Theorem prmdiv 12373

Description: Show an explicit expression for the modular inverse of A mod P. (Contributed by Mario Carneiro, 24-Jan-2015.)

Hypothesis

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

Assertion

Ref	Expression
prmdiv	|- ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) -> ( R e. ( 1 ... ( P - 1 ) ) /\ P || ( ( A x. R ) - 1 ) ) )

Proof of Theorem prmdiv

Step	Hyp	Ref	Expression
1		nprmdvds1 12278	. . . . . 6 |- ( P e. Prime -> -. P || 1 )
2	1	3ad2ant1 1020	. . . . 5 |- ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) -> -. P || 1 )
3		prmz 12249	. . . . . . . . . 10 |- ( P e. Prime -> P e. ZZ )
4	3	3ad2ant1 1020	. . . . . . . . 9 |- ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) -> P e. ZZ )
5		simp2 1000	. . . . . . . . . . 11 |- ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) -> A e. ZZ )
6		phiprm 12361	. . . . . . . . . . . . 13 |- ( P e. Prime -> ( phi ` P ) = ( P - 1 ) )
7	6	3ad2ant1 1020	. . . . . . . . . . . 12 |- ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) -> ( phi ` P ) = ( P - 1 ) )
8		prmnn 12248	. . . . . . . . . . . . . 14 |- ( P e. Prime -> P e. NN )
9	8	3ad2ant1 1020	. . . . . . . . . . . . 13 |- ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) -> P e. NN )
10		nnm1nn0 9281	. . . . . . . . . . . . 13 |- ( P e. NN -> ( P - 1 ) e. NN0 )
11	9, 10	syl 14	. . . . . . . . . . . 12 |- ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) -> ( P - 1 ) e. NN0 )
12	7, 11	eqeltrd 2270	. . . . . . . . . . 11 |- ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) -> ( phi ` P ) e. NN0 )
13		zexpcl 10625	. . . . . . . . . . 11 |- ( ( A e. ZZ /\ ( phi ` P ) e. NN0 ) -> ( A ^ ( phi ` P ) ) e. ZZ )
14	5, 12, 13	syl2anc 411	. . . . . . . . . 10 |- ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) -> ( A ^ ( phi ` P ) ) e. ZZ )
15		1z 9343	. . . . . . . . . 10 |- 1 e. ZZ
16		zsubcl 9358	. . . . . . . . . 10 |- ( ( ( A ^ ( phi ` P ) ) e. ZZ /\ 1 e. ZZ ) -> ( ( A ^ ( phi ` P ) ) - 1 ) e. ZZ )
17	14, 15, 16	sylancl 413	. . . . . . . . 9 |- ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) -> ( ( A ^ ( phi ` P ) ) - 1 ) e. ZZ )
18		prmuz2 12269	. . . . . . . . . . . . . . . 16 |- ( P e. Prime -> P e. ( ZZ>= ` 2 ) )
19	18	3ad2ant1 1020	. . . . . . . . . . . . . . 15 |- ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) -> P e. ( ZZ>= ` 2 ) )
20		uznn0sub 9624	. . . . . . . . . . . . . . 15 |- ( P e. ( ZZ>= ` 2 ) -> ( P - 2 ) e. NN0 )
21	19, 20	syl 14	. . . . . . . . . . . . . 14 |- ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) -> ( P - 2 ) e. NN0 )
22		zexpcl 10625	. . . . . . . . . . . . . 14 |- ( ( A e. ZZ /\ ( P - 2 ) e. NN0 ) -> ( A ^ ( P - 2 ) ) e. ZZ )
23	5, 21, 22	syl2anc 411	. . . . . . . . . . . . 13 |- ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) -> ( A ^ ( P - 2 ) ) e. ZZ )
