Theorem prmdiv 12991

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 12896	. . . . . 6 |- ( P e. Prime -> -. P || 1 )
2	1	3ad2ant1 1049	. . . . 5 |- ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) -> -. P || 1 )
3		prmz 12867	. . . . . . . . . 10 |- ( P e. Prime -> P e. ZZ )
4	3	3ad2ant1 1049	. . . . . . . . 9 |- ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) -> P e. ZZ )
5		simp2 1029	. . . . . . . . . . 11 |- ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) -> A e. ZZ )
6		phiprm 12979	. . . . . . . . . . . . 13 |- ( P e. Prime -> ( phi ` P ) = ( P - 1 ) )
7	6	3ad2ant1 1049	. . . . . . . . . . . 12 |- ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) -> ( phi ` P ) = ( P - 1 ) )
8		prmnn 12866	. . . . . . . . . . . . . 14 |- ( P e. Prime -> P e. NN )
9	8	3ad2ant1 1049	. . . . . . . . . . . . 13 |- ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) -> P e. NN )
10		nnm1nn0 9583	. . . . . . . . . . . . 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 2315	. . . . . . . . . . 11 |- ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) -> ( phi ` P ) e. NN0 )
13		zexpcl 10969	. . . . . . . . . . 11 |- ( ( A e. ZZ /\ ( phi ` P ) e. NN0 ) -> ( A ^ ( phi ` P ) ) e. ZZ )
14	5, 12, 13	syl2anc 415	. . . . . . . . . 10 |- ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) -> ( A ^ ( phi ` P ) ) e. ZZ )
15		1z 9649	. . . . . . . . . 10 |- 1 e. ZZ
16		zsubcl 9664	. . . . . . . . . 10 |- ( ( ( A ^ ( phi ` P ) ) e. ZZ /\ 1 e. ZZ ) -> ( ( A ^ ( phi ` P ) ) - 1 ) e. ZZ )
17	14, 15, 16	sylancl 417	. . . . . . . . 9 |- ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) -> ( ( A ^ ( phi ` P ) ) - 1 ) e. ZZ )
18		prmuz2 12887	. . . . . . . . . . . . . . . 16 |- ( P e. Prime -> P e. ( ZZ>= ` 2 ) )
19	18	3ad2ant1 1049	. . . . . . . . . . . . . . 15 |- ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) -> P e. ( ZZ>= ` 2 ) )
20		uznn0sub 9933	. . . . . . . . . . . . . . 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 10969	. . . . . . . . . . . . . 14 |- ( ( A e. ZZ /\ ( P - 2 ) e. NN0 ) -> ( A ^ ( P - 2 ) ) e. ZZ )
23	5, 21, 22	syl2anc 415	. . . . . . . . . . . . 13 |- ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) -> ( A ^ ( P - 2 ) ) e. ZZ )
24		znq 10003	. . . . . . . . . . . . 13 |- ( ( ( A ^ ( P - 2 ) ) e. ZZ /\ P e. NN ) -> ( ( A ^ ( P - 2 ) ) / P ) e. QQ )
25	23, 9, 24	syl2anc 415	. . . . . . . . . . . 12 |- ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) -> ( ( A ^ ( P - 2 ) ) / P ) e. QQ )
26	25	flqcld 10690	. . . . . . . . . . 11 |- ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) -> ( |_ ` ( ( A ^ ( P - 2 ) ) / P ) ) e. ZZ )
27	5, 26	zmulcld 9753	. . . . . . . . . 10 |- ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) -> ( A x. ( |_ ` ( ( A ^ ( P - 2 ) ) / P ) ) ) e. ZZ )
28	4, 27	zmulcld 9753	. . . . . . . . 9 |- ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) -> ( P x. ( A x. ( |_ ` ( ( A ^ ( P - 2 ) ) / P ) ) ) ) e. ZZ )
29	5, 4	gcdcomd 12729	. . . . . . . . . . . 12 |- ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) -> ( A gcd P ) = ( P gcd A ) )
30		coprm 12900	. . . . . . . . . . . . 13 |- ( ( P e. Prime /\ A e. ZZ ) -> ( -. P || A <-> ( P gcd A ) = 1 ) )
31	30	biimp3a 1386	. . . . . . . . . . . 12 |- ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) -> ( P gcd A ) = 1 )
32	29, 31	eqtrd 2271	. . . . . . . . . . 11 |- ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) -> ( A gcd P ) = 1 )
33		eulerth 12989	. . . . . . . . . . 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 1278	. . . . . . . . . 10 |- ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) -> ( ( A ^ ( phi ` P ) ) mod P ) = ( 1 mod P ) )
35		1zzd 9650	. . . . . . . . . . 11 |- ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) -> 1 e. ZZ )
36		moddvds 12544	. . . . . . . . . . 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 1278	. . . . . . . . . 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 12558	. . . . . . . . . 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 415	. . . . . . . . 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 12572	. . . . . . . 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 9748	. . . . . . . . . . . 12 |- ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) -> A e. CC )
