Theorem domnchr 19880

Description: The characteristic of a domain can only be zero or a prime. (Contributed by Stefan O'Rear, 6-Sep-2015.)

Assertion

Ref	Expression
domnchr	|- ( R e. Domn -> ( (chr ` R ) = 0 \/ (chr ` R ) e. Prime ) )

Proof of Theorem domnchr

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-ne 2795	. . 3 |- ( (chr ` R ) =/= 0 <-> -. (chr ` R ) = 0 )
2		domnring 19296	. . . . . . . . . 10 |- ( R e. Domn -> R e. Ring )
3		eqid 2622	. . . . . . . . . . 11 |- (chr ` R ) = (chr ` R )
4	3	chrcl 19874	. . . . . . . . . 10 |- ( R e. Ring -> (chr ` R ) e. NN0 )
5	2, 4	syl 17	. . . . . . . . 9 |- ( R e. Domn -> (chr ` R ) e. NN0 )
6	5	adantr 481	. . . . . . . 8 |- ( ( R e. Domn /\ (chr ` R ) =/= 0 ) -> (chr ` R ) e. NN0 )
7		simpr 477	. . . . . . . 8 |- ( ( R e. Domn /\ (chr ` R ) =/= 0 ) -> (chr ` R ) =/= 0 )
8		eldifsn 4317	. . . . . . . 8 |- ( (chr ` R ) e. ( NN0 \ { 0 } ) <-> ( (chr ` R ) e. NN0 /\ (chr ` R ) =/= 0 ) )
9	6, 7, 8	sylanbrc 698	. . . . . . 7 |- ( ( R e. Domn /\ (chr ` R ) =/= 0 ) -> (chr ` R ) e. ( NN0 \ { 0 } ) )
10		dfn2 11305	. . . . . . 7 |- NN = ( NN0 \ { 0 } )
11	9, 10	syl6eleqr 2712	. . . . . 6 |- ( ( R e. Domn /\ (chr ` R ) =/= 0 ) -> (chr ` R ) e. NN )
12		domnnzr 19295	. . . . . . . 8 |- ( R e. Domn -> R e. NzRing )
13		nzrring 19261	. . . . . . . . . 10 |- ( R e. NzRing -> R e. Ring )
14		chrnzr 19878	. . . . . . . . . 10 |- ( R e. Ring -> ( R e. NzRing <-> (chr ` R ) =/= 1 ) )
15	13, 14	syl 17	. . . . . . . . 9 |- ( R e. NzRing -> ( R e. NzRing <-> (chr ` R ) =/= 1 ) )
16	15	ibi 256	. . . . . . . 8 |- ( R e. NzRing -> (chr ` R ) =/= 1 )
17	12, 16	syl 17	. . . . . . 7 |- ( R e. Domn -> (chr ` R ) =/= 1 )
18	17	adantr 481	. . . . . 6 |- ( ( R e. Domn /\ (chr ` R ) =/= 0 ) -> (chr ` R ) =/= 1 )
19		eluz2b3 11762	. . . . . 6 |- ( (chr ` R ) e. ( ZZ>= ` 2 ) <-> ( (chr ` R ) e. NN /\ (chr ` R ) =/= 1 ) )
20	11, 18, 19	sylanbrc 698	. . . . 5 |- ( ( R e. Domn /\ (chr ` R ) =/= 0 ) -> (chr ` R ) e. ( ZZ>= ` 2 ) )
21	2	ad2antrr 762	. . . . . . . . . . . 12 |- ( ( ( R e. Domn /\ (chr ` R ) =/= 0 ) /\ ( x e. ZZ /\ y e. ZZ ) ) -> R e. Ring )
22		eqid 2622	. . . . . . . . . . . . 13 |- ( ZRHom ` R ) = ( ZRHom ` R )
23	22	zrhrhm 19860	. . . . . . . . . . . 12 |- ( R e. Ring -> ( ZRHom ` R ) e. (ℤring RingHom R ) )
24	21, 23	syl 17	. . . . . . . . . . 11 |- ( ( ( R e. Domn /\ (chr ` R ) =/= 0 ) /\ ( x e. ZZ /\ y e. ZZ ) ) -> ( ZRHom ` R ) e. (ℤring RingHom R ) )
25		simprl 794	. . . . . . . . . . 11 |- ( ( ( R e. Domn /\ (chr ` R ) =/= 0 ) /\ ( x e. ZZ /\ y e. ZZ ) ) -> x e. ZZ )
26		simprr 796	. . . . . . . . . . 11 |- ( ( ( R e. Domn /\ (chr ` R ) =/= 0 ) /\ ( x e. ZZ /\ y e. ZZ ) ) -> y e. ZZ )
27		zringbas 19824	. . . . . . . . . . . 12 |- ZZ = ( Base ` ℤring )
