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Theorem 4001lem4 15786
Description: Lemma for 4001prm 15787. Calculate the GCD of 2↑800 − 1≡2310 with 𝑁 = 4001. (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
4001lem4 (((2↑800) − 1) gcd 𝑁) = 1

Proof of Theorem 4001lem4
StepHypRef Expression
1 2nn 11137 . . . 4 2 ∈ ℕ
2 8nn0 11267 . . . . . 6 8 ∈ ℕ0
3 0nn0 11259 . . . . . 6 0 ∈ ℕ0
42, 3deccl 11464 . . . . 5 80 ∈ ℕ0
54, 3deccl 11464 . . . 4 800 ∈ ℕ0
6 nnexpcl 12821 . . . 4 ((2 ∈ ℕ ∧ 800 ∈ ℕ0) → (2↑800) ∈ ℕ)
71, 5, 6mp2an 707 . . 3 (2↑800) ∈ ℕ
8 nnm1nn0 11286 . . 3 ((2↑800) ∈ ℕ → ((2↑800) − 1) ∈ ℕ0)
97, 8ax-mp 5 . 2 ((2↑800) − 1) ∈ ℕ0
10 2nn0 11261 . . . . 5 2 ∈ ℕ0
11 3nn0 11262 . . . . 5 3 ∈ ℕ0
1210, 11deccl 11464 . . . 4 23 ∈ ℕ0
13 1nn0 11260 . . . 4 1 ∈ ℕ0
1412, 13deccl 11464 . . 3 231 ∈ ℕ0
1514, 3deccl 11464 . 2 2310 ∈ ℕ0
16 4001prm.1 . . 3 𝑁 = 4001
17 4nn0 11263 . . . . . 6 4 ∈ ℕ0
1817, 3deccl 11464 . . . . 5 40 ∈ ℕ0
1918, 3deccl 11464 . . . 4 400 ∈ ℕ0
20 1nn 10983 . . . 4 1 ∈ ℕ
2119, 20decnncl 11470 . . 3 4001 ∈ ℕ
2216, 21eqeltri 2694 . 2 𝑁 ∈ ℕ
23164001lem2 15784 . . 3 ((2↑800) mod 𝑁) = (2311 mod 𝑁)
24 0p1e1 11084 . . . 4 (0 + 1) = 1
25 eqid 2621 . . . 4 2310 = 2310
2614, 3, 24, 25decsuc 11487 . . 3 (2310 + 1) = 2311
2722, 7, 13, 15, 23, 26modsubi 15711 . 2 (((2↑800) − 1) mod 𝑁) = (2310 mod 𝑁)
28 6nn0 11265 . . . . . 6 6 ∈ ℕ0
2913, 28deccl 11464 . . . . 5 16 ∈ ℕ0
30 9nn0 11268 . . . . 5 9 ∈ ℕ0
3129, 30deccl 11464 . . . 4 169 ∈ ℕ0
3231, 13deccl 11464 . . 3 1691 ∈ ℕ0
3328, 13deccl 11464 . . . . 5 61 ∈ ℕ0
3433, 30deccl 11464 . . . 4 619 ∈ ℕ0
35 5nn0 11264 . . . . . . 7 5 ∈ ℕ0
3617, 35deccl 11464 . . . . . 6 45 ∈ ℕ0
3736, 11deccl 11464 . . . . 5 453 ∈ ℕ0
3829, 28deccl 11464 . . . . . 6 166 ∈ ℕ0
3913, 10deccl 11464 . . . . . . . 8 12 ∈ ℕ0
4039, 13deccl 11464 . . . . . . 7 121 ∈ ℕ0
4111, 13deccl 11464 . . . . . . . . 9 31 ∈ ℕ0
4213, 17deccl 11464 . . . . . . . . . 10 14 ∈ ℕ0
4342nn0zi 11354 . . . . . . . . . . . . 13 14 ∈ ℤ
4411nn0zi 11354 . . . . . . . . . . . . 13 3 ∈ ℤ
45 gcdcom 15170 . . . . . . . . . . . . 13 ((14 ∈ ℤ ∧ 3 ∈ ℤ) → (14 gcd 3) = (3 gcd 14))
4643, 44, 45mp2an 707 . . . . . . . . . . . 12 (14 gcd 3) = (3 gcd 14)
47 3nn 11138 . . . . . . . . . . . . . 14 3 ∈ ℕ
48 4cn 11050 . . . . . . . . . . . . . . . 16 4 ∈ ℂ
49 3cn 11047 . . . . . . . . . . . . . . . 16 3 ∈ ℂ
50 4t3e12 11584 . . . . . . . . . . . . . . . 16 (4 · 3) = 12
5148, 49, 50mulcomli 9999 . . . . . . . . . . . . . . 15 (3 · 4) = 12
52 2p2e4 11096 . . . . . . . . . . . . . . 15 (2 + 2) = 4
