Theorem fmtnofac2 41481

Description: Divisor of Fermat number (Euler's Result refined by François Édouard Anatole Lucas), see fmtnofac1 41482: Let F_n be a Fermat number. Let m be divisor of F_n. Then m is in the form: k*2^(n+2)+1 where k is a nonnegative integer. (Contributed by AV, 30-Jul-2021.)

Assertion

Ref	Expression
fmtnofac2	|- ( ( N e. ( ZZ>= ` 2 ) /\ M e. NN /\ M || (FermatNo ` N ) ) -> E. k e. NN0 M = ( ( k x. ( 2 ^ ( N + 2 ) ) ) + 1 ) )

Distinct variable groups:   k, M   k, N

Proof of Theorem fmtnofac2

Dummy variables x y z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		breq1 4656	. . . . . 6 |- ( x = 1 -> ( x || (FermatNo ` N ) <-> 1 || (FermatNo ` N ) ) )
2	1	anbi2d 740	. . . . 5 |- ( x = 1 -> ( ( N e. ( ZZ>= ` 2 ) /\ x || (FermatNo ` N ) ) <-> ( N e. ( ZZ>= ` 2 ) /\ 1 || (FermatNo ` N ) ) ) )
3		eqeq1 2626	. . . . . 6 |- ( x = 1 -> ( x = ( ( k x. ( 2 ^ ( N + 2 ) ) ) + 1 ) <-> 1 = ( ( k x. ( 2 ^ ( N + 2 ) ) ) + 1 ) ) )
4	3	rexbidv 3052	. . . . 5 |- ( x = 1 -> ( E. k e. NN0 x = ( ( k x. ( 2 ^ ( N + 2 ) ) ) + 1 ) <-> E. k e. NN0 1 = ( ( k x. ( 2 ^ ( N + 2 ) ) ) + 1 ) ) )
5	2, 4	imbi12d 334	. . . 4 |- ( x = 1 -> ( ( ( N e. ( ZZ>= ` 2 ) /\ x || (FermatNo ` N ) ) -> E. k e. NN0 x = ( ( k x. ( 2 ^ ( N + 2 ) ) ) + 1 ) ) <-> ( ( N e. ( ZZ>= ` 2 ) /\ 1 || (FermatNo ` N ) ) -> E. k e. NN0 1 = ( ( k x. ( 2 ^ ( N + 2 ) ) ) + 1 ) ) ) )
6		breq1 4656	. . . . . 6 |- ( x = y -> ( x || (FermatNo ` N ) <-> y || (FermatNo ` N ) ) )
7	6	anbi2d 740	. . . . 5 |- ( x = y -> ( ( N e. ( ZZ>= ` 2 ) /\ x || (FermatNo ` N ) ) <-> ( N e. ( ZZ>= ` 2 ) /\ y || (FermatNo ` N ) ) ) )
8		eqeq1 2626	. . . . . 6 |- ( x = y -> ( x = ( ( k x. ( 2 ^ ( N + 2 ) ) ) + 1 ) <-> y = ( ( k x. ( 2 ^ ( N + 2 ) ) ) + 1 ) ) )
9	8	rexbidv 3052	. . . . 5 |- ( x = y -> ( E. k e. NN0 x = ( ( k x. ( 2 ^ ( N + 2 ) ) ) + 1 ) <-> E. k e. NN0 y = ( ( k x. ( 2 ^ ( N + 2 ) ) ) + 1 ) ) )
10	7, 9	imbi12d 334	. . . 4 |- ( x = y -> ( ( ( N e. ( ZZ>= ` 2 ) /\ x || (FermatNo ` N ) ) -> E. k e. NN0 x = ( ( k x. ( 2 ^ ( N + 2 ) ) ) + 1 ) ) <-> ( ( N e. ( ZZ>= ` 2 ) /\ y || (FermatNo ` N ) ) -> E. k e. NN0 y = ( ( k x. ( 2 ^ ( N + 2 ) ) ) + 1 ) ) ) )
11		breq1 4656	. . . . . 6 |- ( x = z -> ( x || (FermatNo ` N ) <-> z || (FermatNo ` N ) ) )
12	11	anbi2d 740	. . . . 5 |- ( x = z -> ( ( N e. ( ZZ>= ` 2 ) /\ x || (FermatNo ` N ) ) <-> ( N e. ( ZZ>= ` 2 ) /\ z || (FermatNo ` N ) ) ) )
13		eqeq1 2626	. . . . . 6 |- ( x = z -> ( x = ( ( k x. ( 2 ^ ( N + 2 ) ) ) + 1 ) <-> z = ( ( k x. ( 2 ^ ( N + 2 ) ) ) + 1 ) ) )
