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Theorem 1259lem3 13442
Description: Lemma for 1259prm 13445. 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 10227 . . . . . 6  |-  1  e.  NN0
3 2nn0 10228 . . . . . 6  |-  2  e.  NN0
42, 3deccl 10386 . . . . 5  |- ; 1 2  e.  NN0
5 5nn0 10231 . . . . 5  |-  5  e.  NN0
64, 5deccl 10386 . . . 4  |- ;; 1 2 5  e.  NN0
7 9nn 10130 . . . 4  |-  9  e.  NN
86, 7decnncl 10385 . . 3  |- ;;; 1 2 5 9  e.  NN
91, 8eqeltri 2505 . 2  |-  N  e.  NN
10 2nn 10123 . 2  |-  2  e.  NN
11 3nn0 10229 . . 3  |-  3  e.  NN0
12 8nn0 10234 . . 3  |-  8  e.  NN0
1311, 12deccl 10386 . 2  |- ; 3 8  e.  NN0
14 4nn 10125 . . 3  |-  4  e.  NN
1514nnzi 10295 . 2  |-  4  e.  ZZ
16 7nn0 10233 . . 3  |-  7  e.  NN0
1716, 2deccl 10386 . 2  |- ; 7 1  e.  NN0
18 4nn0 10230 . . . 4  |-  4  e.  NN0
1911, 18deccl 10386 . . 3  |- ; 3 4  e.  NN0
202, 2deccl 10386 . . . 4  |- ; 1 1  e.  NN0
2120nn0zi 10296 . . 3  |- ; 1 1  e.  ZZ
2212, 16deccl 10386 . . . 4  |- ; 8 7  e.  NN0
23 0nn0 10226 . . . 4  |-  0  e.  NN0
2422, 23deccl 10386 . . 3  |- ;; 8 7 0  e.  NN0
25 6nn0 10232 . . . 4  |-  6  e.  NN0
262, 25deccl 10386 . . 3  |- ; 1 6  e.  NN0
2711259lem2 13441 . . 3  |-  ( ( 2 ^; 3 4 )  mod 
N )  =  (;; 8 7 0  mod 
N )
28 2exp4 13411 . . . 4  |-  ( 2 ^ 4 )  = ; 1
6
2928oveq1i 6083 . . 3  |-  ( ( 2 ^ 4 )  mod  N )  =  (; 1 6  mod  N
)
30 eqid 2435 . . . 4  |- ; 3 4  = ; 3 4
31 4p4e8 10105 . . . 4  |-  ( 4  +  4 )  =  8
3211, 18, 18, 30, 31decaddi 10416 . . 3  |-  (; 3 4  +  4 )  = ; 3 8
33 9nn0 10235 . . . . 5  |-  9  e.  NN0
34 eqid 2435 . . . . 5  |- ; 7 1  = ; 7 1
35 10nn0 10236 . . . . 5  |-  10  e.  NN0
36 eqid 2435 . . . . . 6  |- ;; 1 2 5  = ;; 1 2 5
3716dec0h 10388 . . . . . . 7  |-  7  = ; 0 7
38 dec10 10402 . . . . . . 7  |-  10  = ; 1 0
39 0p1e1 10083 . . . . . . 7  |-  ( 0  +  1 )  =  1
40 7nn 10128 . . . . . . . . 9  |-  7  e.  NN
4140nncni 10000 . . . . . . . 8  |-  7  e.  CC
4241addid1i 9243 . . . . . . 7  |-  ( 7  +  0 )  =  7
4323, 16, 2, 23, 37, 38, 39, 42decadd 10413 . . . . . 6  |-  ( 7  +  10 )  = ; 1
7
44 eqid 2435 . . . . . . 7  |- ; 1 2  = ; 1 2
45 6nn 10127 . . . . . . . . . 10  |-  6  e.  NN
4645nncni 10000 . . . . . . . . 9  |-  6  e.  CC
