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Theorem domnchr 18364
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 2664 . . 3  |-  ( (chr
`  R )  =/=  0  <->  -.  (chr `  R
)  =  0 )
2 domnrng 17744 . . . . . . . . . 10  |-  ( R  e. Domn  ->  R  e.  Ring )
3 eqid 2467 . . . . . . . . . . 11  |-  (chr `  R )  =  (chr
`  R )
43chrcl 18358 . . . . . . . . . 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 4152 . . . . . . . 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 10808 . . . . . . 7  |-  NN  =  ( NN0  \  { 0 } )
119, 10syl6eleqr 2566 . . . . . 6  |-  ( ( R  e. Domn  /\  (chr `  R )  =/=  0
)  ->  (chr `  R
)  e.  NN )
12 domnnzr 17743 . . . . . . . 8  |-  ( R  e. Domn  ->  R  e. NzRing )
13 nzrrng 17708 . . . . . . . . . 10  |-  ( R  e. NzRing  ->  R  e.  Ring )
14 chrnzr 18362 . . . . . . . . . 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 11155 . . . . . 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 2467 . . . . . . . . . . . . 13  |-  ( ZRHom `  R )  =  ( ZRHom `  R )
2322zrhrhm 18344 . . . . . . . . . . . 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 18290 . . . . . . . . . . . 12  |-  ZZ  =  ( Base ` ring )
28 zringmulr 18293 . . . . . . . . . . . 12  |-  x.  =  ( .r ` ring )
29 eqid 2467 . . . . . . . . . . . 12  |-  ( .r
`  R )  =  ( .r `  R
)
3027, 28, 29rhmmul 17177 . . . . . . . . . . 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 1228 . . . . . . . . . 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 2469 . . . . . . . . 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 2467 . . . . . . . . . . . . 13  |-  ( Base `  R )  =  (
Base `  R )
3527, 34rhmf 17176 . . . . . . . . . . . 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 6022 . . . . . . . . . 10  |-  ( ( ( R  e. Domn  /\  (chr `  R )  =/=  0 )  /\  (
x  e.  ZZ  /\  y  e.  ZZ )
)  ->  ( ( ZRHom `  R ) `  x )  e.  (
Base `  R )
)
3836, 26ffvelrnd 6022 . . . . . . . . . 10  |-  ( ( ( R  e. Domn  /\  (chr `  R )  =/=  0 )  /\  (
x  e.  ZZ  /\  y  e.  ZZ )
)  ->  ( ( ZRHom `  R ) `  y )  e.  (
Base `  R )
)
39 eqid 2467 . . . . . . . . . . 11  |-  ( 0g
`  R )  =  ( 0g `  R
)
4034, 29, 39domneq0 17745 . . . . . . . . . 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 1228 . . . . . . . . 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 10911 . . . . . . . . 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 18360 . . . . . . . 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 18360 . . . . . . . . 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 18360 . . . . . . . . 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 2885 . . . . 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 14109 . . . . 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 1379    e. wcel 1767    =/= wne 2662   A.wral 2814    \ cdif 3473   {csn 4027   class class class wbr 4447   -->wf 5584   ` cfv 5588  (class class class)co 6284   0cc0 9492   1c1 9493    x. cmul 9497   NNcn 10536   2c2 10585   NN0cn0 10795   ZZcz 10864   ZZ>=cuz 11082    || cdivides 13847   Primecprime 14076   Basecbs 14490   .rcmulr 14556   0gc0g 14695   Ringcrg 17000   RingHom crh 17162  NzRingcnzr 17704  Domncdomn 17727  ℤringzring 18284   ZRHomczrh 18332  chrcchr 18334
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6576  ax-inf2 8058  ax-cnex 9548  ax-resscn 9549  ax-1cn 9550  ax-icn 9551  ax-addcl 9552  ax-addrcl 9553  ax-mulcl 9554  ax-mulrcl 9555  ax-mulcom 9556  ax-addass 9557  ax-mulass 9558  ax-distr 9559  ax-i2m1 9560  ax-1ne0 9561  ax-1rid 9562  ax-rnegex 9563  ax-rrecex 9564  ax-cnre 9565  ax-pre-lttri 9566  ax-pre-lttrn 9567  ax-pre-ltadd 9568  ax-pre-mulgt0 9569  ax-pre-sup 9570  ax-addf 9571  ax-mulf 9572
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-riota 6245  df-ov 6287  df-oprab 6288  df-mpt2 6289  df-om 6685  df-1st 6784  df-2nd 6785  df-recs 7042  df-rdg 7076  df-1o 7130  df-2o 7131  df-oadd 7134  df-er 7311  df-map 7422  df-en 7517  df-dom 7518  df-sdom 7519  df-fin 7520  df-sup 7901  df-pnf 9630  df-mnf 9631  df-xr 9632  df-ltxr 9633  df-le 9634  df-sub 9807  df-neg 9808  df-div 10207  df-nn 10537  df-2 10594  df-3 10595  df-4 10596  df-5 10597  df-6 10598  df-7 10599  df-8 10600  df-9 10601  df-10 10602  df-n0 10796  df-z 10865  df-dec 10977  df-uz 11083  df-rp 11221  df-fz 11673  df-fl 11897  df-mod 11965  df-seq 12076  df-exp 12135  df-cj 12895  df-re 12896  df-im 12897  df-sqrt 13031  df-abs 13032  df-dvds 13848  df-gcd 14004  df-prm 14077  df-struct 14492  df-ndx 14493  df-slot 14494  df-base 14495  df-sets 14496  df-ress 14497  df-plusg 14568  df-mulr 14569  df-starv 14570  df-tset 14574  df-ple 14575  df-ds 14577  df-unif 14578  df-0g 14697  df-mnd 15732  df-mhm 15786  df-grp 15867  df-minusg 15868  df-sbg 15869  df-mulg 15870  df-subg 16003  df-ghm 16070  df-od 16359  df-cmn 16606  df-mgp 16944  df-ur 16956  df-rng 17002  df-cring 17003  df-rnghom 17165  df-subrg 17227  df-nzr 17705  df-domn 17731  df-cnfld 18220  df-zring 18285  df-zrh 18336  df-chr 18338
This theorem is referenced by: (None)
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