Theorem 1259lem4 15841

Description: Lemma for 1259prm 15843. 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 11185	. 2 |- 2 e. NN
2		6nn0 11313	. . . 4 |- 6 e. NN0
3		2nn0 11309	. . . 4 |- 2 e. NN0
4	2, 3	deccl 11512	. . 3 |- ; 6 2 e. NN0
5		9nn0 11316	. . 3 |- 9 e. NN0
6	4, 5	deccl 11512	. 2 |- ;; 6 2 9 e. NN0
7		0z 11388	. 2 |- 0 e. ZZ
8		1nn 11031	. 2 |- 1 e. NN
9		1nn0 11308	. 2 |- 1 e. NN0
10	9, 3	deccl 11512	. . . . . . 7 |- ; 1 2 e. NN0
11		5nn0 11312	. . . . . . 7 |- 5 e. NN0
12	10, 11	deccl 11512	. . . . . 6 |- ;; 1 2 5 e. NN0
13		8nn0 11315	. . . . . 6 |- 8 e. NN0
14	12, 13	deccl 11512	. . . . 5 |- ;;; 1 2 5 8 e. NN0
15	14	nn0cni 11304	. . . 4 |- ;;; 1 2 5 8 e. CC
16		ax-1cn 9994	. . . 4 |- 1 e. CC
17		1259prm.1	. . . . 5 |- N = ;;; 1 2 5 9
18		8p1e9 11158	. . . . . 6 |- ( 8 + 1 ) = 9
19		eqid 2622	. . . . . 6 |- ;;; 1 2 5 8 = ;;; 1 2 5 8
20	12, 13, 18, 19	decsuc 11535	. . . . 5 |- (;;; 1 2 5 8 + 1 ) = ;;; 1 2 5 9
21	17, 20	eqtr4i 2647	. . . 4 |- N = (;;; 1 2 5 8 + 1 )
22	15, 16, 21	mvrraddi 10298	. . 3 |- ( N - 1 ) = ;;; 1 2 5 8
23	22, 14	eqeltri 2697	. 2 |- ( N - 1 ) e. NN0
24		9nn 11192	. . . . 5 |- 9 e. NN
25	12, 24	decnncl 11518	. . . 4 |- ;;; 1 2 5 9 e. NN
26	17, 25	eqeltri 2697	. . 3 |- N e. NN
27	2, 9	deccl 11512	. . . 4 |- ; 6 1 e. NN0
28	27, 3	deccl 11512	. . 3 |- ;; 6 1 2 e. NN0
29		3nn0 11310	. . . . 5 |- 3 e. NN0
30		4nn0 11311	. . . . 5 |- 4 e. NN0
31	29, 30	deccl 11512	. . . 4 |- ; 3 4 e. NN0
32	31	nn0zi 11402	. . 3 |- ; 3 4 e. ZZ
33	29, 3	deccl 11512	. . . 4 |- ; 3 2 e. NN0
34	33, 30	deccl 11512	. . 3 |- ;; 3 2 4 e. NN0
35		7nn0 11314	. . . 4 |- 7 e. NN0
36	9, 35	deccl 11512	. . 3 |- ; 1 7 e. NN0
37	9, 29	deccl 11512	. . . 4 |- ; 1 3 e. NN0
38	37, 2	deccl 11512	. . 3 |- ;; 1 3 6 e. NN0
39		0nn0 11307	. . . . . 6 |- 0 e. NN0
40	29, 39	deccl 11512	. . . . 5 |- ; 3 0 e. NN0
41	40, 2	deccl 11512	. . . 4 |- ;; 3 0 6 e. NN0
42		8nn 11191	. . . . 5 |- 8 e. NN
43	9, 42	decnncl 11518	. . . 4 |- ; 1 8 e. NN
44	10, 30	deccl 11512	. . . . 5 |- ;; 1 2 4 e. NN0
45	44, 9	deccl 11512	. . . 4 |- ;;; 1 2 4 1 e. NN0
46	9, 11	deccl 11512	. . . . . 6 |- ; 1 5 e. NN0
47	46, 29	deccl 11512	. . . . 5 |- ;; 1 5 3 e. NN0
48		1z 11407	. . . . 5 |- 1 e. ZZ
49	11, 39	deccl 11512	. . . . 5 |- ; 5 0 e. NN0
50	46, 3	deccl 11512	. . . . . 6 |- ;; 1 5 2 e. NN0
51	3, 11	deccl 11512	. . . . . 6 |- ; 2 5 e. NN0
52	35, 2	deccl 11512	. . . . . . 7 |- ; 7 6 e. NN0
53	17	1259lem3 15840	. . . . . . 7 |- ( ( 2 ^; 7 6 ) mod N ) = ( 5 mod N )
54		eqid 2622	. . . . . . . 8 |- ; 7 6 = ; 7 6
55		4p1e5 11154	. . . . . . . . 9 |- ( 4 + 1 ) = 5
56		7cn 11104	. . . . . . . . . 10 |- 7 e. CC
