Theorem 1259lem4 16835

Description: Lemma for 1259prm 16837. Calculate a power mod. In decimal, we calculate 2↑306 = (2↑76)↑4 · 4≡5↑4 · 4 = 2𝑁 − 18, 2↑612 = (2↑306)↑2≡18↑2 = 324, 2↑629 = 2↑612 · 2↑17≡324 · 136 = 35𝑁 − 1 and finally 2↑(𝑁 − 1) = (2↑629)↑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 = ;;;1259

Assertion

Ref	Expression
1259lem4	⊢ ((2↑(𝑁 − 1)) mod 𝑁) = (1 mod 𝑁)

Proof of Theorem 1259lem4

Step	Hyp	Ref	Expression
1		2nn 12046	. 2 ⊢ 2 ∈ ℕ
2		6nn0 12254	. . . 4 ⊢ 6 ∈ ℕ0
3		2nn0 12250	. . . 4 ⊢ 2 ∈ ℕ0
4	2, 3	deccl 12452	. . 3 ⊢ ;62 ∈ ℕ0
5		9nn0 12257	. . 3 ⊢ 9 ∈ ℕ0
6	4, 5	deccl 12452	. 2 ⊢ ;;629 ∈ ℕ0
7		0z 12330	. 2 ⊢ 0 ∈ ℤ
8		1nn 11984	. 2 ⊢ 1 ∈ ℕ
9		1nn0 12249	. 2 ⊢ 1 ∈ ℕ0
10	9, 3	deccl 12452	. . . . . . 7 ⊢ ;12 ∈ ℕ0
11		5nn0 12253	. . . . . . 7 ⊢ 5 ∈ ℕ0
12	10, 11	deccl 12452	. . . . . 6 ⊢ ;;125 ∈ ℕ0
13		8nn0 12256	. . . . . 6 ⊢ 8 ∈ ℕ0
14	12, 13	deccl 12452	. . . . 5 ⊢ ;;;1258 ∈ ℕ0
15	14	nn0cni 12245	. . . 4 ⊢ ;;;1258 ∈ ℂ
16		ax-1cn 10929	. . . 4 ⊢ 1 ∈ ℂ
17		1259prm.1	. . . . 5 ⊢ 𝑁 = ;;;1259
18		8p1e9 12123	. . . . . 6 ⊢ (8 + 1) = 9
19		eqid 2738	. . . . . 6 ⊢ ;;;1258 = ;;;1258
20	12, 13, 18, 19	decsuc 12468	. . . . 5 ⊢ (;;;1258 + 1) = ;;;1259
21	17, 20	eqtr4i 2769	. . . 4 ⊢ 𝑁 = (;;;1258 + 1)
22	15, 16, 21	mvrraddi 11238	. . 3 ⊢ (𝑁 − 1) = ;;;1258
23	22, 14	eqeltri 2835	. 2 ⊢ (𝑁 − 1) ∈ ℕ0
24		9nn 12071	. . . . 5 ⊢ 9 ∈ ℕ
25	12, 24	decnncl 12457	. . . 4 ⊢ ;;;1259 ∈ ℕ
26	17, 25	eqeltri 2835	. . 3 ⊢ 𝑁 ∈ ℕ
27	2, 9	deccl 12452	. . . 4 ⊢ ;61 ∈ ℕ0
28	27, 3	deccl 12452	. . 3 ⊢ ;;612 ∈ ℕ0
29		3nn0 12251	. . . . 5 ⊢ 3 ∈ ℕ0
30		4nn0 12252	. . . . 5 ⊢ 4 ∈ ℕ0
31	29, 30	deccl 12452	. . . 4 ⊢ ;34 ∈ ℕ0
32	31	nn0zi 12345	. . 3 ⊢ ;34 ∈ ℤ
33	29, 3	deccl 12452	. . . 4 ⊢ ;32 ∈ ℕ0
34	33, 30	deccl 12452	. . 3 ⊢ ;;324 ∈ ℕ0
35		7nn0 12255	. . . 4 ⊢ 7 ∈ ℕ0
36	9, 35	deccl 12452	. . 3 ⊢ ;17 ∈ ℕ0
37	9, 29	deccl 12452	. . . 4 ⊢ ;13 ∈ ℕ0
38	37, 2	deccl 12452	. . 3 ⊢ ;;136 ∈ ℕ0
39		0nn0 12248	. . . . . 6 ⊢ 0 ∈ ℕ0
40	29, 39	deccl 12452	. . . . 5 ⊢ ;30 ∈ ℕ0
41	40, 2	deccl 12452	. . . 4 ⊢ ;;306 ∈ ℕ0
42		8nn 12068	. . . . 5 ⊢ 8 ∈ ℕ
43	9, 42	decnncl 12457	. . . 4 ⊢ ;18 ∈ ℕ
44	10, 30	deccl 12452	. . . . 5 ⊢ ;;124 ∈ ℕ0
45	44, 9	deccl 12452	. . . 4 ⊢ ;;;1241 ∈ ℕ0
46	9, 11	deccl 12452	. . . . . 6 ⊢ ;15 ∈ ℕ0
47	46, 29	deccl 12452	. . . . 5 ⊢ ;;153 ∈ ℕ0
48		1z 12350	. . . . 5 ⊢ 1 ∈ ℤ
49	11, 39	deccl 12452	. . . . 5 ⊢ ;50 ∈ ℕ0
50	46, 3	deccl 12452	. . . . . 6 ⊢ ;;152 ∈ ℕ0
51	3, 11	deccl 12452	. . . . . 6 ⊢ ;25 ∈ ℕ0
52	35, 2	deccl 12452	. . . . . . 7 ⊢ ;76 ∈ ℕ0
53	17	1259lem3 16834	. . . . . . 7 ⊢ ((2↑;76) mod 𝑁) = (5 mod 𝑁)
