Theorem wilthlem1 15974

Description: The only elements that are equal to their own inverses in the multiplicative group of nonzero elements in ZZ / P ZZ are 1 and -u 1 == P - 1. (Note that from prmdiveq 12958, ( N ^ ( P - 2 ) ) mod P is the modular inverse of N in ZZ / P ZZ. (Contributed by Mario Carneiro, 24-Jan-2015.)

Assertion

Ref	Expression
wilthlem1	|- ( ( P e. Prime /\ N e. ( 1 ... ( P - 1 ) ) ) -> ( N = ( ( N ^ ( P - 2 ) ) mod P ) <-> ( N = 1 \/ N = ( P - 1 ) ) ) )

Proof of Theorem wilthlem1

Step	Hyp	Ref	Expression
1		elfzelz 10378	. . . . . . . . . 10 |- ( N e. ( 1 ... ( P - 1 ) ) -> N e. ZZ )
2	1	adantl 277	. . . . . . . . 9 |- ( ( P e. Prime /\ N e. ( 1 ... ( P - 1 ) ) ) -> N e. ZZ )
3		peano2zm 9632	. . . . . . . . 9 |- ( N e. ZZ -> ( N - 1 ) e. ZZ )
4	2, 3	syl 14	. . . . . . . 8 |- ( ( P e. Prime /\ N e. ( 1 ... ( P - 1 ) ) ) -> ( N - 1 ) e. ZZ )
5	4	zcnd 9719	. . . . . . 7 |- ( ( P e. Prime /\ N e. ( 1 ... ( P - 1 ) ) ) -> ( N - 1 ) e. CC )
6	2	peano2zd 9721	. . . . . . . 8 |- ( ( P e. Prime /\ N e. ( 1 ... ( P - 1 ) ) ) -> ( N + 1 ) e. ZZ )
7	6	zcnd 9719	. . . . . . 7 |- ( ( P e. Prime /\ N e. ( 1 ... ( P - 1 ) ) ) -> ( N + 1 ) e. CC )
8	5, 7	mulcomd 8311	. . . . . 6 |- ( ( P e. Prime /\ N e. ( 1 ... ( P - 1 ) ) ) -> ( ( N - 1 ) x. ( N + 1 ) ) = ( ( N + 1 ) x. ( N - 1 ) ) )
9	2	zcnd 9719	. . . . . . 7 |- ( ( P e. Prime /\ N e. ( 1 ... ( P - 1 ) ) ) -> N e. CC )
10		ax-1cn 8236	. . . . . . 7 |- 1 e. CC
11		subsq 11032	. . . . . . 7 |- ( ( N e. CC /\ 1 e. CC ) -> ( ( N ^ 2 ) - ( 1 ^ 2 ) ) = ( ( N + 1 ) x. ( N - 1 ) ) )
12	9, 10, 11	sylancl 413	. . . . . 6 |- ( ( P e. Prime /\ N e. ( 1 ... ( P - 1 ) ) ) -> ( ( N ^ 2 ) - ( 1 ^ 2 ) ) = ( ( N + 1 ) x. ( N - 1 ) ) )
13	9	sqvald 11057	. . . . . . 7 |- ( ( P e. Prime /\ N e. ( 1 ... ( P - 1 ) ) ) -> ( N ^ 2 ) = ( N x. N ) )
14		sq1 11019	. . . . . . . 8 |- ( 1 ^ 2 ) = 1
15	14	a1i 9	. . . . . . 7 |- ( ( P e. Prime /\ N e. ( 1 ... ( P - 1 ) ) ) -> ( 1 ^ 2 ) = 1 )
16	13, 15	oveq12d 6076	. . . . . 6 |- ( ( P e. Prime /\ N e. ( 1 ... ( P - 1 ) ) ) -> ( ( N ^ 2 ) - ( 1 ^ 2 ) ) = ( ( N x. N ) - 1 ) )
17	8, 12, 16	3eqtr2d 2273	. . . . 5 |- ( ( P e. Prime /\ N e. ( 1 ... ( P - 1 ) ) ) -> ( ( N - 1 ) x. ( N + 1 ) ) = ( ( N x. N ) - 1 ) )
