Theorem prmdiveq 12958

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 1027	. . . . . . . . 9 |- ( ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) /\ ( S e. ( 0 ... ( P - 1 ) ) /\ P || ( ( A x. S ) - 1 ) ) ) -> P e. Prime )
2		prmz 12833	. . . . . . . . 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 1028	. . . . . . . . . 10 |- ( ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) /\ ( S e. ( 0 ... ( P - 1 ) ) /\ P || ( ( A x. S ) - 1 ) ) ) -> A e. ZZ )
5		elfzelz 10378	. . . . . . . . . . 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 9724	. . . . . . . . 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 9620	. . . . . . . . 9 |- 1 e. ZZ
9		zsubcl 9635	. . . . . . . . 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 12957	. . . . . . . . . . . . 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 10378	. . . . . . . . . . 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 9724	. . . . . . . . 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 9635	. . . . . . . . 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 533	. . . . . . . 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 12538	. . . . . . 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 9719	. . . . . . . . 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 9719	. . . . . . . . 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 8306	. . . . . . . . 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 8635	. . . . . . . 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 9719	. . . . . . . . 9 |- ( ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) /\ ( S e. ( 0 ... ( P - 1 ) ) /\ P || ( ( A x. S ) - 1 ) ) ) -> A e. CC )
28		elfznn0 10470	. . . . . . . . . . 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 9572	. . . . . . . . 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 9719	. . . . . . . . 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 8704	. . . . . . . 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 2270	. . . . . . 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 4140	. . . . . 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 1029	. . . . . . 7 |- ( ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) /\ ( S e. ( 0 ... ( P - 1 ) ) /\ P || ( ( A x. S ) - 1 ) ) ) -> -. P || A )
36		coprm 12866	. . . . . . . 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 9723	. . . . . . 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 12814	. . . . . . 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 1274	. . . . . 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 12832	. . . . . . 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 12510	. . . . . 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 1274	. . . . 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 9976	. . . . . 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 9983	. . . . . 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 10381	. . . . . 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 10382	. . . . . . 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 9652	. . . . . . 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 10735	. . . . 5 |- ( ( ( S e. QQ /\ P e. QQ ) /\ ( 0 <_ S /\ S < P ) ) -> ( S mod P ) = S )
60	49, 51, 53, 58, 59	syl22anc 1275	. . . 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 12853	. . . . . . . . 9 |- ( P e. Prime -> P e. ( ZZ>= ` 2 ) )
62		uznn0sub 9904	. . . . . . . . 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 10940	. . . . . . . 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 9976	. . . . . . 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 9298	. . . . . 6 |- ( ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) /\ ( S e. ( 0 ... ( P - 1 ) ) /\ P || ( ( A x. S ) - 1 ) ) ) -> 0 < P )
69		modqabs2 10744	. . . . . 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 1274	. . . . 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 6068	. . . . 5 |- ( R mod P ) = ( ( ( A ^ ( P - 2 ) ) mod P ) mod P )
72	70, 71, 11	3eqtr4g 2292	. . . 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 2275	. . 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 10473	. . . . . 6 |- ( 1 ... ( P - 1 ) ) C_ ( 0 ... ( P - 1 ) )
76	75	sseli 3238	. . . . 5 |- ( R e. ( 1 ... ( P - 1 ) ) -> R e. ( 0 ... ( P - 1 ) ) )
77		eleq1 2297	. . . . 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 6066	. . . . . . 7 |- ( S = R -> ( A x. S ) = ( A x. R ) )
80	79	oveq1d 6073	. . . . . 6 |- ( S = R -> ( ( A x. S ) - 1 ) = ( ( A x. R ) - 1 ) )
81	80	breq2d 4126	. . . . 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 1005   = wceq 1398   e. wcel 2205   class class class wbr 4114   ` cfv 5357  (class class class)co 6058   0cc0 8143   1c1 8144   x. cmul 8148   < clt 8324   <_ cle 8325   - cmin 8460   NNcn 9254   2c2 9305   NN0cn0 9513   ZZcz 9594   ZZ>=cuz 9871   QQcq 9969   ...cfz 10361   mod cmo 10708   ^cexp 10924   || cdvds 12498   gcd cgcd 12674   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:  prmdivdiv  12959  modprminveq  12973  wilthlem1  15974