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Theorem 139prm 13125
Description: 139 is a prime number. (Contributed by Mario Carneiro, 19-Feb-2014.) (Proof shortened by Mario Carneiro, 20-Apr-2015.)
Assertion
Ref Expression
139prm  |- ;; 1 3 9  e.  Prime

Proof of Theorem 139prm
StepHypRef Expression
1 1nn0 9981 . . . 4  |-  1  e.  NN0
2 3nn0 9983 . . . 4  |-  3  e.  NN0
31, 2deccl 10138 . . 3  |- ; 1 3  e.  NN0
4 9nn 9884 . . 3  |-  9  e.  NN
53, 4decnncl 10137 . 2  |- ;; 1 3 9  e.  NN
6 8nn0 9988 . . . 4  |-  8  e.  NN0
7 4nn0 9984 . . . 4  |-  4  e.  NN0
86, 7deccl 10138 . . 3  |- ; 8 4  e.  NN0
9 9nn0 9989 . . 3  |-  9  e.  NN0
10 9lt10 9922 . . 3  |-  9  <  10
11 3lt10 9928 . . . 4  |-  3  <  10
12 1lt8 9913 . . . 4  |-  1  <  8
131, 6, 2, 7, 11, 12decltc 10146 . . 3  |- ; 1 3  < ; 8 4
143, 8, 9, 1, 10, 13decltc 10146 . 2  |- ;; 1 3 9  < ;; 8 4 1
15 3nn 9878 . . . 4  |-  3  e.  NN
161, 15decnncl 10137 . . 3  |- ; 1 3  e.  NN
17 1lt10 9930 . . 3  |-  1  <  10
1816, 9, 1, 17declti 10149 . 2  |-  1  < ;; 1 3 9
19 4t2e8 9874 . . 3  |-  ( 4  x.  2 )  =  8
20 df-9 9811 . . 3  |-  9  =  ( 8  +  1 )
213, 7, 19, 20dec2dvds 13078 . 2  |-  -.  2  || ;; 1 3 9
22 6nn0 9986 . . . 4  |-  6  e.  NN0
237, 22deccl 10138 . . 3  |- ; 4 6  e.  NN0
24 1nn 9757 . . 3  |-  1  e.  NN
25 0nn0 9980 . . . 4  |-  0  e.  NN0
26 eqid 2283 . . . 4  |- ; 4 6  = ; 4 6
271dec0h 10140 . . . 4  |-  1  = ; 0 1
28 ax-1cn 8795 . . . . . . 7  |-  1  e.  CC
2928addid2i 9000 . . . . . 6  |-  ( 0  +  1 )  =  1
3029oveq2i 5869 . . . . 5  |-  ( ( 3  x.  4 )  +  ( 0  +  1 ) )  =  ( ( 3  x.  4 )  +  1 )
31 2nn0 9982 . . . . . 6  |-  2  e.  NN0
32 2p1e3 9847 . . . . . 6  |-  ( 2  +  1 )  =  3
337nn0cni 9977 . . . . . . 7  |-  4  e.  CC
34 3cn 9818 . . . . . . 7  |-  3  e.  CC
35 4t3e12 10196 . . . . . . 7  |-  ( 4  x.  3 )  = ; 1
2
3633, 34, 35mulcomli 8844 . . . . . 6  |-  ( 3  x.  4 )  = ; 1
2
371, 31, 32, 36decsuc 10147 . . . . 5  |-  ( ( 3  x.  4 )  +  1 )  = ; 1
3
3830, 37eqtri 2303 . . . 4  |-  ( ( 3  x.  4 )  +  ( 0  +  1 ) )  = ; 1
3
39 8p1e9 9853 . . . . 5  |-  ( 8  +  1 )  =  9
4022nn0cni 9977 . . . . . 6  |-  6  e.  CC
41 6t3e18 10202 . . . . . 6  |-  ( 6  x.  3 )  = ; 1
8
4240, 34, 41mulcomli 8844 . . . . 5  |-  ( 3  x.  6 )  = ; 1
8
431, 6, 39, 42decsuc 10147 . . . 4  |-  ( ( 3  x.  6 )  +  1 )  = ; 1
9
447, 22, 25, 1, 26, 27, 2, 9, 1, 38, 43decma2c 10164 . . 3  |-  ( ( 3  x. ; 4 6 )  +  1 )  = ;; 1 3 9
45 1lt3 9888 . . 3  |-  1  <  3
4615, 23, 24, 44, 45ndvdsi 12609 . 2  |-  -.  3  || ;; 1 3 9
47 4nn 9879 . . 3  |-  4  e.  NN
48 4lt5 9892 . . 3  |-  4  <  5
49 5p4e9 9862 . . 3  |-  ( 5  +  4 )  =  9