28		zringmulr 19827	. . . . . . . . . . . 12 |- x. = ( .r ` ℤring )
29		eqid 2622	. . . . . . . . . . . 12 |- ( .r ` R ) = ( .r ` R )
30	27, 28, 29	rhmmul 18727	. . . . . . . . . . 11 |- ( ( ( ZRHom ` R ) e. (ℤring RingHom R ) /\ x e. ZZ /\ y e. ZZ ) -> ( ( ZRHom ` R ) ` ( x x. y ) ) = ( ( ( ZRHom ` R ) ` x ) ( .r ` R ) ( ( ZRHom ` R ) ` y ) ) )
31	24, 25, 26, 30	syl3anc 1326	. . . . . . . . . 10 |- ( ( ( R e. Domn /\ (chr ` R ) =/= 0 ) /\ ( x e. ZZ /\ y e. ZZ ) ) -> ( ( ZRHom ` R ) ` ( x x. y ) ) = ( ( ( ZRHom ` R ) ` x ) ( .r ` R ) ( ( ZRHom ` R ) ` y ) ) )
32	31	eqeq1d 2624	. . . . . . . . 9 |- ( ( ( R e. Domn /\ (chr ` R ) =/= 0 ) /\ ( x e. ZZ /\ y e. ZZ ) ) -> ( ( ( ZRHom ` R ) ` ( x x. y ) ) = ( 0g ` R ) <-> ( ( ( ZRHom ` R ) ` x ) ( .r ` R ) ( ( ZRHom ` R ) ` y ) ) = ( 0g ` R ) ) )
33		simpll 790	. . . . . . . . . 10 |- ( ( ( R e. Domn /\ (chr ` R ) =/= 0 ) /\ ( x e. ZZ /\ y e. ZZ ) ) -> R e. Domn )
34		eqid 2622	. . . . . . . . . . . . 13 |- ( Base ` R ) = ( Base ` R )
35	27, 34	rhmf 18726	. . . . . . . . . . . 12 |- ( ( ZRHom ` R ) e. (ℤring RingHom R ) -> ( ZRHom ` R ) : ZZ --> ( Base ` R ) )
36	24, 35	syl 17	. . . . . . . . . . 11 |- ( ( ( R e. Domn /\ (chr ` R ) =/= 0 ) /\ ( x e. ZZ /\ y e. ZZ ) ) -> ( ZRHom ` R ) : ZZ --> ( Base ` R ) )
37	36, 25	ffvelrnd 6360	. . . . . . . . . 10 |- ( ( ( R e. Domn /\ (chr ` R ) =/= 0 ) /\ ( x e. ZZ /\ y e. ZZ ) ) -> ( ( ZRHom ` R ) ` x ) e. ( Base ` R ) )
38	36, 26	ffvelrnd 6360	. . . . . . . . . 10 |- ( ( ( R e. Domn /\ (chr ` R ) =/= 0 ) /\ ( x e. ZZ /\ y e. ZZ ) ) -> ( ( ZRHom ` R ) ` y ) e. ( Base ` R ) )
39		eqid 2622	. . . . . . . . . . 11 |- ( 0g ` R ) = ( 0g ` R )
40	34, 29, 39	domneq0 19297	. . . . . . . . . 10 |- ( ( R e. Domn /\ ( ( ZRHom ` R ) ` x ) e. ( Base ` R ) /\ ( ( ZRHom ` R ) ` y ) e. ( Base ` R ) ) -> ( ( ( ( ZRHom ` R ) ` x ) ( .r ` R ) ( ( ZRHom ` R ) ` y ) ) = ( 0g ` R ) <-> ( ( ( ZRHom ` R ) ` x ) = ( 0g ` R ) \/ ( ( ZRHom ` R ) ` y ) = ( 0g ` R ) ) ) )
41	33, 37, 38, 40	syl3anc 1326	. . . . . . . . 9 |- ( ( ( R e. Domn /\ (chr ` R ) =/= 0 ) /\ ( x e. ZZ /\ y e. ZZ ) ) -> ( ( ( ( ZRHom ` R ) ` x ) ( .r ` R ) ( ( ZRHom ` R ) ` y ) ) = ( 0g ` R ) <-> ( ( ( ZRHom ` R ) ` x ) = ( 0g ` R ) \/ ( ( ZRHom ` R ) ` y ) = ( 0g ` R ) ) ) )
42	32, 41	bitrd 268	. . . . . . . 8 |- ( ( ( R e. Domn /\ (chr ` R ) =/= 0 ) /\ ( x e. ZZ /\ y e. ZZ ) ) -> ( ( ( ZRHom ` R ) ` ( x x. y ) ) = ( 0g ` R ) <-> ( ( ( ZRHom ` R ) ` x ) = ( 0g ` R ) \/ ( ( ZRHom ` R ) ` y ) = ( 0g ` R ) ) ) )
43	42	biimpd 219	. . . . . . 7 |- ( ( ( R e. Domn /\ (chr ` R ) =/= 0 ) /\ ( x e. ZZ /\ y e. ZZ ) ) -> ( ( ( ZRHom ` R ) ` ( x x. y ) ) = ( 0g ` R ) -> ( ( ( ZRHom ` R ) ` x ) = ( 0g ` R ) \/ ( ( ZRHom ` R ) ` y ) = ( 0g ` R ) ) ) )
44		zmulcl 11426	. . . . . . . . 9 |- ( ( x e. ZZ /\ y e. ZZ ) -> ( x x. y ) e. ZZ )
