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Theorem 1259lem3 14626
Description: Lemma for 1259prm 14629. 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 10832 . . . . . 6  |-  1  e.  NN0
3 2nn0 10833 . . . . . 6  |-  2  e.  NN0
42, 3deccl 11014 . . . . 5  |- ; 1 2  e.  NN0
5 5nn0 10836 . . . . 5  |-  5  e.  NN0
64, 5deccl 11014 . . . 4  |- ;; 1 2 5  e.  NN0
7 9nn 10721 . . . 4  |-  9  e.  NN
86, 7decnncl 11013 . . 3  |- ;;; 1 2 5 9  e.  NN
91, 8eqeltri 2541 . 2  |-  N  e.  NN
10 2nn 10714 . 2  |-  2  e.  NN
11 3nn0 10834 . . 3  |-  3  e.  NN0
12 8nn0 10839 . . 3  |-  8  e.  NN0
1311, 12deccl 11014 . 2  |- ; 3 8  e.  NN0
14 4z 10919 . 2  |-  4  e.  ZZ
15 7nn0 10838 . . 3  |-  7  e.  NN0
1615, 2deccl 11014 . 2  |- ; 7 1  e.  NN0
17 4nn0 10835 . . . 4  |-  4  e.  NN0
1811, 17deccl 11014 . . 3  |- ; 3 4  e.  NN0
192, 2deccl 11014 . . . 4  |- ; 1 1  e.  NN0
2019nn0zi 10910 . . 3  |- ; 1 1  e.  ZZ
2112, 15deccl 11014 . . . 4  |- ; 8 7  e.  NN0
22 0nn0 10831 . . . 4  |-  0  e.  NN0
2321, 22deccl 11014 . . 3  |- ;; 8 7 0  e.  NN0
24 6nn0 10837 . . . 4  |-  6  e.  NN0
252, 24deccl 11014 . . 3  |- ; 1 6  e.  NN0
2611259lem2 14625 . . 3  |-  ( ( 2 ^; 3 4 )  mod 
N )  =  (;; 8 7 0  mod 
N )
27 2exp4 14582 . . . 4  |-  ( 2 ^ 4 )  = ; 1
6
2827oveq1i 6306 . . 3  |-  ( ( 2 ^ 4 )  mod  N )  =  (; 1 6  mod  N
)
29 eqid 2457 . . . 4  |- ; 3 4  = ; 3 4
30 4p4e8 10693 . . . 4  |-  ( 4  +  4 )  =  8
3111, 17, 17, 29, 30decaddi 11044 . . 3  |-  (; 3 4  +  4 )  = ; 3 8
32 9nn0 10840 . . . . 5  |-  9  e.  NN0
33 eqid 2457 . . . . 5  |- ; 7 1  = ; 7 1
34 10nn0 10841 . . . . 5  |-  10  e.  NN0
35 eqid 2457 . . . . . 6  |- ;; 1 2 5  = ;; 1 2 5
3615dec0h 11016 . . . . . . 7  |-  7  = ; 0 7
37 dec10 11030 . . . . . . 7  |-  10  = ; 1 0
38 0p1e1 10668 . . . . . . 7  |-  ( 0  +  1 )  =  1
39 7cn 10640 . . . . . . . 8  |-  7  e.  CC
4039addid1i 9784 . . . . . . 7  |-  ( 7  +  0 )  =  7
4122, 15, 2, 22, 36, 37, 38, 40decadd 11041 . . . . . 6  |-  ( 7  +  10 )  = ; 1
7
42 eqid 2457 . . . . . . 7  |- ; 1 2  = ; 1 2
43 6cn 10638 . . . . . . . . 9  |-  6  e.  CC
44 ax-1cn 9567 . . . . . . . . 9  |-  1  e.  CC
45 6p1e7 10685 . . . . . . . . 9  |-  ( 6  +  1 )  =  7
4643, 44, 45addcomli 9789 . . . . . . . 8  |-  ( 1  +  6 )  =  7
4746, 36eqtri 2486 . . . . . . 7  |-  ( 1  +  6 )  = ; 0
7
48 eqid 2457 . . . . . . . 8  |- ; 1 1  = ; 1 1
49 2cn 10627 . . . . . . . . . 10  |-  2  e.  CC
5049addid2i 9785 . . . . . . . . 9  |-  ( 0  +  2 )  =  2
513dec0h 11016 . . . . . . . . 9  |-  2  = ; 0 2
5250, 51eqtri 2486 . . . . . . . 8  |-  ( 0  +  2 )  = ; 0
2
