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Theorem 1259lem3 15091
Description: Lemma for 1259prm 15094. 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 10885 . . . . . 6  |-  1  e.  NN0
3 2nn0 10886 . . . . . 6  |-  2  e.  NN0
42, 3deccl 11065 . . . . 5  |- ; 1 2  e.  NN0
5 5nn0 10889 . . . . 5  |-  5  e.  NN0
64, 5deccl 11065 . . . 4  |- ;; 1 2 5  e.  NN0
7 9nn 10774 . . . 4  |-  9  e.  NN
86, 7decnncl 11064 . . 3  |- ;;; 1 2 5 9  e.  NN
91, 8eqeltri 2506 . 2  |-  N  e.  NN
10 2nn 10767 . 2  |-  2  e.  NN
11 3nn0 10887 . . 3  |-  3  e.  NN0
12 8nn0 10892 . . 3  |-  8  e.  NN0
1311, 12deccl 11065 . 2  |- ; 3 8  e.  NN0
14 4z 10971 . 2  |-  4  e.  ZZ
15 7nn0 10891 . . 3  |-  7  e.  NN0
1615, 2deccl 11065 . 2  |- ; 7 1  e.  NN0
17 4nn0 10888 . . . 4  |-  4  e.  NN0
1811, 17deccl 11065 . . 3  |- ; 3 4  e.  NN0
192, 2deccl 11065 . . . 4  |- ; 1 1  e.  NN0
2019nn0zi 10962 . . 3  |- ; 1 1  e.  ZZ
2112, 15deccl 11065 . . . 4  |- ; 8 7  e.  NN0
22 0nn0 10884 . . . 4  |-  0  e.  NN0
2321, 22deccl 11065 . . 3  |- ;; 8 7 0  e.  NN0
24 6nn0 10890 . . . 4  |-  6  e.  NN0
252, 24deccl 11065 . . 3  |- ; 1 6  e.  NN0
2611259lem2 15090 . . 3  |-  ( ( 2 ^; 3 4 )  mod 
N )  =  (;; 8 7 0  mod 
N )
27 2exp4 15044 . . . 4  |-  ( 2 ^ 4 )  = ; 1
6
2827oveq1i 6311 . . 3  |-  ( ( 2 ^ 4 )  mod  N )  =  (; 1 6  mod  N
)
29 eqid 2422 . . . 4  |- ; 3 4  = ; 3 4
30 4p4e8 10746 . . . 4  |-  ( 4  +  4 )  =  8
3111, 17, 17, 29, 30decaddi 11095 . . 3  |-  (; 3 4  +  4 )  = ; 3 8
32 9nn0 10893 . . . . 5  |-  9  e.  NN0
33 eqid 2422 . . . . 5  |- ; 7 1  = ; 7 1
34 10nn0 10894 . . . . 5  |-  10  e.  NN0
35 eqid 2422 . . . . . 6  |- ;; 1 2 5  = ;; 1 2 5
3615dec0h 11067 . . . . . . 7  |-  7  = ; 0 7
37 dec10 11081 . . . . . . 7  |-  10  = ; 1 0
38 0p1e1 10721 . . . . . . 7  |-  ( 0  +  1 )  =  1
39 7cn 10693 . . . . . . . 8  |-  7  e.  CC
4039addid1i 9820 . . . . . . 7  |-  ( 7  +  0 )  =  7
4122, 15, 2, 22, 36, 37, 38, 40decadd 11092 . . . . . 6  |-  ( 7  +  10 )  = ; 1
7
42 eqid 2422 . . . . . . 7  |- ; 1 2  = ; 1 2
43 6cn 10691 . . . . . . . . 9  |-  6  e.  CC
44 ax-1cn 9597 . . . . . . . . 9  |-  1  e.  CC
45 6p1e7 10738 . . . . . . . . 9  |-  ( 6  +  1 )  =  7
4643, 44, 45addcomli 9825 . . . . . . . 8  |-  ( 1  +  6 )  =  7
4746, 36eqtri 2451 . . . . . . 7  |-  ( 1  +  6 )  = ; 0
7
48 eqid 2422 . . . . . . . 8  |- ; 1 1  = ; 1 1
