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Theorem domnchr 16502
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 2461 . . 3  |-  ( (chr
`  R )  =/=  0  <->  -.  (chr `  R
)  =  0 )
2 domnrng 16053 . . . . . . . . . 10  |-  ( R  e. Domn  ->  R  e.  Ring )
3 eqid 2296 . . . . . . . . . . 11  |-  (chr `  R )  =  (chr
`  R )
43chrcl 16496 . . . . . . . . . 10  |-  ( R  e.  Ring  ->  (chr `  R )  e.  NN0 )
52, 4syl 15 . . . . . . . . 9  |-  ( R  e. Domn  ->  (chr `  R
)  e.  NN0 )
65adantr 451 . . . . . . . 8  |-  ( ( R  e. Domn  /\  (chr `  R )  =/=  0
)  ->  (chr `  R
)  e.  NN0 )
7 simpr 447 . . . . . . . 8  |-  ( ( R  e. Domn  /\  (chr `  R )  =/=  0
)  ->  (chr `  R
)  =/=  0 )
8 eldifsn 3762 . . . . . . . 8  |-  ( (chr
`  R )  e.  ( NN0  \  {
0 } )  <->  ( (chr `  R )  e.  NN0  /\  (chr `  R )  =/=  0 ) )
96, 7, 8sylanbrc 645 . . . . . . 7  |-  ( ( R  e. Domn  /\  (chr `  R )  =/=  0
)  ->  (chr `  R
)  e.  ( NN0  \  { 0 } ) )
10 dfn2 9994 . . . . . . 7  |-  NN  =  ( NN0  \  { 0 } )
119, 10syl6eleqr 2387 . . . . . 6  |-  ( ( R  e. Domn  /\  (chr `  R )  =/=  0
)  ->  (chr `  R
)  e.  NN )
12 domnnzr 16052 . . . . . . . 8  |-  ( R  e. Domn  ->  R  e. NzRing )
13 nzrrng 16029 . . . . . . . . . 10  |-  ( R  e. NzRing  ->  R  e.  Ring )
14 chrnzr 16500 . . . . . . . . . 10  |-  ( R  e.  Ring  ->  ( R  e. NzRing 
<->  (chr `  R )  =/=  1 ) )
1513, 14syl 15 . . . . . . . . 9  |-  ( R  e. NzRing  ->  ( R  e. NzRing  <->  (chr
`  R )  =/=  1 ) )
1615ibi 232 . . . . . . . 8  |-  ( R  e. NzRing  ->  (chr `  R
)  =/=  1 )
1712, 16syl 15 . . . . . . 7  |-  ( R  e. Domn  ->  (chr `  R
)  =/=  1 )
1817adantr 451 . . . . . 6  |-  ( ( R  e. Domn  /\  (chr `  R )  =/=  0
)  ->  (chr `  R
)  =/=  1 )
19 eluz2b3 10307 . . . . . 6  |-  ( (chr
`  R )  e.  ( ZZ>= `  2 )  <->  ( (chr `  R )  e.  NN  /\  (chr `  R )  =/=  1
) )
2011, 18, 19sylanbrc 645 . . . . 5  |-  ( ( R  e. Domn  /\  (chr `  R )  =/=  0
)  ->  (chr `  R
)  e.  ( ZZ>= ` 
2 ) )
212ad2antrr 706 . . . . . . . . . . . 12  |-  ( ( ( R  e. Domn  /\  (chr `  R )  =/=  0 )  /\  (
x  e.  ZZ  /\  y  e.  ZZ )
)  ->  R  e.  Ring )
22 eqid 2296 . . . . . . . . . . . . 13  |-  (flds  ZZ )  =  (flds  ZZ )
23 eqid 2296 . . . . . . . . . . . . 13  |-  ( ZRHom `  R )  =  ( ZRHom `  R )
2422, 23zrhrhm 16482 . . . . . . . . . . . 12  |-  ( R  e.  Ring  ->  ( ZRHom `  R )  e.  ( (flds  ZZ ) RingHom  R ) )
2521, 24syl 15 . . . . . . . . . . 11  |-  ( ( ( R  e. Domn  /\  (chr `  R )  =/=  0 )  /\  (
x  e.  ZZ  /\  y  e.  ZZ )
)  ->  ( ZRHom `  R )  e.  ( (flds  ZZ ) RingHom  R ) )
26 simprl 732 . . . . . . . . . . 11  |-  ( ( ( R  e. Domn  /\  (chr `  R )  =/=  0 )  /\  (
x  e.  ZZ  /\  y  e.  ZZ )
)  ->  x  e.  ZZ )
27 simprr 733 . . . . . . . . . . 11  |-  ( ( ( R  e. Domn  /\  (chr `  R )  =/=  0 )  /\  (
x  e.  ZZ  /\  y  e.  ZZ )
)  ->  y  e.  ZZ )
28 zsscn 10048 . . . . . . . . . . . . 13  |-  ZZ  C_  CC
29 cnfldbas 16399 . . . . . . . . . . . . . 14  |-  CC  =  ( Base ` fld )
3022, 29ressbas2 13215 . . . . . . . . . . . . 13  |-  ( ZZ  C_  CC  ->  ZZ  =  ( Base `  (flds  ZZ ) ) )
3128, 30ax-mp 8 . . . . . . . . . . . 12  |-  ZZ  =  ( Base `  (flds  ZZ ) )
