Theorem perfectlem1 15854

Description: Lemma for perfect 15856. (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 9395	. . 3 |- 2 e. NN
2		perfectlem.1	. . . . 5 |- ( ph -> A e. NN )
3	2	nnnn0d 9549	. . . 4 |- ( ph -> A e. NN0 )
4		peano2nn0 9532	. . . 4 |- ( A e. NN0 -> ( A + 1 ) e. NN0 )
5	3, 4	syl 14	. . 3 |- ( ph -> ( A + 1 ) e. NN0 )
6		nnexpcl 10910	. . 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 9303	. . . 4 |- 2 e. RR
9	2	peano2nnd 9248	. . . 4 |- ( ph -> ( A + 1 ) e. NN )
10		1lt2 9403	. . . . 5 |- 1 < 2
11	10	a1i 9	. . . 4 |- ( ph -> 1 < 2 )
12		expgt1 10935	. . . 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 9244	. . . 4 |- 1 e. NN
15		nnsub 9272	. . . 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 9695	. . . . . . 7 |- ( ph -> ( 2 ^ ( A + 1 ) ) e. ZZ )
19		peano2zm 9611	. . . . . . 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 9508	. . . . . . . 8 |- 1 e. NN0
22		perfectlem.2	. . . . . . . 8 |- ( ph -> B e. NN )
23		sgmnncl 15843	. . . . . . . 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 9695	. . . . . 6 |- ( ph -> ( 1 sigma B ) e. ZZ )
26		dvdsmul1 12492	. . . . . 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 9304	. . . . . . . . 9 |- 2 e. CC
29		expp1 10904	. . . . . . . . 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 10910	. . . . . . . . . . 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 9247	. . . . . . . . 9 |- ( ph -> ( 2 ^ A ) e. CC )
34		mulcom 8252	. . . . . . . . 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 2265	. . . . . . 7 |- ( ph -> ( 2 ^ ( A + 1 ) ) = ( 2 x. ( 2 ^ A ) ) )
37	36	oveq1d 6064	. . . . . 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 9247	. . . . . . 7 |- ( ph -> B e. CC )
40	38, 33, 39	mulassd 8293	. . . . . 6 |- ( ph -> ( ( 2 x. ( 2 ^ A ) ) x. B ) = ( 2 x. ( ( 2 ^ A ) x. B ) ) )
41		ax-1cn 8216	. . . . . . . . 9 |- 1 e. CC
42	41	a1i 9	. . . . . . . 8 |- ( ph -> 1 e. CC )
43		perfectlem.3	. . . . . . . . . 10 |- ( ph -> -. 2 || B )
44		2prm 12817	. . . . . . . . . . 11 |- 2 e. Prime
45	22	nnzd 9695	. . . . . . . . . . 11 |- ( ph -> B e. ZZ )
46		coprm 12834	. . . . . . . . . . 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 9601	. . . . . . . . . 10 |- 2 e. ZZ
50		rpexp1i 12844	. . . . . . . . . 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 15851	. . . . . . . 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 9247	. . . . . . . . . . . 12 |- ( ph -> A e. CC )
57		pncan 8475	. . . . . . . . . . . 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 6065	. . . . . . . . . 10 |- ( ph -> ( 2 ^ ( ( A + 1 ) - 1 ) ) = ( 2 ^ A ) )
60	59	oveq2d 6065	. . . . . . . . 9 |- ( ph -> ( 1 sigma ( 2 ^ ( ( A + 1 ) - 1 ) ) ) = ( 1 sigma ( 2 ^ A ) ) )
61		1sgm2ppw 15850	. . . . . . . . . 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 2267	. . . . . . . 8 |- ( ph -> ( 1 sigma ( 2 ^ A ) ) = ( ( 2 ^ ( A + 1 ) ) - 1 ) )
64	63	oveq1d 6064	. . . . . . 7 |- ( ph -> ( ( 1 sigma ( 2 ^ A ) ) x. ( 1 sigma B ) ) = ( ( ( 2 ^ ( A + 1 ) ) - 1 ) x. ( 1 sigma B ) ) )
65	54, 55, 64	3eqtr3d 2273	. . . . . 6 |- ( ph -> ( 2 x. ( ( 2 ^ A ) x. B ) ) = ( ( ( 2 ^ ( A + 1 ) ) - 1 ) x. ( 1 sigma B ) ) )
66	37, 40, 65	3eqtrd 2269	. . . . 5 |- ( ph -> ( ( 2 ^ ( A + 1 ) ) x. B ) = ( ( ( 2 ^ ( A + 1 ) ) - 1 ) x. ( 1 sigma B ) ) )
67	27, 66	breqtrrd 4136	. . . 4 |- ( ph -> ( ( 2 ^ ( A + 1 ) ) - 1 ) || ( ( 2 ^ ( A + 1 ) ) x. B ) )
68	20, 18	gcdcomd 12663	. . . . 5 |- ( ph -> ( ( ( 2 ^ ( A + 1 ) ) - 1 ) gcd ( 2 ^ ( A + 1 ) ) ) = ( ( 2 ^ ( A + 1 ) ) gcd ( ( 2 ^ ( A + 1 ) ) - 1 ) ) )
69		iddvdsexp 12494	. . . . . . . . 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 12591	. . . . . . . . . 10 |- -. 2 || 1
72	49	a1i 9	. . . . . . . . . . . 12 |- ( ph -> 2 e. ZZ )
73		1zzd 9600	. . . . . . . . . . . 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 12514	. . . . . . . . . . 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 12834	. . . . . . . 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 12844	. . . . . . 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 2265	. . . 4 |- ( ph -> ( ( ( 2 ^ ( A + 1 ) ) - 1 ) gcd ( 2 ^ ( A + 1 ) ) ) = 1 )
87		coprmdvds 12782	. . . . 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 12475	. . . 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 2203   class class class wbr 4108  (class class class)co 6049   CCcc 8121   RRcr 8122   1c1 8124   + caddc 8126   x. cmul 8128   < clt 8304   - cmin 8440   / cdiv 8942   NNcn 9233   2c2 9284   NN0cn0 9492   ZZcz 9573   ^cexp 10896   || cdvds 12466   gcd cgcd 12642   Primecprime 12797   sigma csgm 15836