5313, 10, 10, 51, 52decaddi 11531 . . . . . . . . . . . . . 14 ((3 · 4) + 2) = 14
54 2lt3 11147 . . . . . . . . . . . . . 14 2 < 3
5547, 17, 1, 53, 54ndvdsi 15071 . . . . . . . . . . . . 13 ¬ 3 ∥ 14
56 3prm 15341 . . . . . . . . . . . . . 14 3 ∈ ℙ
57 coprm 15358 . . . . . . . . . . . . . 14 ((3 ∈ ℙ ∧ 14 ∈ ℤ) → (¬ 3 ∥ 14 ↔ (3 gcd 14) = 1))
5856, 43, 57mp2an 707 . . . . . . . . . . . . 13 (¬ 3 ∥ 14 ↔ (3 gcd 14) = 1)
5955, 58mpbi 220 . . . . . . . . . . . 12 (3 gcd 14) = 1
6046, 59eqtri 2643 . . . . . . . . . . 11 (14 gcd 3) = 1
61 eqid 2621 . . . . . . . . . . . 12 14 = 14
6211dec0h 11474 . . . . . . . . . . . 12 3 = 03
63 2t1e2 11128 . . . . . . . . . . . . . 14 (2 · 1) = 2
6463, 24oveq12i 6622 . . . . . . . . . . . . 13 ((2 · 1) + (0 + 1)) = (2 + 1)
65 2p1e3 11103 . . . . . . . . . . . . 13 (2 + 1) = 3
6664, 65eqtri 2643 . . . . . . . . . . . 12 ((2 · 1) + (0 + 1)) = 3
67 2cn 11043 . . . . . . . . . . . . . . 15 2 ∈ ℂ
68 4t2e8 11133 . . . . . . . . . . . . . . 15 (4 · 2) = 8
6948, 67, 68mulcomli 9999 . . . . . . . . . . . . . 14 (2 · 4) = 8
7069oveq1i 6620 . . . . . . . . . . . . 13 ((2 · 4) + 3) = (8 + 3)
71 8p3e11 11564 . . . . . . . . . . . . 13 (8 + 3) = 11
7270, 71eqtri 2643 . . . . . . . . . . . 12 ((2 · 4) + 3) = 11
7313, 17, 3, 11, 61, 62, 10, 13, 13, 66, 72decma2c 11520 . . . . . . . . . . 11 ((2 · 14) + 3) = 31
7410, 11, 42, 60, 73gcdi 15712 . . . . . . . . . 10 (31 gcd 14) = 1
75 eqid 2621 . . . . . . . . . . 11 31 = 31
7649mulid2i 9995 . . . . . . . . . . . . 13 (1 · 3) = 3
77 ax-1cn 9946 . . . . . . . . . . . . . 14 1 ∈ ℂ
7877addid1i 10175 . . . . . . . . . . . . 13 (1 + 0) = 1
7976, 78oveq12i 6622 . . . . . . . . . . . 12 ((1 · 3) + (1 + 0)) = (3 + 1)
80 3p1e4 11105 . . . . . . . . . . . 12 (3 + 1) = 4
8179, 80eqtri 2643 . . . . . . . . . . 11 ((1 · 3) + (1 + 0)) = 4
82 1t1e1 11127 . . . . . . . . . . . . 13 (1 · 1) = 1
8382oveq1i 6620 . . . . . . . . . . . 12 ((1 · 1) + 4) = (1 + 4)
84 4p1e5 11106 . . . . . . . . . . . . 13 (4 + 1) = 5
8548, 77, 84addcomli 10180 . . . . . . . . . . . 12 (1 + 4) = 5
8635dec0h 11474 . . . . . . . . . . . 12 5 = 05
8783, 85, 863eqtri 2647 . . . . . . . . . . 11 ((1 · 1) + 4) = 05
8811, 13, 13, 17, 75, 61, 13, 35, 3, 81, 87decma2c 11520 . . . . . . . . . 10 ((1 · 31) + 14) = 45
8913, 42, 41, 74, 88gcdi 15712 . . . . . . . . 9 (45 gcd 31) = 1
90 eqid 2621 . . . . . . . . . 10 45 = 45
9169, 80oveq12i 6622 . . . . . . . . . . 11 ((2 · 4) + (3 + 1)) = (8 + 4)
92 8p4e12 11566 . . . . . . . . . . 11 (8 + 4) = 12
