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Theorem 4001lem2 15773
Description: Lemma for 4001prm 15776. 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 11255 . . . . . 6 4 ∈ ℕ0
3 0nn0 11251 . . . . . 6 0 ∈ ℕ0
42, 3deccl 11456 . . . . 5 40 ∈ ℕ0
54, 3deccl 11456 . . . 4 400 ∈ ℕ0
6 1nn 10975 . . . 4 1 ∈ ℕ
75, 6decnncl 11462 . . 3 4001 ∈ ℕ
81, 7eqeltri 2694 . 2 𝑁 ∈ ℕ
9 2nn 11129 . 2 2 ∈ ℕ
10 9nn0 11260 . . . . 5 9 ∈ ℕ0
112, 10deccl 11456 . . . 4 49 ∈ ℕ0
1211, 3deccl 11456 . . 3 490 ∈ ℕ0
1312nn0zi 11346 . 2 490 ∈ ℤ
14 1nn0 11252 . . . . 5 1 ∈ ℕ0
1514, 2deccl 11456 . . . 4 14 ∈ ℕ0
1615, 3deccl 11456 . . 3 140 ∈ ℕ0
1716, 14deccl 11456 . 2 1401 ∈ ℕ0
18 2nn0 11253 . . . . 5 2 ∈ ℕ0
19 3nn0 11254 . . . . 5 3 ∈ ℕ0
2018, 19deccl 11456 . . . 4 23 ∈ ℕ0
2120, 14deccl 11456 . . 3 231 ∈ ℕ0
2221, 14deccl 11456 . 2 2311 ∈ ℕ0
2318, 3deccl 11456 . . . 4 20 ∈ ℕ0
2423, 3deccl 11456 . . 3 200 ∈ ℕ0
2523, 19deccl 11456 . . . 4 203 ∈ ℕ0
2625nn0zi 11346 . . 3 203 ∈ ℤ
2710, 3deccl 11456 . . . 4 90 ∈ ℕ0
2827, 18deccl 11456 . . 3 902 ∈ ℕ0
2914001lem1 15772 . . 3 ((2↑200) mod 𝑁) = (902 mod 𝑁)
3024nn0cni 11248 . . . 4 200 ∈ ℂ
31 2cn 11035 . . . 4 2 ∈ ℂ
32 eqid 2621 . . . . 5 200 = 200
33 eqid 2621 . . . . . 6 20 = 20
34 2t2e4 11121 . . . . . 6 (2 · 2) = 4
3531mul02i 10169 . . . . . 6 (0 · 2) = 0
3618, 18, 3, 33, 3, 34, 35decmul1 11529 . . . . 5 (20 · 2) = 40
3718, 23, 3, 32, 3, 36, 35decmul1 11529 . . . 4 (200 · 2) = 400
3830, 31, 37mulcomli 9991 . . 3 (2 · 200) = 400
39 eqid 2621 . . . . 5 1401 = 1401
40 6nn0 11257 . . . . . . 7 6 ∈ ℕ0
4114, 40deccl 11456 . . . . . 6 16 ∈ ℕ0
42 eqid 2621 . . . . . 6 400 = 400
43 eqid 2621 . . . . . . 7 140 = 140
44 eqid 2621 . . . . . . . 8 14 = 14
45 4p2e6 11106 . . . . . . . 8 (4 + 2) = 6
4614, 2, 18, 44, 45decaddi 11523 . . . . . . 7 (14 + 2) = 16
47 00id 10155 . . . . . . 7 (0 + 0) = 0
4815, 3, 18, 3, 43, 33, 46, 47decadd 11514 . . . . . 6 (140 + 20) = 160
49 eqid 2621 . . . . . . 7 40 = 40
5041nn0cni 11248 . . . . . . . 8 16 ∈ ℂ
5150addid1i 10167 . . . . . . 7 (16 + 0) = 16
52 eqid 2621 . . . . . . . 8 203 = 203
53 ax-1cn 9938 . . . . . . . . . 10 1 ∈ ℂ
5453addid1i 10167 . . . . . . . . 9 (1 + 0) = 1
5514dec0h 11466 . . . . . . . . 9 1 = 01
5654, 55eqtri 2643 . . . . . . . 8 (1 + 0) = 01
5753addid2i 10168 . . . . . . . . . 10 (0 + 1) = 1
