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Theorem chrrhm 16485
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 15499 . . . . . . 7  |-  ( F  e.  ( R RingHom  S
)  ->  R  e.  Ring )
2 eqid 2283 . . . . . . . 8  |-  (flds  ZZ )  =  (flds  ZZ )
3 eqid 2283 . . . . . . . 8  |-  ( ZRHom `  R )  =  ( ZRHom `  R )
42, 3zrhrhm 16466 . . . . . . 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 10032 . . . . . . . 8  |-  ZZ  C_  CC
7 cnfldbas 16383 . . . . . . . . 9  |-  CC  =  ( Base ` fld )
82, 7ressbas2 13199 . . . . . . . 8  |-  ( ZZ  C_  CC  ->  ZZ  =  ( Base `  (flds  ZZ ) ) )
96, 8ax-mp 8 . . . . . . 7  |-  ZZ  =  ( Base `  (flds  ZZ ) )
10 eqid 2283 . . . . . . 7  |-  ( Base `  R )  =  (
Base `  R )
119, 10rhmf 15504 . . . . . 6  |-  ( ( ZRHom `  R )  e.  ( (flds  ZZ ) RingHom  R )  ->  ( ZRHom `  R ) : ZZ --> ( Base `  R
) )
12 ffn 5389 . . . . . 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 2283 . . . . . . 7  |-  (chr `  R )  =  (chr
`  R )
1514chrcl 16480 . . . . . 6  |-  ( R  e.  Ring  ->  (chr `  R )  e.  NN0 )
16 nn0z 10046 . . . . . 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 5594 . . . . 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 2283 . . . . . . 7  |-  ( 0g
`  R )  =  ( 0g `  R
)
2114, 3, 20chrid 16481 . . . . . 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 5529 . . . 4  |-  ( F  e.  ( R RingHom  S
)  ->  ( F `  ( ( ZRHom `  R ) `  (chr `  R ) ) )  =  ( F `  ( 0g `  R ) ) )
2419, 23eqtrd 2315 . . 3  |-  ( F  e.  ( R RingHom  S
)  ->  ( ( F  o.  ( ZRHom `  R ) ) `  (chr `  R ) )  =  ( F `  ( 0g `  R ) ) )
25 rhmco 15509 . . . . . 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 15500 . . . . . 6  |-  ( F  e.  ( R RingHom  S
)  ->  S  e.  Ring )
28 eqid 2283 . . . . . . 7  |-  ( ZRHom `  S )  =  ( ZRHom `  S )
292, 28zrhrhmb 16465 . . . . . 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 5527 . . 3  |-  ( F  e.  ( R RingHom  S
)  ->  ( ( F  o.  ( ZRHom `  R ) ) `  (chr `  R ) )  =  ( ( ZRHom `  S ) `  (chr `  R ) ) )
33 rhmghm 15503 . . . 4  |-  ( F  e.  ( R RingHom  S
)  ->  F  e.  ( R  GrpHom  S ) )
34 eqid 2283 . . . . 5  |-  ( 0g
`  S )  =  ( 0g `  S
)
3520, 34ghmid 14689 . . . 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 2323 . 2  |-  ( F  e.  ( R RingHom  S
)  ->  ( ( ZRHom `  S ) `  (chr `  R ) )  =  ( 0g `  S ) )
38 eqid 2283 . . . 4  |-  (chr `  S )  =  (chr
`  S )
3938, 28, 34chrdvds 16482 . . 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 1623    e. wcel 1684    C_ wss 3152   class class class wbr 4023    o. ccom 4693    Fn wfn 5250   -->wf 5251   ` cfv 5255  (class class class)co 5858   CCcc 8735   NN0cn0 9965   ZZcz 10024    || cdivides 12531   Basecbs 13148   ↾s cress 13149   0gc0g 13400    GrpHom cghm 14680   Ringcrg 15337   RingHom crh 15494  ℂfldccnfld 16377   ZRHomczrh 16451  chrcchr 16453
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-inf2 7342  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814  ax-pre-sup 8815  ax-addf 8816  ax-mulf 8817
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 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-oadd 6483  df-er 6660  df-map 6774  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-sup 7194  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-nn 9747  df-2 9804  df-3 9805  df-4 9806  df-5 9807  df-6 9808  df-7 9809  df-8 9810  df-9 9811  df-10 9812  df-n0 9966  df-z 10025  df-dec 10125  df-uz 10231  df-rp 10355  df-fz 10783  df-fl 10925  df-mod 10974  df-seq 11047  df-exp 11105  df-cj 11584  df-re 11585  df-im 11586  df-sqr 11720  df-abs 11721  df-dvds 12532  df-struct 13150  df-ndx 13151  df-slot 13152  df-base 13153  df-sets 13154  df-ress 13155  df-plusg 13221  df-mulr 13222  df-starv 13223  df-tset 13227  df-ple 13228  df-ds 13230  df-0g 13404  df-mnd 14367  df-mhm 14415  df-grp 14489  df-minusg 14490  df-sbg 14491  df-mulg 14492  df-subg 14618  df-ghm 14681  df-od 14844  df-cmn 15091  df-mgp 15326  df-rng 15340  df-cring 15341  df-ur 15342  df-rnghom 15496  df-subrg 15543  df-cnfld 16378  df-zrh 16455  df-chr 16457
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