503, 47, 48, 49dec5dvds2 13080 . 2  |-  -.  5  || ;; 1 3 9
51 7nn 9882 . . 3  |-  7  e.  NN
521, 9deccl 10138 . . 3  |- ; 1 9  e.  NN0
53 6nn 9881 . . 3  |-  6  e.  NN
54 eqid 2283 . . . 4  |- ; 1 9  = ; 1 9
5522dec0h 10140 . . . 4  |-  6  = ; 0 6
56 7nn0 9987 . . . 4  |-  7  e.  NN0
5751nncni 9756 . . . . . . 7  |-  7  e.  CC
5857mulid1i 8839 . . . . . 6  |-  ( 7  x.  1 )  =  7
5940addid2i 9000 . . . . . 6  |-  ( 0  +  6 )  =  6
6058, 59oveq12i 5870 . . . . 5  |-  ( ( 7  x.  1 )  +  ( 0  +  6 ) )  =  ( 7  +  6 )
61 7p6e13 10178 . . . . 5  |-  ( 7  +  6 )  = ; 1
3
6260, 61eqtri 2303 . . . 4  |-  ( ( 7  x.  1 )  +  ( 0  +  6 ) )  = ; 1
3
634nncni 9756 . . . . . 6  |-  9  e.  CC
64 9t7e63 10224 . . . . . 6  |-  ( 9  x.  7 )  = ; 6
3
6563, 57, 64mulcomli 8844 . . . . 5  |-  ( 7  x.  9 )  = ; 6
3
66 6p3e9 9865 . . . . . 6  |-  ( 6  +  3 )  =  9
6740, 34, 66addcomli 9004 . . . . 5  |-  ( 3  +  6 )  =  9
6822, 2, 22, 65, 67decaddi 10168 . . . 4  |-  ( ( 7  x.  9 )  +  6 )  = ; 6
9
691, 9, 25, 22, 54, 55, 56, 9, 22, 62, 68decma2c 10164 . . 3  |-  ( ( 7  x. ; 1 9 )  +  6 )  = ;; 1 3 9
70 6lt7 9901 . . 3  |-  6  <  7
7151, 52, 53, 69, 70ndvdsi 12609 . 2  |-  -.  7  || ;; 1 3 9
721, 24decnncl 10137 . . 3  |- ; 1 1  e.  NN
731, 31deccl 10138 . . 3  |- ; 1 2  e.  NN0
74 eqid 2283 . . . 4  |- ; 1 2  = ; 1 2
7556dec0h 10140 . . . 4  |-  7  = ; 0 7
761, 1deccl 10138 . . . 4  |- ; 1 1  e.  NN0
77 2cn 9816 . . . . . . 7  |-  2  e.  CC
7877addid2i 9000 . . . . . 6  |-  ( 0  +  2 )  =  2
7978oveq2i 5869 . . . . 5  |-  ( (; 1
1  x.  1 )  +  ( 0  +  2 ) )  =  ( (; 1 1  x.  1 )  +  2 )
8072nncni 9756 . . . . . . 7  |- ; 1 1  e.  CC
8180mulid1i 8839 . . . . . 6  |-  (; 1 1  x.  1 )  = ; 1 1
8277, 28, 32addcomli 9004 . . . . . 6  |-  ( 1  +  2 )  =  3
831, 1, 31, 81, 82decaddi 10168 . . . . 5  |-  ( (; 1
1  x.  1 )  +  2 )  = ; 1
3
8479, 83eqtri 2303 . . . 4  |-  ( (; 1
1  x.  1 )  +  ( 0  +  2 ) )  = ; 1
3
85 eqid 2283 . . . . 5  |- ; 1 1  = ; 1 1
8677mulid2i 8840 . . . . . . 7  |-  ( 1  x.  2 )  =  2
87 00id 8987 . . . . . . 7  |-  ( 0  +  0 )  =  0
8886, 87oveq12i 5870 . . . . . 6  |-  ( ( 1  x.  2 )  +  ( 0  +  0 ) )  =  ( 2  +  0 )
8977addid1i 8999 . . . . . 6  |-  ( 2  +  0 )  =  2
9088, 89eqtri 2303 . . . . 5  |-  ( ( 1  x.  2 )  +  ( 0  +  0 ) )  =  2
9186oveq1i 5868 . . . . . 6  |-  ( ( 1  x.  2 )  +  7 )  =  ( 2  +  7 )
92 7p2e9 9867 . . . . . . 7  |-  ( 7  +  2 )  =  9
9357, 77, 92addcomli 9004 . . . . . 6  |-  ( 2  +  7 )  =  9
949dec0h 10140 . . . . . 6  |-  9  = ; 0 9
9591, 93, 943eqtri 2307 . . . . 5  |-  ( ( 1  x.  2 )  +  7 )  = ; 0
9
