Theorem wilthlem1 16232

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 13043, ( 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 10445	. . . . . . . . . 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 9693	. . . . . . . . 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 9780	. . . . . . 7 |- ( ( P e. Prime /\ N e. ( 1 ... ( P - 1 ) ) ) -> ( N - 1 ) e. CC )
6	2	peano2zd 9782	. . . . . . . 8 |- ( ( P e. Prime /\ N e. ( 1 ... ( P - 1 ) ) ) -> ( N + 1 ) e. ZZ )
7	6	zcnd 9780	. . . . . . 7 |- ( ( P e. Prime /\ N e. ( 1 ... ( P - 1 ) ) ) -> ( N + 1 ) e. CC )
8	5, 7	mulcomd 8353	. . . . . 6 |- ( ( P e. Prime /\ N e. ( 1 ... ( P - 1 ) ) ) -> ( ( N - 1 ) x. ( N + 1 ) ) = ( ( N + 1 ) x. ( N - 1 ) ) )
9	2	zcnd 9780	. . . . . . 7 |- ( ( P e. Prime /\ N e. ( 1 ... ( P - 1 ) ) ) -> N e. CC )
10		ax-1cn 8278	. . . . . . 7 |- 1 e. CC
11		subsq 11104	. . . . . . 7 |- ( ( N e. CC /\ 1 e. CC ) -> ( ( N ^ 2 ) - ( 1 ^ 2 ) ) = ( ( N + 1 ) x. ( N - 1 ) ) )
12	9, 10, 11	sylancl 417	. . . . . 6 |- ( ( P e. Prime /\ N e. ( 1 ... ( P - 1 ) ) ) -> ( ( N ^ 2 ) - ( 1 ^ 2 ) ) = ( ( N + 1 ) x. ( N - 1 ) ) )
13	9	sqvald 11129	. . . . . . 7 |- ( ( P e. Prime /\ N e. ( 1 ... ( P - 1 ) ) ) -> ( N ^ 2 ) = ( N x. N ) )
14		sq1 11091	. . . . . . . 8 |- ( 1 ^ 2 ) = 1
15	14	a1i 9	. . . . . . 7 |- ( ( P e. Prime /\ N e. ( 1 ... ( P - 1 ) ) ) -> ( 1 ^ 2 ) = 1 )
16	13, 15	oveq12d 6106	. . . . . 6 |- ( ( P e. Prime /\ N e. ( 1 ... ( P - 1 ) ) ) -> ( ( N ^ 2 ) - ( 1 ^ 2 ) ) = ( ( N x. N ) - 1 ) )
17	8, 12, 16	3eqtr2d 2277	. . . . 5 |- ( ( P e. Prime /\ N e. ( 1 ... ( P - 1 ) ) ) -> ( ( N - 1 ) x. ( N + 1 ) ) = ( ( N x. N ) - 1 ) )
18	17	breq2d 4143	. . . 4 |- ( ( P e. Prime /\ N e. ( 1 ... ( P - 1 ) ) ) -> ( P || ( ( N - 1 ) x. ( N + 1 ) ) <-> P || ( ( N x. N ) - 1 ) ) )
19		fz1ssfz0 10541	. . . . . 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 3246	. . . . 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 12950	. . . 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 1278	. . 3 |- ( ( P e. Prime /\ N e. ( 1 ... ( P - 1 ) ) ) -> ( P || ( ( N - 1 ) x. ( N + 1 ) ) <-> ( P || ( N - 1 ) \/ P || ( N + 1 ) ) ) )
27		prmnn 12913	. . . . 5 |- ( P e. Prime -> P e. NN )
28		fzm1ndvds 12648	. . . . 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 2238	. . . . 5 |- ( ( N ^ ( P - 2 ) ) mod P ) = ( ( N ^ ( P - 2 ) ) mod P )
31	30	prmdiveq 13043	. . . 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 1278	. . 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 9682	. . . . 5 |- ( ( P e. Prime /\ N e. ( 1 ... ( P - 1 ) ) ) -> 1 e. ZZ )
36		moddvds 12591	. . . . 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 1278	. . . 4 |- ( ( P e. Prime /\ N e. ( 1 ... ( P - 1 ) ) ) -> ( ( N mod P ) = ( 1 mod P ) <-> P || ( N - 1 ) ) )
38		zq 10042	. . . . . . . 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 12914	. . . . . . . 8 |- ( P e. Prime -> P e. ZZ )
42		zq 10042	. . . . . . . 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 10477	. . . . . . . . 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 9631	. . . . . . 7 |- ( ( P e. Prime /\ N e. ( 1 ... ( P - 1 ) ) ) -> N e. NN0 )
48	47	nn0ge0d 9634	. . . . . 6 |- ( ( P e. Prime /\ N e. ( 1 ... ( P - 1 ) ) ) -> 0 <_ N )
49		elfzle2 10449	. . . . . . . 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 9713	. . . . . . . 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 10807	. . . . . 6 |- ( ( ( N e. QQ /\ P e. QQ ) /\ ( 0 <_ N /\ N < P ) ) -> ( N mod P ) = N )
55	40, 44, 48, 53, 54	syl22anc 1279	. . . . 5 |- ( ( P e. Prime /\ N e. ( 1 ... ( P - 1 ) ) ) -> ( N mod P ) = N )
