Theorem perfectlem1 15726

Description: Lemma for perfect 15728. (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 9305	. . 3 |- 2 e. NN
2		perfectlem.1	. . . . 5 |- ( ph -> A e. NN )
3	2	nnnn0d 9455	. . . 4 |- ( ph -> A e. NN0 )
4		peano2nn0 9442	. . . 4 |- ( A e. NN0 -> ( A + 1 ) e. NN0 )
5	3, 4	syl 14	. . 3 |- ( ph -> ( A + 1 ) e. NN0 )
6		nnexpcl 10815	. . 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 9213	. . . 4 |- 2 e. RR
9	2	peano2nnd 9158	. . . 4 |- ( ph -> ( A + 1 ) e. NN )
10		1lt2 9313	. . . . 5 |- 1 < 2
11	10	a1i 9	. . . 4 |- ( ph -> 1 < 2 )
12		expgt1 10840	. . . 4 |- ( ( 2 e. RR /\ ( A + 1 ) e. NN /\ 1 < 2 ) -> 1 < ( 2 ^ ( A + 1 ) ) )
13	8, 9, 11, 12	mp3an2i 1378	. . 3 |- ( ph -> 1 < ( 2 ^ ( A + 1 ) ) )
14		1nn 9154	. . . 4 |- 1 e. NN
15		nnsub 9182	. . . 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 9601	. . . . . . 7 |- ( ph -> ( 2 ^ ( A + 1 ) ) e. ZZ )
19		peano2zm 9517	. . . . . . 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 9418	. . . . . . . 8 |- 1 e. NN0
22		perfectlem.2	. . . . . . . 8 |- ( ph -> B e. NN )
23		sgmnncl 15715	. . . . . . . 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 9601	. . . . . 6 |- ( ph -> ( 1 sigma B ) e. ZZ )
26		dvdsmul1 12376	. . . . . 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 9214	. . . . . . . . 9 |- 2 e. CC
29		expp1 10809	. . . . . . . . 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 10815	. . . . . . . . . . 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 9157	. . . . . . . . 9 |- ( ph -> ( 2 ^ A ) e. CC )
34		mulcom 8161	. . . . . . . . 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 2264	. . . . . . 7 |- ( ph -> ( 2 ^ ( A + 1 ) ) = ( 2 x. ( 2 ^ A ) ) )
37	36	oveq1d 6033	. . . . . 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 9157	. . . . . . 7 |- ( ph -> B e. CC )
40	38, 33, 39	mulassd 8203	. . . . . 6 |- ( ph -> ( ( 2 x. ( 2 ^ A ) ) x. B ) = ( 2 x. ( ( 2 ^ A ) x. B ) ) )
41		ax-1cn 8125	. . . . . . . . 9 |- 1 e. CC
42	41	a1i 9	. . . . . . . 8 |- ( ph -> 1 e. CC )
43		perfectlem.3	. . . . . . . . . 10 |- ( ph -> -. 2 || B )
44		2prm 12701	. . . . . . . . . . 11 |- 2 e. Prime
45	22	nnzd 9601	. . . . . . . . . . 11 |- ( ph -> B e. ZZ )
46		coprm 12718	. . . . . . . . . . 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 9507	. . . . . . . . . 10 |- 2 e. ZZ
50		rpexp1i 12728	. . . . . . . . . 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 1378	. . . . . . . . 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 15723	. . . . . . . 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 1275	. . . . . . 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 9157	. . . . . . . . . . . 12 |- ( ph -> A e. CC )
57		pncan 8385	. . . . . . . . . . . 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 6034	. . . . . . . . . 10 |- ( ph -> ( 2 ^ ( ( A + 1 ) - 1 ) ) = ( 2 ^ A ) )
60	59	oveq2d 6034	. . . . . . . . 9 |- ( ph -> ( 1 sigma ( 2 ^ ( ( A + 1 ) - 1 ) ) ) = ( 1 sigma ( 2 ^ A ) ) )
61		1sgm2ppw 15722	. . . . . . . . . 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 2266	. . . . . . . 8 |- ( ph -> ( 1 sigma ( 2 ^ A ) ) = ( ( 2 ^ ( A + 1 ) ) - 1 ) )
64	63	oveq1d 6033	. . . . . . 7 |- ( ph -> ( ( 1 sigma ( 2 ^ A ) ) x. ( 1 sigma B ) ) = ( ( ( 2 ^ ( A + 1 ) ) - 1 ) x. ( 1 sigma B ) ) )
65	54, 55, 64	3eqtr3d 2272	. . . . . 6 |- ( ph -> ( 2 x. ( ( 2 ^ A ) x. B ) ) = ( ( ( 2 ^ ( A + 1 ) ) - 1 ) x. ( 1 sigma B ) ) )
66	37, 40, 65	3eqtrd 2268	. . . . 5 |- ( ph -> ( ( 2 ^ ( A + 1 ) ) x. B ) = ( ( ( 2 ^ ( A + 1 ) ) - 1 ) x. ( 1 sigma B ) ) )
67	27, 66	breqtrrd 4116	. . . 4 |- ( ph -> ( ( 2 ^ ( A + 1 ) ) - 1 ) || ( ( 2 ^ ( A + 1 ) ) x. B ) )
68	20, 18	gcdcomd 12547	. . . . 5 |- ( ph -> ( ( ( 2 ^ ( A + 1 ) ) - 1 ) gcd ( 2 ^ ( A + 1 ) ) ) = ( ( 2 ^ ( A + 1 ) ) gcd ( ( 2 ^ ( A + 1 ) ) - 1 ) ) )
69		iddvdsexp 12378	. . . . . . . . 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 12475	. . . . . . . . . 10 |- -. 2 || 1
72	49	a1i 9	. . . . . . . . . . . 12 |- ( ph -> 2 e. ZZ )
73		1zzd 9506	. . . . . . . . . . . 12 |- ( ph -> 1 e. ZZ )
74	72, 18, 73	3jca 1203	. . . . . . . . . . 11 |- ( ph -> ( 2 e. ZZ /\ ( 2 ^ ( A + 1 ) ) e. ZZ /\ 1 e. ZZ ) )
75		dvdssub2 12398	. . . . . . . . . . 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 681	. . . . . . . . 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 12718	. . . . . . . 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 12728	. . . . . . 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 1378	. . . . . 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 2264	. . . 4 |- ( ph -> ( ( ( 2 ^ ( A + 1 ) ) - 1 ) gcd ( 2 ^ ( A + 1 ) ) ) = 1 )
87		coprmdvds 12666	. . . . 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 1273	. . . 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 12359	. . . 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 1203	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 1004   = wceq 1397   e. wcel 2202   class class class wbr 4088  (class class class)co 6018   CCcc 8030   RRcr 8031   1c1 8033   + caddc 8035   x. cmul 8037   < clt 8214   - cmin 8350   / cdiv 8852   NNcn 9143   2c2 9194   NN0cn0 9402   ZZcz 9479   ^cexp 10801   || cdvds 12350   gcd cgcd 12526   Primecprime 12681   sigma csgm 15708

