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Theorem 1259lem3 14261
Description: Lemma for 1259prm 14264. Calculate a power mod. In decimal, we calculate  2 ^ 3 8  =  2 ^ 3 4  x.  2 ^ 4  ==  8
7 0  x.  1 6  =  1 1 N  +  7 1 and  2 ^ 7 6  =  ( 2 ^ 3 4 ) ^ 2  ==  7
1 ^ 2  =  4 N  +  5  ==  5. (Contributed by Mario Carneiro, 22-Feb-2014.) (Revised by Mario Carneiro, 20-Apr-2015.)
Hypothesis
Ref Expression
1259prm.1  |-  N  = ;;; 1 2 5 9
Assertion
Ref Expression
1259lem3  |-  ( ( 2 ^; 7 6 )  mod 
N )  =  ( 5  mod  N )

Proof of Theorem 1259lem3
StepHypRef Expression
1 1259prm.1 . . 3  |-  N  = ;;; 1 2 5 9
2 1nn0 10698 . . . . . 6  |-  1  e.  NN0
3 2nn0 10699 . . . . . 6  |-  2  e.  NN0
42, 3deccl 10872 . . . . 5  |- ; 1 2  e.  NN0
5 5nn0 10702 . . . . 5  |-  5  e.  NN0
64, 5deccl 10872 . . . 4  |- ;; 1 2 5  e.  NN0
7 9nn 10589 . . . 4  |-  9  e.  NN
86, 7decnncl 10871 . . 3  |- ;;; 1 2 5 9  e.  NN
91, 8eqeltri 2535 . 2  |-  N  e.  NN
10 2nn 10582 . 2  |-  2  e.  NN
11 3nn0 10700 . . 3  |-  3  e.  NN0
12 8nn0 10705 . . 3  |-  8  e.  NN0
1311, 12deccl 10872 . 2  |- ; 3 8  e.  NN0
14 4nn 10584 . . 3  |-  4  e.  NN
1514nnzi 10773 . 2  |-  4  e.  ZZ
16 7nn0 10704 . . 3  |-  7  e.  NN0
1716, 2deccl 10872 . 2  |- ; 7 1  e.  NN0
18 4nn0 10701 . . . 4  |-  4  e.  NN0
1911, 18deccl 10872 . . 3  |- ; 3 4  e.  NN0
202, 2deccl 10872 . . . 4  |- ; 1 1  e.  NN0
2120nn0zi 10774 . . 3  |- ; 1 1  e.  ZZ
2212, 16deccl 10872 . . . 4  |- ; 8 7  e.  NN0
23 0nn0 10697 . . . 4  |-  0  e.  NN0
2422, 23deccl 10872 . . 3  |- ;; 8 7 0  e.  NN0
25 6nn0 10703 . . . 4  |-  6  e.  NN0
262, 25deccl 10872 . . 3  |- ; 1 6  e.  NN0
2711259lem2 14260 . . 3  |-  ( ( 2 ^; 3 4 )  mod 
N )  =  (;; 8 7 0  mod 
N )
28 2exp4 14218 . . . 4  |-  ( 2 ^ 4 )  = ; 1
6
2928oveq1i 6202 . . 3  |-  ( ( 2 ^ 4 )  mod  N )  =  (; 1 6  mod  N
)
30 eqid 2451 . . . 4  |- ; 3 4  = ; 3 4
31 4p4e8 10561 . . . 4  |-  ( 4  +  4 )  =  8
3211, 18, 18, 30, 31decaddi 10902 . . 3  |-  (; 3 4  +  4 )  = ; 3 8
33 9nn0 10706 . . . . 5  |-  9  e.  NN0
34 eqid 2451 . . . . 5  |- ; 7 1  = ; 7 1
35 10nn0 10707 . . . . 5  |-  10  e.  NN0
36 eqid 2451 . . . . . 6  |- ;; 1 2 5  = ;; 1 2 5
3716dec0h 10874 . . . . . . 7  |-  7  = ; 0 7
38 dec10 10888 . . . . . . 7  |-  10  = ; 1 0
39 0p1e1 10536 . . . . . . 7  |-  ( 0  +  1 )  =  1
40 7cn 10508 . . . . . . . 8  |-  7  e.  CC
4140addid1i 9659 . . . . . . 7  |-  ( 7  +  0 )  =  7