57		2cn 11091	. . . . . . . . . 10 |- 2 e. CC
58		7t2e14 11648	. . . . . . . . . 10 |- ( 7 x. 2 ) = ; 1 4
59	56, 57, 58	mulcomli 10047	. . . . . . . . 9 |- ( 2 x. 7 ) = ; 1 4
60	9, 30, 55, 59	decsuc 11535	. . . . . . . 8 |- ( ( 2 x. 7 ) + 1 ) = ; 1 5
61		6cn 11102	. . . . . . . . 9 |- 6 e. CC
62		6t2e12 11641	. . . . . . . . 9 |- ( 6 x. 2 ) = ; 1 2
63	61, 57, 62	mulcomli 10047	. . . . . . . 8 |- ( 2 x. 6 ) = ; 1 2
64	3, 35, 2, 54, 3, 9, 60, 63	decmul2c 11589	. . . . . . 7 |- ( 2 x. ; 7 6 ) = ;; 1 5 2
65	51	nn0cni 11304	. . . . . . . . 9 |- ; 2 5 e. CC
66	65	addid2i 10224	. . . . . . . 8 |- ( 0 + ; 2 5 ) = ; 2 5
67	26	nncni 11030	. . . . . . . . . 10 |- N e. CC
68	67	mul02i 10225	. . . . . . . . 9 |- ( 0 x. N ) = 0
69	68	oveq1i 6660	. . . . . . . 8 |- ( ( 0 x. N ) + ; 2 5 ) = ( 0 + ; 2 5 )
70		5t5e25 11639	. . . . . . . 8 |- ( 5 x. 5 ) = ; 2 5
71	66, 69, 70	3eqtr4i 2654	. . . . . . 7 |- ( ( 0 x. N ) + ; 2 5 ) = ( 5 x. 5 )
72	26, 1, 52, 7, 11, 51, 53, 64, 71	mod2xi 15773	. . . . . 6 |- ( ( 2 ^;; 1 5 2 ) mod N ) = (; 2 5 mod N )
73		2p1e3 11151	. . . . . . 7 |- ( 2 + 1 ) = 3
74		eqid 2622	. . . . . . 7 |- ;; 1 5 2 = ;; 1 5 2
75	46, 3, 73, 74	decsuc 11535	. . . . . 6 |- (;; 1 5 2 + 1 ) = ;; 1 5 3
76	49	nn0cni 11304	. . . . . . . 8 |- ; 5 0 e. CC
77	76	addid2i 10224	. . . . . . 7 |- ( 0 + ; 5 0 ) = ; 5 0
78	68	oveq1i 6660	. . . . . . 7 |- ( ( 0 x. N ) + ; 5 0 ) = ( 0 + ; 5 0 )
79		eqid 2622	. . . . . . . 8 |- ; 2 5 = ; 2 5
80		2t2e4 11177	. . . . . . . . . 10 |- ( 2 x. 2 ) = 4
81	80	oveq1i 6660	. . . . . . . . 9 |- ( ( 2 x. 2 ) + 1 ) = ( 4 + 1 )
82	81, 55	eqtri 2644	. . . . . . . 8 |- ( ( 2 x. 2 ) + 1 ) = 5
83		5t2e10 11634	. . . . . . . 8 |- ( 5 x. 2 ) = ; 1 0
84	3, 3, 11, 79, 39, 9, 82, 83	decmul1c 11587	. . . . . . 7 |- (; 2 5 x. 2 ) = ; 5 0
85	77, 78, 84	3eqtr4i 2654	. . . . . 6 |- ( ( 0 x. N ) + ; 5 0 ) = (; 2 5 x. 2 )
86	26, 1, 50, 7, 51, 49, 72, 75, 85	modxp1i 15774	. . . . 5 |- ( ( 2 ^;; 1 5 3 ) mod N ) = (; 5 0 mod N )
87		eqid 2622	. . . . . 6 |- ;; 1 5 3 = ;; 1 5 3
88		eqid 2622	. . . . . . . . 9 |- ; 1 5 = ; 1 5
89	57	mulid1i 10042	. . . . . . . . . . 11 |- ( 2 x. 1 ) = 2
90	89	oveq1i 6660	. . . . . . . . . 10 |- ( ( 2 x. 1 ) + 1 ) = ( 2 + 1 )
91	90, 73	eqtri 2644	. . . . . . . . 9 |- ( ( 2 x. 1 ) + 1 ) = 3
92		5cn 11100	. . . . . . . . . 10 |- 5 e. CC
93	92, 57, 83	mulcomli 10047	. . . . . . . . 9 |- ( 2 x. 5 ) = ; 1 0
94	3, 9, 11, 88, 39, 9, 91, 93	decmul2c 11589	. . . . . . . 8 |- ( 2 x. ; 1 5 ) = ; 3 0
95	94	oveq1i 6660	. . . . . . 7 |- ( ( 2 x. ; 1 5 ) + 0 ) = (; 3 0 + 0 )
96	40	nn0cni 11304	. . . . . . . 8 |- ; 3 0 e. CC
97	96	addid1i 10223	. . . . . . 7 |- (; 3 0 + 0 ) = ; 3 0
