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Theorem chrrhm 16737
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 15751 . . . . . . 7  |-  ( F  e.  ( R RingHom  S
)  ->  R  e.  Ring )
2 eqid 2389 . . . . . . . 8  |-  (flds  ZZ )  =  (flds  ZZ )
3 eqid 2389 . . . . . . . 8  |-  ( ZRHom `  R )  =  ( ZRHom `  R )
42, 3zrhrhm 16718 . . . . . . 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 10224 . . . . . . . 8  |-  ZZ  C_  CC
7 cnfldbas 16632 . . . . . . . . 9  |-  CC  =  ( Base ` fld )
82, 7ressbas2 13449 . . . . . . . 8  |-  ( ZZ  C_  CC  ->  ZZ  =  ( Base `  (flds  ZZ ) ) )
96, 8ax-mp 8 . . . . . . 7  |-  ZZ  =  ( Base `  (flds  ZZ ) )
10 eqid 2389 . . . . . . 7  |-  ( Base `  R )  =  (
Base `  R )
119, 10rhmf 15756 . . . . . 6  |-  ( ( ZRHom `  R )  e.  ( (flds  ZZ ) RingHom  R )  ->  ( ZRHom `  R ) : ZZ --> ( Base `  R
) )
12 ffn 5533 . . . . . 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 2389 . . . . . . 7  |-  (chr `  R )  =  (chr
`  R )
1514chrcl 16732 . . . . . 6  |-  ( R  e.  Ring  ->  (chr `  R )  e.  NN0 )
16 nn0z 10238 . . . . . 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 5739 . . . . 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 2389 . . . . . . 7  |-  ( 0g
`  R )  =  ( 0g `  R
)
2114, 3, 20chrid 16733 . . . . . 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 5674 . . . 4  |-  ( F  e.  ( R RingHom  S
)  ->  ( F `  ( ( ZRHom `  R ) `  (chr `  R ) ) )  =  ( F `  ( 0g `  R ) ) )
2419, 23eqtrd 2421 . . 3  |-  ( F  e.  ( R RingHom  S
)  ->  ( ( F  o.  ( ZRHom `  R ) ) `  (chr `  R ) )  =  ( F `  ( 0g `  R ) ) )
25 rhmco 15761 . . . . . 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 15752 . . . . . 6  |-  ( F  e.  ( R RingHom  S
)  ->  S  e.  Ring )
28 eqid 2389 . . . . . . 7  |-  ( ZRHom `  S )  =  ( ZRHom `  S )
292, 28zrhrhmb 16717 . . . . . 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 5672 . . 3  |-  ( F  e.  ( R RingHom  S
)  ->  ( ( F  o.  ( ZRHom `  R ) ) `  (chr `  R ) )  =  ( ( ZRHom `  S ) `  (chr `  R ) ) )
33 rhmghm 15755 . . . 4  |-  ( F  e.  ( R RingHom  S
)  ->  F  e.  ( R  GrpHom  S ) )
34 eqid 2389 . . . . 5  |-  ( 0g
`  S )  =  ( 0g `  S
)
3520, 34ghmid 14941 . . . 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 2429 . 2  |-  ( F  e.  ( R RingHom  S
)  ->  ( ( ZRHom `  S ) `  (chr `  R ) )  =  ( 0g `  S ) )
38 eqid 2389 . . . 4  |-  (chr `  S )  =  (chr
`  S )
3938, 28, 34chrdvds 16734 . . 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 1649    e. wcel 1717    C_ wss 3265   class class class wbr 4155    o. ccom 4824    Fn wfn 5391   -->wf 5392   ` cfv 5396  (class class class)co 6022   CCcc 8923   NN0cn0 10155   ZZcz 10216    || cdivides 12781   Basecbs 13398   ↾s cress 13399   0gc0g 13652    GrpHom cghm 14932   Ringcrg 15589   RingHom crh 15746  ℂfldccnfld 16628   ZRHomczrh 16703  chrcchr 16705
