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Theorem 1259lem4 17013
Description: Lemma for 1259prm 17015. 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
StepHypRef Expression
1 2nn 12233 . 2 2 ∈ ℕ
2 6nn0 12441 . . . 4 6 ∈ ℕ0
3 2nn0 12437 . . . 4 2 ∈ ℕ0
42, 3deccl 12640 . . 3 62 ∈ ℕ0
5 9nn0 12444 . . 3 9 ∈ ℕ0
64, 5deccl 12640 . 2 629 ∈ ℕ0
7 0z 12517 . 2 0 ∈ ℤ
8 1nn 12171 . 2 1 ∈ ℕ
9 1nn0 12436 . 2 1 ∈ ℕ0
109, 3deccl 12640 . . . . . . 7 12 ∈ ℕ0
11 5nn0 12440 . . . . . . 7 5 ∈ ℕ0
1210, 11deccl 12640 . . . . . 6 125 ∈ ℕ0
13 8nn0 12443 . . . . . 6 8 ∈ ℕ0
1412, 13deccl 12640 . . . . 5 1258 ∈ ℕ0
1514nn0cni 12432 . . . 4 1258 ∈ ℂ
16 ax-1cn 11116 . . . 4 1 ∈ ℂ
17 1259prm.1 . . . . 5 𝑁 = 1259
18 8p1e9 12310 . . . . . 6 (8 + 1) = 9
19 eqid 2737 . . . . . 6 1258 = 1258
2012, 13, 18, 19decsuc 12656 . . . . 5 (1258 + 1) = 1259
2117, 20eqtr4i 2768 . . . 4 𝑁 = (1258 + 1)
2215, 16, 21mvrraddi 11425 . . 3 (𝑁 − 1) = 1258
2322, 14eqeltri 2834 . 2 (𝑁 − 1) ∈ ℕ0
24 9nn 12258 . . . . 5 9 ∈ ℕ
2512, 24decnncl 12645 . . . 4 1259 ∈ ℕ
2617, 25eqeltri 2834 . . 3 𝑁 ∈ ℕ
272, 9deccl 12640 . . . 4 61 ∈ ℕ0
2827, 3deccl 12640 . . 3 612 ∈ ℕ0
29 3nn0 12438 . . . . 5 3 ∈ ℕ0
30 4nn0 12439 . . . . 5 4 ∈ ℕ0
3129, 30deccl 12640 . . . 4 34 ∈ ℕ0
3231nn0zi 12535 . . 3 34 ∈ ℤ
3329, 3deccl 12640 . . . 4 32 ∈ ℕ0
3433, 30deccl 12640 . . 3 324 ∈ ℕ0
35 7nn0 12442 . . . 4 7 ∈ ℕ0
369, 35deccl 12640 . . 3 17 ∈ ℕ0
379, 29deccl 12640 . . . 4 13 ∈ ℕ0
3837, 2deccl 12640 . . 3 136 ∈ ℕ0
39 0nn0 12435 . . . . . 6 0 ∈ ℕ0
4029, 39deccl 12640 . . . . 5 30 ∈ ℕ0
4140, 2deccl 12640 . . . 4 306 ∈ ℕ0
42 8nn 12255 . . . . 5 8 ∈ ℕ
439, 42decnncl 12645 . . . 4 18 ∈ ℕ
4410, 30deccl 12640 . . . . 5 124 ∈ ℕ0
4544, 9deccl 12640 . . . 4 1241 ∈ ℕ0
469, 11deccl 12640 . . . . . 6 15 ∈ ℕ0
4746, 29deccl 12640 . . . . 5 153 ∈ ℕ0
48 1z 12540 . . . . 5 1 ∈ ℤ
4911, 39deccl 12640 . . . . 5 50 ∈ ℕ0
5046, 3deccl 12640 . . . . . 6 152 ∈ ℕ0
513, 11deccl 12640 . . . . . 6 25 ∈ ℕ0
5235, 2deccl 12640 . . . . . . 7 76 ∈ ℕ0
53171259lem3 17012 . . . . . . 7 ((2↑76) mod 𝑁) = (5 mod 𝑁)
54 eqid 2737 . . . . . . . 8 76 = 76
55 4p1e5 12306 . . . . . . . . 9 (4 + 1) = 5
56 7cn 12254 . . . . . . . . . 10 7 ∈ ℂ
57 2cn 12235 . . . . . . . . . 10 2 ∈ ℂ
58 7t2e14 12734 . . . . . . . . . 10 (7 · 2) = 14
5956, 57, 58mulcomli 11171 . . . . . . . . 9 (2 · 7) = 14
609, 30, 55, 59decsuc 12656 . . . . . . . 8 ((2 · 7) + 1) = 15
