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Theorem prmlem2 14586
Description: Our last proving session got as far as 25 because we started with the two "bootstrap" primes 2 and 3, and the next prime is 5, so knowing that 2 and 3 are prime and 4 is not allows us to cover the numbers less than  5 ^ 2  =  2 5. Additionally, nonprimes are "easy", so we can extend this range of known prime/nonprimes all the way until 29, which is the first prime larger than 25. Thus, in this lemma we extend another blanket out to  2 9 ^ 2  =  8 4 1, from which we can prove even more primes. If we wanted, we could keep doing this, but the goal is Bertrand's postulate, and for that we only need a few large primes - we don't need to find them all, as we have been doing thus far. So after this blanket runs out, we'll have to switch to another method (see 1259prm 14599).

As a side note, you can see the pattern of the primes in the indentation pattern of this lemma! (Contributed by Mario Carneiro, 18-Feb-2014.) (Proof shortened by Mario Carneiro, 20-Apr-2015.)

Hypotheses
Ref Expression
prmlem2.n  |-  N  e.  NN
prmlem2.lt  |-  N  < ;; 8 4 1
prmlem2.gt  |-  1  <  N
prmlem2.2  |-  -.  2  ||  N
prmlem2.3  |-  -.  3  ||  N
prmlem2.5  |-  -.  5  ||  N
prmlem2.7  |-  -.  7  ||  N
prmlem2.11  |-  -. ; 1 1  ||  N
prmlem2.13  |-  -. ; 1 3  ||  N
prmlem2.17  |-  -. ; 1 7  ||  N
prmlem2.19  |-  -. ; 1 9  ||  N
prmlem2.23  |-  -. ; 2 3  ||  N
Assertion
Ref Expression
prmlem2  |-  N  e. 
Prime

Proof of Theorem prmlem2
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 prmlem2.n . 2  |-  N  e.  NN
2 prmlem2.gt . 2  |-  1  <  N
3 prmlem2.2 . 2  |-  -.  2  ||  N
4 prmlem2.3 . 2  |-  -.  3  ||  N
5 eluzelre 11101 . . . . . . . . . . . . . . . . . . . 20  |-  ( x  e.  ( ZZ>= ` ; 2 9 )  ->  x  e.  RR )
65resqcld 12317 . . . . . . . . . . . . . . . . . . 19  |-  ( x  e.  ( ZZ>= ` ; 2 9 )  -> 
( x ^ 2 )  e.  RR )
7 eluzle 11103 . . . . . . . . . . . . . . . . . . . 20  |-  ( x  e.  ( ZZ>= ` ; 2 9 )  -> ; 2 9  <_  x )
8 2nn0 10819 . . . . . . . . . . . . . . . . . . . . . . 23  |-  2  e.  NN0
9 9nn0 10826 . . . . . . . . . . . . . . . . . . . . . . 23  |-  9  e.  NN0
108, 9deccl 10999 . . . . . . . . . . . . . . . . . . . . . 22  |- ; 2 9  e.  NN0
1110nn0rei 10813 . . . . . . . . . . . . . . . . . . . . 21  |- ; 2 9  e.  RR
1210nn0ge0i 10830 . . . . . . . . . . . . . . . . . . . . 21  |-  0  <_ ; 2
9
13 le2sq2 12224 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( (; 2 9  e.  RR  /\  0  <_ ; 2 9 )  /\  ( x  e.  RR  /\ ; 2
9  <_  x )
)  ->  (; 2 9 ^ 2 )  <_  ( x ^ 2 ) )
1411, 12, 13mpanl12 682 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( x  e.  RR  /\ ; 2 9  <_  x )  -> 
(; 2 9 ^ 2 )  <_  ( x ^ 2 ) )
155, 7, 14syl2anc 661 . . . . . . . . . . . . . . . . . . 19  |-  ( x  e.  ( ZZ>= ` ; 2 9 )  -> 
