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Theorem 2503lem1 13419
Description: Lemma for 2503prm 13422. 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 10202 . . . . . 6  |-  2  e.  NN0
3 5nn0 10205 . . . . . 6  |-  5  e.  NN0
42, 3deccl 10360 . . . . 5  |- ; 2 5  e.  NN0
5 0nn0 10200 . . . . 5  |-  0  e.  NN0
64, 5deccl 10360 . . . 4  |- ;; 2 5 0  e.  NN0
7 3nn 10098 . . . 4  |-  3  e.  NN
86, 7decnncl 10359 . . 3  |- ;;; 2 5 0 3  e.  NN
91, 8eqeltri 2482 . 2  |-  N  e.  NN
10 2nn 10097 . 2  |-  2  e.  NN
11 9nn0 10209 . 2  |-  9  e.  NN0
12 10nn0 10210 . . . 4  |-  10  e.  NN0
13 4nn0 10204 . . . 4  |-  4  e.  NN0
1412, 13deccl 10360 . . 3  |- ; 10 4  e.  NN0
1514nn0zi 10270 . 2  |- ; 10 4  e.  ZZ
16 1nn0 10201 . . . 4  |-  1  e.  NN0
173, 16deccl 10360 . . 3  |- ; 5 1  e.  NN0
1817, 2deccl 10360 . 2  |- ;; 5 1 2  e.  NN0
19 8nn0 10208 . . . . 5  |-  8  e.  NN0
2016, 19deccl 10360 . . . 4  |- ; 1 8  e.  NN0
21 3nn0 10203 . . . 4  |-  3  e.  NN0
2220, 21deccl 10360 . . 3  |- ;; 1 8 3  e.  NN0
2322, 2deccl 10360 . 2  |- ;;; 1 8 3 2  e.  NN0
24 8p1e9 10073 . . . 4  |-  ( 8  +  1 )  =  9
25 6nn0 10206 . . . . 5  |-  6  e.  NN0
26 2exp8 13386 . . . . 5  |-  ( 2 ^ 8 )  = ;; 2 5 6
27 eqid 2412 . . . . . 6  |- ; 2 5  = ; 2 5
2816dec0h 10362 . . . . . 6  |-  1  = ; 0 1
29 2t2e4 10091 . . . . . . . 8  |-  ( 2  x.  2 )  =  4
30 ax-1cn 9012 . . . . . . . . 9  |-  1  e.  CC
3130addid2i 9218 . . . . . . . 8  |-  ( 0  +  1 )  =  1
3229, 31oveq12i 6060 . . . . . . 7  |-  ( ( 2  x.  2 )  +  ( 0  +  1 ) )  =  ( 4  +  1 )
33 4p1e5 10069 . . . . . . 7  |-  ( 4  +  1 )  =  5
3432, 33eqtri 2432 . . . . . 6  |-  ( ( 2  x.  2 )  +  ( 0  +  1 ) )  =  5
35 5t2e10 10095 . . . . . . . 8  |-  ( 5  x.  2 )  =  10
36 dec10 10376 . . . . . . . 8  |-  10  = ; 1 0
3735, 36eqtri 2432 . . . . . . 7  |-  ( 5  x.  2 )  = ; 1
0
3816, 5, 31, 37decsuc 10369 . . . . . 6  |-  ( ( 5  x.  2 )  +  1 )  = ; 1
1
392, 3, 5, 16, 27, 28, 2, 16, 16, 34, 38decmac 10385 . . . . 5  |-  ( (; 2
5  x.  2 )  +  1 )  = ; 5
1
40 6t2e12 10423 . . . . 5  |-  ( 6  x.  2 )  = ; 1
2
412, 4, 25, 26, 2, 16, 39, 40decmul1c 10393 . . . 4  |-  ( ( 2 ^ 8 )  x.  2 )  = ;; 5 1 2
422, 19, 24, 41numexpp1 13377 . . 3  |-  ( 2 ^ 9 )  = ;; 5 1 2
4342oveq1i 6058 . 2  |-  ( ( 2 ^ 9 )  mod  N )  =  (;; 5 1 2  mod  N )
44 9nn 10104 . . . 4  |-  9  e.  NN
4544nncni 9974 . . 3  |-  9  e.  CC
