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Theorem 4001lem2 16463
Description: Lemma for 4001prm 16466. Calculate a power mod. In decimal, we calculate 2↑400 = (2↑200)↑2≡902↑2 = 203𝑁 + 1401 and 2↑800 = (2↑400)↑2≡1401↑2 = 490𝑁 + 2311 ≡2311. (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
4001lem2 ((2↑800) mod 𝑁) = (2311 mod 𝑁)

Proof of Theorem 4001lem2
StepHypRef Expression
1 4001prm.1 . . 3 𝑁 = 4001
2 4nn0 11904 . . . . . 6 4 ∈ ℕ0
3 0nn0 11900 . . . . . 6 0 ∈ ℕ0
42, 3deccl 12101 . . . . 5 40 ∈ ℕ0
54, 3deccl 12101 . . . 4 400 ∈ ℕ0
6 1nn 11637 . . . 4 1 ∈ ℕ
75, 6decnncl 12106 . . 3 4001 ∈ ℕ
81, 7eqeltri 2906 . 2 𝑁 ∈ ℕ
9 2nn 11698 . 2 2 ∈ ℕ
10 9nn0 11909 . . . . 5 9 ∈ ℕ0
112, 10deccl 12101 . . . 4 49 ∈ ℕ0
1211, 3deccl 12101 . . 3 490 ∈ ℕ0
1312nn0zi 11995 . 2 490 ∈ ℤ
14 1nn0 11901 . . . . 5 1 ∈ ℕ0
1514, 2deccl 12101 . . . 4 14 ∈ ℕ0
1615, 3deccl 12101 . . 3 140 ∈ ℕ0
1716, 14deccl 12101 . 2 1401 ∈ ℕ0
18 2nn0 11902 . . . . 5 2 ∈ ℕ0
19 3nn0 11903 . . . . 5 3 ∈ ℕ0
2018, 19deccl 12101 . . . 4 23 ∈ ℕ0
2120, 14deccl 12101 . . 3 231 ∈ ℕ0
2221, 14deccl 12101 . 2 2311 ∈ ℕ0
2318, 3deccl 12101 . . . 4 20 ∈ ℕ0
2423, 3deccl 12101 . . 3 200 ∈ ℕ0
2523, 19deccl 12101 . . . 4 203 ∈ ℕ0
2625nn0zi 11995 . . 3 203 ∈ ℤ
2710, 3deccl 12101 . . . 4 90 ∈ ℕ0
2827, 18deccl 12101 . . 3 902 ∈ ℕ0
2914001lem1 16462 . . 3 ((2↑200) mod 𝑁) = (902 mod 𝑁)
3024nn0cni 11897 . . . 4 200 ∈ ℂ
31 2cn 11700 . . . 4 2 ∈ ℂ
32 eqid 2818 . . . . 5 200 = 200
33 eqid 2818 . . . . . 6 20 = 20
34 2t2e4 11789 . . . . . 6 (2 · 2) = 4
3531mul02i 10817 . . . . . 6 (0 · 2) = 0
3618, 18, 3, 33, 34, 35decmul1 12150 . . . . 5 (20 · 2) = 40
3718, 23, 3, 32, 36, 35decmul1 12150 . . . 4 (200 · 2) = 400
3830, 31, 37mulcomli 10638 . . 3 (2 · 200) = 400
39 eqid 2818 . . . . 5 1401 = 1401
40 6nn0 11906 . . . . . . 7 6 ∈ ℕ0
4114, 40deccl 12101 . . . . . 6 16 ∈ ℕ0
42 eqid 2818 . . . . . 6 400 = 400
43 eqid 2818 . . . . . . 7 140 = 140
44 eqid 2818 . . . . . . . 8 14 = 14
45 4p2e6 11778 . . . . . . . 8 (4 + 2) = 6
4614, 2, 18, 44, 45decaddi 12146 . . . . . . 7 (14 + 2) = 16
47 00id 10803 . . . . . . 7 (0 + 0) = 0
4815, 3, 18, 3, 43, 33, 46, 47decadd 12140 . . . . . 6 (140 + 20) = 160
49 eqid 2818 . . . . . . 7 40 = 40
5041nn0cni 11897 . . . . . . . 8 16 ∈ ℂ
5150addid1i 10815 . . . . . . 7 (16 + 0) = 16
52 eqid 2818 . . . . . . . 8 203 = 203
53 ax-1cn 10583 . . . . . . . . . 10 1 ∈ ℂ
5453addid1i 10815 . . . . . . . . 9 (1 + 0) = 1
5514dec0h 12108 . . . . . . . . 9 1 = 01
5654, 55eqtri 2841 . . . . . . . 8 (1 + 0) = 01
5753addid2i 10816 . . . . . . . . . 10 (0 + 1) = 1
5857, 14eqeltri 2906 . . . . . . . . 9 (0 + 1) ∈ ℕ0
