Theorem 1259lem4 13265

Description: Lemma for 1259prm 13267. 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.) (Proof shortened by AV, 16-Sep-2021.)

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 9470	. 2 |- 2 e. NN
2		6nn0 9588	. . . 4 |- 6 e. NN0
3		2nn0 9584	. . . 4 |- 2 e. NN0
4	2, 3	deccl 9795	. . 3 |- ; 6 2 e. NN0
5		9nn0 9591	. . 3 |- 9 e. NN0
6	4, 5	deccl 9795	. 2 |- ;; 6 2 9 e. NN0
7		0z 9659	. 2 |- 0 e. ZZ
8		1nn 9317	. 2 |- 1 e. NN
9		1nn0 9583	. 2 |- 1 e. NN0
10		12nn0 9797	. . . . . . 7 |- ; 1 2 e. NN0
11		5nn0 9587	. . . . . . 7 |- 5 e. NN0
12	10, 11	deccl 9795	. . . . . 6 |- ;; 1 2 5 e. NN0
13		8nn0 9590	. . . . . 6 |- 8 e. NN0
14	12, 13	deccl 9795	. . . . 5 |- ;;; 1 2 5 8 e. NN0
15	14	nn0cni 9579	. . . 4 |- ;;; 1 2 5 8 e. CC
16		ax-1cn 8272	. . . 4 |- 1 e. CC
17		1259prm.1	. . . . 5 |- N = ;;; 1 2 5 9
18		8p1e9 9447	. . . . . 6 |- ( 8 + 1 ) = 9
19		eqid 2238	. . . . . 6 |- ;;; 1 2 5 8 = ;;; 1 2 5 8
20	12, 13, 18, 19	decsuc 9816	. . . . 5 |- (;;; 1 2 5 8 + 1 ) = ;;; 1 2 5 9
21	17, 20	eqtr4i 2262	. . . 4 |- N = (;;; 1 2 5 8 + 1 )
22	15, 16, 21	mvrraddi 8544	. . 3 |- ( N - 1 ) = ;;; 1 2 5 8
23	22, 14	eqeltri 2311	. 2 |- ( N - 1 ) e. NN0
24		9nn 9477	. . . . 5 |- 9 e. NN
25	12, 24	decnncl 9804	. . . 4 |- ;;; 1 2 5 9 e. NN
26	17, 25	eqeltri 2311	. . 3 |- N e. NN
27	2, 9	deccl 9795	. . . 4 |- ; 6 1 e. NN0
28	27, 3	deccl 9795	. . 3 |- ;; 6 1 2 e. NN0
29		3nn0 9585	. . . . 5 |- 3 e. NN0
30		4nn0 9586	. . . . 5 |- 4 e. NN0
31	29, 30	deccl 9795	. . . 4 |- ; 3 4 e. NN0
32	31	nn0zi 9670	. . 3 |- ; 3 4 e. ZZ
33	29, 3	deccl 9795	. . . 4 |- ; 3 2 e. NN0
34	33, 30	deccl 9795	. . 3 |- ;; 3 2 4 e. NN0
35		7nn0 9589	. . . 4 |- 7 e. NN0
36	9, 35	deccl 9795	. . 3 |- ; 1 7 e. NN0
37	9, 29	deccl 9795	. . . 4 |- ; 1 3 e. NN0
38	37, 2	deccl 9795	. . 3 |- ;; 1 3 6 e. NN0
39		0nn0 9582	. . . . . 6 |- 0 e. NN0
40	29, 39	deccl 9795	. . . . 5 |- ; 3 0 e. NN0
41	40, 2	deccl 9795	. . . 4 |- ;; 3 0 6 e. NN0
42		8nn 9476	. . . . 5 |- 8 e. NN
43	9, 42	decnncl 9804	. . . 4 |- ; 1 8 e. NN
44	10, 30	deccl 9795	. . . . 5 |- ;; 1 2 4 e. NN0
45	44, 9	deccl 9795	. . . 4 |- ;;; 1 2 4 1 e. NN0
46	9, 11	deccl 9795	. . . . . 6 |- ; 1 5 e. NN0
47	46, 29	deccl 9795	. . . . 5 |- ;; 1 5 3 e. NN0
48		1z 9674	. . . . 5 |- 1 e. ZZ
49	11, 39	deccl 9795	. . . . 5 |- ; 5 0 e. NN0
50	46, 3	deccl 9795	. . . . . 6 |- ;; 1 5 2 e. NN0
51		25nn0 9799	. . . . . 6 |- ; 2 5 e. NN0
52	35, 2	deccl 9795	. . . . . . 7 |- ; 7 6 e. NN0
53	17	1259lem3 13264	. . . . . . 7 |- ( ( 2 ^; 7 6 ) mod N ) = ( 5 mod N )
54		eqid 2238	. . . . . . . 8 |- ; 7 6 = ; 7 6
55		4p1e5 9443	. . . . . . . . 9 |- ( 4 + 1 ) = 5
56		7cn 9390	. . . . . . . . . 10 |- 7 e. CC
57		2cn 9377	. . . . . . . . . 10 |- 2 e. CC
58		7t2e14 9894	. . . . . . . . . 10 |- ( 7 x. 2 ) = ; 1 4