(; 2 9 ^ 2 )  <_  ( x ^ 2 ) )
161nnrei 10552 . . . . . . . . . . . . . . . . . . . 20  |-  N  e.  RR
1711resqcli 12234 . . . . . . . . . . . . . . . . . . . 20  |-  (; 2 9 ^ 2 )  e.  RR
18 prmlem2.lt . . . . . . . . . . . . . . . . . . . . . 22  |-  N  < ;; 8 4 1
1910nn0cni 10814 . . . . . . . . . . . . . . . . . . . . . . . 24  |- ; 2 9  e.  CC
2019sqvali 12228 . . . . . . . . . . . . . . . . . . . . . . 23  |-  (; 2 9 ^ 2 )  =  (; 2 9  x. ; 2 9 )
21 eqid 2443 . . . . . . . . . . . . . . . . . . . . . . . 24  |- ; 2 9  = ; 2 9
22 1nn0 10818 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  1  e.  NN0
23 6nn0 10823 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  6  e.  NN0
248, 23deccl 10999 . . . . . . . . . . . . . . . . . . . . . . . 24  |- ; 2 6  e.  NN0
25 5nn0 10822 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  5  e.  NN0
26 8nn0 10825 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  8  e.  NN0
27192timesi 10663 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  ( 2  x. ; 2 9 )  =  (; 2 9  + ; 2 9 )
28 2p2e4 10660 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29  |-  ( 2  +  2 )  =  4
2928oveq1i 6291 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28  |-  ( ( 2  +  2 )  +  1 )  =  ( 4  +  1 )
30 4p1e5 10669 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28  |-  ( 4  +  1 )  =  5
3129, 30eqtri 2472 . . . . . . . . . . . . . . . . . . . . . . . . . . 27  |-  ( ( 2  +  2 )  +  1 )  =  5
32 9p9e18 11054 . . . . . . . . . . . . . . . . . . . . . . . . . . 27  |-  ( 9  +  9 )  = ; 1
8
338, 9, 8, 9, 21, 21, 31, 26, 32decaddc 11027 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  (; 2 9  + ; 2 9 )  = ; 5
8
3427, 33eqtri 2472 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( 2  x. ; 2 9 )  = ; 5
8
35 eqid 2443 . . . . . . . . . . . . . . . . . . . . . . . . 25  |- ; 2 6  = ; 2 6
36 5p2e7 10680 . . . . . . . . . . . . . . . . . . . . . . . . . . 27  |-  ( 5  +  2 )  =  7
3736oveq1i 6291 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  ( ( 5  +  2 )  +  1 )  =  ( 7  +  1 )
38 7p1e8 10672 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  ( 7  +  1 )  =  8
3937, 38eqtri 2472 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( ( 5  +  2 )  +  1 )  =  8
40 4nn0 10821 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  4  e.  NN0
41 8p6e14 11044 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( 8  +  6 )  = ; 1
4
4225, 26, 8, 23, 34, 35, 39, 40, 41decaddc 11027 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( ( 2  x. ; 2 9 )  + ; 2
6 )  = ; 8 4
43 9t2e18 11080 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  ( 9  x.  2 )  = ; 1
8
44 1p1e2 10656 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  ( 1  +  1 )  =  2
45 8p8e16 11046 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  ( 8  +  8 )  = ; 1
