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Theorem domnchr 17962
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 2607 . . 3  |-  ( (chr
`  R )  =/=  0  <->  -.  (chr `  R
)  =  0 )
2 domnrng 17367 . . . . . . . . . 10  |-  ( R  e. Domn  ->  R  e.  Ring )
3 eqid 2442 . . . . . . . . . . 11  |-  (chr `  R )  =  (chr
`  R )
43chrcl 17956 . . . . . . . . . 10  |-  ( R  e.  Ring  ->  (chr `  R )  e.  NN0 )
52, 4syl 16 . . . . . . . . 9  |-  ( R  e. Domn  ->  (chr `  R
)  e.  NN0 )
65adantr 465 . . . . . . . 8  |-  ( ( R  e. Domn  /\  (chr `  R )  =/=  0
)  ->  (chr `  R
)  e.  NN0 )
7 simpr 461 . . . . . . . 8  |-  ( ( R  e. Domn  /\  (chr `  R )  =/=  0
)  ->  (chr `  R
)  =/=  0 )
8 eldifsn 3999 . . . . . . . 8  |-  ( (chr
`  R )  e.  ( NN0  \  {
0 } )  <->  ( (chr `  R )  e.  NN0  /\  (chr `  R )  =/=  0 ) )
96, 7, 8sylanbrc 664 . . . . . . 7  |-  ( ( R  e. Domn  /\  (chr `  R )  =/=  0
)  ->  (chr `  R
)  e.  ( NN0  \  { 0 } ) )
10 dfn2 10591 . . . . . . 7  |-  NN  =  ( NN0  \  { 0 } )
119, 10syl6eleqr 2533 . . . . . 6  |-  ( ( R  e. Domn  /\  (chr `  R )  =/=  0
)  ->  (chr `  R
)  e.  NN )
12 domnnzr 17366 . . . . . . . 8  |-  ( R  e. Domn  ->  R  e. NzRing )
13 nzrrng 17342 . . . . . . . . . 10  |-  ( R  e. NzRing  ->  R  e.  Ring )
14 chrnzr 17960 . . . . . . . . . 10  |-  ( R  e.  Ring  ->  ( R  e. NzRing 
<->  (chr `  R )  =/=  1 ) )
1513, 14syl 16 . . . . . . . . 9  |-  ( R  e. NzRing  ->  ( R  e. NzRing  <->  (chr
`  R )  =/=  1 ) )
1615ibi 241 . . . . . . . 8  |-  ( R  e. NzRing  ->  (chr `  R
)  =/=  1 )
1712, 16syl 16 . . . . . . 7  |-  ( R  e. Domn  ->  (chr `  R
)  =/=  1 )
1817adantr 465 . . . . . 6  |-  ( ( R  e. Domn  /\  (chr `  R )  =/=  0
)  ->  (chr `  R
)  =/=  1 )
19 eluz2b3 10927 . . . . . 6  |-  ( (chr
`  R )  e.  ( ZZ>= `  2 )  <->  ( (chr `  R )  e.  NN  /\  (chr `  R )  =/=  1
) )
2011, 18, 19sylanbrc 664 . . . . 5  |-  ( ( R  e. Domn  /\  (chr `  R )  =/=  0
)  ->  (chr `  R
)  e.  ( ZZ>= ` 
2 ) )
212ad2antrr 725 . . . . . . . . . . . 12  |-  ( ( ( R  e. Domn  /\  (chr `  R )  =/=  0 )  /\  (
x  e.  ZZ  /\  y  e.  ZZ )
)  ->  R  e.  Ring )
22 eqid 2442 . . . . . . . . . . . . 13  |-  ( ZRHom `  R )  =  ( ZRHom `  R )
2322zrhrhm 17942 . . . . . . . . . . . 12  |-  ( R  e.  Ring  ->  ( ZRHom `  R )  e.  (ring RingHom  R
) )
2421, 23syl 16 . . . . . . . . . . 11  |-  ( ( ( R  e. Domn  /\  (chr `  R )  =/=  0 )  /\  (
x  e.  ZZ  /\  y  e.  ZZ )
)  ->  ( ZRHom `  R )  e.  (ring RingHom  R
) )
25 simprl 755 . . . . . . . . . . 11  |-  ( ( ( R  e. Domn  /\  (chr `  R )  =/=  0 )  /\  (
x  e.  ZZ  /\  y  e.  ZZ )
)  ->  x  e.  ZZ )
26 simprr 756 . . . . . . . . . . 11  |-  ( ( ( R  e. Domn  /\  (chr `  R )  =/=  0 )  /\  (
x  e.  ZZ  /\  y  e.  ZZ )
)  ->  y  e.  ZZ )
27 zringbas 17888 . . . . . . . . . . . 12  |-  ZZ  =  ( Base ` ring )
28 zringmulr 17891 . . . . . . . . . . . 12  |-  x.  =  ( .r ` ring )
29 eqid 2442 . . . . . . . . . . . 12  |-  ( .r
`  R )  =  ( .r `  R
)
3027, 28, 29rhmmul 16816 . . . . . . . . . . 11  |-  ( ( ( ZRHom `  R
)  e.  (ring RingHom  R )  /\  x  e.  ZZ  /\  y  e.  ZZ )  ->  (
( ZRHom `  R
) `  ( x  x.  y ) )  =  ( ( ( ZRHom `  R ) `  x
) ( .r `  R ) ( ( ZRHom `  R ) `  y ) ) )
3124, 25, 26, 30syl3anc 1218 . . . . . . . . . 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 ) ) )
