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Theorem prmirredOLD 18395
Description: The irreducible elements of  ZZ are exactly the prime numbers (and their negatives). (Contributed by Mario Carneiro, 5-Dec-2014.) Obsolete version of prmirred 18392 as of 10-Jun-2019. (New usage is discouraged.) (Proof modification is discouraged.)
Hypotheses
Ref Expression
prmirredOLD.1  |-  Z  =  (flds  ZZ )
prmirredOLD.2  |-  I  =  (Irred `  Z )
Assertion
Ref Expression
prmirredOLD  |-  ( A  e.  I  <->  ( A  e.  ZZ  /\  ( abs `  A )  e.  Prime ) )

Proof of Theorem prmirredOLD
StepHypRef Expression
1 prmirredOLD.2 . . 3  |-  I  =  (Irred `  Z )
2 zsubrg 18339 . . . 4  |-  ZZ  e.  (SubRing ` fld )
3 prmirredOLD.1 . . . . 5  |-  Z  =  (flds  ZZ )
43subrgbas 17307 . . . 4  |-  ( ZZ  e.  (SubRing ` fld )  ->  ZZ  =  ( Base `  Z )
)
52, 4ax-mp 5 . . 3  |-  ZZ  =  ( Base `  Z )
61, 5irredcl 17223 . 2  |-  ( A  e.  I  ->  A  e.  ZZ )
7 elnn0 10809 . . . . . . 7  |-  ( A  e.  NN0  <->  ( A  e.  NN  \/  A  =  0 ) )
8 ax-1 6 . . . . . . . 8  |-  ( A  e.  NN  ->  ( A  e.  I  ->  A  e.  NN ) )
93subrgring 17301 . . . . . . . . . . . 12  |-  ( ZZ  e.  (SubRing ` fld )  ->  Z  e. 
Ring )
102, 9ax-mp 5 . . . . . . . . . . 11  |-  Z  e. 
Ring
11 subrgsubg 17304 . . . . . . . . . . . . . 14  |-  ( ZZ  e.  (SubRing ` fld )  ->  ZZ  e.  (SubGrp ` fld ) )
122, 11ax-mp 5 . . . . . . . . . . . . 13  |-  ZZ  e.  (SubGrp ` fld )
13 cnfld0 18310 . . . . . . . . . . . . . 14  |-  0  =  ( 0g ` fld )
143, 13subg0 16078 . . . . . . . . . . . . 13  |-  ( ZZ  e.  (SubGrp ` fld )  ->  0  =  ( 0g `  Z
) )
1512, 14ax-mp 5 . . . . . . . . . . . 12  |-  0  =  ( 0g `  Z )
161, 15irredn0 17222 . . . . . . . . . . 11  |-  ( ( Z  e.  Ring  /\  A  e.  I )  ->  A  =/=  0 )
1710, 16mpan 670 . . . . . . . . . 10  |-  ( A  e.  I  ->  A  =/=  0 )
1817necon2bi 2704 . . . . . . . . 9  |-  ( A  =  0  ->  -.  A  e.  I )
1918pm2.21d 106 . . . . . . . 8  |-  ( A  =  0  ->  ( A  e.  I  ->  A  e.  NN ) )
208, 19jaoi 379 . . . . . . 7  |-  ( ( A  e.  NN  \/  A  =  0 )  ->  ( A  e.  I  ->  A  e.  NN ) )
217, 20sylbi 195 . . . . . 6  |-  ( A  e.  NN0  ->  ( A  e.  I  ->  A  e.  NN ) )
22 prmnn 14095 . . . . . . 7  |-  ( A  e.  Prime  ->  A  e.  NN )
2322a1i 11 . . . . . 6  |-  ( A  e.  NN0  ->  ( A  e.  Prime  ->  A  e.  NN ) )
243, 1prmirredlemOLD 18393 . . . . . . 7  |-  ( A  e.  NN  ->  ( A  e.  I  <->  A  e.  Prime ) )
2524a1i 11 . . . . . 6  |-  ( A  e.  NN0  ->  ( A  e.  NN  ->  ( A  e.  I  <->  A  e.  Prime ) ) )
2621, 23, 25pm5.21ndd 354 . . . . 5  |-  ( A  e.  NN0  ->  ( A  e.  I  <->  A  e.  Prime ) )
27 nn0re 10816 . . . . . . 7  |-  ( A  e.  NN0  ->  A  e.  RR )
28 nn0ge0 10833 . . . . . . 7  |-  ( A  e.  NN0  ->  0  <_  A )
2927, 28absidd 13233 . . . . . 6  |-  ( A  e.  NN0  ->  ( abs `  A )  =  A )
3029eleq1d 2536 . . . . 5  |-  ( A  e.  NN0  ->  ( ( abs `  A )  e.  Prime  <->  A  e.  Prime ) )
