MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  4001lem1 Structured version   Visualization version   GIF version

Theorem 4001lem1 16246
Description: Lemma for 4001prm 16250. Calculate a power mod. In decimal, we calculate 2↑12 = 4096 = 𝑁 + 95, 2↑24 = (2↑12)↑2≡95↑2 = 2𝑁 + 1023, 2↑25 = 2↑24 · 2≡1023 · 2 = 2046, 2↑50 = (2↑25)↑2≡2046↑2 = 1046𝑁 + 1070, 2↑100 = (2↑50)↑2≡1070↑2 = 286𝑁 + 614 and 2↑200 = (2↑100)↑2≡614↑2 = 94𝑁 + 902 ≡902. (Contributed by Mario Carneiro, 3-Mar-2014.) (Revised by Mario Carneiro, 20-Apr-2015.) (Proof shortened by AV, 16-Sep-2021.)
Hypothesis
Ref Expression
4001prm.1 𝑁 = 4001
Assertion
Ref Expression
4001lem1 ((2↑200) mod 𝑁) = (902 mod 𝑁)

Proof of Theorem 4001lem1
StepHypRef Expression
1 4001prm.1 . . 3 𝑁 = 4001
2 4nn0 11663 . . . . . 6 4 ∈ ℕ0
3 0nn0 11659 . . . . . 6 0 ∈ ℕ0
42, 3deccl 11860 . . . . 5 40 ∈ ℕ0
54, 3deccl 11860 . . . 4 400 ∈ ℕ0
6 1nn 11387 . . . 4 1 ∈ ℕ
75, 6decnncl 11866 . . 3 4001 ∈ ℕ
81, 7eqeltri 2855 . 2 𝑁 ∈ ℕ
9 2nn 11448 . 2 2 ∈ ℕ
10 10nn0 11863 . . 3 10 ∈ ℕ0
1110, 3deccl 11860 . 2 100 ∈ ℕ0
12 9nn0 11668 . . . 4 9 ∈ ℕ0
1312, 2deccl 11860 . . 3 94 ∈ ℕ0
1413nn0zi 11754 . 2 94 ∈ ℤ
15 6nn0 11665 . . . 4 6 ∈ ℕ0
16 1nn0 11660 . . . 4 1 ∈ ℕ0
1715, 16deccl 11860 . . 3 61 ∈ ℕ0
1817, 2deccl 11860 . 2 614 ∈ ℕ0
1912, 3deccl 11860 . . 3 90 ∈ ℕ0
20 2nn0 11661 . . 3 2 ∈ ℕ0
2119, 20deccl 11860 . 2 902 ∈ ℕ0
22 5nn0 11664 . . . 4 5 ∈ ℕ0
2322, 3deccl 11860 . . 3 50 ∈ ℕ0
24 8nn0 11667 . . . . . 6 8 ∈ ℕ0
2520, 24deccl 11860 . . . . 5 28 ∈ ℕ0
2625, 15deccl 11860 . . . 4 286 ∈ ℕ0
2726nn0zi 11754 . . 3 286 ∈ ℤ
28 7nn0 11666 . . . . 5 7 ∈ ℕ0
2910, 28deccl 11860 . . . 4 107 ∈ ℕ0
3029, 3deccl 11860 . . 3 1070 ∈ ℕ0
3120, 22deccl 11860 . . . 4 25 ∈ ℕ0
3210, 2deccl 11860 . . . . . 6 104 ∈ ℕ0
3332, 15deccl 11860 . . . . 5 1046 ∈ ℕ0
3433nn0zi 11754 . . . 4 1046 ∈ ℤ
3520, 3deccl 11860 . . . . . 6 20 ∈ ℕ0
3635, 2deccl 11860 . . . . 5 204 ∈ ℕ0
3736, 15deccl 11860 . . . 4 2046 ∈ ℕ0
3820, 2deccl 11860 . . . . 5 24 ∈ ℕ0
39 0z 11739 . . . . 5 0 ∈ ℤ
4010, 20deccl 11860 . . . . . 6 102 ∈ ℕ0
41 3nn0 11662 . . . . . 6 3 ∈ ℕ0
4240, 41deccl 11860 . . . . 5 1023 ∈ ℕ0
4316, 20deccl 11860 . . . . . 6 12 ∈ ℕ0
44 2z 11761 . . . . . 6 2 ∈ ℤ
4512, 22deccl 11860 . . . . . 6 95 ∈ ℕ0
46 1z 11759 . . . . . . 7 1 ∈ ℤ
4715, 2deccl 11860 . . . . . . 7 64 ∈ ℕ0
48 2exp6 16194 . . . . . . . 8 (2↑6) = 64
4948oveq1i 6932 . . . . . . 7 ((2↑6) mod 𝑁) = (64 mod 𝑁)
50 6cn 11469 . . . . . . . 8 6 ∈ ℂ
51 2cn 11450 . . . . . . . 8 2 ∈ ℂ
52 6t2e12 11951 . . . . . . . 8 (6 · 2) = 12
5350, 51, 52mulcomli 10386 . . . . . . 7 (2 · 6) = 12
54 eqid 2778 . . . . . . . . 9 95 = 95
55 eqid 2778 . . . . . . . . . 10 400 = 400
56 9cn 11481 . . . . . . . . . . . 12 9 ∈ ℂ
5756addid1i 10563 . . . . . . . . . . 11 (9 + 0) = 9
5812dec0h 11868 . . . . . . . . . . 11 9 = 09
5957, 58eqtri 2802 . . . . . . . . . 10 (9 + 0) = 09
60 eqid 2778 . . . . . . . . . . 11 40 = 40
61 00id 10551 . . . . . . . . . . . 12 (0 + 0) = 0
623dec0h 11868 . . . . . . . . . . . 12 0 = 00
6361, 62eqtri 2802 . . . . . . . . . . 11 (0 + 0) = 00
64 4cn 11461 . . . . . . . . . . . . . 14 4 ∈ ℂ
6564mulid2i 10382 . . . . . . . . . . . . 13 (1 · 4) = 4
6665, 61oveq12i 6934 . . . . . . . . . . . 12 ((1 · 4) + (0 + 0)) = (4 + 0)
6764addid1i 10563 . . . . . . . . . . . 12 (4 + 0) = 4
6866, 67eqtri 2802 . . . . . . . . . . 11 ((1 · 4) + (0 + 0)) = 4
69 ax-1cn 10330 . . . . . . . . . . . . . 14 1 ∈ ℂ
