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Theorem 317prm 13450
Description: 317 is a prime number. (Contributed by Mario Carneiro, 19-Feb-2014.) (Proof shortened by Mario Carneiro, 20-Apr-2015.)
Assertion
Ref Expression
317prm  |- ;; 3 1 7  e.  Prime

Proof of Theorem 317prm
StepHypRef Expression
1 3nn0 10241 . . . 4  |-  3  e.  NN0
2 1nn0 10239 . . . 4  |-  1  e.  NN0
31, 2deccl 10398 . . 3  |- ; 3 1  e.  NN0
4 7nn 10140 . . 3  |-  7  e.  NN
53, 4decnncl 10397 . 2  |- ;; 3 1 7  e.  NN
6 8nn0 10246 . . . 4  |-  8  e.  NN0
7 4nn0 10242 . . . 4  |-  4  e.  NN0
86, 7deccl 10398 . . 3  |- ; 8 4  e.  NN0
9 7nn0 10245 . . 3  |-  7  e.  NN0
10 7lt10 10182 . . 3  |-  7  <  10
11 1lt10 10188 . . . 4  |-  1  <  10
12 3lt8 10169 . . . 4  |-  3  <  8
131, 6, 2, 7, 11, 12decltc 10406 . . 3  |- ; 3 1  < ; 8 4
143, 8, 9, 2, 10, 13decltc 10406 . 2  |- ;; 3 1 7  < ;; 8 4 1
15 1nn 10013 . . . 4  |-  1  e.  NN
161, 15decnncl 10397 . . 3  |- ; 3 1  e.  NN
1716, 9, 2, 11declti 10409 . 2  |-  1  < ;; 3 1 7
18 3t2e6 10130 . . 3  |-  ( 3  x.  2 )  =  6
19 df-7 10065 . . 3  |-  7  =  ( 6  +  1 )
203, 1, 18, 19dec2dvds 13401 . 2  |-  -.  2  || ;; 3 1 7
21 3nn 10136 . . 3  |-  3  e.  NN
22 10nn0 10248 . . . 4  |-  10  e.  NN0
23 5nn0 10243 . . . 4  |-  5  e.  NN0
2422, 23deccl 10398 . . 3  |- ; 10 5  e.  NN0
25 2nn 10135 . . 3  |-  2  e.  NN
26 0nn0 10238 . . . 4  |-  0  e.  NN0
27 2nn0 10240 . . . 4  |-  2  e.  NN0
28 eqid 2438 . . . 4  |- ; 10 5  = ; 10 5
2927dec0h 10400 . . . 4  |-  2  = ; 0 2
30 dec10 10414 . . . . 5  |-  10  = ; 1 0
31 ax-1cn 9050 . . . . . . 7  |-  1  e.  CC
3231addid2i 9256 . . . . . 6  |-  ( 0  +  1 )  =  1
332dec0h 10400 . . . . . 6  |-  1  = ; 0 1
3432, 33eqtri 2458 . . . . 5  |-  ( 0  +  1 )  = ; 0
1
35 3cn 10074 . . . . . . . 8  |-  3  e.  CC
3635mulid1i 9094 . . . . . . 7  |-  ( 3  x.  1 )  =  3
37 00id 9243 . . . . . . 7  |-  ( 0  +  0 )  =  0
3836, 37oveq12i 6095 . . . . . 6  |-  ( ( 3  x.  1 )  +  ( 0  +  0 ) )  =  ( 3  +  0 )
3935addid1i 9255 . . . . . 6  |-  ( 3  +  0 )  =  3
4038, 39eqtri 2458 . . . . 5  |-  ( ( 3  x.  1 )  +  ( 0  +  0 ) )  =  3
4135mul01i 9258 . . . . . . . 8  |-  ( 3  x.  0 )  =  0
4241oveq1i 6093 . . . . . . 7  |-  ( ( 3  x.  0 )  +  1 )  =  ( 0  +  1 )
4342, 32eqtri 2458 . . . . . 6  |-  ( ( 3  x.  0 )  +  1 )  =  1
4443, 33eqtri 2458 . . . . 5  |-  ( ( 3  x.  0 )  +  1 )  = ; 0
1
452, 26, 26, 2, 30, 34, 1, 2, 26, 40, 44decma2c 10424 . . . 4  |-  ( ( 3  x.  10 )  +  ( 0  +  1 ) )  = ; 3
1
46 5nn 10138 . . . . . . 7  |-  5  e.  NN
