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Theorem domnchr 16803
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.
StepHypRef Expression
1 df-ne 2600 . . 3  |-  ( (chr
`  R )  =/=  0  <->  -.  (chr `  R
)  =  0 )
2 domnrng 16346 . . . . . . . . . 10  |-  ( R  e. Domn  ->  R  e.  Ring )
3 eqid 2435 . . . . . . . . . . 11  |-  (chr `  R )  =  (chr
`  R )
43chrcl 16797 . . . . . . . . . 10  |-  ( R  e.  Ring  ->  (chr `  R )  e.  NN0 )
52, 4syl 16 . . . . . . . . 9  |-  ( R  e. Domn  ->  (chr `  R
)  e.  NN0 )
65adantr 452 . . . . . . . 8  |-  ( ( R  e. Domn  /\  (chr `  R )  =/=  0
)  ->  (chr `  R
)  e.  NN0 )
7 simpr 448 . . . . . . . 8  |-  ( ( R  e. Domn  /\  (chr `  R )  =/=  0
)  ->  (chr `  R
)  =/=  0 )
8 eldifsn 3919 . . . . . . . 8  |-  ( (chr
`  R )  e.  ( NN0  \  {
0 } )  <->  ( (chr `  R )  e.  NN0  /\  (chr `  R )  =/=  0 ) )
96, 7, 8sylanbrc 646 . . . . . . 7  |-  ( ( R  e. Domn  /\  (chr `  R )  =/=  0
)  ->  (chr `  R
)  e.  ( NN0  \  { 0 } ) )
10 dfn2 10224 . . . . . . 7  |-  NN  =  ( NN0  \  { 0 } )
119, 10syl6eleqr 2526 . . . . . 6  |-  ( ( R  e. Domn  /\  (chr `  R )  =/=  0
)  ->  (chr `  R
)  e.  NN )
12 domnnzr 16345 . . . . . . . 8  |-  ( R  e. Domn  ->  R  e. NzRing )
13 nzrrng 16322 . . . . . . . . . 10  |-  ( R  e. NzRing  ->  R  e.  Ring )
14 chrnzr 16801 . . . . . . . . . 10  |-  ( R  e.  Ring  ->  ( R  e. NzRing 
<->  (chr `  R )  =/=  1 ) )
1513, 14syl 16 . . . . . . . . 9  |-  ( R  e. NzRing  ->  ( R  e. NzRing  <->  (chr
`  R )  =/=  1 ) )
1615ibi 233 . . . . . . . 8  |-  ( R  e. NzRing  ->  (chr `  R
)  =/=  1 )
1712, 16syl 16 . . . . . . 7  |-  ( R  e. Domn  ->  (chr `  R
)  =/=  1 )
1817adantr 452 . . . . . 6  |-  ( ( R  e. Domn  /\  (chr `  R )  =/=  0
)  ->  (chr `  R
)  =/=  1 )
19 eluz2b3 10539 . . . . . 6  |-  ( (chr
`  R )  e.  ( ZZ>= `  2 )  <->  ( (chr `  R )  e.  NN  /\  (chr `  R )  =/=  1
) )
2011, 18, 19sylanbrc 646 . . . . 5  |-  ( ( R  e. Domn  /\  (chr `  R )  =/=  0
)  ->  (chr `  R
)  e.  ( ZZ>= ` 
2 ) )
212ad2antrr 707 . . . . . . . . . . . 12  |-  ( ( ( R  e. Domn  /\  (chr `  R )  =/=  0 )  /\  (
x  e.  ZZ  /\  y  e.  ZZ )
)  ->  R  e.  Ring )
22 eqid 2435 . . . . . . . . . . . . 13  |-  (flds  ZZ )  =  (flds  ZZ )
23 eqid 2435 . . . . . . . . . . . . 13  |-  ( ZRHom `  R )  =  ( ZRHom `  R )
2422, 23zrhrhm 16783 . . . . . . . . . . . 12  |-  ( R  e.  Ring  ->  ( ZRHom `  R )  e.  ( (flds  ZZ ) RingHom  R ) )
2521, 24syl 16 . . . . . . . . . . 11  |-  ( ( ( R  e. Domn  /\  (chr `  R )  =/=  0 )  /\  (