61 6cn 12251 . . . . . . . . 9 6 ∈ ℂ
62 6t2e12 12729 . . . . . . . . 9 (6 · 2) = 12
6361, 57, 62mulcomli 11171 . . . . . . . 8 (2 · 6) = 12
643, 35, 2, 54, 3, 9, 60, 63decmul2c 12691 . . . . . . 7 (2 · 76) = 152
6551nn0cni 12432 . . . . . . . . 9 25 ∈ ℂ
6665addid2i 11350 . . . . . . . 8 (0 + 25) = 25
6726nncni 12170 . . . . . . . . . 10 𝑁 ∈ ℂ
6867mul02i 11351 . . . . . . . . 9 (0 · 𝑁) = 0
6968oveq1i 7372 . . . . . . . 8 ((0 · 𝑁) + 25) = (0 + 25)
70 5t5e25 12728 . . . . . . . 8 (5 · 5) = 25
7166, 69, 703eqtr4i 2775 . . . . . . 7 ((0 · 𝑁) + 25) = (5 · 5)
7226, 1, 52, 7, 11, 51, 53, 64, 71mod2xi 16948 . . . . . 6 ((2↑152) mod 𝑁) = (25 mod 𝑁)
73 2p1e3 12302 . . . . . . 7 (2 + 1) = 3
74 eqid 2737 . . . . . . 7 152 = 152
7546, 3, 73, 74decsuc 12656 . . . . . 6 (152 + 1) = 153
7649nn0cni 12432 . . . . . . . 8 50 ∈ ℂ
7776addid2i 11350 . . . . . . 7 (0 + 50) = 50
7868oveq1i 7372 . . . . . . 7 ((0 · 𝑁) + 50) = (0 + 50)
79 eqid 2737 . . . . . . . 8 25 = 25
80 2t2e4 12324 . . . . . . . . . 10 (2 · 2) = 4
8180oveq1i 7372 . . . . . . . . 9 ((2 · 2) + 1) = (4 + 1)
8281, 55eqtri 2765 . . . . . . . 8 ((2 · 2) + 1) = 5
83 5t2e10 12725 . . . . . . . 8 (5 · 2) = 10
843, 3, 11, 79, 39, 9, 82, 83decmul1c 12690 . . . . . . 7 (25 · 2) = 50
8577, 78, 843eqtr4i 2775 . . . . . 6 ((0 · 𝑁) + 50) = (25 · 2)
8626, 1, 50, 7, 51, 49, 72, 75, 85modxp1i 16949 . . . . 5 ((2↑153) mod 𝑁) = (50 mod 𝑁)
87 eqid 2737 . . . . . 6 153 = 153
88 eqid 2737 . . . . . . . . 9 15 = 15
8957mulid1i 11166 . . . . . . . . . . 11 (2 · 1) = 2
9089oveq1i 7372 . . . . . . . . . 10 ((2 · 1) + 1) = (2 + 1)
9190, 73eqtri 2765 . . . . . . . . 9 ((2 · 1) + 1) = 3
92 5cn 12248 . . . . . . . . . 10 5 ∈ ℂ
9392, 57, 83mulcomli 11171 . . . . . . . . 9 (2 · 5) = 10
943, 9, 11, 88, 39, 9, 91, 93decmul2c 12691 . . . . . . . 8 (2 · 15) = 30
9594oveq1i 7372 . . . . . . 7 ((2 · 15) + 0) = (30 + 0)
9640nn0cni 12432 . . . . . . . 8 30 ∈ ℂ
9796addid1i 11349 . . . . . . 7 (30 + 0) = 30
9895, 97eqtri 2765 . . . . . 6 ((2 · 15) + 0) = 30
99 3cn 12241 . . . . . . . 8 3 ∈ ℂ
100 3t2e6 12326 . . . . . . . 8 (3 · 2) = 6
10199, 57, 100mulcomli 11171 . . . . . . 7 (2 · 3) = 6
1022dec0h 12647 . . . . . . 7 6 = 06
103101, 102eqtri 2765 . . . . . 6 (2 · 3) = 06
1043, 46, 29, 87, 2, 39, 98, 103decmul2c 12691 . . . . 5 (2 · 153) = 306
10567mulid2i 11167 . . . . . . . 8 (1 · 𝑁) = 𝑁