9391, 92eqtri 2643 . . . . . . . . . 10 ((2 · 4) + (3 + 1)) = 12
94 5cn 11052 . . . . . . . . . . . 12 5 ∈ ℂ
95 5t2e10 11586 . . . . . . . . . . . 12 (5 · 2) = 10
9694, 67, 95mulcomli 9999 . . . . . . . . . . 11 (2 · 5) = 10
9713, 3, 24, 96decsuc 11487 . . . . . . . . . 10 ((2 · 5) + 1) = 11
9817, 35, 11, 13, 90, 75, 10, 13, 13, 93, 97decma2c 11520 . . . . . . . . 9 ((2 · 45) + 31) = 121
9910, 41, 36, 89, 98gcdi 15712 . . . . . . . 8 (121 gcd 45) = 1
100 eqid 2621 . . . . . . . . 9 121 = 121
101 eqid 2621 . . . . . . . . . 10 12 = 12
10248addid1i 10175 . . . . . . . . . . 11 (4 + 0) = 4
10317dec0h 11474 . . . . . . . . . . 11 4 = 04
104102, 103eqtri 2643 . . . . . . . . . 10 (4 + 0) = 04
105 00id 10163 . . . . . . . . . . . 12 (0 + 0) = 0
10682, 105oveq12i 6622 . . . . . . . . . . 11 ((1 · 1) + (0 + 0)) = (1 + 0)
107106, 78eqtri 2643 . . . . . . . . . 10 ((1 · 1) + (0 + 0)) = 1
10867mulid2i 9995 . . . . . . . . . . . 12 (1 · 2) = 2
109108oveq1i 6620 . . . . . . . . . . 11 ((1 · 2) + 4) = (2 + 4)
110 4p2e6 11114 . . . . . . . . . . . 12 (4 + 2) = 6
11148, 67, 110addcomli 10180 . . . . . . . . . . 11 (2 + 4) = 6
11228dec0h 11474 . . . . . . . . . . 11 6 = 06
113109, 111, 1123eqtri 2647 . . . . . . . . . 10 ((1 · 2) + 4) = 06
11413, 10, 3, 17, 101, 104, 13, 28, 3, 107, 113decma2c 11520 . . . . . . . . 9 ((1 · 12) + (4 + 0)) = 16
11582oveq1i 6620 . . . . . . . . . 10 ((1 · 1) + 5) = (1 + 5)
116 5p1e6 11107 . . . . . . . . . . 11 (5 + 1) = 6
11794, 77, 116addcomli 10180 . . . . . . . . . 10 (1 + 5) = 6
118115, 117, 1123eqtri 2647 . . . . . . . . 9 ((1 · 1) + 5) = 06
11939, 13, 17, 35, 100, 90, 13, 28, 3, 114, 118decma2c 11520 . . . . . . . 8 ((1 · 121) + 45) = 166
12013, 36, 40, 99, 119gcdi 15712 . . . . . . 7 (166 gcd 121) = 1
121 eqid 2621 . . . . . . . 8 166 = 166
122 eqid 2621 . . . . . . . . 9 16 = 16
12313, 10, 65, 101decsuc 11487 . . . . . . . . 9 (12 + 1) = 13
124 1p1e2 11086 . . . . . . . . . . 11 (1 + 1) = 2
12563, 124oveq12i 6622 . . . . . . . . . 10 ((2 · 1) + (1 + 1)) = (2 + 2)
126125, 52eqtri 2643 . . . . . . . . 9 ((2 · 1) + (1 + 1)) = 4
127 6cn 11054 . . . . . . . . . . 11 6 ∈ ℂ
128 6t2e12 11593 . . . . . . . . . . 11 (6 · 2) = 12
129127, 67, 128mulcomli 9999 . . . . . . . . . 10 (2 · 6) = 12
130 3p2e5 11112 . . . . . . . . . . 11 (3 + 2) = 5
13149, 67, 130addcomli 10180 . . . . . . . . . 10 (2 + 3) = 5
13213, 10, 11, 129, 131decaddi 11531 . . . . . . . . 9 ((2 · 6) + 3) = 15
13313, 28, 13, 11, 122, 123, 10, 35, 13, 126, 132decma2c 11520 . . . . . . . 8 ((2 · 16) + (12 + 1)) = 45
