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Theorem isprm4 11707
Description: The predicate "is a prime number". A prime number is an integer greater than or equal to 2 whose only divisor greater than or equal to 2 is itself. (Contributed by Paul Chapman, 26-Oct-2012.)
Assertion
Ref Expression
isprm4  |-  ( P  e.  Prime  <->  ( P  e.  ( ZZ>= `  2 )  /\  A. z  e.  (
ZZ>= `  2 ) ( z  ||  P  -> 
z  =  P ) ) )
Distinct variable group:    z, P

Proof of Theorem isprm4
StepHypRef Expression
1 isprm2 11705 . 2  |-  ( P  e.  Prime  <->  ( P  e.  ( ZZ>= `  2 )  /\  A. z  e.  NN  ( z  ||  P  ->  ( z  =  1  \/  z  =  P ) ) ) )
2 eluz2nn 9316 . . . . . . . 8  |-  ( z  e.  ( ZZ>= `  2
)  ->  z  e.  NN )
32pm4.71ri 387 . . . . . . 7  |-  ( z  e.  ( ZZ>= `  2
)  <->  ( z  e.  NN  /\  z  e.  ( ZZ>= `  2 )
) )
43imbi1i 237 . . . . . 6  |-  ( ( z  e.  ( ZZ>= ` 
2 )  ->  (
z  ||  P  ->  z  =  P ) )  <-> 
( ( z  e.  NN  /\  z  e.  ( ZZ>= `  2 )
)  ->  ( z  ||  P  ->  z  =  P ) ) )
5 impexp 261 . . . . . 6  |-  ( ( ( z  e.  NN  /\  z  e.  ( ZZ>= ` 
2 ) )  -> 
( z  ||  P  ->  z  =  P ) )  <->  ( z  e.  NN  ->  ( z  e.  ( ZZ>= `  2 )  ->  ( z  ||  P  ->  z  =  P ) ) ) )
64, 5bitri 183 . . . . 5  |-  ( ( z  e.  ( ZZ>= ` 
2 )  ->  (
z  ||  P  ->  z  =  P ) )  <-> 
( z  e.  NN  ->  ( z  e.  (
ZZ>= `  2 )  -> 
( z  ||  P  ->  z  =  P ) ) ) )
7 eluz2b3 9350 . . . . . . . 8  |-  ( z  e.  ( ZZ>= `  2
)  <->  ( z  e.  NN  /\  z  =/=  1 ) )
87imbi1i 237 . . . . . . 7  |-  ( ( z  e.  ( ZZ>= ` 
2 )  ->  (
z  ||  P  ->  z  =  P ) )  <-> 
( ( z  e.  NN  /\  z  =/=  1 )  ->  (
z  ||  P  ->  z  =  P ) ) )
9 impexp 261 . . . . . . . 8  |-  ( ( ( z  e.  NN  /\  z  =/=  1 )  ->  ( z  ||  P  ->  z  =  P ) )  <->  ( z  e.  NN  ->  ( z  =/=  1  ->  ( z 
||  P  ->  z  =  P ) ) ) )
10 bi2.04 247 . . . . . . . . . 10  |-  ( ( z  =/=  1  -> 
( z  ||  P  ->  z  =  P ) )  <->  ( z  ||  P  ->  ( z  =/=  1  ->  z  =  P ) ) )
11 nnz 9027 . . . . . . . . . . . . . 14  |-  ( z  e.  NN  ->  z  e.  ZZ )
12 1zzd 9035 . . . . . . . . . . . . . 14  |-  ( z  e.  NN  ->  1  e.  ZZ )
13 zdceq 9080 . . . . . . . . . . . . . 14  |-  ( ( z  e.  ZZ  /\  1  e.  ZZ )  -> DECID  z  =  1 )
1411, 12, 13syl2anc 406 . . . . . . . . . . . . 13  |-  ( z  e.  NN  -> DECID  z  =  1
)
15 dfordc 860 . . . . . . . . . . . . 13  |-  (DECID  z  =  1  ->  ( (
z  =  1  \/  z  =  P )  <-> 
( -.  z  =  1  ->  z  =  P ) ) )