4746nncni 10012 . . . . . 6  |-  5  e.  CC
48 5t3e15 10458 . . . . . 6  |-  ( 5  x.  3 )  = ; 1
5
4947, 35, 48mulcomli 9099 . . . . 5  |-  ( 3  x.  5 )  = ; 1
5
50 5p2e7 10118 . . . . 5  |-  ( 5  +  2 )  =  7
512, 23, 27, 49, 50decaddi 10428 . . . 4  |-  ( ( 3  x.  5 )  +  2 )  = ; 1
7
5222, 23, 26, 27, 28, 29, 1, 9, 2, 45, 51decma2c 10424 . . 3  |-  ( ( 3  x. ; 10 5 )  +  2 )  = ;; 3 1 7
53 2lt3 10145 . . 3  |-  2  <  3
5421, 24, 25, 52, 53ndvdsi 12932 . 2  |-  -.  3  || ;; 3 1 7
55 2lt5 10152 . . 3  |-  2  <  5
563, 25, 55, 50dec5dvds2 13403 . 2  |-  -.  5  || ;; 3 1 7
577, 23deccl 10398 . . 3  |- ; 4 5  e.  NN0
58 eqid 2438 . . . 4  |- ; 4 5  = ; 4 5
5935addid2i 9256 . . . . . 6  |-  ( 0  +  3 )  =  3
6059oveq2i 6094 . . . . 5  |-  ( ( 7  x.  4 )  +  ( 0  +  3 ) )  =  ( ( 7  x.  4 )  +  3 )
61 7t4e28 10468 . . . . . 6  |-  ( 7  x.  4 )  = ; 2
8
62 2p1e3 10105 . . . . . 6  |-  ( 2  +  1 )  =  3
63 8p3e11 10440 . . . . . 6  |-  ( 8  +  3 )  = ; 1
1
6427, 6, 1, 61, 62, 2, 63decaddci 10429 . . . . 5  |-  ( ( 7  x.  4 )  +  3 )  = ; 3
1
6560, 64eqtri 2458 . . . 4  |-  ( ( 7  x.  4 )  +  ( 0  +  3 ) )  = ; 3
1
66 7t5e35 10469 . . . . 5  |-  ( 7  x.  5 )  = ; 3
5
671, 23, 27, 66, 50decaddi 10428 . . . 4  |-  ( ( 7  x.  5 )  +  2 )  = ; 3
7
687, 23, 26, 27, 58, 29, 9, 9, 1, 65, 67decma2c 10424 . . 3  |-  ( ( 7  x. ; 4 5 )  +  2 )  = ;; 3 1 7
69 2lt7 10163 . . 3  |-  2  <  7
704, 57, 25, 68, 69ndvdsi 12932 . 2  |-  -.  7  || ;; 3 1 7
712, 15decnncl 10397 . . 3  |- ; 1 1  e.  NN
7227, 6deccl 10398 . . 3  |- ; 2 8  e.  NN0
73 9nn 10142 . . 3  |-  9  e.  NN
74 9nn0 10247 . . . 4  |-  9  e.  NN0
75 eqid 2438 . . . 4  |- ; 2 8  = ; 2 8
7674dec0h 10400 . . . 4  |-  9  = ; 0 9
772, 2deccl 10398 . . . 4  |- ; 1 1  e.  NN0
78 eqid 2438 . . . . 5  |- ; 1 1  = ; 1 1
7973nncni 10012 . . . . . . 7  |-  9  e.  CC
8079addid2i 9256 . . . . . 6  |-  ( 0  +  9 )  =  9
8180, 76eqtri 2458 . . . . 5  |-  ( 0  +  9 )  = ; 0
9
82 2cn 10072 . . . . . . . 8  |-  2  e.  CC
8382mulid2i 9095 . . . . . . 7  |-  ( 1  x.  2 )  =  2
8483, 32oveq12i 6095 . . . . . 6  |-  ( ( 1  x.  2 )  +  ( 0  +  1 ) )  =  ( 2  +  1 )
8584, 62eqtri 2458 . . . . 5  |-  ( ( 1  x.  2 )  +  ( 0  +  1 ) )  =  3
8683oveq1i 6093 . . . . . 6  |-  ( ( 1  x.  2 )  +  9 )  =  ( 2  +  9 )
87 9p2e11 10446 . . . . . . 7  |-  ( 9  +  2 )  = ; 1
1
8879, 82, 87addcomli 9260 . . . . . 6  |-  ( 2  +  9 )  = ; 1
1
8986, 88eqtri 2458 . . . . 5  |-  ( ( 1  x.  2 )  +  9 )  = ; 1
1
902, 2, 26, 74, 78, 81, 27, 2, 2, 85, 89decmac 10423 . . . 4  |-  ( (; 1