98	95, 97	eqtri 2644	. . . . . 6 |- ( ( 2 x. ; 1 5 ) + 0 ) = ; 3 0
99		3cn 11095	. . . . . . . 8 |- 3 e. CC
100		3t2e6 11179	. . . . . . . 8 |- ( 3 x. 2 ) = 6
101	99, 57, 100	mulcomli 10047	. . . . . . 7 |- ( 2 x. 3 ) = 6
102	2	dec0h 11522	. . . . . . 7 |- 6 = ; 0 6
103	101, 102	eqtri 2644	. . . . . 6 |- ( 2 x. 3 ) = ; 0 6
104	3, 46, 29, 87, 2, 39, 98, 103	decmul2c 11589	. . . . 5 |- ( 2 x. ;; 1 5 3 ) = ;; 3 0 6
105	67	mulid2i 10043	. . . . . . . 8 |- ( 1 x. N ) = N
106	105, 17	eqtri 2644	. . . . . . 7 |- ( 1 x. N ) = ;;; 1 2 5 9
107		eqid 2622	. . . . . . 7 |- ;;; 1 2 4 1 = ;;; 1 2 4 1
108	3, 30	deccl 11512	. . . . . . . 8 |- ; 2 4 e. NN0
109		eqid 2622	. . . . . . . . 9 |- ; 2 4 = ; 2 4
110	3, 30, 55, 109	decsuc 11535	. . . . . . . 8 |- (; 2 4 + 1 ) = ; 2 5
111		eqid 2622	. . . . . . . . 9 |- ;; 1 2 5 = ;; 1 2 5
112		eqid 2622	. . . . . . . . 9 |- ;; 1 2 4 = ;; 1 2 4
113		eqid 2622	. . . . . . . . . 10 |- ; 1 2 = ; 1 2
114		1p1e2 11134	. . . . . . . . . 10 |- ( 1 + 1 ) = 2
115		2p2e4 11144	. . . . . . . . . 10 |- ( 2 + 2 ) = 4
116	9, 3, 9, 3, 113, 113, 114, 115	decadd 11570	. . . . . . . . 9 |- (; 1 2 + ; 1 2 ) = ; 2 4
117		5p4e9 11167	. . . . . . . . 9 |- ( 5 + 4 ) = 9
118	10, 11, 10, 30, 111, 112, 116, 117	decadd 11570	. . . . . . . 8 |- (;; 1 2 5 + ;; 1 2 4 ) = ;; 2 4 9
119	108, 110, 118	decsucc 11550	. . . . . . 7 |- ( (;; 1 2 5 + ;; 1 2 4 ) + 1 ) = ;; 2 5 0
120		9p1e10 11496	. . . . . . 7 |- ( 9 + 1 ) = ; 1 0
121	12, 5, 44, 9, 106, 107, 119, 120	decaddc2 11575	. . . . . 6 |- ( ( 1 x. N ) + ;;; 1 2 4 1 ) = ;;; 2 5 0 0
122		eqid 2622	. . . . . . 7 |- ; 5 0 = ; 5 0
123	92	mul02i 10225	. . . . . . . . . 10 |- ( 0 x. 5 ) = 0
124	11, 11, 39, 122, 39, 70, 123	decmul1 11585	. . . . . . . . 9 |- (; 5 0 x. 5 ) = ;; 2 5 0
125	124	oveq1i 6660	. . . . . . . 8 |- ( (; 5 0 x. 5 ) + 0 ) = (;; 2 5 0 + 0 )
126	51, 39	deccl 11512	. . . . . . . . . 10 |- ;; 2 5 0 e. NN0
127	126	nn0cni 11304	. . . . . . . . 9 |- ;; 2 5 0 e. CC
128	127	addid1i 10223	. . . . . . . 8 |- (;; 2 5 0 + 0 ) = ;; 2 5 0
129	125, 128	eqtri 2644	. . . . . . 7 |- ( (; 5 0 x. 5 ) + 0 ) = ;; 2 5 0
130	76	mul01i 10226	. . . . . . . 8 |- (; 5 0 x. 0 ) = 0
131	39	dec0h 11522	. . . . . . . 8 |- 0 = ; 0 0
132	130, 131	eqtri 2644	. . . . . . 7 |- (; 5 0 x. 0 ) = ; 0 0
133	49, 11, 39, 122, 39, 39, 129, 132	decmul2c 11589	. . . . . 6 |- (; 5 0 x. ; 5 0 ) = ;;; 2 5 0 0
134	121, 133	eqtr4i 2647	. . . . 5 |- ( ( 1 x. N ) + ;;; 1 2 4 1 ) = (; 5 0 x. ; 5 0 )
135	26, 1, 47, 48, 49, 45, 86, 104, 134	mod2xi 15773	. . . 4 |- ( ( 2 ^;; 3 0 6 ) mod N ) = (;;; 1 2 4 1 mod N )
136		eqid 2622	. . . . 5 |- ;; 3 0 6 = ;; 3 0 6