59	56, 57, 58	mulcomli 8333	. . . . . . . . 9 |- ( 2 x. 7 ) = ; 1 4
60	9, 30, 55, 59	decsuc 9816	. . . . . . . 8 |- ( ( 2 x. 7 ) + 1 ) = ; 1 5
61		6cn 9388	. . . . . . . . 9 |- 6 e. CC
62		6t2e12 9889	. . . . . . . . 9 |- ( 6 x. 2 ) = ; 1 2
63	61, 57, 62	mulcomli 8333	. . . . . . . 8 |- ( 2 x. 6 ) = ; 1 2
64	3, 35, 2, 54, 3, 9, 60, 63	decmul2c 9851	. . . . . . 7 |- ( 2 x. ; 7 6 ) = ;; 1 5 2
65	51	nn0cni 9579	. . . . . . . . 9 |- ; 2 5 e. CC
66	65	addlidi 8470	. . . . . . . 8 |- ( 0 + ; 2 5 ) = ; 2 5
67	26	nncni 9316	. . . . . . . . . 10 |- N e. CC
68	67	mul02i 8718	. . . . . . . . 9 |- ( 0 x. N ) = 0
69	68	oveq1i 6095	. . . . . . . 8 |- ( ( 0 x. N ) + ; 2 5 ) = ( 0 + ; 2 5 )
70		5t5e25 9888	. . . . . . . 8 |- ( 5 x. 5 ) = ; 2 5
71	66, 69, 70	3eqtr4i 2269	. . . . . . 7 |- ( ( 0 x. N ) + ; 2 5 ) = ( 5 x. 5 )
72	26, 1, 52, 7, 11, 51, 53, 64, 71	mod2xi 13216	. . . . . 6 |- ( ( 2 ^;; 1 5 2 ) mod N ) = (; 2 5 mod N )
73		2p1e3 9440	. . . . . . 7 |- ( 2 + 1 ) = 3
74		eqid 2238	. . . . . . 7 |- ;; 1 5 2 = ;; 1 5 2
75	46, 3, 73, 74	decsuc 9816	. . . . . 6 |- (;; 1 5 2 + 1 ) = ;; 1 5 3
76	49	nn0cni 9579	. . . . . . . 8 |- ; 5 0 e. CC
77	76	addlidi 8470	. . . . . . 7 |- ( 0 + ; 5 0 ) = ; 5 0
78	68	oveq1i 6095	. . . . . . 7 |- ( ( 0 x. N ) + ; 5 0 ) = ( 0 + ; 5 0 )
79		eqid 2238	. . . . . . . 8 |- ; 2 5 = ; 2 5
80		2t2e4 9461	. . . . . . . . . 10 |- ( 2 x. 2 ) = 4
81	80	oveq1i 6095	. . . . . . . . 9 |- ( ( 2 x. 2 ) + 1 ) = ( 4 + 1 )
82	81, 55	eqtri 2259	. . . . . . . 8 |- ( ( 2 x. 2 ) + 1 ) = 5
83		5t2e10 9885	. . . . . . . 8 |- ( 5 x. 2 ) = ; 1 0
84	3, 3, 11, 79, 39, 9, 82, 83	decmul1c 9850	. . . . . . 7 |- (; 2 5 x. 2 ) = ; 5 0
85	77, 78, 84	3eqtr4i 2269	. . . . . 6 |- ( ( 0 x. N ) + ; 5 0 ) = (; 2 5 x. 2 )
86	26, 1, 50, 7, 51, 49, 72, 75, 85	modxp1i 13217	. . . . 5 |- ( ( 2 ^;; 1 5 3 ) mod N ) = (; 5 0 mod N )
87		eqid 2238	. . . . . 6 |- ;; 1 5 3 = ;; 1 5 3
88		eqid 2238	. . . . . . . . 9 |- ; 1 5 = ; 1 5
89	57	mulridi 8328	. . . . . . . . . . 11 |- ( 2 x. 1 ) = 2
90	89	oveq1i 6095	. . . . . . . . . 10 |- ( ( 2 x. 1 ) + 1 ) = ( 2 + 1 )
91	90, 73	eqtri 2259	. . . . . . . . 9 |- ( ( 2 x. 1 ) + 1 ) = 3
92		5cn 9386	. . . . . . . . . 10 |- 5 e. CC
93	92, 57, 83	mulcomli 8333	. . . . . . . . 9 |- ( 2 x. 5 ) = ; 1 0
94	3, 9, 11, 88, 39, 9, 91, 93	decmul2c 9851	. . . . . . . 8 |- ( 2 x. ; 1 5 ) = ; 3 0
95	94	oveq1i 6095	. . . . . . 7 |- ( ( 2 x. ; 1 5 ) + 0 ) = (; 3 0 + 0 )
96	40	nn0cni 9579	. . . . . . . 8 |- ; 3 0 e. CC
97	96	addridi 8469	. . . . . . 7 |- (; 3 0 + 0 ) = ; 3 0
98	95, 97	eqtri 2259	. . . . . 6 |- ( ( 2 x. ; 1 5 ) + 0 ) = ; 3 0
99		2t3e6 9464	. . . . . . 7 |- ( 2 x. 3 ) = 6
100	2	dec0h 9807	. . . . . . 7 |- 6 = ; 0 6
101	99, 100	eqtri 2259	. . . . . 6 |- ( 2 x. 3 ) = ; 0 6