54		eqid 2738	. . . . . . . 8 ⊢ ;76 = ;76
55		4p1e5 12119	. . . . . . . . 9 ⊢ (4 + 1) = 5
56		7cn 12067	. . . . . . . . . 10 ⊢ 7 ∈ ℂ
57		2cn 12048	. . . . . . . . . 10 ⊢ 2 ∈ ℂ
58		7t2e14 12546	. . . . . . . . . 10 ⊢ (7 · 2) = ;14
59	56, 57, 58	mulcomli 10984	. . . . . . . . 9 ⊢ (2 · 7) = ;14
60	9, 30, 55, 59	decsuc 12468	. . . . . . . 8 ⊢ ((2 · 7) + 1) = ;15
61		6cn 12064	. . . . . . . . 9 ⊢ 6 ∈ ℂ
62		6t2e12 12541	. . . . . . . . 9 ⊢ (6 · 2) = ;12
63	61, 57, 62	mulcomli 10984	. . . . . . . 8 ⊢ (2 · 6) = ;12
64	3, 35, 2, 54, 3, 9, 60, 63	decmul2c 12503	. . . . . . 7 ⊢ (2 · ;76) = ;;152
65	51	nn0cni 12245	. . . . . . . . 9 ⊢ ;25 ∈ ℂ
66	65	addid2i 11163	. . . . . . . 8 ⊢ (0 + ;25) = ;25
67	26	nncni 11983	. . . . . . . . . 10 ⊢ 𝑁 ∈ ℂ
68	67	mul02i 11164	. . . . . . . . 9 ⊢ (0 · 𝑁) = 0
69	68	oveq1i 7285	. . . . . . . 8 ⊢ ((0 · 𝑁) + ;25) = (0 + ;25)
70		5t5e25 12540	. . . . . . . 8 ⊢ (5 · 5) = ;25
71	66, 69, 70	3eqtr4i 2776	. . . . . . 7 ⊢ ((0 · 𝑁) + ;25) = (5 · 5)
72	26, 1, 52, 7, 11, 51, 53, 64, 71	mod2xi 16770	. . . . . 6 ⊢ ((2↑;;152) mod 𝑁) = (;25 mod 𝑁)
73		2p1e3 12115	. . . . . . 7 ⊢ (2 + 1) = 3
74		eqid 2738	. . . . . . 7 ⊢ ;;152 = ;;152
75	46, 3, 73, 74	decsuc 12468	. . . . . 6 ⊢ (;;152 + 1) = ;;153
76	49	nn0cni 12245	. . . . . . . 8 ⊢ ;50 ∈ ℂ
77	76	addid2i 11163	. . . . . . 7 ⊢ (0 + ;50) = ;50
78	68	oveq1i 7285	. . . . . . 7 ⊢ ((0 · 𝑁) + ;50) = (0 + ;50)
79		eqid 2738	. . . . . . . 8 ⊢ ;25 = ;25
80		2t2e4 12137	. . . . . . . . . 10 ⊢ (2 · 2) = 4
81	80	oveq1i 7285	. . . . . . . . 9 ⊢ ((2 · 2) + 1) = (4 + 1)
82	81, 55	eqtri 2766	. . . . . . . 8 ⊢ ((2 · 2) + 1) = 5
83		5t2e10 12537	. . . . . . . 8 ⊢ (5 · 2) = ;10
84	3, 3, 11, 79, 39, 9, 82, 83	decmul1c 12502	. . . . . . 7 ⊢ (;25 · 2) = ;50
85	77, 78, 84	3eqtr4i 2776	. . . . . 6 ⊢ ((0 · 𝑁) + ;50) = (;25 · 2)
86	26, 1, 50, 7, 51, 49, 72, 75, 85	modxp1i 16771	. . . . 5 ⊢ ((2↑;;153) mod 𝑁) = (;50 mod 𝑁)
87		eqid 2738	. . . . . 6 ⊢ ;;153 = ;;153
88		eqid 2738	. . . . . . . . 9 ⊢ ;15 = ;15
89	57	mulid1i 10979	. . . . . . . . . . 11 ⊢ (2 · 1) = 2
90	89	oveq1i 7285	. . . . . . . . . 10 ⊢ ((2 · 1) + 1) = (2 + 1)
91	90, 73	eqtri 2766	. . . . . . . . 9 ⊢ ((2 · 1) + 1) = 3
92		5cn 12061	. . . . . . . . . 10 ⊢ 5 ∈ ℂ
93	92, 57, 83	mulcomli 10984	. . . . . . . . 9 ⊢ (2 · 5) = ;10
94	3, 9, 11, 88, 39, 9, 91, 93	decmul2c 12503	. . . . . . . 8 ⊢ (2 · ;15) = ;30
95	94	oveq1i 7285	. . . . . . 7 ⊢ ((2 · ;15) + 0) = (;30 + 0)
96	40	nn0cni 12245	. . . . . . . 8 ⊢ ;30 ∈ ℂ
97	96	addid1i 11162	. . . . . . 7 ⊢ (;30 + 0) = ;30
