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Theorem 1259lem3 14140
Description: Lemma for 1259prm 14143. 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 10583 . . . . . 6  |-  1  e.  NN0
3 2nn0 10584 . . . . . 6  |-  2  e.  NN0
42, 3deccl 10757 . . . . 5  |- ; 1 2  e.  NN0
5 5nn0 10587 . . . . 5  |-  5  e.  NN0
64, 5deccl 10757 . . . 4  |- ;; 1 2 5  e.  NN0
7 9nn 10474 . . . 4  |-  9  e.  NN
86, 7decnncl 10756 . . 3  |- ;;; 1 2 5 9  e.  NN
91, 8eqeltri 2503 . 2  |-  N  e.  NN
10 2nn 10467 . 2  |-  2  e.  NN
11 3nn0 10585 . . 3  |-  3  e.  NN0
12 8nn0 10590 . . 3  |-  8  e.  NN0
1311, 12deccl 10757 . 2  |- ; 3 8  e.  NN0
14 4nn 10469 . . 3  |-  4  e.  NN
1514nnzi 10658 . 2  |-  4  e.  ZZ
16 7nn0 10589 . . 3  |-  7  e.  NN0
1716, 2deccl 10757 . 2  |- ; 7 1  e.  NN0
18 4nn0 10586 . . . 4  |-  4  e.  NN0
1911, 18deccl 10757 . . 3  |- ; 3 4  e.  NN0
202, 2deccl 10757 . . . 4  |- ; 1 1  e.  NN0
2120nn0zi 10659 . . 3  |- ; 1 1  e.  ZZ
2212, 16deccl 10757 . . . 4  |- ; 8 7  e.  NN0
23 0nn0 10582 . . . 4  |-  0  e.  NN0
2422, 23deccl 10757 . . 3  |- ;; 8 7 0  e.  NN0
25 6nn0 10588 . . . 4  |-  6  e.  NN0
262, 25deccl 10757 . . 3  |- ; 1 6  e.  NN0
2711259lem2 14139 . . 3  |-  ( ( 2 ^; 3 4 )  mod 
N )  =  (;; 8 7 0  mod 
N )
28 2exp4 14097 . . . 4  |-  ( 2 ^ 4 )  = ; 1
6
2928oveq1i 6090 . . 3  |-  ( ( 2 ^ 4 )  mod  N )  =  (; 1 6  mod  N
)
30 eqid 2433 . . . 4  |- ; 3 4  = ; 3 4
31 4p4e8 10446 . . . 4  |-  ( 4  +  4 )  =  8
3211, 18, 18, 30, 31decaddi 10787 . . 3  |-  (; 3 4  +  4 )  = ; 3 8
33 9nn0 10591 . . . . 5  |-  9  e.  NN0
34 eqid 2433 . . . . 5  |- ; 7 1  = ; 7 1
35 10nn0 10592 . . . . 5  |-  10  e.  NN0
36 eqid 2433 . . . . . 6  |- ;; 1 2 5  = ;; 1 2 5
3716dec0h 10759 . . . . . . 7  |-  7  = ; 0 7
38 dec10 10773 . . . . . . 7  |-  10  = ; 1 0
39 0p1e1 10421 . . . . . . 7  |-  ( 0  +  1 )  =  1
40 7cn 10393 . . . . . . . 8  |-  7  e.  CC
4140addid1i 9544 . . . . . . 7  |-  ( 7  +  0 )  =  7
4223, 16, 2, 23, 37, 38, 39, 41decadd 10784 . . . . . 6  |-  ( 7  +  10 )  = ; 1
7
43 eqid 2433 . . . . . . 7  |- ; 1 2  = ; 1 2
44 6cn 10391 . . . . . . . . 9  |-  6  e.  CC
45 ax-1cn 9328 . . . . . . . . 9  |-  1  e.  CC
46 6p1e7 10438 . . . . . . . . 9  |-  ( 6  +  1 )  =  7
4744, 45, 46addcomli 9549 . . . . . . . 8  |-  ( 1  +  6 )  =  7
4847, 37eqtri 2453 . . . . . . 7  |-  ( 1  +  6 )  = ; 0
7
