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Theorem 2503lem1 13232
Description: Lemma for 2503prm 13235. 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 10074 . . . . . 6  |-  2  e.  NN0
3 5nn0 10077 . . . . . 6  |-  5  e.  NN0
42, 3deccl 10230 . . . . 5  |- ; 2 5  e.  NN0
5 0nn0 10072 . . . . 5  |-  0  e.  NN0
64, 5deccl 10230 . . . 4  |- ;; 2 5 0  e.  NN0
7 3nn 9970 . . . 4  |-  3  e.  NN
86, 7decnncl 10229 . . 3  |- ;;; 2 5 0 3  e.  NN
91, 8eqeltri 2428 . 2  |-  N  e.  NN
10 2nn 9969 . 2  |-  2  e.  NN
11 9nn0 10081 . 2  |-  9  e.  NN0
12 10nn0 10082 . . . 4  |-  10  e.  NN0
13 4nn0 10076 . . . 4  |-  4  e.  NN0
1412, 13deccl 10230 . . 3  |- ; 10 4  e.  NN0
1514nn0zi 10140 . 2  |- ; 10 4  e.  ZZ
16 1nn0 10073 . . . 4  |-  1  e.  NN0
173, 16deccl 10230 . . 3  |- ; 5 1  e.  NN0
1817, 2deccl 10230 . 2  |- ;; 5 1 2  e.  NN0
19 8nn0 10080 . . . . 5  |-  8  e.  NN0
2016, 19deccl 10230 . . . 4  |- ; 1 8  e.  NN0
21 3nn0 10075 . . . 4  |-  3  e.  NN0
2220, 21deccl 10230 . . 3  |- ;; 1 8 3  e.  NN0
2322, 2deccl 10230 . 2  |- ;;; 1 8 3 2  e.  NN0
24 8p1e9 9945 . . . 4  |-  ( 8  +  1 )  =  9
25 6nn0 10078 . . . . 5  |-  6  e.  NN0
26 2exp8 13199 . . . . 5  |-  ( 2 ^ 8 )  = ;; 2 5 6
27 eqid 2358 . . . . . 6  |- ; 2 5  = ; 2 5
2816dec0h 10232 . . . . . 6  |-  1  = ; 0 1
29 2t2e4 9963 . . . . . . . 8  |-  ( 2  x.  2 )  =  4
30 ax-1cn 8885 . . . . . . . . 9  |-  1  e.  CC
3130addid2i 9090 . . . . . . . 8  |-  ( 0  +  1 )  =  1
3229, 31oveq12i 5957 . . . . . . 7  |-  ( ( 2  x.  2 )  +  ( 0  +  1 ) )  =  ( 4  +  1 )
33 4p1e5 9941 . . . . . . 7  |-  ( 4  +  1 )  =  5
3432, 33eqtri 2378 . . . . . 6  |-  ( ( 2  x.  2 )  +  ( 0  +  1 ) )  =  5
35 5t2e10 9967 . . . . . . . 8  |-  ( 5  x.  2 )  =  10
36 dec10 10246 . . . . . . . 8  |-  10  = ; 1 0
3735, 36eqtri 2378 . . . . . . 7  |-  ( 5  x.  2 )  = ; 1
0
3816, 5, 31, 37decsuc 10239 . . . . . 6  |-  ( ( 5  x.  2 )  +  1 )  = ; 1
1
392, 3, 5, 16, 27, 28, 2, 16, 16, 34, 38decmac 10255 . . . . 5  |-  ( (; 2
5  x.  2 )  +  1 )  = ; 5
1
40 6t2e12 10293 . . . . 5  |-  ( 6  x.  2 )  = ; 1
2
412, 4, 25, 26, 2, 16, 39, 40decmul1c 10263 . . . 4  |-  ( ( 2 ^ 8 )  x.  2 )  = ;; 5 1 2
422, 19, 24, 41numexpp1 13190 . . 3  |-  ( 2 ^ 9 )  = ;; 5 1 2
4342oveq1i 5955 . 2  |-  ( ( 2 ^ 9 )  mod  N )  =  (;; 5 1 2  mod  N )
44 9nn 9976 . . . 4  |-  9  e.  NN
4544nncni 9846 . . 3  |-  9  e.  CC