102	3, 46, 29, 87, 2, 39, 98, 101	decmul2c 9851	. . . . 5 |- ( 2 x. ;; 1 5 3 ) = ;; 3 0 6
103	67	mullidi 8329	. . . . . . . 8 |- ( 1 x. N ) = N
104	103, 17	eqtri 2259	. . . . . . 7 |- ( 1 x. N ) = ;;; 1 2 5 9
105		eqid 2238	. . . . . . 7 |- ;;; 1 2 4 1 = ;;; 1 2 4 1
106	3, 30	deccl 9795	. . . . . . . 8 |- ; 2 4 e. NN0
107		eqid 2238	. . . . . . . . 9 |- ; 2 4 = ; 2 4
108	3, 30, 55, 107	decsuc 9816	. . . . . . . 8 |- (; 2 4 + 1 ) = ; 2 5
109		eqid 2238	. . . . . . . . 9 |- ;; 1 2 5 = ;; 1 2 5
110		eqid 2238	. . . . . . . . 9 |- ;; 1 2 4 = ;; 1 2 4
111		eqid 2238	. . . . . . . . . 10 |- ; 1 2 = ; 1 2
112		1p1e2 9423	. . . . . . . . . 10 |- ( 1 + 1 ) = 2
113		2p2e4 9433	. . . . . . . . . 10 |- ( 2 + 2 ) = 4
114	9, 3, 9, 3, 111, 111, 112, 113	decadd 9839	. . . . . . . . 9 |- (; 1 2 + ; 1 2 ) = ; 2 4
115		5p4e9 9455	. . . . . . . . 9 |- ( 5 + 4 ) = 9
116	10, 11, 10, 30, 109, 110, 114, 115	decadd 9839	. . . . . . . 8 |- (;; 1 2 5 + ;; 1 2 4 ) = ;; 2 4 9
117	106, 108, 116	decsucc 9826	. . . . . . 7 |- ( (;; 1 2 5 + ;; 1 2 4 ) + 1 ) = ;; 2 5 0
118		9p1e10 9783	. . . . . . 7 |- ( 9 + 1 ) = ; 1 0
119	12, 5, 44, 9, 104, 105, 117, 118	decaddc2 9841	. . . . . 6 |- ( ( 1 x. N ) + ;;; 1 2 4 1 ) = ;;; 2 5 0 0
120		eqid 2238	. . . . . . 7 |- ; 5 0 = ; 5 0
121	92	mul02i 8718	. . . . . . . . . 10 |- ( 0 x. 5 ) = 0
122	11, 11, 39, 120, 39, 70, 121	decmul1 9849	. . . . . . . . 9 |- (; 5 0 x. 5 ) = ;; 2 5 0
123	122	oveq1i 6095	. . . . . . . 8 |- ( (; 5 0 x. 5 ) + 0 ) = (;; 2 5 0 + 0 )
124	51, 39	deccl 9795	. . . . . . . . . 10 |- ;; 2 5 0 e. NN0
125	124	nn0cni 9579	. . . . . . . . 9 |- ;; 2 5 0 e. CC
126	125	addridi 8469	. . . . . . . 8 |- (;; 2 5 0 + 0 ) = ;; 2 5 0
127	123, 126	eqtri 2259	. . . . . . 7 |- ( (; 5 0 x. 5 ) + 0 ) = ;; 2 5 0
128	76	mul01i 8719	. . . . . . . 8 |- (; 5 0 x. 0 ) = 0
129	39	dec0h 9807	. . . . . . . 8 |- 0 = ; 0 0
130	128, 129	eqtri 2259	. . . . . . 7 |- (; 5 0 x. 0 ) = ; 0 0
131	49, 11, 39, 120, 39, 39, 127, 130	decmul2c 9851	. . . . . 6 |- (; 5 0 x. ; 5 0 ) = ;;; 2 5 0 0
132	119, 131	eqtr4i 2262	. . . . 5 |- ( ( 1 x. N ) + ;;; 1 2 4 1 ) = (; 5 0 x. ; 5 0 )
133	26, 1, 47, 48, 49, 45, 86, 102, 132	mod2xi 13216	. . . 4 |- ( ( 2 ^;; 3 0 6 ) mod N ) = (;;; 1 2 4 1 mod N )
134		eqid 2238	. . . . 5 |- ;; 3 0 6 = ;; 3 0 6
135		eqid 2238	. . . . . 6 |- ; 3 0 = ; 3 0
136	9	dec0h 9807	. . . . . 6 |- 1 = ; 0 1
137		00id 8468	. . . . . . . 8 |- ( 0 + 0 ) = 0
138	99, 137	oveq12i 6097	. . . . . . 7 |- ( ( 2 x. 3 ) + ( 0 + 0 ) ) = ( 6 + 0 )
139	61	addridi 8469	. . . . . . 7 |- ( 6 + 0 ) = 6
140	138, 139	eqtri 2259	. . . . . 6 |- ( ( 2 x. 3 ) + ( 0 + 0 ) ) = 6
141	57	mul01i 8719	. . . . . . . 8 |- ( 2 x. 0 ) = 0
142	141	oveq1i 6095	. . . . . . 7 |- ( ( 2 x. 0 ) + 1 ) = ( 0 + 1 )
