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Theorem chrrhm 16804
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 15814 . . . . . . 7  |-  ( F  e.  ( R RingHom  S
)  ->  R  e.  Ring )
2 eqid 2435 . . . . . . . 8  |-  (flds  ZZ )  =  (flds  ZZ )
3 eqid 2435 . . . . . . . 8  |-  ( ZRHom `  R )  =  ( ZRHom `  R )
42, 3zrhrhm 16785 . . . . . . 7  |-  ( R  e.  Ring  ->  ( ZRHom `  R )  e.  ( (flds  ZZ ) RingHom  R ) )
51, 4syl 16 . . . . . 6  |-  ( F  e.  ( R RingHom  S
)  ->  ( ZRHom `  R )  e.  ( (flds  ZZ ) RingHom  R ) )
6 zsscn 10282 . . . . . . . 8  |-  ZZ  C_  CC
7 cnfldbas 16699 . . . . . . . . 9  |-  CC  =  ( Base ` fld )
82, 7ressbas2 13512 . . . . . . . 8  |-  ( ZZ  C_  CC  ->  ZZ  =  ( Base `  (flds  ZZ ) ) )
96, 8ax-mp 8 . . . . . . 7  |-  ZZ  =  ( Base `  (flds  ZZ ) )
10 eqid 2435 . . . . . . 7  |-  ( Base `  R )  =  (
Base `  R )
119, 10rhmf 15819 . . . . . 6  |-  ( ( ZRHom `  R )  e.  ( (flds  ZZ ) RingHom  R )  ->  ( ZRHom `  R ) : ZZ --> ( Base `  R
) )
12 ffn 5583 . . . . . 6  |-  ( ( ZRHom `  R ) : ZZ --> ( Base `  R
)  ->  ( ZRHom `  R )  Fn  ZZ )
135, 11, 123syl 19 . . . . 5  |-  ( F  e.  ( R RingHom  S
)  ->  ( ZRHom `  R )  Fn  ZZ )
14 eqid 2435 . . . . . . 7  |-  (chr `  R )  =  (chr
`  R )
1514chrcl 16799 . . . . . 6  |-  ( R  e.  Ring  ->  (chr `  R )  e.  NN0 )
16 nn0z 10296 . . . . . 6  |-  ( (chr
`  R )  e. 
NN0  ->  (chr `  R
)  e.  ZZ )
171, 15, 163syl 19 . . . . 5  |-  ( F  e.  ( R RingHom  S
)  ->  (chr `  R
)  e.  ZZ )
18 fvco2 5790 . . . . 5  |-  ( ( ( ZRHom `  R
)  Fn  ZZ  /\  (chr `  R )  e.  ZZ )  ->  (
( F  o.  ( ZRHom `  R ) ) `
 (chr `  R
) )  =  ( F `  ( ( ZRHom `  R ) `  (chr `  R )
) ) )
1913, 17, 18syl2anc 643 . . . 4  |-  ( F  e.  ( R RingHom  S
)  ->  ( ( F  o.  ( ZRHom `  R ) ) `  (chr `  R ) )  =  ( F `  ( ( ZRHom `  R ) `  (chr `  R ) ) ) )
20 eqid 2435 . . . . . . 7  |-  ( 0g
`  R )  =  ( 0g `  R
)
2114, 3, 20chrid 16800 . . . . . 6  |-  ( R  e.  Ring  ->  ( ( ZRHom `  R ) `  (chr `  R )
)  =  ( 0g
`  R ) )
221, 21syl 16 . . . . 5  |-  ( F  e.  ( R RingHom  S
)  ->  ( ( ZRHom `  R ) `  (chr `  R ) )  =  ( 0g `  R ) )
2322fveq2d 5724 . . . 4  |-  ( F  e.  ( R RingHom  S
)  ->  ( F `  ( ( ZRHom `  R ) `  (chr `  R ) ) )  =  ( F `  ( 0g `  R ) ) )
2419, 23eqtrd 2467 . . 3  |-  ( F  e.  ( R RingHom  S
)  ->  ( ( F  o.  ( ZRHom `  R ) ) `  (chr `  R ) )  =  ( F `  ( 0g `  R ) ) )
25 rhmco 15824 . . . . . 6  |-  ( ( F  e.  ( R RingHom  S )  /\  ( ZRHom `  R )  e.  ( (flds  ZZ ) RingHom  R ) )  -> 
( F  o.  ( ZRHom `  R ) )  e.  ( (flds  ZZ ) RingHom  S ) )
265, 25mpdan 650 . . . . 5  |-  ( F  e.  ( R RingHom  S
)  ->  ( F  o.  ( ZRHom `  R
) )  e.  ( (flds  ZZ ) RingHom  S ) )
27 rhmrcl2 15815 . . . . . 6  |-  ( F  e.  ( R RingHom  S
)  ->  S  e.  Ring )
28 eqid 2435 . . . . . . 7  |-  ( ZRHom `  S )  =  ( ZRHom `  S )
292, 28zrhrhmb 16784 . . . . . 6  |-  ( S  e.  Ring  ->  ( ( F  o.  ( ZRHom `  R ) )  e.  ( (flds  ZZ ) RingHom  S )  <->  ( F  o.  ( ZRHom `  R
) )  =  ( ZRHom `  S )
) )
3027, 29syl 16 . . . . 5  |-  ( F  e.  ( R RingHom  S
)  ->  ( ( F  o.  ( ZRHom `  R ) )  e.  ( (flds  ZZ ) RingHom  S )  <->  ( F  o.  ( ZRHom `  R
) )  =  ( ZRHom `  S )
) )
3126, 30mpbid 202 . . . 4  |-  ( F  e.  ( R RingHom  S
)  ->  ( F  o.  ( ZRHom `  R
