Theorem 1259lem4 14618

Description: Lemma for 1259prm 14620. Calculate a power mod. In decimal, we calculate 2 ^ 3 0 6 = ( 2 ^ 7 6 ) ^ 4 x. 4 == 5 ^ 4 x. 4 = 2 N - 1 8, 2 ^ 6 1 2 = ( 2 ^ 3 0 6 ) ^ 2 == 1 8 ^ 2 = 3 2 4, 2 ^ 6 2 9 = 2 ^ 6 1 2 x. 2 ^ 1 7 == 3 2 4 x. 1 3 6 = 3 5 N - 1 and finally 2 ^ ( N - 1 ) = ( 2 ^ 6 2 9 ) ^ 2 == 1 ^ 2 = 1. (Contributed by Mario Carneiro, 22-Feb-2014.) (Revised by Mario Carneiro, 20-Apr-2015.)

Hypothesis

Ref	Expression
1259prm.1	|- N = ;;; 1 2 5 9

Assertion

Ref	Expression
1259lem4	|- ( ( 2 ^ ( N - 1 ) ) mod N ) = ( 1 mod N )

Proof of Theorem 1259lem4

Step	Hyp	Ref	Expression
1		2nn 10610	. 2 |- 2 e. NN
2		6nn0 10733	. . . 4 |- 6 e. NN0
3		2nn0 10729	. . . 4 |- 2 e. NN0
4	2, 3	deccl 10909	. . 3 |- ; 6 2 e. NN0
5		9nn0 10736	. . 3 |- 9 e. NN0
6	4, 5	deccl 10909	. 2 |- ;; 6 2 9 e. NN0
7		0z 10792	. 2 |- 0 e. ZZ
8		1nn 10463	. 2 |- 1 e. NN
9		1nn0 10728	. 2 |- 1 e. NN0
10		1259prm.1	. . . . . 6 |- N = ;;; 1 2 5 9
11	9, 3	deccl 10909	. . . . . . . 8 |- ; 1 2 e. NN0
12		5nn0 10732	. . . . . . . 8 |- 5 e. NN0
13	11, 12	deccl 10909	. . . . . . 7 |- ;; 1 2 5 e. NN0
14		8nn0 10735	. . . . . . 7 |- 8 e. NN0
15		8p1e9 10583	. . . . . . 7 |- ( 8 + 1 ) = 9
16		eqid 2382	. . . . . . 7 |- ;;; 1 2 5 8 = ;;; 1 2 5 8
17	13, 14, 15, 16	decsuc 10918	. . . . . 6 |- (;;; 1 2 5 8 + 1 ) = ;;; 1 2 5 9
18	10, 17	eqtr4i 2414	. . . . 5 |- N = (;;; 1 2 5 8 + 1 )
19	18	oveq1i 6206	. . . 4 |- ( N - 1 ) = ( (;;; 1 2 5 8 + 1 ) - 1 )
20	13, 14	deccl 10909	. . . . . 6 |- ;;; 1 2 5 8 e. NN0
21	20	nn0cni 10724	. . . . 5 |- ;;; 1 2 5 8 e. CC
22		ax-1cn 9461	. . . . 5 |- 1 e. CC
23	21, 22	pncan3oi 9749	. . . 4 |- ( (;;; 1 2 5 8 + 1 ) - 1 ) = ;;; 1 2 5 8
24	19, 23	eqtri 2411	. . 3 |- ( N - 1 ) = ;;; 1 2 5 8
25	24, 20	eqeltri 2466	. 2 |- ( N - 1 ) e. NN0
26		9nn 10617	. . . . 5 |- 9 e. NN
27	13, 26	decnncl 10908	. . . 4 |- ;;; 1 2 5 9 e. NN
28	10, 27	eqeltri 2466	. . 3 |- N e. NN
29	2, 9	deccl 10909	. . . 4 |- ; 6 1 e. NN0
30	29, 3	deccl 10909	. . 3 |- ;; 6 1 2 e. NN0
31		3nn0 10730	. . . . 5 |- 3 e. NN0
32		4nn0 10731	. . . . 5 |- 4 e. NN0
33	31, 32	deccl 10909	. . . 4 |- ; 3 4 e. NN0
34	33	nn0zi 10806	. . 3 |- ; 3 4 e. ZZ
35	31, 3	deccl 10909	. . . 4 |- ; 3 2 e. NN0
36	35, 32	deccl 10909	. . 3 |- ;; 3 2 4 e. NN0
37		7nn0 10734	. . . 4 |- 7 e. NN0
38	9, 37	deccl 10909	. . 3 |- ; 1 7 e. NN0
39	9, 31	deccl 10909	. . . 4 |- ; 1 3 e. NN0
40	39, 2	deccl 10909	. . 3 |- ;; 1 3 6 e. NN0
41		0nn0 10727	. . . . . 6 |- 0 e. NN0
42	31, 41	deccl 10909	. . . . 5 |- ; 3 0 e. NN0
43	42, 2	deccl 10909	. . . 4 |- ;; 3 0 6 e. NN0
44		8nn 10616	. . . . 5 |- 8 e. NN
45	9, 44	decnncl 10908	. . . 4 |- ; 1 8 e. NN
46	11, 32	deccl 10909	. . . . 5 |- ;; 1 2 4 e. NN0
47	46, 9	deccl 10909	. . . 4 |- ;;; 1 2 4 1 e. NN0
48	9, 12	deccl 10909	. . . . . 6 |- ; 1 5 e. NN0
49	48, 31	deccl 10909	. . . . 5 |- ;; 1 5 3 e. NN0
50		1z 10811	. . . . 5 |- 1 e. ZZ
51	12, 41	deccl 10909	. . . . 5 |- ; 5 0 e. NN0
52	48, 3	deccl 10909	. . . . . 6 |- ;; 1 5 2 e. NN0
53	3, 12	deccl 10909	. . . . . 6 |- ; 2 5 e. NN0
54	37, 2	deccl 10909	. . . . . . 7 |- ; 7 6 e. NN0
