Theorem perfectlem1 15319

Description: Lemma for perfect 15321. (Contributed by Mario Carneiro, 7-Jun-2016.)

Hypotheses

Ref	Expression
perfectlem.1	|- ( ph -> A e. NN )
perfectlem.2	|- ( ph -> B e. NN )
perfectlem.3	|- ( ph -> -. 2 || B )
perfectlem.4	|- ( ph -> ( 1 sigma ( ( 2 ^ A ) x. B ) ) = ( 2 x. ( ( 2 ^ A ) x. B ) ) )

Assertion

Ref	Expression
perfectlem1	|- ( ph -> ( ( 2 ^ ( A + 1 ) ) e. NN /\ ( ( 2 ^ ( A + 1 ) ) - 1 ) e. NN /\ ( B / ( ( 2 ^ ( A + 1 ) ) - 1 ) ) e. NN ) )

Proof of Theorem perfectlem1

Step	Hyp	Ref	Expression
1		2nn 9169	. . 3 |- 2 e. NN
2		perfectlem.1	. . . . 5 |- ( ph -> A e. NN )
3	2	nnnn0d 9319	. . . 4 |- ( ph -> A e. NN0 )
4		peano2nn0 9306	. . . 4 |- ( A e. NN0 -> ( A + 1 ) e. NN0 )
5	3, 4	syl 14	. . 3 |- ( ph -> ( A + 1 ) e. NN0 )
6		nnexpcl 10661	. . 3 |- ( ( 2 e. NN /\ ( A + 1 ) e. NN0 ) -> ( 2 ^ ( A + 1 ) ) e. NN )
7	1, 5, 6	sylancr 414	. 2 |- ( ph -> ( 2 ^ ( A + 1 ) ) e. NN )
8		2re 9077	. . . 4 |- 2 e. RR
9	2	peano2nnd 9022	. . . 4 |- ( ph -> ( A + 1 ) e. NN )
10		1lt2 9177	. . . . 5 |- 1 < 2
11	10	a1i 9	. . . 4 |- ( ph -> 1 < 2 )
12		expgt1 10686	. . . 4 |- ( ( 2 e. RR /\ ( A + 1 ) e. NN /\ 1 < 2 ) -> 1 < ( 2 ^ ( A + 1 ) ) )
13	8, 9, 11, 12	mp3an2i 1353	. . 3 |- ( ph -> 1 < ( 2 ^ ( A + 1 ) ) )
14		1nn 9018	. . . 4 |- 1 e. NN
15		nnsub 9046	. . . 4 |- ( ( 1 e. NN /\ ( 2 ^ ( A + 1 ) ) e. NN ) -> ( 1 < ( 2 ^ ( A + 1 ) ) <-> ( ( 2 ^ ( A + 1 ) ) - 1 ) e. NN ) )
16	14, 7, 15	sylancr 414	. . 3 |- ( ph -> ( 1 < ( 2 ^ ( A + 1 ) ) <-> ( ( 2 ^ ( A + 1 ) ) - 1 ) e. NN ) )
17	13, 16	mpbid 147	. 2 |- ( ph -> ( ( 2 ^ ( A + 1 ) ) - 1 ) e. NN )
18	7	nnzd 9464	. . . . . . 7 |- ( ph -> ( 2 ^ ( A + 1 ) ) e. ZZ )
19		peano2zm 9381	. . . . . . 7 |- ( ( 2 ^ ( A + 1 ) ) e. ZZ -> ( ( 2 ^ ( A + 1 ) ) - 1 ) e. ZZ )
20	18, 19	syl 14	. . . . . 6 |- ( ph -> ( ( 2 ^ ( A + 1 ) ) - 1 ) e. ZZ )
21		1nn0 9282	. . . . . . . 8 |- 1 e. NN0
22		perfectlem.2	. . . . . . . 8 |- ( ph -> B e. NN )
23		sgmnncl 15308	. . . . . . . 8 |- ( ( 1 e. NN0 /\ B e. NN ) -> ( 1 sigma B ) e. NN )
