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 = (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 18339	. . . 4 |- ZZ e. (SubRing ` ℂfld )
3		prmirredOLD.1	. . . . 5 |- Z = (ℂfld ↾s ZZ )
4	3	subrgbas 17307	. . . 4 |- ( ZZ e. (SubRing ` ℂfld ) -> ZZ = ( Base ` Z ) )
5	2, 4	ax-mp 5	. . 3 |- ZZ = ( Base ` Z )
6	1, 5	irredcl 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 ) )
9	3	subrgring 17301	. . . . . . . . . . . 12 |- ( ZZ e. (SubRing ` ℂfld ) -> Z e. Ring )
10	2, 9	ax-mp 5	. . . . . . . . . . 11 |- Z e. Ring
11		subrgsubg 17304	. . . . . . . . . . . . . 14 |- ( ZZ e. (SubRing ` ℂfld ) -> ZZ e. (SubGrp ` ℂfld ) )
12	2, 11	ax-mp 5	. . . . . . . . . . . . 13 |- ZZ e. (SubGrp ` ℂfld )
13		cnfld0 18310	. . . . . . . . . . . . . 14 |- 0 = ( 0g ` ℂfld )
14	3, 13	subg0 16078	. . . . . . . . . . . . 13 |- ( ZZ e. (SubGrp ` ℂfld ) -> 0 = ( 0g ` Z ) )
15	12, 14	ax-mp 5	. . . . . . . . . . . 12 |- 0 = ( 0g ` Z )
16	1, 15	irredn0 17222	. . . . . . . . . . 11 |- ( ( Z e. Ring /\ A e. I ) -> A =/= 0 )
17	10, 16	mpan 670	. . . . . . . . . 10 |- ( A e. I -> A =/= 0 )
18	17	necon2bi 2704	. . . . . . . . 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 14095	. . . . . . 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 18393	. . . . . . 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 10816	. . . . . . 7 |- ( A e. NN0 -> A e. RR )
28		nn0ge0 10833	. . . . . . 7 |- ( A e. NN0 -> 0 <_ A )
29	27, 28	absidd 13233	. . . . . 6 |- ( A e. NN0 -> ( abs ` A ) = A )
30	29	eleq1d 2536	. . . . 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 18393	. . . . . 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 2467	. . . . . . . . 9 |- ( invg ` Z ) = ( invg ` Z )
36	1, 35, 5	irrednegb 17230	. . . . . . . 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 2467	. . . . . . . . . . 11 |- ( invg ` ℂfld ) = ( invg ` ℂfld )
39	3, 38, 35	subginv 16079	. . . . . . . . . 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 10881	. . . . . . . . . 10 |- ( A e. ZZ -> A e. CC )
42		cnfldneg 18312	. . . . . . . . . 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 2510	. . . . . . . 8 |- ( A e. ZZ -> ( ( invg ` Z ) ` A ) = -u A )
45	44	eleq1d 2536	. . . . . . 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 10880	. . . . . . . 8 |- ( A e. ZZ -> A e. RR )
49	48	adantr 465	. . . . . . 7 |- ( ( A e. ZZ /\ -u A e. NN ) -> A e. RR )
50		nnnn0 10814	. . . . . . . . . 10 |- ( -u A e. NN -> -u A e. NN0 )
51	50	nn0ge0d 10867	. . . . . . . . 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 10136	. . . . . . . 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 13224	. . . . . 6 |- ( ( A e. ZZ /\ -u A e. NN ) -> ( abs ` A ) = -u A )
56	55	eleq1d 2536	. . . . 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 10890	. . . 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 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  ℂ_fldccnfld 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)