106105, 17eqtri 2765 . . . . . . 7 (1 · 𝑁) = 1259
107 eqid 2737 . . . . . . 7 1241 = 1241
1083, 30deccl 12640 . . . . . . . 8 24 ∈ ℕ0
109 eqid 2737 . . . . . . . . 9 24 = 24
1103, 30, 55, 109decsuc 12656 . . . . . . . 8 (24 + 1) = 25
111 eqid 2737 . . . . . . . . 9 125 = 125
112 eqid 2737 . . . . . . . . 9 124 = 124
113 eqid 2737 . . . . . . . . . 10 12 = 12
114 1p1e2 12285 . . . . . . . . . 10 (1 + 1) = 2
115 2p2e4 12295 . . . . . . . . . 10 (2 + 2) = 4
1169, 3, 9, 3, 113, 113, 114, 115decadd 12679 . . . . . . . . 9 (12 + 12) = 24
117 5p4e9 12318 . . . . . . . . 9 (5 + 4) = 9
11810, 11, 10, 30, 111, 112, 116, 117decadd 12679 . . . . . . . 8 (125 + 124) = 249
119108, 110, 118decsucc 12666 . . . . . . 7 ((125 + 124) + 1) = 250
120 9p1e10 12627 . . . . . . 7 (9 + 1) = 10
12112, 5, 44, 9, 106, 107, 119, 120decaddc2 12681 . . . . . 6 ((1 · 𝑁) + 1241) = 2500
122 eqid 2737 . . . . . . 7 50 = 50
12392mul02i 11351 . . . . . . . . . 10 (0 · 5) = 0
12411, 11, 39, 122, 70, 123decmul1 12689 . . . . . . . . 9 (50 · 5) = 250
125124oveq1i 7372 . . . . . . . 8 ((50 · 5) + 0) = (250 + 0)
12651, 39deccl 12640 . . . . . . . . . 10 250 ∈ ℕ0
127126nn0cni 12432 . . . . . . . . 9 250 ∈ ℂ
128127addid1i 11349 . . . . . . . 8 (250 + 0) = 250
129125, 128eqtri 2765 . . . . . . 7 ((50 · 5) + 0) = 250
13076mul01i 11352 . . . . . . . 8 (50 · 0) = 0
13139dec0h 12647 . . . . . . . 8 0 = 00
132130, 131eqtri 2765 . . . . . . 7 (50 · 0) = 00
13349, 11, 39, 122, 39, 39, 129, 132decmul2c 12691 . . . . . 6 (50 · 50) = 2500
134121, 133eqtr4i 2768 . . . . 5 ((1 · 𝑁) + 1241) = (50 · 50)
13526, 1, 47, 48, 49, 45, 86, 104, 134mod2xi 16948 . . . 4 ((2↑306) mod 𝑁) = (1241 mod 𝑁)
136 eqid 2737 . . . . 5 306 = 306
137 eqid 2737 . . . . . 6 30 = 30
1389dec0h 12647 . . . . . 6 1 = 01
139 00id 11337 . . . . . . . 8 (0 + 0) = 0
140101, 139oveq12i 7374 . . . . . . 7 ((2 · 3) + (0 + 0)) = (6 + 0)
14161addid1i 11349 . . . . . . 7 (6 + 0) = 6
142140, 141eqtri 2765 . . . . . 6 ((2 · 3) + (0 + 0)) = 6
14357mul01i 11352 . . . . . . . 8 (2 · 0) = 0
144143oveq1i 7372 . . . . . . 7 ((2 · 0) + 1) = (0 + 1)
145 0p1e1 12282 . . . . . . 7 (0 + 1) = 1
146144, 145, 1383eqtri 2769 . . . . . 6 ((2 · 0) + 1) = 01
14729, 39, 39, 9, 137, 138, 3, 9, 39, 142, 146decma2c 12678 . . . . 5 ((2 · 30) + 1) = 61
1483, 40, 2, 136, 3, 9, 147, 63decmul2c 12691 . . . 4 (2 · 306) = 612