24	21, 22, 23	sylancr 414	. . . . . . 7 |- ( ph -> ( 1 sigma B ) e. NN )
25	24	nnzd 9464	. . . . . 6 |- ( ph -> ( 1 sigma B ) e. ZZ )
26		dvdsmul1 11995	. . . . . 6 |- ( ( ( ( 2 ^ ( A + 1 ) ) - 1 ) e. ZZ /\ ( 1 sigma B ) e. ZZ ) -> ( ( 2 ^ ( A + 1 ) ) - 1 ) || ( ( ( 2 ^ ( A + 1 ) ) - 1 ) x. ( 1 sigma B ) ) )
27	20, 25, 26	syl2anc 411	. . . . 5 |- ( ph -> ( ( 2 ^ ( A + 1 ) ) - 1 ) || ( ( ( 2 ^ ( A + 1 ) ) - 1 ) x. ( 1 sigma B ) ) )
28		2cn 9078	. . . . . . . . 9 |- 2 e. CC
29		expp1 10655	. . . . . . . . 9 |- ( ( 2 e. CC /\ A e. NN0 ) -> ( 2 ^ ( A + 1 ) ) = ( ( 2 ^ A ) x. 2 ) )
30	28, 3, 29	sylancr 414	. . . . . . . 8 |- ( ph -> ( 2 ^ ( A + 1 ) ) = ( ( 2 ^ A ) x. 2 ) )
31		nnexpcl 10661	. . . . . . . . . . 11 |- ( ( 2 e. NN /\ A e. NN0 ) -> ( 2 ^ A ) e. NN )
32	1, 3, 31	sylancr 414	. . . . . . . . . 10 |- ( ph -> ( 2 ^ A ) e. NN )
33	32	nncnd 9021	. . . . . . . . 9 |- ( ph -> ( 2 ^ A ) e. CC )
34		mulcom 8025	. . . . . . . . 9 |- ( ( ( 2 ^ A ) e. CC /\ 2 e. CC ) -> ( ( 2 ^ A ) x. 2 ) = ( 2 x. ( 2 ^ A ) ) )
35	33, 28, 34	sylancl 413	. . . . . . . 8 |- ( ph -> ( ( 2 ^ A ) x. 2 ) = ( 2 x. ( 2 ^ A ) ) )
36	30, 35	eqtrd 2229	. . . . . . 7 |- ( ph -> ( 2 ^ ( A + 1 ) ) = ( 2 x. ( 2 ^ A ) ) )
37	36	oveq1d 5940	. . . . . 6 |- ( ph -> ( ( 2 ^ ( A + 1 ) ) x. B ) = ( ( 2 x. ( 2 ^ A ) ) x. B ) )
38	28	a1i 9	. . . . . . 7 |- ( ph -> 2 e. CC )
39	22	nncnd 9021	. . . . . . 7 |- ( ph -> B e. CC )
40	38, 33, 39	mulassd 8067	. . . . . 6 |- ( ph -> ( ( 2 x. ( 2 ^ A ) ) x. B ) = ( 2 x. ( ( 2 ^ A ) x. B ) ) )
41		ax-1cn 7989	. . . . . . . . 9 |- 1 e. CC
42	41	a1i 9	. . . . . . . 8 |- ( ph -> 1 e. CC )
43		perfectlem.3	. . . . . . . . . 10 |- ( ph -> -. 2 || B )
44		2prm 12320	. . . . . . . . . . 11 |- 2 e. Prime
45	22	nnzd 9464	. . . . . . . . . . 11 |- ( ph -> B e. ZZ )
46		coprm 12337	. . . . . . . . . . 11 |- ( ( 2 e. Prime /\ B e. ZZ ) -> ( -. 2 || B <-> ( 2 gcd B ) = 1 ) )
47	44, 45, 46	sylancr 414	. . . . . . . . . 10 |- ( ph -> ( -. 2 || B <-> ( 2 gcd B ) = 1 ) )
48	43, 47	mpbid 147	. . . . . . . . 9 |- ( ph -> ( 2 gcd B ) = 1 )