13413, 10, 65, 129decsuc 11487 . . . . . . . 8 ((2 · 6) + 1) = 13
13529, 28, 39, 13, 121, 100, 10, 11, 13, 133, 134decma2c 11520 . . . . . . 7 ((2 · 166) + 121) = 453
13610, 40, 38, 120, 135gcdi 15712 . . . . . 6 (453 gcd 166) = 1
137 eqid 2621 . . . . . . 7 453 = 453
13829nn0cni 11256 . . . . . . . . 9 16 ∈ ℂ
139138addid1i 10175 . . . . . . . 8 (16 + 0) = 16
14048mulid2i 9995 . . . . . . . . . 10 (1 · 4) = 4
141140, 124oveq12i 6622 . . . . . . . . 9 ((1 · 4) + (1 + 1)) = (4 + 2)
142141, 110eqtri 2643 . . . . . . . 8 ((1 · 4) + (1 + 1)) = 6
14394mulid2i 9995 . . . . . . . . . 10 (1 · 5) = 5
144143oveq1i 6620 . . . . . . . . 9 ((1 · 5) + 6) = (5 + 6)
145 6p5e11 11552 . . . . . . . . . 10 (6 + 5) = 11
146127, 94, 145addcomli 10180 . . . . . . . . 9 (5 + 6) = 11
147144, 146eqtri 2643 . . . . . . . 8 ((1 · 5) + 6) = 11
14817, 35, 13, 28, 90, 139, 13, 13, 13, 142, 147decma2c 11520 . . . . . . 7 ((1 · 45) + (16 + 0)) = 61
14976oveq1i 6620 . . . . . . . 8 ((1 · 3) + 6) = (3 + 6)
150 6p3e9 11122 . . . . . . . . 9 (6 + 3) = 9
151127, 49, 150addcomli 10180 . . . . . . . 8 (3 + 6) = 9
15230dec0h 11474 . . . . . . . 8 9 = 09
153149, 151, 1523eqtri 2647 . . . . . . 7 ((1 · 3) + 6) = 09
15436, 11, 29, 28, 137, 121, 13, 30, 3, 148, 153decma2c 11520 . . . . . 6 ((1 · 453) + 166) = 619
15513, 38, 37, 136, 154gcdi 15712 . . . . 5 (619 gcd 453) = 1
156 eqid 2621 . . . . . 6 619 = 619
157 7nn0 11266 . . . . . . 7 7 ∈ ℕ0
158 eqid 2621 . . . . . . 7 61 = 61
159 5p2e7 11117 . . . . . . . 8 (5 + 2) = 7
16017, 35, 10, 90, 159decaddi 11531 . . . . . . 7 (45 + 2) = 47
161102oveq2i 6621 . . . . . . . 8 ((2 · 6) + (4 + 0)) = ((2 · 6) + 4)
16213, 10, 17, 129, 111decaddi 11531 . . . . . . . 8 ((2 · 6) + 4) = 16
163161, 162eqtri 2643 . . . . . . 7 ((2 · 6) + (4 + 0)) = 16
16463oveq1i 6620 . . . . . . . 8 ((2 · 1) + 7) = (2 + 7)
165 7cn 11056 . . . . . . . . 9 7 ∈ ℂ
166 7p2e9 11124 . . . . . . . . 9 (7 + 2) = 9
167165, 67, 166addcomli 10180 . . . . . . . 8 (2 + 7) = 9
168164, 167, 1523eqtri 2647 . . . . . . 7 ((2 · 1) + 7) = 09
16928, 13, 17, 157, 158, 160, 10, 30, 3, 163, 168decma2c 11520 . . . . . 6 ((2 · 61) + (45 + 2)) = 169
170 9cn 11060 . . . . . . . 8 9 ∈ ℂ
171 9t2e18 11615 . . . . . . . 8 (9 · 2) = 18
172170, 67, 171mulcomli 9999 . . . . . . 7 (2 · 9) = 18
17313, 2, 11, 172, 124, 13, 71decaddci 11532 . . . . . 6 ((2 · 9) + 3) = 21
17433, 30, 36, 11, 156, 137, 10, 13, 10, 169, 173decma2c 11520 . . . . 5 ((2 · 619) + 453) = 1691
