MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  1259lem3 Unicode version

Theorem 1259lem3 13131
Description: Lemma for 1259prm 13134. 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 9981 . . . . . 6  |-  1  e.  NN0
3 2nn0 9982 . . . . . 6  |-  2  e.  NN0
42, 3deccl 10138 . . . . 5  |- ; 1 2  e.  NN0
5 5nn0 9985 . . . . 5  |-  5  e.  NN0
64, 5deccl 10138 . . . 4  |- ;; 1 2 5  e.  NN0
7 9nn 9884 . . . 4  |-  9  e.  NN
86, 7decnncl 10137 . . 3  |- ;;; 1 2 5 9  e.  NN
91, 8eqeltri 2353 . 2  |-  N  e.  NN
10 2nn 9877 . 2  |-  2  e.  NN
11 3nn0 9983 . . 3  |-  3  e.  NN0
12 8nn0 9988 . . 3  |-  8  e.  NN0
1311, 12deccl 10138 . 2  |- ; 3 8  e.  NN0
14 4nn 9879 . . 3  |-  4  e.  NN
1514nnzi 10047 . 2  |-  4  e.  ZZ
16 7nn0 9987 . . 3  |-  7  e.  NN0
1716, 2deccl 10138 . 2  |- ; 7 1  e.  NN0
18 4nn0 9984 . . . 4  |-  4  e.  NN0
1911, 18deccl 10138 . . 3  |- ; 3 4  e.  NN0
202, 2deccl 10138 . . . 4  |- ; 1 1  e.  NN0
2120nn0zi 10048 . . 3  |- ; 1 1  e.  ZZ
2212, 16deccl 10138 . . . 4  |- ; 8 7  e.  NN0
23 0nn0 9980 . . . 4  |-  0  e.  NN0
2422, 23deccl 10138 . . 3  |- ;; 8 7 0  e.  NN0
25 6nn0 9986 . . . 4  |-  6  e.  NN0
262, 25deccl 10138 . . 3  |- ; 1 6  e.  NN0
2711259lem2 13130 . . 3  |-  ( ( 2 ^; 3 4 )  mod 
N )  =  (;; 8 7 0  mod 
N )
28 2exp4 13100 . . . 4  |-  ( 2 ^ 4 )  = ; 1
6
2928oveq1i 5868 . . 3  |-  ( ( 2 ^ 4 )  mod  N )  =  (; 1 6  mod  N
)
30 eqid 2283 . . . 4  |- ; 3 4  = ; 3 4
31 4p4e8 9859 . . . 4  |-  ( 4  +  4 )  =  8
3211, 18, 18, 30, 31decaddi 10168 . . 3  |-  (; 3 4  +  4 )  = ; 3 8
33 9nn0 9989 . . . . 5  |-  9  e.  NN0
34 eqid 2283 . . . . 5  |- ; 7 1  = ; 7 1
35 10nn0 9990 . . . . 5  |-  10  e.  NN0
36 eqid 2283 . . . . . 6  |- ;; 1 2 5  = ;; 1 2 5
3716dec0h 10140 . . . . . . 7  |-  7  = ; 0 7
38 dec10 10154 . . . . . . 7  |-  10  = ; 1 0
39 0p1e1 9839 . . . . . . 7  |-  ( 0  +  1 )  =  1
40 7nn 9882 . . . . . . . . 9  |-  7  e.  NN
4140nncni 9756 . . . . . . . 8  |-  7  e.  CC
4241addid1i 8999 . . . . . . 7  |-  ( 7  +  0 )  =  7
4323, 16, 2, 23, 37, 38, 39, 42decadd 10165 . . . . . 6  |-  ( 7  +  10 )  = ; 1
7
44 eqid 2283 . . . . . . 7  |- ; 1 2  = ; 1 2
45 6nn 9881 . . . . . . . . . 10  |-  6  e.  NN
4645nncni 9756 . . . . . . . . 9  |-  6  e.  CC
47 ax-1cn 8795 . . . . . . . . 9  |-  1  e.  CC
48 6p1e7 9851 . . . . . . . . 9  |-  ( 6  +  1 )  =  7
4946, 47, 48addcomli 9004 . . . . . . . 8  |-  ( 1  +  6 )  =  7
5049, 37eqtri 2303 . . . . . . 7  |-  ( 1  +  6 )  = ; 0
7
51 eqid 2283 . . . . . . . 8  |- ; 1 1  = ; 1 1
52 2cn 9816 . . . . . . . . . 10  |-  2  e.  CC