149 eqid 2737 . . . . . 6 18 = 18
15010, 30, 55, 112decsuc 12656 . . . . . 6 (124 + 1) = 125
151 8cn 12257 . . . . . . 7 8 ∈ ℂ
152151, 16, 18addcomli 11354 . . . . . 6 (1 + 8) = 9
15344, 9, 9, 13, 107, 149, 150, 152decadd 12679 . . . . 5 (1241 + 18) = 1259
154153, 17eqtr4i 2768 . . . 4 (1241 + 18) = 𝑁
15534nn0cni 12432 . . . . . 6 324 ∈ ℂ
156155addid2i 11350 . . . . 5 (0 + 324) = 324
15768oveq1i 7372 . . . . 5 ((0 · 𝑁) + 324) = (0 + 324)
1589, 13deccl 12640 . . . . . 6 18 ∈ ℕ0
1599, 30deccl 12640 . . . . . 6 14 ∈ ℕ0
160 eqid 2737 . . . . . . 7 14 = 14
16116mulid1i 11166 . . . . . . . . 9 (1 · 1) = 1
162161, 114oveq12i 7374 . . . . . . . 8 ((1 · 1) + (1 + 1)) = (1 + 2)
163 1p2e3 12303 . . . . . . . 8 (1 + 2) = 3
164162, 163eqtri 2765 . . . . . . 7 ((1 · 1) + (1 + 1)) = 3
165151mulid1i 11166 . . . . . . . . 9 (8 · 1) = 8
166165oveq1i 7372 . . . . . . . 8 ((8 · 1) + 4) = (8 + 4)
167 8p4e12 12707 . . . . . . . 8 (8 + 4) = 12
168166, 167eqtri 2765 . . . . . . 7 ((8 · 1) + 4) = 12
1699, 13, 9, 30, 149, 160, 9, 3, 9, 164, 168decmac 12677 . . . . . 6 ((18 · 1) + 14) = 32
170151mulid2i 11167 . . . . . . . . 9 (1 · 8) = 8
171170oveq1i 7372 . . . . . . . 8 ((1 · 8) + 6) = (8 + 6)
172 8p6e14 12709 . . . . . . . 8 (8 + 6) = 14
173171, 172eqtri 2765 . . . . . . 7 ((1 · 8) + 6) = 14
174 8t8e64 12746 . . . . . . 7 (8 · 8) = 64
17513, 9, 13, 149, 30, 2, 173, 174decmul1c 12690 . . . . . 6 (18 · 8) = 144
176158, 9, 13, 149, 30, 159, 169, 175decmul2c 12691 . . . . 5 (18 · 18) = 324
177156, 157, 1763eqtr4i 2775 . . . 4 ((0 · 𝑁) + 324) = (18 · 18)
1781, 41, 7, 43, 34, 45, 135, 148, 154, 177mod2xnegi 16950 . . 3 ((2↑612) mod 𝑁) = (324 mod 𝑁)
179171259lem1 17010 . . 3 ((2↑17) mod 𝑁) = (136 mod 𝑁)
180 eqid 2737 . . . 4 612 = 612
181 eqid 2737 . . . 4 17 = 17
182 eqid 2737 . . . . 5 61 = 61
1832, 9, 114, 182decsuc 12656 . . . 4 (61 + 1) = 62
184 7p2e9 12321 . . . . 5 (7 + 2) = 9
18556, 57, 184addcomli 11354 . . . 4 (2 + 7) = 9
18627, 3, 9, 35, 180, 181, 183, 185decadd 12679 . . 3 (612 + 17) = 629
18729, 9deccl 12640 . . . . 5 31 ∈ ℕ0
188 eqid 2737 . . . . . . 7 31 = 31
189 3p2e5 12311 . . . . . . . . 9 (3 + 2) = 5
19099, 57, 189addcomli 11354 . . . . . . . 8 (2 + 3) = 5
1919, 3, 29, 113, 190decaddi 12685 . . . . . . 7 (12 + 3) = 15
192 5p1e6 12307 . . . . . . 7 (5 + 1) = 6
19310, 11, 29, 9, 111, 188, 191, 192decadd 12679 . . . . . 6 (125 + 31) = 156
