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Theorem 631prm 13439
Description: 631 is a prime number. (Contributed by Mario Carneiro, 1-Mar-2014.) (Proof shortened by Mario Carneiro, 20-Apr-2015.)
Assertion
Ref Expression
631prm  |- ;; 6 3 1  e.  Prime

Proof of Theorem 631prm
StepHypRef Expression
1 6nn0 10232 . . . 4  |-  6  e.  NN0
2 3nn0 10229 . . . 4  |-  3  e.  NN0
31, 2deccl 10386 . . 3  |- ; 6 3  e.  NN0
4 1nn 10001 . . 3  |-  1  e.  NN
53, 4decnncl 10385 . 2  |- ;; 6 3 1  e.  NN
6 8nn0 10234 . . . 4  |-  8  e.  NN0
7 4nn0 10230 . . . 4  |-  4  e.  NN0
86, 7deccl 10386 . . 3  |- ; 8 4  e.  NN0
9 1nn0 10227 . . 3  |-  1  e.  NN0
10 1lt10 10176 . . 3  |-  1  <  10
11 3lt10 10174 . . . 4  |-  3  <  10
12 6lt8 10154 . . . 4  |-  6  <  8
131, 6, 2, 7, 11, 12decltc 10394 . . 3  |- ; 6 3  < ; 8 4
143, 8, 9, 9, 10, 13decltc 10394 . 2  |- ;; 6 3 1  < ;; 8 4 1
15 3nn 10124 . . . 4  |-  3  e.  NN
161, 15decnncl 10385 . . 3  |- ; 6 3  e.  NN
1716, 9, 9, 10declti 10397 . 2  |-  1  < ;; 6 3 1
18 0nn0 10226 . . 3  |-  0  e.  NN0
19 2cn 10060 . . . 4  |-  2  e.  CC
2019mul02i 9245 . . 3  |-  ( 0  x.  2 )  =  0
21 1e0p1 10400 . . 3  |-  1  =  ( 0  +  1 )
223, 18, 20, 21dec2dvds 13389 . 2  |-  -.  2  || ;; 6 3 1
23 2nn0 10228 . . . . 5  |-  2  e.  NN0
2423, 9deccl 10386 . . . 4  |- ; 2 1  e.  NN0
2524, 18deccl 10386 . . 3  |- ;; 2 1 0  e.  NN0
26 eqid 2435 . . . 4  |- ;; 2 1 0  = ;; 2 1 0
279dec0h 10388 . . . 4  |-  1  = ; 0 1
28 eqid 2435 . . . . 5  |- ; 2 1  = ; 2 1
29 00id 9231 . . . . . 6  |-  ( 0  +  0 )  =  0
3018dec0h 10388 . . . . . 6  |-  0  = ; 0 0
3129, 30eqtri 2455 . . . . 5  |-  ( 0  +  0 )  = ; 0
0
32 3t2e6 10118 . . . . . . 7  |-  ( 3  x.  2 )  =  6
3332, 29oveq12i 6085 . . . . . 6  |-  ( ( 3  x.  2 )  +  ( 0  +  0 ) )  =  ( 6  +  0 )
34 6nn 10127 . . . . . . . 8  |-  6  e.  NN
3534nncni 10000 . . . . . . 7  |-  6  e.  CC
3635addid1i 9243 . . . . . 6  |-  ( 6  +  0 )  =  6
3733, 36eqtri 2455 . . . . 5  |-  ( ( 3  x.  2 )  +  ( 0  +  0 ) )  =  6
38 3cn 10062 . . . . . . . 8  |-  3  e.  CC
3938mulid1i 9082 . . . . . . 7  |-  ( 3  x.  1 )  =  3
4039oveq1i 6083 . . . . . 6  |-  ( ( 3  x.  1 )  +  0 )  =  ( 3  +  0 )
4138addid1i 9243 . . . . . 6  |-  ( 3  +  0 )  =  3
422dec0h 10388 . . . . . 6  |-  3  = ; 0 3
4340, 41, 423eqtri 2459 . . . . 5  |-  ( ( 3  x.  1 )  +  0 )  = ; 0
3
4423, 9, 18, 18, 28, 31, 2, 2, 18, 37, 43decma2c 10412 . . . 4  |-  ( ( 3  x. ; 2 1 )  +  ( 0  +  0 ) )  = ; 6 3
4538mul01i 9246 . . . . . 6  |-  ( 3  x.  0 )  =  0
