Theorem 1259lem4 14473

Description: Lemma for 1259prm 14475. 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 10692	. 2 |- 2 e. NN
2		6nn0 10815	. . . 4 |- 6 e. NN0
3		2nn0 10811	. . . 4 |- 2 e. NN0
4	2, 3	deccl 10989	. . 3 |- ; 6 2 e. NN0
5		9nn0 10818	. . 3 |- 9 e. NN0
6	4, 5	deccl 10989	. 2 |- ;; 6 2 9 e. NN0
7		0z 10874	. 2 |- 0 e. ZZ
8		1nn 10546	. 2 |- 1 e. NN
9		1nn0 10810	. 2 |- 1 e. NN0
10		1259prm.1	. . . . . 6 |- N = ;;; 1 2 5 9
11	9, 3	deccl 10989	. . . . . . . 8 |- ; 1 2 e. NN0
12		5nn0 10814	. . . . . . . 8 |- 5 e. NN0
13	11, 12	deccl 10989	. . . . . . 7 |- ;; 1 2 5 e. NN0
14		8nn0 10817	. . . . . . 7 |- 8 e. NN0
15		8p1e9 10665	. . . . . . 7 |- ( 8 + 1 ) = 9
16		eqid 2467	. . . . . . 7 |- ;;; 1 2 5 8 = ;;; 1 2 5 8
17	13, 14, 15, 16	decsuc 10998	. . . . . 6 |- (;;; 1 2 5 8 + 1 ) = ;;; 1 2 5 9
18	10, 17	eqtr4i 2499	. . . . 5 |- N = (;;; 1 2 5 8 + 1 )
19	18	oveq1i 6293	. . . 4 |- ( N - 1 ) = ( (;;; 1 2 5 8 + 1 ) - 1 )
20	13, 14	deccl 10989	. . . . . 6 |- ;;; 1 2 5 8 e. NN0
21	20	nn0cni 10806	. . . . 5 |- ;;; 1 2 5 8 e. CC
22		ax-1cn 9549	. . . . 5 |- 1 e. CC
23		pncan 9825	. . . . 5 |- ( (;;; 1 2 5 8 e. CC /\ 1 e. CC ) -> ( (;;; 1 2 5 8 + 1 ) - 1 ) = ;;; 1 2 5 8 )
24	21, 22, 23	mp2an 672	. . . 4 |- ( (;;; 1 2 5 8 + 1 ) - 1 ) = ;;; 1 2 5 8
25	19, 24	eqtri 2496	. . 3 |- ( N - 1 ) = ;;; 1 2 5 8
26	25, 20	eqeltri 2551	. 2 |- ( N - 1 ) e. NN0
27		9nn 10699	. . . . 5 |- 9 e. NN
28	13, 27	decnncl 10988	. . . 4 |- ;;; 1 2 5 9 e. NN
29	10, 28	eqeltri 2551	. . 3 |- N e. NN
30	2, 9	deccl 10989	. . . 4 |- ; 6 1 e. NN0
31	30, 3	deccl 10989	. . 3 |- ;; 6 1 2 e. NN0
32		3nn0 10812	. . . . 5 |- 3 e. NN0
33		4nn0 10813	. . . . 5 |- 4 e. NN0
34	32, 33	deccl 10989	. . . 4 |- ; 3 4 e. NN0
35	34	nn0zi 10888	. . 3 |- ; 3 4 e. ZZ
36	32, 3	deccl 10989	. . . 4 |- ; 3 2 e. NN0
37	36, 33	deccl 10989	. . 3 |- ;; 3 2 4 e. NN0
38		7nn0 10816	. . . 4 |- 7 e. NN0
39	9, 38	deccl 10989	. . 3 |- ; 1 7 e. NN0
40	9, 32	deccl 10989	. . . 4 |- ; 1 3 e. NN0
41	40, 2	deccl 10989	. . 3 |- ;; 1 3 6 e. NN0
42		0nn0 10809	. . . . . 6 |- 0 e. NN0
43	32, 42	deccl 10989	. . . . 5 |- ; 3 0 e. NN0
44	43, 2	deccl 10989	. . . 4 |- ;; 3 0 6 e. NN0
45		8nn 10698	. . . . 5 |- 8 e. NN
46	9, 45	decnncl 10988	. . . 4 |- ; 1 8 e. NN
47	11, 33	deccl 10989	. . . . 5 |- ;; 1 2 4 e. NN0
48	47, 9	deccl 10989	. . . 4 |- ;;; 1 2 4 1 e. NN0
49	9, 12	deccl 10989	. . . . . 6 |- ; 1 5 e. NN0
50	49, 32	deccl 10989	. . . . 5 |- ;; 1 5 3 e. NN0
51		1z 10893	. . . . 5 |- 1 e. ZZ
52	12, 42	deccl 10989	. . . . 5 |- ; 5 0 e. NN0
53	49, 3	deccl 10989	. . . . . 6 |- ;; 1 5 2 e. NN0
54	3, 12	deccl 10989	. . . . . 6 |- ; 2 5 e. NN0
55	38, 2	deccl 10989	. . . . . . 7 |- ; 7 6 e. NN0
56	10	1259lem3 14472	. . . . . . 7 |- ( ( 2 ^; 7 6 ) mod N ) = ( 5 mod N )
57		eqid 2467	. . . . . . . 8 |- ; 7 6 = ; 7 6
58		4p1e5 10661	. . . . . . . . 9 |- ( 4 + 1 ) = 5
59		7cn 10618	. . . . . . . . . 10 |- 7 e. CC