5857, 14eqeltri 2694 . . . . . . . . 9 (0 + 1) ∈ ℕ0
59 4cn 11042 . . . . . . . . . 10 4 ∈ ℂ
60 4t2e8 11125 . . . . . . . . . 10 (4 · 2) = 8
6159, 31, 60mulcomli 9991 . . . . . . . . 9 (2 · 4) = 8
6259mul02i 10169 . . . . . . . . . . 11 (0 · 4) = 0
6362, 57oveq12i 6616 . . . . . . . . . 10 ((0 · 4) + (0 + 1)) = (0 + 1)
6463, 57eqtri 2643 . . . . . . . . 9 ((0 · 4) + (0 + 1)) = 1
6518, 3, 58, 33, 2, 61, 64decrmanc 11520 . . . . . . . 8 ((20 · 4) + (0 + 1)) = 81
66 2p1e3 11095 . . . . . . . . 9 (2 + 1) = 3
67 3cn 11039 . . . . . . . . . 10 3 ∈ ℂ
68 4t3e12 11576 . . . . . . . . . 10 (4 · 3) = 12
6959, 67, 68mulcomli 9991 . . . . . . . . 9 (3 · 4) = 12
7014, 18, 66, 69decsuc 11479 . . . . . . . 8 ((3 · 4) + 1) = 13
7123, 19, 3, 14, 52, 56, 2, 19, 14, 65, 70decmac 11510 . . . . . . 7 ((203 · 4) + (1 + 0)) = 813
7225nn0cni 11248 . . . . . . . . . 10 203 ∈ ℂ
7372mul01i 10170 . . . . . . . . 9 (203 · 0) = 0
7473oveq1i 6614 . . . . . . . 8 ((203 · 0) + 6) = (0 + 6)
75 6cn 11046 . . . . . . . . 9 6 ∈ ℂ
7675addid2i 10168 . . . . . . . 8 (0 + 6) = 6
7740dec0h 11466 . . . . . . . 8 6 = 06
7874, 76, 773eqtri 2647 . . . . . . 7 ((203 · 0) + 6) = 06
792, 3, 14, 40, 49, 51, 25, 40, 3, 71, 78decma2c 11512 . . . . . 6 ((203 · 40) + (16 + 0)) = 8136
8073oveq1i 6614 . . . . . . 7 ((203 · 0) + 0) = (0 + 0)
813dec0h 11466 . . . . . . 7 0 = 00
8280, 47, 813eqtri 2647 . . . . . 6 ((203 · 0) + 0) = 00
834, 3, 41, 3, 42, 48, 25, 3, 3, 79, 82decma2c 11512 . . . . 5 ((203 · 400) + (140 + 20)) = 81360
8431mulid1i 9986 . . . . . . 7 (2 · 1) = 2
8553mul02i 10169 . . . . . . 7 (0 · 1) = 0
8614, 18, 3, 33, 3, 84, 85decmul1 11529 . . . . . 6 (20 · 1) = 20
8767mulid1i 9986 . . . . . . . 8 (3 · 1) = 3
8887oveq1i 6614 . . . . . . 7 ((3 · 1) + 1) = (3 + 1)
89 3p1e4 11097 . . . . . . 7 (3 + 1) = 4
9088, 89eqtri 2643 . . . . . 6 ((3 · 1) + 1) = 4
9123, 19, 14, 52, 14, 86, 90decrmanc 11520 . . . . 5 ((203 · 1) + 1) = 204
925, 14, 16, 14, 1, 39, 25, 2, 23, 83, 91decma2c 11512 . . . 4 ((203 · 𝑁) + 1401) = 813604
93 eqid 2621 . . . . 5 902 = 902
94 8nn0 11259 . . . . . . 7 8 ∈ ℕ0
9514, 94deccl 11456 . . . . . 6 18 ∈ ℕ0
9695, 3deccl 11456 . . . . 5 180 ∈ ℕ0
97 eqid 2621 . . . . . 6 90 = 90
98 eqid 2621 . . . . . 6 180 = 180
9995nn0cni 11248 . . . . . . . 8 18 ∈ ℂ
10099addid1i 10167 . . . . . . 7 (18 + 0) = 18
101 1p2e3 11096 . . . . . . . . 9 (1 + 2) = 3