56		prmuz2 12935	. . . . . . . 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 10018	. . . . . . 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 10814	. . . . . 6 |- ( ( P e. QQ /\ 1 < P ) -> ( 1 mod P ) = 1 )
61	44, 59, 60	syl2anc 415	. . . . 5 |- ( ( P e. Prime /\ N e. ( 1 ... ( P - 1 ) ) ) -> ( 1 mod P ) = 1 )
62	55, 61	eqeq12d 2253	. . . 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 9781	. . . . 5 |- ( ( P e. Prime /\ N e. ( 1 ... ( P - 1 ) ) ) -> -u 1 e. ZZ )
65		moddvds 12591	. . . . 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 1278	. . . 4 |- ( ( P e. Prime /\ N e. ( 1 ... ( P - 1 ) ) ) -> ( ( N mod P ) = ( -u 1 mod P ) <-> P || ( N - -u 1 ) ) )
67	34	nncnd 9327	. . . . . . . . . 10 |- ( ( P e. Prime /\ N e. ( 1 ... ( P - 1 ) ) ) -> P e. CC )
68	67	mullidd 8350	. . . . . . . . 9 |- ( ( P e. Prime /\ N e. ( 1 ... ( P - 1 ) ) ) -> ( 1 x. P ) = P )
69	68	oveq2d 6104	. . . . . . . 8 |- ( ( P e. Prime /\ N e. ( 1 ... ( P - 1 ) ) ) -> ( -u 1 + ( 1 x. P ) ) = ( -u 1 + P ) )
70		neg1cn 9418	. . . . . . . . 9 |- -u 1 e. CC
71		addcom 8470	. . . . . . . . 9 |- ( ( -u 1 e. CC /\ P e. CC ) -> ( -u 1 + P ) = ( P + -u 1 ) )
72	70, 67, 71	sylancr 418	. . . . . . . 8 |- ( ( P e. Prime /\ N e. ( 1 ... ( P - 1 ) ) ) -> ( -u 1 + P ) = ( P + -u 1 ) )
73		negsub 8581	. . . . . . . . 9 |- ( ( P e. CC /\ 1 e. CC ) -> ( P + -u 1 ) = ( P - 1 ) )
74	67, 10, 73	sylancl 417	. . . . . . . 8 |- ( ( P e. Prime /\ N e. ( 1 ... ( P - 1 ) ) ) -> ( P + -u 1 ) = ( P - 1 ) )
75	69, 72, 74	3eqtrd 2275	. . . . . . 7 |- ( ( P e. Prime /\ N e. ( 1 ... ( P - 1 ) ) ) -> ( -u 1 + ( 1 x. P ) ) = ( P - 1 ) )
76	75	oveq1d 6103	. . . . . 6 |- ( ( P e. Prime /\ N e. ( 1 ... ( P - 1 ) ) ) -> ( ( -u 1 + ( 1 x. P ) ) mod P ) = ( ( P - 1 ) mod P ) )
77		neg1z 9687	. . . . . . . 8 |- -u 1 e. ZZ
78		zq 10042	. . . . . . . 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 9357	. . . . . . 7 |- ( ( P e. Prime /\ N e. ( 1 ... ( P - 1 ) ) ) -> 0 < P )
81		modqcyc 10817	. . . . . . 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 1279	. . . . . 6 |- ( ( P e. Prime /\ N e. ( 1 ... ( P - 1 ) ) ) -> ( ( -u 1 + ( 1 x. P ) ) mod P ) = ( -u 1 mod P ) )
83		nnm1nn0 9615	. . . . . . . . . 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 9777	. . . . . . . 8 |- ( ( P e. Prime /\ N e. ( 1 ... ( P - 1 ) ) ) -> ( P - 1 ) e. ZZ )
86		zq 10042	. . . . . . . 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 9634	. . . . . . 7 |- ( ( P e. Prime /\ N e. ( 1 ... ( P - 1 ) ) ) -> 0 <_ ( P - 1 ) )
89	34	nnred 9326	. . . . . . . 8 |- ( ( P e. Prime /\ N e. ( 1 ... ( P - 1 ) ) ) -> P e. RR )
90	89	ltm1d 9270	. . . . . . 7 |- ( ( P e. Prime /\ N e. ( 1 ... ( P - 1 ) ) ) -> ( P - 1 ) < P )
91		modqid 10807	. . . . . . 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 1279	. . . . . 6 |- ( ( P e. Prime /\ N e. ( 1 ... ( P - 1 ) ) ) -> ( ( P - 1 ) mod P ) = ( P - 1 ) )
93	76, 82, 92	3eqtr3d 2279	. . . . 5 |- ( ( P e. Prime /\ N e. ( 1 ... ( P - 1 ) ) ) -> ( -u 1 mod P ) = ( P - 1 ) )
94	55, 93	eqeq12d 2253	. . . 4 |- ( ( P e. Prime /\ N e. ( 1 ... ( P - 1 ) ) ) -> ( ( N mod P ) = ( -u 1 mod P ) <-> N = ( P - 1 ) ) )
95		subneg 8582	. . . . . 6 |- ( ( N e. CC /\ 1 e. CC ) -> ( N - -u 1 ) = ( N + 1 ) )
96	9, 10, 95	sylancl 417	. . . . 5 |- ( ( P e. Prime /\ N e. ( 1 ... ( P - 1 ) ) ) -> ( N - -u 1 ) = ( N + 1 ) )
97	96	breq2d 4143	. . . 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 805	. 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 ) ) ) )