18	17	breq2d 4126	. . . 4 |- ( ( P e. Prime /\ N e. ( 1 ... ( P - 1 ) ) ) -> ( P || ( ( N - 1 ) x. ( N + 1 ) ) <-> P || ( ( N x. N ) - 1 ) ) )
19		fz1ssfz0 10473	. . . . . 6 |- ( 1 ... ( P - 1 ) ) C_ ( 0 ... ( P - 1 ) )
20		simpr 110	. . . . . 6 |- ( ( P e. Prime /\ N e. ( 1 ... ( P - 1 ) ) ) -> N e. ( 1 ... ( P - 1 ) ) )
21	19, 20	sselid 3240	. . . . 5 |- ( ( P e. Prime /\ N e. ( 1 ... ( P - 1 ) ) ) -> N e. ( 0 ... ( P - 1 ) ) )
22	21	biantrurd 305	. . . 4 |- ( ( P e. Prime /\ N e. ( 1 ... ( P - 1 ) ) ) -> ( P || ( ( N x. N ) - 1 ) <-> ( N e. ( 0 ... ( P - 1 ) ) /\ P || ( ( N x. N ) - 1 ) ) ) )
23	18, 22	bitrd 188	. . 3 |- ( ( P e. Prime /\ N e. ( 1 ... ( P - 1 ) ) ) -> ( P || ( ( N - 1 ) x. ( N + 1 ) ) <-> ( N e. ( 0 ... ( P - 1 ) ) /\ P || ( ( N x. N ) - 1 ) ) ) )
24		simpl 109	. . . 4 |- ( ( P e. Prime /\ N e. ( 1 ... ( P - 1 ) ) ) -> P e. Prime )
25		euclemma 12868	. . . 4 |- ( ( P e. Prime /\ ( N - 1 ) e. ZZ /\ ( N + 1 ) e. ZZ ) -> ( P || ( ( N - 1 ) x. ( N + 1 ) ) <-> ( P || ( N - 1 ) \/ P || ( N + 1 ) ) ) )
26	24, 4, 6, 25	syl3anc 1274	. . 3 |- ( ( P e. Prime /\ N e. ( 1 ... ( P - 1 ) ) ) -> ( P || ( ( N - 1 ) x. ( N + 1 ) ) <-> ( P || ( N - 1 ) \/ P || ( N + 1 ) ) ) )
27		prmnn 12832	. . . . 5 |- ( P e. Prime -> P e. NN )
28		fzm1ndvds 12567	. . . . 5 |- ( ( P e. NN /\ N e. ( 1 ... ( P - 1 ) ) ) -> -. P || N )
29	27, 28	sylan 283	. . . 4 |- ( ( P e. Prime /\ N e. ( 1 ... ( P - 1 ) ) ) -> -. P || N )
30		eqid 2234	. . . . 5 |- ( ( N ^ ( P - 2 ) ) mod P ) = ( ( N ^ ( P - 2 ) ) mod P )
31	30	prmdiveq 12958	. . . 4 |- ( ( P e. Prime /\ N e. ZZ /\ -. P || N ) -> ( ( N e. ( 0 ... ( P - 1 ) ) /\ P || ( ( N x. N ) - 1 ) ) <-> N = ( ( N ^ ( P - 2 ) ) mod P ) ) )
32	24, 2, 29, 31	syl3anc 1274	. . 3 |- ( ( P e. Prime /\ N e. ( 1 ... ( P - 1 ) ) ) -> ( ( N e. ( 0 ... ( P - 1 ) ) /\ P || ( ( N x. N ) - 1 ) ) <-> N = ( ( N ^ ( P - 2 ) ) mod P ) ) )
33	23, 26, 32	3bitr3rd 219	. 2 |- ( ( P e. Prime /\ N e. ( 1 ... ( P - 1 ) ) ) -> ( N = ( ( N ^ ( P - 2 ) ) mod P ) <-> ( P || ( N - 1 ) \/ P || ( N + 1 ) ) ) )
34	27	adantr 276	. . . . 5 |- ( ( P e. Prime /\ N e. ( 1 ... ( P - 1 ) ) ) -> P e. NN )
35		1zzd 9621	. . . . 5 |- ( ( P e. Prime /\ N e. ( 1 ... ( P - 1 ) ) ) -> 1 e. ZZ )
36		moddvds 12510	. . . . 5 |- ( ( P e. NN /\ N e. ZZ /\ 1 e. ZZ ) -> ( ( N mod P ) = ( 1 mod P ) <-> P || ( N - 1 ) ) )