4645oveq1i 6083 . . . . 5  |-  ( ( 3  x.  0 )  +  1 )  =  ( 0  +  1 )
47 0p1e1 10083 . . . . 5  |-  ( 0  +  1 )  =  1
4846, 47, 273eqtri 2459 . . . 4  |-  ( ( 3  x.  0 )  +  1 )  = ; 0
1
4924, 18, 18, 9, 26, 27, 2, 9, 18, 44, 48decma2c 10412 . . 3  |-  ( ( 3  x. ;; 2 1 0 )  +  1 )  = ;; 6 3 1
50 1lt3 10134 . . 3  |-  1  <  3
5115, 25, 4, 49, 50ndvdsi 12920 . 2  |-  -.  3  || ;; 6 3 1
52 1lt5 10141 . . 3  |-  1  <  5
533, 4, 52dec5dvds 13390 . 2  |-  -.  5  || ;; 6 3 1
54 7nn 10128 . . 3  |-  7  e.  NN
55 9nn0 10235 . . . 4  |-  9  e.  NN0
5655, 18deccl 10386 . . 3  |- ; 9 0  e.  NN0
57 eqid 2435 . . . 4  |- ; 9 0  = ; 9 0
58 7nn0 10233 . . . 4  |-  7  e.  NN0
5929oveq2i 6084 . . . . 5  |-  ( ( 7  x.  9 )  +  ( 0  +  0 ) )  =  ( ( 7  x.  9 )  +  0 )
60 9nn 10130 . . . . . . . 8  |-  9  e.  NN
6160nncni 10000 . . . . . . 7  |-  9  e.  CC
6254nncni 10000 . . . . . . 7  |-  7  e.  CC
63 9t7e63 10472 . . . . . . 7  |-  ( 9  x.  7 )  = ; 6
3
6461, 62, 63mulcomli 9087 . . . . . 6  |-  ( 7  x.  9 )  = ; 6
3
6564oveq1i 6083 . . . . 5  |-  ( ( 7  x.  9 )  +  0 )  =  (; 6 3  +  0 )
663nn0cni 10223 . . . . . 6  |- ; 6 3  e.  CC
6766addid1i 9243 . . . . 5  |-  (; 6 3  +  0 )  = ; 6 3
6859, 65, 673eqtri 2459 . . . 4  |-  ( ( 7  x.  9 )  +  ( 0  +  0 ) )  = ; 6
3
6962mul01i 9246 . . . . . 6  |-  ( 7  x.  0 )  =  0
7069oveq1i 6083 . . . . 5  |-  ( ( 7  x.  0 )  +  1 )  =  ( 0  +  1 )
7170, 47, 273eqtri 2459 . . . 4  |-  ( ( 7  x.  0 )  +  1 )  = ; 0
1
7255, 18, 18, 9, 57, 27, 58, 9, 18, 68, 71decma2c 10412 . . 3  |-  ( ( 7  x. ; 9 0 )  +  1 )  = ;; 6 3 1
73 1lt7 10152 . . 3  |-  1  <  7
7454, 56, 4, 72, 73ndvdsi 12920 . 2  |-  -.  7  || ;; 6 3 1
759, 4decnncl 10385 . . 3  |- ; 1 1  e.  NN
76 5nn0 10231 . . . 4  |-  5  e.  NN0
7776, 58deccl 10386 . . 3  |- ; 5 7  e.  NN0
78 4nn 10125 . . 3  |-  4  e.  NN
79 eqid 2435 . . . 4  |- ; 5 7  = ; 5 7
807dec0h 10388 . . . 4  |-  4  = ; 0 4
819, 9deccl 10386 . . . 4  |- ; 1 1  e.  NN0
82 eqid 2435 . . . . 5  |- ; 1 1  = ; 1 1
83 8nn 10129 . . . . . . . 8  |-  8  e.  NN
8483nncni 10000 . . . . . . 7  |-  8  e.  CC
8584addid2i 9244 . . . . . 6  |-  ( 0  +  8 )  =  8
866dec0h 10388 . . . . . 6  |-  8  = ; 0 8
8785, 86eqtri 2455 . . . . 5  |-  ( 0  +  8 )  = ; 0
8
88 5nn 10126 . . . . . . . . 9  |-  5  e.  NN
8988nncni 10000 . . . . . . . 8  |-  5  e.  CC
9089mulid2i 9083 . . . . . . 7  |-  ( 1  x.  5 )  =  5
9190, 47oveq12i 6085 . . . . . 6  |-  ( ( 1  x.  5 )  +  ( 0  +  1 ) )  =  ( 5  +  1 )
