Theorem 1259lem4 15841

Description: Lemma for 1259prm 15843. 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 11185	. 2 ⊢ 2 ∈ ℕ
2		6nn0 11313	. . . 4 ⊢ 6 ∈ ℕ0
3		2nn0 11309	. . . 4 ⊢ 2 ∈ ℕ0
4	2, 3	deccl 11512	. . 3 ⊢ ;62 ∈ ℕ0
5		9nn0 11316	. . 3 ⊢ 9 ∈ ℕ0
6	4, 5	deccl 11512	. 2 ⊢ ;;629 ∈ ℕ0
7		0z 11388	. 2 ⊢ 0 ∈ ℤ
8		1nn 11031	. 2 ⊢ 1 ∈ ℕ
9		1nn0 11308	. 2 ⊢ 1 ∈ ℕ0
10	9, 3	deccl 11512	. . . . . . 7 ⊢ ;12 ∈ ℕ0
11		5nn0 11312	. . . . . . 7 ⊢ 5 ∈ ℕ0
12	10, 11	deccl 11512	. . . . . 6 ⊢ ;;125 ∈ ℕ0
13		8nn0 11315	. . . . . 6 ⊢ 8 ∈ ℕ0
14	12, 13	deccl 11512	. . . . 5 ⊢ ;;;1258 ∈ ℕ0
15	14	nn0cni 11304	. . . 4 ⊢ ;;;1258 ∈ ℂ
16		ax-1cn 9994	. . . 4 ⊢ 1 ∈ ℂ
17		1259prm.1	. . . . 5 ⊢ 𝑁 = ;;;1259
18		8p1e9 11158	. . . . . 6 ⊢ (8 + 1) = 9
19		eqid 2622	. . . . . 6 ⊢ ;;;1258 = ;;;1258
20	12, 13, 18, 19	decsuc 11535	. . . . 5 ⊢ (;;;1258 + 1) = ;;;1259
21	17, 20	eqtr4i 2647	. . . 4 ⊢ 𝑁 = (;;;1258 + 1)
22	15, 16, 21	mvrraddi 10298	. . 3 ⊢ (𝑁 − 1) = ;;;1258
23	22, 14	eqeltri 2697	. 2 ⊢ (𝑁 − 1) ∈ ℕ0
24		9nn 11192	. . . . 5 ⊢ 9 ∈ ℕ
25	12, 24	decnncl 11518	. . . 4 ⊢ ;;;1259 ∈ ℕ
26	17, 25	eqeltri 2697	. . 3 ⊢ 𝑁 ∈ ℕ
27	2, 9	deccl 11512	. . . 4 ⊢ ;61 ∈ ℕ0
28	27, 3	deccl 11512	. . 3 ⊢ ;;612 ∈ ℕ0
29		3nn0 11310	. . . . 5 ⊢ 3 ∈ ℕ0
30		4nn0 11311	. . . . 5 ⊢ 4 ∈ ℕ0
31	29, 30	deccl 11512	. . . 4 ⊢ ;34 ∈ ℕ0
32	31	nn0zi 11402	. . 3 ⊢ ;34 ∈ ℤ
33	29, 3	deccl 11512	. . . 4 ⊢ ;32 ∈ ℕ0
34	33, 30	deccl 11512	. . 3 ⊢ ;;324 ∈ ℕ0
35		7nn0 11314	. . . 4 ⊢ 7 ∈ ℕ0
36	9, 35	deccl 11512	. . 3 ⊢ ;17 ∈ ℕ0
37	9, 29	deccl 11512	. . . 4 ⊢ ;13 ∈ ℕ0
38	37, 2	deccl 11512	. . 3 ⊢ ;;136 ∈ ℕ0
39		0nn0 11307	. . . . . 6 ⊢ 0 ∈ ℕ0
40	29, 39	deccl 11512	. . . . 5 ⊢ ;30 ∈ ℕ0
41	40, 2	deccl 11512	. . . 4 ⊢ ;;306 ∈ ℕ0
42		8nn 11191	. . . . 5 ⊢ 8 ∈ ℕ
43	9, 42	decnncl 11518	. . . 4 ⊢ ;18 ∈ ℕ
44	10, 30	deccl 11512	. . . . 5 ⊢ ;;124 ∈ ℕ0
45	44, 9	deccl 11512	. . . 4 ⊢ ;;;1241 ∈ ℕ0
46	9, 11	deccl 11512	. . . . . 6 ⊢ ;15 ∈ ℕ0
47	46, 29	deccl 11512	. . . . 5 ⊢ ;;153 ∈ ℕ0
48		1z 11407	. . . . 5 ⊢ 1 ∈ ℤ
49	11, 39	deccl 11512	. . . . 5 ⊢ ;50 ∈ ℕ0
50	46, 3	deccl 11512	. . . . . 6 ⊢ ;;152 ∈ ℕ0
51	3, 11	deccl 11512	. . . . . 6 ⊢ ;25 ∈ ℕ0
52	35, 2	deccl 11512	. . . . . . 7 ⊢ ;76 ∈ ℕ0
53	17	1259lem3 15840	. . . . . . 7 ⊢ ((2↑;76) mod 𝑁) = (5 mod 𝑁)