59 4cn 11710 . . . . . . . . . 10 4 ∈ ℂ
60 4t2e8 11793 . . . . . . . . . 10 (4 · 2) = 8
6159, 31, 60mulcomli 10638 . . . . . . . . 9 (2 · 4) = 8
6259mul02i 10817 . . . . . . . . . . 11 (0 · 4) = 0
6362, 57oveq12i 7157 . . . . . . . . . 10 ((0 · 4) + (0 + 1)) = (0 + 1)
6463, 57eqtri 2841 . . . . . . . . 9 ((0 · 4) + (0 + 1)) = 1
6518, 3, 58, 33, 2, 61, 64decrmanc 12143 . . . . . . . 8 ((20 · 4) + (0 + 1)) = 81
66 2p1e3 11767 . . . . . . . . 9 (2 + 1) = 3
67 3cn 11706 . . . . . . . . . 10 3 ∈ ℂ
68 4t3e12 12184 . . . . . . . . . 10 (4 · 3) = 12
6959, 67, 68mulcomli 10638 . . . . . . . . 9 (3 · 4) = 12
7014, 18, 66, 69decsuc 12117 . . . . . . . 8 ((3 · 4) + 1) = 13
7123, 19, 3, 14, 52, 56, 2, 19, 14, 65, 70decmac 12138 . . . . . . 7 ((203 · 4) + (1 + 0)) = 813
7225nn0cni 11897 . . . . . . . . . 10 203 ∈ ℂ
7372mul01i 10818 . . . . . . . . 9 (203 · 0) = 0
7473oveq1i 7155 . . . . . . . 8 ((203 · 0) + 6) = (0 + 6)
75 6cn 11716 . . . . . . . . 9 6 ∈ ℂ
7675addid2i 10816 . . . . . . . 8 (0 + 6) = 6
7740dec0h 12108 . . . . . . . 8 6 = 06
7874, 76, 773eqtri 2845 . . . . . . 7 ((203 · 0) + 6) = 06
792, 3, 14, 40, 49, 51, 25, 40, 3, 71, 78decma2c 12139 . . . . . 6 ((203 · 40) + (16 + 0)) = 8136
8073oveq1i 7155 . . . . . . 7 ((203 · 0) + 0) = (0 + 0)
813dec0h 12108 . . . . . . 7 0 = 00
8280, 47, 813eqtri 2845 . . . . . 6 ((203 · 0) + 0) = 00
834, 3, 41, 3, 42, 48, 25, 3, 3, 79, 82decma2c 12139 . . . . 5 ((203 · 400) + (140 + 20)) = 81360
8431mulid1i 10633 . . . . . . 7 (2 · 1) = 2
8553mul02i 10817 . . . . . . 7 (0 · 1) = 0
8614, 18, 3, 33, 84, 85decmul1 12150 . . . . . 6 (20 · 1) = 20
8767mulid1i 10633 . . . . . . . 8 (3 · 1) = 3
8887oveq1i 7155 . . . . . . 7 ((3 · 1) + 1) = (3 + 1)
89 3p1e4 11770 . . . . . . 7 (3 + 1) = 4
9088, 89eqtri 2841 . . . . . 6 ((3 · 1) + 1) = 4
9123, 19, 14, 52, 14, 86, 90decrmanc 12143 . . . . 5 ((203 · 1) + 1) = 204
925, 14, 16, 14, 1, 39, 25, 2, 23, 83, 91decma2c 12139 . . . 4 ((203 · 𝑁) + 1401) = 813604
93 eqid 2818 . . . . 5 902 = 902
94 8nn0 11908 . . . . . . 7 8 ∈ ℕ0
9514, 94deccl 12101 . . . . . 6 18 ∈ ℕ0
9695, 3deccl 12101 . . . . 5 180 ∈ ℕ0
97 eqid 2818 . . . . . 6 90 = 90
98 eqid 2818 . . . . . 6 180 = 180
9995nn0cni 11897 . . . . . . . 8 18 ∈ ℂ
10099addid1i 10815 . . . . . . 7 (18 + 0) = 18
101 1p2e3 11768 . . . . . . . . 9 (1 + 2) = 3
102101, 19eqeltri 2906 . . . . . . . 8 (1 + 2) ∈ ℕ0
103 9t9e81 12215 . . . . . . . 8 (9 · 9) = 81
104 9cn 11725 . . . . . . . . . . 11 9 ∈ ℂ
105104mul02i 10817 . . . . . . . . . 10 (0 · 9) = 0
106105, 101oveq12i 7157 . . . . . . . . 9 ((0 · 9) + (1 + 2)) = (0 + 3)
