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Theorem 2503lem1 15101
Description: Lemma for 2503prm 15104. Calculate a power mod. In decimal, we calculate  2 ^ 1 8  =  5 1 2 ^ 2  =  1 0 4 N  +  1 8 3 2  ==  1 8 3 2. (Contributed by Mario Carneiro, 3-Mar-2014.) (Revised by Mario Carneiro, 20-Apr-2015.)
Hypothesis
Ref Expression
2503prm.1  |-  N  = ;;; 2 5 0 3
Assertion
Ref Expression
2503lem1  |-  ( ( 2 ^; 1 8 )  mod 
N )  =  (;;; 1 8 3 2  mod 
N )

Proof of Theorem 2503lem1
StepHypRef Expression
1 2503prm.1 . . 3  |-  N  = ;;; 2 5 0 3
2 2nn0 10888 . . . . . 6  |-  2  e.  NN0
3 5nn0 10891 . . . . . 6  |-  5  e.  NN0
42, 3deccl 11067 . . . . 5  |- ; 2 5  e.  NN0
5 0nn0 10886 . . . . 5  |-  0  e.  NN0
64, 5deccl 11067 . . . 4  |- ;; 2 5 0  e.  NN0
7 3nn 10770 . . . 4  |-  3  e.  NN
86, 7decnncl 11066 . . 3  |- ;;; 2 5 0 3  e.  NN
91, 8eqeltri 2507 . 2  |-  N  e.  NN
10 2nn 10769 . 2  |-  2  e.  NN
11 9nn0 10895 . 2  |-  9  e.  NN0
12 10nn0 10896 . . . 4  |-  10  e.  NN0
13 4nn0 10890 . . . 4  |-  4  e.  NN0
1412, 13deccl 11067 . . 3  |- ; 10 4  e.  NN0
1514nn0zi 10964 . 2  |- ; 10 4  e.  ZZ
16 1nn0 10887 . . . 4  |-  1  e.  NN0
173, 16deccl 11067 . . 3  |- ; 5 1  e.  NN0
1817, 2deccl 11067 . 2  |- ;; 5 1 2  e.  NN0
19 8nn0 10894 . . . . 5  |-  8  e.  NN0
2016, 19deccl 11067 . . . 4  |- ; 1 8  e.  NN0
21 3nn0 10889 . . . 4  |-  3  e.  NN0
2220, 21deccl 11067 . . 3  |- ;; 1 8 3  e.  NN0
2322, 2deccl 11067 . 2  |- ;;; 1 8 3 2  e.  NN0
24 8p1e9 10742 . . . 4  |-  ( 8  +  1 )  =  9
25 6nn0 10892 . . . . 5  |-  6  e.  NN0
26 2exp8 15053 . . . . 5  |-  ( 2 ^ 8 )  = ;; 2 5 6
27 eqid 2423 . . . . . 6  |- ; 2 5  = ; 2 5
2816dec0h 11069 . . . . . 6  |-  1  = ; 0 1
29 2t2e4 10761 . . . . . . . 8  |-  ( 2  x.  2 )  =  4
30 ax-1cn 9599 . . . . . . . . 9  |-  1  e.  CC
3130addid2i 9823 . . . . . . . 8  |-  ( 0  +  1 )  =  1
3229, 31oveq12i 6315 . . . . . . 7  |-  ( ( 2  x.  2 )  +  ( 0  +  1 ) )  =  ( 4  +  1 )
33 4p1e5 10738 . . . . . . 7  |-  ( 4  +  1 )  =  5
3432, 33eqtri 2452 . . . . . 6  |-  ( ( 2  x.  2 )  +  ( 0  +  1 ) )  =  5
35 5t2e10 10766 . . . . . . . 8  |-  ( 5  x.  2 )  =  10
36 dec10 11083 . . . . . . . 8  |-  10  = ; 1 0
3735, 36eqtri 2452 . . . . . . 7  |-  ( 5  x.  2 )  = ; 1
0
3816, 5, 31, 37decsuc 11076 . . . . . 6  |-  ( ( 5  x.  2 )  +  1 )  = ; 1
1
392, 3, 5, 16, 27, 28, 2, 16, 16, 34, 38decmac 11092 . . . . 5  |-  ( (; 2
5  x.  2 )  +  1 )  = ; 5
1
40 6t2e12 11130 . . . . 5  |-  ( 6  x.  2 )  = ; 1
2