49 2cn 10680 . . . . . . . . . 10  |-  2  e.  CC
5049addid2i 9821 . . . . . . . . 9  |-  ( 0  +  2 )  =  2
513dec0h 11067 . . . . . . . . 9  |-  2  = ; 0 2
5250, 51eqtri 2451 . . . . . . . 8  |-  ( 0  +  2 )  = ; 0
2
5344mulid1i 9645 . . . . . . . . . 10  |-  ( 1  x.  1 )  =  1
54 00id 9808 . . . . . . . . . 10  |-  ( 0  +  0 )  =  0
5553, 54oveq12i 6313 . . . . . . . . 9  |-  ( ( 1  x.  1 )  +  ( 0  +  0 ) )  =  ( 1  +  0 )
5644addid1i 9820 . . . . . . . . 9  |-  ( 1  +  0 )  =  1
5755, 56eqtri 2451 . . . . . . . 8  |-  ( ( 1  x.  1 )  +  ( 0  +  0 ) )  =  1
5853oveq1i 6311 . . . . . . . . 9  |-  ( ( 1  x.  1 )  +  2 )  =  ( 1  +  2 )
59 1p2e3 10734 . . . . . . . . 9  |-  ( 1  +  2 )  =  3
6011dec0h 11067 . . . . . . . . 9  |-  3  = ; 0 3
6158, 59, 603eqtri 2455 . . . . . . . 8  |-  ( ( 1  x.  1 )  +  2 )  = ; 0
3
622, 2, 22, 3, 48, 52, 2, 11, 22, 57, 61decmac 11090 . . . . . . 7  |-  ( (; 1
1  x.  1 )  +  ( 0  +  2 ) )  = ; 1
3
6349mulid2i 9646 . . . . . . . . . 10  |-  ( 1  x.  2 )  =  2
6463, 54oveq12i 6313 . . . . . . . . 9  |-  ( ( 1  x.  2 )  +  ( 0  +  0 ) )  =  ( 2  +  0 )
6549addid1i 9820 . . . . . . . . 9  |-  ( 2  +  0 )  =  2
6664, 65eqtri 2451 . . . . . . . 8  |-  ( ( 1  x.  2 )  +  ( 0  +  0 ) )  =  2
6763oveq1i 6311 . . . . . . . . 9  |-  ( ( 1  x.  2 )  +  7 )  =  ( 2  +  7 )
68 7p2e9 10754 . . . . . . . . . 10  |-  ( 7  +  2 )  =  9
6939, 49, 68addcomli 9825 . . . . . . . . 9  |-  ( 2  +  7 )  =  9
7032dec0h 11067 . . . . . . . . 9  |-  9  = ; 0 9
7167, 69, 703eqtri 2455 . . . . . . . 8  |-  ( ( 1  x.  2 )  +  7 )  = ; 0
9
722, 2, 22, 15, 48, 36, 3, 32, 22, 66, 71decmac 11090 . . . . . . 7  |-  ( (; 1
1  x.  2 )  +  7 )  = ; 2
9
732, 3, 22, 15, 42, 47, 19, 32, 3, 62, 72decma2c 11091 . . . . . 6  |-  ( (; 1
1  x. ; 1 2 )  +  ( 1  +  6 ) )  = ;; 1 3 9
74 5cn 10689 . . . . . . . . . 10  |-  5  e.  CC
7574mulid2i 9646 . . . . . . . . 9  |-  ( 1  x.  5 )  =  5
7675, 38oveq12i 6313 . . . . . . . 8  |-  ( ( 1  x.  5 )  +  ( 0  +  1 ) )  =  ( 5  +  1 )
77 5p1e6 10737 . . . . . . . 8  |-  ( 5  +  1 )  =  6
7876, 77eqtri 2451 . . . . . . 7  |-  ( ( 1  x.  5 )  +  ( 0  +  1 ) )  =  6
7975oveq1i 6311 . . . . . . . 8  |-  ( ( 1  x.  5 )  +  7 )  =  ( 5  +  7 )
80 7p5e12 11104 . . . . . . . . 9  |-  ( 7  +  5 )  = ; 1
2
8139, 74, 80addcomli 9825 . . . . . . . 8  |-  ( 5  +  7 )  = ; 1
