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Theorem 317prm 14472
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 10814 . . . 4  |-  3  e.  NN0
2 1nn0 10812 . . . 4  |-  1  e.  NN0
31, 2deccl 10991 . . 3  |- ; 3 1  e.  NN0
4 7nn 10699 . . 3  |-  7  e.  NN
53, 4decnncl 10990 . 2  |- ;; 3 1 7  e.  NN
6 8nn0 10819 . . . 4  |-  8  e.  NN0
7 4nn0 10815 . . . 4  |-  4  e.  NN0
86, 7deccl 10991 . . 3  |- ; 8 4  e.  NN0
9 7nn0 10818 . . 3  |-  7  e.  NN0
10 7lt10 10741 . . 3  |-  7  <  10
11 1lt10 10747 . . . 4  |-  1  <  10
12 3lt8 10728 . . . 4  |-  3  <  8
131, 6, 2, 7, 11, 12decltc 10999 . . 3  |- ; 3 1  < ; 8 4
143, 8, 9, 2, 10, 13decltc 10999 . 2  |- ;; 3 1 7  < ;; 8 4 1
15 1nn 10548 . . . 4  |-  1  e.  NN
161, 15decnncl 10990 . . 3  |- ; 3 1  e.  NN
1716, 9, 2, 11declti 11002 . 2  |-  1  < ;; 3 1 7
18 3t2e6 10688 . . 3  |-  ( 3  x.  2 )  =  6
19 df-7 10600 . . 3  |-  7  =  ( 6  +  1 )
203, 1, 18, 19dec2dvds 14411 . 2  |-  -.  2  || ;; 3 1 7
21 3nn 10695 . . 3  |-  3  e.  NN
22 10nn0 10821 . . . 4  |-  10  e.  NN0
23 5nn0 10816 . . . 4  |-  5  e.  NN0
2422, 23deccl 10991 . . 3  |- ; 10 5  e.  NN0
25 2nn 10694 . . 3  |-  2  e.  NN
26 0nn0 10811 . . . 4  |-  0  e.  NN0
27 2nn0 10813 . . . 4  |-  2  e.  NN0
28 eqid 2467 . . . 4  |- ; 10 5  = ; 10 5
2927dec0h 10993 . . . 4  |-  2  = ; 0 2
30 dec10 11007 . . . . 5  |-  10  = ; 1 0
31 ax-1cn 9551 . . . . . . 7  |-  1  e.  CC
3231addid2i 9768 . . . . . 6  |-  ( 0  +  1 )  =  1
332dec0h 10993 . . . . . 6  |-  1  = ; 0 1
3432, 33eqtri 2496 . . . . 5  |-  ( 0  +  1 )  = ; 0
1
35 3cn 10611 . . . . . . . 8  |-  3  e.  CC
3635mulid1i 9599 . . . . . . 7  |-  ( 3  x.  1 )  =  3
37 00id 9755 . . . . . . 7  |-  ( 0  +  0 )  =  0
3836, 37oveq12i 6297 . . . . . 6  |-  ( ( 3  x.  1 )  +  ( 0  +  0 ) )  =  ( 3  +  0 )
3935addid1i 9767 . . . . . 6  |-  ( 3  +  0 )  =  3
4038, 39eqtri 2496 . . . . 5  |-  ( ( 3  x.  1 )  +  ( 0  +  0 ) )  =  3
4135mul01i 9770 . . . . . . . 8  |-  ( 3  x.  0 )  =  0
4241oveq1i 6295 . . . . . . 7  |-  ( ( 3  x.  0 )  +  1 )  =  ( 0  +  1 )
4342, 32eqtri 2496 . . . . . 6  |-  ( ( 3  x.  0 )  +  1 )  =  1
4443, 33eqtri 2496 . . . . 5  |-  ( ( 3  x.  0 )  +  1 )  = ; 0
1
452, 26, 26, 2, 30, 34, 1, 2, 26, 40, 44decma2c 11017 . . . 4  |-  ( ( 3  x.  10 )  +  ( 0  +  1 ) )  = ; 3
1
46 5cn 10616 . . . . . 6  |-  5  e.  CC
47 5t3e15 11051 . . . . . 6  |-  ( 5  x.  3 )  = ; 1
5
4846, 35, 47mulcomli 9604 . . . . 5  |-  ( 3  x.  5 )  = ; 1