49		2z 9371	. . . . . . . . . 10 |- 2 e. ZZ
50		rpexp1i 12347	. . . . . . . . . 10 |- ( ( 2 e. ZZ /\ B e. ZZ /\ A e. NN0 ) -> ( ( 2 gcd B ) = 1 -> ( ( 2 ^ A ) gcd B ) = 1 ) )
51	49, 45, 3, 50	mp3an2i 1353	. . . . . . . . 9 |- ( ph -> ( ( 2 gcd B ) = 1 -> ( ( 2 ^ A ) gcd B ) = 1 ) )
52	48, 51	mpd 13	. . . . . . . 8 |- ( ph -> ( ( 2 ^ A ) gcd B ) = 1 )
53		sgmmul 15316	. . . . . . . 8 |- ( ( 1 e. CC /\ ( ( 2 ^ A ) e. NN /\ B e. NN /\ ( ( 2 ^ A ) gcd B ) = 1 ) ) -> ( 1 sigma ( ( 2 ^ A ) x. B ) ) = ( ( 1 sigma ( 2 ^ A ) ) x. ( 1 sigma B ) ) )
54	42, 32, 22, 52, 53	syl13anc 1251	. . . . . . 7 |- ( ph -> ( 1 sigma ( ( 2 ^ A ) x. B ) ) = ( ( 1 sigma ( 2 ^ A ) ) x. ( 1 sigma B ) ) )
55		perfectlem.4	. . . . . . 7 |- ( ph -> ( 1 sigma ( ( 2 ^ A ) x. B ) ) = ( 2 x. ( ( 2 ^ A ) x. B ) ) )
56	2	nncnd 9021	. . . . . . . . . . . 12 |- ( ph -> A e. CC )
57		pncan 8249	. . . . . . . . . . . 12 |- ( ( A e. CC /\ 1 e. CC ) -> ( ( A + 1 ) - 1 ) = A )
58	56, 41, 57	sylancl 413	. . . . . . . . . . 11 |- ( ph -> ( ( A + 1 ) - 1 ) = A )
59	58	oveq2d 5941	. . . . . . . . . 10 |- ( ph -> ( 2 ^ ( ( A + 1 ) - 1 ) ) = ( 2 ^ A ) )
60	59	oveq2d 5941	. . . . . . . . 9 |- ( ph -> ( 1 sigma ( 2 ^ ( ( A + 1 ) - 1 ) ) ) = ( 1 sigma ( 2 ^ A ) ) )
61		1sgm2ppw 15315	. . . . . . . . . 10 |- ( ( A + 1 ) e. NN -> ( 1 sigma ( 2 ^ ( ( A + 1 ) - 1 ) ) ) = ( ( 2 ^ ( A + 1 ) ) - 1 ) )
62	9, 61	syl 14	. . . . . . . . 9 |- ( ph -> ( 1 sigma ( 2 ^ ( ( A + 1 ) - 1 ) ) ) = ( ( 2 ^ ( A + 1 ) ) - 1 ) )
63	60, 62	eqtr3d 2231	. . . . . . . 8 |- ( ph -> ( 1 sigma ( 2 ^ A ) ) = ( ( 2 ^ ( A + 1 ) ) - 1 ) )
64	63	oveq1d 5940	. . . . . . 7 |- ( ph -> ( ( 1 sigma ( 2 ^ A ) ) x. ( 1 sigma B ) ) = ( ( ( 2 ^ ( A + 1 ) ) - 1 ) x. ( 1 sigma B ) ) )
65	54, 55, 64	3eqtr3d 2237	. . . . . 6 |- ( ph -> ( 2 x. ( ( 2 ^ A ) x. B ) ) = ( ( ( 2 ^ ( A + 1 ) ) - 1 ) x. ( 1 sigma B ) ) )
66	37, 40, 65	3eqtrd 2233	. . . . 5 |- ( ph -> ( ( 2 ^ ( A + 1 ) ) x. B ) = ( ( ( 2 ^ ( A + 1 ) ) - 1 ) x. ( 1 sigma B ) ) )
67	27, 66	breqtrrd 4062	. . . 4 |- ( ph -> ( ( 2 ^ ( A + 1 ) ) - 1 ) || ( ( 2 ^ ( A + 1 ) ) x. B ) )