1614, 15syl 14 . . . . . . . . . . . 12  |-  ( z  e.  NN  ->  (
( z  =  1  \/  z  =  P )  <->  ( -.  z  =  1  ->  z  =  P ) ) )
17 df-ne 2284 . . . . . . . . . . . . 13  |-  ( z  =/=  1  <->  -.  z  =  1 )
1817imbi1i 237 . . . . . . . . . . . 12  |-  ( ( z  =/=  1  -> 
z  =  P )  <-> 
( -.  z  =  1  ->  z  =  P ) )
1916, 18syl6rbbr 198 . . . . . . . . . . 11  |-  ( z  e.  NN  ->  (
( z  =/=  1  ->  z  =  P )  <-> 
( z  =  1  \/  z  =  P ) ) )
2019imbi2d 229 . . . . . . . . . 10  |-  ( z  e.  NN  ->  (
( z  ||  P  ->  ( z  =/=  1  ->  z  =  P ) )  <->  ( z  ||  P  ->  ( z  =  1  \/  z  =  P ) ) ) )
2110, 20syl5bb 191 . . . . . . . . 9  |-  ( z  e.  NN  ->  (
( z  =/=  1  ->  ( z  ||  P  ->  z  =  P ) )  <->  ( z  ||  P  ->  ( z  =  1  \/  z  =  P ) ) ) )
2221imbi2d 229 . . . . . . . 8  |-  ( z  e.  NN  ->  (
( z  e.  NN  ->  ( z  =/=  1  ->  ( z  ||  P  ->  z  =  P ) ) )  <->  ( z  e.  NN  ->  ( z  ||  P  ->  ( z  =  1  \/  z  =  P ) ) ) ) )
239, 22syl5bb 191 . . . . . . 7  |-  ( z  e.  NN  ->  (
( ( z  e.  NN  /\  z  =/=  1 )  ->  (
z  ||  P  ->  z  =  P ) )  <-> 
( z  e.  NN  ->  ( z  ||  P  ->  ( z  =  1  \/  z  =  P ) ) ) ) )
248, 23syl5bb 191 . . . . . 6  |-  ( z  e.  NN  ->  (
( z  e.  (
ZZ>= `  2 )  -> 
( z  ||  P  ->  z  =  P ) )  <->  ( z  e.  NN  ->  ( z  ||  P  ->  ( z  =  1  \/  z  =  P ) ) ) ) )
2524pm5.74i 179 . . . . 5  |-  ( ( z  e.  NN  ->  ( z  e.  ( ZZ>= ` 
2 )  ->  (
z  ||  P  ->  z  =  P ) ) )  <->  ( z  e.  NN  ->  ( z  e.  NN  ->  ( z  ||  P  ->  ( z  =  1  \/  z  =  P ) ) ) ) )
26 pm5.4 248 . . . . 5  |-  ( ( z  e.  NN  ->  ( z  e.  NN  ->  ( z  ||  P  -> 
( z  =  1  \/  z  =  P ) ) ) )  <-> 
( z  e.  NN  ->  ( z  ||  P  ->  ( z  =  1  \/  z  =  P ) ) ) )
276, 25, 263bitri 205 . . . 4  |-  ( ( z  e.  ( ZZ>= ` 
2 )  ->  (
z  ||  P  ->  z  =  P ) )  <-> 
( z  e.  NN  ->  ( z  ||  P  ->  ( z  =  1  \/  z  =  P ) ) ) )
2827ralbii2 2420 . . 3  |-  ( A. z  e.  ( ZZ>= ` 
2 ) ( z 
||  P  ->  z  =  P )  <->  A. z  e.  NN  ( z  ||  P  ->  ( z  =  1  \/  z  =  P ) ) )
2928anbi2i 450 . 2  |-  ( ( P  e.  ( ZZ>= ` 
2 )  /\  A. z  e.  ( ZZ>= ` 
2 ) ( z 
||  P  ->  z  =  P ) )  <->  ( P  e.  ( ZZ>= `  2 )  /\  A. z  e.  NN  ( z  ||  P  ->  ( z  =  1  \/  z  =  P ) ) ) )
301, 29bitr4i 186 1  |-  ( P  e.  Prime  <->  ( P  e.  ( ZZ>= `  2 )  /\  A. z  e.  (
ZZ>= `  2 ) ( z  ||  P  -> 
