Theorem fmtnofac1 41482

Description: Divisor of Fermat number (Euler's Result), see ProofWiki "Divisor of Fermat Number/Euler's Result", 24-Jul-2021, https://proofwiki.org/wiki/Divisor_of_Fermat_Number/Euler's_Result): "Let F_n be a Fermat number. Let m be divisor of F_n. Then m is in the form: k*2^(n+1)+1 where k is a positive integer." Here, however, k must be a nonnegative integer, because k must be 0 to represent 1 (which is a divisor of F_n ).

Historical Note: In 1747, Leonhard Paul Euler proved that a divisor of a Fermat number F_n is always in the form kx2^(n+1)+1. This was later refined to k*2^(n+2)+1 by François Édouard Anatole Lucas, see fmtnofac2 41481. (Contributed by AV, 30-Jul-2021.)

Assertion

Ref	Expression
fmtnofac1	|- ( ( N e. NN /\ M e. NN /\ M || (FermatNo ` N ) ) -> E. k e. NN0 M = ( ( k x. ( 2 ^ ( N + 1 ) ) ) + 1 ) )

Distinct variable groups:   k, M   k, N

Proof of Theorem fmtnofac1

Dummy variable n is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elnn1uz2 11765	. . 3 |- ( N e. NN <-> ( N = 1 \/ N e. ( ZZ>= ` 2 ) ) )
2		5prm 15815	. . . . . . 7 |- 5 e. Prime
3		dvdsprime 15400	. . . . . . 7 |- ( ( 5 e. Prime /\ M e. NN ) -> ( M || 5 <-> ( M = 5 \/ M = 1 ) ) )
4	2, 3	mpan 706	. . . . . 6 |- ( M e. NN -> ( M || 5 <-> ( M = 5 \/ M = 1 ) ) )
5		1nn0 11308	. . . . . . . . 9 |- 1 e. NN0
6	5	a1i 11	. . . . . . . 8 |- ( M = 5 -> 1 e. NN0 )
7		simpl 473	. . . . . . . . 9 |- ( ( M = 5 /\ k = 1 ) -> M = 5 )
8		oveq1 6657	. . . . . . . . . . 11 |- ( k = 1 -> ( k x. 4 ) = ( 1 x. 4 ) )
9	8	oveq1d 6665	. . . . . . . . . 10 |- ( k = 1 -> ( ( k x. 4 ) + 1 ) = ( ( 1 x. 4 ) + 1 ) )
10	9	adantl 482	. . . . . . . . 9 |- ( ( M = 5 /\ k = 1 ) -> ( ( k x. 4 ) + 1 ) = ( ( 1 x. 4 ) + 1 ) )
11	7, 10	eqeq12d 2637	. . . . . . . 8 |- ( ( M = 5 /\ k = 1 ) -> ( M = ( ( k x. 4 ) + 1 ) <-> 5 = ( ( 1 x. 4 ) + 1 ) ) )
12		df-5 11082	. . . . . . . . . 10 |- 5 = ( 4 + 1 )
13		4cn 11098	. . . . . . . . . . . . 13 |- 4 e. CC
14	13	mulid2i 10043	. . . . . . . . . . . 12 |- ( 1 x. 4 ) = 4
15	14	eqcomi 2631	. . . . . . . . . . 11 |- 4 = ( 1 x. 4 )
16	15	oveq1i 6660	. . . . . . . . . 10 |- ( 4 + 1 ) = ( ( 1 x. 4 ) + 1 )
17	12, 16	eqtri 2644	. . . . . . . . 9 |- 5 = ( ( 1 x. 4 ) + 1 )
18	17	a1i 11	. . . . . . . 8 |- ( M = 5 -> 5 = ( ( 1 x. 4 ) + 1 ) )
19	6, 11, 18	rspcedvd 3317	. . . . . . 7 |- ( M = 5 -> E. k e. NN0 M = ( ( k x. 4 ) + 1 ) )
