Theorem prmdiveq 15491

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		simprr 796	. . . . . . . 8 |- ( ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) /\ ( S e. ( 0 ... ( P - 1 ) ) /\ P || ( ( A x. S ) - 1 ) ) ) -> P || ( ( A x. S ) - 1 ) )
2		prmdiv.1	. . . . . . . . . . 11 |- R = ( ( A ^ ( P - 2 ) ) mod P )
3	2	prmdiv 15490	. . . . . . . . . 10 |- ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) -> ( R e. ( 1 ... ( P - 1 ) ) /\ P || ( ( A x. R ) - 1 ) ) )
4	3	adantr 481	. . . . . . . . 9 |- ( ( ( 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 ) ) )
5	4	simprd 479	. . . . . . . 8 |- ( ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) /\ ( S e. ( 0 ... ( P - 1 ) ) /\ P || ( ( A x. S ) - 1 ) ) ) -> P || ( ( A x. R ) - 1 ) )
6		simpl1 1064	. . . . . . . . . 10 |- ( ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) /\ ( S e. ( 0 ... ( P - 1 ) ) /\ P || ( ( A x. S ) - 1 ) ) ) -> P e. Prime )
7		prmz 15389	. . . . . . . . . 10 |- ( P e. Prime -> P e. ZZ )
8	6, 7	syl 17	. . . . . . . . 9 |- ( ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) /\ ( S e. ( 0 ... ( P - 1 ) ) /\ P || ( ( A x. S ) - 1 ) ) ) -> P e. ZZ )
9		simpl2 1065	. . . . . . . . . . 11 |- ( ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) /\ ( S e. ( 0 ... ( P - 1 ) ) /\ P || ( ( A x. S ) - 1 ) ) ) -> A e. ZZ )
10		elfzelz 12342	. . . . . . . . . . . 12 |- ( S e. ( 0 ... ( P - 1 ) ) -> S e. ZZ )
11	10	ad2antrl 764	. . . . . . . . . . 11 |- ( ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) /\ ( S e. ( 0 ... ( P - 1 ) ) /\ P || ( ( A x. S ) - 1 ) ) ) -> S e. ZZ )
12	9, 11	zmulcld 11488	. . . . . . . . . 10 |- ( ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) /\ ( S e. ( 0 ... ( P - 1 ) ) /\ P || ( ( A x. S ) - 1 ) ) ) -> ( A x. S ) e. ZZ )
13		1z 11407	. . . . . . . . . 10 |- 1 e. ZZ
14		zsubcl 11419	. . . . . . . . . 10 |- ( ( ( A x. S ) e. ZZ /\ 1 e. ZZ ) -> ( ( A x. S ) - 1 ) e. ZZ )
15	12, 13, 14	sylancl 694	. . . . . . . . 9 |- ( ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) /\ ( S e. ( 0 ... ( P - 1 ) ) /\ P || ( ( A x. S ) - 1 ) ) ) -> ( ( A x. S ) - 1 ) e. ZZ )
16	4	simpld 475	. . . . . . . . . . . 12 |- ( ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) /\ ( S e. ( 0 ... ( P - 1 ) ) /\ P || ( ( A x. S ) - 1 ) ) ) -> R e. ( 1 ... ( P - 1 ) ) )
17		elfzelz 12342	. . . . . . . . . . . 12 |- ( R e. ( 1 ... ( P - 1 ) ) -> R e. ZZ )
18	16, 17	syl 17	. . . . . . . . . . 11 |- ( ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) /\ ( S e. ( 0 ... ( P - 1 ) ) /\ P || ( ( A x. S ) - 1 ) ) ) -> R e. ZZ )
19	9, 18	zmulcld 11488	. . . . . . . . . 10 |- ( ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) /\ ( S e. ( 0 ... ( P - 1 ) ) /\ P || ( ( A x. S ) - 1 ) ) ) -> ( A x. R ) e. ZZ )
20		zsubcl 11419	. . . . . . . . . 10 |- ( ( ( A x. R ) e. ZZ /\ 1 e. ZZ ) -> ( ( A x. R ) - 1 ) e. ZZ )
21	19, 13, 20	sylancl 694	. . . . . . . . 9 |- ( ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) /\ ( S e. ( 0 ... ( P - 1 ) ) /\ P || ( ( A x. S ) - 1 ) ) ) -> ( ( A x. R ) - 1 ) e. ZZ )