55	10	1259lem3 14617	. . . . . . 7 |- ( ( 2 ^; 7 6 ) mod N ) = ( 5 mod N )
56		eqid 2382	. . . . . . . 8 |- ; 7 6 = ; 7 6
57		4p1e5 10579	. . . . . . . . 9 |- ( 4 + 1 ) = 5
58		7cn 10536	. . . . . . . . . 10 |- 7 e. CC
59		2cn 10523	. . . . . . . . . 10 |- 2 e. CC
60		7t2e14 10977	. . . . . . . . . 10 |- ( 7 x. 2 ) = ; 1 4
61	58, 59, 60	mulcomli 9514	. . . . . . . . 9 |- ( 2 x. 7 ) = ; 1 4
62	9, 32, 57, 61	decsuc 10918	. . . . . . . 8 |- ( ( 2 x. 7 ) + 1 ) = ; 1 5
63		6cn 10534	. . . . . . . . 9 |- 6 e. CC
64		6t2e12 10972	. . . . . . . . 9 |- ( 6 x. 2 ) = ; 1 2
65	63, 59, 64	mulcomli 9514	. . . . . . . 8 |- ( 2 x. 6 ) = ; 1 2
66	3, 37, 2, 56, 3, 9, 62, 65	decmul2c 10943	. . . . . . 7 |- ( 2 x. ; 7 6 ) = ;; 1 5 2
67	53	nn0cni 10724	. . . . . . . . 9 |- ; 2 5 e. CC
68	67	addid2i 9679	. . . . . . . 8 |- ( 0 + ; 2 5 ) = ; 2 5
69	28	nncni 10462	. . . . . . . . . 10 |- N e. CC
70	69	mul02i 9680	. . . . . . . . 9 |- ( 0 x. N ) = 0
71	70	oveq1i 6206	. . . . . . . 8 |- ( ( 0 x. N ) + ; 2 5 ) = ( 0 + ; 2 5 )
72		5t5e25 10971	. . . . . . . 8 |- ( 5 x. 5 ) = ; 2 5
73	68, 71, 72	3eqtr4i 2421	. . . . . . 7 |- ( ( 0 x. N ) + ; 2 5 ) = ( 5 x. 5 )
74	28, 1, 54, 7, 12, 53, 55, 66, 73	mod2xi 14557	. . . . . 6 |- ( ( 2 ^;; 1 5 2 ) mod N ) = (; 2 5 mod N )
75		2p1e3 10576	. . . . . . 7 |- ( 2 + 1 ) = 3
76		eqid 2382	. . . . . . 7 |- ;; 1 5 2 = ;; 1 5 2
77	48, 3, 75, 76	decsuc 10918	. . . . . 6 |- (;; 1 5 2 + 1 ) = ;; 1 5 3
78	51	nn0cni 10724	. . . . . . . 8 |- ; 5 0 e. CC
79	78	addid2i 9679	. . . . . . 7 |- ( 0 + ; 5 0 ) = ; 5 0
80	70	oveq1i 6206	. . . . . . 7 |- ( ( 0 x. N ) + ; 5 0 ) = ( 0 + ; 5 0 )
81		eqid 2382	. . . . . . . 8 |- ; 2 5 = ; 2 5
82		2t2e4 10602	. . . . . . . . . 10 |- ( 2 x. 2 ) = 4
83	82	oveq1i 6206	. . . . . . . . 9 |- ( ( 2 x. 2 ) + 1 ) = ( 4 + 1 )
84	83, 57	eqtri 2411	. . . . . . . 8 |- ( ( 2 x. 2 ) + 1 ) = 5
85		5t2e10 10607	. . . . . . . . 9 |- ( 5 x. 2 ) = 10
86		dec10 10925	. . . . . . . . 9 |- 10 = ; 1 0
87	85, 86	eqtri 2411	. . . . . . . 8 |- ( 5 x. 2 ) = ; 1 0
88	3, 3, 12, 81, 41, 9, 84, 87	decmul1c 10942	. . . . . . 7 |- (; 2 5 x. 2 ) = ; 5 0
89	79, 80, 88	3eqtr4i 2421	. . . . . 6 |- ( ( 0 x. N ) + ; 5 0 ) = (; 2 5 x. 2 )
90	28, 1, 52, 7, 53, 51, 74, 77, 89	modxp1i 14558	. . . . 5 |- ( ( 2 ^;; 1 5 3 ) mod N ) = (; 5 0 mod N )
91		eqid 2382	. . . . . 6 |- ;; 1 5 3 = ;; 1 5 3
92		eqid 2382	. . . . . . . . 9 |- ; 1 5 = ; 1 5
93	59	mulid1i 9509	. . . . . . . . . . 11 |- ( 2 x. 1 ) = 2
94	93	oveq1i 6206	. . . . . . . . . 10 |- ( ( 2 x. 1 ) + 1 ) = ( 2 + 1 )
95	94, 75	eqtri 2411	. . . . . . . . 9 |- ( ( 2 x. 1 ) + 1 ) = 3
96		5cn 10532	. . . . . . . . . . 11 |- 5 e. CC
97	96, 59, 85	mulcomli 9514	. . . . . . . . . 10 |- ( 2 x. 5 ) = 10
98	97, 86	eqtri 2411	. . . . . . . . 9 |- ( 2 x. 5 ) = ; 1 0
99	3, 9, 12, 92, 41, 9, 95, 98	decmul2c 10943	. . . . . . . 8 |- ( 2 x. ; 1 5 ) = ; 3 0
100	99	oveq1i 6206	. . . . . . 7 |- ( ( 2 x. ; 1 5 ) + 0 ) = (; 3 0 + 0 )
101	42	nn0cni 10724	. . . . . . . 8 |- ; 3 0 e. CC
102	101	addid1i 9678	. . . . . . 7 |- (; 3 0 + 0 ) = ; 3 0