20		0nn0 11307	. . . . . . . . 9 |- 0 e. NN0
21	20	a1i 11	. . . . . . . 8 |- ( M = 1 -> 0 e. NN0 )
22		simpl 473	. . . . . . . . 9 |- ( ( M = 1 /\ k = 0 ) -> M = 1 )
23		oveq1 6657	. . . . . . . . . . 11 |- ( k = 0 -> ( k x. 4 ) = ( 0 x. 4 ) )
24	23	oveq1d 6665	. . . . . . . . . 10 |- ( k = 0 -> ( ( k x. 4 ) + 1 ) = ( ( 0 x. 4 ) + 1 ) )
25	24	adantl 482	. . . . . . . . 9 |- ( ( M = 1 /\ k = 0 ) -> ( ( k x. 4 ) + 1 ) = ( ( 0 x. 4 ) + 1 ) )
26	22, 25	eqeq12d 2637	. . . . . . . 8 |- ( ( M = 1 /\ k = 0 ) -> ( M = ( ( k x. 4 ) + 1 ) <-> 1 = ( ( 0 x. 4 ) + 1 ) ) )
27	13	mul02i 10225	. . . . . . . . . . . 12 |- ( 0 x. 4 ) = 0
28	27	oveq1i 6660	. . . . . . . . . . 11 |- ( ( 0 x. 4 ) + 1 ) = ( 0 + 1 )
29		0p1e1 11132	. . . . . . . . . . 11 |- ( 0 + 1 ) = 1
30	28, 29	eqtri 2644	. . . . . . . . . 10 |- ( ( 0 x. 4 ) + 1 ) = 1
31	30	eqcomi 2631	. . . . . . . . 9 |- 1 = ( ( 0 x. 4 ) + 1 )
32	31	a1i 11	. . . . . . . 8 |- ( M = 1 -> 1 = ( ( 0 x. 4 ) + 1 ) )
33	21, 26, 32	rspcedvd 3317	. . . . . . 7 |- ( M = 1 -> E. k e. NN0 M = ( ( k x. 4 ) + 1 ) )
34	19, 33	jaoi 394	. . . . . 6 |- ( ( M = 5 \/ M = 1 ) -> E. k e. NN0 M = ( ( k x. 4 ) + 1 ) )
35	4, 34	syl6bi 243	. . . . 5 |- ( M e. NN -> ( M || 5 -> E. k e. NN0 M = ( ( k x. 4 ) + 1 ) ) )
36		fveq2 6191	. . . . . . . 8 |- ( N = 1 -> (FermatNo ` N ) = (FermatNo ` 1 ) )
37		fmtno1 41453	. . . . . . . 8 |- (FermatNo ` 1 ) = 5
38	36, 37	syl6eq 2672	. . . . . . 7 |- ( N = 1 -> (FermatNo ` N ) = 5 )
39	38	breq2d 4665	. . . . . 6 |- ( N = 1 -> ( M || (FermatNo ` N ) <-> M || 5 ) )
40		oveq1 6657	. . . . . . . . . . . . 13 |- ( N = 1 -> ( N + 1 ) = ( 1 + 1 ) )
41		1p1e2 11134	. . . . . . . . . . . . 13 |- ( 1 + 1 ) = 2
42	40, 41	syl6eq 2672	. . . . . . . . . . . 12 |- ( N = 1 -> ( N + 1 ) = 2 )
43	42	oveq2d 6666	. . . . . . . . . . 11 |- ( N = 1 -> ( 2 ^ ( N + 1 ) ) = ( 2 ^ 2 ) )
44		sq2 12960	. . . . . . . . . . 11 |- ( 2 ^ 2 ) = 4
45	43, 44	syl6eq 2672	. . . . . . . . . 10 |- ( N = 1 -> ( 2 ^ ( N + 1 ) ) = 4 )
46	45	oveq2d 6666	. . . . . . . . 9 |- ( N = 1 -> ( k x. ( 2 ^ ( N + 1 ) ) ) = ( k x. 4 ) )
47	46	oveq1d 6665	. . . . . . . 8 |- ( N = 1 -> ( ( k x. ( 2 ^ ( N + 1 ) ) ) + 1 ) = ( ( k x. 4 ) + 1 ) )
48	47	eqeq2d 2632	. . . . . . 7 |- ( N = 1 -> ( M = ( ( k x. ( 2 ^ ( N + 1 ) ) ) + 1 ) <-> M = ( ( k x. 4 ) + 1 ) ) )