37	34, 2, 35, 36	syl3anc 1274	. . . 4 |- ( ( P e. Prime /\ N e. ( 1 ... ( P - 1 ) ) ) -> ( ( N mod P ) = ( 1 mod P ) <-> P || ( N - 1 ) ) )
38		zq 9976	. . . . . . . 8 |- ( N e. ZZ -> N e. QQ )
39	1, 38	syl 14	. . . . . . 7 |- ( N e. ( 1 ... ( P - 1 ) ) -> N e. QQ )
40	39	adantl 277	. . . . . 6 |- ( ( P e. Prime /\ N e. ( 1 ... ( P - 1 ) ) ) -> N e. QQ )
41		prmz 12833	. . . . . . . 8 |- ( P e. Prime -> P e. ZZ )
42		zq 9976	. . . . . . . 8 |- ( P e. ZZ -> P e. QQ )
43	41, 42	syl 14	. . . . . . 7 |- ( P e. Prime -> P e. QQ )
44	43	adantr 276	. . . . . 6 |- ( ( P e. Prime /\ N e. ( 1 ... ( P - 1 ) ) ) -> P e. QQ )
45		elfznn 10409	. . . . . . . . 9 |- ( N e. ( 1 ... ( P - 1 ) ) -> N e. NN )
46	45	adantl 277	. . . . . . . 8 |- ( ( P e. Prime /\ N e. ( 1 ... ( P - 1 ) ) ) -> N e. NN )
47	46	nnnn0d 9570	. . . . . . 7 |- ( ( P e. Prime /\ N e. ( 1 ... ( P - 1 ) ) ) -> N e. NN0 )
48	47	nn0ge0d 9573	. . . . . 6 |- ( ( P e. Prime /\ N e. ( 1 ... ( P - 1 ) ) ) -> 0 <_ N )
49		elfzle2 10382	. . . . . . . 8 |- ( N e. ( 1 ... ( P - 1 ) ) -> N <_ ( P - 1 ) )
50	49	adantl 277	. . . . . . 7 |- ( ( P e. Prime /\ N e. ( 1 ... ( P - 1 ) ) ) -> N <_ ( P - 1 ) )
51		zltlem1 9652	. . . . . . . 8 |- ( ( N e. ZZ /\ P e. ZZ ) -> ( N < P <-> N <_ ( P - 1 ) ) )
52	1, 41, 51	syl2anr 290	. . . . . . 7 |- ( ( P e. Prime /\ N e. ( 1 ... ( P - 1 ) ) ) -> ( N < P <-> N <_ ( P - 1 ) ) )
53	50, 52	mpbird 167	. . . . . 6 |- ( ( P e. Prime /\ N e. ( 1 ... ( P - 1 ) ) ) -> N < P )
54		modqid 10735	. . . . . 6 |- ( ( ( N e. QQ /\ P e. QQ ) /\ ( 0 <_ N /\ N < P ) ) -> ( N mod P ) = N )
55	40, 44, 48, 53, 54	syl22anc 1275	. . . . 5 |- ( ( P e. Prime /\ N e. ( 1 ... ( P - 1 ) ) ) -> ( N mod P ) = N )
56		prmuz2 12853	. . . . . . . 8 |- ( P e. Prime -> P e. ( ZZ>= ` 2 ) )
57	56	adantr 276	. . . . . . 7 |- ( ( P e. Prime /\ N e. ( 1 ... ( P - 1 ) ) ) -> P e. ( ZZ>= ` 2 ) )
58		eluz2gt1 9952	. . . . . . 7 |- ( P e. ( ZZ>= ` 2 ) -> 1 < P )
59	57, 58	syl 14	. . . . . 6 |- ( ( P e. Prime /\ N e. ( 1 ... ( P - 1 ) ) ) -> 1 < P )
60		q1mod 10742	. . . . . 6 |- ( ( P e. QQ /\ 1 < P ) -> ( 1 mod P ) = 1 )
61	44, 59, 60	syl2anc 411	. . . . 5 |- ( ( P e. Prime /\ N e. ( 1 ... ( P - 1 ) ) ) -> ( 1 mod P ) = 1 )