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4204  ax-sep 4207  ax-nul 4215  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-iinf 4686  ax-cnex 8123  ax-resscn 8124  ax-1cn 8125  ax-1re 8126  ax-icn 8127  ax-addcl 8128  ax-addrcl 8129  ax-mulcl 8130  ax-mulrcl 8131  ax-addcom 8132  ax-mulcom 8133  ax-addass 8134  ax-mulass 8135  ax-distr 8136  ax-i2m1 8137  ax-0lt1 8138  ax-1rid 8139  ax-0id 8140  ax-rnegex 8141  ax-precex 8142  ax-cnre 8143  ax-pre-ltirr 8144  ax-pre-ltwlin 8145  ax-pre-lttrn 8146  ax-pre-apti 8147  ax-pre-ltadd 8148  ax-pre-mulgt0 8149  ax-pre-mulext 8150  ax-arch 8151  ax-caucvg 8152  ax-pre-suploc 8153  ax-addf 8154  ax-mulf 8155

This theorem depends on definitions:  df-bi 117  df-stab 838  df-dc 842  df-3or 1005  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-nel 2498  df-ral 2515  df-rex 2516  df-reu 2517  df-rmo 2518  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-if 3606  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-iun 3972  df-disj 4065  df-br 4089  df-opab 4151  df-mpt 4152  df-tr 4188  df-id 4390  df-po 4393  df-iso 4394  df-iord 4463  df-on 4465  df-ilim 4466  df-suc 4468  df-iom 4689  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-fo 5332  df-f1o 5333  df-fv 5334  df-isom 5335  df-riota 5971  df-ov 6021  df-oprab 6022  df-mpo 6023  df-of 6235  df-1st 6303  df-2nd 6304  df-recs 6471  df-irdg 6536  df-frec 6557  df-1o 6582  df-2o 6583  df-oadd 6586  df-er 6702  df-map 6819  df-pm 6820  df-en 6910  df-dom 6911  df-fin 6912  df-sup 7183  df-inf 7184  df-pnf 8216  df-mnf 8217  df-xr 8218  df-ltxr 8219  df-le 8220  df-sub 8352  df-neg 8353  df-reap 8755  df-ap 8762  df-div 8853  df-inn 9144  df-2 9202  df-3 9203  df-4 9204  df-n0 9403  df-xnn0 9466  df-z 9480  df-uz 9756  df-q 9854  df-rp 9889  df-xneg 10007  df-xadd 10008  df-ioo 10127  df-ico 10129  df-icc 10130  df-fz 10244  df-fzo 10378  df-fl 10531  df-mod 10586  df-seqfrec 10711  df-exp 10802  df-fac 10989  df-bc 11011  df-ihash 11039  df-shft 11377  df-cj 11404  df-re 11405  df-im 11406  df-rsqrt 11560  df-abs 11561  df-clim 11841  df-sumdc 11916  df-ef 12211  df-e 12212  df-dvds 12351  df-gcd 12527  df-prm 12682  df-pc 12860  df-rest 13326  df-topgen 13345  df-psmet 14560  df-xmet 14561  df-met 14562  df-bl 14563  df-mopn 14564  df-top 14725  df-topon 14738  df-bases 14770  df-ntr 14823  df-cn 14915  df-cnp 14916  df-tx 14980  df-cncf 15298  df-limced 15383  df-dvap 15384  df-relog 15585  df-rpcxp 15586  df-sgm 15709

This theorem is referenced by:  perfectlem2  15727