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Theorem 317prm 15082
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 10887 . . . 4  |-  3  e.  NN0
2 1nn0 10885 . . . 4  |-  1  e.  NN0
31, 2deccl 11065 . . 3  |- ; 3 1  e.  NN0
4 7nn 10772 . . 3  |-  7  e.  NN
53, 4decnncl 11064 . 2  |- ;; 3 1 7  e.  NN
6 8nn0 10892 . . . 4  |-  8  e.  NN0
7 4nn0 10888 . . . 4  |-  4  e.  NN0
86, 7deccl 11065 . . 3  |- ; 8 4  e.  NN0
9 7nn0 10891 . . 3  |-  7  e.  NN0
10 7lt10 10814 . . 3  |-  7  <  10
11 1lt10 10820 . . . 4  |-  1  <  10
12 3lt8 10801 . . . 4  |-  3  <  8
131, 6, 2, 7, 11, 12decltc 11073 . . 3  |- ; 3 1  < ; 8 4
143, 8, 9, 2, 10, 13decltc 11073 . 2  |- ;; 3 1 7  < ;; 8 4 1
15 1nn 10620 . . . 4  |-  1  e.  NN
161, 15decnncl 11064 . . 3  |- ; 3 1  e.  NN
1716, 9, 2, 11declti 11076 . 2  |-  1  < ;; 3 1 7
18 3t2e6 10761 . . 3  |-  ( 3  x.  2 )  =  6
19 df-7 10673 . . 3  |-  7  =  ( 6  +  1 )
203, 1, 18, 19dec2dvds 15020 . 2  |-  -.  2  || ;; 3 1 7
21 3nn 10768 . . 3  |-  3  e.  NN
22 10nn0 10894 . . . 4  |-  10  e.  NN0
23 5nn0 10889 . . . 4  |-  5  e.  NN0
2422, 23deccl 11065 . . 3  |- ; 10 5  e.  NN0
25 2nn 10767 . . 3  |-  2  e.  NN
26 0nn0 10884 . . . 4  |-  0  e.  NN0
27 2nn0 10886 . . . 4  |-  2  e.  NN0
28 eqid 2422 . . . 4  |- ; 10 5  = ; 10 5
2927dec0h 11067 . . . 4  |-  2  = ; 0 2
30 dec10 11081 . . . . 5  |-  10  = ; 1 0
31 ax-1cn 9597 . . . . . . 7  |-  1  e.  CC
3231addid2i 9821 . . . . . 6  |-  ( 0  +  1 )  =  1
332dec0h 11067 . . . . . 6  |-  1  = ; 0 1
3432, 33eqtri 2451 . . . . 5  |-  ( 0  +  1 )  = ; 0
1
35 3cn 10684 . . . . . . . 8  |-  3  e.  CC
3635mulid1i 9645 . . . . . . 7  |-  ( 3  x.  1 )  =  3
37 00id 9808 . . . . . . 7  |-  ( 0  +  0 )  =  0
3836, 37oveq12i 6313 . . . . . 6  |-  ( ( 3  x.  1 )  +  ( 0  +  0 ) )  =  ( 3  +  0 )
3935addid1i 9820 . . . . . 6  |-  ( 3  +  0 )  =  3
4038, 39eqtri 2451 . . . . 5  |-  ( ( 3  x.  1 )  +  ( 0  +  0 ) )  =  3
4135mul01i 9823 . . . . . . . 8  |-  ( 3  x.  0 )  =  0
4241oveq1i 6311 . . . . . . 7  |-  ( ( 3  x.  0 )  +  1 )  =  ( 0  +  1 )
4342, 32eqtri 2451 . . . . . 6  |-  ( ( 3  x.  0 )  +  1 )  =  1
4443, 33eqtri 2451 . . . . 5  |-  ( ( 3  x.  0 )  +  1 )  = ; 0
1
452, 26, 26, 2, 30, 34, 1, 2, 26, 40, 44decma2c 11091 . . . 4  |-  ( ( 3  x.  10 )  +  ( 0  +  1 ) )  = ; 3
1
46 5cn 10689 . . . . . 6  |-  5  e.  CC