7069mul01i 10566 . . . . . . . . . . . . 13 (1 · 0) = 0
7170oveq1i 6932 . . . . . . . . . . . 12 ((1 · 0) + 0) = (0 + 0)
7271, 61, 623eqtri 2806 . . . . . . . . . . 11 ((1 · 0) + 0) = 00
732, 3, 3, 3, 60, 63, 16, 3, 3, 68, 72decma2c 11899 . . . . . . . . . 10 ((1 · 40) + (0 + 0)) = 40
7470oveq1i 6932 . . . . . . . . . . 11 ((1 · 0) + 9) = (0 + 9)
7556addid2i 10564 . . . . . . . . . . 11 (0 + 9) = 9
7674, 75, 583eqtri 2806 . . . . . . . . . 10 ((1 · 0) + 9) = 09
774, 3, 3, 12, 55, 59, 16, 12, 3, 73, 76decma2c 11899 . . . . . . . . 9 ((1 · 400) + (9 + 0)) = 409
7869mulid1i 10381 . . . . . . . . . . 11 (1 · 1) = 1
7978oveq1i 6932 . . . . . . . . . 10 ((1 · 1) + 5) = (1 + 5)
80 5cn 11465 . . . . . . . . . . 11 5 ∈ ℂ
81 5p1e6 11529 . . . . . . . . . . 11 (5 + 1) = 6
8280, 69, 81addcomli 10568 . . . . . . . . . 10 (1 + 5) = 6
8315dec0h 11868 . . . . . . . . . 10 6 = 06
8479, 82, 833eqtri 2806 . . . . . . . . 9 ((1 · 1) + 5) = 06
855, 16, 12, 22, 1, 54, 16, 15, 3, 77, 84decma2c 11899 . . . . . . . 8 ((1 · 𝑁) + 95) = 4096
86 eqid 2778 . . . . . . . . 9 64 = 64
87 eqid 2778 . . . . . . . . . 10 25 = 25
88 2p2e4 11517 . . . . . . . . . . . 12 (2 + 2) = 4
8988oveq2i 6933 . . . . . . . . . . 11 ((6 · 6) + (2 + 2)) = ((6 · 6) + 4)
90 6t6e36 11955 . . . . . . . . . . . 12 (6 · 6) = 36
91 3p1e4 11527 . . . . . . . . . . . 12 (3 + 1) = 4
92 6p4e10 11919 . . . . . . . . . . . 12 (6 + 4) = 10
9341, 15, 2, 90, 91, 92decaddci2 11908 . . . . . . . . . . 11 ((6 · 6) + 4) = 40
9489, 93eqtri 2802 . . . . . . . . . 10 ((6 · 6) + (2 + 2)) = 40
95 6t4e24 11953 . . . . . . . . . . . 12 (6 · 4) = 24
9650, 64, 95mulcomli 10386 . . . . . . . . . . 11 (4 · 6) = 24
97 5p4e9 11540 . . . . . . . . . . . 12 (5 + 4) = 9
9880, 64, 97addcomli 10568 . . . . . . . . . . 11 (4 + 5) = 9
9920, 2, 22, 96, 98decaddi 11906 . . . . . . . . . 10 ((4 · 6) + 5) = 29
10015, 2, 20, 22, 86, 87, 15, 12, 20, 94, 99decmac 11898 . . . . . . . . 9 ((64 · 6) + 25) = 409
101 4p1e5 11528 . . . . . . . . . . 11 (4 + 1) = 5
10220, 2, 101, 95decsuc 11877 . . . . . . . . . 10 ((6 · 4) + 1) = 25
103 4t4e16 11946 . . . . . . . . . 10 (4 · 4) = 16
1042, 15, 2, 86, 15, 16, 102, 103decmul1c 11912 . . . . . . . . 9 (64 · 4) = 256
10547, 15, 2, 86, 15, 31, 100, 104decmul2c 11913 . . . . . . . 8 (64 · 64) = 4096
10685, 105eqtr4i 2805 . . . . . . 7 ((1 · 𝑁) + 95) = (64 · 64)
1078, 9, 15, 46, 47, 45, 49, 53, 106mod2xi 16177 . . . . . 6 ((2↑12) mod 𝑁) = (95 mod 𝑁)
108 eqid 2778 . . . . . . 7 12 = 12
10951mulid1i 10381 . . . . . . . . 9 (2 · 1) = 2
110109oveq1i 6932 . . . . . . . 8 ((2 · 1) + 0) = (2 + 0)
11151addid1i 10563 . . . . . . . 8 (2 + 0) = 2
112110, 111eqtri 2802 . . . . . . 7 ((2 · 1) + 0) = 2
113 2t2e4 11546 . . . . . . . 8 (2 · 2) = 4
1142dec0h 11868 . . . . . . . 8 4 = 04
115113, 114eqtri 2802 . . . . . . 7 (2 · 2) = 04
11620, 16, 20, 108, 2, 3, 112, 115decmul2c 11913 . . . . . 6 (2 · 12) = 24
117 eqid 2778 . . . . . . . 8 1023 = 1023
11840nn0cni 11655 . . . . . . . . . 10 102 ∈ ℂ
119118addid1i 10563 . . . . . . . . 9 (102 + 0) = 102
120 dec10p 11889 . . . . . . . . . 10 (10 + 0) = 10
121 4t2e8 11550 . . . . . . . . . . . . 13 (4 · 2) = 8
12264, 51, 121mulcomli 10386 . . . . . . . . . . . 12 (2 · 4) = 8
12369addid1i 10563 . . . . . . . . . . . 12 (1 + 0) = 1
124122, 123oveq12i 6934 . . . . . . . . . . 11 ((2 · 4) + (1 + 0)) = (8 + 1)