5352addid2i 9000 . . . . . . . . 9  |-  ( 0  +  2 )  =  2
543dec0h 10140 . . . . . . . . 9  |-  2  = ; 0 2
5553, 54eqtri 2303 . . . . . . . 8  |-  ( 0  +  2 )  = ; 0
2
5647mulid1i 8839 . . . . . . . . . 10  |-  ( 1  x.  1 )  =  1
57 00id 8987 . . . . . . . . . 10  |-  ( 0  +  0 )  =  0
5856, 57oveq12i 5870 . . . . . . . . 9  |-  ( ( 1  x.  1 )  +  ( 0  +  0 ) )  =  ( 1  +  0 )
5947addid1i 8999 . . . . . . . . 9  |-  ( 1  +  0 )  =  1
6058, 59eqtri 2303 . . . . . . . 8  |-  ( ( 1  x.  1 )  +  ( 0  +  0 ) )  =  1
6156oveq1i 5868 . . . . . . . . 9  |-  ( ( 1  x.  1 )  +  2 )  =  ( 1  +  2 )
62 2p1e3 9847 . . . . . . . . . 10  |-  ( 2  +  1 )  =  3
6352, 47, 62addcomli 9004 . . . . . . . . 9  |-  ( 1  +  2 )  =  3
6411dec0h 10140 . . . . . . . . 9  |-  3  = ; 0 3
6561, 63, 643eqtri 2307 . . . . . . . 8  |-  ( ( 1  x.  1 )  +  2 )  = ; 0
3
662, 2, 23, 3, 51, 55, 2, 11, 23, 60, 65decmac 10163 . . . . . . 7  |-  ( (; 1
1  x.  1 )  +  ( 0  +  2 ) )  = ; 1
3
6752mulid2i 8840 . . . . . . . . . 10  |-  ( 1  x.  2 )  =  2
6867, 57oveq12i 5870 . . . . . . . . 9  |-  ( ( 1  x.  2 )  +  ( 0  +  0 ) )  =  ( 2  +  0 )
6952addid1i 8999 . . . . . . . . 9  |-  ( 2  +  0 )  =  2
7068, 69eqtri 2303 . . . . . . . 8  |-  ( ( 1  x.  2 )  +  ( 0  +  0 ) )  =  2
7167oveq1i 5868 . . . . . . . . 9  |-  ( ( 1  x.  2 )  +  7 )  =  ( 2  +  7 )
72 7p2e9 9867 . . . . . . . . . 10  |-  ( 7  +  2 )  =  9
7341, 52, 72addcomli 9004 . . . . . . . . 9  |-  ( 2  +  7 )  =  9
7433dec0h 10140 . . . . . . . . 9  |-  9  = ; 0 9
7571, 73, 743eqtri 2307 . . . . . . . 8  |-  ( ( 1  x.  2 )  +  7 )  = ; 0
9
762, 2, 23, 16, 51, 37, 3, 33, 23, 70, 75decmac 10163 . . . . . . 7  |-  ( (; 1
1  x.  2 )  +  7 )  = ; 2
9
772, 3, 23, 16, 44, 50, 20, 33, 3, 66, 76decma2c 10164 . . . . . 6  |-  ( (; 1
1  x. ; 1 2 )  +  ( 1  +  6 ) )  = ;; 1 3 9
78 5nn 9880 . . . . . . . . . . 11  |-  5  e.  NN
7978nncni 9756 . . . . . . . . . 10  |-  5  e.  CC
8079mulid2i 8840 . . . . . . . . 9  |-  ( 1  x.  5 )  =  5
8180, 39oveq12i 5870 . . . . . . . 8  |-  ( ( 1  x.  5 )  +  ( 0  +  1 ) )  =  ( 5  +  1 )
82 5p1e6 9850 . . . . . . . 8  |-  ( 5  +  1 )  =  6
8381, 82eqtri 2303 . . . . . . 7  |-  ( ( 1  x.  5 )  +  ( 0  +  1 ) )  =  6
8480oveq1i 5868 . . . . . . . 8  |-  ( ( 1  x.  5 )  +  7 )  =  ( 5  +  7 )
85 7p5e12 10177 . . . . . . . . 9  |-  ( 7  +  5 )  = ; 1
2
8641, 79, 85addcomli 9004 . . . . . . . 8  |-  ( 5  +  7 )  = ; 1
2
8784, 86eqtri 2303 . . . . . . 7  |-  ( ( 1  x.  5 )  +  7 )  = ; 1
2
882, 2, 23, 16, 51, 37, 5, 3, 2, 83, 87decmac 10163 . . . . . 6  |-  ( (; 1