137		eqid 2622	. . . . . 6 |- ; 3 0 = ; 3 0
138	9	dec0h 11522	. . . . . 6 |- 1 = ; 0 1
139		00id 10211	. . . . . . . 8 |- ( 0 + 0 ) = 0
140	101, 139	oveq12i 6662	. . . . . . 7 |- ( ( 2 x. 3 ) + ( 0 + 0 ) ) = ( 6 + 0 )
141	61	addid1i 10223	. . . . . . 7 |- ( 6 + 0 ) = 6
142	140, 141	eqtri 2644	. . . . . 6 |- ( ( 2 x. 3 ) + ( 0 + 0 ) ) = 6
143	57	mul01i 10226	. . . . . . . 8 |- ( 2 x. 0 ) = 0
144	143	oveq1i 6660	. . . . . . 7 |- ( ( 2 x. 0 ) + 1 ) = ( 0 + 1 )
145		0p1e1 11132	. . . . . . 7 |- ( 0 + 1 ) = 1
146	144, 145, 138	3eqtri 2648	. . . . . 6 |- ( ( 2 x. 0 ) + 1 ) = ; 0 1
147	29, 39, 39, 9, 137, 138, 3, 9, 39, 142, 146	decma2c 11568	. . . . 5 |- ( ( 2 x. ; 3 0 ) + 1 ) = ; 6 1
148	3, 40, 2, 136, 3, 9, 147, 63	decmul2c 11589	. . . 4 |- ( 2 x. ;; 3 0 6 ) = ;; 6 1 2
149		eqid 2622	. . . . . 6 |- ; 1 8 = ; 1 8
150	10, 30, 55, 112	decsuc 11535	. . . . . 6 |- (;; 1 2 4 + 1 ) = ;; 1 2 5
151		8cn 11106	. . . . . . 7 |- 8 e. CC
152	151, 16, 18	addcomli 10228	. . . . . 6 |- ( 1 + 8 ) = 9
153	44, 9, 9, 13, 107, 149, 150, 152	decadd 11570	. . . . 5 |- (;;; 1 2 4 1 + ; 1 8 ) = ;;; 1 2 5 9
154	153, 17	eqtr4i 2647	. . . 4 |- (;;; 1 2 4 1 + ; 1 8 ) = N
155	34	nn0cni 11304	. . . . . 6 |- ;; 3 2 4 e. CC
156	155	addid2i 10224	. . . . 5 |- ( 0 + ;; 3 2 4 ) = ;; 3 2 4
157	68	oveq1i 6660	. . . . 5 |- ( ( 0 x. N ) + ;; 3 2 4 ) = ( 0 + ;; 3 2 4 )
158	9, 13	deccl 11512	. . . . . 6 |- ; 1 8 e. NN0
159	9, 30	deccl 11512	. . . . . 6 |- ; 1 4 e. NN0
160		eqid 2622	. . . . . . 7 |- ; 1 4 = ; 1 4
161	16	mulid1i 10042	. . . . . . . . 9 |- ( 1 x. 1 ) = 1
162	161, 114	oveq12i 6662	. . . . . . . 8 |- ( ( 1 x. 1 ) + ( 1 + 1 ) ) = ( 1 + 2 )
163		1p2e3 11152	. . . . . . . 8 |- ( 1 + 2 ) = 3
164	162, 163	eqtri 2644	. . . . . . 7 |- ( ( 1 x. 1 ) + ( 1 + 1 ) ) = 3
165	151	mulid1i 10042	. . . . . . . . 9 |- ( 8 x. 1 ) = 8
166	165	oveq1i 6660	. . . . . . . 8 |- ( ( 8 x. 1 ) + 4 ) = ( 8 + 4 )
167		8p4e12 11614	. . . . . . . 8 |- ( 8 + 4 ) = ; 1 2
168	166, 167	eqtri 2644	. . . . . . 7 |- ( ( 8 x. 1 ) + 4 ) = ; 1 2
169	9, 13, 9, 30, 149, 160, 9, 3, 9, 164, 168	decmac 11566	. . . . . 6 |- ( (; 1 8 x. 1 ) + ; 1 4 ) = ; 3 2
170	151	mulid2i 10043	. . . . . . . . 9 |- ( 1 x. 8 ) = 8
171	170	oveq1i 6660	. . . . . . . 8 |- ( ( 1 x. 8 ) + 6 ) = ( 8 + 6 )
172		8p6e14 11616	. . . . . . . 8 |- ( 8 + 6 ) = ; 1 4
173	171, 172	eqtri 2644	. . . . . . 7 |- ( ( 1 x. 8 ) + 6 ) = ; 1 4
174		8t8e64 11662	. . . . . . 7 |- ( 8 x. 8 ) = ; 6 4
175	13, 9, 13, 149, 30, 2, 173, 174	decmul1c 11587	. . . . . 6 |- (; 1 8 x. 8 ) = ;; 1 4 4
176	158, 9, 13, 149, 30, 159, 169, 175	decmul2c 11589	. . . . 5 |- (; 1 8 x. ; 1 8 ) = ;; 3 2 4
