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Theorem chrrhm 16501
Description: The characteristic restriction on ring homomorphisms. (Contributed by Stefan O'Rear, 6-Sep-2015.)
Assertion
Ref Expression
chrrhm  |-  ( F  e.  ( R RingHom  S
)  ->  (chr `  S
)  ||  (chr `  R
) )

Proof of Theorem chrrhm
StepHypRef Expression
1 rhmrcl1 15515 . . . . . . 7  |-  ( F  e.  ( R RingHom  S
)  ->  R  e.  Ring )
2 eqid 2296 . . . . . . . 8  |-  (flds  ZZ )  =  (flds  ZZ )
3 eqid 2296 . . . . . . . 8  |-  ( ZRHom `  R )  =  ( ZRHom `  R )
42, 3zrhrhm 16482 . . . . . . 7  |-  ( R  e.  Ring  ->  ( ZRHom `  R )  e.  ( (flds  ZZ ) RingHom  R ) )
51, 4syl 15 . . . . . 6  |-  ( F  e.  ( R RingHom  S
)  ->  ( ZRHom `  R )  e.  ( (flds  ZZ ) RingHom  R ) )
6 zsscn 10048 . . . . . . . 8  |-  ZZ  C_  CC
7 cnfldbas 16399 . . . . . . . . 9  |-  CC  =  ( Base ` fld )
82, 7ressbas2 13215 . . . . . . . 8  |-  ( ZZ  C_  CC  ->  ZZ  =  ( Base `  (flds  ZZ ) ) )
96, 8ax-mp 8 . . . . . . 7  |-  ZZ  =  ( Base `  (flds  ZZ ) )
10 eqid 2296 . . . . . . 7  |-  ( Base `  R )  =  (
Base `  R )
119, 10rhmf 15520 . . . . . 6  |-  ( ( ZRHom `  R )  e.  ( (flds  ZZ ) RingHom  R )  ->  ( ZRHom `  R ) : ZZ --> ( Base `  R
) )
12 ffn 5405 . . . . . 6  |-  ( ( ZRHom `  R ) : ZZ --> ( Base `  R
)  ->  ( ZRHom `  R )  Fn  ZZ )
135, 11, 123syl 18 . . . . 5  |-  ( F  e.  ( R RingHom  S
)  ->  ( ZRHom `  R )  Fn  ZZ )
14 eqid 2296 . . . . . . 7  |-  (chr `  R )  =  (chr
`  R )
1514chrcl 16496 . . . . . 6  |-  ( R  e.  Ring  ->  (chr `  R )  e.  NN0 )
16 nn0z 10062 . . . . . 6  |-  ( (chr
`  R )  e. 
NN0  ->  (chr `  R
)  e.  ZZ )
171, 15, 163syl 18 . . . . 5  |-  ( F  e.  ( R RingHom  S
)  ->  (chr `  R
)  e.  ZZ )
18 fvco2 5610 . . . . 5  |-  ( ( ( ZRHom `  R
)  Fn  ZZ  /\  (chr `  R )  e.  ZZ )  ->  (
( F  o.  ( ZRHom `  R ) ) `
 (chr `  R
) )  =  ( F `  ( ( ZRHom `  R ) `  (chr `  R )
) ) )
1913, 17, 18syl2anc 642 . . . 4  |-  ( F  e.  ( R RingHom  S
)  ->  ( ( F  o.  ( ZRHom `  R ) ) `  (chr `  R ) )  =  ( F `  ( ( ZRHom `  R ) `  (chr `  R ) ) ) )
20 eqid 2296 . . . . . . 7  |-  ( 0g
`  R )  =  ( 0g `  R
)
2114, 3, 20chrid 16497 . . . . . 6  |-  ( R  e.  Ring  ->  ( ( ZRHom `  R ) `  (chr `  R )
)  =  ( 0g
`  R ) )
221, 21syl 15 . . . . 5  |-  ( F  e.  ( R RingHom  S
)  ->  ( ( ZRHom `  R ) `  (chr `  R ) )  =  ( 0g `  R ) )
2322fveq2d 5545 . . . 4  |-  ( F  e.  ( R RingHom  S
)  ->  ( F `  ( ( ZRHom `  R ) `  (chr `  R ) ) )  =  ( F `  ( 0g `  R ) ) )
2419, 23eqtrd 2328 . . 3  |-  ( F  e.  ( R RingHom  S
)  ->  ( ( F  o.  ( ZRHom `  R ) ) `  (chr `  R ) )  =  ( F `  ( 0g `  R ) ) )