98	95, 97	eqtri 2766	. . . . . 6 ⊢ ((2 · ;15) + 0) = ;30
99		3cn 12054	. . . . . . . 8 ⊢ 3 ∈ ℂ
100		3t2e6 12139	. . . . . . . 8 ⊢ (3 · 2) = 6
101	99, 57, 100	mulcomli 10984	. . . . . . 7 ⊢ (2 · 3) = 6
102	2	dec0h 12459	. . . . . . 7 ⊢ 6 = ;06
103	101, 102	eqtri 2766	. . . . . 6 ⊢ (2 · 3) = ;06
104	3, 46, 29, 87, 2, 39, 98, 103	decmul2c 12503	. . . . 5 ⊢ (2 · ;;153) = ;;306
105	67	mulid2i 10980	. . . . . . . 8 ⊢ (1 · 𝑁) = 𝑁
106	105, 17	eqtri 2766	. . . . . . 7 ⊢ (1 · 𝑁) = ;;;1259
107		eqid 2738	. . . . . . 7 ⊢ ;;;1241 = ;;;1241
108	3, 30	deccl 12452	. . . . . . . 8 ⊢ ;24 ∈ ℕ0
109		eqid 2738	. . . . . . . . 9 ⊢ ;24 = ;24
110	3, 30, 55, 109	decsuc 12468	. . . . . . . 8 ⊢ (;24 + 1) = ;25
111		eqid 2738	. . . . . . . . 9 ⊢ ;;125 = ;;125
112		eqid 2738	. . . . . . . . 9 ⊢ ;;124 = ;;124
113		eqid 2738	. . . . . . . . . 10 ⊢ ;12 = ;12
114		1p1e2 12098	. . . . . . . . . 10 ⊢ (1 + 1) = 2
115		2p2e4 12108	. . . . . . . . . 10 ⊢ (2 + 2) = 4
116	9, 3, 9, 3, 113, 113, 114, 115	decadd 12491	. . . . . . . . 9 ⊢ (;12 + ;12) = ;24
117		5p4e9 12131	. . . . . . . . 9 ⊢ (5 + 4) = 9
118	10, 11, 10, 30, 111, 112, 116, 117	decadd 12491	. . . . . . . 8 ⊢ (;;125 + ;;124) = ;;249
119	108, 110, 118	decsucc 12478	. . . . . . 7 ⊢ ((;;125 + ;;124) + 1) = ;;250
120		9p1e10 12439	. . . . . . 7 ⊢ (9 + 1) = ;10
121	12, 5, 44, 9, 106, 107, 119, 120	decaddc2 12493	. . . . . 6 ⊢ ((1 · 𝑁) + ;;;1241) = ;;;2500
122		eqid 2738	. . . . . . 7 ⊢ ;50 = ;50
123	92	mul02i 11164	. . . . . . . . . 10 ⊢ (0 · 5) = 0
124	11, 11, 39, 122, 70, 123	decmul1 12501	. . . . . . . . 9 ⊢ (;50 · 5) = ;;250
125	124	oveq1i 7285	. . . . . . . 8 ⊢ ((;50 · 5) + 0) = (;;250 + 0)
126	51, 39	deccl 12452	. . . . . . . . . 10 ⊢ ;;250 ∈ ℕ0
127	126	nn0cni 12245	. . . . . . . . 9 ⊢ ;;250 ∈ ℂ
128	127	addid1i 11162	. . . . . . . 8 ⊢ (;;250 + 0) = ;;250
129	125, 128	eqtri 2766	. . . . . . 7 ⊢ ((;50 · 5) + 0) = ;;250
130	76	mul01i 11165	. . . . . . . 8 ⊢ (;50 · 0) = 0
131	39	dec0h 12459	. . . . . . . 8 ⊢ 0 = ;00
132	130, 131	eqtri 2766	. . . . . . 7 ⊢ (;50 · 0) = ;00
133	49, 11, 39, 122, 39, 39, 129, 132	decmul2c 12503	. . . . . 6 ⊢ (;50 · ;50) = ;;;2500
134	121, 133	eqtr4i 2769	. . . . 5 ⊢ ((1 · 𝑁) + ;;;1241) = (;50 · ;50)
135	26, 1, 47, 48, 49, 45, 86, 104, 134	mod2xi 16770	. . . 4 ⊢ ((2↑;;306) mod 𝑁) = (;;;1241 mod 𝑁)
136		eqid 2738	. . . . 5 ⊢ ;;306 = ;;306
137		eqid 2738	. . . . . 6 ⊢ ;30 = ;30
138	9	dec0h 12459	. . . . . 6 ⊢ 1 = ;01
139		00id 11150	. . . . . . . 8 ⊢ (0 + 0) = 0
140	101, 139	oveq12i 7287	. . . . . . 7 ⊢ ((2 · 3) + (0 + 0)) = (6 + 0)
141	61	addid1i 11162	. . . . . . 7 ⊢ (6 + 0) = 6