177	156, 157, 176	3eqtr4i 2654	. . . 4 |- ( ( 0 x. N ) + ;; 3 2 4 ) = (; 1 8 x. ; 1 8 )
178	1, 41, 7, 43, 34, 45, 135, 148, 154, 177	mod2xnegi 15775	. . 3 |- ( ( 2 ^;; 6 1 2 ) mod N ) = (;; 3 2 4 mod N )
179	17	1259lem1 15838	. . 3 |- ( ( 2 ^; 1 7 ) mod N ) = (;; 1 3 6 mod N )
180		eqid 2622	. . . 4 |- ;; 6 1 2 = ;; 6 1 2
181		eqid 2622	. . . 4 |- ; 1 7 = ; 1 7
182		eqid 2622	. . . . 5 |- ; 6 1 = ; 6 1
183	2, 9, 114, 182	decsuc 11535	. . . 4 |- (; 6 1 + 1 ) = ; 6 2
184		7p2e9 11172	. . . . 5 |- ( 7 + 2 ) = 9
185	56, 57, 184	addcomli 10228	. . . 4 |- ( 2 + 7 ) = 9
186	27, 3, 9, 35, 180, 181, 183, 185	decadd 11570	. . 3 |- (;; 6 1 2 + ; 1 7 ) = ;; 6 2 9
187	29, 9	deccl 11512	. . . . 5 |- ; 3 1 e. NN0
188		eqid 2622	. . . . . . 7 |- ; 3 1 = ; 3 1
189		3p2e5 11160	. . . . . . . . 9 |- ( 3 + 2 ) = 5
190	99, 57, 189	addcomli 10228	. . . . . . . 8 |- ( 2 + 3 ) = 5
191	9, 3, 29, 113, 190	decaddi 11579	. . . . . . 7 |- (; 1 2 + 3 ) = ; 1 5
192		5p1e6 11155	. . . . . . 7 |- ( 5 + 1 ) = 6
193	10, 11, 29, 9, 111, 188, 191, 192	decadd 11570	. . . . . 6 |- (;; 1 2 5 + ; 3 1 ) = ;; 1 5 6
194	114	oveq1i 6660	. . . . . . . . 9 |- ( ( 1 + 1 ) + 1 ) = ( 2 + 1 )
195	194, 73	eqtri 2644	. . . . . . . 8 |- ( ( 1 + 1 ) + 1 ) = 3
196		7p5e12 11607	. . . . . . . . 9 |- ( 7 + 5 ) = ; 1 2
197	56, 92, 196	addcomli 10228	. . . . . . . 8 |- ( 5 + 7 ) = ; 1 2
198	9, 11, 9, 35, 88, 181, 195, 3, 197	decaddc 11572	. . . . . . 7 |- (; 1 5 + ; 1 7 ) = ; 3 2
199		eqid 2622	. . . . . . . 8 |- ; 3 4 = ; 3 4
200		7p3e10 11603	. . . . . . . . 9 |- ( 7 + 3 ) = ; 1 0
201	56, 99, 200	addcomli 10228	. . . . . . . 8 |- ( 3 + 7 ) = ; 1 0
202	99	mulid1i 10042	. . . . . . . . . 10 |- ( 3 x. 1 ) = 3
203	16	addid1i 10223	. . . . . . . . . 10 |- ( 1 + 0 ) = 1
204	202, 203	oveq12i 6662	. . . . . . . . 9 |- ( ( 3 x. 1 ) + ( 1 + 0 ) ) = ( 3 + 1 )
205		3p1e4 11153	. . . . . . . . 9 |- ( 3 + 1 ) = 4
206	204, 205	eqtri 2644	. . . . . . . 8 |- ( ( 3 x. 1 ) + ( 1 + 0 ) ) = 4
207		4cn 11098	. . . . . . . . . . 11 |- 4 e. CC
208	207	mulid1i 10042	. . . . . . . . . 10 |- ( 4 x. 1 ) = 4
209	208	oveq1i 6660	. . . . . . . . 9 |- ( ( 4 x. 1 ) + 0 ) = ( 4 + 0 )
210	207	addid1i 10223	. . . . . . . . 9 |- ( 4 + 0 ) = 4
211	30	dec0h 11522	. . . . . . . . 9 |- 4 = ; 0 4
212	209, 210, 211	3eqtri 2648	. . . . . . . 8 |- ( ( 4 x. 1 ) + 0 ) = ; 0 4
213	29, 30, 9, 39, 199, 201, 9, 30, 39, 206, 212	decmac 11566	. . . . . . 7 |- ( (; 3 4 x. 1 ) + ( 3 + 7 ) ) = ; 4 4
214	3	dec0h 11522	. . . . . . . 8 |- 2 = ; 0 2
215	100, 145	oveq12i 6662	. . . . . . . . 9 |- ( ( 3 x. 2 ) + ( 0 + 1 ) ) = ( 6 + 1 )
216		6p1e7 11156	. . . . . . . . 9 |- ( 6 + 1 ) = 7
217	215, 216	eqtri 2644	. . . . . . . 8 |- ( ( 3 x. 2 ) + ( 0 + 1 ) ) = 7