143		0p1e1 9420	. . . . . . 7 |- ( 0 + 1 ) = 1
144	142, 143, 136	3eqtri 2263	. . . . . 6 |- ( ( 2 x. 0 ) + 1 ) = ; 0 1
145	29, 39, 39, 9, 135, 136, 3, 9, 39, 140, 144	decma2c 9838	. . . . 5 |- ( ( 2 x. ; 3 0 ) + 1 ) = ; 6 1
146	3, 40, 2, 134, 3, 9, 145, 63	decmul2c 9851	. . . 4 |- ( 2 x. ;; 3 0 6 ) = ;; 6 1 2
147		eqid 2238	. . . . . 6 |- ; 1 8 = ; 1 8
148	10, 30, 55, 110	decsuc 9816	. . . . . 6 |- (;; 1 2 4 + 1 ) = ;; 1 2 5
149		8cn 9392	. . . . . . 7 |- 8 e. CC
150	149, 16, 18	addcomli 8472	. . . . . 6 |- ( 1 + 8 ) = 9
151	44, 9, 9, 13, 105, 147, 148, 150	decadd 9839	. . . . 5 |- (;;; 1 2 4 1 + ; 1 8 ) = ;;; 1 2 5 9
152	151, 17	eqtr4i 2262	. . . 4 |- (;;; 1 2 4 1 + ; 1 8 ) = N
153	34	nn0cni 9579	. . . . . 6 |- ;; 3 2 4 e. CC
154	153	addlidi 8470	. . . . 5 |- ( 0 + ;; 3 2 4 ) = ;; 3 2 4
155	68	oveq1i 6095	. . . . 5 |- ( ( 0 x. N ) + ;; 3 2 4 ) = ( 0 + ;; 3 2 4 )
156	9, 13	deccl 9795	. . . . . 6 |- ; 1 8 e. NN0
157	9, 30	deccl 9795	. . . . . 6 |- ; 1 4 e. NN0
158		eqid 2238	. . . . . . 7 |- ; 1 4 = ; 1 4
159	16	mulridi 8328	. . . . . . . . 9 |- ( 1 x. 1 ) = 1
160	159, 112	oveq12i 6097	. . . . . . . 8 |- ( ( 1 x. 1 ) + ( 1 + 1 ) ) = ( 1 + 2 )
161		1p2e3 9441	. . . . . . . 8 |- ( 1 + 2 ) = 3
162	160, 161	eqtri 2259	. . . . . . 7 |- ( ( 1 x. 1 ) + ( 1 + 1 ) ) = 3
163	149	mulridi 8328	. . . . . . . . 9 |- ( 8 x. 1 ) = 8
164	163	oveq1i 6095	. . . . . . . 8 |- ( ( 8 x. 1 ) + 4 ) = ( 8 + 4 )
165		8p4e12 9867	. . . . . . . 8 |- ( 8 + 4 ) = ; 1 2
166	164, 165	eqtri 2259	. . . . . . 7 |- ( ( 8 x. 1 ) + 4 ) = ; 1 2
167	9, 13, 9, 30, 147, 158, 9, 3, 9, 162, 166	decmac 9837	. . . . . 6 |- ( (; 1 8 x. 1 ) + ; 1 4 ) = ; 3 2
168	149	mullidi 8329	. . . . . . . . 9 |- ( 1 x. 8 ) = 8
169	168	oveq1i 6095	. . . . . . . 8 |- ( ( 1 x. 8 ) + 6 ) = ( 8 + 6 )
170		8p6e14 9869	. . . . . . . 8 |- ( 8 + 6 ) = ; 1 4
171	169, 170	eqtri 2259	. . . . . . 7 |- ( ( 1 x. 8 ) + 6 ) = ; 1 4
172		8t8e64 9906	. . . . . . 7 |- ( 8 x. 8 ) = ; 6 4
173	13, 9, 13, 147, 30, 2, 171, 172	decmul1c 9850	. . . . . 6 |- (; 1 8 x. 8 ) = ;; 1 4 4
174	156, 9, 13, 147, 30, 157, 167, 173	decmul2c 9851	. . . . 5 |- (; 1 8 x. ; 1 8 ) = ;; 3 2 4
175	154, 155, 174	3eqtr4i 2269	. . . 4 |- ( ( 0 x. N ) + ;; 3 2 4 ) = (; 1 8 x. ; 1 8 )
176	1, 41, 7, 43, 34, 45, 133, 146, 152, 175	mod2xnegi 13218	. . 3 |- ( ( 2 ^;; 6 1 2 ) mod N ) = (;; 3 2 4 mod N )
177	17	1259lem1 13262	. . 3 |- ( ( 2 ^; 1 7 ) mod N ) = (;; 1 3 6 mod N )
178		eqid 2238	. . . 4 |- ;; 6 1 2 = ;; 6 1 2
179		eqid 2238	. . . 4 |- ; 1 7 = ; 1 7
180		eqid 2238	. . . . 5 |- ; 6 1 = ; 6 1
181	2, 9, 112, 180	decsuc 9816	. . . 4 |- (; 6 1 + 1 ) = ; 6 2
182		7p2e9 9458	. . . . 5 |- ( 7 + 2 ) = 9
183	56, 57, 182	addcomli 8472	. . . 4 |- ( 2 + 7 ) = 9
184	27, 3, 9, 35, 178, 179, 181, 183	decadd 9839	. . 3 |- (;; 6 1 2 + ; 1 7 ) = ;; 6 2 9