412, 4, 25, 26, 2, 16, 39, 40decmul1c 11100 . . . 4  |-  ( ( 2 ^ 8 )  x.  2 )  = ;; 5 1 2
422, 19, 24, 41numexpp1 15043 . . 3  |-  ( 2 ^ 9 )  = ;; 5 1 2
4342oveq1i 6313 . 2  |-  ( ( 2 ^ 9 )  mod  N )  =  (;; 5 1 2  mod  N )
44 9cn 10699 . . 3  |-  9  e.  CC
45 2cn 10682 . . 3  |-  2  e.  CC
46 9t2e18 11148 . . 3  |-  ( 9  x.  2 )  = ; 1
8
4744, 45, 46mulcomli 9652 . 2  |-  ( 2  x.  9 )  = ; 1
8
48 eqid 2423 . . . 4  |- ;;; 1 8 3 2  = ;;; 1 8 3 2
4921, 16deccl 11067 . . . 4  |- ; 3 1  e.  NN0
502, 16deccl 11067 . . . . 5  |- ; 2 1  e.  NN0
51 eqid 2423 . . . . 5  |- ;; 2 5 0  = ;; 2 5 0
52 eqid 2423 . . . . . 6  |- ;; 1 8 3  = ;; 1 8 3
53 eqid 2423 . . . . . 6  |- ; 3 1  = ; 3 1
54 eqid 2423 . . . . . . 7  |- ; 1 8  = ; 1 8
55 1p1e2 10725 . . . . . . 7  |-  ( 1  +  1 )  =  2
56 8p3e11 11109 . . . . . . 7  |-  ( 8  +  3 )  = ; 1
1
5716, 19, 21, 54, 55, 16, 56decaddci 11098 . . . . . 6  |-  (; 1 8  +  3 )  = ; 2 1
58 3p1e4 10737 . . . . . 6  |-  ( 3  +  1 )  =  4
5920, 21, 21, 16, 52, 53, 57, 58decadd 11094 . . . . 5  |-  (;; 1 8 3  + ; 3 1 )  = ;; 2 1 4
6050nn0cni 10883 . . . . . . 7  |- ; 2 1  e.  CC
6160addid1i 9822 . . . . . 6  |-  (; 2 1  +  0 )  = ; 2 1
623, 2deccl 11067 . . . . . 6  |- ; 5 2  e.  NN0
63 eqid 2423 . . . . . . 7  |- ; 10 4  = ; 10 4
642dec0h 11069 . . . . . . . 8  |-  2  = ; 0 2
65 eqid 2423 . . . . . . . 8  |- ; 5 2  = ; 5 2
66 5cn 10691 . . . . . . . . 9  |-  5  e.  CC
6766addid2i 9823 . . . . . . . 8  |-  ( 0  +  5 )  =  5
68 2p2e4 10729 . . . . . . . 8  |-  ( 2  +  2 )  =  4
695, 2, 3, 2, 64, 65, 67, 68decadd 11094 . . . . . . 7  |-  ( 2  + ; 5 2 )  = ; 5
4
70 5p1e6 10739 . . . . . . . . 9  |-  ( 5  +  1 )  =  6
7125dec0h 11069 . . . . . . . . 9  |-  6  = ; 0 6
7270, 71eqtri 2452 . . . . . . . 8  |-  ( 5  +  1 )  = ; 0
6
7345mulid2i 9648 . . . . . . . . . 10  |-  ( 1  x.  2 )  =  2
74 00id 9810 . . . . . . . . . 10  |-  ( 0  +  0 )  =  0
7573, 74oveq12i 6315 . . . . . . . . 9  |-  ( ( 1  x.  2 )  +  ( 0  +  0 ) )  =  ( 2  +  0 )
7645addid1i 9822 . . . . . . . . 9  |-  ( 2  +  0 )  =  2
7775, 76eqtri 2452 . . . . . . . 8  |-  ( ( 1  x.  2 )  +  ( 0  +  0 ) )  =  2
7845mul02i 9824 . . . . . . . . . 10  |-  ( 0  x.  2 )  =  0
7978oveq1i 6313 . . . . . . . . 9  |-  ( ( 0  x.  2 )  +  6 )  =  ( 0  +  6 )
80 6cn 10693 . . . . . . . . . 10  |-  6  e.  CC
8180addid2i 9823 . . . . . . . . 9  |-  ( 0  +  6 )  =  6
8279, 81, 713eqtri 2456 . . . . . . . 8  |-  ( ( 0  x.  2 )  +  6 )  = ; 0
6