46 2cn 9906 . . 3  |-  2  e.  CC
47 9t2e18 10311 . . 3  |-  ( 9  x.  2 )  = ; 1
8
4845, 46, 47mulcomli 8934 . 2  |-  ( 2  x.  9 )  = ; 1
8
49 eqid 2358 . . . 4  |- ;;; 1 8 3 2  = ;;; 1 8 3 2
5021, 16deccl 10230 . . . 4  |- ; 3 1  e.  NN0
512, 16deccl 10230 . . . . 5  |- ; 2 1  e.  NN0
52 eqid 2358 . . . . 5  |- ;; 2 5 0  = ;; 2 5 0
53 eqid 2358 . . . . . 6  |- ;; 1 8 3  = ;; 1 8 3
54 eqid 2358 . . . . . 6  |- ; 3 1  = ; 3 1
55 eqid 2358 . . . . . . 7  |- ; 1 8  = ; 1 8
56 1p1e2 9930 . . . . . . 7  |-  ( 1  +  1 )  =  2
57 8p3e11 10272 . . . . . . 7  |-  ( 8  +  3 )  = ; 1
1
5816, 19, 21, 55, 56, 16, 57decaddci 10261 . . . . . 6  |-  (; 1 8  +  3 )  = ; 2 1
59 3p1e4 9940 . . . . . 6  |-  ( 3  +  1 )  =  4
6020, 21, 21, 16, 53, 54, 58, 59decadd 10257 . . . . 5  |-  (;; 1 8 3  + ; 3 1 )  = ;; 2 1 4
6151nn0cni 10069 . . . . . . 7  |- ; 2 1  e.  CC
6261addid1i 9089 . . . . . 6  |-  (; 2 1  +  0 )  = ; 2 1
633, 2deccl 10230 . . . . . 6  |- ; 5 2  e.  NN0
64 eqid 2358 . . . . . . 7  |- ; 10 4  = ; 10 4
652dec0h 10232 . . . . . . . 8  |-  2  = ; 0 2
66 eqid 2358 . . . . . . . 8  |- ; 5 2  = ; 5 2
67 5nn 9972 . . . . . . . . . 10  |-  5  e.  NN
6867nncni 9846 . . . . . . . . 9  |-  5  e.  CC
6968addid2i 9090 . . . . . . . 8  |-  ( 0  +  5 )  =  5
70 2p2e4 9934 . . . . . . . 8  |-  ( 2  +  2 )  =  4
715, 2, 3, 2, 65, 66, 69, 70decadd 10257 . . . . . . 7  |-  ( 2  + ; 5 2 )  = ; 5
4
72 5p1e6 9942 . . . . . . . . 9  |-  ( 5  +  1 )  =  6
7325dec0h 10232 . . . . . . . . 9  |-  6  = ; 0 6
7472, 73eqtri 2378 . . . . . . . 8  |-  ( 5  +  1 )  = ; 0
6
7546mulid2i 8930 . . . . . . . . . 10  |-  ( 1  x.  2 )  =  2
76 00id 9077 . . . . . . . . . 10  |-  ( 0  +  0 )  =  0
7775, 76oveq12i 5957 . . . . . . . . 9  |-  ( ( 1  x.  2 )  +  ( 0  +  0 ) )  =  ( 2  +  0 )
7846addid1i 9089 . . . . . . . . 9  |-  ( 2  +  0 )  =  2
7977, 78eqtri 2378 . . . . . . . 8  |-  ( ( 1  x.  2 )  +  ( 0  +  0 ) )  =  2
8046mul02i 9091 . . . . . . . . . 10  |-  ( 0  x.  2 )  =  0
8180oveq1i 5955 . . . . . . . . 9  |-  ( ( 0  x.  2 )  +  6 )  =  ( 0  +  6 )
82 6nn 9973 . . . . . . . . . . 11  |-  6  e.  NN
8382nncni 9846 . . . . . . . . . 10  |-  6  e.  CC
8483addid2i 9090 . . . . . . . . 9  |-  ( 0  +  6 )  =  6
8581, 84, 733eqtri 2382 . . . . . . . 8  |-  ( ( 0  x.  2 )  +  6 )  = ; 0
6
8616, 5, 5, 25, 36, 74, 2, 25, 5, 79, 85decmac 10255 . . . . . . 7  |-  ( ( 10  x.  2 )  +  ( 5  +  1 ) )  = ; 2