92 5p1e6 10096 . . . . . 6  |-  ( 5  +  1 )  =  6
9391, 92eqtri 2455 . . . . 5  |-  ( ( 1  x.  5 )  +  ( 0  +  1 ) )  =  6
9490oveq1i 6083 . . . . . 6  |-  ( ( 1  x.  5 )  +  8 )  =  ( 5  +  8 )
95 8p5e13 10430 . . . . . . 7  |-  ( 8  +  5 )  = ; 1
3
9684, 89, 95addcomli 9248 . . . . . 6  |-  ( 5  +  8 )  = ; 1
3
9794, 96eqtri 2455 . . . . 5  |-  ( ( 1  x.  5 )  +  8 )  = ; 1
3
989, 9, 18, 6, 82, 87, 76, 2, 9, 93, 97decmac 10411 . . . 4  |-  ( (; 1
1  x.  5 )  +  ( 0  +  8 ) )  = ; 6
3
9962mulid2i 9083 . . . . . . 7  |-  ( 1  x.  7 )  =  7
10099, 47oveq12i 6085 . . . . . 6  |-  ( ( 1  x.  7 )  +  ( 0  +  1 ) )  =  ( 7  +  1 )
101 7p1e8 10098 . . . . . 6  |-  ( 7  +  1 )  =  8
102100, 101eqtri 2455 . . . . 5  |-  ( ( 1  x.  7 )  +  ( 0  +  1 ) )  =  8
10399oveq1i 6083 . . . . . 6  |-  ( ( 1  x.  7 )  +  4 )  =  ( 7  +  4 )
104 7p4e11 10424 . . . . . 6  |-  ( 7  +  4 )  = ; 1
1
105103, 104eqtri 2455 . . . . 5  |-  ( ( 1  x.  7 )  +  4 )  = ; 1
1
1069, 9, 18, 7, 82, 80, 58, 9, 9, 102, 105decmac 10411 . . . 4  |-  ( (; 1
1  x.  7 )  +  4 )  = ; 8
1
10776, 58, 18, 7, 79, 80, 81, 9, 6, 98, 106decma2c 10412 . . 3  |-  ( (; 1
1  x. ; 5 7 )  +  4 )  = ;; 6 3 1
108 4lt10 10173 . . . 4  |-  4  <  10
1094, 9, 7, 108declti 10397 . . 3  |-  4  < ; 1
1
11075, 77, 78, 107, 109ndvdsi 12920 . 2  |-  -. ; 1 1  || ;; 6 3 1
1119, 15decnncl 10385 . . 3  |- ; 1 3  e.  NN
1127, 6deccl 10386 . . 3  |- ; 4 8  e.  NN0
113 eqid 2435 . . . 4  |- ; 4 8  = ; 4 8
11458dec0h 10388 . . . 4  |-  7  = ; 0 7
1159, 2deccl 10386 . . . 4  |- ; 1 3  e.  NN0
116 eqid 2435 . . . . 5  |- ; 1 3  = ; 1 3
11781nn0cni 10223 . . . . . 6  |- ; 1 1  e.  CC
118117addid2i 9244 . . . . 5  |-  ( 0  + ; 1 1 )  = ; 1
1
119 4cn 10064 . . . . . . . 8  |-  4  e.  CC
120119mulid2i 9083 . . . . . . 7  |-  ( 1  x.  4 )  =  4
121 1p1e2 10084 . . . . . . 7  |-  ( 1  +  1 )  =  2
122120, 121oveq12i 6085 . . . . . 6  |-  ( ( 1  x.  4 )  +  ( 1  +  1 ) )  =  ( 4  +  2 )
123 4p2e6 10103 . . . . . 6  |-  ( 4  +  2 )  =  6
124122, 123eqtri 2455 . . . . 5  |-  ( ( 1  x.  4 )  +  ( 1  +  1 ) )  =  6
125 4t3e12 10444 . . . . . . 7  |-  ( 4  x.  3 )  = ; 1
2
126119, 38, 125mulcomli 9087 . . . . . 6  |-  ( 3  x.  4 )  = ; 1
2
127 2p1e3 10093 . . . . . 6  |-  ( 2  +  1 )  =  3
1289, 23, 9, 126, 127decaddi 10416 . . . . 5  |-  ( ( 3  x.  4 )  +  1 )  = ; 1
3
1299, 2, 9, 9, 116, 118, 7, 2, 9, 124, 128decmac 10411 . . . 4  |-  ( (; 1
3  x.  4 )  +  ( 0  + ; 1
1 ) )  = ; 6
3