5344mulid1i 9615 . . . . . . . . . 10  |-  ( 1  x.  1 )  =  1
54 00id 9772 . . . . . . . . . 10  |-  ( 0  +  0 )  =  0
5553, 54oveq12i 6308 . . . . . . . . 9  |-  ( ( 1  x.  1 )  +  ( 0  +  0 ) )  =  ( 1  +  0 )
5644addid1i 9784 . . . . . . . . 9  |-  ( 1  +  0 )  =  1
5755, 56eqtri 2486 . . . . . . . 8  |-  ( ( 1  x.  1 )  +  ( 0  +  0 ) )  =  1
5853oveq1i 6306 . . . . . . . . 9  |-  ( ( 1  x.  1 )  +  2 )  =  ( 1  +  2 )
59 1p2e3 10681 . . . . . . . . 9  |-  ( 1  +  2 )  =  3
6011dec0h 11016 . . . . . . . . 9  |-  3  = ; 0 3
6158, 59, 603eqtri 2490 . . . . . . . 8  |-  ( ( 1  x.  1 )  +  2 )  = ; 0
3
622, 2, 22, 3, 48, 52, 2, 11, 22, 57, 61decmac 11039 . . . . . . 7  |-  ( (; 1
1  x.  1 )  +  ( 0  +  2 ) )  = ; 1
3
6349mulid2i 9616 . . . . . . . . . 10  |-  ( 1  x.  2 )  =  2
6463, 54oveq12i 6308 . . . . . . . . 9  |-  ( ( 1  x.  2 )  +  ( 0  +  0 ) )  =  ( 2  +  0 )
6549addid1i 9784 . . . . . . . . 9  |-  ( 2  +  0 )  =  2
6664, 65eqtri 2486 . . . . . . . 8  |-  ( ( 1  x.  2 )  +  ( 0  +  0 ) )  =  2
6763oveq1i 6306 . . . . . . . . 9  |-  ( ( 1  x.  2 )  +  7 )  =  ( 2  +  7 )
68 7p2e9 10701 . . . . . . . . . 10  |-  ( 7  +  2 )  =  9
6939, 49, 68addcomli 9789 . . . . . . . . 9  |-  ( 2  +  7 )  =  9
7032dec0h 11016 . . . . . . . . 9  |-  9  = ; 0 9
7167, 69, 703eqtri 2490 . . . . . . . 8  |-  ( ( 1  x.  2 )  +  7 )  = ; 0
9
722, 2, 22, 15, 48, 36, 3, 32, 22, 66, 71decmac 11039 . . . . . . 7  |-  ( (; 1
1  x.  2 )  +  7 )  = ; 2
9
732, 3, 22, 15, 42, 47, 19, 32, 3, 62, 72decma2c 11040 . . . . . 6  |-  ( (; 1
1  x. ; 1 2 )  +  ( 1  +  6 ) )  = ;; 1 3 9
74 5cn 10636 . . . . . . . . . 10  |-  5  e.  CC
7574mulid2i 9616 . . . . . . . . 9  |-  ( 1  x.  5 )  =  5
7675, 38oveq12i 6308 . . . . . . . 8  |-  ( ( 1  x.  5 )  +  ( 0  +  1 ) )  =  ( 5  +  1 )
77 5p1e6 10684 . . . . . . . 8  |-  ( 5  +  1 )  =  6
7876, 77eqtri 2486 . . . . . . 7  |-  ( ( 1  x.  5 )  +  ( 0  +  1 ) )  =  6
7975oveq1i 6306 . . . . . . . 8  |-  ( ( 1  x.  5 )  +  7 )  =  ( 5  +  7 )
80 7p5e12 11053 . . . . . . . . 9  |-  ( 7  +  5 )  = ; 1
2
8139, 74, 80addcomli 9789 . . . . . . . 8  |-  ( 5  +  7 )  = ; 1
2
8279, 81eqtri 2486 . . . . . . 7  |-  ( ( 1  x.  5 )  +  7 )  = ; 1
2
832, 2, 22, 15, 48, 36, 5, 3, 2, 78, 82decmac 11039 . . . . . 6  |-  ( (; 1
1  x.  5 )  +  7 )  = ; 6
2
844, 5, 2, 15, 35, 41, 19, 3, 24, 73, 83decma2c 11040 . . . . 5  |-  ( (; 1
1  x. ;; 1 2 5 )  +  ( 7  +  10 ) )  = ;;; 1 3 9 2
852dec0h 11016 . . . . . 6  |-  1  = ; 0 1
86 9cn 10644 . . . . . . . . 9  |-  9  e.  CC
8786mulid2i 9616 . . . . . . . 8  |-  ( 1  x.  9 )  =  9