49 eqid 2433 . . . . . . . 8  |- ; 1 1  = ; 1 1
50 2cn 10380 . . . . . . . . . 10  |-  2  e.  CC
5150addid2i 9545 . . . . . . . . 9  |-  ( 0  +  2 )  =  2
523dec0h 10759 . . . . . . . . 9  |-  2  = ; 0 2
5351, 52eqtri 2453 . . . . . . . 8  |-  ( 0  +  2 )  = ; 0
2
5445mulid1i 9376 . . . . . . . . . 10  |-  ( 1  x.  1 )  =  1
55 00id 9532 . . . . . . . . . 10  |-  ( 0  +  0 )  =  0
5654, 55oveq12i 6092 . . . . . . . . 9  |-  ( ( 1  x.  1 )  +  ( 0  +  0 ) )  =  ( 1  +  0 )
5745addid1i 9544 . . . . . . . . 9  |-  ( 1  +  0 )  =  1
5856, 57eqtri 2453 . . . . . . . 8  |-  ( ( 1  x.  1 )  +  ( 0  +  0 ) )  =  1
5954oveq1i 6090 . . . . . . . . 9  |-  ( ( 1  x.  1 )  +  2 )  =  ( 1  +  2 )
60 1p2e3 10434 . . . . . . . . 9  |-  ( 1  +  2 )  =  3
6111dec0h 10759 . . . . . . . . 9  |-  3  = ; 0 3
6259, 60, 613eqtri 2457 . . . . . . . 8  |-  ( ( 1  x.  1 )  +  2 )  = ; 0
3
632, 2, 23, 3, 49, 53, 2, 11, 23, 58, 62decmac 10782 . . . . . . 7  |-  ( (; 1
1  x.  1 )  +  ( 0  +  2 ) )  = ; 1
3
6450mulid2i 9377 . . . . . . . . . 10  |-  ( 1  x.  2 )  =  2
6564, 55oveq12i 6092 . . . . . . . . 9  |-  ( ( 1  x.  2 )  +  ( 0  +  0 ) )  =  ( 2  +  0 )
6650addid1i 9544 . . . . . . . . 9  |-  ( 2  +  0 )  =  2
6765, 66eqtri 2453 . . . . . . . 8  |-  ( ( 1  x.  2 )  +  ( 0  +  0 ) )  =  2
6864oveq1i 6090 . . . . . . . . 9  |-  ( ( 1  x.  2 )  +  7 )  =  ( 2  +  7 )
69 7p2e9 10454 . . . . . . . . . 10  |-  ( 7  +  2 )  =  9
7040, 50, 69addcomli 9549 . . . . . . . . 9  |-  ( 2  +  7 )  =  9
7133dec0h 10759 . . . . . . . . 9  |-  9  = ; 0 9
7268, 70, 713eqtri 2457 . . . . . . . 8  |-  ( ( 1  x.  2 )  +  7 )  = ; 0
9
732, 2, 23, 16, 49, 37, 3, 33, 23, 67, 72decmac 10782 . . . . . . 7  |-  ( (; 1
1  x.  2 )  +  7 )  = ; 2
9
742, 3, 23, 16, 43, 48, 20, 33, 3, 63, 73decma2c 10783 . . . . . 6  |-  ( (; 1
1  x. ; 1 2 )  +  ( 1  +  6 ) )  = ;; 1 3 9
75 5cn 10389 . . . . . . . . . 10  |-  5  e.  CC
7675mulid2i 9377 . . . . . . . . 9  |-  ( 1  x.  5 )  =  5
7776, 39oveq12i 6092 . . . . . . . 8  |-  ( ( 1  x.  5 )  +  ( 0  +  1 ) )  =  ( 5  +  1 )
78 5p1e6 10437 . . . . . . . 8  |-  ( 5  +  1 )  =  6
7977, 78eqtri 2453 . . . . . . 7  |-  ( ( 1  x.  5 )  +  ( 0  +  1 ) )  =  6
8076oveq1i 6090 . . . . . . . 8  |-  ( ( 1  x.  5 )  +  7 )  =  ( 5  +  7 )
81 7p5e12 10796 . . . . . . . . 9  |-  ( 7  +  5 )  = ; 1
2