68	20, 18	gcdcomd 12166	. . . . 5 |- ( ph -> ( ( ( 2 ^ ( A + 1 ) ) - 1 ) gcd ( 2 ^ ( A + 1 ) ) ) = ( ( 2 ^ ( A + 1 ) ) gcd ( ( 2 ^ ( A + 1 ) ) - 1 ) ) )
69		iddvdsexp 11997	. . . . . . . . 9 |- ( ( 2 e. ZZ /\ ( A + 1 ) e. NN ) -> 2 || ( 2 ^ ( A + 1 ) ) )
70	49, 9, 69	sylancr 414	. . . . . . . 8 |- ( ph -> 2 || ( 2 ^ ( A + 1 ) ) )
71		n2dvds1 12094	. . . . . . . . . 10 |- -. 2 || 1
72	49	a1i 9	. . . . . . . . . . . 12 |- ( ph -> 2 e. ZZ )
73		1zzd 9370	. . . . . . . . . . . 12 |- ( ph -> 1 e. ZZ )
74	72, 18, 73	3jca 1179	. . . . . . . . . . 11 |- ( ph -> ( 2 e. ZZ /\ ( 2 ^ ( A + 1 ) ) e. ZZ /\ 1 e. ZZ ) )
75		dvdssub2 12017	. . . . . . . . . . 11 |- ( ( ( 2 e. ZZ /\ ( 2 ^ ( A + 1 ) ) e. ZZ /\ 1 e. ZZ ) /\ 2 || ( ( 2 ^ ( A + 1 ) ) - 1 ) ) -> ( 2 || ( 2 ^ ( A + 1 ) ) <-> 2 || 1 ) )
76	74, 75	sylan 283	. . . . . . . . . 10 |- ( ( ph /\ 2 || ( ( 2 ^ ( A + 1 ) ) - 1 ) ) -> ( 2 || ( 2 ^ ( A + 1 ) ) <-> 2 || 1 ) )
77	71, 76	mtbiri 676	. . . . . . . . 9 |- ( ( ph /\ 2 || ( ( 2 ^ ( A + 1 ) ) - 1 ) ) -> -. 2 || ( 2 ^ ( A + 1 ) ) )
78	77	ex 115	. . . . . . . 8 |- ( ph -> ( 2 || ( ( 2 ^ ( A + 1 ) ) - 1 ) -> -. 2 || ( 2 ^ ( A + 1 ) ) ) )
79	70, 78	mt2d 626	. . . . . . 7 |- ( ph -> -. 2 || ( ( 2 ^ ( A + 1 ) ) - 1 ) )
80		coprm 12337	. . . . . . . 8 |- ( ( 2 e. Prime /\ ( ( 2 ^ ( A + 1 ) ) - 1 ) e. ZZ ) -> ( -. 2 || ( ( 2 ^ ( A + 1 ) ) - 1 ) <-> ( 2 gcd ( ( 2 ^ ( A + 1 ) ) - 1 ) ) = 1 ) )
81	44, 20, 80	sylancr 414	. . . . . . 7 |- ( ph -> ( -. 2 || ( ( 2 ^ ( A + 1 ) ) - 1 ) <-> ( 2 gcd ( ( 2 ^ ( A + 1 ) ) - 1 ) ) = 1 ) )
82	79, 81	mpbid 147	. . . . . 6 |- ( ph -> ( 2 gcd ( ( 2 ^ ( A + 1 ) ) - 1 ) ) = 1 )
83		rpexp1i 12347	. . . . . . 7 |- ( ( 2 e. ZZ /\ ( ( 2 ^ ( A + 1 ) ) - 1 ) e. ZZ /\ ( A + 1 ) e. NN0 ) -> ( ( 2 gcd ( ( 2 ^ ( A + 1 ) ) - 1 ) ) = 1 -> ( ( 2 ^ ( A + 1 ) ) gcd ( ( 2 ^ ( A + 1 ) ) - 1 ) ) = 1 ) )
84	49, 20, 5, 83	mp3an2i 1353	. . . . . 6 |- ( ph -> ( ( 2 gcd ( ( 2 ^ ( A + 1 ) ) - 1 ) ) = 1 -> ( ( 2 ^ ( A + 1 ) ) gcd ( ( 2 ^ ( A + 1 ) ) - 1 ) ) = 1 ) )
