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Theorem isprm4 12119
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 12117 . 2  |-  ( P  e.  Prime  <->  ( P  e.  ( ZZ>= `  2 )  /\  A. z  e.  NN  ( z  ||  P  ->  ( z  =  1  \/  z  =  P ) ) ) )
2 eluz2nn 9566 . . . . . . . 8  |-  ( z  e.  ( ZZ>= `  2
)  ->  z  e.  NN )
32pm4.71ri 392 . . . . . . 7  |-  ( z  e.  ( ZZ>= `  2
)  <->  ( z  e.  NN  /\  z  e.  ( ZZ>= `  2 )
) )
43imbi1i 238 . . . . . 6  |-  ( ( z  e.  ( ZZ>= ` 
2 )  ->  (
z  ||  P  ->  z  =  P ) )  <-> 
( ( z  e.  NN  /\  z  e.  ( ZZ>= `  2 )
)  ->  ( z  ||  P  ->  z  =  P ) ) )
5 impexp 263 . . . . . 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 184 . . . . 5  |-  ( ( z  e.  ( ZZ>= ` 
2 )  ->  (
z  ||  P  ->  z  =  P ) )  <-> 
( z  e.  NN  ->  ( z  e.  (
ZZ>= `  2 )  -> 
( z  ||  P  ->  z  =  P ) ) ) )
7 eluz2b3 9604 . . . . . . . 8  |-  ( z  e.  ( ZZ>= `  2
)  <->  ( z  e.  NN  /\  z  =/=  1 ) )
87imbi1i 238 . . . . . . 7  |-  ( ( z  e.  ( ZZ>= ` 
2 )  ->  (
z  ||  P  ->  z  =  P ) )  <-> 
( ( z  e.  NN  /\  z  =/=  1 )  ->  (
z  ||  P  ->  z  =  P ) ) )
9 impexp 263 . . . . . . . 8  |-  ( ( ( z  e.  NN  /\  z  =/=  1 )  ->  ( z  ||  P  ->  z  =  P ) )  <->  ( z  e.  NN  ->  ( z  =/=  1  ->  ( z 
||  P  ->  z  =  P ) ) ) )
10 bi2.04 248 . . . . . . . . . 10  |-  ( ( z  =/=  1  -> 
( z  ||  P  ->  z  =  P ) )  <->  ( z  ||  P  ->  ( z  =/=  1  ->  z  =  P ) ) )
11 df-ne 2348 . . . . . . . . . . . . 13  |-  ( z  =/=  1  <->  -.  z  =  1 )
1211imbi1i 238 . . . . . . . . . . . 12  |-  ( ( z  =/=  1  -> 
z  =  P )  <-> 
( -.  z  =  1  ->  z  =  P ) )
13 nnz 9272 . . . . . . . . . . . . . 14  |-  ( z  e.  NN  ->  z  e.  ZZ )
14 1zzd 9280 . . . . . . . . . . . . . 14  |-  ( z  e.  NN  ->  1  e.  ZZ )
15 zdceq 9328 . . . . . . . . . . . . . 14  |-  ( ( z  e.  ZZ  /\  1  e.  ZZ )  -> DECID  z  =  1 )
1613, 14, 15syl2anc 411 . . . . . . . . . . . . 13  |-  ( z  e.  NN  -> DECID  z  =  1
)
17 dfordc 892 . . . . . . . . . . . . 13  |-  (DECID  z  =  1  ->  ( (
z  =  1  \/  z  =  P )  <-> 
( -.  z  =  1  ->  z  =  P ) ) )
1816, 17syl 14 . . . . . . . . . . . 12  |-  ( z  e.  NN  ->  (
( z  =  1  \/  z  =  P )  <->  ( -.  z  =  1  ->  z  =  P ) ) )
1912, 18bitr4id 199 . . . . . . . . . . 11  |-  ( z  e.  NN  ->  (
( z  =/=  1  ->  z  =  P )  <-> 
( z  =  1  \/  z  =  P ) ) )
2019imbi2d 230 . . . . . . . . . 10  |-  ( z  e.  NN  ->  (
( z  ||  P  ->  ( z  =/=  1  ->  z  =  P ) )  <->  ( z  ||  P  ->  ( z  =  1  \/  z  =  P ) ) ) )
2110, 20bitrid 192 . . . . . . . . 9  |-  ( z  e.  NN  ->  (
( z  =/=  1  ->  ( z  ||  P  ->  z  =  P ) )  <->  ( z  ||  P  ->  ( z  =  1  \/  z  =  P ) ) ) )
2221imbi2d 230 . . . . . . . 8  |-  ( z  e.  NN  ->  (
( z  e.  NN  ->  ( z  =/=  1  ->  ( z  ||  P  ->  z  =  P ) ) )  <->  ( z  e.  NN  ->  ( z  ||  P  ->  ( z  =  1  \/  z  =  P ) ) ) ) )
239, 22bitrid 192 . . . . . . 7  |-  ( z  e.  NN  ->  (
( ( z  e.  NN  /\  z  =/=  1 )  ->  (
z  ||  P  ->  z  =  P ) )  <-> 