54		eqid 2622	. . . . . . . 8 ⊢ ;76 = ;76
55		4p1e5 11154	. . . . . . . . 9 ⊢ (4 + 1) = 5
56		7cn 11104	. . . . . . . . . 10 ⊢ 7 ∈ ℂ
57		2cn 11091	. . . . . . . . . 10 ⊢ 2 ∈ ℂ
58		7t2e14 11648	. . . . . . . . . 10 ⊢ (7 · 2) = ;14
59	56, 57, 58	mulcomli 10047	. . . . . . . . 9 ⊢ (2 · 7) = ;14
60	9, 30, 55, 59	decsuc 11535	. . . . . . . 8 ⊢ ((2 · 7) + 1) = ;15
61		6cn 11102	. . . . . . . . 9 ⊢ 6 ∈ ℂ
62		6t2e12 11641	. . . . . . . . 9 ⊢ (6 · 2) = ;12
63	61, 57, 62	mulcomli 10047	. . . . . . . 8 ⊢ (2 · 6) = ;12
64	3, 35, 2, 54, 3, 9, 60, 63	decmul2c 11589	. . . . . . 7 ⊢ (2 · ;76) = ;;152
65	51	nn0cni 11304	. . . . . . . . 9 ⊢ ;25 ∈ ℂ
66	65	addid2i 10224	. . . . . . . 8 ⊢ (0 + ;25) = ;25
67	26	nncni 11030	. . . . . . . . . 10 ⊢ 𝑁 ∈ ℂ
68	67	mul02i 10225	. . . . . . . . 9 ⊢ (0 · 𝑁) = 0
69	68	oveq1i 6660	. . . . . . . 8 ⊢ ((0 · 𝑁) + ;25) = (0 + ;25)
70		5t5e25 11639	. . . . . . . 8 ⊢ (5 · 5) = ;25
71	66, 69, 70	3eqtr4i 2654	. . . . . . 7 ⊢ ((0 · 𝑁) + ;25) = (5 · 5)
72	26, 1, 52, 7, 11, 51, 53, 64, 71	mod2xi 15773	. . . . . 6 ⊢ ((2↑;;152) mod 𝑁) = (;25 mod 𝑁)
73		2p1e3 11151	. . . . . . 7 ⊢ (2 + 1) = 3
74		eqid 2622	. . . . . . 7 ⊢ ;;152 = ;;152
75	46, 3, 73, 74	decsuc 11535	. . . . . 6 ⊢ (;;152 + 1) = ;;153
76	49	nn0cni 11304	. . . . . . . 8 ⊢ ;50 ∈ ℂ
77	76	addid2i 10224	. . . . . . 7 ⊢ (0 + ;50) = ;50
78	68	oveq1i 6660	. . . . . . 7 ⊢ ((0 · 𝑁) + ;50) = (0 + ;50)
79		eqid 2622	. . . . . . . 8 ⊢ ;25 = ;25
80		2t2e4 11177	. . . . . . . . . 10 ⊢ (2 · 2) = 4
81	80	oveq1i 6660	. . . . . . . . 9 ⊢ ((2 · 2) + 1) = (4 + 1)
82	81, 55	eqtri 2644	. . . . . . . 8 ⊢ ((2 · 2) + 1) = 5
83		5t2e10 11634	. . . . . . . 8 ⊢ (5 · 2) = ;10
84	3, 3, 11, 79, 39, 9, 82, 83	decmul1c 11587	. . . . . . 7 ⊢ (;25 · 2) = ;50
85	77, 78, 84	3eqtr4i 2654	. . . . . 6 ⊢ ((0 · 𝑁) + ;50) = (;25 · 2)
86	26, 1, 50, 7, 51, 49, 72, 75, 85	modxp1i 15774	. . . . 5 ⊢ ((2↑;;153) mod 𝑁) = (;50 mod 𝑁)
87		eqid 2622	. . . . . 6 ⊢ ;;153 = ;;153
88		eqid 2622	. . . . . . . . 9 ⊢ ;15 = ;15
89	57	mulid1i 10042	. . . . . . . . . . 11 ⊢ (2 · 1) = 2
90	89	oveq1i 6660	. . . . . . . . . 10 ⊢ ((2 · 1) + 1) = (2 + 1)
91	90, 73	eqtri 2644	. . . . . . . . 9 ⊢ ((2 · 1) + 1) = 3
92		5cn 11100	. . . . . . . . . 10 ⊢ 5 ∈ ℂ
93	92, 57, 83	mulcomli 10047	. . . . . . . . 9 ⊢ (2 · 5) = ;10