22		dvds2sub 15016	. . . . . . . . 9 |- ( ( P e. ZZ /\ ( ( A x. S ) - 1 ) e. ZZ /\ ( ( A x. R ) - 1 ) e. ZZ ) -> ( ( P || ( ( A x. S ) - 1 ) /\ P || ( ( A x. R ) - 1 ) ) -> P || ( ( ( A x. S ) - 1 ) - ( ( A x. R ) - 1 ) ) ) )
23	8, 15, 21, 22	syl3anc 1326	. . . . . . . 8 |- ( ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) /\ ( S e. ( 0 ... ( P - 1 ) ) /\ P || ( ( A x. S ) - 1 ) ) ) -> ( ( P || ( ( A x. S ) - 1 ) /\ P || ( ( A x. R ) - 1 ) ) -> P || ( ( ( A x. S ) - 1 ) - ( ( A x. R ) - 1 ) ) ) )
24	1, 5, 23	mp2and 715	. . . . . . 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 ) ) )
25	12	zcnd 11483	. . . . . . . . 9 |- ( ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) /\ ( S e. ( 0 ... ( P - 1 ) ) /\ P || ( ( A x. S ) - 1 ) ) ) -> ( A x. S ) e. CC )
26	19	zcnd 11483	. . . . . . . . 9 |- ( ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) /\ ( S e. ( 0 ... ( P - 1 ) ) /\ P || ( ( A x. S ) - 1 ) ) ) -> ( A x. R ) e. CC )
27		1cnd 10056	. . . . . . . . 9 |- ( ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) /\ ( S e. ( 0 ... ( P - 1 ) ) /\ P || ( ( A x. S ) - 1 ) ) ) -> 1 e. CC )
28	25, 26, 27	nnncan2d 10427	. . . . . . . 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 ) ) )
29	9	zcnd 11483	. . . . . . . . 9 |- ( ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) /\ ( S e. ( 0 ... ( P - 1 ) ) /\ P || ( ( A x. S ) - 1 ) ) ) -> A e. CC )
30		elfznn0 12433	. . . . . . . . . . . 12 |- ( S e. ( 0 ... ( P - 1 ) ) -> S e. NN0 )
31	30	ad2antrl 764	. . . . . . . . . . 11 |- ( ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) /\ ( S e. ( 0 ... ( P - 1 ) ) /\ P || ( ( A x. S ) - 1 ) ) ) -> S e. NN0 )
32	31	nn0red 11352	. . . . . . . . . 10 |- ( ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) /\ ( S e. ( 0 ... ( P - 1 ) ) /\ P || ( ( A x. S ) - 1 ) ) ) -> S e. RR )
33	32	recnd 10068	. . . . . . . . 9 |- ( ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) /\ ( S e. ( 0 ... ( P - 1 ) ) /\ P || ( ( A x. S ) - 1 ) ) ) -> S e. CC )
34	18	zcnd 11483	. . . . . . . . 9 |- ( ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) /\ ( S e. ( 0 ... ( P - 1 ) ) /\ P || ( ( A x. S ) - 1 ) ) ) -> R e. CC )
35	29, 33, 34	subdid 10486	. . . . . . . 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 ) ) )
36	28, 35	eqtr4d 2659	. . . . . . 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 ) ) )
37	24, 36	breqtrd 4679	. . . . . 6 |- ( ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) /\ ( S e. ( 0 ... ( P - 1 ) ) /\ P || ( ( A x. S ) - 1 ) ) ) -> P || ( A x. ( S - R ) ) )
38		simpl3 1066	. . . . . . 7 |- ( ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) /\ ( S e. ( 0 ... ( P - 1 ) ) /\ P || ( ( A x. S ) - 1 ) ) ) -> -. P || A )
39		coprm 15423	. . . . . . . 8 |- ( ( P e. Prime /\ A e. ZZ ) -> ( -. P || A <-> ( P gcd A ) = 1 ) )
40	6, 9, 39	syl2anc 693	. . . . . . 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 ) )
41	38, 40	mpbid 222	. . . . . 6 |- ( ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) /\ ( S e. ( 0 ... ( P - 1 ) ) /\ P || ( ( A x. S ) - 1 ) ) ) -> ( P gcd A ) = 1 )