1  x.  2 )  +  ( 0  +  9 ) )  = ; 3
1
91 8nn 10141 . . . . . . . . 9  |-  8  e.  NN
9291nncni 10012 . . . . . . . 8  |-  8  e.  CC
9392mulid2i 9095 . . . . . . 7  |-  ( 1  x.  8 )  =  8
9493, 32oveq12i 6095 . . . . . 6  |-  ( ( 1  x.  8 )  +  ( 0  +  1 ) )  =  ( 8  +  1 )
95 8p1e9 10111 . . . . . 6  |-  ( 8  +  1 )  =  9
9694, 95eqtri 2458 . . . . 5  |-  ( ( 1  x.  8 )  +  ( 0  +  1 ) )  =  9
9793oveq1i 6093 . . . . . 6  |-  ( ( 1  x.  8 )  +  9 )  =  ( 8  +  9 )
98 9p8e17 10452 . . . . . . 7  |-  ( 9  +  8 )  = ; 1
7
9979, 92, 98addcomli 9260 . . . . . 6  |-  ( 8  +  9 )  = ; 1
7
10097, 99eqtri 2458 . . . . 5  |-  ( ( 1  x.  8 )  +  9 )  = ; 1
7
1012, 2, 26, 74, 78, 76, 6, 9, 2, 96, 100decmac 10423 . . . 4  |-  ( (; 1
1  x.  8 )  +  9 )  = ; 9
7
10227, 6, 26, 74, 75, 76, 77, 9, 74, 90, 101decma2c 10424 . . 3  |-  ( (; 1
1  x. ; 2 8 )  +  9 )  = ;; 3 1 7
103 9lt10 10180 . . . 4  |-  9  <  10
10415, 2, 74, 103declti 10409 . . 3  |-  9  < ; 1
1
10571, 72, 73, 102, 104ndvdsi 12932 . 2  |-  -. ; 1 1  || ;; 3 1 7
1062, 21decnncl 10397 . . 3  |- ; 1 3  e.  NN
10727, 7deccl 10398 . . 3  |- ; 2 4  e.  NN0
108 eqid 2438 . . . 4  |- ; 2 4  = ; 2 4
10923dec0h 10400 . . . 4  |-  5  = ; 0 5
1102, 1deccl 10398 . . . 4  |- ; 1 3  e.  NN0
111 eqid 2438 . . . . 5  |- ; 1 3  = ; 1 3
11247addid2i 9256 . . . . . 6  |-  ( 0  +  5 )  =  5
113112, 109eqtri 2458 . . . . 5  |-  ( 0  +  5 )  = ; 0
5
11418oveq1i 6093 . . . . . 6  |-  ( ( 3  x.  2 )  +  5 )  =  ( 6  +  5 )
115 6p5e11 10434 . . . . . 6  |-  ( 6  +  5 )  = ; 1
1
116114, 115eqtri 2458 . . . . 5  |-  ( ( 3  x.  2 )  +  5 )  = ; 1
1
1172, 1, 26, 23, 111, 113, 27, 2, 2, 85, 116decmac 10423 . . . 4  |-  ( (; 1
3  x.  2 )  +  ( 0  +  5 ) )  = ; 3
1
118 4cn 10076 . . . . . . . 8  |-  4  e.  CC
119118mulid2i 9095 . . . . . . 7  |-  ( 1  x.  4 )  =  4
120119, 32oveq12i 6095 . . . . . 6  |-  ( ( 1  x.  4 )  +  ( 0  +  1 ) )  =  ( 4  +  1 )
121 4p1e5 10107 . . . . . 6  |-  ( 4  +  1 )  =  5
122120, 121eqtri 2458 . . . . 5  |-  ( ( 1  x.  4 )  +  ( 0  +  1 ) )  =  5
123 4t3e12 10456 . . . . . . 7  |-  ( 4  x.  3 )  = ; 1
2
124118, 35, 123mulcomli 9099 . . . . . 6  |-  ( 3  x.  4 )  = ; 1
2
12547, 82, 50addcomli 9260 . . . . . 6  |-  ( 2  +  5 )  =  7
1262, 27, 23, 124, 125decaddi 10428 . . . . 5  |-  ( ( 3  x.  4 )  +  5 )  = ; 1
7
1272, 1, 26, 23, 111, 109, 7, 9, 2, 122, 126decmac 10423 . . . 4  |-  ( (; 1
3  x.  4 )  +  5 )  = ; 5
7
12827, 7, 26, 23, 108, 109, 110, 9, 23, 117, 127decma2c 10424 . . 3  |-  ( (; 1