103	100, 102	eqtri 2411	. . . . . 6 |- ( ( 2 x. ; 1 5 ) + 0 ) = ; 3 0
104		3cn 10527	. . . . . . . 8 |- 3 e. CC
105		3t2e6 10604	. . . . . . . 8 |- ( 3 x. 2 ) = 6
106	104, 59, 105	mulcomli 9514	. . . . . . 7 |- ( 2 x. 3 ) = 6
107	2	dec0h 10911	. . . . . . 7 |- 6 = ; 0 6
108	106, 107	eqtri 2411	. . . . . 6 |- ( 2 x. 3 ) = ; 0 6
109	3, 48, 31, 91, 2, 41, 103, 108	decmul2c 10943	. . . . 5 |- ( 2 x. ;; 1 5 3 ) = ;; 3 0 6
110	69	mulid2i 9510	. . . . . . . 8 |- ( 1 x. N ) = N
111	110, 10	eqtri 2411	. . . . . . 7 |- ( 1 x. N ) = ;;; 1 2 5 9
112		eqid 2382	. . . . . . 7 |- ;;; 1 2 4 1 = ;;; 1 2 4 1
113	3, 32	deccl 10909	. . . . . . . 8 |- ; 2 4 e. NN0
114		eqid 2382	. . . . . . . . 9 |- ; 2 4 = ; 2 4
115	3, 32, 57, 114	decsuc 10918	. . . . . . . 8 |- (; 2 4 + 1 ) = ; 2 5
116		eqid 2382	. . . . . . . . 9 |- ;; 1 2 5 = ;; 1 2 5
117		eqid 2382	. . . . . . . . 9 |- ;; 1 2 4 = ;; 1 2 4
118		eqid 2382	. . . . . . . . . 10 |- ; 1 2 = ; 1 2
119		1p1e2 10566	. . . . . . . . . 10 |- ( 1 + 1 ) = 2
120		2p2e4 10570	. . . . . . . . . 10 |- ( 2 + 2 ) = 4
121	9, 3, 9, 3, 118, 118, 119, 120	decadd 10936	. . . . . . . . 9 |- (; 1 2 + ; 1 2 ) = ; 2 4
122		5p4e9 10592	. . . . . . . . 9 |- ( 5 + 4 ) = 9
123	11, 12, 11, 32, 116, 117, 121, 122	decadd 10936	. . . . . . . 8 |- (;; 1 2 5 + ;; 1 2 4 ) = ;; 2 4 9
124	113, 115, 123	decsucc 10922	. . . . . . 7 |- ( (;; 1 2 5 + ;; 1 2 4 ) + 1 ) = ;; 2 5 0
125		9p1e10 10584	. . . . . . 7 |- ( 9 + 1 ) = 10
126	13, 5, 46, 9, 111, 112, 124, 125	decaddc2 10938	. . . . . 6 |- ( ( 1 x. N ) + ;;; 1 2 4 1 ) = ;;; 2 5 0 0
127		eqid 2382	. . . . . . 7 |- ; 5 0 = ; 5 0
128	72	oveq1i 6206	. . . . . . . . . . 11 |- ( ( 5 x. 5 ) + 0 ) = (; 2 5 + 0 )
129	67	addid1i 9678	. . . . . . . . . . 11 |- (; 2 5 + 0 ) = ; 2 5
130	128, 129	eqtri 2411	. . . . . . . . . 10 |- ( ( 5 x. 5 ) + 0 ) = ; 2 5
131	96	mul02i 9680	. . . . . . . . . . 11 |- ( 0 x. 5 ) = 0
132	41	dec0h 10911	. . . . . . . . . . 11 |- 0 = ; 0 0
133	131, 132	eqtri 2411	. . . . . . . . . 10 |- ( 0 x. 5 ) = ; 0 0
134	12, 12, 41, 127, 41, 41, 130, 133	decmul1c 10942	. . . . . . . . 9 |- (; 5 0 x. 5 ) = ;; 2 5 0
135	134	oveq1i 6206	. . . . . . . 8 |- ( (; 5 0 x. 5 ) + 0 ) = (;; 2 5 0 + 0 )
136	53, 41	deccl 10909	. . . . . . . . . 10 |- ;; 2 5 0 e. NN0
137	136	nn0cni 10724	. . . . . . . . 9 |- ;; 2 5 0 e. CC
138	137	addid1i 9678	. . . . . . . 8 |- (;; 2 5 0 + 0 ) = ;; 2 5 0
139	135, 138	eqtri 2411	. . . . . . 7 |- ( (; 5 0 x. 5 ) + 0 ) = ;; 2 5 0
140	78	mul01i 9681	. . . . . . . 8 |- (; 5 0 x. 0 ) = 0
141	140, 132	eqtri 2411	. . . . . . 7 |- (; 5 0 x. 0 ) = ; 0 0
142	51, 12, 41, 127, 41, 41, 139, 141	decmul2c 10943	. . . . . 6 |- (; 5 0 x. ; 5 0 ) = ;;; 2 5 0 0
143	126, 142	eqtr4i 2414	. . . . 5 |- ( ( 1 x. N ) + ;;; 1 2 4 1 ) = (; 5 0 x. ; 5 0 )
144	28, 1, 49, 50, 51, 47, 90, 109, 143	mod2xi 14557	. . . 4 |- ( ( 2 ^;; 3 0 6 ) mod N ) = (;;; 1 2 4 1 mod N )
145		eqid 2382	. . . . 5 |- ;; 3 0 6 = ;; 3 0 6
146		eqid 2382	. . . . . 6 |- ; 3 0 = ; 3 0
147	9	dec0h 10911	. . . . . 6 |- 1 = ; 0 1
148		00id 9666	. . . . . . . 8 |- ( 0 + 0 ) = 0
