Theorem prmdiv 15490

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 15418	. . . . . 6 |- ( P e. Prime -> -. P || 1 )
2	1	3ad2ant1 1082	. . . . 5 |- ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) -> -. P || 1 )
3		prmnn 15388	. . . . . . . . . . . 12 |- ( P e. Prime -> P e. NN )
4	3	3ad2ant1 1082	. . . . . . . . . . 11 |- ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) -> P e. NN )
5		simp2 1062	. . . . . . . . . . 11 |- ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) -> A e. ZZ )
6		prmz 15389	. . . . . . . . . . . . . 14 |- ( P e. Prime -> P e. ZZ )
7	6	3ad2ant1 1082	. . . . . . . . . . . . 13 |- ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) -> P e. ZZ )
8		gcdcom 15235	. . . . . . . . . . . . 13 |- ( ( A e. ZZ /\ P e. ZZ ) -> ( A gcd P ) = ( P gcd A ) )
9	5, 7, 8	syl2anc 693	. . . . . . . . . . . 12 |- ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) -> ( A gcd P ) = ( P gcd A ) )
10		coprm 15423	. . . . . . . . . . . . 13 |- ( ( P e. Prime /\ A e. ZZ ) -> ( -. P || A <-> ( P gcd A ) = 1 ) )
11	10	biimp3a 1432	. . . . . . . . . . . 12 |- ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) -> ( P gcd A ) = 1 )
12	9, 11	eqtrd 2656	. . . . . . . . . . 11 |- ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) -> ( A gcd P ) = 1 )
13		eulerth 15488	. . . . . . . . . . 11 |- ( ( P e. NN /\ A e. ZZ /\ ( A gcd P ) = 1 ) -> ( ( A ^ ( phi ` P ) ) mod P ) = ( 1 mod P ) )
14	4, 5, 12, 13	syl3anc 1326	. . . . . . . . . 10 |- ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) -> ( ( A ^ ( phi ` P ) ) mod P ) = ( 1 mod P ) )
15		phiprm 15482	. . . . . . . . . . . . . 14 |- ( P e. Prime -> ( phi ` P ) = ( P - 1 ) )
16	15	3ad2ant1 1082	. . . . . . . . . . . . 13 |- ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) -> ( phi ` P ) = ( P - 1 ) )
17		nnm1nn0 11334	. . . . . . . . . . . . . 14 |- ( P e. NN -> ( P - 1 ) e. NN0 )
18	4, 17	syl 17	. . . . . . . . . . . . 13 |- ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) -> ( P - 1 ) e. NN0 )
19	16, 18	eqeltrd 2701	. . . . . . . . . . . 12 |- ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) -> ( phi ` P ) e. NN0 )
20		zexpcl 12875	. . . . . . . . . . . 12 |- ( ( A e. ZZ /\ ( phi ` P ) e. NN0 ) -> ( A ^ ( phi ` P ) ) e. ZZ )
21	5, 19, 20	syl2anc 693	. . . . . . . . . . 11 |- ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) -> ( A ^ ( phi ` P ) ) e. ZZ )
22		1zzd 11408	. . . . . . . . . . 11 |- ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) -> 1 e. ZZ )
23		moddvds 14991	. . . . . . . . . . 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 ) ) )
24	4, 21, 22, 23	syl3anc 1326	. . . . . . . . . 10 |- ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) -> ( ( ( A ^ ( phi ` P ) ) mod P ) = ( 1 mod P ) <-> P || ( ( A ^ ( phi ` P ) ) - 1 ) ) )
25	14, 24	mpbid 222	. . . . . . . . 9 |- ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) -> P || ( ( A ^ ( phi ` P ) ) - 1 ) )
26		prmuz2 15408	. . . . . . . . . . . . . . . . 17 |- ( P e. Prime -> P e. ( ZZ>= ` 2 ) )
27	26	3ad2ant1 1082	. . . . . . . . . . . . . . . 16 |- ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) -> P e. ( ZZ>= ` 2 ) )
28		uznn0sub 11719	. . . . . . . . . . . . . . . 16 |- ( P e. ( ZZ>= ` 2 ) -> ( P - 2 ) e. NN0 )
29	27, 28	syl 17	. . . . . . . . . . . . . . 15 |- ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) -> ( P - 2 ) e. NN0 )
30		zexpcl 12875	. . . . . . . . . . . . . . 15 |- ( ( A e. ZZ /\ ( P - 2 ) e. NN0 ) -> ( A ^ ( P - 2 ) ) e. ZZ )