6
4622, 26, 26, 43, 44, 23, 45decaddci 11030 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( ( 9  x.  2 )  +  8 )  = ; 2
6
47 9t9e81 11087 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( 9  x.  9 )  = ; 8
1
489, 8, 9, 21, 22, 26, 46, 47decmul2c 11033 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( 9  x. ; 2 9 )  = ;; 2 6 1
4910, 8, 9, 21, 22, 24, 42, 48decmul1c 11032 . . . . . . . . . . . . . . . . . . . . . . 23  |-  (; 2 9  x. ; 2 9 )  = ;; 8 4 1
5020, 49eqtri 2472 . . . . . . . . . . . . . . . . . . . . . 22  |-  (; 2 9 ^ 2 )  = ;; 8 4 1
5118, 50breqtrri 4462 . . . . . . . . . . . . . . . . . . . . 21  |-  N  < 
(; 2 9 ^ 2 )
52 ltletr 9679 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( N  e.  RR  /\  (; 2 9 ^ 2 )  e.  RR  /\  (
x ^ 2 )  e.  RR )  -> 
( ( N  < 
(; 2 9 ^ 2 )  /\  (; 2 9 ^ 2 )  <_  ( x ^ 2 ) )  ->  N  <  (
x ^ 2 ) ) )
5351, 52mpani 676 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( N  e.  RR  /\  (; 2 9 ^ 2 )  e.  RR  /\  (
x ^ 2 )  e.  RR )  -> 
( (; 2 9 ^ 2 )  <_  ( x ^ 2 )  ->  N  <  ( x ^
2 ) ) )
5416, 17, 53mp3an12 1315 . . . . . . . . . . . . . . . . . . 19  |-  ( ( x ^ 2 )  e.  RR  ->  (
(; 2 9 ^ 2 )  <_  ( x ^ 2 )  ->  N  <  ( x ^
2 ) ) )
556, 15, 54sylc 60 . . . . . . . . . . . . . . . . . 18  |-  ( x  e.  ( ZZ>= ` ; 2 9 )  ->  N  <  ( x ^
2 ) )
56 ltnle 9667 . . . . . . . . . . . . . . . . . . 19  |-  ( ( N  e.  RR  /\  ( x ^ 2 )  e.  RR )  ->  ( N  < 
( x ^ 2 )  <->  -.  ( x ^ 2 )  <_  N ) )
5716, 6, 56sylancr 663 . . . . . . . . . . . . . . . . . 18  |-  ( x  e.  ( ZZ>= ` ; 2 9 )  -> 
( N  <  (
x ^ 2 )  <->  -.  ( x ^ 2 )  <_  N )
)
5855, 57mpbid 210 . . . . . . . . . . . . . . . . 17  |-  ( x  e.  ( ZZ>= ` ; 2 9 )  ->  -.  ( x ^ 2 )  <_  N )
5958pm2.21d 106 . . . . . . . . . . . . . . . 16  |-  ( x  e.  ( ZZ>= ` ; 2 9 )  -> 
( ( x ^
2 )  <_  N  ->  -.  x  ||  N
) )
6059adantld 467 . . . . . . . . . . . . . . 15  |-  ( x  e.  ( ZZ>= ` ; 2 9 )  -> 
( ( x  e.  ( Prime  \  { 2 } )  /\  (
x ^ 2 )  <_  N )  ->  -.  x  ||  N ) )
6160adantl 466 . . . . . . . . . . . . . 14  |-  ( ( -.  2  || ; 2 9  /\  x  e.  ( ZZ>= ` ; 2 9 ) )  ->  ( ( x  e.  ( Prime  \  {
2 } )  /\  ( x ^ 2 )  <_  N )  ->  -.  x  ||  N
) )
62 9nn 10707 . . . . . . . . . . . . . . . 16  |-  9  e.  NN
63 3nn 10701 . . . . . . . . . . . . . . . 16  |-  3  e.  NN
64 1lt9 10744 . . . . . . . . . . . . . . . 16  |-  1  <  9
65 1lt3 10711 . . . . . . . . . . . . . . . 16  |-  1  <  3
66 9t3e27 11081 . . . . . . . . . . . . . . . 16  |-  ( 9  x.  3 )  = ; 2
7
6762, 63, 64, 65, 66nprmi 14213 . . . . . . . . . . . . . . 15  |-  -. ; 2 7  e.  Prime
6867pm2.21i 131 . . . . . . . . . . . . . 14  |-  (; 2 7  e.  Prime  ->  -. ; 2 7  ||  N )
69 7nn0 10824 . . . . . . . . . . . . . . 15  |-  7  e.  NN0
70 eqid 2443 . . . . . . . . . . . . . . 15  |- ; 2 7  = ; 2 7