149	106, 148	oveq12i 6208	. . . . . . 7 |- ( ( 2 x. 3 ) + ( 0 + 0 ) ) = ( 6 + 0 )
150	63	addid1i 9678	. . . . . . 7 |- ( 6 + 0 ) = 6
151	149, 150	eqtri 2411	. . . . . 6 |- ( ( 2 x. 3 ) + ( 0 + 0 ) ) = 6
152	59	mul01i 9681	. . . . . . . 8 |- ( 2 x. 0 ) = 0
153	152	oveq1i 6206	. . . . . . 7 |- ( ( 2 x. 0 ) + 1 ) = ( 0 + 1 )
154		0p1e1 10564	. . . . . . 7 |- ( 0 + 1 ) = 1
155	153, 154, 147	3eqtri 2415	. . . . . 6 |- ( ( 2 x. 0 ) + 1 ) = ; 0 1
156	31, 41, 41, 9, 146, 147, 3, 9, 41, 151, 155	decma2c 10935	. . . . 5 |- ( ( 2 x. ; 3 0 ) + 1 ) = ; 6 1
157	3, 42, 2, 145, 3, 9, 156, 65	decmul2c 10943	. . . 4 |- ( 2 x. ;; 3 0 6 ) = ;; 6 1 2
158		eqid 2382	. . . . . 6 |- ; 1 8 = ; 1 8
159	11, 32, 57, 117	decsuc 10918	. . . . . 6 |- (;; 1 2 4 + 1 ) = ;; 1 2 5
160		8cn 10538	. . . . . . 7 |- 8 e. CC
161	160, 22, 15	addcomli 9683	. . . . . 6 |- ( 1 + 8 ) = 9
162	46, 9, 9, 14, 112, 158, 159, 161	decadd 10936	. . . . 5 |- (;;; 1 2 4 1 + ; 1 8 ) = ;;; 1 2 5 9
163	162, 10	eqtr4i 2414	. . . 4 |- (;;; 1 2 4 1 + ; 1 8 ) = N
164	36	nn0cni 10724	. . . . . 6 |- ;; 3 2 4 e. CC
165	164	addid2i 9679	. . . . 5 |- ( 0 + ;; 3 2 4 ) = ;; 3 2 4
166	70	oveq1i 6206	. . . . 5 |- ( ( 0 x. N ) + ;; 3 2 4 ) = ( 0 + ;; 3 2 4 )
167	9, 14	deccl 10909	. . . . . 6 |- ; 1 8 e. NN0
168	9, 32	deccl 10909	. . . . . 6 |- ; 1 4 e. NN0
169		eqid 2382	. . . . . . 7 |- ; 1 4 = ; 1 4
170	22	mulid1i 9509	. . . . . . . . 9 |- ( 1 x. 1 ) = 1
171	170, 119	oveq12i 6208	. . . . . . . 8 |- ( ( 1 x. 1 ) + ( 1 + 1 ) ) = ( 1 + 2 )
172		1p2e3 10577	. . . . . . . 8 |- ( 1 + 2 ) = 3
173	171, 172	eqtri 2411	. . . . . . 7 |- ( ( 1 x. 1 ) + ( 1 + 1 ) ) = 3
174	160	mulid1i 9509	. . . . . . . . 9 |- ( 8 x. 1 ) = 8
175	174	oveq1i 6206	. . . . . . . 8 |- ( ( 8 x. 1 ) + 4 ) = ( 8 + 4 )
176		8p4e12 10952	. . . . . . . 8 |- ( 8 + 4 ) = ; 1 2
177	175, 176	eqtri 2411	. . . . . . 7 |- ( ( 8 x. 1 ) + 4 ) = ; 1 2
178	9, 14, 9, 32, 158, 169, 9, 3, 9, 173, 177	decmac 10934	. . . . . 6 |- ( (; 1 8 x. 1 ) + ; 1 4 ) = ; 3 2
179	160	mulid2i 9510	. . . . . . . . 9 |- ( 1 x. 8 ) = 8
180	179	oveq1i 6206	. . . . . . . 8 |- ( ( 1 x. 8 ) + 6 ) = ( 8 + 6 )
181		8p6e14 10954	. . . . . . . 8 |- ( 8 + 6 ) = ; 1 4
182	180, 181	eqtri 2411	. . . . . . 7 |- ( ( 1 x. 8 ) + 6 ) = ; 1 4
183		8t8e64 10989	. . . . . . 7 |- ( 8 x. 8 ) = ; 6 4
184	14, 9, 14, 158, 32, 2, 182, 183	decmul1c 10942	. . . . . 6 |- (; 1 8 x. 8 ) = ;; 1 4 4
185	167, 9, 14, 158, 32, 168, 178, 184	decmul2c 10943	. . . . 5 |- (; 1 8 x. ; 1 8 ) = ;; 3 2 4
186	165, 166, 185	3eqtr4i 2421	. . . 4 |- ( ( 0 x. N ) + ;; 3 2 4 ) = (; 1 8 x. ; 1 8 )
187	1, 43, 7, 45, 36, 47, 144, 157, 163, 186	mod2xnegi 14559	. . 3 |- ( ( 2 ^;; 6 1 2 ) mod N ) = (;; 3 2 4 mod N )
188	10	1259lem1 14615	. . 3 |- ( ( 2 ^; 1 7 ) mod N ) = (;; 1 3 6 mod N )
189		eqid 2382	. . . 4 |- ;; 6 1 2 = ;; 6 1 2
190		eqid 2382	. . . 4 |- ; 1 7 = ; 1 7
191		eqid 2382	. . . . 5 |- ; 6 1 = ; 6 1
192	2, 9, 119, 191	decsuc 10918	. . . 4 |- (; 6 1 + 1 ) = ; 6 2
193		7p2e9 10597	. . . . 5 |- ( 7 + 2 ) = 9
194	58, 59, 193	addcomli 9683	. . . 4 |- ( 2 + 7 ) = 9