218		4t2e8 11181	. . . . . . . . . 10 |- ( 4 x. 2 ) = 8
219	218	oveq1i 6660	. . . . . . . . 9 |- ( ( 4 x. 2 ) + 2 ) = ( 8 + 2 )
220		8p2e10 11610	. . . . . . . . 9 |- ( 8 + 2 ) = ; 1 0
221	219, 220	eqtri 2644	. . . . . . . 8 |- ( ( 4 x. 2 ) + 2 ) = ; 1 0
222	29, 30, 39, 3, 199, 214, 3, 39, 9, 217, 221	decmac 11566	. . . . . . 7 |- ( (; 3 4 x. 2 ) + 2 ) = ; 7 0
223	9, 3, 29, 3, 113, 198, 31, 39, 35, 213, 222	decma2c 11568	. . . . . 6 |- ( (; 3 4 x. ; 1 2 ) + (; 1 5 + ; 1 7 ) ) = ;; 4 4 0
224		5t3e15 11635	. . . . . . . . 9 |- ( 5 x. 3 ) = ; 1 5
225	92, 99, 224	mulcomli 10047	. . . . . . . 8 |- ( 3 x. 5 ) = ; 1 5
226		5p2e7 11165	. . . . . . . 8 |- ( 5 + 2 ) = 7
227	9, 11, 3, 225, 226	decaddi 11579	. . . . . . 7 |- ( ( 3 x. 5 ) + 2 ) = ; 1 7
228		5t4e20 11637	. . . . . . . . 9 |- ( 5 x. 4 ) = ; 2 0
229	92, 207, 228	mulcomli 10047	. . . . . . . 8 |- ( 4 x. 5 ) = ; 2 0
230	61	addid2i 10224	. . . . . . . 8 |- ( 0 + 6 ) = 6
231	3, 39, 2, 229, 230	decaddi 11579	. . . . . . 7 |- ( ( 4 x. 5 ) + 6 ) = ; 2 6
232	29, 30, 2, 199, 11, 2, 3, 227, 231	decrmac 11577	. . . . . 6 |- ( (; 3 4 x. 5 ) + 6 ) = ;; 1 7 6
233	10, 11, 46, 2, 111, 193, 31, 2, 36, 223, 232	decma2c 11568	. . . . 5 |- ( (; 3 4 x. ;; 1 2 5 ) + (;; 1 2 5 + ; 3 1 ) ) = ;;; 4 4 0 6
234		9cn 11108	. . . . . . . 8 |- 9 e. CC
235		9t3e27 11664	. . . . . . . 8 |- ( 9 x. 3 ) = ; 2 7
236	234, 99, 235	mulcomli 10047	. . . . . . 7 |- ( 3 x. 9 ) = ; 2 7
237		7p4e11 11605	. . . . . . 7 |- ( 7 + 4 ) = ; 1 1
238	3, 35, 30, 236, 73, 9, 237	decaddci 11580	. . . . . 6 |- ( ( 3 x. 9 ) + 4 ) = ; 3 1
239		9t4e36 11665	. . . . . . . 8 |- ( 9 x. 4 ) = ; 3 6
240	234, 207, 239	mulcomli 10047	. . . . . . 7 |- ( 4 x. 9 ) = ; 3 6
241	151, 61, 172	addcomli 10228	. . . . . . 7 |- ( 6 + 8 ) = ; 1 4
242	29, 2, 13, 240, 205, 30, 241	decaddci 11580	. . . . . 6 |- ( ( 4 x. 9 ) + 8 ) = ; 4 4
243	29, 30, 13, 199, 5, 30, 30, 238, 242	decrmac 11577	. . . . 5 |- ( (; 3 4 x. 9 ) + 8 ) = ;; 3 1 4
244	12, 5, 12, 13, 17, 22, 31, 30, 187, 233, 243	decma2c 11568	. . . 4 |- ( (; 3 4 x. N ) + ( N - 1 ) ) = ;;;; 4 4 0 6 4
245		eqid 2622	. . . . 5 |- ;; 1 3 6 = ;; 1 3 6
246	9, 5	deccl 11512	. . . . . 6 |- ; 1 9 e. NN0
247	246, 30	deccl 11512	. . . . 5 |- ;; 1 9 4 e. NN0
248		eqid 2622	. . . . . 6 |- ; 1 3 = ; 1 3
249		eqid 2622	. . . . . 6 |- ;; 1 9 4 = ;; 1 9 4
250	5, 35	deccl 11512	. . . . . 6 |- ; 9 7 e. NN0
251	9, 9	deccl 11512	. . . . . . 7 |- ; 1 1 e. NN0
252		eqid 2622	. . . . . . 7 |- ;; 3 2 4 = ;; 3 2 4
253		eqid 2622	. . . . . . . 8 |- ; 1 9 = ; 1 9
254		eqid 2622	. . . . . . . 8 |- ; 9 7 = ; 9 7
255	234, 16, 120	addcomli 10228	. . . . . . . . 9 |- ( 1 + 9 ) = ; 1 0