2
8279, 81eqtri 2451 . . . . . . 7  |-  ( ( 1  x.  5 )  +  7 )  = ; 1
2
832, 2, 22, 15, 48, 36, 5, 3, 2, 78, 82decmac 11090 . . . . . 6  |-  ( (; 1
1  x.  5 )  +  7 )  = ; 6
2
844, 5, 2, 15, 35, 41, 19, 3, 24, 73, 83decma2c 11091 . . . . 5  |-  ( (; 1
1  x. ;; 1 2 5 )  +  ( 7  +  10 ) )  = ;;; 1 3 9 2
852dec0h 11067 . . . . . 6  |-  1  = ; 0 1
86 9cn 10697 . . . . . . . . 9  |-  9  e.  CC
8786mulid2i 9646 . . . . . . . 8  |-  ( 1  x.  9 )  =  9
8887, 38oveq12i 6313 . . . . . . 7  |-  ( ( 1  x.  9 )  +  ( 0  +  1 ) )  =  ( 9  +  1 )
89 9p1e10 10741 . . . . . . 7  |-  ( 9  +  1 )  =  10
9088, 89eqtri 2451 . . . . . 6  |-  ( ( 1  x.  9 )  +  ( 0  +  1 ) )  =  10
9187oveq1i 6311 . . . . . . 7  |-  ( ( 1  x.  9 )  +  1 )  =  ( 9  +  1 )
9291, 89, 373eqtri 2455 . . . . . 6  |-  ( ( 1  x.  9 )  +  1 )  = ; 1
0
932, 2, 22, 2, 48, 85, 32, 22, 2, 90, 92decmac 11090 . . . . 5  |-  ( (; 1
1  x.  9 )  +  1 )  = ; 10 0
946, 32, 15, 2, 1, 33, 19, 22, 34, 84, 93decma2c 11091 . . . 4  |-  ( (; 1
1  x.  N )  + ; 7 1 )  = ;;;; 1 3 9 2 0
95 eqid 2422 . . . . 5  |- ; 1 6  = ; 1 6
965, 3deccl 11065 . . . . . 6  |- ; 5 2  e.  NN0
9796, 3deccl 11065 . . . . 5  |- ;; 5 2 2  e.  NN0
98 eqid 2422 . . . . . 6  |- ;; 8 7 0  = ;; 8 7 0
99 eqid 2422 . . . . . 6  |- ;; 5 2 2  = ;; 5 2 2
100 eqid 2422 . . . . . . 7  |- ; 8 7  = ; 8 7
10196nn0cni 10881 . . . . . . . 8  |- ; 5 2  e.  CC
102101addid1i 9820 . . . . . . 7  |-  (; 5 2  +  0 )  = ; 5 2
103 8cn 10695 . . . . . . . . . 10  |-  8  e.  CC
104103mulid1i 9645 . . . . . . . . 9  |-  ( 8  x.  1 )  =  8
10574addid1i 9820 . . . . . . . . 9  |-  ( 5  +  0 )  =  5
106104, 105oveq12i 6313 . . . . . . . 8  |-  ( ( 8  x.  1 )  +  ( 5  +  0 ) )  =  ( 8  +  5 )
107 8p5e13 11109 . . . . . . . 8  |-  ( 8  +  5 )  = ; 1
3
108106, 107eqtri 2451 . . . . . . 7  |-  ( ( 8  x.  1 )  +  ( 5  +  0 ) )  = ; 1
3
10939mulid1i 9645 . . . . . . . . 9  |-  ( 7  x.  1 )  =  7
110109oveq1i 6311 . . . . . . . 8  |-  ( ( 7  x.  1 )  +  2 )  =  ( 7  +  2 )
111110, 68, 703eqtri 2455 . . . . . . 7  |-  ( ( 7  x.  1 )  +  2 )  = ; 0
9
11212, 15, 5, 3, 100, 102, 2, 32, 22, 108, 111decmac 11090 . . . . . 6  |-  ( (; 8
7  x.  1 )  +  (; 5 2  +  0 ) )  = ;; 1 3 9
11344mul02i 9822 . . . . . . . 8  |-  ( 0  x.  1 )  =  0
114113oveq1i 6311 . . . . . . 7  |-  ( ( 0  x.  1 )  +  2 )  =  ( 0  +  2 )