24		znq 9689	. . . . . . . . . . . . 13 |- ( ( ( A ^ ( P - 2 ) ) e. ZZ /\ P e. NN ) -> ( ( A ^ ( P - 2 ) ) / P ) e. QQ )
25	23, 9, 24	syl2anc 411	. . . . . . . . . . . 12 |- ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) -> ( ( A ^ ( P - 2 ) ) / P ) e. QQ )
26	25	flqcld 10346	. . . . . . . . . . 11 |- ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) -> ( |_ ` ( ( A ^ ( P - 2 ) ) / P ) ) e. ZZ )
27	5, 26	zmulcld 9445	. . . . . . . . . 10 |- ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) -> ( A x. ( |_ ` ( ( A ^ ( P - 2 ) ) / P ) ) ) e. ZZ )
28	4, 27	zmulcld 9445	. . . . . . . . 9 |- ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) -> ( P x. ( A x. ( |_ ` ( ( A ^ ( P - 2 ) ) / P ) ) ) ) e. ZZ )
29	5, 4	gcdcomd 12111	. . . . . . . . . . . 12 |- ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) -> ( A gcd P ) = ( P gcd A ) )
30		coprm 12282	. . . . . . . . . . . . 13 |- ( ( P e. Prime /\ A e. ZZ ) -> ( -. P || A <-> ( P gcd A ) = 1 ) )
31	30	biimp3a 1356	. . . . . . . . . . . 12 |- ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) -> ( P gcd A ) = 1 )
32	29, 31	eqtrd 2226	. . . . . . . . . . 11 |- ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) -> ( A gcd P ) = 1 )
33		eulerth 12371	. . . . . . . . . . 11 |- ( ( P e. NN /\ A e. ZZ /\ ( A gcd P ) = 1 ) -> ( ( A ^ ( phi ` P ) ) mod P ) = ( 1 mod P ) )
34	9, 5, 32, 33	syl3anc 1249	. . . . . . . . . 10 |- ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) -> ( ( A ^ ( phi ` P ) ) mod P ) = ( 1 mod P ) )
35		1zzd 9344	. . . . . . . . . . 11 |- ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) -> 1 e. ZZ )
36		moddvds 11942	. . . . . . . . . . 11 |- ( ( P e. NN /\ ( A ^ ( phi ` P ) ) e. ZZ /\ 1 e. ZZ ) -> ( ( ( A ^ ( phi ` P ) ) mod P ) = ( 1 mod P ) <-> P || ( ( A ^ ( phi ` P ) ) - 1 ) ) )
37	9, 14, 35, 36	syl3anc 1249	. . . . . . . . . 10 |- ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) -> ( ( ( A ^ ( phi ` P ) ) mod P ) = ( 1 mod P ) <-> P || ( ( A ^ ( phi ` P ) ) - 1 ) ) )
38	34, 37	mpbid 147	. . . . . . . . 9 |- ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) -> P || ( ( A ^ ( phi ` P ) ) - 1 ) )
39		dvdsmul1 11956	. . . . . . . . . 10 |- ( ( P e. ZZ /\ ( A x. ( |_ ` ( ( A ^ ( P - 2 ) ) / P ) ) ) e. ZZ ) -> P || ( P x. ( A x. ( |_ ` ( ( A ^ ( P - 2 ) ) / P ) ) ) ) )
40	4, 27, 39	syl2anc 411	. . . . . . . . 9 |- ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) -> P || ( P x. ( A x. ( |_ ` ( ( A ^ ( P - 2 ) ) / P ) ) ) ) )
41	4, 17, 28, 38, 40	dvds2subd 11970	. . . . . . . 8 |- ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) -> P || ( ( ( A ^ ( phi ` P ) ) - 1 ) - ( P x. ( A x. ( |_ ` ( ( A ^ ( P - 2 ) ) / P ) ) ) ) ) )
42	5	zcnd 9440	. . . . . . . . . . . 12 |- ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) -> A e. CC )
43	23	zcnd 9440	. . . . . . . . . . . 12 |- ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) -> ( A ^ ( P - 2 ) ) e. CC )
