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Theorem 4001lem4 16465
Description: Lemma for 4001prm 16466. 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 11698 . . . 4 2 ∈ ℕ
2 8nn0 11908 . . . . . 6 8 ∈ ℕ0
3 0nn0 11900 . . . . . 6 0 ∈ ℕ0
42, 3deccl 12101 . . . . 5 80 ∈ ℕ0
54, 3deccl 12101 . . . 4 800 ∈ ℕ0
6 nnexpcl 13430 . . . 4 ((2 ∈ ℕ ∧ 800 ∈ ℕ0) → (2↑800) ∈ ℕ)
71, 5, 6mp2an 688 . . 3 (2↑800) ∈ ℕ
8 nnm1nn0 11926 . . 3 ((2↑800) ∈ ℕ → ((2↑800) − 1) ∈ ℕ0)
97, 8ax-mp 5 . 2 ((2↑800) − 1) ∈ ℕ0
10 2nn0 11902 . . . . 5 2 ∈ ℕ0
11 3nn0 11903 . . . . 5 3 ∈ ℕ0
1210, 11deccl 12101 . . . 4 23 ∈ ℕ0
13 1nn0 11901 . . . 4 1 ∈ ℕ0
1412, 13deccl 12101 . . 3 231 ∈ ℕ0
1514, 3deccl 12101 . 2 2310 ∈ ℕ0
16 4001prm.1 . . 3 𝑁 = 4001
17 4nn0 11904 . . . . . 6 4 ∈ ℕ0
1817, 3deccl 12101 . . . . 5 40 ∈ ℕ0
1918, 3deccl 12101 . . . 4 400 ∈ ℕ0
20 1nn 11637 . . . 4 1 ∈ ℕ
2119, 20decnncl 12106 . . 3 4001 ∈ ℕ
2216, 21eqeltri 2906 . 2 𝑁 ∈ ℕ
23164001lem2 16463 . . 3 ((2↑800) mod 𝑁) = (2311 mod 𝑁)
24 0p1e1 11747 . . . 4 (0 + 1) = 1
25 eqid 2818 . . . 4 2310 = 2310
2614, 3, 24, 25decsuc 12117 . . 3 (2310 + 1) = 2311
2722, 7, 13, 15, 23, 26modsubi 16396 . 2 (((2↑800) − 1) mod 𝑁) = (2310 mod 𝑁)
28 6nn0 11906 . . . . . 6 6 ∈ ℕ0
2913, 28deccl 12101 . . . . 5 16 ∈ ℕ0
30 9nn0 11909 . . . . 5 9 ∈ ℕ0
3129, 30deccl 12101 . . . 4 169 ∈ ℕ0
3231, 13deccl 12101 . . 3 1691 ∈ ℕ0
3328, 13deccl 12101 . . . . 5 61 ∈ ℕ0
3433, 30deccl 12101 . . . 4 619 ∈ ℕ0
35 5nn0 11905 . . . . . . 7 5 ∈ ℕ0
3617, 35deccl 12101 . . . . . 6 45 ∈ ℕ0
3736, 11deccl 12101 . . . . 5 453 ∈ ℕ0
3829, 28deccl 12101 . . . . . 6 166 ∈ ℕ0
3913, 10deccl 12101 . . . . . . . 8 12 ∈ ℕ0
4039, 13deccl 12101 . . . . . . 7 121 ∈ ℕ0
4111, 13deccl 12101 . . . . . . . . 9 31 ∈ ℕ0
4213, 17deccl 12101 . . . . . . . . . 10 14 ∈ ℕ0
4342nn0zi 11995 . . . . . . . . . . . . 13 14 ∈ ℤ
4411nn0zi 11995 . . . . . . . . . . . . 13 3 ∈ ℤ
45 gcdcom 15850 . . . . . . . . . . . . 13 ((14 ∈ ℤ ∧ 3 ∈ ℤ) → (14 gcd 3) = (3 gcd 14))
4643, 44, 45mp2an 688 . . . . . . . . . . . 12 (14 gcd 3) = (3 gcd 14)
47 3nn 11704 . . . . . . . . . . . . . 14 3 ∈ ℕ
48 4cn 11710 . . . . . . . . . . . . . . . 16 4 ∈ ℂ
49 3cn 11706 . . . . . . . . . . . . . . . 16 3 ∈ ℂ
50 4t3e12 12184 . . . . . . . . . . . . . . . 16 (4 · 3) = 12
5148, 49, 50mulcomli 10638 . . . . . . . . . . . . . . 15 (3 · 4) = 12
52 2p2e4 11760 . . . . . . . . . . . . . . 15 (2 + 2) = 4
5313, 10, 10, 51, 52decaddi 12146 . . . . . . . . . . . . . 14 ((3 · 4) + 2) = 14
