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Theorem 317prm 14274
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 10711 . . . 4  |-  3  e.  NN0
2 1nn0 10709 . . . 4  |-  1  e.  NN0
31, 2deccl 10883 . . 3  |- ; 3 1  e.  NN0
4 7nn 10598 . . 3  |-  7  e.  NN
53, 4decnncl 10882 . 2  |- ;; 3 1 7  e.  NN
6 8nn0 10716 . . . 4  |-  8  e.  NN0
7 4nn0 10712 . . . 4  |-  4  e.  NN0
86, 7deccl 10883 . . 3  |- ; 8 4  e.  NN0
9 7nn0 10715 . . 3  |-  7  e.  NN0
10 7lt10 10640 . . 3  |-  7  <  10
11 1lt10 10646 . . . 4  |-  1  <  10
12 3lt8 10627 . . . 4  |-  3  <  8
131, 6, 2, 7, 11, 12decltc 10891 . . 3  |- ; 3 1  < ; 8 4
143, 8, 9, 2, 10, 13decltc 10891 . 2  |- ;; 3 1 7  < ;; 8 4 1
15 1nn 10447 . . . 4  |-  1  e.  NN
161, 15decnncl 10882 . . 3  |- ; 3 1  e.  NN
1716, 9, 2, 11declti 10894 . 2  |-  1  < ;; 3 1 7
18 3t2e6 10587 . . 3  |-  ( 3  x.  2 )  =  6
19 df-7 10499 . . 3  |-  7  =  ( 6  +  1 )
203, 1, 18, 19dec2dvds 14213 . 2  |-  -.  2  || ;; 3 1 7
21 3nn 10594 . . 3  |-  3  e.  NN
22 10nn0 10718 . . . 4  |-  10  e.  NN0
23 5nn0 10713 . . . 4  |-  5  e.  NN0
2422, 23deccl 10883 . . 3  |- ; 10 5  e.  NN0
25 2nn 10593 . . 3  |-  2  e.  NN
26 0nn0 10708 . . . 4  |-  0  e.  NN0
27 2nn0 10710 . . . 4  |-  2  e.  NN0
28 eqid 2454 . . . 4  |- ; 10 5  = ; 10 5
2927dec0h 10885 . . . 4  |-  2  = ; 0 2
30 dec10 10899 . . . . 5  |-  10  = ; 1 0
31 ax-1cn 9454 . . . . . . 7  |-  1  e.  CC
3231addid2i 9671 . . . . . 6  |-  ( 0  +  1 )  =  1
332dec0h 10885 . . . . . 6  |-  1  = ; 0 1
3432, 33eqtri 2483 . . . . 5  |-  ( 0  +  1 )  = ; 0
1
35 3cn 10510 . . . . . . . 8  |-  3  e.  CC
3635mulid1i 9502 . . . . . . 7  |-  ( 3  x.  1 )  =  3
37 00id 9658 . . . . . . 7  |-  ( 0  +  0 )  =  0
3836, 37oveq12i 6215 . . . . . 6  |-  ( ( 3  x.  1 )  +  ( 0  +  0 ) )  =  ( 3  +  0 )
3935addid1i 9670 . . . . . 6  |-  ( 3  +  0 )  =  3
4038, 39eqtri 2483 . . . . 5  |-  ( ( 3  x.  1 )  +  ( 0  +  0 ) )  =  3
4135mul01i 9673 . . . . . . . 8  |-  ( 3  x.  0 )  =  0
4241oveq1i 6213 . . . . . . 7  |-  ( ( 3  x.  0 )  +  1 )  =  ( 0  +  1 )
4342, 32eqtri 2483 . . . . . 6  |-  ( ( 3  x.  0 )  +  1 )  =  1
4443, 33eqtri 2483 . . . . 5  |-  ( ( 3  x.  0 )  +  1 )  = ; 0
1
452, 26, 26, 2, 30, 34, 1, 2, 26, 40, 44decma2c 10909 . . . 4  |-  ( ( 3  x.  10 )  +  ( 0  +  1 ) )  = ; 3
1
46 5cn 10515 . . . . . 6  |-  5  e.  CC
47 5t3e15 10943 . . . . . 6  |-  ( 5  x.  3 )  = ; 1
5
4846, 35, 47mulcomli 9507 . . . . 5  |-  ( 3  x.  5 )  = ; 1
5
49 5p2e7 10573 . . . . 5  |-  ( 5  +  2 )  =  7
502, 23, 27, 48, 49decaddi 10913 . . . 4  |-  ( ( 3  x.  5 )  +  2 )  = ; 1
