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Theorem prmirred 16665
Description: The irreducible elements of  ZZ are exactly the prime numbers (and their negatives). (Contributed by Mario Carneiro, 5-Dec-2014.)
Hypotheses
Ref Expression
prmirred.1  |-  Z  =  (flds  ZZ )
prmirred.2  |-  I  =  (Irred `  Z )
Assertion
Ref Expression
prmirred  |-  ( A  e.  I  <->  ( A  e.  ZZ  /\  ( abs `  A )  e.  Prime ) )

Proof of Theorem prmirred
StepHypRef Expression
1 prmirred.2 . . 3  |-  I  =  (Irred `  Z )
2 zsubrg 16642 . . . 4  |-  ZZ  e.  (SubRing ` fld )
3 prmirred.1 . . . . 5  |-  Z  =  (flds  ZZ )
43subrgbas 15764 . . . 4  |-  ( ZZ  e.  (SubRing ` fld )  ->  ZZ  =  ( Base `  Z )
)
52, 4ax-mp 8 . . 3  |-  ZZ  =  ( Base `  Z )
61, 5irredcl 15696 . 2  |-  ( A  e.  I  ->  A  e.  ZZ )
7 elnn0 10116 . . . . . . 7  |-  ( A  e.  NN0  <->  ( A  e.  NN  \/  A  =  0 ) )
8 ax-1 5 . . . . . . . 8  |-  ( A  e.  NN  ->  ( A  e.  I  ->  A  e.  NN ) )
93subrgrng 15758 . . . . . . . . . . . 12  |-  ( ZZ  e.  (SubRing ` fld )  ->  Z  e. 
Ring )
102, 9ax-mp 8 . . . . . . . . . . 11  |-  Z  e. 
Ring
11 subrgsubg 15761 . . . . . . . . . . . . . 14  |-  ( ZZ  e.  (SubRing ` fld )  ->  ZZ  e.  (SubGrp ` fld ) )
122, 11ax-mp 8 . . . . . . . . . . . . 13  |-  ZZ  e.  (SubGrp ` fld )
13 cnfld0 16615 . . . . . . . . . . . . . 14  |-  0  =  ( 0g ` fld )
143, 13subg0 14837 . . . . . . . . . . . . 13  |-  ( ZZ  e.  (SubGrp ` fld )  ->  0  =  ( 0g `  Z
) )
1512, 14ax-mp 8 . . . . . . . . . . . 12  |-  0  =  ( 0g `  Z )
161, 15irredn0 15695 . . . . . . . . . . 11  |-  ( ( Z  e.  Ring  /\  A  e.  I )  ->  A  =/=  0 )
1710, 16mpan 651 . . . . . . . . . 10  |-  ( A  e.  I  ->  A  =/=  0 )
1817necon2bi 2575 . . . . . . . . 9  |-  ( A  =  0  ->  -.  A  e.  I )
1918pm2.21d 98 . . . . . . . 8  |-  ( A  =  0  ->  ( A  e.  I  ->  A  e.  NN ) )
208, 19jaoi 368 . . . . . . 7  |-  ( ( A  e.  NN  \/  A  =  0 )  ->  ( A  e.  I  ->  A  e.  NN ) )
217, 20sylbi 187 . . . . . 6  |-  ( A  e.  NN0  ->  ( A  e.  I  ->  A  e.  NN ) )
22 prmnn 12969 . . . . . . 7  |-  ( A  e.  Prime  ->  A  e.  NN )
2322a1i 10 . . . . . 6  |-  ( A  e.  NN0  ->  ( A  e.  Prime  ->  A  e.  NN ) )
243, 1prmirredlem 16663 . . . . . . 7  |-  ( A  e.  NN  ->  ( A  e.  I  <->  A  e.  Prime ) )
2524a1i 10 . . . . . 6  |-  ( A  e.  NN0  ->  ( A  e.  NN  ->  ( A  e.  I  <->  A  e.  Prime ) ) )
2621, 23, 25pm5.21ndd 343 . . . . 5  |-  ( A  e.  NN0  ->  ( A  e.  I  <->  A  e.  Prime ) )
27 nn0re 10123 . . . . . . 7  |-  ( A  e.  NN0  ->  A  e.  RR )
28 nn0ge0 10140 . . . . . . 7  |-  ( A  e.  NN0  ->  0  <_  A )
2927, 28absidd 12112 . . . . . 6  |-  ( A  e.  NN0  ->  ( abs `  A )  =  A )
3029eleq1d 2432 . . . . 5  |-  ( A  e.  NN0  ->  ( ( abs `  A )  e.  Prime  <->  A  e.  Prime ) )
3126, 30bitr4d 247 . . . 4  |-  ( A  e.  NN0  ->  ( A  e.  I  <->  ( abs `  A )  e.  Prime ) )
3231adantl 452 . . 3  |-  ( ( A  e.  ZZ  /\  A  e.  NN0 )  -> 
( A  e.  I  <->  ( abs `  A )  e.  Prime ) )
333, 1prmirredlem 16663 . . . . . 6  |-  ( -u A  e.  NN  ->  (
-u A  e.  I  <->  -u A  e.  Prime )