194114oveq1i 7372 . . . . . . . . 9 ((1 + 1) + 1) = (2 + 1)
195194, 73eqtri 2765 . . . . . . . 8 ((1 + 1) + 1) = 3
196 7p5e12 12702 . . . . . . . . 9 (7 + 5) = 12
19756, 92, 196addcomli 11354 . . . . . . . 8 (5 + 7) = 12
1989, 11, 9, 35, 88, 181, 195, 3, 197decaddc 12680 . . . . . . 7 (15 + 17) = 32
199 eqid 2737 . . . . . . . 8 34 = 34
200 7p3e10 12700 . . . . . . . . 9 (7 + 3) = 10
20156, 99, 200addcomli 11354 . . . . . . . 8 (3 + 7) = 10
20299mulid1i 11166 . . . . . . . . . 10 (3 · 1) = 3
20316addid1i 11349 . . . . . . . . . 10 (1 + 0) = 1
204202, 203oveq12i 7374 . . . . . . . . 9 ((3 · 1) + (1 + 0)) = (3 + 1)
205 3p1e4 12305 . . . . . . . . 9 (3 + 1) = 4
206204, 205eqtri 2765 . . . . . . . 8 ((3 · 1) + (1 + 0)) = 4
207 4cn 12245 . . . . . . . . . . 11 4 ∈ ℂ
208207mulid1i 11166 . . . . . . . . . 10 (4 · 1) = 4
209208oveq1i 7372 . . . . . . . . 9 ((4 · 1) + 0) = (4 + 0)
210207addid1i 11349 . . . . . . . . 9 (4 + 0) = 4
21130dec0h 12647 . . . . . . . . 9 4 = 04
212209, 210, 2113eqtri 2769 . . . . . . . 8 ((4 · 1) + 0) = 04
21329, 30, 9, 39, 199, 201, 9, 30, 39, 206, 212decmac 12677 . . . . . . 7 ((34 · 1) + (3 + 7)) = 44
2143dec0h 12647 . . . . . . . 8 2 = 02
215100, 145oveq12i 7374 . . . . . . . . 9 ((3 · 2) + (0 + 1)) = (6 + 1)
216 6p1e7 12308 . . . . . . . . 9 (6 + 1) = 7
217215, 216eqtri 2765 . . . . . . . 8 ((3 · 2) + (0 + 1)) = 7
218 4t2e8 12328 . . . . . . . . . 10 (4 · 2) = 8
219218oveq1i 7372 . . . . . . . . 9 ((4 · 2) + 2) = (8 + 2)
220 8p2e10 12705 . . . . . . . . 9 (8 + 2) = 10
221219, 220eqtri 2765 . . . . . . . 8 ((4 · 2) + 2) = 10
22229, 30, 39, 3, 199, 214, 3, 39, 9, 217, 221decmac 12677 . . . . . . 7 ((34 · 2) + 2) = 70
2239, 3, 29, 3, 113, 198, 31, 39, 35, 213, 222decma2c 12678 . . . . . 6 ((34 · 12) + (15 + 17)) = 440
224 5t3e15 12726 . . . . . . . . 9 (5 · 3) = 15
22592, 99, 224mulcomli 11171 . . . . . . . 8 (3 · 5) = 15
226 5p2e7 12316 . . . . . . . 8 (5 + 2) = 7
2279, 11, 3, 225, 226decaddi 12685 . . . . . . 7 ((3 · 5) + 2) = 17
228 5t4e20 12727 . . . . . . . . 9 (5 · 4) = 20
22992, 207, 228mulcomli 11171 . . . . . . . 8 (4 · 5) = 20
23061addid2i 11350 . . . . . . . 8 (0 + 6) = 6
2313, 39, 2, 229, 230decaddi 12685 . . . . . . 7 ((4 · 5) + 6) = 26
23229, 30, 2, 199, 11, 2, 3, 227, 231decrmac 12683 . . . . . 6 ((34 · 5) + 6) = 176