47 ax-1cn 9038 . . . . . . . . 9  |-  1  e.  CC
48 6p1e7 10097 . . . . . . . . 9  |-  ( 6  +  1 )  =  7
4946, 47, 48addcomli 9248 . . . . . . . 8  |-  ( 1  +  6 )  =  7
5049, 37eqtri 2455 . . . . . . 7  |-  ( 1  +  6 )  = ; 0
7
51 eqid 2435 . . . . . . . 8  |- ; 1 1  = ; 1 1
52 2cn 10060 . . . . . . . . . 10  |-  2  e.  CC
5352addid2i 9244 . . . . . . . . 9  |-  ( 0  +  2 )  =  2
543dec0h 10388 . . . . . . . . 9  |-  2  = ; 0 2
5553, 54eqtri 2455 . . . . . . . 8  |-  ( 0  +  2 )  = ; 0
2
5647mulid1i 9082 . . . . . . . . . 10  |-  ( 1  x.  1 )  =  1
57 00id 9231 . . . . . . . . . 10  |-  ( 0  +  0 )  =  0
5856, 57oveq12i 6085 . . . . . . . . 9  |-  ( ( 1  x.  1 )  +  ( 0  +  0 ) )  =  ( 1  +  0 )
5947addid1i 9243 . . . . . . . . 9  |-  ( 1  +  0 )  =  1
6058, 59eqtri 2455 . . . . . . . 8  |-  ( ( 1  x.  1 )  +  ( 0  +  0 ) )  =  1
6156oveq1i 6083 . . . . . . . . 9  |-  ( ( 1  x.  1 )  +  2 )  =  ( 1  +  2 )
62 2p1e3 10093 . . . . . . . . . 10  |-  ( 2  +  1 )  =  3
6352, 47, 62addcomli 9248 . . . . . . . . 9  |-  ( 1  +  2 )  =  3
6411dec0h 10388 . . . . . . . . 9  |-  3  = ; 0 3
6561, 63, 643eqtri 2459 . . . . . . . 8  |-  ( ( 1  x.  1 )  +  2 )  = ; 0
3
662, 2, 23, 3, 51, 55, 2, 11, 23, 60, 65decmac 10411 . . . . . . 7  |-  ( (; 1
1  x.  1 )  +  ( 0  +  2 ) )  = ; 1
3
6752mulid2i 9083 . . . . . . . . . 10  |-  ( 1  x.  2 )  =  2
6867, 57oveq12i 6085 . . . . . . . . 9  |-  ( ( 1  x.  2 )  +  ( 0  +  0 ) )  =  ( 2  +  0 )
6952addid1i 9243 . . . . . . . . 9  |-  ( 2  +  0 )  =  2
7068, 69eqtri 2455 . . . . . . . 8  |-  ( ( 1  x.  2 )  +  ( 0  +  0 ) )  =  2
7167oveq1i 6083 . . . . . . . . 9  |-  ( ( 1  x.  2 )  +  7 )  =  ( 2  +  7 )
72 7p2e9 10113 . . . . . . . . . 10  |-  ( 7  +  2 )  =  9
7341, 52, 72addcomli 9248 . . . . . . . . 9  |-  ( 2  +  7 )  =  9
7433dec0h 10388 . . . . . . . . 9  |-  9  = ; 0 9
7571, 73, 743eqtri 2459 . . . . . . . 8  |-  ( ( 1  x.  2 )  +  7 )  = ; 0
9
762, 2, 23, 16, 51, 37, 3, 33, 23, 70, 75decmac 10411 . . . . . . 7  |-  ( (; 1
1  x.  2 )  +  7 )  = ; 2
9
772, 3, 23, 16, 44, 50, 20, 33, 3, 66, 76decma2c 10412 . . . . . 6  |-  ( (; 1
1  x. ; 1 2 )  +  ( 1  +  6 ) )  = ;; 1 3 9
78 5nn 10126 . . . . . . . . . . 11  |-  5  e.  NN
7978nncni 10000 . . . . . . . . . 10  |-  5  e.  CC