49	48	rexbidv 3052	. . . . . 6 |- ( N = 1 -> ( E. k e. NN0 M = ( ( k x. ( 2 ^ ( N + 1 ) ) ) + 1 ) <-> E. k e. NN0 M = ( ( k x. 4 ) + 1 ) ) )
50	39, 49	imbi12d 334	. . . . 5 |- ( N = 1 -> ( ( M || (FermatNo ` N ) -> E. k e. NN0 M = ( ( k x. ( 2 ^ ( N + 1 ) ) ) + 1 ) ) <-> ( M || 5 -> E. k e. NN0 M = ( ( k x. 4 ) + 1 ) ) ) )
51	35, 50	syl5ibr 236	. . . 4 |- ( N = 1 -> ( M e. NN -> ( M || (FermatNo ` N ) -> E. k e. NN0 M = ( ( k x. ( 2 ^ ( N + 1 ) ) ) + 1 ) ) ) )
52		fmtnofac2 41481	. . . . . 6 |- ( ( N e. ( ZZ>= ` 2 ) /\ M e. NN /\ M || (FermatNo ` N ) ) -> E. n e. NN0 M = ( ( n x. ( 2 ^ ( N + 2 ) ) ) + 1 ) )
53		id 22	. . . . . . . . . . . 12 |- ( n e. NN0 -> n e. NN0 )
54		2nn0 11309	. . . . . . . . . . . . 13 |- 2 e. NN0
55	54	a1i 11	. . . . . . . . . . . 12 |- ( n e. NN0 -> 2 e. NN0 )
56	53, 55	nn0mulcld 11356	. . . . . . . . . . 11 |- ( n e. NN0 -> ( n x. 2 ) e. NN0 )
57	56	adantl 482	. . . . . . . . . 10 |- ( ( ( N e. ( ZZ>= ` 2 ) /\ M e. NN /\ M || (FermatNo ` N ) ) /\ n e. NN0 ) -> ( n x. 2 ) e. NN0 )
58	57	adantr 481	. . . . . . . . 9 |- ( ( ( ( N e. ( ZZ>= ` 2 ) /\ M e. NN /\ M || (FermatNo ` N ) ) /\ n e. NN0 ) /\ M = ( ( n x. ( 2 ^ ( N + 2 ) ) ) + 1 ) ) -> ( n x. 2 ) e. NN0 )
59		simpr 477	. . . . . . . . . 10 |- ( ( ( ( N e. ( ZZ>= ` 2 ) /\ M e. NN /\ M || (FermatNo ` N ) ) /\ n e. NN0 ) /\ M = ( ( n x. ( 2 ^ ( N + 2 ) ) ) + 1 ) ) -> M = ( ( n x. ( 2 ^ ( N + 2 ) ) ) + 1 ) )
60		oveq1 6657	. . . . . . . . . . 11 |- ( k = ( n x. 2 ) -> ( k x. ( 2 ^ ( N + 1 ) ) ) = ( ( n x. 2 ) x. ( 2 ^ ( N + 1 ) ) ) )
61	60	oveq1d 6665	. . . . . . . . . 10 |- ( k = ( n x. 2 ) -> ( ( k x. ( 2 ^ ( N + 1 ) ) ) + 1 ) = ( ( ( n x. 2 ) x. ( 2 ^ ( N + 1 ) ) ) + 1 ) )
62	59, 61	eqeqan12d 2638	. . . . . . . . 9 |- ( ( ( ( ( N e. ( ZZ>= ` 2 ) /\ M e. NN /\ M || (FermatNo ` N ) ) /\ n e. NN0 ) /\ M = ( ( n x. ( 2 ^ ( N + 2 ) ) ) + 1 ) ) /\ k = ( n x. 2 ) ) -> ( M = ( ( k x. ( 2 ^ ( N + 1 ) ) ) + 1 ) <-> ( ( n x. ( 2 ^ ( N + 2 ) ) ) + 1 ) = ( ( ( n x. 2 ) x. ( 2 ^ ( N + 1 ) ) ) + 1 ) ) )
63		eluzge2nn0 11727	. . . . . . . . . . . . . . . . . . . 20 |- ( N e. ( ZZ>= ` 2 ) -> N e. NN0 )
64	63	nn0cnd 11353	. . . . . . . . . . . . . . . . . . 19 |- ( N e. ( ZZ>= ` 2 ) -> N e. CC )
65		add1p1 11283	. . . . . . . . . . . . . . . . . . 19 |- ( N e. CC -> ( ( N + 1 ) + 1 ) = ( N + 2 ) )
66	64, 65	syl 17	. . . . . . . . . . . . . . . . . 18 |- ( N e. ( ZZ>= ` 2 ) -> ( ( N + 1 ) + 1 ) = ( N + 2 ) )