94	3, 9, 11, 88, 39, 9, 91, 93	decmul2c 11589	. . . . . . . 8 ⊢ (2 · ;15) = ;30
95	94	oveq1i 6660	. . . . . . 7 ⊢ ((2 · ;15) + 0) = (;30 + 0)
96	40	nn0cni 11304	. . . . . . . 8 ⊢ ;30 ∈ ℂ
97	96	addid1i 10223	. . . . . . 7 ⊢ (;30 + 0) = ;30
98	95, 97	eqtri 2644	. . . . . 6 ⊢ ((2 · ;15) + 0) = ;30
99		3cn 11095	. . . . . . . 8 ⊢ 3 ∈ ℂ
100		3t2e6 11179	. . . . . . . 8 ⊢ (3 · 2) = 6
101	99, 57, 100	mulcomli 10047	. . . . . . 7 ⊢ (2 · 3) = 6
102	2	dec0h 11522	. . . . . . 7 ⊢ 6 = ;06
103	101, 102	eqtri 2644	. . . . . 6 ⊢ (2 · 3) = ;06
104	3, 46, 29, 87, 2, 39, 98, 103	decmul2c 11589	. . . . 5 ⊢ (2 · ;;153) = ;;306
105	67	mulid2i 10043	. . . . . . . 8 ⊢ (1 · 𝑁) = 𝑁
106	105, 17	eqtri 2644	. . . . . . 7 ⊢ (1 · 𝑁) = ;;;1259
107		eqid 2622	. . . . . . 7 ⊢ ;;;1241 = ;;;1241
108	3, 30	deccl 11512	. . . . . . . 8 ⊢ ;24 ∈ ℕ0
109		eqid 2622	. . . . . . . . 9 ⊢ ;24 = ;24
110	3, 30, 55, 109	decsuc 11535	. . . . . . . 8 ⊢ (;24 + 1) = ;25
111		eqid 2622	. . . . . . . . 9 ⊢ ;;125 = ;;125
112		eqid 2622	. . . . . . . . 9 ⊢ ;;124 = ;;124
113		eqid 2622	. . . . . . . . . 10 ⊢ ;12 = ;12
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 ⊢ (;12 + ;12) = ;24
117		5p4e9 11167	. . . . . . . . 9 ⊢ (5 + 4) = 9
118	10, 11, 10, 30, 111, 112, 116, 117	decadd 11570	. . . . . . . 8 ⊢ (;;125 + ;;124) = ;;249
119	108, 110, 118	decsucc 11550	. . . . . . 7 ⊢ ((;;125 + ;;124) + 1) = ;;250
120		9p1e10 11496	. . . . . . 7 ⊢ (9 + 1) = ;10
121	12, 5, 44, 9, 106, 107, 119, 120	decaddc2 11575	. . . . . 6 ⊢ ((1 · 𝑁) + ;;;1241) = ;;;2500
122		eqid 2622	. . . . . . 7 ⊢ ;50 = ;50
123	92	mul02i 10225	. . . . . . . . . 10 ⊢ (0 · 5) = 0
124	11, 11, 39, 122, 39, 70, 123	decmul1 11585	. . . . . . . . 9 ⊢ (;50 · 5) = ;;250
125	124	oveq1i 6660	. . . . . . . 8 ⊢ ((;50 · 5) + 0) = (;;250 + 0)
126	51, 39	deccl 11512	. . . . . . . . . 10 ⊢ ;;250 ∈ ℕ0
127	126	nn0cni 11304	. . . . . . . . 9 ⊢ ;;250 ∈ ℂ
128	127	addid1i 10223	. . . . . . . 8 ⊢ (;;250 + 0) = ;;250
129	125, 128	eqtri 2644	. . . . . . 7 ⊢ ((;50 · 5) + 0) = ;;250
130	76	mul01i 10226	. . . . . . . 8 ⊢ (;50 · 0) = 0
131	39	dec0h 11522	. . . . . . . 8 ⊢ 0 = ;00
132	130, 131	eqtri 2644	. . . . . . 7 ⊢ (;50 · 0) = ;00
133	49, 11, 39, 122, 39, 39, 129, 132	decmul2c 11589	. . . . . 6 ⊢ (;50 · ;50) = ;;;2500