3126, 30bitr4d 256 . . . 4  |-  ( A  e.  NN0  ->  ( A  e.  I  <->  ( abs `  A )  e.  Prime ) )
3231adantl 466 . . 3  |-  ( ( A  e.  ZZ  /\  A  e.  NN0 )  -> 
( A  e.  I  <->  ( abs `  A )  e.  Prime ) )
333, 1prmirredlemOLD 18393 . . . . . 6  |-  ( -u A  e.  NN  ->  (
-u A  e.  I  <->  -u A  e.  Prime )
)
3433adantl 466 . . . . 5  |-  ( ( A  e.  ZZ  /\  -u A  e.  NN )  ->  ( -u A  e.  I  <->  -u A  e.  Prime ) )
35 eqid 2467 . . . . . . . . 9  |-  ( invg `  Z )  =  ( invg `  Z )
361, 35, 5irrednegb 17230 . . . . . . . 8  |-  ( ( Z  e.  Ring  /\  A  e.  ZZ )  ->  ( A  e.  I  <->  ( ( invg `  Z ) `
 A )  e.  I ) )
3710, 36mpan 670 . . . . . . 7  |-  ( A  e.  ZZ  ->  ( A  e.  I  <->  ( ( invg `  Z ) `
 A )  e.  I ) )
38 eqid 2467 . . . . . . . . . . 11  |-  ( invg ` fld )  =  ( invg ` fld )
393, 38, 35subginv 16079 . . . . . . . . . 10  |-  ( ( ZZ  e.  (SubGrp ` fld )  /\  A  e.  ZZ )  ->  ( ( invg ` fld ) `  A )  =  ( ( invg `  Z ) `
 A ) )
4012, 39mpan 670 . . . . . . . . 9  |-  ( A  e.  ZZ  ->  (
( invg ` fld ) `  A )  =  ( ( invg `  Z ) `  A
) )
41 zcn 10881 . . . . . . . . . 10  |-  ( A  e.  ZZ  ->  A  e.  CC )
42 cnfldneg 18312 . . . . . . . . . 10  |-  ( A  e.  CC  ->  (
( invg ` fld ) `  A )  =  -u A )
4341, 42syl 16 . . . . . . . . 9  |-  ( A  e.  ZZ  ->  (
( invg ` fld ) `  A )  =  -u A )
4440, 43eqtr3d 2510 . . . . . . . 8  |-  ( A  e.  ZZ  ->  (
( invg `  Z ) `  A
)  =  -u A
)
4544eleq1d 2536 . . . . . . 7  |-  ( A  e.  ZZ  ->  (
( ( invg `  Z ) `  A
)  e.  I  <->  -u A  e.  I ) )
4637, 45bitrd 253 . . . . . 6  |-  ( A  e.  ZZ  ->  ( A  e.  I  <->  -u A  e.  I ) )
4746adantr 465 . . . . 5  |-  ( ( A  e.  ZZ  /\  -u A  e.  NN )  ->  ( A  e.  I  <->  -u A  e.  I
) )
48 zre 10880 . . . . . . . 8  |-  ( A  e.  ZZ  ->  A  e.  RR )
4948adantr 465 . . . . . . 7  |-  ( ( A  e.  ZZ  /\  -u A  e.  NN )  ->  A  e.  RR )
50 nnnn0 10814 . . . . . . . . . 10  |-  ( -u A  e.  NN  ->  -u A  e.  NN0 )
5150nn0ge0d 10867 . . . . . . . . 9  |-  ( -u A  e.  NN  ->  0  <_  -u A )
5251adantl 466 . . . . . . . 8  |-  ( ( A  e.  ZZ  /\  -u A  e.  NN )  ->  0  <_  -u A
)
5349le0neg1d 10136 . . . . . . . 8  |-  ( ( A  e.  ZZ  /\  -u A  e.  NN )  ->  ( A  <_ 
0  <->  0  <_  -u A
) )
5452, 53mpbird 232 . . . . . . 7  |-  ( ( A  e.  ZZ  /\  -u A  e.  NN )  ->  A  <_  0
)
5549, 54absnidd 13224 . . . . . 6  |-  ( ( A  e.  ZZ  /\  -u A  e.  NN )  ->  ( abs `  A
)  =  -u A
)
5655eleq1d 2536 . . . . 5  |-  ( ( A  e.  ZZ  /\  -u A  e.  NN )  ->  ( ( abs `  A )  e.  Prime  <->  -u A  e.  Prime ) )
5734, 47, 563bitr4d 285 . . . 4  |-  ( ( A  e.  ZZ  /\  -u A  e.  NN )  ->  ( A  e.  I  <->  ( abs `  A
)  e.  Prime )
)
5857adantrl 715 . . 3  |-  ( ( A  e.  ZZ  /\  ( A  e.  RR  /\  -u A  e.  NN ) )  ->  ( A  e.  I  <->  ( abs `  A )  e.  Prime ) )
59 elznn0nn 10890 . . . 4  |-  ( A  e.  ZZ  <->  ( A  e.  NN0  \/  ( A  e.  RR  /\  -u A  e.  NN ) ) )
6059biimpi 194 . . 3  |-  ( A  e.  ZZ  ->  ( A  e.  NN0  \/  ( A  e.  RR  /\  -u A  e.  NN ) ) )