14	13	rexbidv 3052	. . . . 5 |- ( x = z -> ( E. k e. NN0 x = ( ( k x. ( 2 ^ ( N + 2 ) ) ) + 1 ) <-> E. k e. NN0 z = ( ( k x. ( 2 ^ ( N + 2 ) ) ) + 1 ) ) )
15	12, 14	imbi12d 334	. . . 4 |- ( x = z -> ( ( ( N e. ( ZZ>= ` 2 ) /\ x || (FermatNo ` N ) ) -> E. k e. NN0 x = ( ( k x. ( 2 ^ ( N + 2 ) ) ) + 1 ) ) <-> ( ( N e. ( ZZ>= ` 2 ) /\ z || (FermatNo ` N ) ) -> E. k e. NN0 z = ( ( k x. ( 2 ^ ( N + 2 ) ) ) + 1 ) ) ) )
16		breq1 4656	. . . . . 6 |- ( x = ( y x. z ) -> ( x || (FermatNo ` N ) <-> ( y x. z ) || (FermatNo ` N ) ) )
17	16	anbi2d 740	. . . . 5 |- ( x = ( y x. z ) -> ( ( N e. ( ZZ>= ` 2 ) /\ x || (FermatNo ` N ) ) <-> ( N e. ( ZZ>= ` 2 ) /\ ( y x. z ) || (FermatNo ` N ) ) ) )
18		eqeq1 2626	. . . . . 6 |- ( x = ( y x. z ) -> ( x = ( ( k x. ( 2 ^ ( N + 2 ) ) ) + 1 ) <-> ( y x. z ) = ( ( k x. ( 2 ^ ( N + 2 ) ) ) + 1 ) ) )
19	18	rexbidv 3052	. . . . 5 |- ( x = ( y x. z ) -> ( E. k e. NN0 x = ( ( k x. ( 2 ^ ( N + 2 ) ) ) + 1 ) <-> E. k e. NN0 ( y x. z ) = ( ( k x. ( 2 ^ ( N + 2 ) ) ) + 1 ) ) )
20	17, 19	imbi12d 334	. . . 4 |- ( x = ( y x. z ) -> ( ( ( N e. ( ZZ>= ` 2 ) /\ x || (FermatNo ` N ) ) -> E. k e. NN0 x = ( ( k x. ( 2 ^ ( N + 2 ) ) ) + 1 ) ) <-> ( ( N e. ( ZZ>= ` 2 ) /\ ( y x. z ) || (FermatNo ` N ) ) -> E. k e. NN0 ( y x. z ) = ( ( k x. ( 2 ^ ( N + 2 ) ) ) + 1 ) ) ) )
21		breq1 4656	. . . . . 6 |- ( x = M -> ( x || (FermatNo ` N ) <-> M || (FermatNo ` N ) ) )
22	21	anbi2d 740	. . . . 5 |- ( x = M -> ( ( N e. ( ZZ>= ` 2 ) /\ x || (FermatNo ` N ) ) <-> ( N e. ( ZZ>= ` 2 ) /\ M || (FermatNo ` N ) ) ) )
23		eqeq1 2626	. . . . . 6 |- ( x = M -> ( x = ( ( k x. ( 2 ^ ( N + 2 ) ) ) + 1 ) <-> M = ( ( k x. ( 2 ^ ( N + 2 ) ) ) + 1 ) ) )
24	23	rexbidv 3052	. . . . 5 |- ( x = M -> ( E. k e. NN0 x = ( ( k x. ( 2 ^ ( N + 2 ) ) ) + 1 ) <-> E. k e. NN0 M = ( ( k x. ( 2 ^ ( N + 2 ) ) ) + 1 ) ) )
25	22, 24	imbi12d 334	. . . 4 |- ( x = M -> ( ( ( N e. ( ZZ>= ` 2 ) /\ x || (FermatNo ` N ) ) -> E. k e. NN0 x = ( ( k x. ( 2 ^ ( N + 2 ) ) ) + 1 ) ) <-> ( ( N e. ( ZZ>= ` 2 ) /\ M || (FermatNo ` N ) ) -> E. k e. NN0 M = ( ( k x. ( 2 ^ ( N + 2 ) ) ) + 1 ) ) ) )
26		0nn0 11307	. . . . . . 7 |- 0 e. NN0
27	26	a1i 11	. . . . . 6 |- ( N e. ( ZZ>= ` 2 ) -> 0 e. NN0 )
28		oveq1 6657	. . . . . . . . 9 |- ( k = 0 -> ( k x. ( 2 ^ ( N + 2 ) ) ) = ( 0 x. ( 2 ^ ( N + 2 ) ) ) )
29	28	oveq1d 6665	. . . . . . . 8 |- ( k = 0 -> ( ( k x. ( 2 ^ ( N + 2 ) ) ) + 1 ) = ( ( 0 x. ( 2 ^ ( N + 2 ) ) ) + 1 ) )