134	121, 133	eqtr4i 2647	. . . . 5 ⊢ ((1 · 𝑁) + ;;;1241) = (;50 · ;50)
135	26, 1, 47, 48, 49, 45, 86, 104, 134	mod2xi 15773	. . . 4 ⊢ ((2↑;;306) mod 𝑁) = (;;;1241 mod 𝑁)
136		eqid 2622	. . . . 5 ⊢ ;;306 = ;;306
137		eqid 2622	. . . . . 6 ⊢ ;30 = ;30
138	9	dec0h 11522	. . . . . 6 ⊢ 1 = ;01
139		00id 10211	. . . . . . . 8 ⊢ (0 + 0) = 0
140	101, 139	oveq12i 6662	. . . . . . 7 ⊢ ((2 · 3) + (0 + 0)) = (6 + 0)
141	61	addid1i 10223	. . . . . . 7 ⊢ (6 + 0) = 6
142	140, 141	eqtri 2644	. . . . . 6 ⊢ ((2 · 3) + (0 + 0)) = 6
143	57	mul01i 10226	. . . . . . . 8 ⊢ (2 · 0) = 0
144	143	oveq1i 6660	. . . . . . 7 ⊢ ((2 · 0) + 1) = (0 + 1)
145		0p1e1 11132	. . . . . . 7 ⊢ (0 + 1) = 1
146	144, 145, 138	3eqtri 2648	. . . . . 6 ⊢ ((2 · 0) + 1) = ;01
147	29, 39, 39, 9, 137, 138, 3, 9, 39, 142, 146	decma2c 11568	. . . . 5 ⊢ ((2 · ;30) + 1) = ;61
148	3, 40, 2, 136, 3, 9, 147, 63	decmul2c 11589	. . . 4 ⊢ (2 · ;;306) = ;;612
149		eqid 2622	. . . . . 6 ⊢ ;18 = ;18
150	10, 30, 55, 112	decsuc 11535	. . . . . 6 ⊢ (;;124 + 1) = ;;125
151		8cn 11106	. . . . . . 7 ⊢ 8 ∈ ℂ
152	151, 16, 18	addcomli 10228	. . . . . 6 ⊢ (1 + 8) = 9
153	44, 9, 9, 13, 107, 149, 150, 152	decadd 11570	. . . . 5 ⊢ (;;;1241 + ;18) = ;;;1259
154	153, 17	eqtr4i 2647	. . . 4 ⊢ (;;;1241 + ;18) = 𝑁
155	34	nn0cni 11304	. . . . . 6 ⊢ ;;324 ∈ ℂ
156	155	addid2i 10224	. . . . 5 ⊢ (0 + ;;324) = ;;324
157	68	oveq1i 6660	. . . . 5 ⊢ ((0 · 𝑁) + ;;324) = (0 + ;;324)
158	9, 13	deccl 11512	. . . . . 6 ⊢ ;18 ∈ ℕ0
159	9, 30	deccl 11512	. . . . . 6 ⊢ ;14 ∈ ℕ0
160		eqid 2622	. . . . . . 7 ⊢ ;14 = ;14
161	16	mulid1i 10042	. . . . . . . . 9 ⊢ (1 · 1) = 1
162	161, 114	oveq12i 6662	. . . . . . . 8 ⊢ ((1 · 1) + (1 + 1)) = (1 + 2)
163		1p2e3 11152	. . . . . . . 8 ⊢ (1 + 2) = 3
164	162, 163	eqtri 2644	. . . . . . 7 ⊢ ((1 · 1) + (1 + 1)) = 3
165	151	mulid1i 10042	. . . . . . . . 9 ⊢ (8 · 1) = 8
166	165	oveq1i 6660	. . . . . . . 8 ⊢ ((8 · 1) + 4) = (8 + 4)
167		8p4e12 11614	. . . . . . . 8 ⊢ (8 + 4) = ;12
168	166, 167	eqtri 2644	. . . . . . 7 ⊢ ((8 · 1) + 4) = ;12
169	9, 13, 9, 30, 149, 160, 9, 3, 9, 164, 168	decmac 11566	. . . . . 6 ⊢ ((;18 · 1) + ;14) = ;32
170	151	mulid2i 10043	. . . . . . . . 9 ⊢ (1 · 8) = 8
171	170	oveq1i 6660	. . . . . . . 8 ⊢ ((1 · 8) + 6) = (8 + 6)
