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Theorem domnchr 16736
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 2552 . . 3  |-  ( (chr
`  R )  =/=  0  <->  -.  (chr `  R
)  =  0 )
2 domnrng 16283 . . . . . . . . . 10  |-  ( R  e. Domn  ->  R  e.  Ring )
3 eqid 2387 . . . . . . . . . . 11  |-  (chr `  R )  =  (chr
`  R )
43chrcl 16730 . . . . . . . . . 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 3870 . . . . . . . 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 10166 . . . . . . 7  |-  NN  =  ( NN0  \  { 0 } )
119, 10syl6eleqr 2478 . . . . . 6  |-  ( ( R  e. Domn  /\  (chr `  R )  =/=  0
)  ->  (chr `  R
)  e.  NN )
12 domnnzr 16282 . . . . . . . 8  |-  ( R  e. Domn  ->  R  e. NzRing )
13 nzrrng 16259 . . . . . . . . . 10  |-  ( R  e. NzRing  ->  R  e.  Ring )
14 chrnzr 16734 . . . . . . . . . 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 10481 . . . . . 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 2387 . . . . . . . . . . . . 13  |-  (flds  ZZ )  =  (flds  ZZ )
23 eqid 2387 . . . . . . . . . . . . 13  |-  ( ZRHom `  R )  =  ( ZRHom `  R )
2422, 23zrhrhm 16716 . . . . . . . . . . . 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 10222 . . . . . . . . . . . . 13  |-  ZZ  C_  CC
29 cnfldbas 16630 . . . . . . . . . . . . . 14  |-  CC  =  ( Base ` fld )
3022, 29ressbas2 13447 . . . . . . . . . . . . 13  |-  ( ZZ  C_  CC  ->  ZZ  =  ( Base `  (flds  ZZ ) ) )
3128, 30ax-mp 8 . . . . . . . . . . . 12  |-  ZZ  =  ( Base `  (flds  ZZ ) )
32 zex 10223 . . . . . . . . . . . . 13  |-  ZZ  e.  _V
33 cnfldmul 16632 . . . . . . . . . . . . . 14  |-  x.  =  ( .r ` fld )
3422, 33ressmulr 13509 . . . . . . . . . . . . 13  |-  ( ZZ  e.  _V  ->  x.  =  ( .r `  (flds  ZZ ) ) )
3532, 34ax-mp 8 . . . . . . . . . . . 12  |-  x.  =  ( .r `  (flds  ZZ ) )
36 eqid 2387 . . . . . . . . . . . 12  |-  ( .r
`  R )  =  ( .r `  R
)
3731, 35, 36rhmmul 15755 . . . . . . . . . . 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 2395 . . . . . . . . 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 2387 . . . . . . . . . . . . 13  |-  ( Base `  R )  =  (
Base `  R )
4231, 41rhmf 15754 . . . . . . . . . . . 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 5810 . . . . . . . . . 10  |-  ( ( ( R  e. Domn  /\  (chr `  R )  =/=  0 )  /\  (
x  e.  ZZ  /\  y  e.  ZZ )
)  ->  ( ( ZRHom `  R ) `  x )  e.  (
Base `  R )
)
4543, 27ffvelrnd 5810 . . . . . . . . . 10  |-  ( ( ( R  e. Domn  /\  (chr `  R )  =/=  0 )  /\  (
x  e.  ZZ  /\  y  e.  ZZ )
)  ->  ( ( ZRHom `  R ) `  y )  e.  (
Base `  R )
)
46 eqid 2387 . . . . . . . . . . 11  |-  ( 0g
`  R )  =  ( 0g `  R
)
4741, 36, 46domneq0 16284 . . . . . . . . . 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 10256 . . . . . . . . 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 16732 . . . . . . . 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 16732 . . . . . . . . 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 16732 . . . . . . . . 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 2741 . . . . 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 13036 . . . . 