3  x. ; 2 4 )  +  5 )  = ;; 3 1 7
129 5lt10 10184 . . . 4  |-  5  <  10
13015, 1, 23, 129declti 10409 . . 3  |-  5  < ; 1
3
131106, 107, 46, 128, 130ndvdsi 12932 . 2  |-  -. ; 1 3  || ;; 3 1 7
1322, 4decnncl 10397 . . 3  |- ; 1 7  e.  NN
1332, 6deccl 10398 . . 3  |- ; 1 8  e.  NN0
134 eqid 2438 . . . 4  |- ; 1 8  = ; 1 8
1352, 9deccl 10398 . . . 4  |- ; 1 7  e.  NN0
136 eqid 2438 . . . . 5  |- ; 1 7  = ; 1 7
137 3p1e4 10106 . . . . . . 7  |-  ( 3  +  1 )  =  4
13835, 31, 137addcomli 9260 . . . . . 6  |-  ( 1  +  3 )  =  4
13926, 2, 2, 1, 33, 111, 32, 138decadd 10425 . . . . 5  |-  ( 1  + ; 1 3 )  = ; 1
4
14031mulid1i 9094 . . . . . . 7  |-  ( 1  x.  1 )  =  1
141 1p1e2 10096 . . . . . . 7  |-  ( 1  +  1 )  =  2
142140, 141oveq12i 6095 . . . . . 6  |-  ( ( 1  x.  1 )  +  ( 1  +  1 ) )  =  ( 1  +  2 )
14382, 31, 62addcomli 9260 . . . . . 6  |-  ( 1  +  2 )  =  3
144142, 143eqtri 2458 . . . . 5  |-  ( ( 1  x.  1 )  +  ( 1  +  1 ) )  =  3
1454nncni 10012 . . . . . . . 8  |-  7  e.  CC
146145mulid1i 9094 . . . . . . 7  |-  ( 7  x.  1 )  =  7
147146oveq1i 6093 . . . . . 6  |-  ( ( 7  x.  1 )  +  4 )  =  ( 7  +  4 )
148 7p4e11 10436 . . . . . 6  |-  ( 7  +  4 )  = ; 1
1
149147, 148eqtri 2458 . . . . 5  |-  ( ( 7  x.  1 )  +  4 )  = ; 1
1
1502, 9, 2, 7, 136, 139, 2, 2, 2, 144, 149decmac 10423 . . . 4  |-  ( (; 1
7  x.  1 )  +  ( 1  + ; 1
3 ) )  = ; 3
1
15193, 112oveq12i 6095 . . . . . 6  |-  ( ( 1  x.  8 )  +  ( 0  +  5 ) )  =  ( 8  +  5 )
152 8p5e13 10442 . . . . . 6  |-  ( 8  +  5 )  = ; 1
3
153151, 152eqtri 2458 . . . . 5  |-  ( ( 1  x.  8 )  +  ( 0  +  5 ) )  = ; 1
3
154 6nn0 10244 . . . . . 6  |-  6  e.  NN0
155 6p1e7 10109 . . . . . 6  |-  ( 6  +  1 )  =  7
156 8t7e56 10477 . . . . . . 7  |-  ( 8  x.  7 )  = ; 5
6
15792, 145, 156mulcomli 9099 . . . . . 6  |-  ( 7  x.  8 )  = ; 5
6
15823, 154, 155, 157decsuc 10407 . . . . 5  |-  ( ( 7  x.  8 )  +  1 )  = ; 5
7
1592, 9, 26, 2, 136, 33, 6, 9, 23, 153, 158decmac 10423 . . . 4  |-  ( (; 1
7  x.  8 )  +  1 )  = ;; 1 3 7
1602, 6, 2, 2, 134, 78, 135, 9, 110, 150, 159decma2c 10424 . . 3  |-  ( (; 1
7  x. ; 1 8 )  + ; 1
1 )  = ;; 3 1 7
161 1lt7 10164 . . . 4  |-  1  <  7
1622, 2, 4, 161declt 10405 . . 3  |- ; 1 1  < ; 1 7
163132, 133, 71, 160, 162ndvdsi 12932 . 2  |-  -. ; 1 7  || ;; 3 1 7
1642, 73decnncl 10397 . . 3  |- ; 1 9  e.  NN
1652, 154deccl 10398 . . 3  |- ; 1 6  e.  NN0
166 eqid 2438 . . . 4  |- ; 1 6  = ; 1 6
1672, 74deccl 10398 . . . 4  |- ; 1 9  e.  NN0
168 eqid 2438 . . . . 5  |- ; 1 9  = ; 1 9