5
49 5p2e7 10674 . . . . 5  |-  ( 5  +  2 )  =  7
502, 23, 27, 48, 49decaddi 11021 . . . 4  |-  ( ( 3  x.  5 )  +  2 )  = ; 1
7
5122, 23, 26, 27, 28, 29, 1, 9, 2, 45, 50decma2c 11017 . . 3  |-  ( ( 3  x. ; 10 5 )  +  2 )  = ;; 3 1 7
52 2lt3 10704 . . 3  |-  2  <  3
5321, 24, 25, 51, 52ndvdsi 13930 . 2  |-  -.  3  || ;; 3 1 7
54 2lt5 10711 . . 3  |-  2  <  5
553, 25, 54, 49dec5dvds2 14413 . 2  |-  -.  5  || ;; 3 1 7
567, 23deccl 10991 . . 3  |- ; 4 5  e.  NN0
57 eqid 2467 . . . 4  |- ; 4 5  = ; 4 5
5835addid2i 9768 . . . . . 6  |-  ( 0  +  3 )  =  3
5958oveq2i 6296 . . . . 5  |-  ( ( 7  x.  4 )  +  ( 0  +  3 ) )  =  ( ( 7  x.  4 )  +  3 )
60 7t4e28 11061 . . . . . 6  |-  ( 7  x.  4 )  = ; 2
8
61 2p1e3 10660 . . . . . 6  |-  ( 2  +  1 )  =  3
62 8p3e11 11033 . . . . . 6  |-  ( 8  +  3 )  = ; 1
1
6327, 6, 1, 60, 61, 2, 62decaddci 11022 . . . . 5  |-  ( ( 7  x.  4 )  +  3 )  = ; 3
1
6459, 63eqtri 2496 . . . 4  |-  ( ( 7  x.  4 )  +  ( 0  +  3 ) )  = ; 3
1
65 7t5e35 11062 . . . . 5  |-  ( 7  x.  5 )  = ; 3
5
661, 23, 27, 65, 49decaddi 11021 . . . 4  |-  ( ( 7  x.  5 )  +  2 )  = ; 3
7
677, 23, 26, 27, 57, 29, 9, 9, 1, 64, 66decma2c 11017 . . 3  |-  ( ( 7  x. ; 4 5 )  +  2 )  = ;; 3 1 7
68 2lt7 10722 . . 3  |-  2  <  7
694, 56, 25, 67, 68ndvdsi 13930 . 2  |-  -.  7  || ;; 3 1 7
702, 15decnncl 10990 . . 3  |- ; 1 1  e.  NN
7127, 6deccl 10991 . . 3  |- ; 2 8  e.  NN0
72 9nn 10701 . . 3  |-  9  e.  NN
73 9nn0 10820 . . . 4  |-  9  e.  NN0
74 eqid 2467 . . . 4  |- ; 2 8  = ; 2 8
7573dec0h 10993 . . . 4  |-  9  = ; 0 9
762, 2deccl 10991 . . . 4  |- ; 1 1  e.  NN0
77 eqid 2467 . . . . 5  |- ; 1 1  = ; 1 1
78 9cn 10624 . . . . . . 7  |-  9  e.  CC
7978addid2i 9768 . . . . . 6  |-  ( 0  +  9 )  =  9
8079, 75eqtri 2496 . . . . 5  |-  ( 0  +  9 )  = ; 0
9
81 2cn 10607 . . . . . . . 8  |-  2  e.  CC
8281mulid2i 9600 . . . . . . 7  |-  ( 1  x.  2 )  =  2
8382, 32oveq12i 6297 . . . . . 6  |-  ( ( 1  x.  2 )  +  ( 0  +  1 ) )  =  ( 2  +  1 )
8483, 61eqtri 2496 . . . . 5  |-  ( ( 1  x.  2 )  +  ( 0  +  1 ) )  =  3
8582oveq1i 6295 . . . . . 6  |-  ( ( 1  x.  2 )  +  9 )  =  ( 2  +  9 )
86 9p2e11 11039 . . . . . . 7  |-  ( 9  +  2 )  = ; 1
1
8778, 81, 86addcomli 9772 . . . . . 6  |-  ( 2  +  9 )  = ; 1
1
8885, 87eqtri 2496 . . . . 5  |-  ( ( 1  x.  2 )  +  9 )  = ; 1
1
892, 2, 26, 73, 77, 80, 27, 2, 2, 84, 88decmac 11016 . . . 4  |-  ( (; 1
1  x.  2 )  +  ( 0  +  9 ) )  = ; 3
1
