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Theorem isprm4 11995
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 11993 . 2  |-  ( P  e.  Prime  <->  ( P  e.  ( ZZ>= `  2 )  /\  A. z  e.  NN  ( z  ||  P  ->  ( z  =  1  \/  z  =  P ) ) ) )
2 eluz2nn 9477 . . . . . . . 8  |-  ( z  e.  ( ZZ>= `  2
)  ->  z  e.  NN )
32pm4.71ri 390 . . . . . . 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 9515 . . . . . . . 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 df-ne 2328 . . . . . . . . . . . . 13  |-  ( z  =/=  1  <->  -.  z  =  1 )
1211imbi1i 237 . . . . . . . . . . . 12  |-  ( ( z  =/=  1  -> 
z  =  P )  <-> 
( -.  z  =  1  ->  z  =  P ) )
13 nnz 9186 . . . . . . . . . . . . . 14  |-  ( z  e.  NN  ->  z  e.  ZZ )
14 1zzd 9194 . . . . . . . . . . . . . 14  |-  ( z  e.  NN  ->  1  e.  ZZ )
15 zdceq 9239 . . . . . . . . . . . . . 14  |-  ( ( z  e.  ZZ  /\  1  e.  ZZ )  -> DECID  z  =  1 )
1613, 14, 15syl2anc 409 . . . . . . . . . . . . 13  |-  ( z  e.  NN  -> DECID  z  =  1
)
17 dfordc 878 . . . . . . . . . . . . 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 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 2467 . . 3  |-  ( A. z  e.  ( ZZ>= ` 
2 ) ( z 
||  P  ->  z  =  P )  <->  A. z  e.  NN  ( z  ||  P  ->  ( z  =  1  \/  z  =  P ) ) )
2928anbi2i 453 . 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 698  DECID wdc 820    = wceq 1335    e. wcel 2128    =/= wne 2327   A.wral 2435   class class class wbr 3965   ` cfv 5170   1c1 7733   NNcn 8833   2c2 8884   ZZcz 9167   ZZ>=cuz 9439    || cdvds 11683   Primecprime 11983
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 604  ax-in2 605  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-13 2130  ax-14 2131  ax-ext 2139  ax-coll 4079  ax-sep 4082  ax-nul 4090  ax-pow 4135  ax-pr 4169  ax-un 4393  ax-setind 4496  ax-iinf 4547  ax-cnex 7823  ax-resscn 7824  ax-1cn 7825  ax-1re 7826  ax-icn 7827  ax-addcl 7828  ax-addrcl 7829  ax-mulcl 7830  ax-mulrcl 7831  ax-addcom 7832  ax-mulcom 7833  ax-addass 7834  ax-mulass 7835  ax-distr 7836  ax-i2m1 7837  ax-0lt1 7838  ax-1rid 7839  ax-0id 7840  ax-rnegex 7841  ax-precex 7842  ax-cnre 7843  ax-pre-ltirr 7844  ax-pre-ltwlin 7845  ax-pre-lttrn 7846  ax-pre-apti 7847  ax-pre-ltadd 7848  ax-pre-mulgt0 7849  ax-pre-mulext 7850  ax-arch 7851  ax-caucvg 7852
This theorem depends on definitions:  df-bi 116  df-dc 821  df-3or 964  df-3an 965  df-tru 1338  df-fal 1341  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ne 2328  df-nel 2423  df-ral 2440  df-rex 2441  df-reu 2442  df-rmo 2443  df-rab 2444  df-v 2714  df-sbc 2938  df-csb 3032  df-dif 3104  df-un 3106  df-in 3108  df-ss 3115  df-nul 3395  df-if 3506  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-uni 3773  df-int 3808  df-iun 3851  df-br 3966  df-opab 4026  df-mpt 4027  df-tr 4063  df-id 4253  df-po 4256  df-iso 4257  df-iord 4326  df-on 4328  df-ilim 4329  df-suc 4331  df-iom 4550  df-xp 4592  df-rel 4593  df-cnv 4594  df-co 4595  df-dm 4596  df-rn 4597  df-res 4598  df-ima 4599  df-iota 5135  df-fun 5172  df-fn 5173  df-f 5174  df-f1 5175  df-fo 5176  df-f1o 5177  df-fv 5178  df-riota 5780  df-ov 5827  df-oprab 5828  df-mpo 5829  df-1st 6088  df-2nd 6089  df-recs 6252  df-frec 6338  df-1o 6363  df-2o 6364  df-er 6480  df-en 6686  df-pnf 7914  df-mnf 7915  df-xr 7916  df-ltxr 7917  df-le 7918  df-sub 8048  df-neg 8049  df-reap 8450  df-ap 8457  df-div 8546  df-inn 8834  df-2 8892  df-3 8893  df-4 8894  df-n0 9091  df-z 9168  df-uz 9440  df-q 9529  df-rp 9561  df-seqfrec 10345  df-exp 10419  df-cj 10742  df-re 10743  df-im 10744  df-rsqrt 10898  df-abs 10899  df-dvds 11684  df-prm 11984
This theorem is referenced by:  nprm  11999  prmuz2  12007  dvdsprm  12013
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