47 5t3e15 11125 . . . . . 6  |-  ( 5  x.  3 )  = ; 1
5
4846, 35, 47mulcomli 9650 . . . . 5  |-  ( 3  x.  5 )  = ; 1
5
49 5p2e7 10747 . . . . 5  |-  ( 5  +  2 )  =  7
502, 23, 27, 48, 49decaddi 11095 . . . 4  |-  ( ( 3  x.  5 )  +  2 )  = ; 1
7
5122, 23, 26, 27, 28, 29, 1, 9, 2, 45, 50decma2c 11091 . . 3  |-  ( ( 3  x. ; 10 5 )  +  2 )  = ;; 3 1 7
52 2lt3 10777 . . 3  |-  2  <  3
5321, 24, 25, 51, 52ndvdsi 14376 . 2  |-  -.  3  || ;; 3 1 7
54 2lt5 10784 . . 3  |-  2  <  5
553, 25, 54, 49dec5dvds2 15022 . 2  |-  -.  5  || ;; 3 1 7
567, 23deccl 11065 . . 3  |- ; 4 5  e.  NN0
57 eqid 2422 . . . 4  |- ; 4 5  = ; 4 5
5835addid2i 9821 . . . . . 6  |-  ( 0  +  3 )  =  3
5958oveq2i 6312 . . . . 5  |-  ( ( 7  x.  4 )  +  ( 0  +  3 ) )  =  ( ( 7  x.  4 )  +  3 )
60 7t4e28 11135 . . . . . 6  |-  ( 7  x.  4 )  = ; 2
8
61 2p1e3 10733 . . . . . 6  |-  ( 2  +  1 )  =  3
62 8p3e11 11107 . . . . . 6  |-  ( 8  +  3 )  = ; 1
1
6327, 6, 1, 60, 61, 2, 62decaddci 11096 . . . . 5  |-  ( ( 7  x.  4 )  +  3 )  = ; 3
1
6459, 63eqtri 2451 . . . 4  |-  ( ( 7  x.  4 )  +  ( 0  +  3 ) )  = ; 3
1
65 7t5e35 11136 . . . . 5  |-  ( 7  x.  5 )  = ; 3
5
661, 23, 27, 65, 49decaddi 11095 . . . 4  |-  ( ( 7  x.  5 )  +  2 )  = ; 3
7
677, 23, 26, 27, 57, 29, 9, 9, 1, 64, 66decma2c 11091 . . 3  |-  ( ( 7  x. ; 4 5 )  +  2 )  = ;; 3 1 7
68 2lt7 10795 . . 3  |-  2  <  7
694, 56, 25, 67, 68ndvdsi 14376 . 2  |-  -.  7  || ;; 3 1 7
702, 15decnncl 11064 . . 3  |- ; 1 1  e.  NN
7127, 6deccl 11065 . . 3  |- ; 2 8  e.  NN0
72 9nn 10774 . . 3  |-  9  e.  NN
73 9nn0 10893 . . . 4  |-  9  e.  NN0
74 eqid 2422 . . . 4  |- ; 2 8  = ; 2 8
7573dec0h 11067 . . . 4  |-  9  = ; 0 9
762, 2deccl 11065 . . . 4  |- ; 1 1  e.  NN0
77 eqid 2422 . . . . 5  |- ; 1 1  = ; 1 1
78 9cn 10697 . . . . . . 7  |-  9  e.  CC
7978addid2i 9821 . . . . . 6  |-  ( 0  +  9 )  =  9
8079, 75eqtri 2451 . . . . 5  |-  ( 0  +  9 )  = ; 0
9
81 2cn 10680 . . . . . . . 8  |-  2  e.  CC
8281mulid2i 9646 . . . . . . 7  |-  ( 1  x.  2 )  =  2
8382, 32oveq12i 6313 . . . . . 6  |-  ( ( 1  x.  2 )  +  ( 0  +  1 ) )  =  ( 2  +  1 )
8483, 61eqtri 2451 . . . . 5  |-  ( ( 1  x.  2 )  +  ( 0  +  1 ) )  =  3
8582oveq1i 6311 . . . . . 6  |-  ( ( 1  x.  2 )  +  9 )  =  ( 2  +  9 )
86 9p2e11 11113 . . . . . . 7  |-  ( 9  +  2 )  = ; 1
1