125 8p1e9 11532 . . . . . . . . . . 11 (8 + 1) = 9
126124, 125eqtri 2802 . . . . . . . . . 10 ((2 · 4) + (1 + 0)) = 9
12751mul01i 10566 . . . . . . . . . . . 12 (2 · 0) = 0
128127oveq1i 6932 . . . . . . . . . . 11 ((2 · 0) + 0) = (0 + 0)
129128, 61, 623eqtri 2806 . . . . . . . . . 10 ((2 · 0) + 0) = 00
1302, 3, 16, 3, 60, 120, 20, 3, 3, 126, 129decma2c 11899 . . . . . . . . 9 ((2 · 40) + (10 + 0)) = 90
131127oveq1i 6932 . . . . . . . . . 10 ((2 · 0) + 2) = (0 + 2)
13251addid2i 10564 . . . . . . . . . 10 (0 + 2) = 2
13320dec0h 11868 . . . . . . . . . 10 2 = 02
134131, 132, 1333eqtri 2806 . . . . . . . . 9 ((2 · 0) + 2) = 02
1354, 3, 10, 20, 55, 119, 20, 20, 3, 130, 134decma2c 11899 . . . . . . . 8 ((2 · 400) + (102 + 0)) = 902
136109oveq1i 6932 . . . . . . . . 9 ((2 · 1) + 3) = (2 + 3)
137 3cn 11456 . . . . . . . . . 10 3 ∈ ℂ
138 3p2e5 11533 . . . . . . . . . 10 (3 + 2) = 5
139137, 51, 138addcomli 10568 . . . . . . . . 9 (2 + 3) = 5
14022dec0h 11868 . . . . . . . . 9 5 = 05
141136, 139, 1403eqtri 2806 . . . . . . . 8 ((2 · 1) + 3) = 05
1425, 16, 40, 41, 1, 117, 20, 22, 3, 135, 141decma2c 11899 . . . . . . 7 ((2 · 𝑁) + 1023) = 9025
1432, 28deccl 11860 . . . . . . . 8 47 ∈ ℕ0
144 eqid 2778 . . . . . . . . 9 47 = 47
14598oveq2i 6933 . . . . . . . . . 10 ((9 · 9) + (4 + 5)) = ((9 · 9) + 9)
146 9t9e81 11976 . . . . . . . . . . 11 (9 · 9) = 81
147 9p1e10 11847 . . . . . . . . . . . 12 (9 + 1) = 10
14856, 69, 147addcomli 10568 . . . . . . . . . . 11 (1 + 9) = 10
14924, 16, 12, 146, 125, 148decaddci2 11908 . . . . . . . . . 10 ((9 · 9) + 9) = 90
150145, 149eqtri 2802 . . . . . . . . 9 ((9 · 9) + (4 + 5)) = 90
151 9t5e45 11972 . . . . . . . . . . 11 (9 · 5) = 45
15256, 80, 151mulcomli 10386 . . . . . . . . . 10 (5 · 9) = 45
153 7cn 11473 . . . . . . . . . . 11 7 ∈ ℂ
154 7p5e12 11924 . . . . . . . . . . 11 (7 + 5) = 12
155153, 80, 154addcomli 10568 . . . . . . . . . 10 (5 + 7) = 12
1562, 22, 28, 152, 101, 20, 155decaddci 11907 . . . . . . . . 9 ((5 · 9) + 7) = 52
15712, 22, 2, 28, 54, 144, 12, 20, 22, 150, 156decmac 11898 . . . . . . . 8 ((95 · 9) + 47) = 902
158 5p2e7 11538 . . . . . . . . . 10 (5 + 2) = 7
1592, 22, 20, 151, 158decaddi 11906 . . . . . . . . 9 ((9 · 5) + 2) = 47
160 5t5e25 11950 . . . . . . . . 9 (5 · 5) = 25
16122, 12, 22, 54, 22, 20, 159, 160decmul1c 11912 . . . . . . . 8 (95 · 5) = 475
16245, 12, 22, 54, 22, 143, 157, 161decmul2c 11913 . . . . . . 7 (95 · 95) = 9025
163142, 162eqtr4i 2805 . . . . . 6 ((2 · 𝑁) + 1023) = (95 · 95)
1648, 9, 43, 44, 45, 42, 107, 116, 163mod2xi 16177 . . . . 5 ((2↑24) mod 𝑁) = (1023 mod 𝑁)
165 eqid 2778 . . . . . 6 24 = 24
16620, 2, 101, 165decsuc 11877 . . . . 5 (24 + 1) = 25
16737nn0cni 11655 . . . . . . 7 2046 ∈ ℂ
168167addid2i 10564 . . . . . 6 (0 + 2046) = 2046
1698nncni 11385 . . . . . . . 8 𝑁 ∈ ℂ
170169mul02i 10565 . . . . . . 7 (0 · 𝑁) = 0
171170oveq1i 6932 . . . . . 6 ((0 · 𝑁) + 2046) = (0 + 2046)
172 eqid 2778 . . . . . . . 8 102 = 102
17320dec0u 11867 . . . . . . . 8 (10 · 2) = 20
17420, 10, 20, 172, 173, 113decmul1 11910 . . . . . . 7 (102 · 2) = 204
175 3t2e6 11548 . . . . . . 7 (3 · 2) = 6
17620, 40, 41, 117, 174, 175decmul1 11910 . . . . . 6 (1023 · 2) = 2046
177168, 171, 1763eqtr4i 2812 . . . . 5 ((0 · 𝑁) + 2046) = (1023 · 2)
1788, 9, 38, 39, 42, 37, 164, 166, 177modxp1i 16178 . . . 4 ((2↑25) mod 𝑁) = (2046 mod 𝑁)
179113oveq1i 6932 . . . . . 6 ((2 · 2) + 1) = (4 + 1)
