Theorem prmdiv 12237

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 12142	. . . . . 6 |- ( P e. Prime -> -. P || 1 )
2	1	3ad2ant1 1018	. . . . 5 |- ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) -> -. P || 1 )
3		prmz 12113	. . . . . . . . . 10 |- ( P e. Prime -> P e. ZZ )
4	3	3ad2ant1 1018	. . . . . . . . 9 |- ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) -> P e. ZZ )
5		simp2 998	. . . . . . . . . . 11 |- ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) -> A e. ZZ )
6		phiprm 12225	. . . . . . . . . . . . 13 |- ( P e. Prime -> ( phi ` P ) = ( P - 1 ) )
7	6	3ad2ant1 1018	. . . . . . . . . . . 12 |- ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) -> ( phi ` P ) = ( P - 1 ) )
8		prmnn 12112	. . . . . . . . . . . . . 14 |- ( P e. Prime -> P e. NN )
9	8	3ad2ant1 1018	. . . . . . . . . . . . 13 |- ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) -> P e. NN )
10		nnm1nn0 9219	. . . . . . . . . . . . 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 2254	. . . . . . . . . . 11 |- ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) -> ( phi ` P ) e. NN0 )
13		zexpcl 10537	. . . . . . . . . . 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 9281	. . . . . . . . . 10 |- 1 e. ZZ
16		zsubcl 9296	. . . . . . . . . 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 12133	. . . . . . . . . . . . . . . 16 |- ( P e. Prime -> P e. ( ZZ>= ` 2 ) )
19	18	3ad2ant1 1018	. . . . . . . . . . . . . . 15 |- ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) -> P e. ( ZZ>= ` 2 ) )
20		uznn0sub 9561	. . . . . . . . . . . . . . 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 10537	. . . . . . . . . . . . . 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 9626	. . . . . . . . . . . . 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 10279	. . . . . . . . . . 11 |- ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) -> ( |_ ` ( ( A ^ ( P - 2 ) ) / P ) ) e. ZZ )
27	5, 26	zmulcld 9383	. . . . . . . . . 10 |- ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) -> ( A x. ( |_ ` ( ( A ^ ( P - 2 ) ) / P ) ) ) e. ZZ )
28	4, 27	zmulcld 9383	. . . . . . . . 9 |- ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) -> ( P x. ( A x. ( |_ ` ( ( A ^ ( P - 2 ) ) / P ) ) ) ) e. ZZ )
29	5, 4	gcdcomd 11977	. . . . . . . . . . . 12 |- ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) -> ( A gcd P ) = ( P gcd A ) )
30		coprm 12146	. . . . . . . . . . . . 13 |- ( ( P e. Prime /\ A e. ZZ ) -> ( -. P || A <-> ( P gcd A ) = 1 ) )
31	30	biimp3a 1345	. . . . . . . . . . . 12 |- ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) -> ( P gcd A ) = 1 )
32	29, 31	eqtrd 2210	. . . . . . . . . . 11 |- ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) -> ( A gcd P ) = 1 )
33		eulerth 12235	. . . . . . . . . . 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 1238	. . . . . . . . . 10 |- ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) -> ( ( A ^ ( phi ` P ) ) mod P ) = ( 1 mod P ) )
35		1zzd 9282	. . . . . . . . . . 11 |- ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) -> 1 e. ZZ )
36		moddvds 11808	. . . . . . . . . . 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 1238	. . . . . . . . . 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 11822	. . . . . . . . . 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 11836	. . . . . . . 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 9378	. . . . . . . . . . . 12 |- ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) -> A e. CC )
43	23	zcnd 9378	. . . . . . . . . . . 12 |- ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) -> ( A ^ ( P - 2 ) ) e. CC )
44	4, 26	zmulcld 9383	. . . . . . . . . . . . 13 |- ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) -> ( P x. ( |_ ` ( ( A ^ ( P - 2 ) ) / P ) ) ) e. ZZ )