16926, 2, 2, 2, 33, 78, 32, 141decadd 10425 . . . . 5  |-  ( 1  + ; 1 1 )  = ; 1
2
17079mulid1i 9094 . . . . . . 7  |-  ( 9  x.  1 )  =  9
171170oveq1i 6093 . . . . . 6  |-  ( ( 9  x.  1 )  +  2 )  =  ( 9  +  2 )
172171, 87eqtri 2458 . . . . 5  |-  ( ( 9  x.  1 )  +  2 )  = ; 1
1
1732, 74, 2, 27, 168, 169, 2, 2, 2, 144, 172decmac 10423 . . . 4  |-  ( (; 1
9  x.  1 )  +  ( 1  + ; 1
1 ) )  = ; 3
1
1741dec0h 10400 . . . . 5  |-  3  = ; 0 3
175 6nn 10139 . . . . . . . . 9  |-  6  e.  NN
176175nncni 10012 . . . . . . . 8  |-  6  e.  CC
177176mulid2i 9095 . . . . . . 7  |-  ( 1  x.  6 )  =  6
178177, 112oveq12i 6095 . . . . . 6  |-  ( ( 1  x.  6 )  +  ( 0  +  5 ) )  =  ( 6  +  5 )
179178, 115eqtri 2458 . . . . 5  |-  ( ( 1  x.  6 )  +  ( 0  +  5 ) )  = ; 1
1
180 9t6e54 10483 . . . . . 6  |-  ( 9  x.  6 )  = ; 5
4
181 4p3e7 10116 . . . . . 6  |-  ( 4  +  3 )  =  7
18223, 7, 1, 180, 181decaddi 10428 . . . . 5  |-  ( ( 9  x.  6 )  +  3 )  = ; 5
7
1832, 74, 26, 1, 168, 174, 154, 9, 23, 179, 182decmac 10423 . . . 4  |-  ( (; 1
9  x.  6 )  +  3 )  = ;; 1 1 7
1842, 154, 2, 1, 166, 111, 167, 9, 77, 173, 183decma2c 10424 . . 3  |-  ( (; 1
9  x. ; 1 6 )  + ; 1
3 )  = ;; 3 1 7
185 3lt9 10177 . . . 4  |-  3  <  9
1862, 1, 73, 185declt 10405 . . 3  |- ; 1 3  < ; 1 9
187164, 165, 106, 184, 186ndvdsi 12932 . 2  |-  -. ; 1 9  || ;; 3 1 7
18827, 21decnncl 10397 . . 3  |- ; 2 3  e.  NN
189106nnnn0i 10231 . . 3  |- ; 1 3  e.  NN0
1902, 91decnncl 10397 . . 3  |- ; 1 8  e.  NN
19127, 1deccl 10398 . . . 4  |- ; 2 3  e.  NN0
192 eqid 2438 . . . . 5  |- ; 2 3  = ; 2 3
193 7p1e8 10110 . . . . . . 7  |-  ( 7  +  1 )  =  8
194145, 31, 193addcomli 9260 . . . . . 6  |-  ( 1  +  7 )  =  8
1956dec0h 10400 . . . . . 6  |-  8  = ; 0 8
196194, 195eqtri 2458 . . . . 5  |-  ( 1  +  7 )  = ; 0
8
19782mulid1i 9094 . . . . . . 7  |-  ( 2  x.  1 )  =  2
198197, 32oveq12i 6095 . . . . . 6  |-  ( ( 2  x.  1 )  +  ( 0  +  1 ) )  =  ( 2  +  1 )
199198, 62eqtri 2458 . . . . 5  |-  ( ( 2  x.  1 )  +  ( 0  +  1 ) )  =  3
20036oveq1i 6093 . . . . . 6  |-  ( ( 3  x.  1 )  +  8 )  =  ( 3  +  8 )
20192, 35, 63addcomli 9260 . . . . . 6  |-  ( 3  +  8 )  = ; 1
1
202200, 201eqtri 2458 . . . . 5  |-  ( ( 3  x.  1 )  +  8 )  = ; 1
1
20327, 1, 26, 6, 192, 196, 2, 2, 2, 199, 202decmac 10423 . . . 4  |-  ( (; 2
3  x.  1 )  +  ( 1  +  7 ) )  = ; 3
1
20435, 82, 18mulcomli 9099 . . . . . . 7  |-  ( 2  x.  3 )  =  6
205204, 32oveq12i 6095 . . . . . 6  |-  ( ( 2  x.  3 )  +  ( 0  +  1 ) )  =  ( 6  +  1 )
206205, 155eqtri 2458 . . . . 5  |-  ( ( 2  x.  3 )  +  ( 0  +  1 ) )  =  7