142	140, 141	eqtri 2766	. . . . . 6 ⊢ ((2 · 3) + (0 + 0)) = 6
143	57	mul01i 11165	. . . . . . . 8 ⊢ (2 · 0) = 0
144	143	oveq1i 7285	. . . . . . 7 ⊢ ((2 · 0) + 1) = (0 + 1)
145		0p1e1 12095	. . . . . . 7 ⊢ (0 + 1) = 1
146	144, 145, 138	3eqtri 2770	. . . . . 6 ⊢ ((2 · 0) + 1) = ;01
147	29, 39, 39, 9, 137, 138, 3, 9, 39, 142, 146	decma2c 12490	. . . . 5 ⊢ ((2 · ;30) + 1) = ;61
148	3, 40, 2, 136, 3, 9, 147, 63	decmul2c 12503	. . . 4 ⊢ (2 · ;;306) = ;;612
149		eqid 2738	. . . . . 6 ⊢ ;18 = ;18
150	10, 30, 55, 112	decsuc 12468	. . . . . 6 ⊢ (;;124 + 1) = ;;125
151		8cn 12070	. . . . . . 7 ⊢ 8 ∈ ℂ
152	151, 16, 18	addcomli 11167	. . . . . 6 ⊢ (1 + 8) = 9
153	44, 9, 9, 13, 107, 149, 150, 152	decadd 12491	. . . . 5 ⊢ (;;;1241 + ;18) = ;;;1259
154	153, 17	eqtr4i 2769	. . . 4 ⊢ (;;;1241 + ;18) = 𝑁
155	34	nn0cni 12245	. . . . . 6 ⊢ ;;324 ∈ ℂ
156	155	addid2i 11163	. . . . 5 ⊢ (0 + ;;324) = ;;324
157	68	oveq1i 7285	. . . . 5 ⊢ ((0 · 𝑁) + ;;324) = (0 + ;;324)
158	9, 13	deccl 12452	. . . . . 6 ⊢ ;18 ∈ ℕ0
159	9, 30	deccl 12452	. . . . . 6 ⊢ ;14 ∈ ℕ0
160		eqid 2738	. . . . . . 7 ⊢ ;14 = ;14
161	16	mulid1i 10979	. . . . . . . . 9 ⊢ (1 · 1) = 1
162	161, 114	oveq12i 7287	. . . . . . . 8 ⊢ ((1 · 1) + (1 + 1)) = (1 + 2)
163		1p2e3 12116	. . . . . . . 8 ⊢ (1 + 2) = 3
164	162, 163	eqtri 2766	. . . . . . 7 ⊢ ((1 · 1) + (1 + 1)) = 3
165	151	mulid1i 10979	. . . . . . . . 9 ⊢ (8 · 1) = 8
166	165	oveq1i 7285	. . . . . . . 8 ⊢ ((8 · 1) + 4) = (8 + 4)
167		8p4e12 12519	. . . . . . . 8 ⊢ (8 + 4) = ;12
168	166, 167	eqtri 2766	. . . . . . 7 ⊢ ((8 · 1) + 4) = ;12
169	9, 13, 9, 30, 149, 160, 9, 3, 9, 164, 168	decmac 12489	. . . . . 6 ⊢ ((;18 · 1) + ;14) = ;32
170	151	mulid2i 10980	. . . . . . . . 9 ⊢ (1 · 8) = 8
171	170	oveq1i 7285	. . . . . . . 8 ⊢ ((1 · 8) + 6) = (8 + 6)
172		8p6e14 12521	. . . . . . . 8 ⊢ (8 + 6) = ;14
173	171, 172	eqtri 2766	. . . . . . 7 ⊢ ((1 · 8) + 6) = ;14
174		8t8e64 12558	. . . . . . 7 ⊢ (8 · 8) = ;64
175	13, 9, 13, 149, 30, 2, 173, 174	decmul1c 12502	. . . . . 6 ⊢ (;18 · 8) = ;;144
176	158, 9, 13, 149, 30, 159, 169, 175	decmul2c 12503	. . . . 5 ⊢ (;18 · ;18) = ;;324
177	156, 157, 176	3eqtr4i 2776	. . . 4 ⊢ ((0 · 𝑁) + ;;324) = (;18 · ;18)
178	1, 41, 7, 43, 34, 45, 135, 148, 154, 177	mod2xnegi 16772	. . 3 ⊢ ((2↑;;612) mod 𝑁) = (;;324 mod 𝑁)
179	17	1259lem1 16832	. . 3 ⊢ ((2↑;17) mod 𝑁) = (;;136 mod 𝑁)
180		eqid 2738	. . . 4 ⊢ ;;612 = ;;612
181		eqid 2738	. . . 4 ⊢ ;17 = ;17
182		eqid 2738	. . . . 5 ⊢ ;61 = ;61
183	2, 9, 114, 182	decsuc 12468	. . . 4 ⊢ (;61 + 1) = ;62