180179, 101eqtri 2802 . . . . 5 ((2 · 2) + 1) = 5
181 5t2e10 11947 . . . . . 6 (5 · 2) = 10
18280, 51, 181mulcomli 10386 . . . . 5 (2 · 5) = 10
18320, 20, 22, 87, 3, 16, 180, 182decmul2c 11913 . . . 4 (2 · 25) = 50
184 eqid 2778 . . . . . 6 1070 = 1070
18520, 16deccl 11860 . . . . . . 7 21 ∈ ℕ0
186 eqid 2778 . . . . . . . 8 107 = 107
187 eqid 2778 . . . . . . . 8 104 = 104
188 0p1e1 11504 . . . . . . . . 9 (0 + 1) = 1
189 10p10e20 11942 . . . . . . . . 9 (10 + 10) = 20
19020, 3, 188, 189decsuc 11877 . . . . . . . 8 ((10 + 10) + 1) = 21
191 7p4e11 11923 . . . . . . . 8 (7 + 4) = 11
19210, 28, 10, 2, 186, 187, 190, 16, 191decaddc 11901 . . . . . . 7 (107 + 104) = 211
193185nn0cni 11655 . . . . . . . . 9 21 ∈ ℂ
194193addid1i 10563 . . . . . . . 8 (21 + 0) = 21
195111, 20eqeltri 2855 . . . . . . . . 9 (2 + 0) ∈ ℕ0
196 eqid 2778 . . . . . . . . 9 1046 = 1046
197 dfdec10 11848 . . . . . . . . . . 11 41 = ((10 · 4) + 1)
198197eqcomi 2787 . . . . . . . . . 10 ((10 · 4) + 1) = 41
199 6p2e8 11541 . . . . . . . . . . 11 (6 + 2) = 8
20016, 15, 20, 103, 199decaddi 11906 . . . . . . . . . 10 ((4 · 4) + 2) = 18
20110, 2, 20, 187, 2, 24, 16, 198, 200decrmac 11904 . . . . . . . . 9 ((104 · 4) + 2) = 418
20295, 111oveq12i 6934 . . . . . . . . . 10 ((6 · 4) + (2 + 0)) = (24 + 2)
203 4p2e6 11535 . . . . . . . . . . 11 (4 + 2) = 6
20420, 2, 20, 165, 203decaddi 11906 . . . . . . . . . 10 (24 + 2) = 26
205202, 204eqtri 2802 . . . . . . . . 9 ((6 · 4) + (2 + 0)) = 26
20632, 15, 195, 196, 2, 15, 20, 201, 205decrmac 11904 . . . . . . . 8 ((1046 · 4) + (2 + 0)) = 4186
20733nn0cni 11655 . . . . . . . . . . 11 1046 ∈ ℂ
208207mul01i 10566 . . . . . . . . . 10 (1046 · 0) = 0
209208oveq1i 6932 . . . . . . . . 9 ((1046 · 0) + 1) = (0 + 1)
21016dec0h 11868 . . . . . . . . 9 1 = 01
211209, 188, 2103eqtri 2806 . . . . . . . 8 ((1046 · 0) + 1) = 01
2122, 3, 20, 16, 60, 194, 33, 16, 3, 206, 211decma2c 11899 . . . . . . 7 ((1046 · 40) + (21 + 0)) = 41861
2134, 3, 185, 16, 55, 192, 33, 16, 3, 212, 211decma2c 11899 . . . . . 6 ((1046 · 400) + (107 + 104)) = 418611
214207mulid1i 10381 . . . . . . . 8 (1046 · 1) = 1046
215214oveq1i 6932 . . . . . . 7 ((1046 · 1) + 0) = (1046 + 0)
216207addid1i 10563 . . . . . . 7 (1046 + 0) = 1046
217215, 216eqtri 2802 . . . . . 6 ((1046 · 1) + 0) = 1046
2185, 16, 29, 3, 1, 184, 33, 15, 32, 213, 217decma2c 11899 . . . . 5 ((1046 · 𝑁) + 1070) = 4186116
219 eqid 2778 . . . . . 6 2046 = 2046
22043, 20deccl 11860 . . . . . . 7 122 ∈ ℕ0
221220, 28deccl 11860 . . . . . 6 1227 ∈ ℕ0
222 eqid 2778 . . . . . . 7 204 = 204
223 eqid 2778 . . . . . . 7 1227 = 1227
22424, 16deccl 11860 . . . . . . . 8 81 ∈ ℕ0
225224, 12deccl 11860 . . . . . . 7 819 ∈ ℕ0
226 eqid 2778 . . . . . . . 8 20 = 20
227 eqid 2778 . . . . . . . . 9 122 = 122
228 eqid 2778 . . . . . . . . 9 819 = 819
229 eqid 2778 . . . . . . . . . . 11 81 = 81
230 8cn 11477 . . . . . . . . . . . 12 8 ∈ ℂ
231230, 69, 125addcomli 10568 . . . . . . . . . . 11 (1 + 8) = 9
232 2p1e3 11524 . . . . . . . . . . 11 (2 + 1) = 3
23316, 20, 24, 16, 108, 229, 231, 232decadd 11900 . . . . . . . . . 10 (12 + 81) = 93
23412, 41, 91, 233decsuc 11877 . . . . . . . . 9 ((12 + 81) + 1) = 94
235 9p2e11 11934 . . . . . . . . . 10 (9 + 2) = 11
23656, 51, 235addcomli 10568 . . . . . . . . 9 (2 + 9) = 11
23743, 20, 224, 12, 227, 228, 234, 16, 236decaddc 11901 . . . . . . . 8 (122 + 819) = 941