115114, 50, 513eqtri 2455 . . . . . 6  |-  ( ( 0  x.  1 )  +  2 )  = ; 0
2
11621, 22, 96, 3, 98, 99, 2, 3, 22, 112, 115decmac 11090 . . . . 5  |-  ( (;; 8 7 0  x.  1 )  + ;; 5 2 2 )  = ;;; 1 3 9 2
117 8t6e48 11143 . . . . . . . . . 10  |-  ( 8  x.  6 )  = ; 4
8
118 4p1e5 10736 . . . . . . . . . 10  |-  ( 4  +  1 )  =  5
119 8p4e12 11108 . . . . . . . . . 10  |-  ( 8  +  4 )  = ; 1
2
12017, 12, 17, 117, 118, 3, 119decaddci 11096 . . . . . . . . 9  |-  ( ( 8  x.  6 )  +  4 )  = ; 5
2
121 7t6e42 11137 . . . . . . . . 9  |-  ( 7  x.  6 )  = ; 4
2
12224, 12, 15, 100, 3, 17, 120, 121decmul1c 11098 . . . . . . . 8  |-  (; 8 7  x.  6 )  = ;; 5 2 2
123122oveq1i 6311 . . . . . . 7  |-  ( (; 8
7  x.  6 )  +  0 )  =  (;; 5 2 2  +  0 )
12497nn0cni 10881 . . . . . . . 8  |- ;; 5 2 2  e.  CC
125124addid1i 9820 . . . . . . 7  |-  (;; 5 2 2  +  0 )  = ;; 5 2 2
126123, 125eqtri 2451 . . . . . 6  |-  ( (; 8
7  x.  6 )  +  0 )  = ;; 5 2 2
12743mul02i 9822 . . . . . . 7  |-  ( 0  x.  6 )  =  0
12822dec0h 11067 . . . . . . 7  |-  0  = ; 0 0
129127, 128eqtri 2451 . . . . . 6  |-  ( 0  x.  6 )  = ; 0
0
13024, 21, 22, 98, 22, 22, 126, 129decmul1c 11098 . . . . 5  |-  (;; 8 7 0  x.  6 )  = ;;; 5 2 2 0
13123, 2, 24, 95, 22, 97, 116, 130decmul2c 11099 . . . 4  |-  (;; 8 7 0  x. ; 1 6 )  = ;;;; 1 3 9 2 0
13294, 131eqtr4i 2454 . . 3  |-  ( (; 1
1  x.  N )  + ; 7 1 )  =  (;; 8 7 0  x. ; 1 6 )
1339, 10, 18, 20, 23, 16, 17, 25, 26, 28, 31, 132modxai 15027 . 2  |-  ( ( 2 ^; 3 8 )  mod 
N )  =  (; 7
1  mod  N )
134 eqid 2422 . . 3  |- ; 3 8  = ; 3 8
135 3cn 10684 . . . . . 6  |-  3  e.  CC
136 3t2e6 10761 . . . . . 6  |-  ( 3  x.  2 )  =  6
137135, 49, 136mulcomli 9650 . . . . 5  |-  ( 2  x.  3 )  =  6
138137oveq1i 6311 . . . 4  |-  ( ( 2  x.  3 )  +  1 )  =  ( 6  +  1 )
139138, 45eqtri 2451 . . 3  |-  ( ( 2  x.  3 )  +  1 )  =  7
140 8t2e16 11139 . . . 4  |-  ( 8  x.  2 )  = ; 1
6
141103, 49, 140mulcomli 9650 . . 3  |-  ( 2  x.  8 )  = ; 1
6
1423, 11, 12, 134, 24, 2, 139, 141decmul2c 11099 . 2  |-  ( 2  x. ; 3 8 )  = ; 7
6
1435dec0h 11067 . . . 4  |-  5  = ; 0 5
144 4cn 10687 . . . . . . 7  |-  4  e.  CC
145144addid2i 9821 . . . . . 6  |-  ( 0  +  4 )  =  4
14617dec0h 11067 . . . . . 6  |-  4  = ; 0 4
147145, 146eqtri 2451 . . . . 5  |-  ( 0  +  4 )  = ; 0
4
148144mulid1i 9645 . . . . . . . 8  |-  ( 4  x.  1 )  =  4