30	29	eqeq2d 2632	. . . . . . 7 |- ( k = 0 -> ( 1 = ( ( k x. ( 2 ^ ( N + 2 ) ) ) + 1 ) <-> 1 = ( ( 0 x. ( 2 ^ ( N + 2 ) ) ) + 1 ) ) )
31	30	adantl 482	. . . . . 6 |- ( ( N e. ( ZZ>= ` 2 ) /\ k = 0 ) -> ( 1 = ( ( k x. ( 2 ^ ( N + 2 ) ) ) + 1 ) <-> 1 = ( ( 0 x. ( 2 ^ ( N + 2 ) ) ) + 1 ) ) )
32		2nn0 11309	. . . . . . . . . . . 12 |- 2 e. NN0
33	32	a1i 11	. . . . . . . . . . 11 |- ( N e. ( ZZ>= ` 2 ) -> 2 e. NN0 )
34		eluzge2nn0 11727	. . . . . . . . . . . 12 |- ( N e. ( ZZ>= ` 2 ) -> N e. NN0 )
35	34, 33	nn0addcld 11355	. . . . . . . . . . 11 |- ( N e. ( ZZ>= ` 2 ) -> ( N + 2 ) e. NN0 )
36	33, 35	nn0expcld 13031	. . . . . . . . . 10 |- ( N e. ( ZZ>= ` 2 ) -> ( 2 ^ ( N + 2 ) ) e. NN0 )
37	36	nn0cnd 11353	. . . . . . . . 9 |- ( N e. ( ZZ>= ` 2 ) -> ( 2 ^ ( N + 2 ) ) e. CC )
38	37	mul02d 10234	. . . . . . . 8 |- ( N e. ( ZZ>= ` 2 ) -> ( 0 x. ( 2 ^ ( N + 2 ) ) ) = 0 )
39	38	oveq1d 6665	. . . . . . 7 |- ( N e. ( ZZ>= ` 2 ) -> ( ( 0 x. ( 2 ^ ( N + 2 ) ) ) + 1 ) = ( 0 + 1 ) )
40		0p1e1 11132	. . . . . . 7 |- ( 0 + 1 ) = 1
41	39, 40	syl6req 2673	. . . . . 6 |- ( N e. ( ZZ>= ` 2 ) -> 1 = ( ( 0 x. ( 2 ^ ( N + 2 ) ) ) + 1 ) )
42	27, 31, 41	rspcedvd 3317	. . . . 5 |- ( N e. ( ZZ>= ` 2 ) -> E. k e. NN0 1 = ( ( k x. ( 2 ^ ( N + 2 ) ) ) + 1 ) )
43	42	adantr 481	. . . 4 |- ( ( N e. ( ZZ>= ` 2 ) /\ 1 || (FermatNo ` N ) ) -> E. k e. NN0 1 = ( ( k x. ( 2 ^ ( N + 2 ) ) ) + 1 ) )
44		simpl 473	. . . . . . 7 |- ( ( N e. ( ZZ>= ` 2 ) /\ x || (FermatNo ` N ) ) -> N e. ( ZZ>= ` 2 ) )
45	44	adantl 482	. . . . . 6 |- ( ( x e. Prime /\ ( N e. ( ZZ>= ` 2 ) /\ x || (FermatNo ` N ) ) ) -> N e. ( ZZ>= ` 2 ) )
46		simpl 473	. . . . . 6 |- ( ( x e. Prime /\ ( N e. ( ZZ>= ` 2 ) /\ x || (FermatNo ` N ) ) ) -> x e. Prime )
47		simprr 796	. . . . . 6 |- ( ( x e. Prime /\ ( N e. ( ZZ>= ` 2 ) /\ x || (FermatNo ` N ) ) ) -> x || (FermatNo ` N ) )
48		nnssnn0 11295	. . . . . . 7 |- NN C_ NN0
49		fmtnoprmfac2 41479	. . . . . . 7 |- ( ( N e. ( ZZ>= ` 2 ) /\ x e. Prime /\ x || (FermatNo ` N ) ) -> E. k e. NN x = ( ( k x. ( 2 ^ ( N + 2 ) ) ) + 1 ) )
50		ssrexv 3667	. . . . . . 7 |- ( NN C_ NN0 -> ( E. k e. NN x = ( ( k x. ( 2 ^ ( N + 2 ) ) ) + 1 ) -> E. k e. NN0 x = ( ( k x. ( 2 ^ ( N + 2 ) ) ) + 1 ) ) )
51	48, 49, 50	mpsyl 68	. . . . . 6 |- ( ( N e. ( ZZ>= ` 2 ) /\ x e. Prime /\ x || (FermatNo ` N ) ) -> E. k e. NN0 x = ( ( k x. ( 2 ^ ( N + 2 ) ) ) + 1 ) )
52	45, 46, 47, 51	syl3anc 1326	. . . . 5 |- ( ( x e. Prime /\ ( N e. ( ZZ>= ` 2 ) /\ x || (FermatNo ` N ) ) ) -> E. k e. NN0 x = ( ( k x. ( 2 ^ ( N + 2 ) ) ) + 1 ) )