8316, 5, 5, 25, 36, 72, 2, 25, 5, 77, 82decmac 11092 . . . . . . 7  |-  ( ( 10  x.  2 )  +  ( 5  +  1 ) )  = ; 2
6
84 4t2e8 10765 . . . . . . . . 9  |-  ( 4  x.  2 )  =  8
8584oveq1i 6313 . . . . . . . 8  |-  ( ( 4  x.  2 )  +  4 )  =  ( 8  +  4 )
86 8p4e12 11110 . . . . . . . 8  |-  ( 8  +  4 )  = ; 1
2
8785, 86eqtri 2452 . . . . . . 7  |-  ( ( 4  x.  2 )  +  4 )  = ; 1
2
8812, 13, 3, 13, 63, 69, 2, 2, 16, 83, 87decmac 11092 . . . . . 6  |-  ( (; 10 4  x.  2 )  +  ( 2  + ; 5
2 ) )  = ;; 2 6 2
8945addid2i 9823 . . . . . . . . 9  |-  ( 0  +  2 )  =  2
9089, 64eqtri 2452 . . . . . . . 8  |-  ( 0  +  2 )  = ; 0
2
9166mulid2i 9648 . . . . . . . . . 10  |-  ( 1  x.  5 )  =  5
9291, 74oveq12i 6315 . . . . . . . . 9  |-  ( ( 1  x.  5 )  +  ( 0  +  0 ) )  =  ( 5  +  0 )
9366addid1i 9822 . . . . . . . . 9  |-  ( 5  +  0 )  =  5
9492, 93eqtri 2452 . . . . . . . 8  |-  ( ( 1  x.  5 )  +  ( 0  +  0 ) )  =  5
9566mul02i 9824 . . . . . . . . . 10  |-  ( 0  x.  5 )  =  0
9695oveq1i 6313 . . . . . . . . 9  |-  ( ( 0  x.  5 )  +  2 )  =  ( 0  +  2 )
9796, 89, 643eqtri 2456 . . . . . . . 8  |-  ( ( 0  x.  5 )  +  2 )  = ; 0
2
9816, 5, 5, 2, 36, 90, 3, 2, 5, 94, 97decmac 11092 . . . . . . 7  |-  ( ( 10  x.  5 )  +  ( 0  +  2 ) )  = ; 5
2
99 4cn 10689 . . . . . . . . 9  |-  4  e.  CC
100 5t4e20 11128 . . . . . . . . 9  |-  ( 5  x.  4 )  = ; 2
0
10166, 99, 100mulcomli 9652 . . . . . . . 8  |-  ( 4  x.  5 )  = ; 2
0
1022, 5, 31, 101decsuc 11076 . . . . . . 7  |-  ( ( 4  x.  5 )  +  1 )  = ; 2
1
10312, 13, 5, 16, 63, 28, 3, 16, 2, 98, 102decmac 11092 . . . . . 6  |-  ( (; 10 4  x.  5 )  +  1 )  = ;; 5 2 1
1042, 3, 2, 16, 27, 61, 14, 16, 62, 88, 103decma2c 11093 . . . . 5  |-  ( (; 10 4  x. ; 2 5 )  +  (; 2 1  +  0 ) )  = ;;; 2 6 2 1
10514nn0cni 10883 . . . . . . . 8  |- ; 10 4  e.  CC
106105mul01i 9825 . . . . . . 7  |-  (; 10 4  x.  0 )  =  0
107106oveq1i 6313 . . . . . 6  |-  ( (; 10 4  x.  0 )  +  4 )  =  ( 0  +  4 )
10899addid2i 9823 . . . . . 6  |-  ( 0  +  4 )  =  4
10913dec0h 11069 . . . . . 6  |-  4  = ; 0 4
110107, 108, 1093eqtri 2456 . . . . 5  |-  ( (; 10 4  x.  0 )  +  4 )  = ; 0
4
1114, 5, 50, 13, 51, 59, 14, 13, 5, 104, 110decma2c 11093 . . . 4  |-  ( (; 10 4  x. ;; 2
5 0 )  +  (;; 1 8 3  + ; 3 1 ) )  = ;;;; 2 6 2 1 4
11231, 28eqtri 2452 . . . . . 6  |-  ( 0  +  1 )  = ; 0
1
113 3cn 10686 . . . . . . . . 9  |-  3  e.  CC
114113mulid2i 9648 . . . . . . . 8  |-  ( 1  x.  3 )  =  3
115114, 74oveq12i 6315 . . . . . . 7  |-  ( ( 1  x.  3 )  +  ( 0  +  0 ) )  =  ( 3  +  0 )