195	29, 3, 9, 37, 189, 190, 192, 194	decadd 10936	. . 3 |- (;; 6 1 2 + ; 1 7 ) = ;; 6 2 9
196	31, 9	deccl 10909	. . . . 5 |- ; 3 1 e. NN0
197		eqid 2382	. . . . . . 7 |- ; 3 1 = ; 3 1
198		3p2e5 10585	. . . . . . . . 9 |- ( 3 + 2 ) = 5
199	104, 59, 198	addcomli 9683	. . . . . . . 8 |- ( 2 + 3 ) = 5
200	9, 3, 31, 118, 199	decaddi 10939	. . . . . . 7 |- (; 1 2 + 3 ) = ; 1 5
201		5p1e6 10580	. . . . . . 7 |- ( 5 + 1 ) = 6
202	11, 12, 31, 9, 116, 197, 200, 201	decadd 10936	. . . . . 6 |- (;; 1 2 5 + ; 3 1 ) = ;; 1 5 6
203	119	oveq1i 6206	. . . . . . . . 9 |- ( ( 1 + 1 ) + 1 ) = ( 2 + 1 )
204	203, 75	eqtri 2411	. . . . . . . 8 |- ( ( 1 + 1 ) + 1 ) = 3
205		7p5e12 10948	. . . . . . . . 9 |- ( 7 + 5 ) = ; 1 2
206	58, 96, 205	addcomli 9683	. . . . . . . 8 |- ( 5 + 7 ) = ; 1 2
207	9, 12, 9, 37, 92, 190, 204, 3, 206	decaddc 10937	. . . . . . 7 |- (; 1 5 + ; 1 7 ) = ; 3 2
208		eqid 2382	. . . . . . . 8 |- ; 3 4 = ; 3 4
209		7p3e10 10598	. . . . . . . . . 10 |- ( 7 + 3 ) = 10
210	58, 104, 209	addcomli 9683	. . . . . . . . 9 |- ( 3 + 7 ) = 10
211	210, 86	eqtri 2411	. . . . . . . 8 |- ( 3 + 7 ) = ; 1 0
212	104	mulid1i 9509	. . . . . . . . . 10 |- ( 3 x. 1 ) = 3
213	22	addid1i 9678	. . . . . . . . . 10 |- ( 1 + 0 ) = 1
214	212, 213	oveq12i 6208	. . . . . . . . 9 |- ( ( 3 x. 1 ) + ( 1 + 0 ) ) = ( 3 + 1 )
215		3p1e4 10578	. . . . . . . . 9 |- ( 3 + 1 ) = 4
216	214, 215	eqtri 2411	. . . . . . . 8 |- ( ( 3 x. 1 ) + ( 1 + 0 ) ) = 4
217		4cn 10530	. . . . . . . . . . 11 |- 4 e. CC
218	217	mulid1i 9509	. . . . . . . . . 10 |- ( 4 x. 1 ) = 4
219	218	oveq1i 6206	. . . . . . . . 9 |- ( ( 4 x. 1 ) + 0 ) = ( 4 + 0 )
220	217	addid1i 9678	. . . . . . . . 9 |- ( 4 + 0 ) = 4
221	32	dec0h 10911	. . . . . . . . 9 |- 4 = ; 0 4
222	219, 220, 221	3eqtri 2415	. . . . . . . 8 |- ( ( 4 x. 1 ) + 0 ) = ; 0 4
223	31, 32, 9, 41, 208, 211, 9, 32, 41, 216, 222	decmac 10934	. . . . . . 7 |- ( (; 3 4 x. 1 ) + ( 3 + 7 ) ) = ; 4 4
224	3	dec0h 10911	. . . . . . . 8 |- 2 = ; 0 2
225	105, 154	oveq12i 6208	. . . . . . . . 9 |- ( ( 3 x. 2 ) + ( 0 + 1 ) ) = ( 6 + 1 )
226		6p1e7 10581	. . . . . . . . 9 |- ( 6 + 1 ) = 7
227	225, 226	eqtri 2411	. . . . . . . 8 |- ( ( 3 x. 2 ) + ( 0 + 1 ) ) = 7
228		4t2e8 10606	. . . . . . . . . 10 |- ( 4 x. 2 ) = 8
229	228	oveq1i 6206	. . . . . . . . 9 |- ( ( 4 x. 2 ) + 2 ) = ( 8 + 2 )
230		8p2e10 10599	. . . . . . . . 9 |- ( 8 + 2 ) = 10
231	229, 230, 86	3eqtri 2415	. . . . . . . 8 |- ( ( 4 x. 2 ) + 2 ) = ; 1 0
232	31, 32, 41, 3, 208, 224, 3, 41, 9, 227, 231	decmac 10934	. . . . . . 7 |- ( (; 3 4 x. 2 ) + 2 ) = ; 7 0
233	9, 3, 31, 3, 118, 207, 33, 41, 37, 223, 232	decma2c 10935	. . . . . 6 |- ( (; 3 4 x. ; 1 2 ) + (; 1 5 + ; 1 7 ) ) = ;; 4 4 0
234	59	addid2i 9679	. . . . . . . . 9 |- ( 0 + 2 ) = 2
235	234	oveq2i 6207	. . . . . . . 8 |- ( ( 3 x. 5 ) + ( 0 + 2 ) ) = ( ( 3 x. 5 ) + 2 )
236		5t3e15 10969	. . . . . . . . . 10 |- ( 5 x. 3 ) = ; 1 5
237	96, 104, 236	mulcomli 9514	. . . . . . . . 9 |- ( 3 x. 5 ) = ; 1 5
238		5p2e7 10590	. . . . . . . . 9 |- ( 5 + 2 ) = 7
239	9, 12, 3, 237, 238	decaddi 10939	. . . . . . . 8 |- ( ( 3 x. 5 ) + 2 ) = ; 1 7
