Theorem prmdiveq 12992

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 1031	. . . . . . . . 9 |- ( ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) /\ ( S e. ( 0 ... ( P - 1 ) ) /\ P || ( ( A x. S ) - 1 ) ) ) -> P e. Prime )
2		prmz 12867	. . . . . . . . 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 1032	. . . . . . . . . 10 |- ( ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) /\ ( S e. ( 0 ... ( P - 1 ) ) /\ P || ( ( A x. S ) - 1 ) ) ) -> A e. ZZ )
5		elfzelz 10407	. . . . . . . . . . 11 |- ( S e. ( 0 ... ( P - 1 ) ) -> S e. ZZ )
6	5	ad2antrl 494	. . . . . . . . . 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 9753	. . . . . . . . 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 9649	. . . . . . . . 9 |- 1 e. ZZ
9		zsubcl 9664	. . . . . . . . 9 |- ( ( ( A x. S ) e. ZZ /\ 1 e. ZZ ) -> ( ( A x. S ) - 1 ) e. ZZ )
10	7, 8, 9	sylancl 417	. . . . . . . 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 12991	. . . . . . . . . . . . 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 10407	. . . . . . . . . . 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 9753	. . . . . . . . 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 9664	. . . . . . . . 9 |- ( ( ( A x. R ) e. ZZ /\ 1 e. ZZ ) -> ( ( A x. R ) - 1 ) e. ZZ )
19	17, 8, 18	sylancl 417	. . . . . . . 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 537	. . . . . . . 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 12572	. . . . . . 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 9748	. . . . . . . . 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 9748	. . . . . . . . 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 8332	. . . . . . . . 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 8662	. . . . . . . 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 9748	. . . . . . . . 9 |- ( ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) /\ ( S e. ( 0 ... ( P - 1 ) ) /\ P || ( ( A x. S ) - 1 ) ) ) -> A e. CC )
28		elfznn0 10499	. . . . . . . . . . 11 |- ( S e. ( 0 ... ( P - 1 ) ) -> S e. NN0 )
29	28	ad2antrl 494	. . . . . . . . . 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 9601	. . . . . . . . 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 9748	. . . . . . . . 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 8731	. . . . . . . 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 2274	. . . . . . 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 4151	. . . . . 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 1033	. . . . . . 7 |- ( ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) /\ ( S e. ( 0 ... ( P - 1 ) ) /\ P || ( ( A x. S ) - 1 ) ) ) -> -. P || A )
36		coprm 12900	. . . . . . . 8 |- ( ( P e. Prime /\ A e. ZZ ) -> ( -. P || A <-> ( P gcd A ) = 1 ) )
37	1, 4, 36	syl2anc 415	. . . . . . 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 9752	. . . . . . 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 12848	. . . . . . 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 1278	. . . . . 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 437	. . . . 5 |- ( ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) /\ ( S e. ( 0 ... ( P - 1 ) ) /\ P || ( ( A x. S ) - 1 ) ) ) -> P || ( S - R ) )
43		prmnn 12866	. . . . . . 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 12544	. . . . . 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 1278	. . . . 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 10005	. . . . . 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 10012	. . . . . 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 10410	. . . . . 6 |- ( S e. ( 0 ... ( P - 1 ) ) -> 0 <_ S )
53	52	ad2antrl 494	. . . . 5 |- ( ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) /\ ( S e. ( 0 ... ( P - 1 ) ) /\ P || ( ( A x. S ) - 1 ) ) ) -> 0 <_ S )
54		elfzle2 10411	. . . . . . 7 |- ( S e. ( 0 ... ( P - 1 ) ) -> S <_ ( P - 1 ) )
55	54	ad2antrl 494	. . . . . 6 |- ( ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) /\ ( S e. ( 0 ... ( P - 1 ) ) /\ P || ( ( A x. S ) - 1 ) ) ) -> S <_ ( P - 1 ) )
56		zltlem1 9681	. . . . . . 7 |- ( ( S e. ZZ /\ P e. ZZ ) -> ( S < P <-> S <_ ( P - 1 ) ) )
57	6, 3, 56	syl2anc 415	. . . . . 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 10764	. . . . 5 |- ( ( ( S e. QQ /\ P e. QQ ) /\ ( 0 <_ S /\ S < P ) ) -> ( S mod P ) = S )
60	49, 51, 53, 58, 59	syl22anc 1279	. . . 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 12887	. . . . . . . . 9 |- ( P e. Prime -> P e. ( ZZ>= ` 2 ) )
62		uznn0sub 9933	. . . . . . . . 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 10969	. . . . . . . 8 |- ( ( A e. ZZ /\ ( P - 2 ) e. NN0 ) -> ( A ^ ( P - 2 ) ) e. ZZ )
65	4, 63, 64	syl2anc 415	. . . . . . 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 10005	. . . . . . 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 9327	. . . . . 6 |- ( ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) /\ ( S e. ( 0 ... ( P - 1 ) ) /\ P || ( ( A x. S ) - 1 ) ) ) -> 0 < P )
69		modqabs2 10773	. . . . . 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 1278	. . . . 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 6085	. . . . 5 |- ( R mod P ) = ( ( ( A ^ ( P - 2 ) ) mod P ) mod P )
72	70, 71, 11	3eqtr4g 2296	. . . 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 2279	. . 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 10502	. . . . . 6 |- ( 1 ... ( P - 1 ) ) C_ ( 0 ... ( P - 1 ) )
76	75	sseli 3244	. . . . 5 |- ( R e. ( 1 ... ( P - 1 ) ) -> R e. ( 0 ... ( P - 1 ) ) )
77		eleq1 2301	. . . . 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 6083	. . . . . . 7 |- ( S = R -> ( A x. S ) = ( A x. R ) )
80	79	oveq1d 6090	. . . . . 6 |- ( S = R -> ( ( A x. S ) - 1 ) = ( ( A x. R ) - 1 ) )
81	80	breq2d 4137	. . . . 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 1009   = wceq 1402   e. wcel 2209   class class class wbr 4125   ` cfv 5372  (class class class)co 6075   0cc0 8169   1c1 8170   x. cmul 8174   < clt 8350   <_ cle 8351   - cmin 8487   NNcn 9283   2c2 9334   NN0cn0 9542   ZZcz 9623   ZZ>=cuz 9900   QQcq 9998   ...cfz 10390   mod cmo 10737   ^cexp 10953   || cdvds 12532   gcd cgcd 12708   Primecprime 12863