6
87 4t2e8 9966 . . . . . . . . 9  |-  ( 4  x.  2 )  =  8
8887oveq1i 5955 . . . . . . . 8  |-  ( ( 4  x.  2 )  +  4 )  =  ( 8  +  4 )
89 8p4e12 10273 . . . . . . . 8  |-  ( 8  +  4 )  = ; 1
2
9088, 89eqtri 2378 . . . . . . 7  |-  ( ( 4  x.  2 )  +  4 )  = ; 1
2
9112, 13, 3, 13, 64, 71, 2, 2, 16, 86, 90decmac 10255 . . . . . 6  |-  ( (; 10 4  x.  2 )  +  ( 2  + ; 5
2 ) )  = ;; 2 6 2
9246addid2i 9090 . . . . . . . . 9  |-  ( 0  +  2 )  =  2
9392, 65eqtri 2378 . . . . . . . 8  |-  ( 0  +  2 )  = ; 0
2
9468mulid2i 8930 . . . . . . . . . 10  |-  ( 1  x.  5 )  =  5
9594, 76oveq12i 5957 . . . . . . . . 9  |-  ( ( 1  x.  5 )  +  ( 0  +  0 ) )  =  ( 5  +  0 )
9668addid1i 9089 . . . . . . . . 9  |-  ( 5  +  0 )  =  5
9795, 96eqtri 2378 . . . . . . . 8  |-  ( ( 1  x.  5 )  +  ( 0  +  0 ) )  =  5
9868mul02i 9091 . . . . . . . . . 10  |-  ( 0  x.  5 )  =  0
9998oveq1i 5955 . . . . . . . . 9  |-  ( ( 0  x.  5 )  +  2 )  =  ( 0  +  2 )
10099, 92, 653eqtri 2382 . . . . . . . 8  |-  ( ( 0  x.  5 )  +  2 )  = ; 0
2
10116, 5, 5, 2, 36, 93, 3, 2, 5, 97, 100decmac 10255 . . . . . . 7  |-  ( ( 10  x.  5 )  +  ( 0  +  2 ) )  = ; 5
2
102 4cn 9910 . . . . . . . . 9  |-  4  e.  CC
103 5t4e20 10291 . . . . . . . . 9  |-  ( 5  x.  4 )  = ; 2
0
10468, 102, 103mulcomli 8934 . . . . . . . 8  |-  ( 4  x.  5 )  = ; 2
0
1052, 5, 31, 104decsuc 10239 . . . . . . 7  |-  ( ( 4  x.  5 )  +  1 )  = ; 2
1
10612, 13, 5, 16, 64, 28, 3, 16, 2, 101, 105decmac 10255 . . . . . 6  |-  ( (; 10 4  x.  5 )  +  1 )  = ;; 5 2 1
1072, 3, 2, 16, 27, 62, 14, 16, 63, 91, 106decma2c 10256 . . . . 5  |-  ( (; 10 4  x. ; 2 5 )  +  (; 2 1  +  0 ) )  = ;;; 2 6 2 1
10814nn0cni 10069 . . . . . . . 8  |- ; 10 4  e.  CC
109108mul01i 9092 . . . . . . 7  |-  (; 10 4  x.  0 )  =  0
110109oveq1i 5955 . . . . . 6  |-  ( (; 10 4  x.  0 )  +  4 )  =  ( 0  +  4 )
111102addid2i 9090 . . . . . 6  |-  ( 0  +  4 )  =  4
11213dec0h 10232 . . . . . 6  |-  4  = ; 0 4
113110, 111, 1123eqtri 2382 . . . . 5  |-  ( (; 10 4  x.  0 )  +  4 )  = ; 0
4
1144, 5, 51, 13, 52, 60, 14, 13, 5, 107, 113decma2c 10256 . . . 4  |-  ( (; 10 4  x. ;; 2
5 0 )  +  (;; 1 8 3  + ; 3 1 ) )  = ;;;; 2 6 2 1 4
11531, 28eqtri 2378 . . . . . 6  |-  ( 0  +  1 )  = ; 0
1
116 3cn 9908 . . . . . . . . 9  |-  3  e.  CC
117116mulid2i 8930 . . . . . . . 8  |-  ( 1  x.  3 )  =  3