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 1649    e. wcel 1717    =/= wne 2550   A.wral 2649   _Vcvv 2899    \ cdif 3260    C_ wss 3263   {csn 3757   class class class wbr 4153   -->wf 5390   ` cfv 5394  (class class class)co 6020   CCcc 8921   0cc0 8923   1c1 8924    x. cmul 8928   NNcn 9932   2c2 9981   NN0cn0 10153   ZZcz 10214   ZZ>=cuz 10420    || cdivides 12779   Primecprime 13006   Basecbs 13396   ↾s cress 13397   .rcmulr 13457   0gc0g 13650   Ringcrg 15587   RingHom crh 15744  NzRingcnzr 16255  Domncdomn 16267  ℂfldccnfld 16626   ZRHomczrh 16701  chrcchr 16703
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-rep 4261  ax-sep 4271  ax-nul 4279  ax-pow 4318  ax-pr 4344  ax-un 4641  ax-inf2 7529  ax-cnex 8979  ax-resscn 8980  ax-1cn 8981  ax-icn 8982  ax-addcl 8983  ax-addrcl 8984  ax-mulcl 8985  ax-mulrcl 8986  ax-mulcom 8987  ax-addass 8988  ax-mulass 8989  ax-distr 8990  ax-i2m1 8991  ax-1ne0 8992  ax-1rid 8993  ax-rnegex 8994  ax-rrecex 8995  ax-cnre 8996  ax-pre-lttri 8997  ax-pre-lttrn 8998  ax-pre-ltadd 8999  ax-pre-mulgt0 9000  ax-pre-sup 9001  ax-addf 9002  ax-mulf 9003
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-nel 2553  df-ral 2654  df-rex 2655  df-reu 2656  df-rmo 2657  df-rab 2658  df-v 2901  df-sbc 3105  df-csb 3195  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-pss 3279  df-nul 3572  df-if 3683  df-pw 3744  df-sn 3763  df-pr 3764  df-tp 3765  df-op 3766  df-uni 3958  df-int 3993  df-iun 4037  df-br 4154  df-opab 4208  df-mpt 4209  df-tr 4244  df-eprel 4435  df-id 4439  df-po 4444  df-so 4445  df-fr 4482  df-we 4484  df-ord 4525  df-on 4526  df-lim 4527  df-suc 4528  df-om 4786  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-rn 4829  df-res 4830  df-ima 4831  df-iota 5358  df-fun 5396  df-fn 5397  df-f 5398  df-f1 5399  df-fo 5400  df-f1o 5401  df-fv 5402  df-ov 6023  df-oprab 6024  df-mpt2 6025  df-1st 6288  df-2nd 6289  df-riota 6485  df-recs 6569  df-rdg 6604  df-1o 6660  df-2o 6661  df-oadd 6664  df-er 6841  df-map 6956  df-en 7046  df-dom 7047  df-sdom 7048  df-fin 7049  df-sup 7381  df-pnf 9055  df-mnf 9056  df-xr 9057  df-ltxr 9058  df-le 9059  df-sub 9225  df-neg 9226  df-div 9610  df-nn 9933  df-2 9990  df-3 9991  df-4 9992  df-5 9993  df-6 9994  df-7 9995  df-8 9996  df-9 9997  df-10 9998  df-n0 10154  df-z 10215  df-dec 10315  df-uz 10421  df-rp 10545  df-fz 10976  df-fl 11129  df-mod 11178  df-seq 11251  df-exp 11310  df-cj 11831  df-re 11832  df-im 11833  df-sqr 11967  df-abs 11968  df-dvds 12780  df-gcd 12934  df-prm 13007  df-struct 13398  df-ndx 13399  df-slot 13400  df-base 13401  df-sets 13402  df-ress 13403  df-plusg 13469  df-mulr 13470  df-starv 13471  df-tset 13475  df-ple 13476  df-ds 13478  df-unif 13479  df-0g 13654  df-mnd 14617  df-mhm 14665  df-grp 14739  df-minusg 14740  df-sbg 14741  df-mulg 14742  df-subg 14868  df-ghm 14931  df-od 15094  df-cmn 15341  df-mgp 15576  df-rng 15590  df-cring 15591  df-ur 15592  df-rnghom 15746  df-subrg 15793  df-nzr 16256  df-domn 16271  df-cnfld 16627  df-zrh 16705  df-chr 16707
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