46 2cn 10034 . . 3  |-  2  e.  CC
47 9t2e18 10441 . . 3  |-  ( 9  x.  2 )  = ; 1
8
4845, 46, 47mulcomli 9061 . 2  |-  ( 2  x.  9 )  = ; 1
8
49 eqid 2412 . . . 4  |- ;;; 1 8 3 2  = ;;; 1 8 3 2
5021, 16deccl 10360 . . . 4  |- ; 3 1  e.  NN0
512, 16deccl 10360 . . . . 5  |- ; 2 1  e.  NN0
52 eqid 2412 . . . . 5  |- ;; 2 5 0  = ;; 2 5 0
53 eqid 2412 . . . . . 6  |- ;; 1 8 3  = ;; 1 8 3
54 eqid 2412 . . . . . 6  |- ; 3 1  = ; 3 1
55 eqid 2412 . . . . . . 7  |- ; 1 8  = ; 1 8
56 1p1e2 10058 . . . . . . 7  |-  ( 1  +  1 )  =  2
57 8p3e11 10402 . . . . . . 7  |-  ( 8  +  3 )  = ; 1
1
5816, 19, 21, 55, 56, 16, 57decaddci 10391 . . . . . 6  |-  (; 1 8  +  3 )  = ; 2 1
59 3p1e4 10068 . . . . . 6  |-  ( 3  +  1 )  =  4
6020, 21, 21, 16, 53, 54, 58, 59decadd 10387 . . . . 5  |-  (;; 1 8 3  + ; 3 1 )  = ;; 2 1 4
6151nn0cni 10197 . . . . . . 7  |- ; 2 1  e.  CC
6261addid1i 9217 . . . . . 6  |-  (; 2 1  +  0 )  = ; 2 1
633, 2deccl 10360 . . . . . 6  |- ; 5 2  e.  NN0
64 eqid 2412 . . . . . . 7  |- ; 10 4  = ; 10 4
652dec0h 10362 . . . . . . . 8  |-  2  = ; 0 2
66 eqid 2412 . . . . . . . 8  |- ; 5 2  = ; 5 2
67 5nn 10100 . . . . . . . . . 10  |-  5  e.  NN
6867nncni 9974 . . . . . . . . 9  |-  5  e.  CC
6968addid2i 9218 . . . . . . . 8  |-  ( 0  +  5 )  =  5
70 2p2e4 10062 . . . . . . . 8  |-  ( 2  +  2 )  =  4
715, 2, 3, 2, 65, 66, 69, 70decadd 10387 . . . . . . 7  |-  ( 2  + ; 5 2 )  = ; 5
4
72 5p1e6 10070 . . . . . . . . 9  |-  ( 5  +  1 )  =  6
7325dec0h 10362 . . . . . . . . 9  |-  6  = ; 0 6
7472, 73eqtri 2432 . . . . . . . 8  |-  ( 5  +  1 )  = ; 0
6
7546mulid2i 9057 . . . . . . . . . 10  |-  ( 1  x.  2 )  =  2
76 00id 9205 . . . . . . . . . 10  |-  ( 0  +  0 )  =  0
7775, 76oveq12i 6060 . . . . . . . . 9  |-  ( ( 1  x.  2 )  +  ( 0  +  0 ) )  =  ( 2  +  0 )
7846addid1i 9217 . . . . . . . . 9  |-  ( 2  +  0 )  =  2
7977, 78eqtri 2432 . . . . . . . 8  |-  ( ( 1  x.  2 )  +  ( 0  +  0 ) )  =  2
8046mul02i 9219 . . . . . . . . . 10  |-  ( 0  x.  2 )  =  0
8180oveq1i 6058 . . . . . . . . 9  |-  ( ( 0  x.  2 )  +  6 )  =  ( 0  +  6 )
82 6nn 10101 . . . . . . . . . . 11  |-  6  e.  NN
8382nncni 9974 . . . . . . . . . 10  |-  6  e.  CC
8483addid2i 9218 . . . . . . . . 9  |-  ( 0  +  6 )  =  6
8581, 84, 733eqtri 2436 . . . . . . . 8  |-  ( ( 0  x.  2 )  +  6 )  = ; 0
6
8616, 5, 5, 25, 36, 74, 2, 25, 5, 79, 85decmac 10385 . . . . . . 7  |-  ( ( 10  x.  2 )  +  ( 5  +  1 ) )  = ; 2