116113addid1i 9822 . . . . . . 7  |-  ( 3  +  0 )  =  3
117115, 116eqtri 2452 . . . . . 6  |-  ( ( 1  x.  3 )  +  ( 0  +  0 ) )  =  3
118113mul02i 9824 . . . . . . . 8  |-  ( 0  x.  3 )  =  0
119118oveq1i 6313 . . . . . . 7  |-  ( ( 0  x.  3 )  +  1 )  =  ( 0  +  1 )
120119, 31, 283eqtri 2456 . . . . . 6  |-  ( ( 0  x.  3 )  +  1 )  = ; 0
1
12116, 5, 5, 16, 36, 112, 21, 16, 5, 117, 120decmac 11092 . . . . 5  |-  ( ( 10  x.  3 )  +  ( 0  +  1 ) )  = ; 3
1
122 4t3e12 11125 . . . . . 6  |-  ( 4  x.  3 )  = ; 1
2
12316, 2, 2, 122, 68decaddi 11097 . . . . 5  |-  ( ( 4  x.  3 )  +  2 )  = ; 1
4
12412, 13, 5, 2, 63, 64, 21, 13, 16, 121, 123decmac 11092 . . . 4  |-  ( (; 10 4  x.  3 )  +  2 )  = ;; 3 1 4
1256, 21, 22, 2, 1, 48, 14, 13, 49, 111, 124decma2c 11093 . . 3  |-  ( (; 10 4  x.  N )  + ;;; 1 8 3 2 )  = ;;;;; 2 6 2 1 4 4
126 eqid 2423 . . . 4  |- ;; 5 1 2  = ;; 5 1 2
12712, 2deccl 11067 . . . 4  |- ; 10 2  e.  NN0
128 eqid 2423 . . . . 5  |- ; 5 1  = ; 5 1
129 eqid 2423 . . . . 5  |- ; 10 2  = ; 10 2
13066, 30, 70addcomli 9827 . . . . . . 7  |-  ( 1  +  5 )  =  6
13116, 5, 3, 16, 36, 128, 130, 31decadd 11094 . . . . . 6  |-  ( 10  + ; 5 1 )  = ; 6
1
132 7nn0 10893 . . . . . . 7  |-  7  e.  NN0
133 6p1e7 10740 . . . . . . . 8  |-  ( 6  +  1 )  =  7
134132dec0h 11069 . . . . . . . 8  |-  7  = ; 0 7
135133, 134eqtri 2452 . . . . . . 7  |-  ( 6  +  1 )  = ; 0
7
13631oveq2i 6314 . . . . . . . 8  |-  ( ( 5  x.  5 )  +  ( 0  +  1 ) )  =  ( ( 5  x.  5 )  +  1 )
137 5t5e25 11129 . . . . . . . . 9  |-  ( 5  x.  5 )  = ; 2
5
1382, 3, 70, 137decsuc 11076 . . . . . . . 8  |-  ( ( 5  x.  5 )  +  1 )  = ; 2
6
139136, 138eqtri 2452 . . . . . . 7  |-  ( ( 5  x.  5 )  +  ( 0  +  1 ) )  = ; 2
6
14091oveq1i 6313 . . . . . . . 8  |-  ( ( 1  x.  5 )  +  7 )  =  ( 5  +  7 )
141 7cn 10695 . . . . . . . . 9  |-  7  e.  CC
142 7p5e12 11106 . . . . . . . . 9  |-  ( 7  +  5 )  = ; 1
2
143141, 66, 142addcomli 9827 . . . . . . . 8  |-  ( 5  +  7 )  = ; 1
2
144140, 143eqtri 2452 . . . . . . 7  |-  ( ( 1  x.  5 )  +  7 )  = ; 1
2
1453, 16, 5, 132, 128, 135, 3, 2, 16, 139, 144decmac 11092 . . . . . 6  |-  ( (; 5
1  x.  5 )  +  ( 6  +  1 ) )  = ;; 2 6 2
14666, 45, 35mulcomli 9652 . . . . . . . 8  |-  ( 2  x.  5 )  =  10
147146, 36eqtri 2452 . . . . . . 7  |-  ( 2  x.  5 )  = ; 1
0
14816, 5, 31, 147decsuc 11076 . . . . . 6  |-  ( ( 2  x.  5 )  +  1 )  = ; 1
1
14917, 2, 25, 16, 126, 131, 3, 16, 16, 145, 148decmac 11092 . . . . 5  |-  ( (;; 5 1 2  x.  5 )  +  ( 10  + ; 5 1 ) )  = ;;; 2 6 2 1