54 2lt3 11797 . . . . . . . . . . . . . 14 2 < 3
5547, 17, 1, 53, 54ndvdsi 15751 . . . . . . . . . . . . 13 ¬ 3 ∥ 14
56 3prm 16026 . . . . . . . . . . . . . 14 3 ∈ ℙ
57 coprm 16043 . . . . . . . . . . . . . 14 ((3 ∈ ℙ ∧ 14 ∈ ℤ) → (¬ 3 ∥ 14 ↔ (3 gcd 14) = 1))
5856, 43, 57mp2an 688 . . . . . . . . . . . . 13 (¬ 3 ∥ 14 ↔ (3 gcd 14) = 1)
5955, 58mpbi 231 . . . . . . . . . . . 12 (3 gcd 14) = 1
6046, 59eqtri 2841 . . . . . . . . . . 11 (14 gcd 3) = 1
61 eqid 2818 . . . . . . . . . . . 12 14 = 14
6211dec0h 12108 . . . . . . . . . . . 12 3 = 03
63 2t1e2 11788 . . . . . . . . . . . . . 14 (2 · 1) = 2
6463, 24oveq12i 7157 . . . . . . . . . . . . 13 ((2 · 1) + (0 + 1)) = (2 + 1)
65 2p1e3 11767 . . . . . . . . . . . . 13 (2 + 1) = 3
6664, 65eqtri 2841 . . . . . . . . . . . 12 ((2 · 1) + (0 + 1)) = 3
67 2cn 11700 . . . . . . . . . . . . . . 15 2 ∈ ℂ
68 4t2e8 11793 . . . . . . . . . . . . . . 15 (4 · 2) = 8
6948, 67, 68mulcomli 10638 . . . . . . . . . . . . . 14 (2 · 4) = 8
7069oveq1i 7155 . . . . . . . . . . . . 13 ((2 · 4) + 3) = (8 + 3)
71 8p3e11 12167 . . . . . . . . . . . . 13 (8 + 3) = 11
7270, 71eqtri 2841 . . . . . . . . . . . 12 ((2 · 4) + 3) = 11
7313, 17, 3, 11, 61, 62, 10, 13, 13, 66, 72decma2c 12139 . . . . . . . . . . 11 ((2 · 14) + 3) = 31
7410, 11, 42, 60, 73gcdi 16397 . . . . . . . . . 10 (31 gcd 14) = 1
75 eqid 2818 . . . . . . . . . . 11 31 = 31
7649mulid2i 10634 . . . . . . . . . . . . 13 (1 · 3) = 3
77 ax-1cn 10583 . . . . . . . . . . . . . 14 1 ∈ ℂ
7877addid1i 10815 . . . . . . . . . . . . 13 (1 + 0) = 1
7976, 78oveq12i 7157 . . . . . . . . . . . 12 ((1 · 3) + (1 + 0)) = (3 + 1)
80 3p1e4 11770 . . . . . . . . . . . 12 (3 + 1) = 4
8179, 80eqtri 2841 . . . . . . . . . . 11 ((1 · 3) + (1 + 0)) = 4
82 1t1e1 11787 . . . . . . . . . . . . 13 (1 · 1) = 1
8382oveq1i 7155 . . . . . . . . . . . 12 ((1 · 1) + 4) = (1 + 4)
84 4p1e5 11771 . . . . . . . . . . . . 13 (4 + 1) = 5
8548, 77, 84addcomli 10820 . . . . . . . . . . . 12 (1 + 4) = 5
8635dec0h 12108 . . . . . . . . . . . 12 5 = 05
8783, 85, 863eqtri 2845 . . . . . . . . . . 11 ((1 · 1) + 4) = 05
8811, 13, 13, 17, 75, 61, 13, 35, 3, 81, 87decma2c 12139 . . . . . . . . . 10 ((1 · 31) + 14) = 45
8913, 42, 41, 74, 88gcdi 16397 . . . . . . . . 9 (45 gcd 31) = 1
90 eqid 2818 . . . . . . . . . 10 45 = 45
9169, 80oveq12i 7157 . . . . . . . . . . 11 ((2 · 4) + (3 + 1)) = (8 + 4)
92 8p4e12 12168 . . . . . . . . . . 11 (8 + 4) = 12
9391, 92eqtri 2841 . . . . . . . . . 10 ((2 · 4) + (3 + 1)) = 12
