Theorem prmdiveq 12429

Description: The modular inverse of A mod P is unique. (Contributed by Mario Carneiro, 24-Jan-2015.)

Hypothesis

Ref	Expression
prmdiv.1	|- R = ( ( A ^ ( P - 2 ) ) mod P )

Assertion

Ref	Expression
prmdiveq	|- ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) -> ( ( S e. ( 0 ... ( P - 1 ) ) /\ P || ( ( A x. S ) - 1 ) ) <-> S = R ) )

Proof of Theorem prmdiveq

Step	Hyp	Ref	Expression
1		simpl1 1002	. . . . . . . . 9 |- ( ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) /\ ( S e. ( 0 ... ( P - 1 ) ) /\ P || ( ( A x. S ) - 1 ) ) ) -> P e. Prime )
2		prmz 12304	. . . . . . . . 9 |- ( P e. Prime -> P e. ZZ )
3	1, 2	syl 14	. . . . . . . 8 |- ( ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) /\ ( S e. ( 0 ... ( P - 1 ) ) /\ P || ( ( A x. S ) - 1 ) ) ) -> P e. ZZ )
4		simpl2 1003	. . . . . . . . . 10 |- ( ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) /\ ( S e. ( 0 ... ( P - 1 ) ) /\ P || ( ( A x. S ) - 1 ) ) ) -> A e. ZZ )
5		elfzelz 10117	. . . . . . . . . . 11 |- ( S e. ( 0 ... ( P - 1 ) ) -> S e. ZZ )
6	5	ad2antrl 490	. . . . . . . . . 10 |- ( ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) /\ ( S e. ( 0 ... ( P - 1 ) ) /\ P || ( ( A x. S ) - 1 ) ) ) -> S e. ZZ )
7	4, 6	zmulcld 9471	. . . . . . . . 9 |- ( ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) /\ ( S e. ( 0 ... ( P - 1 ) ) /\ P || ( ( A x. S ) - 1 ) ) ) -> ( A x. S ) e. ZZ )
8		1z 9369	. . . . . . . . 9 |- 1 e. ZZ
9		zsubcl 9384	. . . . . . . . 9 |- ( ( ( A x. S ) e. ZZ /\ 1 e. ZZ ) -> ( ( A x. S ) - 1 ) e. ZZ )
10	7, 8, 9	sylancl 413	. . . . . . . 8 |- ( ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) /\ ( S e. ( 0 ... ( P - 1 ) ) /\ P || ( ( A x. S ) - 1 ) ) ) -> ( ( A x. S ) - 1 ) e. ZZ )
11		prmdiv.1	. . . . . . . . . . . . . 14 |- R = ( ( A ^ ( P - 2 ) ) mod P )
12	11	prmdiv 12428	. . . . . . . . . . . . 13 |- ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) -> ( R e. ( 1 ... ( P - 1 ) ) /\ P || ( ( A x. R ) - 1 ) ) )
13	12	adantr 276	. . . . . . . . . . . 12 |- ( ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) /\ ( S e. ( 0 ... ( P - 1 ) ) /\ P || ( ( A x. S ) - 1 ) ) ) -> ( R e. ( 1 ... ( P - 1 ) ) /\ P || ( ( A x. R ) - 1 ) ) )
14	13	simpld 112	. . . . . . . . . . 11 |- ( ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) /\ ( S e. ( 0 ... ( P - 1 ) ) /\ P || ( ( A x. S ) - 1 ) ) ) -> R e. ( 1 ... ( P - 1 ) ) )
15		elfzelz 10117	. . . . . . . . . . 11 |- ( R e. ( 1 ... ( P - 1 ) ) -> R e. ZZ )
16	14, 15	syl 14	. . . . . . . . . 10 |- ( ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) /\ ( S e. ( 0 ... ( P - 1 ) ) /\ P || ( ( A x. S ) - 1 ) ) ) -> R e. ZZ )