1  x.  5 )  +  7 )  = ; 6
2
894, 5, 2, 16, 36, 43, 20, 3, 25, 77, 88decma2c 10164 . . . . 5  |-  ( (; 1
1  x. ;; 1 2 5 )  +  ( 7  +  10 ) )  = ;;; 1 3 9 2
902dec0h 10140 . . . . . 6  |-  1  = ; 0 1
917nncni 9756 . . . . . . . . 9  |-  9  e.  CC
9291mulid2i 8840 . . . . . . . 8  |-  ( 1  x.  9 )  =  9
9392, 39oveq12i 5870 . . . . . . 7  |-  ( ( 1  x.  9 )  +  ( 0  +  1 ) )  =  ( 9  +  1 )
94 9p1e10 9854 . . . . . . 7  |-  ( 9  +  1 )  =  10
9593, 94eqtri 2303 . . . . . 6  |-  ( ( 1  x.  9 )  +  ( 0  +  1 ) )  =  10
9692oveq1i 5868 . . . . . . 7  |-  ( ( 1  x.  9 )  +  1 )  =  ( 9  +  1 )
9796, 94, 383eqtri 2307 . . . . . 6  |-  ( ( 1  x.  9 )  +  1 )  = ; 1
0
982, 2, 23, 2, 51, 90, 33, 23, 2, 95, 97decmac 10163 . . . . 5  |-  ( (; 1
1  x.  9 )  +  1 )  = ; 10 0
996, 33, 16, 2, 1, 34, 20, 23, 35, 89, 98decma2c 10164 . . . 4  |-  ( (; 1
1  x.  N )  + ; 7 1 )  = ;;;; 1 3 9 2 0
100 eqid 2283 . . . . 5  |- ; 1 6  = ; 1 6
1015, 3deccl 10138 . . . . . 6  |- ; 5 2  e.  NN0
102101, 3deccl 10138 . . . . 5  |- ;; 5 2 2  e.  NN0
103 eqid 2283 . . . . . 6  |- ;; 8 7 0  = ;; 8 7 0
104 eqid 2283 . . . . . 6  |- ;; 5 2 2  = ;; 5 2 2
105 eqid 2283 . . . . . . 7  |- ; 8 7  = ; 8 7
106101nn0cni 9977 . . . . . . . 8  |- ; 5 2  e.  CC
107106addid1i 8999 . . . . . . 7  |-  (; 5 2  +  0 )  = ; 5 2
108 8nn 9883 . . . . . . . . . . 11  |-  8  e.  NN
109108nncni 9756 . . . . . . . . . 10  |-  8  e.  CC
110109mulid1i 8839 . . . . . . . . 9  |-  ( 8  x.  1 )  =  8
11179addid1i 8999 . . . . . . . . 9  |-  ( 5  +  0 )  =  5
112110, 111oveq12i 5870 . . . . . . . 8  |-  ( ( 8  x.  1 )  +  ( 5  +  0 ) )  =  ( 8  +  5 )
113 8p5e13 10182 . . . . . . . 8  |-  ( 8  +  5 )  = ; 1
3
114112, 113eqtri 2303 . . . . . . 7  |-  ( ( 8  x.  1 )  +  ( 5  +  0 ) )  = ; 1
3
11541mulid1i 8839 . . . . . . . . 9  |-  ( 7  x.  1 )  =  7
116115oveq1i 5868 . . . . . . . 8  |-  ( ( 7  x.  1 )  +  2 )  =  ( 7  +  2 )
117116, 72, 743eqtri 2307 . . . . . . 7  |-  ( ( 7  x.  1 )  +  2 )  = ; 0
9
11812, 16, 5, 3, 105, 107, 2, 33, 23, 114, 117decmac 10163 . . . . . 6  |-  ( (; 8
7  x.  1 )  +  (; 5 2  +  0 ) )  = ;; 1 3 9
11947mul02i 9001 . . . . . . . 8  |-  ( 0  x.  1 )  =  0
120119oveq1i 5868 . . . . . . 7  |-  ( ( 0  x.  1 )  +  2 )  =  ( 0  +  2 )
121120, 53, 543eqtri 2307 . . . . . 6  |-  ( ( 0  x.  1 )  +  2 )  = ; 0
2
12222, 23, 101, 3, 103, 104, 2, 3, 23, 118, 121decmac 10163 . . . . 5  |-  ( (;; 8 7 0  x.  1 )  + ;; 5 2 2 )  = ;;; 1 3 9 2
123 8t6e48 10216 . . . . . . . . . 10  |-  ( 8  x.  6 )  = ; 4
8
124 4p1e5 9849 . . . . . . . . . 10  |-  ( 4  +  1 )  =  5