60		2cn 10605	. . . . . . . . . 10 |- 2 e. CC
61		7t2e14 11057	. . . . . . . . . 10 |- ( 7 x. 2 ) = ; 1 4
62	59, 60, 61	mulcomli 9602	. . . . . . . . 9 |- ( 2 x. 7 ) = ; 1 4
63	9, 33, 58, 62	decsuc 10998	. . . . . . . 8 |- ( ( 2 x. 7 ) + 1 ) = ; 1 5
64		6cn 10616	. . . . . . . . 9 |- 6 e. CC
65		6t2e12 11052	. . . . . . . . 9 |- ( 6 x. 2 ) = ; 1 2
66	64, 60, 65	mulcomli 9602	. . . . . . . 8 |- ( 2 x. 6 ) = ; 1 2
67	3, 38, 2, 57, 3, 9, 63, 66	decmul2c 11023	. . . . . . 7 |- ( 2 x. ; 7 6 ) = ;; 1 5 2
68	54	nn0cni 10806	. . . . . . . . 9 |- ; 2 5 e. CC
69	68	addid2i 9766	. . . . . . . 8 |- ( 0 + ; 2 5 ) = ; 2 5
70	29	nncni 10545	. . . . . . . . . 10 |- N e. CC
71	70	mul02i 9767	. . . . . . . . 9 |- ( 0 x. N ) = 0
72	71	oveq1i 6293	. . . . . . . 8 |- ( ( 0 x. N ) + ; 2 5 ) = ( 0 + ; 2 5 )
73		5t5e25 11051	. . . . . . . 8 |- ( 5 x. 5 ) = ; 2 5
74	69, 72, 73	3eqtr4i 2506	. . . . . . 7 |- ( ( 0 x. N ) + ; 2 5 ) = ( 5 x. 5 )
75	29, 1, 55, 7, 12, 54, 56, 67, 74	mod2xi 14413	. . . . . 6 |- ( ( 2 ^;; 1 5 2 ) mod N ) = (; 2 5 mod N )
76		2p1e3 10658	. . . . . . 7 |- ( 2 + 1 ) = 3
77		eqid 2467	. . . . . . 7 |- ;; 1 5 2 = ;; 1 5 2
78	49, 3, 76, 77	decsuc 10998	. . . . . 6 |- (;; 1 5 2 + 1 ) = ;; 1 5 3
79	52	nn0cni 10806	. . . . . . . 8 |- ; 5 0 e. CC
80	79	addid2i 9766	. . . . . . 7 |- ( 0 + ; 5 0 ) = ; 5 0
81	71	oveq1i 6293	. . . . . . 7 |- ( ( 0 x. N ) + ; 5 0 ) = ( 0 + ; 5 0 )
82		eqid 2467	. . . . . . . 8 |- ; 2 5 = ; 2 5
83		2t2e4 10684	. . . . . . . . . 10 |- ( 2 x. 2 ) = 4
84	83	oveq1i 6293	. . . . . . . . 9 |- ( ( 2 x. 2 ) + 1 ) = ( 4 + 1 )
85	84, 58	eqtri 2496	. . . . . . . 8 |- ( ( 2 x. 2 ) + 1 ) = 5
86		5t2e10 10689	. . . . . . . . 9 |- ( 5 x. 2 ) = 10
87		dec10 11005	. . . . . . . . 9 |- 10 = ; 1 0
88	86, 87	eqtri 2496	. . . . . . . 8 |- ( 5 x. 2 ) = ; 1 0
89	3, 3, 12, 82, 42, 9, 85, 88	decmul1c 11022	. . . . . . 7 |- (; 2 5 x. 2 ) = ; 5 0
90	80, 81, 89	3eqtr4i 2506	. . . . . 6 |- ( ( 0 x. N ) + ; 5 0 ) = (; 2 5 x. 2 )
91	29, 1, 53, 7, 54, 52, 75, 78, 90	modxp1i 14414	. . . . 5 |- ( ( 2 ^;; 1 5 3 ) mod N ) = (; 5 0 mod N )
92		eqid 2467	. . . . . 6 |- ;; 1 5 3 = ;; 1 5 3
93		eqid 2467	. . . . . . . . 9 |- ; 1 5 = ; 1 5
94	60	mulid1i 9597	. . . . . . . . . . 11 |- ( 2 x. 1 ) = 2
95	94	oveq1i 6293	. . . . . . . . . 10 |- ( ( 2 x. 1 ) + 1 ) = ( 2 + 1 )
96	95, 76	eqtri 2496	. . . . . . . . 9 |- ( ( 2 x. 1 ) + 1 ) = 3
97		5cn 10614	. . . . . . . . . . 11 |- 5 e. CC
98	97, 60, 86	mulcomli 9602	. . . . . . . . . 10 |- ( 2 x. 5 ) = 10
99	98, 87	eqtri 2496	. . . . . . . . 9 |- ( 2 x. 5 ) = ; 1 0
100	3, 9, 12, 93, 42, 9, 96, 99	decmul2c 11023	. . . . . . . 8 |- ( 2 x. ; 1 5 ) = ; 3 0
101	100	oveq1i 6293	. . . . . . 7 |- ( ( 2 x. ; 1 5 ) + 0 ) = (; 3 0 + 0 )
102	43	nn0cni 10806	. . . . . . . 8 |- ; 3 0 e. CC
103	102	addid1i 9765	. . . . . . 7 |- (; 3 0 + 0 ) = ; 3 0
104	101, 103	eqtri 2496	. . . . . 6 |- ( ( 2 x. ; 1 5 ) + 0 ) = ; 3 0
105		3cn 10609	. . . . . . . 8 |- 3 e. CC