32 zex 10049 . . . . . . . . . . . . 13  |-  ZZ  e.  _V
33 cnfldmul 16401 . . . . . . . . . . . . . 14  |-  x.  =  ( .r ` fld )
3422, 33ressmulr 13277 . . . . . . . . . . . . 13  |-  ( ZZ  e.  _V  ->  x.  =  ( .r `  (flds  ZZ ) ) )
3532, 34ax-mp 8 . . . . . . . . . . . 12  |-  x.  =  ( .r `  (flds  ZZ ) )
36 eqid 2296 . . . . . . . . . . . 12  |-  ( .r
`  R )  =  ( .r `  R
)
3731, 35, 36rhmmul 15521 . . . . . . . . . . 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 1182 . . . . . . . . . 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 2304 . . . . . . . . 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 730 . . . . . . . . . 10  |-  ( ( ( R  e. Domn  /\  (chr `  R )  =/=  0 )  /\  (
x  e.  ZZ  /\  y  e.  ZZ )
)  ->  R  e. Domn )
41 eqid 2296 . . . . . . . . . . . . 13  |-  ( Base `  R )  =  (
Base `  R )
4231, 41rhmf 15520 . . . . . . . . . . . 12  |-  ( ( ZRHom `  R )  e.  ( (flds  ZZ ) RingHom  R )  ->  ( ZRHom `  R ) : ZZ --> ( Base `  R
) )
4325, 42syl 15 . . . . . . . . . . 11  |-  ( ( ( R  e. Domn  /\  (chr `  R )  =/=  0 )  /\  (
x  e.  ZZ  /\  y  e.  ZZ )
)  ->  ( ZRHom `  R ) : ZZ --> ( Base `  R )
)
44 ffvelrn 5679 . . . . . . . . . . 11  |-  ( ( ( ZRHom `  R
) : ZZ --> ( Base `  R )  /\  x  e.  ZZ )  ->  (
( ZRHom `  R
) `  x )  e.  ( Base `  R
) )
4543, 26, 44syl2anc 642 . . . . . . . . . 10  |-  ( ( ( R  e. Domn  /\  (chr `  R )  =/=  0 )  /\  (
x  e.  ZZ  /\  y  e.  ZZ )
)  ->  ( ( ZRHom `  R ) `  x )  e.  (
Base `  R )
)
46 ffvelrn 5679 . . . . . . . . . . 11  |-  ( ( ( ZRHom `  R
) : ZZ --> ( Base `  R )  /\  y  e.  ZZ )  ->  (
( ZRHom `  R
) `  y )  e.  ( Base `  R
) )
4743, 27, 46syl2anc 642 . . . . . . . . . 10  |-  ( ( ( R  e. Domn  /\  (chr `  R )  =/=  0 )  /\  (
x  e.  ZZ  /\  y  e.  ZZ )
)  ->  ( ( ZRHom `  R ) `  y )  e.  (
Base `  R )
)
48 eqid 2296 . . . . . . . . . . 11  |-  ( 0g
`  R )  =  ( 0g `  R
)
4941, 36, 48domneq0 16054 . . . . . . . . . 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 ) ) ) )
5040, 45, 47, 49syl3anc 1182 . . . . . . . . 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 ) ) ) )
5139, 50bitrd 244 . . . . . . . 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 ) ) ) )
5251biimpd 198 . . . . . . 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 ) ) ) )
53 zmulcl 10082 . . . . . . . . 9  |-  ( ( x  e.  ZZ  /\  y  e.  ZZ )  ->  ( x  x.  y
)  e.  ZZ )
5453adantl 452 . . . . . . . 8  |-  ( ( ( R  e. Domn  /\  (chr `  R )  =/=  0 )  /\  (
x  e.  ZZ  /\  y  e.  ZZ )
)  ->  ( x  x.  y )  e.  ZZ )
553, 23, 48chrdvds 16498 . . . . . . . 8  |-  ( ( R  e.  Ring  /\  (
x  x.  y )  e.  ZZ )  -> 
( (chr `  R
)  ||  ( x  x.  y )  <->  ( ( ZRHom `  R ) `  ( x  x.  y
) )  =  ( 0g `  R ) ) )
5621, 54, 55syl2anc 642 . . . . . . 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 ) ) )
573, 23, 48chrdvds 16498 . . . . . . . . 9  |-  ( ( R  e.  Ring  /\  x  e.  ZZ )  ->  (
(chr `  R )  ||  x  <->  ( ( ZRHom `  R ) `  x
)  =  ( 0g
`  R ) ) )
5821, 26, 57syl2anc 642 . . . . . . . 8  |-  ( ( ( R  e. Domn  /\  (chr `  R )  =/=  0 )  /\  (
x  e.  ZZ  /\  y  e.  ZZ )
)  ->  ( (chr `  R )  ||  x  <->  ( ( ZRHom `  R
) `  x )  =  ( 0g `  R ) ) )