8240, 75, 81addcomli 9549 . . . . . . . 8  |-  ( 5  +  7 )  = ; 1
2
8380, 82eqtri 2453 . . . . . . 7  |-  ( ( 1  x.  5 )  +  7 )  = ; 1
2
842, 2, 23, 16, 49, 37, 5, 3, 2, 79, 83decmac 10782 . . . . . 6  |-  ( (; 1
1  x.  5 )  +  7 )  = ; 6
2
854, 5, 2, 16, 36, 42, 20, 3, 25, 74, 84decma2c 10783 . . . . 5  |-  ( (; 1
1  x. ;; 1 2 5 )  +  ( 7  +  10 ) )  = ;;; 1 3 9 2
862dec0h 10759 . . . . . 6  |-  1  = ; 0 1
87 9cn 10397 . . . . . . . . 9  |-  9  e.  CC
8887mulid2i 9377 . . . . . . . 8  |-  ( 1  x.  9 )  =  9
8988, 39oveq12i 6092 . . . . . . 7  |-  ( ( 1  x.  9 )  +  ( 0  +  1 ) )  =  ( 9  +  1 )
90 9p1e10 10441 . . . . . . 7  |-  ( 9  +  1 )  =  10
9189, 90eqtri 2453 . . . . . 6  |-  ( ( 1  x.  9 )  +  ( 0  +  1 ) )  =  10
9288oveq1i 6090 . . . . . . 7  |-  ( ( 1  x.  9 )  +  1 )  =  ( 9  +  1 )
9392, 90, 383eqtri 2457 . . . . . 6  |-  ( ( 1  x.  9 )  +  1 )  = ; 1
0
942, 2, 23, 2, 49, 86, 33, 23, 2, 91, 93decmac 10782 . . . . 5  |-  ( (; 1
1  x.  9 )  +  1 )  = ; 10 0
956, 33, 16, 2, 1, 34, 20, 23, 35, 85, 94decma2c 10783 . . . 4  |-  ( (; 1
1  x.  N )  + ; 7 1 )  = ;;;; 1 3 9 2 0
96 eqid 2433 . . . . 5  |- ; 1 6  = ; 1 6
975, 3deccl 10757 . . . . . 6  |- ; 5 2  e.  NN0
9897, 3deccl 10757 . . . . 5  |- ;; 5 2 2  e.  NN0
99 eqid 2433 . . . . . 6  |- ;; 8 7 0  = ;; 8 7 0
100 eqid 2433 . . . . . 6  |- ;; 5 2 2  = ;; 5 2 2
101 eqid 2433 . . . . . . 7  |- ; 8 7  = ; 8 7
10297nn0cni 10579 . . . . . . . 8  |- ; 5 2  e.  CC
103102addid1i 9544 . . . . . . 7  |-  (; 5 2  +  0 )  = ; 5 2
104 8cn 10395 . . . . . . . . . 10  |-  8  e.  CC
105104mulid1i 9376 . . . . . . . . 9  |-  ( 8  x.  1 )  =  8
10675addid1i 9544 . . . . . . . . 9  |-  ( 5  +  0 )  =  5
107105, 106oveq12i 6092 . . . . . . . 8  |-  ( ( 8  x.  1 )  +  ( 5  +  0 ) )  =  ( 8  +  5 )
108 8p5e13 10801 . . . . . . . 8  |-  ( 8  +  5 )  = ; 1
3
109107, 108eqtri 2453 . . . . . . 7  |-  ( ( 8  x.  1 )  +  ( 5  +  0 ) )  = ; 1
3
11040mulid1i 9376 . . . . . . . . 9  |-  ( 7  x.  1 )  =  7
111110oveq1i 6090 . . . . . . . 8  |-  ( ( 7  x.  1 )  +  2 )  =  ( 7  +  2 )
112111, 69, 713eqtri 2457 . . . . . . 7  |-  ( ( 7  x.  1 )  +  2 )  = ; 0
9
11312, 16, 5, 3, 101, 103, 2, 33, 23, 109, 112decmac 10782 . . . . . 6  |-  ( (; 8
7  x.  1 )  +  (; 5 2  +  0 ) )  = ;; 1 3 9
11445mul02i 9546 . . . . . . . 8  |-  ( 0  x.  1 )  =  0
115114oveq1i 6090 . . . . . . 7  |-  ( ( 0  x.  1 )  +  2 )  =  ( 0  +  2 )