4223, 16, 2, 23, 37, 38, 39, 41decadd 10899 . . . . . 6  |-  ( 7  +  10 )  = ; 1
7
43 eqid 2451 . . . . . . 7  |- ; 1 2  = ; 1 2
44 6cn 10506 . . . . . . . . 9  |-  6  e.  CC
45 ax-1cn 9443 . . . . . . . . 9  |-  1  e.  CC
46 6p1e7 10553 . . . . . . . . 9  |-  ( 6  +  1 )  =  7
4744, 45, 46addcomli 9664 . . . . . . . 8  |-  ( 1  +  6 )  =  7
4847, 37eqtri 2480 . . . . . . 7  |-  ( 1  +  6 )  = ; 0
7
49 eqid 2451 . . . . . . . 8  |- ; 1 1  = ; 1 1
50 2cn 10495 . . . . . . . . . 10  |-  2  e.  CC
5150addid2i 9660 . . . . . . . . 9  |-  ( 0  +  2 )  =  2
523dec0h 10874 . . . . . . . . 9  |-  2  = ; 0 2
5351, 52eqtri 2480 . . . . . . . 8  |-  ( 0  +  2 )  = ; 0
2
5445mulid1i 9491 . . . . . . . . . 10  |-  ( 1  x.  1 )  =  1
55 00id 9647 . . . . . . . . . 10  |-  ( 0  +  0 )  =  0
5654, 55oveq12i 6204 . . . . . . . . 9  |-  ( ( 1  x.  1 )  +  ( 0  +  0 ) )  =  ( 1  +  0 )
5745addid1i 9659 . . . . . . . . 9  |-  ( 1  +  0 )  =  1
5856, 57eqtri 2480 . . . . . . . 8  |-  ( ( 1  x.  1 )  +  ( 0  +  0 ) )  =  1
5954oveq1i 6202 . . . . . . . . 9  |-  ( ( 1  x.  1 )  +  2 )  =  ( 1  +  2 )
60 1p2e3 10549 . . . . . . . . 9  |-  ( 1  +  2 )  =  3
6111dec0h 10874 . . . . . . . . 9  |-  3  = ; 0 3
6259, 60, 613eqtri 2484 . . . . . . . 8  |-  ( ( 1  x.  1 )  +  2 )  = ; 0
3
632, 2, 23, 3, 49, 53, 2, 11, 23, 58, 62decmac 10897 . . . . . . 7  |-  ( (; 1
1  x.  1 )  +  ( 0  +  2 ) )  = ; 1
3
6450mulid2i 9492 . . . . . . . . . 10  |-  ( 1  x.  2 )  =  2
6564, 55oveq12i 6204 . . . . . . . . 9  |-  ( ( 1  x.  2 )  +  ( 0  +  0 ) )  =  ( 2  +  0 )
6650addid1i 9659 . . . . . . . . 9  |-  ( 2  +  0 )  =  2
6765, 66eqtri 2480 . . . . . . . 8  |-  ( ( 1  x.  2 )  +  ( 0  +  0 ) )  =  2
6864oveq1i 6202 . . . . . . . . 9  |-  ( ( 1  x.  2 )  +  7 )  =  ( 2  +  7 )
69 7p2e9 10569 . . . . . . . . . 10  |-  ( 7  +  2 )  =  9
7040, 50, 69addcomli 9664 . . . . . . . . 9  |-  ( 2  +  7 )  =  9
7133dec0h 10874 . . . . . . . . 9  |-  9  = ; 0 9
7268, 70, 713eqtri 2484 . . . . . . . 8  |-  ( ( 1  x.  2 )  +  7 )  = ; 0
9
732, 2, 23, 16, 49, 37, 3, 33, 23, 67, 72decmac 10897 . . . . . . 7  |-  ( (; 1
1  x.  2 )  +  7 )  = ; 2
9
742, 3, 23, 16, 43, 48, 20, 33, 3, 63, 73decma2c 10898 . . . . . 6  |-  ( (; 1
1  x. ; 1 2 )  +  ( 1  +  6 ) )  = ;; 1 3 9
75 5cn 10504 . . . . . . . . . 10  |-  5  e.  CC
7675mulid2i 9492 . . . . . . . . 9  |-  ( 1  x.  5 )  =  5
