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Theorem isprm4 11366
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 11364 . 2  |-  ( P  e.  Prime  <->  ( P  e.  ( ZZ>= `  2 )  /\  A. z  e.  NN  ( z  ||  P  ->  ( z  =  1  \/  z  =  P ) ) ) )
2 eluz2nn 9047 . . . . . . . 8  |-  ( z  e.  ( ZZ>= `  2
)  ->  z  e.  NN )
32pm4.71ri 384 . . . . . . 7  |-  ( z  e.  ( ZZ>= `  2
)  <->  ( z  e.  NN  /\  z  e.  ( ZZ>= `  2 )
) )
43imbi1i 236 . . . . . 6  |-  ( ( z  e.  ( ZZ>= ` 
2 )  ->  (
z  ||  P  ->  z  =  P ) )  <-> 
( ( z  e.  NN  /\  z  e.  ( ZZ>= `  2 )
)  ->  ( z  ||  P  ->  z  =  P ) ) )
5 impexp 259 . . . . . 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 182 . . . . 5  |-  ( ( z  e.  ( ZZ>= ` 
2 )  ->  (
z  ||  P  ->  z  =  P ) )  <-> 
( z  e.  NN  ->  ( z  e.  (
ZZ>= `  2 )  -> 
( z  ||  P  ->  z  =  P ) ) ) )
7 eluz2b3 9081 . . . . . . . 8  |-  ( z  e.  ( ZZ>= `  2
)  <->  ( z  e.  NN  /\  z  =/=  1 ) )
87imbi1i 236 . . . . . . 7  |-  ( ( z  e.  ( ZZ>= ` 
2 )  ->  (
z  ||  P  ->  z  =  P ) )  <-> 
( ( z  e.  NN  /\  z  =/=  1 )  ->  (
z  ||  P  ->  z  =  P ) ) )
9 impexp 259 . . . . . . . 8  |-  ( ( ( z  e.  NN  /\  z  =/=  1 )  ->  ( z  ||  P  ->  z  =  P ) )  <->  ( z  e.  NN  ->  ( z  =/=  1  ->  ( z 
||  P  ->  z  =  P ) ) ) )
10 bi2.04 246 . . . . . . . . . 10  |-  ( ( z  =/=  1  -> 
( z  ||  P  ->  z  =  P ) )  <->  ( z  ||  P  ->  ( z  =/=  1  ->  z  =  P ) ) )
11 nnz 8759 . . . . . . . . . . . . . 14  |-  ( z  e.  NN  ->  z  e.  ZZ )
12 1zzd 8767 . . . . . . . . . . . . . 14  |-  ( z  e.  NN  ->  1  e.  ZZ )
13 zdceq 8812 . . . . . . . . . . . . . 14  |-  ( ( z  e.  ZZ  /\  1  e.  ZZ )  -> DECID  z  =  1 )
1411, 12, 13syl2anc 403 . . . . . . . . . . . . 13  |-  ( z  e.  NN  -> DECID  z  =  1
)
15 dfordc 829 . . . . . . . . . . . . 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 2256 . . . . . . . . . . . . 13  |-  ( z  =/=  1  <->  -.  z  =  1 )
1817imbi1i 236 . . . . . . . . . . . 12  |-  ( ( z  =/=  1  -> 
z  =  P )  <-> 
( -.  z  =  1  ->  z  =  P ) )
1916, 18syl6rbbr 197 . . . . . . . . . . 11  |-  ( z  e.  NN  ->  (
( z  =/=  1  ->  z  =  P )  <-> 
( z  =  1  \/  z  =  P ) ) )
2019imbi2d 228 . . . . . . . . . 10  |-  ( z  e.  NN  ->  (
( z  ||  P  ->  ( z  =/=  1  ->  z  =  P ) )  <->  ( z  ||  P  ->  ( z  =  1  \/  z  =  P ) ) ) )
2110, 20syl5bb 190 . . . . . . . . 9  |-  ( z  e.  NN  ->  (
( z  =/=  1  ->  ( z  ||  P  ->  z  =  P ) )  <->  ( z  ||  P  ->  ( z  =  1  \/  z  =  P ) ) ) )
2221imbi2d 228 . . . . . . . 8  |-  ( z  e.  NN  ->  (
( z  e.  NN  ->  ( z  =/=  1  ->  ( z  ||  P  ->  z  =  P ) ) )  <->  ( z  e.  NN  ->  ( z  ||  P  ->  ( z  =  1  \/  z  =  P ) ) ) ) )
239, 22syl5bb 190 . . . . . . 7  |-  ( z  e.  NN  ->  (
( ( z  e.  NN  /\  z  =/=  1 )  ->  (
z  ||  P  ->  z  =  P ) )  <-> 
( z  e.  NN  ->  ( z  ||  P  ->  ( z  =  1  \/  z  =  P ) ) ) ) )
248, 23syl5bb 190 . . . . . 6  |-  ( z  e.  NN  ->  (
( z  e.  (
ZZ>= `  2 )  -> 