45	44	adantl 482	. . . . . . . 8 |- ( ( ( R e. Domn /\ (chr ` R ) =/= 0 ) /\ ( x e. ZZ /\ y e. ZZ ) ) -> ( x x. y ) e. ZZ )
46	3, 22, 39	chrdvds 19876	. . . . . . . 8 |- ( ( R e. Ring /\ ( x x. y ) e. ZZ ) -> ( (chr ` R ) || ( x x. y ) <-> ( ( ZRHom ` R ) ` ( x x. y ) ) = ( 0g ` R ) ) )
47	21, 45, 46	syl2anc 693	. . . . . . 7 |- ( ( ( R e. Domn /\ (chr ` R ) =/= 0 ) /\ ( x e. ZZ /\ y e. ZZ ) ) -> ( (chr ` R ) || ( x x. y ) <-> ( ( ZRHom ` R ) ` ( x x. y ) ) = ( 0g ` R ) ) )
48	3, 22, 39	chrdvds 19876	. . . . . . . . 9 |- ( ( R e. Ring /\ x e. ZZ ) -> ( (chr ` R ) || x <-> ( ( ZRHom ` R ) ` x ) = ( 0g ` R ) ) )
49	21, 25, 48	syl2anc 693	. . . . . . . 8 |- ( ( ( R e. Domn /\ (chr ` R ) =/= 0 ) /\ ( x e. ZZ /\ y e. ZZ ) ) -> ( (chr ` R ) || x <-> ( ( ZRHom ` R ) ` x ) = ( 0g ` R ) ) )
50	3, 22, 39	chrdvds 19876	. . . . . . . . 9 |- ( ( R e. Ring /\ y e. ZZ ) -> ( (chr ` R ) || y <-> ( ( ZRHom ` R ) ` y ) = ( 0g ` R ) ) )
51	21, 26, 50	syl2anc 693	. . . . . . . 8 |- ( ( ( R e. Domn /\ (chr ` R ) =/= 0 ) /\ ( x e. ZZ /\ y e. ZZ ) ) -> ( (chr ` R ) || y <-> ( ( ZRHom ` R ) ` y ) = ( 0g ` R ) ) )
52	49, 51	orbi12d 746	. . . . . . 7 |- ( ( ( R e. Domn /\ (chr ` R ) =/= 0 ) /\ ( x e. ZZ /\ y e. ZZ ) ) -> ( ( (chr ` R ) || x \/ (chr ` R ) || y ) <-> ( ( ( ZRHom ` R ) ` x ) = ( 0g ` R ) \/ ( ( ZRHom ` R ) ` y ) = ( 0g ` R ) ) ) )
53	43, 47, 52	3imtr4d 283	. . . . . 6 |- ( ( ( R e. Domn /\ (chr ` R ) =/= 0 ) /\ ( x e. ZZ /\ y e. ZZ ) ) -> ( (chr ` R ) || ( x x. y ) -> ( (chr ` R ) || x \/ (chr ` R ) || y ) ) )
54	53	ralrimivva 2971	. . . . 5 |- ( ( R e. Domn /\ (chr ` R ) =/= 0 ) -> A. x e. ZZ A. y e. ZZ ( (chr ` R ) || ( x x. y ) -> ( (chr ` R ) || x \/ (chr ` R ) || y ) ) )
55		isprm6 15426	. . . . 5 |- ( (chr ` R ) e. Prime <-> ( (chr ` R ) e. ( ZZ>= ` 2 ) /\ A. x e. ZZ A. y e. ZZ ( (chr ` R ) || ( x x. y ) -> ( (chr ` R ) || x \/ (chr ` R ) || y ) ) ) )
56	20, 54, 55	sylanbrc 698	. . . 4 |- ( ( R e. Domn /\ (chr ` R ) =/= 0 ) -> (chr ` R ) e. Prime )
57	56	ex 450	. . 3 |- ( R e. Domn -> ( (chr ` R ) =/= 0 -> (chr ` R ) e. Prime ) )
58	1, 57	syl5bir 233	. 2 |- ( R e. Domn -> ( -. (chr ` R ) = 0 -> (chr ` R ) e. Prime ) )
59	58	orrd 393	1 |- ( R e. Domn -> ( (chr ` R ) = 0 \/ (chr ` R ) e. Prime ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   \/ wo 383   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   A.wral 2912   \ cdif 3571   {csn 4177   class class class wbr 4653   -->wf 5884   ` cfv 5888  (class class class)co 6650   0cc0 9936   1c1 9937   x. cmul 9941   NNcn 11020   2c2 11070   NN0cn0 11292   ZZcz 11377   ZZ>=cuz 11687   || cdvds 14983   Primecprime 15385   Basecbs 15857   .rcmulr 15942   0gc0g 16100   Ringcrg 18547   RingHom crh 18712  NzRingcnzr 19257  Domncdomn 19280  ℤ_ringzring 19818   ZRHomczrh 19848  chrcchr 19850