102101, 19eqeltri 2694 . . . . . . . 8 (1 + 2) ∈ ℕ0
103 9t9e81 11614 . . . . . . . 8 (9 · 9) = 81
104 9cn 11052 . . . . . . . . . . 11 9 ∈ ℂ
105104mul02i 10169 . . . . . . . . . 10 (0 · 9) = 0
106105, 101oveq12i 6616 . . . . . . . . 9 ((0 · 9) + (1 + 2)) = (0 + 3)
10767addid2i 10168 . . . . . . . . 9 (0 + 3) = 3
108106, 107eqtri 2643 . . . . . . . 8 ((0 · 9) + (1 + 2)) = 3
10910, 3, 102, 97, 10, 103, 108decrmanc 11520 . . . . . . 7 ((90 · 9) + (1 + 2)) = 813
110 9t2e18 11607 . . . . . . . . 9 (9 · 2) = 18
111104, 31, 110mulcomli 9991 . . . . . . . 8 (2 · 9) = 18
112 1p1e2 11078 . . . . . . . 8 (1 + 1) = 2
113 8p8e16 11562 . . . . . . . 8 (8 + 8) = 16
11414, 94, 94, 111, 112, 40, 113decaddci 11524 . . . . . . 7 ((2 · 9) + 8) = 26
11527, 18, 14, 94, 93, 100, 10, 40, 18, 109, 114decmac 11510 . . . . . 6 ((902 · 9) + (18 + 0)) = 8136
11628nn0cni 11248 . . . . . . . . 9 902 ∈ ℂ
117116mul01i 10170 . . . . . . . 8 (902 · 0) = 0
118117oveq1i 6614 . . . . . . 7 ((902 · 0) + 0) = (0 + 0)
119118, 47, 813eqtri 2647 . . . . . 6 ((902 · 0) + 0) = 00
12010, 3, 95, 3, 97, 98, 28, 3, 3, 115, 119decma2c 11512 . . . . 5 ((902 · 90) + 180) = 81360
12118, 10, 3, 97, 3, 110, 35decmul1 11529 . . . . . 6 (90 · 2) = 180
12218, 27, 18, 93, 2, 121, 34decmul1 11529 . . . . 5 (902 · 2) = 1804
12328, 27, 18, 93, 2, 96, 120, 122decmul2c 11533 . . . 4 (902 · 902) = 813604
12492, 123eqtr4i 2646 . . 3 ((203 · 𝑁) + 1401) = (902 · 902)
1258, 9, 24, 26, 28, 17, 29, 38, 124mod2xi 15697 . 2 ((2↑400) mod 𝑁) = (1401 mod 𝑁)
1265nn0cni 11248 . . 3 400 ∈ ℂ
12718, 2, 3, 49, 3, 60, 35decmul1 11529 . . . 4 (40 · 2) = 80
12818, 4, 3, 42, 3, 127, 35decmul1 11529 . . 3 (400 · 2) = 800
129126, 31, 128mulcomli 9991 . 2 (2 · 400) = 800
130 eqid 2621 . . . 4 2311 = 2311
13118, 94deccl 11456 . . . . 5 28 ∈ ℕ0
132 eqid 2621 . . . . . 6 231 = 231
133 eqid 2621 . . . . . 6 49 = 49
134 7nn0 11258 . . . . . . 7 7 ∈ ℕ0
135 7p1e8 11101 . . . . . . 7 (7 + 1) = 8
136 eqid 2621 . . . . . . . 8 23 = 23
137 4p3e7 11107 . . . . . . . . 9 (4 + 3) = 7
13859, 67, 137addcomli 10172 . . . . . . . 8 (3 + 4) = 7
13918, 19, 2, 136, 138decaddi 11523 . . . . . . 7 (23 + 4) = 27
14018, 134, 135, 139decsuc 11479 . . . . . 6 ((23 + 4) + 1) = 28
141 9p1e10 11440 . . . . . . 7 (9 + 1) = 10
142104, 53, 141addcomli 10172 . . . . . 6 (1 + 9) = 10