43	23	zcnd 9748	. . . . . . . . . . . 12 |- ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) -> ( A ^ ( P - 2 ) ) e. CC )
44	4, 26	zmulcld 9753	. . . . . . . . . . . . 13 |- ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) -> ( P x. ( |_ ` ( ( A ^ ( P - 2 ) ) / P ) ) ) e. ZZ )
45	44	zcnd 9748	. . . . . . . . . . . 12 |- ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) -> ( P x. ( |_ ` ( ( A ^ ( P - 2 ) ) / P ) ) ) e. CC )
46	42, 43, 45	subdid 8731	. . . . . . . . . . 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 10005	. . . . . . . . . . . . . . 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 10012	. . . . . . . . . . . . . . 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 9327	. . . . . . . . . . . . . 14 |- ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) -> 0 < P )
53		modqval 10739	. . . . . . . . . . . . . 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 1278	. . . . . . . . . . . . 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 2283	. . . . . . . . . . . 12 |- ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) -> R = ( ( A ^ ( P - 2 ) ) - ( P x. ( |_ ` ( ( A ^ ( P - 2 ) ) / P ) ) ) ) )
56	55	oveq2d 6091	. . . . . . . . . . 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 9401	. . . . . . . . . . . . . . . . 17 |- ( 2 - 1 ) = 1
58	57	oveq2i 6086	. . . . . . . . . . . . . . . 16 |- ( P - ( 2 - 1 ) ) = ( P - 1 )
59	7, 58	eqtr4di 2289	. . . . . . . . . . . . . . 15 |- ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) -> ( phi ` P ) = ( P - ( 2 - 1 ) ) )
60	9	nncnd 9297	. . . . . . . . . . . . . . . 16 |- ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) -> P e. CC )
61		2cnd 9356	. . . . . . . . . . . . . . . 16 |- ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) -> 2 e. CC )
62		1cnd 8332	. . . . . . . . . . . . . . . 16 |- ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) -> 1 e. CC )
63	60, 61, 62	subsubd 8655	. . . . . . . . . . . . . . 15 |- ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) -> ( P - ( 2 - 1 ) ) = ( ( P - 2 ) + 1 ) )
64	59, 63	eqtrd 2271	. . . . . . . . . . . . . 14 |- ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) -> ( phi ` P ) = ( ( P - 2 ) + 1 ) )
65	64	oveq2d 6091	. . . . . . . . . . . . 13 |- ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) -> ( A ^ ( phi ` P ) ) = ( A ^ ( ( P - 2 ) + 1 ) ) )
66	42, 21	expp1d 11090	. . . . . . . . . . . . 13 |- ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) -> ( A ^ ( ( P - 2 ) + 1 ) ) = ( ( A ^ ( P - 2 ) ) x. A ) )
67	43, 42	mulcomd 8337	. . . . . . . . . . . . 13 |- ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) -> ( ( A ^ ( P - 2 ) ) x. A ) = ( A x. ( A ^ ( P - 2 ) ) ) )
68	65, 66, 67	3eqtrd 2275	. . . . . . . . . . . 12 |- ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) -> ( A ^ ( phi ` P ) ) = ( A x. ( A ^ ( P - 2 ) ) ) )
69	26	zcnd 9748	. . . . . . . . . . . . 13 |- ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) -> ( |_ ` ( ( A ^ ( P - 2 ) ) / P ) ) e. CC )
70	60, 42, 69	mul12d 8468	. . . . . . . . . . . 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 6093	. . . . . . . . . . 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 2281	. . . . . . . . . 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 6090	. . . . . . . . 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 9748	. . . . . . . . . 10 |- ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) -> ( A ^ ( phi ` P ) ) e. CC )
75	28	zcnd 9748	. . . . . . . . . 10 |- ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) -> ( P x. ( A x. ( |_ ` ( ( A ^ ( P - 2 ) ) / P ) ) ) ) e. CC )
76	74, 75, 62	sub32d 8659	. . . . . . . . 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 2271	. . . . . . . 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 4153	. . . . . . 7 |- ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) -> P || ( ( A x. R ) - 1 ) )
79		oveq2 6083	. . . . . . . . 9 |- ( R = 0 -> ( A x. R ) = ( A x. 0 ) )
80	79	oveq1d 6090	. . . . . . . 8 |- ( R = 0 -> ( ( A x. R ) - 1 ) = ( ( A x. 0 ) - 1 ) )
