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Theorem isprm4 12768
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 12766 . 2  |-  ( P  e.  Prime  <->  ( P  e.  ( ZZ>= `  2 )  /\  A. z  e.  NN  ( z  ||  P  ->  ( z  =  1  \/  z  =  P ) ) ) )
2 eluz2b3 10291 . . . . . 6  |-  ( z  e.  ( ZZ>= `  2
)  <->  ( z  e.  NN  /\  z  =/=  1 ) )
32imbi1i 315 . . . . 5  |-  ( ( z  e.  ( ZZ>= ` 
2 )  ->  (
z  ||  P  ->  z  =  P ) )  <-> 
( ( z  e.  NN  /\  z  =/=  1 )  ->  (
z  ||  P  ->  z  =  P ) ) )
4 impexp 433 . . . . . 6  |-  ( ( ( z  e.  NN  /\  z  =/=  1 )  ->  ( z  ||  P  ->  z  =  P ) )  <->  ( z  e.  NN  ->  ( z  =/=  1  ->  ( z 
||  P  ->  z  =  P ) ) ) )
5 bi2.04 350 . . . . . . . 8  |-  ( ( z  =/=  1  -> 
( z  ||  P  ->  z  =  P ) )  <->  ( z  ||  P  ->  ( z  =/=  1  ->  z  =  P ) ) )
6 df-ne 2448 . . . . . . . . . . 11  |-  ( z  =/=  1  <->  -.  z  =  1 )
76imbi1i 315 . . . . . . . . . 10  |-  ( ( z  =/=  1  -> 
z  =  P )  <-> 
( -.  z  =  1  ->  z  =  P ) )
8 df-or 359 . . . . . . . . . 10  |-  ( ( z  =  1  \/  z  =  P )  <-> 
( -.  z  =  1  ->  z  =  P ) )
97, 8bitr4i 243 . . . . . . . . 9  |-  ( ( z  =/=  1  -> 
z  =  P )  <-> 
( z  =  1  \/  z  =  P ) )
109imbi2i 303 . . . . . . . 8  |-  ( ( z  ||  P  -> 
( z  =/=  1  ->  z  =  P ) )  <->  ( z  ||  P  ->  ( z  =  1  \/  z  =  P ) ) )
115, 10bitri 240 . . . . . . 7  |-  ( ( z  =/=  1  -> 
( z  ||  P  ->  z  =  P ) )  <->  ( z  ||  P  ->  ( z  =  1  \/  z  =  P ) ) )
1211imbi2i 303 . . . . . 6  |-  ( ( z  e.  NN  ->  ( z  =/=  1  -> 
( z  ||  P  ->  z  =  P ) ) )  <->  ( z  e.  NN  ->  ( z  ||  P  ->  ( z  =  1  \/  z  =  P ) ) ) )
134, 12bitri 240 . . . . 5  |-  ( ( ( z  e.  NN  /\  z  =/=  1 )  ->  ( z  ||  P  ->  z  =  P ) )  <->  ( z  e.  NN  ->  ( z  ||  P  ->  ( z  =  1  \/  z  =  P ) ) ) )
143, 13bitri 240 . . . 4  |-  ( ( z  e.  ( ZZ>= ` 
2 )  ->  (
z  ||  P  ->  z  =  P ) )  <-> 
( z  e.  NN  ->  ( z  ||  P  ->  ( z  =  1  \/  z  =  P ) ) ) )
1514ralbii2 2571 . . 3  |-  ( A. z  e.  ( ZZ>= ` 
2 ) ( z 
||  P  ->  z  =  P )  <->  A. z  e.  NN  ( z  ||  P  ->  ( z  =  1  \/  z  =  P ) ) )
1615anbi2i 675 . 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 ) ) ) )
171, 16bitr4i 243 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    <-> wb 176    \/ wo 357    /\ wa 358    = wceq 1623    e. wcel 1684    =/= wne 2446   A.wral 2543   class class class wbr 4023   ` cfv 5255   1c1 8738   NNcn 9746   2c2 9795   ZZ>=cuz 10230    || cdivides 12531   Primecprime 12758
This theorem is referenced by:  nprm  12772  prmuz2  12776  dvdsprm  12778  isprm5  12791
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814
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 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-2o 6480  df-oadd 6483  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-nn 9747  df-2 9804  df-n0 9966  df-z 10025  df-uz 10231  df-dvds 12532  df-prm 12759
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