7776, 39oveq12i 6204 . . . . . . . 8  |-  ( ( 1  x.  5 )  +  ( 0  +  1 ) )  =  ( 5  +  1 )
78 5p1e6 10552 . . . . . . . 8  |-  ( 5  +  1 )  =  6
7977, 78eqtri 2480 . . . . . . 7  |-  ( ( 1  x.  5 )  +  ( 0  +  1 ) )  =  6
8076oveq1i 6202 . . . . . . . 8  |-  ( ( 1  x.  5 )  +  7 )  =  ( 5  +  7 )
81 7p5e12 10911 . . . . . . . . 9  |-  ( 7  +  5 )  = ; 1
2
8240, 75, 81addcomli 9664 . . . . . . . 8  |-  ( 5  +  7 )  = ; 1
2
8380, 82eqtri 2480 . . . . . . 7  |-  ( ( 1  x.  5 )  +  7 )  = ; 1
2
842, 2, 23, 16, 49, 37, 5, 3, 2, 79, 83decmac 10897 . . . . . 6  |-  ( (; 1
1  x.  5 )  +  7 )  = ; 6
2
854, 5, 2, 16, 36, 42, 20, 3, 25, 74, 84decma2c 10898 . . . . 5  |-  ( (; 1
1  x. ;; 1 2 5 )  +  ( 7  +  10 ) )  = ;;; 1 3 9 2
862dec0h 10874 . . . . . 6  |-  1  = ; 0 1
87 9cn 10512 . . . . . . . . 9  |-  9  e.  CC
8887mulid2i 9492 . . . . . . . 8  |-  ( 1  x.  9 )  =  9
8988, 39oveq12i 6204 . . . . . . 7  |-  ( ( 1  x.  9 )  +  ( 0  +  1 ) )  =  ( 9  +  1 )
90 9p1e10 10556 . . . . . . 7  |-  ( 9  +  1 )  =  10
9189, 90eqtri 2480 . . . . . 6  |-  ( ( 1  x.  9 )  +  ( 0  +  1 ) )  =  10
9288oveq1i 6202 . . . . . . 7  |-  ( ( 1  x.  9 )  +  1 )  =  ( 9  +  1 )
9392, 90, 383eqtri 2484 . . . . . 6  |-  ( ( 1  x.  9 )  +  1 )  = ; 1
0
942, 2, 23, 2, 49, 86, 33, 23, 2, 91, 93decmac 10897 . . . . 5  |-  ( (; 1
1  x.  9 )  +  1 )  = ; 10 0
956, 33, 16, 2, 1, 34, 20, 23, 35, 85, 94decma2c 10898 . . . 4  |-  ( (; 1
1  x.  N )  + ; 7 1 )  = ;;;; 1 3 9 2 0
96 eqid 2451 . . . . 5  |- ; 1 6  = ; 1 6
975, 3deccl 10872 . . . . . 6  |- ; 5 2  e.  NN0
9897, 3deccl 10872 . . . . 5  |- ;; 5 2 2  e.  NN0
99 eqid 2451 . . . . . 6  |- ;; 8 7 0  = ;; 8 7 0
100 eqid 2451 . . . . . 6  |- ;; 5 2 2  = ;; 5 2 2
101 eqid 2451 . . . . . . 7  |- ; 8 7  = ; 8 7
10297nn0cni 10694 . . . . . . . 8  |- ; 5 2  e.  CC
103102addid1i 9659 . . . . . . 7  |-  (; 5 2  +  0 )  = ; 5 2
104 8cn 10510 . . . . . . . . . 10  |-  8  e.  CC
105104mulid1i 9491 . . . . . . . . 9  |-  ( 8  x.  1 )  =  8
10675addid1i 9659 . . . . . . . . 9  |-  ( 5  +  0 )  =  5
107105, 106oveq12i 6204 . . . . . . . 8  |-  ( ( 8  x.  1 )  +  ( 5  +  0 ) )  =  ( 8  +  5 )
108 8p5e13 10916 . . . . . . . 8  |-  ( 8  +  5 )  = ; 1
3
109107, 108eqtri 2480 . . . . . . 7  |-  ( ( 8  x.  1 )  +  ( 5  +  0 ) )  = ; 1
3
11040mulid1i 9491 . . . . . . . . 9  |-  ( 7  x.  1 )  =  7
111110oveq1i 6202 . . . . . . . 8  |-  ( ( 7  x.  1 )  +  2 )  =  ( 7  +  2 )
