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Theorem isprm2 12782
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 12779 . . . . 5  |-  -.  1  e.  Prime
2 eleq1 2356 . . . . . 6  |-  ( P  =  1  ->  ( P  e.  Prime  <->  1  e.  Prime ) )
32biimpcd 215 . . . . 5  |-  ( P  e.  Prime  ->  ( P  =  1  ->  1  e.  Prime ) )
41, 3mtoi 169 . . . 4  |-  ( P  e.  Prime  ->  -.  P  =  1 )
5 df-ne 2461 . . . 4  |-  ( P  =/=  1  <->  -.  P  =  1 )
64, 5sylibr 203 . . 3  |-  ( P  e.  Prime  ->  P  =/=  1 )
76pm4.71i 613 . 2  |-  ( P  e.  Prime  <->  ( P  e. 
Prime  /\  P  =/=  1
) )
8 isprm 12776 . . . 4  |-  ( P  e.  Prime  <->  ( P  e.  NN  /\  { n  e.  NN  |  n  ||  P }  ~~  2o ) )
9 isprm2lem 12781 . . . . . . 7  |-  ( ( P  e.  NN  /\  P  =/=  1 )  -> 
( { n  e.  NN  |  n  ||  P }  ~~  2o  <->  { n  e.  NN  |  n  ||  P }  =  {
1 ,  P }
) )
10 eqss 3207 . . . . . . . . . . 11  |-  ( { n  e.  NN  |  n  ||  P }  =  { 1 ,  P } 
<->  ( { n  e.  NN  |  n  ||  P }  C_  { 1 ,  P }  /\  { 1 ,  P }  C_ 
{ n  e.  NN  |  n  ||  P }
) )
1110imbi2i 303 . . . . . . . . . 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 } ) ) )
12 1idssfct 12780 . . . . . . . . . . 11  |-  ( P  e.  NN  ->  { 1 ,  P }  C_  { n  e.  NN  |  n  ||  P } )
13 jcab 833 . . . . . . . . . . 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 } ) ) )
1412, 13mpbiran2 885 . . . . . . . . . 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 } ) )
1511, 14bitri 240 . . . . . . . . 9  |-  ( ( P  e.  NN  ->  { n  e.  NN  |  n  ||  P }  =  { 1 ,  P } )  <->  ( P  e.  NN  ->  { n  e.  NN  |  n  ||  P }  C_  { 1 ,  P } ) )
1615pm5.74ri 237 . . . . . . . 8  |-  ( P  e.  NN  ->  ( { n  e.  NN  |  n  ||  P }  =  { 1 ,  P } 
<->  { n  e.  NN  |  n  ||  P }  C_ 
{ 1 ,  P } ) )
1716adantr 451 . . . . . . 7  |-  ( ( P  e.  NN  /\  P  =/=  1 )  -> 
( { n  e.  NN  |  n  ||  P }  =  {
1 ,  P }  <->  { n  e.  NN  |  n  ||  P }  C_  { 1 ,  P }
) )
189, 17bitrd 244 . . . . . 6  |-  ( ( P  e.  NN  /\  P  =/=  1 )  -> 
( { n  e.  NN  |  n  ||  P }  ~~  2o  <->  { n  e.  NN  |  n  ||  P }  C_  { 1 ,  P } ) )
1918expcom 424 . . . . 5  |-  ( P  =/=  1  ->  ( P  e.  NN  ->  ( { n  e.  NN  |  n  ||  P }  ~~  2o  <->  { n  e.  NN  |  n  ||  P }  C_ 
{ 1 ,  P } ) ) )
2019pm5.32d 620 . . . 4  |-  ( P  =/=  1  ->  (
( P  e.  NN  /\ 
{ n  e.  NN  |  n  ||  P }  ~~  2o )  <->  ( P  e.  NN  /\  { n  e.  NN  |  n  ||  P }  C_  { 1 ,  P } ) ) )
218, 20syl5bb 248 . . 3  |-  ( P  =/=  1  ->  ( P  e.  Prime  <->  ( P  e.  NN  /\  { n  e.  NN  |  n  ||  P }  C_  { 1 ,  P } ) ) )
2221pm5.32ri 619 . 2  |-  ( ( P  e.  Prime  /\  P  =/=  1 )  <->  ( ( P  e.  NN  /\  {
n  e.  NN  |  n  ||  P }  C_  { 1 ,  P }
)  /\  P  =/=  1 ) )
23 ancom 437 . . . 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 }
) ) )
24 anass 630 . . . 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 } ) ) )
2523, 24bitr4i 243 . . 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 }
) )
26 ancom 437 . . . . 5  |-  ( ( P  =/=  1  /\  P  e.  NN )  <-> 
( P  e.  NN  /\  P  =/=  1 ) )
27 eluz2b3 10307 . . . . 5  |-  ( P  e.  ( ZZ>= `  2
)  <->  ( P  e.  NN  /\  P  =/=  1 ) )
2826, 27bitr4i 243 . . . 4  |-  ( ( P  =/=  1  /\  P  e.  NN )  <-> 
P  e.  ( ZZ>= ` 
2 ) )
2928anbi1i 676 . . 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 }
) )
30 dfss2 3182 . . . . 5  |-  ( { n  e.  NN  |  n  ||  P }  C_  { 1 ,  P }  <->  A. z ( z  e. 