42	11, 18	zsubcld 11487	. . . . . . 7 |- ( ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) /\ ( S e. ( 0 ... ( P - 1 ) ) /\ P || ( ( A x. S ) - 1 ) ) ) -> ( S - R ) e. ZZ )
43		coprmdvds 15366	. . . . . . 7 |- ( ( P e. ZZ /\ A e. ZZ /\ ( S - R ) e. ZZ ) -> ( ( P || ( A x. ( S - R ) ) /\ ( P gcd A ) = 1 ) -> P || ( S - R ) ) )
44	8, 9, 42, 43	syl3anc 1326	. . . . . 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 ) ) )
45	37, 41, 44	mp2and 715	. . . . 5 |- ( ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) /\ ( S e. ( 0 ... ( P - 1 ) ) /\ P || ( ( A x. S ) - 1 ) ) ) -> P || ( S - R ) )
46		prmnn 15388	. . . . . . 7 |- ( P e. Prime -> P e. NN )
47	6, 46	syl 17	. . . . . 6 |- ( ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) /\ ( S e. ( 0 ... ( P - 1 ) ) /\ P || ( ( A x. S ) - 1 ) ) ) -> P e. NN )
48		moddvds 14991	. . . . . 6 |- ( ( P e. NN /\ S e. ZZ /\ R e. ZZ ) -> ( ( S mod P ) = ( R mod P ) <-> P || ( S - R ) ) )
49	47, 11, 18, 48	syl3anc 1326	. . . . 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 ) ) )
50	45, 49	mpbird 247	. . . 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 ) )
51	47	nnrpd 11870	. . . . 5 |- ( ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) /\ ( S e. ( 0 ... ( P - 1 ) ) /\ P || ( ( A x. S ) - 1 ) ) ) -> P e. RR+ )
52		elfzle1 12344	. . . . . 6 |- ( S e. ( 0 ... ( P - 1 ) ) -> 0 <_ S )
53	52	ad2antrl 764	. . . . 5 |- ( ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) /\ ( S e. ( 0 ... ( P - 1 ) ) /\ P || ( ( A x. S ) - 1 ) ) ) -> 0 <_ S )
54		elfzle2 12345	. . . . . . 7 |- ( S e. ( 0 ... ( P - 1 ) ) -> S <_ ( P - 1 ) )
55	54	ad2antrl 764	. . . . . 6 |- ( ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) /\ ( S e. ( 0 ... ( P - 1 ) ) /\ P || ( ( A x. S ) - 1 ) ) ) -> S <_ ( P - 1 ) )
56		zltlem1 11430	. . . . . . 7 |- ( ( S e. ZZ /\ P e. ZZ ) -> ( S < P <-> S <_ ( P - 1 ) ) )
57	11, 8, 56	syl2anc 693	. . . . . 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 247	. . . . 5 |- ( ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) /\ ( S e. ( 0 ... ( P - 1 ) ) /\ P || ( ( A x. S ) - 1 ) ) ) -> S < P )
59		modid 12695	. . . . 5 |- ( ( ( S e. RR /\ P e. RR+ ) /\ ( 0 <_ S /\ S < P ) ) -> ( S mod P ) = S )
60	32, 51, 53, 58, 59	syl22anc 1327	. . . 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 15408	. . . . . . . . 9 |- ( P e. Prime -> P e. ( ZZ>= ` 2 ) )
62		uznn0sub 11719	. . . . . . . . 9 |- ( P e. ( ZZ>= ` 2 ) -> ( P - 2 ) e. NN0 )
63	6, 61, 62	3syl 18	. . . . . . . 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 12875	. . . . . . . 8 |- ( ( A e. ZZ /\ ( P - 2 ) e. NN0 ) -> ( A ^ ( P - 2 ) ) e. ZZ )
65	9, 63, 64	syl2anc 693	. . . . . . 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	65	zred 11482	. . . . . 6 |- ( ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) /\ ( S e. ( 0 ... ( P - 1 ) ) /\ P || ( ( A x. S ) - 1 ) ) ) -> ( A ^ ( P - 2 ) ) e. RR )
67		modabs2 12704	. . . . . 6 |- ( ( ( A ^ ( P - 2 ) ) e. RR /\ P e. RR+ ) -> ( ( ( A ^ ( P - 2 ) ) mod P ) mod P ) = ( ( A ^ ( P - 2 ) ) mod P ) )
68	66, 51, 67	syl2anc 693	. . . . 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 ) )