17	4, 16	zmulcld 9471	. . . . . . . . 9 |- ( ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) /\ ( S e. ( 0 ... ( P - 1 ) ) /\ P || ( ( A x. S ) - 1 ) ) ) -> ( A x. R ) e. ZZ )
18		zsubcl 9384	. . . . . . . . 9 |- ( ( ( A x. R ) e. ZZ /\ 1 e. ZZ ) -> ( ( A x. R ) - 1 ) e. ZZ )
19	17, 8, 18	sylancl 413	. . . . . . . 8 |- ( ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) /\ ( S e. ( 0 ... ( P - 1 ) ) /\ P || ( ( A x. S ) - 1 ) ) ) -> ( ( A x. R ) - 1 ) e. ZZ )
20		simprr 531	. . . . . . . 8 |- ( ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) /\ ( S e. ( 0 ... ( P - 1 ) ) /\ P || ( ( A x. S ) - 1 ) ) ) -> P || ( ( A x. S ) - 1 ) )
21	13	simprd 114	. . . . . . . 8 |- ( ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) /\ ( S e. ( 0 ... ( P - 1 ) ) /\ P || ( ( A x. S ) - 1 ) ) ) -> P || ( ( A x. R ) - 1 ) )
22	3, 10, 19, 20, 21	dvds2subd 12009	. . . . . . 7 |- ( ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) /\ ( S e. ( 0 ... ( P - 1 ) ) /\ P || ( ( A x. S ) - 1 ) ) ) -> P || ( ( ( A x. S ) - 1 ) - ( ( A x. R ) - 1 ) ) )
23	7	zcnd 9466	. . . . . . . . 9 |- ( ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) /\ ( S e. ( 0 ... ( P - 1 ) ) /\ P || ( ( A x. S ) - 1 ) ) ) -> ( A x. S ) e. CC )
24	17	zcnd 9466	. . . . . . . . 9 |- ( ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) /\ ( S e. ( 0 ... ( P - 1 ) ) /\ P || ( ( A x. S ) - 1 ) ) ) -> ( A x. R ) e. CC )
25		1cnd 8059	. . . . . . . . 9 |- ( ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) /\ ( S e. ( 0 ... ( P - 1 ) ) /\ P || ( ( A x. S ) - 1 ) ) ) -> 1 e. CC )
26	23, 24, 25	nnncan2d 8389	. . . . . . . 8 |- ( ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) /\ ( S e. ( 0 ... ( P - 1 ) ) /\ P || ( ( A x. S ) - 1 ) ) ) -> ( ( ( A x. S ) - 1 ) - ( ( A x. R ) - 1 ) ) = ( ( A x. S ) - ( A x. R ) ) )
27	4	zcnd 9466	. . . . . . . . 9 |- ( ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) /\ ( S e. ( 0 ... ( P - 1 ) ) /\ P || ( ( A x. S ) - 1 ) ) ) -> A e. CC )
28		elfznn0 10206	. . . . . . . . . . 11 |- ( S e. ( 0 ... ( P - 1 ) ) -> S e. NN0 )
29	28	ad2antrl 490	. . . . . . . . . 10 |- ( ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) /\ ( S e. ( 0 ... ( P - 1 ) ) /\ P || ( ( A x. S ) - 1 ) ) ) -> S e. NN0 )
30	29	nn0cnd 9321	. . . . . . . . 9 |- ( ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) /\ ( S e. ( 0 ... ( P - 1 ) ) /\ P || ( ( A x. S ) - 1 ) ) ) -> S e. CC )