44	4, 26	zmulcld 9445	. . . . . . . . . . . . 13 |- ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) -> ( P x. ( |_ ` ( ( A ^ ( P - 2 ) ) / P ) ) ) e. ZZ )
45	44	zcnd 9440	. . . . . . . . . . . 12 |- ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) -> ( P x. ( |_ ` ( ( A ^ ( P - 2 ) ) / P ) ) ) e. CC )
46	42, 43, 45	subdid 8433	. . . . . . . . . . 11 |- ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) -> ( A x. ( ( A ^ ( P - 2 ) ) - ( P x. ( |_ ` ( ( A ^ ( P - 2 ) ) / P ) ) ) ) ) = ( ( A x. ( A ^ ( P - 2 ) ) ) - ( A x. ( P x. ( |_ ` ( ( A ^ ( P - 2 ) ) / P ) ) ) ) ) )
47		prmdiv.1	. . . . . . . . . . . . 13 |- R = ( ( A ^ ( P - 2 ) ) mod P )
48		zq 9691	. . . . . . . . . . . . . . 15 |- ( ( A ^ ( P - 2 ) ) e. ZZ -> ( A ^ ( P - 2 ) ) e. QQ )
49	23, 48	syl 14	. . . . . . . . . . . . . 14 |- ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) -> ( A ^ ( P - 2 ) ) e. QQ )
50		nnq 9698	. . . . . . . . . . . . . . 15 |- ( P e. NN -> P e. QQ )
51	9, 50	syl 14	. . . . . . . . . . . . . 14 |- ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) -> P e. QQ )
52	9	nngt0d 9026	. . . . . . . . . . . . . 14 |- ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) -> 0 < P )
53		modqval 10395	. . . . . . . . . . . . . 14 |- ( ( ( A ^ ( P - 2 ) ) e. QQ /\ P e. QQ /\ 0 < P ) -> ( ( A ^ ( P - 2 ) ) mod P ) = ( ( A ^ ( P - 2 ) ) - ( P x. ( |_ ` ( ( A ^ ( P - 2 ) ) / P ) ) ) ) )
54	49, 51, 52, 53	syl3anc 1249	. . . . . . . . . . . . 13 |- ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) -> ( ( A ^ ( P - 2 ) ) mod P ) = ( ( A ^ ( P - 2 ) ) - ( P x. ( |_ ` ( ( A ^ ( P - 2 ) ) / P ) ) ) ) )
55	47, 54	eqtrid 2238	. . . . . . . . . . . 12 |- ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) -> R = ( ( A ^ ( P - 2 ) ) - ( P x. ( |_ ` ( ( A ^ ( P - 2 ) ) / P ) ) ) ) )
56	55	oveq2d 5934	. . . . . . . . . . 11 |- ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) -> ( A x. R ) = ( A x. ( ( A ^ ( P - 2 ) ) - ( P x. ( |_ ` ( ( A ^ ( P - 2 ) ) / P ) ) ) ) ) )
57		2m1e1 9100	. . . . . . . . . . . . . . . . 17 |- ( 2 - 1 ) = 1
58	57	oveq2i 5929	. . . . . . . . . . . . . . . 16 |- ( P - ( 2 - 1 ) ) = ( P - 1 )
59	7, 58	eqtr4di 2244	. . . . . . . . . . . . . . 15 |- ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) -> ( phi ` P ) = ( P - ( 2 - 1 ) ) )
60	9	nncnd 8996	. . . . . . . . . . . . . . . 16 |- ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) -> P e. CC )
61		2cnd 9055	. . . . . . . . . . . . . . . 16 |- ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) -> 2 e. CC )
62		1cnd 8035	. . . . . . . . . . . . . . . 16 |- ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) -> 1 e. CC )
63	60, 61, 62	subsubd 8358	. . . . . . . . . . . . . . 15 |- ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) -> ( P - ( 2 - 1 ) ) = ( ( P - 2 ) + 1 ) )
