Theorem perfectlem1 15515

Description: Lemma for perfect 15517. (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 9205	. . 3 |- 2 e. NN
2		perfectlem.1	. . . . 5 |- ( ph -> A e. NN )
3	2	nnnn0d 9355	. . . 4 |- ( ph -> A e. NN0 )
4		peano2nn0 9342	. . . 4 |- ( A e. NN0 -> ( A + 1 ) e. NN0 )
5	3, 4	syl 14	. . 3 |- ( ph -> ( A + 1 ) e. NN0 )
6		nnexpcl 10704	. . 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 9113	. . . 4 |- 2 e. RR
9	2	peano2nnd 9058	. . . 4 |- ( ph -> ( A + 1 ) e. NN )
10		1lt2 9213	. . . . 5 |- 1 < 2
11	10	a1i 9	. . . 4 |- ( ph -> 1 < 2 )
12		expgt1 10729	. . . 4 |- ( ( 2 e. RR /\ ( A + 1 ) e. NN /\ 1 < 2 ) -> 1 < ( 2 ^ ( A + 1 ) ) )
13	8, 9, 11, 12	mp3an2i 1355	. . 3 |- ( ph -> 1 < ( 2 ^ ( A + 1 ) ) )
14		1nn 9054	. . . 4 |- 1 e. NN
15		nnsub 9082	. . . 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 9501	. . . . . . 7 |- ( ph -> ( 2 ^ ( A + 1 ) ) e. ZZ )
19		peano2zm 9417	. . . . . . 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 9318	. . . . . . . 8 |- 1 e. NN0
22		perfectlem.2	. . . . . . . 8 |- ( ph -> B e. NN )
23		sgmnncl 15504	. . . . . . . 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 9501	. . . . . 6 |- ( ph -> ( 1 sigma B ) e. ZZ )
26		dvdsmul1 12168	. . . . . 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 9114	. . . . . . . . 9 |- 2 e. CC
29		expp1 10698	. . . . . . . . 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 10704	. . . . . . . . . . 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 9057	. . . . . . . . 9 |- ( ph -> ( 2 ^ A ) e. CC )
34		mulcom 8061	. . . . . . . . 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 2239	. . . . . . 7 |- ( ph -> ( 2 ^ ( A + 1 ) ) = ( 2 x. ( 2 ^ A ) ) )
37	36	oveq1d 5966	. . . . . 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 9057	. . . . . . 7 |- ( ph -> B e. CC )
40	38, 33, 39	mulassd 8103	. . . . . 6 |- ( ph -> ( ( 2 x. ( 2 ^ A ) ) x. B ) = ( 2 x. ( ( 2 ^ A ) x. B ) ) )
41		ax-1cn 8025	. . . . . . . . 9 |- 1 e. CC
42	41	a1i 9	. . . . . . . 8 |- ( ph -> 1 e. CC )
43		perfectlem.3	. . . . . . . . . 10 |- ( ph -> -. 2 || B )
44		2prm 12493	. . . . . . . . . . 11 |- 2 e. Prime
45	22	nnzd 9501	. . . . . . . . . . 11 |- ( ph -> B e. ZZ )
46		coprm 12510	. . . . . . . . . . 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 9407	. . . . . . . . . 10 |- 2 e. ZZ
50		rpexp1i 12520	. . . . . . . . . 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 1355	. . . . . . . . 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 15512	. . . . . . . 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 1252	. . . . . . 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 9057	. . . . . . . . . . . 12 |- ( ph -> A e. CC )
57		pncan 8285	. . . . . . . . . . . 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 5967	. . . . . . . . . 10 |- ( ph -> ( 2 ^ ( ( A + 1 ) - 1 ) ) = ( 2 ^ A ) )
60	59	oveq2d 5967	. . . . . . . . 9 |- ( ph -> ( 1 sigma ( 2 ^ ( ( A + 1 ) - 1 ) ) ) = ( 1 sigma ( 2 ^ A ) ) )
61		1sgm2ppw 15511	. . . . . . . . . 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 2241	. . . . . . . 8 |- ( ph -> ( 1 sigma ( 2 ^ A ) ) = ( ( 2 ^ ( A + 1 ) ) - 1 ) )
64	63	oveq1d 5966	. . . . . . 7 |- ( ph -> ( ( 1 sigma ( 2 ^ A ) ) x. ( 1 sigma B ) ) = ( ( ( 2 ^ ( A + 1 ) ) - 1 ) x. ( 1 sigma B ) ) )
65	54, 55, 64	3eqtr3d 2247	. . . . . 6 |- ( ph -> ( 2 x. ( ( 2 ^ A ) x. B ) ) = ( ( ( 2 ^ ( A + 1 ) ) - 1 ) x. ( 1 sigma B ) ) )
66	37, 40, 65	3eqtrd 2243	. . . . 5 |- ( ph -> ( ( 2 ^ ( A + 1 ) ) x. B ) = ( ( ( 2 ^ ( A + 1 ) ) - 1 ) x. ( 1 sigma B ) ) )
67	27, 66	breqtrrd 4075	. . . 4 |- ( ph -> ( ( 2 ^ ( A + 1 ) ) - 1 ) || ( ( 2 ^ ( A + 1 ) ) x. B ) )
68	20, 18	gcdcomd 12339	. . . . 5 |- ( ph -> ( ( ( 2 ^ ( A + 1 ) ) - 1 ) gcd ( 2 ^ ( A + 1 ) ) ) = ( ( 2 ^ ( A + 1 ) ) gcd ( ( 2 ^ ( A + 1 ) ) - 1 ) ) )
69		iddvdsexp 12170	. . . . . . . . 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 12267	. . . . . . . . . 10 |- -. 2 || 1
72	49	a1i 9	. . . . . . . . . . . 12 |- ( ph -> 2 e. ZZ )
73		1zzd 9406	. . . . . . . . . . . 12 |- ( ph -> 1 e. ZZ )
74	72, 18, 73	3jca 1180	. . . . . . . . . . 11 |- ( ph -> ( 2 e. ZZ /\ ( 2 ^ ( A + 1 ) ) e. ZZ /\ 1 e. ZZ ) )
75		dvdssub2 12190	. . . . . . . . . . 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 677	. . . . . . . . 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 12510	. . . . . . . 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 12520	. . . . . . 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 1355	. . . . . 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 2239	. . . 4 |- ( ph -> ( ( ( 2 ^ ( A + 1 ) ) - 1 ) gcd ( 2 ^ ( A + 1 ) ) ) = 1 )
87		coprmdvds 12458	. . . . 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 1250	. . . 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 12151	. . . 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 1180	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 981   = wceq 1373   e. wcel 2177   class class class wbr 4047  (class class class)co 5951   CCcc 7930   RRcr 7931   1c1 7933   + caddc 7935   x. cmul 7937   < clt 8114   - cmin 8250   / cdiv 8752   NNcn 9043   2c2 9094   NN0cn0 9302   ZZcz 9379   ^cexp 10690   || cdvds 12142   gcd cgcd 12318   Primecprime 12473   sigma csgm 15497