6
87 4t2e8 10094 . . . . . . . . 9  |-  ( 4  x.  2 )  =  8
8887oveq1i 6058 . . . . . . . 8  |-  ( ( 4  x.  2 )  +  4 )  =  ( 8  +  4 )
89 8p4e12 10403 . . . . . . . 8  |-  ( 8  +  4 )  = ; 1
2
9088, 89eqtri 2432 . . . . . . 7  |-  ( ( 4  x.  2 )  +  4 )  = ; 1
2
9112, 13, 3, 13, 64, 71, 2, 2, 16, 86, 90decmac 10385 . . . . . 6  |-  ( (; 10 4  x.  2 )  +  ( 2  + ; 5
2 ) )  = ;; 2 6 2
9246addid2i 9218 . . . . . . . . 9  |-  ( 0  +  2 )  =  2
9392, 65eqtri 2432 . . . . . . . 8  |-  ( 0  +  2 )  = ; 0
2
9468mulid2i 9057 . . . . . . . . . 10  |-  ( 1  x.  5 )  =  5
9594, 76oveq12i 6060 . . . . . . . . 9  |-  ( ( 1  x.  5 )  +  ( 0  +  0 ) )  =  ( 5  +  0 )
9668addid1i 9217 . . . . . . . . 9  |-  ( 5  +  0 )  =  5
9795, 96eqtri 2432 . . . . . . . 8  |-  ( ( 1  x.  5 )  +  ( 0  +  0 ) )  =  5
9868mul02i 9219 . . . . . . . . . 10  |-  ( 0  x.  5 )  =  0
9998oveq1i 6058 . . . . . . . . 9  |-  ( ( 0  x.  5 )  +  2 )  =  ( 0  +  2 )
10099, 92, 653eqtri 2436 . . . . . . . 8  |-  ( ( 0  x.  5 )  +  2 )  = ; 0
2
10116, 5, 5, 2, 36, 93, 3, 2, 5, 97, 100decmac 10385 . . . . . . 7  |-  ( ( 10  x.  5 )  +  ( 0  +  2 ) )  = ; 5
2
102 4cn 10038 . . . . . . . . 9  |-  4  e.  CC
103 5t4e20 10421 . . . . . . . . 9  |-  ( 5  x.  4 )  = ; 2
0
10468, 102, 103mulcomli 9061 . . . . . . . 8  |-  ( 4  x.  5 )  = ; 2
0
1052, 5, 31, 104decsuc 10369 . . . . . . 7  |-  ( ( 4  x.  5 )  +  1 )  = ; 2
1
10612, 13, 5, 16, 64, 28, 3, 16, 2, 101, 105decmac 10385 . . . . . 6  |-  ( (; 10 4  x.  5 )  +  1 )  = ;; 5 2 1
1072, 3, 2, 16, 27, 62, 14, 16, 63, 91, 106decma2c 10386 . . . . 5  |-  ( (; 10 4  x. ; 2 5 )  +  (; 2 1  +  0 ) )  = ;;; 2 6 2 1
10814nn0cni 10197 . . . . . . . 8  |- ; 10 4  e.  CC
109108mul01i 9220 . . . . . . 7  |-  (; 10 4  x.  0 )  =  0
110109oveq1i 6058 . . . . . 6  |-  ( (; 10 4  x.  0 )  +  4 )  =  ( 0  +  4 )
111102addid2i 9218 . . . . . 6  |-  ( 0  +  4 )  =  4
11213dec0h 10362 . . . . . 6  |-  4  = ; 0 4
113110, 111, 1123eqtri 2436 . . . . 5  |-  ( (; 10 4  x.  0 )  +  4 )  = ; 0
4
1144, 5, 51, 13, 52, 60, 14, 13, 5, 107, 113decma2c 10386 . . . 4  |-  ( (; 10 4  x. ;; 2
5 0 )  +  (;; 1 8 3  + ; 3 1 ) )  = ;;;; 2 6 2 1 4
11531, 28eqtri 2432 . . . . . 6  |-  ( 0  +  1 )  = ; 0
1
116 3cn 10036 . . . . . . . . 9  |-  3  e.  CC
117116mulid2i 9057 . . . . . . . 8  |-  ( 1  x.  3 )  =  3