31	5, 29, 30	syl2anc 693	. . . . . . . . . . . . . 14 |- ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) -> ( A ^ ( P - 2 ) ) e. ZZ )
32	31	zred 11482	. . . . . . . . . . . . 13 |- ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) -> ( A ^ ( P - 2 ) ) e. RR )
33	32, 4	nndivred 11069	. . . . . . . . . . . 12 |- ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) -> ( ( A ^ ( P - 2 ) ) / P ) e. RR )
34	33	flcld 12599	. . . . . . . . . . 11 |- ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) -> ( |_ ` ( ( A ^ ( P - 2 ) ) / P ) ) e. ZZ )
35	5, 34	zmulcld 11488	. . . . . . . . . 10 |- ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) -> ( A x. ( |_ ` ( ( A ^ ( P - 2 ) ) / P ) ) ) e. ZZ )
36		dvdsmul1 15003	. . . . . . . . . 10 |- ( ( P e. ZZ /\ ( A x. ( |_ ` ( ( A ^ ( P - 2 ) ) / P ) ) ) e. ZZ ) -> P || ( P x. ( A x. ( |_ ` ( ( A ^ ( P - 2 ) ) / P ) ) ) ) )
37	7, 35, 36	syl2anc 693	. . . . . . . . 9 |- ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) -> P || ( P x. ( A x. ( |_ ` ( ( A ^ ( P - 2 ) ) / P ) ) ) ) )
38		1z 11407	. . . . . . . . . . 11 |- 1 e. ZZ
39		zsubcl 11419	. . . . . . . . . . 11 |- ( ( ( A ^ ( phi ` P ) ) e. ZZ /\ 1 e. ZZ ) -> ( ( A ^ ( phi ` P ) ) - 1 ) e. ZZ )
40	21, 38, 39	sylancl 694	. . . . . . . . . 10 |- ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) -> ( ( A ^ ( phi ` P ) ) - 1 ) e. ZZ )
41	7, 35	zmulcld 11488	. . . . . . . . . 10 |- ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) -> ( P x. ( A x. ( |_ ` ( ( A ^ ( P - 2 ) ) / P ) ) ) ) e. ZZ )
42		dvds2sub 15016	. . . . . . . . . 10 |- ( ( P e. ZZ /\ ( ( A ^ ( phi ` P ) ) - 1 ) e. ZZ /\ ( P x. ( A x. ( |_ ` ( ( A ^ ( P - 2 ) ) / P ) ) ) ) e. ZZ ) -> ( ( P || ( ( A ^ ( phi ` P ) ) - 1 ) /\ P || ( P x. ( A x. ( |_ ` ( ( A ^ ( P - 2 ) ) / P ) ) ) ) ) -> P || ( ( ( A ^ ( phi ` P ) ) - 1 ) - ( P x. ( A x. ( |_ ` ( ( A ^ ( P - 2 ) ) / P ) ) ) ) ) ) )
43	7, 40, 41, 42	syl3anc 1326	. . . . . . . . 9 |- ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) -> ( ( P || ( ( A ^ ( phi ` P ) ) - 1 ) /\ P || ( P x. ( A x. ( |_ ` ( ( A ^ ( P - 2 ) ) / P ) ) ) ) ) -> P || ( ( ( A ^ ( phi ` P ) ) - 1 ) - ( P x. ( A x. ( |_ ` ( ( A ^ ( P - 2 ) ) / P ) ) ) ) ) ) )
44	25, 37, 43	mp2and 715	. . . . . . . 8 |- ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) -> P || ( ( ( A ^ ( phi ` P ) ) - 1 ) - ( P x. ( A x. ( |_ ` ( ( A ^ ( P - 2 ) ) / P ) ) ) ) ) )
45	5	zcnd 11483	. . . . . . . . . . . 12 |- ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) -> A e. CC )
46	31	zcnd 11483	. . . . . . . . . . . 12 |- ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) -> ( A ^ ( P - 2 ) ) e. CC )
47	7, 34	zmulcld 11488	. . . . . . . . . . . . 13 |- ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) -> ( P x. ( |_ ` ( ( A ^ ( P - 2 ) ) / P ) ) ) e. ZZ )
48	47	zcnd 11483	. . . . . . . . . . . 12 |- ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) -> ( P x. ( |_ ` ( ( A ^ ( P - 2 ) ) / P ) ) ) e. CC )
49	45, 46, 48	subdid 10486	. . . . . . . . . . 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 ) ) ) ) ) )
50		prmdiv.1	. . . . . . . . . . . . 13 |- R = ( ( A ^ ( P - 2 ) ) mod P )
51	4	nnrpd 11870	. . . . . . . . . . . . . 14 |- ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) -> P e. RR+ )
52		modval 12670	. . . . . . . . . . . . . 14 |- ( ( ( A ^ ( P - 2 ) ) e. RR /\ P e. RR+ ) -> ( ( A ^ ( P - 2 ) ) mod P ) = ( ( A ^ ( P - 2 ) ) - ( P x. ( |_ ` ( ( A ^ ( P - 2 ) ) / P ) ) ) ) )
53	32, 51, 52	syl2anc 693	. . . . . . . . . . . . 13 |- ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) -> ( ( A ^ ( P - 2 ) ) mod P ) = ( ( A ^ ( P - 2 ) ) - ( P x. ( |_ ` ( ( A ^ ( P - 2 ) ) / P ) ) ) ) )