185	29, 9	deccl 9795	. . . . 5 |- ; 3 1 e. NN0
186		eqid 2238	. . . . . . 7 |- ; 3 1 = ; 3 1
187		3cn 9381	. . . . . . . . 9 |- 3 e. CC
188		3p2e5 9448	. . . . . . . . 9 |- ( 3 + 2 ) = 5
189	187, 57, 188	addcomli 8472	. . . . . . . 8 |- ( 2 + 3 ) = 5
190	9, 3, 29, 111, 189	decaddi 9845	. . . . . . 7 |- (; 1 2 + 3 ) = ; 1 5
191		5p1e6 9444	. . . . . . 7 |- ( 5 + 1 ) = 6
192	10, 11, 29, 9, 109, 186, 190, 191	decadd 9839	. . . . . 6 |- (;; 1 2 5 + ; 3 1 ) = ;; 1 5 6
193	112	oveq1i 6095	. . . . . . . . 9 |- ( ( 1 + 1 ) + 1 ) = ( 2 + 1 )
194	193, 73	eqtri 2259	. . . . . . . 8 |- ( ( 1 + 1 ) + 1 ) = 3
195		7p5e12 9862	. . . . . . . . 9 |- ( 7 + 5 ) = ; 1 2
196	56, 92, 195	addcomli 8472	. . . . . . . 8 |- ( 5 + 7 ) = ; 1 2
197	9, 11, 9, 35, 88, 179, 194, 3, 196	decaddc 9840	. . . . . . 7 |- (; 1 5 + ; 1 7 ) = ; 3 2
198		eqid 2238	. . . . . . . 8 |- ; 3 4 = ; 3 4
199		7p3e10 9860	. . . . . . . . 9 |- ( 7 + 3 ) = ; 1 0
200	56, 187, 199	addcomli 8472	. . . . . . . 8 |- ( 3 + 7 ) = ; 1 0
201	187	mulridi 8328	. . . . . . . . . 10 |- ( 3 x. 1 ) = 3
202	16	addridi 8469	. . . . . . . . . 10 |- ( 1 + 0 ) = 1
203	201, 202	oveq12i 6097	. . . . . . . . 9 |- ( ( 3 x. 1 ) + ( 1 + 0 ) ) = ( 3 + 1 )
204		3p1e4 9442	. . . . . . . . 9 |- ( 3 + 1 ) = 4
205	203, 204	eqtri 2259	. . . . . . . 8 |- ( ( 3 x. 1 ) + ( 1 + 0 ) ) = 4
206		4cn 9384	. . . . . . . . . . 11 |- 4 e. CC
207	206	mulridi 8328	. . . . . . . . . 10 |- ( 4 x. 1 ) = 4
208	207	oveq1i 6095	. . . . . . . . 9 |- ( ( 4 x. 1 ) + 0 ) = ( 4 + 0 )
209	206	addridi 8469	. . . . . . . . 9 |- ( 4 + 0 ) = 4
210	30	dec0h 9807	. . . . . . . . 9 |- 4 = ; 0 4
211	208, 209, 210	3eqtri 2263	. . . . . . . 8 |- ( ( 4 x. 1 ) + 0 ) = ; 0 4
212	29, 30, 9, 39, 198, 200, 9, 30, 39, 205, 211	decmac 9837	. . . . . . 7 |- ( (; 3 4 x. 1 ) + ( 3 + 7 ) ) = ; 4 4
213	3	dec0h 9807	. . . . . . . 8 |- 2 = ; 0 2
214		3t2e6 9463	. . . . . . . . . 10 |- ( 3 x. 2 ) = 6
215	214, 143	oveq12i 6097	. . . . . . . . 9 |- ( ( 3 x. 2 ) + ( 0 + 1 ) ) = ( 6 + 1 )
216		6p1e7 9445	. . . . . . . . 9 |- ( 6 + 1 ) = 7
217	215, 216	eqtri 2259	. . . . . . . 8 |- ( ( 3 x. 2 ) + ( 0 + 1 ) ) = 7
218		4t2e8 9466	. . . . . . . . . 10 |- ( 4 x. 2 ) = 8
219	218	oveq1i 6095	. . . . . . . . 9 |- ( ( 4 x. 2 ) + 2 ) = ( 8 + 2 )
220		8p2e10 9865	. . . . . . . . 9 |- ( 8 + 2 ) = ; 1 0
221	219, 220	eqtri 2259	. . . . . . . 8 |- ( ( 4 x. 2 ) + 2 ) = ; 1 0
222	29, 30, 39, 3, 198, 213, 3, 39, 9, 217, 221	decmac 9837	. . . . . . 7 |- ( (; 3 4 x. 2 ) + 2 ) = ; 7 0
223	9, 3, 29, 3, 111, 197, 31, 39, 35, 212, 222	decma2c 9838	. . . . . 6 |- ( (; 3 4 x. ; 1 2 ) + (; 1 5 + ; 1 7 ) ) = ;; 4 4 0
224		5t3e15 9886	. . . . . . . . 9 |- ( 5 x. 3 ) = ; 1 5
225	92, 187, 224	mulcomli 8333	. . . . . . . 8 |- ( 3 x. 5 ) = ; 1 5