10767addid2i 10816 . . . . . . . . 9 (0 + 3) = 3
108106, 107eqtri 2841 . . . . . . . 8 ((0 · 9) + (1 + 2)) = 3
10910, 3, 102, 97, 10, 103, 108decrmanc 12143 . . . . . . 7 ((90 · 9) + (1 + 2)) = 813
110 9t2e18 12208 . . . . . . . . 9 (9 · 2) = 18
111104, 31, 110mulcomli 10638 . . . . . . . 8 (2 · 9) = 18
112 1p1e2 11750 . . . . . . . 8 (1 + 1) = 2
113 8p8e16 12172 . . . . . . . 8 (8 + 8) = 16
11414, 94, 94, 111, 112, 40, 113decaddci 12147 . . . . . . 7 ((2 · 9) + 8) = 26
11527, 18, 14, 94, 93, 100, 10, 40, 18, 109, 114decmac 12138 . . . . . 6 ((902 · 9) + (18 + 0)) = 8136
11628nn0cni 11897 . . . . . . . . 9 902 ∈ ℂ
117116mul01i 10818 . . . . . . . 8 (902 · 0) = 0
118117oveq1i 7155 . . . . . . 7 ((902 · 0) + 0) = (0 + 0)
119118, 47, 813eqtri 2845 . . . . . 6 ((902 · 0) + 0) = 00
12010, 3, 95, 3, 97, 98, 28, 3, 3, 115, 119decma2c 12139 . . . . 5 ((902 · 90) + 180) = 81360
12118, 10, 3, 97, 110, 35decmul1 12150 . . . . . 6 (90 · 2) = 180
12218, 27, 18, 93, 121, 34decmul1 12150 . . . . 5 (902 · 2) = 1804
12328, 27, 18, 93, 2, 96, 120, 122decmul2c 12152 . . . 4 (902 · 902) = 813604
12492, 123eqtr4i 2844 . . 3 ((203 · 𝑁) + 1401) = (902 · 902)
1258, 9, 24, 26, 28, 17, 29, 38, 124mod2xi 16393 . 2 ((2↑400) mod 𝑁) = (1401 mod 𝑁)
1265nn0cni 11897 . . 3 400 ∈ ℂ
12718, 2, 3, 49, 60, 35decmul1 12150 . . . 4 (40 · 2) = 80
12818, 4, 3, 42, 127, 35decmul1 12150 . . 3 (400 · 2) = 800
129126, 31, 128mulcomli 10638 . 2 (2 · 400) = 800
130 eqid 2818 . . . 4 2311 = 2311
13118, 94deccl 12101 . . . . 5 28 ∈ ℕ0
132 eqid 2818 . . . . . 6 231 = 231
133 eqid 2818 . . . . . 6 49 = 49
134 7nn0 11907 . . . . . . 7 7 ∈ ℕ0
135 7p1e8 11774 . . . . . . 7 (7 + 1) = 8
136 eqid 2818 . . . . . . . 8 23 = 23
137 4p3e7 11779 . . . . . . . . 9 (4 + 3) = 7
13859, 67, 137addcomli 10820 . . . . . . . 8 (3 + 4) = 7
13918, 19, 2, 136, 138decaddi 12146 . . . . . . 7 (23 + 4) = 27
14018, 134, 135, 139decsuc 12117 . . . . . 6 ((23 + 4) + 1) = 28
141 9p1e10 12088 . . . . . . 7 (9 + 1) = 10
142104, 53, 141addcomli 10820 . . . . . 6 (1 + 9) = 10
14320, 14, 2, 10, 132, 133, 140, 142decaddc2 12142 . . . . 5 (231 + 49) = 280
144131nn0cni 11897 . . . . . . 7 28 ∈ ℂ
145144addid1i 10815 . . . . . 6 (28 + 0) = 28
14631addid1i 10815 . . . . . . . 8 (2 + 0) = 2
147146, 18eqeltri 2906 . . . . . . 7 (2 + 0) ∈ ℕ0
148 eqid 2818 . . . . . . 7 490 = 490
149 4t4e16 12185 . . . . . . . . 9 (4 · 4) = 16
150 6p3e9 11785 . . . . . . . . 9 (6 + 3) = 9
15114, 40, 19, 149, 150decaddi 12146 . . . . . . . 8 ((4 · 4) + 3) = 19
152 9t4e36 12210 . . . . . . . 8 (9 · 4) = 36
1532, 2, 10, 133, 40, 19, 151, 152decmul1c 12151 . . . . . . 7 (49 · 4) = 196