23813nn0cni 11655 . . . . . . . . . 10 94 ∈ ℂ
239238addid1i 10563 . . . . . . . . 9 (94 + 0) = 94
240123, 16eqeltri 2855 . . . . . . . . . . 11 (1 + 0) ∈ ℕ0
24151mul02i 10565 . . . . . . . . . . . . 13 (0 · 2) = 0
242241, 123oveq12i 6934 . . . . . . . . . . . 12 ((0 · 2) + (1 + 0)) = (0 + 1)
243242, 188eqtri 2802 . . . . . . . . . . 11 ((0 · 2) + (1 + 0)) = 1
24420, 3, 240, 226, 20, 113, 243decrmanc 11903 . . . . . . . . . 10 ((20 · 2) + (1 + 0)) = 41
245121oveq1i 6932 . . . . . . . . . . 11 ((4 · 2) + 0) = (8 + 0)
246230addid1i 10563 . . . . . . . . . . 11 (8 + 0) = 8
24724dec0h 11868 . . . . . . . . . . 11 8 = 08
248245, 246, 2473eqtri 2806 . . . . . . . . . 10 ((4 · 2) + 0) = 08
24935, 2, 16, 3, 222, 147, 20, 24, 3, 244, 248decmac 11898 . . . . . . . . 9 ((204 · 2) + (9 + 1)) = 418
25064, 51, 203addcomli 10568 . . . . . . . . . 10 (2 + 4) = 6
25116, 20, 2, 52, 250decaddi 11906 . . . . . . . . 9 ((6 · 2) + 4) = 16
25236, 15, 12, 2, 219, 239, 20, 15, 16, 249, 251decmac 11898 . . . . . . . 8 ((2046 · 2) + (94 + 0)) = 4186
253167mul01i 10566 . . . . . . . . . 10 (2046 · 0) = 0
254253oveq1i 6932 . . . . . . . . 9 ((2046 · 0) + 1) = (0 + 1)
255254, 188, 2103eqtri 2806 . . . . . . . 8 ((2046 · 0) + 1) = 01
25620, 3, 13, 16, 226, 237, 37, 16, 3, 252, 255decma2c 11899 . . . . . . 7 ((2046 · 20) + (122 + 819)) = 41861
25741dec0h 11868 . . . . . . . . 9 3 = 03
258188, 16eqeltri 2855 . . . . . . . . . 10 (0 + 1) ∈ ℕ0
25964mul02i 10565 . . . . . . . . . . . 12 (0 · 4) = 0
260259, 188oveq12i 6934 . . . . . . . . . . 11 ((0 · 4) + (0 + 1)) = (0 + 1)
261260, 188eqtri 2802 . . . . . . . . . 10 ((0 · 4) + (0 + 1)) = 1
26220, 3, 258, 226, 2, 122, 261decrmanc 11903 . . . . . . . . 9 ((20 · 4) + (0 + 1)) = 81
263 6p3e9 11542 . . . . . . . . . 10 (6 + 3) = 9
26416, 15, 41, 103, 263decaddi 11906 . . . . . . . . 9 ((4 · 4) + 3) = 19
26535, 2, 3, 41, 222, 257, 2, 12, 16, 262, 264decmac 11898 . . . . . . . 8 ((204 · 4) + 3) = 819
266153, 64, 191addcomli 10568 . . . . . . . . 9 (4 + 7) = 11
26720, 2, 28, 95, 232, 16, 266decaddci 11907 . . . . . . . 8 ((6 · 4) + 7) = 31
26836, 15, 28, 219, 2, 16, 41, 265, 267decrmac 11904 . . . . . . 7 ((2046 · 4) + 7) = 8191
26935, 2, 220, 28, 222, 223, 37, 16, 225, 256, 268decma2c 11899 . . . . . 6 ((2046 · 204) + 1227) = 418611
27050mul02i 10565 . . . . . . . . . . 11 (0 · 6) = 0
271270oveq1i 6932 . . . . . . . . . 10 ((0 · 6) + 2) = (0 + 2)
272271, 132eqtri 2802 . . . . . . . . 9 ((0 · 6) + 2) = 2
27320, 3, 20, 226, 15, 53, 272decrmanc 11903 . . . . . . . 8 ((20 · 6) + 2) = 122
274 4p3e7 11536 . . . . . . . . 9 (4 + 3) = 7
27520, 2, 41, 96, 274decaddi 11906 . . . . . . . 8 ((4 · 6) + 3) = 27
27635, 2, 41, 222, 15, 28, 20, 273, 275decrmac 11904 . . . . . . 7 ((204 · 6) + 3) = 1227
27715, 36, 15, 219, 15, 41, 276, 90decmul1c 11912 . . . . . 6 (2046 · 6) = 12276
27837, 36, 15, 219, 15, 221, 269, 277decmul2c 11913 . . . . 5 (2046 · 2046) = 4186116
279218, 278eqtr4i 2805 . . . 4 ((1046 · 𝑁) + 1070) = (2046 · 2046)
2808, 9, 31, 34, 37, 30, 178, 183, 279mod2xi 16177 . . 3 ((2↑50) mod 𝑁) = (1070 mod 𝑁)
28123nn0cni 11655 . . . 4 50 ∈ ℂ
282 eqid 2778 . . . . 5 50 = 50
28320, 22, 3, 282, 181, 241decmul1 11910 . . . 4 (50 · 2) = 100
284281, 51, 283mulcomli 10386 . . 3 (2 · 50) = 100
285 eqid 2778 . . . . 5 614 = 614
28620, 12deccl 11860 . . . . 5 29 ∈ ℕ0
287 eqid 2778 . . . . . . 7 61 = 61