90 8cn 10622 . . . . . . . 8  |-  8  e.  CC
9190mulid2i 9600 . . . . . . 7  |-  ( 1  x.  8 )  =  8
9291, 32oveq12i 6297 . . . . . 6  |-  ( ( 1  x.  8 )  +  ( 0  +  1 ) )  =  ( 8  +  1 )
93 8p1e9 10667 . . . . . 6  |-  ( 8  +  1 )  =  9
9492, 93eqtri 2496 . . . . 5  |-  ( ( 1  x.  8 )  +  ( 0  +  1 ) )  =  9
9591oveq1i 6295 . . . . . 6  |-  ( ( 1  x.  8 )  +  9 )  =  ( 8  +  9 )
96 9p8e17 11045 . . . . . . 7  |-  ( 9  +  8 )  = ; 1
7
9778, 90, 96addcomli 9772 . . . . . 6  |-  ( 8  +  9 )  = ; 1
7
9895, 97eqtri 2496 . . . . 5  |-  ( ( 1  x.  8 )  +  9 )  = ; 1
7
992, 2, 26, 73, 77, 75, 6, 9, 2, 94, 98decmac 11016 . . . 4  |-  ( (; 1
1  x.  8 )  +  9 )  = ; 9
7
10027, 6, 26, 73, 74, 75, 76, 9, 73, 89, 99decma2c 11017 . . 3  |-  ( (; 1
1  x. ; 2 8 )  +  9 )  = ;; 3 1 7
101 9lt10 10739 . . . 4  |-  9  <  10
10215, 2, 73, 101declti 11002 . . 3  |-  9  < ; 1
1
10370, 71, 72, 100, 102ndvdsi 13930 . 2  |-  -. ; 1 1  || ;; 3 1 7
1042, 21decnncl 10990 . . 3  |- ; 1 3  e.  NN
10527, 7deccl 10991 . . 3  |- ; 2 4  e.  NN0
106 5nn 10697 . . 3  |-  5  e.  NN
107 eqid 2467 . . . 4  |- ; 2 4  = ; 2 4
10823dec0h 10993 . . . 4  |-  5  = ; 0 5
1092, 1deccl 10991 . . . 4  |- ; 1 3  e.  NN0
110 eqid 2467 . . . . 5  |- ; 1 3  = ; 1 3
11146addid2i 9768 . . . . . 6  |-  ( 0  +  5 )  =  5
112111, 108eqtri 2496 . . . . 5  |-  ( 0  +  5 )  = ; 0
5
11318oveq1i 6295 . . . . . 6  |-  ( ( 3  x.  2 )  +  5 )  =  ( 6  +  5 )
114 6p5e11 11027 . . . . . 6  |-  ( 6  +  5 )  = ; 1
1
115113, 114eqtri 2496 . . . . 5  |-  ( ( 3  x.  2 )  +  5 )  = ; 1
1
1162, 1, 26, 23, 110, 112, 27, 2, 2, 84, 115decmac 11016 . . . 4  |-  ( (; 1
3  x.  2 )  +  ( 0  +  5 ) )  = ; 3
1
117 4cn 10614 . . . . . . . 8  |-  4  e.  CC
118117mulid2i 9600 . . . . . . 7  |-  ( 1  x.  4 )  =  4
119118, 32oveq12i 6297 . . . . . 6  |-  ( ( 1  x.  4 )  +  ( 0  +  1 ) )  =  ( 4  +  1 )
120 4p1e5 10663 . . . . . 6  |-  ( 4  +  1 )  =  5
121119, 120eqtri 2496 . . . . 5  |-  ( ( 1  x.  4 )  +  ( 0  +  1 ) )  =  5
122 4t3e12 11049 . . . . . . 7  |-  ( 4  x.  3 )  = ; 1
2
123117, 35, 122mulcomli 9604 . . . . . 6  |-  ( 3  x.  4 )  = ; 1
2
12446, 81, 49addcomli 9772 . . . . . 6  |-  ( 2  +  5 )  =  7
1252, 27, 23, 123, 124decaddi 11021 . . . . 5  |-  ( ( 3  x.  4 )  +  5 )  = ; 1
7
1262, 1, 26, 23, 110, 108, 7, 9, 2, 121, 125decmac 11016 . . . 4  |-  ( (; 1
3  x.  4 )  +  5 )  = ; 5
7
12727, 7, 26, 23, 107, 108, 109, 9, 23, 116, 126decma2c 11017 . . 3  |-  ( (; 1