118117, 76oveq12i 6060 . . . . . . 7  |-  ( ( 1  x.  3 )  +  ( 0  +  0 ) )  =  ( 3  +  0 )
119116addid1i 9217 . . . . . . 7  |-  ( 3  +  0 )  =  3
120118, 119eqtri 2432 . . . . . 6  |-  ( ( 1  x.  3 )  +  ( 0  +  0 ) )  =  3
121116mul02i 9219 . . . . . . . 8  |-  ( 0  x.  3 )  =  0
122121oveq1i 6058 . . . . . . 7  |-  ( ( 0  x.  3 )  +  1 )  =  ( 0  +  1 )
123122, 31, 283eqtri 2436 . . . . . 6  |-  ( ( 0  x.  3 )  +  1 )  = ; 0
1
12416, 5, 5, 16, 36, 115, 21, 16, 5, 120, 123decmac 10385 . . . . 5  |-  ( ( 10  x.  3 )  +  ( 0  +  1 ) )  = ; 3
1
125 4t3e12 10418 . . . . . 6  |-  ( 4  x.  3 )  = ; 1
2
12616, 2, 2, 125, 70decaddi 10390 . . . . 5  |-  ( ( 4  x.  3 )  +  2 )  = ; 1
4
12712, 13, 5, 2, 64, 65, 21, 13, 16, 124, 126decmac 10385 . . . 4  |-  ( (; 10 4  x.  3 )  +  2 )  = ;; 3 1 4
1286, 21, 22, 2, 1, 49, 14, 13, 50, 114, 127decma2c 10386 . . 3  |-  ( (; 10 4  x.  N )  + ;;; 1 8 3 2 )  = ;;;;; 2 6 2 1 4 4
129 eqid 2412 . . . 4  |- ;; 5 1 2  = ;; 5 1 2
13012, 2deccl 10360 . . . 4  |- ; 10 2  e.  NN0
131 eqid 2412 . . . . 5  |- ; 5 1  = ; 5 1
132 eqid 2412 . . . . 5  |- ; 10 2  = ; 10 2
13368, 30, 72addcomli 9222 . . . . . . 7  |-  ( 1  +  5 )  =  6
13416, 5, 3, 16, 36, 131, 133, 31decadd 10387 . . . . . 6  |-  ( 10  + ; 5 1 )  = ; 6
1
135 7nn0 10207 . . . . . . 7  |-  7  e.  NN0
136 6p1e7 10071 . . . . . . . 8  |-  ( 6  +  1 )  =  7
137135dec0h 10362 . . . . . . . 8  |-  7  = ; 0 7
138136, 137eqtri 2432 . . . . . . 7  |-  ( 6  +  1 )  = ; 0
7
13931oveq2i 6059 . . . . . . . 8  |-  ( ( 5  x.  5 )  +  ( 0  +  1 ) )  =  ( ( 5  x.  5 )  +  1 )
140 5t5e25 10422 . . . . . . . . 9  |-  ( 5  x.  5 )  = ; 2
5
1412, 3, 72, 140decsuc 10369 . . . . . . . 8  |-  ( ( 5  x.  5 )  +  1 )  = ; 2
6
142139, 141eqtri 2432 . . . . . . 7  |-  ( ( 5  x.  5 )  +  ( 0  +  1 ) )  = ; 2
6
14394oveq1i 6058 . . . . . . . 8  |-  ( ( 1  x.  5 )  +  7 )  =  ( 5  +  7 )
144 7nn 10102 . . . . . . . . . 10  |-  7  e.  NN
145144nncni 9974 . . . . . . . . 9  |-  7  e.  CC
146 7p5e12 10399 . . . . . . . . 9  |-  ( 7  +  5 )  = ; 1
2
147145, 68, 146addcomli 9222 . . . . . . . 8  |-  ( 5  +  7 )  = ; 1
2
148143, 147eqtri 2432 . . . . . . 7  |-  ( ( 1  x.  5 )  +  7 )  = ; 1
2
1493, 16, 5, 135, 131, 138, 3, 2, 16, 142, 148decmac 10385 . . . . . 6  |-  ( (; 5
1  x.  5 )  +  ( 6  +  1 ) )  = ;; 2 6 2
15068, 46, 35mulcomli 9061 . . . . . . . 8  |-  ( 2  x.  5 )  =  10