116115, 51, 523eqtri 2457 . . . . . 6  |-  ( ( 0  x.  1 )  +  2 )  = ; 0
2
11722, 23, 97, 3, 99, 100, 2, 3, 23, 113, 116decmac 10782 . . . . 5  |-  ( (;; 8 7 0  x.  1 )  + ;; 5 2 2 )  = ;;; 1 3 9 2
118 8t6e48 10835 . . . . . . . . . 10  |-  ( 8  x.  6 )  = ; 4
8
119 4p1e5 10436 . . . . . . . . . 10  |-  ( 4  +  1 )  =  5
120 8p4e12 10800 . . . . . . . . . 10  |-  ( 8  +  4 )  = ; 1
2
12118, 12, 18, 118, 119, 3, 120decaddci 10788 . . . . . . . . 9  |-  ( ( 8  x.  6 )  +  4 )  = ; 5
2
122 7t6e42 10829 . . . . . . . . 9  |-  ( 7  x.  6 )  = ; 4
2
12325, 12, 16, 101, 3, 18, 121, 122decmul1c 10790 . . . . . . . 8  |-  (; 8 7  x.  6 )  = ;; 5 2 2
124123oveq1i 6090 . . . . . . 7  |-  ( (; 8
7  x.  6 )  +  0 )  =  (;; 5 2 2  +  0 )
12598nn0cni 10579 . . . . . . . 8  |- ;; 5 2 2  e.  CC
126125addid1i 9544 . . . . . . 7  |-  (;; 5 2 2  +  0 )  = ;; 5 2 2
127124, 126eqtri 2453 . . . . . 6  |-  ( (; 8
7  x.  6 )  +  0 )  = ;; 5 2 2
12844mul02i 9546 . . . . . . 7  |-  ( 0  x.  6 )  =  0
12923dec0h 10759 . . . . . . 7  |-  0  = ; 0 0
130128, 129eqtri 2453 . . . . . 6  |-  ( 0  x.  6 )  = ; 0
0
13125, 22, 23, 99, 23, 23, 127, 130decmul1c 10790 . . . . 5  |-  (;; 8 7 0  x.  6 )  = ;;; 5 2 2 0
13224, 2, 25, 96, 23, 98, 117, 131decmul2c 10791 . . . 4  |-  (;; 8 7 0  x. ; 1 6 )  = ;;;; 1 3 9 2 0
13395, 132eqtr4i 2456 . . 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 14080 . 2  |-  ( ( 2 ^; 3 8 )  mod 
N )  =  (; 7
1  mod  N )
135 eqid 2433 . . 3  |- ; 3 8  = ; 3 8
136 3cn 10384 . . . . . 6  |-  3  e.  CC
137 3t2e6 10461 . . . . . 6  |-  ( 3  x.  2 )  =  6
138136, 50, 137mulcomli 9381 . . . . 5  |-  ( 2  x.  3 )  =  6
139138oveq1i 6090 . . . 4  |-  ( ( 2  x.  3 )  +  1 )  =  ( 6  +  1 )
140139, 46eqtri 2453 . . 3  |-  ( ( 2  x.  3 )  +  1 )  =  7
141 8t2e16 10831 . . . 4  |-  ( 8  x.  2 )  = ; 1
6
142104, 50, 141mulcomli 9381 . . 3  |-  ( 2  x.  8 )  = ; 1
6
1433, 11, 12, 135, 25, 2, 140, 142decmul2c 10791 . 2  |-  ( 2  x. ; 3 8 )  = ; 7
6
1445dec0h 10759 . . . 4  |-  5  = ; 0 5
145 4cn 10387 . . . . . . 7  |-  4  e.  CC
146145addid2i 9545 . . . . . 6  |-  ( 0  +  4 )  =  4
14718dec0h 10759 . . . . . 6  |-  4  = ; 0 4
148146, 147eqtri 2453 . . . . 5  |-  ( 0  +  4 )  = ; 0
4
149145mulid1i 9376 . . . . . . . 8  |-  ( 4  x.  1 )  =  4
150149, 39oveq12i 6092 . . . . . . 7  |-  ( ( 4  x.  1 )  +  ( 0  +  1 ) )  =  ( 4  +  1 )