8778, 81, 86addcomli 9825 . . . . . 6  |-  ( 2  +  9 )  = ; 1
1
8885, 87eqtri 2451 . . . . 5  |-  ( ( 1  x.  2 )  +  9 )  = ; 1
1
892, 2, 26, 73, 77, 80, 27, 2, 2, 84, 88decmac 11090 . . . 4  |-  ( (; 1
1  x.  2 )  +  ( 0  +  9 ) )  = ; 3
1
90 8cn 10695 . . . . . . . 8  |-  8  e.  CC
9190mulid2i 9646 . . . . . . 7  |-  ( 1  x.  8 )  =  8
9291, 32oveq12i 6313 . . . . . 6  |-  ( ( 1  x.  8 )  +  ( 0  +  1 ) )  =  ( 8  +  1 )
93 8p1e9 10740 . . . . . 6  |-  ( 8  +  1 )  =  9
9492, 93eqtri 2451 . . . . 5  |-  ( ( 1  x.  8 )  +  ( 0  +  1 ) )  =  9
9591oveq1i 6311 . . . . . 6  |-  ( ( 1  x.  8 )  +  9 )  =  ( 8  +  9 )
96 9p8e17 11119 . . . . . . 7  |-  ( 9  +  8 )  = ; 1
7
9778, 90, 96addcomli 9825 . . . . . 6  |-  ( 8  +  9 )  = ; 1
7
9895, 97eqtri 2451 . . . . 5  |-  ( ( 1  x.  8 )  +  9 )  = ; 1
7
992, 2, 26, 73, 77, 75, 6, 9, 2, 94, 98decmac 11090 . . . 4  |-  ( (; 1
1  x.  8 )  +  9 )  = ; 9
7
10027, 6, 26, 73, 74, 75, 76, 9, 73, 89, 99decma2c 11091 . . 3  |-  ( (; 1
1  x. ; 2 8 )  +  9 )  = ;; 3 1 7
101 9lt10 10812 . . . 4  |-  9  <  10
10215, 2, 73, 101declti 11076 . . 3  |-  9  < ; 1
1
10370, 71, 72, 100, 102ndvdsi 14376 . 2  |-  -. ; 1 1  || ;; 3 1 7
1042, 21decnncl 11064 . . 3  |- ; 1 3  e.  NN
10527, 7deccl 11065 . . 3  |- ; 2 4  e.  NN0
106 5nn 10770 . . 3  |-  5  e.  NN
107 eqid 2422 . . . 4  |- ; 2 4  = ; 2 4
10823dec0h 11067 . . . 4  |-  5  = ; 0 5
1092, 1deccl 11065 . . . 4  |- ; 1 3  e.  NN0
110 eqid 2422 . . . . 5  |- ; 1 3  = ; 1 3
11146addid2i 9821 . . . . . 6  |-  ( 0  +  5 )  =  5
112111, 108eqtri 2451 . . . . 5  |-  ( 0  +  5 )  = ; 0
5
11318oveq1i 6311 . . . . . 6  |-  ( ( 3  x.  2 )  +  5 )  =  ( 6  +  5 )
114 6p5e11 11101 . . . . . 6  |-  ( 6  +  5 )  = ; 1
1
115113, 114eqtri 2451 . . . . 5  |-  ( ( 3  x.  2 )  +  5 )  = ; 1
1
1162, 1, 26, 23, 110, 112, 27, 2, 2, 84, 115decmac 11090 . . . 4  |-  ( (; 1
3  x.  2 )  +  ( 0  +  5 ) )  = ; 3
1
117 4cn 10687 . . . . . . . 8  |-  4  e.  CC
118117mulid2i 9646 . . . . . . 7  |-  ( 1  x.  4 )  =  4
119118, 32oveq12i 6313 . . . . . 6  |-  ( ( 1  x.  4 )  +  ( 0  +  1 ) )  =  ( 4  +  1 )
120 4p1e5 10736 . . . . . 6  |-  ( 4  +  1 )  =  5
121119, 120eqtri 2451 . . . . 5  |-  ( ( 1  x.  4 )  +  ( 0  +  1 ) )  =  5
122 4t3e12 11123 . . . . . . 7  |-  ( 4  x.  3 )  = ; 1
2
123117, 35, 122mulcomli 9650 . . . . . 6  |-  ( 3  x.  4 )  = ; 1