149148, 38oveq12i 6313 . . . . . . 7  |-  ( ( 4  x.  1 )  +  ( 0  +  1 ) )  =  ( 4  +  1 )
150149, 118eqtri 2451 . . . . . 6  |-  ( ( 4  x.  1 )  +  ( 0  +  1 ) )  =  5
151 4t2e8 10763 . . . . . . . 8  |-  ( 4  x.  2 )  =  8
152151oveq1i 6311 . . . . . . 7  |-  ( ( 4  x.  2 )  +  2 )  =  ( 8  +  2 )
153 8p2e10 10756 . . . . . . 7  |-  ( 8  +  2 )  =  10
154152, 153, 373eqtri 2455 . . . . . 6  |-  ( ( 4  x.  2 )  +  2 )  = ; 1
0
1552, 3, 22, 3, 42, 52, 17, 22, 2, 150, 154decma2c 11091 . . . . 5  |-  ( ( 4  x. ; 1 2 )  +  ( 0  +  2 ) )  = ; 5 0
156 5t4e20 11126 . . . . . . 7  |-  ( 5  x.  4 )  = ; 2
0
15774, 144, 156mulcomli 9650 . . . . . 6  |-  ( 4  x.  5 )  = ; 2
0
1583, 22, 17, 157, 145decaddi 11095 . . . . 5  |-  ( ( 4  x.  5 )  +  4 )  = ; 2
4
1594, 5, 22, 17, 35, 147, 17, 17, 3, 155, 158decma2c 11091 . . . 4  |-  ( ( 4  x. ;; 1 2 5 )  +  ( 0  +  4 ) )  = ;; 5 0 4
160 9t4e36 11148 . . . . . 6  |-  ( 9  x.  4 )  = ; 3
6
16186, 144, 160mulcomli 9650 . . . . 5  |-  ( 4  x.  9 )  = ; 3
6
162 3p1e4 10735 . . . . 5  |-  ( 3  +  1 )  =  4
163 6p5e11 11101 . . . . 5  |-  ( 6  +  5 )  = ; 1
1
16411, 24, 5, 161, 162, 2, 163decaddci 11096 . . . 4  |-  ( ( 4  x.  9 )  +  5 )  = ; 4
1
1656, 32, 22, 5, 1, 143, 17, 2, 17, 159, 164decma2c 11091 . . 3  |-  ( ( 4  x.  N )  +  5 )  = ;;; 5 0 4 1
16638oveq2i 6312 . . . . . 6  |-  ( ( 7  x.  7 )  +  ( 0  +  1 ) )  =  ( ( 7  x.  7 )  +  1 )
167 7t7e49 11138 . . . . . . 7  |-  ( 7  x.  7 )  = ; 4
9
16817, 118, 167decsucc 11078 . . . . . 6  |-  ( ( 7  x.  7 )  +  1 )  = ; 5
0
169166, 168eqtri 2451 . . . . 5  |-  ( ( 7  x.  7 )  +  ( 0  +  1 ) )  = ; 5
0
17039mulid2i 9646 . . . . . . 7  |-  ( 1  x.  7 )  =  7
171170oveq1i 6311 . . . . . 6  |-  ( ( 1  x.  7 )  +  7 )  =  ( 7  +  7 )
172 7p7e14 11106 . . . . . 6  |-  ( 7  +  7 )  = ; 1
4
173171, 172eqtri 2451 . . . . 5  |-  ( ( 1  x.  7 )  +  7 )  = ; 1
4
17415, 2, 22, 15, 33, 36, 15, 17, 2, 169, 173decmac 11090 . . . 4  |-  ( (; 7
1  x.  7 )  +  7 )  = ;; 5 0 4
17516nn0cni 10881 . . . . 5  |- ; 7 1  e.  CC
176175mulid1i 9645 . . . 4  |-  (; 7 1  x.  1 )  = ; 7 1
17716, 15, 2, 33, 2, 15, 174, 176decmul2c 11099 . . 3  |-  (; 7 1  x. ; 7 1 )  = ;;; 5 0 4 1
178165, 177eqtr4i 2454 . 2  |-  ( ( 4  x.  N )  +  5 )  =  (; 7 1  x. ; 7 1 )
1799, 10, 13, 14, 16, 5, 133, 142, 178mod2xi 15028 1  |-  ( ( 2 ^; 7 6 )  mod 
N )  =  ( 5  mod  N )
