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Theorem rhmima 15891
Description: The homomorphic image of a subring is a subring. (Contributed by Stefan O'Rear, 10-Mar-2015.) (Revised by Mario Carneiro, 6-May-2015.)
Assertion
Ref Expression
rhmima  |-  ( ( F  e.  ( M RingHom  N )  /\  X  e.  (SubRing `  M )
)  ->  ( F " X )  e.  (SubRing `  N ) )

Proof of Theorem rhmima
StepHypRef Expression
1 rhmghm 15818 . . 3  |-  ( F  e.  ( M RingHom  N
)  ->  F  e.  ( M  GrpHom  N ) )
2 subrgsubg 15866 . . 3  |-  ( X  e.  (SubRing `  M
)  ->  X  e.  (SubGrp `  M ) )
3 ghmima 15018 . . 3  |-  ( ( F  e.  ( M 
GrpHom  N )  /\  X  e.  (SubGrp `  M )
)  ->  ( F " X )  e.  (SubGrp `  N ) )
41, 2, 3syl2an 464 . 2  |-  ( ( F  e.  ( M RingHom  N )  /\  X  e.  (SubRing `  M )
)  ->  ( F " X )  e.  (SubGrp `  N ) )
5 eqid 2435 . . . 4  |-  (mulGrp `  M )  =  (mulGrp `  M )
6 eqid 2435 . . . 4  |-  (mulGrp `  N )  =  (mulGrp `  N )
75, 6rhmmhm 15817 . . 3  |-  ( F  e.  ( M RingHom  N
)  ->  F  e.  ( (mulGrp `  M ) MndHom  (mulGrp `  N ) ) )
85subrgsubm 15873 . . 3  |-  ( X  e.  (SubRing `  M
)  ->  X  e.  (SubMnd `  (mulGrp `  M
) ) )
9 mhmima 14755 . . 3  |-  ( ( F  e.  ( (mulGrp `  M ) MndHom  (mulGrp `  N ) )  /\  X  e.  (SubMnd `  (mulGrp `  M ) ) )  ->  ( F " X )  e.  (SubMnd `  (mulGrp `  N )
) )
107, 8, 9syl2an 464 . 2  |-  ( ( F  e.  ( M RingHom  N )  /\  X  e.  (SubRing `  M )
)  ->  ( F " X )  e.  (SubMnd `  (mulGrp `  N )
) )
11 rhmrcl2 15815 . . . 4  |-  ( F  e.  ( M RingHom  N
)  ->  N  e.  Ring )
1211adantr 452 . . 3  |-  ( ( F  e.  ( M RingHom  N )  /\  X  e.  (SubRing `  M )
)  ->  N  e.  Ring )
136issubrg3 15888 . . 3  |-  ( N  e.  Ring  ->  ( ( F " X )  e.  (SubRing `  N
)  <->  ( ( F
" X )  e.  (SubGrp `  N )  /\  ( F " X
)  e.  (SubMnd `  (mulGrp `  N ) ) ) ) )
1412, 13syl 16 . 2  |-  ( ( F  e.  ( M RingHom  N )  /\  X  e.  (SubRing `  M )
)  ->  ( ( F " X )  e.  (SubRing `  N )  <->  ( ( F " X
)  e.  (SubGrp `  N )  /\  ( F " X )  e.  (SubMnd `  (mulGrp `  N
) ) ) ) )
154, 10, 14mpbir2and 889 1  |-  ( ( F  e.  ( M RingHom  N )  /\  X  e.  (SubRing `  M )
)  ->  ( F " X )  e.  (SubRing `  N ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    e. wcel 1725   "cima 4873   ` cfv 5446  (class class class)co 6073   MndHom cmhm 14728  SubMndcsubmnd 14729  SubGrpcsubg 14930    GrpHom cghm 14995  mulGrpcmgp 15640   Ringcrg 15652   RingHom crh 15809  SubRingcsubrg 15856
This theorem is referenced by:  mpfsubrg  19953  pf1subrg  19960  plypf1  20123
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-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
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-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-riota 6541  df-recs 6625  df-rdg 6660  df-er 6897  df-map 7012  df-en 7102  df-dom 7103  df-sdom 7104  df-pnf 9114  df-mnf 9115  df-xr 9116  df-ltxr 9117  df-le 9118  df-sub 9285  df-neg 9286  df-nn 9993  df-2 10050  df-3 10051  df-ndx 13464  df-slot 13465  df-base 13466  df-sets 13467  df-ress 13468  df-plusg 13534  df-mulr 13535  df-0g 13719  df-mnd 14682  df-mhm 14730  df-submnd 14731  df-grp 14804  df-minusg 14805  df-subg 14933  df-ghm 14996  df-mgp 15641  df-rng 15655  df-ur 15657  df-rnghom 15811  df-subrg 15858
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