nul 3497  df-if 3608  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-int 3934  df-iun 3977  df-disj 4070  df-br 4094  df-opab 4156  df-mpt 4157  df-tr 4193  df-id 4396  df-po 4399  df-iso 4400  df-iord 4469  df-on 4471  df-ilim 4472  df-suc 4474  df-iom 4695  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744  df-iota 5293  df-fun 5335  df-fn 5336  df-f 5337  df-f1 5338  df-fo 5339  df-f1o 5340  df-fv 5341  df-isom 5342  df-riota 5981  df-ov 6031  df-oprab 6032  df-mpo 6033  df-of 6244  df-1st 6312  df-2nd 6313  df-recs 6514  df-irdg 6579  df-frec 6600  df-1o 6625  df-2o 6626  df-oadd 6629  df-er 6745  df-map 6862  df-pm 6863  df-en 6953  df-dom 6954  df-fin 6955  df-sup 7226  df-inf 7227  df-pnf 8258  df-mnf 8259  df-xr 8260  df-ltxr 8261  df-le 8262  df-sub 8394  df-neg 8395  df-reap 8797  df-ap 8804  df-div 8895  df-inn 9186  df-2 9244  df-3 9245  df-4 9246  df-n0 9445  df-xnn0 9510  df-z 9524  df-uz 9800  df-q 9898  df-rp 9933  df-xneg 10051  df-xadd 10052  df-ioo 10171  df-ico 10173  df-icc 10174  df-fz 10289  df-fzo 10423  df-fl 10576  df-mod 10631  df-seqfrec 10756  df-exp 10847  df-fac 11034  df-bc 11056  df-ihash 11084  df-shft 11438  df-cj 11465  df-re 11466  df-im 11467  df-rsqrt 11621  df-abs 11622  df-clim 11902  df-sumdc 11977  df-ef 12272  df-e 12273  df-dvds 12412  df-gcd 12588  df-prm 12743  df-pc 12921  df-rest 13387  df-topgen 13406  df-psmet 14622  df-xmet 14623  df-met 14624  df-bl 14625  df-mopn 14626  df-top 14792  df-topon 14805  df-bases 14837  df-ntr 14890  df-cn 14982  df-cnp 14983  df-tx 15047  df-cncf 15365  df-limced 15450  df-dvap 15451  df-relog 15652  df-rpcxp 15653  df-sgm 15779

This theorem is referenced by:  perfect  15798