94 5cn 11713 . . . . . . . . . . . 12 5 ∈ ℂ
95 5t2e10 12186 . . . . . . . . . . . 12 (5 · 2) = 10
9694, 67, 95mulcomli 10638 . . . . . . . . . . 11 (2 · 5) = 10
9713, 3, 24, 96decsuc 12117 . . . . . . . . . 10 ((2 · 5) + 1) = 11
9817, 35, 11, 13, 90, 75, 10, 13, 13, 93, 97decma2c 12139 . . . . . . . . 9 ((2 · 45) + 31) = 121
9910, 41, 36, 89, 98gcdi 16397 . . . . . . . 8 (121 gcd 45) = 1
100 eqid 2818 . . . . . . . . 9 121 = 121
101 eqid 2818 . . . . . . . . . 10 12 = 12
10248addid1i 10815 . . . . . . . . . . 11 (4 + 0) = 4
10317dec0h 12108 . . . . . . . . . . 11 4 = 04
104102, 103eqtri 2841 . . . . . . . . . 10 (4 + 0) = 04
105 00id 10803 . . . . . . . . . . . 12 (0 + 0) = 0
10682, 105oveq12i 7157 . . . . . . . . . . 11 ((1 · 1) + (0 + 0)) = (1 + 0)
107106, 78eqtri 2841 . . . . . . . . . 10 ((1 · 1) + (0 + 0)) = 1
10867mulid2i 10634 . . . . . . . . . . . 12 (1 · 2) = 2
109108oveq1i 7155 . . . . . . . . . . 11 ((1 · 2) + 4) = (2 + 4)
110 4p2e6 11778 . . . . . . . . . . . 12 (4 + 2) = 6
11148, 67, 110addcomli 10820 . . . . . . . . . . 11 (2 + 4) = 6
11228dec0h 12108 . . . . . . . . . . 11 6 = 06
113109, 111, 1123eqtri 2845 . . . . . . . . . 10 ((1 · 2) + 4) = 06
11413, 10, 3, 17, 101, 104, 13, 28, 3, 107, 113decma2c 12139 . . . . . . . . 9 ((1 · 12) + (4 + 0)) = 16
11582oveq1i 7155 . . . . . . . . . 10 ((1 · 1) + 5) = (1 + 5)
116 5p1e6 11772 . . . . . . . . . . 11 (5 + 1) = 6
11794, 77, 116addcomli 10820 . . . . . . . . . 10 (1 + 5) = 6
118115, 117, 1123eqtri 2845 . . . . . . . . 9 ((1 · 1) + 5) = 06
11939, 13, 17, 35, 100, 90, 13, 28, 3, 114, 118decma2c 12139 . . . . . . . 8 ((1 · 121) + 45) = 166
12013, 36, 40, 99, 119gcdi 16397 . . . . . . 7 (166 gcd 121) = 1
121 eqid 2818 . . . . . . . 8 166 = 166
122 eqid 2818 . . . . . . . . 9 16 = 16
12313, 10, 65, 101decsuc 12117 . . . . . . . . 9 (12 + 1) = 13
124 1p1e2 11750 . . . . . . . . . . 11 (1 + 1) = 2
12563, 124oveq12i 7157 . . . . . . . . . 10 ((2 · 1) + (1 + 1)) = (2 + 2)
126125, 52eqtri 2841 . . . . . . . . 9 ((2 · 1) + (1 + 1)) = 4
127 6cn 11716 . . . . . . . . . . 11 6 ∈ ℂ
128 6t2e12 12190 . . . . . . . . . . 11 (6 · 2) = 12
129127, 67, 128mulcomli 10638 . . . . . . . . . 10 (2 · 6) = 12
130 3p2e5 11776 . . . . . . . . . . 11 (3 + 2) = 5
13149, 67, 130addcomli 10820 . . . . . . . . . 10 (2 + 3) = 5
13213, 10, 11, 129, 131decaddi 12146 . . . . . . . . 9 ((2 · 6) + 3) = 15
13313, 28, 13, 11, 122, 123, 10, 35, 13, 126, 132decma2c 12139 . . . . . . . 8 ((2 · 16) + (12 + 1)) = 45
