Theorem prmdiv 12957

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 12862	. . . . . 6 |- ( P e. Prime -> -. P || 1 )
2	1	3ad2ant1 1045	. . . . 5 |- ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) -> -. P || 1 )
3		prmz 12833	. . . . . . . . . 10 |- ( P e. Prime -> P e. ZZ )
4	3	3ad2ant1 1045	. . . . . . . . 9 |- ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) -> P e. ZZ )
5		simp2 1025	. . . . . . . . . . 11 |- ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) -> A e. ZZ )
6		phiprm 12945	. . . . . . . . . . . . 13 |- ( P e. Prime -> ( phi ` P ) = ( P - 1 ) )
7	6	3ad2ant1 1045	. . . . . . . . . . . 12 |- ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) -> ( phi ` P ) = ( P - 1 ) )
8		prmnn 12832	. . . . . . . . . . . . . 14 |- ( P e. Prime -> P e. NN )
9	8	3ad2ant1 1045	. . . . . . . . . . . . 13 |- ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) -> P e. NN )
10		nnm1nn0 9554	. . . . . . . . . . . . 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 2311	. . . . . . . . . . 11 |- ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) -> ( phi ` P ) e. NN0 )
13		zexpcl 10940	. . . . . . . . . . 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 9620	. . . . . . . . . 10 |- 1 e. ZZ
16		zsubcl 9635	. . . . . . . . . 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 12853	. . . . . . . . . . . . . . . 16 |- ( P e. Prime -> P e. ( ZZ>= ` 2 ) )
19	18	3ad2ant1 1045	. . . . . . . . . . . . . . 15 |- ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) -> P e. ( ZZ>= ` 2 ) )
20		uznn0sub 9904	. . . . . . . . . . . . . . 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 10940	. . . . . . . . . . . . . 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 9974	. . . . . . . . . . . . 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 10661	. . . . . . . . . . 11 |- ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) -> ( |_ ` ( ( A ^ ( P - 2 ) ) / P ) ) e. ZZ )
27	5, 26	zmulcld 9724	. . . . . . . . . 10 |- ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) -> ( A x. ( |_ ` ( ( A ^ ( P - 2 ) ) / P ) ) ) e. ZZ )
28	4, 27	zmulcld 9724	. . . . . . . . 9 |- ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) -> ( P x. ( A x. ( |_ ` ( ( A ^ ( P - 2 ) ) / P ) ) ) ) e. ZZ )
29	5, 4	gcdcomd 12695	. . . . . . . . . . . 12 |- ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) -> ( A gcd P ) = ( P gcd A ) )
30		coprm 12866	. . . . . . . . . . . . 13 |- ( ( P e. Prime /\ A e. ZZ ) -> ( -. P || A <-> ( P gcd A ) = 1 ) )
31	30	biimp3a 1382	. . . . . . . . . . . 12 |- ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) -> ( P gcd A ) = 1 )
32	29, 31	eqtrd 2267	. . . . . . . . . . 11 |- ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) -> ( A gcd P ) = 1 )
33		eulerth 12955	. . . . . . . . . . 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 1274	. . . . . . . . . 10 |- ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) -> ( ( A ^ ( phi ` P ) ) mod P ) = ( 1 mod P ) )
35		1zzd 9621	. . . . . . . . . . 11 |- ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) -> 1 e. ZZ )
36		moddvds 12510	. . . . . . . . . . 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 1274	. . . . . . . . . 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 12524	. . . . . . . . . 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 12538	. . . . . . . 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 9719	. . . . . . . . . . . 12 |- ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) -> A e. CC )
43	23	zcnd 9719	. . . . . . . . . . . 12 |- ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) -> ( A ^ ( P - 2 ) ) e. CC )
44	4, 26	zmulcld 9724	. . . . . . . . . . . . 13 |- ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) -> ( P x. ( |_ ` ( ( A ^ ( P - 2 ) ) / P ) ) ) e. ZZ )
45	44	zcnd 9719	. . . . . . . . . . . 12 |- ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) -> ( P x. ( |_ ` ( ( A ^ ( P - 2 ) ) / P ) ) ) e. CC )
46	42, 43, 45	subdid 8704	. . . . . . . . . . 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 9976	. . . . . . . . . . . . . . 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 9983	. . . . . . . . . . . . . . 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 9298	. . . . . . . . . . . . . 14 |- ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) -> 0 < P )
53		modqval 10710	. . . . . . . . . . . . . 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 1274	. . . . . . . . . . . . 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 2279	. . . . . . . . . . . 12 |- ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) -> R = ( ( A ^ ( P - 2 ) ) - ( P x. ( |_ ` ( ( A ^ ( P - 2 ) ) / P ) ) ) ) )
56	55	oveq2d 6074	. . . . . . . . . . 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 9372	. . . . . . . . . . . . . . . . 17 |- ( 2 - 1 ) = 1