106		3t2e6 10686	. . . . . . . 8 |- ( 3 x. 2 ) = 6
107	105, 60, 106	mulcomli 9602	. . . . . . 7 |- ( 2 x. 3 ) = 6
108	2	dec0h 10991	. . . . . . 7 |- 6 = ; 0 6
109	107, 108	eqtri 2496	. . . . . 6 |- ( 2 x. 3 ) = ; 0 6
110	3, 49, 32, 92, 2, 42, 104, 109	decmul2c 11023	. . . . 5 |- ( 2 x. ;; 1 5 3 ) = ;; 3 0 6
111	70	mulid2i 9598	. . . . . . . 8 |- ( 1 x. N ) = N
112	111, 10	eqtri 2496	. . . . . . 7 |- ( 1 x. N ) = ;;; 1 2 5 9
113		eqid 2467	. . . . . . 7 |- ;;; 1 2 4 1 = ;;; 1 2 4 1
114	3, 33	deccl 10989	. . . . . . . 8 |- ; 2 4 e. NN0
115		eqid 2467	. . . . . . . . 9 |- ; 2 4 = ; 2 4
116	3, 33, 58, 115	decsuc 10998	. . . . . . . 8 |- (; 2 4 + 1 ) = ; 2 5
117		eqid 2467	. . . . . . . . 9 |- ;; 1 2 5 = ;; 1 2 5
118		eqid 2467	. . . . . . . . 9 |- ;; 1 2 4 = ;; 1 2 4
119		eqid 2467	. . . . . . . . . 10 |- ; 1 2 = ; 1 2
120		1p1e2 10648	. . . . . . . . . 10 |- ( 1 + 1 ) = 2
121		2p2e4 10652	. . . . . . . . . 10 |- ( 2 + 2 ) = 4
122	9, 3, 9, 3, 119, 119, 120, 121	decadd 11016	. . . . . . . . 9 |- (; 1 2 + ; 1 2 ) = ; 2 4
123		5p4e9 10674	. . . . . . . . 9 |- ( 5 + 4 ) = 9
124	11, 12, 11, 33, 117, 118, 122, 123	decadd 11016	. . . . . . . 8 |- (;; 1 2 5 + ;; 1 2 4 ) = ;; 2 4 9
125	114, 116, 124	decsucc 11002	. . . . . . 7 |- ( (;; 1 2 5 + ;; 1 2 4 ) + 1 ) = ;; 2 5 0
126		9p1e10 10666	. . . . . . 7 |- ( 9 + 1 ) = 10
127	13, 5, 47, 9, 112, 113, 125, 126	decaddc2 11018	. . . . . 6 |- ( ( 1 x. N ) + ;;; 1 2 4 1 ) = ;;; 2 5 0 0
128		eqid 2467	. . . . . . 7 |- ; 5 0 = ; 5 0
129	73	oveq1i 6293	. . . . . . . . . . 11 |- ( ( 5 x. 5 ) + 0 ) = (; 2 5 + 0 )
130	68	addid1i 9765	. . . . . . . . . . 11 |- (; 2 5 + 0 ) = ; 2 5
131	129, 130	eqtri 2496	. . . . . . . . . 10 |- ( ( 5 x. 5 ) + 0 ) = ; 2 5
132	97	mul02i 9767	. . . . . . . . . . 11 |- ( 0 x. 5 ) = 0
133	42	dec0h 10991	. . . . . . . . . . 11 |- 0 = ; 0 0
134	132, 133	eqtri 2496	. . . . . . . . . 10 |- ( 0 x. 5 ) = ; 0 0
135	12, 12, 42, 128, 42, 42, 131, 134	decmul1c 11022	. . . . . . . . 9 |- (; 5 0 x. 5 ) = ;; 2 5 0
136	135	oveq1i 6293	. . . . . . . 8 |- ( (; 5 0 x. 5 ) + 0 ) = (;; 2 5 0 + 0 )
137	54, 42	deccl 10989	. . . . . . . . . 10 |- ;; 2 5 0 e. NN0
138	137	nn0cni 10806	. . . . . . . . 9 |- ;; 2 5 0 e. CC
139	138	addid1i 9765	. . . . . . . 8 |- (;; 2 5 0 + 0 ) = ;; 2 5 0
140	136, 139	eqtri 2496	. . . . . . 7 |- ( (; 5 0 x. 5 ) + 0 ) = ;; 2 5 0
141	79	mul01i 9768	. . . . . . . 8 |- (; 5 0 x. 0 ) = 0
142	141, 133	eqtri 2496	. . . . . . 7 |- (; 5 0 x. 0 ) = ; 0 0
143	52, 12, 42, 128, 42, 42, 140, 142	decmul2c 11023	. . . . . 6 |- (; 5 0 x. ; 5 0 ) = ;;; 2 5 0 0
144	127, 143	eqtr4i 2499	. . . . 5 |- ( ( 1 x. N ) + ;;; 1 2 4 1 ) = (; 5 0 x. ; 5 0 )
145	29, 1, 50, 51, 52, 48, 91, 110, 144	mod2xi 14413	. . . 4 |- ( ( 2 ^;; 3 0 6 ) mod N ) = (;;; 1 2 4 1 mod N )
146		eqid 2467	. . . . 5 |- ;; 3 0 6 = ;; 3 0 6
147		eqid 2467	. . . . . 6 |- ; 3 0 = ; 3 0
148	9	dec0h 10991	. . . . . 6 |- 1 = ; 0 1