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2179  ax-14 2180  ax-ext 2188  ax-coll 4163  ax-sep 4166  ax-nul 4174  ax-pow 4222  ax-pr 4257  ax-un 4484  ax-setind 4589  ax-iinf 4640  ax-cnex 8023  ax-resscn 8024  ax-1cn 8025  ax-1re 8026  ax-icn 8027  ax-addcl 8028  ax-addrcl 8029  ax-mulcl 8030  ax-mulrcl 8031  ax-addcom 8032  ax-mulcom 8033  ax-addass 8034  ax-mulass 8035  ax-distr 8036  ax-i2m1 8037  ax-0lt1 8038  ax-1rid 8039  ax-0id 8040  ax-rnegex 8041  ax-precex 8042  ax-cnre 8043  ax-pre-ltirr 8044  ax-pre-ltwlin 8045  ax-pre-lttrn 8046  ax-pre-apti 8047  ax-pre-ltadd 8048  ax-pre-mulgt0 8049  ax-pre-mulext 8050  ax-arch 8051  ax-caucvg 8052  ax-pre-suploc 8053  ax-addf 8054  ax-mulf 8055

This theorem depends on definitions:  df-bi 117  df-stab 833  df-dc 837  df-3or 982  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ne 2378  df-nel 2473  df-ral 2490  df-rex 2491  df-reu 2492  df-rmo 2493  df-rab 2494  df-v 2775  df-sbc 3000  df-csb 3095  df-dif 3169  df-un 3171  df-in 3173  df-ss 3180  df-nul 3462  df-if 3573  df-pw 3619  df-sn 3640  df-pr 3641  df-op 3643  df-uni 3853  df-int 3888  df-iun 3931  df-disj 4024  df-br 4048  df-opab 4110  df-mpt 4111  df-tr 4147  df-id 4344  df-po 4347  df-iso 4348  df-iord 4417  df-on 4419  df-ilim 4420  df-suc 4422  df-iom 4643  df-xp 4685  df-rel 4686  df-cnv 4687  df-co 4688  df-dm 4689  df-rn 4690  df-res 4691  df-ima 4692  df-iota 5237  df-fun 5278  df-fn 5279  df-f 5280  df-f1 5281  df-fo 5282  df-f1o 5283  df-fv 5284  df-isom 5285  df-riota 5906  df-ov 5954  df-oprab 5955  df-mpo 5956  df-of 6165  df-1st 6233  df-2nd 6234  df-recs 6398  df-irdg 6463  df-frec 6484  df-1o 6509  df-2o 6510  df-oadd 6513  df-er 6627  df-map 6744  df-pm 6745  df-en 6835  df-dom 6836  df-fin 6837  df-sup 7093  df-inf 7094  df-pnf 8116  df-mnf 8117  df-xr 8118  df-ltxr 8119  df-le 8120  df-sub 8252  df-neg 8253  df-reap 8655  df-ap 8662  df-div 8753  df-inn 9044  df-2 9102  df-3 9103  df-4 9104  df-n0 9303  df-xnn0 9366  df-z 9380  df-uz 9656  df-q 9748  df-rp 9783  df-xneg 9901  df-xadd 9902  df-ioo 10021  df-ico 10023  df-icc 10024  df-fz 10138  df-fzo 10272  df-fl 10420  df-mod 10475  df-seqfrec 10600  df-exp 10691  df-fac 10878  df-bc 10900  df-ihash 10928  df-shft 11170  df-cj 11197  df-re 11198  df-im 11199  df-rsqrt 11353  df-abs 11354  df-clim 11634  df-sumdc 11709  df-ef 12003  df-e 12004  df-dvds 12143  df-gcd 12319  df-prm 12474  df-pc 12652  df-rest 13117  df-topgen 13136  df-psmet 14349  df-xmet 14350  df-met 14351  df-bl 14352  df-mopn 14353  df-top 14514  df-topon 14527  df-bases 14559  df-ntr 14612  df-cn 14704  df-cnp 14705  df-tx 14769  df-cncf 15087  df-limced 15172  df-dvap 15173  df-relog 15374  df-rpcxp 15375  df-sgm 15498

This theorem is referenced by:  perfectlem2  15516