14320, 14, 2, 10, 132, 133, 140, 142decaddc2 11519 . . . . 5 (231 + 49) = 280
144131nn0cni 11248 . . . . . . 7 28 ∈ ℂ
145144addid1i 10167 . . . . . 6 (28 + 0) = 28
14631addid1i 10167 . . . . . . . 8 (2 + 0) = 2
147146, 18eqeltri 2694 . . . . . . 7 (2 + 0) ∈ ℕ0
148 eqid 2621 . . . . . . 7 490 = 490
149 4t4e16 11577 . . . . . . . . 9 (4 · 4) = 16
150 6p3e9 11114 . . . . . . . . 9 (6 + 3) = 9
15114, 40, 19, 149, 150decaddi 11523 . . . . . . . 8 ((4 · 4) + 3) = 19
152 9t4e36 11609 . . . . . . . 8 (9 · 4) = 36
1532, 2, 10, 133, 40, 19, 151, 152decmul1c 11531 . . . . . . 7 (49 · 4) = 196
15462, 146oveq12i 6616 . . . . . . . 8 ((0 · 4) + (2 + 0)) = (0 + 2)
15531addid2i 10168 . . . . . . . 8 (0 + 2) = 2
156154, 155eqtri 2643 . . . . . . 7 ((0 · 4) + (2 + 0)) = 2
15711, 3, 147, 148, 2, 153, 156decrmanc 11520 . . . . . 6 ((490 · 4) + (2 + 0)) = 1962
15812nn0cni 11248 . . . . . . . . 9 490 ∈ ℂ
159158mul01i 10170 . . . . . . . 8 (490 · 0) = 0
160159oveq1i 6614 . . . . . . 7 ((490 · 0) + 8) = (0 + 8)
161 8cn 11050 . . . . . . . 8 8 ∈ ℂ
162161addid2i 10168 . . . . . . 7 (0 + 8) = 8
16394dec0h 11466 . . . . . . 7 8 = 08
164160, 162, 1633eqtri 2647 . . . . . 6 ((490 · 0) + 8) = 08
1652, 3, 18, 94, 49, 145, 12, 94, 3, 157, 164decma2c 11512 . . . . 5 ((490 · 40) + (28 + 0)) = 19628
166159oveq1i 6614 . . . . . 6 ((490 · 0) + 0) = (0 + 0)
167166, 47, 813eqtri 2647 . . . . 5 ((490 · 0) + 0) = 00
1684, 3, 131, 3, 42, 143, 12, 3, 3, 165, 167decma2c 11512 . . . 4 ((490 · 400) + (231 + 49)) = 196280
16959mulid1i 9986 . . . . . 6 (4 · 1) = 4
170104mulid1i 9986 . . . . . 6 (9 · 1) = 9
17114, 2, 10, 133, 10, 169, 170decmul1 11529 . . . . 5 (49 · 1) = 49
17285oveq1i 6614 . . . . . 6 ((0 · 1) + 1) = (0 + 1)
173172, 57eqtri 2643 . . . . 5 ((0 · 1) + 1) = 1
17411, 3, 14, 148, 14, 171, 173decrmanc 11520 . . . 4 ((490 · 1) + 1) = 491
1755, 14, 21, 14, 1, 130, 12, 14, 11, 168, 174decma2c 11512 . . 3 ((490 · 𝑁) + 2311) = 1962801
17615nn0cni 11248 . . . . . . 7 14 ∈ ℂ
177176addid1i 10167 . . . . . 6 (14 + 0) = 14
178 5nn0 11256 . . . . . . . 8 5 ∈ ℕ0
179178, 40deccl 11456 . . . . . . 7 56 ∈ ℕ0
180179, 3deccl 11456 . . . . . 6 560 ∈ ℕ0
181 eqid 2621 . . . . . . . 8 560 = 560
182179nn0cni 11248 . . . . . . . . 9 56 ∈ ℂ
183182addid2i 10168 . . . . . . . 8 (0 + 56) = 56
1843, 14, 179, 3, 55, 181, 183, 54decadd 11514 . . . . . . 7 (1 + 560) = 561