31	16	zcnd 9466	. . . . . . . . 9 |- ( ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) /\ ( S e. ( 0 ... ( P - 1 ) ) /\ P || ( ( A x. S ) - 1 ) ) ) -> R e. CC )
32	27, 30, 31	subdid 8457	. . . . . . . 8 |- ( ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) /\ ( S e. ( 0 ... ( P - 1 ) ) /\ P || ( ( A x. S ) - 1 ) ) ) -> ( A x. ( S - R ) ) = ( ( A x. S ) - ( A x. R ) ) )
33	26, 32	eqtr4d 2232	. . . . . . 7 |- ( ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) /\ ( S e. ( 0 ... ( P - 1 ) ) /\ P || ( ( A x. S ) - 1 ) ) ) -> ( ( ( A x. S ) - 1 ) - ( ( A x. R ) - 1 ) ) = ( A x. ( S - R ) ) )
34	22, 33	breqtrd 4060	. . . . . 6 |- ( ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) /\ ( S e. ( 0 ... ( P - 1 ) ) /\ P || ( ( A x. S ) - 1 ) ) ) -> P || ( A x. ( S - R ) ) )
35		simpl3 1004	. . . . . . 7 |- ( ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) /\ ( S e. ( 0 ... ( P - 1 ) ) /\ P || ( ( A x. S ) - 1 ) ) ) -> -. P || A )
36		coprm 12337	. . . . . . . 8 |- ( ( P e. Prime /\ A e. ZZ ) -> ( -. P || A <-> ( P gcd A ) = 1 ) )
37	1, 4, 36	syl2anc 411	. . . . . . 7 |- ( ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) /\ ( S e. ( 0 ... ( P - 1 ) ) /\ P || ( ( A x. S ) - 1 ) ) ) -> ( -. P || A <-> ( P gcd A ) = 1 ) )
38	35, 37	mpbid 147	. . . . . 6 |- ( ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) /\ ( S e. ( 0 ... ( P - 1 ) ) /\ P || ( ( A x. S ) - 1 ) ) ) -> ( P gcd A ) = 1 )
39	6, 16	zsubcld 9470	. . . . . . 7 |- ( ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) /\ ( S e. ( 0 ... ( P - 1 ) ) /\ P || ( ( A x. S ) - 1 ) ) ) -> ( S - R ) e. ZZ )
40		coprmdvds 12285	. . . . . . 7 |- ( ( P e. ZZ /\ A e. ZZ /\ ( S - R ) e. ZZ ) -> ( ( P || ( A x. ( S - R ) ) /\ ( P gcd A ) = 1 ) -> P || ( S - R ) ) )
41	3, 4, 39, 40	syl3anc 1249	. . . . . 6 |- ( ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) /\ ( S e. ( 0 ... ( P - 1 ) ) /\ P || ( ( A x. S ) - 1 ) ) ) -> ( ( P || ( A x. ( S - R ) ) /\ ( P gcd A ) = 1 ) -> P || ( S - R ) ) )
42	34, 38, 41	mp2and 433	. . . . 5 |- ( ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) /\ ( S e. ( 0 ... ( P - 1 ) ) /\ P || ( ( A x. S ) - 1 ) ) ) -> P || ( S - R ) )
43		prmnn 12303	. . . . . . 7 |- ( P e. Prime -> P e. NN )
44	1, 43	syl 14	. . . . . 6 |- ( ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) /\ ( S e. ( 0 ... ( P - 1 ) ) /\ P || ( ( A x. S ) - 1 ) ) ) -> P e. NN )
45		moddvds 11981	. . . . . 6 |- ( ( P e. NN /\ S e. ZZ /\ R e. ZZ ) -> ( ( S mod P ) = ( R mod P ) <-> P || ( S - R ) ) )
46	44, 6, 16, 45	syl3anc 1249	. . . . 5 |- ( ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) /\ ( S e. ( 0 ... ( P - 1 ) ) /\ P || ( ( A x. S ) - 1 ) ) ) -> ( ( S mod P ) = ( R mod P ) <-> P || ( S - R ) ) )