8079mulid2i 9083 . . . . . . . . 9  |-  ( 1  x.  5 )  =  5
8180, 39oveq12i 6085 . . . . . . . 8  |-  ( ( 1  x.  5 )  +  ( 0  +  1 ) )  =  ( 5  +  1 )
82 5p1e6 10096 . . . . . . . 8  |-  ( 5  +  1 )  =  6
8381, 82eqtri 2455 . . . . . . 7  |-  ( ( 1  x.  5 )  +  ( 0  +  1 ) )  =  6
8480oveq1i 6083 . . . . . . . 8  |-  ( ( 1  x.  5 )  +  7 )  =  ( 5  +  7 )
85 7p5e12 10425 . . . . . . . . 9  |-  ( 7  +  5 )  = ; 1
2
8641, 79, 85addcomli 9248 . . . . . . . 8  |-  ( 5  +  7 )  = ; 1
2
8784, 86eqtri 2455 . . . . . . 7  |-  ( ( 1  x.  5 )  +  7 )  = ; 1
2
882, 2, 23, 16, 51, 37, 5, 3, 2, 83, 87decmac 10411 . . . . . 6  |-  ( (; 1
1  x.  5 )  +  7 )  = ; 6
2
894, 5, 2, 16, 36, 43, 20, 3, 25, 77, 88decma2c 10412 . . . . 5  |-  ( (; 1
1  x. ;; 1 2 5 )  +  ( 7  +  10 ) )  = ;;; 1 3 9 2
902dec0h 10388 . . . . . 6  |-  1  = ; 0 1
917nncni 10000 . . . . . . . . 9  |-  9  e.  CC
9291mulid2i 9083 . . . . . . . 8  |-  ( 1  x.  9 )  =  9
9392, 39oveq12i 6085 . . . . . . 7  |-  ( ( 1  x.  9 )  +  ( 0  +  1 ) )  =  ( 9  +  1 )
94 9p1e10 10100 . . . . . . 7  |-  ( 9  +  1 )  =  10
9593, 94eqtri 2455 . . . . . 6  |-  ( ( 1  x.  9 )  +  ( 0  +  1 ) )  =  10
9692oveq1i 6083 . . . . . . 7  |-  ( ( 1  x.  9 )  +  1 )  =  ( 9  +  1 )
9796, 94, 383eqtri 2459 . . . . . 6  |-  ( ( 1  x.  9 )  +  1 )  = ; 1
0
982, 2, 23, 2, 51, 90, 33, 23, 2, 95, 97decmac 10411 . . . . 5  |-  ( (; 1
1  x.  9 )  +  1 )  = ; 10 0
996, 33, 16, 2, 1, 34, 20, 23, 35, 89, 98decma2c 10412 . . . 4  |-  ( (; 1
1  x.  N )  + ; 7 1 )  = ;;;; 1 3 9 2 0
100 eqid 2435 . . . . 5  |- ; 1 6  = ; 1 6
1015, 3deccl 10386 . . . . . 6  |- ; 5 2  e.  NN0
102101, 3deccl 10386 . . . . 5  |- ;; 5 2 2  e.  NN0
103 eqid 2435 . . . . . 6  |- ;; 8 7 0  = ;; 8 7 0
104 eqid 2435 . . . . . 6  |- ;; 5 2 2  = ;; 5 2 2
105 eqid 2435 . . . . . . 7  |- ; 8 7  = ; 8 7
106101nn0cni 10223 . . . . . . . 8  |- ; 5 2  e.  CC
107106addid1i 9243 . . . . . . 7  |-  (; 5 2  +  0 )  = ; 5 2
108 8nn 10129 . . . . . . . . . . 11  |-  8  e.  NN
109108nncni 10000 . . . . . . . . . 10  |-  8  e.  CC
110109mulid1i 9082 . . . . . . . . 9  |-  ( 8  x.  1 )  =  8
11179addid1i 9243 . . . . . . . . 9  |-  ( 5  +  0 )  =  5
112110, 111oveq12i 6085 . . . . . . . 8  |-  ( ( 8  x.  1 )  +  ( 5  +  0 ) )  =  ( 8  +  5 )