288 eqid 2778 . . . . . . 7 29 = 29
289199oveq1i 6932 . . . . . . . 8 ((6 + 2) + 1) = (8 + 1)
290289, 125eqtri 2802 . . . . . . 7 ((6 + 2) + 1) = 9
29115, 16, 20, 12, 287, 288, 290, 148decaddc2 11902 . . . . . 6 (61 + 29) = 90
29261, 3eqeltri 2855 . . . . . . . 8 (0 + 0) ∈ ℕ0
293 eqid 2778 . . . . . . . 8 286 = 286
294 eqid 2778 . . . . . . . . 9 28 = 28
295122oveq1i 6932 . . . . . . . . . 10 ((2 · 4) + 3) = (8 + 3)
296 8p3e11 11928 . . . . . . . . . 10 (8 + 3) = 11
297295, 296eqtri 2802 . . . . . . . . 9 ((2 · 4) + 3) = 11
298 8t4e32 11964 . . . . . . . . . 10 (8 · 4) = 32
29941, 20, 20, 298, 88decaddi 11906 . . . . . . . . 9 ((8 · 4) + 2) = 34
30020, 24, 20, 294, 2, 2, 41, 297, 299decrmac 11904 . . . . . . . 8 ((28 · 4) + 2) = 114
30195, 61oveq12i 6934 . . . . . . . . 9 ((6 · 4) + (0 + 0)) = (24 + 0)
30238nn0cni 11655 . . . . . . . . . 10 24 ∈ ℂ
303302addid1i 10563 . . . . . . . . 9 (24 + 0) = 24
304301, 303eqtri 2802 . . . . . . . 8 ((6 · 4) + (0 + 0)) = 24
30525, 15, 292, 293, 2, 2, 20, 300, 304decrmac 11904 . . . . . . 7 ((286 · 4) + (0 + 0)) = 1144
30626nn0cni 11655 . . . . . . . . . 10 286 ∈ ℂ
307306mul01i 10566 . . . . . . . . 9 (286 · 0) = 0
308307oveq1i 6932 . . . . . . . 8 ((286 · 0) + 9) = (0 + 9)
309308, 75, 583eqtri 2806 . . . . . . 7 ((286 · 0) + 9) = 09
3102, 3, 3, 12, 60, 59, 26, 12, 3, 305, 309decma2c 11899 . . . . . 6 ((286 · 40) + (9 + 0)) = 11449
311307oveq1i 6932 . . . . . . 7 ((286 · 0) + 0) = (0 + 0)
312311, 61, 623eqtri 2806 . . . . . 6 ((286 · 0) + 0) = 00
3134, 3, 12, 3, 55, 291, 26, 3, 3, 310, 312decma2c 11899 . . . . 5 ((286 · 400) + (61 + 29)) = 114490
314230mulid1i 10381 . . . . . . . 8 (8 · 1) = 8
31516, 20, 24, 294, 109, 314decmul1 11910 . . . . . . 7 (28 · 1) = 28
31620, 24, 125, 315decsuc 11877 . . . . . 6 ((28 · 1) + 1) = 29
31750mulid1i 10381 . . . . . . . 8 (6 · 1) = 6
318317oveq1i 6932 . . . . . . 7 ((6 · 1) + 4) = (6 + 4)
319318, 92eqtri 2802 . . . . . 6 ((6 · 1) + 4) = 10
32025, 15, 2, 293, 16, 3, 16, 316, 319decrmac 11904 . . . . 5 ((286 · 1) + 4) = 290
3215, 16, 17, 2, 1, 285, 26, 3, 286, 313, 320decma2c 11899 . . . 4 ((286 · 𝑁) + 614) = 1144900
32216, 16deccl 11860 . . . . . . . . 9 11 ∈ ℕ0
323322, 2deccl 11860 . . . . . . . 8 114 ∈ ℕ0
324323, 2deccl 11860 . . . . . . 7 1144 ∈ ℕ0
325324, 12deccl 11860 . . . . . 6 11449 ∈ ℕ0
32628, 2deccl 11860 . . . . . . . 8 74 ∈ ℕ0
327326, 12deccl 11860 . . . . . . 7 749 ∈ ℕ0
328 eqid 2778 . . . . . . . 8 10 = 10
329 eqid 2778 . . . . . . . 8 749 = 749
330326nn0cni 11655 . . . . . . . . . 10 74 ∈ ℂ
331330addid1i 10563 . . . . . . . . 9 (74 + 0) = 74
332153addid1i 10563 . . . . . . . . . . 11 (7 + 0) = 7
333332, 28eqeltri 2855 . . . . . . . . . 10 (7 + 0) ∈ ℕ0
33410nn0cni 11655 . . . . . . . . . . . 12 10 ∈ ℂ
335334mulid1i 10381 . . . . . . . . . . 11 (10 · 1) = 10
33616, 3, 188, 335decsuc 11877 . . . . . . . . . 10 ((10 · 1) + 1) = 11
337153mulid1i 10381 . . . . . . . . . . . 12 (7 · 1) = 7
338337, 332oveq12i 6934 . . . . . . . . . . 11 ((7 · 1) + (7 + 0)) = (7 + 7)
339 7p7e14 11926 . . . . . . . . . . 11 (7 + 7) = 14
340338, 339eqtri 2802 . . . . . . . . . 10 ((7 · 1) + (7 + 0)) = 14
34110, 28, 333, 186, 16, 2, 16, 336, 340decrmac 11904 . . . . . . . . 9 ((107 · 1) + (7 + 0)) = 114
34269mul02i 10565 . . . . . . . . . . 11 (0 · 1) = 0
343342oveq1i 6932 . . . . . . . . . 10 ((0 · 1) + 4) = (0 + 4)
34464addid2i 10564 . . . . . . . . . 10 (0 + 4) = 4