172		8p6e14 11616	. . . . . . . 8 ⊢ (8 + 6) = ;14
173	171, 172	eqtri 2644	. . . . . . 7 ⊢ ((1 · 8) + 6) = ;14
174		8t8e64 11662	. . . . . . 7 ⊢ (8 · 8) = ;64
175	13, 9, 13, 149, 30, 2, 173, 174	decmul1c 11587	. . . . . 6 ⊢ (;18 · 8) = ;;144
176	158, 9, 13, 149, 30, 159, 169, 175	decmul2c 11589	. . . . 5 ⊢ (;18 · ;18) = ;;324
177	156, 157, 176	3eqtr4i 2654	. . . 4 ⊢ ((0 · 𝑁) + ;;324) = (;18 · ;18)
178	1, 41, 7, 43, 34, 45, 135, 148, 154, 177	mod2xnegi 15775	. . 3 ⊢ ((2↑;;612) mod 𝑁) = (;;324 mod 𝑁)
179	17	1259lem1 15838	. . 3 ⊢ ((2↑;17) mod 𝑁) = (;;136 mod 𝑁)
180		eqid 2622	. . . 4 ⊢ ;;612 = ;;612
181		eqid 2622	. . . 4 ⊢ ;17 = ;17
182		eqid 2622	. . . . 5 ⊢ ;61 = ;61
183	2, 9, 114, 182	decsuc 11535	. . . 4 ⊢ (;61 + 1) = ;62
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 ⊢ (;;612 + ;17) = ;;629
187	29, 9	deccl 11512	. . . . 5 ⊢ ;31 ∈ ℕ0
188		eqid 2622	. . . . . . 7 ⊢ ;31 = ;31
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 ⊢ (;12 + 3) = ;15
192		5p1e6 11155	. . . . . . 7 ⊢ (5 + 1) = 6
193	10, 11, 29, 9, 111, 188, 191, 192	decadd 11570	. . . . . 6 ⊢ (;;125 + ;31) = ;;156
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) = ;12
197	56, 92, 196	addcomli 10228	. . . . . . . 8 ⊢ (5 + 7) = ;12
198	9, 11, 9, 35, 88, 181, 195, 3, 197	decaddc 11572	. . . . . . 7 ⊢ (;15 + ;17) = ;32
199		eqid 2622	. . . . . . . 8 ⊢ ;34 = ;34
200		7p3e10 11603	. . . . . . . . 9 ⊢ (7 + 3) = ;10
201	56, 99, 200	addcomli 10228	. . . . . . . 8 ⊢ (3 + 7) = ;10
202	99	mulid1i 10042	. . . . . . . . . 10 ⊢ (3 · 1) = 3
203	16	addid1i 10223	. . . . . . . . . 10 ⊢ (1 + 0) = 1
204	202, 203	oveq12i 6662	. . . . . . . . 9 ⊢ ((3 · 1) + (1 + 0)) = (3 + 1)
205		3p1e4 11153	. . . . . . . . 9 ⊢ (3 + 1) = 4
206	204, 205	eqtri 2644	. . . . . . . 8 ⊢ ((3 · 1) + (1 + 0)) = 4
207		4cn 11098	. . . . . . . . . . 11 ⊢ 4 ∈ ℂ
208	207	mulid1i 10042	. . . . . . . . . 10 ⊢ (4 · 1) = 4
209	208	oveq1i 6660	. . . . . . . . 9 ⊢ ((4 · 1) + 0) = (4 + 0)
210	207	addid1i 10223	. . . . . . . . 9 ⊢ (4 + 0) = 4
211	30	dec0h 11522	. . . . . . . . 9 ⊢ 4 = ;04
212	209, 210, 211	3eqtri 2648	. . . . . . . 8 ⊢ ((4 · 1) + 0) = ;04
213	29, 30, 9, 39, 199, 201, 9, 30, 39, 206, 212	decmac 11566	. . . . . . 7 ⊢ ((;34 · 1) + (3 + 7)) = ;44