13084mulid2i 9083 . . . . . . 7  |-  ( 1  x.  8 )  =  8
13138addid2i 9244 . . . . . . 7  |-  ( 0  +  3 )  =  3
132130, 131oveq12i 6085 . . . . . 6  |-  ( ( 1  x.  8 )  +  ( 0  +  3 ) )  =  ( 8  +  3 )
133 8p3e11 10428 . . . . . 6  |-  ( 8  +  3 )  = ; 1
1
134132, 133eqtri 2455 . . . . 5  |-  ( ( 1  x.  8 )  +  ( 0  +  3 ) )  = ; 1
1
135 8t3e24 10461 . . . . . . 7  |-  ( 8  x.  3 )  = ; 2
4
13684, 38, 135mulcomli 9087 . . . . . 6  |-  ( 3  x.  8 )  = ; 2
4
13762, 119, 104addcomli 9248 . . . . . 6  |-  ( 4  +  7 )  = ; 1
1
13823, 7, 58, 136, 127, 9, 137decaddci 10417 . . . . 5  |-  ( ( 3  x.  8 )  +  7 )  = ; 3
1
1399, 2, 18, 58, 116, 114, 6, 9, 2, 134, 138decmac 10411 . . . 4  |-  ( (; 1
3  x.  8 )  +  7 )  = ;; 1 1 1
1407, 6, 18, 58, 113, 114, 115, 9, 81, 129, 139decma2c 10412 . . 3  |-  ( (; 1
3  x. ; 4 8 )  +  7 )  = ;; 6 3 1
141 7lt10 10170 . . . 4  |-  7  <  10
1424, 2, 58, 141declti 10397 . . 3  |-  7  < ; 1
3
143111, 112, 54, 140, 142ndvdsi 12920 . 2  |-  -. ; 1 3  || ;; 6 3 1
1449, 54decnncl 10385 . . 3  |- ; 1 7  e.  NN
1452, 58deccl 10386 . . 3  |- ; 3 7  e.  NN0
146 2nn 10123 . . 3  |-  2  e.  NN
147 eqid 2435 . . . 4  |- ; 3 7  = ; 3 7
14823dec0h 10388 . . . 4  |-  2  = ; 0 2
1499, 58deccl 10386 . . . 4  |- ; 1 7  e.  NN0
1509, 23deccl 10386 . . . 4  |- ; 1 2  e.  NN0
151 eqid 2435 . . . . 5  |- ; 1 7  = ; 1 7
152150nn0cni 10223 . . . . . 6  |- ; 1 2  e.  CC
153152addid2i 9244 . . . . 5  |-  ( 0  + ; 1 2 )  = ; 1
2
15438mulid2i 9083 . . . . . . 7  |-  ( 1  x.  3 )  =  3
155 ax-1cn 9038 . . . . . . . 8  |-  1  e.  CC
15619, 155, 127addcomli 9248 . . . . . . 7  |-  ( 1  +  2 )  =  3
157154, 156oveq12i 6085 . . . . . 6  |-  ( ( 1  x.  3 )  +  ( 1  +  2 ) )  =  ( 3  +  3 )
158 3p3e6 10102 . . . . . 6  |-  ( 3  +  3 )  =  6
159157, 158eqtri 2455 . . . . 5  |-  ( ( 1  x.  3 )  +  ( 1  +  2 ) )  =  6
160 7t3e21 10455 . . . . . 6  |-  ( 7  x.  3 )  = ; 2
1
16123, 9, 23, 160, 156decaddi 10416 . . . . 5  |-  ( ( 7  x.  3 )  +  2 )  = ; 2
3
1629, 58, 9, 23, 151, 153, 2, 2, 23, 159, 161decmac 10411 . . . 4  |-  ( (; 1
7  x.  3 )  +  ( 0  + ; 1
2 ) )  = ; 6
3
16389addid2i 9244 . . . . . . 7  |-  ( 0  +  5 )  =  5
16499, 163oveq12i 6085 . . . . . 6  |-  ( ( 1  x.  7 )  +  ( 0  +  5 ) )  =  ( 7  +  5 )
165 7p5e12 10425 . . . . . 6  |-  ( 7  +  5 )  = ; 1
2
166164, 165eqtri 2455 . . . . 5  |-  ( ( 1  x.  7 )  +  ( 0  +  5 ) )  = ; 1
2
167 7t7e49 10459 . . . . . 6  |-  ( 7  x.  7 )  = ; 4
9
168 4p1e5 10095 . . . . . 6  |-  ( 4  +  1 )  =  5