149		00id 9753	. . . . . . . 8 |- ( 0 + 0 ) = 0
150	107, 149	oveq12i 6295	. . . . . . 7 |- ( ( 2 x. 3 ) + ( 0 + 0 ) ) = ( 6 + 0 )
151	64	addid1i 9765	. . . . . . 7 |- ( 6 + 0 ) = 6
152	150, 151	eqtri 2496	. . . . . 6 |- ( ( 2 x. 3 ) + ( 0 + 0 ) ) = 6
153	60	mul01i 9768	. . . . . . . 8 |- ( 2 x. 0 ) = 0
154	153	oveq1i 6293	. . . . . . 7 |- ( ( 2 x. 0 ) + 1 ) = ( 0 + 1 )
155		0p1e1 10646	. . . . . . 7 |- ( 0 + 1 ) = 1
156	154, 155, 148	3eqtri 2500	. . . . . 6 |- ( ( 2 x. 0 ) + 1 ) = ; 0 1
157	32, 42, 42, 9, 147, 148, 3, 9, 42, 152, 156	decma2c 11015	. . . . 5 |- ( ( 2 x. ; 3 0 ) + 1 ) = ; 6 1
158	3, 43, 2, 146, 3, 9, 157, 66	decmul2c 11023	. . . 4 |- ( 2 x. ;; 3 0 6 ) = ;; 6 1 2
159		eqid 2467	. . . . . 6 |- ; 1 8 = ; 1 8
160	11, 33, 58, 118	decsuc 10998	. . . . . 6 |- (;; 1 2 4 + 1 ) = ;; 1 2 5
161		8cn 10620	. . . . . . 7 |- 8 e. CC
162	161, 22, 15	addcomli 9770	. . . . . 6 |- ( 1 + 8 ) = 9
163	47, 9, 9, 14, 113, 159, 160, 162	decadd 11016	. . . . 5 |- (;;; 1 2 4 1 + ; 1 8 ) = ;;; 1 2 5 9
164	163, 10	eqtr4i 2499	. . . 4 |- (;;; 1 2 4 1 + ; 1 8 ) = N
165	37	nn0cni 10806	. . . . . 6 |- ;; 3 2 4 e. CC
166	165	addid2i 9766	. . . . 5 |- ( 0 + ;; 3 2 4 ) = ;; 3 2 4
167	71	oveq1i 6293	. . . . 5 |- ( ( 0 x. N ) + ;; 3 2 4 ) = ( 0 + ;; 3 2 4 )
168	9, 14	deccl 10989	. . . . . 6 |- ; 1 8 e. NN0
169	9, 33	deccl 10989	. . . . . 6 |- ; 1 4 e. NN0
170		eqid 2467	. . . . . . 7 |- ; 1 4 = ; 1 4
171	22	mulid1i 9597	. . . . . . . . 9 |- ( 1 x. 1 ) = 1
172	171, 120	oveq12i 6295	. . . . . . . 8 |- ( ( 1 x. 1 ) + ( 1 + 1 ) ) = ( 1 + 2 )
173		1p2e3 10659	. . . . . . . 8 |- ( 1 + 2 ) = 3
174	172, 173	eqtri 2496	. . . . . . 7 |- ( ( 1 x. 1 ) + ( 1 + 1 ) ) = 3
175	161	mulid1i 9597	. . . . . . . . 9 |- ( 8 x. 1 ) = 8
176	175	oveq1i 6293	. . . . . . . 8 |- ( ( 8 x. 1 ) + 4 ) = ( 8 + 4 )
177		8p4e12 11032	. . . . . . . 8 |- ( 8 + 4 ) = ; 1 2
178	176, 177	eqtri 2496	. . . . . . 7 |- ( ( 8 x. 1 ) + 4 ) = ; 1 2
179	9, 14, 9, 33, 159, 170, 9, 3, 9, 174, 178	decmac 11014	. . . . . 6 |- ( (; 1 8 x. 1 ) + ; 1 4 ) = ; 3 2
180	161	mulid2i 9598	. . . . . . . . 9 |- ( 1 x. 8 ) = 8
181	180	oveq1i 6293	. . . . . . . 8 |- ( ( 1 x. 8 ) + 6 ) = ( 8 + 6 )
182		8p6e14 11034	. . . . . . . 8 |- ( 8 + 6 ) = ; 1 4
183	181, 182	eqtri 2496	. . . . . . 7 |- ( ( 1 x. 8 ) + 6 ) = ; 1 4
184		8t8e64 11069	. . . . . . 7 |- ( 8 x. 8 ) = ; 6 4
185	14, 9, 14, 159, 33, 2, 183, 184	decmul1c 11022	. . . . . 6 |- (; 1 8 x. 8 ) = ;; 1 4 4
186	168, 9, 14, 159, 33, 169, 179, 185	decmul2c 11023	. . . . 5 |- (; 1 8 x. ; 1 8 ) = ;; 3 2 4
187	166, 167, 186	3eqtr4i 2506	. . . 4 |- ( ( 0 x. N ) + ;; 3 2 4 ) = (; 1 8 x. ; 1 8 )
188	1, 44, 7, 46, 37, 48, 145, 158, 164, 187	mod2xnegi 14415	. . 3 |- ( ( 2 ^;; 6 1 2 ) mod N ) = (;; 3 2 4 mod N )
189	10	1259lem1 14470	. . 3 |- ( ( 2 ^; 1 7 ) mod N ) = (;; 1 3 6 mod N )
190		eqid 2467	. . . 4 |- ;; 6 1 2 = ;; 6 1 2
191		eqid 2467	. . . 4 |- ; 1 7 = ; 1 7