214	3	dec0h 11522	. . . . . . . 8 ⊢ 2 = ;02
215	100, 145	oveq12i 6662	. . . . . . . . 9 ⊢ ((3 · 2) + (0 + 1)) = (6 + 1)
216		6p1e7 11156	. . . . . . . . 9 ⊢ (6 + 1) = 7
217	215, 216	eqtri 2644	. . . . . . . 8 ⊢ ((3 · 2) + (0 + 1)) = 7
218		4t2e8 11181	. . . . . . . . . 10 ⊢ (4 · 2) = 8
219	218	oveq1i 6660	. . . . . . . . 9 ⊢ ((4 · 2) + 2) = (8 + 2)
220		8p2e10 11610	. . . . . . . . 9 ⊢ (8 + 2) = ;10
221	219, 220	eqtri 2644	. . . . . . . 8 ⊢ ((4 · 2) + 2) = ;10
222	29, 30, 39, 3, 199, 214, 3, 39, 9, 217, 221	decmac 11566	. . . . . . 7 ⊢ ((;34 · 2) + 2) = ;70
223	9, 3, 29, 3, 113, 198, 31, 39, 35, 213, 222	decma2c 11568	. . . . . 6 ⊢ ((;34 · ;12) + (;15 + ;17)) = ;;440
224		5t3e15 11635	. . . . . . . . 9 ⊢ (5 · 3) = ;15
225	92, 99, 224	mulcomli 10047	. . . . . . . 8 ⊢ (3 · 5) = ;15
226		5p2e7 11165	. . . . . . . 8 ⊢ (5 + 2) = 7
227	9, 11, 3, 225, 226	decaddi 11579	. . . . . . 7 ⊢ ((3 · 5) + 2) = ;17
228		5t4e20 11637	. . . . . . . . 9 ⊢ (5 · 4) = ;20
229	92, 207, 228	mulcomli 10047	. . . . . . . 8 ⊢ (4 · 5) = ;20
230	61	addid2i 10224	. . . . . . . 8 ⊢ (0 + 6) = 6
231	3, 39, 2, 229, 230	decaddi 11579	. . . . . . 7 ⊢ ((4 · 5) + 6) = ;26
232	29, 30, 2, 199, 11, 2, 3, 227, 231	decrmac 11577	. . . . . 6 ⊢ ((;34 · 5) + 6) = ;;176
233	10, 11, 46, 2, 111, 193, 31, 2, 36, 223, 232	decma2c 11568	. . . . 5 ⊢ ((;34 · ;;125) + (;;125 + ;31)) = ;;;4406
234		9cn 11108	. . . . . . . 8 ⊢ 9 ∈ ℂ
235		9t3e27 11664	. . . . . . . 8 ⊢ (9 · 3) = ;27
236	234, 99, 235	mulcomli 10047	. . . . . . 7 ⊢ (3 · 9) = ;27
237		7p4e11 11605	. . . . . . 7 ⊢ (7 + 4) = ;11
238	3, 35, 30, 236, 73, 9, 237	decaddci 11580	. . . . . 6 ⊢ ((3 · 9) + 4) = ;31
239		9t4e36 11665	. . . . . . . 8 ⊢ (9 · 4) = ;36
240	234, 207, 239	mulcomli 10047	. . . . . . 7 ⊢ (4 · 9) = ;36
241	151, 61, 172	addcomli 10228	. . . . . . 7 ⊢ (6 + 8) = ;14
242	29, 2, 13, 240, 205, 30, 241	decaddci 11580	. . . . . 6 ⊢ ((4 · 9) + 8) = ;44
243	29, 30, 13, 199, 5, 30, 30, 238, 242	decrmac 11577	. . . . 5 ⊢ ((;34 · 9) + 8) = ;;314
244	12, 5, 12, 13, 17, 22, 31, 30, 187, 233, 243	decma2c 11568	. . . 4 ⊢ ((;34 · 𝑁) + (𝑁 − 1)) = ;;;;44064
245		eqid 2622	. . . . 5 ⊢ ;;136 = ;;136
246	9, 5	deccl 11512	. . . . . 6 ⊢ ;19 ∈ ℕ0
247	246, 30	deccl 11512	. . . . 5 ⊢ ;;194 ∈ ℕ0