125 8p4e12 10181 . . . . . . . . . 10  |-  ( 8  +  4 )  = ; 1
2
12618, 12, 18, 123, 124, 3, 125decaddci 10169 . . . . . . . . 9  |-  ( ( 8  x.  6 )  +  4 )  = ; 5
2
127 7t6e42 10210 . . . . . . . . 9  |-  ( 7  x.  6 )  = ; 4
2
12825, 12, 16, 105, 3, 18, 126, 127decmul1c 10171 . . . . . . . 8  |-  (; 8 7  x.  6 )  = ;; 5 2 2
129128oveq1i 5868 . . . . . . 7  |-  ( (; 8
7  x.  6 )  +  0 )  =  (;; 5 2 2  +  0 )
130102nn0cni 9977 . . . . . . . 8  |- ;; 5 2 2  e.  CC
131130addid1i 8999 . . . . . . 7  |-  (;; 5 2 2  +  0 )  = ;; 5 2 2
132129, 131eqtri 2303 . . . . . 6  |-  ( (; 8
7  x.  6 )  +  0 )  = ;; 5 2 2
13346mul02i 9001 . . . . . . 7  |-  ( 0  x.  6 )  =  0
13423dec0h 10140 . . . . . . 7  |-  0  = ; 0 0
135133, 134eqtri 2303 . . . . . 6  |-  ( 0  x.  6 )  = ; 0
0
13625, 22, 23, 103, 23, 23, 132, 135decmul1c 10171 . . . . 5  |-  (;; 8 7 0  x.  6 )  = ;;; 5 2 2 0
13724, 2, 25, 100, 23, 102, 122, 136decmul2c 10172 . . . 4  |-  (;; 8 7 0  x. ; 1 6 )  = ;;;; 1 3 9 2 0
13899, 137eqtr4i 2306 . . 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 13083 . 2  |-  ( ( 2 ^; 3 8 )  mod 
N )  =  (; 7
1  mod  N )
140 eqid 2283 . . 3  |- ; 3 8  = ; 3 8
141 3cn 9818 . . . . . 6  |-  3  e.  CC
142 3t2e6 9872 . . . . . 6  |-  ( 3  x.  2 )  =  6
143141, 52, 142mulcomli 8844 . . . . 5  |-  ( 2  x.  3 )  =  6
144143oveq1i 5868 . . . 4  |-  ( ( 2  x.  3 )  +  1 )  =  ( 6  +  1 )
145144, 48eqtri 2303 . . 3  |-  ( ( 2  x.  3 )  +  1 )  =  7
146 8t2e16 10212 . . . 4  |-  ( 8  x.  2 )  = ; 1
6
147109, 52, 146mulcomli 8844 . . 3  |-  ( 2  x.  8 )  = ; 1
6
1483, 11, 12, 140, 25, 2, 145, 147decmul2c 10172 . 2  |-  ( 2  x. ; 3 8 )  = ; 7
6
1495dec0h 10140 . . . 4  |-  5  = ; 0 5
150 4cn 9820 . . . . . . 7  |-  4  e.  CC
151150addid2i 9000 . . . . . 6  |-  ( 0  +  4 )  =  4
15218dec0h 10140 . . . . . 6  |-  4  = ; 0 4
153151, 152eqtri 2303 . . . . 5  |-  ( 0  +  4 )  = ; 0
4
154150mulid1i 8839 . . . . . . . 8  |-  ( 4  x.  1 )  =  4
155154, 39oveq12i 5870 . . . . . . 7  |-  ( ( 4  x.  1 )  +  ( 0  +  1 ) )  =  ( 4  +  1 )
156155, 124eqtri 2303 . . . . . 6  |-  ( ( 4  x.  1 )  +  ( 0  +  1 ) )  =  5
157 4t2e8 9874 . . . . . . . 8  |-  ( 4  x.  2 )  =  8
158157oveq1i 5868 . . . . . . 7  |-  ( ( 4  x.  2 )  +  2 )  =  ( 8  +  2 )
159 8p2e10 9869 . . . . . . 7  |-  ( 8  +  2 )  =  10
160158, 159, 383eqtri 2307 . . . . . 6  |-  ( ( 4  x.  2 )  +  2 )  = ; 1
0
1612, 3, 23, 3, 44, 55, 18, 23, 2, 156, 160decma2c 10164 . . . . 5  |-  ( ( 4  x. ; 1 2 )  +  ( 0  +  2 ) )  = ; 5 0
162 5t4e20 10199 . . . . . . 7  |-  ( 5  x.  4 )  = ; 2
0