6132, 58, 60mpjaodan 784 . 2  |-  ( A  e.  ZZ  ->  ( A  e.  I  <->  ( abs `  A )  e.  Prime ) )
626, 61biadan2 642 1  |-  ( A  e.  I  <->  ( A  e.  ZZ  /\  ( abs `  A )  e.  Prime ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    \/ wo 368    /\ wa 369    = wceq 1379    e. wcel 1767    =/= wne 2662   class class class wbr 4453   ` cfv 5594  (class class class)co 6295   CCcc 9502   RRcr 9503   0cc0 9504    <_ cle 9641   -ucneg 9818   NNcn 10548   NN0cn0 10807   ZZcz 10876   abscabs 13046   Primecprime 14092   Basecbs 14506   ↾s cress 14507   0gc0g 14711   invgcminusg 15925  SubGrpcsubg 16066   Ringcrg 17068  Irredcir 17159  SubRingcsubrg 17294  ℂfldccnfld 18288
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4564  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587  ax-cnex 9560  ax-resscn 9561  ax-1cn 9562  ax-icn 9563  ax-addcl 9564  ax-addrcl 9565  ax-mulcl 9566  ax-mulrcl 9567  ax-mulcom 9568  ax-addass 9569  ax-mulass 9570  ax-distr 9571  ax-i2m1 9572  ax-1ne0 9573  ax-1rid 9574  ax-rnegex 9575  ax-rrecex 9576  ax-cnre 9577  ax-pre-lttri 9578  ax-pre-lttrn 9579  ax-pre-ltadd 9580  ax-pre-mulgt0 9581  ax-pre-sup 9582  ax-addf 9583  ax-mulf 9584
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2822  df-rex 2823  df-reu 2824  df-rmo 2825  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-tp 4038  df-op 4040  df-uni 4252  df-int 4289  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-tr 4547  df-eprel 4797  df-id 4801  df-po 4806  df-so 4807  df-fr 4844  df-we 4846  df-ord 4887  df-on 4888  df-lim 4889  df-suc 4890  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6256  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-om 6696  df-1st 6795  df-2nd 6796  df-tpos 6967  df-recs 7054  df-rdg 7088  df-1o 7142  df-2o 7143  df-oadd 7146  df-er 7323  df-en 7529  df-dom 7530  df-sdom 7531  df-fin 7532  df-sup 7913  df-pnf 9642  df-mnf 9643  df-xr 9644  df-ltxr 9645  df-le 9646  df-sub 9819  df-neg 9820  df-div 10219  df-nn 10549  df-2 10606  df-3 10607  df-4 10608  df-5 10609  df-6 10610  df-7 10611  df-8 10612  df-9 10613  df-10 10614  df-n0 10808  df-z 10877  df-dec 10989  df-uz 11095  df-rp 11233  df-fz 11685  df-seq 12088  df-exp 12147  df-cj 12911  df-re 12912  df-im 12913  df-sqrt 13047  df-abs 13048  df-dvds 13864  df-prm 14093  df-gz 14323  df-struct 14508  df-ndx 14509  df-slot 14510  df-base 14511  df-sets 14512  df-ress 14513  df-plusg 14584  df-mulr 14585  df-starv 14586  df-tset 14590  df-ple 14591  df-ds 14593  df-unif 14594  df-0g 14713  df-mgm 15745  df-sgrp 15784  df-mnd 15794  df-grp 15928  df-minusg 15929  df-subg 16069  df-cmn 16671  df-mgp 17012  df-ur 17024  df-ring 17070  df-cring 17071  df-oppr 17142  df-dvdsr 17160  df-unit 17161  df-irred 17162  df-invr 17191  df-dvr 17202  df-drng 17267  df-subrg 17296  df-cnfld 18289
This theorem is referenced by: (None)
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