x  e.  ZZ  /\  y  e.  ZZ )
)  ->  ( ZRHom `  R )  e.  ( (flds  ZZ ) RingHom  R ) )
26 simprl 733 . . . . . . . . . . 11  |-  ( ( ( R  e. Domn  /\  (chr `  R )  =/=  0 )  /\  (
x  e.  ZZ  /\  y  e.  ZZ )
)  ->  x  e.  ZZ )
27 simprr 734 . . . . . . . . . . 11  |-  ( ( ( R  e. Domn  /\  (chr `  R )  =/=  0 )  /\  (
x  e.  ZZ  /\  y  e.  ZZ )
)  ->  y  e.  ZZ )
28 zsscn 10280 . . . . . . . . . . . . 13  |-  ZZ  C_  CC
29 cnfldbas 16697 . . . . . . . . . . . . . 14  |-  CC  =  ( Base ` fld )
3022, 29ressbas2 13510 . . . . . . . . . . . . 13  |-  ( ZZ  C_  CC  ->  ZZ  =  ( Base `  (flds  ZZ ) ) )
3128, 30ax-mp 8 . . . . . . . . . . . 12  |-  ZZ  =  ( Base `  (flds  ZZ ) )
32 zex 10281 . . . . . . . . . . . . 13  |-  ZZ  e.  _V
33 cnfldmul 16699 . . . . . . . . . . . . . 14  |-  x.  =  ( .r ` fld )
3422, 33ressmulr 13572 . . . . . . . . . . . . 13  |-  ( ZZ  e.  _V  ->  x.  =  ( .r `  (flds  ZZ ) ) )
3532, 34ax-mp 8 . . . . . . . . . . . 12  |-  x.  =  ( .r `  (flds  ZZ ) )
36 eqid 2435 . . . . . . . . . . . 12  |-  ( .r
`  R )  =  ( .r `  R
)
3731, 35, 36rhmmul 15818 . . . . . . . . . . 11  |-  ( ( ( ZRHom `  R
)  e.  ( (flds  ZZ ) RingHom  R )  /\  x  e.  ZZ  /\  y  e.  ZZ )  ->  (
( ZRHom `  R
) `  ( x  x.  y ) )  =  ( ( ( ZRHom `  R ) `  x
) ( .r `  R ) ( ( ZRHom `  R ) `  y ) ) )
3825, 26, 27, 37syl3anc 1184 . . . . . . . . . 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 ) ) )
3938eqeq1d 2443 . . . . . . . . 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 ) ) )
40 simpll 731 . . . . . . . . . 10  |-  ( ( ( R  e. Domn  /\  (chr `  R )  =/=  0 )  /\  (
x  e.  ZZ  /\  y  e.  ZZ )
)  ->  R  e. Domn )
41 eqid 2435 . . . . . . . . . . . . 13  |-  ( Base `  R )  =  (
Base `  R )
4231, 41rhmf 15817 . . . . . . . . . . . 12  |-  ( ( ZRHom `  R )  e.  ( (flds  ZZ ) RingHom  R )  ->  ( ZRHom `  R ) : ZZ --> ( Base `  R
) )
4325, 42syl 16 . . . . . . . . . . 11  |-  ( ( ( R  e. Domn  /\  (chr `  R )  =/=  0 )  /\  (
x  e.  ZZ  /\  y  e.  ZZ )
)  ->  ( ZRHom `  R ) : ZZ --> ( Base `  R )
)
4443, 26ffvelrnd 5863 . . . . . . . . . 10  |-  ( ( ( R  e. Domn  /\  (chr `  R )  =/=  0 )  /\  (
x  e.  ZZ  /\  y  e.  ZZ )
)  ->  ( ( ZRHom `  R ) `  x )  e.  (
Base `  R )
)
4543, 27ffvelrnd 5863 . . . . . . . . . 10  |-  ( ( ( R  e. Domn  /\  (chr `  R )  =/=  0 )  /\  (
x  e.  ZZ  /\  y  e.  ZZ )
)  ->  ( ( ZRHom `  R ) `  y )  e.  (
Base `  R )
)
46 eqid 2435 . . . . . . . . . . 11  |-  ( 0g