1505dec0h 11069 . . . . . . . 8  |-  0  = ; 0 0
15174, 150eqtri 2452 . . . . . . 7  |-  ( 0  +  0 )  = ; 0
0
15266mulid1i 9647 . . . . . . . . 9  |-  ( 5  x.  1 )  =  5
153152, 74oveq12i 6315 . . . . . . . 8  |-  ( ( 5  x.  1 )  +  ( 0  +  0 ) )  =  ( 5  +  0 )
154153, 93eqtri 2452 . . . . . . 7  |-  ( ( 5  x.  1 )  +  ( 0  +  0 ) )  =  5
15530mulid1i 9647 . . . . . . . . 9  |-  ( 1  x.  1 )  =  1
156155oveq1i 6313 . . . . . . . 8  |-  ( ( 1  x.  1 )  +  0 )  =  ( 1  +  0 )
15730addid1i 9822 . . . . . . . 8  |-  ( 1  +  0 )  =  1
158156, 157, 283eqtri 2456 . . . . . . 7  |-  ( ( 1  x.  1 )  +  0 )  = ; 0
1
1593, 16, 5, 5, 128, 151, 16, 16, 5, 154, 158decmac 11092 . . . . . 6  |-  ( (; 5
1  x.  1 )  +  ( 0  +  0 ) )  = ; 5
1
16045mulid1i 9647 . . . . . . . 8  |-  ( 2  x.  1 )  =  2
161160oveq1i 6313 . . . . . . 7  |-  ( ( 2  x.  1 )  +  2 )  =  ( 2  +  2 )
162161, 68, 1093eqtri 2456 . . . . . 6  |-  ( ( 2  x.  1 )  +  2 )  = ; 0
4
16317, 2, 5, 2, 126, 64, 16, 13, 5, 159, 162decmac 11092 . . . . 5  |-  ( (;; 5 1 2  x.  1 )  +  2 )  = ;; 5 1 4
1643, 16, 12, 2, 128, 129, 18, 13, 17, 149, 163decma2c 11093 . . . 4  |-  ( (;; 5 1 2  x. ; 5
1 )  + ; 10 2 )  = ;;;; 2 6 2 1 4
16535oveq1i 6313 . . . . . . . . 9  |-  ( ( 5  x.  2 )  +  0 )  =  ( 10  +  0 )
166 10nn 10777 . . . . . . . . . . 11  |-  10  e.  NN
167166nncni 10621 . . . . . . . . . 10  |-  10  e.  CC
168167addid1i 9822 . . . . . . . . 9  |-  ( 10  +  0 )  =  10
169165, 168eqtri 2452 . . . . . . . 8  |-  ( ( 5  x.  2 )  +  0 )  =  10
17073, 64eqtri 2452 . . . . . . . 8  |-  ( 1  x.  2 )  = ; 0
2
1712, 3, 16, 128, 2, 5, 169, 170decmul1c 11100 . . . . . . 7  |-  (; 5 1  x.  2 )  = ; 10 2
172171oveq1i 6313 . . . . . 6  |-  ( (; 5
1  x.  2 )  +  0 )  =  (; 10 2  +  0
)
173127nn0cni 10883 . . . . . . 7  |- ; 10 2  e.  CC
174173addid1i 9822 . . . . . 6  |-  (; 10 2  +  0 )  = ; 10 2
175172, 174eqtri 2452 . . . . 5  |-  ( (; 5
1  x.  2 )  +  0 )  = ; 10 2
17629, 109eqtri 2452 . . . . 5  |-  ( 2  x.  2 )  = ; 0
4
1772, 17, 2, 126, 13, 5, 175, 176decmul1c 11100 . . . 4  |-  (;; 5 1 2  x.  2 )  = ;; 10 2 4
17818, 17, 2, 126, 13, 127, 164, 177decmul2c 11101 . . 3  |-  (;; 5 1 2  x. ;; 5 1 2 )  = ;;;;; 2 6 2 1 4 4
179125, 178eqtr4i 2455 . 2  |-  ( (; 10 4  x.  N )  + ;;; 1 8 3 2 )  =  (;; 5 1 2  x. ;; 5 1 2 )
1809, 10, 11, 15, 18, 23, 43, 47, 179mod2xi 15034 1  |-  ( ( 2 ^; 1 8 )  mod 
N )  =  (;;; 1 8 3 2  mod 
N )