192		eqid 2467	. . . . 5 |- ; 6 1 = ; 6 1
193	2, 9, 120, 192	decsuc 10998	. . . 4 |- (; 6 1 + 1 ) = ; 6 2
194		7p2e9 10679	. . . . 5 |- ( 7 + 2 ) = 9
195	59, 60, 194	addcomli 9770	. . . 4 |- ( 2 + 7 ) = 9
196	30, 3, 9, 38, 190, 191, 193, 195	decadd 11016	. . 3 |- (;; 6 1 2 + ; 1 7 ) = ;; 6 2 9
197	32, 9	deccl 10989	. . . . 5 |- ; 3 1 e. NN0
198		eqid 2467	. . . . . . 7 |- ; 3 1 = ; 3 1
199		3p2e5 10667	. . . . . . . . 9 |- ( 3 + 2 ) = 5
200	105, 60, 199	addcomli 9770	. . . . . . . 8 |- ( 2 + 3 ) = 5
201	9, 3, 32, 119, 200	decaddi 11019	. . . . . . 7 |- (; 1 2 + 3 ) = ; 1 5
202		5p1e6 10662	. . . . . . 7 |- ( 5 + 1 ) = 6
203	11, 12, 32, 9, 117, 198, 201, 202	decadd 11016	. . . . . 6 |- (;; 1 2 5 + ; 3 1 ) = ;; 1 5 6
204	120	oveq1i 6293	. . . . . . . . 9 |- ( ( 1 + 1 ) + 1 ) = ( 2 + 1 )
205	204, 76	eqtri 2496	. . . . . . . 8 |- ( ( 1 + 1 ) + 1 ) = 3
206		7p5e12 11028	. . . . . . . . 9 |- ( 7 + 5 ) = ; 1 2
207	59, 97, 206	addcomli 9770	. . . . . . . 8 |- ( 5 + 7 ) = ; 1 2
208	9, 12, 9, 38, 93, 191, 205, 3, 207	decaddc 11017	. . . . . . 7 |- (; 1 5 + ; 1 7 ) = ; 3 2
209		eqid 2467	. . . . . . . 8 |- ; 3 4 = ; 3 4
210		7p3e10 10680	. . . . . . . . . 10 |- ( 7 + 3 ) = 10
211	59, 105, 210	addcomli 9770	. . . . . . . . 9 |- ( 3 + 7 ) = 10
212	211, 87	eqtri 2496	. . . . . . . 8 |- ( 3 + 7 ) = ; 1 0
213	105	mulid1i 9597	. . . . . . . . . 10 |- ( 3 x. 1 ) = 3
214	22	addid1i 9765	. . . . . . . . . 10 |- ( 1 + 0 ) = 1
215	213, 214	oveq12i 6295	. . . . . . . . 9 |- ( ( 3 x. 1 ) + ( 1 + 0 ) ) = ( 3 + 1 )
216		3p1e4 10660	. . . . . . . . 9 |- ( 3 + 1 ) = 4
217	215, 216	eqtri 2496	. . . . . . . 8 |- ( ( 3 x. 1 ) + ( 1 + 0 ) ) = 4
218		4cn 10612	. . . . . . . . . . 11 |- 4 e. CC
219	218	mulid1i 9597	. . . . . . . . . 10 |- ( 4 x. 1 ) = 4
220	219	oveq1i 6293	. . . . . . . . 9 |- ( ( 4 x. 1 ) + 0 ) = ( 4 + 0 )
221	218	addid1i 9765	. . . . . . . . 9 |- ( 4 + 0 ) = 4
222	33	dec0h 10991	. . . . . . . . 9 |- 4 = ; 0 4
223	220, 221, 222	3eqtri 2500	. . . . . . . 8 |- ( ( 4 x. 1 ) + 0 ) = ; 0 4
224	32, 33, 9, 42, 209, 212, 9, 33, 42, 217, 223	decmac 11014	. . . . . . 7 |- ( (; 3 4 x. 1 ) + ( 3 + 7 ) ) = ; 4 4
225	3	dec0h 10991	. . . . . . . 8 |- 2 = ; 0 2
226	106, 155	oveq12i 6295	. . . . . . . . 9 |- ( ( 3 x. 2 ) + ( 0 + 1 ) ) = ( 6 + 1 )
227		6p1e7 10663	. . . . . . . . 9 |- ( 6 + 1 ) = 7
228	226, 227	eqtri 2496	. . . . . . . 8 |- ( ( 3 x. 2 ) + ( 0 + 1 ) ) = 7
229		4t2e8 10688	. . . . . . . . . 10 |- ( 4 x. 2 ) = 8
230	229	oveq1i 6293	. . . . . . . . 9 |- ( ( 4 x. 2 ) + 2 ) = ( 8 + 2 )
231		8p2e10 10681	. . . . . . . . 9 |- ( 8 + 2 ) = 10
232	230, 231, 87	3eqtri 2500	. . . . . . . 8 |- ( ( 4 x. 2 ) + 2 ) = ; 1 0
233	32, 33, 42, 3, 209, 225, 3, 42, 9, 228, 232	decmac 11014	. . . . . . 7 |- ( (; 3 4 x. 2 ) + 2 ) = ; 7 0
234	9, 3, 32, 3, 119, 208, 34, 42, 38, 224, 233	decma2c 11015	. . . . . 6 |- ( (; 3 4 x. ; 1 2 ) + (; 1 5 + ; 1 7 ) ) = ;; 4 4 0
235	60	addid2i 9766	. . . . . . . . 9 |- ( 0 + 2 ) = 2