345343, 344, 1143eqtri 2806 . . . . . . . . 9 ((0 · 1) + 4) = 04
34629, 3, 28, 2, 184, 331, 16, 2, 3, 341, 345decmac 11898 . . . . . . . 8 ((1070 · 1) + (74 + 0)) = 1144
34730nn0cni 11655 . . . . . . . . . . 11 1070 ∈ ℂ
348347mul01i 10566 . . . . . . . . . 10 (1070 · 0) = 0
349348oveq1i 6932 . . . . . . . . 9 ((1070 · 0) + 9) = (0 + 9)
350349, 75, 583eqtri 2806 . . . . . . . 8 ((1070 · 0) + 9) = 09
35116, 3, 326, 12, 328, 329, 30, 12, 3, 346, 350decma2c 11899 . . . . . . 7 ((1070 · 10) + 749) = 11449
352 dfdec10 11848 . . . . . . . . . 10 74 = ((10 · 7) + 4)
353352eqcomi 2787 . . . . . . . . 9 ((10 · 7) + 4) = 74
354 7t7e49 11961 . . . . . . . . 9 (7 · 7) = 49
35528, 10, 28, 186, 12, 2, 353, 354decmul1c 11912 . . . . . . . 8 (107 · 7) = 749
356153mul02i 10565 . . . . . . . 8 (0 · 7) = 0
35728, 29, 3, 184, 355, 356decmul1 11910 . . . . . . 7 (1070 · 7) = 7490
35830, 10, 28, 186, 3, 327, 351, 357decmul2c 11913 . . . . . 6 (1070 · 107) = 114490
359325, 3, 3, 358, 61decaddi 11906 . . . . 5 ((1070 · 107) + 0) = 114490
360348, 62eqtri 2802 . . . . 5 (1070 · 0) = 00
36130, 29, 3, 184, 3, 3, 359, 360decmul2c 11913 . . . 4 (1070 · 1070) = 1144900
362321, 361eqtr4i 2805 . . 3 ((286 · 𝑁) + 614) = (1070 · 1070)
3638, 9, 23, 27, 30, 18, 280, 284, 362mod2xi 16177 . 2 ((2↑100) mod 𝑁) = (614 mod 𝑁)
36411nn0cni 11655 . . 3 100 ∈ ℂ
365 eqid 2778 . . . 4 100 = 100
36620, 10, 3, 365, 173, 241decmul1 11910 . . 3 (100 · 2) = 200
367364, 51, 366mulcomli 10386 . 2 (2 · 100) = 200
368 eqid 2778 . . . 4 902 = 902
369 eqid 2778 . . . . . 6 90 = 90
37012, 3, 12, 369, 75decaddi 11906 . . . . 5 (90 + 9) = 99
371 eqid 2778 . . . . . . 7 94 = 94
372 6p1e7 11530 . . . . . . . 8 (6 + 1) = 7
373 9t4e36 11971 . . . . . . . 8 (9 · 4) = 36
37441, 15, 372, 373decsuc 11877 . . . . . . 7 ((9 · 4) + 1) = 37
375103, 61oveq12i 6934 . . . . . . . 8 ((4 · 4) + (0 + 0)) = (16 + 0)
37616, 15deccl 11860 . . . . . . . . . 10 16 ∈ ℕ0
377376nn0cni 11655 . . . . . . . . 9 16 ∈ ℂ
378377addid1i 10563 . . . . . . . 8 (16 + 0) = 16
379375, 378eqtri 2802 . . . . . . 7 ((4 · 4) + (0 + 0)) = 16
38012, 2, 292, 371, 2, 15, 16, 374, 379decrmac 11904 . . . . . 6 ((94 · 4) + (0 + 0)) = 376
381238mul01i 10566 . . . . . . . 8 (94 · 0) = 0
382381oveq1i 6932 . . . . . . 7 ((94 · 0) + 9) = (0 + 9)
383382, 75, 583eqtri 2806 . . . . . 6 ((94 · 0) + 9) = 09
3842, 3, 3, 12, 60, 59, 13, 12, 3, 380, 383decma2c 11899 . . . . 5 ((94 · 40) + (9 + 0)) = 3769
3854, 3, 12, 12, 55, 370, 13, 12, 3, 384, 383decma2c 11899 . . . 4 ((94 · 400) + (90 + 9)) = 37699
38656mulid1i 10381 . . . . 5 (9 · 1) = 9
38764mulid1i 10381 . . . . . . 7 (4 · 1) = 4
388387oveq1i 6932 . . . . . 6 ((4 · 1) + 2) = (4 + 2)
389388, 203eqtri 2802 . . . . 5 ((4 · 1) + 2) = 6
39012, 2, 20, 371, 16, 386, 389decrmanc 11903 . . . 4 ((94 · 1) + 2) = 96
3915, 16, 19, 20, 1, 368, 13, 15, 12, 385, 390decma2c 11899 . . 3 ((94 · 𝑁) + 902) = 376996
39238, 22deccl 11860 . . . 4 245 ∈ ℕ0
393 eqid 2778 . . . . 5 245 = 245
39450, 51, 199addcomli 10568 . . . . . . 7 (2 + 6) = 8
39520, 2, 15, 16, 165, 287, 394, 101decadd 11900 . . . . . 6 (24 + 61) = 85
396 8p2e10 11927 . . . . . . 7 (8 + 2) = 10
39741, 15, 372, 90decsuc 11877 . . . . . . 7 ((6 · 6) + 1) = 37
39850mulid2i 10382 . . . . . . . . 9 (1 · 6) = 6
399398oveq1i 6932 . . . . . . . 8 ((1 · 6) + 0) = (6 + 0)
40050addid1i 10563 . . . . . . . 8 (6 + 0) = 6