15462, 146oveq12i 7157 . . . . . . . 8 ((0 · 4) + (2 + 0)) = (0 + 2)
15531addid2i 10816 . . . . . . . 8 (0 + 2) = 2
156154, 155eqtri 2841 . . . . . . 7 ((0 · 4) + (2 + 0)) = 2
15711, 3, 147, 148, 2, 153, 156decrmanc 12143 . . . . . 6 ((490 · 4) + (2 + 0)) = 1962
15812nn0cni 11897 . . . . . . . . 9 490 ∈ ℂ
159158mul01i 10818 . . . . . . . 8 (490 · 0) = 0
160159oveq1i 7155 . . . . . . 7 ((490 · 0) + 8) = (0 + 8)
161 8cn 11722 . . . . . . . 8 8 ∈ ℂ
162161addid2i 10816 . . . . . . 7 (0 + 8) = 8
16394dec0h 12108 . . . . . . 7 8 = 08
164160, 162, 1633eqtri 2845 . . . . . 6 ((490 · 0) + 8) = 08
1652, 3, 18, 94, 49, 145, 12, 94, 3, 157, 164decma2c 12139 . . . . 5 ((490 · 40) + (28 + 0)) = 19628
166159oveq1i 7155 . . . . . 6 ((490 · 0) + 0) = (0 + 0)
167166, 47, 813eqtri 2845 . . . . 5 ((490 · 0) + 0) = 00
1684, 3, 131, 3, 42, 143, 12, 3, 3, 165, 167decma2c 12139 . . . 4 ((490 · 400) + (231 + 49)) = 196280
16959mulid1i 10633 . . . . . 6 (4 · 1) = 4
170104mulid1i 10633 . . . . . 6 (9 · 1) = 9
17114, 2, 10, 133, 169, 170decmul1 12150 . . . . 5 (49 · 1) = 49
17285oveq1i 7155 . . . . . 6 ((0 · 1) + 1) = (0 + 1)
173172, 57eqtri 2841 . . . . 5 ((0 · 1) + 1) = 1
17411, 3, 14, 148, 14, 171, 173decrmanc 12143 . . . 4 ((490 · 1) + 1) = 491
1755, 14, 21, 14, 1, 130, 12, 14, 11, 168, 174decma2c 12139 . . 3 ((490 · 𝑁) + 2311) = 1962801
17615nn0cni 11897 . . . . . . 7 14 ∈ ℂ
177176addid1i 10815 . . . . . 6 (14 + 0) = 14
178 5nn0 11905 . . . . . . . 8 5 ∈ ℕ0
179178, 40deccl 12101 . . . . . . 7 56 ∈ ℕ0
180179, 3deccl 12101 . . . . . 6 560 ∈ ℕ0
181 eqid 2818 . . . . . . . 8 560 = 560
182179nn0cni 11897 . . . . . . . . 9 56 ∈ ℂ
183182addid2i 10816 . . . . . . . 8 (0 + 56) = 56
1843, 14, 179, 3, 55, 181, 183, 54decadd 12140 . . . . . . 7 (1 + 560) = 561
185182addid1i 10815 . . . . . . . 8 (56 + 0) = 56
186 5cn 11713 . . . . . . . . . . 11 5 ∈ ℂ
187186addid1i 10815 . . . . . . . . . 10 (5 + 0) = 5
188187, 178eqeltri 2906 . . . . . . . . 9 (5 + 0) ∈ ℕ0
18953mulid1i 10633 . . . . . . . . 9 (1 · 1) = 1
190169, 187oveq12i 7157 . . . . . . . . . 10 ((4 · 1) + (5 + 0)) = (4 + 5)
191 5p4e9 11783 . . . . . . . . . . 11 (5 + 4) = 9
192186, 59, 191addcomli 10820 . . . . . . . . . 10 (4 + 5) = 9
193190, 192eqtri 2841 . . . . . . . . 9 ((4 · 1) + (5 + 0)) = 9
19414, 2, 188, 44, 14, 189, 193decrmanc 12143 . . . . . . . 8 ((14 · 1) + (5 + 0)) = 19
19585oveq1i 7155 . . . . . . . . 9 ((0 · 1) + 6) = (0 + 6)
196195, 76, 773eqtri 2845 . . . . . . . 8 ((0 · 1) + 6) = 06
19715, 3, 178, 40, 43, 185, 14, 40, 3, 194, 196decmac 12138 . . . . . . 7 ((140 · 1) + (56 + 0)) = 196