67	66	eqcomd 2628	. . . . . . . . . . . . . . . . 17 |- ( N e. ( ZZ>= ` 2 ) -> ( N + 2 ) = ( ( N + 1 ) + 1 ) )
68	67	oveq2d 6666	. . . . . . . . . . . . . . . 16 |- ( N e. ( ZZ>= ` 2 ) -> ( 2 ^ ( N + 2 ) ) = ( 2 ^ ( ( N + 1 ) + 1 ) ) )
69		2cnd 11093	. . . . . . . . . . . . . . . . 17 |- ( N e. ( ZZ>= ` 2 ) -> 2 e. CC )
70		peano2nn0 11333	. . . . . . . . . . . . . . . . . 18 |- ( N e. NN0 -> ( N + 1 ) e. NN0 )
71	63, 70	syl 17	. . . . . . . . . . . . . . . . 17 |- ( N e. ( ZZ>= ` 2 ) -> ( N + 1 ) e. NN0 )
72	69, 71	expp1d 13009	. . . . . . . . . . . . . . . 16 |- ( N e. ( ZZ>= ` 2 ) -> ( 2 ^ ( ( N + 1 ) + 1 ) ) = ( ( 2 ^ ( N + 1 ) ) x. 2 ) )
73	54	a1i 11	. . . . . . . . . . . . . . . . . . 19 |- ( N e. ( ZZ>= ` 2 ) -> 2 e. NN0 )
74	73, 71	nn0expcld 13031	. . . . . . . . . . . . . . . . . 18 |- ( N e. ( ZZ>= ` 2 ) -> ( 2 ^ ( N + 1 ) ) e. NN0 )
75	74	nn0cnd 11353	. . . . . . . . . . . . . . . . 17 |- ( N e. ( ZZ>= ` 2 ) -> ( 2 ^ ( N + 1 ) ) e. CC )
76	75, 69	mulcomd 10061	. . . . . . . . . . . . . . . 16 |- ( N e. ( ZZ>= ` 2 ) -> ( ( 2 ^ ( N + 1 ) ) x. 2 ) = ( 2 x. ( 2 ^ ( N + 1 ) ) ) )
77	68, 72, 76	3eqtrd 2660	. . . . . . . . . . . . . . 15 |- ( N e. ( ZZ>= ` 2 ) -> ( 2 ^ ( N + 2 ) ) = ( 2 x. ( 2 ^ ( N + 1 ) ) ) )
78	77	adantr 481	. . . . . . . . . . . . . 14 |- ( ( N e. ( ZZ>= ` 2 ) /\ n e. NN0 ) -> ( 2 ^ ( N + 2 ) ) = ( 2 x. ( 2 ^ ( N + 1 ) ) ) )
79	78	oveq2d 6666	. . . . . . . . . . . . 13 |- ( ( N e. ( ZZ>= ` 2 ) /\ n e. NN0 ) -> ( n x. ( 2 ^ ( N + 2 ) ) ) = ( n x. ( 2 x. ( 2 ^ ( N + 1 ) ) ) ) )
80		nn0cn 11302	. . . . . . . . . . . . . . 15 |- ( n e. NN0 -> n e. CC )
81	80	adantl 482	. . . . . . . . . . . . . 14 |- ( ( N e. ( ZZ>= ` 2 ) /\ n e. NN0 ) -> n e. CC )
82		2cnd 11093	. . . . . . . . . . . . . 14 |- ( ( N e. ( ZZ>= ` 2 ) /\ n e. NN0 ) -> 2 e. CC )
83	75	adantr 481	. . . . . . . . . . . . . 14 |- ( ( N e. ( ZZ>= ` 2 ) /\ n e. NN0 ) -> ( 2 ^ ( N + 1 ) ) e. CC )
84	81, 82, 83	mulassd 10063	. . . . . . . . . . . . 13 |- ( ( N e. ( ZZ>= ` 2 ) /\ n e. NN0 ) -> ( ( n x. 2 ) x. ( 2 ^ ( N + 1 ) ) ) = ( n x. ( 2 x. ( 2 ^ ( N + 1 ) ) ) ) )
85	79, 84	eqtr4d 2659	. . . . . . . . . . . 12 |- ( ( N e. ( ZZ>= ` 2 ) /\ n e. NN0 ) -> ( n x. ( 2 ^ ( N + 2 ) ) ) = ( ( n x. 2 ) x. ( 2 ^ ( N + 1 ) ) ) )
86	85	3ad2antl1 1223	. . . . . . . . . . 11 |- ( ( ( N e. ( ZZ>= ` 2 ) /\ M e. NN /\ M || (FermatNo ` N ) ) /\ n e. NN0 ) -> ( n x. ( 2 ^ ( N + 2 ) ) ) = ( ( n x. 2 ) x. ( 2 ^ ( N + 1 ) ) ) )