62	55, 61	eqeq12d 2249	. . . 4 |- ( ( P e. Prime /\ N e. ( 1 ... ( P - 1 ) ) ) -> ( ( N mod P ) = ( 1 mod P ) <-> N = 1 ) )
63	37, 62	bitr3d 190	. . 3 |- ( ( P e. Prime /\ N e. ( 1 ... ( P - 1 ) ) ) -> ( P || ( N - 1 ) <-> N = 1 ) )
64	35	znegcld 9720	. . . . 5 |- ( ( P e. Prime /\ N e. ( 1 ... ( P - 1 ) ) ) -> -u 1 e. ZZ )
65		moddvds 12510	. . . . 5 |- ( ( P e. NN /\ N e. ZZ /\ -u 1 e. ZZ ) -> ( ( N mod P ) = ( -u 1 mod P ) <-> P || ( N - -u 1 ) ) )
66	34, 2, 64, 65	syl3anc 1274	. . . 4 |- ( ( P e. Prime /\ N e. ( 1 ... ( P - 1 ) ) ) -> ( ( N mod P ) = ( -u 1 mod P ) <-> P || ( N - -u 1 ) ) )
67	34	nncnd 9268	. . . . . . . . . 10 |- ( ( P e. Prime /\ N e. ( 1 ... ( P - 1 ) ) ) -> P e. CC )
68	67	mullidd 8308	. . . . . . . . 9 |- ( ( P e. Prime /\ N e. ( 1 ... ( P - 1 ) ) ) -> ( 1 x. P ) = P )
69	68	oveq2d 6074	. . . . . . . 8 |- ( ( P e. Prime /\ N e. ( 1 ... ( P - 1 ) ) ) -> ( -u 1 + ( 1 x. P ) ) = ( -u 1 + P ) )
70		neg1cn 9359	. . . . . . . . 9 |- -u 1 e. CC
71		addcom 8426	. . . . . . . . 9 |- ( ( -u 1 e. CC /\ P e. CC ) -> ( -u 1 + P ) = ( P + -u 1 ) )
72	70, 67, 71	sylancr 414	. . . . . . . 8 |- ( ( P e. Prime /\ N e. ( 1 ... ( P - 1 ) ) ) -> ( -u 1 + P ) = ( P + -u 1 ) )
73		negsub 8537	. . . . . . . . 9 |- ( ( P e. CC /\ 1 e. CC ) -> ( P + -u 1 ) = ( P - 1 ) )
74	67, 10, 73	sylancl 413	. . . . . . . 8 |- ( ( P e. Prime /\ N e. ( 1 ... ( P - 1 ) ) ) -> ( P + -u 1 ) = ( P - 1 ) )
75	69, 72, 74	3eqtrd 2271	. . . . . . 7 |- ( ( P e. Prime /\ N e. ( 1 ... ( P - 1 ) ) ) -> ( -u 1 + ( 1 x. P ) ) = ( P - 1 ) )
76	75	oveq1d 6073	. . . . . 6 |- ( ( P e. Prime /\ N e. ( 1 ... ( P - 1 ) ) ) -> ( ( -u 1 + ( 1 x. P ) ) mod P ) = ( ( P - 1 ) mod P ) )
77		neg1z 9626	. . . . . . . 8 |- -u 1 e. ZZ
78		zq 9976	. . . . . . . 8 |- ( -u 1 e. ZZ -> -u 1 e. QQ )
79	77, 78	mp1i 10	. . . . . . 7 |- ( ( P e. Prime /\ N e. ( 1 ... ( P - 1 ) ) ) -> -u 1 e. QQ )
80	34	nngt0d 9298	. . . . . . 7 |- ( ( P e. Prime /\ N e. ( 1 ... ( P - 1 ) ) ) -> 0 < P )
81		modqcyc 10745	. . . . . . 7 |- ( ( ( -u 1 e. QQ /\ 1 e. ZZ ) /\ ( P e. QQ /\ 0 < P ) ) -> ( ( -u 1 + ( 1 x. P ) ) mod P ) = ( -u 1 mod P ) )
82	79, 35, 44, 80, 81	syl22anc 1275	. . . . . 6 |- ( ( P e. Prime /\ N e. ( 1 ... ( P - 1 ) ) ) -> ( ( -u 1 + ( 1 x. P ) ) mod P ) = ( -u 1 mod P ) )
83		nnm1nn0 9554	. . . . . . . . . 10 |- ( P e. NN -> ( P - 1 ) e. NN0 )