( z  ||  P  ->  z  =  P ) )  <->  ( z  e.  NN  ->  ( z  ||  P  ->  ( z  =  1  \/  z  =  P ) ) ) ) )
2524pm5.74i 178 . . . . 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 pm5.4 247 . . . . 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 204 . . . 4  |-  ( ( z  e.  ( ZZ>= ` 
2 )  ->  (
z  ||  P  ->  z  =  P ) )  <-> 
( z  e.  NN  ->  ( z  ||  P  ->  ( z  =  1  \/  z  =  P ) ) ) )
2827ralbii2 2388 . . 3  |-  ( A. z  e.  ( ZZ>= ` 
2 ) ( z 
||  P  ->  z  =  P )  <->  A. z  e.  NN  ( z  ||  P  ->  ( z  =  1  \/  z  =  P ) ) )
2928anbi2i 445 . 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 185 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 102    <-> wb 103    \/ wo 664  DECID wdc 780    = wceq 1289    e. wcel 1438    =/= wne 2255   A.wral 2359   class class class wbr 3843   ` cfv 5010   1c1 7341   NNcn 8412   2c2 8463   ZZcz 8740   ZZ>=cuz 9009    || cdvds 11061   Primecprime 11354
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-coll 3952  ax-sep 3955  ax-nul 3963  ax-pow 4007  ax-pr 4034  ax-un 4258  ax-setind 4351  ax-iinf 4401  ax-cnex 7426  ax-resscn 7427  ax-1cn 7428  ax-1re 7429  ax-icn 7430  ax-addcl 7431  ax-addrcl 7432  ax-mulcl 7433  ax-mulrcl 7434  ax-addcom 7435  ax-mulcom 7436  ax-addass 7437  ax-mulass 7438  ax-distr 7439  ax-i2m1 7440  ax-0lt1 7441  ax-1rid 7442  ax-0id 7443  ax-rnegex 7444  ax-precex 7445  ax-cnre 7446  ax-pre-ltirr 7447  ax-pre-ltwlin 7448  ax-pre-lttrn 7449  ax-pre-apti 7450  ax-pre-ltadd 7451  ax-pre-mulgt0 7452  ax-pre-mulext 7453  ax-arch 7454  ax-caucvg 7455
This theorem depends on definitions:  df-bi 115  df-dc 781  df-3or 925  df-3an 926  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ne 2256  df-nel 2351  df-ral 2364  df-rex 2365  df-reu 2366  df-rmo 2367  df-rab 2368  df-v 2621  df-sbc 2841  df-csb 2934  df-dif 3001  df-un 3003  df-in 3005  df-ss 3012  df-nul 3287  df-if 3392  df-pw 3429  df-sn 3450  df-pr 3451  df-op 3453  df-uni 3652  df-int 3687  df-iun 3730  df-br 3844  df-opab 3898  df-mpt 3899  df-tr 3935  df-id 4118  df-po 4121  df-iso 4122  df-iord 4191  df-on 4193  df-ilim 4194  df-suc 4196  df-iom 4404  df-xp 4442  df-rel 4443  df-cnv 4444  df-co 4445  df-dm 4446  df-rn 4447  df-res 4448  df-ima 4449  df-iota 4975  df-fun 5012  df-fn 5013  df-f 5014  df-f1 5015  df-fo 5016  df-f1o 5017  df-fv 5018  df-riota 5600  df-ov 5647  df-oprab 5648  df-mpt2 5649  df-1st 5903  df-2nd 5904  df-recs 6062  df-frec 6148  df-1o 6173  df-2o 6174  df-er 6282  df-en 6448  df-pnf 7514  df-mnf 7515  df-xr 7516  df-ltxr 7517  df-le 7518  df-sub 7645  df-neg 7646  df-reap 8042  df-ap 8049  df-div 8130  df-inn 8413  df-2 8471  df-3 8472  df-4 8473  df-n0 8664  df-z 8741  df-uz 9010  df-q 9095  df-rp 9125  df-iseq 9841  df-seq3 9842  df-exp 9943  df-cj 10264  df-re 10265  df-im 10266  df-rsqrt 10419  df-abs 10420  df-dvds 11062  df-prm 11355
This theorem is referenced by:  nprm  11370  prmuz2  11377  dvdsprm  11383
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