112111, 69, 713eqtri 2484 . . . . . . 7  |-  ( ( 7  x.  1 )  +  2 )  = ; 0
9
11312, 16, 5, 3, 101, 103, 2, 33, 23, 109, 112decmac 10897 . . . . . 6  |-  ( (; 8
7  x.  1 )  +  (; 5 2  +  0 ) )  = ;; 1 3 9
11445mul02i 9661 . . . . . . . 8  |-  ( 0  x.  1 )  =  0
115114oveq1i 6202 . . . . . . 7  |-  ( ( 0  x.  1 )  +  2 )  =  ( 0  +  2 )
116115, 51, 523eqtri 2484 . . . . . 6  |-  ( ( 0  x.  1 )  +  2 )  = ; 0
2
11722, 23, 97, 3, 99, 100, 2, 3, 23, 113, 116decmac 10897 . . . . 5  |-  ( (;; 8 7 0  x.  1 )  + ;; 5 2 2 )  = ;;; 1 3 9 2
118 8t6e48 10950 . . . . . . . . . 10  |-  ( 8  x.  6 )  = ; 4
8
119 4p1e5 10551 . . . . . . . . . 10  |-  ( 4  +  1 )  =  5
120 8p4e12 10915 . . . . . . . . . 10  |-  ( 8  +  4 )  = ; 1
2
12118, 12, 18, 118, 119, 3, 120decaddci 10903 . . . . . . . . 9  |-  ( ( 8  x.  6 )  +  4 )  = ; 5
2
122 7t6e42 10944 . . . . . . . . 9  |-  ( 7  x.  6 )  = ; 4
2
12325, 12, 16, 101, 3, 18, 121, 122decmul1c 10905 . . . . . . . 8  |-  (; 8 7  x.  6 )  = ;; 5 2 2
124123oveq1i 6202 . . . . . . 7  |-  ( (; 8
7  x.  6 )  +  0 )  =  (;; 5 2 2  +  0 )
12598nn0cni 10694 . . . . . . . 8  |- ;; 5 2 2  e.  CC
126125addid1i 9659 . . . . . . 7  |-  (;; 5 2 2  +  0 )  = ;; 5 2 2
127124, 126eqtri 2480 . . . . . 6  |-  ( (; 8
7  x.  6 )  +  0 )  = ;; 5 2 2
12844mul02i 9661 . . . . . . 7  |-  ( 0  x.  6 )  =  0
12923dec0h 10874 . . . . . . 7  |-  0  = ; 0 0
130128, 129eqtri 2480 . . . . . 6  |-  ( 0  x.  6 )  = ; 0
0
13125, 22, 23, 99, 23, 23, 127, 130decmul1c 10905 . . . . 5  |-  (;; 8 7 0  x.  6 )  = ;;; 5 2 2 0
13224, 2, 25, 96, 23, 98, 117, 131decmul2c 10906 . . . 4  |-  (;; 8 7 0  x. ; 1 6 )  = ;;;; 1 3 9 2 0
13395, 132eqtr4i 2483 . . 3  |-  ( (; 1
1  x.  N )  + ; 7 1 )  =  (;; 8 7 0  x. ; 1 6 )
1349, 10, 19, 21, 24, 17, 18, 26, 27, 29, 32, 133modxai 14201 . 2  |-  ( ( 2 ^; 3 8 )  mod 
N )  =  (; 7
1  mod  N )
135 eqid 2451 . . 3  |- ; 3 8  = ; 3 8
136 3cn 10499 . . . . . 6  |-  3  e.  CC
137 3t2e6 10576 . . . . . 6  |-  ( 3  x.  2 )  =  6
138136, 50, 137mulcomli 9496 . . . . 5  |-  ( 2  x.  3 )  =  6
139138oveq1i 6202 . . . 4  |-  ( ( 2  x.  3 )  +  1 )  =  ( 6  +  1 )
140139, 46eqtri 2480 . . 3  |-  ( ( 2  x.  3 )  +  1 )  =  7
141 8t2e16 10946 . . . 4  |-  ( 8  x.  2 )  = ; 1
6
142104, 50, 141mulcomli 9496 . . 3  |-  ( 2  x.  8 )  = ; 1
6
1433, 11, 12, 135, 25, 2, 140, 142decmul2c 10906 . 2  |-  ( 2  x. ; 3 8 )  = ; 7
6