2
12446, 81, 49addcomli 9825 . . . . . 6  |-  ( 2  +  5 )  =  7
1252, 27, 23, 123, 124decaddi 11095 . . . . 5  |-  ( ( 3  x.  4 )  +  5 )  = ; 1
7
1262, 1, 26, 23, 110, 108, 7, 9, 2, 121, 125decmac 11090 . . . 4  |-  ( (; 1
3  x.  4 )  +  5 )  = ; 5
7
12727, 7, 26, 23, 107, 108, 109, 9, 23, 116, 126decma2c 11091 . . 3  |-  ( (; 1
3  x. ; 2 4 )  +  5 )  = ;; 3 1 7
128 5lt10 10816 . . . 4  |-  5  <  10
12915, 1, 23, 128declti 11076 . . 3  |-  5  < ; 1
3
130104, 105, 106, 127, 129ndvdsi 14376 . 2  |-  -. ; 1 3  || ;; 3 1 7
1312, 4decnncl 11064 . . 3  |- ; 1 7  e.  NN
1322, 6deccl 11065 . . 3  |- ; 1 8  e.  NN0
133 eqid 2422 . . . 4  |- ; 1 8  = ; 1 8
1342, 9deccl 11065 . . . 4  |- ; 1 7  e.  NN0
135 eqid 2422 . . . . 5  |- ; 1 7  = ; 1 7
136 3p1e4 10735 . . . . . . 7  |-  ( 3  +  1 )  =  4
13735, 31, 136addcomli 9825 . . . . . 6  |-  ( 1  +  3 )  =  4
13826, 2, 2, 1, 33, 110, 32, 137decadd 11092 . . . . 5  |-  ( 1  + ; 1 3 )  = ; 1
4
13931mulid1i 9645 . . . . . . 7  |-  ( 1  x.  1 )  =  1
140 1p1e2 10723 . . . . . . 7  |-  ( 1  +  1 )  =  2
141139, 140oveq12i 6313 . . . . . 6  |-  ( ( 1  x.  1 )  +  ( 1  +  1 ) )  =  ( 1  +  2 )
142 1p2e3 10734 . . . . . 6  |-  ( 1  +  2 )  =  3
143141, 142eqtri 2451 . . . . 5  |-  ( ( 1  x.  1 )  +  ( 1  +  1 ) )  =  3
144 7cn 10693 . . . . . . . 8  |-  7  e.  CC
145144mulid1i 9645 . . . . . . 7  |-  ( 7  x.  1 )  =  7
146145oveq1i 6311 . . . . . 6  |-  ( ( 7  x.  1 )  +  4 )  =  ( 7  +  4 )
147 7p4e11 11103 . . . . . 6  |-  ( 7  +  4 )  = ; 1
1
148146, 147eqtri 2451 . . . . 5  |-  ( ( 7  x.  1 )  +  4 )  = ; 1
1
1492, 9, 2, 7, 135, 138, 2, 2, 2, 143, 148decmac 11090 . . . 4  |-  ( (; 1
7  x.  1 )  +  ( 1  + ; 1
3 ) )  = ; 3
1
15091, 111oveq12i 6313 . . . . . 6  |-  ( ( 1  x.  8 )  +  ( 0  +  5 ) )  =  ( 8  +  5 )
151 8p5e13 11109 . . . . . 6  |-  ( 8  +  5 )  = ; 1
3
152150, 151eqtri 2451 . . . . 5  |-  ( ( 1  x.  8 )  +  ( 0  +  5 ) )  = ; 1
3
153 6nn0 10890 . . . . . 6  |-  6  e.  NN0
154 6p1e7 10738 . . . . . 6  |-  ( 6  +  1 )  =  7
155 8t7e56 11144 . . . . . . 7  |-  ( 8  x.  7 )  = ; 5
6
15690, 144, 155mulcomli 9650 . . . . . 6  |-  ( 7  x.  8 )  = ; 5
6
15723, 153, 154, 156decsuc 11074 . . . . 5  |-  ( ( 7  x.  8 )  +  1 )  = ; 5
7
1582, 9, 26, 2, 135, 33, 6, 9, 23, 152, 157decmac 11090 . . . 4  |-  ( (; 1
7  x.  8 )  +  1 )  = ;; 1 3 7