47	42, 46	mpbird 167	. . . 4 |- ( ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) /\ ( S e. ( 0 ... ( P - 1 ) ) /\ P || ( ( A x. S ) - 1 ) ) ) -> ( S mod P ) = ( R mod P ) )
48		zq 9717	. . . . . 6 |- ( S e. ZZ -> S e. QQ )
49	6, 48	syl 14	. . . . 5 |- ( ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) /\ ( S e. ( 0 ... ( P - 1 ) ) /\ P || ( ( A x. S ) - 1 ) ) ) -> S e. QQ )
50		nnq 9724	. . . . . 6 |- ( P e. NN -> P e. QQ )
51	44, 50	syl 14	. . . . 5 |- ( ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) /\ ( S e. ( 0 ... ( P - 1 ) ) /\ P || ( ( A x. S ) - 1 ) ) ) -> P e. QQ )
52		elfzle1 10119	. . . . . 6 |- ( S e. ( 0 ... ( P - 1 ) ) -> 0 <_ S )
53	52	ad2antrl 490	. . . . 5 |- ( ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) /\ ( S e. ( 0 ... ( P - 1 ) ) /\ P || ( ( A x. S ) - 1 ) ) ) -> 0 <_ S )
54		elfzle2 10120	. . . . . . 7 |- ( S e. ( 0 ... ( P - 1 ) ) -> S <_ ( P - 1 ) )
55	54	ad2antrl 490	. . . . . 6 |- ( ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) /\ ( S e. ( 0 ... ( P - 1 ) ) /\ P || ( ( A x. S ) - 1 ) ) ) -> S <_ ( P - 1 ) )
56		zltlem1 9400	. . . . . . 7 |- ( ( S e. ZZ /\ P e. ZZ ) -> ( S < P <-> S <_ ( P - 1 ) ) )
57	6, 3, 56	syl2anc 411	. . . . . 6 |- ( ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) /\ ( S e. ( 0 ... ( P - 1 ) ) /\ P || ( ( A x. S ) - 1 ) ) ) -> ( S < P <-> S <_ ( P - 1 ) ) )
58	55, 57	mpbird 167	. . . . 5 |- ( ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) /\ ( S e. ( 0 ... ( P - 1 ) ) /\ P || ( ( A x. S ) - 1 ) ) ) -> S < P )
59		modqid 10458	. . . . 5 |- ( ( ( S e. QQ /\ P e. QQ ) /\ ( 0 <_ S /\ S < P ) ) -> ( S mod P ) = S )
60	49, 51, 53, 58, 59	syl22anc 1250	. . . 4 |- ( ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) /\ ( S e. ( 0 ... ( P - 1 ) ) /\ P || ( ( A x. S ) - 1 ) ) ) -> ( S mod P ) = S )
61		prmuz2 12324	. . . . . . . . 9 |- ( P e. Prime -> P e. ( ZZ>= ` 2 ) )
62		uznn0sub 9650	. . . . . . . . 9 |- ( P e. ( ZZ>= ` 2 ) -> ( P - 2 ) e. NN0 )
63	1, 61, 62	3syl 17	. . . . . . . 8 |- ( ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) /\ ( S e. ( 0 ... ( P - 1 ) ) /\ P || ( ( A x. S ) - 1 ) ) ) -> ( P - 2 ) e. NN0 )
64		zexpcl 10663	. . . . . . . 8 |- ( ( A e. ZZ /\ ( P - 2 ) e. NN0 ) -> ( A ^ ( P - 2 ) ) e. ZZ )
65	4, 63, 64	syl2anc 411	. . . . . . 7 |- ( ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) /\ ( S e. ( 0 ... ( P - 1 ) ) /\ P || ( ( A x. S ) - 1 ) ) ) -> ( A ^ ( P - 2 ) ) e. ZZ )