`  R )  =  ( 0g `  R
)
4741, 36, 46domneq0 16347 . . . . . . . . . 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 ) ) ) )
4840, 44, 45, 47syl3anc 1184 . . . . . . . . 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 ) ) ) )
4939, 48bitrd 245 . . . . . . . 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 ) ) ) )
5049biimpd 199 . . . . . . 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 ) ) ) )
51 zmulcl 10314 . . . . . . . . 9  |-  ( ( x  e.  ZZ  /\  y  e.  ZZ )  ->  ( x  x.  y
)  e.  ZZ )
5251adantl 453 . . . . . . . 8  |-  ( ( ( R  e. Domn  /\  (chr `  R )  =/=  0 )  /\  (
x  e.  ZZ  /\  y  e.  ZZ )
)  ->  ( x  x.  y )  e.  ZZ )
533, 23, 46chrdvds 16799 . . . . . . . 8  |-  ( ( R  e.  Ring  /\  (
x  x.  y )  e.  ZZ )  -> 
( (chr `  R
)  ||  ( x  x.  y )  <->  ( ( ZRHom `  R ) `  ( x  x.  y
) )  =  ( 0g `  R ) ) )
5421, 52, 53syl2anc 643 . . . . . . 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 ) ) )
553, 23, 46chrdvds 16799 . . . . . . . . 9  |-  ( ( R  e.  Ring  /\  x  e.  ZZ )  ->  (
(chr `  R )  ||  x  <->  ( ( ZRHom `  R ) `  x
)  =  ( 0g
`  R ) ) )
5621, 26, 55syl2anc 643 . . . . . . . 8  |-  ( ( ( R  e. Domn  /\  (chr `  R )  =/=  0 )  /\  (
x  e.  ZZ  /\  y  e.  ZZ )
)  ->  ( (chr `  R )  ||  x  <->  ( ( ZRHom `  R
) `  x )  =  ( 0g `  R ) ) )
573, 23, 46chrdvds 16799 . . . . . . . . 9  |-  ( ( R  e.  Ring  /\  y  e.  ZZ )  ->  (
(chr `  R )  ||  y  <->  ( ( ZRHom `  R ) `  y
)  =  ( 0g
`  R ) ) )
5821, 27, 57syl2anc 643 . . . . . . . 8  |-  ( ( ( R  e. Domn  /\  (chr `  R )  =/=  0 )  /\  (
x  e.  ZZ  /\  y  e.  ZZ )
)  ->  ( (chr `  R )  ||  y  <->  ( ( ZRHom `  R
) `  y )  =  ( 0g `  R ) ) )
5956, 58orbi12d 691 . . . . . . 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 ) ) ) )
6050, 54, 593imtr4d 260 . . . . . 6  |-  ( ( ( R  e. Domn  /\  (chr `  R )  =/=  0 )  /\  (
x  e.  ZZ  /\  y  e.  ZZ )
)  ->  ( (chr `  R )  ||  (
x  x.  y )  ->  ( (chr `  R )  ||  x  \/  (chr `  R )  ||  y ) ) )
6160ralrimivva 2790 . . . . 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 ) ) )
62 isprm6 13099 . . . . 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 ) ) ) )
6320, 61, 62sylanbrc 646 . . . 4  |-  ( ( R  e. Domn  /\  (chr `  R )  =/=  0
)  ->  (chr `  R
)  e.  Prime )
6463ex 424 . . 3  |-  ( R  e. Domn  ->  ( (chr `  R )  =/=  0  ->  (chr `  R )  e.  Prime ) )
651, 64syl5bir 210 . 2  |-  ( R  e. Domn  ->  ( -.  (chr `  R )  =  0  ->  (chr `  R
)  e.  Prime )
)