3231eqeq1d 2450 . . . . . . . . 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 753 . . . . . . . . . 10  |-  ( ( ( R  e. Domn  /\  (chr `  R )  =/=  0 )  /\  (
x  e.  ZZ  /\  y  e.  ZZ )
)  ->  R  e. Domn )
34 eqid 2442 . . . . . . . . . . . . 13  |-  ( Base `  R )  =  (
Base `  R )
3527, 34rhmf 16815 . . . . . . . . . . . 12  |-  ( ( ZRHom `  R )  e.  (ring RingHom  R )  ->  ( ZRHom `  R ) : ZZ --> ( Base `  R
) )
3624, 35syl 16 . . . . . . . . . . 11  |-  ( ( ( R  e. Domn  /\  (chr `  R )  =/=  0 )  /\  (
x  e.  ZZ  /\  y  e.  ZZ )
)  ->  ( ZRHom `  R ) : ZZ --> ( Base `  R )
)
3736, 25ffvelrnd 5843 . . . . . . . . . 10  |-  ( ( ( R  e. Domn  /\  (chr `  R )  =/=  0 )  /\  (
x  e.  ZZ  /\  y  e.  ZZ )
)  ->  ( ( ZRHom `  R ) `  x )  e.  (
Base `  R )
)
3836, 26ffvelrnd 5843 . . . . . . . . . 10  |-  ( ( ( R  e. Domn  /\  (chr `  R )  =/=  0 )  /\  (
x  e.  ZZ  /\  y  e.  ZZ )
)  ->  ( ( ZRHom `  R ) `  y )  e.  (
Base `  R )
)
39 eqid 2442 . . . . . . . . . . 11  |-  ( 0g
`  R )  =  ( 0g `  R
)
4034, 29, 39domneq0 17368 . . . . . . . . . 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 ) ) ) )
4133, 37, 38, 40syl3anc 1218 . . . . . . . . 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 ) ) ) )
4232, 41bitrd 253 . . . . . . . 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 ) ) ) )
4342biimpd 207 . . . . . . 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 10692 . . . . . . . . 9  |-  ( ( x  e.  ZZ  /\  y  e.  ZZ )  ->  ( x  x.  y
)  e.  ZZ )
4544adantl 466 . . . . . . . 8  |-  ( ( ( R  e. Domn  /\  (chr `  R )  =/=  0 )  /\  (
x  e.  ZZ  /\  y  e.  ZZ )
)  ->  ( x  x.  y )  e.  ZZ )
463, 22, 39chrdvds 17958 . . . . . . . 8  |-  ( ( R  e.  Ring  /\  (
x  x.  y )  e.  ZZ )  -> 
( (chr `  R
)  ||  ( x  x.  y )  <->  ( ( ZRHom `  R ) `  ( x  x.  y
) )  =  ( 0g `  R ) ) )
4721, 45, 46syl2anc 661 . . . . . . 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 ) ) )
483, 22, 39chrdvds 17958 . . . . . . . . 9  |-  ( ( R  e.  Ring  /\  x  e.  ZZ )  ->  (
(chr `  R )  ||  x  <->  ( ( ZRHom `  R ) `  x
)  =  ( 0g
`  R ) ) )
4921, 25, 48syl2anc 661 . . . . . . . 8  |-  ( ( ( R  e. Domn  /\  (chr `  R )  =/=  0 )  /\  (
x  e.  ZZ  /\  y  e.  ZZ )
)  ->  ( (chr `  R )  ||  x  <->  ( ( ZRHom `  R
) `  x )  =  ( 0g `  R ) ) )
503, 22, 39chrdvds 17958 . . . . . . . . 9  |-  ( ( R  e.  Ring  /\  y  e.  ZZ )  ->  (
(chr `  R )  ||  y  <->  ( ( ZRHom `  R ) `  y
)  =  ( 0g
`  R ) ) )
5121, 26, 50syl2anc 661 . . . . . . . 8  |-  ( ( ( R  e. Domn  /\  (chr `  R )  =/=  0 )  /\  (
x  e.  ZZ  /\  y  e.  ZZ )
)  ->  ( (chr `  R )  ||  y  <->  ( ( ZRHom `  R
) `  y )  =  ( 0g `  R ) ) )
5249, 51orbi12d 709 . . . . . . 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 ) ) ) )
5343, 47, 523imtr4d 268 . . . . . 6  |-  ( ( ( R  e. Domn  /\  (chr `  R )  =/=  0 )  /\  (
x  e.  ZZ  /\  y  e.  ZZ )
)  ->  ( (chr `  R )  ||  (
x  x.  y )  ->  ( (chr `  R )  ||  x  \/  (chr `  R )  ||  y ) ) )
5453ralrimivva 2807 . . . . 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 13794 . . . . 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 ) ) ) )
5620, 54, 55sylanbrc 664 . . . 4  |-  ( ( R  e. Domn  /\  (chr `  R )  =/=  0
)  ->  (chr `  R
)  e.  Prime )
5756ex 434 . . 3  |-  ( R  e. Domn  ->  ( (chr `  R )  =/=  0  ->  (chr `  R )  e.  Prime ) )