66		zq 9717	. . . . . . 7 |- ( ( A ^ ( P - 2 ) ) e. ZZ -> ( A ^ ( P - 2 ) ) e. QQ )
67	65, 66	syl 14	. . . . . 6 |- ( ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) /\ ( S e. ( 0 ... ( P - 1 ) ) /\ P || ( ( A x. S ) - 1 ) ) ) -> ( A ^ ( P - 2 ) ) e. QQ )
68	44	nngt0d 9051	. . . . . 6 |- ( ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) /\ ( S e. ( 0 ... ( P - 1 ) ) /\ P || ( ( A x. S ) - 1 ) ) ) -> 0 < P )
69		modqabs2 10467	. . . . . 6 |- ( ( ( A ^ ( P - 2 ) ) e. QQ /\ P e. QQ /\ 0 < P ) -> ( ( ( A ^ ( P - 2 ) ) mod P ) mod P ) = ( ( A ^ ( P - 2 ) ) mod P ) )
70	67, 51, 68, 69	syl3anc 1249	. . . . 5 |- ( ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) /\ ( S e. ( 0 ... ( P - 1 ) ) /\ P || ( ( A x. S ) - 1 ) ) ) -> ( ( ( A ^ ( P - 2 ) ) mod P ) mod P ) = ( ( A ^ ( P - 2 ) ) mod P ) )
71	11	oveq1i 5935	. . . . 5 |- ( R mod P ) = ( ( ( A ^ ( P - 2 ) ) mod P ) mod P )
72	70, 71, 11	3eqtr4g 2254	. . . 4 |- ( ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) /\ ( S e. ( 0 ... ( P - 1 ) ) /\ P || ( ( A x. S ) - 1 ) ) ) -> ( R mod P ) = R )
73	47, 60, 72	3eqtr3d 2237	. . 3 |- ( ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) /\ ( S e. ( 0 ... ( P - 1 ) ) /\ P || ( ( A x. S ) - 1 ) ) ) -> S = R )
74	73	ex 115	. 2 |- ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) -> ( ( S e. ( 0 ... ( P - 1 ) ) /\ P || ( ( A x. S ) - 1 ) ) -> S = R ) )
75		fz1ssfz0 10209	. . . . . 6 |- ( 1 ... ( P - 1 ) ) C_ ( 0 ... ( P - 1 ) )
76	75	sseli 3180	. . . . 5 |- ( R e. ( 1 ... ( P - 1 ) ) -> R e. ( 0 ... ( P - 1 ) ) )
77		eleq1 2259	. . . . 5 |- ( S = R -> ( S e. ( 0 ... ( P - 1 ) ) <-> R e. ( 0 ... ( P - 1 ) ) ) )
78	76, 77	imbitrrid 156	. . . 4 |- ( S = R -> ( R e. ( 1 ... ( P - 1 ) ) -> S e. ( 0 ... ( P - 1 ) ) ) )
79		oveq2 5933	. . . . . . 7 |- ( S = R -> ( A x. S ) = ( A x. R ) )
80	79	oveq1d 5940	. . . . . 6 |- ( S = R -> ( ( A x. S ) - 1 ) = ( ( A x. R ) - 1 ) )
81	80	breq2d 4046	. . . . 5 |- ( S = R -> ( P || ( ( A x. S ) - 1 ) <-> P || ( ( A x. R ) - 1 ) ) )
82	81	biimprd 158	. . . 4 |- ( S = R -> ( P || ( ( A x. R ) - 1 ) -> P || ( ( A x. S ) - 1 ) ) )
83	78, 82	anim12d 335	. . 3 |- ( S = R -> ( ( R e. ( 1 ... ( P - 1 ) ) /\ P || ( ( A x. R ) - 1 ) ) -> ( S e. ( 0 ... ( P - 1 ) ) /\ P || ( ( A x. S ) - 1 ) ) ) )
84	12, 83	syl5com 29	. 2 |- ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) -> ( S = R -> ( S e. ( 0 ... ( P - 1 ) ) /\ P || ( ( A x. S ) - 1 ) ) ) )