8887, 38oveq12i 6308 . . . . . . 7  |-  ( ( 1  x.  9 )  +  ( 0  +  1 ) )  =  ( 9  +  1 )
89 9p1e10 10688 . . . . . . 7  |-  ( 9  +  1 )  =  10
9088, 89eqtri 2486 . . . . . 6  |-  ( ( 1  x.  9 )  +  ( 0  +  1 ) )  =  10
9187oveq1i 6306 . . . . . . 7  |-  ( ( 1  x.  9 )  +  1 )  =  ( 9  +  1 )
9291, 89, 373eqtri 2490 . . . . . 6  |-  ( ( 1  x.  9 )  +  1 )  = ; 1
0
932, 2, 22, 2, 48, 85, 32, 22, 2, 90, 92decmac 11039 . . . . 5  |-  ( (; 1
1  x.  9 )  +  1 )  = ; 10 0
946, 32, 15, 2, 1, 33, 19, 22, 34, 84, 93decma2c 11040 . . . 4  |-  ( (; 1
1  x.  N )  + ; 7 1 )  = ;;;; 1 3 9 2 0
95 eqid 2457 . . . . 5  |- ; 1 6  = ; 1 6
965, 3deccl 11014 . . . . . 6  |- ; 5 2  e.  NN0
9796, 3deccl 11014 . . . . 5  |- ;; 5 2 2  e.  NN0
98 eqid 2457 . . . . . 6  |- ;; 8 7 0  = ;; 8 7 0
99 eqid 2457 . . . . . 6  |- ;; 5 2 2  = ;; 5 2 2
100 eqid 2457 . . . . . . 7  |- ; 8 7  = ; 8 7
10196nn0cni 10828 . . . . . . . 8  |- ; 5 2  e.  CC
102101addid1i 9784 . . . . . . 7  |-  (; 5 2  +  0 )  = ; 5 2
103 8cn 10642 . . . . . . . . . 10  |-  8  e.  CC
104103mulid1i 9615 . . . . . . . . 9  |-  ( 8  x.  1 )  =  8
10574addid1i 9784 . . . . . . . . 9  |-  ( 5  +  0 )  =  5
106104, 105oveq12i 6308 . . . . . . . 8  |-  ( ( 8  x.  1 )  +  ( 5  +  0 ) )  =  ( 8  +  5 )
107 8p5e13 11058 . . . . . . . 8  |-  ( 8  +  5 )  = ; 1
3
108106, 107eqtri 2486 . . . . . . 7  |-  ( ( 8  x.  1 )  +  ( 5  +  0 ) )  = ; 1
3
10939mulid1i 9615 . . . . . . . . 9  |-  ( 7  x.  1 )  =  7
110109oveq1i 6306 . . . . . . . 8  |-  ( ( 7  x.  1 )  +  2 )  =  ( 7  +  2 )
111110, 68, 703eqtri 2490 . . . . . . 7  |-  ( ( 7  x.  1 )  +  2 )  = ; 0
9
11212, 15, 5, 3, 100, 102, 2, 32, 22, 108, 111decmac 11039 . . . . . 6  |-  ( (; 8
7  x.  1 )  +  (; 5 2  +  0 ) )  = ;; 1 3 9
11344mul02i 9786 . . . . . . . 8  |-  ( 0  x.  1 )  =  0
114113oveq1i 6306 . . . . . . 7  |-  ( ( 0  x.  1 )  +  2 )  =  ( 0  +  2 )
115114, 50, 513eqtri 2490 . . . . . 6  |-  ( ( 0  x.  1 )  +  2 )  = ; 0
2
11621, 22, 96, 3, 98, 99, 2, 3, 22, 112, 115decmac 11039 . . . . 5  |-  ( (;; 8 7 0  x.  1 )  + ;; 5 2 2 )  = ;;; 1 3 9 2
117 8t6e48 11092 . . . . . . . . . 10  |-  ( 8  x.  6 )  = ; 4
8
118 4p1e5 10683 . . . . . . . . . 10  |-  ( 4  +  1 )  =  5
119 8p4e12 11057 . . . . . . . . . 10  |-  ( 8  +  4 )  = ; 1
2
12017, 12, 17, 117, 118, 3, 119decaddci 11045 . . . . . . . . 9  |-  ( ( 8  x.  6 )  +  4 )  = ; 5
2
121 7t6e42 11086 . . . . . . . . 9  |-  ( 7  x.  6 )  = ; 4
2
12224, 12, 15, 100, 3, 17, 120, 121decmul1c 11047 . . . . . . . 8  |-  (; 8 7  x.  6 )  = ;; 5 2 2
123122oveq1i 6306 . . . . . . 7  |-  ( (; 8
7  x.  6 )  +  0 )  =  (;; 5 2 2  +  0 )