256	9, 39, 145, 255	decsuc 11535	. . . . . . . 8 |- ( ( 1 + 9 ) + 1 ) = ; 1 1
257		9p7e16 11625	. . . . . . . 8 |- ( 9 + 7 ) = ; 1 6
258	9, 5, 5, 35, 253, 254, 256, 2, 257	decaddc 11572	. . . . . . 7 |- (; 1 9 + ; 9 7 ) = ;; 1 1 6
259		eqid 2622	. . . . . . . 8 |- ; 3 2 = ; 3 2
260		eqid 2622	. . . . . . . . 9 |- ; 1 1 = ; 1 1
261	9, 9, 114, 260	decsuc 11535	. . . . . . . 8 |- (; 1 1 + 1 ) = ; 1 2
262	89	oveq1i 6660	. . . . . . . . 9 |- ( ( 2 x. 1 ) + 2 ) = ( 2 + 2 )
263	262, 115, 211	3eqtri 2648	. . . . . . . 8 |- ( ( 2 x. 1 ) + 2 ) = ; 0 4
264	29, 3, 9, 3, 259, 261, 9, 30, 39, 206, 263	decmac 11566	. . . . . . 7 |- ( (; 3 2 x. 1 ) + (; 1 1 + 1 ) ) = ; 4 4
265	208	oveq1i 6660	. . . . . . . 8 |- ( ( 4 x. 1 ) + 6 ) = ( 4 + 6 )
266		6p4e10 11598	. . . . . . . . 9 |- ( 6 + 4 ) = ; 1 0
267	61, 207, 266	addcomli 10228	. . . . . . . 8 |- ( 4 + 6 ) = ; 1 0
268	265, 267	eqtri 2644	. . . . . . 7 |- ( ( 4 x. 1 ) + 6 ) = ; 1 0
269	33, 30, 251, 2, 252, 258, 9, 39, 9, 264, 268	decmac 11566	. . . . . 6 |- ( (;; 3 2 4 x. 1 ) + (; 1 9 + ; 9 7 ) ) = ;; 4 4 0
270	145, 138	eqtri 2644	. . . . . . . 8 |- ( 0 + 1 ) = ; 0 1
271		3t3e9 11180	. . . . . . . . . 10 |- ( 3 x. 3 ) = 9
272	271, 139	oveq12i 6662	. . . . . . . . 9 |- ( ( 3 x. 3 ) + ( 0 + 0 ) ) = ( 9 + 0 )
273	234	addid1i 10223	. . . . . . . . 9 |- ( 9 + 0 ) = 9
274	272, 273	eqtri 2644	. . . . . . . 8 |- ( ( 3 x. 3 ) + ( 0 + 0 ) ) = 9
275	101	oveq1i 6660	. . . . . . . . 9 |- ( ( 2 x. 3 ) + 1 ) = ( 6 + 1 )
276	35	dec0h 11522	. . . . . . . . 9 |- 7 = ; 0 7
277	275, 216, 276	3eqtri 2648	. . . . . . . 8 |- ( ( 2 x. 3 ) + 1 ) = ; 0 7
278	29, 3, 39, 9, 259, 270, 29, 35, 39, 274, 277	decmac 11566	. . . . . . 7 |- ( (; 3 2 x. 3 ) + ( 0 + 1 ) ) = ; 9 7
279		4t3e12 11632	. . . . . . . 8 |- ( 4 x. 3 ) = ; 1 2
280		4p2e6 11162	. . . . . . . . 9 |- ( 4 + 2 ) = 6
281	207, 57, 280	addcomli 10228	. . . . . . . 8 |- ( 2 + 4 ) = 6
282	9, 3, 30, 279, 281	decaddi 11579	. . . . . . 7 |- ( ( 4 x. 3 ) + 4 ) = ; 1 6
283	33, 30, 39, 30, 252, 211, 29, 2, 9, 278, 282	decmac 11566	. . . . . 6 |- ( (;; 3 2 4 x. 3 ) + 4 ) = ;; 9 7 6
284	9, 29, 246, 30, 248, 249, 34, 2, 250, 269, 283	decma2c 11568	. . . . 5 |- ( (;; 3 2 4 x. ; 1 3 ) + ;; 1 9 4 ) = ;;; 4 4 0 6
285		6t3e18 11642	. . . . . . . . 9 |- ( 6 x. 3 ) = ; 1 8
286	61, 99, 285	mulcomli 10047	. . . . . . . 8 |- ( 3 x. 6 ) = ; 1 8
287	9, 13, 18, 286	decsuc 11535	. . . . . . 7 |- ( ( 3 x. 6 ) + 1 ) = ; 1 9
288	9, 3, 3, 63, 115	decaddi 11579	. . . . . . 7 |- ( ( 2 x. 6 ) + 2 ) = ; 1 4
289	29, 3, 3, 259, 2, 30, 9, 287, 288	decrmac 11577	. . . . . 6 |- ( (; 3 2 x. 6 ) + 2 ) = ;; 1 9 4