Colors of variables: wff setvar class
Syntax hints:    = wceq 1438  (class class class)co 6303   0cc0 9541   1c1 9542    + caddc 9544    x. cmul 9546   NNcn 10611   2c2 10661   3c3 10662   4c4 10663   5c5 10664   6c6 10665   7c7 10666   8c8 10667   9c9 10668   10c10 10669  ;cdc 11053    mod cmo 12097   ^cexp 12273
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1666  ax-4 1679  ax-5 1749  ax-6 1795  ax-7 1840  ax-8 1871  ax-9 1873  ax-10 1888  ax-11 1893  ax-12 1906  ax-13 2054  ax-ext 2401  ax-sep 4544  ax-nul 4553  ax-pow 4600  ax-pr 4658  ax-un 6595  ax-cnex 9597  ax-resscn 9598  ax-1cn 9599  ax-icn 9600  ax-addcl 9601  ax-addrcl 9602  ax-mulcl 9603  ax-mulrcl 9604  ax-mulcom 9605  ax-addass 9606  ax-mulass 9607  ax-distr 9608  ax-i2m1 9609  ax-1ne0 9610  ax-1rid 9611  ax-rnegex 9612  ax-rrecex 9613  ax-cnre 9614  ax-pre-lttri 9615  ax-pre-lttrn 9616  ax-pre-ltadd 9617  ax-pre-mulgt0 9618  ax-pre-sup 9619
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 984  df-3an 985  df-tru 1441  df-ex 1661  df-nf 1665  df-sb 1788  df-eu 2270  df-mo 2271  df-clab 2409  df-cleq 2415  df-clel 2418  df-nfc 2573  df-ne 2621  df-nel 2622  df-ral 2781  df-rex 2782  df-reu 2783  df-rmo 2784  df-rab 2785  df-v 3084  df-sbc 3301  df-csb 3397  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-pss 3453  df-nul 3763  df-if 3911  df-pw 3982  df-sn 3998  df-pr 4000  df-tp 4002  df-op 4004  df-uni 4218  df-iun 4299  df-br 4422  df-opab 4481  df-mpt 4482  df-tr 4517  df-eprel 4762  df-id 4766  df-po 4772  df-so 4773  df-fr 4810  df-we 4812  df-xp 4857  df-rel 4858  df-cnv 4859  df-co 4860  df-dm 4861  df-rn 4862  df-res 4863  df-ima 4864  df-pred 5397  df-ord 5443  df-on 5444  df-lim 5445  df-suc 5446  df-iota 5563  df-fun 5601  df-fn 5602  df-f 5603  df-f1 5604  df-fo 5605  df-f1o 5606  df-fv 5607  df-riota 6265  df-ov 6306  df-oprab 6307  df-mpt2 6308  df-om 6705  df-2nd 6806  df-wrecs 7034  df-recs 7096  df-rdg 7134  df-er 7369  df-en 7576  df-dom 7577  df-sdom 7578  df-sup 7960  df-inf 7961  df-pnf 9679  df-mnf 9680  df-xr 9681  df-ltxr 9682  df-le 9683  df-sub 9864  df-neg 9865  df-div 10272  df-nn 10612  df-2 10670  df-3 10671  df-4 10672  df-5 10673  df-6 10674  df-7 10675  df-8 10676  df-9 10677  df-10 10678  df-n0 10872  df-z 10940  df-dec 11054  df-uz 11162  df-rp 11305  df-fl 12029  df-mod 12098  df-seq 12215  df-exp 12274
This theorem is referenced by:  2503lem2  15102  2503lem3  15103
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