207 3t3e9 10131 . . . . . . 7  |-  ( 3  x.  3 )  =  9
208207oveq1i 6093 . . . . . 6  |-  ( ( 3  x.  3 )  +  8 )  =  ( 9  +  8 )
209208, 98eqtri 2458 . . . . 5  |-  ( ( 3  x.  3 )  +  8 )  = ; 1
7
21027, 1, 26, 6, 192, 195, 1, 9, 2, 206, 209decmac 10423 . . . 4  |-  ( (; 2
3  x.  3 )  +  8 )  = ; 7
7
2112, 1, 2, 6, 111, 134, 191, 9, 9, 203, 210decma2c 10424 . . 3  |-  ( (; 2
3  x. ; 1 3 )  + ; 1
8 )  = ;; 3 1 7
212 8lt10 10181 . . . 4  |-  8  <  10
213 1lt2 10144 . . . 4  |-  1  <  2
2142, 27, 6, 1, 212, 213decltc 10406 . . 3  |- ; 1 8  < ; 2 3
215188, 189, 190, 211, 214ndvdsi 12932 . 2  |-  -. ; 2 3  || ;; 3 1 7
2165, 14, 17, 20, 54, 56, 70, 105, 131, 163, 187, 215prmlem2 13444 1  |- ;; 3 1 7  e.  Prime
Colors of variables: wff set class
Syntax hints:    e. wcel 1726  (class class class)co 6083   0cc0 8992   1c1 8993    + caddc 8995    x. cmul 8997   2c2 10051   3c3 10052   4c4 10053   5c5 10054   6c6 10055   7c7 10056   8c8 10057   9c9 10058   10c10 10059  ;cdc 10384   Primecprime 13081
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703  ax-cnex 9048  ax-resscn 9049  ax-1cn 9050  ax-icn 9051  ax-addcl 9052  ax-addrcl 9053  ax-mulcl 9054  ax-mulrcl 9055  ax-mulcom 9056  ax-addass 9057  ax-mulass 9058  ax-distr 9059  ax-i2m1 9060  ax-1ne0 9061  ax-1rid 9062  ax-rnegex 9063  ax-rrecex 9064  ax-cnre 9065  ax-pre-lttri 9066  ax-pre-lttrn 9067  ax-pre-ltadd 9068  ax-pre-mulgt0 9069  ax-pre-sup 9070
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rmo 2715  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-int 4053  df-iun 4097  df-br 4215  df-opab 4269  df-mpt 4270  df-tr 4305  df-eprel 4496  df-id 4500  df-po 4505  df-so 4506  df-fr 4543  df-we 4545  df-ord 4586  df-on 4587  df-lim 4588  df-suc 4589  df-om 4848  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-ov 6086  df-oprab 6087  df-mpt2 6088  df-1st 6351  df-2nd 6352  df-riota 6551  df-recs 6635  df-rdg 6670  df-1o 6726  df-2o 6727  df-oadd 6730  df-er 6907  df-en 7112  df-dom 7113  df-sdom 7114  df-fin 7115  df-sup 7448  df-pnf 9124  df-mnf 9125  df-xr 9126  df-ltxr 9127  df-le 9128  df-sub 9295  df-neg 9296  df-div 9680  df-nn 10003  df-2 10060  df-3 10061  df-4 10062  df-5 10063  df-6 10064  df-7 10065  df-8 10066  df-9 10067  df-10 10068  df-n0 10224  df-z 10285  df-dec 10385  df-uz 10491  df-rp 10615  df-fz 11046  df-seq 11326  df-exp 11385  df-cj 11906  df-re 11907  df-im 11908  df-sqr 12042  df-abs 12043  df-dvds 12855  df-prm 13082
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