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Theorem prmirred 16448
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 16425 . . . 4  |-  ZZ  e.  (SubRing ` fld )
3 prmirred.1 . . . . 5  |-  Z  =  (flds  ZZ )
43subrgbas 15554 . . . 4  |-  ( ZZ  e.  (SubRing ` fld )  ->  ZZ  =  ( Base `  Z )
)
52, 4ax-mp 8 . . 3  |-  ZZ  =  ( Base `  Z )
61, 5irredcl 15486 . 2  |-  ( A  e.  I  ->  A  e.  ZZ )
7 elnn0 9967 . . . . . . 7  |-  ( A  e.  NN0  <->  ( A  e.  NN  \/  A  =  0 ) )
8 ax-1 5 . . . . . . . 8  |-  ( A  e.  NN  ->  ( A  e.  I  ->  A  e.  NN ) )
93subrgrng 15548 . . . . . . . . . . . 12  |-  ( ZZ  e.  (SubRing ` fld )  ->  Z  e. 
Ring )
102, 9ax-mp 8 . . . . . . . . . . 11  |-  Z  e. 
Ring
11 subrgsubg 15551 . . . . . . . . . . . . . 14  |-  ( ZZ  e.  (SubRing ` fld )  ->  ZZ  e.  (SubGrp ` fld ) )
122, 11ax-mp 8 . . . . . . . . . . . . 13  |-  ZZ  e.  (SubGrp ` fld )
13 cnfld0 16398 . . . . . . . . . . . . . 14  |-  0  =  ( 0g ` fld )
143, 13subg0 14627 . . . . . . . . . . . . 13  |-  ( ZZ  e.  (SubGrp ` fld )  ->  0  =  ( 0g `  Z
) )
1512, 14ax-mp 8 . . . . . . . . . . . 12  |-  0  =  ( 0g `  Z )
161, 15irredn0 15485 . . . . . . . . . . 11  |-  ( ( Z  e.  Ring  /\  A  e.  I )  ->  A  =/=  0 )
1710, 16mpan 651 . . . . . . . . . 10  |-  ( A  e.  I  ->  A  =/=  0 )
1817necon2bi 2492 . . . . . . . . 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 12761 . . . . . . 7  |-  ( A  e.  Prime  ->  A  e.  NN )
2322a1i 10 . . . . . 6  |-  ( A  e.  NN0  ->  ( A  e.  Prime  ->  A  e.  NN ) )
243, 1prmirredlem 16446 . . . . . . 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 9974 . . . . . . 7  |-  ( A  e.  NN0  ->  A  e.  RR )
28 nn0ge0 9991 . . . . . . 7  |-  ( A  e.  NN0  ->  0  <_  A )
2927, 28absidd 11905 . . . . . 6  |-  ( A  e.  NN0  ->  ( abs `  A )  =  A )
3029eleq1d 2349 . . . . 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 16446 . . . . . 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 2283 . . . . . . . . 9  |-  ( inv g `  Z )  =  ( inv g `  Z )
361, 35, 5irrednegb 15493 . . . . . . . 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 2283 . . . . . . . . . . 11  |-  ( inv g ` fld )  =  ( inv g ` fld )
393, 38, 35subginv 14628 . . . . . . . . . 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 10029 . . . . . . . . . 10  |-  ( A  e.  ZZ  ->  A  e.  CC )
42 cnfldneg 16400 . . . . . . . . . 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 2317 . . . . . . . 8  |-  ( A  e.  ZZ  ->  (
( inv g `  Z ) `  A
)  =  -u A
)
4544eleq1d 2349 . . . . . . 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 10028 . . . . . . . 8  |-  ( A  e.  ZZ  ->  A  e.  RR )
4948adantr 451 . . . . . . 7  |-  ( ( A  e.  ZZ  /\  -u A  e.  NN )  ->  A  e.  RR )
50 nnnn0 9972 . . . . . . . . . 10  |-  ( -u A  e.  NN  ->  -u A  e.  NN0 )
5150nn0ge0d 10021 . . . . . . . . 9  |-  ( -u A  e.  NN  ->  0  <_  -u A )
5251adantl 452 . . . . . . . 8  |-  ( ( A  e.  ZZ  /\  -u A  e.  NN )  ->  0  <_  -u A
)
5349le0neg1d 9344 . . . . . . . 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 11896 . . . . . 6  |-  ( ( A  e.  ZZ  /\  -u A  e.  NN )  ->  ( abs `  A
)  =  -u A
)
5655eleq1d 2349 . . . . 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 10037 . . . 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 1623    e. wcel 1684    =/= wne 2446   class class class wbr 4023   ` cfv 5255  (class class class)co 5858   CCcc 8735   RRcr 8736   0cc0 8737    <_ cle 8868   -ucneg 9038   NNcn 9746   NN0cn0 9965   ZZcz 10024   abscabs 11719   Primecprime 12758   Basecbs 13148   ↾s cress 13149   0gc0g 13400   inv gcminusg 14363  SubGrpcsubg 14615   Ringcrg 15337  Irredcir 15422  SubRingcsubrg 15541  ℂfldccnfld 16377
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-rep 4131  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  ax-pre-sup 8815  ax-addf 8816  ax-mulf 8817
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-rmo 2551  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-1st 6122  df-2nd 6123  df-tpos 6234  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-sup 7194  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-nn 9747  df-2 9804  df-3 9805  df-4 9806  df-5 9807  df-6 9808  df-7 9809  df-8 9810  df-9 9811  df-10 9812  df-n0 9966  df-z 10025  df-dec 10125  df-uz 10231  df-rp 10355  df-fz 10783  df-seq 11047  df-exp 11105  df-cj 11584  df-re 11585  df-im 11586  df-sqr 11720  df-abs 11721  df-dvds 12532  df-prm 12759  df-gz 12977  df-struct 13150  df-ndx 13151  df-slot 13152  df-base 13153  df-sets 13154  df-ress 13155  df-plusg 13221  df-mulr 13222  df-starv 13223  df-tset 13227  df-ple 13228  df-ds 13230  df-0g 13404  df-mnd 14367  df-grp 14489  df-minusg 14490  df-subg 14618  df-cmn 15091  df-mgp 15326  df-rng 15340  df-cring 15341  df-ur 15342  df-oppr 15405  df-dvdsr 15423  df-unit 15424  df-irred 15425  df-invr 15454  df-dvr 15465  df-drng 15514  df-subrg 15543  df-cnfld 16378
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