240	235, 239	eqtri 2411	. . . . . . 7 |- ( ( 3 x. 5 ) + ( 0 + 2 ) ) = ; 1 7
241		5t4e20 10970	. . . . . . . . 9 |- ( 5 x. 4 ) = ; 2 0
242	96, 217, 241	mulcomli 9514	. . . . . . . 8 |- ( 4 x. 5 ) = ; 2 0
243	63	addid2i 9679	. . . . . . . 8 |- ( 0 + 6 ) = 6
244	3, 41, 2, 242, 243	decaddi 10939	. . . . . . 7 |- ( ( 4 x. 5 ) + 6 ) = ; 2 6
245	31, 32, 41, 2, 208, 107, 12, 2, 3, 240, 244	decmac 10934	. . . . . 6 |- ( (; 3 4 x. 5 ) + 6 ) = ;; 1 7 6
246	11, 12, 48, 2, 116, 202, 33, 2, 38, 233, 245	decma2c 10935	. . . . 5 |- ( (; 3 4 x. ;; 1 2 5 ) + (;; 1 2 5 + ; 3 1 ) ) = ;;; 4 4 0 6
247	14	dec0h 10911	. . . . . 6 |- 8 = ; 0 8
248	217	addid2i 9679	. . . . . . . 8 |- ( 0 + 4 ) = 4
249	248	oveq2i 6207	. . . . . . 7 |- ( ( 3 x. 9 ) + ( 0 + 4 ) ) = ( ( 3 x. 9 ) + 4 )
250		9cn 10540	. . . . . . . . 9 |- 9 e. CC
251		9t3e27 10991	. . . . . . . . 9 |- ( 9 x. 3 ) = ; 2 7
252	250, 104, 251	mulcomli 9514	. . . . . . . 8 |- ( 3 x. 9 ) = ; 2 7
253		7p4e11 10947	. . . . . . . 8 |- ( 7 + 4 ) = ; 1 1
254	3, 37, 32, 252, 75, 9, 253	decaddci 10940	. . . . . . 7 |- ( ( 3 x. 9 ) + 4 ) = ; 3 1
255	249, 254	eqtri 2411	. . . . . 6 |- ( ( 3 x. 9 ) + ( 0 + 4 ) ) = ; 3 1
256		9t4e36 10992	. . . . . . . 8 |- ( 9 x. 4 ) = ; 3 6
257	250, 217, 256	mulcomli 9514	. . . . . . 7 |- ( 4 x. 9 ) = ; 3 6
258	160, 63, 181	addcomli 9683	. . . . . . 7 |- ( 6 + 8 ) = ; 1 4
259	31, 2, 14, 257, 215, 32, 258	decaddci 10940	. . . . . 6 |- ( ( 4 x. 9 ) + 8 ) = ; 4 4
260	31, 32, 41, 14, 208, 247, 5, 32, 32, 255, 259	decmac 10934	. . . . 5 |- ( (; 3 4 x. 9 ) + 8 ) = ;; 3 1 4
261	13, 5, 13, 14, 10, 24, 33, 32, 196, 246, 260	decma2c 10935	. . . 4 |- ( (; 3 4 x. N ) + ( N - 1 ) ) = ;;;; 4 4 0 6 4
262		eqid 2382	. . . . 5 |- ;; 1 3 6 = ;; 1 3 6
263	9, 5	deccl 10909	. . . . . 6 |- ; 1 9 e. NN0
264	263, 32	deccl 10909	. . . . 5 |- ;; 1 9 4 e. NN0
265		eqid 2382	. . . . . 6 |- ; 1 3 = ; 1 3
266		eqid 2382	. . . . . 6 |- ;; 1 9 4 = ;; 1 9 4
267	5, 37	deccl 10909	. . . . . 6 |- ; 9 7 e. NN0
268	9, 9	deccl 10909	. . . . . . 7 |- ; 1 1 e. NN0
269		eqid 2382	. . . . . . 7 |- ;; 3 2 4 = ;; 3 2 4
270		eqid 2382	. . . . . . . 8 |- ; 1 9 = ; 1 9
271		eqid 2382	. . . . . . . 8 |- ; 9 7 = ; 9 7
272	250, 22, 125	addcomli 9683	. . . . . . . . . 10 |- ( 1 + 9 ) = 10
273	272, 86	eqtri 2411	. . . . . . . . 9 |- ( 1 + 9 ) = ; 1 0
274	9, 41, 154, 273	decsuc 10918	. . . . . . . 8 |- ( ( 1 + 9 ) + 1 ) = ; 1 1
275		9p7e16 10962	. . . . . . . 8 |- ( 9 + 7 ) = ; 1 6
276	9, 5, 5, 37, 270, 271, 274, 2, 275	decaddc 10937	. . . . . . 7 |- (; 1 9 + ; 9 7 ) = ;; 1 1 6
277		eqid 2382	. . . . . . . 8 |- ; 3 2 = ; 3 2
278		eqid 2382	. . . . . . . . 9 |- ; 1 1 = ; 1 1
279	9, 9, 119, 278	decsuc 10918	. . . . . . . 8 |- (; 1 1 + 1 ) = ; 1 2
280	93	oveq1i 6206	. . . . . . . . 9 |- ( ( 2 x. 1 ) + 2 ) = ( 2 + 2 )
281	280, 120, 221	3eqtri 2415	. . . . . . . 8 |- ( ( 2 x. 1 ) + 2 ) = ; 0 4
282	31, 3, 9, 3, 277, 279, 9, 32, 41, 216, 281	decmac 10934	. . . . . . 7 |- ( (; 3 2 x. 1 ) + (; 1 1 + 1 ) ) = ; 4 4
283	218	oveq1i 6206	. . . . . . . 8 |- ( ( 4 x. 1 ) + 6 ) = ( 4 + 6 )
284		6p4e10 10596	. . . . . . . . 9 |- ( 6 + 4 ) = 10