236	235	oveq2i 6294	. . . . . . . 8 |- ( ( 3 x. 5 ) + ( 0 + 2 ) ) = ( ( 3 x. 5 ) + 2 )
237		5t3e15 11049	. . . . . . . . . 10 |- ( 5 x. 3 ) = ; 1 5
238	97, 105, 237	mulcomli 9602	. . . . . . . . 9 |- ( 3 x. 5 ) = ; 1 5
239		5p2e7 10672	. . . . . . . . 9 |- ( 5 + 2 ) = 7
240	9, 12, 3, 238, 239	decaddi 11019	. . . . . . . 8 |- ( ( 3 x. 5 ) + 2 ) = ; 1 7
241	236, 240	eqtri 2496	. . . . . . 7 |- ( ( 3 x. 5 ) + ( 0 + 2 ) ) = ; 1 7
242		5t4e20 11050	. . . . . . . . 9 |- ( 5 x. 4 ) = ; 2 0
243	97, 218, 242	mulcomli 9602	. . . . . . . 8 |- ( 4 x. 5 ) = ; 2 0
244	64	addid2i 9766	. . . . . . . 8 |- ( 0 + 6 ) = 6
245	3, 42, 2, 243, 244	decaddi 11019	. . . . . . 7 |- ( ( 4 x. 5 ) + 6 ) = ; 2 6
246	32, 33, 42, 2, 209, 108, 12, 2, 3, 241, 245	decmac 11014	. . . . . 6 |- ( (; 3 4 x. 5 ) + 6 ) = ;; 1 7 6
247	11, 12, 49, 2, 117, 203, 34, 2, 39, 234, 246	decma2c 11015	. . . . 5 |- ( (; 3 4 x. ;; 1 2 5 ) + (;; 1 2 5 + ; 3 1 ) ) = ;;; 4 4 0 6
248	14	dec0h 10991	. . . . . 6 |- 8 = ; 0 8
249	218	addid2i 9766	. . . . . . . 8 |- ( 0 + 4 ) = 4
250	249	oveq2i 6294	. . . . . . 7 |- ( ( 3 x. 9 ) + ( 0 + 4 ) ) = ( ( 3 x. 9 ) + 4 )
251		9cn 10622	. . . . . . . . 9 |- 9 e. CC
252		9t3e27 11071	. . . . . . . . 9 |- ( 9 x. 3 ) = ; 2 7
253	251, 105, 252	mulcomli 9602	. . . . . . . 8 |- ( 3 x. 9 ) = ; 2 7
254		7p4e11 11027	. . . . . . . 8 |- ( 7 + 4 ) = ; 1 1
255	3, 38, 33, 253, 76, 9, 254	decaddci 11020	. . . . . . 7 |- ( ( 3 x. 9 ) + 4 ) = ; 3 1
256	250, 255	eqtri 2496	. . . . . 6 |- ( ( 3 x. 9 ) + ( 0 + 4 ) ) = ; 3 1
257		9t4e36 11072	. . . . . . . 8 |- ( 9 x. 4 ) = ; 3 6
258	251, 218, 257	mulcomli 9602	. . . . . . 7 |- ( 4 x. 9 ) = ; 3 6
259	161, 64, 182	addcomli 9770	. . . . . . 7 |- ( 6 + 8 ) = ; 1 4
260	32, 2, 14, 258, 216, 33, 259	decaddci 11020	. . . . . 6 |- ( ( 4 x. 9 ) + 8 ) = ; 4 4
261	32, 33, 42, 14, 209, 248, 5, 33, 33, 256, 260	decmac 11014	. . . . 5 |- ( (; 3 4 x. 9 ) + 8 ) = ;; 3 1 4
262	13, 5, 13, 14, 10, 25, 34, 33, 197, 247, 261	decma2c 11015	. . . 4 |- ( (; 3 4 x. N ) + ( N - 1 ) ) = ;;;; 4 4 0 6 4
263		eqid 2467	. . . . 5 |- ;; 1 3 6 = ;; 1 3 6
264	9, 5	deccl 10989	. . . . . 6 |- ; 1 9 e. NN0
265	264, 33	deccl 10989	. . . . 5 |- ;; 1 9 4 e. NN0
266		eqid 2467	. . . . . 6 |- ; 1 3 = ; 1 3
267		eqid 2467	. . . . . 6 |- ;; 1 9 4 = ;; 1 9 4
268	5, 38	deccl 10989	. . . . . 6 |- ; 9 7 e. NN0
269	9, 9	deccl 10989	. . . . . . 7 |- ; 1 1 e. NN0
270		eqid 2467	. . . . . . 7 |- ;; 3 2 4 = ;; 3 2 4
271		eqid 2467	. . . . . . . 8 |- ; 1 9 = ; 1 9
272		eqid 2467	. . . . . . . 8 |- ; 9 7 = ; 9 7
273	251, 22, 126	addcomli 9770	. . . . . . . . . 10 |- ( 1 + 9 ) = 10
274	273, 87	eqtri 2496	. . . . . . . . 9 |- ( 1 + 9 ) = ; 1 0
275	9, 42, 155, 274	decsuc 10998	. . . . . . . 8 |- ( ( 1 + 9 ) + 1 ) = ; 1 1
276		9p7e16 11042	. . . . . . . 8 |- ( 9 + 7 ) = ; 1 6
277	9, 5, 5, 38, 271, 272, 275, 2, 276	decaddc 11017	. . . . . . 7 |- (; 1 9 + ; 9 7 ) = ;; 1 1 6