184		7p2e9 12134	. . . . 5 ⊢ (7 + 2) = 9
185	56, 57, 184	addcomli 11167	. . . 4 ⊢ (2 + 7) = 9
186	27, 3, 9, 35, 180, 181, 183, 185	decadd 12491	. . 3 ⊢ (;;612 + ;17) = ;;629
187	29, 9	deccl 12452	. . . . 5 ⊢ ;31 ∈ ℕ0
188		eqid 2738	. . . . . . 7 ⊢ ;31 = ;31
189		3p2e5 12124	. . . . . . . . 9 ⊢ (3 + 2) = 5
190	99, 57, 189	addcomli 11167	. . . . . . . 8 ⊢ (2 + 3) = 5
191	9, 3, 29, 113, 190	decaddi 12497	. . . . . . 7 ⊢ (;12 + 3) = ;15
192		5p1e6 12120	. . . . . . 7 ⊢ (5 + 1) = 6
193	10, 11, 29, 9, 111, 188, 191, 192	decadd 12491	. . . . . 6 ⊢ (;;125 + ;31) = ;;156
194	114	oveq1i 7285	. . . . . . . . 9 ⊢ ((1 + 1) + 1) = (2 + 1)
195	194, 73	eqtri 2766	. . . . . . . 8 ⊢ ((1 + 1) + 1) = 3
196		7p5e12 12514	. . . . . . . . 9 ⊢ (7 + 5) = ;12
197	56, 92, 196	addcomli 11167	. . . . . . . 8 ⊢ (5 + 7) = ;12
198	9, 11, 9, 35, 88, 181, 195, 3, 197	decaddc 12492	. . . . . . 7 ⊢ (;15 + ;17) = ;32
199		eqid 2738	. . . . . . . 8 ⊢ ;34 = ;34
200		7p3e10 12512	. . . . . . . . 9 ⊢ (7 + 3) = ;10
201	56, 99, 200	addcomli 11167	. . . . . . . 8 ⊢ (3 + 7) = ;10
202	99	mulid1i 10979	. . . . . . . . . 10 ⊢ (3 · 1) = 3
203	16	addid1i 11162	. . . . . . . . . 10 ⊢ (1 + 0) = 1
204	202, 203	oveq12i 7287	. . . . . . . . 9 ⊢ ((3 · 1) + (1 + 0)) = (3 + 1)
205		3p1e4 12118	. . . . . . . . 9 ⊢ (3 + 1) = 4
206	204, 205	eqtri 2766	. . . . . . . 8 ⊢ ((3 · 1) + (1 + 0)) = 4
207		4cn 12058	. . . . . . . . . . 11 ⊢ 4 ∈ ℂ
208	207	mulid1i 10979	. . . . . . . . . 10 ⊢ (4 · 1) = 4
209	208	oveq1i 7285	. . . . . . . . 9 ⊢ ((4 · 1) + 0) = (4 + 0)
210	207	addid1i 11162	. . . . . . . . 9 ⊢ (4 + 0) = 4
211	30	dec0h 12459	. . . . . . . . 9 ⊢ 4 = ;04
212	209, 210, 211	3eqtri 2770	. . . . . . . 8 ⊢ ((4 · 1) + 0) = ;04
213	29, 30, 9, 39, 199, 201, 9, 30, 39, 206, 212	decmac 12489	. . . . . . 7 ⊢ ((;34 · 1) + (3 + 7)) = ;44
214	3	dec0h 12459	. . . . . . . 8 ⊢ 2 = ;02
215	100, 145	oveq12i 7287	. . . . . . . . 9 ⊢ ((3 · 2) + (0 + 1)) = (6 + 1)
216		6p1e7 12121	. . . . . . . . 9 ⊢ (6 + 1) = 7
217	215, 216	eqtri 2766	. . . . . . . 8 ⊢ ((3 · 2) + (0 + 1)) = 7
218		4t2e8 12141	. . . . . . . . . 10 ⊢ (4 · 2) = 8
219	218	oveq1i 7285	. . . . . . . . 9 ⊢ ((4 · 2) + 2) = (8 + 2)
220		8p2e10 12517	. . . . . . . . 9 ⊢ (8 + 2) = ;10
221	219, 220	eqtri 2766	. . . . . . . 8 ⊢ ((4 · 2) + 2) = ;10
222	29, 30, 39, 3, 199, 214, 3, 39, 9, 217, 221	decmac 12489	. . . . . . 7 ⊢ ((;34 · 2) + 2) = ;70
223	9, 3, 29, 3, 113, 198, 31, 39, 35, 213, 222	decma2c 12490	. . . . . 6 ⊢ ((;34 · ;12) + (;15 + ;17)) = ;;440
224		5t3e15 12538	. . . . . . . . 9 ⊢ (5 · 3) = ;15