23310, 11, 46, 2, 111, 193, 31, 2, 36, 223, 232decma2c 12678 . . . . 5 ((34 · 125) + (125 + 31)) = 4406
234 9cn 12260 . . . . . . . 8 9 ∈ ℂ
235 9t3e27 12748 . . . . . . . 8 (9 · 3) = 27
236234, 99, 235mulcomli 11171 . . . . . . 7 (3 · 9) = 27
237 7p4e11 12701 . . . . . . 7 (7 + 4) = 11
2383, 35, 30, 236, 73, 9, 237decaddci 12686 . . . . . 6 ((3 · 9) + 4) = 31
239 9t4e36 12749 . . . . . . . 8 (9 · 4) = 36
240234, 207, 239mulcomli 11171 . . . . . . 7 (4 · 9) = 36
241151, 61, 172addcomli 11354 . . . . . . 7 (6 + 8) = 14
24229, 2, 13, 240, 205, 30, 241decaddci 12686 . . . . . 6 ((4 · 9) + 8) = 44
24329, 30, 13, 199, 5, 30, 30, 238, 242decrmac 12683 . . . . 5 ((34 · 9) + 8) = 314
24412, 5, 12, 13, 17, 22, 31, 30, 187, 233, 243decma2c 12678 . . . 4 ((34 · 𝑁) + (𝑁 − 1)) = 44064
245 eqid 2737 . . . . 5 136 = 136
2469, 5deccl 12640 . . . . . 6 19 ∈ ℕ0
247246, 30deccl 12640 . . . . 5 194 ∈ ℕ0
248 eqid 2737 . . . . . 6 13 = 13
249 eqid 2737 . . . . . 6 194 = 194
2505, 35deccl 12640 . . . . . 6 97 ∈ ℕ0
2519, 9deccl 12640 . . . . . . 7 11 ∈ ℕ0
252 eqid 2737 . . . . . . 7 324 = 324
253 eqid 2737 . . . . . . . 8 19 = 19
254 eqid 2737 . . . . . . . 8 97 = 97
255234, 16, 120addcomli 11354 . . . . . . . . 9 (1 + 9) = 10
2569, 39, 145, 255decsuc 12656 . . . . . . . 8 ((1 + 9) + 1) = 11
257 9p7e16 12717 . . . . . . . 8 (9 + 7) = 16
2589, 5, 5, 35, 253, 254, 256, 2, 257decaddc 12680 . . . . . . 7 (19 + 97) = 116
259 eqid 2737 . . . . . . . 8 32 = 32
260 eqid 2737 . . . . . . . . 9 11 = 11
2619, 9, 114, 260decsuc 12656 . . . . . . . 8 (11 + 1) = 12
26289oveq1i 7372 . . . . . . . . 9 ((2 · 1) + 2) = (2 + 2)
263262, 115, 2113eqtri 2769 . . . . . . . 8 ((2 · 1) + 2) = 04
26429, 3, 9, 3, 259, 261, 9, 30, 39, 206, 263decmac 12677 . . . . . . 7 ((32 · 1) + (11 + 1)) = 44
265208oveq1i 7372 . . . . . . . 8 ((4 · 1) + 6) = (4 + 6)
266 6p4e10 12697 . . . . . . . . 9 (6 + 4) = 10
26761, 207, 266addcomli 11354 . . . . . . . 8 (4 + 6) = 10
268265, 267eqtri 2765 . . . . . . 7 ((4 · 1) + 6) = 10
26933, 30, 251, 2, 252, 258, 9, 39, 9, 264, 268decmac 12677 . . . . . 6 ((324 · 1) + (19 + 97)) = 440
270145, 138eqtri 2765 . . . . . . . 8 (0 + 1) = 01
271 3t3e9 12327 . . . . . . . . . 10 (3 · 3) = 9
272271, 139oveq12i 7374 . . . . . . . . 9 ((3 · 3) + (0 + 0)) = (9 + 0)
273234addid1i 11349 . . . . . . . . 9 (9 + 0) = 9
274272, 273eqtri 2765 . . . . . . . 8 ((3 · 3) + (0 + 0)) = 9