16379, 150, 162mulcomli 8844 . . . . . 6  |-  ( 4  x.  5 )  = ; 2
0
1643, 23, 18, 163, 151decaddi 10168 . . . . 5  |-  ( ( 4  x.  5 )  +  4 )  = ; 2
4
1654, 5, 23, 18, 36, 153, 18, 18, 3, 161, 164decma2c 10164 . . . 4  |-  ( ( 4  x. ;; 1 2 5 )  +  ( 0  +  4 ) )  = ;; 5 0 4
166 9t4e36 10221 . . . . . 6  |-  ( 9  x.  4 )  = ; 3
6
16791, 150, 166mulcomli 8844 . . . . 5  |-  ( 4  x.  9 )  = ; 3
6
168 3p1e4 9848 . . . . 5  |-  ( 3  +  1 )  =  4
169 6p5e11 10174 . . . . 5  |-  ( 6  +  5 )  = ; 1
1
17011, 25, 5, 167, 168, 2, 169decaddci 10169 . . . 4  |-  ( ( 4  x.  9 )  +  5 )  = ; 4
1
1716, 33, 23, 5, 1, 149, 18, 2, 18, 165, 170decma2c 10164 . . 3  |-  ( ( 4  x.  N )  +  5 )  = ;;; 5 0 4 1
17239oveq2i 5869 . . . . . 6  |-  ( ( 7  x.  7 )  +  ( 0  +  1 ) )  =  ( ( 7  x.  7 )  +  1 )
173 7t7e49 10211 . . . . . . 7  |-  ( 7  x.  7 )  = ; 4
9
17418, 124, 173decsucc 10151 . . . . . 6  |-  ( ( 7  x.  7 )  +  1 )  = ; 5
0
175172, 174eqtri 2303 . . . . 5  |-  ( ( 7  x.  7 )  +  ( 0  +  1 ) )  = ; 5
0
17641mulid2i 8840 . . . . . . 7  |-  ( 1  x.  7 )  =  7
177176oveq1i 5868 . . . . . 6  |-  ( ( 1  x.  7 )  +  7 )  =  ( 7  +  7 )
178 7p7e14 10179 . . . . . 6  |-  ( 7  +  7 )  = ; 1
4
179177, 178eqtri 2303 . . . . 5  |-  ( ( 1  x.  7 )  +  7 )  = ; 1
4
18016, 2, 23, 16, 34, 37, 16, 18, 2, 175, 179decmac 10163 . . . 4  |-  ( (; 7
1  x.  7 )  +  7 )  = ;; 5 0 4
18117nn0cni 9977 . . . . 5  |- ; 7 1  e.  CC
182181mulid1i 8839 . . . 4  |-  (; 7 1  x.  1 )  = ; 7 1
18317, 16, 2, 34, 2, 16, 180, 182decmul2c 10172 . . 3  |-  (; 7 1  x. ; 7 1 )  = ;;; 5 0 4 1
184171, 183eqtr4i 2306 . 2  |-  ( ( 4  x.  N )  +  5 )  =  (; 7 1  x. ; 7 1 )
1859, 10, 13, 15, 17, 5, 139, 148, 184mod2xi 13084 1  |-  ( ( 2 ^; 7 6 )  mod 
N )  =  ( 5  mod  N )
Colors of variables: wff set class
Syntax hints:    = wceq 1623  (class class class)co 5858   0cc0 8737   1c1 8738    + caddc 8740    x. cmul 8742   NNcn 9746   2c2 9795   3c3 9796   4c4 9797   5c5 9798   6c6 9799   7c7 9800   8c8 9801   9c9 9802   10c10 9803  ;cdc 10124    mod cmo 10973   ^cexp 11104
This theorem is referenced by:  1259lem4  13132
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814  ax-pre-sup 8815
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-sup 7194  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-nn 9747  df-2 9804  df-3 9805  df-4 9806  df-5 9807  df-6 9808  df-7 9809  df-8 9810  df-9 9811  df-10 9812  df-n0 9966  df-z 10025  df-dec 10125  df-uz 10231  df-rp 10355  df-fl 10925  df-mod 10974  df-seq 11047  df-exp 11105
  Copyright terms: Public domain W3C validator