248		eqid 2622	. . . . . 6 ⊢ ;13 = ;13
249		eqid 2622	. . . . . 6 ⊢ ;;194 = ;;194
250	5, 35	deccl 11512	. . . . . 6 ⊢ ;97 ∈ ℕ0
251	9, 9	deccl 11512	. . . . . . 7 ⊢ ;11 ∈ ℕ0
252		eqid 2622	. . . . . . 7 ⊢ ;;324 = ;;324
253		eqid 2622	. . . . . . . 8 ⊢ ;19 = ;19
254		eqid 2622	. . . . . . . 8 ⊢ ;97 = ;97
255	234, 16, 120	addcomli 10228	. . . . . . . . 9 ⊢ (1 + 9) = ;10
256	9, 39, 145, 255	decsuc 11535	. . . . . . . 8 ⊢ ((1 + 9) + 1) = ;11
257		9p7e16 11625	. . . . . . . 8 ⊢ (9 + 7) = ;16
258	9, 5, 5, 35, 253, 254, 256, 2, 257	decaddc 11572	. . . . . . 7 ⊢ (;19 + ;97) = ;;116
259		eqid 2622	. . . . . . . 8 ⊢ ;32 = ;32
260		eqid 2622	. . . . . . . . 9 ⊢ ;11 = ;11
261	9, 9, 114, 260	decsuc 11535	. . . . . . . 8 ⊢ (;11 + 1) = ;12
262	89	oveq1i 6660	. . . . . . . . 9 ⊢ ((2 · 1) + 2) = (2 + 2)
263	262, 115, 211	3eqtri 2648	. . . . . . . 8 ⊢ ((2 · 1) + 2) = ;04
264	29, 3, 9, 3, 259, 261, 9, 30, 39, 206, 263	decmac 11566	. . . . . . 7 ⊢ ((;32 · 1) + (;11 + 1)) = ;44
265	208	oveq1i 6660	. . . . . . . 8 ⊢ ((4 · 1) + 6) = (4 + 6)
266		6p4e10 11598	. . . . . . . . 9 ⊢ (6 + 4) = ;10
267	61, 207, 266	addcomli 10228	. . . . . . . 8 ⊢ (4 + 6) = ;10
268	265, 267	eqtri 2644	. . . . . . 7 ⊢ ((4 · 1) + 6) = ;10
269	33, 30, 251, 2, 252, 258, 9, 39, 9, 264, 268	decmac 11566	. . . . . 6 ⊢ ((;;324 · 1) + (;19 + ;97)) = ;;440
270	145, 138	eqtri 2644	. . . . . . . 8 ⊢ (0 + 1) = ;01
271		3t3e9 11180	. . . . . . . . . 10 ⊢ (3 · 3) = 9
272	271, 139	oveq12i 6662	. . . . . . . . 9 ⊢ ((3 · 3) + (0 + 0)) = (9 + 0)
273	234	addid1i 10223	. . . . . . . . 9 ⊢ (9 + 0) = 9
274	272, 273	eqtri 2644	. . . . . . . 8 ⊢ ((3 · 3) + (0 + 0)) = 9
275	101	oveq1i 6660	. . . . . . . . 9 ⊢ ((2 · 3) + 1) = (6 + 1)
276	35	dec0h 11522	. . . . . . . . 9 ⊢ 7 = ;07
277	275, 216, 276	3eqtri 2648	. . . . . . . 8 ⊢ ((2 · 3) + 1) = ;07
278	29, 3, 39, 9, 259, 270, 29, 35, 39, 274, 277	decmac 11566	. . . . . . 7 ⊢ ((;32 · 3) + (0 + 1)) = ;97
279		4t3e12 11632	. . . . . . . 8 ⊢ (4 · 3) = ;12
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 · 3) + 4) = ;16
283	33, 30, 39, 30, 252, 211, 29, 2, 9, 278, 282	decmac 11566	. . . . . 6 ⊢ ((;;324 · 3) + 4) = ;;976
284	9, 29, 246, 30, 248, 249, 34, 2, 250, 269, 283	decma2c 11568	. . . . 5 ⊢ ((;;324 · ;13) + ;;194) = ;;;4406