3  x. ; 2 4 )  +  5 )  = ;; 3 1 7
128 5lt10 10743 . . . 4  |-  5  <  10
12915, 1, 23, 128declti 11002 . . 3  |-  5  < ; 1
3
130104, 105, 106, 127, 129ndvdsi 13930 . 2  |-  -. ; 1 3  || ;; 3 1 7
1312, 4decnncl 10990 . . 3  |- ; 1 7  e.  NN
1322, 6deccl 10991 . . 3  |- ; 1 8  e.  NN0
133 eqid 2467 . . . 4  |- ; 1 8  = ; 1 8
1342, 9deccl 10991 . . . 4  |- ; 1 7  e.  NN0
135 eqid 2467 . . . . 5  |- ; 1 7  = ; 1 7
136 3p1e4 10662 . . . . . . 7  |-  ( 3  +  1 )  =  4
13735, 31, 136addcomli 9772 . . . . . 6  |-  ( 1  +  3 )  =  4
13826, 2, 2, 1, 33, 110, 32, 137decadd 11018 . . . . 5  |-  ( 1  + ; 1 3 )  = ; 1
4
13931mulid1i 9599 . . . . . . 7  |-  ( 1  x.  1 )  =  1
140 1p1e2 10650 . . . . . . 7  |-  ( 1  +  1 )  =  2
141139, 140oveq12i 6297 . . . . . 6  |-  ( ( 1  x.  1 )  +  ( 1  +  1 ) )  =  ( 1  +  2 )
142 1p2e3 10661 . . . . . 6  |-  ( 1  +  2 )  =  3
143141, 142eqtri 2496 . . . . 5  |-  ( ( 1  x.  1 )  +  ( 1  +  1 ) )  =  3
144 7cn 10620 . . . . . . . 8  |-  7  e.  CC
145144mulid1i 9599 . . . . . . 7  |-  ( 7  x.  1 )  =  7
146145oveq1i 6295 . . . . . 6  |-  ( ( 7  x.  1 )  +  4 )  =  ( 7  +  4 )
147 7p4e11 11029 . . . . . 6  |-  ( 7  +  4 )  = ; 1
1
148146, 147eqtri 2496 . . . . 5  |-  ( ( 7  x.  1 )  +  4 )  = ; 1
1
1492, 9, 2, 7, 135, 138, 2, 2, 2, 143, 148decmac 11016 . . . 4  |-  ( (; 1
7  x.  1 )  +  ( 1  + ; 1
3 ) )  = ; 3
1
15091, 111oveq12i 6297 . . . . . 6  |-  ( ( 1  x.  8 )  +  ( 0  +  5 ) )  =  ( 8  +  5 )
151 8p5e13 11035 . . . . . 6  |-  ( 8  +  5 )  = ; 1
3
152150, 151eqtri 2496 . . . . 5  |-  ( ( 1  x.  8 )  +  ( 0  +  5 ) )  = ; 1
3
153 6nn0 10817 . . . . . 6  |-  6  e.  NN0
154 6p1e7 10665 . . . . . 6  |-  ( 6  +  1 )  =  7
155 8t7e56 11070 . . . . . . 7  |-  ( 8  x.  7 )  = ; 5
6
15690, 144, 155mulcomli 9604 . . . . . 6  |-  ( 7  x.  8 )  = ; 5
6
15723, 153, 154, 156decsuc 11000 . . . . 5  |-  ( ( 7  x.  8 )  +  1 )  = ; 5
7
1582, 9, 26, 2, 135, 33, 6, 9, 23, 152, 157decmac 11016 . . . 4  |-  ( (; 1
7  x.  8 )  +  1 )  = ;; 1 3 7
1592, 6, 2, 2, 133, 77, 134, 9, 109, 149, 158decma2c 11017 . . 3  |-  ( (; 1
7  x. ; 1 8 )  + ; 1
1 )  = ;; 3 1 7
160 1lt7 10723 . . . 4  |-  1  <  7
1612, 2, 4, 160declt 10998 . . 3  |- ; 1 1  < ; 1 7
162131, 132, 70, 159, 161ndvdsi 13930 . 2  |-  -. ; 1 7  || ;; 3 1 7
1632, 72decnncl 10990 . . 3  |- ; 1 9  e.  NN
1642, 153deccl 10991 . . 3  |- ; 1 6  e.  NN0
165 eqid 2467 . . . 4  |- ; 1 6  = ; 1 6
1662, 73deccl 10991 . . . 4  |- ; 1 9  e.  NN0