226		5p2e7 9453	. . . . . . . 8 |- ( 5 + 2 ) = 7
227	9, 11, 3, 225, 226	decaddi 9845	. . . . . . 7 |- ( ( 3 x. 5 ) + 2 ) = ; 1 7
228		5t4e20 9887	. . . . . . . . 9 |- ( 5 x. 4 ) = ; 2 0
229	92, 206, 228	mulcomli 8333	. . . . . . . 8 |- ( 4 x. 5 ) = ; 2 0
230	61	addlidi 8470	. . . . . . . 8 |- ( 0 + 6 ) = 6
231	3, 39, 2, 229, 230	decaddi 9845	. . . . . . 7 |- ( ( 4 x. 5 ) + 6 ) = ; 2 6
232	29, 30, 2, 198, 11, 2, 3, 227, 231	decrmac 9843	. . . . . 6 |- ( (; 3 4 x. 5 ) + 6 ) = ;; 1 7 6
233	10, 11, 46, 2, 109, 192, 31, 2, 36, 223, 232	decma2c 9838	. . . . 5 |- ( (; 3 4 x. ;; 1 2 5 ) + (;; 1 2 5 + ; 3 1 ) ) = ;;; 4 4 0 6
234		9cn 9394	. . . . . . . 8 |- 9 e. CC
235		9t3e27 9908	. . . . . . . 8 |- ( 9 x. 3 ) = ; 2 7
236	234, 187, 235	mulcomli 8333	. . . . . . 7 |- ( 3 x. 9 ) = ; 2 7
237		7p4e11 9861	. . . . . . 7 |- ( 7 + 4 ) = ; 1 1
238	3, 35, 30, 236, 73, 9, 237	decaddci 9846	. . . . . 6 |- ( ( 3 x. 9 ) + 4 ) = ; 3 1
239		9t4e36 9909	. . . . . . . 8 |- ( 9 x. 4 ) = ; 3 6
240	234, 206, 239	mulcomli 8333	. . . . . . 7 |- ( 4 x. 9 ) = ; 3 6
241	149, 61, 170	addcomli 8472	. . . . . . 7 |- ( 6 + 8 ) = ; 1 4
242	29, 2, 13, 240, 204, 30, 241	decaddci 9846	. . . . . 6 |- ( ( 4 x. 9 ) + 8 ) = ; 4 4
243	29, 30, 13, 198, 5, 30, 30, 238, 242	decrmac 9843	. . . . 5 |- ( (; 3 4 x. 9 ) + 8 ) = ;; 3 1 4
244	12, 5, 12, 13, 17, 22, 31, 30, 185, 233, 243	decma2c 9838	. . . 4 |- ( (; 3 4 x. N ) + ( N - 1 ) ) = ;;;; 4 4 0 6 4
245		eqid 2238	. . . . 5 |- ;; 1 3 6 = ;; 1 3 6
246	9, 5	deccl 9795	. . . . . 6 |- ; 1 9 e. NN0
247	246, 30	deccl 9795	. . . . 5 |- ;; 1 9 4 e. NN0
248		eqid 2238	. . . . . 6 |- ; 1 3 = ; 1 3
249		eqid 2238	. . . . . 6 |- ;; 1 9 4 = ;; 1 9 4
250	5, 35	deccl 9795	. . . . . 6 |- ; 9 7 e. NN0
251	9, 9	deccl 9795	. . . . . . 7 |- ; 1 1 e. NN0
252		eqid 2238	. . . . . . 7 |- ;; 3 2 4 = ;; 3 2 4
253		eqid 2238	. . . . . . . 8 |- ; 1 9 = ; 1 9
254		eqid 2238	. . . . . . . 8 |- ; 9 7 = ; 9 7
255	234, 16, 118	addcomli 8472	. . . . . . . . 9 |- ( 1 + 9 ) = ; 1 0
256	9, 39, 143, 255	decsuc 9816	. . . . . . . 8 |- ( ( 1 + 9 ) + 1 ) = ; 1 1
257		9p7e16 9877	. . . . . . . 8 |- ( 9 + 7 ) = ; 1 6
258	9, 5, 5, 35, 253, 254, 256, 2, 257	decaddc 9840	. . . . . . 7 |- (; 1 9 + ; 9 7 ) = ;; 1 1 6
259		eqid 2238	. . . . . . . 8 |- ; 3 2 = ; 3 2
260		eqid 2238	. . . . . . . . 9 |- ; 1 1 = ; 1 1
261	9, 9, 112, 260	decsuc 9816	. . . . . . . 8 |- (; 1 1 + 1 ) = ; 1 2
262	89	oveq1i 6095	. . . . . . . . 9 |- ( ( 2 x. 1 ) + 2 ) = ( 2 + 2 )
263	262, 113, 210	3eqtri 2263	. . . . . . . 8 |- ( ( 2 x. 1 ) + 2 ) = ; 0 4
264	29, 3, 9, 3, 259, 261, 9, 30, 39, 205, 263	decmac 9837	. . . . . . 7 |- ( (; 3 2 x. 1 ) + (; 1 1 + 1 ) ) = ; 4 4
265	207	oveq1i 6095	. . . . . . . 8 |- ( ( 4 x. 1 ) + 6 ) = ( 4 + 6 )
266		6p4e10 9857	. . . . . . . . 9 |- ( 6 + 4 ) = ; 1 0