)
3433adantl 452 . . . . 5  |-  ( ( A  e.  ZZ  /\  -u A  e.  NN )  ->  ( -u A  e.  I  <->  -u A  e.  Prime ) )
35 eqid 2366 . . . . . . . . 9  |-  ( inv g `  Z )  =  ( inv g `  Z )
361, 35, 5irrednegb 15703 . . . . . . . 8  |-  ( ( Z  e.  Ring  /\  A  e.  ZZ )  ->  ( A  e.  I  <->  ( ( inv g `  Z ) `
 A )  e.  I ) )
3710, 36mpan 651 . . . . . . 7  |-  ( A  e.  ZZ  ->  ( A  e.  I  <->  ( ( inv g `  Z ) `
 A )  e.  I ) )
38 eqid 2366 . . . . . . . . . . 11  |-  ( inv g ` fld )  =  ( inv g ` fld )
393, 38, 35subginv 14838 . . . . . . . . . 10  |-  ( ( ZZ  e.  (SubGrp ` fld )  /\  A  e.  ZZ )  ->  ( ( inv g ` fld ) `  A )  =  ( ( inv g `  Z ) `
 A ) )
4012, 39mpan 651 . . . . . . . . 9  |-  ( A  e.  ZZ  ->  (
( inv g ` fld ) `  A )  =  ( ( inv g `  Z ) `  A
) )
41 zcn 10180 . . . . . . . . . 10  |-  ( A  e.  ZZ  ->  A  e.  CC )
42 cnfldneg 16617 . . . . . . . . . 10  |-  ( A  e.  CC  ->  (
( inv g ` fld ) `  A )  =  -u A )
4341, 42syl 15 . . . . . . . . 9  |-  ( A  e.  ZZ  ->  (
( inv g ` fld ) `  A )  =  -u A )
4440, 43eqtr3d 2400 . . . . . . . 8  |-  ( A  e.  ZZ  ->  (
( inv g `  Z ) `  A
)  =  -u A
)
4544eleq1d 2432 . . . . . . 7  |-  ( A  e.  ZZ  ->  (
( ( inv g `  Z ) `  A
)  e.  I  <->  -u A  e.  I ) )
4637, 45bitrd 244 . . . . . 6  |-  ( A  e.  ZZ  ->  ( A  e.  I  <->  -u A  e.  I ) )
4746adantr 451 . . . . 5  |-  ( ( A  e.  ZZ  /\  -u A  e.  NN )  ->  ( A  e.  I  <->  -u A  e.  I
) )
48 zre 10179 . . . . . . . 8  |-  ( A  e.  ZZ  ->  A  e.  RR )
4948adantr 451 . . . . . . 7  |-  ( ( A  e.  ZZ  /\  -u A  e.  NN )  ->  A  e.  RR )
50 nnnn0 10121 . . . . . . . . . 10  |-  ( -u A  e.  NN  ->  -u A  e.  NN0 )
5150nn0ge0d 10170 . . . . . . . . 9  |-  ( -u A  e.  NN  ->  0  <_  -u A )
5251adantl 452 . . . . . . . 8  |-  ( ( A  e.  ZZ  /\  -u A  e.  NN )  ->  0  <_  -u A
)
5349le0neg1d 9491 . . . . . . . 8  |-  ( ( A  e.  ZZ  /\  -u A  e.  NN )  ->  ( A  <_ 
0  <->  0  <_  -u A
) )
5452, 53mpbird 223 . . . . . . 7  |-  ( ( A  e.  ZZ  /\  -u A  e.  NN )  ->  A  <_  0
)
5549, 54absnidd 12103 . . . . . 6  |-  ( ( A  e.  ZZ  /\  -u A  e.  NN )  ->  ( abs `  A
)  =  -u A
)
5655eleq1d 2432 . . . . 5  |-  ( ( A  e.  ZZ  /\  -u A  e.  NN )  ->  ( ( abs `  A )  e.  Prime  <->  -u A  e.  Prime ) )
5734, 47, 563bitr4d 276 . . . 4  |-  ( ( A  e.  ZZ  /\  -u A  e.  NN )  ->  ( A  e.  I  <->  ( abs `  A
)  e.  Prime )
)
5857adantrl 696 . . 3  |-  ( ( A  e.  ZZ  /\  ( A  e.  RR  /\  -u A  e.  NN ) )  ->  ( A  e.  I  <->  ( abs `  A )  e.  Prime ) )
59 elznn0nn 10188 . . . 4  |-  ( A  e.  ZZ  <->  ( A  e.  NN0  \/  ( A  e.  RR  /\  -u A  e.  NN ) ) )
6059biimpi 186 . . 3  |-  ( A  e.  ZZ  ->  ( A  e.  NN0  \/  ( A  e.  RR  /\  -u A  e.  NN ) ) )
6132, 58, 60mpjaodan 761 . 2  |-  ( A  e.  ZZ  ->  ( A  e.  I  <->  ( abs `  A )  e.  Prime ) )