225	92, 99, 224	mulcomli 10984	. . . . . . . 8 ⊢ (3 · 5) = ;15
226		5p2e7 12129	. . . . . . . 8 ⊢ (5 + 2) = 7
227	9, 11, 3, 225, 226	decaddi 12497	. . . . . . 7 ⊢ ((3 · 5) + 2) = ;17
228		5t4e20 12539	. . . . . . . . 9 ⊢ (5 · 4) = ;20
229	92, 207, 228	mulcomli 10984	. . . . . . . 8 ⊢ (4 · 5) = ;20
230	61	addid2i 11163	. . . . . . . 8 ⊢ (0 + 6) = 6
231	3, 39, 2, 229, 230	decaddi 12497	. . . . . . 7 ⊢ ((4 · 5) + 6) = ;26
232	29, 30, 2, 199, 11, 2, 3, 227, 231	decrmac 12495	. . . . . 6 ⊢ ((;34 · 5) + 6) = ;;176
233	10, 11, 46, 2, 111, 193, 31, 2, 36, 223, 232	decma2c 12490	. . . . 5 ⊢ ((;34 · ;;125) + (;;125 + ;31)) = ;;;4406
234		9cn 12073	. . . . . . . 8 ⊢ 9 ∈ ℂ
235		9t3e27 12560	. . . . . . . 8 ⊢ (9 · 3) = ;27
236	234, 99, 235	mulcomli 10984	. . . . . . 7 ⊢ (3 · 9) = ;27
237		7p4e11 12513	. . . . . . 7 ⊢ (7 + 4) = ;11
238	3, 35, 30, 236, 73, 9, 237	decaddci 12498	. . . . . 6 ⊢ ((3 · 9) + 4) = ;31
239		9t4e36 12561	. . . . . . . 8 ⊢ (9 · 4) = ;36
240	234, 207, 239	mulcomli 10984	. . . . . . 7 ⊢ (4 · 9) = ;36
241	151, 61, 172	addcomli 11167	. . . . . . 7 ⊢ (6 + 8) = ;14
242	29, 2, 13, 240, 205, 30, 241	decaddci 12498	. . . . . 6 ⊢ ((4 · 9) + 8) = ;44
243	29, 30, 13, 199, 5, 30, 30, 238, 242	decrmac 12495	. . . . 5 ⊢ ((;34 · 9) + 8) = ;;314
244	12, 5, 12, 13, 17, 22, 31, 30, 187, 233, 243	decma2c 12490	. . . 4 ⊢ ((;34 · 𝑁) + (𝑁 − 1)) = ;;;;44064
245		eqid 2738	. . . . 5 ⊢ ;;136 = ;;136
246	9, 5	deccl 12452	. . . . . 6 ⊢ ;19 ∈ ℕ0
247	246, 30	deccl 12452	. . . . 5 ⊢ ;;194 ∈ ℕ0
248		eqid 2738	. . . . . 6 ⊢ ;13 = ;13
249		eqid 2738	. . . . . 6 ⊢ ;;194 = ;;194
250	5, 35	deccl 12452	. . . . . 6 ⊢ ;97 ∈ ℕ0
251	9, 9	deccl 12452	. . . . . . 7 ⊢ ;11 ∈ ℕ0
252		eqid 2738	. . . . . . 7 ⊢ ;;324 = ;;324
253		eqid 2738	. . . . . . . 8 ⊢ ;19 = ;19
254		eqid 2738	. . . . . . . 8 ⊢ ;97 = ;97
255	234, 16, 120	addcomli 11167	. . . . . . . . 9 ⊢ (1 + 9) = ;10
256	9, 39, 145, 255	decsuc 12468	. . . . . . . 8 ⊢ ((1 + 9) + 1) = ;11
257		9p7e16 12529	. . . . . . . 8 ⊢ (9 + 7) = ;16
258	9, 5, 5, 35, 253, 254, 256, 2, 257	decaddc 12492	. . . . . . 7 ⊢ (;19 + ;97) = ;;116
259		eqid 2738	. . . . . . . 8 ⊢ ;32 = ;32
260		eqid 2738	. . . . . . . . 9 ⊢ ;11 = ;11
261	9, 9, 114, 260	decsuc 12468	. . . . . . . 8 ⊢ (;11 + 1) = ;12
262	89	oveq1i 7285	. . . . . . . . 9 ⊢ ((2 · 1) + 2) = (2 + 2)
263	262, 115, 211	3eqtri 2770	. . . . . . . 8 ⊢ ((2 · 1) + 2) = ;04
264	29, 3, 9, 3, 259, 261, 9, 30, 39, 206, 263	decmac 12489	. . . . . . 7 ⊢ ((;32 · 1) + (;11 + 1)) = ;44
265	208	oveq1i 7285	. . . . . . . 8 ⊢ ((4 · 1) + 6) = (4 + 6)
266		6p4e10 12509	. . . . . . . . 9 ⊢ (6 + 4) = ;10