87	86	adantr 481	. . . . . . . . . 10 |- ( ( ( ( N e. ( ZZ>= ` 2 ) /\ M e. NN /\ M || (FermatNo ` N ) ) /\ n e. NN0 ) /\ M = ( ( n x. ( 2 ^ ( N + 2 ) ) ) + 1 ) ) -> ( n x. ( 2 ^ ( N + 2 ) ) ) = ( ( n x. 2 ) x. ( 2 ^ ( N + 1 ) ) ) )
88	87	oveq1d 6665	. . . . . . . . 9 |- ( ( ( ( N e. ( ZZ>= ` 2 ) /\ M e. NN /\ M || (FermatNo ` N ) ) /\ n e. NN0 ) /\ M = ( ( n x. ( 2 ^ ( N + 2 ) ) ) + 1 ) ) -> ( ( n x. ( 2 ^ ( N + 2 ) ) ) + 1 ) = ( ( ( n x. 2 ) x. ( 2 ^ ( N + 1 ) ) ) + 1 ) )
89	58, 62, 88	rspcedvd 3317	. . . . . . . 8 |- ( ( ( ( N e. ( ZZ>= ` 2 ) /\ M e. NN /\ M || (FermatNo ` N ) ) /\ n e. NN0 ) /\ M = ( ( n x. ( 2 ^ ( N + 2 ) ) ) + 1 ) ) -> E. k e. NN0 M = ( ( k x. ( 2 ^ ( N + 1 ) ) ) + 1 ) )
90	89	ex 450	. . . . . . 7 |- ( ( ( N e. ( ZZ>= ` 2 ) /\ M e. NN /\ M || (FermatNo ` N ) ) /\ n e. NN0 ) -> ( M = ( ( n x. ( 2 ^ ( N + 2 ) ) ) + 1 ) -> E. k e. NN0 M = ( ( k x. ( 2 ^ ( N + 1 ) ) ) + 1 ) ) )
91	90	rexlimdva 3031	. . . . . 6 |- ( ( N e. ( ZZ>= ` 2 ) /\ M e. NN /\ M || (FermatNo ` N ) ) -> ( E. n e. NN0 M = ( ( n x. ( 2 ^ ( N + 2 ) ) ) + 1 ) -> E. k e. NN0 M = ( ( k x. ( 2 ^ ( N + 1 ) ) ) + 1 ) ) )
92	52, 91	mpd 15	. . . . 5 |- ( ( N e. ( ZZ>= ` 2 ) /\ M e. NN /\ M || (FermatNo ` N ) ) -> E. k e. NN0 M = ( ( k x. ( 2 ^ ( N + 1 ) ) ) + 1 ) )
93	92	3exp 1264	. . . 4 |- ( N e. ( ZZ>= ` 2 ) -> ( M e. NN -> ( M || (FermatNo ` N ) -> E. k e. NN0 M = ( ( k x. ( 2 ^ ( N + 1 ) ) ) + 1 ) ) ) )
94	51, 93	jaoi 394	. . 3 |- ( ( N = 1 \/ N e. ( ZZ>= ` 2 ) ) -> ( M e. NN -> ( M || (FermatNo ` N ) -> E. k e. NN0 M = ( ( k x. ( 2 ^ ( N + 1 ) ) ) + 1 ) ) ) )
95	1, 94	sylbi 207	. 2 |- ( N e. NN -> ( M e. NN -> ( M || (FermatNo ` N ) -> E. k e. NN0 M = ( ( k x. ( 2 ^ ( N + 1 ) ) ) + 1 ) ) ) )
96	95	3imp 1256	1 |- ( ( N e. NN /\ M e. NN /\ M || (FermatNo ` N ) ) -> E. k e. NN0 M = ( ( k x. ( 2 ^ ( N + 1 ) ) ) + 1 ) )

Syntax hints:   -> wi 4   <-> wb 196   \/ wo 383   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   E.wrex 2913   class class class wbr 4653   ` cfv 5888  (class class class)co 6650   CCcc 9934   0cc0 9936   1c1 9937   + caddc 9939   x. cmul 9941   NNcn 11020   2c2 11070   4c4 11072   5c5 11073   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-9 11086  df-n0 11293  df-xnn0 11364  df-z 11378  df-dec 11494  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: (None)