1592, 6, 2, 2, 133, 77, 134, 9, 109, 149, 158decma2c 11091 . . 3  |-  ( (; 1
7  x. ; 1 8 )  + ; 1
1 )  = ;; 3 1 7
160 1lt7 10796 . . . 4  |-  1  <  7
1612, 2, 4, 160declt 11072 . . 3  |- ; 1 1  < ; 1 7
162131, 132, 70, 159, 161ndvdsi 14376 . 2  |-  -. ; 1 7  || ;; 3 1 7
1632, 72decnncl 11064 . . 3  |- ; 1 9  e.  NN
1642, 153deccl 11065 . . 3  |- ; 1 6  e.  NN0
165 eqid 2422 . . . 4  |- ; 1 6  = ; 1 6
1662, 73deccl 11065 . . . 4  |- ; 1 9  e.  NN0
167 eqid 2422 . . . . 5  |- ; 1 9  = ; 1 9
16826, 2, 2, 2, 33, 77, 32, 140decadd 11092 . . . . 5  |-  ( 1  + ; 1 1 )  = ; 1
2
16978mulid1i 9645 . . . . . . 7  |-  ( 9  x.  1 )  =  9
170169oveq1i 6311 . . . . . 6  |-  ( ( 9  x.  1 )  +  2 )  =  ( 9  +  2 )
171170, 86eqtri 2451 . . . . 5  |-  ( ( 9  x.  1 )  +  2 )  = ; 1
1
1722, 73, 2, 27, 167, 168, 2, 2, 2, 143, 171decmac 11090 . . . 4  |-  ( (; 1
9  x.  1 )  +  ( 1  + ; 1
1 ) )  = ; 3
1
1731dec0h 11067 . . . . 5  |-  3  = ; 0 3
174 6cn 10691 . . . . . . . 8  |-  6  e.  CC
175174mulid2i 9646 . . . . . . 7  |-  ( 1  x.  6 )  =  6
176175, 111oveq12i 6313 . . . . . 6  |-  ( ( 1  x.  6 )  +  ( 0  +  5 ) )  =  ( 6  +  5 )
177176, 114eqtri 2451 . . . . 5  |-  ( ( 1  x.  6 )  +  ( 0  +  5 ) )  = ; 1
1
178 9t6e54 11150 . . . . . 6  |-  ( 9  x.  6 )  = ; 5
4
179 4p3e7 10745 . . . . . 6  |-  ( 4  +  3 )  =  7
18023, 7, 1, 178, 179decaddi 11095 . . . . 5  |-  ( ( 9  x.  6 )  +  3 )  = ; 5
7
1812, 73, 26, 1, 167, 173, 153, 9, 23, 177, 180decmac 11090 . . . 4  |-  ( (; 1
9  x.  6 )  +  3 )  = ;; 1 1 7
1822, 153, 2, 1, 165, 110, 166, 9, 76, 172, 181decma2c 11091 . . 3  |-  ( (; 1
9  x. ; 1 6 )  + ; 1
3 )  = ;; 3 1 7
183 3lt9 10809 . . . 4  |-  3  <  9
1842, 1, 72, 183declt 11072 . . 3  |- ; 1 3  < ; 1 9
185163, 164, 104, 182, 184ndvdsi 14376 . 2  |-  -. ; 1 9  || ;; 3 1 7
18627, 21decnncl 11064 . . 3  |- ; 2 3  e.  NN
187104nnnn0i 10877 . . 3  |- ; 1 3  e.  NN0
188 8nn 10773 . . . 4  |-  8  e.  NN
1892, 188decnncl 11064 . . 3  |- ; 1 8  e.  NN
19027, 1deccl 11065 . . . 4  |- ; 2 3  e.  NN0
191 eqid 2422 . . . . 5  |- ; 2 3  = ; 2 3
192 7p1e8 10739 . . . . . . 7  |-  ( 7  +  1 )  =  8
193144, 31, 192addcomli 9825 . . . . . 6  |-  ( 1  +  7 )  =  8
1946dec0h 11067 . . . . . 6  |-  8  = ; 0 8
195193, 194eqtri 2451 . . . . 5  |-  ( 1  +  7 )  = ; 0
8
19681mulid1i 9645 . . . . . . 7  |-  ( 2  x.  1 )  =  2