71 7p2e9 10687 . . . . . . . . . . . . . . 15  |-  ( 7  +  2 )  =  9
728, 69, 8, 70, 71decaddi 11029 . . . . . . . . . . . . . 14  |-  (; 2 7  +  2 )  = ; 2 9
7361, 68, 72prmlem0 14572 . . . . . . . . . . . . 13  |-  ( ( -.  2  || ; 2 7  /\  x  e.  ( ZZ>= ` ; 2 7 ) )  ->  ( ( x  e.  ( Prime  \  {
2 } )  /\  ( x ^ 2 )  <_  N )  ->  -.  x  ||  N
) )
74 5nn 10703 . . . . . . . . . . . . . . 15  |-  5  e.  NN
75 1lt5 10718 . . . . . . . . . . . . . . 15  |-  1  <  5
76 5t5e25 11061 . . . . . . . . . . . . . . 15  |-  ( 5  x.  5 )  = ; 2
5
7774, 74, 75, 75, 76nprmi 14213 . . . . . . . . . . . . . 14  |-  -. ; 2 5  e.  Prime
7877pm2.21i 131 . . . . . . . . . . . . 13  |-  (; 2 5  e.  Prime  ->  -. ; 2 5  ||  N )
79 eqid 2443 . . . . . . . . . . . . . 14  |- ; 2 5  = ; 2 5
808, 25, 8, 79, 36decaddi 11029 . . . . . . . . . . . . 13  |-  (; 2 5  +  2 )  = ; 2 7
8173, 78, 80prmlem0 14572 . . . . . . . . . . . 12  |-  ( ( -.  2  || ; 2 5  /\  x  e.  ( ZZ>= ` ; 2 5 ) )  ->  ( ( x  e.  ( Prime  \  {
2 } )  /\  ( x ^ 2 )  <_  N )  ->  -.  x  ||  N
) )
82 prmlem2.23 . . . . . . . . . . . . 13  |-  -. ; 2 3  ||  N
8382a1i 11 . . . . . . . . . . . 12  |-  (; 2 3  e.  Prime  ->  -. ; 2 3  ||  N )
84 3nn0 10820 . . . . . . . . . . . . 13  |-  3  e.  NN0
85 eqid 2443 . . . . . . . . . . . . 13  |- ; 2 3  = ; 2 3
86 3p2e5 10675 . . . . . . . . . . . . 13  |-  ( 3  +  2 )  =  5
878, 84, 8, 85, 86decaddi 11029 . . . . . . . . . . . 12  |-  (; 2 3  +  2 )  = ; 2 5
8881, 83, 87prmlem0 14572 . . . . . . . . . . 11  |-  ( ( -.  2  || ; 2 3  /\  x  e.  ( ZZ>= ` ; 2 3 ) )  ->  ( ( x  e.  ( Prime  \  {
2 } )  /\  ( x ^ 2 )  <_  N )  ->  -.  x  ||  N
) )
89 7nn 10705 . . . . . . . . . . . . 13  |-  7  e.  NN
90 1lt7 10729 . . . . . . . . . . . . 13  |-  1  <  7
91 7t3e21 11068 . . . . . . . . . . . . 13  |-  ( 7  x.  3 )  = ; 2
1
9289, 63, 90, 65, 91nprmi 14213 . . . . . . . . . . . 12  |-  -. ; 2 1  e.  Prime
9392pm2.21i 131 . . . . . . . . . . 11  |-  (; 2 1  e.  Prime  ->  -. ; 2 1  ||  N )
94 eqid 2443 . . . . . . . . . . . 12  |- ; 2 1  = ; 2 1
95 1p2e3 10667 . . . . . . . . . . . 12  |-  ( 1  +  2 )  =  3
968, 22, 8, 94, 95decaddi 11029 . . . . . . . . . . 11  |-  (; 2 1  +  2 )  = ; 2 3
9788, 93, 96prmlem0 14572 . . . . . . . . . 10  |-  ( ( -.  2  || ; 2 1  /\  x  e.  ( ZZ>= ` ; 2 1 ) )  ->  ( ( x  e.  ( Prime  \  {
2 } )  /\  ( x ^ 2 )  <_  N )  ->  -.  x  ||  N
) )
98 prmlem2.19 . . . . . . . . . . 11  |-  -. ; 1 9  ||  N
9998a1i 11 . . . . . . . . . 10  |-  (; 1 9  e.  Prime  ->  -. ; 1 9  ||  N )
100 eqid 2443 . . . . . . . . . . 11  |- ; 1 9  = ; 1 9
101 9p2e11 11047 . . . . . . . . . . 11  |-  ( 9  +  2 )  = ; 1
1
10222, 9, 8, 100, 44, 22, 101decaddci 11030 . . . . . . . . . 10  |-  (; 1 9  +  2 )  = ; 2 1
10397, 99, 102prmlem0 14572 . . . . . . . . 9  |-  ( ( -.  2  || ; 1 9  /\  x  e.  ( ZZ>= ` ; 1 9 ) )  ->  ( ( x  e.  ( Prime  \  {