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-inf2 8538  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014  ax-addf 10015  ax-mulf 10016

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-2o 7561  df-oadd 7564  df-er 7742  df-map 7859  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-sup 8348  df-inf 8349  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-3 11080  df-4 11081  df-5 11082  df-6 11083  df-7 11084  df-8 11085  df-9 11086  df-n0 11293  df-z 11378  df-dec 11494  df-uz 11688  df-rp 11833  df-fz 12327  df-fl 12593  df-mod 12669  df-seq 12802  df-exp 12861  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-dvds 14984  df-gcd 15217  df-prm 15386  df-struct 15859  df-ndx 15860  df-slot 15861  df-base 15863  df-sets 15864  df-ress 15865  df-plusg 15954  df-mulr 15955  df-starv 15956  df-tset 15960  df-ple 15961  df-ds 15964  df-unif 15965  df-0g 16102  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-mhm 17335  df-grp 17425  df-minusg 17426  df-sbg 17427  df-mulg 17541  df-subg 17591  df-ghm 17658  df-od 17948  df-cmn 18195  df-mgp 18490  df-ur 18502  df-ring 18549  df-cring 18550  df-rnghom 18715  df-subrg 18778  df-nzr 19258  df-domn 19284  df-cnfld 19747  df-zring 19819  df-zrh 19852  df-chr 19854

This theorem is referenced by: (None)