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 2370  ax-rep 4263  ax-sep 4273  ax-nul 4281  ax-pow 4320  ax-pr 4346  ax-un 4643  ax-inf2 7531  ax-cnex 8981  ax-resscn 8982  ax-1cn 8983  ax-icn 8984  ax-addcl 8985  ax-addrcl 8986  ax-mulcl 8987  ax-mulrcl 8988  ax-mulcom 8989  ax-addass 8990  ax-mulass 8991  ax-distr 8992  ax-i2m1 8993  ax-1ne0 8994  ax-1rid 8995  ax-rnegex 8996  ax-rrecex 8997  ax-cnre 8998  ax-pre-lttri 8999  ax-pre-lttrn 9000  ax-pre-ltadd 9001  ax-pre-mulgt0 9002  ax-pre-sup 9003  ax-addf 9004  ax-mulf 9005
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 2244  df-mo 2245  df-clab 2376  df-cleq 2382  df-clel 2385  df-nfc 2514  df-ne 2554  df-nel 2555  df-ral 2656  df-rex 2657  df-reu 2658  df-rmo 2659  df-rab 2660  df-v 2903  df-sbc 3107  df-csb 3197  df-dif 3268  df-un 3270  df-in 3272  df-ss 3279  df-pss 3281  df-nul 3574  df-if 3685  df-pw 3746  df-sn 3765  df-pr 3766  df-tp 3767  df-op 3768  df-uni 3960  df-int 3995  df-iun 4039  df-br 4156  df-opab 4210  df-mpt 4211  df-tr 4246  df-eprel 4437  df-id 4441  df-po 4446  df-so 4447  df-fr 4484  df-we 4486  df-ord 4527  df-on 4528  df-lim 4529  df-suc 4530  df-om 4788  df-xp 4826  df-rel 4827  df-cnv 4828  df-co 4829  df-dm 4830  df-rn 4831  df-res 4832  df-ima 4833  df-iota 5360  df-fun 5398  df-fn 5399  df-f 5400  df-f1 5401  df-fo 5402  df-f1o 5403  df-fv 5404  df-ov 6025  df-oprab 6026  df-mpt2 6027  df-1st 6290  df-2nd 6291  df-riota 6487  df-recs 6571  df-rdg 6606  df-1o 6662  df-oadd 6666  df-er 6843  df-map 6958  df-en 7048  df-dom 7049  df-sdom 7050  df-fin 7051  df-sup 7383  df-pnf 9057  df-mnf 9058  df-xr 9059  df-ltxr 9060  df-le 9061  df-sub 9227  df-neg 9228  df-div 9612  df-nn 9935  df-2 9992  df-3 9993  df-4 9994  df-5 9995  df-6 9996  df-7 9997  df-8 9998  df-9 9999  df-10 10000  df-n0 10156  df-z 10217  df-dec 10317  df-uz 10423  df-rp 10547  df-fz 10978  df-fl 11131  df-mod 11180  df-seq 11253  df-exp 11312  df-cj 11833  df-re 11834  df-im 11835  df-sqr 11969  df-abs 11970  df-dvds 12782  df-struct 13400  df-ndx 13401  df-slot 13402  df-base 13403  df-sets 13404  df-ress 13405  df-plusg 13471  df-mulr 13472  df-starv 13473  df-tset 13477  df-ple 13478  df-ds 13480  df-unif 13481  df-0g 13656  df-mnd 14619  df-mhm 14667  df-grp 14741  df-minusg 14742  df-sbg 14743  df-mulg 14744  df-subg 14870  df-ghm 14933  df-od 15096  df-cmn 15343  df-mgp 15578  df-rng 15592  df-cring 15593  df-ur 15594  df-rnghom 15748  df-subrg 15795  df-cnfld 16629  df-zrh 16707  df-chr 16709
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