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 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 4241  ax-sep 4244  ax-nul 4254  ax-pow 4306  ax-pr 4341  ax-un 4573  ax-setind 4679  ax-iinf 4730  ax-cnex 8260  ax-resscn 8261  ax-1cn 8262  ax-1re 8263  ax-icn 8264  ax-addcl 8265  ax-addrcl 8266  ax-mulcl 8267  ax-mulrcl 8268  ax-addcom 8269  ax-mulcom 8270  ax-addass 8271  ax-mulass 8272  ax-distr 8273  ax-i2m1 8274  ax-0lt1 8275  ax-1rid 8276  ax-0id 8277  ax-rnegex 8278  ax-precex 8279  ax-cnre 8280  ax-pre-ltirr 8281  ax-pre-ltwlin 8282  ax-pre-lttrn 8283  ax-pre-apti 8284  ax-pre-ltadd 8285  ax-pre-mulgt0 8286  ax-pre-mulext 8287  ax-arch 8288  ax-caucvg 8289

This theorem 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 3636  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-int 3966  df-iun 4009  df-br 4126  df-opab 4188  df-mpt 4189  df-tr 4225  df-id 4433  df-po 4436  df-iso 4437  df-iord 4506  df-on 4508  df-ilim 4509  df-suc 4511  df-iom 4733  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-res 4781  df-ima 4782  df-iota 5332  df-fun 5374  df-fn 5375  df-f 5376  df-f1 5377  df-fo 5378  df-f1o 5379  df-fv 5380  df-isom 5381  df-riota 6028  df-ov 6078  df-oprab 6079  df-mpo 6080  df-1st 6364  df-2nd 6365  df-recs 6566  df-irdg 6631  df-frec 6652  df-1o 6677  df-2o 6678  df-oadd 6681  df-er 6797  df-en 7013  df-dom 7014  df-fin 7015  df-sup 7314  df-pnf 8352  df-mnf 8353  df-xr 8354  df-ltxr 8355  df-le 8356  df-sub 8489  df-neg 8490  df-reap 8893  df-ap 8900  df-div 8993  df-inn 9284  df-2 9342  df-3 9343  df-4 9344  df-n0 9543  df-z 9624  df-uz 9901  df-q 9999  df-rp 10034  df-fz 10391  df-fzo 10528  df-fl 10683  df-mod 10738  df-seqfrec 10863  df-exp 10954  df-ihash 11193  df-cj 11585  df-re 11586  df-im 11587  df-rsqrt 11742  df-abs 11743  df-clim 12023  df-proddc 12296  df-dvds 12533  df-gcd 12709  df-prm 12864  df-phi 12967

This theorem is referenced by:  prmdivdiv  12993  modprminveq  13007  wilthlem1  16008