85	74, 84	impbid 129	1 |- ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) -> ( ( S e. ( 0 ... ( P - 1 ) ) /\ P || ( ( A x. S ) - 1 ) ) <-> S = R ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 980   = wceq 1364   e. wcel 2167   class class class wbr 4034   ` cfv 5259  (class class class)co 5925   0cc0 7896   1c1 7897   x. cmul 7901   < clt 8078   <_ cle 8079   - cmin 8214   NNcn 9007   2c2 9058   NN0cn0 9266   ZZcz 9343   ZZ>=cuz 9618   QQcq 9710   ...cfz 10100   mod cmo 10431   ^cexp 10647   || cdvds 11969   gcd cgcd 12145   Primecprime 12300

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 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-coll 4149  ax-sep 4152  ax-nul 4160  ax-pow 4208  ax-pr 4243  ax-un 4469  ax-setind 4574  ax-iinf 4625  ax-cnex 7987  ax-resscn 7988  ax-1cn 7989  ax-1re 7990  ax-icn 7991  ax-addcl 7992  ax-addrcl 7993  ax-mulcl 7994  ax-mulrcl 7995  ax-addcom 7996  ax-mulcom 7997  ax-addass 7998  ax-mulass 7999  ax-distr 8000  ax-i2m1 8001  ax-0lt1 8002  ax-1rid 8003  ax-0id 8004  ax-rnegex 8005  ax-precex 8006  ax-cnre 8007  ax-pre-ltirr 8008  ax-pre-ltwlin 8009  ax-pre-lttrn 8010  ax-pre-apti 8011  ax-pre-ltadd 8012  ax-pre-mulgt0 8013  ax-pre-mulext 8014  ax-arch 8015  ax-caucvg 8016

This theorem depends on definitions:  df-bi 117  df-stab 832  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-nel 2463  df-ral 2480  df-rex 2481  df-reu 2482  df-rmo 2483  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3452  df-if 3563  df-pw 3608  df-sn 3629  df-pr 3630  df-op 3632  df-uni 3841  df-int 3876  df-iun 3919  df-br 4035  df-opab 4096  df-mpt 4097  df-tr 4133  df-id 4329  df-po 4332  df-iso 4333  df-iord 4402  df-on 4404  df-ilim 4405  df-suc 4407  df-iom 4628  df-xp 4670  df-rel 4671  df-cnv 4672  df-co 4673  df-dm 4674  df-rn 4675  df-res 4676  df-ima 4677  df-iota 5220  df-fun 5261  df-fn 5262  df-f 5263  df-f1 5264  df-fo 5265  df-f1o 5266  df-fv 5267  df-isom 5268  df-riota 5880  df-ov 5928  df-oprab 5929  df-mpo 5930  df-1st 6207  df-2nd 6208  df-recs 6372  df-irdg 6437  df-frec 6458  df-1o 6483  df-2o 6484  df-oadd 6487  df-er 6601  df-en 6809  df-dom 6810  df-fin 6811  df-sup 7059  df-pnf 8080  df-mnf 8081  df-xr 8082  df-ltxr 8083  df-le 8084  df-sub 8216  df-neg 8217  df-reap 8619  df-ap 8626  df-div 8717  df-inn 9008  df-2 9066  df-3 9067  df-4 9068  df-n0 9267  df-z 9344  df-uz 9619  df-q 9711  df-rp 9746  df-fz 10101  df-fzo 10235  df-fl 10377  df-mod 10432  df-seqfrec 10557  df-exp 10648  df-ihash 10885  df-cj 11024  df-re 11025  df-im 11026  df-rsqrt 11180  df-abs 11181  df-clim 11461  df-proddc 11733  df-dvds 11970  df-gcd 12146  df-prm 12301  df-phi 12404

This theorem is referenced by:  prmdivdiv  12430  modprminveq  12444  wilthlem1  15300