17510, 37, 34, 155, 174gcdi 15712 . . . 4 (1691 gcd 619) = 1
176 eqid 2621 . . . . 5 1691 = 1691
177 eqid 2621 . . . . . 6 169 = 169
17828, 13, 124, 158decsuc 11487 . . . . . 6 (61 + 1) = 62
179 6p1e7 11108 . . . . . . . 8 (6 + 1) = 7
180157dec0h 11474 . . . . . . . 8 7 = 07
181179, 180eqtri 2643 . . . . . . 7 (6 + 1) = 07
18282, 24oveq12i 6622 . . . . . . . 8 ((1 · 1) + (0 + 1)) = (1 + 1)
183182, 124eqtri 2643 . . . . . . 7 ((1 · 1) + (0 + 1)) = 2
184127mulid2i 9995 . . . . . . . . 9 (1 · 6) = 6
185184oveq1i 6620 . . . . . . . 8 ((1 · 6) + 7) = (6 + 7)
186 7p6e13 11560 . . . . . . . . 9 (7 + 6) = 13
187165, 127, 186addcomli 10180 . . . . . . . 8 (6 + 7) = 13
188185, 187eqtri 2643 . . . . . . 7 ((1 · 6) + 7) = 13
18913, 28, 3, 157, 122, 181, 13, 11, 13, 183, 188decma2c 11520 . . . . . 6 ((1 · 16) + (6 + 1)) = 23
190170mulid2i 9995 . . . . . . . 8 (1 · 9) = 9
191190oveq1i 6620 . . . . . . 7 ((1 · 9) + 2) = (9 + 2)
192 9p2e11 11571 . . . . . . 7 (9 + 2) = 11
193191, 192eqtri 2643 . . . . . 6 ((1 · 9) + 2) = 11
19429, 30, 28, 10, 177, 178, 13, 13, 13, 189, 193decma2c 11520 . . . . 5 ((1 · 169) + (61 + 1)) = 231
19582oveq1i 6620 . . . . . 6 ((1 · 1) + 9) = (1 + 9)
196 9p1e10 11448 . . . . . . 7 (9 + 1) = 10
197170, 77, 196addcomli 10180 . . . . . 6 (1 + 9) = 10
198195, 197eqtri 2643 . . . . 5 ((1 · 1) + 9) = 10
19931, 13, 33, 30, 176, 156, 13, 3, 13, 194, 198decma2c 11520 . . . 4 ((1 · 1691) + 619) = 2310
20013, 34, 32, 175, 199gcdi 15712 . . 3 (2310 gcd 1691) = 1
201 eqid 2621 . . . . . 6 231 = 231
20231nn0cni 11256 . . . . . . 7 169 ∈ ℂ
203202addid1i 10175 . . . . . 6 (169 + 0) = 169
204 eqid 2621 . . . . . . 7 23 = 23
20513, 28, 179, 122decsuc 11487 . . . . . . 7 (16 + 1) = 17
206108, 124oveq12i 6622 . . . . . . . 8 ((1 · 2) + (1 + 1)) = (2 + 2)
207206, 52eqtri 2643 . . . . . . 7 ((1 · 2) + (1 + 1)) = 4
20876oveq1i 6620 . . . . . . . 8 ((1 · 3) + 7) = (3 + 7)
209 7p3e10 11555 . . . . . . . . 9 (7 + 3) = 10
210165, 49, 209addcomli 10180 . . . . . . . 8 (3 + 7) = 10
211208, 210eqtri 2643 . . . . . . 7 ((1 · 3) + 7) = 10
21210, 11, 13, 157, 204, 205, 13, 3, 13, 207, 211decma2c 11520 . . . . . 6 ((1 · 23) + (16 + 1)) = 40
21312, 13, 29, 30, 201, 203, 13, 3, 13, 212, 198decma2c 11520 . . . . 5 ((1 · 231) + (169 + 0)) = 400
21477mul01i 10178 . . . . . . 7 (1 · 0) = 0
215214oveq1i 6620 . . . . . 6 ((1 · 0) + 1) = (0 + 1)
21613dec0h 11474 . . . . . 6 1 = 01
217215, 24, 2163eqtri 2647 . . . . 5 ((1 · 0) + 1) = 01
21814, 3, 31, 13, 25, 176, 13, 13, 3, 213, 217decma2c 11520 . . . 4 ((1 · 2310) + 1691) = 4001