69	2	oveq1i 6660	. . . . 5 |- ( R mod P ) = ( ( ( A ^ ( P - 2 ) ) mod P ) mod P )
70	68, 69, 2	3eqtr4g 2681	. . . 4 |- ( ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) /\ ( S e. ( 0 ... ( P - 1 ) ) /\ P || ( ( A x. S ) - 1 ) ) ) -> ( R mod P ) = R )
71	50, 60, 70	3eqtr3d 2664	. . 3 |- ( ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) /\ ( S e. ( 0 ... ( P - 1 ) ) /\ P || ( ( A x. S ) - 1 ) ) ) -> S = R )
72	71	ex 450	. 2 |- ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) -> ( ( S e. ( 0 ... ( P - 1 ) ) /\ P || ( ( A x. S ) - 1 ) ) -> S = R ) )
73		1e0p1 11552	. . . . . . . 8 |- 1 = ( 0 + 1 )
74	73	oveq1i 6660	. . . . . . 7 |- ( 1 ... ( P - 1 ) ) = ( ( 0 + 1 ) ... ( P - 1 ) )
75		0z 11388	. . . . . . . 8 |- 0 e. ZZ
76		fzp1ss 12392	. . . . . . . 8 |- ( 0 e. ZZ -> ( ( 0 + 1 ) ... ( P - 1 ) ) C_ ( 0 ... ( P - 1 ) ) )
77	75, 76	ax-mp 5	. . . . . . 7 |- ( ( 0 + 1 ) ... ( P - 1 ) ) C_ ( 0 ... ( P - 1 ) )
78	74, 77	eqsstri 3635	. . . . . 6 |- ( 1 ... ( P - 1 ) ) C_ ( 0 ... ( P - 1 ) )
79	78	sseli 3599	. . . . 5 |- ( R e. ( 1 ... ( P - 1 ) ) -> R e. ( 0 ... ( P - 1 ) ) )
80		eleq1 2689	. . . . 5 |- ( S = R -> ( S e. ( 0 ... ( P - 1 ) ) <-> R e. ( 0 ... ( P - 1 ) ) ) )
81	79, 80	syl5ibr 236	. . . 4 |- ( S = R -> ( R e. ( 1 ... ( P - 1 ) ) -> S e. ( 0 ... ( P - 1 ) ) ) )
82		oveq2 6658	. . . . . . 7 |- ( S = R -> ( A x. S ) = ( A x. R ) )
83	82	oveq1d 6665	. . . . . 6 |- ( S = R -> ( ( A x. S ) - 1 ) = ( ( A x. R ) - 1 ) )
84	83	breq2d 4665	. . . . 5 |- ( S = R -> ( P || ( ( A x. S ) - 1 ) <-> P || ( ( A x. R ) - 1 ) ) )
85	84	biimprd 238	. . . 4 |- ( S = R -> ( P || ( ( A x. R ) - 1 ) -> P || ( ( A x. S ) - 1 ) ) )
86	81, 85	anim12d 586	. . 3 |- ( S = R -> ( ( R e. ( 1 ... ( P - 1 ) ) /\ P || ( ( A x. R ) - 1 ) ) -> ( S e. ( 0 ... ( P - 1 ) ) /\ P || ( ( A x. S ) - 1 ) ) ) )
87	3, 86	syl5com 31	. 2 |- ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) -> ( S = R -> ( S e. ( 0 ... ( P - 1 ) ) /\ P || ( ( A x. S ) - 1 ) ) ) )
88	72, 87	impbid 202	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   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   C_ wss 3574   class class class wbr 4653   ` cfv 5888  (class class class)co 6650   RRcr 9935   0cc0 9936   1c1 9937   + caddc 9939   x. cmul 9941   < clt 10074   <_ cle 10075   - cmin 10266   NNcn 11020   2c2 11070   NN0cn0 11292   ZZcz 11377   ZZ>=cuz 11687   RR+crp 11832   ...cfz 12326   mod cmo 12668   ^cexp 12860   || cdvds 14983   gcd cgcd 15216   Primecprime 15385

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-2o 7561  df-oadd 7564  df-er 7742  df-map 7859  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-sup 8348  df-inf 8349  df-card 8765  df-cda 8990  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-3 11080  df-n0 11293  df-xnn0 11364  df-z 11378  df-uz 11688  df-rp 11833  df-fz 12327  df-fzo 12466  df-fl 12593  df-mod 12669  df-seq 12802  df-exp 12861  df-hash 13118  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-dvds 14984  df-gcd 15217  df-prm 15386  df-phi 15471

This theorem is referenced by:  prmdivdiv  15492  modprminveq  15505  wilthlem1  24794  wilthlem2  24795