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 2205  ax-14 2206  ax-ext 2214  ax-coll 4224  ax-sep 4227  ax-nul 4235  ax-pow 4286  ax-pr 4321  ax-un 4553  ax-setind 4658  ax-iinf 4709  ax-cnex 8214  ax-resscn 8215  ax-1cn 8216  ax-1re 8217  ax-icn 8218  ax-addcl 8219  ax-addrcl 8220  ax-mulcl 8221  ax-mulrcl 8222  ax-addcom 8223  ax-mulcom 8224  ax-addass 8225  ax-mulass 8226  ax-distr 8227  ax-i2m1 8228  ax-0lt1 8229  ax-1rid 8230  ax-0id 8231  ax-rnegex 8232  ax-precex 8233  ax-cnre 8234  ax-pre-ltirr 8235  ax-pre-ltwlin 8236  ax-pre-lttrn 8237  ax-pre-apti 8238  ax-pre-ltadd 8239  ax-pre-mulgt0 8240  ax-pre-mulext 8241  ax-arch 8242  ax-caucvg 8243  ax-pre-suploc 8244  ax-addf 8245  ax-mulf 8246

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 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-nel 2508  df-ral 2525  df-rex 2526  df-reu 2527  df-rmo 2528  df-rab 2529  df-v 2814  df-sbc 3042  df-csb 3138  df-dif 3212  df-un 3214  df-in 3216  df-ss 3223  df-nul 3508  df-if 3620  df-pw 3670  df-sn 3694  df-pr 3695  df-op 3697  df-uni 3914  df-int 3949  df-iun 3992  df-disj 4085  df-br 4109  df-opab 4171  df-mpt 4172  df-tr 4208  df-id 4413  df-po 4416  df-iso 4417  df-iord 4486  df-on 4488  df-ilim 4489  df-suc 4491  df-iom 4712  df-xp 4754  df-rel 4755  df-cnv 4756  df-co 4757  df-dm 4758  df-rn 4759  df-res 4760  df-ima 4761  df-iota 5311  df-fun 5353  df-fn 5354  df-f 5355  df-f1 5356  df-fo 5357  df-f1o 5358  df-fv 5359  df-isom 5360  df-riota 6002  df-ov 6052  df-oprab 6053  df-mpo 6054  df-of 6265  df-1st 6333  df-2nd 6334  df-recs 6535  df-irdg 6600  df-frec 6621  df-1o 6646  df-2o 6647  df-oadd 6650  df-er 6766  df-map 6883  df-pm 6884  df-en 6975  df-dom 6976  df-fin 6977  df-sup 7274  df-inf 7275  df-pnf 8306  df-mnf 8307  df-xr 8308  df-ltxr 8309  df-le 8310  df-sub 8442  df-neg 8443  df-reap 8845  df-ap 8852  df-div 8943  df-inn 9234  df-2 9292  df-3 9293  df-4 9294  df-n0 9493  df-xnn0 9560  df-z 9574  df-uz 9850  df-q 9948  df-rp 9983  df-xneg 10101  df-xadd 10102  df-ioo 10221  df-ico 10223  df-icc 10224  df-fz 10339  df-fzo 10473  df-fl 10626  df-mod 10681  df-seqfrec 10806  df-exp 10897  df-fac 11084  df-bc 11106  df-ihash 11134  df-shft 11493  df-cj 11520  df-re 11521  df-im 11522  df-rsqrt 11676  df-abs 11677  df-clim 11957  df-sumdc 12032  df-ef 12327  df-e 12328  df-dvds 12467  df-gcd 12643  df-prm 12798  df-pc 12976  df-rest 13443  df-topgen 13462  df-psmet 14678  df-xmet 14679  df-met 14680  df-bl 14681  df-mopn 14682  df-top 14850  df-topon 14863  df-bases 14895  df-ntr 14948  df-cn 15040  df-cnp 15041  df-tx 15105  df-cncf 15423  df-limced 15508  df-dvap 15509  df-relog 15710  df-rpcxp 15711  df-sgm 15837

This theorem is referenced by:  perfectlem2  15855