Colors of variables: wff setvar class
Syntax hints:    = wceq 1437  (class class class)co 6301   0cc0 9539   1c1 9540    + caddc 9542    x. cmul 9544   NNcn 10609   2c2 10659   3c3 10660   4c4 10661   5c5 10662   6c6 10663   7c7 10664   8c8 10665   9c9 10666   10c10 10667  ;cdc 11051    mod cmo 12095   ^cexp 12271
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1839  ax-8 1870  ax-9 1872  ax-10 1887  ax-11 1892  ax-12 1905  ax-13 2053  ax-ext 2400  ax-sep 4543  ax-nul 4551  ax-pow 4598  ax-pr 4656  ax-un 6593  ax-cnex 9595  ax-resscn 9596  ax-1cn 9597  ax-icn 9598  ax-addcl 9599  ax-addrcl 9600  ax-mulcl 9601  ax-mulrcl 9602  ax-mulcom 9603  ax-addass 9604  ax-mulass 9605  ax-distr 9606  ax-i2m1 9607  ax-1ne0 9608  ax-1rid 9609  ax-rnegex 9610  ax-rrecex 9611  ax-cnre 9612  ax-pre-lttri 9613  ax-pre-lttrn 9614  ax-pre-ltadd 9615  ax-pre-mulgt0 9616  ax-pre-sup 9617
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2269  df-mo 2270  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2572  df-ne 2620  df-nel 2621  df-ral 2780  df-rex 2781  df-reu 2782  df-rmo 2783  df-rab 2784  df-v 3083  df-sbc 3300  df-csb 3396  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-pss 3452  df-nul 3762  df-if 3910  df-pw 3981  df-sn 3997  df-pr 3999  df-tp 4001  df-op 4003  df-uni 4217  df-iun 4298  df-br 4421  df-opab 4480  df-mpt 4481  df-tr 4516  df-eprel 4760  df-id 4764  df-po 4770  df-so 4771  df-fr 4808  df-we 4810  df-xp 4855  df-rel 4856  df-cnv 4857  df-co 4858  df-dm 4859  df-rn 4860  df-res 4861  df-ima 4862  df-pred 5395  df-ord 5441  df-on 5442  df-lim 5443  df-suc 5444  df-iota 5561  df-fun 5599  df-fn 5600  df-f 5601  df-f1 5602  df-fo 5603  df-f1o 5604  df-fv 5605  df-riota 6263  df-ov 6304  df-oprab 6305  df-mpt2 6306  df-om 6703  df-2nd 6804  df-wrecs 7032  df-recs 7094  df-rdg 7132  df-er 7367  df-en 7574  df-dom 7575  df-sdom 7576  df-sup 7958  df-inf 7959  df-pnf 9677  df-mnf 9678  df-xr 9679  df-ltxr 9680  df-le 9681  df-sub 9862  df-neg 9863  df-div 10270  df-nn 10610  df-2 10668  df-3 10669  df-4 10670  df-5 10671  df-6 10672  df-7 10673  df-8 10674  df-9 10675  df-10 10676  df-n0 10870  df-z 10938  df-dec 11052  df-uz 11160  df-rp 11303  df-fl 12027  df-mod 12096  df-seq 12213  df-exp 12272
This theorem is referenced by:  1259lem4  15092
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