197196, 32oveq12i 6313 . . . . . 6  |-  ( ( 2  x.  1 )  +  ( 0  +  1 ) )  =  ( 2  +  1 )
198197, 61eqtri 2451 . . . . 5  |-  ( ( 2  x.  1 )  +  ( 0  +  1 ) )  =  3
19936oveq1i 6311 . . . . . 6  |-  ( ( 3  x.  1 )  +  8 )  =  ( 3  +  8 )
20090, 35, 62addcomli 9825 . . . . . 6  |-  ( 3  +  8 )  = ; 1
1
201199, 200eqtri 2451 . . . . 5  |-  ( ( 3  x.  1 )  +  8 )  = ; 1
1
20227, 1, 26, 6, 191, 195, 2, 2, 2, 198, 201decmac 11090 . . . 4  |-  ( (; 2
3  x.  1 )  +  ( 1  +  7 ) )  = ; 3
1
20335, 81, 18mulcomli 9650 . . . . . . 7  |-  ( 2  x.  3 )  =  6
204203, 32oveq12i 6313 . . . . . 6  |-  ( ( 2  x.  3 )  +  ( 0  +  1 ) )  =  ( 6  +  1 )
205204, 154eqtri 2451 . . . . 5  |-  ( ( 2  x.  3 )  +  ( 0  +  1 ) )  =  7
206 3t3e9 10762 . . . . . . 7  |-  ( 3  x.  3 )  =  9
207206oveq1i 6311 . . . . . 6  |-  ( ( 3  x.  3 )  +  8 )  =  ( 9  +  8 )
208207, 96eqtri 2451 . . . . 5  |-  ( ( 3  x.  3 )  +  8 )  = ; 1
7
20927, 1, 26, 6, 191, 194, 1, 9, 2, 205, 208decmac 11090 . . . 4  |-  ( (; 2
3  x.  3 )  +  8 )  = ; 7
7
2102, 1, 2, 6, 110, 133, 190, 9, 9, 202, 209decma2c 11091 . . 3  |-  ( (; 2
3  x. ; 1 3 )  + ; 1
8 )  = ;; 3 1 7
211 8lt10 10813 . . . 4  |-  8  <  10
212 1lt2 10776 . . . 4  |-  1  <  2
2132, 27, 6, 1, 211, 212decltc 11073 . . 3  |- ; 1 8  < ; 2 3
214186, 187, 189, 210, 213ndvdsi 14376 . 2  |-  -. ; 2 3  || ;; 3 1 7
2155, 14, 17, 20, 53, 55, 69, 103, 130, 162, 185, 214prmlem2 15076 1  |- ;; 3 1 7  e.  Prime
Colors of variables: wff setvar class
Syntax hints:    e. wcel 1868  (class class class)co 6301   0cc0 9539   1c1 9540    + caddc 9542    x. cmul 9544   2c2 10659   3c3 10660   4c4 10661   5c5 10662   6c6 10663   7c7 10664   8c8 10665   9c9 10666   10c10 10667  ;cdc 11051   Primecprime 14607
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-int 4253  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-1st 6803  df-2nd 6804  df-wrecs 7032  df-recs 7094  df-rdg 7132  df-1o 7186  df-2o 7187  df-oadd 7190  df-er 7367  df-en 7574  df-dom 7575  df-sdom 7576  df-fin 7577  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-fz 11785  df-seq 12213  df-exp 12272  df-cj 13148  df-re 13149  df-im 13150  df-sqrt 13284  df-abs 13285  df-dvds 14291  df-prm 14608
This theorem is referenced by: (None)
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