13413, 10, 65, 129decsuc 12117 . . . . . . . 8 ((2 · 6) + 1) = 13
13529, 28, 39, 13, 121, 100, 10, 11, 13, 133, 134decma2c 12139 . . . . . . 7 ((2 · 166) + 121) = 453
13610, 40, 38, 120, 135gcdi 16397 . . . . . 6 (453 gcd 166) = 1
137 eqid 2818 . . . . . . 7 453 = 453
13829nn0cni 11897 . . . . . . . . 9 16 ∈ ℂ
139138addid1i 10815 . . . . . . . 8 (16 + 0) = 16
14048mulid2i 10634 . . . . . . . . . 10 (1 · 4) = 4
141140, 124oveq12i 7157 . . . . . . . . 9 ((1 · 4) + (1 + 1)) = (4 + 2)
142141, 110eqtri 2841 . . . . . . . 8 ((1 · 4) + (1 + 1)) = 6
14394mulid2i 10634 . . . . . . . . . 10 (1 · 5) = 5
144143oveq1i 7155 . . . . . . . . 9 ((1 · 5) + 6) = (5 + 6)
145 6p5e11 12159 . . . . . . . . . 10 (6 + 5) = 11
146127, 94, 145addcomli 10820 . . . . . . . . 9 (5 + 6) = 11
147144, 146eqtri 2841 . . . . . . . 8 ((1 · 5) + 6) = 11
14817, 35, 13, 28, 90, 139, 13, 13, 13, 142, 147decma2c 12139 . . . . . . 7 ((1 · 45) + (16 + 0)) = 61
14976oveq1i 7155 . . . . . . . 8 ((1 · 3) + 6) = (3 + 6)
150 6p3e9 11785 . . . . . . . . 9 (6 + 3) = 9
151127, 49, 150addcomli 10820 . . . . . . . 8 (3 + 6) = 9
15230dec0h 12108 . . . . . . . 8 9 = 09
153149, 151, 1523eqtri 2845 . . . . . . 7 ((1 · 3) + 6) = 09
15436, 11, 29, 28, 137, 121, 13, 30, 3, 148, 153decma2c 12139 . . . . . 6 ((1 · 453) + 166) = 619
15513, 38, 37, 136, 154gcdi 16397 . . . . 5 (619 gcd 453) = 1
156 eqid 2818 . . . . . 6 619 = 619
157 7nn0 11907 . . . . . . 7 7 ∈ ℕ0
158 eqid 2818 . . . . . . 7 61 = 61
159 5p2e7 11781 . . . . . . . 8 (5 + 2) = 7
16017, 35, 10, 90, 159decaddi 12146 . . . . . . 7 (45 + 2) = 47
161102oveq2i 7156 . . . . . . . 8 ((2 · 6) + (4 + 0)) = ((2 · 6) + 4)
16213, 10, 17, 129, 111decaddi 12146 . . . . . . . 8 ((2 · 6) + 4) = 16
163161, 162eqtri 2841 . . . . . . 7 ((2 · 6) + (4 + 0)) = 16
16463oveq1i 7155 . . . . . . . 8 ((2 · 1) + 7) = (2 + 7)
165 7cn 11719 . . . . . . . . 9 7 ∈ ℂ
166 7p2e9 11786 . . . . . . . . 9 (7 + 2) = 9
167165, 67, 166addcomli 10820 . . . . . . . 8 (2 + 7) = 9
168164, 167, 1523eqtri 2845 . . . . . . 7 ((2 · 1) + 7) = 09
16928, 13, 17, 157, 158, 160, 10, 30, 3, 163, 168decma2c 12139 . . . . . 6 ((2 · 61) + (45 + 2)) = 169
170 9cn 11725 . . . . . . . 8 9 ∈ ℂ
171 9t2e18 12208 . . . . . . . 8 (9 · 2) = 18
172170, 67, 171mulcomli 10638 . . . . . . 7 (2 · 9) = 18
17313, 2, 11, 172, 124, 13, 71decaddci 12147 . . . . . 6 ((2 · 9) + 3) = 21
17433, 30, 36, 11, 156, 137, 10, 13, 10, 169, 173decma2c 12139 . . . . 5 ((2 · 619) + 453) = 1691
