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Theorem prmirredOLD 18017
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 18014 as of 10-Jun-2019. (New usage 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 17961 . . . 4  |-  ZZ  e.  (SubRing ` fld )
3 prmirredOLD.1 . . . . 5  |-  Z  =  (flds  ZZ )
43subrgbas 16966 . . . 4  |-  ( ZZ  e.  (SubRing ` fld )  ->  ZZ  =  ( Base `  Z )
)
52, 4ax-mp 5 . . 3  |-  ZZ  =  ( Base `  Z )
61, 5irredcl 16888 . 2  |-  ( A  e.  I  ->  A  e.  ZZ )
7 elnn0 10668 . . . . . . 7  |-  ( A  e.  NN0  <->  ( A  e.  NN  \/  A  =  0 ) )
8 ax-1 6 . . . . . . . 8  |-  ( A  e.  NN  ->  ( A  e.  I  ->  A  e.  NN ) )
93subrgrng 16960 . . . . . . . . . . . 12  |-  ( ZZ  e.  (SubRing ` fld )  ->  Z  e. 
Ring )
102, 9ax-mp 5 . . . . . . . . . . 11  |-  Z  e. 
Ring
11 subrgsubg 16963 . . . . . . . . . . . . . 14  |-  ( ZZ  e.  (SubRing ` fld )  ->  ZZ  e.  (SubGrp ` fld ) )
122, 11ax-mp 5 . . . . . . . . . . . . 13  |-  ZZ  e.  (SubGrp ` fld )
13 cnfld0 17935 . . . . . . . . . . . . . 14  |-  0  =  ( 0g ` fld )
143, 13subg0 15775 . . . . . . . . . . . . 13  |-  ( ZZ  e.  (SubGrp ` fld )  ->  0  =  ( 0g `  Z
) )
1512, 14ax-mp 5 . . . . . . . . . . . 12  |-  0  =  ( 0g `  Z )
161, 15irredn0 16887 . . . . . . . . . . 11  |-  ( ( Z  e.  Ring  /\  A  e.  I )  ->  A  =/=  0 )
1710, 16mpan 670 . . . . . . . . . 10  |-  ( A  e.  I  ->  A  =/=  0 )
1817necon2bi 2682 . . . . . . . . 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 13854 . . . . . . 7  |-  ( A  e.  Prime  ->  A  e.  NN )
2322a1i 11 . . . . . 6  |-  ( A  e.  NN0  ->  ( A  e.  Prime  ->  A  e.  NN ) )
243, 1prmirredlemOLD 18015 . . . . . . 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 10675 . . . . . . 7  |-  ( A  e.  NN0  ->  A  e.  RR )
28 nn0ge0 10692 . . . . . . 7  |-  ( A  e.  NN0  ->  0  <_  A )
2927, 28absidd 12997 . . . . . 6  |-  ( A  e.  NN0  ->  ( abs `  A )  =  A )
3029eleq1d 2518 . . . . 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 18015 . . . . . 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 2450 . . . . . . . . 9  |-  ( invg `  Z )  =  ( invg `  Z )
361, 35, 5irrednegb 16895 . . . . . . . 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 2450 . . . . . . . . . . 11  |-  ( invg ` fld )  =  ( invg ` fld )
393, 38, 35subginv 15776 . . . . . . . . . 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 10738 . . . . . . . . . 10  |-  ( A  e.  ZZ  ->  A  e.  CC )
42 cnfldneg 17937 . . . . . . . . . 10  |-  ( A  e.  CC  ->  (
( invg ` fld ) `  A )  =  -u A )
4341, 42syl 16 . . . . . . . . 9  |-  ( A  e.  ZZ  ->  (
( invg ` fld ) `  A )  =  -u A )
4440, 43eqtr3d 2492 . . . . . . . 8  |-  ( A  e.  ZZ  ->  (
( invg `  Z ) `  A
)  =  -u A
)
4544eleq1d 2518 . . . . . . 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 10737 . . . . . . . 8  |-  ( A  e.  ZZ  ->  A  e.  RR )
4948adantr 465 . . . . . . 7  |-  ( ( A  e.  ZZ  /\  -u A  e.  NN )  ->  A  e.  RR )
50 nnnn0 10673 . . . . . . . . . 10  |-  ( -u A  e.  NN  ->  -u A  e.  NN0 )
5150nn0ge0d 10726 . . . . . . . . 9  |-  ( -u A  e.  NN  ->  0  <_  -u A )
5251adantl 466 . . . . . . . 8  |-  ( ( A  e.  ZZ  /\  -u A  e.  NN )  ->  0  <_  -u A
)
5349le0neg1d 9998 . . . . . . . 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 12988 . . . . . 6  |-  ( ( A  e.  ZZ  /\  -u A  e.  NN )  ->  ( abs `  A