54	50, 53	syl5eq 2668	. . . . . . . . . . . 12 |- ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) -> R = ( ( A ^ ( P - 2 ) ) - ( P x. ( |_ ` ( ( A ^ ( P - 2 ) ) / P ) ) ) ) )
55	54	oveq2d 6666	. . . . . . . . . . 11 |- ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) -> ( A x. R ) = ( A x. ( ( A ^ ( P - 2 ) ) - ( P x. ( |_ ` ( ( A ^ ( P - 2 ) ) / P ) ) ) ) ) )
56		2m1e1 11135	. . . . . . . . . . . . . . . . 17 |- ( 2 - 1 ) = 1
57	56	oveq2i 6661	. . . . . . . . . . . . . . . 16 |- ( P - ( 2 - 1 ) ) = ( P - 1 )
58	16, 57	syl6eqr 2674	. . . . . . . . . . . . . . 15 |- ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) -> ( phi ` P ) = ( P - ( 2 - 1 ) ) )
59	4	nncnd 11036	. . . . . . . . . . . . . . . 16 |- ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) -> P e. CC )
60		2cnd 11093	. . . . . . . . . . . . . . . 16 |- ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) -> 2 e. CC )
61		1cnd 10056	. . . . . . . . . . . . . . . 16 |- ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) -> 1 e. CC )
62	59, 60, 61	subsubd 10420	. . . . . . . . . . . . . . 15 |- ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) -> ( P - ( 2 - 1 ) ) = ( ( P - 2 ) + 1 ) )
63	58, 62	eqtrd 2656	. . . . . . . . . . . . . 14 |- ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) -> ( phi ` P ) = ( ( P - 2 ) + 1 ) )
64	63	oveq2d 6666	. . . . . . . . . . . . 13 |- ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) -> ( A ^ ( phi ` P ) ) = ( A ^ ( ( P - 2 ) + 1 ) ) )
65	45, 29	expp1d 13009	. . . . . . . . . . . . 13 |- ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) -> ( A ^ ( ( P - 2 ) + 1 ) ) = ( ( A ^ ( P - 2 ) ) x. A ) )
66	46, 45	mulcomd 10061	. . . . . . . . . . . . 13 |- ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) -> ( ( A ^ ( P - 2 ) ) x. A ) = ( A x. ( A ^ ( P - 2 ) ) ) )
67	64, 65, 66	3eqtrd 2660	. . . . . . . . . . . 12 |- ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) -> ( A ^ ( phi ` P ) ) = ( A x. ( A ^ ( P - 2 ) ) ) )
68	34	zcnd 11483	. . . . . . . . . . . . 13 |- ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) -> ( |_ ` ( ( A ^ ( P - 2 ) ) / P ) ) e. CC )
69	59, 45, 68	mul12d 10245	. . . . . . . . . . . 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 ) ) ) ) )
70	67, 69	oveq12d 6668	. . . . . . . . . . 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 ) ) ) ) ) )
71	49, 55, 70	3eqtr4d 2666	. . . . . . . . . 10 |- ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) -> ( A x. R ) = ( ( A ^ ( phi ` P ) ) - ( P x. ( A x. ( |_ ` ( ( A ^ ( P - 2 ) ) / P ) ) ) ) ) )
72	71	oveq1d 6665	. . . . . . . . 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 ) )
73	21	zcnd 11483	. . . . . . . . . 10 |- ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) -> ( A ^ ( phi ` P ) ) e. CC )
74	41	zcnd 11483	. . . . . . . . . 10 |- ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) -> ( P x. ( A x. ( |_ ` ( ( A ^ ( P - 2 ) ) / P ) ) ) ) e. CC )
75	73, 74, 61	sub32d 10424	. . . . . . . . 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 ) ) ) ) ) )
76	72, 75	eqtrd 2656	. . . . . . . 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 ) ) ) ) ) )
77	44, 76	breqtrrd 4681	. . . . . . 7 |- ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) -> P || ( ( A x. R ) - 1 ) )
78		oveq2 6658	. . . . . . . . 9 |- ( R = 0 -> ( A x. R ) = ( A x. 0 ) )
79	78	oveq1d 6665	. . . . . . . 8 |- ( R = 0 -> ( ( A x. R ) - 1 ) = ( ( A x. 0 ) - 1 ) )