401399, 400eqtri 2802 . . . . . . 7 ((1 · 6) + 0) = 6
40215, 16, 16, 3, 287, 396, 15, 397, 401decma 11897 . . . . . 6 ((61 · 6) + (8 + 2)) = 376
40317, 2, 24, 22, 285, 395, 15, 12, 20, 402, 99decmac 11898 . . . . 5 ((614 · 6) + (24 + 61)) = 3769
40416, 15, 16, 287, 317, 78decmul1 11910 . . . . . 6 (61 · 1) = 61
405387oveq1i 6932 . . . . . . 7 ((4 · 1) + 5) = (4 + 5)
406405, 98eqtri 2802 . . . . . 6 ((4 · 1) + 5) = 9
40717, 2, 22, 285, 16, 404, 406decrmanc 11903 . . . . 5 ((614 · 1) + 5) = 619
40815, 16, 38, 22, 287, 393, 18, 12, 17, 403, 407decma2c 11899 . . . 4 ((614 · 61) + 245) = 37699
40965oveq1i 6932 . . . . . . 7 ((1 · 4) + 1) = (4 + 1)
410409, 101eqtri 2802 . . . . . 6 ((1 · 4) + 1) = 5
41115, 16, 16, 287, 2, 95, 410decrmanc 11903 . . . . 5 ((61 · 4) + 1) = 245
4122, 17, 2, 285, 15, 16, 411, 103decmul1c 11912 . . . 4 (614 · 4) = 2456
41318, 17, 2, 285, 15, 392, 408, 412decmul2c 11913 . . 3 (614 · 614) = 376996
414391, 413eqtr4i 2805 . 2 ((94 · 𝑁) + 902) = (614 · 614)
4158, 9, 11, 14, 18, 21, 363, 367, 414mod2xi 16177 1 ((2↑200) mod 𝑁) = (902 mod 𝑁)
Colors of variables: wff setvar class
Syntax hints:   = wceq 1601  (class class class)co 6922  0cc0 10272  1c1 10273   + caddc 10275   · cmul 10277  cn 11374  2c2 11430  3c3 11431  4c4 11432  5c5 11433  6c6 11434  7c7 11435  8c8 11436  9c9 11437  0cn0 11642  cdc 11845   mod cmo 12987  cexp 13178
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-8 2109  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-sep 5017  ax-nul 5025  ax-pow 5077  ax-pr 5138  ax-un 7226  ax-cnex 10328  ax-resscn 10329  ax-1cn 10330  ax-icn 10331  ax-addcl 10332  ax-addrcl 10333  ax-mulcl 10334  ax-mulrcl 10335  ax-mulcom 10336  ax-addass 10337  ax-mulass 10338  ax-distr 10339  ax-i2m1 10340  ax-1ne0 10341  ax-1rid 10342  ax-rnegex 10343  ax-rrecex 10344  ax-cnre 10345  ax-pre-lttri 10346  ax-pre-lttrn 10347  ax-pre-ltadd 10348  ax-pre-mulgt0 10349  ax-pre-sup 10350
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3or 1072  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ne 2970  df-nel 3076  df-ral 3095  df-rex 3096  df-reu 3097  df-rmo 3098  df-rab 3099  df-v 3400  df-sbc 3653  df-csb 3752  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-pss 3808  df-nul 4142  df-if 4308  df-pw 4381  df-sn 4399  df-pr 4401  df-tp 4403  df-op 4405  df-uni 4672  df-iun 4755  df-br 4887  df-opab 4949  df-mpt 4966  df-tr 4988  df-id 5261  df-eprel 5266  df-po 5274  df-so 5275  df-fr 5314  df-we 5316  df-xp 5361  df-rel 5362  df-cnv 5363  df-co 5364  df-dm 5365  df-rn 5366  df-res 5367  df-ima 5368  df-pred 5933  df-ord 5979  df-on 5980  df-lim 5981  df-suc 5982  df-iota 6099  df-fun 6137  df-fn 6138  df-f 6139  df-f1 6140  df-fo 6141  df-f1o 6142  df-fv 6143  df-riota 6883  df-ov 6925  df-oprab 6926  df-mpt2 6927  df-om 7344  df-2nd 7446  df-wrecs 7689  df-recs 7751  df-rdg 7789  df-er 8026  df-en 8242  df-dom 8243  df-sdom 8244  df-sup 8636  df-inf 8637  df-pnf 10413  df-mnf 10414  df-xr 10415  df-ltxr 10416  df-le 10417  df-sub 10608  df-neg 10609  df-div 11033  df-nn 11375  df-2 11438  df-3 11439  df-4 11440  df-5 11441  df-6 11442  df-7 11443  df-8 11444  df-9 11445  df-n0 11643  df-z 11729  df-dec 11846  df-uz 11993  df-rp 12138  df-fl 12912  df-mod 12988  df-seq 13120  df-exp 13179
This theorem is referenced by:  4001lem2  16247  4001lem3  16248
  Copyright terms: Public domain W3C validator