275101oveq1i 7372 . . . . . . . . 9 ((2 · 3) + 1) = (6 + 1)
27635dec0h 12647 . . . . . . . . 9 7 = 07
277275, 216, 2763eqtri 2769 . . . . . . . 8 ((2 · 3) + 1) = 07
27829, 3, 39, 9, 259, 270, 29, 35, 39, 274, 277decmac 12677 . . . . . . 7 ((32 · 3) + (0 + 1)) = 97
279 4t3e12 12723 . . . . . . . 8 (4 · 3) = 12
280 4p2e6 12313 . . . . . . . . 9 (4 + 2) = 6
281207, 57, 280addcomli 11354 . . . . . . . 8 (2 + 4) = 6
2829, 3, 30, 279, 281decaddi 12685 . . . . . . 7 ((4 · 3) + 4) = 16
28333, 30, 39, 30, 252, 211, 29, 2, 9, 278, 282decmac 12677 . . . . . 6 ((324 · 3) + 4) = 976
2849, 29, 246, 30, 248, 249, 34, 2, 250, 269, 283decma2c 12678 . . . . 5 ((324 · 13) + 194) = 4406
285 6t3e18 12730 . . . . . . . . 9 (6 · 3) = 18
28661, 99, 285mulcomli 11171 . . . . . . . 8 (3 · 6) = 18
2879, 13, 18, 286decsuc 12656 . . . . . . 7 ((3 · 6) + 1) = 19
2889, 3, 3, 63, 115decaddi 12685 . . . . . . 7 ((2 · 6) + 2) = 14
28929, 3, 3, 259, 2, 30, 9, 287, 288decrmac 12683 . . . . . 6 ((32 · 6) + 2) = 194
290 6t4e24 12731 . . . . . . 7 (6 · 4) = 24
29161, 207, 290mulcomli 11171 . . . . . 6 (4 · 6) = 24
2922, 33, 30, 252, 30, 3, 289, 291decmul1c 12690 . . . . 5 (324 · 6) = 1944
29334, 37, 2, 245, 30, 247, 284, 292decmul2c 12691 . . . 4 (324 · 136) = 44064
294244, 293eqtr4i 2768 . . 3 ((34 · 𝑁) + (𝑁 − 1)) = (324 · 136)
29526, 1, 28, 32, 34, 23, 36, 38, 178, 179, 186, 294modxai 16947 . 2 ((2↑629) mod 𝑁) = ((𝑁 − 1) mod 𝑁)
296 eqid 2737 . . . 4 629 = 629
297 eqid 2737 . . . . 5 62 = 62
298139oveq2i 7373 . . . . . 6 ((2 · 6) + (0 + 0)) = ((2 · 6) + 0)
29963oveq1i 7372 . . . . . 6 ((2 · 6) + 0) = (12 + 0)
30010nn0cni 12432 . . . . . . 7 12 ∈ ℂ
301300addid1i 11349 . . . . . 6 (12 + 0) = 12
302298, 299, 3013eqtri 2769 . . . . 5 ((2 · 6) + (0 + 0)) = 12
30311dec0h 12647 . . . . . 6 5 = 05
30481, 55, 3033eqtri 2769 . . . . 5 ((2 · 2) + 1) = 05
3052, 3, 39, 9, 297, 138, 3, 11, 39, 302, 304decma2c 12678 . . . 4 ((2 · 62) + 1) = 125
306 9t2e18 12747 . . . . 5 (9 · 2) = 18
307234, 57, 306mulcomli 11171 . . . 4 (2 · 9) = 18
3083, 4, 5, 296, 13, 9, 305, 307decmul2c 12691 . . 3 (2 · 629) = 1258
309308, 22eqtr4i 2768 . 2 (2 · 629) = (𝑁 − 1)
310 npcan 11417 . . 3 ((𝑁 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝑁 − 1) + 1) = 𝑁)
31167, 16, 310mp2an 691 . 2 ((𝑁 − 1) + 1) = 𝑁
31268oveq1i 7372 . . 3 ((0 · 𝑁) + 1) = (0 + 1)