81	80	breq2d 4137	. . . . . . 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 8710	. . . . . . . . . 10 |- ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) -> ( A x. 0 ) = 0 )
84	83	oveq1d 6090	. . . . . . . . 9 |- ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) -> ( ( A x. 0 ) - 1 ) = ( 0 - 1 ) )
85		df-neg 8490	. . . . . . . . 9 |- -u 1 = ( 0 - 1 )
86	84, 85	eqtr4di 2289	. . . . . . . 8 |- ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) -> ( ( A x. 0 ) - 1 ) = -u 1 )
87	86	breq2d 4137	. . . . . . 7 |- ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) -> ( P || ( ( A x. 0 ) - 1 ) <-> P || -u 1 ) )
88		dvdsnegb 12553	. . . . . . . 8 |- ( ( P e. ZZ /\ 1 e. ZZ ) -> ( P || 1 <-> P || -u 1 ) )
89	4, 15, 88	sylancl 417	. . . . . . 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 673	. . . 4 |- ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) -> -. R = 0 )
93		zmodfz 10761	. . . . . . . 8 |- ( ( ( A ^ ( P - 2 ) ) e. ZZ /\ P e. NN ) -> ( ( A ^ ( P - 2 ) ) mod P ) e. ( 0 ... ( P - 1 ) ) )
94	23, 9, 93	syl2anc 415	. . . . . . 7 |- ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) -> ( ( A ^ ( P - 2 ) ) mod P ) e. ( 0 ... ( P - 1 ) ) )
95	47, 94	eqeltrid 2325	. . . . . 6 |- ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) -> R e. ( 0 ... ( P - 1 ) ) )
96		nn0uz 9936	. . . . . . . 8 |- NN0 = ( ZZ>= ` 0 )
97	11, 96	eleqtrdi 2331	. . . . . . 7 |- ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) -> ( P - 1 ) e. ( ZZ>= ` 0 ) )
98		elfzp12 10484	. . . . . . 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 736	. . . 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 9797	. . . 4 |- 1 = ( 0 + 1 )
104	103	oveq1i 6085	. . 3 |- ( 1 ... ( P - 1 ) ) = ( ( 0 + 1 ) ... ( P - 1 ) )
105	102, 104	eleqtrrdi 2332	. 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 720   /\ w3a 1009   = wceq 1402   e. wcel 2209   class class class wbr 4125   ` cfv 5372  (class class class)co 6075   0cc0 8169   1c1 8170   + caddc 8172   x. cmul 8174   < clt 8350   - cmin 8487   -ucneg 8488   / cdiv 8992   NNcn 9283   2c2 9334   NN0cn0 9542   ZZcz 9623   ZZ>=cuz 9900   QQcq 9998   ...cfz 10390   |_cfl 10681   mod cmo 10737   ^cexp 10953   || cdvds 12532   gcd cgcd 12708   Primecprime 12863   phicphi 12965

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4241  ax-sep 4244  ax-nul 4254  ax-pow 4306  ax-pr 4341  ax-un 4573  ax-setind 4679  ax-iinf 4730  ax-cnex 8260  ax-resscn 8261  ax-1cn 8262  ax-1re 8263  ax-icn 8264  ax-addcl 8265  ax-addrcl 8266  ax-mulcl 8267  ax-mulrcl 8268  ax-addcom 8269  ax-mulcom 8270  ax-addass 8271  ax-mulass 8272  ax-distr 8273  ax-i2m1 8274  ax-0lt1 8275  ax-1rid 8276  ax-0id 8277  ax-rnegex 8278  ax-precex 8279  ax-cnre 8280  ax-pre-ltirr 8281  ax-pre-ltwlin 8282  ax-pre-lttrn 8283  ax-pre-apti 8284  ax-pre-ltadd 8285  ax-pre-mulgt0 8286  ax-pre-mulext 8287  ax-arch 8288  ax-caucvg 8289

This theorem depends on definitions:  df-bi 117  df-stab 843  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-if 3636  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-int 3966  df-iun 4009  df-br 4126  df-opab 4188  df-mpt 4189  df-tr 4225  df-id 4433  df-po 4436  df-iso 4437  df-iord 4506  df-on 4508  df-ilim 4509  df-suc 4511  df-iom 4733  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-res 4781  df-ima 4782  df-iota 5332  df-fun 5374  df-fn 5375  df-f 5376  df-f1 5377  df-fo 5378  df-f1o 5379  df-fv 5380  df-isom 5381  df-riota 6028  df-ov 6078  df-oprab 6079  df-mpo 6080  df-1st 6364  df-2nd 6365  df-recs 6566  df-irdg 6631  df-frec 6652  df-1o 6677  df-2o 6678  df-oadd 6681  df-er 6797  df-en 7013  df-dom 7014  df-fin 7015  df-sup 7314  df-pnf 8352  df-mnf 8353  df-xr 8354  df-ltxr 8355  df-le 8356  df-sub 8489  df-neg 8490  df-reap 8893  df-ap 8900  df-div 8993  df-inn 9284  df-2 9342  df-3 9343  df-4 9344  df-n0 9543  df-z 9624  df-uz 9901  df-q 9999  df-rp 10034  df-fz 10391  df-fzo 10528  df-fl 10683  df-mod 10738  df-seqfrec 10863  df-exp 10954  df-ihash 11193  df-cj 11585  df-re 11586  df-im 11587  df-rsqrt 11742  df-abs 11743  df-clim 12023  df-proddc 12296  df-dvds 12533  df-gcd 12709  df-prm 12864  df-phi 12967

This theorem is referenced by:  prmdiveq  12992  prmdivdiv  12993  modprminv  13006