7
5122, 23, 26, 27, 28, 29, 1, 9, 2, 45, 50decma2c 10909 . . 3  |-  ( ( 3  x. ; 10 5 )  +  2 )  = ;; 3 1 7
52 2lt3 10603 . . 3  |-  2  <  3
5321, 24, 25, 51, 52ndvdsi 13735 . 2  |-  -.  3  || ;; 3 1 7
54 2lt5 10610 . . 3  |-  2  <  5
553, 25, 54, 49dec5dvds2 14215 . 2  |-  -.  5  || ;; 3 1 7
567, 23deccl 10883 . . 3  |- ; 4 5  e.  NN0
57 eqid 2454 . . . 4  |- ; 4 5  = ; 4 5
5835addid2i 9671 . . . . . 6  |-  ( 0  +  3 )  =  3
5958oveq2i 6214 . . . . 5  |-  ( ( 7  x.  4 )  +  ( 0  +  3 ) )  =  ( ( 7  x.  4 )  +  3 )
60 7t4e28 10953 . . . . . 6  |-  ( 7  x.  4 )  = ; 2
8
61 2p1e3 10559 . . . . . 6  |-  ( 2  +  1 )  =  3
62 8p3e11 10925 . . . . . 6  |-  ( 8  +  3 )  = ; 1
1
6327, 6, 1, 60, 61, 2, 62decaddci 10914 . . . . 5  |-  ( ( 7  x.  4 )  +  3 )  = ; 3
1
6459, 63eqtri 2483 . . . 4  |-  ( ( 7  x.  4 )  +  ( 0  +  3 ) )  = ; 3
1
65 7t5e35 10954 . . . . 5  |-  ( 7  x.  5 )  = ; 3
5
661, 23, 27, 65, 49decaddi 10913 . . . 4  |-  ( ( 7  x.  5 )  +  2 )  = ; 3
7
677, 23, 26, 27, 57, 29, 9, 9, 1, 64, 66decma2c 10909 . . 3  |-  ( ( 7  x. ; 4 5 )  +  2 )  = ;; 3 1 7
68 2lt7 10621 . . 3  |-  2  <  7
694, 56, 25, 67, 68ndvdsi 13735 . 2  |-  -.  7  || ;; 3 1 7
702, 15decnncl 10882 . . 3  |- ; 1 1  e.  NN
7127, 6deccl 10883 . . 3  |- ; 2 8  e.  NN0
72 9nn 10600 . . 3  |-  9  e.  NN
73 9nn0 10717 . . . 4  |-  9  e.  NN0
74 eqid 2454 . . . 4  |- ; 2 8  = ; 2 8
7573dec0h 10885 . . . 4  |-  9  = ; 0 9
762, 2deccl 10883 . . . 4  |- ; 1 1  e.  NN0
77 eqid 2454 . . . . 5  |- ; 1 1  = ; 1 1
78 9cn 10523 . . . . . . 7  |-  9  e.  CC
7978addid2i 9671 . . . . . 6  |-  ( 0  +  9 )  =  9
8079, 75eqtri 2483 . . . . 5  |-  ( 0  +  9 )  = ; 0
9
81 2cn 10506 . . . . . . . 8  |-  2  e.  CC
8281mulid2i 9503 . . . . . . 7  |-  ( 1  x.  2 )  =  2
8382, 32oveq12i 6215 . . . . . 6  |-  ( ( 1  x.  2 )  +  ( 0  +  1 ) )  =  ( 2  +  1 )
8483, 61eqtri 2483 . . . . 5  |-  ( ( 1  x.  2 )  +  ( 0  +  1 ) )  =  3
8582oveq1i 6213 . . . . . 6  |-  ( ( 1  x.  2 )  +  9 )  =  ( 2  +  9 )
86 9p2e11 10931 . . . . . . 7  |-  ( 9  +  2 )  = ; 1
1
8778, 81, 86addcomli 9675 . . . . . 6  |-  ( 2  +  9 )  = ; 1
1
8885, 87eqtri 2483 . . . . 5  |-  ( ( 1  x.  2 )  +  9 )  = ; 1
1
892, 2, 26, 73, 77, 80, 27, 2, 2, 84, 88decmac 10908 . . . 4  |-  ( (; 1
1  x.  2 )  +  ( 0  +  9 ) )  = ; 3
1
90 8cn 10521 . . . . . . . 8  |-  8  e.  CC
9190mulid2i 9503 . . . . . . 7  |-  ( 1  x.  8 )  =  8
9291, 32oveq12i 6215 . . . . . 6  |-  ( ( 1  x.  8 )  +  ( 0  +  1 ) )  =  ( 8  +  1 )
93 8p1e9 10566 . . . . . 6  |-  ( 8  +  1 )  =  9