)  =  -u A
)
5655eleq1d 2518 . . . . 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 10747 . . . 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 1370    e. wcel 1757    =/= wne 2641   class class class wbr 4376   ` cfv 5502  (class class class)co 6176   CCcc 9367   RRcr 9368   0cc0 9369    <_ cle 9506   -ucneg 9683   NNcn 10409   NN0cn0 10666   ZZcz 10733   abscabs 12811   Primecprime 13851   Basecbs 14262   ↾s cress 14263   0gc0g 14466   invgcminusg 15499  SubGrpcsubg 15763   Ringcrg 16737  Irredcir 16824  SubRingcsubrg 16953  ℂfldccnfld 17913
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1709  ax-7 1729  ax-8 1759  ax-9 1761  ax-10 1776  ax-11 1781  ax-12 1793  ax-13 1944  ax-ext 2429  ax-rep 4487  ax-sep 4497  ax-nul 4505  ax-pow 4554  ax-pr 4615  ax-un 6458  ax-cnex 9425  ax-resscn 9426  ax-1cn 9427  ax-icn 9428  ax-addcl 9429  ax-addrcl 9430  ax-mulcl 9431  ax-mulrcl 9432  ax-mulcom 9433  ax-addass 9434  ax-mulass 9435  ax-distr 9436  ax-i2m1 9437  ax-1ne0 9438  ax-1rid 9439  ax-rnegex 9440  ax-rrecex 9441  ax-cnre 9442  ax-pre-lttri 9443  ax-pre-lttrn 9444  ax-pre-ltadd 9445  ax-pre-mulgt0 9446  ax-pre-sup 9447  ax-addf 9448  ax-mulf 9449
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1702  df-eu 2263  df-mo 2264  df-clab 2436  df-cleq 2442  df-clel 2445  df-nfc 2598  df-ne 2643  df-nel 2644  df-ral 2797  df-rex 2798  df-reu 2799  df-rmo 2800  df-rab 2801  df-v 3056  df-sbc 3271  df-csb 3373  df-dif 3415  df-un 3417  df-in 3419  df-ss 3426  df-pss 3428  df-nul 3722  df-if 3876  df-pw 3946  df-sn 3962  df-pr 3964  df-tp 3966  df-op 3968  df-uni 4176  df-int 4213  df-iun 4257  df-br 4377  df-opab 4435  df-mpt 4436  df-tr 4470  df-eprel 4716  df-id 4720  df-po 4725  df-so 4726  df-fr 4763  df-we 4765  df-ord 4806  df-on 4807  df-lim 4808  df-suc 4809  df-xp 4930  df-rel 4931  df-cnv 4932  df-co 4933  df-dm 4934  df-rn 4935  df-res 4936  df-ima 4937  df-iota 5465  df-fun 5504  df-fn 5505  df-f 5506  df-f1 5507  df-fo 5508  df-f1o 5509  df-fv 5510  df-riota 6137  df-ov 6179  df-oprab 6180  df-mpt2 6181  df-om 6563  df-1st 6663  df-2nd 6664  df-tpos 6831  df-recs 6918  df-rdg 6952  df-1o 7006  df-2o 7007  df-oadd 7010  df-er 7187  df-en 7397  df-dom 7398  df-sdom 7399  df-fin 7400  df-sup 7778  df-pnf 9507  df-mnf 9508  df-xr 9509  df-ltxr 9510  df-le 9511  df-sub 9684  df-neg 9685  df-div 10081  df-nn 10410  df-2 10467  df-3 10468  df-4 10469  df-5 10470  df-6 10471  df-7 10472  df-8 10473  df-9 10474  df-10 10475  df-n0 10667  df-z 10734  df-dec 10843  df-uz 10949  df-rp 11079  df-fz 11525  df-seq 11894  df-exp 11953  df-cj 12676  df-re 12677  df-im 12678  df-sqr 12812  df-abs 12813  df-dvds 13624  df-prm 13852  df-gz 14079  df-struct 14264  df-ndx 14265  df-slot 14266  df-base 14267  df-sets 14268  df-ress 14269  df-plusg 14339  df-mulr 14340  df-starv 14341  df-tset 14345  df-ple 14346  df-ds 14348  df-unif 14349  df-0g 14468  df-mnd 15503  df-grp 15633  df-minusg 15634  df-subg 15766  df-cmn 16369  df-mgp 16683  df-ur 16695  df-rng 16739  df-cring 16740  df-oppr 16807  df-dvdsr 16825  df-unit 16826  df-irred 16827  df-invr 16856  df-dvr 16867  df-drng 16926  df-subrg 16955  df-cnfld 17914
This theorem is referenced by: (None)
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