17510, 37, 34, 155, 174gcdi 16397 . . . 4 (1691 gcd 619) = 1
176 eqid 2818 . . . . 5 1691 = 1691
177 eqid 2818 . . . . . 6 169 = 169
17828, 13, 124, 158decsuc 12117 . . . . . 6 (61 + 1) = 62
179 6p1e7 11773 . . . . . . . 8 (6 + 1) = 7
180157dec0h 12108 . . . . . . . 8 7 = 07
181179, 180eqtri 2841 . . . . . . 7 (6 + 1) = 07
18282, 24oveq12i 7157 . . . . . . . 8 ((1 · 1) + (0 + 1)) = (1 + 1)
183182, 124eqtri 2841 . . . . . . 7 ((1 · 1) + (0 + 1)) = 2
184127mulid2i 10634 . . . . . . . . 9 (1 · 6) = 6
185184oveq1i 7155 . . . . . . . 8 ((1 · 6) + 7) = (6 + 7)
186 7p6e13 12164 . . . . . . . . 9 (7 + 6) = 13
187165, 127, 186addcomli 10820 . . . . . . . 8 (6 + 7) = 13
188185, 187eqtri 2841 . . . . . . 7 ((1 · 6) + 7) = 13
18913, 28, 3, 157, 122, 181, 13, 11, 13, 183, 188decma2c 12139 . . . . . 6 ((1 · 16) + (6 + 1)) = 23
190170mulid2i 10634 . . . . . . . 8 (1 · 9) = 9
191190oveq1i 7155 . . . . . . 7 ((1 · 9) + 2) = (9 + 2)
192 9p2e11 12173 . . . . . . 7 (9 + 2) = 11
193191, 192eqtri 2841 . . . . . 6 ((1 · 9) + 2) = 11
19429, 30, 28, 10, 177, 178, 13, 13, 13, 189, 193decma2c 12139 . . . . 5 ((1 · 169) + (61 + 1)) = 231
19582oveq1i 7155 . . . . . 6 ((1 · 1) + 9) = (1 + 9)
196 9p1e10 12088 . . . . . . 7 (9 + 1) = 10
197170, 77, 196addcomli 10820 . . . . . 6 (1 + 9) = 10
198195, 197eqtri 2841 . . . . 5 ((1 · 1) + 9) = 10
19931, 13, 33, 30, 176, 156, 13, 3, 13, 194, 198decma2c 12139 . . . 4 ((1 · 1691) + 619) = 2310
20013, 34, 32, 175, 199gcdi 16397 . . 3 (2310 gcd 1691) = 1
201 eqid 2818 . . . . . 6 231 = 231
20231nn0cni 11897 . . . . . . 7 169 ∈ ℂ
203202addid1i 10815 . . . . . 6 (169 + 0) = 169
204 eqid 2818 . . . . . . 7 23 = 23
20513, 28, 179, 122decsuc 12117 . . . . . . 7 (16 + 1) = 17
206108, 124oveq12i 7157 . . . . . . . 8 ((1 · 2) + (1 + 1)) = (2 + 2)
207206, 52eqtri 2841 . . . . . . 7 ((1 · 2) + (1 + 1)) = 4
20876oveq1i 7155 . . . . . . . 8 ((1 · 3) + 7) = (3 + 7)
209 7p3e10 12161 . . . . . . . . 9 (7 + 3) = 10
210165, 49, 209addcomli 10820 . . . . . . . 8 (3 + 7) = 10
211208, 210eqtri 2841 . . . . . . 7 ((1 · 3) + 7) = 10
21210, 11, 13, 157, 204, 205, 13, 3, 13, 207, 211decma2c 12139 . . . . . 6 ((1 · 23) + (16 + 1)) = 40
21312, 13, 29, 30, 201, 203, 13, 3, 13, 212, 198decma2c 12139 . . . . 5 ((1 · 231) + (169 + 0)) = 400
21477mul01i 10818 . . . . . . 7 (1 · 0) = 0
215214oveq1i 7155 . . . . . 6 ((1 · 0) + 1) = (0 + 1)
21613dec0h 12108 . . . . . 6 1 = 01
217215, 24, 2163eqtri 2845 . . . . 5 ((1 · 0) + 1) = 01
