Metamath Proof Explorer


Theorem 139prmALT

Description: 139 is a prime number. In contrast to 139prm , the proof of this theorem uses 3dvds2dec for checking the divisibility by 3. Although the proof using 3dvds2dec is longer (regarding size: 1849 characters compared with 1809 for 139prm ), the number of essential steps is smaller (301 compared with 327 for 139prm ). (Contributed by Mario Carneiro, 19-Feb-2014) (Revised by AV, 18-Aug-2021) (New usage is discouraged.) (Proof modification is discouraged.)

Ref Expression
Assertion 139prmALT
|- ; ; 1 3 9 e. Prime

Proof

Step Hyp Ref Expression
1 1nn0
 |-  1 e. NN0
2 3nn0
 |-  3 e. NN0
3 1 2 deccl
 |-  ; 1 3 e. NN0
4 9nn
 |-  9 e. NN
5 3 4 decnncl
 |-  ; ; 1 3 9 e. NN
6 8nn0
 |-  8 e. NN0
7 4nn0
 |-  4 e. NN0
8 9nn0
 |-  9 e. NN0
9 1lt8
 |-  1 < 8
10 3lt10
 |-  3 < ; 1 0
11 9lt10
 |-  9 < ; 1 0
12 1 6 2 7 8 1 9 10 11 3decltc
 |-  ; ; 1 3 9 < ; ; 8 4 1
13 3nn
 |-  3 e. NN
14 1 13 decnncl
 |-  ; 1 3 e. NN
15 1lt10
 |-  1 < ; 1 0
16 14 8 1 15 declti
 |-  1 < ; ; 1 3 9
17 4t2e8
 |-  ( 4 x. 2 ) = 8
18 df-9
 |-  9 = ( 8 + 1 )
19 3 7 17 18 dec2dvds
 |-  -. 2 || ; ; 1 3 9
20 3ndvds4
 |-  -. 3 || 4
21 1 2 3dvdsdec
 |-  ( 3 || ; 1 3 <-> 3 || ( 1 + 3 ) )
22 3cn
 |-  3 e. CC
23 ax-1cn
 |-  1 e. CC
24 3p1e4
 |-  ( 3 + 1 ) = 4
25 22 23 24 addcomli
 |-  ( 1 + 3 ) = 4
26 25 breq2i
 |-  ( 3 || ( 1 + 3 ) <-> 3 || 4 )
27 21 26 bitri
 |-  ( 3 || ; 1 3 <-> 3 || 4 )
28 20 27 mtbir
 |-  -. 3 || ; 1 3
29 1 2 8 3dvds2dec
 |-  ( 3 || ; ; 1 3 9 <-> 3 || ( ( 1 + 3 ) + 9 ) )
30 25 oveq1i
 |-  ( ( 1 + 3 ) + 9 ) = ( 4 + 9 )
31 9cn
 |-  9 e. CC
32 4cn
 |-  4 e. CC
33 9p4e13
 |-  ( 9 + 4 ) = ; 1 3
34 31 32 33 addcomli
 |-  ( 4 + 9 ) = ; 1 3
35 30 34 eqtri
 |-  ( ( 1 + 3 ) + 9 ) = ; 1 3
36 35 breq2i
 |-  ( 3 || ( ( 1 + 3 ) + 9 ) <-> 3 || ; 1 3 )
37 29 36 bitri
 |-  ( 3 || ; ; 1 3 9 <-> 3 || ; 1 3 )
38 28 37 mtbir
 |-  -. 3 || ; ; 1 3 9
39 4nn
 |-  4 e. NN
40 4lt5
 |-  4 < 5
41 5p4e9
 |-  ( 5 + 4 ) = 9
42 3 39 40 41 dec5dvds2
 |-  -. 5 || ; ; 1 3 9
43 7nn
 |-  7 e. NN
44 1 8 deccl
 |-  ; 1 9 e. NN0
45 6nn
 |-  6 e. NN
46 0nn0
 |-  0 e. NN0
47 6nn0
 |-  6 e. NN0
48 eqid
 |-  ; 1 9 = ; 1 9
49 47 dec0h
 |-  6 = ; 0 6
50 7nn0