64	59, 63	eqtrd 2226	. . . . . . . . . . . . . 14 |- ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) -> ( phi ` P ) = ( ( P - 2 ) + 1 ) )
65	64	oveq2d 5934	. . . . . . . . . . . . 13 |- ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) -> ( A ^ ( phi ` P ) ) = ( A ^ ( ( P - 2 ) + 1 ) ) )
66	42, 21	expp1d 10745	. . . . . . . . . . . . 13 |- ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) -> ( A ^ ( ( P - 2 ) + 1 ) ) = ( ( A ^ ( P - 2 ) ) x. A ) )
67	43, 42	mulcomd 8041	. . . . . . . . . . . . 13 |- ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) -> ( ( A ^ ( P - 2 ) ) x. A ) = ( A x. ( A ^ ( P - 2 ) ) ) )
68	65, 66, 67	3eqtrd 2230	. . . . . . . . . . . 12 |- ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) -> ( A ^ ( phi ` P ) ) = ( A x. ( A ^ ( P - 2 ) ) ) )
69	26	zcnd 9440	. . . . . . . . . . . . 13 |- ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) -> ( |_ ` ( ( A ^ ( P - 2 ) ) / P ) ) e. CC )
70	60, 42, 69	mul12d 8171	. . . . . . . . . . . 12 |- ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) -> ( P x. ( A x. ( |_ ` ( ( A ^ ( P - 2 ) ) / P ) ) ) ) = ( A x. ( P x. ( |_ ` ( ( A ^ ( P - 2 ) ) / P ) ) ) ) )
71	68, 70	oveq12d 5936	. . . . . . . . . . 11 |- ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) -> ( ( A ^ ( phi ` P ) ) - ( P x. ( A x. ( |_ ` ( ( A ^ ( P - 2 ) ) / P ) ) ) ) ) = ( ( A x. ( A ^ ( P - 2 ) ) ) - ( A x. ( P x. ( |_ ` ( ( A ^ ( P - 2 ) ) / P ) ) ) ) ) )
72	46, 56, 71	3eqtr4d 2236	. . . . . . . . . 10 |- ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) -> ( A x. R ) = ( ( A ^ ( phi ` P ) ) - ( P x. ( A x. ( |_ ` ( ( A ^ ( P - 2 ) ) / P ) ) ) ) ) )
73	72	oveq1d 5933	. . . . . . . . 9 |- ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) -> ( ( A x. R ) - 1 ) = ( ( ( A ^ ( phi ` P ) ) - ( P x. ( A x. ( |_ ` ( ( A ^ ( P - 2 ) ) / P ) ) ) ) ) - 1 ) )
74	14	zcnd 9440	. . . . . . . . . 10 |- ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) -> ( A ^ ( phi ` P ) ) e. CC )
75	28	zcnd 9440	. . . . . . . . . 10 |- ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) -> ( P x. ( A x. ( |_ ` ( ( A ^ ( P - 2 ) ) / P ) ) ) ) e. CC )
76	74, 75, 62	sub32d 8362	. . . . . . . . 9 |- ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) -> ( ( ( A ^ ( phi ` P ) ) - ( P x. ( A x. ( |_ ` ( ( A ^ ( P - 2 ) ) / P ) ) ) ) ) - 1 ) = ( ( ( A ^ ( phi ` P ) ) - 1 ) - ( P x. ( A x. ( |_ ` ( ( A ^ ( P - 2 ) ) / P ) ) ) ) ) )
77	73, 76	eqtrd 2226	. . . . . . . 8 |- ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) -> ( ( A x. R ) - 1 ) = ( ( ( A ^ ( phi ` P ) ) - 1 ) - ( P x. ( A x. ( |_ ` ( ( A ^ ( P - 2 ) ) / P ) ) ) ) ) )
78	41, 77	breqtrrd 4057	. . . . . . 7 |- ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) -> P || ( ( A x. R ) - 1 ) )
79		oveq2 5926	. . . . . . . . 9 |- ( R = 0 -> ( A x. R ) = ( A x. 0 ) )
80	79	oveq1d 5933	. . . . . . . 8 |- ( R = 0 -> ( ( A x. R ) - 1 ) = ( ( A x. 0 ) - 1 ) )