21814, 3, 31, 13, 25, 176, 13, 13, 3, 213, 217decma2c 12139 . . . 4 ((1 · 2310) + 1691) = 4001
219218, 16eqtr4i 2844 . . 3 ((1 · 2310) + 1691) = 𝑁
22013, 32, 15, 200, 219gcdi 16397 . 2 (𝑁 gcd 2310) = 1
2219, 15, 22, 27, 220gcdmodi 16398 1 (((2↑800) − 1) gcd 𝑁) = 1
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 207   = wceq 1528  wcel 2105   class class class wbr 5057  (class class class)co 7145  0cc0 10525  1c1 10526   + caddc 10528   · cmul 10530  cmin 10858  cn 11626  2c2 11680  3c3 11681  4c4 11682  5c5 11683  6c6 11684  7c7 11685  8c8 11686  9c9 11687  0cn0 11885  cz 11969  cdc 12086  cexp 13417  cdvds 15595   gcd cgcd 15831  cprime 16003
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450  ax-cnex 10581  ax-resscn 10582  ax-1cn 10583  ax-icn 10584  ax-addcl 10585  ax-addrcl 10586  ax-mulcl 10587  ax-mulrcl 10588  ax-mulcom 10589  ax-addass 10590  ax-mulass 10591  ax-distr 10592  ax-i2m1 10593  ax-1ne0 10594  ax-1rid 10595  ax-rnegex 10596  ax-rrecex 10597  ax-cnre 10598  ax-pre-lttri 10599  ax-pre-lttrn 10600  ax-pre-ltadd 10601  ax-pre-mulgt0 10602  ax-pre-sup 10603
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3or 1080  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-nel 3121  df-ral 3140  df-rex 3141  df-reu 3142  df-rmo 3143  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-pss 3951  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-tp 4562  df-op 4564  df-uni 4831  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-tr 5164  df-id 5453  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-pred 6141  df-ord 6187  df-on 6188  df-lim 6189  df-suc 6190  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-riota 7103  df-ov 7148  df-oprab 7149  df-mpo 7150  df-om 7570  df-1st 7678  df-2nd 7679  df-wrecs 7936  df-recs 7997  df-rdg 8035  df-1o 8091  df-2o 8092  df-er 8278  df-en 8498  df-dom 8499  df-sdom 8500  df-fin 8501  df-sup 8894  df-inf 8895  df-pnf 10665  df-mnf 10666  df-xr 10667  df-ltxr 10668  df-le 10669  df-sub 10860  df-neg 10861  df-div 11286  df-nn 11627  df-2 11688  df-3 11689  df-4 11690  df-5 11691  df-6 11692  df-7 11693  df-8 11694  df-9 11695  df-n0 11886  df-z 11970  df-dec 12087  df-uz 12232  df-rp 12378  df-fz 12881  df-fl 13150  df-mod 13226  df-seq 13358  df-exp 13418  df-cj 14446  df-re 14447  df-im 14448  df-sqrt 14582  df-abs 14583  df-dvds 15596  df-gcd 15832  df-prm 16004
This theorem is referenced by:  4001prm  16466
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