This proof depends on syntax axioms:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 720   = wceq 1402   e. wcel 2209   class class class wbr 4131   ` cfv 5380  (class class class)co 6088   CCcc 8183   0cc0 8185   1c1 8186   + caddc 8188   x. cmul 8190   < clt 8366   <_ cle 8367   - cmin 8504   -ucneg 8505   NNcn 9313   2c2 9364   NN0cn0 9574   ZZcz 9655   ZZ>=cuz 9937   QQcq 10035   ...cfz 10428   mod cmo 10780   ^cexp 10996   || cdvds 12579   Primecprime 12910

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4248  ax-sep 4251  ax-nul 4261  ax-pow 4313  ax-pr 4348  ax-un 4580  ax-setind 4686  ax-iinf 4737  ax-cnex 8276  ax-resscn 8277  ax-1cn 8278  ax-1re 8279  ax-icn 8280  ax-addcl 8281  ax-addrcl 8282  ax-mulcl 8283  ax-mulrcl 8284  ax-addcom 8285  ax-mulcom 8286  ax-addass 8287  ax-mulass 8288  ax-distr 8289  ax-i2m1 8290  ax-0lt1 8291  ax-1rid 8292  ax-0id 8293  ax-rnegex 8294  ax-precex 8295  ax-cnre 8296  ax-pre-ltirr 8297  ax-pre-ltwlin 8298  ax-pre-lttrn 8299  ax-pre-apti 8300  ax-pre-ltadd 8301  ax-pre-mulgt0 8302  ax-pre-mulext 8303  ax-arch 8304  ax-caucvg 8305

This proof depends on definitions:  df-bi 117  df-stab 843  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-if 3639  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4132  df-opab 4194  df-mpt 4195  df-tr 4232  df-id 4440  df-po 4443  df-iso 4444  df-iord 4513  df-on 4515  df-ilim 4516  df-suc 4518  df-iom 4740  df-xp 4782  df-rel 4783  df-cnv 4784  df-co 4785  df-dm 4786  df-rn 4787  df-res 4788  df-ima 4789  df-iota 5340  df-fun 5382  df-fn 5383  df-f 5384  df-f1 5385  df-fo 5386  df-f1o 5387  df-fv 5388  df-isom 5389  df-riota 6041  df-ov 6091  df-oprab 6092  df-mpo 6093  df-1st 6378  df-2nd 6379  df-recs 6580  df-irdg 6645  df-frec 6666  df-1o 6691  df-2o 6692  df-oadd 6695  df-er 6811  df-en 7027  df-dom 7028  df-fin 7029  df-sup 7330  df-pnf 8368  df-mnf 8369  df-xr 8370  df-ltxr 8371  df-le 8372  df-sub 8506  df-neg 8507  df-reap 8911  df-ap 8918  df-div 9011  df-inn 9314  df-2 9372  df-3 9373  df-4 9374  df-n0 9575  df-z 9656  df-uz 9938  df-q 10036  df-rp 10072  df-fz 10429  df-fzo 10567  df-fl 10722  df-mod 10781  df-seqfrec 10906  df-exp 10997  df-ihash 11237  df-cj 11629  df-re 11630  df-im 11631  df-rsqrt 11786  df-abs 11787  df-clim 12070  df-proddc 12343  df-dvds 12580  df-gcd 12756  df-prm 12911  df-phi 13018

This theorem is used by: (None)