25 rhmco 15525 . . . . . 6  |-  ( ( F  e.  ( R RingHom  S )  /\  ( ZRHom `  R )  e.  ( (flds  ZZ ) RingHom  R ) )  -> 
( F  o.  ( ZRHom `  R ) )  e.  ( (flds  ZZ ) RingHom  S ) )
265, 25mpdan 649 . . . . 5  |-  ( F  e.  ( R RingHom  S
)  ->  ( F  o.  ( ZRHom `  R
) )  e.  ( (flds  ZZ ) RingHom  S ) )
27 rhmrcl2 15516 . . . . . 6  |-  ( F  e.  ( R RingHom  S
)  ->  S  e.  Ring )
28 eqid 2296 . . . . . . 7  |-  ( ZRHom `  S )  =  ( ZRHom `  S )
292, 28zrhrhmb 16481 . . . . . 6  |-  ( S  e.  Ring  ->  ( ( F  o.  ( ZRHom `  R ) )  e.  ( (flds  ZZ ) RingHom  S )  <->  ( F  o.  ( ZRHom `  R
) )  =  ( ZRHom `  S )
) )
3027, 29syl 15 . . . . 5  |-  ( F  e.  ( R RingHom  S
)  ->  ( ( F  o.  ( ZRHom `  R ) )  e.  ( (flds  ZZ ) RingHom  S )  <->  ( F  o.  ( ZRHom `  R
) )  =  ( ZRHom `  S )
) )
3126, 30mpbid 201 . . . 4  |-  ( F  e.  ( R RingHom  S
)  ->  ( F  o.  ( ZRHom `  R
) )  =  ( ZRHom `  S )
)
3231fveq1d 5543 . . 3  |-  ( F  e.  ( R RingHom  S
)  ->  ( ( F  o.  ( ZRHom `  R ) ) `  (chr `  R ) )  =  ( ( ZRHom `  S ) `  (chr `  R ) ) )
33 rhmghm 15519 . . . 4  |-  ( F  e.  ( R RingHom  S
)  ->  F  e.  ( R  GrpHom  S ) )
34 eqid 2296 . . . . 5  |-  ( 0g
`  S )  =  ( 0g `  S
)
3520, 34ghmid 14705 . . . 4  |-  ( F  e.  ( R  GrpHom  S )  ->  ( F `  ( 0g `  R
) )  =  ( 0g `  S ) )
3633, 35syl 15 . . 3  |-  ( F  e.  ( R RingHom  S
)  ->  ( F `  ( 0g `  R
) )  =  ( 0g `  S ) )
3724, 32, 363eqtr3d 2336 . 2  |-  ( F  e.  ( R RingHom  S
)  ->  ( ( ZRHom `  S ) `  (chr `  R ) )  =  ( 0g `  S ) )
38 eqid 2296 . . . 4  |-  (chr `  S )  =  (chr
`  S )
3938, 28, 34chrdvds 16498 . . 3  |-  ( ( S  e.  Ring  /\  (chr `  R )  e.  ZZ )  ->  ( (chr `  S )  ||  (chr `  R )  <->  ( ( ZRHom `  S ) `  (chr `  R ) )  =  ( 0g `  S ) ) )
4027, 17, 39syl2anc 642 . 2  |-  ( F  e.  ( R RingHom  S
)  ->  ( (chr `  S )  ||  (chr `  R )  <->  ( ( ZRHom `  S ) `  (chr `  R ) )  =  ( 0g `  S ) ) )
4137, 40mpbird 223 1  |-  ( F  e.  ( R RingHom  S
)  ->  (chr `  S
)  ||  (chr `  R
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    = wceq 1632    e. wcel 1696    C_ wss 3165   class class class wbr 4039    o. ccom 4709    Fn wfn 5266   -->wf 5267   ` cfv 5271  (class class class)co 5874   CCcc 8751   NN0cn0 9981   ZZcz 10040    || cdivides 12547   Basecbs 13164   ↾s cress 13165   0gc0g 13416    GrpHom cghm 14696   Ringcrg 15353   RingHom crh 15510  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-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-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-cnfld 16394  df-zrh 16471  df-chr 16473
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