Theorem perfectlem1 15916

Description: Lemma for perfect 15918. (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 9404	. . 3 |- 2 e. NN
2		perfectlem.1	. . . . 5 |- ( ph -> A e. NN )
3	2	nnnn0d 9558	. . . 4 |- ( ph -> A e. NN0 )
4		peano2nn0 9541	. . . 4 |- ( A e. NN0 -> ( A + 1 ) e. NN0 )
5	3, 4	syl 14	. . 3 |- ( ph -> ( A + 1 ) e. NN0 )
6		nnexpcl 10921	. . 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 9312	. . . 4 |- 2 e. RR
9	2	peano2nnd 9257	. . . 4 |- ( ph -> ( A + 1 ) e. NN )
10		1lt2 9412	. . . . 5 |- 1 < 2
11	10	a1i 9	. . . 4 |- ( ph -> 1 < 2 )
12		expgt1 10946	. . . 4 |- ( ( 2 e. RR /\ ( A + 1 ) e. NN /\ 1 < 2 ) -> 1 < ( 2 ^ ( A + 1 ) ) )
13	8, 9, 11, 12	mp3an2i 1379	. . 3 |- ( ph -> 1 < ( 2 ^ ( A + 1 ) ) )
14		1nn 9253	. . . 4 |- 1 e. NN
15		nnsub 9281	. . . 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 9705	. . . . . . 7 |- ( ph -> ( 2 ^ ( A + 1 ) ) e. ZZ )
19		peano2zm 9620	. . . . . . 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 9517	. . . . . . . 8 |- 1 e. NN0
22		perfectlem.2	. . . . . . . 8 |- ( ph -> B e. NN )
23		sgmnncl 15905	. . . . . . . 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 9705	. . . . . 6 |- ( ph -> ( 1 sigma B ) e. ZZ )
26		dvdsmul1 12507	. . . . . 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 9313	. . . . . . . . 9 |- 2 e. CC
29		expp1 10915	. . . . . . . . 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 10921	. . . . . . . . . . 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 9256	. . . . . . . . 9 |- ( ph -> ( 2 ^ A ) e. CC )
34		mulcom 8261	. . . . . . . . 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 2267	. . . . . . 7 |- ( ph -> ( 2 ^ ( A + 1 ) ) = ( 2 x. ( 2 ^ A ) ) )
37	36	oveq1d 6067	. . . . . 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 9256	. . . . . . 7 |- ( ph -> B e. CC )
40	38, 33, 39	mulassd 8302	. . . . . 6 |- ( ph -> ( ( 2 x. ( 2 ^ A ) ) x. B ) = ( 2 x. ( ( 2 ^ A ) x. B ) ) )
41		ax-1cn 8225	. . . . . . . . 9 |- 1 e. CC
42	41	a1i 9	. . . . . . . 8 |- ( ph -> 1 e. CC )
43		perfectlem.3	. . . . . . . . . 10 |- ( ph -> -. 2 || B )
44		2prm 12832	. . . . . . . . . . 11 |- 2 e. Prime
45	22	nnzd 9705	. . . . . . . . . . 11 |- ( ph -> B e. ZZ )
46		coprm 12849	. . . . . . . . . . 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 9610	. . . . . . . . . 10 |- 2 e. ZZ
50		rpexp1i 12859	. . . . . . . . . 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 1379	. . . . . . . . 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 15913	. . . . . . . 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 1276	. . . . . . 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 9256	. . . . . . . . . . . 12 |- ( ph -> A e. CC )
57		pncan 8484	. . . . . . . . . . . 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 6068	. . . . . . . . . 10 |- ( ph -> ( 2 ^ ( ( A + 1 ) - 1 ) ) = ( 2 ^ A ) )
60	59	oveq2d 6068	. . . . . . . . 9 |- ( ph -> ( 1 sigma ( 2 ^ ( ( A + 1 ) - 1 ) ) ) = ( 1 sigma ( 2 ^ A ) ) )
61		1sgm2ppw 15912	. . . . . . . . . 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 2269	. . . . . . . 8 |- ( ph -> ( 1 sigma ( 2 ^ A ) ) = ( ( 2 ^ ( A + 1 ) ) - 1 ) )
64	63	oveq1d 6067	. . . . . . 7 |- ( ph -> ( ( 1 sigma ( 2 ^ A ) ) x. ( 1 sigma B ) ) = ( ( ( 2 ^ ( A + 1 ) ) - 1 ) x. ( 1 sigma B ) ) )
65	54, 55, 64	3eqtr3d 2275	. . . . . 6 |- ( ph -> ( 2 x. ( ( 2 ^ A ) x. B ) ) = ( ( ( 2 ^ ( A + 1 ) ) - 1 ) x. ( 1 sigma B ) ) )
66	37, 40, 65	3eqtrd 2271	. . . . 5 |- ( ph -> ( ( 2 ^ ( A + 1 ) ) x. B ) = ( ( ( 2 ^ ( A + 1 ) ) - 1 ) x. ( 1 sigma B ) ) )
67	27, 66	breqtrrd 4139	. . . 4 |- ( ph -> ( ( 2 ^ ( A + 1 ) ) - 1 ) || ( ( 2 ^ ( A + 1 ) ) x. B ) )
68	20, 18	gcdcomd 12678	. . . . 5 |- ( ph -> ( ( ( 2 ^ ( A + 1 ) ) - 1 ) gcd ( 2 ^ ( A + 1 ) ) ) = ( ( 2 ^ ( A + 1 ) ) gcd ( ( 2 ^ ( A + 1 ) ) - 1 ) ) )
69		iddvdsexp 12509	. . . . . . . . 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 12606	. . . . . . . . . 10 |- -. 2 || 1
72	49	a1i 9	. . . . . . . . . . . 12 |- ( ph -> 2 e. ZZ )
73		1zzd 9609	. . . . . . . . . . . 12 |- ( ph -> 1 e. ZZ )
74	72, 18, 73	3jca 1204	. . . . . . . . . . 11 |- ( ph -> ( 2 e. ZZ /\ ( 2 ^ ( A + 1 ) ) e. ZZ /\ 1 e. ZZ ) )
75		dvdssub2 12529	. . . . . . . . . . 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 682	. . . . . . . . 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 630	. . . . . . 7 |- ( ph -> -. 2 || ( ( 2 ^ ( A + 1 ) ) - 1 ) )
80		coprm 12849	. . . . . . . 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 12859	. . . . . . 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 1379	. . . . . 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 2267	. . . 4 |- ( ph -> ( ( ( 2 ^ ( A + 1 ) ) - 1 ) gcd ( 2 ^ ( A + 1 ) ) ) = 1 )
87		coprmdvds 12797	. . . . 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 1274	. . . 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 12490	. . . 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 1204	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 1005   = wceq 1398   e. wcel 2205   class class class wbr 4111  (class class class)co 6052   CCcc 8130   RRcr 8131   1c1 8133   + caddc 8135   x. cmul 8137   < clt 8313   - cmin 8449   / cdiv 8951   NNcn 9242   2c2 9293   NN0cn0 9501   ZZcz 9582   ^cexp 10907   || cdvds 12481   gcd cgcd 12657   Primecprime 12812   sigma csgm 15898