167 eqid 2467 . . . . 5  |- ; 1 9  = ; 1 9
16826, 2, 2, 2, 33, 77, 32, 140decadd 11018 . . . . 5  |-  ( 1  + ; 1 1 )  = ; 1
2
16978mulid1i 9599 . . . . . . 7  |-  ( 9  x.  1 )  =  9
170169oveq1i 6295 . . . . . 6  |-  ( ( 9  x.  1 )  +  2 )  =  ( 9  +  2 )
171170, 86eqtri 2496 . . . . 5  |-  ( ( 9  x.  1 )  +  2 )  = ; 1
1
1722, 73, 2, 27, 167, 168, 2, 2, 2, 143, 171decmac 11016 . . . 4  |-  ( (; 1
9  x.  1 )  +  ( 1  + ; 1
1 ) )  = ; 3
1
1731dec0h 10993 . . . . 5  |-  3  = ; 0 3
174 6cn 10618 . . . . . . . 8  |-  6  e.  CC
175174mulid2i 9600 . . . . . . 7  |-  ( 1  x.  6 )  =  6
176175, 111oveq12i 6297 . . . . . 6  |-  ( ( 1  x.  6 )  +  ( 0  +  5 ) )  =  ( 6  +  5 )
177176, 114eqtri 2496 . . . . 5  |-  ( ( 1  x.  6 )  +  ( 0  +  5 ) )  = ; 1
1
178 9t6e54 11076 . . . . . 6  |-  ( 9  x.  6 )  = ; 5
4
179 4p3e7 10672 . . . . . 6  |-  ( 4  +  3 )  =  7
18023, 7, 1, 178, 179decaddi 11021 . . . . 5  |-  ( ( 9  x.  6 )  +  3 )  = ; 5
7
1812, 73, 26, 1, 167, 173, 153, 9, 23, 177, 180decmac 11016 . . . 4  |-  ( (; 1
9  x.  6 )  +  3 )  = ;; 1 1 7
1822, 153, 2, 1, 165, 110, 166, 9, 76, 172, 181decma2c 11017 . . 3  |-  ( (; 1
9  x. ; 1 6 )  + ; 1
3 )  = ;; 3 1 7
183 3lt9 10736 . . . 4  |-  3  <  9
1842, 1, 72, 183declt 10998 . . 3  |- ; 1 3  < ; 1 9
185163, 164, 104, 182, 184ndvdsi 13930 . 2  |-  -. ; 1 9  || ;; 3 1 7
18627, 21decnncl 10990 . . 3  |- ; 2 3  e.  NN
187104nnnn0i 10804 . . 3  |- ; 1 3  e.  NN0
188 8nn 10700 . . . 4  |-  8  e.  NN
1892, 188decnncl 10990 . . 3  |- ; 1 8  e.  NN
19027, 1deccl 10991 . . . 4  |- ; 2 3  e.  NN0
191 eqid 2467 . . . . 5  |- ; 2 3  = ; 2 3
192 7p1e8 10666 . . . . . . 7  |-  ( 7  +  1 )  =  8
193144, 31, 192addcomli 9772 . . . . . 6  |-  ( 1  +  7 )  =  8
1946dec0h 10993 . . . . . 6  |-  8  = ; 0 8
195193, 194eqtri 2496 . . . . 5  |-  ( 1  +  7 )  = ; 0
8
19681mulid1i 9599 . . . . . . 7  |-  ( 2  x.  1 )  =  2
197196, 32oveq12i 6297 . . . . . 6  |-  ( ( 2  x.  1 )  +  ( 0  +  1 ) )  =  ( 2  +  1 )
198197, 61eqtri 2496 . . . . 5  |-  ( ( 2  x.  1 )  +  ( 0  +  1 ) )  =  3
19936oveq1i 6295 . . . . . 6  |-  ( ( 3  x.  1 )  +  8 )  =  ( 3  +  8 )
20090, 35, 62addcomli 9772 . . . . . 6  |-  ( 3  +  8 )  = ; 1
1
201199, 200eqtri 2496 . . . . 5  |-  ( ( 3  x.  1 )  +  8 )  = ; 1
1
20227, 1, 26, 6, 191, 195, 2, 2, 2, 198, 201decmac 11016 . . . 4  |-  ( (; 2
3  x.  1 )  +  ( 1  +  7 ) )  = ; 3
1
20335, 81, 18mulcomli 9604 . . . . . . 7  |-  ( 2  x.  3 )  =  6