285	63, 217, 284	addcomli 9683	. . . . . . . 8 |- ( 4 + 6 ) = 10
286	283, 285, 86	3eqtri 2415	. . . . . . 7 |- ( ( 4 x. 1 ) + 6 ) = ; 1 0
287	35, 32, 268, 2, 269, 276, 9, 41, 9, 282, 286	decmac 10934	. . . . . 6 |- ( (;; 3 2 4 x. 1 ) + (; 1 9 + ; 9 7 ) ) = ;; 4 4 0
288	154, 147	eqtri 2411	. . . . . . . 8 |- ( 0 + 1 ) = ; 0 1
289		3t3e9 10605	. . . . . . . . . 10 |- ( 3 x. 3 ) = 9
290	289, 148	oveq12i 6208	. . . . . . . . 9 |- ( ( 3 x. 3 ) + ( 0 + 0 ) ) = ( 9 + 0 )
291	250	addid1i 9678	. . . . . . . . 9 |- ( 9 + 0 ) = 9
292	290, 291	eqtri 2411	. . . . . . . 8 |- ( ( 3 x. 3 ) + ( 0 + 0 ) ) = 9
293	106	oveq1i 6206	. . . . . . . . 9 |- ( ( 2 x. 3 ) + 1 ) = ( 6 + 1 )
294	37	dec0h 10911	. . . . . . . . 9 |- 7 = ; 0 7
295	293, 226, 294	3eqtri 2415	. . . . . . . 8 |- ( ( 2 x. 3 ) + 1 ) = ; 0 7
296	31, 3, 41, 9, 277, 288, 31, 37, 41, 292, 295	decmac 10934	. . . . . . 7 |- ( (; 3 2 x. 3 ) + ( 0 + 1 ) ) = ; 9 7
297		4t3e12 10967	. . . . . . . 8 |- ( 4 x. 3 ) = ; 1 2
298		4p2e6 10587	. . . . . . . . 9 |- ( 4 + 2 ) = 6
299	217, 59, 298	addcomli 9683	. . . . . . . 8 |- ( 2 + 4 ) = 6
300	9, 3, 32, 297, 299	decaddi 10939	. . . . . . 7 |- ( ( 4 x. 3 ) + 4 ) = ; 1 6
301	35, 32, 41, 32, 269, 221, 31, 2, 9, 296, 300	decmac 10934	. . . . . 6 |- ( (;; 3 2 4 x. 3 ) + 4 ) = ;; 9 7 6
302	9, 31, 263, 32, 265, 266, 36, 2, 267, 287, 301	decma2c 10935	. . . . 5 |- ( (;; 3 2 4 x. ; 1 3 ) + ;; 1 9 4 ) = ;;; 4 4 0 6
303	154	oveq2i 6207	. . . . . . . 8 |- ( ( 3 x. 6 ) + ( 0 + 1 ) ) = ( ( 3 x. 6 ) + 1 )
304		6t3e18 10973	. . . . . . . . . 10 |- ( 6 x. 3 ) = ; 1 8
305	63, 104, 304	mulcomli 9514	. . . . . . . . 9 |- ( 3 x. 6 ) = ; 1 8
306	9, 14, 15, 305	decsuc 10918	. . . . . . . 8 |- ( ( 3 x. 6 ) + 1 ) = ; 1 9
307	303, 306	eqtri 2411	. . . . . . 7 |- ( ( 3 x. 6 ) + ( 0 + 1 ) ) = ; 1 9
308	9, 3, 3, 65, 120	decaddi 10939	. . . . . . 7 |- ( ( 2 x. 6 ) + 2 ) = ; 1 4
309	31, 3, 41, 3, 277, 224, 2, 32, 9, 307, 308	decmac 10934	. . . . . 6 |- ( (; 3 2 x. 6 ) + 2 ) = ;; 1 9 4
310		6t4e24 10974	. . . . . . 7 |- ( 6 x. 4 ) = ; 2 4
311	63, 217, 310	mulcomli 9514	. . . . . 6 |- ( 4 x. 6 ) = ; 2 4
312	2, 35, 32, 269, 32, 3, 309, 311	decmul1c 10942	. . . . 5 |- (;; 3 2 4 x. 6 ) = ;;; 1 9 4 4
313	36, 39, 2, 262, 32, 264, 302, 312	decmul2c 10943	. . . 4 |- (;; 3 2 4 x. ;; 1 3 6 ) = ;;;; 4 4 0 6 4
314	261, 313	eqtr4i 2414	. . 3 |- ( (; 3 4 x. N ) + ( N - 1 ) ) = (;; 3 2 4 x. ;; 1 3 6 )
315	28, 1, 30, 34, 36, 25, 38, 40, 187, 188, 195, 314	modxai 14556	. 2 |- ( ( 2 ^;; 6 2 9 ) mod N ) = ( ( N - 1 ) mod N )
316		eqid 2382	. . . 4 |- ;; 6 2 9 = ;; 6 2 9
317		eqid 2382	. . . . 5 |- ; 6 2 = ; 6 2
318	148	oveq2i 6207	. . . . . 6 |- ( ( 2 x. 6 ) + ( 0 + 0 ) ) = ( ( 2 x. 6 ) + 0 )
319	65	oveq1i 6206	. . . . . 6 |- ( ( 2 x. 6 ) + 0 ) = (; 1 2 + 0 )
320	11	nn0cni 10724	. . . . . . 7 |- ; 1 2 e. CC
321	320	addid1i 9678	. . . . . 6 |- (; 1 2 + 0 ) = ; 1 2
322	318, 319, 321	3eqtri 2415	. . . . 5 |- ( ( 2 x. 6 ) + ( 0 + 0 ) ) = ; 1 2
323	12	dec0h 10911	. . . . . 6 |- 5 = ; 0 5
324	83, 57, 323	3eqtri 2415	. . . . 5 |- ( ( 2 x. 2 ) + 1 ) = ; 0 5