6665orrd 368 1  |-  ( R  e. Domn  ->  ( (chr `  R )  =  0  \/  (chr `  R
)  e.  Prime )
)
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    \/ wo 358    /\ wa 359    = wceq 1652    e. wcel 1725    =/= wne 2598   A.wral 2697   _Vcvv 2948    \ cdif 3309    C_ wss 3312   {csn 3806   class class class wbr 4204   -->wf 5442   ` cfv 5446  (class class class)co 6073   CCcc 8978   0cc0 8980   1c1 8981    x. cmul 8985   NNcn 9990   2c2 10039   NN0cn0 10211   ZZcz 10272   ZZ>=cuz 10478    || cdivides 12842   Primecprime 13069   Basecbs 13459   ↾s cress 13460   .rcmulr 13520   0gc0g 13713   Ringcrg 15650   RingHom crh 15807  NzRingcnzr 16318  Domncdomn 16330  ℂfldccnfld 16693   ZRHomczrh 16768  chrcchr 16770
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-inf2 7586  ax-cnex 9036  ax-resscn 9037  ax-1cn 9038  ax-icn 9039  ax-addcl 9040  ax-addrcl 9041  ax-mulcl 9042  ax-mulrcl 9043  ax-mulcom 9044  ax-addass 9045  ax-mulass 9046  ax-distr 9047  ax-i2m1 9048  ax-1ne0 9049  ax-1rid 9050  ax-rnegex 9051  ax-rrecex 9052  ax-cnre 9053  ax-pre-lttri 9054  ax-pre-lttrn 9055  ax-pre-ltadd 9056  ax-pre-mulgt0 9057  ax-pre-sup 9058  ax-addf 9059  ax-mulf 9060
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-int 4043  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-riota 6541  df-recs 6625  df-rdg 6660  df-1o 6716  df-2o 6717  df-oadd 6720  df-er 6897  df-map 7012  df-en 7102  df-dom 7103  df-sdom 7104  df-fin 7105  df-sup 7438  df-pnf 9112  df-mnf 9113  df-xr 9114  df-ltxr 9115  df-le 9116  df-sub 9283  df-neg 9284  df-div 9668  df-nn 9991  df-2 10048  df-3 10049  df-4 10050  df-5 10051  df-6 10052  df-7 10053  df-8 10054  df-9 10055  df-10 10056  df-n0 10212  df-z 10273  df-dec 10373  df-uz 10479  df-rp 10603  df-fz 11034  df-fl 11192  df-mod 11241  df-seq 11314  df-exp 11373  df-cj 11894  df-re 11895  df-im 11896  df-sqr 12030  df-abs 12031  df-dvds 12843  df-gcd 12997  df-prm 13070  df-struct 13461  df-ndx 13462  df-slot 13463  df-base 13464  df-sets 13465  df-ress 13466  df-plusg 13532  df-mulr 13533  df-starv 13534  df-tset 13538  df-ple 13539  df-ds 13541  df-unif 13542  df-0g 13717  df-mnd 14680  df-mhm 14728  df-grp 14802  df-minusg 14803  df-sbg 14804  df-mulg 14805  df-subg 14931  df-ghm 14994  df-od 15157  df-cmn 15404  df-mgp 15639  df-rng 15653  df-cring 15654  df-ur 15655  df-rnghom 15809  df-subrg 15856  df-nzr 16319  df-domn 16334  df-cnfld 16694  df-zrh 16772  df-chr 16774
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