81	80	breq2d 4041	. . . . . . 7 |- ( R = 0 -> ( P || ( ( A x. R ) - 1 ) <-> P || ( ( A x. 0 ) - 1 ) ) )
82	78, 81	syl5ibcom 155	. . . . . 6 |- ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) -> ( R = 0 -> P || ( ( A x. 0 ) - 1 ) ) )
83	42	mul01d 8412	. . . . . . . . . 10 |- ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) -> ( A x. 0 ) = 0 )
84	83	oveq1d 5933	. . . . . . . . 9 |- ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) -> ( ( A x. 0 ) - 1 ) = ( 0 - 1 ) )
85		df-neg 8193	. . . . . . . . 9 |- -u 1 = ( 0 - 1 )
86	84, 85	eqtr4di 2244	. . . . . . . 8 |- ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) -> ( ( A x. 0 ) - 1 ) = -u 1 )
87	86	breq2d 4041	. . . . . . 7 |- ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) -> ( P || ( ( A x. 0 ) - 1 ) <-> P || -u 1 ) )
88		dvdsnegb 11951	. . . . . . . 8 |- ( ( P e. ZZ /\ 1 e. ZZ ) -> ( P || 1 <-> P || -u 1 ) )
89	4, 15, 88	sylancl 413	. . . . . . 7 |- ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) -> ( P || 1 <-> P || -u 1 ) )
90	87, 89	bitr4d 191	. . . . . 6 |- ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) -> ( P || ( ( A x. 0 ) - 1 ) <-> P || 1 ) )
91	82, 90	sylibd 149	. . . . 5 |- ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) -> ( R = 0 -> P || 1 ) )
92	2, 91	mtod 664	. . . 4 |- ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) -> -. R = 0 )
93		zmodfz 10417	. . . . . . . 8 |- ( ( ( A ^ ( P - 2 ) ) e. ZZ /\ P e. NN ) -> ( ( A ^ ( P - 2 ) ) mod P ) e. ( 0 ... ( P - 1 ) ) )
94	23, 9, 93	syl2anc 411	. . . . . . 7 |- ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) -> ( ( A ^ ( P - 2 ) ) mod P ) e. ( 0 ... ( P - 1 ) ) )
95	47, 94	eqeltrid 2280	. . . . . 6 |- ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) -> R e. ( 0 ... ( P - 1 ) ) )
96		nn0uz 9627	. . . . . . . 8 |- NN0 = ( ZZ>= ` 0 )
97	11, 96	eleqtrdi 2286	. . . . . . 7 |- ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) -> ( P - 1 ) e. ( ZZ>= ` 0 ) )
98		elfzp12 10165	. . . . . . 7 |- ( ( P - 1 ) e. ( ZZ>= ` 0 ) -> ( R e. ( 0 ... ( P - 1 ) ) <-> ( R = 0 \/ R e. ( ( 0 + 1 ) ... ( P - 1 ) ) ) ) )
99	97, 98	syl 14	. . . . . 6 |- ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) -> ( R e. ( 0 ... ( P - 1 ) ) <-> ( R = 0 \/ R e. ( ( 0 + 1 ) ... ( P - 1 ) ) ) ) )
100	95, 99	mpbid 147	. . . . 5 |- ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) -> ( R = 0 \/ R e. ( ( 0 + 1 ) ... ( P - 1 ) ) ) )
101	100	ord 725	. . . 4 |- ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) -> ( -. R = 0 -> R e. ( ( 0 + 1 ) ... ( P - 1 ) ) ) )
102	92, 101	mpd 13	. . 3 |- ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) -> R e. ( ( 0 + 1 ) ... ( P - 1 ) ) )
103		1e0p1 9489	. . . 4 |- 1 = ( 0 + 1 )
104	103	oveq1i 5928	. . 3 |- ( 1 ... ( P - 1 ) ) = ( ( 0 + 1 ) ... ( P - 1 ) )
