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Theorem isprm2 13015
Description: The predicate "is a prime number". A prime number is an integer greater than or equal to 2 whose only positive divisors are 1 and itself. (Contributed by Paul Chapman, 26-Oct-2012.)
Assertion
Ref Expression
isprm2  |-  ( P  e.  Prime  <->  ( P  e.  ( ZZ>= `  2 )  /\  A. z  e.  NN  ( z  ||  P  ->  ( z  =  1  \/  z  =  P ) ) ) )
Distinct variable group:    z, P

Proof of Theorem isprm2
Dummy variable  n is distinct from all other variables.
StepHypRef Expression
1 1nprm 13012 . . . . 5  |-  -.  1  e.  Prime
2 eleq1 2448 . . . . . 6  |-  ( P  =  1  ->  ( P  e.  Prime  <->  1  e.  Prime ) )
32biimpcd 216 . . . . 5  |-  ( P  e.  Prime  ->  ( P  =  1  ->  1  e.  Prime ) )
41, 3mtoi 171 . . . 4  |-  ( P  e.  Prime  ->  -.  P  =  1 )
54neneqad 2621 . . 3  |-  ( P  e.  Prime  ->  P  =/=  1 )
65pm4.71i 614 . 2  |-  ( P  e.  Prime  <->  ( P  e. 
Prime  /\  P  =/=  1
) )
7 isprm 13009 . . . 4  |-  ( P  e.  Prime  <->  ( P  e.  NN  /\  { n  e.  NN  |  n  ||  P }  ~~  2o ) )
8 isprm2lem 13014 . . . . . . 7  |-  ( ( P  e.  NN  /\  P  =/=  1 )  -> 
( { n  e.  NN  |  n  ||  P }  ~~  2o  <->  { n  e.  NN  |  n  ||  P }  =  {
1 ,  P }
) )
9 eqss 3307 . . . . . . . . . . 11  |-  ( { n  e.  NN  |  n  ||  P }  =  { 1 ,  P } 
<->  ( { n  e.  NN  |  n  ||  P }  C_  { 1 ,  P }  /\  { 1 ,  P }  C_ 
{ n  e.  NN  |  n  ||  P }
) )
109imbi2i 304 . . . . . . . . . 10  |-  ( ( P  e.  NN  ->  { n  e.  NN  |  n  ||  P }  =  { 1 ,  P } )  <->  ( P  e.  NN  ->  ( {
n  e.  NN  |  n  ||  P }  C_  { 1 ,  P }  /\  { 1 ,  P }  C_  { n  e.  NN  |  n  ||  P } ) ) )
11 1idssfct 13013 . . . . . . . . . . 11  |-  ( P  e.  NN  ->  { 1 ,  P }  C_  { n  e.  NN  |  n  ||  P } )
12 jcab 834 . . . . . . . . . . 11  |-  ( ( P  e.  NN  ->  ( { n  e.  NN  |  n  ||  P }  C_ 
{ 1 ,  P }  /\  { 1 ,  P }  C_  { n  e.  NN  |  n  ||  P } ) )  <->  ( ( P  e.  NN  ->  { n  e.  NN  |  n  ||  P }  C_  { 1 ,  P }
)  /\  ( P  e.  NN  ->  { 1 ,  P }  C_  { n  e.  NN  |  n  ||  P } ) ) )
1311, 12mpbiran2 886 . . . . . . . . . 10  |-  ( ( P  e.  NN  ->  ( { n  e.  NN  |  n  ||  P }  C_ 
{ 1 ,  P }  /\  { 1 ,  P }  C_  { n  e.  NN  |  n  ||  P } ) )  <->  ( P  e.  NN  ->  { n  e.  NN  |  n  ||  P }  C_  { 1 ,  P } ) )
1410, 13bitri 241 . . . . . . . . 9  |-  ( ( P  e.  NN  ->  { n  e.  NN  |  n  ||  P }  =  { 1 ,  P } )  <->  ( P  e.  NN  ->  { n  e.  NN  |  n  ||  P }  C_  { 1 ,  P } ) )
1514pm5.74ri 238 . . . . . . . 8  |-  ( P  e.  NN  ->  ( { n  e.  NN  |  n  ||  P }  =  { 1 ,  P } 
<->  { n  e.  NN  |  n  ||  P }  C_ 
{ 1 ,  P } ) )
1615adantr 452 . . . . . . 7  |-  ( ( P  e.  NN  /\  P  =/=  1 )  -> 
( { n  e.  NN  |  n  ||  P }  =  {
1 ,  P }  <->  { n  e.  NN  |  n  ||  P }  C_  { 1 ,  P }
) )
178, 16bitrd 245 . . . . . 6  |-  ( ( P  e.  NN  /\  P  =/=  1 )  -> 