9492, 93eqtri 2483 . . . . 5  |-  ( ( 1  x.  8 )  +  ( 0  +  1 ) )  =  9
9591oveq1i 6213 . . . . . 6  |-  ( ( 1  x.  8 )  +  9 )  =  ( 8  +  9 )
96 9p8e17 10937 . . . . . . 7  |-  ( 9  +  8 )  = ; 1
7
9778, 90, 96addcomli 9675 . . . . . 6  |-  ( 8  +  9 )  = ; 1
7
9895, 97eqtri 2483 . . . . 5  |-  ( ( 1  x.  8 )  +  9 )  = ; 1
7
992, 2, 26, 73, 77, 75, 6, 9, 2, 94, 98decmac 10908 . . . 4  |-  ( (; 1
1  x.  8 )  +  9 )  = ; 9
7
10027, 6, 26, 73, 74, 75, 76, 9, 73, 89, 99decma2c 10909 . . 3  |-  ( (; 1
1  x. ; 2 8 )  +  9 )  = ;; 3 1 7
101 9lt10 10638 . . . 4  |-  9  <  10
10215, 2, 73, 101declti 10894 . . 3  |-  9  < ; 1
1
10370, 71, 72, 100, 102ndvdsi 13735 . 2  |-  -. ; 1 1  || ;; 3 1 7
1042, 21decnncl 10882 . . 3  |- ; 1 3  e.  NN
10527, 7deccl 10883 . . 3  |- ; 2 4  e.  NN0
106 5nn 10596 . . 3  |-  5  e.  NN
107 eqid 2454 . . . 4  |- ; 2 4  = ; 2 4
10823dec0h 10885 . . . 4  |-  5  = ; 0 5
1092, 1deccl 10883 . . . 4  |- ; 1 3  e.  NN0
110 eqid 2454 . . . . 5  |- ; 1 3  = ; 1 3
11146addid2i 9671 . . . . . 6  |-  ( 0  +  5 )  =  5
112111, 108eqtri 2483 . . . . 5  |-  ( 0  +  5 )  = ; 0
5
11318oveq1i 6213 . . . . . 6  |-  ( ( 3  x.  2 )  +  5 )  =  ( 6  +  5 )
114 6p5e11 10919 . . . . . 6  |-  ( 6  +  5 )  = ; 1
1
115113, 114eqtri 2483 . . . . 5  |-  ( ( 3  x.  2 )  +  5 )  = ; 1
1
1162, 1, 26, 23, 110, 112, 27, 2, 2, 84, 115decmac 10908 . . . 4  |-  ( (; 1
3  x.  2 )  +  ( 0  +  5 ) )  = ; 3
1
117 4cn 10513 . . . . . . . 8  |-  4  e.  CC
118117mulid2i 9503 . . . . . . 7  |-  ( 1  x.  4 )  =  4
119118, 32oveq12i 6215 . . . . . 6  |-  ( ( 1  x.  4 )  +  ( 0  +  1 ) )  =  ( 4  +  1 )
120 4p1e5 10562 . . . . . 6  |-  ( 4  +  1 )  =  5
121119, 120eqtri 2483 . . . . 5  |-  ( ( 1  x.  4 )  +  ( 0  +  1 ) )  =  5
122 4t3e12 10941 . . . . . . 7  |-  ( 4  x.  3 )  = ; 1
2
123117, 35, 122mulcomli 9507 . . . . . 6  |-  ( 3  x.  4 )  = ; 1
2
12446, 81, 49addcomli 9675 . . . . . 6  |-  ( 2  +  5 )  =  7
1252, 27, 23, 123, 124decaddi 10913 . . . . 5  |-  ( ( 3  x.  4 )  +  5 )  = ; 1
7
1262, 1, 26, 23, 110, 108, 7, 9, 2, 121, 125decmac 10908 . . . 4  |-  ( (; 1
3  x.  4 )  +  5 )  = ; 5
7
12727, 7, 26, 23, 107, 108, 109, 9, 23, 116, 126decma2c 10909 . . 3  |-  ( (; 1
3  x. ; 2 4 )  +  5 )  = ;; 3 1 7
128 5lt10 10642 . . . 4  |-  5  <  10
12915, 1, 23, 128declti 10894 . . 3  |-  5  < ; 1
3
130104, 105, 106, 127, 129ndvdsi 13735 . 2  |-  -. ; 1 3  || ;; 3 1 7
1312, 4decnncl 10882 . . 3  |- ; 1 7  e.  NN
1322, 6deccl 10883 . . 3  |- ; 1 8  e.  NN0
133 eqid 2454 . . . 4  |- ; 1 8  = ; 1 8