278		eqid 2467	. . . . . . . 8 |- ; 3 2 = ; 3 2
279		eqid 2467	. . . . . . . . 9 |- ; 1 1 = ; 1 1
280	9, 9, 120, 279	decsuc 10998	. . . . . . . 8 |- (; 1 1 + 1 ) = ; 1 2
281	94	oveq1i 6293	. . . . . . . . 9 |- ( ( 2 x. 1 ) + 2 ) = ( 2 + 2 )
282	281, 121, 222	3eqtri 2500	. . . . . . . 8 |- ( ( 2 x. 1 ) + 2 ) = ; 0 4
283	32, 3, 9, 3, 278, 280, 9, 33, 42, 217, 282	decmac 11014	. . . . . . 7 |- ( (; 3 2 x. 1 ) + (; 1 1 + 1 ) ) = ; 4 4
284	219	oveq1i 6293	. . . . . . . 8 |- ( ( 4 x. 1 ) + 6 ) = ( 4 + 6 )
285		6p4e10 10678	. . . . . . . . 9 |- ( 6 + 4 ) = 10
286	64, 218, 285	addcomli 9770	. . . . . . . 8 |- ( 4 + 6 ) = 10
287	284, 286, 87	3eqtri 2500	. . . . . . 7 |- ( ( 4 x. 1 ) + 6 ) = ; 1 0
288	36, 33, 269, 2, 270, 277, 9, 42, 9, 283, 287	decmac 11014	. . . . . 6 |- ( (;; 3 2 4 x. 1 ) + (; 1 9 + ; 9 7 ) ) = ;; 4 4 0
289	155, 148	eqtri 2496	. . . . . . . 8 |- ( 0 + 1 ) = ; 0 1
290		3t3e9 10687	. . . . . . . . . 10 |- ( 3 x. 3 ) = 9
291	290, 149	oveq12i 6295	. . . . . . . . 9 |- ( ( 3 x. 3 ) + ( 0 + 0 ) ) = ( 9 + 0 )
292	251	addid1i 9765	. . . . . . . . 9 |- ( 9 + 0 ) = 9
293	291, 292	eqtri 2496	. . . . . . . 8 |- ( ( 3 x. 3 ) + ( 0 + 0 ) ) = 9
294	107	oveq1i 6293	. . . . . . . . 9 |- ( ( 2 x. 3 ) + 1 ) = ( 6 + 1 )
295	38	dec0h 10991	. . . . . . . . 9 |- 7 = ; 0 7
296	294, 227, 295	3eqtri 2500	. . . . . . . 8 |- ( ( 2 x. 3 ) + 1 ) = ; 0 7
297	32, 3, 42, 9, 278, 289, 32, 38, 42, 293, 296	decmac 11014	. . . . . . 7 |- ( (; 3 2 x. 3 ) + ( 0 + 1 ) ) = ; 9 7
298		4t3e12 11047	. . . . . . . 8 |- ( 4 x. 3 ) = ; 1 2
299		4p2e6 10669	. . . . . . . . 9 |- ( 4 + 2 ) = 6
300	218, 60, 299	addcomli 9770	. . . . . . . 8 |- ( 2 + 4 ) = 6
301	9, 3, 33, 298, 300	decaddi 11019	. . . . . . 7 |- ( ( 4 x. 3 ) + 4 ) = ; 1 6
302	36, 33, 42, 33, 270, 222, 32, 2, 9, 297, 301	decmac 11014	. . . . . 6 |- ( (;; 3 2 4 x. 3 ) + 4 ) = ;; 9 7 6
303	9, 32, 264, 33, 266, 267, 37, 2, 268, 288, 302	decma2c 11015	. . . . 5 |- ( (;; 3 2 4 x. ; 1 3 ) + ;; 1 9 4 ) = ;;; 4 4 0 6
304	155	oveq2i 6294	. . . . . . . 8 |- ( ( 3 x. 6 ) + ( 0 + 1 ) ) = ( ( 3 x. 6 ) + 1 )
305		6t3e18 11053	. . . . . . . . . 10 |- ( 6 x. 3 ) = ; 1 8
306	64, 105, 305	mulcomli 9602	. . . . . . . . 9 |- ( 3 x. 6 ) = ; 1 8
307	9, 14, 15, 306	decsuc 10998	. . . . . . . 8 |- ( ( 3 x. 6 ) + 1 ) = ; 1 9
308	304, 307	eqtri 2496	. . . . . . 7 |- ( ( 3 x. 6 ) + ( 0 + 1 ) ) = ; 1 9
309	9, 3, 3, 66, 121	decaddi 11019	. . . . . . 7 |- ( ( 2 x. 6 ) + 2 ) = ; 1 4
310	32, 3, 42, 3, 278, 225, 2, 33, 9, 308, 309	decmac 11014	. . . . . 6 |- ( (; 3 2 x. 6 ) + 2 ) = ;; 1 9 4
311		6t4e24 11054	. . . . . . 7 |- ( 6 x. 4 ) = ; 2 4
312	64, 218, 311	mulcomli 9602	. . . . . 6 |- ( 4 x. 6 ) = ; 2 4
313	2, 36, 33, 270, 33, 3, 310, 312	decmul1c 11022	. . . . 5 |- (;; 3 2 4 x. 6 ) = ;;; 1 9 4 4
314	37, 40, 2, 263, 33, 265, 303, 313	decmul2c 11023	. . . 4 |- (;; 3 2 4 x. ;; 1 3 6 ) = ;;;; 4 4 0 6 4
315	262, 314	eqtr4i 2499	. . 3 |- ( (; 3 4 x. N ) + ( N - 1 ) ) = (;; 3 2 4 x. ;; 1 3 6 )