( z  e.  NN  ->  ( z  ||  P  ->  ( z  =  1  \/  z  =  P ) ) ) ) )
248, 23bitrid 192 . . . . . 6  |-  ( z  e.  NN  ->  (
( z  e.  (
ZZ>= `  2 )  -> 
( z  ||  P  ->  z  =  P ) )  <->  ( z  e.  NN  ->  ( z  ||  P  ->  ( z  =  1  \/  z  =  P ) ) ) ) )
2524pm5.74i 180 . . . . 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 pm5.4 249 . . . . 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 206 . . . 4  |-  ( ( z  e.  ( ZZ>= ` 
2 )  ->  (
z  ||  P  ->  z  =  P ) )  <-> 
( z  e.  NN  ->  ( z  ||  P  ->  ( z  =  1  \/  z  =  P ) ) ) )
2827ralbii2 2487 . . 3  |-  ( A. z  e.  ( ZZ>= ` 
2 ) ( z 
||  P  ->  z  =  P )  <->  A. z  e.  NN  ( z  ||  P  ->  ( z  =  1  \/  z  =  P ) ) )
2928anbi2i 457 . 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 187 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 104    <-> wb 105    \/ wo 708  DECID wdc 834    = wceq 1353    e. wcel 2148    =/= wne 2347   A.wral 2455   class class class wbr 4004   ` cfv 5217   1c1 7812   NNcn 8919   2c2 8970   ZZcz 9253   ZZ>=cuz 9528    || cdvds 11794   Primecprime 12107
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4119  ax-sep 4122  ax-nul 4130  ax-pow 4175  ax-pr 4210  ax-un 4434  ax-setind 4537  ax-iinf 4588  ax-cnex 7902  ax-resscn 7903  ax-1cn 7904  ax-1re 7905  ax-icn 7906  ax-addcl 7907  ax-addrcl 7908  ax-mulcl 7909  ax-mulrcl 7910  ax-addcom 7911  ax-mulcom 7912  ax-addass 7913  ax-mulass 7914  ax-distr 7915  ax-i2m1 7916  ax-0lt1 7917  ax-1rid 7918  ax-0id 7919  ax-rnegex 7920  ax-precex 7921  ax-cnre 7922  ax-pre-ltirr 7923  ax-pre-ltwlin 7924  ax-pre-lttrn 7925  ax-pre-apti 7926  ax-pre-ltadd 7927  ax-pre-mulgt0 7928  ax-pre-mulext 7929  ax-arch 7930  ax-caucvg 7931
This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rmo 2463  df-rab 2464  df-v 2740  df-sbc 2964  df-csb 3059  df-dif 3132  df-un 3134  df-in 3136  df-ss 3143  df-nul 3424  df-if 3536  df-pw 3578  df-sn 3599  df-pr 3600  df-op 3602  df-uni 3811  df-int 3846  df-iun 3889  df-br 4005  df-opab 4066  df-mpt 4067  df-tr 4103  df-id 4294  df-po 4297  df-iso 4298  df-iord 4367  df-on 4369  df-ilim 4370  df-suc 4372  df-iom 4591  df-xp 4633  df-rel 4634  df-cnv 4635  df-co 4636  df-dm 4637  df-rn 4638  df-res 4639  df-ima 4640  df-iota 5179  df-fun 5219  df-fn 5220  df-f 5221  df-f1 5222  df-fo 5223  df-f1o 5224  df-fv 5225  df-riota 5831  df-ov 5878  df-oprab 5879  df-mpo 5880  df-1st 6141  df-2nd 6142  df-recs 6306  df-frec 6392  df-1o 6417  df-2o 6418  df-er 6535  df-en 6741  df-pnf 7994  df-mnf 7995  df-xr 7996  df-ltxr 7997  df-le 7998  df-sub 8130  df-neg 8131  df-reap 8532  df-ap 8539  df-div 8630  df-inn 8920  df-2 8978  df-3 8979  df-4 8980  df-n0 9177  df-z 9254  df-uz 9529  df-q 9620  df-rp 9654  df-seqfrec 10446  df-exp 10520  df-cj 10851  df-re 10852  df-im 10853  df-rsqrt 11007  df-abs 11008  df-dvds 11795  df-prm 12108
This theorem is referenced by:  nprm  12123  prmuz2  12131  dvdsprm  12137
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