151150, 36eqtri 2432 . . . . . . 7  |-  ( 2  x.  5 )  = ; 1
0
15216, 5, 31, 151decsuc 10369 . . . . . 6  |-  ( ( 2  x.  5 )  +  1 )  = ; 1
1
15317, 2, 25, 16, 129, 134, 3, 16, 16, 149, 152decmac 10385 . . . . 5  |-  ( (;; 5 1 2  x.  5 )  +  ( 10  + ; 5 1 ) )  = ;;; 2 6 2 1
1545dec0h 10362 . . . . . . . 8  |-  0  = ; 0 0
15576, 154eqtri 2432 . . . . . . 7  |-  ( 0  +  0 )  = ; 0
0
15668mulid1i 9056 . . . . . . . . 9  |-  ( 5  x.  1 )  =  5
157156, 76oveq12i 6060 . . . . . . . 8  |-  ( ( 5  x.  1 )  +  ( 0  +  0 ) )  =  ( 5  +  0 )
158157, 96eqtri 2432 . . . . . . 7  |-  ( ( 5  x.  1 )  +  ( 0  +  0 ) )  =  5
15930mulid1i 9056 . . . . . . . . 9  |-  ( 1  x.  1 )  =  1
160159oveq1i 6058 . . . . . . . 8  |-  ( ( 1  x.  1 )  +  0 )  =  ( 1  +  0 )
16130addid1i 9217 . . . . . . . 8  |-  ( 1  +  0 )  =  1
162160, 161, 283eqtri 2436 . . . . . . 7  |-  ( ( 1  x.  1 )  +  0 )  = ; 0
1
1633, 16, 5, 5, 131, 155, 16, 16, 5, 158, 162decmac 10385 . . . . . 6  |-  ( (; 5
1  x.  1 )  +  ( 0  +  0 ) )  = ; 5
1
16446mulid1i 9056 . . . . . . . 8  |-  ( 2  x.  1 )  =  2
165164oveq1i 6058 . . . . . . 7  |-  ( ( 2  x.  1 )  +  2 )  =  ( 2  +  2 )
166165, 70, 1123eqtri 2436 . . . . . 6  |-  ( ( 2  x.  1 )  +  2 )  = ; 0
4
16717, 2, 5, 2, 129, 65, 16, 13, 5, 163, 166decmac 10385 . . . . 5  |-  ( (;; 5 1 2  x.  1 )  +  2 )  = ;; 5 1 4
1683, 16, 12, 2, 131, 132, 18, 13, 17, 153, 167decma2c 10386 . . . 4  |-  ( (;; 5 1 2  x. ; 5
1 )  + ; 10 2 )  = ;;;; 2 6 2 1 4
16935oveq1i 6058 . . . . . . . . 9  |-  ( ( 5  x.  2 )  +  0 )  =  ( 10  +  0 )
170 10nn 10105 . . . . . . . . . . 11  |-  10  e.  NN
171170nncni 9974 . . . . . . . . . 10  |-  10  e.  CC
172171addid1i 9217 . . . . . . . . 9  |-  ( 10  +  0 )  =  10
173169, 172eqtri 2432 . . . . . . . 8  |-  ( ( 5  x.  2 )  +  0 )  =  10
17475, 65eqtri 2432 . . . . . . . 8  |-  ( 1  x.  2 )  = ; 0
2
1752, 3, 16, 131, 2, 5, 173, 174decmul1c 10393 . . . . . . 7  |-  (; 5 1  x.  2 )  = ; 10 2
176175oveq1i 6058 . . . . . 6  |-  ( (; 5
1  x.  2 )  +  0 )  =  (; 10 2  +  0
)
177130nn0cni 10197 . . . . . . 7  |- ; 10 2  e.  CC
178177addid1i 9217 . . . . . 6  |-  (; 10 2  +  0 )  = ; 10 2
179176, 178eqtri 2432 . . . . 5  |-  ( (; 5
1  x.  2 )  +  0 )  = ; 10 2
18029, 112eqtri 2432 . . . . 5  |-  ( 2  x.  2 )  = ; 0
4
1812, 17, 2, 129, 13, 5, 179, 180decmul1c 10393 . . . 4  |-  (;; 5 1 2  x.  2 )  = ;; 10 2 4