( { n  e.  NN  |  n  ||  P }  ~~  2o  <->  { n  e.  NN  |  n  ||  P }  C_  { 1 ,  P } ) )
1817expcom 425 . . . . 5  |-  ( P  =/=  1  ->  ( P  e.  NN  ->  ( { n  e.  NN  |  n  ||  P }  ~~  2o  <->  { n  e.  NN  |  n  ||  P }  C_ 
{ 1 ,  P } ) ) )
1918pm5.32d 621 . . . 4  |-  ( P  =/=  1  ->  (
( P  e.  NN  /\ 
{ n  e.  NN  |  n  ||  P }  ~~  2o )  <->  ( P  e.  NN  /\  { n  e.  NN  |  n  ||  P }  C_  { 1 ,  P } ) ) )
207, 19syl5bb 249 . . 3  |-  ( P  =/=  1  ->  ( P  e.  Prime  <->  ( P  e.  NN  /\  { n  e.  NN  |  n  ||  P }  C_  { 1 ,  P } ) ) )
2120pm5.32ri 620 . 2  |-  ( ( P  e.  Prime  /\  P  =/=  1 )  <->  ( ( P  e.  NN  /\  {
n  e.  NN  |  n  ||  P }  C_  { 1 ,  P }
)  /\  P  =/=  1 ) )
22 ancom 438 . . . 4  |-  ( ( ( P  e.  NN  /\ 
{ n  e.  NN  |  n  ||  P }  C_ 
{ 1 ,  P } )  /\  P  =/=  1 )  <->  ( P  =/=  1  /\  ( P  e.  NN  /\  {
n  e.  NN  |  n  ||  P }  C_  { 1 ,  P }
) ) )
23 anass 631 . . . 4  |-  ( ( ( P  =/=  1  /\  P  e.  NN )  /\  { n  e.  NN  |  n  ||  P }  C_  { 1 ,  P } )  <-> 
( P  =/=  1  /\  ( P  e.  NN  /\ 
{ n  e.  NN  |  n  ||  P }  C_ 
{ 1 ,  P } ) ) )
2422, 23bitr4i 244 . . 3  |-  ( ( ( P  e.  NN  /\ 
{ n  e.  NN  |  n  ||  P }  C_ 
{ 1 ,  P } )  /\  P  =/=  1 )  <->  ( ( P  =/=  1  /\  P  e.  NN )  /\  {
n  e.  NN  |  n  ||  P }  C_  { 1 ,  P }
) )
25 ancom 438 . . . . 5  |-  ( ( P  =/=  1  /\  P  e.  NN )  <-> 
( P  e.  NN  /\  P  =/=  1 ) )
26 eluz2b3 10482 . . . . 5  |-  ( P  e.  ( ZZ>= `  2
)  <->  ( P  e.  NN  /\  P  =/=  1 ) )
2725, 26bitr4i 244 . . . 4  |-  ( ( P  =/=  1  /\  P  e.  NN )  <-> 
P  e.  ( ZZ>= ` 
2 ) )
2827anbi1i 677 . . 3  |-  ( ( ( P  =/=  1  /\  P  e.  NN )  /\  { n  e.  NN  |  n  ||  P }  C_  { 1 ,  P } )  <-> 
( P  e.  (
ZZ>= `  2 )  /\  { n  e.  NN  |  n  ||  P }  C_  { 1 ,  P }
) )
29 dfss2 3281 . . . . 5  |-  ( { n  e.  NN  |  n  ||  P }  C_  { 1 ,  P }  <->  A. z ( z  e. 
{ n  e.  NN  |  n  ||  P }  ->  z  e.  { 1 ,  P } ) )
30 breq1 4157 . . . . . . . . . 10  |-  ( n  =  z  ->  (
n  ||  P  <->  z  ||  P ) )
3130elrab 3036 . . . . . . . . 9  |-  ( z  e.  { n  e.  NN  |  n  ||  P }  <->  ( z  e.  NN  /\  z  ||  P ) )
32 vex 2903 . . . . . . . . . 10  |-  z  e. 
_V
3332elpr 3776 . . . . . . . . 9  |-  ( z  e.  { 1 ,  P }  <->  ( z  =  1  \/  z  =  P ) )
3431, 33imbi12i 317 . . . . . . . 8  |-  ( ( z  e.  { n  e.  NN  |  n  ||  P }  ->  z  e. 
{ 1 ,  P } )  <->  ( (
z  e.  NN  /\  z  ||  P )  -> 
( z  =  1  \/  z  =  P ) ) )
35 impexp 434 . . . . . . . 8  |-  ( ( ( z  e.  NN  /\  z  ||  P )  ->  ( z  =  1  \/  z  =  P ) )  <->  ( z  e.  NN  ->  ( z  ||  P  ->  ( z  =  1  \/  z  =  P ) ) ) )
3634, 35bitri 241 . . . . . . 7  |-  ( ( z  e.  { n  e.  NN  |  n  ||  P }  ->  z  e. 
{ 1 ,  P } )  <->  ( z  e.  NN  ->  ( z  ||  P  ->  ( z  =  1  \/  z  =  P ) ) ) )
3736albii 1572 . . . . . 6  |-  ( A. z ( z  e. 