285		6t3e18 11642	. . . . . . . . 9 ⊢ (6 · 3) = ;18
286	61, 99, 285	mulcomli 10047	. . . . . . . 8 ⊢ (3 · 6) = ;18
287	9, 13, 18, 286	decsuc 11535	. . . . . . 7 ⊢ ((3 · 6) + 1) = ;19
288	9, 3, 3, 63, 115	decaddi 11579	. . . . . . 7 ⊢ ((2 · 6) + 2) = ;14
289	29, 3, 3, 259, 2, 30, 9, 287, 288	decrmac 11577	. . . . . 6 ⊢ ((;32 · 6) + 2) = ;;194
290		6t4e24 11643	. . . . . . 7 ⊢ (6 · 4) = ;24
291	61, 207, 290	mulcomli 10047	. . . . . 6 ⊢ (4 · 6) = ;24
292	2, 33, 30, 252, 30, 3, 289, 291	decmul1c 11587	. . . . 5 ⊢ (;;324 · 6) = ;;;1944
293	34, 37, 2, 245, 30, 247, 284, 292	decmul2c 11589	. . . 4 ⊢ (;;324 · ;;136) = ;;;;44064
294	244, 293	eqtr4i 2647	. . 3 ⊢ ((;34 · 𝑁) + (𝑁 − 1)) = (;;324 · ;;136)
295	26, 1, 28, 32, 34, 23, 36, 38, 178, 179, 186, 294	modxai 15772	. 2 ⊢ ((2↑;;629) mod 𝑁) = ((𝑁 − 1) mod 𝑁)
296		eqid 2622	. . . 4 ⊢ ;;629 = ;;629
297		eqid 2622	. . . . 5 ⊢ ;62 = ;62
298	139	oveq2i 6661	. . . . . 6 ⊢ ((2 · 6) + (0 + 0)) = ((2 · 6) + 0)
299	63	oveq1i 6660	. . . . . 6 ⊢ ((2 · 6) + 0) = (;12 + 0)
300	10	nn0cni 11304	. . . . . . 7 ⊢ ;12 ∈ ℂ
301	300	addid1i 10223	. . . . . 6 ⊢ (;12 + 0) = ;12
302	298, 299, 301	3eqtri 2648	. . . . 5 ⊢ ((2 · 6) + (0 + 0)) = ;12
303	11	dec0h 11522	. . . . . 6 ⊢ 5 = ;05
304	81, 55, 303	3eqtri 2648	. . . . 5 ⊢ ((2 · 2) + 1) = ;05
305	2, 3, 39, 9, 297, 138, 3, 11, 39, 302, 304	decma2c 11568	. . . 4 ⊢ ((2 · ;62) + 1) = ;;125
306		9t2e18 11663	. . . . 5 ⊢ (9 · 2) = ;18
307	234, 57, 306	mulcomli 10047	. . . 4 ⊢ (2 · 9) = ;18
308	3, 4, 5, 296, 13, 9, 305, 307	decmul2c 11589	. . 3 ⊢ (2 · ;;629) = ;;;1258
309	308, 22	eqtr4i 2647	. 2 ⊢ (2 · ;;629) = (𝑁 − 1)
310		npcan 10290	. . 3 ⊢ ((𝑁 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝑁 − 1) + 1) = 𝑁)
311	67, 16, 310	mp2an 708	. 2 ⊢ ((𝑁 − 1) + 1) = 𝑁
312	68	oveq1i 6660	. . 3 ⊢ ((0 · 𝑁) + 1) = (0 + 1)
313	145, 312, 161	3eqtr4i 2654	. 2 ⊢ ((0 · 𝑁) + 1) = (1 · 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   ∈ wcel 1990  (class class class)co 6650  ℂcc 9934  0cc0 9936  1c1 9937   + caddc 9939   · cmul 9941   − cmin 10266  ℕcn 11020  2c2 11070  3c3 11071  4c4 11072  5c5 11073  6c6 11074  7c7 11075  8c8 11076  9c9 11077  ℕ₀cn0 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