12497nn0cni 10828 . . . . . . . 8  |- ;; 5 2 2  e.  CC
125124addid1i 9784 . . . . . . 7  |-  (;; 5 2 2  +  0 )  = ;; 5 2 2
126123, 125eqtri 2486 . . . . . 6  |-  ( (; 8
7  x.  6 )  +  0 )  = ;; 5 2 2
12743mul02i 9786 . . . . . . 7  |-  ( 0  x.  6 )  =  0
12822dec0h 11016 . . . . . . 7  |-  0  = ; 0 0
129127, 128eqtri 2486 . . . . . 6  |-  ( 0  x.  6 )  = ; 0
0
13024, 21, 22, 98, 22, 22, 126, 129decmul1c 11047 . . . . 5  |-  (;; 8 7 0  x.  6 )  = ;;; 5 2 2 0
13123, 2, 24, 95, 22, 97, 116, 130decmul2c 11048 . . . 4  |-  (;; 8 7 0  x. ; 1 6 )  = ;;;; 1 3 9 2 0
13294, 131eqtr4i 2489 . . 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 14565 . 2  |-  ( ( 2 ^; 3 8 )  mod 
N )  =  (; 7
1  mod  N )
134 eqid 2457 . . 3  |- ; 3 8  = ; 3 8
135 3cn 10631 . . . . . 6  |-  3  e.  CC
136 3t2e6 10708 . . . . . 6  |-  ( 3  x.  2 )  =  6
137135, 49, 136mulcomli 9620 . . . . 5  |-  ( 2  x.  3 )  =  6
138137oveq1i 6306 . . . 4  |-  ( ( 2  x.  3 )  +  1 )  =  ( 6  +  1 )
139138, 45eqtri 2486 . . 3  |-  ( ( 2  x.  3 )  +  1 )  =  7
140 8t2e16 11088 . . . 4  |-  ( 8  x.  2 )  = ; 1
6
141103, 49, 140mulcomli 9620 . . 3  |-  ( 2  x.  8 )  = ; 1
6
1423, 11, 12, 134, 24, 2, 139, 141decmul2c 11048 . 2  |-  ( 2  x. ; 3 8 )  = ; 7
6
1435dec0h 11016 . . . 4  |-  5  = ; 0 5
144 4cn 10634 . . . . . . 7  |-  4  e.  CC
145144addid2i 9785 . . . . . 6  |-  ( 0  +  4 )  =  4
14617dec0h 11016 . . . . . 6  |-  4  = ; 0 4
147145, 146eqtri 2486 . . . . 5  |-  ( 0  +  4 )  = ; 0
4
148144mulid1i 9615 . . . . . . . 8  |-  ( 4  x.  1 )  =  4
149148, 38oveq12i 6308 . . . . . . 7  |-  ( ( 4  x.  1 )  +  ( 0  +  1 ) )  =  ( 4  +  1 )
150149, 118eqtri 2486 . . . . . 6  |-  ( ( 4  x.  1 )  +  ( 0  +  1 ) )  =  5
151 4t2e8 10710 . . . . . . . 8  |-  ( 4  x.  2 )  =  8
152151oveq1i 6306 . . . . . . 7  |-  ( ( 4  x.  2 )  +  2 )  =  ( 8  +  2 )
153 8p2e10 10703 . . . . . . 7  |-  ( 8  +  2 )  =  10
154152, 153, 373eqtri 2490 . . . . . 6  |-  ( ( 4  x.  2 )  +  2 )  = ; 1
0
1552, 3, 22, 3, 42, 52, 17, 22, 2, 150, 154decma2c 11040 . . . . 5  |-  ( ( 4  x. ; 1 2 )  +  ( 0  +  2 ) )  = ; 5 0
156 5t4e20 11075 . . . . . . 7  |-  ( 5  x.  4 )  = ; 2
0
15774, 144, 156mulcomli 9620 . . . . . 6  |-  ( 4  x.  5 )  = ; 2
0
1583, 22, 17, 157, 145decaddi 11044 . . . . 5  |-  ( ( 4  x.  5 )  +  4 )  = ; 2
4
1594, 5, 22, 17, 35, 147, 17, 17, 3, 155, 158decma2c 11040 . . . 4  |-  ( ( 4  x. ;; 1 2 5 )  +  ( 0  +  4 ) )  = ;; 5 0 4
160 9t4e36 11097 . . . . . 6  |-  ( 9  x.  4 )  = ; 3
6
16186, 144, 160mulcomli 9620 . . . . 5  |-  ( 4  x.  9 )  = ; 3
6
162 3p1e4 10682 . . . . 5  |-  ( 3  +  1 )  =  4