581, 57syl5bir 218 . 2  |-  ( R  e. Domn  ->  ( -.  (chr `  R )  =  0  ->  (chr `  R
)  e.  Prime )
)
5958orrd 378 1  |-  ( R  e. Domn  ->  ( (chr `  R )  =  0  \/  (chr `  R
)  e.  Prime )
)
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    \/ wo 368    /\ wa 369    = wceq 1369    e. wcel 1756    =/= wne 2605   A.wral 2714    \ cdif 3324   {csn 3876   class class class wbr 4291   -->wf 5413   ` cfv 5417  (class class class)co 6090   0cc0 9281   1c1 9282    x. cmul 9286   NNcn 10321   2c2 10370   NN0cn0 10578   ZZcz 10645   ZZ>=cuz 10860    || cdivides 13534   Primecprime 13762   Basecbs 14173   .rcmulr 14238   0gc0g 14377   Ringcrg 16644   RingHom crh 16803  NzRingcnzr 17338  Domncdomn 17350  ℤringzring 17882   ZRHomczrh 17930  chrcchr 17932
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4402  ax-sep 4412  ax-nul 4420  ax-pow 4469  ax-pr 4530  ax-un 6371  ax-inf2 7846  ax-cnex 9337  ax-resscn 9338  ax-1cn 9339  ax-icn 9340  ax-addcl 9341  ax-addrcl 9342  ax-mulcl 9343  ax-mulrcl 9344  ax-mulcom 9345  ax-addass 9346  ax-mulass 9347  ax-distr 9348  ax-i2m1 9349  ax-1ne0 9350  ax-1rid 9351  ax-rnegex 9352  ax-rrecex 9353  ax-cnre 9354  ax-pre-lttri 9355  ax-pre-lttrn 9356  ax-pre-ltadd 9357  ax-pre-mulgt0 9358  ax-pre-sup 9359  ax-addf 9360  ax-mulf 9361
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-nel 2608  df-ral 2719  df-rex 2720  df-reu 2721  df-rmo 2722  df-rab 2723  df-v 2973  df-sbc 3186  df-csb 3288  df-dif 3330  df-un 3332  df-in 3334  df-ss 3341  df-pss 3343  df-nul 3637  df-if 3791  df-pw 3861  df-sn 3877  df-pr 3879  df-tp 3881  df-op 3883  df-uni 4091  df-int 4128  df-iun 4172  df-br 4292  df-opab 4350  df-mpt 4351  df-tr 4385  df-eprel 4631  df-id 4635  df-po 4640  df-so 4641  df-fr 4678  df-we 4680  df-ord 4721  df-on 4722  df-lim 4723  df-suc 4724  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-iota 5380  df-fun 5419  df-fn 5420  df-f 5421  df-f1 5422  df-fo 5423  df-f1o 5424  df-fv 5425  df-riota 6051  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-om 6476  df-1st 6576  df-2nd 6577  df-recs 6831  df-rdg 6865  df-1o 6919  df-2o 6920  df-oadd 6923  df-er 7100  df-map 7215  df-en 7310  df-dom 7311  df-sdom 7312  df-fin 7313  df-sup 7690  df-pnf 9419  df-mnf 9420  df-xr 9421  df-ltxr 9422  df-le 9423  df-sub 9596  df-neg 9597  df-div 9993  df-nn 10322  df-2 10379  df-3 10380  df-4 10381  df-5 10382  df-6 10383  df-7 10384  df-8 10385  df-9 10386  df-10 10387  df-n0 10579  df-z 10646  df-dec 10755  df-uz 10861  df-rp 10991  df-fz 11437  df-fl 11641  df-mod 11708  df-seq 11806  df-exp 11865  df-cj 12587  df-re 12588  df-im 12589  df-sqr 12723  df-abs 12724  df-dvds 13535  df-gcd 13690  df-prm 13763  df-struct 14175  df-ndx 14176  df-slot 14177  df-base 14178  df-sets 14179  df-ress 14180  df-plusg 14250  df-mulr 14251  df-starv 14252  df-tset 14256  df-ple 14257  df-ds 14259  df-unif 14260  df-0g 14379  df-mnd 15414  df-mhm 15463  df-grp 15544  df-minusg 15545  df-sbg 15546  df-mulg 15547  df-subg 15677  df-ghm 15744  df-od 16031  df-cmn 16278  df-mgp 16591  df-ur 16603  df-rng 16646  df-cring 16647  df-rnghom 16805  df-subrg 16862  df-nzr 17339  df-domn 17354  df-cnfld 17818  df-zring 17883  df-zrh 17934  df-chr 17936
This theorem is referenced by: (None)
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