961, 1, 25, 56, 85, 75, 31, 9, 25, 90, 95decmac 10163 . . . 4  |-  ( (; 1
1  x.  2 )  +  7 )  = ; 2
9
971, 31, 25, 56, 74, 75, 76, 9, 31, 84, 96decma2c 10164 . . 3  |-  ( (; 1
1  x. ; 1 2 )  +  7 )  = ;; 1 3 9
98 7lt10 9924 . . . 4  |-  7  <  10
9924, 1, 56, 98declti 10149 . . 3  |-  7  < ; 1
1
10072, 73, 51, 97, 99ndvdsi 12609 . 2  |-  -. ; 1 1  || ;; 1 3 9
101 10nn0 9990 . . 3  |-  10  e.  NN0
102 dec10 10154 . . . 4  |-  10  = ; 1 0
103 eqid 2283 . . . . 5  |- ; 1 3  = ; 1 3
10425dec0h 10140 . . . . . 6  |-  0  = ; 0 0
10587, 104eqtri 2303 . . . . 5  |-  ( 0  +  0 )  = ; 0
0
10628mulid1i 8839 . . . . . . 7  |-  ( 1  x.  1 )  =  1
107106, 87oveq12i 5870 . . . . . 6  |-  ( ( 1  x.  1 )  +  ( 0  +  0 ) )  =  ( 1  +  0 )
10828addid1i 8999 . . . . . 6  |-  ( 1  +  0 )  =  1
109107, 108eqtri 2303 . . . . 5  |-  ( ( 1  x.  1 )  +  ( 0  +  0 ) )  =  1
11034mulid1i 8839 . . . . . . 7  |-  ( 3  x.  1 )  =  3
111110oveq1i 5868 . . . . . 6  |-  ( ( 3  x.  1 )  +  0 )  =  ( 3  +  0 )
11234addid1i 8999 . . . . . 6  |-  ( 3  +  0 )  =  3
1132dec0h 10140 . . . . . 6  |-  3  = ; 0 3
114111, 112, 1133eqtri 2307 . . . . 5  |-  ( ( 3  x.  1 )  +  0 )  = ; 0
3
1151, 2, 25, 25, 103, 105, 1, 2, 25, 109, 114decmac 10163 . . . 4  |-  ( (; 1
3  x.  1 )  +  ( 0  +  0 ) )  = ; 1
3
1163nn0cni 9977 . . . . . . 7  |- ; 1 3  e.  CC
117116mul01i 9002 . . . . . 6  |-  (; 1 3  x.  0 )  =  0
118117oveq1i 5868 . . . . 5  |-  ( (; 1
3  x.  0 )  +  9 )  =  ( 0  +  9 )
11963addid2i 9000 . . . . 5  |-  ( 0  +  9 )  =  9
120118, 119, 943eqtri 2307 . . . 4  |-  ( (; 1
3  x.  0 )  +  9 )  = ; 0
9
1211, 25, 25, 9, 102, 94, 3, 9, 25, 115, 120decma2c 10164 . . 3  |-  ( (; 1
3  x.  10 )  +  9 )  = ;; 1 3 9
12224, 2, 9, 10declti 10149 . . 3  |-  9  < ; 1
3
12316, 101, 4, 121, 122ndvdsi 12609 . 2  |-  -. ; 1 3  || ;; 1 3 9
1241, 51decnncl 10137 . . 3  |- ; 1 7  e.  NN
125 eqid 2283 . . . 4  |- ; 1 7  = ; 1 7
126 5nn0 9985 . . . 4  |-  5  e.  NN0
127 8nn 9883 . . . . . . . 8  |-  8  e.  NN
128127nncni 9756 . . . . . . 7  |-  8  e.  CC
129128mulid2i 8840 . . . . . 6  |-  ( 1  x.  8 )  =  8
130 5nn 9880 . . . . . . . 8  |-  5  e.  NN
131130nncni 9756 . . . . . . 7  |-  5  e.  CC
132131addid2i 9000 . . . . . 6  |-  ( 0  +  5 )  =  5
133129, 132oveq12i 5870 . . . . 5  |-  ( ( 1  x.  8 )  +  ( 0  +  5 ) )  =  ( 8  +  5 )
134 8p5e13 10182 . . . . 5  |-  ( 8  +  5 )  = ; 1
3
135133, 134eqtri 2303 . . . 4  |-  ( ( 1  x.  8 )  +  ( 0  +  5 ) )  = ; 1
3
136 8t7e56 10217 . . . . . 6  |-  ( 8  x.  7 )  = ; 5
6
137128, 57, 136mulcomli 8844 . . . . 5  |-  ( 7  x.  8 )  = ; 5
6
138126, 22, 2, 137, 66decaddi 10168 . . . 4  |-  ( ( 7  x.  8 )  +  3 )  = ; 5