z  =  P ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 103    <-> wb 104    \/ wo 680  DECID wdc 802    = wceq 1314    e. wcel 1463    =/= wne 2283   A.wral 2391   class class class wbr 3897   ` cfv 5091   1c1 7585   NNcn 8680   2c2 8731   ZZcz 9008   ZZ>=cuz 9278    || cdvds 11400   Primecprime 11695
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 586  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-13 1474  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-coll 4011  ax-sep 4014  ax-nul 4022  ax-pow 4066  ax-pr 4099  ax-un 4323  ax-setind 4420  ax-iinf 4470  ax-cnex 7675  ax-resscn 7676  ax-1cn 7677  ax-1re 7678  ax-icn 7679  ax-addcl 7680  ax-addrcl 7681  ax-mulcl 7682  ax-mulrcl 7683  ax-addcom 7684  ax-mulcom 7685  ax-addass 7686  ax-mulass 7687  ax-distr 7688  ax-i2m1 7689  ax-0lt1 7690  ax-1rid 7691  ax-0id 7692  ax-rnegex 7693  ax-precex 7694  ax-cnre 7695  ax-pre-ltirr 7696  ax-pre-ltwlin 7697  ax-pre-lttrn 7698  ax-pre-apti 7699  ax-pre-ltadd 7700  ax-pre-mulgt0 7701  ax-pre-mulext 7702  ax-arch 7703  ax-caucvg 7704
This theorem depends on definitions:  df-bi 116  df-dc 803  df-3or 946  df-3an 947  df-tru 1317  df-fal 1320  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ne 2284  df-nel 2379  df-ral 2396  df-rex 2397  df-reu 2398  df-rmo 2399  df-rab 2400  df-v 2660  df-sbc 2881  df-csb 2974  df-dif 3041  df-un 3043  df-in 3045  df-ss 3052  df-nul 3332  df-if 3443  df-pw 3480  df-sn 3501  df-pr 3502  df-op 3504  df-uni 3705  df-int 3740  df-iun 3783  df-br 3898  df-opab 3958  df-mpt 3959  df-tr 3995  df-id 4183  df-po 4186  df-iso 4187  df-iord 4256  df-on 4258  df-ilim 4259  df-suc 4261  df-iom 4473  df-xp 4513  df-rel 4514  df-cnv 4515  df-co 4516  df-dm 4517  df-rn 4518  df-res 4519  df-ima 4520  df-iota 5056  df-fun 5093  df-fn 5094  df-f 5095  df-f1 5096  df-fo 5097  df-f1o 5098  df-fv 5099  df-riota 5696  df-ov 5743  df-oprab 5744  df-mpo 5745  df-1st 6004  df-2nd 6005  df-recs 6168  df-frec 6254  df-1o 6279  df-2o 6280  df-er 6395  df-en 6601  df-pnf 7766  df-mnf 7767  df-xr 7768  df-ltxr 7769  df-le 7770  df-sub 7899  df-neg 7900  df-reap 8300  df-ap 8307  df-div 8396  df-inn 8681  df-2 8739  df-3 8740  df-4 8741  df-n0 8932  df-z 9009  df-uz 9279  df-q 9364  df-rp 9394  df-seqfrec 10170  df-exp 10244  df-cj 10565  df-re 10566  df-im 10567  df-rsqrt 10721  df-abs 10722  df-dvds 11401  df-prm 11696
This theorem is referenced by:  nprm  11711  prmuz2  11718  dvdsprm  11724
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