85	82, 84	mpd 13	. . . . 5 |- ( ph -> ( ( 2 ^ ( A + 1 ) ) gcd ( ( 2 ^ ( A + 1 ) ) - 1 ) ) = 1 )
86	68, 85	eqtrd 2229	. . . 4 |- ( ph -> ( ( ( 2 ^ ( A + 1 ) ) - 1 ) gcd ( 2 ^ ( A + 1 ) ) ) = 1 )
87		coprmdvds 12285	. . . . 5 |- ( ( ( ( 2 ^ ( A + 1 ) ) - 1 ) e. ZZ /\ ( 2 ^ ( A + 1 ) ) e. ZZ /\ B e. ZZ ) -> ( ( ( ( 2 ^ ( A + 1 ) ) - 1 ) || ( ( 2 ^ ( A + 1 ) ) x. B ) /\ ( ( ( 2 ^ ( A + 1 ) ) - 1 ) gcd ( 2 ^ ( A + 1 ) ) ) = 1 ) -> ( ( 2 ^ ( A + 1 ) ) - 1 ) || B ) )
88	20, 18, 45, 87	syl3anc 1249	. . . 4 |- ( ph -> ( ( ( ( 2 ^ ( A + 1 ) ) - 1 ) || ( ( 2 ^ ( A + 1 ) ) x. B ) /\ ( ( ( 2 ^ ( A + 1 ) ) - 1 ) gcd ( 2 ^ ( A + 1 ) ) ) = 1 ) -> ( ( 2 ^ ( A + 1 ) ) - 1 ) || B ) )
89	67, 86, 88	mp2and 433	. . 3 |- ( ph -> ( ( 2 ^ ( A + 1 ) ) - 1 ) || B )
90		nndivdvds 11978	. . . 4 |- ( ( B e. NN /\ ( ( 2 ^ ( A + 1 ) ) - 1 ) e. NN ) -> ( ( ( 2 ^ ( A + 1 ) ) - 1 ) || B <-> ( B / ( ( 2 ^ ( A + 1 ) ) - 1 ) ) e. NN ) )
91	22, 17, 90	syl2anc 411	. . 3 |- ( ph -> ( ( ( 2 ^ ( A + 1 ) ) - 1 ) || B <-> ( B / ( ( 2 ^ ( A + 1 ) ) - 1 ) ) e. NN ) )
92	89, 91	mpbid 147	. 2 |- ( ph -> ( B / ( ( 2 ^ ( A + 1 ) ) - 1 ) ) e. NN )
93	7, 17, 92	3jca 1179	1 |- ( ph -> ( ( 2 ^ ( A + 1 ) ) e. NN /\ ( ( 2 ^ ( A + 1 ) ) - 1 ) e. NN /\ ( B / ( ( 2 ^ ( A + 1 ) ) - 1 ) ) e. NN ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 980   = wceq 1364   e. wcel 2167   class class class wbr 4034  (class class class)co 5925   CCcc 7894   RRcr 7895   1c1 7897   + caddc 7899   x. cmul 7901   < clt 8078   - cmin 8214   / cdiv 8716   NNcn 9007   2c2 9058   NN0cn0 9266   ZZcz 9343   ^cexp 10647   || cdvds 11969   gcd cgcd 12145   Primecprime 12300   sigma csgm 15301

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 4149  ax-sep 4152  ax-nul 4160  ax-pow 4208  ax-pr 4243  ax-un 4469  ax-setind 4574  ax-iinf 4625  ax-cnex 7987  ax-resscn 7988  ax-1cn 7989  ax-1re 7990  ax-icn 7991  ax-addcl 7992  ax-addrcl 7993  ax-mulcl 7994  ax-mulrcl 7995  ax-addcom 7996  ax-mulcom 7997  ax-addass 7998  ax-mulass 7999  ax-distr 8000  ax-i2m1 8001  ax-0lt1 8002  ax-1rid 8003  ax-0id 8004  ax-rnegex 8005  ax-precex 8006  ax-cnre 8007  ax-pre-ltirr 8008  ax-pre-ltwlin 8009  ax-pre-lttrn 8010  ax-pre-apti 8011  ax-pre-ltadd 8012  ax-pre-mulgt0 8013  ax-pre-mulext 8014  ax-arch 8015  ax-caucvg 8016  ax-pre-suploc 8017  ax-addf 8018  ax-mulf 8019