113 8p5e13 10430 . . . . . . . 8  |-  ( 8  +  5 )  = ; 1
3
114112, 113eqtri 2455 . . . . . . 7  |-  ( ( 8  x.  1 )  +  ( 5  +  0 ) )  = ; 1
3
11541mulid1i 9082 . . . . . . . . 9  |-  ( 7  x.  1 )  =  7
116115oveq1i 6083 . . . . . . . 8  |-  ( ( 7  x.  1 )  +  2 )  =  ( 7  +  2 )
117116, 72, 743eqtri 2459 . . . . . . 7  |-  ( ( 7  x.  1 )  +  2 )  = ; 0
9
11812, 16, 5, 3, 105, 107, 2, 33, 23, 114, 117decmac 10411 . . . . . 6  |-  ( (; 8
7  x.  1 )  +  (; 5 2  +  0 ) )  = ;; 1 3 9
11947mul02i 9245 . . . . . . . 8  |-  ( 0  x.  1 )  =  0
120119oveq1i 6083 . . . . . . 7  |-  ( ( 0  x.  1 )  +  2 )  =  ( 0  +  2 )
121120, 53, 543eqtri 2459 . . . . . 6  |-  ( ( 0  x.  1 )  +  2 )  = ; 0
2
12222, 23, 101, 3, 103, 104, 2, 3, 23, 118, 121decmac 10411 . . . . 5  |-  ( (;; 8 7 0  x.  1 )  + ;; 5 2 2 )  = ;;; 1 3 9 2
123 8t6e48 10464 . . . . . . . . . 10  |-  ( 8  x.  6 )  = ; 4
8
124 4p1e5 10095 . . . . . . . . . 10  |-  ( 4  +  1 )  =  5
125 8p4e12 10429 . . . . . . . . . 10  |-  ( 8  +  4 )  = ; 1
2
12618, 12, 18, 123, 124, 3, 125decaddci 10417 . . . . . . . . 9  |-  ( ( 8  x.  6 )  +  4 )  = ; 5
2
127 7t6e42 10458 . . . . . . . . 9  |-  ( 7  x.  6 )  = ; 4
2
12825, 12, 16, 105, 3, 18, 126, 127decmul1c 10419 . . . . . . . 8  |-  (; 8 7  x.  6 )  = ;; 5 2 2
129128oveq1i 6083 . . . . . . 7  |-  ( (; 8
7  x.  6 )  +  0 )  =  (;; 5 2 2  +  0 )
130102nn0cni 10223 . . . . . . . 8  |- ;; 5 2 2  e.  CC
131130addid1i 9243 . . . . . . 7  |-  (;; 5 2 2  +  0 )  = ;; 5 2 2
132129, 131eqtri 2455 . . . . . 6  |-  ( (; 8
7  x.  6 )  +  0 )  = ;; 5 2 2
13346mul02i 9245 . . . . . . 7  |-  ( 0  x.  6 )  =  0
13423dec0h 10388 . . . . . . 7  |-  0  = ; 0 0
135133, 134eqtri 2455 . . . . . 6  |-  ( 0  x.  6 )  = ; 0
0
13625, 22, 23, 103, 23, 23, 132, 135decmul1c 10419 . . . . 5  |-  (;; 8 7 0  x.  6 )  = ;;; 5 2 2 0
13724, 2, 25, 100, 23, 102, 122, 136decmul2c 10420 . . . 4  |-  (;; 8 7 0  x. ; 1 6 )  = ;;;; 1 3 9 2 0
13899, 137eqtr4i 2458 . . 3  |-  ( (; 1
1  x.  N )  + ; 7 1 )  =  (;; 8 7 0  x. ; 1 6 )
1399, 10, 19, 21, 24, 17, 18, 26, 27, 29, 32, 138modxai 13394 . 2  |-  ( ( 2 ^; 3 8 )  mod 
N )  =  (; 7
1  mod  N )
140 eqid 2435 . . 3  |- ; 3 8  = ; 3 8
141 3cn 10062 . . . . . 6  |-  3  e.  CC