163 6p5e11 11050 . . . . 5  |-  ( 6  +  5 )  = ; 1
1
16411, 24, 5, 161, 162, 2, 163decaddci 11045 . . . 4  |-  ( ( 4  x.  9 )  +  5 )  = ; 4
1
1656, 32, 22, 5, 1, 143, 17, 2, 17, 159, 164decma2c 11040 . . 3  |-  ( ( 4  x.  N )  +  5 )  = ;;; 5 0 4 1
16638oveq2i 6307 . . . . . 6  |-  ( ( 7  x.  7 )  +  ( 0  +  1 ) )  =  ( ( 7  x.  7 )  +  1 )
167 7t7e49 11087 . . . . . . 7  |-  ( 7  x.  7 )  = ; 4
9
16817, 118, 167decsucc 11027 . . . . . 6  |-  ( ( 7  x.  7 )  +  1 )  = ; 5
0
169166, 168eqtri 2486 . . . . 5  |-  ( ( 7  x.  7 )  +  ( 0  +  1 ) )  = ; 5
0
17039mulid2i 9616 . . . . . . 7  |-  ( 1  x.  7 )  =  7
171170oveq1i 6306 . . . . . 6  |-  ( ( 1  x.  7 )  +  7 )  =  ( 7  +  7 )
172 7p7e14 11055 . . . . . 6  |-  ( 7  +  7 )  = ; 1
4
173171, 172eqtri 2486 . . . . 5  |-  ( ( 1  x.  7 )  +  7 )  = ; 1
4
17415, 2, 22, 15, 33, 36, 15, 17, 2, 169, 173decmac 11039 . . . 4  |-  ( (; 7
1  x.  7 )  +  7 )  = ;; 5 0 4
17516nn0cni 10828 . . . . 5  |- ; 7 1  e.  CC
176175mulid1i 9615 . . . 4  |-  (; 7 1  x.  1 )  = ; 7 1
17716, 15, 2, 33, 2, 15, 174, 176decmul2c 11048 . . 3  |-  (; 7 1  x. ; 7 1 )  = ;;; 5 0 4 1
178165, 177eqtr4i 2489 . 2  |-  ( ( 4  x.  N )  +  5 )  =  (; 7 1  x. ; 7 1 )
1799, 10, 13, 14, 16, 5, 133, 142, 178mod2xi 14566 1  |-  ( ( 2 ^; 7 6 )  mod 
N )  =  ( 5  mod  N )
Colors of variables: wff setvar class
Syntax hints:    = wceq 1395  (class class class)co 6296   0cc0 9509   1c1 9510    + caddc 9512    x. cmul 9514   NNcn 10556   2c2 10606   3c3 10607   4c4 10608   5c5 10609   6c6 10610   7c7 10611   8c8 10612   9c9 10613   10c10 10614  ;cdc 11000    mod cmo 11998   ^cexp 12168
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-cnex 9565  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-mulcom 9573  ax-addass 9574  ax-mulass 9575  ax-distr 9576  ax-i2m1 9577  ax-1ne0 9578  ax-1rid 9579  ax-rnegex 9580  ax-rrecex 9581  ax-cnre 9582  ax-pre-lttri 9583  ax-pre-lttrn 9584  ax-pre-ltadd 9585  ax-pre-mulgt0 9586  ax-pre-sup 9587
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-om 6700  df-2nd 6800  df-recs 7060  df-rdg 7094  df-er 7329  df-en 7536  df-dom 7537  df-sdom 7538  df-sup 7919  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-le 9651  df-sub 9826  df-neg 9827  df-div 10228  df-nn 10557  df-2 10615  df-3 10616  df-4 10617  df-5 10618  df-6 10619  df-7 10620  df-8 10621  df-9 10622  df-10 10623  df-n0 10817  df-z 10886  df-dec 11001  df-uz 11107  df-rp 11246  df-fl 11931  df-mod 11999  df-seq 12110  df-exp 12169
This theorem is referenced by:  1259lem4  14627
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