) )  =  ( ZRHom `  S )
)
3231fveq1d 5722 . . 3  |-  ( F  e.  ( R RingHom  S
)  ->  ( ( F  o.  ( ZRHom `  R ) ) `  (chr `  R ) )  =  ( ( ZRHom `  S ) `  (chr `  R ) ) )
33 rhmghm 15818 . . . 4  |-  ( F  e.  ( R RingHom  S
)  ->  F  e.  ( R  GrpHom  S ) )
34 eqid 2435 . . . . 5  |-  ( 0g
`  S )  =  ( 0g `  S
)
3520, 34ghmid 15004 . . . 4  |-  ( F  e.  ( R  GrpHom  S )  ->  ( F `  ( 0g `  R
) )  =  ( 0g `  S ) )
3633, 35syl 16 . . 3  |-  ( F  e.  ( R RingHom  S
)  ->  ( F `  ( 0g `  R
) )  =  ( 0g `  S ) )
3724, 32, 363eqtr3d 2475 . 2  |-  ( F  e.  ( R RingHom  S
)  ->  ( ( ZRHom `  S ) `  (chr `  R ) )  =  ( 0g `  S ) )
38 eqid 2435 . . . 4  |-  (chr `  S )  =  (chr
`  S )
3938, 28, 34chrdvds 16801 . . 3  |-  ( ( S  e.  Ring  /\  (chr `  R )  e.  ZZ )  ->  ( (chr `  S )  ||  (chr `  R )  <->  ( ( ZRHom `  S ) `  (chr `  R ) )  =  ( 0g `  S ) ) )
4027, 17, 39syl2anc 643 . 2  |-  ( F  e.  ( R RingHom  S
)  ->  ( (chr `  S )  ||  (chr `  R )  <->  ( ( ZRHom `  S ) `  (chr `  R ) )  =  ( 0g `  S ) ) )
4137, 40mpbird 224 1  |-  ( F  e.  ( R RingHom  S
)  ->  (chr `  S
)  ||  (chr `  R
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    = wceq 1652    e. wcel 1725    C_ wss 3312   class class class wbr 4204    o. ccom 4874    Fn wfn 5441   -->wf 5442   ` cfv 5446  (class class class)co 6073   CCcc 8980   NN0cn0 10213   ZZcz 10274    || cdivides 12844   Basecbs 13461   ↾s cress 13462   0gc0g 13715    GrpHom cghm 14995   Ringcrg 15652   RingHom crh 15809  ℂfldccnfld 16695   ZRHomczrh 16770  chrcchr 16772
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-inf2 7588  ax-cnex 9038  ax-resscn 9039  ax-1cn 9040  ax-icn 9041  ax-addcl 9042  ax-addrcl 9043  ax-mulcl 9044  ax-mulrcl 9045  ax-mulcom 9046  ax-addass 9047  ax-mulass 9048  ax-distr 9049  ax-i2m1 9050  ax-1ne0 9051  ax-1rid 9052  ax-rnegex 9053  ax-rrecex 9054  ax-cnre 9055  ax-pre-lttri 9056  ax-pre-lttrn 9057  ax-pre-ltadd 9058  ax-pre-mulgt0 9059  ax-pre-sup 9060  ax-addf 9061  ax-mulf 9062
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-int 4043  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-riota 6541  df-recs 6625  df-rdg 6660  df-1o 6716  df-oadd 6720  df-er 6897  df-map 7012  df-en 7102  df-dom 7103  df-sdom 7104  df-fin 7105  df-sup 7438  df-pnf 9114  df-mnf 9115  df-xr 9116  df-ltxr 9117  df-le 9118  df-sub 9285  df-neg 9286  df-div 9670  df-nn 9993  df-2 10050  df-3 10051  df-4 10052  df-5 10053  df-6 10054  df-7 10055  df-8 10056  df-9 10057  df-10 10058  df-n0 10214  df-z 10275  df-dec 10375  df-uz 10481  df-rp 10605  df-fz 11036  df-fl 11194  df-mod 11243  df-seq 11316  df-exp 11375  df-cj 11896  df-re 11897  df-im 11898  df-sqr 12032  df-abs 12033  df-dvds 12845  df-struct 13463  df-ndx 13464  df-slot 13465  df-base 13466  df-sets 13467  df-ress 13468  df-plusg 13534  df-mulr 13535  df-starv 13536  df-tset 13540  df-ple 13541  df-ds 13543  df-unif 13544  df-0g 13719  df-mnd 14682  df-mhm 14730  df-grp 14804  df-minusg 14805  df-sbg 14806  df-mulg 14807  df-subg 14933  df-ghm 14996  df-od 15159  df-cmn 15406  df-mgp 15641  df-rng 15655  df-cring 15656  df-ur 15657  df-rnghom 15811  df-subrg 15858  df-cnfld 16696  df-zrh 16774  df-chr 16776
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