58	57	oveq2i 6069	. . . . . . . . . . . . . . . 16 |- ( P - ( 2 - 1 ) ) = ( P - 1 )
59	7, 58	eqtr4di 2285	. . . . . . . . . . . . . . 15 |- ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) -> ( phi ` P ) = ( P - ( 2 - 1 ) ) )
60	9	nncnd 9268	. . . . . . . . . . . . . . . 16 |- ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) -> P e. CC )
61		2cnd 9327	. . . . . . . . . . . . . . . 16 |- ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) -> 2 e. CC )
62		1cnd 8306	. . . . . . . . . . . . . . . 16 |- ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) -> 1 e. CC )
63	60, 61, 62	subsubd 8628	. . . . . . . . . . . . . . 15 |- ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) -> ( P - ( 2 - 1 ) ) = ( ( P - 2 ) + 1 ) )
64	59, 63	eqtrd 2267	. . . . . . . . . . . . . 14 |- ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) -> ( phi ` P ) = ( ( P - 2 ) + 1 ) )
65	64	oveq2d 6074	. . . . . . . . . . . . 13 |- ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) -> ( A ^ ( phi ` P ) ) = ( A ^ ( ( P - 2 ) + 1 ) ) )
66	42, 21	expp1d 11061	. . . . . . . . . . . . 13 |- ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) -> ( A ^ ( ( P - 2 ) + 1 ) ) = ( ( A ^ ( P - 2 ) ) x. A ) )
67	43, 42	mulcomd 8311	. . . . . . . . . . . . 13 |- ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) -> ( ( A ^ ( P - 2 ) ) x. A ) = ( A x. ( A ^ ( P - 2 ) ) ) )
68	65, 66, 67	3eqtrd 2271	. . . . . . . . . . . 12 |- ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) -> ( A ^ ( phi ` P ) ) = ( A x. ( A ^ ( P - 2 ) ) ) )
69	26	zcnd 9719	. . . . . . . . . . . . 13 |- ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) -> ( |_ ` ( ( A ^ ( P - 2 ) ) / P ) ) e. CC )
70	60, 42, 69	mul12d 8441	. . . . . . . . . . . 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 6076	. . . . . . . . . . 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 2277	. . . . . . . . . 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 6073	. . . . . . . . 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 9719	. . . . . . . . . 10 |- ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) -> ( A ^ ( phi ` P ) ) e. CC )
75	28	zcnd 9719	. . . . . . . . . 10 |- ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) -> ( P x. ( A x. ( |_ ` ( ( A ^ ( P - 2 ) ) / P ) ) ) ) e. CC )
76	74, 75, 62	sub32d 8632	. . . . . . . . 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 2267	. . . . . . . 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 4142	. . . . . . 7 |- ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) -> P || ( ( A x. R ) - 1 ) )
79		oveq2 6066	. . . . . . . . 9 |- ( R = 0 -> ( A x. R ) = ( A x. 0 ) )
80	79	oveq1d 6073	. . . . . . . 8 |- ( R = 0 -> ( ( A x. R ) - 1 ) = ( ( A x. 0 ) - 1 ) )
81	80	breq2d 4126	. . . . . . 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 8683	. . . . . . . . . 10 |- ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) -> ( A x. 0 ) = 0 )
84	83	oveq1d 6073	. . . . . . . . 9 |- ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) -> ( ( A x. 0 ) - 1 ) = ( 0 - 1 ) )
85		df-neg 8463	. . . . . . . . 9 |- -u 1 = ( 0 - 1 )
86	84, 85	eqtr4di 2285	. . . . . . . 8 |- ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) -> ( ( A x. 0 ) - 1 ) = -u 1 )
87	86	breq2d 4126	. . . . . . 7 |- ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) -> ( P || ( ( A x. 0 ) - 1 ) <-> P || -u 1 ) )
88		dvdsnegb 12519	. . . . . . . 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 669	. . . 4 |- ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) -> -. R = 0 )
93		zmodfz 10732	. . . . . . . 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 2321	. . . . . 6 |- ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) -> R e. ( 0 ... ( P - 1 ) ) )
96		nn0uz 9907	. . . . . . . 8 |- NN0 = ( ZZ>= ` 0 )
97	11, 96	eleqtrdi 2327	. . . . . . 7 |- ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) -> ( P - 1 ) e. ( ZZ>= ` 0 ) )
98		elfzp12 10455	. . . . . . 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 732	. . . 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 9768	. . . 4 |- 1 = ( 0 + 1 )
104	103	oveq1i 6068	. . 3 |- ( 1 ... ( P - 1 ) ) = ( ( 0 + 1 ) ... ( P - 1 ) )
105	102, 104	eleqtrrdi 2328	. 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 716   /\ w3a 1005   = wceq 1398   e. wcel 2205   class class class wbr 4114   ` cfv 5357  (class class class)co 6058   0cc0 8143   1c1 8144   + caddc 8146   x. cmul 8148   < clt 8324   - cmin 8460   -ucneg 8461   / cdiv 8963   NNcn 9254   2c2 9305   NN0cn0 9513   ZZcz 9594   ZZ>=cuz 9871   QQcq 9969   ...cfz 10361   |_cfl 10652   mod cmo 10708   ^cexp 10924   || cdvds 12498   gcd cgcd 12674   Primecprime 12829   phicphi 12931

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:  prmdiveq  12958  prmdivdiv  12959  modprminv  12972