2 } )  /\  ( x ^ 2 )  <_  N )  ->  -.  x  ||  N
) )
104 prmlem2.17 . . . . . . . . . 10  |-  -. ; 1 7  ||  N
105104a1i 11 . . . . . . . . 9  |-  (; 1 7  e.  Prime  ->  -. ; 1 7  ||  N )
106 eqid 2443 . . . . . . . . . 10  |- ; 1 7  = ; 1 7
10722, 69, 8, 106, 71decaddi 11029 . . . . . . . . 9  |-  (; 1 7  +  2 )  = ; 1 9
108103, 105, 107prmlem0 14572 . . . . . . . 8  |-  ( ( -.  2  || ; 1 7  /\  x  e.  ( ZZ>= ` ; 1 7 ) )  ->  ( ( x  e.  ( Prime  \  {
2 } )  /\  ( x ^ 2 )  <_  N )  ->  -.  x  ||  N
) )
109 5t3e15 11059 . . . . . . . . . 10  |-  ( 5  x.  3 )  = ; 1
5
11074, 63, 75, 65, 109nprmi 14213 . . . . . . . . 9  |-  -. ; 1 5  e.  Prime
111110pm2.21i 131 . . . . . . . 8  |-  (; 1 5  e.  Prime  ->  -. ; 1 5  ||  N )
112 eqid 2443 . . . . . . . . 9  |- ; 1 5  = ; 1 5
11322, 25, 8, 112, 36decaddi 11029 . . . . . . . 8  |-  (; 1 5  +  2 )  = ; 1 7
114108, 111, 113prmlem0 14572 . . . . . . 7  |-  ( ( -.  2  || ; 1 5  /\  x  e.  ( ZZ>= ` ; 1 5 ) )  ->  ( ( x  e.  ( Prime  \  {
2 } )  /\  ( x ^ 2 )  <_  N )  ->  -.  x  ||  N
) )
115 prmlem2.13 . . . . . . . 8  |-  -. ; 1 3  ||  N
116115a1i 11 . . . . . . 7  |-  (; 1 3  e.  Prime  ->  -. ; 1 3  ||  N )
117 eqid 2443 . . . . . . . 8  |- ; 1 3  = ; 1 3
11822, 84, 8, 117, 86decaddi 11029 . . . . . . 7  |-  (; 1 3  +  2 )  = ; 1 5
119114, 116, 118prmlem0 14572 . . . . . 6  |-  ( ( -.  2  || ; 1 3  /\  x  e.  ( ZZ>= ` ; 1 3 ) )  ->  ( ( x  e.  ( Prime  \  {
2 } )  /\  ( x ^ 2 )  <_  N )  ->  -.  x  ||  N
) )
120 prmlem2.11 . . . . . . 7  |-  -. ; 1 1  ||  N
121120a1i 11 . . . . . 6  |-  (; 1 1  e.  Prime  ->  -. ; 1 1  ||  N )
122 eqid 2443 . . . . . . 7  |- ; 1 1  = ; 1 1
12322, 22, 8, 122, 95decaddi 11029 . . . . . 6  |-  (; 1 1  +  2 )  = ; 1 3
124119, 121, 123prmlem0 14572 . . . . 5  |-  ( ( -.  2  || ; 1 1  /\  x  e.  ( ZZ>= ` ; 1 1 ) )  ->  ( ( x  e.  ( Prime  \  {
2 } )  /\  ( x ^ 2 )  <_  N )  ->  -.  x  ||  N
) )
125 9nprm 14579 . . . . . 6  |-  -.  9  e.  Prime
126125pm2.21i 131 . . . . 5  |-  ( 9  e.  Prime  ->  -.  9  ||  N )
127124, 126, 101prmlem0 14572 . . . 4  |-  ( ( -.  2  ||  9  /\  x  e.  ( ZZ>=
`  9 ) )  ->  ( ( x  e.  ( Prime  \  {
2 } )  /\  ( x ^ 2 )  <_  N )  ->  -.  x  ||  N
) )
128 prmlem2.7 . . . . 5  |-  -.  7  ||  N
129128a1i 11 . . . 4  |-  ( 7  e.  Prime  ->  -.  7  ||  N )
130127, 129, 71prmlem0 14572 . . 3  |-  ( ( -.  2  ||  7  /\  x  e.  ( ZZ>=
`  7 ) )  ->  ( ( x  e.  ( Prime  \  {
2 } )  /\  ( x ^ 2 )  <_  N )  ->  -.  x  ||  N
) )
131 prmlem2.5 . . . 4  |-  -.  5  ||  N
132131a1i 11 . . 3  |-  ( 5  e.  Prime  ->  -.  5  ||  N )
133130, 132, 36prmlem0 14572 . 2  |-  ( ( -.  2  ||  5  /\  x  e.  ( ZZ>=
`  5 ) )  ->  ( ( x  e.  ( Prime  \  {
2 } )  /\  ( x ^ 2 )  <_  N )  ->  -.  x  ||  N
) )
1341, 2, 3, 4, 133prmlem1a 14573 1  |-  N  e. 