313145, 312, 1613eqtr4i 2775 . 2 ((0 · 𝑁) + 1) = (1 · 1)
3141, 6, 7, 8, 9, 23, 295, 309, 311, 313mod2xnegi 16950 1 ((2↑(𝑁 − 1)) mod 𝑁) = (1 mod 𝑁)
Colors of variables: wff setvar class
Syntax hints:   = wceq 1542  wcel 2107  (class class class)co 7362  cc 11056  0cc0 11058  1c1 11059   + caddc 11061   · cmul 11063  cmin 11392  cn 12160  2c2 12215  3c3 12216  4c4 12217  5c5 12218  6c6 12219  7c7 12220  8c8 12221  9c9 12222  0cn0 12420  cdc 12625   mod cmo 13781  cexp 13974
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 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-sep 5261  ax-nul 5268  ax-pow 5325  ax-pr 5389  ax-un 7677  ax-cnex 11114  ax-resscn 11115  ax-1cn 11116  ax-icn 11117  ax-addcl 11118  ax-addrcl 11119  ax-mulcl 11120  ax-mulrcl 11121  ax-mulcom 11122  ax-addass 11123  ax-mulass 11124  ax-distr 11125  ax-i2m1 11126  ax-1ne0 11127  ax-1rid 11128  ax-rnegex 11129  ax-rrecex 11130  ax-cnre 11131  ax-pre-lttri 11132  ax-pre-lttrn 11133  ax-pre-ltadd 11134  ax-pre-mulgt0 11135  ax-pre-sup 11136
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-nel 3051  df-ral 3066  df-rex 3075  df-rmo 3356  df-reu 3357  df-rab 3411  df-v 3450  df-sbc 3745  df-csb 3861  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-pss 3934  df-nul 4288  df-if 4492  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-iun 4961  df-br 5111  df-opab 5173  df-mpt 5194  df-tr 5228  df-id 5536  df-eprel 5542  df-po 5550  df-so 5551  df-fr 5593  df-we 5595  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-pred 6258  df-ord 6325  df-on 6326  df-lim 6327  df-suc 6328  df-iota 6453  df-fun 6503  df-fn 6504  df-f 6505  df-f1 6506  df-fo 6507  df-f1o 6508  df-fv 6509  df-riota 7318  df-ov 7365  df-oprab 7366  df-mpo 7367  df-om 7808  df-2nd 7927  df-frecs 8217  df-wrecs 8248  df-recs 8322  df-rdg 8361  df-er 8655  df-en 8891  df-dom 8892  df-sdom 8893  df-sup 9385  df-inf 9386  df-pnf 11198  df-mnf 11199  df-xr 11200  df-ltxr 11201  df-le 11202  df-sub 11394  df-neg 11395  df-div 11820  df-nn 12161  df-2 12223  df-3 12224  df-4 12225  df-5 12226  df-6 12227  df-7 12228  df-8 12229  df-9 12230  df-n0 12421  df-z 12507  df-dec 12626  df-uz 12771  df-rp 12923  df-fl 13704  df-mod 13782  df-seq 13914  df-exp 13975
This theorem is referenced by:  1259prm  17015
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