1342, 9deccl 10883 . . . 4  |- ; 1 7  e.  NN0
135 eqid 2454 . . . . 5  |- ; 1 7  = ; 1 7
136 3p1e4 10561 . . . . . . 7  |-  ( 3  +  1 )  =  4
13735, 31, 136addcomli 9675 . . . . . 6  |-  ( 1  +  3 )  =  4
13826, 2, 2, 1, 33, 110, 32, 137decadd 10910 . . . . 5  |-  ( 1  + ; 1 3 )  = ; 1
4
13931mulid1i 9502 . . . . . . 7  |-  ( 1  x.  1 )  =  1
140 1p1e2 10549 . . . . . . 7  |-  ( 1  +  1 )  =  2
141139, 140oveq12i 6215 . . . . . 6  |-  ( ( 1  x.  1 )  +  ( 1  +  1 ) )  =  ( 1  +  2 )
142 1p2e3 10560 . . . . . 6  |-  ( 1  +  2 )  =  3
143141, 142eqtri 2483 . . . . 5  |-  ( ( 1  x.  1 )  +  ( 1  +  1 ) )  =  3
144 7cn 10519 . . . . . . . 8  |-  7  e.  CC
145144mulid1i 9502 . . . . . . 7  |-  ( 7  x.  1 )  =  7
146145oveq1i 6213 . . . . . 6  |-  ( ( 7  x.  1 )  +  4 )  =  ( 7  +  4 )
147 7p4e11 10921 . . . . . 6  |-  ( 7  +  4 )  = ; 1
1
148146, 147eqtri 2483 . . . . 5  |-  ( ( 7  x.  1 )  +  4 )  = ; 1
1
1492, 9, 2, 7, 135, 138, 2, 2, 2, 143, 148decmac 10908 . . . 4  |-  ( (; 1
7  x.  1 )  +  ( 1  + ; 1
3 ) )  = ; 3
1
15091, 111oveq12i 6215 . . . . . 6  |-  ( ( 1  x.  8 )  +  ( 0  +  5 ) )  =  ( 8  +  5 )
151 8p5e13 10927 . . . . . 6  |-  ( 8  +  5 )  = ; 1
3
152150, 151eqtri 2483 . . . . 5  |-  ( ( 1  x.  8 )  +  ( 0  +  5 ) )  = ; 1
3
153 6nn0 10714 . . . . . 6  |-  6  e.  NN0
154 6p1e7 10564 . . . . . 6  |-  ( 6  +  1 )  =  7
155 8t7e56 10962 . . . . . . 7  |-  ( 8  x.  7 )  = ; 5
6
15690, 144, 155mulcomli 9507 . . . . . 6  |-  ( 7  x.  8 )  = ; 5
6
15723, 153, 154, 156decsuc 10892 . . . . 5  |-  ( ( 7  x.  8 )  +  1 )  = ; 5
7
1582, 9, 26, 2, 135, 33, 6, 9, 23, 152, 157decmac 10908 . . . 4  |-  ( (; 1
7  x.  8 )  +  1 )  = ;; 1 3 7
1592, 6, 2, 2, 133, 77, 134, 9, 109, 149, 158decma2c 10909 . . 3  |-  ( (; 1
7  x. ; 1 8 )  + ; 1
1 )  = ;; 3 1 7
160 1lt7 10622 . . . 4  |-  1  <  7
1612, 2, 4, 160declt 10890 . . 3  |- ; 1 1  < ; 1 7
162131, 132, 70, 159, 161ndvdsi 13735 . 2  |-  -. ; 1 7  || ;; 3 1 7
1632, 72decnncl 10882 . . 3  |- ; 1 9  e.  NN
1642, 153deccl 10883 . . 3  |- ; 1 6  e.  NN0
165 eqid 2454 . . . 4  |- ; 1 6  = ; 1 6
1662, 73deccl 10883 . . . 4  |- ; 1 9  e.  NN0
167 eqid 2454 . . . . 5  |- ; 1 9  = ; 1 9
16826, 2, 2, 2, 33, 77, 32, 140decadd 10910 . . . . 5  |-  ( 1  + ; 1 1 )  = ; 1
2
16978mulid1i 9502 . . . . . . 7  |-  ( 9  x.  1 )  =  9
170169oveq1i 6213 . . . . . 6  |-  ( ( 9  x.  1 )  +  2 )  =  ( 9  +  2 )
171170, 86eqtri 2483 . . . . 5  |-  ( ( 9  x.  1 )  +  2 )  = ; 1
1
1722, 73, 2, 27, 167, 168, 2, 2, 2, 143, 171decmac 10908 . . . 4  |-  ( (; 1
9  x.  1 )  +  ( 1  + ; 1