118117, 76oveq12i 5957 . . . . . . 7  |-  ( ( 1  x.  3 )  +  ( 0  +  0 ) )  =  ( 3  +  0 )
119116addid1i 9089 . . . . . . 7  |-  ( 3  +  0 )  =  3
120118, 119eqtri 2378 . . . . . 6  |-  ( ( 1  x.  3 )  +  ( 0  +  0 ) )  =  3
121116mul02i 9091 . . . . . . . 8  |-  ( 0  x.  3 )  =  0
122121oveq1i 5955 . . . . . . 7  |-  ( ( 0  x.  3 )  +  1 )  =  ( 0  +  1 )
123122, 31, 283eqtri 2382 . . . . . 6  |-  ( ( 0  x.  3 )  +  1 )  = ; 0
1
12416, 5, 5, 16, 36, 115, 21, 16, 5, 120, 123decmac 10255 . . . . 5  |-  ( ( 10  x.  3 )  +  ( 0  +  1 ) )  = ; 3
1
125 4t3e12 10288 . . . . . 6  |-  ( 4  x.  3 )  = ; 1
2
12616, 2, 2, 125, 70decaddi 10260 . . . . 5  |-  ( ( 4  x.  3 )  +  2 )  = ; 1
4
12712, 13, 5, 2, 64, 65, 21, 13, 16, 124, 126decmac 10255 . . . 4  |-  ( (; 10 4  x.  3 )  +  2 )  = ;; 3 1 4
1286, 21, 22, 2, 1, 49, 14, 13, 50, 114, 127decma2c 10256 . . 3  |-  ( (; 10 4  x.  N )  + ;;; 1 8 3 2 )  = ;;;;; 2 6 2 1 4 4
129 eqid 2358 . . . 4  |- ;; 5 1 2  = ;; 5 1 2
13012, 2deccl 10230 . . . 4  |- ; 10 2  e.  NN0
131 eqid 2358 . . . . 5  |- ; 5 1  = ; 5 1
132 eqid 2358 . . . . 5  |- ; 10 2  = ; 10 2
13368, 30, 72addcomli 9094 . . . . . . 7  |-  ( 1  +  5 )  =  6
13416, 5, 3, 16, 36, 131, 133, 31decadd 10257 . . . . . 6  |-  ( 10  + ; 5 1 )  = ; 6
1
135 7nn0 10079 . . . . . . 7  |-  7  e.  NN0
136 6p1e7 9943 . . . . . . . 8  |-  ( 6  +  1 )  =  7
137135dec0h 10232 . . . . . . . 8  |-  7  = ; 0 7
138136, 137eqtri 2378 . . . . . . 7  |-  ( 6  +  1 )  = ; 0
7
13931oveq2i 5956 . . . . . . . 8  |-  ( ( 5  x.  5 )  +  ( 0  +  1 ) )  =  ( ( 5  x.  5 )  +  1 )
140 5t5e25 10292 . . . . . . . . 9  |-  ( 5  x.  5 )  = ; 2
5
1412, 3, 72, 140decsuc 10239 . . . . . . . 8  |-  ( ( 5  x.  5 )  +  1 )  = ; 2
6
142139, 141eqtri 2378 . . . . . . 7  |-  ( ( 5  x.  5 )  +  ( 0  +  1 ) )  = ; 2
6
14394oveq1i 5955 . . . . . . . 8  |-  ( ( 1  x.  5 )  +  7 )  =  ( 5  +  7 )
144 7nn 9974 . . . . . . . . . 10  |-  7  e.  NN
145144nncni 9846 . . . . . . . . 9  |-  7  e.  CC
146 7p5e12 10269 . . . . . . . . 9  |-  ( 7  +  5 )  = ; 1
2
147145, 68, 146addcomli 9094 . . . . . . . 8  |-  ( 5  +  7 )  = ; 1
2
148143, 147eqtri 2378 . . . . . . 7  |-  ( ( 1  x.  5 )  +  7 )  = ; 1
2
1493, 16, 5, 135, 131, 138, 3, 2, 16, 142, 148decmac 10255 . . . . . 6  |-  ( (; 5
1  x.  5 )  +  ( 6  +  1 ) )  = ;; 2 6 2
15068, 46, 35mulcomli 8934 . . . . . . . 8  |-  ( 2  x.  5 )  =  10