9
1391, 56, 25, 2, 125, 113, 6, 9, 126, 135, 138decmac 10163 . . 3  |-  ( (; 1
7  x.  8 )  +  3 )  = ;; 1 3 9
14024, 56, 2, 11declti 10149 . . 3  |-  3  < ; 1
7
141124, 6, 15, 139, 140ndvdsi 12609 . 2  |-  -. ; 1 7  || ;; 1 3 9
1421, 4decnncl 10137 . . 3  |- ; 1 9  e.  NN
14357mulid2i 8840 . . . . . 6  |-  ( 1  x.  7 )  =  7
144143, 59oveq12i 5870 . . . . 5  |-  ( ( 1  x.  7 )  +  ( 0  +  6 ) )  =  ( 7  +  6 )
145144, 61eqtri 2303 . . . 4  |-  ( ( 1  x.  7 )  +  ( 0  +  6 ) )  = ; 1
3
14622, 2, 22, 64, 67decaddi 10168 . . . 4  |-  ( ( 9  x.  7 )  +  6 )  = ; 6
9
1471, 9, 25, 22, 54, 55, 56, 9, 22, 145, 146decmac 10163 . . 3  |-  ( (; 1
9  x.  7 )  +  6 )  = ;; 1 3 9
148 6lt10 9925 . . . 4  |-  6  <  10
14924, 9, 22, 148declti 10149 . . 3  |-  6  < ; 1
9
150142, 56, 53, 147, 149ndvdsi 12609 . 2  |-  -. ; 1 9  || ;; 1 3 9
15131, 15decnncl 10137 . . 3  |- ; 2 3  e.  NN
152 eqid 2283 . . . 4  |- ; 2 3  = ; 2 3
15329oveq2i 5869 . . . . 5  |-  ( ( 2  x.  6 )  +  ( 0  +  1 ) )  =  ( ( 2  x.  6 )  +  1 )
154 6t2e12 10201 . . . . . . 7  |-  ( 6  x.  2 )  = ; 1
2
15540, 77, 154mulcomli 8844 . . . . . 6  |-  ( 2  x.  6 )  = ; 1
2
1561, 31, 32, 155decsuc 10147 . . . . 5  |-  ( ( 2  x.  6 )  +  1 )  = ; 1
3
157153, 156eqtri 2303 . . . 4  |-  ( ( 2  x.  6 )  +  ( 0  +  1 ) )  = ; 1
3
15831, 2, 25, 1, 152, 27, 22, 9, 1, 157, 43decmac 10163 . . 3  |-  ( (; 2
3  x.  6 )  +  1 )  = ;; 1 3 9
159 2nn 9877 . . . 4  |-  2  e.  NN
160159, 2, 1, 17declti 10149 . . 3  |-  1  < ; 2
3
161151, 22, 24, 158, 160ndvdsi 12609 . 2  |-  -. ; 2 3  || ;; 1 3 9
1625, 14, 18, 21, 46, 50, 71, 100, 123, 141, 150, 161prmlem2 13121 1  |- ;; 1 3 9  e.  Prime
Colors of variables: wff set class
Syntax hints:    e. wcel 1684  (class class class)co 5858   0cc0 8737   1c1 8738    + caddc 8740    x. cmul 8742   2c2 9795   3c3 9796   4c4 9797   5c5 9798   6c6 9799   7c7 9800   8c8 9801   9c9 9802   10c10 9803  ;cdc 10124   Primecprime 12758
This theorem is referenced by:  2503prm  13138
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814  ax-pre-sup 8815
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-2o 6480  df-oadd 6483  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-sup 7194  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-nn 9747  df-2 9804  df-3 9805  df-4 9806  df-5 9807  df-6 9808  df-7 9809  df-8 9810  df-9 9811  df-10 9812  df-n0 9966  df-z 10025  df-dec 10125  df-uz 10231  df-rp 10355  df-fz 10783  df-seq 11047  df-exp 11105  df-cj 11584  df-re 11585  df-im 11586  df-sqr 11720  df-abs 11721  df-dvds 12532  df-prm 12759
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