267	61, 206, 266	addcomli 8472	. . . . . . . 8 |- ( 4 + 6 ) = ; 1 0
268	265, 267	eqtri 2259	. . . . . . 7 |- ( ( 4 x. 1 ) + 6 ) = ; 1 0
269	33, 30, 251, 2, 252, 258, 9, 39, 9, 264, 268	decmac 9837	. . . . . 6 |- ( (;; 3 2 4 x. 1 ) + (; 1 9 + ; 9 7 ) ) = ;; 4 4 0
270	143, 136	eqtri 2259	. . . . . . . 8 |- ( 0 + 1 ) = ; 0 1
271		3t3e9 9465	. . . . . . . . . 10 |- ( 3 x. 3 ) = 9
272	271, 137	oveq12i 6097	. . . . . . . . 9 |- ( ( 3 x. 3 ) + ( 0 + 0 ) ) = ( 9 + 0 )
273	234	addridi 8469	. . . . . . . . 9 |- ( 9 + 0 ) = 9
274	272, 273	eqtri 2259	. . . . . . . 8 |- ( ( 3 x. 3 ) + ( 0 + 0 ) ) = 9
275	99	oveq1i 6095	. . . . . . . . 9 |- ( ( 2 x. 3 ) + 1 ) = ( 6 + 1 )
276	35	dec0h 9807	. . . . . . . . 9 |- 7 = ; 0 7
277	275, 216, 276	3eqtri 2263	. . . . . . . 8 |- ( ( 2 x. 3 ) + 1 ) = ; 0 7
278	29, 3, 39, 9, 259, 270, 29, 35, 39, 274, 277	decmac 9837	. . . . . . 7 |- ( (; 3 2 x. 3 ) + ( 0 + 1 ) ) = ; 9 7
279		4t3e12 9883	. . . . . . . 8 |- ( 4 x. 3 ) = ; 1 2
280		4p2e6 9450	. . . . . . . . 9 |- ( 4 + 2 ) = 6
281	206, 57, 280	addcomli 8472	. . . . . . . 8 |- ( 2 + 4 ) = 6
282	9, 3, 30, 279, 281	decaddi 9845	. . . . . . 7 |- ( ( 4 x. 3 ) + 4 ) = ; 1 6
283	33, 30, 39, 30, 252, 210, 29, 2, 9, 278, 282	decmac 9837	. . . . . 6 |- ( (;; 3 2 4 x. 3 ) + 4 ) = ;; 9 7 6
284	9, 29, 246, 30, 248, 249, 34, 2, 250, 269, 283	decma2c 9838	. . . . 5 |- ( (;; 3 2 4 x. ; 1 3 ) + ;; 1 9 4 ) = ;;; 4 4 0 6
285		6t3e18 9890	. . . . . . . . 9 |- ( 6 x. 3 ) = ; 1 8
286	61, 187, 285	mulcomli 8333	. . . . . . . 8 |- ( 3 x. 6 ) = ; 1 8
287	9, 13, 18, 286	decsuc 9816	. . . . . . 7 |- ( ( 3 x. 6 ) + 1 ) = ; 1 9
288	9, 3, 3, 63, 113	decaddi 9845	. . . . . . 7 |- ( ( 2 x. 6 ) + 2 ) = ; 1 4
289	29, 3, 3, 259, 2, 30, 9, 287, 288	decrmac 9843	. . . . . 6 |- ( (; 3 2 x. 6 ) + 2 ) = ;; 1 9 4
290		6t4e24 9891	. . . . . . 7 |- ( 6 x. 4 ) = ; 2 4
291	61, 206, 290	mulcomli 8333	. . . . . 6 |- ( 4 x. 6 ) = ; 2 4
292	2, 33, 30, 252, 30, 3, 289, 291	decmul1c 9850	. . . . 5 |- (;; 3 2 4 x. 6 ) = ;;; 1 9 4 4
293	34, 37, 2, 245, 30, 247, 284, 292	decmul2c 9851	. . . 4 |- (;; 3 2 4 x. ;; 1 3 6 ) = ;;;; 4 4 0 6 4
294	244, 293	eqtr4i 2262	. . 3 |- ( (; 3 4 x. N ) + ( N - 1 ) ) = (;; 3 2 4 x. ;; 1 3 6 )
295	26, 1, 28, 32, 34, 23, 36, 38, 176, 177, 184, 294	modxai 13215	. 2 |- ( ( 2 ^;; 6 2 9 ) mod N ) = ( ( N - 1 ) mod N )
296		eqid 2238	. . . 4 |- ;; 6 2 9 = ;; 6 2 9
297		eqid 2238	. . . . 5 |- ; 6 2 = ; 6 2
298	137	oveq2i 6096	. . . . . 6 |- ( ( 2 x. 6 ) + ( 0 + 0 ) ) = ( ( 2 x. 6 ) + 0 )
299	63	oveq1i 6095	. . . . . 6 |- ( ( 2 x. 6 ) + 0 ) = (; 1 2 + 0 )
300	10	nn0cni 9579	. . . . . . 7 |- ; 1 2 e. CC
301	300	addridi 8469	. . . . . 6 |- (; 1 2 + 0 ) = ; 1 2
302	298, 299, 301	3eqtri 2263	. . . . 5 |- ( ( 2 x. 6 ) + ( 0 + 0 ) ) = ; 1 2