142 3t2e6 10118 . . . . . 6  |-  ( 3  x.  2 )  =  6
143141, 52, 142mulcomli 9087 . . . . 5  |-  ( 2  x.  3 )  =  6
144143oveq1i 6083 . . . 4  |-  ( ( 2  x.  3 )  +  1 )  =  ( 6  +  1 )
145144, 48eqtri 2455 . . 3  |-  ( ( 2  x.  3 )  +  1 )  =  7
146 8t2e16 10460 . . . 4  |-  ( 8  x.  2 )  = ; 1
6
147109, 52, 146mulcomli 9087 . . 3  |-  ( 2  x.  8 )  = ; 1
6
1483, 11, 12, 140, 25, 2, 145, 147decmul2c 10420 . 2  |-  ( 2  x. ; 3 8 )  = ; 7
6
1495dec0h 10388 . . . 4  |-  5  = ; 0 5
150 4cn 10064 . . . . . . 7  |-  4  e.  CC
151150addid2i 9244 . . . . . 6  |-  ( 0  +  4 )  =  4
15218dec0h 10388 . . . . . 6  |-  4  = ; 0 4
153151, 152eqtri 2455 . . . . 5  |-  ( 0  +  4 )  = ; 0
4
154150mulid1i 9082 . . . . . . . 8  |-  ( 4  x.  1 )  =  4
155154, 39oveq12i 6085 . . . . . . 7  |-  ( ( 4  x.  1 )  +  ( 0  +  1 ) )  =  ( 4  +  1 )
156155, 124eqtri 2455 . . . . . 6  |-  ( ( 4  x.  1 )  +  ( 0  +  1 ) )  =  5
157 4t2e8 10120 . . . . . . . 8  |-  ( 4  x.  2 )  =  8
158157oveq1i 6083 . . . . . . 7  |-  ( ( 4  x.  2 )  +  2 )  =  ( 8  +  2 )
159 8p2e10 10115 . . . . . . 7  |-  ( 8  +  2 )  =  10
160158, 159, 383eqtri 2459 . . . . . 6  |-  ( ( 4  x.  2 )  +  2 )  = ; 1
0
1612, 3, 23, 3, 44, 55, 18, 23, 2, 156, 160decma2c 10412 . . . . 5  |-  ( ( 4  x. ; 1 2 )  +  ( 0  +  2 ) )  = ; 5 0
162 5t4e20 10447 . . . . . . 7  |-  ( 5  x.  4 )  = ; 2
0
16379, 150, 162mulcomli 9087 . . . . . 6  |-  ( 4  x.  5 )  = ; 2
0
1643, 23, 18, 163, 151decaddi 10416 . . . . 5  |-  ( ( 4  x.  5 )  +  4 )  = ; 2
4
1654, 5, 23, 18, 36, 153, 18, 18, 3, 161, 164decma2c 10412 . . . 4  |-  ( ( 4  x. ;; 1 2 5 )  +  ( 0  +  4 ) )  = ;; 5 0 4
166 9t4e36 10469 . . . . . 6  |-  ( 9  x.  4 )  = ; 3
6
16791, 150, 166mulcomli 9087 . . . . 5  |-  ( 4  x.  9 )  = ; 3
6
168 3p1e4 10094 . . . . 5  |-  ( 3  +  1 )  =  4
169 6p5e11 10422 . . . . 5  |-  ( 6  +  5 )  = ; 1
1
17011, 25, 5, 167, 168, 2, 169decaddci 10417 . . . 4  |-  ( ( 4  x.  9 )  +  5 )  = ; 4
1
1716, 33, 23, 5, 1, 149, 18, 2, 18, 165, 170decma2c 10412 . . 3  |-  ( ( 4  x.  N )  +  5 )  = ;;; 5 0 4 1
17239oveq2i 6084 . . . . . 6  |-  ( ( 7  x.  7 )  +  ( 0  +  1 ) )  =  ( ( 7  x.  7 )  +  1 )
173 7t7e49 10459 . . . . . . 7  |-  ( 7  x.  7 )  = ; 4
9