18218, 17, 2, 129, 13, 130, 168, 181decmul2c 10394 . . 3  |-  (;; 5 1 2  x. ;; 5 1 2 )  = ;;;;; 2 6 2 1 4 4
183128, 182eqtr4i 2435 . 2  |-  ( (; 10 4  x.  N )  + ;;; 1 8 3 2 )  =  (;; 5 1 2  x. ;; 5 1 2 )
1849, 10, 11, 15, 18, 23, 43, 48, 183mod2xi 13368 1  |-  ( ( 2 ^; 1 8 )  mod 
N )  =  (;;; 1 8 3 2  mod 
N )
Colors of variables: wff set class
Syntax hints:    = wceq 1649  (class class class)co 6048   0cc0 8954   1c1 8955    + caddc 8957    x. cmul 8959   NNcn 9964   2c2 10013   3c3 10014   4c4 10015   5c5 10016   6c6 10017   7c7 10018   8c8 10019   9c9 10020   10c10 10021  ;cdc 10346    mod cmo 11213   ^cexp 11345
This theorem is referenced by:  2503lem2  13420  2503lem3  13421
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393  ax-sep 4298  ax-nul 4306  ax-pow 4345  ax-pr 4371  ax-un 4668  ax-cnex 9010  ax-resscn 9011  ax-1cn 9012  ax-icn 9013  ax-addcl 9014  ax-addrcl 9015  ax-mulcl 9016  ax-mulrcl 9017  ax-mulcom 9018  ax-addass 9019  ax-mulass 9020  ax-distr 9021  ax-i2m1 9022  ax-1ne0 9023  ax-1rid 9024  ax-rnegex 9025  ax-rrecex 9026  ax-cnre 9027  ax-pre-lttri 9028  ax-pre-lttrn 9029  ax-pre-ltadd 9030  ax-pre-mulgt0 9031  ax-pre-sup 9032
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-nel 2578  df-ral 2679  df-rex 2680  df-reu 2681  df-rmo 2682  df-rab 2683  df-v 2926  df-sbc 3130  df-csb 3220  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-pss 3304  df-nul 3597  df-if 3708  df-pw 3769  df-sn 3788  df-pr 3789  df-tp 3790  df-op 3791  df-uni 3984  df-iun 4063  df-br 4181  df-opab 4235  df-mpt 4236  df-tr 4271  df-eprel 4462  df-id 4466  df-po 4471  df-so 4472  df-fr 4509  df-we 4511  df-ord 4552  df-on 4553  df-lim 4554  df-suc 4555  df-om 4813  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5385  df-fun 5423  df-fn 5424  df-f 5425  df-f1 5426  df-fo 5427  df-f1o 5428  df-fv 5429  df-ov 6051  df-oprab 6052  df-mpt2 6053  df-2nd 6317  df-riota 6516  df-recs 6600  df-rdg 6635  df-er 6872  df-en 7077  df-dom 7078  df-sdom 7079  df-sup 7412  df-pnf 9086  df-mnf 9087  df-xr 9088  df-ltxr 9089  df-le 9090  df-sub 9257  df-neg 9258  df-div 9642  df-nn 9965  df-2 10022  df-3 10023  df-4 10024  df-5 10025  df-6 10026  df-7 10027  df-8 10028  df-9 10029  df-10 10030  df-n0 10186  df-z 10247  df-dec 10347  df-uz 10453  df-rp 10577  df-fl 11165  df-mod 11214  df-seq 11287  df-exp 11346
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