303	11	dec0h 9807	. . . . . 6 |- 5 = ; 0 5
304	81, 55, 303	3eqtri 2263	. . . . 5 |- ( ( 2 x. 2 ) + 1 ) = ; 0 5
305	2, 3, 39, 9, 297, 136, 3, 11, 39, 302, 304	decma2c 9838	. . . 4 |- ( ( 2 x. ; 6 2 ) + 1 ) = ;; 1 2 5
306		9t2e18 9907	. . . . 5 |- ( 9 x. 2 ) = ; 1 8
307	234, 57, 306	mulcomli 8333	. . . 4 |- ( 2 x. 9 ) = ; 1 8
308	3, 4, 5, 296, 13, 9, 305, 307	decmul2c 9851	. . 3 |- ( 2 x. ;; 6 2 9 ) = ;;; 1 2 5 8
309	308, 22	eqtr4i 2262	. 2 |- ( 2 x. ;; 6 2 9 ) = ( N - 1 )
310		npcan 8536	. . 3 |- ( ( N e. CC /\ 1 e. CC ) -> ( ( N - 1 ) + 1 ) = N )
311	67, 16, 310	mp2an 430	. 2 |- ( ( N - 1 ) + 1 ) = N
312	68	oveq1i 6095	. . 3 |- ( ( 0 x. N ) + 1 ) = ( 0 + 1 )
313	143, 312, 159	3eqtr4i 2269	. 2 |- ( ( 0 x. N ) + 1 ) = ( 1 x. 1 )
314	1, 6, 7, 8, 9, 23, 295, 309, 311, 313	mod2xnegi 13218	1 |- ( ( 2 ^ ( N - 1 ) ) mod N ) = ( 1 mod N )

This proof depends on syntax axioms:   = wceq 1402   e. wcel 2209  (class class class)co 6085   CCcc 8177   0cc0 8179   1c1 8180   + caddc 8182   x. cmul 8184   - cmin 8498   NNcn 9306   2c2 9357   3c3 9358   4c4 9359   5c5 9360   6c6 9361   7c7 9362   8c8 9363   9c9 9364   NN0cn0 9567  ;cdc 9781   mod cmo 10772   ^cexp 10988

This proof depends on 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 4246  ax-sep 4249  ax-nul 4259  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684  ax-iinf 4735  ax-cnex 8270  ax-resscn 8271  ax-1cn 8272  ax-1re 8273  ax-icn 8274  ax-addcl 8275  ax-addrcl 8276  ax-mulcl 8277  ax-mulrcl 8278  ax-addcom 8279  ax-mulcom 8280  ax-addass 8281  ax-mulass 8282  ax-distr 8283  ax-i2m1 8284  ax-0lt1 8285  ax-1rid 8286  ax-0id 8287  ax-rnegex 8288  ax-precex 8289  ax-cnre 8290  ax-pre-ltirr 8291  ax-pre-ltwlin 8292  ax-pre-lttrn 8293  ax-pre-apti 8294  ax-pre-ltadd 8295  ax-pre-mulgt0 8296  ax-pre-mulext 8297  ax-arch 8298

This proof depends on definitions:  df-bi 117  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 3639  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4131  df-opab 4193  df-mpt 4194  df-tr 4230  df-id 4438  df-po 4441  df-iso 4442  df-iord 4511  df-on 4513  df-ilim 4514  df-suc 4516  df-iom 4738  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-f1 5382  df-fo 5383  df-f1o 5384  df-fv 5385  df-riota 6038  df-ov 6088  df-oprab 6089  df-mpo 6090  df-1st 6374  df-2nd 6375  df-recs 6576  df-frec 6662  df-pnf 8362  df-mnf 8363  df-xr 8364  df-ltxr 8365  df-le 8366  df-sub 8500  df-neg 8501  df-reap 8905  df-ap 8912  df-div 9005  df-inn 9307  df-2 9365  df-3 9366  df-4 9367  df-5 9368  df-6 9369  df-7 9370  df-8 9371  df-9 9372  df-n0 9568  df-z 9649  df-dec 9782  df-uz 9931  df-q 10029  df-rp 10065  df-fl 10715  df-mod 10773  df-seqfrec 10898  df-exp 10989

This theorem is used by:  1259prm  13267