325	2, 3, 41, 9, 317, 147, 3, 12, 41, 322, 324	decma2c 10935	. . . 4 |- ( ( 2 x. ; 6 2 ) + 1 ) = ;; 1 2 5
326		9t2e18 10990	. . . . 5 |- ( 9 x. 2 ) = ; 1 8
327	250, 59, 326	mulcomli 9514	. . . 4 |- ( 2 x. 9 ) = ; 1 8
328	3, 4, 5, 316, 14, 9, 325, 327	decmul2c 10943	. . 3 |- ( 2 x. ;; 6 2 9 ) = ;;; 1 2 5 8
329	328, 24	eqtr4i 2414	. 2 |- ( 2 x. ;; 6 2 9 ) = ( N - 1 )
330		npcan 9742	. . 3 |- ( ( N e. CC /\ 1 e. CC ) -> ( ( N - 1 ) + 1 ) = N )
331	69, 22, 330	mp2an 670	. 2 |- ( ( N - 1 ) + 1 ) = N
332	70	oveq1i 6206	. . 3 |- ( ( 0 x. N ) + 1 ) = ( 0 + 1 )
333	154, 332, 170	3eqtr4i 2421	. 2 |- ( ( 0 x. N ) + 1 ) = ( 1 x. 1 )
334	1, 6, 7, 8, 9, 25, 315, 329, 331, 333	mod2xnegi 14559	1 |- ( ( 2 ^ ( N - 1 ) ) mod N ) = ( 1 mod N )

Syntax hints:   = wceq 1399   e. wcel 1826  (class class class)co 6196   CCcc 9401   0cc0 9403   1c1 9404   + caddc 9406   x. cmul 9408   - cmin 9718   NNcn 10452   2c2 10502   3c3 10503   4c4 10504   5c5 10505   6c6 10506   7c7 10507   8c8 10508   9c9 10509   10c10 10510   NN0cn0 10712  ;cdc 10895   mod cmo 11896   ^cexp 12069

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-8 1828  ax-9 1830  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006  ax-ext 2360  ax-sep 4488  ax-nul 4496  ax-pow 4543  ax-pr 4601  ax-un 6491  ax-cnex 9459  ax-resscn 9460  ax-1cn 9461  ax-icn 9462  ax-addcl 9463  ax-addrcl 9464  ax-mulcl 9465  ax-mulrcl 9466  ax-mulcom 9467  ax-addass 9468  ax-mulass 9469  ax-distr 9470  ax-i2m1 9471  ax-1ne0 9472  ax-1rid 9473  ax-rnegex 9474  ax-rrecex 9475  ax-cnre 9476  ax-pre-lttri 9477  ax-pre-lttrn 9478  ax-pre-ltadd 9479  ax-pre-mulgt0 9480  ax-pre-sup 9481

This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1402  df-ex 1621  df-nf 1625  df-sb 1748  df-eu 2222  df-mo 2223  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2532  df-ne 2579  df-nel 2580  df-ral 2737  df-rex 2738  df-reu 2739  df-rmo 2740  df-rab 2741  df-v 3036  df-sbc 3253  df-csb 3349  df-dif 3392  df-un 3394  df-in 3396  df-ss 3403  df-pss 3405  df-nul 3712  df-if 3858  df-pw 3929  df-sn 3945  df-pr 3947  df-tp 3949  df-op 3951  df-uni 4164  df-iun 4245  df-br 4368  df-opab 4426  df-mpt 4427  df-tr 4461  df-eprel 4705  df-id 4709  df-po 4714  df-so 4715  df-fr 4752  df-we 4754  df-ord 4795  df-on 4796  df-lim 4797  df-suc 4798  df-xp 4919  df-rel 4920  df-cnv 4921  df-co 4922  df-dm 4923  df-rn 4924  df-res 4925  df-ima 4926  df-iota 5460  df-fun 5498  df-fn 5499  df-f 5500  df-f1 5501  df-fo 5502  df-f1o 5503  df-fv 5504  df-riota 6158  df-ov 6199  df-oprab 6200  df-mpt2 6201  df-om 6600  df-2nd 6700  df-recs 6960  df-rdg 6994  df-er 7229  df-en 7436  df-dom 7437  df-sdom 7438  df-sup 7816  df-pnf 9541  df-mnf 9542  df-xr 9543  df-ltxr 9544  df-le 9545  df-sub 9720  df-neg 9721  df-div 10124  df-nn 10453  df-2 10511  df-3 10512  df-4 10513  df-5 10514  df-6 10515  df-7 10516  df-8 10517  df-9 10518  df-10 10519  df-n0 10713  df-z 10782  df-dec 10896  df-uz 11002  df-rp 11140  df-fl 11828  df-mod 11897  df-seq 12011  df-exp 12070

This theorem is referenced by:  1259prm  14620