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 3452  df-if 3563  df-pw 3608  df-sn 3629  df-pr 3630  df-op 3632  df-uni 3841  df-int 3876  df-iun 3919  df-disj 4012  df-br 4035  df-opab 4096  df-mpt 4097  df-tr 4133  df-id 4329  df-po 4332  df-iso 4333  df-iord 4402  df-on 4404  df-ilim 4405  df-suc 4407  df-iom 4628  df-xp 4670  df-rel 4671  df-cnv 4672  df-co 4673  df-dm 4674  df-rn 4675  df-res 4676  df-ima 4677  df-iota 5220  df-fun 5261  df-fn 5262  df-f 5263  df-f1 5264  df-fo 5265  df-f1o 5266  df-fv 5267  df-isom 5268  df-riota 5880  df-ov 5928  df-oprab 5929  df-mpo 5930  df-of 6139  df-1st 6207  df-2nd 6208  df-recs 6372  df-irdg 6437  df-frec 6458  df-1o 6483  df-2o 6484  df-oadd 6487  df-er 6601  df-map 6718  df-pm 6719  df-en 6809  df-dom 6810  df-fin 6811  df-sup 7059  df-inf 7060  df-pnf 8080  df-mnf 8081  df-xr 8082  df-ltxr 8083  df-le 8084  df-sub 8216  df-neg 8217  df-reap 8619  df-ap 8626  df-div 8717  df-inn 9008  df-2 9066  df-3 9067  df-4 9068  df-n0 9267  df-xnn0 9330  df-z 9344  df-uz 9619  df-q 9711  df-rp 9746  df-xneg 9864  df-xadd 9865  df-ioo 9984  df-ico 9986  df-icc 9987  df-fz 10101  df-fzo 10235  df-fl 10377  df-mod 10432  df-seqfrec 10557  df-exp 10648  df-fac 10835  df-bc 10857  df-ihash 10885  df-shft 10997  df-cj 11024  df-re 11025  df-im 11026  df-rsqrt 11180  df-abs 11181  df-clim 11461  df-sumdc 11536  df-ef 11830  df-e 11831  df-dvds 11970  df-gcd 12146  df-prm 12301  df-pc 12479  df-rest 12943  df-topgen 12962  df-psmet 14175  df-xmet 14176  df-met 14177  df-bl 14178  df-mopn 14179  df-top 14318  df-topon 14331  df-bases 14363  df-ntr 14416  df-cn 14508  df-cnp 14509  df-tx 14573  df-cncf 14891  df-limced 14976  df-dvap 14977  df-relog 15178  df-rpcxp 15179  df-sgm 15302

This theorem is referenced by:  perfectlem2  15320