626, 61biadan2 623 1  |-  ( A  e.  I  <->  ( A  e.  ZZ  /\  ( abs `  A )  e.  Prime ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    \/ wo 357    /\ wa 358    = wceq 1647    e. wcel 1715    =/= wne 2529   class class class wbr 4125   ` cfv 5358  (class class class)co 5981   CCcc 8882   RRcr 8883   0cc0 8884    <_ cle 9015   -ucneg 9185   NNcn 9893   NN0cn0 10114   ZZcz 10175   abscabs 11926   Primecprime 12966   Basecbs 13356   ↾s cress 13357   0gc0g 13610   inv gcminusg 14573  SubGrpcsubg 14825   Ringcrg 15547  Irredcir 15632  SubRingcsubrg 15751  ℂfldccnfld 16593
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1551  ax-5 1562  ax-17 1621  ax-9 1659  ax-8 1680  ax-13 1717  ax-14 1719  ax-6 1734  ax-7 1739  ax-11 1751  ax-12 1937  ax-ext 2347  ax-rep 4233  ax-sep 4243  ax-nul 4251  ax-pow 4290  ax-pr 4316  ax-un 4615  ax-cnex 8940  ax-resscn 8941  ax-1cn 8942  ax-icn 8943  ax-addcl 8944  ax-addrcl 8945  ax-mulcl 8946  ax-mulrcl 8947  ax-mulcom 8948  ax-addass 8949  ax-mulass 8950  ax-distr 8951  ax-i2m1 8952  ax-1ne0 8953  ax-1rid 8954  ax-rnegex 8955  ax-rrecex 8956  ax-cnre 8957  ax-pre-lttri 8958  ax-pre-lttrn 8959  ax-pre-ltadd 8960  ax-pre-mulgt0 8961  ax-pre-sup 8962  ax-addf 8963  ax-mulf 8964
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 936  df-3an 937  df-tru 1324  df-ex 1547  df-nf 1550  df-sb 1654  df-eu 2221  df-mo 2222  df-clab 2353  df-cleq 2359  df-clel 2362  df-nfc 2491  df-ne 2531  df-nel 2532  df-ral 2633  df-rex 2634  df-reu 2635  df-rmo 2636  df-rab 2637  df-v 2875  df-sbc 3078  df-csb 3168  df-dif 3241  df-un 3243  df-in 3245  df-ss 3252  df-pss 3254  df-nul 3544  df-if 3655  df-pw 3716  df-sn 3735  df-pr 3736  df-tp 3737  df-op 3738  df-uni 3930  df-int 3965  df-iun 4009  df-br 4126  df-opab 4180  df-mpt 4181  df-tr 4216  df-eprel 4408  df-id 4412  df-po 4417  df-so 4418  df-fr 4455  df-we 4457  df-ord 4498  df-on 4499  df-lim 4500  df-suc 4501  df-om 4760  df-xp 4798  df-rel 4799  df-cnv 4800  df-co 4801  df-dm 4802  df-rn 4803  df-res 4804  df-ima 4805  df-iota 5322  df-fun 5360  df-fn 5361  df-f 5362  df-f1 5363  df-fo 5364  df-f1o 5365  df-fv 5366  df-ov 5984  df-oprab 5985  df-mpt2 5986  df-1st 6249  df-2nd 6250  df-tpos 6376  df-riota 6446  df-recs 6530  df-rdg 6565  df-1o 6621  df-2o 6622  df-oadd 6625  df-er 6802  df-en 7007  df-dom 7008  df-sdom 7009  df-fin 7010  df-sup 7341  df-pnf 9016  df-mnf 9017  df-xr 9018  df-ltxr 9019  df-le 9020  df-sub 9186  df-neg 9187  df-div 9571  df-nn 9894  df-2 9951  df-3 9952  df-4 9953  df-5 9954  df-6 9955  df-7 9956  df-8 9957  df-9 9958  df-10 9959  df-n0 10115  df-z 10176  df-dec 10276  df-uz 10382  df-rp 10506  df-fz 10936  df-seq 11211  df-exp 11270  df-cj 11791  df-re 11792  df-im 11793  df-sqr 11927  df-abs 11928  df-dvds 12740  df-prm 12967  df-gz 13185  df-struct 13358  df-ndx 13359  df-slot 13360  df-base 13361  df-sets 13362  df-ress 13363  df-plusg 13429  df-mulr 13430  df-starv 13431  df-tset 13435  df-ple 13436  df-ds 13438  df-unif 13439  df-0g 13614  df-mnd 14577  df-grp 14699  df-minusg 14700  df-subg 14828  df-cmn 15301  df-mgp 15536  df-rng 15550  df-cring 15551  df-ur 15552  df-oppr 15615  df-dvdsr 15633  df-unit 15634  df-irred 15635  df-invr 15664  df-dvr 15675  df-drng 15724  df-subrg 15753  df-cnfld 16594
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