204203, 32oveq12i 6297 . . . . . 6  |-  ( ( 2  x.  3 )  +  ( 0  +  1 ) )  =  ( 6  +  1 )
205204, 154eqtri 2496 . . . . 5  |-  ( ( 2  x.  3 )  +  ( 0  +  1 ) )  =  7
206 3t3e9 10689 . . . . . . 7  |-  ( 3  x.  3 )  =  9
207206oveq1i 6295 . . . . . 6  |-  ( ( 3  x.  3 )  +  8 )  =  ( 9  +  8 )
208207, 96eqtri 2496 . . . . 5  |-  ( ( 3  x.  3 )  +  8 )  = ; 1
7
20927, 1, 26, 6, 191, 194, 1, 9, 2, 205, 208decmac 11016 . . . 4  |-  ( (; 2
3  x.  3 )  +  8 )  = ; 7
7
2102, 1, 2, 6, 110, 133, 190, 9, 9, 202, 209decma2c 11017 . . 3  |-  ( (; 2
3  x. ; 1 3 )  + ; 1
8 )  = ;; 3 1 7
211 8lt10 10740 . . . 4  |-  8  <  10
212 1lt2 10703 . . . 4  |-  1  <  2
2132, 27, 6, 1, 211, 212decltc 10999 . . 3  |- ; 1 8  < ; 2 3
214186, 187, 189, 210, 213ndvdsi 13930 . 2  |-  -. ; 2 3  || ;; 3 1 7
2155, 14, 17, 20, 53, 55, 69, 103, 130, 162, 185, 214prmlem2 14466 1  |- ;; 3 1 7  e.  Prime
Colors of variables: wff setvar class
Syntax hints:    e. wcel 1767  (class class class)co 6285   0cc0 9493   1c1 9494    + caddc 9496    x. cmul 9498   2c2 10586   3c3 10587   4c4 10588   5c5 10589   6c6 10590   7c7 10591   8c8 10592   9c9 10593   10c10 10594  ;cdc 10977   Primecprime 14079
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6577  ax-cnex 9549  ax-resscn 9550  ax-1cn 9551  ax-icn 9552  ax-addcl 9553  ax-addrcl 9554  ax-mulcl 9555  ax-mulrcl 9556  ax-mulcom 9557  ax-addass 9558  ax-mulass 9559  ax-distr 9560  ax-i2m1 9561  ax-1ne0 9562  ax-1rid 9563  ax-rnegex 9564  ax-rrecex 9565  ax-cnre 9566  ax-pre-lttri 9567  ax-pre-lttrn 9568  ax-pre-ltadd 9569  ax-pre-mulgt0 9570  ax-pre-sup 9571
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-riota 6246  df-ov 6288  df-oprab 6289  df-mpt2 6290  df-om 6686  df-1st 6785  df-2nd 6786  df-recs 7043  df-rdg 7077  df-1o 7131  df-2o 7132  df-oadd 7135  df-er 7312  df-en 7518  df-dom 7519  df-sdom 7520  df-fin 7521  df-sup 7902  df-pnf 9631  df-mnf 9632  df-xr 9633  df-ltxr 9634  df-le 9635  df-sub 9808  df-neg 9809  df-div 10208  df-nn 10538  df-2 10595  df-3 10596  df-4 10597  df-5 10598  df-6 10599  df-7 10600  df-8 10601  df-9 10602  df-10 10603  df-n0 10797  df-z 10866  df-dec 10978  df-uz 11084  df-rp 11222  df-fz 11674  df-seq 12077  df-exp 12136  df-cj 12898  df-re 12899  df-im 12900  df-sqrt 13034  df-abs 13035  df-dvds 13851  df-prm 14080
This theorem is referenced by: (None)
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