267	61, 207, 266	addcomli 11167	. . . . . . . 8 ⊢ (4 + 6) = ;10
268	265, 267	eqtri 2766	. . . . . . 7 ⊢ ((4 · 1) + 6) = ;10
269	33, 30, 251, 2, 252, 258, 9, 39, 9, 264, 268	decmac 12489	. . . . . 6 ⊢ ((;;324 · 1) + (;19 + ;97)) = ;;440
270	145, 138	eqtri 2766	. . . . . . . 8 ⊢ (0 + 1) = ;01
271		3t3e9 12140	. . . . . . . . . 10 ⊢ (3 · 3) = 9
272	271, 139	oveq12i 7287	. . . . . . . . 9 ⊢ ((3 · 3) + (0 + 0)) = (9 + 0)
273	234	addid1i 11162	. . . . . . . . 9 ⊢ (9 + 0) = 9
274	272, 273	eqtri 2766	. . . . . . . 8 ⊢ ((3 · 3) + (0 + 0)) = 9
275	101	oveq1i 7285	. . . . . . . . 9 ⊢ ((2 · 3) + 1) = (6 + 1)
276	35	dec0h 12459	. . . . . . . . 9 ⊢ 7 = ;07
277	275, 216, 276	3eqtri 2770	. . . . . . . 8 ⊢ ((2 · 3) + 1) = ;07
278	29, 3, 39, 9, 259, 270, 29, 35, 39, 274, 277	decmac 12489	. . . . . . 7 ⊢ ((;32 · 3) + (0 + 1)) = ;97
279		4t3e12 12535	. . . . . . . 8 ⊢ (4 · 3) = ;12
280		4p2e6 12126	. . . . . . . . 9 ⊢ (4 + 2) = 6
281	207, 57, 280	addcomli 11167	. . . . . . . 8 ⊢ (2 + 4) = 6
282	9, 3, 30, 279, 281	decaddi 12497	. . . . . . 7 ⊢ ((4 · 3) + 4) = ;16
283	33, 30, 39, 30, 252, 211, 29, 2, 9, 278, 282	decmac 12489	. . . . . 6 ⊢ ((;;324 · 3) + 4) = ;;976
284	9, 29, 246, 30, 248, 249, 34, 2, 250, 269, 283	decma2c 12490	. . . . 5 ⊢ ((;;324 · ;13) + ;;194) = ;;;4406
285		6t3e18 12542	. . . . . . . . 9 ⊢ (6 · 3) = ;18
286	61, 99, 285	mulcomli 10984	. . . . . . . 8 ⊢ (3 · 6) = ;18
287	9, 13, 18, 286	decsuc 12468	. . . . . . 7 ⊢ ((3 · 6) + 1) = ;19
288	9, 3, 3, 63, 115	decaddi 12497	. . . . . . 7 ⊢ ((2 · 6) + 2) = ;14
289	29, 3, 3, 259, 2, 30, 9, 287, 288	decrmac 12495	. . . . . 6 ⊢ ((;32 · 6) + 2) = ;;194
290		6t4e24 12543	. . . . . . 7 ⊢ (6 · 4) = ;24
291	61, 207, 290	mulcomli 10984	. . . . . 6 ⊢ (4 · 6) = ;24
292	2, 33, 30, 252, 30, 3, 289, 291	decmul1c 12502	. . . . 5 ⊢ (;;324 · 6) = ;;;1944
293	34, 37, 2, 245, 30, 247, 284, 292	decmul2c 12503	. . . 4 ⊢ (;;324 · ;;136) = ;;;;44064
294	244, 293	eqtr4i 2769	. . 3 ⊢ ((;34 · 𝑁) + (𝑁 − 1)) = (;;324 · ;;136)
295	26, 1, 28, 32, 34, 23, 36, 38, 178, 179, 186, 294	modxai 16769	. 2 ⊢ ((2↑;;629) mod 𝑁) = ((𝑁 − 1) mod 𝑁)
296		eqid 2738	. . . 4 ⊢ ;;629 = ;;629
297		eqid 2738	. . . . 5 ⊢ ;62 = ;62
298	139	oveq2i 7286	. . . . . 6 ⊢ ((2 · 6) + (0 + 0)) = ((2 · 6) + 0)
299	63	oveq1i 7285	. . . . . 6 ⊢ ((2 · 6) + 0) = (;12 + 0)
300	10	nn0cni 12245	. . . . . . 7 ⊢ ;12 ∈ ℂ
301	300	addid1i 11162	. . . . . 6 ⊢ (;12 + 0) = ;12
302	298, 299, 301	3eqtri 2770	. . . . 5 ⊢ ((2 · 6) + (0 + 0)) = ;12
303	11	dec0h 12459	. . . . . 6 ⊢ 5 = ;05