151150, 36eqtri 2378 . . . . . . 7  |-  ( 2  x.  5 )  = ; 1
0
15216, 5, 31, 151decsuc 10239 . . . . . 6  |-  ( ( 2  x.  5 )  +  1 )  = ; 1
1
15317, 2, 25, 16, 129, 134, 3, 16, 16, 149, 152decmac 10255 . . . . 5  |-  ( (;; 5 1 2  x.  5 )  +  ( 10  + ; 5 1 ) )  = ;;; 2 6 2 1
1545dec0h 10232 . . . . . . . 8  |-  0  = ; 0 0
15576, 154eqtri 2378 . . . . . . 7  |-  ( 0  +  0 )  = ; 0
0
15668mulid1i 8929 . . . . . . . . 9  |-  ( 5  x.  1 )  =  5
157156, 76oveq12i 5957 . . . . . . . 8  |-  ( ( 5  x.  1 )  +  ( 0  +  0 ) )  =  ( 5  +  0 )
158157, 96eqtri 2378 . . . . . . 7  |-  ( ( 5  x.  1 )  +  ( 0  +  0 ) )  =  5
15930mulid1i 8929 . . . . . . . . 9  |-  ( 1  x.  1 )  =  1
160159oveq1i 5955 . . . . . . . 8  |-  ( ( 1  x.  1 )  +  0 )  =  ( 1  +  0 )
16130addid1i 9089 . . . . . . . 8  |-  ( 1  +  0 )  =  1
162160, 161, 283eqtri 2382 . . . . . . 7  |-  ( ( 1  x.  1 )  +  0 )  = ; 0
1
1633, 16, 5, 5, 131, 155, 16, 16, 5, 158, 162decmac 10255 . . . . . 6  |-  ( (; 5
1  x.  1 )  +  ( 0  +  0 ) )  = ; 5
1
16446mulid1i 8929 . . . . . . . 8  |-  ( 2  x.  1 )  =  2
165164oveq1i 5955 . . . . . . 7  |-  ( ( 2  x.  1 )  +  2 )  =  ( 2  +  2 )
166165, 70, 1123eqtri 2382 . . . . . 6  |-  ( ( 2  x.  1 )  +  2 )  = ; 0
4
16717, 2, 5, 2, 129, 65, 16, 13, 5, 163, 166decmac 10255 . . . . 5  |-  ( (;; 5 1 2  x.  1 )  +  2 )  = ;; 5 1 4
1683, 16, 12, 2, 131, 132, 18, 13, 17, 153, 167decma2c 10256 . . . 4  |-  ( (;; 5 1 2  x. ; 5
1 )  + ; 10 2 )  = ;;;; 2 6 2 1 4
16935oveq1i 5955 . . . . . . . . 9  |-  ( ( 5  x.  2 )  +  0 )  =  ( 10  +  0 )
170 10nn 9977 . . . . . . . . . . 11  |-  10  e.  NN
171170nncni 9846 . . . . . . . . . 10  |-  10  e.  CC
172171addid1i 9089 . . . . . . . . 9  |-  ( 10  +  0 )  =  10
173169, 172eqtri 2378 . . . . . . . 8  |-  ( ( 5  x.  2 )  +  0 )  =  10
17475, 65eqtri 2378 . . . . . . . 8  |-  ( 1  x.  2 )  = ; 0
2
1752, 3, 16, 131, 2, 5, 173, 174decmul1c 10263 . . . . . . 7  |-  (; 5 1  x.  2 )  = ; 10 2
176175oveq1i 5955 . . . . . 6  |-  ( (; 5
1  x.  2 )  +  0 )  =  (; 10 2  +  0
)
177130nn0cni 10069 . . . . . . 7  |- ; 10 2  e.  CC
178177addid1i 9089 . . . . . 6  |-  (; 10 2  +  0 )  = ; 10 2
179176, 178eqtri 2378 . . . . 5  |-  ( (; 5
1  x.  2 )  +  0 )  = ; 10 2
18029, 112eqtri 2378 . . . . 5  |-  ( 2  x.  2 )  = ; 0
4
1812, 17, 2, 129, 13, 5, 179, 180decmul1c 10263 . . . 4  |-  (;; 5 1 2  x.  2 )  = ;; 10 2 4