53	52	ex 450	. . . 4 |- ( x e. Prime -> ( ( N e. ( ZZ>= ` 2 ) /\ x || (FermatNo ` N ) ) -> E. k e. NN0 x = ( ( k x. ( 2 ^ ( N + 2 ) ) ) + 1 ) ) )
54		fmtnofac2lem 41480	. . . 4 |- ( ( y e. ( ZZ>= ` 2 ) /\ z e. ( ZZ>= ` 2 ) ) -> ( ( ( ( N e. ( ZZ>= ` 2 ) /\ y || (FermatNo ` N ) ) -> E. k e. NN0 y = ( ( k x. ( 2 ^ ( N + 2 ) ) ) + 1 ) ) /\ ( ( N e. ( ZZ>= ` 2 ) /\ z || (FermatNo ` N ) ) -> E. k e. NN0 z = ( ( k x. ( 2 ^ ( N + 2 ) ) ) + 1 ) ) ) -> ( ( N e. ( ZZ>= ` 2 ) /\ ( y x. z ) || (FermatNo ` N ) ) -> E. k e. NN0 ( y x. z ) = ( ( k x. ( 2 ^ ( N + 2 ) ) ) + 1 ) ) ) )
55	5, 10, 15, 20, 25, 43, 53, 54	prmind 15399	. . 3 |- ( M e. NN -> ( ( N e. ( ZZ>= ` 2 ) /\ M || (FermatNo ` N ) ) -> E. k e. NN0 M = ( ( k x. ( 2 ^ ( N + 2 ) ) ) + 1 ) ) )
56	55	expd 452	. 2 |- ( M e. NN -> ( N e. ( ZZ>= ` 2 ) -> ( M || (FermatNo ` N ) -> E. k e. NN0 M = ( ( k x. ( 2 ^ ( N + 2 ) ) ) + 1 ) ) ) )
57	56	3imp21 1277	1 |- ( ( N e. ( ZZ>= ` 2 ) /\ M e. NN /\ M || (FermatNo ` N ) ) -> E. k e. NN0 M = ( ( k x. ( 2 ^ ( N + 2 ) ) ) + 1 ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   E.wrex 2913   C_ wss 3574   class class class wbr 4653   ` cfv 5888  (class class class)co 6650   0cc0 9936   1c1 9937   + caddc 9939   x. cmul 9941   NNcn 11020   2c2 11070   NN0cn0 11292   ZZ>=cuz 11687   ^cexp 12860   || cdvds 14983   Primecprime 15385  FermatNocfmtno 41439

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-inf2 8538  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-fal 1489  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-se 5074  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-isom 5897  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-2o 7561  df-oadd 7564  df-er 7742  df-map 7859  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-sup 8348  df-inf 8349  df-oi 8415  df-card 8765  df-cda 8990  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-3 11080  df-4 11081  df-5 11082  df-6 11083  df-7 11084  df-8 11085  df-n0 11293  df-xnn0 11364  df-z 11378  df-uz 11688  df-q 11789  df-rp 11833  df-ioo 12179  df-ico 12181  df-fz 12327  df-fzo 12466  df-fl 12593  df-mod 12669  df-seq 12802  df-exp 12861  df-fac 13061  df-hash 13118  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-clim 14219  df-prod 14636  df-dvds 14984  df-gcd 15217  df-prm 15386  df-odz 15470  df-phi 15471  df-pc 15542  df-lgs 25020  df-fmtno 41440

This theorem is referenced by:  fmtnofac1  41482