Prime
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 974    e. wcel 1804    \ cdif 3458   {csn 4014   class class class wbr 4437   ` cfv 5578  (class class class)co 6281   RRcr 9494   0cc0 9495   1c1 9496    + caddc 9498    x. cmul 9500    < clt 9631    <_ cle 9632   NNcn 10543   2c2 10592   3c3 10593   4c4 10594   5c5 10595   6c6 10596   7c7 10597   8c8 10598   9c9 10599  ;cdc 10985   ZZ>=cuz 11091   ^cexp 12147    || cdvds 13967   Primecprime 14198
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676  ax-un 6577  ax-cnex 9551  ax-resscn 9552  ax-1cn 9553  ax-icn 9554  ax-addcl 9555  ax-addrcl 9556  ax-mulcl 9557  ax-mulrcl 9558  ax-mulcom 9559  ax-addass 9560  ax-mulass 9561  ax-distr 9562  ax-i2m1 9563  ax-1ne0 9564  ax-1rid 9565  ax-rnegex 9566  ax-rrecex 9567  ax-cnre 9568  ax-pre-lttri 9569  ax-pre-lttrn 9570  ax-pre-ltadd 9571  ax-pre-mulgt0 9572
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 975  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-nel 2641  df-ral 2798  df-rex 2799  df-reu 2800  df-rmo 2801  df-rab 2802  df-v 3097  df-sbc 3314  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3771  df-if 3927  df-pw 3999  df-sn 4015  df-pr 4017  df-tp 4019  df-op 4021  df-uni 4235  df-int 4272  df-iun 4317  df-br 4438  df-opab 4496  df-mpt 4497  df-tr 4531  df-eprel 4781  df-id 4785  df-po 4790  df-so 4791  df-fr 4828  df-we 4830  df-ord 4871  df-on 4872  df-lim 4873  df-suc 4874  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002  df-iota 5541  df-fun 5580  df-fn 5581  df-f 5582  df-f1 5583  df-fo 5584  df-f1o 5585  df-fv 5586  df-riota 6242  df-ov 6284  df-oprab 6285  df-mpt2 6286  df-om 6686  df-1st 6785  df-2nd 6786  df-recs 7044  df-rdg 7078  df-1o 7132  df-2o 7133  df-oadd 7136  df-er 7313  df-en 7519  df-dom 7520  df-sdom 7521  df-fin 7522  df-pnf 9633  df-mnf 9634  df-xr 9635  df-ltxr 9636  df-le 9637  df-sub 9812  df-neg 9813  df-div 10214  df-nn 10544  df-2 10601  df-3 10602  df-4 10603  df-5 10604  df-6 10605  df-7 10606  df-8 10607  df-9 10608  df-10 10609  df-n0 10803  df-z 10872  df-dec 10986  df-uz 11092  df-rp 11231  df-fz 11683  df-seq 12089  df-exp 12148  df-dvds 13968  df-prm 14199
This theorem is referenced by:  37prm  14587  43prm  14588  83prm  14589  139prm  14590  163prm  14591  317prm  14592  631prm  14593
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