80	79	breq2d 4665	. . . . . . 7 |- ( R = 0 -> ( P || ( ( A x. R ) - 1 ) <-> P || ( ( A x. 0 ) - 1 ) ) )
81	77, 80	syl5ibcom 235	. . . . . 6 |- ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) -> ( R = 0 -> P || ( ( A x. 0 ) - 1 ) ) )
82	45	mul01d 10235	. . . . . . . . . 10 |- ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) -> ( A x. 0 ) = 0 )
83	82	oveq1d 6665	. . . . . . . . 9 |- ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) -> ( ( A x. 0 ) - 1 ) = ( 0 - 1 ) )
84		df-neg 10269	. . . . . . . . 9 |- -u 1 = ( 0 - 1 )
85	83, 84	syl6eqr 2674	. . . . . . . 8 |- ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) -> ( ( A x. 0 ) - 1 ) = -u 1 )
86	85	breq2d 4665	. . . . . . 7 |- ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) -> ( P || ( ( A x. 0 ) - 1 ) <-> P || -u 1 ) )
87		dvdsnegb 14999	. . . . . . . 8 |- ( ( P e. ZZ /\ 1 e. ZZ ) -> ( P || 1 <-> P || -u 1 ) )
88	7, 38, 87	sylancl 694	. . . . . . 7 |- ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) -> ( P || 1 <-> P || -u 1 ) )
89	86, 88	bitr4d 271	. . . . . 6 |- ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) -> ( P || ( ( A x. 0 ) - 1 ) <-> P || 1 ) )
90	81, 89	sylibd 229	. . . . 5 |- ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) -> ( R = 0 -> P || 1 ) )
91	2, 90	mtod 189	. . . 4 |- ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) -> -. R = 0 )
92		zmodfz 12692	. . . . . . . 8 |- ( ( ( A ^ ( P - 2 ) ) e. ZZ /\ P e. NN ) -> ( ( A ^ ( P - 2 ) ) mod P ) e. ( 0 ... ( P - 1 ) ) )
93	31, 4, 92	syl2anc 693	. . . . . . 7 |- ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) -> ( ( A ^ ( P - 2 ) ) mod P ) e. ( 0 ... ( P - 1 ) ) )
94	50, 93	syl5eqel 2705	. . . . . 6 |- ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) -> R e. ( 0 ... ( P - 1 ) ) )
95		nn0uz 11722	. . . . . . . 8 |- NN0 = ( ZZ>= ` 0 )
96	18, 95	syl6eleq 2711	. . . . . . 7 |- ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) -> ( P - 1 ) e. ( ZZ>= ` 0 ) )
97		elfzp12 12419	. . . . . . 7 |- ( ( P - 1 ) e. ( ZZ>= ` 0 ) -> ( R e. ( 0 ... ( P - 1 ) ) <-> ( R = 0 \/ R e. ( ( 0 + 1 ) ... ( P - 1 ) ) ) ) )
98	96, 97	syl 17	. . . . . 6 |- ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) -> ( R e. ( 0 ... ( P - 1 ) ) <-> ( R = 0 \/ R e. ( ( 0 + 1 ) ... ( P - 1 ) ) ) ) )
99	94, 98	mpbid 222	. . . . 5 |- ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) -> ( R = 0 \/ R e. ( ( 0 + 1 ) ... ( P - 1 ) ) ) )
100	99	ord 392	. . . 4 |- ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) -> ( -. R = 0 -> R e. ( ( 0 + 1 ) ... ( P - 1 ) ) ) )
101	91, 100	mpd 15	. . 3 |- ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) -> R e. ( ( 0 + 1 ) ... ( P - 1 ) ) )
102		1e0p1 11552	. . . 4 |- 1 = ( 0 + 1 )
103	102	oveq1i 6660	. . 3 |- ( 1 ... ( P - 1 ) ) = ( ( 0 + 1 ) ... ( P - 1 ) )
104	101, 103	syl6eleqr 2712	. 2 |- ( ( P e. Prime /\ A e. ZZ /\ -. P || A ) -> R e. ( 1 ... ( P - 1 ) ) )
105	104, 77	jca 554	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   <-> wb 196   \/ wo 383   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   class class class wbr 4653   ` cfv 5888  (class class class)co 6650   RRcr 9935   0cc0 9936   1c1 9937   + caddc 9939   x. cmul 9941   - cmin 10266   -ucneg 10267   / cdiv 10684   NNcn 11020   2c2 11070   NN0cn0 11292   ZZcz 11377   ZZ>=cuz 11687   RR+crp 11832   ...cfz 12326   |_cfl 12591   mod cmo 12668   ^cexp 12860   || cdvds 14983   gcd cgcd 15216   Primecprime 15385   phicphi 15469

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:  prmdiveq  15491  prmdivdiv  15492  modprminv  15504  wilthlem2  24795  wilthlem3  24796