185182addid1i 10167 . . . . . . . 8 (56 + 0) = 56
186 5cn 11044 . . . . . . . . . . 11 5 ∈ ℂ
187186addid1i 10167 . . . . . . . . . 10 (5 + 0) = 5
188187, 178eqeltri 2694 . . . . . . . . 9 (5 + 0) ∈ ℕ0
18953mulid1i 9986 . . . . . . . . 9 (1 · 1) = 1
190169, 187oveq12i 6616 . . . . . . . . . 10 ((4 · 1) + (5 + 0)) = (4 + 5)
191 5p4e9 11111 . . . . . . . . . . 11 (5 + 4) = 9
192186, 59, 191addcomli 10172 . . . . . . . . . 10 (4 + 5) = 9
193190, 192eqtri 2643 . . . . . . . . 9 ((4 · 1) + (5 + 0)) = 9
19414, 2, 188, 44, 14, 189, 193decrmanc 11520 . . . . . . . 8 ((14 · 1) + (5 + 0)) = 19
19585oveq1i 6614 . . . . . . . . 9 ((0 · 1) + 6) = (0 + 6)
196195, 76, 773eqtri 2647 . . . . . . . 8 ((0 · 1) + 6) = 06
19715, 3, 178, 40, 43, 185, 14, 40, 3, 194, 196decmac 11510 . . . . . . 7 ((140 · 1) + (56 + 0)) = 196
198189oveq1i 6614 . . . . . . . 8 ((1 · 1) + 1) = (1 + 1)
19918dec0h 11466 . . . . . . . 8 2 = 02
200198, 112, 1993eqtri 2647 . . . . . . 7 ((1 · 1) + 1) = 02
20116, 14, 179, 14, 39, 184, 14, 18, 3, 197, 200decmac 11510 . . . . . 6 ((1401 · 1) + (1 + 560)) = 1962
20259mulid2i 9987 . . . . . . . . . . . 12 (1 · 4) = 4
203202oveq1i 6614 . . . . . . . . . . 11 ((1 · 4) + 1) = (4 + 1)
204 4p1e5 11098 . . . . . . . . . . 11 (4 + 1) = 5
205203, 204eqtri 2643 . . . . . . . . . 10 ((1 · 4) + 1) = 5
2062, 14, 2, 44, 40, 14, 205, 149decmul1c 11531 . . . . . . . . 9 (14 · 4) = 56
20775addid1i 10167 . . . . . . . . 9 (6 + 0) = 6
208178, 40, 3, 206, 207decaddi 11523 . . . . . . . 8 ((14 · 4) + 0) = 56
209 0cn 9976 . . . . . . . . 9 0 ∈ ℂ
21059mul01i 10170 . . . . . . . . . 10 (4 · 0) = 0
211210, 81eqtri 2643 . . . . . . . . 9 (4 · 0) = 00
21259, 209, 211mulcomli 9991 . . . . . . . 8 (0 · 4) = 00
2132, 15, 3, 43, 3, 3, 208, 212decmul1c 11531 . . . . . . 7 (140 · 4) = 560
214202oveq1i 6614 . . . . . . . 8 ((1 · 4) + 4) = (4 + 4)
215 4p4e8 11108 . . . . . . . 8 (4 + 4) = 8
216214, 215eqtri 2643 . . . . . . 7 ((1 · 4) + 4) = 8
21716, 14, 2, 39, 2, 213, 216decrmanc 11520 . . . . . 6 ((1401 · 4) + 4) = 5608
21814, 2, 14, 2, 44, 177, 17, 94, 180, 201, 217decma2c 11512 . . . . 5 ((1401 · 14) + (14 + 0)) = 19628
21917nn0cni 11248 . . . . . . . 8 1401 ∈ ℂ
220219mul01i 10170 . . . . . . 7 (1401 · 0) = 0
221220oveq1i 6614 . . . . . 6 ((1401 · 0) + 0) = (0 + 0)
222221, 47, 813eqtri 2647 . . . . 5 ((1401 · 0) + 0) = 00