{ n  e.  NN  |  n  ||  P }  ->  z  e.  { 1 ,  P } ) )
31 breq1 4042 . . . . . . . . . 10  |-  ( n  =  z  ->  (
n  ||  P  <->  z  ||  P ) )
3231elrab 2936 . . . . . . . . 9  |-  ( z  e.  { n  e.  NN  |  n  ||  P }  <->  ( z  e.  NN  /\  z  ||  P ) )
33 vex 2804 . . . . . . . . . 10  |-  z  e. 
_V
3433elpr 3671 . . . . . . . . 9  |-  ( z  e.  { 1 ,  P }  <->  ( z  =  1  \/  z  =  P ) )
3532, 34imbi12i 316 . . . . . . . 8  |-  ( ( z  e.  { n  e.  NN  |  n  ||  P }  ->  z  e. 
{ 1 ,  P } )  <->  ( (
z  e.  NN  /\  z  ||  P )  -> 
( z  =  1  \/  z  =  P ) ) )
36 impexp 433 . . . . . . . 8  |-  ( ( ( z  e.  NN  /\  z  ||  P )  ->  ( z  =  1  \/  z  =  P ) )  <->  ( z  e.  NN  ->  ( z  ||  P  ->  ( z  =  1  \/  z  =  P ) ) ) )
3735, 36bitri 240 . . . . . . 7  |-  ( ( z  e.  { n  e.  NN  |  n  ||  P }  ->  z  e. 
{ 1 ,  P } )  <->  ( z  e.  NN  ->  ( z  ||  P  ->  ( z  =  1  \/  z  =  P ) ) ) )
3837albii 1556 . . . . . 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 ) ) ) )
39 df-ral 2561 . . . . . 6  |-  ( A. z  e.  NN  (
z  ||  P  ->  ( z  =  1  \/  z  =  P ) )  <->  A. z ( z  e.  NN  ->  (
z  ||  P  ->  ( z  =  1  \/  z  =  P ) ) ) )
4038, 39bitr4i 243 . . . . 5  |-  ( A. z ( z  e. 
{ n  e.  NN  |  n  ||  P }  ->  z  e.  { 1 ,  P } )  <->  A. z  e.  NN  ( z  ||  P  ->  ( z  =  1  \/  z  =  P ) ) )
4130, 40bitri 240 . . . 4  |-  ( { n  e.  NN  |  n  ||  P }  C_  { 1 ,  P }  <->  A. z  e.  NN  (
z  ||  P  ->  ( z  =  1  \/  z  =  P ) ) )
4241anbi2i 675 . . 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 ) ) ) )
4325, 29, 423bitri 262 . 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 ) ) ) )
447, 22, 433bitri 262 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:   -. wn 3    -> wi 4    <-> wb 176    \/ wo 357    /\ wa 358   A.wal 1530    = wceq 1632    e. wcel 1696    =/= wne 2459   A.wral 2556   {crab 2560    C_ wss 3165   {cpr 3654   class class class wbr 4039   ` cfv 5271   2oc2o 6489    ~~ cen 6876   1c1 8754   NNcn 9762   2c2 9811   ZZ>=cuz 10246    || cdivides 12547   Primecprime 12774
This theorem is referenced by:  isprm3  12783  isprm4  12784  dvdsprime  12787  coprm  12795  isprm6  12804  prmirredlem  16462  znidomb  16531  perfectlem2  20485
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-riota 6320  df-recs 6404  df-rdg 6439  df-1o 6495  df-2o 6496  df-oadd 6499  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-nn 9763  df-2 9820  df-n0 9982  df-z 10041  df-uz 10247  df-dvds 12548  df-prm 12775
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