198189oveq1i 7155 . . . . . . . 8 ((1 · 1) + 1) = (1 + 1)
19918dec0h 12108 . . . . . . . 8 2 = 02
200198, 112, 1993eqtri 2845 . . . . . . 7 ((1 · 1) + 1) = 02
20116, 14, 179, 14, 39, 184, 14, 18, 3, 197, 200decmac 12138 . . . . . 6 ((1401 · 1) + (1 + 560)) = 1962
20259mulid2i 10634 . . . . . . . . . . . 12 (1 · 4) = 4
203202oveq1i 7155 . . . . . . . . . . 11 ((1 · 4) + 1) = (4 + 1)
204 4p1e5 11771 . . . . . . . . . . 11 (4 + 1) = 5
205203, 204eqtri 2841 . . . . . . . . . 10 ((1 · 4) + 1) = 5
2062, 14, 2, 44, 40, 14, 205, 149decmul1c 12151 . . . . . . . . 9 (14 · 4) = 56
20775addid1i 10815 . . . . . . . . 9 (6 + 0) = 6
208178, 40, 3, 206, 207decaddi 12146 . . . . . . . 8 ((14 · 4) + 0) = 56
209 0cn 10621 . . . . . . . . 9 0 ∈ ℂ
21059mul01i 10818 . . . . . . . . . 10 (4 · 0) = 0
211210, 81eqtri 2841 . . . . . . . . 9 (4 · 0) = 00
21259, 209, 211mulcomli 10638 . . . . . . . 8 (0 · 4) = 00
2132, 15, 3, 43, 3, 3, 208, 212decmul1c 12151 . . . . . . 7 (140 · 4) = 560
214202oveq1i 7155 . . . . . . . 8 ((1 · 4) + 4) = (4 + 4)
215 4p4e8 11780 . . . . . . . 8 (4 + 4) = 8
216214, 215eqtri 2841 . . . . . . 7 ((1 · 4) + 4) = 8
21716, 14, 2, 39, 2, 213, 216decrmanc 12143 . . . . . 6 ((1401 · 4) + 4) = 5608
21814, 2, 14, 2, 44, 177, 17, 94, 180, 201, 217decma2c 12139 . . . . 5 ((1401 · 14) + (14 + 0)) = 19628
21917nn0cni 11897 . . . . . . . 8 1401 ∈ ℂ
220219mul01i 10818 . . . . . . 7 (1401 · 0) = 0
221220oveq1i 7155 . . . . . 6 ((1401 · 0) + 0) = (0 + 0)
222221, 47, 813eqtri 2845 . . . . 5 ((1401 · 0) + 0) = 00
22315, 3, 15, 3, 43, 43, 17, 3, 3, 218, 222decma2c 12139 . . . 4 ((1401 · 140) + 140) = 196280
224219mulid1i 10633 . . . 4 (1401 · 1) = 1401
22517, 16, 14, 39, 14, 16, 223, 224decmul2c 12152 . . 3 (1401 · 1401) = 1962801
226175, 225eqtr4i 2844 . 2 ((490 · 𝑁) + 2311) = (1401 · 1401)
2278, 9, 5, 13, 17, 22, 125, 129, 226mod2xi 16393 1 ((2↑800) mod 𝑁) = (2311 mod 𝑁)
Colors of variables: wff setvar class
Syntax hints:   = wceq 1528  (class class class)co 7145  0cc0 10525  1c1 10526   + caddc 10528   · cmul 10530  cn 11626  2c2 11680  3c3 11681  4c4 11682  5c5 11683  6c6 11684  7c7 11685  8c8 11686  9c9 11687  0cn0 11885  cdc 12086   mod cmo 13225  cexp 13417
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-2nd 7679  df-wrecs 7936  df-recs 7997  df-rdg 8035  df-er 8278  df-en 8498  df-dom 8499  df-sdom 8500  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-fl 13150  df-mod 13226  df-seq 13358  df-exp 13418
This theorem is referenced by:  4001lem3  16464  4001lem4  16465
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