151150, 119eqtri 2453 . . . . . 6  |-  ( ( 4  x.  1 )  +  ( 0  +  1 ) )  =  5
152 4t2e8 10463 . . . . . . . 8  |-  ( 4  x.  2 )  =  8
153152oveq1i 6090 . . . . . . 7  |-  ( ( 4  x.  2 )  +  2 )  =  ( 8  +  2 )
154 8p2e10 10456 . . . . . . 7  |-  ( 8  +  2 )  =  10
155153, 154, 383eqtri 2457 . . . . . 6  |-  ( ( 4  x.  2 )  +  2 )  = ; 1
0
1562, 3, 23, 3, 43, 53, 18, 23, 2, 151, 155decma2c 10783 . . . . 5  |-  ( ( 4  x. ; 1 2 )  +  ( 0  +  2 ) )  = ; 5 0
157 5t4e20 10818 . . . . . . 7  |-  ( 5  x.  4 )  = ; 2
0
15875, 145, 157mulcomli 9381 . . . . . 6  |-  ( 4  x.  5 )  = ; 2
0
1593, 23, 18, 158, 146decaddi 10787 . . . . 5  |-  ( ( 4  x.  5 )  +  4 )  = ; 2
4
1604, 5, 23, 18, 36, 148, 18, 18, 3, 156, 159decma2c 10783 . . . 4  |-  ( ( 4  x. ;; 1 2 5 )  +  ( 0  +  4 ) )  = ;; 5 0 4
161 9t4e36 10840 . . . . . 6  |-  ( 9  x.  4 )  = ; 3
6
16287, 145, 161mulcomli 9381 . . . . 5  |-  ( 4  x.  9 )  = ; 3
6
163 3p1e4 10435 . . . . 5  |-  ( 3  +  1 )  =  4
164 6p5e11 10793 . . . . 5  |-  ( 6  +  5 )  = ; 1
1
16511, 25, 5, 162, 163, 2, 164decaddci 10788 . . . 4  |-  ( ( 4  x.  9 )  +  5 )  = ; 4
1
1666, 33, 23, 5, 1, 144, 18, 2, 18, 160, 165decma2c 10783 . . 3  |-  ( ( 4  x.  N )  +  5 )  = ;;; 5 0 4 1
16739oveq2i 6091 . . . . . 6  |-  ( ( 7  x.  7 )  +  ( 0  +  1 ) )  =  ( ( 7  x.  7 )  +  1 )
168 7t7e49 10830 . . . . . . 7  |-  ( 7  x.  7 )  = ; 4
9
16918, 119, 168decsucc 10770 . . . . . 6  |-  ( ( 7  x.  7 )  +  1 )  = ; 5
0
170167, 169eqtri 2453 . . . . 5  |-  ( ( 7  x.  7 )  +  ( 0  +  1 ) )  = ; 5
0
17140mulid2i 9377 . . . . . . 7  |-  ( 1  x.  7 )  =  7
172171oveq1i 6090 . . . . . 6  |-  ( ( 1  x.  7 )  +  7 )  =  ( 7  +  7 )
173 7p7e14 10798 . . . . . 6  |-  ( 7  +  7 )  = ; 1
4
174172, 173eqtri 2453 . . . . 5  |-  ( ( 1  x.  7 )  +  7 )  = ; 1
4
17516, 2, 23, 16, 34, 37, 16, 18, 2, 170, 174decmac 10782 . . . 4  |-  ( (; 7
1  x.  7 )  +  7 )  = ;; 5 0 4
17617nn0cni 10579 . . . . 5  |- ; 7 1  e.  CC
177176mulid1i 9376 . . . 4  |-  (; 7 1  x.  1 )  = ; 7 1
17817, 16, 2, 34, 2, 16, 175, 177decmul2c 10791 . . 3  |-  (; 7 1  x. ; 7 1 )  = ;;; 5 0 4 1
179166, 178eqtr4i 2456 . 2  |-  ( ( 4  x.  N )  +  5 )  =  (; 7 1  x. ; 7 1 )
1809, 10, 13, 15, 17, 5, 134, 143, 179mod2xi 14081 1  |-  ( ( 2 ^; 7 6 )  mod 
N )  =  ( 5  mod  N )
Colors of variables: wff setvar class