304	81, 55, 303	3eqtri 2770	. . . . 5 ⊢ ((2 · 2) + 1) = ;05
305	2, 3, 39, 9, 297, 138, 3, 11, 39, 302, 304	decma2c 12490	. . . 4 ⊢ ((2 · ;62) + 1) = ;;125
306		9t2e18 12559	. . . . 5 ⊢ (9 · 2) = ;18
307	234, 57, 306	mulcomli 10984	. . . 4 ⊢ (2 · 9) = ;18
308	3, 4, 5, 296, 13, 9, 305, 307	decmul2c 12503	. . 3 ⊢ (2 · ;;629) = ;;;1258
309	308, 22	eqtr4i 2769	. 2 ⊢ (2 · ;;629) = (𝑁 − 1)
310		npcan 11230	. . 3 ⊢ ((𝑁 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝑁 − 1) + 1) = 𝑁)
311	67, 16, 310	mp2an 689	. 2 ⊢ ((𝑁 − 1) + 1) = 𝑁
312	68	oveq1i 7285	. . 3 ⊢ ((0 · 𝑁) + 1) = (0 + 1)
313	145, 312, 161	3eqtr4i 2776	. 2 ⊢ ((0 · 𝑁) + 1) = (1 · 1)
314	1, 6, 7, 8, 9, 23, 295, 309, 311, 313	mod2xnegi 16772	1 ⊢ ((2↑(𝑁 − 1)) mod 𝑁) = (1 mod 𝑁)

Syntax hints:   = wceq 1539   ∈ wcel 2106  (class class class)co 7275  ℂcc 10869  0cc0 10871  1c1 10872   + caddc 10874   · cmul 10876   − cmin 11205  ℕcn 11973  2c2 12028  3c3 12029  4c4 12030  5c5 12031  6c6 12032  7c7 12033  8c8 12034  9c9 12035  ℕ₀cn0 12233  ;cdc 12437   mod cmo 13589  ↑cexp 13782

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588  ax-cnex 10927  ax-resscn 10928  ax-1cn 10929  ax-icn 10930  ax-addcl 10931  ax-addrcl 10932  ax-mulcl 10933  ax-mulrcl 10934  ax-mulcom 10935  ax-addass 10936  ax-mulass 10937  ax-distr 10938  ax-i2m1 10939  ax-1ne0 10940  ax-1rid 10941  ax-rnegex 10942  ax-rrecex 10943  ax-cnre 10944  ax-pre-lttri 10945  ax-pre-lttrn 10946  ax-pre-ltadd 10947  ax-pre-mulgt0 10948  ax-pre-sup 10949

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3069  df-rex 3070  df-rmo 3071  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-pss 3906  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-tr 5192  df-id 5489  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-we 5546  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-pred 6202  df-ord 6269  df-on 6270  df-lim 6271  df-suc 6272  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-riota 7232  df-ov 7278  df-oprab 7279  df-mpo 7280  df-om 7713  df-2nd 7832  df-frecs 8097  df-wrecs 8128  df-recs 8202  df-rdg 8241  df-er 8498  df-en 8734  df-dom 8735  df-sdom 8736  df-sup 9201  df-inf 9202  df-pnf 11011  df-mnf 11012  df-xr 11013  df-ltxr 11014  df-le 11015  df-sub 11207  df-neg 11208  df-div 11633  df-nn 11974  df-2 12036  df-3 12037  df-4 12038  df-5 12039  df-6 12040  df-7 12041  df-8 12042  df-9 12043  df-n0 12234  df-z 12320  df-dec 12438  df-uz 12583  df-rp 12731  df-fl 13512  df-mod 13590  df-seq 13722  df-exp 13783

This theorem is referenced by:  1259prm  16837