17418, 124, 173decsucc 10399 . . . . . 6  |-  ( ( 7  x.  7 )  +  1 )  = ; 5
0
175172, 174eqtri 2455 . . . . 5  |-  ( ( 7  x.  7 )  +  ( 0  +  1 ) )  = ; 5
0
17641mulid2i 9083 . . . . . . 7  |-  ( 1  x.  7 )  =  7
177176oveq1i 6083 . . . . . 6  |-  ( ( 1  x.  7 )  +  7 )  =  ( 7  +  7 )
178 7p7e14 10427 . . . . . 6  |-  ( 7  +  7 )  = ; 1
4
179177, 178eqtri 2455 . . . . 5  |-  ( ( 1  x.  7 )  +  7 )  = ; 1
4
18016, 2, 23, 16, 34, 37, 16, 18, 2, 175, 179decmac 10411 . . . 4  |-  ( (; 7
1  x.  7 )  +  7 )  = ;; 5 0 4
18117nn0cni 10223 . . . . 5  |- ; 7 1  e.  CC
182181mulid1i 9082 . . . 4  |-  (; 7 1  x.  1 )  = ; 7 1
18317, 16, 2, 34, 2, 16, 180, 182decmul2c 10420 . . 3  |-  (; 7 1  x. ; 7 1 )  = ;;; 5 0 4 1
184171, 183eqtr4i 2458 . 2  |-  ( ( 4  x.  N )  +  5 )  =  (; 7 1  x. ; 7 1 )
1859, 10, 13, 15, 17, 5, 139, 148, 184mod2xi 13395 1  |-  ( ( 2 ^; 7 6 )  mod 
N )  =  ( 5  mod  N )
Colors of variables: wff set class
Syntax hints:    = wceq 1652  (class class class)co 6073   0cc0 8980   1c1 8981    + caddc 8983    x. cmul 8985   NNcn 9990   2c2 10039   3c3 10040   4c4 10041   5c5 10042   6c6 10043   7c7 10044   8c8 10045   9c9 10046   10c10 10047  ;cdc 10372    mod cmo 11240   ^cexp 11372
This theorem is referenced by:  1259lem4  13443
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-cnex 9036  ax-resscn 9037  ax-1cn 9038  ax-icn 9039  ax-addcl 9040  ax-addrcl 9041  ax-mulcl 9042  ax-mulrcl 9043  ax-mulcom 9044  ax-addass 9045  ax-mulass 9046  ax-distr 9047  ax-i2m1 9048  ax-1ne0 9049  ax-1rid 9050  ax-rnegex 9051  ax-rrecex 9052  ax-cnre 9053  ax-pre-lttri 9054  ax-pre-lttrn 9055  ax-pre-ltadd 9056  ax-pre-mulgt0 9057  ax-pre-sup 9058
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-2nd 6342  df-riota 6541  df-recs 6625  df-rdg 6660  df-er 6897  df-en 7102  df-dom 7103  df-sdom 7104  df-sup 7438  df-pnf 9112  df-mnf 9113  df-xr 9114  df-ltxr 9115  df-le 9116  df-sub 9283  df-neg 9284  df-div 9668  df-nn 9991  df-2 10048  df-3 10049  df-4 10050  df-5 10051  df-6 10052  df-7 10053  df-8 10054  df-9 10055  df-10 10056  df-n0 10212  df-z 10273  df-dec 10373  df-uz 10479  df-rp 10603  df-fl 11192  df-mod 11241  df-seq 11314  df-exp 11373
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