219218, 16eqtr4i 2646 . . 3 ((1 · 2310) + 1691) = 𝑁
22013, 32, 15, 200, 219gcdi 15712 . 2 (𝑁 gcd 2310) = 1
2219, 15, 22, 27, 220gcdmodi 15713 1 (((2↑800) − 1) gcd 𝑁) = 1
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 196   = wceq 1480  wcel 1987   class class class wbr 4618  (class class class)co 6610  0cc0 9888  1c1 9889   + caddc 9891   · cmul 9893  cmin 10218  cn 10972  2c2 11022  3c3 11023  4c4 11024  5c5 11025  6c6 11026  7c7 11027  8c8 11028  9c9 11029  0cn0 11244  cz 11329  cdc 11445  cexp 12808  cdvds 14918   gcd cgcd 15151  cprime 15320
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6909  ax-cnex 9944  ax-resscn 9945  ax-1cn 9946  ax-icn 9947  ax-addcl 9948  ax-addrcl 9949  ax-mulcl 9950  ax-mulrcl 9951  ax-mulcom 9952  ax-addass 9953  ax-mulass 9954  ax-distr 9955  ax-i2m1 9956  ax-1ne0 9957  ax-1rid 9958  ax-rnegex 9959  ax-rrecex 9960  ax-cnre 9961  ax-pre-lttri 9962  ax-pre-lttrn 9963  ax-pre-ltadd 9964  ax-pre-mulgt0 9965  ax-pre-sup 9966
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2912  df-rex 2913  df-reu 2914  df-rmo 2915  df-rab 2916  df-v 3191  df-sbc 3422  df-csb 3519  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-pss 3575  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-tp 4158  df-op 4160  df-uni 4408  df-int 4446  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-tr 4718  df-eprel 4990  df-id 4994  df-po 5000  df-so 5001  df-fr 5038  df-we 5040  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-pred 5644  df-ord 5690  df-on 5691  df-lim 5692  df-suc 5693  df-iota 5815  df-fun 5854  df-fn 5855  df-f 5856  df-f1 5857  df-fo 5858  df-f1o 5859  df-fv 5860  df-riota 6571  df-ov 6613  df-oprab 6614  df-mpt2 6615  df-om 7020  df-1st 7120  df-2nd 7121  df-wrecs 7359  df-recs 7420  df-rdg 7458  df-1o 7512  df-2o 7513  df-oadd 7516  df-er 7694  df-en 7908  df-dom 7909  df-sdom 7910  df-fin 7911  df-sup 8300  df-inf 8301  df-pnf 10028  df-mnf 10029  df-xr 10030  df-ltxr 10031  df-le 10032  df-sub 10220  df-neg 10221  df-div 10637  df-nn 10973  df-2 11031  df-3 11032  df-4 11033  df-5 11034  df-6 11035  df-7 11036  df-8 11037  df-9 11038  df-n0 11245  df-z 11330  df-dec 11446  df-uz 11640  df-rp 11785  df-fz 12277  df-fl 12541  df-mod 12617  df-seq 12750  df-exp 12809  df-cj 13781  df-re 13782  df-im 13783  df-sqrt 13917  df-abs 13918  df-dvds 14919  df-gcd 15152  df-prm 15321
This theorem is referenced by:  4001prm  15787
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