593, 23, 48chrdvds 16498 . . . . . . . . 9  |-  ( ( R  e.  Ring  /\  y  e.  ZZ )  ->  (
(chr `  R )  ||  y  <->  ( ( ZRHom `  R ) `  y
)  =  ( 0g
`  R ) ) )
6021, 27, 59syl2anc 642 . . . . . . . 8  |-  ( ( ( R  e. Domn  /\  (chr `  R )  =/=  0 )  /\  (
x  e.  ZZ  /\  y  e.  ZZ )
)  ->  ( (chr `  R )  ||  y  <->  ( ( ZRHom `  R
) `  y )  =  ( 0g `  R ) ) )
6158, 60orbi12d 690 . . . . . . 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 ) ) ) )
6252, 56, 613imtr4d 259 . . . . . 6  |-  ( ( ( R  e. Domn  /\  (chr `  R )  =/=  0 )  /\  (
x  e.  ZZ  /\  y  e.  ZZ )
)  ->  ( (chr `  R )  ||  (
x  x.  y )  ->  ( (chr `  R )  ||  x  \/  (chr `  R )  ||  y ) ) )
6362ralrimivva 2648 . . . . 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 ) ) )
64 isprm6 12804 . . . . 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 ) ) ) )
6520, 63, 64sylanbrc 645 . . . 4  |-  ( ( R  e. Domn  /\  (chr `  R )  =/=  0
)  ->  (chr `  R
)  e.  Prime )
6665ex 423 . . 3  |-  ( R  e. Domn  ->  ( (chr `  R )  =/=  0  ->  (chr `  R )  e.  Prime ) )
671, 66syl5bir 209 . 2  |-  ( R  e. Domn  ->  ( -.  (chr `  R )  =  0  ->  (chr `  R
)  e.  Prime )
)
6867orrd 367 1  |-  ( R  e. Domn  ->  ( (chr `  R )  =  0  \/  (chr `  R
)  e.  Prime )
)
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    \/ wo 357    /\ wa 358    = wceq 1632    e. wcel 1696    =/= wne 2459   A.wral 2556   _Vcvv 2801    \ cdif 3162    C_ wss 3165   {csn 3653   class class class wbr 4039   -->wf 5267   ` cfv 5271  (class class class)co 5874   CCcc 8751   0cc0 8753   1c1 8754    x. cmul 8758   NNcn 9762   2c2 9811   NN0cn0 9981   ZZcz 10040   ZZ>=cuz 10246    || cdivides 12547   Primecprime 12774   Basecbs 13164   ↾s cress 13165   .rcmulr 13225   0gc0g 13416   Ringcrg 15353   RingHom crh 15510  NzRingcnzr 16025  Domncdomn 16037  ℂfldccnfld 16393   ZRHomczrh 16467  chrcchr 16469
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-inf2 7358  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830  ax-pre-sup 8831  ax-addf 8832  ax-mulf 8833
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-1o 6495  df-2o 6496  df-oadd 6499  df-er 6676  df-map 6790  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-sup 7210  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-div 9440  df-nn 9763  df-2 9820  df-3 9821  df-4 9822  df-5 9823  df-6 9824  df-7 9825  df-8 9826  df-9 9827  df-10 9828  df-n0 9982  df-z 10041  df-dec 10141  df-uz 10247  df-rp 10371  df-fz 10799  df-fl 10941  df-mod 10990  df-seq 11063  df-exp 11121  df-cj 11600  df-re 11601  df-im 11602  df-sqr 11736  df-abs 11737  df-dvds 12548  df-gcd 12702  df-prm 12775  df-struct 13166  df-ndx 13167  df-slot 13168  df-base 13169  df-sets 13170  df-ress 13171  df-plusg 13237  df-mulr 13238  df-starv 13239  df-tset 13243  df-ple 13244  df-ds 13246  df-0g 13420  df-mnd 14383  df-mhm 14431  df-grp 14505  df-minusg 14506  df-sbg 14507  df-mulg 14508  df-subg 14634  df-ghm 14697  df-od 14860  df-cmn 15107  df-mgp 15342  df-rng 15356  df-cring 15357  df-ur 15358  df-rnghom 15512  df-subrg 15559  df-nzr 16026  df-domn 16041  df-cnfld 16394  df-zrh 16471  df-chr 16473
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