22315, 3, 15, 3, 43, 43, 17, 3, 3, 218, 222decma2c 11512 . . . 4 ((1401 · 140) + 140) = 196280
224219mulid1i 9986 . . . 4 (1401 · 1) = 1401
22517, 16, 14, 39, 14, 16, 223, 224decmul2c 11533 . . 3 (1401 · 1401) = 1962801
226175, 225eqtr4i 2646 . 2 ((490 · 𝑁) + 2311) = (1401 · 1401)
2278, 9, 5, 13, 17, 22, 125, 129, 226mod2xi 15697 1 ((2↑800) mod 𝑁) = (2311 mod 𝑁)
Colors of variables: wff setvar class
Syntax hints:   = wceq 1480  (class class class)co 6604  0cc0 9880  1c1 9881   + caddc 9883   · cmul 9885  cn 10964  2c2 11014  3c3 11015  4c4 11016  5c5 11017  6c6 11018  7c7 11019  8c8 11020  9c9 11021  0cn0 11236  cdc 11437   mod cmo 12608  cexp 12800
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 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902  ax-cnex 9936  ax-resscn 9937  ax-1cn 9938  ax-icn 9939  ax-addcl 9940  ax-addrcl 9941  ax-mulcl 9942  ax-mulrcl 9943  ax-mulcom 9944  ax-addass 9945  ax-mulass 9946  ax-distr 9947  ax-i2m1 9948  ax-1ne0 9949  ax-1rid 9950  ax-rnegex 9951  ax-rrecex 9952  ax-cnre 9953  ax-pre-lttri 9954  ax-pre-lttrn 9955  ax-pre-ltadd 9956  ax-pre-mulgt0 9957  ax-pre-sup 9958
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 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-pss 3571  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-tp 4153  df-op 4155  df-uni 4403  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-tr 4713  df-eprel 4985  df-id 4989  df-po 4995  df-so 4996  df-fr 5033  df-we 5035  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-pred 5639  df-ord 5685  df-on 5686  df-lim 5687  df-suc 5688  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-riota 6565  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-om 7013  df-2nd 7114  df-wrecs 7352  df-recs 7413  df-rdg 7451  df-er 7687  df-en 7900  df-dom 7901  df-sdom 7902  df-sup 8292  df-inf 8293  df-pnf 10020  df-mnf 10021  df-xr 10022  df-ltxr 10023  df-le 10024  df-sub 10212  df-neg 10213  df-div 10629  df-nn 10965  df-2 11023  df-3 11024  df-4 11025  df-5 11026  df-6 11027  df-7 11028  df-8 11029  df-9 11030  df-n0 11237  df-z 11322  df-dec 11438  df-uz 11632  df-rp 11777  df-fl 12533  df-mod 12609  df-seq 12742  df-exp 12801
This theorem is referenced by:  4001lem3  15774  4001lem4  15775
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