84	34, 83	syl 14	. . . . . . . . 9 |- ( ( P e. Prime /\ N e. ( 1 ... ( P - 1 ) ) ) -> ( P - 1 ) e. NN0 )
85	84	nn0zd 9716	. . . . . . . 8 |- ( ( P e. Prime /\ N e. ( 1 ... ( P - 1 ) ) ) -> ( P - 1 ) e. ZZ )
86		zq 9976	. . . . . . . 8 |- ( ( P - 1 ) e. ZZ -> ( P - 1 ) e. QQ )
87	85, 86	syl 14	. . . . . . 7 |- ( ( P e. Prime /\ N e. ( 1 ... ( P - 1 ) ) ) -> ( P - 1 ) e. QQ )
88	84	nn0ge0d 9573	. . . . . . 7 |- ( ( P e. Prime /\ N e. ( 1 ... ( P - 1 ) ) ) -> 0 <_ ( P - 1 ) )
89	34	nnred 9267	. . . . . . . 8 |- ( ( P e. Prime /\ N e. ( 1 ... ( P - 1 ) ) ) -> P e. RR )
90	89	ltm1d 9223	. . . . . . 7 |- ( ( P e. Prime /\ N e. ( 1 ... ( P - 1 ) ) ) -> ( P - 1 ) < P )
91		modqid 10735	. . . . . . 7 |- ( ( ( ( P - 1 ) e. QQ /\ P e. QQ ) /\ ( 0 <_ ( P - 1 ) /\ ( P - 1 ) < P ) ) -> ( ( P - 1 ) mod P ) = ( P - 1 ) )
92	87, 44, 88, 90, 91	syl22anc 1275	. . . . . 6 |- ( ( P e. Prime /\ N e. ( 1 ... ( P - 1 ) ) ) -> ( ( P - 1 ) mod P ) = ( P - 1 ) )
93	76, 82, 92	3eqtr3d 2275	. . . . 5 |- ( ( P e. Prime /\ N e. ( 1 ... ( P - 1 ) ) ) -> ( -u 1 mod P ) = ( P - 1 ) )
94	55, 93	eqeq12d 2249	. . . 4 |- ( ( P e. Prime /\ N e. ( 1 ... ( P - 1 ) ) ) -> ( ( N mod P ) = ( -u 1 mod P ) <-> N = ( P - 1 ) ) )
95		subneg 8538	. . . . . 6 |- ( ( N e. CC /\ 1 e. CC ) -> ( N - -u 1 ) = ( N + 1 ) )
96	9, 10, 95	sylancl 413	. . . . 5 |- ( ( P e. Prime /\ N e. ( 1 ... ( P - 1 ) ) ) -> ( N - -u 1 ) = ( N + 1 ) )
97	96	breq2d 4126	. . . 4 |- ( ( P e. Prime /\ N e. ( 1 ... ( P - 1 ) ) ) -> ( P || ( N - -u 1 ) <-> P || ( N + 1 ) ) )
98	66, 94, 97	3bitr3rd 219	. . 3 |- ( ( P e. Prime /\ N e. ( 1 ... ( P - 1 ) ) ) -> ( P || ( N + 1 ) <-> N = ( P - 1 ) ) )
99	63, 98	orbi12d 801	. 2 |- ( ( P e. Prime /\ N e. ( 1 ... ( P - 1 ) ) ) -> ( ( P || ( N - 1 ) \/ P || ( N + 1 ) ) <-> ( N = 1 \/ N = ( P - 1 ) ) ) )
100	33, 99	bitrd 188	1 |- ( ( P e. Prime /\ N e. ( 1 ... ( P - 1 ) ) ) -> ( N = ( ( N ^ ( P - 2 ) ) mod P ) <-> ( N = 1 \/ N = ( P - 1 ) ) ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 716   = wceq 1398   e. wcel 2205   class class class wbr 4114   ` cfv 5357  (class class class)co 6058   CCcc 8141   0cc0 8143   1c1 8144   + caddc 8146   x. cmul 8148   < clt 8324   <_ cle 8325   - cmin 8460   -ucneg 8461   NNcn 9254   2c2 9305   NN0cn0 9513   ZZcz 9594   ZZ>=cuz 9871   QQcq 9969   ...cfz 10361   mod cmo 10708   ^cexp 10924   || cdvds 12498   Primecprime 12829