45	44	zcnd 9378	. . . . . . . . . . . 12 |- ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) -> ( P x. ( |_ ` ( ( A ^ ( P - 2 ) ) / P ) ) ) e. CC )
46	42, 43, 45	subdid 8373	. . . . . . . . . . 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 9628	. . . . . . . . . . . . . . 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 9635	. . . . . . . . . . . . . . 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 8965	. . . . . . . . . . . . . 14 |- ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) -> 0 < P )
53		modqval 10326	. . . . . . . . . . . . . 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 1238	. . . . . . . . . . . . 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 2222	. . . . . . . . . . . 12 |- ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) -> R = ( ( A ^ ( P - 2 ) ) - ( P x. ( |_ ` ( ( A ^ ( P - 2 ) ) / P ) ) ) ) )
56	55	oveq2d 5893	. . . . . . . . . . 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 9039	. . . . . . . . . . . . . . . . 17 |- ( 2 - 1 ) = 1
58	57	oveq2i 5888	. . . . . . . . . . . . . . . 16 |- ( P - ( 2 - 1 ) ) = ( P - 1 )
59	7, 58	eqtr4di 2228	. . . . . . . . . . . . . . 15 |- ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) -> ( phi ` P ) = ( P - ( 2 - 1 ) ) )
60	9	nncnd 8935	. . . . . . . . . . . . . . . 16 |- ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) -> P e. CC )
61		2cnd 8994	. . . . . . . . . . . . . . . 16 |- ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) -> 2 e. CC )
62		1cnd 7975	. . . . . . . . . . . . . . . 16 |- ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) -> 1 e. CC )
63	60, 61, 62	subsubd 8298	. . . . . . . . . . . . . . 15 |- ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) -> ( P - ( 2 - 1 ) ) = ( ( P - 2 ) + 1 ) )
64	59, 63	eqtrd 2210	. . . . . . . . . . . . . 14 |- ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) -> ( phi ` P ) = ( ( P - 2 ) + 1 ) )
65	64	oveq2d 5893	. . . . . . . . . . . . 13 |- ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) -> ( A ^ ( phi ` P ) ) = ( A ^ ( ( P - 2 ) + 1 ) ) )
66	42, 21	expp1d 10657	. . . . . . . . . . . . 13 |- ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) -> ( A ^ ( ( P - 2 ) + 1 ) ) = ( ( A ^ ( P - 2 ) ) x. A ) )
67	43, 42	mulcomd 7981	. . . . . . . . . . . . 13 |- ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) -> ( ( A ^ ( P - 2 ) ) x. A ) = ( A x. ( A ^ ( P - 2 ) ) ) )
68	65, 66, 67	3eqtrd 2214	. . . . . . . . . . . 12 |- ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) -> ( A ^ ( phi ` P ) ) = ( A x. ( A ^ ( P - 2 ) ) ) )
69	26	zcnd 9378	. . . . . . . . . . . . 13 |- ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) -> ( |_ ` ( ( A ^ ( P - 2 ) ) / P ) ) e. CC )
70	60, 42, 69	mul12d 8111	. . . . . . . . . . . 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 5895	. . . . . . . . . . 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 2220	. . . . . . . . . 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 5892	. . . . . . . . 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 9378	. . . . . . . . . 10 |- ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) -> ( A ^ ( phi ` P ) ) e. CC )
75	28	zcnd 9378	. . . . . . . . . 10 |- ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) -> ( P x. ( A x. ( |_ ` ( ( A ^ ( P - 2 ) ) / P ) ) ) ) e. CC )
76	74, 75, 62	sub32d 8302	. . . . . . . . 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 2210	. . . . . . . 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 4033	. . . . . . 7 |- ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) -> P || ( ( A x. R ) - 1 ) )
79		oveq2 5885	. . . . . . . . 9 |- ( R = 0 -> ( A x. R ) = ( A x. 0 ) )
80	79	oveq1d 5892	. . . . . . . 8 |- ( R = 0 -> ( ( A x. R ) - 1 ) = ( ( A x. 0 ) - 1 ) )
81	80	breq2d 4017	. . . . . . 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 8352	. . . . . . . . . 10 |- ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) -> ( A x. 0 ) = 0 )