169 9p2e11 10434 . . . . . 6  |-  ( 9  +  2 )  = ; 1
1
1707, 55, 23, 167, 168, 9, 169decaddci 10417 . . . . 5  |-  ( ( 7  x.  7 )  +  2 )  = ; 5
1
1719, 58, 18, 23, 151, 148, 58, 9, 76, 166, 170decmac 10411 . . . 4  |-  ( (; 1
7  x.  7 )  +  2 )  = ;; 1 2 1
1722, 58, 18, 23, 147, 148, 149, 9, 150, 162, 171decma2c 10412 . . 3  |-  ( (; 1
7  x. ; 3 7 )  +  2 )  = ;; 6 3 1
173 2lt10 10175 . . . 4  |-  2  <  10
1744, 58, 23, 173declti 10397 . . 3  |-  2  < ; 1
7
175144, 145, 146, 172, 174ndvdsi 12920 . 2  |-  -. ; 1 7  || ;; 6 3 1
1769, 60decnncl 10385 . . 3  |- ; 1 9  e.  NN
1772, 2deccl 10386 . . 3  |- ; 3 3  e.  NN0
178 eqid 2435 . . . 4  |- ; 3 3  = ; 3 3
1799, 55deccl 10386 . . . 4  |- ; 1 9  e.  NN0
180 eqid 2435 . . . . 5  |- ; 1 9  = ; 1 9
18135addid2i 9244 . . . . . 6  |-  ( 0  +  6 )  =  6
1821dec0h 10388 . . . . . 6  |-  6  = ; 0 6
183181, 182eqtri 2455 . . . . 5  |-  ( 0  +  6 )  = ; 0
6
184154, 131oveq12i 6085 . . . . . 6  |-  ( ( 1  x.  3 )  +  ( 0  +  3 ) )  =  ( 3  +  3 )
185184, 158eqtri 2455 . . . . 5  |-  ( ( 1  x.  3 )  +  ( 0  +  3 ) )  =  6
186 9t3e27 10468 . . . . . 6  |-  ( 9  x.  3 )  = ; 2
7
187 7p6e13 10426 . . . . . 6  |-  ( 7  +  6 )  = ; 1
3
18823, 58, 1, 186, 127, 2, 187decaddci 10417 . . . . 5  |-  ( ( 9  x.  3 )  +  6 )  = ; 3
3
1899, 55, 18, 1, 180, 183, 2, 2, 2, 185, 188decmac 10411 . . . 4  |-  ( (; 1
9  x.  3 )  +  ( 0  +  6 ) )  = ; 6
3
19023, 58, 7, 186, 127, 9, 104decaddci 10417 . . . . 5  |-  ( ( 9  x.  3 )  +  4 )  = ; 3
1
1919, 55, 18, 7, 180, 80, 2, 9, 2, 185, 190decmac 10411 . . . 4  |-  ( (; 1
9  x.  3 )  +  4 )  = ; 6
1
1922, 2, 18, 7, 178, 80, 179, 9, 1, 189, 191decma2c 10412 . . 3  |-  ( (; 1
9  x. ; 3 3 )  +  4 )  = ;; 6 3 1
1934, 55, 7, 108declti 10397 . . 3  |-  4  < ; 1
9
194176, 177, 78, 192, 193ndvdsi 12920 . 2  |-  -. ; 1 9  || ;; 6 3 1
19523, 15decnncl 10385 . . 3  |- ; 2 3  e.  NN
19623, 58deccl 10386 . . 3  |- ; 2 7  e.  NN0
197 10nn 10131 . . 3  |-  10  e.  NN
198 eqid 2435 . . . 4  |- ; 2 7  = ; 2 7
199 dec10 10402 . . . 4  |-  10  = ; 1 0
20023, 2deccl 10386 . . . 4  |- ; 2 3  e.  NN0
2019, 1deccl 10386 . . . 4  |- ; 1 6  e.  NN0
202 eqid 2435 . . . . 5  |- ; 2 3  = ; 2 3
203 eqid 2435 . . . . . 6  |- ; 1 6  = ; 1 6
204 6p1e7 10097 . . . . . . 7  |-  ( 6  +  1 )  =  7
20535, 155, 204addcomli 9248 . . . . . 6  |-  ( 1  +  6 )  =  7
20618, 9, 9, 1, 27, 203, 47, 205decadd 10413 . . . . 5  |-  ( 1  + ; 1 6 )  = ; 1
7
207 2t2e4 10117 . . . . . . 7  |-  ( 2  x.  2 )  =  4