1445dec0h 10874 . . . 4  |-  5  = ; 0 5
145 4cn 10502 . . . . . . 7  |-  4  e.  CC
146145addid2i 9660 . . . . . 6  |-  ( 0  +  4 )  =  4
14718dec0h 10874 . . . . . 6  |-  4  = ; 0 4
148146, 147eqtri 2480 . . . . 5  |-  ( 0  +  4 )  = ; 0
4
149145mulid1i 9491 . . . . . . . 8  |-  ( 4  x.  1 )  =  4
150149, 39oveq12i 6204 . . . . . . 7  |-  ( ( 4  x.  1 )  +  ( 0  +  1 ) )  =  ( 4  +  1 )
151150, 119eqtri 2480 . . . . . 6  |-  ( ( 4  x.  1 )  +  ( 0  +  1 ) )  =  5
152 4t2e8 10578 . . . . . . . 8  |-  ( 4  x.  2 )  =  8
153152oveq1i 6202 . . . . . . 7  |-  ( ( 4  x.  2 )  +  2 )  =  ( 8  +  2 )
154 8p2e10 10571 . . . . . . 7  |-  ( 8  +  2 )  =  10
155153, 154, 383eqtri 2484 . . . . . 6  |-  ( ( 4  x.  2 )  +  2 )  = ; 1
0
1562, 3, 23, 3, 43, 53, 18, 23, 2, 151, 155decma2c 10898 . . . . 5  |-  ( ( 4  x. ; 1 2 )  +  ( 0  +  2 ) )  = ; 5 0
157 5t4e20 10933 . . . . . . 7  |-  ( 5  x.  4 )  = ; 2
0
15875, 145, 157mulcomli 9496 . . . . . 6  |-  ( 4  x.  5 )  = ; 2
0
1593, 23, 18, 158, 146decaddi 10902 . . . . 5  |-  ( ( 4  x.  5 )  +  4 )  = ; 2
4
1604, 5, 23, 18, 36, 148, 18, 18, 3, 156, 159decma2c 10898 . . . 4  |-  ( ( 4  x. ;; 1 2 5 )  +  ( 0  +  4 ) )  = ;; 5 0 4
161 9t4e36 10955 . . . . . 6  |-  ( 9  x.  4 )  = ; 3
6
16287, 145, 161mulcomli 9496 . . . . 5  |-  ( 4  x.  9 )  = ; 3
6
163 3p1e4 10550 . . . . 5  |-  ( 3  +  1 )  =  4
164 6p5e11 10908 . . . . 5  |-  ( 6  +  5 )  = ; 1
1
16511, 25, 5, 162, 163, 2, 164decaddci 10903 . . . 4  |-  ( ( 4  x.  9 )  +  5 )  = ; 4
1
1666, 33, 23, 5, 1, 144, 18, 2, 18, 160, 165decma2c 10898 . . 3  |-  ( ( 4  x.  N )  +  5 )  = ;;; 5 0 4 1
16739oveq2i 6203 . . . . . 6  |-  ( ( 7  x.  7 )  +  ( 0  +  1 ) )  =  ( ( 7  x.  7 )  +  1 )
168 7t7e49 10945 . . . . . . 7  |-  ( 7  x.  7 )  = ; 4
9
16918, 119, 168decsucc 10885 . . . . . 6  |-  ( ( 7  x.  7 )  +  1 )  = ; 5
0
170167, 169eqtri 2480 . . . . 5  |-  ( ( 7  x.  7 )  +  ( 0  +  1 ) )  = ; 5
0
17140mulid2i 9492 . . . . . . 7  |-  ( 1  x.  7 )  =  7
172171oveq1i 6202 . . . . . 6  |-  ( ( 1  x.  7 )  +  7 )  =  ( 7  +  7 )
173 7p7e14 10913 . . . . . 6  |-  ( 7  +  7 )  = ; 1
4
174172, 173eqtri 2480 . . . . 5  |-  ( ( 1  x.  7 )  +  7 )  = ; 1
4
17516, 2, 23, 16, 34, 37, 16, 18, 2, 170, 174decmac 10897 . . . 4  |-  ( (; 7
1  x.  7 )  +  7 )  = ;; 5 0 4
17617nn0cni 10694 . . . . 5  |- ; 7 1  e.  CC
177176mulid1i 9491 . . . 4  |-  (; 7 1  x.  1 )  = ; 7 1