105	102, 104	eleqtrrdi 2287	. 2 |- ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) -> R e. ( 1 ... ( P - 1 ) ) )
106	105, 78	jca 306	1 |- ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) -> ( R e. ( 1 ... ( P - 1 ) ) /\ P || ( ( A x. R ) - 1 ) ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 709   /\ w3a 980   = wceq 1364   e. wcel 2164   class class class wbr 4029   ` cfv 5254  (class class class)co 5918   0cc0 7872   1c1 7873   + caddc 7875   x. cmul 7877   < clt 8054   - cmin 8190   -ucneg 8191   / cdiv 8691   NNcn 8982   2c2 9033   NN0cn0 9240   ZZcz 9317   ZZ>=cuz 9592   QQcq 9684   ...cfz 10074   |_cfl 10337   mod cmo 10393   ^cexp 10609   || cdvds 11930   gcd cgcd 12079   Primecprime 12245   phicphi 12347

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-coll 4144  ax-sep 4147  ax-nul 4155  ax-pow 4203  ax-pr 4238  ax-un 4464  ax-setind 4569  ax-iinf 4620  ax-cnex 7963  ax-resscn 7964  ax-1cn 7965  ax-1re 7966  ax-icn 7967  ax-addcl 7968  ax-addrcl 7969  ax-mulcl 7970  ax-mulrcl 7971  ax-addcom 7972  ax-mulcom 7973  ax-addass 7974  ax-mulass 7975  ax-distr 7976  ax-i2m1 7977  ax-0lt1 7978  ax-1rid 7979  ax-0id 7980  ax-rnegex 7981  ax-precex 7982  ax-cnre 7983  ax-pre-ltirr 7984  ax-pre-ltwlin 7985  ax-pre-lttrn 7986  ax-pre-apti 7987  ax-pre-ltadd 7988  ax-pre-mulgt0 7989  ax-pre-mulext 7990  ax-arch 7991  ax-caucvg 7992

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 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-nel 2460  df-ral 2477  df-rex 2478  df-reu 2479  df-rmo 2480  df-rab 2481  df-v 2762  df-sbc 2986  df-csb 3081  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3447  df-if 3558  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-int 3871  df-iun 3914  df-br 4030  df-opab 4091  df-mpt 4092  df-tr 4128  df-id 4324  df-po 4327  df-iso 4328  df-iord 4397  df-on 4399  df-ilim 4400  df-suc 4402  df-iom 4623  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-rn 4670  df-res 4671  df-ima 4672  df-iota 5215  df-fun 5256  df-fn 5257  df-f 5258  df-f1 5259  df-fo 5260  df-f1o 5261  df-fv 5262  df-isom 5263  df-riota 5873  df-ov 5921  df-oprab 5922  df-mpo 5923  df-1st 6193  df-2nd 6194  df-recs 6358  df-irdg 6423  df-frec 6444  df-1o 6469  df-2o 6470  df-oadd 6473  df-er 6587  df-en 6795  df-dom 6796  df-fin 6797  df-sup 7043  df-pnf 8056  df-mnf 8057  df-xr 8058  df-ltxr 8059  df-le 8060  df-sub 8192  df-neg 8193  df-reap 8594  df-ap 8601  df-div 8692  df-inn 8983  df-2 9041  df-3 9042  df-4 9043  df-n0 9241  df-z 9318  df-uz 9593  df-q 9685  df-rp 9720  df-fz 10075  df-fzo 10209  df-fl 10339  df-mod 10394  df-seqfrec 10519  df-exp 10610  df-ihash 10847  df-cj 10986  df-re 10987  df-im 10988  df-rsqrt 11142  df-abs 11143  df-clim 11422  df-proddc 11694  df-dvds 11931  df-gcd 12080  df-prm 12246  df-phi 12349

This theorem is referenced by:  prmdiveq  12374  prmdivdiv  12375  modprminv  12387