208207, 121oveq12i 6085 . . . . . 6  |-  ( ( 2  x.  2 )  +  ( 1  +  1 ) )  =  ( 4  +  2 )
209208, 123eqtri 2455 . . . . 5  |-  ( ( 2  x.  2 )  +  ( 1  +  1 ) )  =  6
21032oveq1i 6083 . . . . . 6  |-  ( ( 3  x.  2 )  +  7 )  =  ( 6  +  7 )
21162, 35, 187addcomli 9248 . . . . . 6  |-  ( 6  +  7 )  = ; 1
3
212210, 211eqtri 2455 . . . . 5  |-  ( ( 3  x.  2 )  +  7 )  = ; 1
3
21323, 2, 9, 58, 202, 206, 23, 2, 9, 209, 212decmac 10411 . . . 4  |-  ( (; 2
3  x.  2 )  +  ( 1  + ; 1
6 ) )  = ; 6
3
214 7t2e14 10454 . . . . . . . . 9  |-  ( 7  x.  2 )  = ; 1
4
21562, 19, 214mulcomli 9087 . . . . . . . 8  |-  ( 2  x.  7 )  = ; 1
4
2169, 7, 23, 215, 123decaddi 10416 . . . . . . 7  |-  ( ( 2  x.  7 )  +  2 )  = ; 1
6
21762, 38, 160mulcomli 9087 . . . . . . 7  |-  ( 3  x.  7 )  = ; 2
1
21858, 23, 2, 202, 9, 23, 216, 217decmul1c 10419 . . . . . 6  |-  (; 2 3  x.  7 )  = ;; 1 6 1
219218oveq1i 6083 . . . . 5  |-  ( (; 2
3  x.  7 )  +  0 )  =  (;; 1 6 1  +  0 )
220201, 9deccl 10386 . . . . . . 7  |- ;; 1 6 1  e.  NN0
221220nn0cni 10223 . . . . . 6  |- ;; 1 6 1  e.  CC
222221addid1i 9243 . . . . 5  |-  (;; 1 6 1  +  0 )  = ;; 1 6 1
223219, 222eqtri 2455 . . . 4  |-  ( (; 2
3  x.  7 )  +  0 )  = ;; 1 6 1
22423, 58, 9, 18, 198, 199, 200, 9, 201, 213, 223decma2c 10412 . . 3  |-  ( (; 2
3  x. ; 2 7 )  +  10 )  = ;; 6 3 1
225 10pos 10082 . . . . 5  |-  0  <  10
226 1lt2 10132 . . . . 5  |-  1  <  2
2279, 23, 18, 2, 225, 226decltc 10394 . . . 4  |- ; 1 0  < ; 2 3
228199, 227eqbrtri 4223 . . 3  |-  10  < ; 2 3
229195, 196, 197, 224, 228ndvdsi 12920 . 2  |-  -. ; 2 3  || ;; 6 3 1
2305, 14, 17, 22, 51, 53, 74, 110, 143, 175, 194, 229prmlem2 13432 1  |- ;; 6 3 1  e.  Prime
Colors of variables: wff set class
Syntax hints:    e. wcel 1725  (class class class)co 6073   0cc0 8980   1c1 8981    + caddc 8983    x. cmul 8985    < clt 9110   2c2 10039   3c3 10040   4c4 10041   5c5 10042   6c6 10043   7c7 10044   8c8 10045   9c9 10046   10c10 10047  ;cdc 10372   Primecprime 13069
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-int 4043  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-1st 6341  df-2nd 6342  df-riota 6541  df-recs 6625  df-rdg 6660  df-1o 6716  df-2o 6717  df-oadd 6720  df-er 6897  df-en 7102  df-dom 7103  df-sdom 7104  df-fin 7105  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-fz 11034  df-seq 11314  df-exp 11373  df-cj 11894  df-re 11895  df-im 11896  df-sqr 12030  df-abs 12031  df-dvds 12843  df-prm 13070
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