17817, 16, 2, 34, 2, 16, 175, 177decmul2c 10906 . . 3  |-  (; 7 1  x. ; 7 1 )  = ;;; 5 0 4 1
179166, 178eqtr4i 2483 . 2  |-  ( ( 4  x.  N )  +  5 )  =  (; 7 1  x. ; 7 1 )
1809, 10, 13, 15, 17, 5, 134, 143, 179mod2xi 14202 1  |-  ( ( 2 ^; 7 6 )  mod 
N )  =  ( 5  mod  N )
Colors of variables: wff setvar class
Syntax hints:    = wceq 1370  (class class class)co 6192   0cc0 9385   1c1 9386    + caddc 9388    x. cmul 9390   NNcn 10425   2c2 10474   3c3 10475   4c4 10476   5c5 10477   6c6 10478   7c7 10479   8c8 10480   9c9 10481   10c10 10482  ;cdc 10858    mod cmo 11811   ^cexp 11968
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-sep 4513  ax-nul 4521  ax-pow 4570  ax-pr 4631  ax-un 6474  ax-cnex 9441  ax-resscn 9442  ax-1cn 9443  ax-icn 9444  ax-addcl 9445  ax-addrcl 9446  ax-mulcl 9447  ax-mulrcl 9448  ax-mulcom 9449  ax-addass 9450  ax-mulass 9451  ax-distr 9452  ax-i2m1 9453  ax-1ne0 9454  ax-1rid 9455  ax-rnegex 9456  ax-rrecex 9457  ax-cnre 9458  ax-pre-lttri 9459  ax-pre-lttrn 9460  ax-pre-ltadd 9461  ax-pre-mulgt0 9462  ax-pre-sup 9463
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-nel 2647  df-ral 2800  df-rex 2801  df-reu 2802  df-rmo 2803  df-rab 2804  df-v 3072  df-sbc 3287  df-csb 3389  df-dif 3431  df-un 3433  df-in 3435  df-ss 3442  df-pss 3444  df-nul 3738  df-if 3892  df-pw 3962  df-sn 3978  df-pr 3980  df-tp 3982  df-op 3984  df-uni 4192  df-iun 4273  df-br 4393  df-opab 4451  df-mpt 4452  df-tr 4486  df-eprel 4732  df-id 4736  df-po 4741  df-so 4742  df-fr 4779  df-we 4781  df-ord 4822  df-on 4823  df-lim 4824  df-suc 4825  df-xp 4946  df-rel 4947  df-cnv 4948  df-co 4949  df-dm 4950  df-rn 4951  df-res 4952  df-ima 4953  df-iota 5481  df-fun 5520  df-fn 5521  df-f 5522  df-f1 5523  df-fo 5524  df-f1o 5525  df-fv 5526  df-riota 6153  df-ov 6195  df-oprab 6196  df-mpt2 6197  df-om 6579  df-2nd 6680  df-recs 6934  df-rdg 6968  df-er 7203  df-en 7413  df-dom 7414  df-sdom 7415  df-sup 7794  df-pnf 9523  df-mnf 9524  df-xr 9525  df-ltxr 9526  df-le 9527  df-sub 9700  df-neg 9701  df-div 10097  df-nn 10426  df-2 10483  df-3 10484  df-4 10485  df-5 10486  df-6 10487  df-7 10488  df-8 10489  df-9 10490  df-10 10491  df-n0 10683  df-z 10750  df-dec 10859  df-uz 10965  df-rp 11095  df-fl 11745  df-mod 11812  df-seq 11910  df-exp 11969
This theorem is referenced by:  1259lem4  14262
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