1 ) )  = ; 3
1
1731dec0h 10885 . . . . 5  |-  3  = ; 0 3
174 6cn 10517 . . . . . . . 8  |-  6  e.  CC
175174mulid2i 9503 . . . . . . 7  |-  ( 1  x.  6 )  =  6
176175, 111oveq12i 6215 . . . . . 6  |-  ( ( 1  x.  6 )  +  ( 0  +  5 ) )  =  ( 6  +  5 )
177176, 114eqtri 2483 . . . . 5  |-  ( ( 1  x.  6 )  +  ( 0  +  5 ) )  = ; 1
1
178 9t6e54 10968 . . . . . 6  |-  ( 9  x.  6 )  = ; 5
4
179 4p3e7 10571 . . . . . 6  |-  ( 4  +  3 )  =  7
18023, 7, 1, 178, 179decaddi 10913 . . . . 5  |-  ( ( 9  x.  6 )  +  3 )  = ; 5
7
1812, 73, 26, 1, 167, 173, 153, 9, 23, 177, 180decmac 10908 . . . 4  |-  ( (; 1
9  x.  6 )  +  3 )  = ;; 1 1 7
1822, 153, 2, 1, 165, 110, 166, 9, 76, 172, 181decma2c 10909 . . 3  |-  ( (; 1
9  x. ; 1 6 )  + ; 1
3 )  = ;; 3 1 7
183 3lt9 10635 . . . 4  |-  3  <  9
1842, 1, 72, 183declt 10890 . . 3  |- ; 1 3  < ; 1 9
185163, 164, 104, 182, 184ndvdsi 13735 . 2  |-  -. ; 1 9  || ;; 3 1 7
18627, 21decnncl 10882 . . 3  |- ; 2 3  e.  NN
187104nnnn0i 10701 . . 3  |- ; 1 3  e.  NN0
188 8nn 10599 . . . 4  |-  8  e.  NN
1892, 188decnncl 10882 . . 3  |- ; 1 8  e.  NN
19027, 1deccl 10883 . . . 4  |- ; 2 3  e.  NN0
191 eqid 2454 . . . . 5  |- ; 2 3  = ; 2 3
192 7p1e8 10565 . . . . . . 7  |-  ( 7  +  1 )  =  8
193144, 31, 192addcomli 9675 . . . . . 6  |-  ( 1  +  7 )  =  8
1946dec0h 10885 . . . . . 6  |-  8  = ; 0 8
195193, 194eqtri 2483 . . . . 5  |-  ( 1  +  7 )  = ; 0
8
19681mulid1i 9502 . . . . . . 7  |-  ( 2  x.  1 )  =  2
197196, 32oveq12i 6215 . . . . . 6  |-  ( ( 2  x.  1 )  +  ( 0  +  1 ) )  =  ( 2  +  1 )
198197, 61eqtri 2483 . . . . 5  |-  ( ( 2  x.  1 )  +  ( 0  +  1 ) )  =  3
19936oveq1i 6213 . . . . . 6  |-  ( ( 3  x.  1 )  +  8 )  =  ( 3  +  8 )
20090, 35, 62addcomli 9675 . . . . . 6  |-  ( 3  +  8 )  = ; 1
1
201199, 200eqtri 2483 . . . . 5  |-  ( ( 3  x.  1 )  +  8 )  = ; 1
1
20227, 1, 26, 6, 191, 195, 2, 2, 2, 198, 201decmac 10908 . . . 4  |-  ( (; 2
3  x.  1 )  +  ( 1  +  7 ) )  = ; 3
1
20335, 81, 18mulcomli 9507 . . . . . . 7  |-  ( 2  x.  3 )  =  6
204203, 32oveq12i 6215 . . . . . 6  |-  ( ( 2  x.  3 )  +  ( 0  +  1 ) )  =  ( 6  +  1 )
205204, 154eqtri 2483 . . . . 5  |-  ( ( 2  x.  3 )  +  ( 0  +  1 ) )  =  7
206 3t3e9 10588 . . . . . . 7  |-  ( 3  x.  3 )  =  9
207206oveq1i 6213 . . . . . 6  |-  ( ( 3  x.  3 )  +  8 )  =  ( 9  +  8 )
208207, 96eqtri 2483 . . . . 5  |-  ( ( 3  x.  3 )  +  8 )  = ; 1
7
20927, 1, 26, 6, 191, 194, 1, 9, 2, 205, 208decmac 10908 . . . 4  |-  ( (; 2
3  x.  3 )  +  8 )  = ; 7
7
2102, 1, 2, 6, 110, 133, 190, 9, 9, 202, 209decma2c 10909 . . 3  |-  ( (; 2