{ n  e.  NN  |  n  ||  P }  ->  z  e.  { 1 ,  P } )  <->  A. z ( z  e.  NN  ->  ( z  ||  P  ->  ( z  =  1  \/  z  =  P ) ) ) )
38 df-ral 2655 . . . . . 6  |-  ( A. z  e.  NN  (
z  ||  P  ->  ( z  =  1  \/  z  =  P ) )  <->  A. z ( z  e.  NN  ->  (
z  ||  P  ->  ( z  =  1  \/  z  =  P ) ) ) )
3937, 38bitr4i 244 . . . . 5  |-  ( A. z ( z  e. 
{ n  e.  NN  |  n  ||  P }  ->  z  e.  { 1 ,  P } )  <->  A. z  e.  NN  ( z  ||  P  ->  ( z  =  1  \/  z  =  P ) ) )
4029, 39bitri 241 . . . 4  |-  ( { n  e.  NN  |  n  ||  P }  C_  { 1 ,  P }  <->  A. z  e.  NN  (
z  ||  P  ->  ( z  =  1  \/  z  =  P ) ) )
4140anbi2i 676 . . 3  |-  ( ( P  e.  ( ZZ>= ` 
2 )  /\  {
n  e.  NN  |  n  ||  P }  C_  { 1 ,  P }
)  <->  ( P  e.  ( ZZ>= `  2 )  /\  A. z  e.  NN  ( z  ||  P  ->  ( z  =  1  \/  z  =  P ) ) ) )
4224, 28, 413bitri 263 . 2  |-  ( ( ( P  e.  NN  /\ 
{ n  e.  NN  |  n  ||  P }  C_ 
{ 1 ,  P } )  /\  P  =/=  1 )  <->  ( P  e.  ( ZZ>= `  2 )  /\  A. z  e.  NN  ( z  ||  P  ->  ( z  =  1  \/  z  =  P ) ) ) )
436, 21, 423bitri 263 1  |-  ( P  e.  Prime  <->  ( P  e.  ( ZZ>= `  2 )  /\  A. z  e.  NN  ( z  ||  P  ->  ( z  =  1  \/  z  =  P ) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    \/ wo 358    /\ wa 359   A.wal 1546    = wceq 1649    e. wcel 1717    =/= wne 2551   A.wral 2650   {crab 2654    C_ wss 3264   {cpr 3759   class class class wbr 4154   ` cfv 5395   2oc2o 6655    ~~ cen 7043   1c1 8925   NNcn 9933   2c2 9982   ZZ>=cuz 10421    || cdivides 12780   Primecprime 13007
This theorem is referenced by:  isprm3  13016  isprm4  13017  dvdsprime  13020  coprm  13028  isprm6  13037  prmirredlem  16697  znidomb  16766  perfectlem2  20882
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369  ax-sep 4272  ax-nul 4280  ax-pow 4319  ax-pr 4345  ax-un 4642  ax-cnex 8980  ax-resscn 8981  ax-1cn 8982  ax-icn 8983  ax-addcl 8984  ax-addrcl 8985  ax-mulcl 8986  ax-mulrcl 8987  ax-mulcom 8988  ax-addass 8989  ax-mulass 8990  ax-distr 8991  ax-i2m1 8992  ax-1ne0 8993  ax-1rid 8994  ax-rnegex 8995  ax-rrecex 8996  ax-cnre 8997  ax-pre-lttri 8998  ax-pre-lttrn 8999  ax-pre-ltadd 9000  ax-pre-mulgt0 9001
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2243  df-mo 2244  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ne 2553  df-nel 2554  df-ral 2655  df-rex 2656  df-reu 2657  df-rab 2659  df-v 2902  df-sbc 3106  df-csb 3196  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-pss 3280  df-nul 3573  df-if 3684  df-pw 3745  df-sn 3764  df-pr 3765  df-tp 3766  df-op 3767  df-uni 3959  df-int 3994  df-iun 4038  df-br 4155  df-opab 4209  df-mpt 4210  df-tr 4245  df-eprel 4436  df-id 4440  df-po 4445  df-so 4446  df-fr 4483  df-we 4485  df-ord 4526  df-on 4527  df-lim 4528  df-suc 4529  df-om 4787  df-xp 4825  df-rel 4826  df-cnv 4827  df-co 4828  df-dm 4829  df-rn 4830  df-res 4831  df-ima 4832  df-iota 5359  df-fun 5397  df-fn 5398  df-f 5399  df-f1 5400  df-fo 5401  df-f1o 5402  df-fv 5403  df-ov 6024  df-oprab 6025  df-mpt2 6026  df-riota 6486  df-recs 6570  df-rdg 6605  df-1o 6661  df-2o 6662  df-oadd 6665  df-er 6842  df-en 7047  df-dom 7048  df-sdom 7049  df-fin 7050  df-pnf 9056  df-mnf 9057  df-xr 9058  df-ltxr 9059  df-le 9060  df-sub 9226  df-neg 9227  df-nn 9934  df-2 9991  df-n0 10155  df-z 10216  df-uz 10422  df-dvds 12781  df-prm 13008
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