 |-  7 e. NN0
51 7cn
 |-  7 e. CC
52 51 mulid1i
 |-  ( 7 x. 1 ) = 7
53 6cn
 |-  6 e. CC
54 53 addid2i
 |-  ( 0 + 6 ) = 6
55 52 54 oveq12i
 |-  ( ( 7 x. 1 ) + ( 0 + 6 ) ) = ( 7 + 6 )
56 7p6e13
 |-  ( 7 + 6 ) = ; 1 3
57 55 56 eqtri
 |-  ( ( 7 x. 1 ) + ( 0 + 6 ) ) = ; 1 3
58 9t7e63
 |-  ( 9 x. 7 ) = ; 6 3
59 31 51 58 mulcomli
 |-  ( 7 x. 9 ) = ; 6 3
60 6p3e9
 |-  ( 6 + 3 ) = 9
61 53 22 60 addcomli
 |-  ( 3 + 6 ) = 9
62 47 2 47 59 61 decaddi
 |-  ( ( 7 x. 9 ) + 6 ) = ; 6 9
63 1 8 46 47 48 49 50 8 47 57 62 decma2c
 |-  ( ( 7 x. ; 1 9 ) + 6 ) = ; ; 1 3 9
64 6lt7
 |-  6 < 7
65 43 44 45 63 64 ndvdsi
 |-  -. 7 || ; ; 1 3 9
66 1nn
 |-  1 e. NN
67 1 66 decnncl
 |-  ; 1 1 e. NN
68 2nn0
 |-  2 e. NN0
69 1 68 deccl
 |-  ; 1 2 e. NN0
70 eqid
 |-  ; 1 2 = ; 1 2
71 50 dec0h
 |-  7 = ; 0 7
72 1 1 deccl
 |-  ; 1 1 e. NN0
73 2cn
 |-  2 e. CC
74 73 addid2i
 |-  ( 0 + 2 ) = 2
75 74 oveq2i
 |-  ( ( ; 1 1 x. 1 ) + ( 0 + 2 ) ) = ( ( ; 1 1 x. 1 ) + 2 )
76 67 nncni
 |-  ; 1 1 e. CC
77 76 mulid1i
 |-  ( ; 1 1 x. 1 ) = ; 1 1
78 1p2e3
 |-  ( 1 + 2 ) = 3
79 1 1 68 77 78 decaddi
 |-  ( ( ; 1 1 x. 1 ) + 2 ) = ; 1 3
80 75 79 eqtri
 |-  ( ( ; 1 1 x. 1 ) + ( 0 + 2 ) ) = ; 1 3
81 eqid
 |-  ; 1 1 = ; 1 1
82 73 mulid2i
 |-  ( 1 x. 2 ) = 2
83 00id
 |-  ( 0 + 0 ) = 0
84 82 83 oveq12i
 |-  ( ( 1 x. 2 ) + ( 0 + 0 ) ) = ( 2 + 0 )
85 73 addid1i
 |-  ( 2 + 0 ) = 2
86 84 85 eqtri
 |-  ( ( 1 x. 2 ) + ( 0 + 0 ) ) = 2
87 82 oveq1i
 |-  ( ( 1 x. 2 ) + 7 ) = ( 2 + 7 )
88 7p2e9
 |-  ( 7 + 2 ) = 9
89 51 73 88 addcomli
 |-  ( 2 + 7 ) = 9
90 8 dec0h
 |-  9 = ; 0 9
91 87 89 90 3eqtri
 |-  ( ( 1 x. 2 ) + 7 ) = ; 0 9
92 1 1 46 50 81 71 68 8 46 86 91 decmac
 |-  ( ( ; 1 1 x. 2 ) + 7 ) = ; 2 9
93 1 68 46 50 70 71 72 8 68 80 92 decma2c
 |-  ( ( ; 1 1 x. ; 1 2 ) + 7 ) = ; ; 1 3 9
94 7lt10
 |-  7 < ; 1 0
95 66 1 50 94 declti
 |-  7 < ; 1 1
96 67 69 43 93 95 ndvdsi
 |-  -. ; 1 1 || ; ; 1 3 9
97 1 46 deccl
 |-  ; 1 0 e. NN0
98 eqid
 |-  ; 1 0 = ; 1 0
99 3 nn0cni
 |-  ; 1 3 e. CC
100 99 mulid1i
 |-  ( ; 1 3 x. 1 ) = ; 1 3
101 100 83 oveq12i
 |-  ( ( ; 1 3 x. 1 ) + ( 0 + 0 ) ) = ( ; 1 3 + 0 )
102 99 addid1i
 |-  ( ; 1 3 + 0 ) = ; 1 3
103 101 102 eqtri
 |-  ( ( ; 1 3 x. 1 ) + ( 0 + 0 ) ) = ; 1 3