3  x. ; 1 3 )  + ; 1
8 )  = ;; 3 1 7
211 8lt10 10639 . . . 4  |-  8  <  10
212 1lt2 10602 . . . 4  |-  1  <  2
2132, 27, 6, 1, 211, 212decltc 10891 . . 3  |- ; 1 8  < ; 2 3
214186, 187, 189, 210, 213ndvdsi 13735 . 2  |-  -. ; 2 3  || ;; 3 1 7
2155, 14, 17, 20, 53, 55, 69, 103, 130, 162, 185, 214prmlem2 14268 1  |- ;; 3 1 7  e.  Prime
Colors of variables: wff setvar class
Syntax hints:    e. wcel 1758  (class class class)co 6203   0cc0 9396   1c1 9397    + caddc 9399    x. cmul 9401   2c2 10485   3c3 10486   4c4 10487   5c5 10488   6c6 10489   7c7 10490   8c8 10491   9c9 10492   10c10 10493  ;cdc 10869   Primecprime 13884
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642  ax-un 6485  ax-cnex 9452  ax-resscn 9453  ax-1cn 9454  ax-icn 9455  ax-addcl 9456  ax-addrcl 9457  ax-mulcl 9458  ax-mulrcl 9459  ax-mulcom 9460  ax-addass 9461  ax-mulass 9462  ax-distr 9463  ax-i2m1 9464  ax-1ne0 9465  ax-1rid 9466  ax-rnegex 9467  ax-rrecex 9468  ax-cnre 9469  ax-pre-lttri 9470  ax-pre-lttrn 9471  ax-pre-ltadd 9472  ax-pre-mulgt0 9473  ax-pre-sup 9474
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-nel 2651  df-ral 2804  df-rex 2805  df-reu 2806  df-rmo 2807  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3399  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-pss 3455  df-nul 3749  df-if 3903  df-pw 3973  df-sn 3989  df-pr 3991  df-tp 3993  df-op 3995  df-uni 4203  df-int 4240  df-iun 4284  df-br 4404  df-opab 4462  df-mpt 4463  df-tr 4497  df-eprel 4743  df-id 4747  df-po 4752  df-so 4753  df-fr 4790  df-we 4792  df-ord 4833  df-on 4834  df-lim 4835  df-suc 4836  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-res 4963  df-ima 4964  df-iota 5492  df-fun 5531  df-fn 5532  df-f 5533  df-f1 5534  df-fo 5535  df-f1o 5536  df-fv 5537  df-riota 6164  df-ov 6206  df-oprab 6207  df-mpt2 6208  df-om 6590  df-1st 6690  df-2nd 6691  df-recs 6945  df-rdg 6979  df-1o 7033  df-2o 7034  df-oadd 7037  df-er 7214  df-en 7424  df-dom 7425  df-sdom 7426  df-fin 7427  df-sup 7805  df-pnf 9534  df-mnf 9535  df-xr 9536  df-ltxr 9537  df-le 9538  df-sub 9711  df-neg 9712  df-div 10108  df-nn 10437  df-2 10494  df-3 10495  df-4 10496  df-5 10497  df-6 10498  df-7 10499  df-8 10500  df-9 10501  df-10 10502  df-n0 10694  df-z 10761  df-dec 10870  df-uz 10976  df-rp 11106  df-fz 11558  df-seq 11927  df-exp 11986  df-cj 12709  df-re 12710  df-im 12711  df-sqr 12845  df-abs 12846  df-dvds 13657  df-prm 13885
This theorem is referenced by: (None)
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