290		6t4e24 11643	. . . . . . 7 |- ( 6 x. 4 ) = ; 2 4
291	61, 207, 290	mulcomli 10047	. . . . . 6 |- ( 4 x. 6 ) = ; 2 4
292	2, 33, 30, 252, 30, 3, 289, 291	decmul1c 11587	. . . . 5 |- (;; 3 2 4 x. 6 ) = ;;; 1 9 4 4
293	34, 37, 2, 245, 30, 247, 284, 292	decmul2c 11589	. . . 4 |- (;; 3 2 4 x. ;; 1 3 6 ) = ;;;; 4 4 0 6 4
294	244, 293	eqtr4i 2647	. . 3 |- ( (; 3 4 x. N ) + ( N - 1 ) ) = (;; 3 2 4 x. ;; 1 3 6 )
295	26, 1, 28, 32, 34, 23, 36, 38, 178, 179, 186, 294	modxai 15772	. 2 |- ( ( 2 ^;; 6 2 9 ) mod N ) = ( ( N - 1 ) mod N )
296		eqid 2622	. . . 4 |- ;; 6 2 9 = ;; 6 2 9
297		eqid 2622	. . . . 5 |- ; 6 2 = ; 6 2
298	139	oveq2i 6661	. . . . . 6 |- ( ( 2 x. 6 ) + ( 0 + 0 ) ) = ( ( 2 x. 6 ) + 0 )
299	63	oveq1i 6660	. . . . . 6 |- ( ( 2 x. 6 ) + 0 ) = (; 1 2 + 0 )
300	10	nn0cni 11304	. . . . . . 7 |- ; 1 2 e. CC
301	300	addid1i 10223	. . . . . 6 |- (; 1 2 + 0 ) = ; 1 2
302	298, 299, 301	3eqtri 2648	. . . . 5 |- ( ( 2 x. 6 ) + ( 0 + 0 ) ) = ; 1 2
303	11	dec0h 11522	. . . . . 6 |- 5 = ; 0 5
304	81, 55, 303	3eqtri 2648	. . . . 5 |- ( ( 2 x. 2 ) + 1 ) = ; 0 5
305	2, 3, 39, 9, 297, 138, 3, 11, 39, 302, 304	decma2c 11568	. . . 4 |- ( ( 2 x. ; 6 2 ) + 1 ) = ;; 1 2 5
306		9t2e18 11663	. . . . 5 |- ( 9 x. 2 ) = ; 1 8
307	234, 57, 306	mulcomli 10047	. . . 4 |- ( 2 x. 9 ) = ; 1 8
308	3, 4, 5, 296, 13, 9, 305, 307	decmul2c 11589	. . 3 |- ( 2 x. ;; 6 2 9 ) = ;;; 1 2 5 8
309	308, 22	eqtr4i 2647	. 2 |- ( 2 x. ;; 6 2 9 ) = ( N - 1 )
310		npcan 10290	. . 3 |- ( ( N e. CC /\ 1 e. CC ) -> ( ( N - 1 ) + 1 ) = N )
311	67, 16, 310	mp2an 708	. 2 |- ( ( N - 1 ) + 1 ) = N
312	68	oveq1i 6660	. . 3 |- ( ( 0 x. N ) + 1 ) = ( 0 + 1 )
313	145, 312, 161	3eqtr4i 2654	. 2 |- ( ( 0 x. N ) + 1 ) = ( 1 x. 1 )
314	1, 6, 7, 8, 9, 23, 295, 309, 311, 313	mod2xnegi 15775	1 |- ( ( 2 ^ ( N - 1 ) ) mod N ) = ( 1 mod N )

Syntax hints:   = wceq 1483   e. wcel 1990  (class class class)co 6650   CCcc 9934   0cc0 9936   1c1 9937   + caddc 9939   x. cmul 9941   - cmin 10266   NNcn 11020   2c2 11070   3c3 11071   4c4 11072   5c5 11073   6c6 11074   7c7 11075   8c8 11076   9c9 11077   NN0cn0 11292  ;cdc 11493   mod cmo 12668   ^cexp 12860

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-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-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-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-sup 8348  df-inf 8349  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-4 11081  df-5 11082  df-6 11083  df-7 11084  df-8 11085  df-9 11086  df-n0 11293  df-z 11378  df-dec 11494  df-uz 11688  df-rp 11833  df-fl 12593  df-mod 12669  df-seq 12802  df-exp 12861

This theorem is referenced by:  1259prm  15843