104 99 mul01i
 |-  ( ; 1 3 x. 0 ) = 0
105 104 oveq1i
 |-  ( ( ; 1 3 x. 0 ) + 9 ) = ( 0 + 9 )
106 31 addid2i
 |-  ( 0 + 9 ) = 9
107 105 106 90 3eqtri
 |-  ( ( ; 1 3 x. 0 ) + 9 ) = ; 0 9
108 1 46 46 8 98 90 3 8 46 103 107 decma2c
 |-  ( ( ; 1 3 x. ; 1 0 ) + 9 ) = ; ; 1 3 9
109 66 2 8 11 declti
 |-  9 < ; 1 3
110 14 97 4 108 109 ndvdsi
 |-  -. ; 1 3 || ; ; 1 3 9
111 1 43 decnncl
 |-  ; 1 7 e. NN
112 eqid
 |-  ; 1 7 = ; 1 7
113 2 dec0h
 |-  3 = ; 0 3
114 5nn0
 |-  5 e. NN0
115 8cn
 |-  8 e. CC
116 115 mulid2i
 |-  ( 1 x. 8 ) = 8
117 5cn
 |-  5 e. CC
118 117 addid2i
 |-  ( 0 + 5 ) = 5
119 116 118 oveq12i
 |-  ( ( 1 x. 8 ) + ( 0 + 5 ) ) = ( 8 + 5 )
120 8p5e13
 |-  ( 8 + 5 ) = ; 1 3
121 119 120 eqtri
 |-  ( ( 1 x. 8 ) + ( 0 + 5 ) ) = ; 1 3
122 8t7e56
 |-  ( 8 x. 7 ) = ; 5 6
123 115 51 122 mulcomli
 |-  ( 7 x. 8 ) = ; 5 6
124 114 47 2 123 60 decaddi
 |-  ( ( 7 x. 8 ) + 3 ) = ; 5 9
125 1 50 46 2 112 113 6 8 114 121 124 decmac
 |-  ( ( ; 1 7 x. 8 ) + 3 ) = ; ; 1 3 9
126 66 50 2 10 declti
 |-  3 < ; 1 7
127 111 6 13 125 126 ndvdsi
 |-  -. ; 1 7 || ; ; 1 3 9
128 1 4 decnncl
 |-  ; 1 9 e. NN
129 51 mulid2i
 |-  ( 1 x. 7 ) = 7
130 129 54 oveq12i
 |-  ( ( 1 x. 7 ) + ( 0 + 6 ) ) = ( 7 + 6 )
131 130 56 eqtri
 |-  ( ( 1 x. 7 ) + ( 0 + 6 ) ) = ; 1 3
132 47 2 47 58 61 decaddi
 |-  ( ( 9 x. 7 ) + 6 ) = ; 6 9
133 1 8 46 47 48 49 50 8 47 131 132 decmac
 |-  ( ( ; 1 9 x. 7 ) + 6 ) = ; ; 1 3 9
134 6lt10
 |-  6 < ; 1 0
135 66 8 47 134 declti
 |-  6 < ; 1 9
136 128 50 45 133 135 ndvdsi
 |-  -. ; 1 9 || ; ; 1 3 9
137 68 13 decnncl
 |-  ; 2 3 e. NN
138 eqid
 |-  ; 2 3 = ; 2 3
139 2p1e3
 |-  ( 2 + 1 ) = 3
140 6t2e12
 |-  ( 6 x. 2 ) = ; 1 2
141 53 73 140 mulcomli
 |-  ( 2 x. 6 ) = ; 1 2
142 1 68 139 141 decsuc
 |-  ( ( 2 x. 6 ) + 1 ) = ; 1 3
143 8p1e9
 |-  ( 8 + 1 ) = 9
144 6t3e18
 |-  ( 6 x. 3 ) = ; 1 8
145 53 22 144 mulcomli
 |-  ( 3 x. 6 ) = ; 1 8
146 1 6 143 145 decsuc
 |-  ( ( 3 x. 6 ) + 1 ) = ; 1 9
147 68 2 1 138 47 8 1 142 146 decrmac
 |-  ( ( ; 2 3 x. 6 ) + 1 ) = ; ; 1 3 9
148 2nn
 |-  2 e. NN
149 148 2 1 15 declti
 |-  1 < ; 2 3
150 137 47 66 147 149 ndvdsi
 |-  -. ; 2 3 || ; ; 1 3 9
151 5 12 16 19 38 42 65 96 110 127 136 150 prmlem2
 |-  ; ; 1 3 9 e. Prime