Syntax hints:    = wceq 1362  (class class class)co 6080   0cc0 9270   1c1 9271    + caddc 9273    x. cmul 9275   NNcn 10310   2c2 10359   3c3 10360   4c4 10361   5c5 10362   6c6 10363   7c7 10364   8c8 10365   9c9 10366   10c10 10367  ;cdc 10743    mod cmo 11692   ^cexp 11849
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1594  ax-4 1605  ax-5 1669  ax-6 1707  ax-7 1727  ax-8 1757  ax-9 1759  ax-10 1774  ax-11 1779  ax-12 1791  ax-13 1942  ax-ext 2414  ax-sep 4401  ax-nul 4409  ax-pow 4458  ax-pr 4519  ax-un 6361  ax-cnex 9326  ax-resscn 9327  ax-1cn 9328  ax-icn 9329  ax-addcl 9330  ax-addrcl 9331  ax-mulcl 9332  ax-mulrcl 9333  ax-mulcom 9334  ax-addass 9335  ax-mulass 9336  ax-distr 9337  ax-i2m1 9338  ax-1ne0 9339  ax-1rid 9340  ax-rnegex 9341  ax-rrecex 9342  ax-cnre 9343  ax-pre-lttri 9344  ax-pre-lttrn 9345  ax-pre-ltadd 9346  ax-pre-mulgt0 9347  ax-pre-sup 9348
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 959  df-3an 960  df-tru 1365  df-ex 1590  df-nf 1593  df-sb 1700  df-eu 2258  df-mo 2259  df-clab 2420  df-cleq 2426  df-clel 2429  df-nfc 2558  df-ne 2598  df-nel 2599  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2964  df-sbc 3176  df-csb 3277  df-dif 3319  df-un 3321  df-in 3323  df-ss 3330  df-pss 3332  df-nul 3626  df-if 3780  df-pw 3850  df-sn 3866  df-pr 3868  df-tp 3870  df-op 3872  df-uni 4080  df-iun 4161  df-br 4281  df-opab 4339  df-mpt 4340  df-tr 4374  df-eprel 4619  df-id 4623  df-po 4628  df-so 4629  df-fr 4666  df-we 4668  df-ord 4709  df-on 4710  df-lim 4711  df-suc 4712  df-xp 4833  df-rel 4834  df-cnv 4835  df-co 4836  df-dm 4837  df-rn 4838  df-res 4839  df-ima 4840  df-iota 5369  df-fun 5408  df-fn 5409  df-f 5410  df-f1 5411  df-fo 5412  df-f1o 5413  df-fv 5414  df-riota 6039  df-ov 6083  df-oprab 6084  df-mpt2 6085  df-om 6466  df-2nd 6567  df-recs 6818  df-rdg 6852  df-er 7089  df-en 7299  df-dom 7300  df-sdom 7301  df-sup 7679  df-pnf 9408  df-mnf 9409  df-xr 9410  df-ltxr 9411  df-le 9412  df-sub 9585  df-neg 9586  df-div 9982  df-nn 10311  df-2 10368  df-3 10369  df-4 10370  df-5 10371  df-6 10372  df-7 10373  df-8 10374  df-9 10375  df-10 10376  df-n0 10568  df-z 10635  df-dec 10744  df-uz 10850  df-rp 10980  df-fl 11626  df-mod 11693  df-seq 11791  df-exp 11850
This theorem is referenced by:  1259lem4  14141
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