84	83	oveq1d 5892	. . . . . . . . 9 |- ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) -> ( ( A x. 0 ) - 1 ) = ( 0 - 1 ) )
85		df-neg 8133	. . . . . . . . 9 |- -u 1 = ( 0 - 1 )
86	84, 85	eqtr4di 2228	. . . . . . . 8 |- ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) -> ( ( A x. 0 ) - 1 ) = -u 1 )
87	86	breq2d 4017	. . . . . . 7 |- ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) -> ( P || ( ( A x. 0 ) - 1 ) <-> P || -u 1 ) )
88		dvdsnegb 11817	. . . . . . . 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 663	. . . 4 |- ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) -> -. R = 0 )
93		zmodfz 10348	. . . . . . . 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 2264	. . . . . 6 |- ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) -> R e. ( 0 ... ( P - 1 ) ) )
96		nn0uz 9564	. . . . . . . 8 |- NN0 = ( ZZ>= ` 0 )
97	11, 96	eleqtrdi 2270	. . . . . . 7 |- ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) -> ( P - 1 ) e. ( ZZ>= ` 0 ) )
98		elfzp12 10101	. . . . . . 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 724	. . . 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 9427	. . . 4 |- 1 = ( 0 + 1 )
104	103	oveq1i 5887	. . 3 |- ( 1 ... ( P - 1 ) ) = ( ( 0 + 1 ) ... ( P - 1 ) )
105	102, 104	eleqtrrdi 2271	. 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 708   /\ w3a 978   = wceq 1353   e. wcel 2148   class class class wbr 4005   ` cfv 5218  (class class class)co 5877   0cc0 7813   1c1 7814   + caddc 7816   x. cmul 7818   < clt 7994   - cmin 8130   -ucneg 8131   / cdiv 8631   NNcn 8921   2c2 8972   NN0cn0 9178   ZZcz 9255   ZZ>=cuz 9530   QQcq 9621   ...cfz 10010   |_cfl 10270   mod cmo 10324   ^cexp 10521   || cdvds 11796   gcd cgcd 11945   Primecprime 12109   phicphi 12211

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4120  ax-sep 4123  ax-nul 4131  ax-pow 4176  ax-pr 4211  ax-un 4435  ax-setind 4538  ax-iinf 4589  ax-cnex 7904  ax-resscn 7905  ax-1cn 7906  ax-1re 7907  ax-icn 7908  ax-addcl 7909  ax-addrcl 7910  ax-mulcl 7911  ax-mulrcl 7912  ax-addcom 7913  ax-mulcom 7914  ax-addass 7915  ax-mulass 7916  ax-distr 7917  ax-i2m1 7918  ax-0lt1 7919  ax-1rid 7920  ax-0id 7921  ax-rnegex 7922  ax-precex 7923  ax-cnre 7924  ax-pre-ltirr 7925  ax-pre-ltwlin 7926  ax-pre-lttrn 7927  ax-pre-apti 7928  ax-pre-ltadd 7929  ax-pre-mulgt0 7930  ax-pre-mulext 7931  ax-arch 7932  ax-caucvg 7933

This theorem depends on definitions:  df-bi 117  df-stab 831  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rmo 2463  df-rab 2464  df-v 2741  df-sbc 2965  df-csb 3060  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-nul 3425  df-if 3537  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-int 3847  df-iun 3890  df-br 4006  df-opab 4067  df-mpt 4068  df-tr 4104  df-id 4295  df-po 4298  df-iso 4299  df-iord 4368  df-on 4370  df-ilim 4371  df-suc 4373  df-iom 4592  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-res 4640  df-ima 4641  df-iota 5180  df-fun 5220  df-fn 5221  df-f 5222  df-f1 5223  df-fo 5224  df-f1o 5225  df-fv 5226  df-isom 5227  df-riota 5833  df-ov 5880  df-oprab 5881  df-mpo 5882  df-1st 6143  df-2nd 6144  df-recs 6308  df-irdg 6373  df-frec 6394  df-1o 6419  df-2o 6420  df-oadd 6423  df-er 6537  df-en 6743  df-dom 6744  df-fin 6745  df-sup 6985  df-pnf 7996  df-mnf 7997  df-xr 7998  df-ltxr 7999  df-le 8000  df-sub 8132  df-neg 8133  df-reap 8534  df-ap 8541  df-div 8632  df-inn 8922  df-2 8980  df-3 8981  df-4 8982  df-n0 9179  df-z 9256  df-uz 9531  df-q 9622  df-rp 9656  df-fz 10011  df-fzo 10145  df-fl 10272  df-mod 10325  df-seqfrec 10448  df-exp 10522  df-ihash 10758  df-cj 10853  df-re 10854  df-im 10855  df-rsqrt 11009  df-abs 11010  df-clim 11289  df-proddc 11561  df-dvds 11797  df-gcd 11946  df-prm 12110  df-phi 12213

This theorem is referenced by:  prmdiveq  12238  prmdivdiv  12239  modprminv  12251