18218, 17, 2, 129, 13, 130, 168, 181decmul2c 10264 . . 3  |-  (;; 5 1 2  x. ;; 5 1 2 )  = ;;;;; 2 6 2 1 4 4
183128, 182eqtr4i 2381 . 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 13181 1  |-  ( ( 2 ^; 1 8 )  mod 
N )  =  (;;; 1 8 3 2  mod 
N )
Colors of variables: wff set class
Syntax hints:    = wceq 1642  (class class class)co 5945   0cc0 8827   1c1 8828    + caddc 8830    x. cmul 8832   NNcn 9836   2c2 9885   3c3 9886   4c4 9887   5c5 9888   6c6 9889   7c7 9890   8c8 9891   9c9 9892   10c10 9893  ;cdc 10216    mod cmo 11065   ^cexp 11197
This theorem is referenced by:  2503lem2  13233  2503lem3  13234
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-sep 4222  ax-nul 4230  ax-pow 4269  ax-pr 4295  ax-un 4594  ax-cnex 8883  ax-resscn 8884  ax-1cn 8885  ax-icn 8886  ax-addcl 8887  ax-addrcl 8888  ax-mulcl 8889  ax-mulrcl 8890  ax-mulcom 8891  ax-addass 8892  ax-mulass 8893  ax-distr 8894  ax-i2m1 8895  ax-1ne0 8896  ax-1rid 8897  ax-rnegex 8898  ax-rrecex 8899  ax-cnre 8900  ax-pre-lttri 8901  ax-pre-lttrn 8902  ax-pre-ltadd 8903  ax-pre-mulgt0 8904  ax-pre-sup 8905
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-nel 2524  df-ral 2624  df-rex 2625  df-reu 2626  df-rmo 2627  df-rab 2628  df-v 2866  df-sbc 3068  df-csb 3158  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-pss 3244  df-nul 3532  df-if 3642  df-pw 3703  df-sn 3722  df-pr 3723  df-tp 3724  df-op 3725  df-uni 3909  df-iun 3988  df-br 4105  df-opab 4159  df-mpt 4160  df-tr 4195  df-eprel 4387  df-id 4391  df-po 4396  df-so 4397  df-fr 4434  df-we 4436  df-ord 4477  df-on 4478  df-lim 4479  df-suc 4480  df-om 4739  df-xp 4777  df-rel 4778  df-cnv 4779  df-co 4780  df-dm 4781  df-rn 4782  df-res 4783  df-ima 4784  df-iota 5301  df-fun 5339  df-fn 5340  df-f 5341  df-f1 5342  df-fo 5343  df-f1o 5344  df-fv 5345  df-ov 5948  df-oprab 5949  df-mpt2 5950  df-2nd 6210  df-riota 6391  df-recs 6475  df-rdg 6510  df-er 6747  df-en 6952  df-dom 6953  df-sdom 6954  df-sup 7284  df-pnf 8959  df-mnf 8960  df-xr 8961  df-ltxr 8962  df-le 8963  df-sub 9129  df-neg 9130  df-div 9514  df-nn 9837  df-2 9894  df-3 9895  df-4 9896  df-5 9897  df-6 9898  df-7 9899  df-8 9900  df-9 9901  df-10 9902  df-n0 10058  df-z 10117  df-dec 10217  df-uz 10323  df-rp 10447  df-fl 11017  df-mod 11066  df-seq 11139  df-exp 11198
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