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 = (ℂfld ↾s ZZ )
prmirredOLD.2	|- I = (Irred ` Z )

Assertion

Ref	Expression
prmirredOLD	|- ( A e. I <-> ( A e. ZZ /\ ( abs ` A ) e. Prime ) )

Proof of Theorem prmirredOLD

Step	Hyp	Ref	Expression
1		prmirredOLD.2	. . 3 |- I = (Irred ` Z )
2		zsubrg 17961	. . . 4 |- ZZ e. (SubRing ` ℂfld )
3		prmirredOLD.1	. . . . 5 |- Z = (ℂfld ↾s ZZ )
4	3	subrgbas 16966	. . . 4 |- ( ZZ e. (SubRing ` ℂfld ) -> ZZ = ( Base ` Z ) )
5	2, 4	ax-mp 5	. . 3 |- ZZ = ( Base ` Z )
6	1, 5	irredcl 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 ) )
9	3	subrgrng 16960	. . . . . . . . . . . 12 |- ( ZZ e. (SubRing ` ℂfld ) -> Z e. Ring )
10	2, 9	ax-mp 5	. . . . . . . . . . 11 |- Z e. Ring
11		subrgsubg 16963	. . . . . . . . . . . . . 14 |- ( ZZ e. (SubRing ` ℂfld ) -> ZZ e. (SubGrp ` ℂfld ) )
12	2, 11	ax-mp 5	. . . . . . . . . . . . 13 |- ZZ e. (SubGrp ` ℂfld )
13		cnfld0 17935	. . . . . . . . . . . . . 14 |- 0 = ( 0g ` ℂfld )
14	3, 13	subg0 15775	. . . . . . . . . . . . 13 |- ( ZZ e. (SubGrp ` ℂfld ) -> 0 = ( 0g ` Z ) )
15	12, 14	ax-mp 5	. . . . . . . . . . . 12 |- 0 = ( 0g ` Z )
16	1, 15	irredn0 16887	. . . . . . . . . . 11 |- ( ( Z e. Ring /\ A e. I ) -> A =/= 0 )
17	10, 16	mpan 670	. . . . . . . . . 10 |- ( A e. I -> A =/= 0 )
18	17	necon2bi 2682	. . . . . . . . 9 |- ( A = 0 -> -. A e. I )
19	18	pm2.21d 106	. . . . . . . 8 |- ( A = 0 -> ( A e. I -> A e. NN ) )
20	8, 19	jaoi 379	. . . . . . 7 |- ( ( A e. NN \/ A = 0 ) -> ( A e. I -> A e. NN ) )
21	7, 20	sylbi 195	. . . . . 6 |- ( A e. NN0 -> ( A e. I -> A e. NN ) )
22		prmnn 13854	. . . . . . 7 |- ( A e. Prime -> A e. NN )
23	22	a1i 11	. . . . . 6 |- ( A e. NN0 -> ( A e. Prime -> A e. NN ) )
24	3, 1	prmirredlemOLD 18015	. . . . . . 7 |- ( A e. NN -> ( A e. I <-> A e. Prime ) )
25	24	a1i 11	. . . . . 6 |- ( A e. NN0 -> ( A e. NN -> ( A e. I <-> A e. Prime ) ) )
26	21, 23, 25	pm5.21ndd 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 )
29	27, 28	absidd 12997	. . . . . 6 |- ( A e. NN0 -> ( abs ` A ) = A )
30	29	eleq1d 2518	. . . . 5 |- ( A e. NN0 -> ( ( abs ` A ) e. Prime <-> A e. Prime ) )
31	26, 30	bitr4d 256	. . . 4 |- ( A e. NN0 -> ( A e. I <-> ( abs ` A ) e. Prime ) )
32	31	adantl 466	. . 3 |- ( ( A e. ZZ /\ A e. NN0 ) -> ( A e. I <-> ( abs ` A ) e. Prime ) )
33	3, 1	prmirredlemOLD 18015	. . . . . 6 |- ( -u A e. NN -> ( -u A e. I <-> -u A e. Prime ) )