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 2207  ax-14 2208  ax-ext 2216  ax-coll 4230  ax-sep 4233  ax-nul 4241  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-setind 4664  ax-iinf 4715  ax-cnex 8234  ax-resscn 8235  ax-1cn 8236  ax-1re 8237  ax-icn 8238  ax-addcl 8239  ax-addrcl 8240  ax-mulcl 8241  ax-mulrcl 8242  ax-addcom 8243  ax-mulcom 8244  ax-addass 8245  ax-mulass 8246  ax-distr 8247  ax-i2m1 8248  ax-0lt1 8249  ax-1rid 8250  ax-0id 8251  ax-rnegex 8252  ax-precex 8253  ax-cnre 8254  ax-pre-ltirr 8255  ax-pre-ltwlin 8256  ax-pre-lttrn 8257  ax-pre-apti 8258  ax-pre-ltadd 8259  ax-pre-mulgt0 8260  ax-pre-mulext 8261  ax-arch 8262  ax-caucvg 8263

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 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-nel 2510  df-ral 2527  df-rex 2528  df-reu 2529  df-rmo 2530  df-rab 2531  df-v 2817  df-sbc 3046  df-csb 3142  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-nul 3513  df-if 3625  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-int 3955  df-iun 3998  df-br 4115  df-opab 4177  df-mpt 4178  df-tr 4214  df-id 4419  df-po 4422  df-iso 4423  df-iord 4492  df-on 4494  df-ilim 4495  df-suc 4497  df-iom 4718  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-ima 4767  df-iota 5317  df-fun 5359  df-fn 5360  df-f 5361  df-f1 5362  df-fo 5363  df-f1o 5364  df-fv 5365  df-isom 5366  df-riota 6011  df-ov 6061  df-oprab 6062  df-mpo 6063  df-1st 6347  df-2nd 6348  df-recs 6549  df-irdg 6614  df-frec 6635  df-1o 6660  df-2o 6661  df-oadd 6664  df-er 6780  df-en 6989  df-dom 6990  df-fin 6991  df-sup 7288  df-pnf 8326  df-mnf 8327  df-xr 8328  df-ltxr 8329  df-le 8330  df-sub 8462  df-neg 8463  df-reap 8866  df-ap 8873  df-div 8964  df-inn 9255  df-2 9313  df-3 9314  df-4 9315  df-n0 9514  df-z 9595  df-uz 9872  df-q 9970  df-rp 10005  df-fz 10362  df-fzo 10499  df-fl 10654  df-mod 10709  df-seqfrec 10834  df-exp 10925  df-ihash 11164  df-cj 11552  df-re 11553  df-im 11554  df-rsqrt 11708  df-abs 11709  df-clim 11989  df-proddc 12262  df-dvds 12499  df-gcd 12675  df-prm 12830  df-phi 12933

This theorem is referenced by: (None)