316	29, 1, 31, 35, 37, 26, 39, 41, 188, 189, 196, 315	modxai 14412	. 2 |- ( ( 2 ^;; 6 2 9 ) mod N ) = ( ( N - 1 ) mod N )
317		eqid 2467	. . . 4 |- ;; 6 2 9 = ;; 6 2 9
318		eqid 2467	. . . . 5 |- ; 6 2 = ; 6 2
319	149	oveq2i 6294	. . . . . 6 |- ( ( 2 x. 6 ) + ( 0 + 0 ) ) = ( ( 2 x. 6 ) + 0 )
320	66	oveq1i 6293	. . . . . 6 |- ( ( 2 x. 6 ) + 0 ) = (; 1 2 + 0 )
321	11	nn0cni 10806	. . . . . . 7 |- ; 1 2 e. CC
322	321	addid1i 9765	. . . . . 6 |- (; 1 2 + 0 ) = ; 1 2
323	319, 320, 322	3eqtri 2500	. . . . 5 |- ( ( 2 x. 6 ) + ( 0 + 0 ) ) = ; 1 2
324	12	dec0h 10991	. . . . . 6 |- 5 = ; 0 5
325	84, 58, 324	3eqtri 2500	. . . . 5 |- ( ( 2 x. 2 ) + 1 ) = ; 0 5
326	2, 3, 42, 9, 318, 148, 3, 12, 42, 323, 325	decma2c 11015	. . . 4 |- ( ( 2 x. ; 6 2 ) + 1 ) = ;; 1 2 5
327		9t2e18 11070	. . . . 5 |- ( 9 x. 2 ) = ; 1 8
328	251, 60, 327	mulcomli 9602	. . . 4 |- ( 2 x. 9 ) = ; 1 8
329	3, 4, 5, 317, 14, 9, 326, 328	decmul2c 11023	. . 3 |- ( 2 x. ;; 6 2 9 ) = ;;; 1 2 5 8
330	329, 25	eqtr4i 2499	. 2 |- ( 2 x. ;; 6 2 9 ) = ( N - 1 )
331		npcan 9828	. . 3 |- ( ( N e. CC /\ 1 e. CC ) -> ( ( N - 1 ) + 1 ) = N )
332	70, 22, 331	mp2an 672	. 2 |- ( ( N - 1 ) + 1 ) = N
333	71	oveq1i 6293	. . 3 |- ( ( 0 x. N ) + 1 ) = ( 0 + 1 )
334	155, 333, 171	3eqtr4i 2506	. 2 |- ( ( 0 x. N ) + 1 ) = ( 1 x. 1 )
335	1, 6, 7, 8, 9, 26, 316, 330, 332, 334	mod2xnegi 14415	1 |- ( ( 2 ^ ( N - 1 ) ) mod N ) = ( 1 mod N )

Syntax hints:   = wceq 1379   e. wcel 1767  (class class class)co 6283   CCcc 9489   0cc0 9491   1c1 9492   + caddc 9494   x. cmul 9496   - cmin 9804   NNcn 10535   2c2 10584   3c3 10585   4c4 10586   5c5 10587   6c6 10588   7c7 10589   8c8 10590   9c9 10591   10c10 10592   NN0cn0 10794  ;cdc 10975   mod cmo 11963   ^cexp 12133

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6575  ax-cnex 9547  ax-resscn 9548  ax-1cn 9549  ax-icn 9550  ax-addcl 9551  ax-addrcl 9552  ax-mulcl 9553  ax-mulrcl 9554  ax-mulcom 9555  ax-addass 9556  ax-mulass 9557  ax-distr 9558  ax-i2m1 9559  ax-1ne0 9560  ax-1rid 9561  ax-rnegex 9562  ax-rrecex 9563  ax-cnre 9564  ax-pre-lttri 9565  ax-pre-lttrn 9566  ax-pre-ltadd 9567  ax-pre-mulgt0 9568  ax-pre-sup 9569

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5550  df-fun 5589  df-fn 5590  df-f 5591  df-f1 5592  df-fo 5593  df-f1o 5594  df-fv 5595  df-riota 6244  df-ov 6286  df-oprab 6287  df-mpt2 6288  df-om 6680  df-2nd 6785  df-recs 7042  df-rdg 7076  df-er 7311  df-en 7517  df-dom 7518  df-sdom 7519  df-sup 7900  df-pnf 9629  df-mnf 9630  df-xr 9631  df-ltxr 9632  df-le 9633  df-sub 9806  df-neg 9807  df-div 10206  df-nn 10536  df-2 10593  df-3 10594  df-4 10595  df-5 10596  df-6 10597  df-7 10598  df-8 10599  df-9 10600  df-10 10601  df-n0 10795  df-z 10864  df-dec 10976  df-uz 11082  df-rp 11220  df-fl 11896  df-mod 11964  df-seq 12075  df-exp 12134

This theorem is referenced by:  1259prm  14475