34	33	adantl 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 )
36	1, 35, 5	irrednegb 16895	. . . . . . . 8 |- ( ( Z e. Ring /\ A e. ZZ ) -> ( A e. I <-> ( ( invg ` Z ) ` A ) e. I ) )
37	10, 36	mpan 670	. . . . . . 7 |- ( A e. ZZ -> ( A e. I <-> ( ( invg ` Z ) ` A ) e. I ) )
38		eqid 2450	. . . . . . . . . . 11 |- ( invg ` ℂfld ) = ( invg ` ℂfld )
39	3, 38, 35	subginv 15776	. . . . . . . . . 10 |- ( ( ZZ e. (SubGrp ` ℂfld ) /\ A e. ZZ ) -> ( ( invg ` ℂfld ) ` A ) = ( ( invg ` Z ) ` A ) )
40	12, 39	mpan 670	. . . . . . . . 9 |- ( A e. ZZ -> ( ( invg ` ℂfld ) ` A ) = ( ( invg ` Z ) ` A ) )
41		zcn 10738	. . . . . . . . . 10 |- ( A e. ZZ -> A e. CC )
42		cnfldneg 17937	. . . . . . . . . 10 |- ( A e. CC -> ( ( invg ` ℂfld ) ` A ) = -u A )
43	41, 42	syl 16	. . . . . . . . 9 |- ( A e. ZZ -> ( ( invg ` ℂfld ) ` A ) = -u A )
44	40, 43	eqtr3d 2492	. . . . . . . 8 |- ( A e. ZZ -> ( ( invg ` Z ) ` A ) = -u A )
45	44	eleq1d 2518	. . . . . . 7 |- ( A e. ZZ -> ( ( ( invg ` Z ) ` A ) e. I <-> -u A e. I ) )
46	37, 45	bitrd 253	. . . . . 6 |- ( A e. ZZ -> ( A e. I <-> -u A e. I ) )
47	46	adantr 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 )
49	48	adantr 465	. . . . . . 7 |- ( ( A e. ZZ /\ -u A e. NN ) -> A e. RR )
50		nnnn0 10673	. . . . . . . . . 10 |- ( -u A e. NN -> -u A e. NN0 )
51	50	nn0ge0d 10726	. . . . . . . . 9 |- ( -u A e. NN -> 0 <_ -u A )
52	51	adantl 466	. . . . . . . 8 |- ( ( A e. ZZ /\ -u A e. NN ) -> 0 <_ -u A )
53	49	le0neg1d 9998	. . . . . . . 8 |- ( ( A e. ZZ /\ -u A e. NN ) -> ( A <_ 0 <-> 0 <_ -u A ) )
54	52, 53	mpbird 232	. . . . . . 7 |- ( ( A e. ZZ /\ -u A e. NN ) -> A <_ 0 )
55	49, 54	absnidd 12988	. . . . . 6 |- ( ( A e. ZZ /\ -u A e. NN ) -> ( abs ` A ) = -u A )
56	55	eleq1d 2518	. . . . 5 |- ( ( A e. ZZ /\ -u A e. NN ) -> ( ( abs ` A ) e. Prime <-> -u A e. Prime ) )
57	34, 47, 56	3bitr4d 285	. . . 4 |- ( ( A e. ZZ /\ -u A e. NN ) -> ( A e. I <-> ( abs ` A ) e. Prime ) )
58	57	adantrl 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 ) ) )
60	59	biimpi 194	. . 3 |- ( A e. ZZ -> ( A e. NN0 \/ ( A e. RR /\ -u A e. NN ) ) )
61	32, 58, 60	mpjaodan 784	. 2 |- ( A e. ZZ -> ( A e. I <-> ( abs ` A ) e. Prime ) )
62	6, 61	biadan2 642	1 |- ( A e. I <-> ( A e. ZZ /\ ( abs ` A ) e. Prime ) )

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  ℂ_fldccnfld 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)