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 4227  ax-sep 4230  ax-nul 4238  ax-pow 4289  ax-pr 4324  ax-un 4556  ax-setind 4661  ax-iinf 4712  ax-cnex 8223  ax-resscn 8224  ax-1cn 8225  ax-1re 8226  ax-icn 8227  ax-addcl 8228  ax-addrcl 8229  ax-mulcl 8230  ax-mulrcl 8231  ax-addcom 8232  ax-mulcom 8233  ax-addass 8234  ax-mulass 8235  ax-distr 8236  ax-i2m1 8237  ax-0lt1 8238  ax-1rid 8239  ax-0id 8240  ax-rnegex 8241  ax-precex 8242  ax-cnre 8243  ax-pre-ltirr 8244  ax-pre-ltwlin 8245  ax-pre-lttrn 8246  ax-pre-apti 8247  ax-pre-ltadd 8248  ax-pre-mulgt0 8249  ax-pre-mulext 8250  ax-arch 8251  ax-caucvg 8252  ax-pre-suploc 8253  ax-addf 8254  ax-mulf 8255

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 3045  df-csb 3141  df-dif 3215  df-un 3217  df-in 3219  df-ss 3226  df-nul 3511  df-if 3623  df-pw 3673  df-sn 3697  df-pr 3698  df-op 3700  df-uni 3917  df-int 3952  df-iun 3995  df-disj 4088  df-br 4112  df-opab 4174  df-mpt 4175  df-tr 4211  df-id 4416  df-po 4419  df-iso 4420  df-iord 4489  df-on 4491  df-ilim 4492  df-suc 4494  df-iom 4715  df-xp 4757  df-rel 4758  df-cnv 4759  df-co 4760  df-dm 4761  df-rn 4762  df-res 4763  df-ima 4764  df-iota 5314  df-fun 5356  df-fn 5357  df-f 5358  df-f1 5359  df-fo 5360  df-f1o 5361  df-fv 5362  df-isom 5363  df-riota 6005  df-ov 6055  df-oprab 6056  df-mpo 6057  df-of 6268  df-1st 6336  df-2nd 6337  df-recs 6538  df-irdg 6603  df-frec 6624  df-1o 6649  df-2o 6650  df-oadd 6653  df-er 6769  df-map 6886  df-pm 6887  df-en 6978  df-dom 6979  df-fin 6980  df-sup 7277  df-inf 7278  df-pnf 8315  df-mnf 8316  df-xr 8317  df-ltxr 8318  df-le 8319  df-sub 8451  df-neg 8452  df-reap 8854  df-ap 8861  df-div 8952  df-inn 9243  df-2 9301  df-3 9302  df-4 9303  df-n0 9502  df-xnn0 9569  df-z 9583  df-uz 9860  df-q 9958  df-rp 9993  df-xneg 10111  df-xadd 10112  df-ioo 10231  df-ico 10233  df-icc 10234  df-fz 10349  df-fzo 10484  df-fl 10637  df-mod 10692  df-seqfrec 10817  df-exp 10908  df-fac 11096  df-bc 11118  df-ihash 11147  df-shft 11508  df-cj 11535  df-re 11536  df-im 11537  df-rsqrt 11691  df-abs 11692  df-clim 11972  df-sumdc 12047  df-ef 12342  df-e 12343  df-dvds 12482  df-gcd 12658  df-prm 12813  df-pc 12991  df-rest 13475  df-topgen 13494  df-psmet 14740  df-xmet 14741  df-met 14742  df-bl 14743  df-mopn 14744  df-top 14912  df-topon 14925  df-bases 14957  df-ntr 15010  df-cn 15102  df-cnp 15103  df-tx 15167  df-cncf 15485  df-limced 15570  df-dvap 15571  df-relog 15772  df-rpcxp 15773  df-sgm 15899

This theorem is referenced by:  perfectlem2  15917